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Burnup and feasibility study of low power density PWR's

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Title:
Burnup and feasibility study of low power density PWR's
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Molins-Bartra, Cesar, 1953-
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English
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xiii, 306 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Boron ( jstor )
Coolants ( jstor )
Fuel cells ( jstor )
Geometry ( jstor )
Isotopes ( jstor )
Neutrons ( jstor )
Plutonium ( jstor )
Poisons ( jstor )
Reactivity ( jstor )
Refueling ( jstor )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF ( lcsh )
Fuel burnup (Nuclear engineering) ( lcsh )
Nuclear Engineering Sciences thesis Ph. D ( lcsh )
Pressurized water reactors ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1981.
Bibliography:
Bibliography: leaves 303-305.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Cesar Molins-Bartra.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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07895224 ( OCLC )

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BtJPNUP AND FEASIBILITY STUDY OF,
LOW POWER DENSITY PWR' S











BY

CESAR MOLINS-BA1RRA
















A DISSEEMPMTON PRESEM TO THE'GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF TH~E REQUIEENTIS FOR THE DEGREE OF DOCIOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA 1981


































Tb Him who cares for all

and who gave us this world,

as an insignificant sample of my deep appreciation























ACKNCWL)GEM2ENTS


The author wishes to express his sincere appreciation to the members of his supervisory committee N.J. Diaz, E.E. Carroll, E.T. Dugan, C.C. Oliver and M.L. Muga for their interest and help in preparing this dissertation. Special thanks are given to Dr. N.J. Diaz for originating the dissertation subject and providing direction of the research project. The author is deeply indebted to Dr. E.T. Dugan for providing continued technical guidance and criticism and invaluable help in obtaining necessary calculational tools, together with endless patience in correcting this report.

The author is indebted to J.A. Wethington, Jr., M.J. Ohanian, G.R. Dalton, W.H. Ellis, G.S. Roessler and W.G. Vernetson for the friendship and support given through the years of this research.

The main body of the calculations of this study were performed at Oak Ridge National Laboratory, Tennessee, where the author met a number of great friends who were instrumental in the execution of this work. Special thanks are given to J.C. Cleveland for making the necessary arrangements that made the stay at O.R.N.L. possible and for his technical advice and assistance; to Rafael Perez and iii











Pedro Otaduy (the Godfather and the executive coordinator of the Spanish colony at O.R.N.L., respectively) and to the rest of the Spanish group for their continued moral support and friendship; and to F. Clark and T. Prinm for their administrative and technical advice.

The author acknowledges that during his tenure at the

University of Florida he was financially supported by the National Science Foundation, the Hauck fund, Oak Ridge National Laboratory and two consecutive University of Florida Graduate School Fellowships.

The author keeps a special thought for Keith Johnson for his untiring help in computing matters, and to Pedro Salas for the many hours of friendship and sharing in happiness and in distress.

























iv
















TABLE OF CONTENTS


RAME

ACNOWLEDGEMENTS * * * * * * iii

LIST OF TABLES . .. . * * v * iii

LIST OF FIGURES ..................... ix

ABSTRACT ......................... xii

I. INTRODUCTION . . . .. . . .. . . . 1

1.1. Background . . . . ......... 1
1.2. The SHARP Concept . . . . 4
1.3. Previous Studies of Low Power Density Cores . 9
1.4. Purpose of this Study .... ... .. 10

II. SCOPING WORK . . ... . . . . . 14

2.1. Problem Framing . ... .. . .. 14
2.2. Calculational Methods for the Scoping Study . 25
2.2.1. Heat Transfer Calculations . . . 25 2.2.2. Neutron Cross Section Calculations . 31 2.2.3. Criticality and Burnup Methods . . 33
2.3. Results of Preliminary Work . . . . . 41
2.3.1. Reactivity Coefficients . . . 41 2.3.2. Neutronics and Safety Considerations 51
2.3.3. Burnup Achievements and Isotopic
Inventories . . . . . . . 63
2.4. Scoping Work Conclusions . .. . ... . 70

III. BURNUP CALCULATION METHODS a . .. . .. 72

3.1. Burnup Calculational Methods and Present Needs 72
3.2. Method Developed for this Study . a . . 99
3.2.1. The Fuel-burning Codes . . . . 99
3.2.2. The CRIBUR Core Model a . . . 106
3.3. Benchmarking of the Burnup Calculations . 132
3.3.1. Available References for Comparison . 132 3.3.2. Criticality Benchmarking . . . 135 3.3.3. Power Distribution Benchmarking .... 140




v












IV. DETAILED SHARP BURNUP OPTIMIZATION STUDIES . . 143

4.1. Cases Chosen for Final Study . . . .. 143
4.2. Results of Optimization. ......... . 147
4.2.1. Neutronics and Isotopic Results . . 147
4.2.2. Plant Operational Data Variations . 160
4.2.3. Ore Usage and Enrichment Needs . . 175

V. THERMAL-HYDRAULICS AND ECONOMIC CONSIDERATIONS .... 179

5.1. Safety Related Thermal-hydraulic Considerations 179
5.1.1. Motivation .............. 179
5.1.2. Thermal-hydraulic Studies . o . 180
5.2. Economic Evaluation ............. 190
5.2.1. Introduction ............. 190
5.2.2. The Economic Comparison Studies . . 191

VI. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 201

6.1. Introduction ...... ...... 201
6.2. Neutronics, Burnup and Ore Usage .. .... 204
6.3. Plant Operations Considerations . . . . 207
6.4. Economic Effects ..... .. 209
6.5. Suggestions for Future Rearch ........ 211

APPENDICES
A. METHODS OF IMPROVING BURNUP IN PWR'S ....... 213

A.1. Introduction ................ 213
A.1.1. Motivation and Constraints . . 213
A.1.2. Schemes for Improving Burnup . . 215
A.2. Techniques for Improvement of Burnup o . 219'
A.2.1. Increasing Number of Batches . . 219
A.2.2. Burnable Poisons o.__ . 223
A.2.3. Low-Leakage Fuel Management . o 229
A.2.4. Alterations of Fuel-to-Water Ratio . 233
A.2.5. Low Power Density . 0 0 . 237
A.2.6. Flattening Axial Power Distribution 239
A.2.7. Increasing Enrichment . 0 . . 240
A.2.8. End of Cycle Coastdown . . . 0 242
A.2.9. Other Possibilities of Minor Importance 244

Bi. BRT CODE DESCRIPTION ............. .. 248

B2. PHRO ODE DESCRIPTION ............. 251

B3. MONA CODE DESCRIPTION .. ............. 253

B4. LEOPARD CODE DESCRIPTION ............... 255


vi















B&. NTPUNamCOE DR IPTION . TIO9...s..*.*.* 263

B7. PDO-7 CODE DESCRIIONO... .. .. .. .. .. 265

B8. TEMPRrTCM~E DESCRIPTION. o..o... . .. 269

B9. CNCEPT,-IVCODE DESCIPTION . . .. ......4 271

B10. GEM CODE DESCRIPTION s.. *... o*. ..*.9 273

Bli. POWCx OCME DESCRIPTION . .. .. ... 275

Cl. CRIBUR COE SOUJRCELISTING . .. .. .. ... 277

C2 SAM'PLE RUN OF CRIBEJR e o * * . a a 293 D. ISOTOPIC AND SPECTRAL DATA FROM BURNUP CALCULATIONS. 299 RETE'NCES *. . . . . . . . 303






























vii
















LIST OF TABLES



2.1.1. Main Core Parameters for the Standard Reactor. . 19

2.1.2. Basic Fuel Cell of the Standard Reactor. . . . 19

2.1.3. Basic Cell Composition. Core #1 . . . . 21

2.1.4. Identification of Cases for the Scoping Study . 23

2.1.5. Tabulation of Case Number Densities Which Differ
from Those of Case #1 ............... 24
2.2.1. Basic Cell Temperatures vs. Power Level . . . 30

2.3.1. Core Reactivities at B.O.L . . . .. .... 50

2.3.2. Scoping Burnup and Isotopic Results .. ..... 65

3.3.1. CRIBUR and Batch-averaging Method Criticality Errors 137

3.3.2. Effects of Code and Data Alterations on Core K-eff. 139

4.2.1. Burnup Achievements, Fissile Isotope Usage and
Ore Usage . .. .. .. .. . . 149

4.2.2. Plant Operational Data Comparison . . . . 161

4.2.3. Ore, Enrichment and Plutonium Normalized to 1125 MWe 176

5.2.1. Capital Costs (Mills/KWh) and Percent Increases
over Standard Plant .......... ..... 193
5.2.2. Fuel Costs (Mills/KWh) and Percent Increases over
Standard Plant .................. 193
5.2.3. Generation Costs (Mills/KWh) and Percent Increases
over Standard Plant ...... ........... 199

A.2.1. Discharge Burnups of Low Power Density Cores . . 239

D.l. Cell-Homogenized Nupber Densities
(Atoms/Barn cm X 10F) ............... 301

D.2. Average Cell Neutron Velocities (eV) . . . . 302

viii

















LIST OF FIGURES



2.2.1. Pin Cell Gemetry ................. 26

2.2.2. Geometry Used for MONA Criticality Calculations . 35 2.2.3. Code and Data Flow for Preliminary Burnup Calculation 37 2.2.4. Comparison of Boron Letdown from Leopard and Burnup. 40 2.3.1. Soluble Boron Worth at Critical vs. Burnup . . 46 2.3.2. Soluble Boron Worth at Critical. Changing Pitch . 47 2.3.3. Infinite Multiplication Factor vs. Pitch . . . 53 2.3.4. Eta and Fast Fission Factor vs. Pitch . . . 56

2.3.5. Resonance Escape Probability and Thermal Utilization 58 2.3.6. Thermal-to-Fast Flux Ratio . 0 . . 0 0 60

2.3.7. Discharge Burnup of 25% Power Density Core vs. Pitch 64 2.3.8. U-235 and Plutonium Usage vs. Pitch . . . . 69

3.1.1. Basic Steps in a Burnup Calculation . . . . 75

3.1.2. Examples of Burnup Conditions and Timestep
Specification for Fuel Cell Burnup Calculation . 78

3.1.3. Geometry and Composition Specification for a PDQ-7
1/4-assembly Burnup Calculation . . o . . 81 3.1.4. Grouping of Pin Cell Data into Core Data . . . 85

3.1.5. Constant-boron Pin Cell Burning with Boron-free
Pseudo-burnup Steps ................ 87





ix













3.1.6. Pin Cell Burning with Soluble Boron Letdown Curve
and Power Level Following .. . . 90
3.1.7.* Effect of Time-dependent Power Level and Soluble
Boron Concentration Specification . . . . 93 3.2.1. Estimates of Boron Letdown and Relative Power ... 105

3.2.2. Flowchart of Data for a Fuel Burnup Calculation
Using PDQ-7 1/4-Assembly Geometry . . . . 107

3.2.3. Core Geometry Used in CIBUR. 109

3.2.4. Bessel-shaped Flux Distribution . . . . . 116

3.2.5. Fake-burnup Steps for Soluble Boron Worth . . 124 3.2.6. Problem of Erroneous Cycle Length . . . . 126

3.2.7. Flowchart of CR2EBUR Code . a 0 . 0 0 a 130 4.2.1. Recyclable Fissile and Net Fissile Usage vs. Pitch 155 4.2.2. Plutonium Discharge and U98 Use vs. Pitch . . 158 4.2.3. Real Cycle Length vs. Pitch . . . . . . 167

4.2.4. Total Life Refueling Time vs. Pitch . . . . 169

4.2.5. Availability and Capacity Factors vs. Pitch . . 172

5.1.1. Fuel Centerline Temperature vs. Relative
Mass Flow Rate a 6 0 0 . . 0 0 a a 183 5.1.2. Fuel Average Temperature vs. Relative Mass Flow Rate 185 5.1.3. Clad Surface Temperature vs. Relative Mass Flow Rate 185 5.1.4. DNBR-minimum vs. Relative Mass Flow Rate . . . 186

5.1.5. Percent Savings in DNBR-minimum and Fuel Temperature 188 5.2.1. Capital Cost vs. Plant Power .. . . . o . 194

5.2.2. Fuel Cycle Cost vs. Plant Power . . . . . 196

5.2.3. Total Generation Cost vs. Plant Power . . . 200



x












6.2.1. Discharge Burnup vs. Pitch . . . . .* . 205

A.2.1. Increase in Fuel Burnup due to Partial Refueling . 220 A.2.2. Ore Utilization vs. Enrichment and Cycle Length . 222 A.2.3. Orientations of Non-synnetric Burnable Poisons . 227 A.2.4. Batch Distribution in Core . .. . . .. 230

A.2.5. Discharge Burnup vs. Enrichment at Several
Fuel Densities ................... 235

A.2.6. Ore Utilization vs. Enrichment at Several
Fuel Densities ................... 236

A.2.7. Power Shaping Effect of Partial Length Rod . . 241 A.2.8. Ore Utilization vs. Fuel Enrichment and Burnup . 243






























xi

















Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy BJNPAND FEASIBILITY STUDY OF
LOR POWER DENSITY PWR' S

BY

Cesar Molins-Bartra

June 1981

Chairman: Nils J. Diaz
Major Department: Nuclear Engineering Sciences

Operational and safety problems of current Pressurized Water Reactors are often associated with the high power density level of the cores. An alternate use of current-design cores is proposed by reducing the power density. The effects should be improved safety. improved ore utilization, and improved operational characteristics.

A scoping study is performed in order to define core parameters suitable for optimization under the low power density characteristics, while minimizing redesign requirements.

A neutronic optimization study of the reactor cores is performed by systematic changes in the fuel lattice pitch. A new core burnup computational model (CRIBUR) is developed, which allows calculation of the burnup and isotopic analysis of a multi-batch core in its equilibrium cycle with a moderate computational and human effort. CRIBtIR provides better accuracy and sensitivity than other known existing models of comparable scope. with a moderate



xii












computational. effort. The code is benchmarked against actual core data and against multi-dimensional diffusion theory core calculations. and its sensitivity to several of the calculational parameters is also tested.

The thermal-hydraulic behavior of the low-power cores is compared to that of the standard reactors. and their enhanced safety margins are clearly demonstrated.

The low-power cores yield higher burnup, levels than the standard reactors. Ore utilization is also improved in a

once-through fuel management policy. Isotopic comparisons are

presented. Core cycles are drastically increased- Plant availability and capacity factors are also increased as a result of both the reduced impact of refueling downtimes and the reduced forced outage time resulting from the improvement of operational characteristics.

An economic comparison of the low-power cores is presented as a function of the core power level and the level of inflation. Low-power cores are at an economic disadvantage when compared to the standard reactor primarily because of the impact of the increased capital cost. The fuel cycle cost is also higher because of the long fuel core residence time. This economic disadvantage needs to be weighed against the impr oved safety and operational reliability to determine the commercial feasibility of the concept.




xiii
















CHAPTER I
RODUMICIN


Background

The most widely used reactor system for present and near future commercial production of nuclear energy is the Light Water Reactor (MR). The LWR's were developed originally as compact#, light-weight, high-power units suitable for ship propulsion and power, and they indeed performed as intended. The large research and development efforts of the U.S. military establishment for the LWR systems made them attractive for commercial power generation, and thus, they were deployed in land-based electric power plants.

LWRIs have been generating commercial electricity for a

considerable amount of time, totaling over 500 reactor-years of operation in the United States alone. However the safety systems and engineering safeguards intended to prevent or to contain and reduce the impact of accidents have become extremely sophisticated and costly. This is due to several cumulative reasons: 1) the fact that LWR's work under conditions that are close to technological limits, 2) the high power density typical of these compact reactors, 3) the fuel and coolant conditions, and 4) the inherent "core-cooling" problems in case of accidents or severe transients.








2


The many operational transients saiety-related incidents and the rare but significant accidents resulting in extended reactor shutdown, e.g. Three Mile Island and others, clearly substantiate this assertion.

A key dilemma appears immediately: a very high power density reactor has intrinsic serious safety problems; however, economic considerations have resulted in larger, high power density reactors working closer to their technological limits and requiring improved engineering safeguards. The industry has naturally worked towards extracting the maximum power from a given core, and also towards extending the reactor fuel cycle time in an attempt to reduce the economic impact of refueling outages and fuel carrying charges. Both these goals tend to demand performance from the reactors in a manner that is not always compatible with strict safety

considerations.

Several questions need to be asked at this point. First: Is there a way of relieving the LWR's cores from these limiting situations without incurring an unacceptable economic penalty? Second: Will the new concept or approach need extensive (and thus costly and long) research and development and therefore have no impact in the short term? Third: Will the new concept be easily licensable, or will it require completely new licensing regulations and processes that would take many years to develop and create significant uncertainties for its deployment?

The first part of the first question has a relatively obvious answer from the technical point of view: reducing the power density









3


of the cores and making other technological fixes while maintaining the basic reactor design and technology will increase safety margins in the critical operational variables. However, the plant economics could suffer a significant penalty since a reduced power density requires a higher capital investment per unit of installed power, and this is the major part of the cost of nuclear-produced energy. On the other hand, the reduction of power density might allow a better fuel economy, reduce refueling times, reduce

personnel radiation exposures and waste handling and allow a better overall plant capacity, thus utilizing better the invested capital.

Favorable answers to the second and third questions strongly dictate that presently known and proven technology be used if the solution is to be regarded as a viable alternative by the electric power generating industry.

The Safer Highly Available Reactor Plant (SHARP) concept described below appears to be a solution meeting these overall demands. It would definitely enhance safety by improving the core power distribution, thermal-hydraulics conditions, heat removal requirements and overall plant operations. It would not require

significant new engineering and technical innovations, since minimal variations would be applied to the currently used reactor designs. These new plants will probably be easier to license than the current plants. The economic aspect is the only phase on which a conclusion is not easily forthcoming, and it requires in-depth analysis before reasonable cost comparisons can be established. The purpose of this work is to conduct an in-depth comparative and








4


optimization study of the fuel utilization capabilities of the Safer Highly Available Reactor Plants versus the standard MR, plants currently in operation or under construction, and concurrently, to develop calculational tools that will facilitate and enhance the accuracy of scoping-type burnup studies. The primary study must be complemented with indications of the

comparative safety and economic performance of the SHARP with respect to standard PWR plants.

1.2. The SUM Concept

The main goals pursued by the "Safer Highly Available Reactor Plantu (SHARP) are the following:

Enhance reactor safety.

Reduce the safety and operational problems caused by the operating conditions prevalent in current LWR systems.

Reduce lengthy and costly research and licensing procedures.
Offer a solution which utilizes current technology and can be implemented in a relatively short time.

Reduce economic penalties to make a commercially viable operation.

The dominant feature of the SHARP is the use of a standard PWR core, without core changes requiring technological modifications, at a reduced power density to reduce safety-related and operational problems associated with standard full power operating conditions. The single low power density core may then be employed in an essentially standard plant configuration, yielding a reduced power









5


rating plant. Alternatively#, it could be employed in parallel with several other low power density cores powering a single standard secondary side so as to yield a plant of normal or full power rating. This multi-core arrangement was first considered attractive to maintain the overall capital cost down and maintain large plant output. However, from the commercial point of view, it is obviously more reasonable to study the single-reactor reduced power rating plant, whose design and construction are well known. It is important to consider also that the economy of scale afforded by the present large plants lies in the reduced number of critical, expensive components used for the production of a large amount of power. The low power density concept breaks away from this constraint and attempts to demonstrate that there is no large advantage in designing large power output plants as compared to smaller sizes,, when all factors influencing plant economics are considered. Furthermore, it is frequent to find electrical demand areas where the large power rating plants are oversized and a smaller plant is better suited for such places.

A single-core SHARP, which is the main concept analyzed in this work, is a plant using a standard, full-sized reactor vessel operating at a reduced power level, with the balance of plant dimensioned according to the desired rated output power (50% of standard, 33% of standard, etc.). The use of a single full-sized core for the low power density system accomplishes the dual goal of avoiding extensive plant redesign, and operating a core at a power level clearly within its technological limits with enhanced safety











and operational characteristics. A higher plant capital cost per unit of installed power is expected (1) because of the losses associated with the economics of size, i.e. because of the additional investment in the "oversize" Nuclear Stemn Supply System (NSSS) components However, the overall cost of energy production would be competitive because of the improved operating conditions and safety.

Note that although the power density (and thus, the heat flux across the fuel elements of the core) is reduced, the thermodynamic conditions of the coolant would be maintained at the same level as in the standard plants to avoid loss of thermodynamic efficiency of the plant. This can be easily achieved by reducing the coolant flow across the core, within established heat transfer limitations. Reducing the heat flux in the fuel while maintaining standard coolant conditions reduces the temperature of the fuel,, which improves fuel/clad thermal conditions, reduces stresses in the fuel, and hopefully reduces fuel pin failures.

The advantages that can be obtained from the SHARP are:

a). From the heat transfer point of view:

Lower temperatures in th fuel pellets, due to the lower heat flux. This would imply reduced thermal-related damage to the pellets and to the cladding, and therefore, reduced pin failures.

Reduced probability of reaching critical heat transfer conditions.








7

-Reduced heat stored in the fuel, and increased available heat capacity, i.e., in case of an accident the core is capable of retaining more heat before suffering damages.

Milder accident conditions and emergency cooling requirements, due to the lower fuel temperature and lower power density (which means reduced decay heat generation).

b). From the neutronics point of view:

Less Doppler broadening of resonances, and therefore extra reactivity available.

Reduced xenon concentration due to the lower neutron flux, which means again some extra reactivity available.

Reduced xenon oscillation problems due to the lower, overall xenon concentration.

c0. From the fuel cycle point of view:

Longer burnup achievable from the same initial fuel, due to the extra reactivities mentioned above.

Largely increased time between refuelings due to the double effect of larger burnups achievable and the lower power generation of each core. This would reduce the impact of refueling outages on plant availability tending to increase it. The increased availability would result in proportionally increased energy generation in a given time period, thus reducing the impact of capital (which would be a basically fixed total cost) on energy generation cost.

Reduced relative activity of the fuel at discharge per MWd generated due to the largely increased residence of the








8


fuel in the core, which would allow more of the mid-life fission products to decay while in the reactor.

Reduced ore requirements due to the larger burnups achievable from the same initial cores.

Reduced enrichment needs.

d). Fran the operations point of view:

Increased plant availability and capacity factor due to the longer inherent fuel cycle.

Operations well within technological limits with reduced failure of components and reduced forced outage periods.

Decreased operational transients and constraints.

Better load following capability because of the large technological margins available for maneuvering.

Reduced overall personnel radiation exposure.

Reduced fuel handling cost because of less material being handled and because of its lower specific (per MWd generated) radioactivity (which results in reduced personnel radiation exposure).
Possibility of reduced spent fuel storage and

transportation requirements.

Easier inventory control and reduced risk of proliferation because of the reduction of fuel handling operations.

All these advantages appear to be qualitatively obvious, but the question remains whether or not they can outweigh the economic disadvantage mentioned before.








9


The present study conducted on the low power density

single-core, reduced power rating plant using essentially the standard plant configuration is also timely because of the present worldwide interest towards building new, small size plants. There are many situations where a 1000+ We plant is just too large, and so is the capital investment associated with it.

1.3. Previous Studies of Low Power Density CQres

The general trend followed by industry since the application of nuclear reactors for commercial production of energy has been to increase plant size and-power densities in an effort to reduce the capital cost per KW installed. The trend of higher power densities reduced both the materials involved in the reactor construction and the fuel inventory necessary for a given plant power rating.

A few commercial reactor concepts having significantly lower power densities than the LWR's are in operation, but they present higher capital cost and reduced operating experience. Such is the case, 'for example, of the Magnox reactors.

Lower power densities for LWR's were, thus, not considered seriously until recently under the NLW(2) project

(Nonproliferation Alternative Systems Assessment Program) where low power density cores were investigated primarily as a means for increasing the fuel utilization in a once-through fuel management scheme. This would reduce fuel handling operations, and therefore proliferation risks, while also providing additional safety

margins.









10


This low power density study for the NASAP project was carried out by Westinghouse Corp., and their approach to low power density was that of increasing somewhat the overall core dimensions,, while maintaining or even augmenting the output power. This approach meant only a moderate reduction in power density (about 23% as compared to the 50 to 75% reductions considered in the SHARP concept). This system was found to be advantageous from the points of view of safety and fuel utilization when compared to the standard high power plants. On the other hand, the need for redesign of the core, the pressure vessel, fuel handling mechanisms, etc., was found to require large investments both in time and in capital, and the idea was not considered practical by the private industry unless governmental support was provided.

1.4. Purpose of this Study

The main purpose of this study is to research and develop the neutronic and fuel utilization characteristics of the SHARP

concept, and for purposes of completeness, to assess its overall safety and economic capabilities in comparison with standard PWR

plants.

Several steps are necessary in order to accomplish this task:

a). Definition of what constitutes a SHARP for the purpose of this study (ranges of power densities, safety considerations and/or goals, etc.).

b). Definition of a set of possible reactor cores on which parametric studies can be effectively conducted.











c). Preliminary neutronic and fuel cycle study of the defined cores with known and reliable calculational tools in order to have a good estimate of the performance characteristics of each core. This would include obtaining first estimates of basic core

parameters (such as reactivity worth of boron, coolant temperature, Doppler reactivity coefficient, etc.) and of the expected variation of core and fuel cycle parameters (burnup, cycle length, etc.).

d). Exploration and evaluation of currently available calculational tools that may be suitable for this type of study, and development of new calculational schemes that may accomplish the established goals with the optimum cost/results ratio and serve the nuclear industry as an accurate scoping tool.

For example, this study needs a good set of calculational tools for the neutronics calculations, where power levels, fuel

temperatures, etc. can be easily specified and changed, since these are some of the main parameters that will differentiate the low-power core from the standard core. It also requires reliable and economical means of calculating core burnup distribution, core life time, etc.

e). Detailed neutronic and fuel cycle studies of the cores selected from the parametric variations performed in the preliminary study. This study must result in an optimization of the SHARP cores from fuel-cycle and plant operation points of view. The results of the SHARP study must be compared to those of

standard PWR's analyzed with the same calculational tools, in order to obtain relative figures of merit with a minimum of









12


methodological errors that could bias the estimates towards either

system.

V. overview of economic evaluation of the SHARP concept as compared to the standard PWR to establish the overall economic advantages or disadvantages that may be expected from the SHARP concept. The economic evaluation must consider the variations in the costs of interest and escalation, as well as the possible size or rating variations of the plant components which depend on the plant power rating. For example, when studying a single-core SHARP for a power level of 50% that of a standard plant, the NSSS is dimensioned equal to that of the standard, full-power plant, but the BOP is dimensioned for only the new 50% power rating.

The next chapter describes the SHARP parameters and the preliminary calculations carried out in order to establish the basic expected performance of the low power cores.

Chapter III describes the burnup calculational methods used in the industry; they are compared to the needs of this study, and a new method is developed, which best suits the scope of this work and results in reduced computational effort. The new method should serve as a valuable industry-wide burnup calculational tool because of its accuracy, ease of utilization and low computational cost.

Chapter IV shows the cases chosen for in-depth study and the results obtained from the burnup calculations, plus some data referred to the expected comparative plant performance.








13


Chapter V is a brief description of the thermal-hydraulics safety"related aspects of the SHARP as compared to the standard

plant, and an insight into the comparative economic behavior of the SHARP.

Chapter VI contains the main conclusions of this study and recommendations for future research on the topic, considering points that have appeared as unresolved and potentially advantageous questions concerning the SHARP concept.

















CHAPTER II
SCOPING him

2-1. Problem Eraming

The purpose of this chapter is to study a sequence of different low-power cores in order to calculate first estimates of the low-power cores characteristics. A reference standard core must be defined against which one may compare the characteristics of the low-power cores. Finally, the results of the comparisons of the different reduced power density cores versus the standard one will be used to define the characteristics of selected cores that will undergo an in-depth study. The possible need for new

calculational tools to perform such in depth study will also be examined. This chapter will, therefore, study a series of low-power cores. The results of this study will be used to select the better candidate cores, and a range of their expected

characteristics, for further analysis.

The study of the SHARP requires an initial definition of the parameters of the reactor and/or of the fuel cycle that may be used as a means of comparing the SHARP with the standard PWR plant. These parameters may be classified into three main categories of interest: Safety, Neutronics, and Fuel Cycle. These are key areas of investigation which have both independent and interrelated problems.



14












Parameters of interest from the safety point of view would be:

a). The moderator temperature reactivity coefficient (MTrC).

b) The Doppler reactivity coefficient.

c). The fuel average temperature.

d) The power peaking factors.

e). The soluble boron reactivity worth (at the concentrations needed throughout a cycle life for each particular reactor system).

fV. Power density and linear power (KW/ft).

From the neutronics point of view, some of the more important parameters to be observed are:

a).* The changes of Doppler reactivity coefficient at the different power density levels.

b). xenon concentration.

c).* Core effective multiplication factor at beginning of life.

c). Evolution of isotopics.

d). N"-rtron energy spectrum variations.

e) Effects of varying enrichments if enrichments different from those normally used for present PWR's can be considered.

From the fuel cycle point of view, there are two main variables that have to be optimized with regard to energy

production, but which are bound by conditions like the maximum power peaking factor, fuel enrichment costs, thermal-hydraulic safety aspects, etc. These two main variables are the following:

a?). The total energy obtainable per ton of uranium ore.

b). The core cycle length.








16


Obviously, one is concerned with obtaining the maximum energy from "each ton of ore. In addition to this, utilities are also interested in having long core cycles. Presently there is a trend to change from a 12-month to an 18-month cycle scheme because of the reduction in refueling outage time and personnel radiation exposure, which might result in attractive savings in power generation cost.

If the currently standard core structure were maintained

(3-batch, out-in scheme) it would only take an increased fuel

enrichment and some help from burnable poisons in order to have an 18-month cycle instead of a 12-month one. By so doing, the ore

utilization is favored by the fact that the fuel is discharged at a higher burnup. A study by Combustion Engineering(3) shows that ore utilization is improved for increasing burnup levels up to about 50 GWd/MTU and fuel enrichments of about 4.5%. The problem, however, is that an 18-month cycle would yield high discharge burnup levels as compared to what is acceptable based on the present level of

fuel technology.

In order to keep the discharge burnup in an acceptable range,

given the present fuel technology, and to take advantage of the reduced refueling outages of an 18-month scheme, utilities are forced to switch to larger batch sizes. This enables the power generation cost to be reduced, but with worse ore economy than that

obtained in a 12-month core cycle. This is of some concern from a fuel resources viewpoint, mainly if reprocessing is not considered.








17

The optimum fuel cycle appears to be one which obtains maximum energy per ton of ore and which also has long core cycles. Discharge burnups, however, must be limited, considering present fuel technology, to somewhere between 30 and at the most 40 G~d/MTrJ. In addition, the size of the batches should be kept small (i.e. the number of batches should not be reduced) in order to maintain good ore utilization.

The reduced power density cores are expected to yield

increased burnups given a certain core management scheme, but they are not expected to vary in extremely large proportions. This would probably make the low-power cores able to comply with all the desirable conditions mentioned above: long core cycles because of the reduced power density and the associated additional burnup, and better ore utilization because of the extended burnup obtained from the same initial core load.

The comparison of the fuel cycle performance of the SHARP versus a standard PWR plant can be done from many different frames of reference, each of which would enhance the comparison of the cores in a particular aspect. However, the two most significant ones would probably be the following:

a). Maintaining feed enrichment and core structure, evaluate the differences in burnup achievable, cycle length, and ore utilization.

b). Maintaining core structure and discharge burnup, evaluate the differences in ore requirements,, enrichment needs, and cycle

lengths.








18


Another of the other possible scenarios for canparison could be to maintain the discharge burnup constant and allow fuel enrichment, cycle length and core structure to vary. However, it is the feeling of the author that these cases would not aid significantly in demonstrating the differences of a SHARP as compared to a standard plant.

Once the main parameters of interest have been defined, it is necessary to define a reference core that represents a standard reactor plant, and whose characteristics and performance is known and usable as a frame of reference for comparison of the SHARP characteristics and as a benchmark of the calculational methods used in the study.

After the reference core is defined, it is necessary to define a basic SHARP core and a series of "variational" cores that will be used in order to obtain the coefficients, parameters, and comarative results defined above. After the main SHARP characteristics are obtained, another set of cores will be selected for the in-depth study and comparison with the standard plant. The calculational tools needed for the in-depth study will be defined based on the requirements and restrictions observed in the scoping calculations.

The core chosen for the standard reference plant is a Westinghouse four-loop, 3400 DlWth. PWR, with 17x17 pin fuel assemblies(4) whose main parameters appear on table 2.1.1. Table 2.1.2 describes the main characteristics of the basic fuel cell of this standard reactor.








19


Table 2.1.1. Main Core Parameters for the Standard Reactor.


Core Shape Cylindrical

Radius 168.53 cm

Active Height 365.00 cm

Reflector Thickness 34.00 cm

Active volume 3.2568 E+7 cm3

Heavy Metal Loading 94.418 rMTU

Array Geometry Rectangular

Pitch 1.2573 cm

Coolant Pressure 2250 psia

Avg. Coolant Temperature 583 K

Thermal Power 3400 Mth

Pin Average Linear Power 6 1cM/ft






Table 2.1.2. Basic Fuel Cell of the Standard Reactor.


Region Material Radius (cm) Thick. (cm) Vol. Fraction


1 Fuel 0.4096 0.4096 0.3334

2 Gap 0.4178 0.0082 0.0135

3 Clad 0.4750 0.0572 0.1015

4 Water 0.7094 0.2344 0.5516









20

The basic reference core used in the scoping study is described in Table 2.1.1; its fuel cell characteristics are described in Table 2.1.2. The fuel used for the reference core is uranium dioxide, enriched to 3% in U-235 isotope; the moderator is considered at standard operating pressure (2250 psia.), but at roon temperature (293 M). No soluble or lumped poisons are considered, nor are any fission products present for all the scoping beginning of life (BOL) calculations, unless otherwise specified.

Under these conditions, the composition of the basic fuel cell is as shown in Table 2.1.3. Region 1 corresponds, as in Table 2.1.2, to the fuel pellet; Region 2 is the gap between pellet and clad; Region 3 is the Zircaloy,-4 clad and Region 4 is the light water moderator-coolant. This reference core will be named

core #1.

The pin-average linear power for the standard reactor is 6 KW/ft. A pin-average linear power of 1.5 KI"/ft. is used for the scoping studies of the SHARP. This is one fourth the linear power of the standard Westinghouse core.

In an attempt to cover a wider scope and range of possible applications of low-power reactors, the scoping study includes sane exploration of a very low power, low enrichment core which could be used as a preheater in a multi-core configuration similar to the one illustrated in Figure 1.2.2. This core would have lower than standard moderator temperature and a very low power (20% that of the standard core). Such a low power density, low moderator temperature core might possibly be fueled with spent fuel








21












Table 2.1.3. Basic Cell Composition. Core #1.


Isotope Region Pure # Dens. (*) Cell Avg. # Dens. (*)


H 4 6.7 E-2 3.6957 E-2

O 4 3.35E-2

O 1 4.4009 E-2 3.3152 E-2

U-235 1 6.6830 E-4 2.2281 E-4

U-238 1 2.1337 E-2 7.1137 E-3

Zr 3 4.2808 E-2 4.345 E-3

Ni 3 0... E-10 0... E-10

Sn 3 4.8556 E-4 4.9285 E-5

Fe 3 1.4946 E-4 1.517 E-5

Cr 3 7.6426 E-5 7.7573 E-6

He 2 1.9 E-3 2.565 E-5


Units are atoms per barn-cm.








22


discharged from standard PWR plants. This core was thought of as a possibility for further use of standard plant spent fuel.

Table 2.1.4 identifies the cores used for the scoping study

with their main distinguishing characteristics. The

characteristics of these cores were selected in order to obtain indicative figures on reactivity coefficients and burnup variations. Each core has a case I.D. assigned to it, which is used for future reference. The power levels indicated are in percent relative to the standard core's full power (6 KW'/ft. or 3400 MWth. total core power).

In all cases, both the fuel and the moderator are treated as having a uniform temperature throughout their masses. In the reduced power density system, the moderator temperature is always kept at the average coolant temperature of the standard plant core, in order to maintain the thermodynamic characteristics of the steam cycle, and the thermal efficiency of the plant. The only cores with different coolant temperature are the ones intended for

preheater operation. The fuel average temperatures depend

obviously on the linear power density, and the values used for each of the scoping study cores are shown in next section.

Table 2.1.4 shows also the core effective neutron multiplication factor (K-eff.) obtained for each of the scoping

study cores in a BOL, clean, unrodded situation. The calculational procedures used for the scoping study are different than the ones used for the detailed burnup study, and they are specified later in this chapter.









23














Table 2.1.4. Identification of Cases for the Scoping Study.


Case # Enrich. Mod.Temp. Fuel Temp. Poisons Power K-eff(*)


1 3% 293 K 293 K 0 0 1.3532

2 3% 583 K 293 K 0 0 1.2933

3 3% 583 K 1005 K 0 100% 1.2675

3A 2.6% 583 K 1005 K 0 100% 1.233

4 3% 583 K 700 K 0 25% 1.2770

4A 2.6% 583 K 700 K 0 25% 1.242

5 2.6% 293 K 293 K 0 0 1.3184

6 3% 293 K 293 K 1300 PPM 0 1.1579

7 3% 420 K 520 K 0 20% 1.3304

8 2% 420 K 520 K 0 20% 1.2260

9 3% 583 K 1005 K 1856 PPM 100% 1.0781


* Calculated with PHROG-BRI'-MONA as shown in Sections 2.2.2 and

2.2.3.








24


Table 2.1.5. Tabulation of Case Number Densities Which Differ from Those of Case #1.


Case # Isotope Region Pure #Dens(* Cell Hom. #Dens(*

2 0 4 2.346 E-2 2.76149 E-2

2 H 4 4.6926 E-2 2.58845 E-2

3 Same as core #2

4 Same as core #2

5 U-235 1 5.7924 E-4 1.9312 E-4

5 U-238 1 2.1426 E-2 7.14335 E-3

6 B 4 7.258 E-5 4.0037 E-5

7 0 4 3.0982 E-2 3.17624 E-2

7 H 4 6.1964 E-2 3.41795 E-2

8 U-235 1 4.4561 E-4 1.48565 E-4

8 U-238 1 2.1559 E-2 7.18790 E-3

8 0 4 3.0982 E-2 3.17624 E-2

8 H 4 6.1964 E-2 3.41795 E-2

9 B 4 7.258 E-5 4.0037 E-5

9 0 4 2.346 E-2 2.76149 E-2

9 H 4 4.6926 E-2 2.58845 E-2

3A U-235 1 5.7924 E-4 1.9312 E-4

3A U-238 1 2.1426 E-2 7.14335 E-3

3A 0 4 2.346 E-2 2.76149 E-2

3A H 4 4.6926 E-2 2.58845 E-2

4A Same as core #3A


*Units are atoms per barn--cm.








25


Same of the cores identified in Table 2.1.4 show variations in moderator temperature or fuel enrichment that imply changes in the cell number densities as compared to the reference cell. The coolant and fuel densities for the different temperature situations were calculated with the help of standard tables for each material(5,6). Table 2.1.5 shows the changed number densities for the affected isotopes and applicable cases. For the rest of the isotopes and/or cases, the reference core number densities apply.

2.2. Calculational Methods for the Scopingt Study


2.2-1. Heat Tranfer Calculations.

Some of the main differences between a standard reactor and a SHARP are related to the differences in the fuel temperature. It is therefore necessary to know the fuel temperatures that correspond to several different power-density cores before their neutronic study can be started. However, for the purpose of the present study, it is not necessary fo obtain extremely accurate results, since the data will be used for obtaining varying nuclear properties of the fuel, which are not drastically affected by a few degrees of uncertainty in the temperature.

The calculational. scheme used may be found in any complete Nuclear Heat Transfer or Reactor Analysis text (7) It is assumed that a uniform volumetric heat source exists in the fuel region; that the bulk moderator operating conditions are kept constant for all the different cores (583 K, 2250 psia.) except for the preheater cores (cases #7 and #8) which have their moderator at 420








26


deg. 'K but at the same 2250 psia pressure. For all the other cores, subcooled nucleate boiling has been assumed at the pin surface, while cores #7 and #8 were assumed to have subcooled

convection, due to the much lower bulk coolant temperature and to the very low heat flux rate. Figure 2.2.1 shows the geometry assumed in the pin for the heat transfer calculations.



Fuel Ga Clad Coolant









I.o



11 \I\\
.41 .008 .057 .234

(Dimensions in cm)





Figure 2.2.1. Pin Cell Geometry.


Given the heat transfer conditions found at the pin surface,

the temperature increment between the bulk coolant and the clad surface may be obtained by the Jens & Lottes expression:








27


Ts = Tc + 60 ( 21' 10 6 )0.25 exp ( p / 900 )


where Ts = pin surface temperature, deg. F

Tc = bulk coolant temperature, deg. F
V = surface heat flow rate, Btu/hr sq.ft

p = coolant pressure, psia


The temperature increment across the clad is given by the expression:


ATcl =' do In ( do di
2 *K S


where &Tcl = increment of temperature across the clad, deg. F

q' = linear heat flow rate, Btu/hr ft

do = clad outer diameter, ft di = clad inner diameter, ft

K = clad thermal conductivity, Btu/hr ft deg.F

S = clad surface per unit pin length, ft


Since the thickness of the gap is extremely small, the

temperature change across the gap may be obtained as A Tg =
K*S


where ATg = increment of temperature across the gap, deg. F

q' = linear heat flow rate, Btu/hr ft

K = thermal gap conductance, Btu/hr sq.ft deg.F

S = surface of heat transfer per unit pin length, ft.









28


With the previous expressions, it is possible to obtain the pellet surface temperature. Since the temperature of interest for nuclear calculations is the average temperature of the pellet, it is necessary to know the temperature profile in the pellet, as a function of the distance to the centerline, so that volumetric weighting of the temperature can be done.

The temperature profile inside the pellet (assuming uniform heat generation) is given by T(r) = To r2
7T Rz 4 ifwhere T(r) = temperature of the pellet at radius r deg. F

To = temperature of the pellet centerline, deg. F

qI = linear heat flow rate, Btu/hr ft

R = pellet outer radius,, ft

Kf = pellet thermal conductivity, Btu/hr ft deg. F

r = distance from centerline, ft


One more quantity is needed at this point: the pellet

centerline temperature, which is given by To = Ts + at
4 Tr Kf

where To = pellet centerline temperature, deg. F

Ts = pellet surface temperature, deg. F








29


With the pellet temperature profile expression in hand, the pellet average temperature may be obtained as


Tae = (2 / R2) *T(r) r dr = To (qI / 8 r, Kf) J0


For the 20% linear power, preheater cores, the calculational scheme was the same except for the temperature step from coolant to clad surface, in which case a subcooled convection heat transfer expression was used, such as the following: AT=q'' / h


where q'" = surface heat flow rate, Btu/hr sq.ft

h = forced convection coefficient, Btu/hr sq.ft deg. F


The forced convection coefficient may be calculated as


h = 0.0023 Re0"8 Pr04 (K/D)


where K = coolant thermal conductivity, Btu/hr ft deg.F
D = hydraulic diameter of the coolant channel, ft

Re = Reynolds number Pr = Prandti number


Once the thermal calculations were completed, the standard

core fuel average temperature was checked against the vendor's literature, and found to differ by a few degrees; this difference is probably due to slightly different conductivity coefficients across the core (due to temperature changes), which are not taken









30












Table 2.2.1. Basic Cell Temperatures vs. Power Level.


Power level % 100 50 33 25 20 (pre)


Clad surface 593.3 592.8 592.5 592.4 304.2

Clad increment 46.4 23.2 15.5 11.6 9.3

Gap increment 120.1 60.1 40.0 30.0 24.0

Fuel surface 759.9 676.1 648.1 634.0 337.5

Fuel centerline 1846.7 1219.5 1010.3 905.7 555.0

Fuel average 1303.3 947.8 829.2 769.9 446.3

Fuel average M 979.0 781.9 716.0 682.0 503.2

Vendor normalized W 1005.0 805.0 735.0 700.0 520.0



Note: Temperatures obtained from thermal calculations as described in Section 2.2.1.
Temperatures expressed in degrees Fahrenheit, unless otherwise specified as Kelvin W.
Column 20 (pre) refers to the 20% power, preheated core.








31


into account in the thermal calculations. Since this difference was not significant from the nuclear point of view, all the results were modified slightly to maintain consistency with the vendor's data in the neutronic calculations. T!he consistent data permit the further benchmarking of the nuclear calculations. Table 2.2.1 shows the main temperatures involved in the calculation for all the different power levels studied. All temperatures are in degrees Fahrenheit, unless otherwise specified.


2.2.2. Neutron Cross Section Calculations.


Section 1 of this Chapter described the set of cores chosen

for the scoping observation of the differences between a standard core and a low power density core. It is necessary to gather a series of codes of known reliability which can yield an accurate neutronic picture of a LWR core. Parametric studies can then be

conducted which will define significant factors differentiating a SHARP from a standard plant. In these parametric studies, the absolute accuracy of the results is not as significant as their relative values which are used to establish figures of merit.

The best neutron cross section calculationa. method available for the scoping study involved the use of several standard codes. This method allows great flexibility in specifying isotopic number densities, geometries, etc., but it requires a relatively large

amount of manual data handling from code to code. This section briefly describes these codes.








32



Two codes are used for the calculation of cross sections. PHROGM is used for calculation of the fast-group neutron cross sections, and the original library, consisting of a 68 energy group cross section data set is collapsed into three fast groups which are used in the core model code. A more detailed explanation of PHROG can be found in Appendix B2. As is shown in the next

section, the core criticality calculations are done with a model which defines two distinct regions: fuel and reflector. Fast cross section calculations need to be run for both of these regions. The fuel region is run first, and the group-wise fluxes and currents existing in the fuel are used as weighting functions for the ,-calculation of the reflector constants.

The thermal-group cross sections are calculated with the Battelle-Revised Thermos (BRT) code(9) using a 30-group cross section library. Thermal cross sections are collapsed into a single thermal group. it is also necessary to obtain separate

cross sections for the fuel region and for the reflector. The fuel region does not present any problem, since it can be well represented by the calculation of a unit fuel cell. However, BRT does not allow an intrinsic representation of the reflector region without an adjacent core. The reflector region is then calculated from a two-region slab reactor configuration, where one of the regions has the average core region characteristics, while the other represents the reflector. The geometry chosen was that of a slab. Because of limitations in the number of mesh spaces allowed by BRT, inaccuracies at the core-reflector boundary arise when a








33



cylindrical shape is considered. The basic features of M~ are described in Appendix El.

Four sets of cross sections are obtained with the calculations described above: one set with three-group fast cross sections for the core region; another set with three-group fast cross sections for the reflector; a third set with thermal cross sections for the core, and finally a set with thermal cross sections for a slab core-reflector configuration, from which only the reflector

constants are used. These cross section sets are then organized for input to the core models, which are discussed in the next section.

2.2.3. Criticality and Burnup Methods,

The first objective of the scoping study is to obtain the neutron multiplication factors of the cores described in Table 2.1.4. These multiplication factors are used to estimate the main reactivity coefficients involved in the neutronic aspects that differentiate the SHARP core from the standard core. The second goal of the scoping study is to obtain a first estimate of the burnup levels achievable by each particular core, as well as the isotopics associated with them. Following is the description of the methods used for achieving both of these purposes.

The core modeling for the criticality calculations is done with the Z4ONAMlO) code which is described in more detail in Appendix B33. MNA accepts the cross section input prepared by PHROEG and BRT~, and information about the core geometry and material region copositions. The code can perform a number of









34


calculations#, such as buckling searches,, poison searches, etc. In this case, a normal effective multiplication factor (K-eff.)

calculation is requested.

The code accepts a cylindrical geometry, but it is a

one-dimensional diffusion-theory code. In order to obtain accurate results for the neutron multiplication factor in one-dimensional cylindrical geometry, accurate values for the perpendicular buckling to account for perpendicular leakage are required. In

the absence of this information, it is known from previous

experience that an "equivalent" spherical system can yield accurate results for the neutron multiplication factor. For the "equivalent" spherical configuration, the core volume is kept equal to that of the cylindrical reactor, while for the reflector, the thickness is kept equal to the cylindrical reactor value. The core region has 199 mesh points allocated, and 34 mesh points are assigned to the reflector, thus assuring that the mesh spacing is smaller than the neutron diffusion length. This is necessary for these calculations if neutron diffusion theory is expected to describe the core fluxes and currents with an acceptable level of accuracy, An extrapolated zero-flux boundary condition is specified for the outer boundary. Figure 2.2.2 illustrates the geometric model used with MONA for the criticality calculations.

Two methods were used for the scoping burnup calculations. The first method involves the use of the calculational scheme already used for the criticality calculations. Once the reactor criticality status is established soluble boron is added to the








35











i/ Reflector





Core




mesh: 199 pts.



HF2ogenized Fuel Mixture Cl34 uts .






Water + Structural Materials




Figure 2.2.2. Geometry Used for MONA Criticality Calculations.








36


mo derator in the amount necessary to force the core to be exactly critical.

With the reactor critical, the fuel-region homogenized composition that was input to MONA and the flux spectrum generated by MONA for the fuel region are input into a BURUP code, which performs the isotopic burning of the mixture to the burnup degree specified, and at the power level desired. The BUBNUP code used in this case is geometrically non-dimensional, but it allows for the specification of a number of time-steps after which the homogenized composition of the fuel region is recalculated. The flux spectrum is assumed to remain constant throughout the burnup calculation.

After the burnup calculation, the homogenized fuel-region composition is used again for a new core criticality evaluation, in the way explained above. Figure 2.2.3 shows a schematic diagram of the flow of data involved in this type of burnup calculation.

This burnup calculation scheme has some very obvious drawbacks:

a). Each burnup timestep requires an inordinate amount of

effort.

b). There is a large amount of punched card handling and typing, which largely increases the probability of human error in the calculations.

c) The BUEIUM code used has a rather low degree of sophistication, resulting in a reduced number and complexity of radioactive chains.








37




Cell PHROG Core
Geometry Fuel 3-group
&Compos. Cell Fast X-sect.

spectrum

Reflector PHROG flector
imposition Reflector 3-group
Fast X-sec


BRT Core
Fuel Thermal
Cell X-sect.


Slab Reac BRT Reflector
Geometry Slab Thermal
& Compos. Reac, X-sect,


Sphere MONA
Geometry Spherical
Reactor


Spectrum
K-eff.



< End of> New BURNUP YES K-eff .
Life ? Composition =1 ?

YS N

END )Adjust
C k / Soluble Boron


Pigure 2.2.3. Code and Data Flow for Preliminary Burnue Calculation.









38

This method was, however, used for some cases, but a faster and at least equally accurate method had to be obtained. The
second burnup method employed the LEOPARD program(ll). This program, which is discussed in more detail in Appendix B4, makes an automated chain of calculations involving cross sections, spectra

and burruip for an elementary fuel cell. The cross sections calculations are based on the MUFT(12) and SOFOCATE(13) codes. The geometry is always that of a fuel pin, which may be surrounded by a buffer zone simulating the structural materials present in the core but not associated with the elementary fuel cell.

LEOPARD requires an initial pin composition, a power level, a soluble boron concentration history, and a burnup timestep structure. The code calculates each isotope's cross sections

collapsed to a specified number of broad groups, performs a multiplication factor calculation, burns the fuel, recalculates the pin composition, and resumes the loop until the completion of all the burnup timesteps.

Although the cross section generation in LEOPARD is not quite as accurate as the one used for the criticality calculations, the burnup calculational structure is far superior to the one in the BURNUP code, and the automatic coupling of the calculations allows for the effortless and error free handling of a much larger number of isotopes, which definitely adds to the accuracy of the overall calculation. It is important to note the drastic reduction of effort required for similar results between the first

(PHROG-BRT-MONA-BURNOP) and the second (LEOPARD) method of burnup








39


calculation. Obviously, LEOPARD does not model the core and the reflector since it is just a pin cell, one-dimensional code, but neutron leakage effects are considered by means of a perpendicular buckling, which enables the code to perform rough estimates of criticality.

In an effort to compare the two burnup calculational methods, parallel calculations were run for one core representing a standard reactor's first load. Since no fuel regions were considered in the MONA core model, the fuel enrichment was specified as an average of

the concentrations of the three real batches. This was also the only way of simulating the situation in LEOPARD, because it obviously can not accept several enrichments simultaneously. Note,, however, that this is a rather crude way of representing a core, because it is very different to have three distinct regions with various enrichments or burnup levels than to have a large region with averaged characteristics. However, the procedure was deemed

adequate for a scoping comparative study.

The results from the LEOPARD and MONA-et-seq. comparison showed a discharge burnup discrepancy of about 13%. Figure 2.2.4 shows the estimated soluble boron letdown curve obtained from each calculational scheme. The agreement is very good up to about two thirds of the core life, but the separation increases thereafter. This is most probably due to the absence of proper treatment of the fission products in the MONA method, because they could not be included in the fuel pellet region, and therefore were only accounted for as a reduction of core reactivity, but their









40











Critical Boron Concentration ppm 2500


2000






1000







0 5 10 15 Burnup
G~d/T



Figure 2.2.4. Ccuparison of Boron Letdlown fran LEOPARD and BUPNUP.








41


neutronic effects on the other isotopes (due to spectral effects) could not be simulated. However, the agreement between the two methods can still be considered good, considering the different conceptual approaches, and the small amount of core information included in them. The most reassuring fact is that the quoted beginning of life (BOL) core reactivity and the quoted end of life (MOL) burnup level of the first core for the standard reactor lie just between the values obtained from the two models, with LEOPARD giving a better result for the EOL burnup.

With all these considerations in mind, LEOPARD was used with the "average fuel" enrichment for the scoping burnup, calculations. These calculations used a fuel pin with core-averaged enrichment, a perpendicular buckling corresponding to the axial leakage of a standard core, and an approximately critical boron letdown curve. Calculations were performed for the standard core, a 25% power density core using varying lattice pitches, and a 20% power density core with reduced moderator temperature (the preheater core, designated as core #8 in Table 2.1.4).* The results of the scoping calculations are shown in the next Section.


2.3. Results of Preliminary Work

2.3.1. Reactivity Coefficients.

Table 2.1.4 shows the main parameters characterizing the

different cores that were chosen for the scoping study of the SHARP. The table includes the effective multiplication factors (K-eff.) of the cores at beginning of life (B.O.L.) with no xenon








42



or control rods, calculated by the PHROG-BRT-MONA scheme, as explained in sections 2.2.2 and 2.2.3.

The core effective multiplication factors obtained for the different cores are used to calculate bulk reactivity coefficients. Thus, the average moderator temperature reactivity coefficient (M.T.C.) is -20.67 p.c.m./deg.K of moderator temperature variation for the temperature range between 293 K and 583 K. This rather

large reactivity coefficient would be the maln asset of a core such as core #8, which would use very low enriched fuel or even spent fuel discharged from a standard plant. A reduction of average moderator temperature of 163 Kj, as described on table 2.1.4, would provide 3.37 % of extra reactivity as compared to an identical core working with normal moderator temperature. This excess

reactivity would enable the core to work with the less reactive fuel or burn the standard fuel beyond the current discharge burnup levels. However, it should be kept in mind that these preheated cores are in some way a technical speculation, and that the real interest of this study is centered in the low-power cores with standard moderator conditions, which enables them to drive a standard balance of plant in a single-core arrangement.

As the average linear power produced by the fuel is varied according to the different power densities of the SHARP cores, so is the average fuel temperature, even though the moderator

conditions remain unchanged. This variation of fuel temperature causes a change of the absorption resonances width, due to the Doppler effect, and this causes a change of the core reactivity.









43


The Doppler coefficient of reactivity obtained from Table 2.1.4 is

-3.99 p.c.m./deg.K of fuel temperature change, for a variation between~ 293 K and 700 K (which correspond to 0 power and 25% power respectively). The coefficient drops to -3.12 p.c.m./deg.K for the temperature range between 700 K and 1005 K (25% power to full power range). This shows the well known behavior of saturation of the Doppler effect for increasing temperatures of the fuel. The average coefficient that applies for the total range from 0 power to full power is -3.62 p.c.m./deg.K The reduced fuel temperature is one of the main effects contributing to an increase in the reactivity of a low-power core with respect to the standard core. As was mentioned before, the moderator conditions would remain unchanged for the different power level cores for thermodynamic .reasons, and only the fuel temperature would experience a variation due to the change of the heat production rate.

Fuel enrichment obviously has an effect on core reactivity, and therefore, it is also Possible to define a reactivity coefficient for it. For enrichments between 2.6% and 3% in U-235,

the reactivity coefficient oscillates between 8.705% / 1% enrichment variation for a cold, clean reactor and 8.643% /1%

enrichment variation for a hot, clean, full power reactor. For the enrichment step between 2% and 3% of U-235, the 1/5 power, preheater core shows a reactivity coefficient of 10.44% / 1% enrichment variation. This larger magnitude may be due to the

double effect of bracketing a lower enrichment end, and having a

better moderated configuration due to the low moderator temperature








44



which implies higher moderator density. This rather large

reactivity coefficient associated with the fuel enrichment could be used in order to stretch burnup of the cores by increasing the fuel enrichment, which up to a certain point will result in a better ore utilization (14).

Higher fuel enrichment would, however, bring problems associated with higher power peaking requiring the permanent use of burnable poisons and the technological problem of loss of clad integrity at very high burnup levels.

The reactivity coefficient of soluble boron in the moderator is given an in-depth study for two main reasons: First, the soluble boron concentration that makes a reactor critical at the different points through the core's life is a very useful and realistic way of estimating the cycle life of the core. However, for this to be feasible, it is important to know the reactivity worth of soluble boron at all times through the core's life. This coefficient may be expected to vary, depending mainly on boron concentration and

core burnup status; these two variables are somewhat related, but they certainly don't follow any exact relationship. Second, just reducing the power density may cause the standard reactor lattice geometry not to be the optimal one from the burnup point of view, since the nuclear characteristics of the fuel are somewhat perturbed. Some changes in lattice geometry, such as fuel pitch may be required then in order to optimize the core for the new low-power situation. If this is the case, the ratio of moderator-to--fuel is likely to be changed, and then, a given








45



variation of soluble boron concentration would mean different total amounts of boron being added to or retrieved from the core, which implies that the soluble boron reactivity coefficient may also change for different lattice geometries.

Soluble boron worth variations are analyzed as a function of boron concentration, core burnup, core power level,, moderator density, and basic cell geometry. These calculations were based on the LEOPARD scoping core burnup calculations.

When adjusting soluble boron concentration to keep the reactor critical through life, the worth of boron decreases slightly from the B.O.L. until about 2000 or 3000 MWd/MTU and increases thereafter, with a very slowly increasing slope. This variation is shown in Figure 2.3.1. This figure shows also that the variation of power level, without variation of moderator conditions, causes practically no change in the boron reactivity coefficient. On the other hand, either a variation of pitch or a change of moderator density (temperature) cause sharp changes in boron reactivity worth. In both cases, as expected, boron reactivity worth

increases for the changes that imply a larger absolute mmunt of boron present in the core (i.e. for an increased pitch or for an increased moderator density).

Figure 2.3.2 shows more clearly the effect of lattice pitch on boron worth. It is also more evident that the variation of boron worth at critical through life is larger for larger pitches. The cause of the monotonical increase of boron worth with life is due to two adding effects: the self-shielding effect of boron itself








46

.Boron Worth paVppn

Data from LEOPARD Calculations.


2 Core 4A Core 3A





1.6 cm pitch, 583 K Moderator




1.5

C o re 8

.25 cm pitch, 420 K Moderator













1.25 cm pitch, Core 4A 583 K Moderator

0.5
0 5 I I I ........... .I
0 6 12 18 Burnup
Gd/MT

Figure 2.3.1. Soluble Boron Worth at Critical vs. Burnup.








47

Boron Worth pcm/PPn

Data fran LEOPARD Calculations.
Pitch, cm

1.6



21







1.5.5






1.4



1.35


1
1.3


1.25



0 6 12 '18 Burnup
GWd/MT Figure 2.3.2. Soluble Boron Worth at Critical. Changing Pitch.








48


and the buildup of Pu-239 As core life advances, the boron concentration is steadily reduced;~ the boron self-shielding also decreases, causing an increase in boron worth. The contribution of Pu-239 buildup to the increase of the boron worth increases with core life. It should be noted that by the end of the core life, about 40 % of the core power is generated by Pu-239 that has been bred through parasitic captures in U-238.* Pti-239 has a higher thermal fission cross section than U-235, and it has a resonance peak in the thermal energy region (tenths of an-eV.). This causes the reactor to increase its sensitivity to the presence of a thermal poison (such as boron) that can remove neutrons from the high fission cross section energy region.

The rate of increase of boron worth with core life is faster for larger lattices, because of the effect of increased boron inventory in larger lattices.

Thus, from the point of view of the SHARP, it is possible to conclude that the reduced power density will not affect boron worth to a significant extent as long as the moderator conditions and cell geometry characteristics remain unchanged. However, if geometry changes are performed in order to optimize the cell behavior for burnup or fuel utilization, boron worth may be expected to show a significant variation, tending to increase for larger pitches and tending to increase towards E.O.L., mainly if the burnup levels achieved are high.

An important goal of the scoping study, is to make a first estimate of the burnup advantages that can be expected from the low








49


power cores relative to the standard core. One simple way of making such estimates is to calculate the total reactivity of the cores at B.O.L., and to assume that reactivity is lost linearly as

core burnup increases, which is a reasonable approximation for a first-estimate calculation.

The most significant parameters altering the core reactivity at B.O.L. when the fuel enrichment, core geometry and moderator conditions are fixed,, are the fuel temperature and the equilibrium xenon concentration. Fuel temperature affects the core reactivity in the amounts determined by the Doppler coefficient, explained earlier in this section. The reactivity worth of xenon in the different cores can not be obtained from the LEOPARD burnup calculations performed in the scoping study, but it can be obtained from the vendor's literature (4). Combining these xenon reactivity modifications with the core effective multiplication factors listed in Table 2.1.4, it is possible to obtain a comparison of the reactivity available at B.O.L. for each core, and therefore, an expectation of their comparative discharge burnup levels. These

figures appear on Table 2.3.1 for the standard core, the 25% power level core, and the 20% power level preheater core. Two fuel enrichments are considered for each power level.

using the linear reactivity assumption (15), Table 2.3.1 shows that a 25% power SHARP may yield a discharge burnup 10.6% above that of a standard reactor, for an average fuel enrichment of 2.6% U-235 in both cores. The burnup increase of 9% appears for a fuel enrichment of 3.1%. These burnup increases are desirable from two








50



main points of view: they represent additional energy obtained from the same initial ore, and they entail longer core cycles, which tends to reduce costs of refueling outages.

The expected burnups shown by the 20% power, preheater core are really surprising. The better moderation boosts reactivity in such a way that for 3% enriched fuel, a burnup increase of 32% above that of the standard core would be expected. If the preheater core is loaded with low enriched fuel, e.g. an enrichment of 2%, it would still yield 88% of the burnup expected from a standard reactor loaded with 3% enriched fuel.

However, it should be kept in mind that the burnup results obtained from these LEOPARD calculations do not take into account the significantly different neutronic situation of the various batches present in the core, which will definitely affect the isotopic dynamics, and therefore the core life. In any case, these figures give good hope for better fuel utilization by the SHARP's.


Table 2.3.1. Core Reactivities at B.O.L.


Case # ENRCH K-eff. Xe worth BOL reactivity

3 3% 1.268 -3.5% 23.3%

3A 2.6% 1.233 -3.5% 19.8%

4 3% 1.277 -2.3% 25.4%

4A 2.6% 1.242 -2.3% 21.9%

7 3% 1.330 -2.1% 30.9%

8 2% 1.226 -2.1% 20.5%








51


2.3,2. NeUtronics and Safety Considerations.

The reduction of power density in the cores of the SHARP causes, among other effects,, a reduction of the Doppler effect, a reduction of xenon concentration, and a reduction of fuel temperature. These changes entail unavoidable differences in the neutronic characteristics of the SHARP as compared to the standard reactor. Thus, it is necessary to check the neutronic changes associated with the power reduction, because some of their aspects are closely related to plant safety, while others are related to fuel performance or isotopic evolution.

As was pointed out in the preceding section, the neutronic differences existing between the standard reactor and the low-power reactors may result in a difference in the optimum fuel arrangement in the core. In this study, the variation of fuel arrangement is limited to the variation of fuel lattice pitch. This variation implies a change of the thermal-hydraulic characteristics of the

fuel cell, as well as a change of the fuel-to-moderator ratio, whose heavy effect on the core neutronics is well known. The pin geometry is kept constant and equal to the standard core design.

The immediate effect of changing the pitch or fuel-to-moderator ratio is a change in the neutron energy spectrum, which in turn causes a change of cell reactivity, and as fuel burnup progresses, it affects the isotopics evolution. There is an

optimum pitch which yields the maximum reactivity for a given fuel pin. However, such a configuration is not safe from the operational point of view. At the optimum point or in an








52


over-moderated configuration, the moderator temperature coefficient (DffC) becomes zero or positive, making the core inherently unstable. Stability and safety require a negative MflC, which means a pitch must be selected which yields an undermoderated

configuration.

The scoping study included an exploration of the reactivity changes occurring in the fuel cell as pitch is varied, for some of the more representative core cases being considered. Figure 2.3.3 shows the plots of the infinite multiplication factor versus pitch for the full power standard core (core 3A), the 25% power core (core 4A) and the 20% power core with lower moderator temperature (core 8) as obtained from PH1ROG-BRP-MONA. The plots correspond to BOL, with no poisons, control rods or xenon present in the core, but with the temperatures corresponding to the assigned core power levels.

The curves show a uniform increase of reactivity between core 3A and core 4A, due to the reduced Doppler effect. Remember that no xenon effect is taken into account in these curves, since the cores are clean and with zero burnup. Curves 3A and 4A do not show, however, any visible change of shape; there is only a vertical shift due to the gained reactivity. This means that the best moderated pitch is the same independently of the power density (under equal moderator conditions). With this in mind, the

overmoderation / undermoderation safety criteria for the low-power cores would admit some relaxation as compared to the standard core, due to the inherently safer operating conditions of the SHAR.








53



K.
standard. Core 4A
Pitdi

1.3. 583 K




1.2
420 K




1.1 I Data fran PHROG-BRT-MONA Calculations.





1.0 0.9




0.8


I



1.0 1.2 1.4 1.6 1.8 Pitch
an

Figure 2.3.3. Infinite Multiplication Factor vs. Pitch.








54


Figure 2.3.3 shows the pitch used by the standard reactor. Note that it complies with the safety criterion of working in an undermoderated configuration. However, it seems that the pitch could still be allowed to increase somewhat without compromising the safety, since the operation point of the standard reactor is rather far from the peak of the curve. Later in this section it is explained why the standard reactor can not take advantage of the extra reactivity achievable from a slight increase of pitch, while the low-power core can.

When examining the curve for core #8 on Figure 2.3.3, a sharp change of shape can be noticed as compared to the other curves. Reactor 8's curve has a larger curvature; its reactivity is much more sensitive to the variations of pitch. This fact is easily explainable: Reactor 8 works with reduced moderator temperature, but at the same pressure as reactors 3A and 4A. The moderator has, therefore, a higher density. This can be observed on Table 2.1.5. The higher moderator density causes a given change of pitch to represent a larger increase in moderation, i.e. there are more mean free paths or more interactions taking place in the moderator region between pins at higher moderator density. it is as if the x-axis of the curve for core #8 has been compressed as compared to the standard moderator temperature ones.

One of the consequences of the higher moderator density is a reduction of the distance between the standard reactor working pitch and the optimum moderation point. This effectively raises the multiplication factor of core 8 when using the standard pitch,








55


and it is one of the main factors responsible for the high reactivity found for core 8 in spite of its low fuel enrichment (see Table 2.3.1). However, it is also important to note that this same curve shift prevents considering a different pitch for core 8, because it is already working at a point much closer to the optimum moderation ratio, and any further change of pitch would bring the core too close to a possibly overmoderated configuration, which as explained above, is not acceptable from a safety point of view.

Figures 2.3.4 and 2.3.5 show the changes in the four factors of the "four-factor formula" for cores 3A, 4A and 8, as pitches are varied. These are the results of the criticality calculations performed for BOL with PHPOG-BRT-MONA. It is apparent that the neutronic effect of the higher moderator density of core 8 is far larger than the change caused by the Doppler effect difference between cores 3A and 4A. None of the four factors exhibit a difference of more than one percent in their corresponding values between cores 3A and 4A. However, the largest differences are observed in the resonance escape probability for very undermoderated lattices. This could be expected because this is the configuration and the factor that give the maximum enhancement to the Doppler effect variation.

The change of moderator density causes a sharp difference of spectra which is reflected in all the four factors. The fast fission factor, epsilon, is consistently lower for core 8 than for the other two. Note that the fast fission factor experiences a sharp rise when the lattice pitch reaches very small values and








56










2.8











2.2
Data frcm PHROG-BRT-MONA Calculations.








Cores 3A, 4A Core 8
1.6








Cores 3A, 4A Core 8
1. 0 1 .... ..... .
0.85 1.1 1.35 1.6 Pitch
cm
Figure 2.3.4. Eta and Fast Fission Factor vs. Pitch.








57


there is almost no moderator in the cell. In the range of pitches that can be reasonably considered for practical purposes (between 1.2 and 1.4 an) the fast fission factor slowly decreases as more moderator is brought into the cell.

Eta, the number of neutrons produced per thermal neutron absorbed in the fuel#, is the least sensitive of the factors, as it remains almost flat through the whole range of lattice pitches; it only shows a fall for very tight configurations. However, it is important to note that there is still a difference between core 8 and cores 3A and 4A. This shows that more important than the

amount of moderator present in the cell is the variation of mean free path associated with the change of moderator density.

The resonance escape probability F p, shows a very predictable pattern, increasing as the amount of moderator increases, but showing a very clear saturation effect as large pitches are reached. However, in the range of practically reasonable pitches, p shows a rather steep positive slope; this factor is the one which is primarily responsible for the increase of reactivity achieved by increases of pitch in this range. Note that core 8 stays consistently above cores 3A and 4A throughout the range of pitches examined in spite of the larger relative number of resonance absorber atoms (mainly U-238) present in core 8. This is a direct effect of the different mean free path caused by the increased moderator density. If only the better moderation effect and saturation effect of p for larger pitches existed the distance between the curves for core 8 and cores 3A and 4A could be expected








58

f p


f Coe 3A, 4A






0.75









p


0.5


Data fran OHG-BRT-ON Calculations.








0.25







0.9 1.0 1.1 1.2 1.3 1.4 1.5 Pitch
cm
Figure 2.3.5. Resonance Escape Probability and Thermal Utilization.








59


to decrease significantly as larger pitches are reached, but the reduction of distance is insignificant, indicating again the key role of the changed mean free path.

Finally, the thermal utilization factor, f, or ratio of thermal neutrons absorbed in the fuel to total thermal absorptions in the cell, shows a drop as pitch is increased. This effect is directly related to the amount of absorbing moderator atoms present in the fuel cell. Note that as pitch increases, the curve for core 8, which is naturally below that of cores 3A and 4A, drops faster than that of cores 3A and 4A; the reason being simply that the same increase of pitch adds more moderator atoms into core 8 than for the other cores, because of the higher moderator density of core 8.

Many of the reactivity and isotopic differences between the standard core and the low power density cores have been attributed to neutron energy spectral effects. Figure 2.3.6 illustrates some aspects of the spectral changes. The graph represents the ratio of thermal flux to first fast group (of the three fast groups used for the criticality calculations) for cores 3A, 4A and 8, and for short burnup times, up to about 50 days of equivalent full power operation of the standard reactor. These figures were obtained from short burnup calculations performed with the

PHROG-Brdl--MONA-BURNUP scheme. The effect of the increased moderator density of core 8 is much larger than all the other effects differentiating cores 3A and 4A. However, all cores show a parallel behavior as burnup increases: there is an immediate drop

of thermal flux right at the beginning of life. This may be








60












Data fran PHR1-BRT-MONA-B1RNUJP.Calculations.
0.9
'**- Core 8.


0.8




0.7




0.6Coe4




0.5

0 20 40 60 Time
100% Full-power Days Figure 2.3.6. Thermal-to-Fast Flux Ratio.








61


attributed to the appearance of xenon. Note that the cores are always burned in the critical condition, which means that at BOL there is a high concentration of soluble boron keeping the core at critical. This boron concentration is rapidly reduced as xenon builds up to an equilibrium level. It would seem then, that the core is changing one thermal poison for another, and therefore no significant change of neutron spectrum would occur, but this is not the case. The fact that the thermal poison is in the fuel (in the case of xenon) as opposed to being in the moderator (in the case of boron) has a significant effect on the neutron spectrum. This is due to the different relative changes of thermal absorption cross section occurring in the fuel and in the moderator. Thus, as

burnup increases, the thermal fraction of the neutron spectrum continues to drop slowly (the spectrum hardens) due to the

generation of fission products, which are thermal poisons located within the fuel, as is the case with xenon.

A key point in the safety of operation of a new reactor is its thermal-hydraulic behavior. It has been stated before that the low power reactors should deliver the outgoing coolant in the same thermodynamic conditions as the standard reactor, because this is a basic factor in the thermal efficiency of the plant. It is obvious that if a reactor delivers a reduced amount of power, it is also forced to deliver a reduced coolant mass flow in order to keep the thermodynamic conditions at the outlet unchanged, and this flow reduction encompasses different characteristics in the heat transfer conditions.








62



One of the main parameters of concern when analyzing the heat transfer conditions in a PWR is the minimum DNBR (Departure from Nucleate Boiling Ratio) existing in the core. The minimum DNBR indicates how far the core is from reaching a film boiling heat transfer condition, which would result in heavy overheating and damage of the fuel.

Mark Miller performed calculations(16) on the subject of the DNBR for several power densities and several flow rates and compared the results to the DNBR actually found in the standard reference core. A more detailed analysis of the thermal-hydraulic performance of the low-power cores is presented in Chapter V. It is, however, important to point out here that all the reduced-power cored show a higher DNBR than the standard core, under equal outlet enthalpy condition, and using the same fuel lattice geometry. For larger pitches (which may be of interest from the discharge burnup viewpoint) the low-power cores can accept some pitch increase and maintain the outlet enthalpy and still stay at a better DNBR than the standard reactor. However, the standard core can not use larger pitches without seriously reducing its thermal-hydraulic safety margins, due to the reduction of coolant velocity associated with the increase of fuel lattice pitch. This is an unequivocal and key point proving the substantially higher safety level of the low power reactors as compared to the standard ones.








63


2.3.3. rnup Achievments .and Isotgpic InveXntorie* As explained in Section 2.2.3, the preliminary burnup

calculations were performed with the LEOPAIRD code, using a pin enrichment equal to the average enrichment of the core, and burning it with a varying soluble boron concentration that would keep the core as close as possible to criticality. Eurnup explorations were done for the standard reference core 3A, for the 25% power density core 4A, and as a scope extension, for the 20% power density, preheater core 8. All of them were examined at the standard core's pitch of 1.2573 cn; cores 3A and 8 were also investigated at one larger pitch, and core 4IA was studied for a wide range of pitches, since it is the most representative low power density core. The multiple burnup calculations performed for core 4A have the purpose of determining the parametric effect of pitch on the discharge

burnup level, as well as giving an indication of the optimum pitch value.

Table 2.3.2 shows the main burnup and isotopic results obtained from the preliminary burnup calculations.

Reducing power density to 25% of the standard level while keeping the standard core's pitch results in an increment of burnup of 9%. However, the burnup increment can be improved to about 16% by increasing the pitch of the low power reactor to about 1.3 or 1.4 cm. Figure 2.3.7 shows the burnup levels achievable as pitch is varied on core 4IA. Note the important fact that the maximum burnup is not achieved at the pitch value having maximum BOL reactivity (shown in Figure 2.3.3) but at a smaller pitch. This








64





Discharge
Burnup Gd/MT




21.5






21.0 Data fran LEOPARD Calculations. 20.5






20.0



Std.Core


]1 I i i i iii iiiii
1.25 1.35 1.45 Pitch
cm


Figure 2.3.7. Discharge Burnup of 25% lbwer Density Core vs. Pitch.








65



Table 2.3.2. Scoping Burnup and Isotopic Results.


Case Pitch Energy Cycle U-235 (Kg/GWd) U-238 Fiss Pu (Kg/G~d) I.D. (cm) (Gwd) Life Fiss. Disch. (Kg/@(d) Fiss. DISCH. 3A 1.25 1737 ly 316d 0.6343 0.6319 0.0805 0.3291 0.4064

3A 1.60 1849 ly 360d 0.7073 0.4779 0.0495 0.2686 0.2726


4A 1.25 1893 8y 049d 0.6218 0.5293 0.0854 0.3519 0.3895 4A 1.30 2022 8y 252d 0.6210 0.4519 0.0787 0.3512 0.3474 4A 1.35 2033 8y 269d 0.6370 0.4307 0.0708 0.3355 0.3169 4A 1.40 2028 8y 261d 0.6518 0.4195 0.0647 0.3207 0.2969 4A 1.45 2031 8y 266d 0.6587 0.4112 0.0613 0.3127 0.2844 4A 1.60 1881 8y 031d 0.7033 0.4604 0.0499 0.2676 0.2610


8 1.25 1511 8y 043d 0.6387 0.4785 0.0707 0.3227 0.3795 8 1.40 1362 7y 116d 0.7092 0.5386 0.0545 0.2688 0.3448



Note: Ave. fuel enrichment is 2.6% for all cores except for core 8

it is 2%.

Total energy generated assumes a core with 94.42 MTU loading.

Cycle life is calculated assuming 75% availability.

For Core 3A, pitch = 1.2573 which is the Westinghouse value

for a standard core using 17X17 assemblies.

Data obtained from LEOPARD burnup calculations.











means that the isotopic and spectral effects occurring during the life of the core cause a shift of the EML reactivity, favoring less moderated configurations. The main reason for this effect is a conflict between optimum moderation and enhanced conversion ratio: Conversion ratio is enhanced when the resonance absorption in U-238 is increased, but this is obviously an effect which reduces the immediate reactivity of the core. The trade-off appears when realizing that at E.O.L., a large proportion of the power is generated by the converted plutonium. Thus, if conversion ratio is reduced in trying to optimize moderation (and thus increasing BOL reactivity), the core reaches EOL with a reduced amount ohf plutonium, and therefore becomes subcritical. at a lower burnup level.

Note on Table 2.3.2 the relatively high level of burnup achieved by core 8, in spite of being fueled with only 2% enriched fuel. The effect must be attributed primarily to the different moderator characteristics, in addition to the effect of the reduced power level.

It appears at this point that the best pitch from the burnup viewpoint is larger than the one used in the standard reactor, but smaller than the one yielding maximum reactivity at BCL. However, the model used for these scoping calculations can not be assumed to accurately represent a reactor in an equilibrium cycle since BOL compositions with no burned fuel were employed. The question is then whether the optimum burnup pitch obtained in these studies will still be the best for an "equilibrium cycle" core or if it








67


will be a different one. In order to answer this question it is

necessary to perform burnup calculations with a model that can reasonably represent a core in an equilibrium cycle, i.e. there must be some burned fuel present at BOL for each reload core. These scoping studies, however#, can serve as guidelines for

"equilibrium cycle" calculations.

There are a number of criteria that can be considered

important when examining the results of a burnup analysis from an isotopic or fuel utilization point of view:

a). Search for the maximum energy output from the same load of fuel.

b). Search for the minimum discharge of plutonium, for

non-proliferation purposes.

c). Search for the minimum amount of net fissile material being consumed per unit of energy produced. This would account for the fissile species being discharged with the spent fuel, and therefore would assume a spent fuel reprocessing policy.

d). In the same way as the criterion above looked for the best resource utilization, it might also be important to look for the best economical combination of resource utilization and cost of fuel cycle (including fuel enrichment, fabrication, storage, reprocessing, etc.).

Although the criteria that involve reprocessing appear more meaningful from the scientific or technological point of view than those which do not include it, the present political situation in the U.S. prevents commercial reprocessing, and this must be








68


considered when comparing results of burnup or uranium resources utilization.

Since the present study assumes all fresh fuel to have the same characteristics, increasing the discharge burnup implies

improving ore utilization. In this aspect, core 4A has a better fuel utilization than core 3A. Although core 8 is able to achieve a high burnup for a low-enriched fuel, the total use of U-235 per unit of energy produced appears to be somewhat worse than that of cores 3A and 4A. However, core 8 is not starting from the same type of fuel, and therefore this result should not be considered as a negative point for core 8. In addition, core 8 could be used to further burn the fuel that is ordinarily discharged from standard PWR's which would in fact represent an improvement of overall ore utilization if fuel reprocessing is still not being considered.

Figure 2.3.8 shows the use of U-235 per unit of energy produced, and the relative amounts of plutonium-to-U-235 burned for the different cores, and for some varying pitches. Note that the amount of plutonium burned is maximum for the tightest pitches, due to the better conversion ratio associated with the hard-flux, undermoderated configurations. As a logical result, U-235 consumption per unit energy produced tends to increase with pitch. It is interesting to realize that for a slight increase of pitch above the standard (from 1.25 to 1.3 an), the U-235 specific consumption actually decreases slightly. This is due to the reactivity enhancement which occurs in going to larger pitches. The poorer plutonium production quickly offsets this factor and the








69









0.7 0,..
Core 8 ****o
/ Core 3A Core 4A

*** U-235 Utilization
Kg/GWd
0.6


Data from LEOPARD Calculations.Core 8\ Core 3A re 4A


0.4


Relative Pu-to-U-235 Burnup Kg Pu / Kg U-235

0.3






1.3 1.4 1.5 1.6 1.7 Pitch
cm

Figure 2.3.8. U-235 and Plutonium usage.vs. Pitch.







70



U-235 specific consumption begins to increase. It is thus seen that both U-235 specific usage and plutonium relative burning remain fairly close to the optimum values for the pitches that allow highest burnup and therefore longest cycle times, while worsening rapidly after the region of interest.

2.4. Scoping Work Conclusions

The scoping calculations show that there is good hope for a significant increase of discharge burnup from the same initial fuel by using low power density cores. The main effects contributing to the necessary extra reactivity are the reduced level of xenon and the reduced effect of Doppler broadening of resonances. A 9%

increase of burnup seems to be achievable by going to a 25% power density core, while up to 16% improvement appears feasible by varying the cell pitch. However, these calculations were performed with a calculational tool which is not well-suited for the burnup analysis of a core consisting of fuel of various enrichments and exposure levels. Some changes are surely to be expected when using an adequate core model for these burnup calculations. The burnup calculations made in this scoping study are indicative of a first-load core behavior rather than of an equilibrium core; the latter is the configuration of most interest for burnup or economic studies.

It has become evident that a maximum level of automatization is necessary for the calculational techniques to be used in

detailed burnup calculations. The adoption of LEOPARD to substitute the chain of PHEROG BRT MONA BURNUP proved to be







71



most advantageous, and a similarly automnated scheme should be used for the detailed calculations. However, a new burnup calculational scheme must be developed which allows the adequate representation of a several-batch reactor core in its equilibrium cycle.

After these scoping studies are completed, the in-depth study should be centered on the cores that can best conduce to short-term, commercially feasible low power density systems. The in-depth study is, therefore, centered on a single-core concept having a reduced power density in the 25 to 60% of the standard core. The cores selected for the in-depth study will be discussed in Chapter IV.

The safety aspects of the low power cores are extremely favorable. The fuel average temperature is drastically reduced, indicating a reduced risk of pin damage. The heat transfer conditions have their safety margins, e.g. the DNBR, significantly improved, which makes the probability of critical heat transfer conditions much lower than it is in the standard reactor.
















CHAPTER III
BURNUP CALCULATION METHODS

3.1. Burnup Calculational Methods and Present Needs

Once the philosophy behind the development of the SHARP is established and a power density or a range of power densities are chosen, the key work to be done in the area of fuel utilization is the optimization of the pin-cell of the SHARP. The fuel

utilization and performance (burnup and isotopic results) are then to be compared with those obtained from a standard power density core to determine its relative merit. This optimization and

comparison require a large number of reactor life calculations, where items such as batch-wise burnup, discharge burnup and

physical cycle life, etc. are of interest among others.

Many different organizations are interested in developing and performing reactor fuel cycle and burnup calculations. The wide range of organizations and specific interests has caused the development of different calculational schemes, in order to accommodate the different cost/effort/accuracy ratios desired for each particular type of study. Some of the more important types of burnup studies are the following:

a). Vendor burnup studies and core optimization, performed as design work. These studies are most complete, and they generally include three-dimensional, pin-by-pin studies which are closely


72








73



coupled with thermo-hydraulics models. TIhese type of studies require large computational efforts and highly specialized personnel, which makes them extremely costly.

b). Burnup studies performed by utilities, either for core following or for licensing. These also have to be detailed

studies, because they require strong economical decisions affecting the core cycle, or have significant safety considerations affecting the licensing procedure. However, there is presently a trend by utilities to use more epirical codes (such as nodal codes) which run at lower costs than the fully detailed pin-by6-pin studies. Nodal codes generally need base detailed diffusion theory calculations for normalization purposes, but still result in an overall lower cost than the fine mesh neutron diffusion theory codes, while being able to maintain the information necessary for adequate core management.

c). Studies performed by engineering companies or government agencies for assessment of energy policies or new reactor concepts or fuel cycles. These studies fall in a whole new class. They are not directly concerned with the behavior of each particular pin, not even of assembly-wise details. They look at overall results of burnup, time scales, and monetary volumes. Although they need to yield reasonable results, the numerical results themselves are normally less critical, and emphasis is placed on the comparative results of two or more different concepts. These studies generally use some type of empirical formula or simplified core model which








74


does not require detailed specification of compositions and

geometries, as the two previous types of studies did.

d). Burnup studies performed in universities or

research-oriented organizations for developmental studies of new reactor concepts, for preliminary reactor design or fuel management. These studies may range in their specificity anywhere from the very rough empirical formulas used for fuel resources utilization and policy studies to a level close to the vendor or utility calculations. However, they tend to be more concerned with

particular theoretical details, which call for calculational methods specific to each study. These studies normally follow standard industrial calculational procedures until a certain point where the particular aspect of the study calls for a specific

calculational tool, which is often developed for the occasion.

A wide range of other types of burnup or fuel cycle studies exist, but the four categories stated above illustrate the different levels of accuracy or detail that may be sought depending

on the purpose of the study, and how they require different

calculational costs, manpower needs and calculational tools.

For every burnup study (except for the very simplest ones, where just an empirical formula is applied),two main calculational steps are performed. Each of these steps can have different levels of sophistication depending on the particular needs of the study, and in some cases one or both of the steps may need to be repeated iteratively in order to obtain the accuracy required by the study. Figure 3.1.1 shows the basic flow of data between the two steps.







75











-Cell Geometry Fuel Cell Burnup Code
-Temperatures
-Power Level Generates Neutronic
-Initial Isotopics and isotopic parameters
-Operating Conditions of Fuel through Life
S-Time Steps ...._-lOptional Opional Feedback for
Coupling d- justment of
Codes Burnup Parameters


-Core Geometry M 1ELII
-Fuel Arrangement
-Control Materials Power Distributions,

-Time steps Criticality, Burnup, etc.



Figure 3.1.1. Basic Steps in a Burnup Calculation.








76


The first step involves using a code in which a representative unit cell of the reactor is burned. Macroscopic core geometry is avoided; the emphasis is placed in burning a unit fuel cell and surrounding it with neutronic conditions as close as possible to those to be encountered by the "average" fuel element during the real core life. The purpose of this calculational step is to obtain neutronic characteristics of the fuel as it undergoes burnup, which will then be used in the second step of the burnup calculations.

Input to these first-step codes normally includes the

following:

a). Fuel cell geometry and initial isotopics.

b). Temperatures of fuel and moderator.

c). Power density (normally linear power density).

d).* Definition of a buffer region surrounding the fuel cell in order to modify the neutron spectrum for the presence of structural materials (guide thimtbles, fuel assembly cans, spacer grids, water holes, etc).

e). Specification of a series of burnup timesteps that will determine the points where the code has to recalculate the

neutronic status of the fuel cell, which has been modified by the burnup process.

fl. Specification of controllable poisons present in the fuel cell. This is normally specified as soluble boron concentration in the moderator, and may be specified for each of the burnurp tiuesteps defined for the calculation.








77



g). Some models may allow for the specification of variable power levels for the different timesteps, the specification of Dancoff factors or some geometric data that may allow the code to calculate them, the specification of a perpendicular buckling, or a buckling search in order to obtain a critical spectrum, etc.

Most of these first-step pin cell codes are self-chaining, in the sense that the user specifies the cell geometry, isotopics and

burnup history at the beginning of the code execution, and the code performs the burnup steps and the cross section evaluations after each tiniestep without user intervention. Other schemes, like the one used for some of the preliminary calculations in the SHARP study, involve several codes to perform the pin cell burnup, and they need the user to manually handle the cross section libraries and the burnup isotopics back and forth between one code and another.* The advantages and disadvantages of each method rely mainly on the flexibility and freedom of data handling and geometry specification versus convenience and man time requirements. Figure 3.1.2 shows some possible schematics of the data specification for this first step of burnup calculations.

The output of these first-step codes normally includes the

following data:

a). Microscopic cross sections for each isotope present in the fuel cell, for the number of neutron energy groups specified in the input (normally between two and five groups) and for each specified timestep.





78
Soluble
Boron
Concentration
:*:- Different Specifications

1200
-J1 L

I I I I 1 I I I




FL0l -I---I--j.4-I-- 4-ir -ycl Timep
It I I III I I III I I I Ii I I I! 1 1 II I I I I
II I !! Ii I I I 1 I I 1
Relative 1 1 1 1 11 1 1. I I I I I
Pow r 1I I 1 I I I I I I 1 I1
1.1 II IJ .L.L.L.. 1 1 I I
1.0 j f ITime
0.9I II I I I
First clel I SJcnd CycL I_ irI Cycle +ITIe-tep


Figure 3.1.2. Examples of Burnup Conditions and Timestep
Specification for Fuel Cell Burnup Calculation.









79


b). Macroscopic cross sections of the fuel and the whole cell, weighted with the neutron spectrum present at each tiiestep, and with the number of energy groups requested.

c). Infinite multiplication factor of the fuel cell, and effective multiplication factor if a perpendicular buckling was specified.

6). Average neutron speed.

e). Concentration of each isotope present in the cell, regionwise and cell homogenized.

V). Cumulative burnup level since the beginning of the burnup calculation.

g). Other data depending on each particular model, such as the calculated Dancoff factors, fraction of the total core power produced by each fissile isotope, cumulative fission densities, conversion factors, etc.

The second step of the burnup calculations involves taking the data generated by the first code and using it in a core model which accepts overall core information regarding geometry and core operation in order to simulate the actual life of the reactor. This step is the one showing the widest variations from one type of burnup study to another. The most sophisticated models are able to

follow the core life without further iterations, while others need to feed their data back to step one of the burnup calculation, for a new iteration. The flow of data from step one to two and back depends on the sophistication of each of the models and the burnup data sought from the overall study.








80


The second step uses such a wide range of calculational approaches or methods, that it is difficult to specify the general input requirements and output data. As a general rule, the

explicit core model (in which the fuel is represented either pin by pin or in relatively small nodes, and control materials are specifically treated) needs the initial isotopics information, the microscopic cross section libraries generated by the first step of the burnup calculation, the core geometry data, the power history, the control materials history, and general editing information, while the output includes core isotopics in zone averages, pin by pin, or whatever region type is used by the code versus core history; criticality evaluations, power distributions; burnup information in average or explicit for each unit considered in the core, region averaged multigroup neutronics data, etc. Codes of this type are for example PDQ-7(17), SIMULIArEMl), etc.

Figure 3.1.3 shows the geometry and material zones specifications needed for a two-dimensional calculation of a quarter assembly of a PWR with PDQ-7. Note the extreme detail of

the geometrical description, and the relatively large number of different compositions considered. It is easy to imagine the extreme complexity of the calculation of a whole core in three dimensions, with the necessity to identify different fuel batches, control rod assemblies, burnable poison rods assemblies, assemblies with water holes, etc. Nodal codes avoid some of the complexity and reduce computational time by lumping each fuel assembly into a few nodes, but they require node-interaction parameters and albedos








81









U.
*- -* --. -... ..
























Diffusion Calculation Mesh a Fuel Pins
ITT Interassembly Water Gap
Water Holes (or LBP's)
Instrument Thimble



Figure 3.1.3. Geometry and Composition Specification for a PDQ-7 1/4-assembly Burnup Calculation.








82


which must be normalized with a detailed core calculation, as well as requiring adjustment of a large number of empirical factors.

There are other computational models which run at lower costs and may not require the technical expertise needed for running full blown PDQ-7 models or three-dimensional nodal codes. These models normally accept a lower degree of geometrical sophistication, or they may do just a part of the burnup calculation. For example, they may perform the criticality and eigenvalue calculations, but not be able to perform the actual core burning, which then has to be done by a separate code. This is the case of MONA, which was used in some of the preliminary studies of the SHARP. This type of codes are useful for criticality studies or spectral effects analysis, but they are of little use for a real life burnup, study

because of the enormous manual data handling required. However, they have the advantage of the low cost and the relatively simple input.

Other types of calculational models are often used for scoping or medium accuracy burnup, calculations. They normally are simplified codes which use part of the data obtained from the first step of the burnup. calculations, and then introduce an approximate core model, using empirical factors or formulas whenever the available data is insufficient for a rigorous treatment of the items needed for the calculation. However, these types of codes cover a wide range of sophistication and thus their accuracy and the information that can be obtained from them is largely variable. The core model used for the SHARP can be included in this









83


classification, evolving frau a well known calculational model. The innovative core model used for the SHARP is an improvement, on the accuracy of the results, resembling closer the real reactor life. It is achieved by the proper handling of significant core history data, which appear to have been previously neglected. An explanation of the data handling by these codes follows.

These single calculational models are normally based on a pin cell burnup calculation and often use the infinite multiplication factor calculations in order to obtain an estimate of the core criticality state or its reactivity.

In the simplest model, a pin cell would be burned for its whole life without any poisoning, and at a constant power level. The characteristics of a pin are a good representation of the fuel batch to which it belongs, since the pin is surrounded by many fuel pins alike. The only exceptions are the pins facing a fuel assembly of a different batch (and therefore having a sharply different burnup status) or facing the reflector. Then, if the fuel pin is burned for its entire life, it certainly passes through the stages corresponding to each batch in the core. This means

that it should be possible to take some of the neutronic properties of the pin being burned in the cell burnup code and obtain the characteristics of each batch present in the core at any particular time, which should allow in some way to calculate the status of the whole core. if the core contains three batches, the cell burnup calculation should be performed in such a way that the timesteps chosen would be repeated three times, and each of the three sets of








84


timesteps would span exactly the expected duration of one core cycle. This would allow the data from the corresponding timesteps in each cycle to be grouped into a core timestep. Figure 3.1.4 illustrates this "timestep synthesis" from a hypothetical cell burnup scheme in which five timesteps were allowed for each of the three cycles of core residence of the fuel. The data sought is the infinite multiplication factor of the core, based on the unpoisoned multiplication factor of the pin through its life. The figure shows the conceptual grouping of each timestep's data, but not the calculations involved in it, which will be dicussed later. Note that the timestep pattern is repeated in each cycle of the cell burnup scheme, in order to make the timesteps correspond to the same time of core life.

However, there are several ways of specifying the pin cell

burnup and of doing the calculations for collapsing the cell timesteps into core life timesteps. Some of the systems used for pin cell burnup specification will now be examined.

The simplest scheme, as mentioned before, burns the fuel cell without any poisoning and at the nominal power, for the length of time that is estimated that the fuel will remain in the core. Then the multiplication factors of the cell timesteps are collapsed in order to obtain the core multiplication factor through the core life. An interpolation or extrapolation of the curve of core multiplication factor as it reaches a value of unity determines the end of the core cycle. The core cycle length is then optionally used to perform new iterations of the fuel cell burnup calculation,







85






Pin Cell K00




1.30 FsCycle Second Cycle 1 Third Cycle




1.30







K .







1.15


1.00 LL F Tie
E Dt



Figure 3.1.4. Grouping of Pin Cell Data into Core Data.








86

until it agrees with the cycle length estimated by the timestep collapsing procedure.

This scheme, however, involves gross errors because the

absence of soluble boron causes variations of some percent in the cross sections of some isotopes such as U-235 and Pu-239. This entails errors in isotope concentrations and as a result, errors in the multiplication factor calculations.

Another cell burnup method, which partially solves this

problem, specifies a soluble boron concentration in the moderator with a value similar to the time-averaged boron concentration of a real core. This concentration is approximately 400 ppmo But then

the fuel cell multiplication factors can not be directly used for the calculation of the core multiplication factor, because they involve the soluble boron poisoning. The problem is solved by using a very short "pseudo-burnup" timestep, with no soluble boron, after each real burnup timestep. A "pseudo-burnup" timestep is a very short timestep which practically adds no burnup to the fuel, so that the isotopics are not altered, but still allows for the eigenvalue calculation in the boron-free configuration. This way, the fuel is always burned with a spectrum influenced by the 400 pPM of soluble boron, but the multiplication factors are obtained from the pseudo-burnup timesteps which contain no boron, and the "clean" batch reactivity can still be obtained through the fuel life.

Figure 3.1.5 illustrates this method, showing the boron specification and the multiplication factors that may be obtained through the core life.




Full Text
LIST OF FIGURES
Page
2.2.1. Pin Cell Geometry 26
2.2.2. Geometry Used for MONA Criticality Calculations . 35
2.2.3. Code and Data Flow for Preliminary Burnup Calculation 37
2.2.4. Comparison of Boron Letdown from Leopard and Burnup. 40
2.3.1. Soluble Boron Worth at Critical vs. Burnup 46
2.3.2. Soluble Boron Worth at Critical. Changing Pitch . 47
2.3.3. Infinite Multiplication Factor vs. Pitch 53
2.3.4. Eta and Fast Fission Factor vs. Pitch 56
2.3.5. Resonance Escape Probability and Thermal Utilization 58
2.3.6. Thermal-to-Fast Flux Ratio 60
2.3.7. Discharge Burnup of 25% Power Density Core vs. Pitch 64
2.3.8. U-235 and Plutonium Usage vs. Pitch 69
3.1.1. Basic Steps in a Burnup Calculation 75
3.1.2. Examples of Burnup Conditions and Timestep
Specification for Fuel Cell Burnup Calculation ... 78
3.1.3. Geometry and Composition Specification for a PDQ-7
1/4-assembly Burnup Calculation 81
3.1.4. Grouping of Pin Cell Data into Core Data 85
3.1.5. Constant-boron Pin Cell Burning with Boron-free
Pseudo-burnup Steps 87
ix


83
classification, evolving from a well known calculational model.
The innovative core model used for the SHARP is an improvement on
the accuracy of the results, resembling closer the real reactor
life. It is achieved by the proper handling of significant core
history data, which appear to have been previously neglected. An
explanation of the data handling by these codes follows.
These simple calculational models are normally based on a pin
cell burnup calculation and often use the infinite multiplication
factor calculations in order to obtain an estimate of the core
criticality state or its reactivity.
In the simplest model, a pin cell would be burned for its
whole life without any poisoning, and at a constant power level.
The characteristics of a pin are a good representation of the fuel
batch to which it belongs, since the pin is surrounded by many fuel
pins alike. The only exceptions are the pins facing a fuel
assembly of a different batch (and therefore having a sharply
different burnup status) or facing the reflector. Then, if the
fuel pin is burned for its entire life, it certainly passes through
the stages corresponding to each batch in the core. This means
that it should be possible to take seme of the neutronic properties
of the pin being burned in the cell burnup code and obtain the
characteristics of each batch present in the core at any particular
time, which should allow in some way to calculate the status of the
whole core. If the core contains three batches, the cell burnup
calculation should be performed in such a way that the timesteps
chosen would be repeated three times, and each of the three sets of


123
The way in which CRIBUR accounts for the reactivity worth of
soluble boron is also a unique feature in a core model of this
simplicity. To the best of the author's knowledge, any code of a
complexity level comparable to CRIBUR's considers soluble boron
worth as an empirical constant, hardwired in the code; frequently
this variable is not established or used, as discussed in Section
3.1. However, an analysis of EPRI-CELL calculations show that
soluble boron worth may vary as much as 25% in the range of
concentrations that may be normally encountered in a real PWR. In
addition to that, soluble boron worth was shown to depend heavily
on the fuel cell geometry (see Chapter 2). This suggested the need
to evaluate soluble boron worth for each particular reactor case,
and if at all possible, it should be evaluated for different
concentration levels.
CRIBUR will accept a soluble boron worth constant if so
indicated by the operator, or may even provide a default constant
which is representative of the average value found in a standard
PWR. However, it also provides the opportunity of calculating it
for each case and to make it a function of the boron concentration.
If a soluble boron worth calculation is desired, it is
necessary to specify some "fake-burnup" steps in one of the
fuel-burnup cycles, in a similar way as explained on Figure 3.1.5.
Since the fuel-burnup calculation performed to provide data to
CRIBUR follows the soluble boron letdown curve for each core cycle
through he whole fuel life, the calculation has seme burnup steps
in which the boron concentration is high, some in which it is near


281
780 FORI^ITOCT: PRINT USING
A$;I?RC(I,3)?RC(I,4);RC(I,5);RC(I,6);RC(I,7);RC(I,8):NEXTI
790 INPUT"ANY CHANGED STEP AND COLUMN";CH,CC:IFCH=0THEN810
800 COCC+2: INPUT "NEW VALUE";RC(CH,CC) :CH=0:GOT0760
810 COO :CLS: PRINT "CHECK BATCH WITH LBP AND LBP'S BATCHWISE WORTH
(%K/K) AT BOC, EOC ": PRINT "BATCH "BP, "BOC %K/K="BB,"EOC %K/K="BN
820 INPUT "NEW VALUES FOR ALL, OR HIT ENTER; CC, CH, A: IFCOOTHEN8 40
830 BP=CC:BB=CH:BN=A:GCT0810
840 CO0:CLS:PRINTnCHECK POWER LEVEL AND FUEL
LOADING":PRINT"POWER="PL"MW THERMAL","FUEL LOADING="FL"MTU"
850 INPUT "NEW VALUES FOR BOTH, OR HIT 'ENTER' ";CC,CH:IFCO0THEN870
860 PL=CC:FL=CH:GOT0840
870 CLS:PRINT"CHECK BORON WORTH EVALUATION POINTS": IFBEO0THEN900
880 PR3NT"NO BORON WORTH EVALUATION POINTS. BORON
WORTH="BW:INPUT"NEW VALUE, OR JUST 'ENTER;CH:IFCH=0THEN980
890 BW=CH:CH=0:GOT0870
900 IFCA=0THEN930ELSEPRINT"STEPS TO BE KILLED AFTER B-WORTH
EVALUATION (IN INCREASING ORDER)
910 FORI=lTOCA:PRINTBK(I),;iNEXTI
920 PRINT" ":PRINT" "
930 PRINT"TIMESTEPS FOR EVALUATION :"
940 F0RI=1T0BE/2:PRINTBW(I*2-1),BW(I*2) :NEXTI
950 INPUT"WISH TO INPUT A NEW SET (1), OR NO (0) ";CH:IFCH=0THEN980
960 FORI=lTOBE/2;PRINT"PAIR #"I:INPUTBW(I*2-1) ,BW(I*2) :NEXTI
970 CH=0:GOT0870
980 REM ****************************************


286
1920 LPRINT" "¡LPRINT"NEW STEPS ENDING TIMES ";
1930 FORJ=lTOCT:K=RC(J,I+2):LPRINT USING B$;R(K,9);:NEXTJ
1940 LPRINT" LPRINT "NEW TIMESTEP DURATION ";
1950 FORJ=lTOCT:K=RC(Jf 1+2) ¡X=R(Kf9)-OK¡LPRINT USING
B$;X;¡OK=R(K,9)¡NEXTJ¡B$="###.#####"¡GOTO1980
1960 B$="###.#####"¡LPRINT" LPRINT"INITIAL STEPS K-INF.
1970 FORJ=lTOCT:K=RC(J,I+2):LPRINT USING B$;R(K,1);¡NEXTJ¡GOTO2000
1980 LPRINT" ":LPRINT"RMCTIVITY-CORRECTED K-INF";
1990 FORJ=lTOCT:K=RC(JrI+2):LPRINT USING B$?R(K,0);¡NEXTJ¡GOTO2040
2000 LPRINT" ":LPRINT"CLEAN K-INF.
2010 FORJ=lTOCT:K=RC(J,1+2)¡LPRINT USING B$;R(K-lfll);NEXTJ
2020 LPRINT" "LPRINT"INITIAL BORON PPM. ";
2030 FORJ=lTOCT:K=RC(J, 1+2) LPRINT USING A$;R(Kf6) ; NEXTJ¡G0TO2060
2040 LPRINT" ¡LPRINT"NEW BORON PPM.
2050 FORJ=1TOCTK=RC(J,I+2) ¡LPRINT USING A$;R(Kf7); ¡NEXrj¡GarO2080
2060 LPRINT" ";LPRINT"INITIAL RELATIVE POWERS
2070 FORJ=lTOCT:K=RC(J,1+2)¡LPRINT USING
B$;R(Kf2);¡NEXTJ¡B$="######.##"¡GOTO1920
2080 LPRINT" LPRINT "NEW RELATIVE POWERS ";
2090 FORJ=lTOCT¡K=RC(J, 1+2) ¡LPRINT USING B$;R(K,5) ; ¡NEXTJ
2100 LPRINT" "¡LPRINT" "¡LPRINT" "
2110 NEXTI
2120 LPRINT" "¡LPRINT" "¡X=0
2130 FORI=lTONB¡LPRINT"ESTIMATED BURNUP FOR
CYCLE"I"IS"RR(I,0) "MWD/MTU"¡X=X+RR(Ir0) ¡NEXTI
2140 K=RC(CT,3) ¡LERINT"END OF LIFE BURNUP IS ESTIMATED


74
does not require detailed specification of compositions and
geometries, as the two previous types of studies did.
d). Burnup studies performed in universities or
research-oriented organizations for developmental studies of new
reactor concepts, for preliminary reactor design or fuel
management. These studies may range in their specificity anywhere
from the very rough empirical formulas used for fuel resources
utilization and policy studies to a level close to the vendor or
utility calculations. However, they tend to be more concerned with
particular theoretical details, which call for calculational
methods specific to each study. These studies normally follow
standard industrial calculational procedures until a certain point
where the particular aspect of the study calls for a specific
calculational tool, which is often developed for the occasion.
A wide range of other types of burnup or fuel cycle studies
exist, but the four categories stated above illustrate the
different levels of accuracy or detail that may be sought depending
on the purpose of the study, and how they require different
calculational costs, manpower needs and calculational tools.
For every burnup study (except for the very simplest ones,
where just an empirical formula is applied),two main calculational
steps are performed. Each of these steps can have different levels
of sophistication depending on the particular needs of the study,
and in some cases one or both of the steps may need to be repeated
iteratively in order to obtain the accuracy required by the study.
Figure 3.1.1 shows the basic flow of data between the two steps.


7
- Reduced heat stored in the fuel, and increased
available heat capacity, i.e., in case of an accident the core is
capable of retaining more heat before suffering damages.
- Milder accident conditions and emergency cooling
requirements, due to the lower fuel temperature and lower power
density (which means reduced decay heat generation).
b). Frcm the neutronics point of view:
- Less Doppler broadening of resonances, and therefore
extra reactivity available.
- Reduced xenon concentration due to the lower neutron
flux, which means again some extra reactivity available.
- Reduced xenon oscillation problems due to the lower,
overall xenon concentration.
c). Frcm the fuel cycle point of view:
- Longer burnup achievable from the same initial fuel,
due to the extra reactivities mentioned above.
- Largely increased time between refuelings due to the
double effect of larger burnups achievable and the lower power
generation of each core. This would reduce the impact of refueling
outages on plant availability tending to increase it. The increased
availability would result in proportionally increased energy
generation in a given time period, thus reducing the impact of
capital (which would be a basically fixed total cost) on energy
generation cost.
- Reduced relative activity of the fuel at discharge
per MWd generated due to the largely increased residence of the


148
processing to analyze the key issues and the significant
differences between the various cases studied.
The results shown in this section are directly obtained from
the FDQ-7 CRIBUR burnup optimization calculations or frcan simple
manipulations of their output data.
The first set of data which is needed for core optimization
analysis is the U-235 and plutonium isotopes number densities at
loading and discharge. However, the number densities per se are
not a common means of comparing core performances and they are
listed in Appendix D1 for reference purposes. The figures are used
later in this chapter in order to obtain other commonly used
comparative indices. As stated before, all the burnup studies
were done under the assumption of once-through fuel management, and
thus, reprocessing was not considered. In these studies, the only
fissile material loaded into the core was U-235. However, for the
sake of completeness and because it will be necessary to reprocess
spent fuel to recycle fissile species, some figures concerning
these discharged isotopes have been calculated and are presented in
this section.
Table 4.2.1 presents the comparison of burnup achievements for
the different cores, fissile species accounting, and ore usage.
The first two columns in Table 4.2.1 define the core case studied.
Note that the two parameters varied through the study are the power
level and the fuel cell pitch. The full-power, standard pitch core
results are included as the reference case for comparison purposes.
The third column shows the effective core cycle duration as


117
it is then possible to express the ratio S1/S1+S2) as
SI/(S1+S2) = 2.405 J, (2.405) Xo J, (Xo) or
(2.405 Jt (2.405))
SI/(S1+S2) = (1 (n-l)/n ) J, (2.405 (n-l)/n)
J, (2.405)
where the Jt Bessel function can be expressed in a series
development whose first three terms are
J, (X) = (X/2) (X /16) + (X /384)
This completes the calculation of SI/(S1+S2). If the whole
shape factor is now included in the peripheral batch non-leakage
probability calculation, the expression becomes
Pnlb = 1 ((1-Pnlr) *Ei$/$b) ($1/1-$) Pc
(SI/(S1+S2))
where $¡ are the batch fluxes of the ideal reactor (following
Bessel's Jo) and $b are the actual batch fluxes, obtained from the
fuel burning calculations. Pc is the proportionality constant or
scaling factor, which value was adjusted for the present system to
0.35"and it can be reevaluated for different core configurations
using simple diffusion theory codes. Note that in the expression
above, S1/(S1+S2) is equivalent to since these were the
fluxes associated with the equivalent homogeneous bare reactor.
Simplifying assumptions are made at this point in order to
reduce data input to CRIBUR. The assumptions and approximations


63
2.3.3. Burnup Achievements and Isotopic Inventories.
As explained in Section 2.2.3, the preliminary burnup
calculations were performed with the LEOPARD code, using a pin
enrichment equal to the average enrichment of the core, and burning
it with a varying soluble boron concentration that would keep the
core as close as possible to criticality. Burnup explorations were
done for the standard reference core 3A, for the 25% power density
core 4A, and as a scope extension, for the 20% power density,
preheater core 8. All of them were examined at the standard core's
pitch of 1.2573 cm; cores 3A and 8 were also investigated at one
larger pitch, and core 4A was studied for a wide range of pitches,
since it is the most representative low power density core. The
multiple burnup calculations performed for core 4A have the purpose
of determining the parametric effect of pitch on the discharge
burnup level, as well as giving an indication of the optimum pitch
value.
Table 2.3.2 shows the main burnup and isotopic results
obtained iron the preliminary burnup calculations.
Reducing power density to 25% of the standard level while
keeping the standard core's pitch results in an increment of burnup
of 9%. However, the burnup increment can be improved to about 16%
by increasing the pitch of the low power reactor to about 1.3 or
1.4 cm. Figure 2.3.7 shows the burnup levels achievable as pitch
is varied on core 4A. Note the important fact that the maximum
burnup is not achieved at the pitch value having maximum BGL
reactivity (shown in Figure 2.3.3) but at a smaller pitch. This


150
obtained by CRIBUR calculations. The figures are, thus, full-power
hours of operation before E.O.L. is reached. Of course, in this
context "full power" refers to the total power level assigned to
each particular core. Note that for all three reduced power levels
studied, the maximum cycle length is obtained for a pitch of 1.35
cm as compared to the 1.25 cm of the standard core. It seems
reasonable to assume that the standard power core would also
present a longer cycle life with a pitch of 1.35 cm, but
thermal-hydraulics and safety limitations prevent the use of such
pitch, as explained in Chapter V. The improved burnups achievable
with the different pitches for the different fractions of full
power translate into cycle lengths higher than the factor of two-,
three- or four-fold that could be expected from the cores operating
at the 50, 33 or 25 percent power level with respect to the
standard core. Thus, the 50%, 33% and 25% power cores with the
1.35 cm pitch show a cycle length which is respectively 2.11, 3.18
and 4.26 times the cycle length of the standard core. This means
that for the 50% power core (which is the most commercially
attractive of the cases considered) the same initial fuel load
would provide not only the natural double core life as compared to
the standard plant but an additional 11%. The 33% and the 25%
power cores would provide additional cycle lengths of 18% and 26%
respectively. However, this additional cycle lengths, although
very important from the plant cycle and refueling outage schedule
viewpoints, should not be confused with the additional burnup
achieved by each core, which is discussed below. The basic


304
13. Amster, H., and Suarez, R., "The Calculation of Thermal
Constants Averaged over a Wigner-Wilkins Flux Spectrum:
Description of the SOFOCATE Code," WAPD-TM-39, Atomic Power
Division, Westinghouse Electric Corp. (1957).
14. Macnabb, W.V., "Two Near-term Alternatives for Improved
Nuclear Fuel Utilization," Nuclear Technology, Vol. 49 (1980).
15. Driscoll, M.J., "Improved FWR Core Designs, Bimonthly Progress
Report No. 2," IPWRCD-2, MIT Nuclear Engineering Department
(1980).
16. Miller, M., "Thermalhydraulics Analysis of Alternative PWR
Core Designs," Master's Thesis, Nuclear Engineering
Department, University of Florida (1981).
17. Cadwell, W.R., "PDQ-7 Reference Manual," WAPD-TM-678, Atomic
Power Division, Westinghouse Electric Corp. (1967).
18. Ver Planck, D.M., "Manual for the Reactor Analysis Program
SIMULATE," YAEC 1158, Electric Power Research Institute
(1978).
19. Eich, W.A., Cobb, W.R., and Tivel, D.E., "EPRI-CELL Code
Description," EPRI ARMP System Documentation, Electric Power
Research Institute (1975).
20. Joanou, G.D., and Dudek, J.S., "GAM-I: A consistent P-1
Multigroup Code for the Calculation of Fast Neutron Spectra
and Multigroup Constants," GA-1850 (1961).
21. Honeck, H.C., "THERMOS, A Thermalization Transport Theory Code
for Reactor Lattice Calculations," BNL 5826 (1961).
22. England, T.R., "CINDER: A One-point Depletion and Fission
Product Program," WAPD-TM-334, Atomic Power Division,
Westinghouse Electric Corp. (1964).
23. Rothleder, B.M., "NUPUNCHER Code Description," EPRI ARMP
System Documentation, Electric Power Research Institute
(1975).
24. Breen, R.J., Marlowe, O.J., and Pfeifer, C.J., "HARMONY:
System for Nuclear Reactor Depletion Computation,"
WAPD-TM-478, Atomic Power Division, Westinghouse Electric
Corp. (1965).
25. Rothleder, B.M., Blake, R.A., Fisher, J.R., and Kendrik, E.D.,
"PWR Core Modeling Procedures for Advanced Recycle Methodology
Program," Research Project 976-1, Electric Power Research
Institute (1979).


105
very strong thermal neutron absorber. Furthermore, the power level
of the assembly through life affects in a significant way the
concentration of most isotopes through the core life, which is a
further reason calling for an accurate specification of the power
level at all times.
Thus, when the quarter-assembly modeled in PDQ-7 is burned for
its complete core life (a series of as many consecutive cycles as
fuel batches are present in the core), an estimate of the soluble
boron concentration versus time is specified for each of the
cycles, as well as the power level of the batch at each specific
moment. A first guess of possible soluble boron concentration and
power level evolutions through the whole life of an assembly are
shown in Figure 3.2.1.
Soluble
Boron
Figure 3.2.1. Estimates of Boron Letdown and Relative Power.


103
estimate of the batch multiplication factor should be closer than
EPRI-CELL's, because of its more realistic modeling of the really
repeating pattern in the fuel batch, which is the fuel assembly,
rather than the fuel pin.
It should be pointed out that the use of "simplified"
geometries for the PDQ-7 calculation could result in severely
distorted results. The 1/4 assembly requires the specification of
different compositions due to the presence of the water holes and
instrument thimble, and one might be tempted to use a simple 2X2
pin geometry, trying to take advantage of PDQ-7's flexibility while
avoiding the additional data handling complexity of the complete
1/4 assembly. The results of a 2X2 pin array burning can be just
disastrous. Not only it does not take into account the
differential effects of the unevenly distributed water holes and
interassembly gaps on the different fuel pins of the assembly, but
being composed of just fuel element cells, it does not include any
material simulating the effects of the structural materials, which
in EPRI-CELL are represented by the buffer region. Such a burnup
calculation may result in isotopic errors of the order of 13% for
-235 and 20% for Pu-239 at a fuel burnup level of 40 GWd/MTU, with
the associated errors in the calculation of batch multiplication
factors.
When PDQ-7 is finally selected for the fuel-burning
calculation, a third code from the ARMP package needs also to be
used in order to simplify the point-burnup calculation. This code
is NUPUNCHER(23) and its function is to translate the


131
L
¡continuation
Icalculation of Stems' Relative Powers!
¡Calculation of Core Effective Multiplication Factor
:Jf
Calculation of New Boron Concentrations for Criticality
2jl
Calculation of New Core Li^e and New Timestep Lengths
Iflpply Convergence Accelerator
¡Core K's Close Enough_.to iLBefore, Boron Adjustment ?lN0-1go to A
YES
ICalculation_of S.teMdse Jurnups With New Set of Relative Powers
New Cycle Burnups Agree With Old Iteration's ?1 YES-
NO
i.
AERly, Conv-ejcgence.. Accelerator for Relative Powersl
1
Calculation of "Present Core Status" Burnups
Storage as "Old Iteration" Burnups
J
!GQ..tP.Bl-
Erint Results
Figure 3.2.7. Continuation.


149
Table 4.2.1. Burnup Achievements, Fissile Isotope Usage and Ore Usage.
Pite
cm
i Power
%
Cycle
F.P.H.
Dis. BU.
MWd/MTU
Dis. Fis.
MF/MWd
(a)
Rec. Fis.
% Dis/Ld.
Net Fis.
MT/MWd
(a)
Pu Dis.
MT/MWd
(a)
U308
OT/MWd
(b)
1.20
50
15588
35172
4.195
47.16
4.619
2.751
1.899
1.20
33
23751
35725
4.020
45.90
4.658
2.685
1.870
1.20
25
32058
36166
3.914
45.24
4.657
2.651
1.847
1.25
100(c)
8018
36183
3.637
42.06
4.931
2.465
1.846
1.25
50
16529
37295
3.418
40.73
4.894
2.414
1.791
1.25
33
24837
37359
3.357
40.07
4.941
2.381
1.788
1.25
25
33359
37634
3.287
39.52
4.950
2.354
1.775
1.30
50
16704
37689
3.054
36.77
5.171
2.218
1.772
1.30
33
25329
38099
2.949
35.90
5.187
2.176
1.753
1.30
25
33924
38271
2.897
35.42
5.203
2.154
1.745
1.35
50
16901
38134
2.720
33.14
5.409
2.071
1.752
1.35
33
25526
38397
2.672
32.77
5.402
2.033
1.740
1.35
25
34162
38541
2.628
32.36
5.415
2.013
1.733
1.40
50
16534
37306
2.636
31.43
5.673
1.967
1.791
1.40
33
24994
37595
2.555
30.69
5.691
1.935
1.777
1.40
25
33415
37698
2.514
30.29
5.710
1.917
1.772
(a):* 1.0 E-7 (b):* 1.0 E-4 (c): Reference Case.


292
3220 FORJ=3TONEH-2
3230 P0RK=CMK)1STEP-1 sIFRC (I, J) >BK(K) THENRC(I, J) =RC (I, J) -K:G0T0325Q
3240 NEXTK
3250 NEXTJrl
3260 RETURN
3270 CLS
3280 PRINT"CYCLES1 BURNUPS MATRIX"
3290 F0RI=1T0NB
3300 FORJ=0TO2:PRINTRR(I, J),;:NEXTJ
3310 PRINT" ":NEXTI
3320 RETURN
3330 CLS:PRINTnSTEPn,nEND TIME","WORK K"f"CORE K"
3340 FORI=1TOTS:PRINTI,R(I,9),R(I,0),;
3350 IFI<=CTTHENPRINTRC(1,0)ELSEPRINT" "
3360 NEXTI
3370 RETURN


Page
IV.DETAILED SHARP BUHNOP OPTIMIZATION STUDIES 143
4.1. Cases Chosen for Final Study 143
4.2. Results of Optimization 147
4.2.1. Neutronics and Isotopic Results .... 147
4.2.2. Plant Operational Data Variations . 160
4.2.3. Ore Usage and Enrichment Needs .... 175
V. THERMAL-HYDRAULICS AND ECONOMIC CONSIDERATIONS .... 179
5.1. Safety Related Thermal-hydraulic Considerations 179
5.1.1. Motivation 179
5.1.2. Thermal-hydraulic Studies 180
5.2. Economic Evaluation 190
5.2.1. Introduction 190
5.2.2. The Economic Comparison Studies .... 191
VI.CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 201
6.1. Introduction 201
6.2. Neutronics, Burnup and Ore Usage 204
6.3. Plant Operations Considerations 207
6.4. Economic Effects 209
6.5. Suggestions for Future Research 211
APPENDICES
A. METHODS OF IMPROVING BURNUP IN PWR'S 213
A.l. Introduction 213
A.1.1. Motivation and Constraints ..... 213
A.1.2. Schemes for Inproving Burnup ..... 215
A.2. Techniques for Improvement of Burnup ..... 219
A.2.1. Increasing Number of Batches 219
A.2.2. Burnable Poisons 223
A.2.3. Low-Leakage Fuel Management 229
A.2.4. Alterations of Fuel-to-Water Ratio . 233
A.2.5. Low Power Density 237
A.2.6. Flattening Axial Power Distribution 239
A.2.7. Increasing Enrichment 240
A.2.8. End of Cycle Coastdown 242
A.2.9. Other Possibilities of Minor Importance 244
Bl. ERT CODE DESCRIPTION 248
B2. EHROG CODE DESCRIPTION 251
B3. MONA CODE DESCRIPTION 253
B4. LEOPARD CODE DESCRIPTION 255
vi


291
2960 P0RI=1T0NB:CH=ABS((RR(I,0)-FR(I,2) )/RR(I,2)) :IFCH>XXTHENXX=CH
2970 NEXTI
2980 IFKX<=0.01THEN1850
2990 FORI=LTOTS:R(I,8) = (R(I,5)+R(I,8))/2:NEXTI
3000 RP=8:GOSUB2380
3010 FORI=1TONB:RR(I,2)=RR(I,0):NEXTI
3020 GOTO1340
3030 REM ASSIGNING NEW STEP ENDING TIMES
3040 X=0:Y=R(RC(CT,3),9)
3050 FORJ=2TONB:CH=RC (CT,J+D+l:CC=RC (CT,J+2)
3060 FORI=CHTOCC:R(If9)=R(If9)+(A-Y)*(J-1) :NEXTI,J
3070 F0RI=1T0NB
3080 FORJ=lTOCT:CH=RC(J,I+2) :Y=R(CH,9) :IFY>=AfXTHEN3100
3090 IFJ=CTTHEN3100ELSEIFY 3100 R(CH,9)=X+A:NEXTJ
3110 X=X+A:NEXTI
3120 F0RI=1T0NB:CC=CT
3130 CH=RC(CC,I+2):IFR(CH-lf9)<>A*ITHEN3150
3140 R(CH,7) =0:COCC-1:IFCC=0THEN3150ELSE3130
3150 NEXTI
3160 IFMK<0.01THEN2930ELSE1330
3170 F0RI=CAT01STEP-1
3180 FORJ=BK(I)TOTS-1
3190 PQRK=0TO12:R(J,K)=R(J+1,K):NEXTK,J,I
3200 TS=TS-CA
3210 F0RI=1T0CT


272
d). Date of reception of the construction permit.
e). Date of commercial operation.
f). Interest rate.
The output of the code is extremely flexible, and is basically
defined by the user. It can be reduced to a short one-page summary
of the major accounts, or it can include detailed listings of all
the subaccounts. The code provides also for the possibility of
printing a total cumulative cash flow curve.


114
the reflector savings to the core dimensions to obtain an
"equivalent" bare system. In any case, the leakage calculation is
still based on a homogeneous, cylindrical core whose flux magnitude
in the radial direction is assumed to follow a Jo Bessel function.
It is obvious that real reactors do not have such a flux shape,
mainly because of the geometric disposition of the different fuel
batches in the core.
Given a certain total neutron population of a reactor, the
probability of leakage must obviously depend on the geometric
distribution of the neutrons inside the core. In other words, for
a given reactor type, if the neutron population is concentrated
near the centerline, there should be a lower escape probability
than if most of the neutrons live near the periphery. This
consideration suggested the need of modifying the non-leakage
probability with a factor that would somehow take into account the
geometric difference between the real flux distribution of the core
being studied and the theoretical Jo Bessel shape. For this reason
the author called this factor the "Shape Factor".
One assumption used in calculating the shape factor is the
previous assumption that all radial leakage comes from the
peripheral fuel batch. A second simplifying assumption states that
the actual radial leakage from the reactor is proportional to the
ratio of the neutron population of the outer batch over the
population of the whole core. In other words, it is assumed that a
reactor having 50% of the total core population in the outer batch
presents more radial leakage probability than a core having only


187
that burnup-optimized pitch configuration, as well as at the
standard value. Figure 5.1.5 shows the results of these
calculations.
Figure 5.1.5 is a plot of the percent savings relative to
standard core values achieved in the minimum DNBR, and in the
maximum and average fuel temperatures as a function of the core
power density level. Both the standard fuel pitch and the
burnup-optimized fuel pitch have been considered. These
calculations assume a coolant flow rate that maintains the outlet
coolant enthalpy at a the nominal standard core level.
When the standard pitch is considered, reducing the power
density causes a constant improvement in the minimum DNBR, tending
to an infinite value as the "zero-power" reactor is approached.
Obviously, the savings for the 100% power level is zero. Both the
maximum and the average fuel temperatures show an improvement, but
it is clearly bound, and tends to level off. Note that the fuel
temperatures (maximum or average) are not affected by the pitch
value.
When the burnup-optimized pitch is considered, the same
coolant mass flow requires a lower fluid velocity, which tends to
worsen the heat transfer conditions. This fact is apparent in the
corresponding DNBR curve, which appears shifted downwards from the
one corresponding to the standard pitch. This is the fact that
prevents the standard full-power reactor from operating at the
burnup-optimized fuel pitch, as was indicated in the preliminary
calculations chapter. The minimum DNBR would be 20% worse than it


BUKNUP AND FEASIBILITY' STUDY OF
LOW PCWER DENSITY FWRS
BY
CESAR MOLINS-BARTRA
A DISSERTATION PRESENTED TO IRE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1981

r
k

To Him who cares for all
and who gave us this world,
as an insignificant sample of my deep appreciation

ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation to the
members of his supervisory committee N.J. Diaz, E.E. Carroll,
E.T. Dugan, C.C. Oliver and M.L. Muga for their interest and help
in preparing this dissertation. Special thanks are given to Dr.
N.J. Diaz for originating the dissertation subject and providing
direction of the research project. The author is deeply indebted
to Dr. E.T. Dugan for providing continued technical guidance and
criticism and invaluable help in obtaining necessary calculational
tools, together with endless patience in correcting this report.
The author is indebted to J.A. Wethington, Jr., M.J. Ohanian,
G.R. Dalton, W.H. Ellis, G.S. Roessler and W.G. Vernetson for the
friendship and support given through the years of this research.
The main body of the calculations of this study were performed
at Oak Ridge National Laboratory, Tennessee, where the author met a
number of great friends who were instrumental in the execution of
this work. Special thanks are given to J.C. Cleveland for making
the necessary arrangements that made the stay at O.R.N.L. possible
and for his technical advice and assistance; to Rafael Perez and
iii

Pedro Otaduy (the Godfather and the executive coordinator of the
Spanish colony at O.R.N.L., respectively) and to the rest of the
Spanish group for their continued moral support and friendship; and
to F. Clark and T. Priiran for their administrative and technical
advice.
The author acknowledges that during his tenure at the
University of Florida he was financially supported by the National
Science Foundation, the Hauck fund, Oak Ridge National Laboratory
and two consecutive University of Florida Graduate School
Fellowships.
The author keeps a special thought for Keith Johnson for his
untiring help in computing matters, and to Pedro Salas for the many
hours of friendship and sharing in happiness and in distress.
iv

TABLE OF CONTENTS
gage.
ACKNOWLEDGEMENTS iii
LIST OF TABLES viii
LIST OF FIGURES ix
ABSTRACT xii
I.INTRODUCTION 1
1.1. Background 1
1.2. The SHARP Concept 4
1.3. Previous Studies of Low Power Density Cores . 9
1.4. Purpose of this Study 10
II.SCOPING WORK 14
2.1. Problem Framing 14
2.2. Calculational Methods for the Scoping Study . 25
2.2.1. Heat Transfer Calculations 25
2.2.2. Neutron Cross Section Calculations . 31
2.2.3. Criticality and Burnup Methods .... 33
2.3. Results of Preliminary Work 41
2.3.1. Reactivity Coefficients 41
2.3.2. Neutronics and Safety Considerations 51
2.3.3. Burnup Achievements and Isotopic
Inventories 63
2.4. Scoping Work Conclusions 70
III.BURNUP CALCULATION METHODS 72
3.1. Burnup Calculational Methods and Present Needs 72
3.2. Method Developed for this Study 99
3.2.1. The Fuel-burning Codes 99
3.2.2. The CRIBUR Core Model 106
3.3. Benchmarking of the Burnup Calculations .... 132
3.3.1. Available References for Comparison . 132
3.3.2. Criticality Benchmarking 135
3.3.3. Power Distribution Benchmarking .... 140
v

Page
IV.DETAILED SHARP BUHNOP OPTIMIZATION STUDIES 143
4.1. Cases Chosen for Final Study 143
4.2. Results of Optimization 147
4.2.1. Neutronics and Isotopic Results .... 147
4.2.2. Plant Operational Data Variations . 160
4.2.3. Ore Usage and Enrichment Needs .... 175
V. THERMAL-HYDRAULICS AND ECONOMIC CONSIDERATIONS .... 179
5.1. Safety Related Thermal-hydraulic Considerations 179
5.1.1. Motivation 179
5.1.2. Thermal-hydraulic Studies 180
5.2. Economic Evaluation 190
5.2.1. Introduction 190
5.2.2. The Economic Comparison Studies .... 191
VI.CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 201
6.1. Introduction 201
6.2. Neutronics, Burnup and Ore Usage 204
6.3. Plant Operations Considerations 207
6.4. Economic Effects 209
6.5. Suggestions for Future Research 211
APPENDICES
A. METHODS OF IMPROVING BURNUP IN PWR'S 213
A.l. Introduction 213
A.1.1. Motivation and Constraints ..... 213
A.1.2. Schemes for Inproving Burnup ..... 215
A.2. Techniques for Improvement of Burnup ..... 219
A.2.1. Increasing Number of Batches 219
A.2.2. Burnable Poisons 223
A.2.3. Low-Leakage Fuel Management 229
A.2.4. Alterations of Fuel-to-Water Ratio . 233
A.2.5. Low Power Density 237
A.2.6. Flattening Axial Power Distribution 239
A.2.7. Increasing Enrichment 240
A.2.8. End of Cycle Coastdown 242
A.2.9. Other Possibilities of Minor Importance 244
Bl. ERT CODE DESCRIPTION 248
B2. EHROG CODE DESCRIPTION 251
B3. MONA CODE DESCRIPTION 253
B4. LEOPARD CODE DESCRIPTION 255
vi

Page.
B5. EERI-CELL CODE DESCRIPTION ............. 258
B6. NPUNCHER CODE DESCRIPTION 263
B7. PDQ-7 CODE DESCRIPTION 265
B8. TEMPRET CODE DESCRIPTION .............. 269
B9. CONCBFT-IV CODE DESCRIPTION 271
BIO. GEM CODE DESCRIPTION 273
Bll. FQWERCO CODE DESCRIPTION 275
Cl. CRIBDR CODE SOURCE LISTING 277
C2. SAMPLE RUN OF CRIBUR 293
D. ISOTOPIC AND SPECTRAL DATA FROM BURNUP CALCULATIONS. 299
REFERENCES 303
BIOGRAPHICAL SKETCH 306
vii

LIST OF TABLES
Page
2.1.1. Main Core Parameters for the Standard Reactor. ... 19
2.1.2. Basic Fuel Cell of the Standard Reactor 19
2.1.3. Basic Cell Composition. Core #1 21
2.1.4. Identification of Cases for the Scoping Study ... 23
2.1.5. Tabulation of Case Number Densities Which Differ
from Those of Case #1 24
2.2.1. Basic Cell Temperatures vs. Power Level 30
2.3.1. Core Reactivities at B.O.L 50
2.3.2. Scoping Burnup and Isotopic Results 65
3.3.1. CRIBUR and Batch-averaging Method Criticality Errors 137
3.3.2. Effects of Code and Data Alterations on Core K-eff. 139
4.2.1. Burnup Achievements, Fissile Isotope Usage and
Ore Usage ...... 149
4.2.2. Plant Operational Data Comparison 161
4.2.3. Ore, Enrichment and Plutonium Normalized to 1125 MWe 176
5.2.1. Capital Costs (Mills/KWh) and Percent Increases
over Standard Plant 193
5.2.2. Fuel Costs (Mills/KWh) and Percent Increases over
Standard Plant 193
5.2.3. Generation Costs (Mills/KWh) and Percent Increases
over Standard Plant 199
A.2.1. Discharge Burnups of Low Power Density Cores .... 239
D.l. Cell-Homogenized Number Densities
(Atoms/Barn cm X l(r) 301
D.2. Average Cell Neutron Velocities (eV) 302
viii

LIST OF FIGURES
Page
2.2.1. Pin Cell Geometry 26
2.2.2. Geometry Used for MONA Criticality Calculations . 35
2.2.3. Code and Data Flow for Preliminary Burnup Calculation 37
2.2.4. Comparison of Boron Letdown from Leopard and Burnup. 40
2.3.1. Soluble Boron Worth at Critical vs. Burnup 46
2.3.2. Soluble Boron Worth at Critical. Changing Pitch . 47
2.3.3. Infinite Multiplication Factor vs. Pitch 53
2.3.4. Eta and Fast Fission Factor vs. Pitch 56
2.3.5. Resonance Escape Probability and Thermal Utilization 58
2.3.6. Thermal-to-Fast Flux Ratio 60
2.3.7. Discharge Burnup of 25% Power Density Core vs. Pitch 64
2.3.8. U-235 and Plutonium Usage vs. Pitch 69
3.1.1. Basic Steps in a Burnup Calculation 75
3.1.2. Examples of Burnup Conditions and Timestep
Specification for Fuel Cell Burnup Calculation ... 78
3.1.3. Geometry and Composition Specification for a PDQ-7
1/4-assembly Burnup Calculation 81
3.1.4. Grouping of Pin Cell Data into Core Data 85
3.1.5. Constant-boron Pin Cell Burning with Boron-free
Pseudo-burnup Steps 87
ix

Page
3.1.6. Pin Cell Burning with Soluble Boron Letdown Curve
and Power Level Following ... 90
3.1.7. Effect of Time-dependent Power Level and Soluble
Boron Concentration Specification . 93
3.2.1. Estimates of Boron Letdown and Relative Power . 105
3.2.2. Flowchart of Data for a Fuel Burnup Calculation
Using PDQ-7 1/4-Assembly Geometry 107
3.2.3. Core Geometry Used in CRIBUR 109
3.2.4. Bessel-shaped Flux Distribution 116
3.2.5. Fake-burnup Steps for Soluble Boron Worth 124
3.2.6. Problem of Erroneous Cycle Length 126
3.2.7. Flowchart of CRIBUR Code 130
4.2.1. Recyclable Fissile and Net Fissile Usage vs. Pitch 155
4.2.2. Plutonium Discharge and U^DgUse vs. Pitch 158
4.2.3. Real Cycle Length vs. Pitch 167
4.2.4. Total Life Refueling Time vs. Pitch 169
4.2.5. Availability and Capacity Factors vs. Pitch .... 172
5.1.1. Fuel Centerline Temperature vs. Relative
Mass Flow Rate 183
5.1.2. Fuel Average Temperature vs. Relative Mass Flow Rate 185
5.1.3. Clad Surface Temperature vs. Relative Mass Flow Rate 185
5.1.4. DNBR-minimum vs. Relative Mass Flow Rate 186
5.1.5. Percent Savings in DNBR-minimum and Fuel Temperature 188
5.2.1. Capital Cost vs. Plant Power 194
5.2.2. Fuel Cycle Cost vs. Plant Power . 196
5.2.3. Total Generation Cost vs. Plant Power 200
x

*
Page
6.2.1. Discharge Burnup vs. Pitch 205
A.2.1. Increase in Fuel Burnup due to Partial Refueling . 220
A.2.2. Ore Utilization vs. Enrichment and Cycle Length . 222
A.2.3. Orientations of Non-symmetric Burnable Poisons . 227
A.2.4. Batch Distribution in Core 230
A.2.5. Discharge Burnup vs. Enrichment at Several
Fuel Densities .... ...... 235
A.2.6. Ore Utilization vs. Enrichment at Several
Fuel Densities 236
A.2.7. Power Shaping Effect of Partial Length Rod .... 241
A.2.8. Ore Utilization vs. Fuel Enrichment and Burnup . 243
xi

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BURMJP AND FEASIBILITY STUDY OF
LOW POWER DENSITY PWR'S
By
Cesar Molins-Bartra
June 1981
Chairman: Nils J- Diaz
Major Department: Nuclear Engineering Sciences
Operational and safety problems of current Pressurized Water
Reactors are often associated with the high power density level of
the cores. An alternate use of current-design cores is proposed by
reducing the power density. The effects should be improved safety,
improved ore utilization, and improved operational characteristics.
A scoping study is performed in order to define core
parameters suitable for optimization under the low power density
characteristics, while minimizing redesign requirements.
A neutronic optimization study of the reactor cores is
performed by systematic changes in the fuel lattice pitch. A new
core burnup computational model (CRIBUR) is developed, which allows
calculation of the burnup and isotopic analysis of a multi-batch
core in its equilibrium cycle with a moderate computational and
human effort. CRIBUR provides better accuracy and sensitivity than
other known existing models of comparable scope, with a moderate
xii

computational effort- The code is benchmarked against actual core
data and against multi-dimensional diffusion theory core
calculations, and its sensitivity to several of the calculational
parameters is also tested-
The thermal-hydraulic behavior of the low-power cores is
compared to that of the standard reactors, and their enhanced
safety margins are clearly demonstrated.
The low-power cores yield higher burnup levels than the
standard reactors. Ore utilization is also improved in a
once-through fuel management policy- Isotopic comparisons are
presented. Core cycles are drastically increased- Plant
availability and capacity factors are also increased as a result of
both the reduced impact of refueling downtimes and the reduced
forced outage time resulting from the improvement of operational
characteristics.
An economic comparison of the low-power cores is presented as
a function of the core power level and the level of inflation.
Low-power cores are at an economic disadvantage when compared to
the standard reactor primarily because of the impact of the
increased capital cost- The fuel cycle cost is also higher because
of the long fuel core residence time- This economic disadvantage
needs to be weighed against the iirproved safety and operational
reliability to determine the commercial feasibility of the concept.
xiii

CHAPTER I
INTRCOUCTXON
1.1. Background
The most widely used reactor system for present and near
future commercial production of nuclear energy is the Light Water
Reactor (LWR). The LWR's were developed originally as compact,
light-weight, high-power units suitable for ship propulsion and
power, and they indeed performed as intended. The large research
and development efforts of the U.S. military establishment for the
LWR systems made them attractive for commercial power generation,
and thus, they were deployed in land-based electric power plants.
LWR's have been generating commercial electricity for a
considerable amount of time, totaling over 500 reactor-years of
operation in the United States alone. However, the safety systems
and engineering safeguards intended to prevent or to contain and
reduce the impact of accidents have become extremely sophisticated
and costly. This is due to several cumulative reasons: 1) the fact
that LWR's work under conditions that are close to technological
limits, 2) the high power density typical of these compact
reactors, 3) the fuel and coolant conditions, and 4) the inherent
"core-cooling" problems in case of accidents or severe transients.
1

2
The many operational transients? safety-related incidents and
the rare but significant accidents resulting in extended reactor
shutdown, e.g. Three Mile Island and others, clearly substantiate
this assertion.
A key dilemma appears immediately: a very high power density
reactor has intrinsic serious safety problems; however, economic
considerations have resulted in larger, high power density reactors
working closer to their technological limits and requiring improved
engineering safeguards. The industry has naturally worked towards
extracting the maximum power from a given core, and also towards
extending the reactor fuel cycle time in an attempt to reduce the
economic impact of refueling outages and fuel carrying charges.
Both these goals tend to demand performance from the reactors in a
manner that is not always compatible with strict safety
considerations.
Several questions need to be asked at this point. First: Is
there a way of relieving the LWR's cores from these limiting
situations without incurring an unacceptable economic penalty?
Second: Will the new concept or approach need extensive (and thus
costly and long) research and development and therefore have no
impact in the short term? Third: Will the new concept be easily
licensable, or will it require completely new licensing regulations
and processes that would take many years to develop and create
significant uncertainties for its deployment?
The first part of the first question has a relatively obvious
answer from the technical point of view: reducing the power density

3
of the cores and making other technological fixes while maintaining
the basic reactor design and technology will increase safety
margins in the critical operational variables. However, the plant
economics could suffer a significant penalty since a reduced power
density requires a higher capital investment per unit of installed
power, and this is the major part of the cost of nuclear-produced
energy. On the other hand, the reduction of power density might
allow a better fuel economy, reduce refueling times, reduce
personnel radiation exposures and waste handling and allow a better
overall plant capacity, thus utilizing better the invested capital.
Favorable answers to the second and third questions strongly
dictate that presently known and proven technology be used if the
solution is to be regarded as a viable alternative by the electric
power generating industry.
The Safer Highly Available Reactor Plant (SHARP) concept
described below appears to be a solution meeting these overall
demands., It would definitely enhance safety by improving the core
power distribution, thermal-hydraulics conditions, heat removal
requirements and overall plant operations. It would not require
significant new engineering and technical innovations, since
minimal variations would be applied to the currently used reactor
designs. These new plants will probably be easier to license than
the current plants. The economic aspect is the only phase on which
a conclusion is not easily forthcoming, and it requires in-depth
analysis before reasonable cost comparisons can be established.
The purpose of this work is to conduct an in-depth comparative and

4
optimization study of the fuel utilization capabilities of the
Safer Highly Available Reactor Plants versus the standard LWR,
plants currently in operation or under construction, and
concurrently, to develop calculational tools that will facilitate
and enhance the accuracy of scoping-type burnup studies. The
primary study must be complemented with indications of the
comparative safety and economic performance of the SHARP with
respect to standard FWR plants.
1.2. The SHARP Concept
The main goals pursued by the "Safer Highly Available
Reactor Plant" (SHARP) are the following:
- Enhance reactor safety.
- Reduce the safety and operational problems caused by
the operating conditions prevalent in current LWR systems.
- Reduce lengthy and costly research and licensing
procedures.
- Offer a solution which utilizes current technology
and can be implemented in a relatively short time.
- Reduce economic penalties to make a commercially
viable operation.
The dominant feature of the SHARP is the use of a standard PWR
core, without core changes requiring technological modifications,
at a reduced power density to reduce safety-related and operational
problems associated with standard full power operating conditions.
The single low power density core may then be employed in an
essentially standard plant configuration, yielding a reduced power

5
rating plant. Alternatively, it could be employed in parallel with
several other low power density cores powering a single standard
secondary side so as to yield a plant of normal or full power
rating. This multi-core arrangement was first considered
attractive to maintain the overall capital cost down and maintain
large plant output. However, from the commercial point of view, it
is obviously more reasonable to study the single-reactor reduced
power rating plant, whose design and construction are well known.
It is important to consider also that the economy of scale afforded
by the present large plants lies in the reduced number of critical,
expensive components used for the production of a large amount of
power. The low power density concept breaks away from this
constraint and attempts to demonstrate that there is no large
advantage in designing large power output plants as compared to
smaller sizes, when all factors influencing plant economics are
considered. Furthermore, it is frequent to find electrical demand
areas where the large power rating plants are oversized and a
smaller plant is better suited for such places.
A single-core SHARP, which is the main concept analyzed in
this work, is a plant using a standard, full-sized reactor vessel
operating at a reduced power level, with the balance of plant
dimensioned according to the desired rated output power (50% of
standard, 33% of standard, etc.). The use of a single full-sized
core for the low power density system accomplishes the dual goal of
avoiding extensive plant redesign, and operating a core at a power
level clearly within its technological limits with enhanced safety

6
and operational characteristics. A higher plant capital cost per
unit of installed power is expected (1) because of the losses
associated with the economics of size, i.e. because of the
additional investment in the "oversize" Nuclear Steam Supply System
(NSSS) components However, the overall cost of energy production
would be competitive because of the improved operating conditions
and safety.
Note that although the power density (and thus, the heat flux
across the fuel elements of the core) is reduced, the thermodynamic
conditions of the coolant would be maintained at the same level as
in the standard plants to avoid loss of thermodynamic efficiency of
the plant. This can be easily achieved by reducing the coolant
flow across the core, within established heat transfer limitations.
Reducing the heat flux in the fuel while maintaining standard
coolant conditions reduces the temperature of the fuel, which
improves fuel/clad thermal conditions, reduces stresses in the
fuel, and hopefully reduces fuel pin failures.
The advantages that can be obtained from the SHARP are:
a). From the heat transfer point of view:
- Lower temperatures in the fuel pellets, due to the
lower heat flux. This would imply reduced thermal-related damage
to the pellets and to the cladding, and therefore, reduced pin
failures.
- Reduced probability of reaching critical heat
transfer conditions

7
- Reduced heat stored in the fuel, and increased
available heat capacity, i.e., in case of an accident the core is
capable of retaining more heat before suffering damages.
- Milder accident conditions and emergency cooling
requirements, due to the lower fuel temperature and lower power
density (which means reduced decay heat generation).
b). Frcm the neutronics point of view:
- Less Doppler broadening of resonances, and therefore
extra reactivity available.
- Reduced xenon concentration due to the lower neutron
flux, which means again some extra reactivity available.
- Reduced xenon oscillation problems due to the lower,
overall xenon concentration.
c). Frcm the fuel cycle point of view:
- Longer burnup achievable from the same initial fuel,
due to the extra reactivities mentioned above.
- Largely increased time between refuelings due to the
double effect of larger burnups achievable and the lower power
generation of each core. This would reduce the impact of refueling
outages on plant availability tending to increase it. The increased
availability would result in proportionally increased energy
generation in a given time period, thus reducing the impact of
capital (which would be a basically fixed total cost) on energy
generation cost.
- Reduced relative activity of the fuel at discharge
per MWd generated due to the largely increased residence of the

8
fuel in the core, which would allow more of the mid-life fission
products to decay while in the reactor.
- Reduced ore requirements due to the larger burnups
achievable from the same initial cores.
- Reduced enrichment needs.
d). From the operations point of view:
- Increased plant availability and capacity factor due
to the longer inherent fuel cycle.
- Operations well within technological limits with
reduced failure of components and reduced forced outage periods.
- Decreased operational transients and constraints.
- Better load following capability because of the
large technological margins available for maneuvering.
- Reduced overall personnel radiation exposure.
- Reduced fuel handling cost because of less material
being handled and because of its lower specific (per MWd generated)
radioactivity (which results in reduced personnel radiation
ejqoosure).
- Possibility of reduced spent fuel storage and
transportation requirements.
- Easier inventory control and reduced risk of
proliferation because of the reduction of fuel handling operations.
All these advantages appear to be qualitatively obvious, but
the question remains whether or not they can outweigh the economic
disadvantage mentioned before.

9
The present study conducted on the low power density
single-core, reduced power rating plant using essentially the
standard plant configuration is also timely because of the present
worldwide interest towards building new, small size plants. There
are many situations where a 1000+ Me plant is just too large, and
so is the capital investment associated with it.
1.3. Previous Studies of Low Power Density Cores
The general trend followed by industry since the application
of nuclear reactors for commercial production of energy has been to
increase plant size and power densities in an effort to reduce the
capital cost per KW installed. The trend of higher power densities
reduced both the materials involved in the reactor construction and
the fuel inventory necessary for a given plant power rating.
A few commercial jreactor concepts having significantly lower
power densities than the LWR's are in operation, but they present
higher capital cost and reduced operating experience. Such is the
case, for example, of the Magnox reactors.
Lower power densities for LWR's were, thus, not considered
seriously until recently under the NSAP(2) project
(Nonproliferation Alternative Systems Assessment Program) where low
power density cores were investigated primarily as a means for
increasing the fuel utilization in a once-through fuel management
scheme. This would reduce fuel handling operations, and therefore
proliferation risks, while also providing additional safety
margins.
fc*.

10
This low power density study for the NASAP project was carried
out by Westinghouse Corp., and their approach to low power density
was that of increasing somewhat the overall core dimensions, while
maintaining or even augmenting the output power. This approach
meant only a moderate reduction in power density (about 23% as
compared to the 50 to 75% reductions considered in the SHARP
concept). This system was found to be advantageous from the points
of view of safety and fuel utilization when compared to the
standard high power plants. On the other hand, the need for
redesign of the core, the pressure vessel, fuel handling
mechanisms, etc., was found to require large investments both in
time and in capital, and the idea was not considered practical by
the private industry unless governmental support was provided.
1^4. Purpose of this Study
The main purpose of this study is to research and develop the
neutronic and fuel utilization characteristics of the SHARP
concept, and for purposes of completeness, to assess its overall
safety and economic capabilities in comparison with standard PWR
plants.
Several steps are necessary in order to accomplish this task:
a). Definition of what constitues a SHARP for the purpose of
this study (ranges of power densities, safety considerations and/or
goals, etc.).
b). Definition of a set of possible reactor cores on which
parametric studies can be effectively conducted.

11
c). Preliminary neutronic and fuel cycle study of the defined
cores with known and reliable calculational tools in order to have
a good estimate of the performance characteristics of each core.
This would include obtaining first estimates of basic core
parameters (such as reactivity worth of boron, coolant temperature,
Doppler reactivity coefficient, etc.) and of the expected variation
of core and fuel cycle parameters (burnup, cycle length, etc.).
d). Exploration and evaluation of currently available
calculational tools that may be suitable for this type of study,
and development of new calculational schemes that may accomplish
the established goals with the optimum cost/results ratio and serve
the nuclear industry as an accurate scoping tool.
For example, this study needs a good set of calculational
tools for the neutronics calculations, where power levels, fuel
temperatures, etc. can be easily specified and changed, since
these are some of the main parameters that will differentiate the
low-power core from the standard core. It also requires reliable
and economical means of calculating core burnup distribution, core
life time, etc.
e). Detailed neutronic and fuel cycle studies of the cores
selected from the parametric variations performed in the
preliminary study. This study must result in an optimization of
the SHARP cores from fuel-cycle and plant operation points of view.
The results of the SHARP study must be compared to those of
standard FWR's analyzed with the same calculational tools, in order
to obtain relative figures of merit with a minimum of

12
methodological errors that could bias the estimates towards either
system.
f). Overview of economic evaluation of the SHARP concept as
compared to the standard PWR to establish the overall economic
advantages or disadvantages that may be expected from the SHARP
concept. The economic evaluation must consider the variations in
the costs of interest and escalation, as well as the possible size
or rating variations of the plant components which depend on the
plant power rating. For example, when studying a single-core SHARP
for a power level of 50% that of a standard plant, the NSSS is
dimensioned equal to that of the standard, full-power plant, but
the BOP is dimensioned for only the new 50% power rating.
The next chapter describes the SHARP parameters and the
preliminary calculations carried out in order to establish the
basic expected performance of the low power cores.
Chapter III describes the burnup calculational methods used in
the industry; they are compared to the needs of this study, and a
new method is developed, which best suits the scope of this work
and results in reduced computational effort. The new method should
serve as a valuable industry-wide burnup calculational tool because
of its accuracy, ease of utilization and low computational cost.
Chapter IV shows the cases chosen for in-depth study and the
results obtained from the burnup calculations, plus some data
referred to the expected comparative plant performance.

13
Chapter V is a brief description of the thermal-hydraulics
safety-related aspects of the SHARP as compared to the standard
plant, and an insight into the comparative economic behavior of the
SHARP.
Chapter VI contains the main conclusions of this study and
recommendations for future research on the topic, considering
points that have appeared as unresolved and potentially
advantageous questions concerning the SHARP concept.

CHAPTER II
SCOPING WORK
2.1. Problem Framing
The purpose of this chapter is to study a sequence of
different low-power cores in order to calculate first estimates of
the low-power cores characteristics. A reference standard core
must be defined against which one may compare the characteristics
of the low-power cores. Finally, the results of the comparisons of
the different reduced power density cores versus the standard one
will be used to define the characteristics of selected cores that
will undergo an in-depth study. The possible need for new
calculational tools to perform such in depth study will also be
examined. This chapter will, therefore, study a series of
low-power cores. The results of this study will be used to select
the better candidate cores, and a range of their expected
characteristics, for further analysis.
The study of the SHARP requires an initial definition of the
parameters of the reactor and/or of the fuel cycle that may be used
as a means of comparing the SHARP with the standard PWR plant.
These parameters may be classified into three main categories of
interest: Safety, Neutronics, and Fuel Cycle. These are key areas
of investigation which have both independent and interrelated
problems.
14

15
Parameters of interest from the safety point of view would be:
a). The moderator temperature reactivity coefficient (MTC).
b). The Doppler reactivity coefficient.
c). The fuel average temperature.
d). The power peaking factors.
e). The soluble boron reactivity worth (at the concentrations
needed throughout a cycle life for each particular reactor system).
f). Power density and linear power (KW/'ft).
Fi:om the neutronics point of view, some of the more important
parameters to be observed are:
a). The changes of Doppler reactivity coefficient at the
different power density levels.
b). Xenon concentration.
c). Core effective multiplication factor at beginning of
life.
c). Evolution of isotopics.
d). Neutron energy spectrum variations.
e). Effects of varying enrichments if enrichments different
from those normally used for present FWR's can be considered.
Pnom the fuel cycle point of view, there are two main
variables that have to be optimized with regard to energy
production, but which are bound by conditions like the maximum
power peaking factor, fuel enrichment costs, thermal-hydraulic
safety aspects, etc. These two main variables are the following:
a). The total energy obtainable per ton of uranium ore.
b]. The core cycle length.

16
Obviously, one is concerned with obtaining the maximum energy
from each ton of ore. In addition to this, utilities are also
interested in having long core cycles. Presently there is a trend
to change from a 12-month to an 18-month cycle scheme because of
the reduction in refueling outage time and personnel radiation
exposure, which might result in attractive savings in power
generation cost.
If the currently standard core structure were maintained
(3-batch, out-in scheme) it would only take an increased fuel
enrichment and some help from burnable poisons in order to have an
18-month cycle instead of a 12-month one. By so doing, the ore
utilization is favored by the fact that the fuel is discharged at a
higher burnup. A study by Combustion Engineering(3) shows that ore
utilization is improved for increasing burnup levels up to about 50
GWd/MTU and fuel enrichments of about 4.5%. The problem, however,
is that an 18-month cycle would yield high discharge burnup levels
as compared to what is acceptable based on the present level of
fuel technology.
In order to keep the discharge burnup in an acceptable range,
given the present fuel technology, and to take advantage of the
reduced refueling outages of an 18-month scheme, utilities are
forced to switch to larger batch sizes. This enables the power
generation cost to be reduced, but with worse ore economy than that
obtained in a 12-month core cycle. This is of some concern from a
fuel resources viewpoint, mainly if reprocessing is not considered.

17
Hie optimum fuel cycle appears to be one which obtains maximum
energy per ton of ore and which also has long core cycles.
Discharge burnups, however, must be limited, considering present
fuel technology, to somewhere between 30 and at the most 40
GWd/MTU. In addition, the size of the batches should be kept small
(i.e. the number of batches should not be reduced) in order to
maintain good ore utilization.
The reduced power density cores are expected to yield
increased burnups given a certain core management scheme, but they
are not expected to vary in extremely large proportions. This
would probably make the low-power cores able to comply with all the
desirable conditions mentioned above: long core cycles because of
the reduced power density and the associated additional burnup, and
better ore utilization because of the extended burnup obtained from
the same initial core load.
The comparison of the fuel cycle performance of the SHARP
versus a standard FWR plant can be done from many different frames
of reference, each of which would enhance the comparison of the
cores in a particular aspect. However, the two most significant
ones would probably be the following:
a). Maintaining feed enrichment and core structure, evaluate
the differences in burnup achievable, cycle length, and ore
utilization.
b), Maintaining core structure and discharge burnup, evaluate
the differences in ore requirements, enrichment needs, and cycle
lengths,,

18
Mother of the other possible scenarios for comparison could
be to maintain the discharge burnup constant and allow fuel
enrichment, cycle length and core structure to vary. However, it
is the feeling of the author that these cases would not aid
significantly in demonstrating the differences of a SHARP as
compared to a standard plant.
Once the main parameters of interest have been defined, it is
necessary to define a reference core that represents a standard
reactor plant, and whose characteristics and performance is known
and usable as a frame of reference for comparison of the SHARP
characteristics and as a benchmark of the calculational methods
used in the study.
After the reference core is defined, it is necessary to define
a basic SHARP core and a series of "variational" cores that will be
used in order to obtain the coefficients, parameters, and
comparative results defined above. After the main SHARP
characteristics are obtained, another set of cores will be selected
for the in-depth study and comparison with the standard plant. The
calculational tools needed for the in-depth study will be defined
based on the requirements and restrictions observed in the scoping
calculations.
The core chosen for the standard reference plant is a
Westinghouse four-loop, 3400 MWth. FWR, with 17x17 pin fuel
assemblies(4) whose main parameters appear on table 2.1.1. Table
2.1.2 describes the main characteristics of the basic fuel cell of
this standard reactor.

19
Table 2.1.1. Main Core Parameters for the Standard Reactor.
Core Shape
Cylindrical
Radius
168.53 cm
Active Height
365.00 cm
Reflector Thickness
34.00 cm
Active Volume
3.2568 E+7 cm'
Heavy Metal Loading
94.418 WTO
Array Geometry
Rectangular
Pitch
1.2573 cm
Coolant Pressure
2250 psia
Avg. Coolant Temperature
583 K
Thermal Power
3400 MWth
Pin Average Linear Power
6 KW/ft
Table 2.1.2. Basic Fuel Cell of the Standard Reactor.
Region
Material
Radius(cm)
Thick.(cm)
Vol. Fraction
1
Fuel
0.4096
0.4096
0.3334
2
Gap
0.4178
0.0082
0.0135
3
Clad
0.4750
0.0572
0.1015
4
Water
0.7094
0.2344
0.5516

20
The basic reference core used in the scoping study is
described in Table 2.1.1; its fuel cell characteristics are
described in Table 2.1.2. The fuel used for the reference core is
uranium dioxide, enriched to 3% in U-235 isotope; the moderator is
considered at standard operating pressure (2250 psia.), but at rocm
temperature (293 K). No soluble or lumped poisons are considered,
nor are any fission products present for all the scoping beginning
of life (BOL) calculations, unless otherwise specified.
Under these conditions, the composition of the basic fuel cell
is as shown in Table 2.1.3. Region 1 corresponds, as in Table
2.1.2, to the fuel pellet; Region 2 is the gap between pellet and
clad; Region 3 is the Zircaloy-4 clad and Region 4 is the light
water moderator-coolant. This reference core will be named
core #1.
The pin-average linear power for the standard reactor is
6 KW/ft. A pin-average linear power of 1.5 KW/ft. is used for the
scoping studies of the SHARP. This is one fourth the linear power
of the standard Westinghouse core.
In an attempt to cover a wider scope and range of possible
applications of low-power reactors, the scoping study includes some
exploration of a very low power, low enrichment core which could be
used as a preheater in a multi-core configuration similar to the
one illustrated in Figure 1.2.2. This core would have lower than
standard moderator temperature and a very low power (20% that of
the standard core). Such a low power density, low moderator
temperature core might possibly be fueled with spent fuel

21
Table 2.1.3. Basic Cell Composition. Core #1.
Isotope
Region
Pure # Dens.(*)
Cell Avg.
# Dens.(*)
H
4
6.7 E-2
3.6957
E-2
0
4
3.35E-2
0
1
4.4009 E-2
3.3152
E-2
U-235
1
6.6830 E-4
2.2281
E-4
U-238
1
2.1337 E-2
7.1137
E-3
Zr
3
4.2808 E-2
4.345
E-3
Ni
3
0... E-10
0...
E-10
Sn
3
4.8556 E-4
4.9285
E-5
Fe
3
1.4946 E-4
1.517
E-5
Cr
3
7.6426 E-5
7.7573
E-6
He
2
1.9 E-3
2.565
E-5
* Units are atoms per barn-cm.

22
discharged from standard FWR plants. This core was thought of as a
possibility for further use of standard plant spent fuel.
Table 2.1.4 identifies the cores used for the scoping study
with their main distinguishing characteristics. The
characteristics of these cores were selected in order to obtain
indicative figures on reactivity coefficients and burnup
variations. Each core has a case I.D. assigned to it, which is
used for future reference. The power levels indicated are in
percent relative to the standard core's full power (6 KW/ft. or
3400 MWth. total core power).
In all cases, both the fuel and the moderator are treated as
having a uniform temperature throughout their masses. In the
reduced power density systems, the moderator temperature is always
kept at the average coolant temperature of the standard plant core,
in order to maintain the thermodynamic characteristics of the steam
cycle, and the thermal efficiency of the plant. The only cores
with different coolant temperature are the ones intended for
preheater operation. The fuel average temperatures depend
obviously on the linear power density, and the values used for each
of the scoping study cores are shown in next section.
Table 2.1.4 shows also the core effective neutron
multiplication factor (K-eff.) obtained for each of the scoping
study cores in a BOL, clean, unrodded situation. The calculational
procedures used for the scoping study are different than the ones
used for the detailed burnup study, and they are specified later in
this chapter.

23
Table 2.1.4. Identification of Cases for the Scoping Study.
Cass
# Enrich.
Mod.Temp.
Fuel Temp.
Poisons
Power
K-eff(*)
1
3%
293 K
293 K
0
0
1.3532
2
3%
583 K
293 K
0
0
1.2933
3
3%
583 K
1005 K
0
100%
1.2675
3A
2.6%
583 K
1005 K
0
100%
1.233
4
3%
583 K
700 K
0
25%
1.2770
4A
2.6%
583 K
700 K
0
25%
1.242
5
2.6%
293 K
293 K
0
0
1.3184
6
3%
293 K
293 K
1300 pm
0
1.1579
7
3%
420 K
520 K
0
20%
1.3304
8
2%
420 K
520 K
0
20%
1.2260
9
3%
583 K
1005 K
1856 PPM
100%
1.0781
* Calculated with PHFOG-BRT-MONA as shown in Sections 2.2.2
and
2.2.3

24
Table 2.1.5. Tabulation of Case Number Densities Which Differ fresa
Those of Case #1.
Case # Isotope
Region
Pure #Dens(*)
Cell Hem.
#Dens(*)
2
0
4
2.346 E-2
2.76149
E-2
2
H
4
4.6926 E-2
2.58845
E-2
3
Same as
core #2
4
Same as
core #2
5
U-235
1
5.7924 E-4
1.9312
E-4
5
U-238
1
2.1426 E-2
7.14335
E-3
6
B
4
7.258 E-5
4.0037
E-5
7
0
4
3.0982 E-2
3.17624
E-2
7
H
4
6.1964 E-2
3.41795
E-2
8
U-235
1
4.4561 E-4
1.48565
E-4
8
U-238
1
2.1559 E-2
7.18790
E-3
8
0
4
3.0982 E-2
3.17624
E-2
8
H
4
6.1964 E-2
3.41795
E-2
9
B
4
7.258 E-5
4.0037
E-5
9
0
4
2.346 E-2
2.76149
E-2
9
H
4
4.6926 E-2
2.58845
E-2
3A
U-235
1
5.7924 E-4
1.9312
E-4
3A
U-238
1
2.1426 E-2
7.14335
E-3
3A
0
4
2.346 E-2
2.76149
E-2
3A
H
4
4.6926 E-2
2.58845
E-2
4A
Same as
core #3A
* Units are atoms per barn-cm.

25
Sane of the cores identified in Table 2.1.4 show variations in
moderator temperature or fuel enrichment that imply changes in the
cell number densities as compared to the reference cell. The
coolant and fuel densities for the different temperature situations
were calculated with the help of standard tables for each
materials,6). Table 2.1.5 shows the changed number densities for
the affacted isotopes and applicable cases. For the rest of the
isotopes and/or cases, the reference core number densities apply.
2.2. Calculational Methods for the Scoping Study.
2.2.1. Heat Transfer Calculations.
Some of the main differences between a standard reactor and a
SHARP are related to the differences in the fuel temperature. It
is therefore necessary to know the fuel temperatures that
correspond to several different power-density cores before their
neutronic study can be started. However, for the purpose of the
present study, it is not necessary to obtain extremely accurate
results, since the data will be used for obtaining varying nuclear
properties of the fuel, which are not drastically affected by a few
degrees of uncertainty in the temperature.
The calculational scheme used may be found in any complete
Nuclear Heat Transfer or Reactor Analysis text(7) It is assumed
that a uniform volumetric heat source exists in the fuel region;
that the bulk moderator operating conditions are kept constant for
all the different cores (583 K, 2250 psia.) except for the
preheater cores (cases #7 and #8) which have their moderator at 420

26
deg. K but at the same 2250 psia pressure. For all the other
cores, subcooled nucleate boiling has been assumed at the pin
surface, while cores #7 and #8 were assumed to have subcooled
convection, due to the much lower bulk coolant temperature and to
the very low heat flux rate. Figure 2.2.1 shows the geometry
assumed in the pin for the heat transfer calculations.
Figure 2.2.1. Pin Cell Geometry.
Given the heat transfer conditions found at the pin surface,
the temperature increment between the bulk coolant and the clad
surface may be obtained by the Jens & Lottes expression:

27
Ts Te + 60 ( a" / IQ6 )25
exp ( p / 900 )
where Ts = pin surface temperature, deg. F
Tc = bulk coolant temperature, deg. F
q'' = surface heat flow rate, Btu/hr sq.ft
p = coolant pressure, psia
The temperature increment across the clad is given by the
expression:
A Tel = q' do In ( do / di )
2 K S
where ATcl = increment of temperature across the clad, deg. F
q' = linear heat flow rate, Btu/hr ft
do = clad outer diameter, ft
di = clad inner diameter, ft
K = clad thermal conductivity, Btu/hr ft deg.F
S = clad surface per unit pin length, ft
Since the thickness of the gap is extremely small, the
temperature change across the gap may be obtained as
A Tg =
K S
where ATg = increment of temperature across the gap, deg. F
q1 = linear heat flow rate, Btu/hr ft
K = thermal gap conductance, Btu/hr sq.ft deg.F
= surface of heat transfer per unit pin length, ft.
S

28
With the previous expressions, it is possible to obtain the
pellet surface temperature. Since the temperature of interest for
nuclear calculations is the average temperature of the pellet, it
is necessary to know the temperature profile in the pellet, as a
function of the distance to the centerline, so that volumetric
weighting of the temperature can be done.
The temperature profile inside the pellet (assuming uniform
heat generation) is given by
T(r) = To o r2
7T R* 4 Kf
where T(r) = temperature of the pellet at radius r deg. F
To = temperature of the pellet centerline, deg. F
q' = linear heat flow rate, Btu/hr ft
R = pellet outer radius, ft
Kf = pellet thermal conductivity, Btu/hr ft deg. F
r = distance from centerline, ft
One more quantity is needed at this point: the pellet
centerline temperature, which is given by
To = Ts + g1
4 TT Kf
where To = pellet centerline temperature, deg. F
Ts = pellet surface temperature, deg. F

29
With the pellet temperature profile expression in hand, the
pellet average temperature may be obtained as
o fR
Tave = (2 / Rz) T(r) r dr = To Jo
For the 20% linear power, preheater cores, the calculational
scheme was the same except for the temperature step from coolant to
clad surface, in which case a subcooled convection heat transfer
expression was used, such as the following:
AT = q" / h
where q1' = surface heat flow rate, Btu/hr sq.ft
h = forced convection coefficient, Btu/hr sq.ft deg. F
The forced convection coefficient may be calculated as
h = 0.0023 Re0*8 Pr0*4 (K/D)
where K = coolant thermal conductivity, Btu/hr ft deg.F
D = hydraulic diameter of the coolant channel, ft
Re = Reynolds number
Pr = Prandti number
Once the thermal calculations were completed, the standard
core fuel average temperature was checked against the vendor's
literature, and found to differ by a few degrees; this difference
is probably due to slightly different conductivity coefficients
across the core (due to temperature changes), which are not taken

30
Table 2.2.1. Basic Cell Temperatures vs. Power Level.
Power level % 100 50 33 25 20 (pre)
Clad surface 593.3
Clad increment 46.4
Gap increment 120.1
Fuel surface 759.9
Fuel centerline 1846.7
Fuel average 1303.3
Fuel average (K) 979.0
Vendor normalized (K) 1005.0
592.8
592.5
592.4
304.2
23.2
15.5
11.6
9.3
60.1
40.0
30.0
24.0
676.1
648.1
634.0
337.5
1219.5
1010.3
905.7
555.0
947.8
829.2
769.9
446.3
781.9
716.0
682.0
503.2
805.0
735.0
700.0
520.0
Note: Temperatures obtained from thermal calculations as described
in Section 2.2.1.
Temperatures expressed in degrees Fahrenheit, unless
otherwise specified as Kelvin Column 20 (pre) refers to the 20% power, preheater core.

31
into account in the thermal calculations. Since this difference
was not significant from the nuclear point of view, all the results
were modified slightly to maintain consistency with the vendor's
data in the neutronic calculations. The consistent data permit the
further benchmarking of the nuclear calculations. Table 2.2.1
shows the main temperatures involved in the calculation for all the
different power levels studied. All temperatures are in degrees
Fahrenheit, unless otherwise specified.
2.2.2. Neutron Cross Section Calculations.
Section 1 of this Chapter described the set of cores chosen
for the scoping observation of the differences between a standard
core and a low power density core. It is necessary to gather a
series of codes of known reliability which can yield an accurate
neutronic picture of a LWR core. Parametric studies can then be
conducted which will define significant factors differentiating a
SHARP from a standard plant. In these parametric studies, the
absolute accuracy of the results is not as significant as their
relative values which are used to establish figures of merit.
The best neutron cross section calculational method available
for the scoping study involved the use of several standard codes.
This method allows great flexibility in specifying isotopic number
densities, geometries, etc., but it requires a relatively large
amount of manual data handling from code to code. This section
briefly describes these codes.

32
Two codes are used for the calculation of cross sections.
PHRQG8) is used for calculation of the fast-group neutron cross
sections, and the original library, consisting of a 68 energy group
cross section data set is collapsed into three fast groups which
are used in the core model code. A more detailed explanation of
PHROG can be found in Appendix B2. As is shown in the next
section, the core criticality calculations are done with a model
which defines two distinct regions: fuel and reflector. Fast cross
section calculations need to be run for both of these regions. The
fuel region is run first, and the group-wise fluxes and currents
existing in the fuel are used as weighting functions for the
-calculation of the reflector constants.
The thermal-group cross sections are calculated with the
Battelle-Revised Thermos (BRT) code(9) using a 30-group cross
section library. Thermal cross sections are collapsed into a
single thermal group. It is also necessary to obtain separate
cross sections for the fuel region and for the reflector. The fuel
region does not present any problem, since it can be well
represented by the calculation of a unit fuel cell. However, BRT
does not allow an intrinsic representation of the reflector region
without an adjacent core. The reflector region is then calculated
from a two-region slab reactor configuration, where one of the
regions has the average core region characteristics, while the
other represents the reflector. The geometry chosen was that of a
slab. Because of limitations in the number of mesh spaces allowed
by BRT, inaccuracies at the core-reflector boundary arise when a

33
cylindrical shape is considered. The basic features of ERT are
described in Appendix Bl.
Four sets of cross sections are obtained with the calculations
described above: one set with three-group fast cross sections for
the core region; another set with three-group fast cross sections
for the reflector; a third set with thermal cross sections for the
core, and finally a set with thermal cross sections for a slab
core-reflector configuration, from which only the reflector
constants are used. These cross section sets are then organized
for input to the core models, which are discussed in the next
section.
2.2.3. Criticality and Burnup Methods.
The first objective of the scoping study is to obtain the
neutron multiplication factors of the cores described in Table
2.1.4. These multiplication factors are used to estimate the main
reactivity coefficients involved in the neutronic aspects that
differentiate the SHARP core from the standard core. The second
goal of the scoping study is to obtain a first estimate of the
burnup levels achievable by each particular core, as well as the
isotopics associated with them. Following is the description of
the methods used for achieving both of these purposes.
The core modeling for the criticality calculations is done
with the HONA(IO) code which is described in more detail in
Appendix B3. MONA accepts the cross section input prepared by
PHROG and BRT, and information about the core geometry and material
region compositions. The code can perform a number of

34
calculations, such as buckling searches, poison searches, etc. In
this case, a normal effective multiplication factor (K-eff.)
calculation is requested.
The code accepts a cylindrical geometry, but it is a
one-dimensional diffusion-theory code. In order to obtain accurate
results for the neutron multiplication factor in one-dimensional
cylindrical geometry, accurate values for the perpendicular
bucklings to account for perpendicular leakage are required. In
the absence of this information, it is known from previous
experience that an "equivalent" spherical system can yield accurate
results for the neutron multiplication factor. For the
"equivalent" spherical configuration, the core volume is kept equal
to that of the cylindrical reactor, while for the reflector, the
thickness is kept equal to the cylindrical reactor value. The core
region has 199 mesh points allocated, and 34 mesh points are
assigned to the reflector, thus assuring that the mesh spacing is
smaller than the neutron diffusion length. This is necessary for
these calculations if neutron diffusion theory is expected to
describe the core fluxes and currents with an acceptable level of
accuracy. An extrapolated zero-flux boundary condition is
specified for the outer boundary. Figure 2.2.2 illustrates the
geometric model used with MONA, for the criticality calculations.
Two methods were used for the scoping burnup calculations.
The first method involves the use of the calculational scheme
already used for the criticality calculations. Once the reactor
criticality status is established, soluble boron is added to the

35
Figure 2.2.2. Geometry Used for MOM Criticality Calculations.

36
moderator in the amount necessary to force the core to be exactly
critical.
With the reactor critical, the fuel-region homogenized
composition that was input to MONA and the flux spectrum generated
by MONA for the fuel region are input into a BURNUP code, which
performs the isotopic burning of the mixture to the burnup degree
specified, and at the power level desired. The BURNUP code used in
this case is geometrically non-dimensional, but it allows for the
specification of a number of time-steps after which the homogenized
composition of the fuel region is recalculated. The flux spectrum
is assumed to remain constant throughout the burnup calculation.
After the burnup calculation, the homogenized fuel-region
composition is used again for a new core criticality evaluation, in
the way explained above. Figure 2.2.3 shows a schematic diagram of
the flow of data involved in this type of burnup calculation.
This burnup calculation scheme has some very obvious
drawbacks:
a). Each burnup timestep requires an inordinate amount of
effort.
b). There is a large amount of punched card handling and
typing, which largely increases the probability of human error in
the calculations.
c). The BURNUP code used has a rather low degree of
sophistication, resulting in a reduced number and complexity of
radioactive chains.

37
Figure 2.2.3. Code and Data Flow for Preliminary Burnup Calculation.

38
This method was, however, used for some cases, but a faster
and at least equally accurate method had to be obtained. The
second burnup method employed the LEOPARD program(11). This
program, which is discussed in more detail in Appendix B4, makes an
automated chain of calculations involving cross sections, spectra
and burr'tp for an elementary fuel cell. The cross sections
calculations are based on the MUETQ2) and SOFOCATE(13) codes. The
geometry is always that of a fuel pin, which may be surrounded by a
buffer zone simulating the structural materials present in the core
but not associated with the elementary fuel cell.
LEOPARD requires an initial pin composition, a power level, a
soluble boron concentration history, and a burnup timestep
structure. The code calculates each isotope's cross sections
collapsed to a specified number of broad groups, performs a
multiplication factor calculation, burns the fuel, recalculates the
pin composition, and resumes the loop until the completion of all
the burnup timesteps.
Although the cross section generation in LEOPARD is not quite
as accurate as the one used for the criticality calculations, the
burnup calculational structure is far superior to the one in the
BURNUP code, and the automatic coupling of the calculations allows
for the effortless and error free handling of a much larger number
of isotopes, which definitely adds to the accuracy of the overall
calculation. It is important to note the drastic reduction of
effort required for similar results between the first
(PHRDG-BRT-MCNA-BUFNUP) and the second (LEOPARD) method of burnup

39
calculation. Obviously, LEOPARD does not model the core and the
reflector since it is just a pin cell, one-dimensional code, but
neutron leakage effects are considered by means of a perpendicular
buckling, which enables the code to perform rough estimates of
criticality.
In an effort to compare the two burnup calculational methods,
parallel calculations were run for one core representing a standard
reactor's first load. Since no fuel regions were considered in the
MONA core model, the fuel enrichment was specified as an average of
the concentrations of the three real batches. This was also the
only v/ay of simulating the situation in LEOPARD, because it
obviously can not accept several enrichments simultaneously. Note,
however, that this is a rather crude way of representing a core,
because it is very different to have three distinct regions with
various enrichments or burnup levels than to have a large region
with averaged characteristics. However, the procedure was deemed
adequate for a scoping comparative study.
The results from the LEOPARD and MGNA-et-seq. comparison
showed a discharge burnup discrepancy of about 13%. Figure 2.2.4
shows the estimated soluble boron letdown curve obtained from each
calculational scheme. The agreement is very good up to about two
thirds of the core life, but the separation increases thereafter.
This is most probably due to the absence of proper treatment of the
fission products in the MONA method, because they could not be
included in the fuel pellet region, and therefore were only
accounted for as a reduction of core reactivity,
but their

40
Critical
Boron
Concentration
ppn
2500
2000
1000
GWd/MT
Figure 2-2.4. Comparison of Boron Letdown frcm LEOPARD and BURNCJP.

41
neutronic effects on the other isotopes (due to spectral effects)
could not be simulated. However, the agreement between the two
methods can still be considered good, considering the different
conceptual approaches, and the small amount of core information
included in them. The most reassuring fact is that the quoted
beginning of life (BGL) core reactivity and the quoted end of life
(EOL) burnup level of the first core for the standard reactor lie
just between the values obtained from the two models, with LEOPARD
giving a better result for the EOL burnup.
With all these considerations in mind, LEOPARD was used with
the "average fuel" enrichment for the scoping burnup calculations.
These calculations used a fuel pin with core-averaged enrichment, a
perpendicular buckling corresponding to the axial leakage of a
standard core, and an approximately critical boron letdown curve.
Calculations were performed for the standard core, a 25% power
density core using varying lattice pitches, and a 20% power density
core with reduced moderator temperature (the preheater core,
designated as core #8 in Table 2.1.4). The results of the scoping
calculations are shown in the next Section.
2.3. Results of Preliminary Work
2.3.1. Reactivity Coefficients.
Table 2.1.4 shows the main parameters characterizing the
different cores that were chosen for the scoping study of the
SHARP. The table includes the effective multiplication factors
(K-eff.) of the cores at beginning of life (B.O.L.) with no xenon

42
or control rods, calculated by the PHROG-BRT-MONA scheme, as
explained in sections 2.2.2 and 2.2.3.
The core effective multiplication factors obtained for the
different cores are used to calculate bulk reactivity coefficients.
Thus, the average moderator temperature reactivity coefficient
(M.T.C.) is -20.67 p.c.m./deg.K of moderator temperature variation
for the temperature range between 293 K and 583 K. This rather
large reactivity coefficient would be the main asset of a core such
as core #8, which would use very lew enriched fuel or even spent
fuel discharged from a standard plant. A reduction of average
moderator temperature of 163 K, as described on table 2.1.4, would
provide 3.37 % of extra reactivity as compared to an identical
core working with normal moderator temperature. This excess
reactivity would enable the core to work with the less reactive
fuel or burn the standard fuel beyond the current discharge burnup
levels. However, it should be kept in mind that these preheater
cores are in some way a technical speculation, and that the real
interest of this study is centered in the low-power cores with
standard moderator conditions, which enables than to drive a
standard balance of plant in a single-core arrangement.
As the average linear power produced by the fuel is varied
according to the different power densities of the SHARP cores, so
is the average fuel temperature, even though the moderator
conditions remain unchanged. This variation of fuel temperature
causes a change of the absorption resonances width, due to the
Doppler effect, and this causes a change of the core reactivity.

43
The Doppler coefficient of reactivity obtained from Table 2.1.4 is
-3.99 p.c.m./deg.K of fuel temperature change, for a variation
between 293 K and 700 K (which correspond to 0 power and 25% power
respectively). The coefficient drops to -3.12 p.c.m./deg.K for the
temperature range between 700 K and 1005 K (25% power to full power
range). This shows the well known behavior of saturation of the
Doppler effect for increasing temperatures of the fuel. The
average coefficient that applies for the total range from 0 power
to full power is -3.62 p.c.m./deg.K The reduced fuel temperature
is one of the main effects contributing to an increase in the
reactivity of a low-power core with respect to the standard core.
As was mentioned before, the moderator conditions would remain
unchanged for the different power level cores for thermodynamic
reasons, and only the fuel temperature would experience a variation
due to the change of the heat production rate.
Fuel enrichment obviously has an effect on core reactivity,
and therefore, it is also possible to define a reactivity
coefficient for it. For enrichments between 2.6% and 3% in U-235,
the reactivity coefficient oscillates between 8.705% / 1%
enrichment variation for a cold, clean reactor and 8.643% / 1%
enrichment variation for a hot, clean, full power reactor. For the
enrichment step between 2% and 3% of 0-235, the 1/5 power,
preheater core shows a reactivity coefficient of 10.44% / 1%
enrichment variation. This larger magnitude may be due to the
double effect of bracketing a lower enrichment end, and having a
better moderated configuration due to the low moderator temperature

44
which implies higher moderator density. This rather large
reactivity coefficient associated with the fuel enrichment could be
used in order to stretch burnup of the cores by increasing the fuel
enrichment, which up to a certain point will result in a better ore
utilization (14).
Higher fuel enrichment would, however, bring problems
associated with higher power peaking requiring the permanent use of
burnable poisons and the technological problem of loss of clad
integrity at very high burnup levels.
The reactivity coefficient of soluble boron in the moderator
is given an in-depth study for two main reasons: First, the soluble
boron concentration that makes a reactor critical at the different
points through the core's life is a very useful and realistic way
of estimating the cycle life of the core. However, for this to be
feasible, it is important to know the reactivity worth of soluble
boron at all times through the core's life. This coefficient may
be expected to vary, depending mainly on boron concentration and
core burnup status; these two variables are somewhat related, but
they certainly don't follow any exact relationship. Second, just
reducing the power density may cause the standard reactor lattice
geometry not to be the optimal one from the burnup point of view,
since the nuclear characteristics of the fuel are somewhat
perturbed. Some changes in lattice geometry, such as fuel pitch may
be required then in order to optimize the core for the new
low-power situation. If this is the case, the ratio of
moderator-to-fuel is likely to be changed, and then, a given

45
variation of soluble boron concentration would mean different total
amounts of boron being added to or retrieved from the core, which
implies that the soluble boron reactivity coefficient may also
change for different lattice geometries.
Soluble boron worth variations are analyzed as a function of
boron concentration, core burnup, core power level, moderator
density, and basic cell geometry. These calculations were based on
the LEOPARD scoping core burnup calculations.
When adjusting soluble boron concentration to keep the reactor
critical through life, the worth of boron decreases slightly from
the B.O.L. until about 2000 or 3000 MWd/MTU and increases
thereafter, with a very slowly increasing slope. This variation is
shown in Figure 2.3.1. This figure shows also that the variation
of power level, without variation of moderator conditions, causes
practically no change in the boron reactivity coefficient. On the
other hand, either a variation of pitch or a change of moderator
density (temperature) cause sharp changes in boron reactivity
worth. In both cases, as expected, boron reactivity worth
increases for the changes that imply a larger absolute amount of
boron present in the core (i.e. for an increased pitch or for an
increased moderator density).
Figure 2.3.2 shows more clearly the effect of lattice pitch on
boron worth. It is also more evident that the variation of boron
worth at critical through life is larger for larger pitches. The
cause of the monotonical increase of boron worth with life is due
to two adding effects: the self-shielding effect of boron itself

46
Boron Worth
pcm/pptn
Figure 2.3.1. Soluble Boron Worth at Critical vs. Burnup.

47
Boron Worth
Figure 2.3.2. Soluble Boron Worth at Critical. Changing Pitch.

48
and the buildup of Pu-239. As core life advances, the boron
concentration is steadily reduced; the boron self-shielding also
decreases, causing an increase in boron worth. The contribution of
Pu-239 buildup to the increase of the boron worth increases with
core life. It should be noted that by the end of the core life,
about 40 % of the core power is generated by Pu-239 that has been
bred through parasitic captures in U-238. Pu-239 has a higher
thermal fission cross section than U-235, and it has a resonance
peak in the thermal energy region (tenths of an eV.). This causes
the reactor to increase its sensitivity to the presence of a
thermal poison (such as boron) that can remove neutrons from the
high fission cross section energy region.
The rate of increase of boron worth with core life is faster
for larger lattices, because of the effect of increased boron
inventory in larger lattices.
Thus, iron the point of view of the SHARP, it is possible to
conclude that the reduced power density will not affect boron worth
to a significant extent as long as the moderator conditions and
cell geometry characteristics remain unchanged. However, if
geometry changes are performed in order to optimize the cell
behavior for burnup or fuel utilization, boron worth may be
expected to show a significant variation, tending to increase for
larger pitches and tending to increase towards E.O.L., mainly if
the burnup levels achieved are high.
An important goal of the scoping study, is to make a first
estimate of the burnup advantages that can be expected from the low

49
power cores relative to the standard core. One simple way of
making such estimates is to calculate the total reactivity of the
cores at B.O.L., and to assume that reactivity is lost linearly as
core burnup increases, which is a reasonable approximation for a
first-estimate calculation.
The most significant parameters altering the core reactivity
at B.O.L. when the fuel enrichment, core geometry and moderator
conditions are fixed, are the fuel temperature and the equilibrium
xenon concentration. Fuel temperature affects the core reactivity
in the amounts determined by the Doppler coefficient, explained
earlier in this section. The reactivity worth of xenon in the
different cores can not be obtained from the LEOPARD burnup
calculations performed in the scoping study, but it can be obtained
from the vendors literature (4). Combining these xenon reactivity
modifications with the core effective multiplication factors listed
in Table 2.1.4, it is possible to obtain a comparison of the
reactivity available at B.O.L. for each core, and therefore, an
expectation of their comparative discharge burnup levels. These
figures appear on Table 2.3.1 for the standard core, the 25% power
level core, and the 20% power level preheater core. Two fuel
enrichments are considered for each power level.
Using the linear reactivity assumption (15), Table 2.3.1 shows
that a 25% power SHARP may yield a discharge burnup 10.6% above
that of a standard reactor, for an average fuel enrichment of 2.6%
U-235 in both cores. The burnup increase of 9% appears for a fuel
enrichment of 3.1%. These burnup increases are desirable from two
*u.\

50
main points of view: they represent additional energy obtained from
the same initial ore, and they entail longer core cycles, which
tends to reduce costs of refueling outages.
The expected burnups shown by the 20% power, preheater core
are really surprising. The better moderation boosts reactivity in
such a way that for 3% enriched fuel, a burnup increase of 32%
above that of the standard core would be expected. If the
preheater core is loaded with low enriched fuel, e.g. an enrichment
of 2%, it would still yield 88% of the burnup expected from a
standard reactor loaded with 3% enriched fuel.
However, it should be kept in mind that the burnup results
obtained from these LEOPARD calculations do not take into account
the significantly different neutronic situation of the various
batches present in the core, which will definitely affect the
isotopic dynamics, and therefore the core life. In any case, these
figures give good hope for better fuel utilization by the SHARP'S.
Table 2.3.1. Core Reactivities at B.O.L.
Case #
ENRICH
K-eff.
Xe worth
BOL reactivity
3
3%
1.268
-3.5%
23.3%
3A
2.6%
1.233
-3.5%
19.8%
4
3%
1.277
-2.3%
25.4%
4A
2.6%
1.242
-2.3%
21.9%
7
3%
1.330
-2.1%
30.9%
8
2%
1.226
-2.1%
20.5%
9>.

51
2.3.2. Neutronics and Safety Considerations.
The reduction of power density in the cores of the SHARP
causes, among other effects, a reduction of the Doppler effect, a
reduction of xenon concentration, and a reduction of fuel
temperature. These changes entail unavoidable differences in the
neutronic characteristics of the SHARP as compared to the standard
reactor. Thus, it is necessary to check the neutronic changes
associated with the power reduction, because some of their aspects
are closely related to plant safety, while others are related to
fuel performance or isotopic evolution.
As was pointed out in the preceding section, the neutronic
differences existing between the standard reactor and the low-power
reactors may result in a difference in the optimum fuel arrangement
in the core. In this study, the variation of fuel arrangement is
limited to the variation of fuel lattice pitch. This variation
implies a change of the thermal-hydraulic characteristics of the
fuel cell, as well as a change of the fuel-to-moderator ratio,
whose heavy effect on the core neutronics is well known. The pin
geometry is kept constant and equal to the standard core design.
The immediate effect of changing the pitch or
fuel-to-moderator ratio is a change in the neutron energy spectrum,
which in turn causes a change of cell reactivity, and as fuel
burnup progresses, it affects the isotopics evolution. There is an
optimum pitch which yields the maximum reactivity for a given fuel
pin. However, such a configuration is not safe from the
operational point of view. At the optimum point or in an

52
over-moderated configuration, the moderator temperature coefficient
(WTC) becomes zero or positive, making the core inherently
unstable. Stability and safety require a negative MTC, which means
a pitch must be selected which yields an undermoderated
configuration.
The scoping study included an exploration of the reactivity
changes occurring in the fuel cell as pitch is varied, for seme of
the more representative core cases being considered. Figure 2.3.3
shows the plots of the infinite multiplication factor versus pitch
for the full power standard core (core 3A), the 25% power core
(core 4A) and the 20% power core with lower moderator temperature
(core 8) as obtained from PHROG-BRT-MONA. The plots correspond to
BOL, with no poisons, control rods or xenon present in the core,
but with the temperatures corresponding to the assigned core power
levels.
The curves show a uniform increase of reactivity between core
3A and core 4A, due to the reduced Doppler effect. Remember that
no xenon effect is taken into account in these curves, since the
cores are clean and with zero burnup. Curves 3A and 4A do not
show, however, any visible change of shape; there is only a
vertical shift due to the gained reactivity. This means that the
best moderated pitch is the same independently of the power density
(under equal moderator conditions). With this in mind, the
overmoderation / undermoderation safety criteria for the low-power
cores would admit some relaxation as compared to the standard core,
due to the inherently safer operating conditions of the SHARP.

53
Figure 2.3.3. Infinite Multiplication Factor vs. Pitch.

54
Figure 2.3.3 shows the pitch used by the standard reactor. Note
that it complies with the safety criterion of working in an
undemoderated configuration. However, it seems that the pitch
could still be allowed to increase somewhat without compromising
the safety, since the operation point of the standard reactor is
rather far from the peak of the curve. Later in this section it is
explained why the standard reactor can not take advantage of the
extra reactivity achievable from a slight increase of pitch, while
the low-power core can.
When examining the curve for core #8 on Figure 2.3.3, a sharp
change of shape can be noticed as compared to the other curves.
Reactor 8's curve has a larger curvature; its reactivity is much
more sensitive to the variations of pitch. This fact is easily
explainable: Reactor 8 works with reduced moderator temperature,
but at the same pressure as reactors 3A and 4A. The moderator has,
therefore, a higher density. This can be observed on Table 2.1.5.
The higher moderator density causes a given change of pitch to
represent a larger increase in moderation, i.e. there are more mean
free paths or more interactions taking place in the moderator
region between pins at higher moderator density. It is as if the
x-axis of the curve for core #8 has been compressed as compared to
the standard moderator temperature ones.
One of the consequences of the higher moderator density is a
reduction of the distance between the standard reactor working
pitch and the optimum moderation point. This effectively raises
the multiplication factor of core 8 when using the standard pitch.

55
and it is one of the main factors responsible for the high
reactivity found for core 8 in spite of its low fuel enrichment
(see Table 2.3.1). However, it is also important to note that this
same curve shift prevents considering a different pitch for core 8,
because it is already working at a point much closer to the optimum
moderation ratio, and any further change of pitch would bring the
core too close to a possibly overmoderated configuration, which as
explained above, is not acceptable from a safety point of view.
Figures 2.3.4 and 2.3.5 show the changes in the four factors
of the "four-factor formula" for cores 3A, 4A and 8, as pitches are
varied. These are the results of the criticality calculations
performed for BOL with PHROG-BRT-MONA. It is apparent that the
neutronic effect of the higher moderator density of core 8 is far
larger than the change caused by the Doppler effect difference
between cores 3A and 4A. None of the four factors exhibit a
difference of more than one percent in their corresponding values
between cores 3A and 4A. However, the largest differences are
observed in the resonance escape probability for very
undermoderated lattices. This could be expected because this is
the configuration and the factor that give the maximum enhancement
to the Doppler effect variation.
The change of moderator density causes a sharp difference of
spectra which is reflected in all the four factors. The fast
fission factor, epsilon, is consistently lower for core 8 than for
the other two. Note that the fast fission factor experiences a
sharp rise when the lattice pitch reaches very small values and

56
Figure 2.3.4. Eta and Fast Fission Factor vs. Pitch

57
there is almost no moderator in the cell. In the range of pitches
that can be reasonably considered for practical purposes (between
1.2 and 1.4 cm) the fast fission factor slowly decreases as more
moderator is brought into the cell.
Eta, the number of neutrons produced per thermal neutron
absorbed in the fuel, is the least sensitive of the factors, as it
remains almost flat through the whole range of lattice pitches; it
only shows a fall for very tight configurations. However, it is
important to note that there is still a difference between core 8
and cores 3A and 4A. This shows that more important than the
amount of moderator present in the cell is the variation of mean
free path associated with the change of moderator density.
The resonance escape probability p, shows a very predictable
pattern, increasing as the amount of moderator increases, but
showing a very clear saturation effect as large pitches are
reached, However, in the range of practically reasonable pitches,
p shows a rather steep positive slope; this factor is the one which
is primarily responsible for the increase of reactivity achieved by
increases of pitch in this range. Note that core 8 stays
consistently above cores 3A and 4A throughout the range of pitches
examined in spite of the larger relative number of resonance
absorber atoms (mainly U-238) present in core 8. This is a direct
effect of the different mean free path caused by the increased
moderator density. If only the better moderation effect and
saturation effect of p for larger pitches existed, the distance
between the curves for core 8 and cores 3A and 4A could be expected

58

59
to decrease significantly as larger pitches are reached, but the
reduction of distance is insignificant, indicating again the key
role of the changed mean free path.
Finally, the thermal utilization factor, f, or ratio of
thermal neutrons absorbed in the fuel to total thermal absorptions
in the cell, shows a drop as pitch is increased. This effect is
directly related to the amount of absorbing moderator atoms present
in the fuel cell. Note that as pitch increases, the curve for core
8, which is naturally below that of cores 3A and 4A, drops faster
than that of cores 3A and 4A; the reason being simply that the same
increase of pitch adds more moderator atoms into core 8 than for
the other cores, because of the higher moderator density of core 8.
Many of the reactivity and isotopic differences between the
standard core and the low power density cores have been attributed
to neutron energy spectral effects. Figure 2.3.6 illustrates some
aspects of the spectral changes. The graph represents the ratio of
thermal flux to first fast group (of the three fast groups used for
the criticality calculations) for cores 3A, 4A and 8, and for short
burnup times, up to about 50 days of equivalent full power
operation of the standard reactor. These figures were obtained
from short burnup calculations performed with the
PHFOG-KOT-MONA-BURNUP scheme. The effect of the increased
moderator density of core 8 is much larger than all the other
effects differentiating cores 3A and 4A. However, all cores show a
parallel behavior as burnup increases: there is an immediate drop
of thermal flux right at the beginning of life. This may be

60
Figure 2.3.6. Thermal-to-Fast Flux Ratio.

61
attributed to the appearance of xenon. Note that the cores are
always burned in the critical condition, which means that at BOL
there is a high concentration of soluble boron keeping the core at
critical. This boron concentration is rapidly reduced as xenon
builds up to an equilibrium level. It would seem then, that the
core is changing one thermal poison for another, and therefore no
significant change of neutron spectrum would occur, but this is not
the case. The fact that the thermal poison is in the fuel (in the
case of xenon) as opposed to being in the moderator (in the case of
boron) has a significant effect on the neutron spectrum. This is
due to the different relative changes of thermal absorption cross
section occurring in the fuel and in the moderator. Thus, as
burnup increases, the thermal fraction of the neutron spectrum
continues to drop slowly (the spectrum hardens) due to the
generation of fission products, which are thermal poisons located
within the fuel, as is the case with xenon.
A key point in the safety of operation of a new reactor is its
thermal-hydraulic behavior. It has been stated before that the low
power reactors should deliver the outgoing coolant in the same
thermodynamic conditions as the standard reactor, because this is a
basic factor in the thermal efficiency of the plant. It is obvious
that if a reactor delivers a reduced amount of power, it is also
forced to deliver a reduced coolant mass flow in order to keep the
thermodynamic conditions at the outlet unchanged, and this flow
reduction encompasses different characteristics
transfer conditions.
in the heat

62
One of the main parameters of concern when analyzing the heat
transfer conditions in a EWR is the minimum DNBR (Departure from
Nucleate Boiling Ratio) existing in the core. The minimum DNBR
indicates how far the core is from reaching a film boiling heat
transfer condition, which would result in heavy overheating and
damage of the fuel.
Mark Miller performed calculations(16) on the subject of the
DNBR for several power densities and several flow rates and
compared the results to the DNBR actually found in the standard
reference core. A more detailed analysis of the thermal-hydraulic
performance of the low-power cores is presented in Chapter V. It
is, however, important to point out here that all the reduced-power
cores show a higher DNBR than the standard core, under equal outlet
enthalpy condition, and using the same fuel lattice geometry. For
larger pitches (which may be of interest from the discharge burnup
viewpoint) the low-power cores can accept some pitch increase and
maintain the outlet enthalpy and still stay at a better DNBR than
the standard reactor. However, the standard core can not use
larger pitches without seriously reducing its thermal-hydraulic
safety margins, due to the reduction of coolant velocity associated
with the increase of fuel lattice pitch. This is an unequivocal
and key point proving the substantially higher safety level of the
low power reactors as compared to the standard ones.

63
2.3.3. Burnup Achievements and Isotopic Inventories.
As explained in Section 2.2.3, the preliminary burnup
calculations were performed with the LEOPARD code, using a pin
enrichment equal to the average enrichment of the core, and burning
it with a varying soluble boron concentration that would keep the
core as close as possible to criticality. Burnup explorations were
done for the standard reference core 3A, for the 25% power density
core 4A, and as a scope extension, for the 20% power density,
preheater core 8. All of them were examined at the standard core's
pitch of 1.2573 cm; cores 3A and 8 were also investigated at one
larger pitch, and core 4A was studied for a wide range of pitches,
since it is the most representative low power density core. The
multiple burnup calculations performed for core 4A have the purpose
of determining the parametric effect of pitch on the discharge
burnup level, as well as giving an indication of the optimum pitch
value.
Table 2.3.2 shows the main burnup and isotopic results
obtained iron the preliminary burnup calculations.
Reducing power density to 25% of the standard level while
keeping the standard core's pitch results in an increment of burnup
of 9%. However, the burnup increment can be improved to about 16%
by increasing the pitch of the low power reactor to about 1.3 or
1.4 cm. Figure 2.3.7 shows the burnup levels achievable as pitch
is varied on core 4A. Note the important fact that the maximum
burnup is not achieved at the pitch value having maximum BGL
reactivity (shown in Figure 2.3.3) but at a smaller pitch. This

64
Discharge
Burnup
GWd/MT
21.5
21.0
20.5
20.0
Figure 2.3.7. Discharge Burnup of 25% Ibwer Density Core vs. Pitch.

65
Table 2.3.2. Scoping Burnup and Isotopic Results.
Case
Pitch
Energy
Cycle
U-235
(Kq/GWd)
U-238 Fiss
Pu
(Kg/GWd)
I.D.
(cm)
(GWd)
Life
Fiss.
Disch.
(Kg/GWd)
Fiss.
DISCH.
3A
1.25
1737
ly 316d
0.6343
0.6319
0.0805
0.3291
0.4064
3A
1,60
1849
ly 360d
0.7073
0.4779
0.0495
0.2686
0.2726
4A
1.25
1893
8y 049d
0.6218
0.5293
0.0854
0.3519
0.3895
4A
1.30
2022
8y 252d
0.6210
0.4519
0.0787
0.3512
0.3474
4A
1.35
2033
8y 269d
0.6370
0.4307
0.0708
0.3355
0.3169
4A
1.40
2028
8y 261d
0.6518
0.4195
0.0647
0.3207
0.2969
4A
1.45
2031
8y 266d
0.6587
0.4112
0.0613
0.3127
0.2844
4A
1.60
1881
8y 031d
0.7033
0.4604
0.0499
0.2676
0.2610
8
1.25
1511
8y 043d
0.6387
0.4785
0.0707
0.3227
0.3795
8
1.40
1362
7y 116d
0.7092
0.5386
0.0545
0.2688
0.3448
Note: Ave. fuel enrichment is 2.6% for all cores except for core 8
it is 2%.
Total energy generated assumes a core with 94.42 MTU loading.
Cycle life is calculated assuming 75% availability.
For Core 3A, pitch = 1.2573 which is the Westinghouse value
for a standard core using 17X17 assemblies.
Data obtained from LEOPARD burnup calculations.

66
means that the isotopic and spectral effects occurring during the
life of the core cause a shift of the EQL reactivity, favoring less
moderated configurations. The main reason for this effect is a
conflict between optimum moderation and enhanced conversion ratio:
Conversion ratio is enhanced when the resonance absorption in U-238
is increased, but this is obviously an effect which reduces the
immediate reactivity of the core. The trade-off appears when
realizing that at E.O.L., a large proportion of the power is
generated by the converted plutonium. Thus, if conversion ratio is
reduced in trying to optimize moderation (and thus increasing BOL
reactivity), the core reaches EQL with a reduced amount of
plutonium, and therefore becomes subcritical at a lower burnup
level.
Note on Table 2.3.2 the relatively high level of burnup
achieved by core 8, in spite of being fueled with only 2% enriched
fuel. The effect must be attributed primarily to the different
moderator characteristics, in addition to the effect of the reduced
power level.
It appears at this point that the best pitch from the burnup
viewpoint is larger than the one used in the standard reactor, but
smaller than the one yielding maximum reactivity at BOL. However,
the model used for these scoping calculations can not be assumed to
accurately represent a reactor in an equilibrium cycle since BOL
compositions with no burned fuel were employed. The question is
then whether the optimum burnup pitch obtained in these studies
will still be the best for an "equilibrium cycle" core or if it

67
will be a different one. In order to answer this question it is
necessary to perform burnup calculations with a model that can
reasonably represent a core in an equilibrium cycle, i.e. there
must be some burned fuel present at BOL for each reload core.
These scoping studies, however, can serve as guidelines for
"equilibrium cycle" calculations.
There are a number of criteria that can be considered
important when examining the results of a burnup analysis from an
isotopic or fuel utilization point of view:
a). Search for the maximum energy output from the same load
of fuel.
b). Search for the minimum discharge of plutonium, for
non-proliferation purposes.
c). Search for the minimum amount of net fissile material
being consumed per unit of energy produced. This would account for
the fissile species being discharged with the spent fuel, and
therefore would assume a spent fuel reprocessing policy.
d). In the same way as the criterion above looked for the
best resource utilization, it might also be important to look for
the best economical combination of resource utilization and cost of
fuel cycle (including fuel enrichment, fabrication, storage,
reprocessing, etc.).
Although the criteria that involve reprocessing appear more
meaningful from the scientific or technological point of view than
those which do not include it, the present political situation in
the U.S. prevents commercial reprocessing, and this must be

68
considered when comparing results of burnup or uranium resources
utilization.
Since the present study assumes all fresh fuel to have the
same characteristics, increasing the discharge burnup implies
improving ore utilization. In this aspect, core 4A has a better
fuel utilization than core 3A. Although core 8 is able to achieve
a high burnup for a low-enriched fuel, the total use of U-235 per
unit of energy produced appears to be somewhat worse than that of
cores 3A and 4A. However, core 8 is not starting from the same
type of fuel, and therefore this result should not be considered as
a negative point for core 8. In addition, core 8 could be used to
further burn the fuel that is ordinarily discharged from standard
PWR's which would in fact represent an improvement of overall ore
utilization if fuel reprocessing is still not being considered.
Figure 2.3.8 shows the use of U-235 per unit of energy
produced, and the relative amounts of plutonium-to-U-235 burned for
the different cores, and for some varying pitches. Note that the
amount of plutonium burned is maximum for the tightest pitches, due
to the better conversion ratio associated with the hard-flux,
undermoderated configurations. As a logical result, U-235
consumption per unit energy produced tends to increase with pitch.
It is interesting to realize that for a slight increase of pitch
above the standard (from 1.25 to 1.3 cm), the U-235 specific
consumption actually decreases slightly. This is due to the
reactivity enhancement which occurs in going to larger pitches.
The poorer plutonium production quickly offsets this factor and the

69

70
U-235 specific consumption begins to increase. It is thus seen
that both U-235 specific usage and plutonium relative burning
remain fairly close to the optimum values for the pitches that
allow highest burnup and therefore longest cycle times, while
worsening rapidly after the region of interest.
2.4. Scoping Work Conclusions
The scoping calculations show that there is good hope for a
significant increase of discharge burnup from the same initial fuel
by using low power density cores. The main effects contributing to
the necessary extra reactivity are the reduced level of xenon and
the reduced effect of Doppler broadening of resonances. A 9%
increase of burnup seems to be achievable by going to a 25% power
density core, while up to 16% improvement appears feasible by
varying the cell pitch. However, these calculations were performed
with a calculational tool which is not well-suited for the burnup
analysis of a core consisting of fuel of various enrichments and
exposure levels. Some changes are surely to be expected when using
an adequate core model for these burnup calculations. The burnup
calculations made in this scoping study are indicative of a
first-load core behavior rather than of an equilibrium core; the
latter is the configuration of most interest for burnup or economic
studies.
It has become evident that a maximum level of automatization
is necessary for the calculational techniques to be used in
detailed burnup calculations. The adoption of LEOPARD to
substitute the chain of PHROG BRT MONA BURNUP proved to be

71
most advantageous, and a similarly automated scheme should be used
for the detailed calculations. However, a new burnup calculational
scheme must be developed which allows the adequate representation
of a several-batch reactor core in its equilibrium cycle.
After these scoping studies are completed, the in-depth study
should be centered on the cores that can best conduce to
short-term, commercially feasible low power density systems. The
in-depth study is, therefore, centered on a single-core concept
having a reduced power density in the 25 to 60% of the standard
core. The cores selected for the in-depth study will be discussed
in Chapter IV.
The safety aspects of the low power cores are extremely
favorable. The fuel average temperature is drastically reduced,
indicating a reduced risk of pin damage. The heat transfer
conditions have their safety margins, e.g. the DNBR, significantly
improved, which makes the probability of critical heat transfer
conditions much lower than it is in the standard reactor.

CHAPTER III
BURNUP CALCULATION METHODS
3.1. Burnup Calculational Methods and Present Needs
Once the philosophy behind the development of the SHARP is
established and a power density or a range of power densities are
chosen, the key work to be done in the area of fuel utilization is
the optimization of the pin-cell of the SHARP. The fuel
utilization and performance (burnup and isotopic results) are then
to be compared with those obtained from a standard power density
core to determine its relative merit. This optimization and
comparison require a large number of reactor life calculations,
where items such as batch-wise burnup, discharge burnup and
physical cycle life, etc. are of interest among others.
Many different organizations are interested in developing and
performing reactor fuel cycle and burnup calculations. The wide
range of organizations and specific interests has caused the
development of different calculational schemes, in order to
accomodate the different cost/effort/accuracy ratios desired for
each particular type of study. Some of the more important types of
burnup studies are the following:
a). Vendor burnup studies and core optimization, performed as
design work. These studies are most complete, and they generally
include three-dimensional, pin-by-pin studies which are closely
72

73
coupled with thermo-hydraulics models. These type of studies
require large computational efforts and highly specialized
personnel, which makes them extremely costly.
b). Burnup studies performed by utilities, either for core
following or for licensing. These also have to be detailed
studies, because they require strong economical decisions affecting
the core cycle, or have significant safety considerations affecting
the licensing procedure. However, there is presently a trend by
utilities to use more empirical codes (such as nodal codes) which
run at lower costs than the fully detailed pin-by-pin studies.
Nodal codes generally need base detailed diffusion theory
calculations for normalization purposes, but still result in an
overall lower cost than the fine mesh neutron diffusion theory
codes, while being able to maintain the information necessary for
adequate core management.
c). Studies performed by engineering companies or government
agencies for assessment of energy policies or new reactor concepts
or fuel cycles. These studies fall in a whole new class. They are
not directly concerned with the behavior of each particular pin,
not even of assembly-wise details. They look at overall results of
burnup, time scales, and monetary volumes. Although they need to
yield reasonable results, the numerical results themselves are
normally less critical, and emphasis is placed on the comparative
results of two or more different concepts. These studies generally
use some type of empirical formula or simplified core model which

74
does not require detailed specification of compositions and
geometries, as the two previous types of studies did.
d). Burnup studies performed in universities or
research-oriented organizations for developmental studies of new
reactor concepts, for preliminary reactor design or fuel
management. These studies may range in their specificity anywhere
from the very rough empirical formulas used for fuel resources
utilization and policy studies to a level close to the vendor or
utility calculations. However, they tend to be more concerned with
particular theoretical details, which call for calculational
methods specific to each study. These studies normally follow
standard industrial calculational procedures until a certain point
where the particular aspect of the study calls for a specific
calculational tool, which is often developed for the occasion.
A wide range of other types of burnup or fuel cycle studies
exist, but the four categories stated above illustrate the
different levels of accuracy or detail that may be sought depending
on the purpose of the study, and how they require different
calculational costs, manpower needs and calculational tools.
For every burnup study (except for the very simplest ones,
where just an empirical formula is applied),two main calculational
steps are performed. Each of these steps can have different levels
of sophistication depending on the particular needs of the study,
and in some cases one or both of the steps may need to be repeated
iteratively in order to obtain the accuracy required by the study.
Figure 3.1.1 shows the basic flow of data between the two steps.

75
Figure 3.1.1. Basic Steps in a Burnup Calculation

76
The first step involves using a code in which a
representative unit cell of the reactor is burned. Macroscopic
core geometry is avoided; the emphasis is placed in burning a unit
fuel cell and surrounding it with neutronic conditions as close as
possible to those to be encountered by the "average" fuel element
during the real core life. The purpose of this calculational step
is to obtain neutronic characteristics of the fuel as it undergoes
burnup, which will then be used in the second step of the burnup
calculations.
Input to these first-step codes normally includes the
following:
a). Fuel cell geometry and initial isotopics.
b). Temperatures of fuel and moderator.
c). Power density (normally linear power density).
d). Definition of a buffer region surrounding the fuel cell
in order to modify the neutron spectrum for the presence of
structural materials (guide thimbles, fuel assembly cans, spacer
grids, water holes, etc).
e). Specification of a series of burnup timesteps that will
determine the points where the code has to recalculate the
neutronic status of the fuel cell, which has been modified by the
burnup process.
f). Specification of controllable poisons present in the fuel
cell. This is normally specified as soluble boron concentration in
the moderator, and may be specified for each of the burnup
timesteps defined for the calculation.

77
g). Some models may allow for the specification of variable
power levels for the different timesteps, the specification of
Dancoff factors or some geometric data that may allow the code to
calculate than, the specification of a perpendicular buckling, or a
buckling search in order to obtain a critical spectrum, etc.
Most of these first-step pin cell codes are self-chaining, in
the sense that the user specifies the cell geometry, isotopics and
burnup history at the beginning of the code execution, and the code
performs the burnup steps and the cross section evaluations after
each timestep without user intervention. Other schemes, like the
one used for some of the preliminary calculations in the SHARP
study, involve several codes to perform the pin cell burnup, and
they need the user to manually handle the cross section libraries
and the burnup isotopics back and forth between one code and
another. The advantages and disadvantages of each method rely
mainly on the flexibility and freedom of data handling and geometry
specification versus convenience and man time requirements. Figure
3.1.2 shows some possible schematics of the data specification for
this first step of burnup calculations.
The output of these first-step codes normally includes the
following data:
a) Microscopic cross sections for each isotope present in
the fuel cell, for the number of neutron energy groups specified in
the input (normally between two and five groups) and for each
specified timestep.

78
Soluble
Boron
Concentration
Figure 3.1.2. Exanples of Burnup Conditions and Timestep
Specification for Fuel Cell Burnup Calculation.

79
b). Macroscopic cross sections of the fuel and the whole
cell, weighted with the neutron spectrum present at each timestep,
and with the number of energy groups requested.
c). Infinite multiplication factor of the fuel cell, and
effective multiplication factor if a perpendicular buckling was
specified.
d). Average neutron speed.
e). Concentration of each isotope present in the cell,
regionwise and cell homogenized.
f). Cumulative burnup level since the beginning of the burnup
calculation.
g). Other data depending on each particular model, such as
the calculated Dancoff factors, fraction of the total core power
produced by each fissile isotope, cumulative fission densities,
conversion factors, etc.
The second step of the burnup calculations involves taking the
data generated by the first code and using it in a core model which
accepts overall core information regarding geometry and core
operation in order to simulate the actual life of the reactor.
This step is the one showing the widest variations from one type of
burnup study to another. The most sophisticated models are able to
follow the core life without further iterations, while others need
to feed their data back to step one of the burnup calculation, for
a new iteration. The flow of data from step one to two and back
depends on the sophistication of each of the models and the burnup
data sought from the overall study.

80
The second step uses such a wide range of calculational
approaches or methods, that it is difficult to specify the general
input requirements and output data. As a general rule, the
explicit core model (in which the fuel is represented either pin by
pin or in relatively small nodes, and control materials are
specifically treated) needs the initial isotopics information, the
microscopic cross section libraries generated by the first step of
the burnup calculation, the core geometry data, the power history,
the control materials history, and general editing information,
while the output includes core isotopics in zone averages, pin by
pin, or whatever region type is used by the code versus core
history; criticality evaluations, power distributions, burnup
information in average or explicit for each unit considered in the
core, region averaged multigroup neutronics data, etc. Codes of
this type are for example FDQ-7(17), SIMULATE(18), etc.
Figure 3.1.3 shows the geometry and material zones
specifications needed for a two-dimensional calculation of a
quarter assembly of a FWR with FDQ-7. Note the extreme detail of
the geometrical description, and the relatively large number of
different compositions considered. It is easy to imagine the
extreme complexity of the calculation of a whole core in three
dimensions, with the necessity to identify different fuel batches,
control rod assemblies, burnable poison rods assemblies, assemblies
with water holes, etc. Nodal codes avoid some of the complexity
and reduce computational time by lumping each fuel assembly into a
few nodes, but they require node-interaction parameters and albedos

81
Diffusion Calculation Mesh
l~" I Fuel Pins
i'x V\ \ i Interassembly Water Gap
fyx;::~i Water Holes (or LBP's)
Instrument Thimble
Figure 3.1.3. Geometry and Composition Specification for a FDQ-7
1/4-assembly Burnup Calculation.

82
which must be normalized with a detailed core calculation, as well
as requiring adjustment of a large number of empirical factors.
There are other computational models which run at lower costs
and may not require the technical expertise needed for running full
blown PDQ-7 models or three-dimensional nodal codes. These models
normally accept a lower degree of geometrical sophistication, or
they may do just a part of the burnup calculation. For example,
they may perform the criticality and eigenvalue calculations, but
not be able to perform the actual core burning, which then has to
be done by a separate code. This is the case of MONA, which was
used in some of the preliminary studies of the SHARP. This type of
codes are useful for criticality studies or spectral effects
analysis, but they are of little use for a real life burnup study
because of the enormous manual data handling required. However,
they have the advantage of the low cost and the relatively simple
input.
Other types of calculational models are often used for scoping
or medium accuracy burnup calculations. They normally are
simplified codes which use part of the data obtained from the first
step of the burnup. calculations, and then introduce an approximate
core model, using empirical factors or formulas whenever the
available data is insufficient for a rigorous treatment of the
items needed for the calculation. However, these types of codes
cover a wide range of sophistication and thus their accuracy and
the information that can be obtained from them is largely variable.
The core model used for the SHARP can be included in this

83
classification, evolving from a well known calculational model.
The innovative core model used for the SHARP is an improvement on
the accuracy of the results, resembling closer the real reactor
life. It is achieved by the proper handling of significant core
history data, which appear to have been previously neglected. An
explanation of the data handling by these codes follows.
These simple calculational models are normally based on a pin
cell burnup calculation and often use the infinite multiplication
factor calculations in order to obtain an estimate of the core
criticality state or its reactivity.
In the simplest model, a pin cell would be burned for its
whole life without any poisoning, and at a constant power level.
The characteristics of a pin are a good representation of the fuel
batch to which it belongs, since the pin is surrounded by many fuel
pins alike. The only exceptions are the pins facing a fuel
assembly of a different batch (and therefore having a sharply
different burnup status) or facing the reflector. Then, if the
fuel pin is burned for its entire life, it certainly passes through
the stages corresponding to each batch in the core. This means
that it should be possible to take seme of the neutronic properties
of the pin being burned in the cell burnup code and obtain the
characteristics of each batch present in the core at any particular
time, which should allow in some way to calculate the status of the
whole core. If the core contains three batches, the cell burnup
calculation should be performed in such a way that the timesteps
chosen would be repeated three times, and each of the three sets of

84
timesteps would span exactly the expected duration of one core
cycle. This would allow the data from the corresponding timesteps
in each cycle to be grouped into a core timestep. Figure 3.1.4
illustrates this "timestep synthesis" from a hypothetical cell
burnup scheme in which five timesteps were allowed for each of the
three cycles of core residence of the fuel. The data sought is the
infinite multiplication factor of the core, based on the unpoisoned
multiplication factor of the pin through its life. The figure
shows the conceptual grouping of each timestep's data, but not the
calculations involved in it, which will be dicussed later. Note
that the timestep pattern is repeated in each cycle of the cell
burnup scheme, in order to make the timesteps correspond to the
same time of core life.
However, there are several ways of specifying the pin cell
burnup and of doing the calculations for collapsing the cell
timesteps into core life timesteps. Some of the systems used for
pin cell burnup specification will now be examined.
The simplest scheme, as mentioned before, burns the fuel cell
without any poisoning and at the nominal power, for the length of
time that is estimated that the fuel will remain in the core. Then
the multiplication factors of the cell timesteps are collapsed in
order to obtain the core multiplication factor through the core
life. An interpolation or extrapolation of the curve of core
multiplication factor as it reaches a value of unity determines the
end of the core cycle. The core cycle length is then optionally
used to perform new iterations of the fuel cell burnup calculation.

85
Pin Cell
K oo
Figure 3.1.4. Grouping of Pin Cell Data into Core Data.

86
until it agrees with the cycle length estimated by the timestep
collapsing procedure.
This scheme, however, involves gross errors because the
absence of soluble boron causes variations of some percent in the
cross sections of some isotopes such as U-235 and Pu-239. This
entails errors in isotope concentrations and as a result, errors in
the multiplication factor calculations.
Another cell burnup method, which partially solves this
problen, specifies a soluble boron concentration in the moderator
with a value similar to the time-averaged boron concentration of a
real core. This concentration is approximately 400 ppn. But then
the fuel cell multiplication factors can not be directly used for
the calculation of the core multiplication factor, because they
involve the soluble boron poisoning. The problem is solved by
using a very short "pseudo-burnup" timestep, with no soluble boron,
after each real burnup timestep. A "pseudo-burnup" timestep is a
very short timestep which practically adds no burnup to the fuel,
so that the isotopics are not altered, but still allows for the
eigenvalue calculation in the boron-free configuration. This way,
the fuel is always burned with a spectrum influenced by the 400 ppn
of soluble boron, but the multiplication factors are obtained from
the pseudo-burnup timesteps which contain no boron, and the "clean"
batch reactivity can still be obtained through the fuel life.
Figure 3.1.5 illustrates this method, showing the boron
specification and the multiplication factors that may be obtained
through the core life.

87
Soluble
Boron
Pin Cell
First Cycle
Second Cycle
Third Cycle
Figure 3.1.5. Constant-boron Pin Cell Burning with Boron-free
Pseudo-burnup Steps.

88
The advantage of this method is that the core isotopics do not
contain the heavy errors due to errors in the neutron energy
spectrum, and that the method is still rather simple. On the other
hand, the pin cell burnup calculations require a much larger number
of timesteps, which increases the computational costs, and
although the final isotopics may not be grossly wrong, there is no
guarantee that the chosen boron average concentration is really the
average for the real reactor life.
PDQ-7 assembly burnup calculations conducted have shown that
an average error of 50 ppm in boron concentration (even in
real-time concentration following calculations) causes isotopic
errors of 1.2% in Pu-239 and 1.27% in Pu-241 for a burnup level of
34000 MWd in a fuel cell configuration of a tight 1.20 cm pitch
(containing very small amounts of moderator and its associated
soluble boron). For a large pitch cell of 1.60 cm, isotopic
differences of 3% in Pu-239 were found after only 9000 MWd burnup,
for an average error in boron concentration of 100 ppm. These
facts require a very precise specification of soluble boron
concentration if accurate isotopics are needed, and this is the
case if one is concerned about core life determination. The system
described on Figure 3.1.5 has the further disadvantage that the
fuel is burned without the critical spectrum, and although the
discharge isotopics may not be grossly in error, the errors
existing at any particular time during the core life are unknown,
and so are the errors incurred in calculating the core
multiplication factors, which in turn will determine the core life.

'
89
The above reasons induced the author to consider that the
specification of the boron level that would keep the reactor
critical at all times is of significant importance for an accurate
determination of the core life and should not be neglected, as is
commonly done in simplified burnup calculations.
Following the soluble boron concentration through the life of
the fuel implies a fuel cell burnup boron specificaton such as the
one shown on Figure 3.1.6. This solves the problen of the
erroneous neutron spectrum, but creates the new problem of the
boron letdown curve needing careful adjustment, mainly concerning
the core life duration. This is needed because specifying too
short or too long of a cycle time on the boron letdown curve could
cause the period of fuel burning to be done with a very low boron
concentration when the highest concentration would be required, or
vice versa, and this would bring back the problems of erroneous
isotopics and eigenvalue estimates because of inexact spectra.
This boron determination requirement is solved by the calculational
method described later in this chapter. This new method contains
important differences with the schemes described above. The
previous schemes relied on an extrapolation or interpolation of the
core multiplication factor (or sometimes on the core reactivity,
which is supposed to follow a more linear behavior) for the
determination of the core life. The scheme used in this work,
where soluble boron is constantly adjusted to its real value, can
not rely on these eigenvalues since the core eigenvalue should by
definition be kept at a value of unity throughout the core life.

90
Soluble
Boron
First Cycle T Second Cycle T Third Cycle
Relative
Power
Figure 3.1.6. Pin Cell Burning with Soluble Boron Letdown Curve
and Power Level Following.

91
It would still be possible to use the "clean" pseudo-burnup steps,
but the author decided that the adjustment of the boron letdown
curve to the point where its value should be zero was best suited
for this purpose. The advantage of using the boron curve for the
EOL estimate is that it contains the constantly varying spectral
and self-shielding changes, which depend on core burnup status and
boron concentration, while other popular models neglect these
aspects.
The above paragraphs discussed the need for an accurate
determination of the soluble boron concentration level in the fuel
cell burnup code, but almost as important as the boron is the
specification of the true power density level through the fuel
life.
It is frequent to see that the simple burnup schemes do not
specify changing power density levels for the fuel as it progresses
through burnup. It is even frequent to see almost an equivalence
being used when referring to real-time core life and burnup since
they are considered to be so closely equivalent. However, it is
well known that the different batches in the core do not share
equal power levels in spite of the efforts to flatten the power
distributions. As a general rule, the batches being burned for the
first and second cycles generally hold a higher power level than
the third-cycle batch.
When assigning a boron letdown curve for the fuel cell burnup,
one is not only assigning a certain boron concentration to a core
life-time, but also to each burnup point through the fuel life.

92
This means that a certain boron poisoning is assigned for each
isotopic status of the fuel. Figure 3.1.7 illustrates the
consequences of not specifying the different power levels of the
fuel as it advances through its life.
If the power level is maintained constant, as is the case with
most of the simple burnup calculational schemes, the end of core
cycles correspond to 1/3 of the discharge burnup, 2/3 of the
discharge burnup, and the total discharge burnup, in a three-batch
core. Thus, the burnup levels slightly after 1/3 and 2/3 of the
discharge burnup are burned with a very high boron level, which
corresponds to the beginning of the core life. However, the real
core situation has a higher burnup at the beginning of the second
and third cycles, because of the higher power level of the fuel
during the first two cycles of core residence. This burnup-time
mismatch of the flat-powered model as compared to the real core
burnup situation may cause a sustained error in the correspondence
of soluble boron to burnup level, as shown on Figure 3.1.7, with
the resulting errors in batch multiplication factors and neutron
spectrum.
Figure 3.1.7 shows a hypothetical boron letdown curve for a
core cycle, and using the same time axis, the fuel burnup is
represented on the ordinates for both a case where the power level
is kept constant and for a case where cycles 1 and 2 have a
somewhat larger power level than cycle 3, as is the case in a real
core. It is possible to see the sustained boron concentration
mismatch, which becomes most accentuated in the vicinity of the

93
Soluble
Figure 3.1.7. Effect of Time-dependent Power Level and
Soluble Boron Concentration Specification.

94
cycle ends, due to the fact of having the two burnup schemes start
their second and third cycles with different burnups.
This boron mismatch causes yet a further problem: at the BOL,
the constant-power scheme forms the core with batches having
burnups of 0, 1/3 discharge and 2/3 discharge. Meanwhile, the
varying-power scheme starts the cycle with burnup of 0 for the
first batch, somewhat more than 1/3 of discharge burnup for the
second batch, and somewhat more than 2/3 of the discharge burnup
for the third batch. These increments are of the order of 5% of
the burnup level for the first batch, and of the order of 8% of the
burnup level for the second batch, which represent non-negligible
differences in the batches' reactivities. Having a higher level of
burnup in two out of the three batches, the varying-power scheme
presents a lower reactivity than the constant-power scheme, thus
requiring a lower level of soluble boron in order to achieve
criticality. This situation is maintained, with different sizes of
mismatch, through the whole core cycle, thus causing evidently a
different critical boron letdown curve for each of the schemes, and
yielding a necessarily different core cycle length. This, together
with the spectral effects of soluble boron are the two main causes
that require maintaining the real power levels during the fuel cell
burnup calculations.
Once it is clear how the simplified burnup models can use a
fuel cell burnup scheme with its chained timesteps and all the
different adjustable data in order to obtain the core behavior
through life, it is necessary to look at the different methods

95
available for combining the several fuel cell timesteps that form
the complete core at each particular time.
The core is composed, at any timestep of its life, of a
certain number of fuel batches. Each of these fuel batches is
represented in the simplified burnup core model by one timestep
from each of the cycles specified in the fuel cell burnup
calculation, as shown on Figure 3.1.4. The problem is, at this
point, to determine the core multiplication factor from the
multiplication factors of each batch. This can also be done in a
number of ways, with varying degrees of complexity. The simplest
model gathers the batches' infinity multiplication factors and
averages them. The result is considered to be the multiplication
factor of the core. That is,
Kc =
n
where Kc = Core effective multiplication factor.
Ki = Infinite multiplication factor of batch "i".
n = number of batches in the core.
Note that the fuel cell burnup schemes often include a
perpendicular leakage factor, so that what is referred here as
"batch infinite multiplication factor" may well be already
corrected for axial leakage, which makes the core multiplication
factor appear much more realistic when compared to the more
sophisticated core criticality calculations.

96
The above estimate of the core multiplication factor, although
frequently used for rough criticality estimates, is rather
simple-minded, since it does not take into account the fact that
there are batches which for their combination of power sharing and
geometrical situation represent a heavier weight in the core
multiplication factor. It does not take into account either the
fact that the geometrical disposition of the batches in the core
causes some batches to experience a much larger radial neutron
leakage than others.
More complex type of "simple" burnup calculation core models
use weighting factors from the batches' power sharing in the
calculation of the core multiplication factor. For example,
VK = Zj (Fi/Ki)
where K = Core effective multiplication factor.
Fi = Fraction of the core power produced by batch "i".
' Ki = Multiplication factor of batch "i".
In this case, the power fraction generated by each batch in
the core may be obtained as
Fi = Ki"
( ZiKin)
This calculational scheme is significantly more complete than
the simple K-averaging one described before, but still does not
take into account the very important fact that one of the batches

97
lies on the core periphery, thus experiencing a radial leakage
which is quite different than that of the internal batches. There
is a model which corrects for this effect by multiplying the
peripheral batch multiplication factor by an empirical non-leakage
constant, which is again an improvement of the core model.
However, the radial leakage of the peripheral batch is mainly
affected by the reflector characteristics, and since the
reflector's main component is cooling water, the neutronic changes
experienced by the coolant during the core life must heavily affect
the reflector's performance. Indeed, the changes in boron
concentration in the water from B.O.L. through E.O.L. cause the
thermal absorption cross section to vary by a factor of about
three, and even the transport cross section is affected by some
percent. This suggests that if soluble boron concentration is
important for the spectral effects in the core, it is also
important in its effect on the reflector characteristics which
directly affect the radial leakage, and this effect should be also
taken into account if at all possible. It appears that this effect
has not been taken into account in known simplified burnup models;
it has been included in the model used in this work.
Whenever a simplified model for the representation of a
complex system is used, it is necessary to reach some compromise
and take into account only those factors which are considered of
first importance, while bypassing others of lower importance. The
factors considered of importance in the model used in this work are
summarized in the following specifications:

se
a). The fuel burned in the "basic" codes must be imbedded in
a flux spectrum which is as close as possible to the one existing
in the actual reactor for the fuel being studied.
b). Boron concentration must be kept as close as possible to
its true value throughout the core life, in order to avoid
erroneous spectra that would affect both the criticality
calculations and the isotopic evolution.
c). The model must be able to follow a "real time" evolution
of the core, as opposed to using burnup as a time measurement,
because burnup is not uniform through the core life, and some
isotope appearing or being burned earlier or later in the core life
affects the spectrum, the criticality study, the isotopics, and
eventually the cycle length.
d). In order to accomplish the previous point, the model must
be able to assign the right power to the fuel at each point in
life, i.e., if a "real time" model is sought, it is essential to
have a correct time-power-soluble boron correspondence.
e). In order to obtain the correct batch power assignment at
all times during the fuel life, it is necessary to have a fair
estimate of neutron non-leakage probability for each batch, which
if at all possible, should be made automatically adjustable through
life.
With all these ideas in mind, the model was developed as
described in the next sections, starting with a fuel pin code,
going to a fuel assembly burnup calculation, and finally feeding
the data to the core-simulation code.

99
1.2. Method Developed for this Study
3.2.1. The Fuel-burning Codes.
As it was explained in the preceding section, any burnup
calculation must start with a fuel-burning scheme which provides
the necessary data for the subsequent core model. In the case of
the SHARP study, the number of initial calculations needed was
rather large, due to the significant number of different power
levels and pitches that had to be studied. This required a rather
automatized calculational procedure which should be of a relatively
low cost. In addition, since the differences between the several
cores to be studied might not be too large, calculations need to
retain maximum accuracy while again not incurring in excessive
costs. The first task was to choose the code or codes to be used
for the fuel-burning step.
One of the more classic pin-burn codes is LEOPARD(11), which
is described in more detail in Appendix B.5. LEOPARD uses a
MUFT-SGFOCATE(12,13) scheme for the calculation of cross sections
for a pin-cell geometry, and it has been and is being widely used
in industry for many of the burnup study types described on the
preceding section. LEOPARD offers flexibility, low cost and simple
input requirements; however, LEOPARD presents sane problems for the
SHARP study:
a). It does not allow for a change of the pin power during
the burnup process, thus making specification 4 of the model
requirements hard to keep.

100
b). It has known restrictions concerning the self-shielding
calculations. These restrictions might be detrimental in the SHARP
study since the somewhat high burnups expected entail
higher-than-normal concentration of highly absorbing nuclides such
as fission products and plutonium.
The EPRI-CELL(19) code was also considered. Although
EPRI-CELL is not a "classical" code yet because it is fairly new
(released to industry users around 1977), the calculational scheme
it uses is based on well-proven calculational methods, and the
benchmarking done to date proves its high accuracy. EPRI-CELL uses
a GAM-THERMOS-CINDER(20-22) scheme for cross section evaluation and
fuel burning isotopic chains. It presents basically all the
advantages of LEOPARD (namely, it is a pin-cell code of simple
input and automatically chains spectrum and burnup calculations)
and it does not present the two limitations of LEOPARD, thus
allowing a higher flexibility and accuracy in the pin-burn
calculation. The only disadvantage of EPRI-CELL when compared to
LEOPARD is its running cost, which is about one order of magnitude
higher. Despite this drawback, EPRI-CELL was chosen as the
pin-burn code, with the initial intention of using it as the only
calculational tool external to the core model.
However, when examining the documentation of the EPRI ARMP
code package (of which EPRI-CELL is a part), it is evident that the
purpose of EPRI-CELL is to generate sets of cross-sections through
the life of the fuel for use in generating the HARMONY tables for
PDQ-7. PDQ-7 is then expected to be used for the actual core-life

101
calculations, since it allows for multi-dimensional and complex
geometry specifications. PDQ-7 calculations are long and
cumbersome if one is trying to model a two or three-dimensional
large grid, but it may be used with reasonable ease for a small
geometry (a 1/4 fuel assembly in two dimensions, for example).
This last fact suggested that PDQ-7 could be used instead of
EFRI-CELL in order to perform the fuel burnup calculations. This
may appear awkward since it first takes an EPRI-CELL run to feed
the cross section tables to PDQ-7. The system requires more codes
in order to obtain apparently the same data for the core model, and
manual data handling with PDQ-7 is far more cumbersome than with
EPRI-CELL, even for a simple geometry case. However, there are a
number of advantages in using PDQ-7 instead of EPRI-CELL for the
fuel-burning calculations. Some of these advantages are given
below:
a). Given the type of core model that must be used for the
burnup calculations of the SHARP study, it normally takes more than
one iteration to adjust all the core life parameters described in
the previous section (soluble boron letdown curve, batch relative
powers, etc.) and therefore, each case studied requires several
fuel burnup calculations. If PDQ-7 is used, the iteration only
requires repeating the PDQ-7 calculation and not the EPRI-CELL one,
if the fuel cell characteristics are not altered. Each 1/4
assembly EDQ-7 calculation costs only one fourth as much as an
EPRI-CELL one, because PDQ-7 is a diffusion-theory code working
with pre-tabulated few-group cross section libraries, as opposed to

102
EPRI-CELL, which is a neutron transport theory code, working with
raw fine-group libraries. However, PDQ-7's calculations may be as
accurate or even more accurate than EPRI-CELL's as will be
explained below.
b). For a well-versed user, PDQ-7 has almost no restrictions.
The code allows for extraordinary flexibility in the specification
of the parameters for simulation of the core environment for the
fuel burning process, and it has very powerful editing
capabilities.
c). EPRI-CELL simulates the reactor environment affecting the
fuel pin by placing a buffer region around the moderator ring.
This buffer region is composed of coolant and a certain proportion
of structural materials which modify the fuel cell neutron
spectrum. This system yields reasonably accurate average isotopics
for the fuel batches, but there are really very few pins in any
assembly which would behave as a "batch average" pin. PDQ-7, on
the other hand, can specify the geometry of a 1/4 fuel assembly as
described on Figure 3.1.3 in the previous section. That
description treats each fuel pin in the assembly as a separate
unit, and it is obvious that the pins neighboring water holes or
facing the interassembly water gap do not see the same spectrum as
a pin surrounded by identical pins. These spectral differences
among fuel pins may translate into slight differences in the
calculation of batch-wise multiplication factors due to the
combination of different spectra and different isotopic composition
of each pin in the fuel assembly. If this is the case, PDQ-7's

103
estimate of the batch multiplication factor should be closer than
EPRI-CELL's, because of its more realistic modeling of the really
repeating pattern in the fuel batch, which is the fuel assembly,
rather than the fuel pin.
It should be pointed out that the use of "simplified"
geometries for the PDQ-7 calculation could result in severely
distorted results. The 1/4 assembly requires the specification of
different compositions due to the presence of the water holes and
instrument thimble, and one might be tempted to use a simple 2X2
pin geometry, trying to take advantage of PDQ-7's flexibility while
avoiding the additional data handling complexity of the complete
1/4 assembly. The results of a 2X2 pin array burning can be just
disastrous. Not only it does not take into account the
differential effects of the unevenly distributed water holes and
interassembly gaps on the different fuel pins of the assembly, but
being composed of just fuel element cells, it does not include any
material simulating the effects of the structural materials, which
in EPRI-CELL are represented by the buffer region. Such a burnup
calculation may result in isotopic errors of the order of 13% for
-235 and 20% for Pu-239 at a fuel burnup level of 40 GWd/MTU, with
the associated errors in the calculation of batch multiplication
factors.
When PDQ-7 is finally selected for the fuel-burning
calculation, a third code from the ARMP package needs also to be
used in order to simplify the point-burnup calculation. This code
is NUPUNCHER(23) and its function is to translate the

104
burnup-dependent cross sections generated by EPRI-CELL into a card
deck with an adequate format for the HARMONY(24) part of PDQ-7.
The deck includes the description of the isotopic chains and their
constants. It also allows for the cross sections to be fitted
against any variable that can be deemed significant on any given
cross section, for any particular isotope (for example, Pu-240
thermal absorption cross section is generally not expressed as a
function of the pin burnup, but rather as a function of Pu-240
concentration itself). NUPUNCHER greatly simplifies the task of
preparing input to PDQ-7.
Once the quarter-assembly geometry is set in PDQ-7, it is
necessary to burn the fuel following the circumstances encountered
in an actual reactor core as closely as possible. One of the main
parameters that influences the evolution of the fuel is the neutron
energy spectrum, as was explained above. The neutron energy
spectrum existing in the reactor at any given time depends on many
factors, some of which are related to the physical design of the
core, and therefore are implicit in the geometric and compositional
description of the quarter-assembly, while others depend on factors
that may vary during the core life. Two of the main variable
parameters influencing the neutron spectrum are the soluble boron
concentration and the power level. The soluble boron concentration
directly affects the neutron spectrum because it is a strong
thermal-neutron absorber. The power level affects the neutron
spectrum in an indirect way, because it determines the
concentration of xenon present in the fuel, and xenon is also a

105
very strong thermal neutron absorber. Furthermore, the power level
of the assembly through life affects in a significant way the
concentration of most isotopes through the core life, which is a
further reason calling for an accurate specification of the power
level at all times.
Thus, when the quarter-assembly modeled in PDQ-7 is burned for
its complete core life (a series of as many consecutive cycles as
fuel batches are present in the core), an estimate of the soluble
boron concentration versus time is specified for each of the
cycles, as well as the power level of the batch at each specific
moment. A first guess of possible soluble boron concentration and
power level evolutions through the whole life of an assembly are
shown in Figure 3.2.1.
Soluble
Boron
Figure 3.2.1. Estimates of Boron Letdown and Relative Power.

106
Note that the point in Figure 3.2.1 where the soluble boron
concentration reaches a zero value and jumps back up to the value
it had at BOL signifies the core cycle duration. Once the assembly
is burned according to the power and boron concentrations assumed
for all its life, all these burnup environment data together with
the multiplication factors of the assembly at each time step are
fed into the core model code. The core model code must evaluate
the multiplication factor of the whole core, the power level of
each batch, and the core cycle life duration; it thus furnishes
data for a better estimate of the real behavior of the core through
life. If the newly calculated data from the core model code agree
closely with the power levels, boron concentrations and cycle
lengths previously input to PDQ-7, the guess is final, and burnup,
core life and isotopics can be obtained from PDQ-7 and the core
model. If the core model calculated data differ significantly from
the data input to PDQ-7, it is necessary to modify the assembly
burnup data according to the core model, and run a new PDQ-7 burnup
case, starting a new iteration.
Figure 3.2.2 shows a flow chart of the codes and data handling
necessary in order to obtain the fuel burnup data to be input to
the core model code. The core model code is described in the next
section.
3,2.2. The_CRIBUR Core Model.
The CRIBUR code was developed as a simple model for core life
calculations which would include all the points outlined in Section

107
USER-PROVIDED DATA
CODES
LIBRARIES
Figure 3.2.2. Flowchart of Data for a Fuel Burnup Calculation Using
EDQ-7 1/4-assembly Geometry.

108
3.1 as being of significant importance for the determination of the
core life. Consequently. CRiBUR needs the input of a string of
infinite multiplication factors of a fuel batch burned with a
scheme in which the soluble boron letdown curve has been specified,
as well as the power levels of the batch through its life. CRIBUR
calculations include the following:
a). The criticality status of the core at each timestep
through its life, by combining the multiplication factors of the
batches in the corresponding time.
b). The necessary adjustments to the soluble boron letdown
curve in order to maintain core criticality.
c). A new estimate of the core life by interpolation or
extrapolation of the soluble boron letdown curve.
d). The power level of each batch during its core residence.
The fact that the core model emphasizes the core criticality
through life gave it the name of CRIBUR (CRItical BURning).
CRIBUR considers the core as an infinite cylinder surrounded
by a reflector- Radial leakage is obviously not the same for the
peripheral batch as for the inner batches, and thus it requires a
special treatment; on the other hand, the axial leakage is common
to all the batches, and is directly accounted for in the
2-dimensional PDQ-7 assembly calculation. thus making it
unnecessary to keep track of it in the core model.
In this study, the core model represents a Westinghouse 3400
MWth. four-loop system with a core radius of 168.53 cm. This

109
dimension is used for radial leakage calculations, and although it
is hardwired into the program, it may be easily changed if a
different core radius is considered. Figure 3.2.3 shows a
schematic drawing of the geometry used in the core model.
Reflector
Outer Batch
Inner Batch
Figure 3.2.3. Core Geometry Used in CRIBUR
Note that the core is considered as being composed of two fuel
regions: an inner zone containing all the fuel batches except the
one placed at the core periphery, and a peripheral region
containing only the peripheral batch. The reflector then surrounds
the fuel region. This configuration is used in order to account
for the different radial leakage effects on the different batches.
In a large FWR core such as the one being studied, it is reasonable
to assume that no significant net radial leakage occurs from the
non-peripheral batches, and therefore, all net neutron leakage in
the radial direction is associated with the peripheral batch. This
neutron leakage of the peripheral batch is accounted for in the
core criticality calculations by modifying the infinite

no
multiplication factor of the batch in the way described below.
Since the reflector is composed of cooling water, it contains
varying amounts of soluble boron at different points in life, which
makes its reflecting characteristics variable with time. This
effect is accounted for in CRIBUR as described below.
Due to the fact that the peripheral batch has some neutron
leakage into the reflector, its multiplication factor is affected
by the probability of a neutron leaking out of the fuel zone-
Expressed in mathematical terms, this would be:
Kb = Kib Pnlb
where Kb = K of the peripheral batch, after radial leakage
modification.
Kib = K inf- of the batch, as obtained from the point-burnup
calculation.
Pnlb = Probability of non-leakage of neutrons from the
peripheral batch.
By definition of probability, it is possible to write:
Pnlb = 1 Plb
where Plb = probability of neutron leakage from the external batch.
And similarly, it is possible to write for the whole reactor:
Pnlr = 1 Plr
where Plr = Probability of radial leakage for the whole reactor.

Ill
The problem is now to obtain a reasonable value for Pnlb.
Considering that the zone occupied by the peripheral batch is very
thick in terns of the neutron diffusion length, it is possible to
assume that none of the neutrons leaking from the core into the
reflector come, from the inner region batches, and therefore all
net losses are from the peripheral batch population. Since in the
present large FWR cores the radial power distribution is maintained
fairly flat, it is also reasonable to assume that there is no
significant net neutron current between the outer and the inner
core zones. In this case, the probability of leakage of a neutron
belonging to the peripheral batch can be expressed as the
probability of leakage of a neutron from the whole reactor, scaled
up by the ratio of population of the whole core to that of the
peripheral batch. This can be mathematically expressed as follows:
Plr Nr = Plb Nb
where Nr = reactor neutron population.
Nb = peripheral batch neutron population.
Recalling that the probability of a neutron leaking can be
expressed as
Plb = Nleak / Ntotal
where Nleak = number of neutrons leaked.
Ntotal = neutron population.
It is possible to rewrite the peripheral batch neutron leakage
probability as

112
Plb = Nleak / Ntotal = Plr Nr / Nb
This would require knowledge of the total core and peripheral
batch's neutron populations, but since the parameter of interest is
really the ratio of these populations rather than the individual
values of any of than, it is possible to make a further
approximation that will simplify the calculation. The
approximation assumes that the neutron population in the peripheral
batch has approximately the same average speed as the whole core
population. This is not completely exact, because the different
isotopic compositions of the various batches, and the fact that
fast neutrons tend to leak in a larger proportion than thermal
neutrons, cause slight differences in the batch averaged neutron
speeds but the error incurred is of negligible importance. Once
this assumption is accepted, since the neutron flux is
(p = n v
where n = neutron density.
v = average neutron speed.
it is possible to express the ratio of neutron populations of the
core and the peripheral batch as
Nr = 2-1 $1 Vi
Nb Vb
where Vi = volume of batch 1i' in the core.
Vb = volume of the peripheral batch.

113
Since normally all the batches in the core have the same
number of assemblies, all Vi's are equal, and can be factored out
of the summation and cancelled with the Vb of the denominator. The
probability of non-leakage from the peripheral batch becomes
Pnlb = 1 ((1-Pnlr) *£i /$b)
It is now necessary to find some way of obtaining the
probability of neutron non-leakage from the reactor and the
batches' average fluxes with the minimum data handling possible,
since all these data have to be input to the core model.
Hiere are well known expressions that determine, as a good
approximation, the probability of neutron non-leakage from an
homogeneous reactor, the reflector effect being accounted for with
the use of the "reflector savings". The batches' fluxes can be
obtained from the fuel burnup calculations. Considering that
neutrons can leak during slowing down or after becoming thermal,
the probability of non-leakage can be expressed as
Pnlr = (1 + L2* B2) expi-TT* b2)
where L = diffusion length of the core.
B = geometrical buckling.
V = neutron age.
However, it is important to note that the above expression
used for determining Pnlr is correct when applied to a bare
homogeneous reactor. The reflector effect is mocked up by adding

114
the reflector savings to the core dimensions to obtain an
"equivalent" bare system. In any case, the leakage calculation is
still based on a homogeneous, cylindrical core whose flux magnitude
in the radial direction is assumed to follow a Jo Bessel function.
It is obvious that real reactors do not have such a flux shape,
mainly because of the geometric disposition of the different fuel
batches in the core.
Given a certain total neutron population of a reactor, the
probability of leakage must obviously depend on the geometric
distribution of the neutrons inside the core. In other words, for
a given reactor type, if the neutron population is concentrated
near the centerline, there should be a lower escape probability
than if most of the neutrons live near the periphery. This
consideration suggested the need of modifying the non-leakage
probability with a factor that would somehow take into account the
geometric difference between the real flux distribution of the core
being studied and the theoretical Jo Bessel shape. For this reason
the author called this factor the "Shape Factor".
One assumption used in calculating the shape factor is the
previous assumption that all radial leakage comes from the
peripheral fuel batch. A second simplifying assumption states that
the actual radial leakage from the reactor is proportional to the
ratio of the neutron population of the outer batch over the
population of the whole core. In other words, it is assumed that a
reactor having 50% of the total core population in the outer batch
presents more radial leakage probability than a core having only

115
25% of the total population present in the outer batch. The
proportionality constant can be easily obtained with a few simple
calculations using a diffusion theory code. This procedure gives a
simple way of scaling the leakage according to the population or
flux distribution. Another aproximation needed (which was also
used before) is that of considering the average speed of neutrons
being constant across the reactor; this allows the use of
volumetric flux averages instead of neutron populations for the
leakage calculations.
With these assumptions in mind, the shape factor can be
expressed as
SF = ($b/£i)
(SI / (S1+S2))
where b= actual flux average of the peripheral batch.
<|)i= actual flux average of batch "i" in the core.
Sl= neutron population of the peripheral batch in the
Jo-shaped core.
S2= neutron population of the internal core zone in the
Jo-shaped core.
For any given number of batches, SI and S2 would be constant
values, but since CRIBUR was designed to handle cores with any
number of batches up to five, it is necessary to express SI and S2
as a function of the number of batches present in the core, since
the peripheral batch will correspond to a varying thickness of the
peripheral shell.

116
If n is the number of batches present in the core, recalling
that the first zero of the Jo function is at 2.405 and that all the
batches in the core are assumed to have the same volume, it is
possible to express the radius Xo which will separate the inner
zone from the peripheral batch (see Figure 3.2.4) as:
Xo = \l(7T*(2.405)2 /n) ((n-l)/7T) or
Xo = 2.405 \](n-l)/n
Figure 3.2.4. Bessel-shaped Flux Distribution.
It is now necessary to obtain the ratio of SI to S1+S2.
Recalling that the integral of the Jo Bessel function can be
-x
2 7T x Jo(X) dX = 2 It x Jx (X)
obtained as

117
it is then possible to express the ratio S1/S1+S2) as
SI/(S1+S2) = 2.405 J, (2.405) Xo J, (Xo) or
(2.405 Jt (2.405))
SI/(S1+S2) = (1 (n-l)/n ) J, (2.405 (n-l)/n)
J, (2.405)
where the Jt Bessel function can be expressed in a series
development whose first three terms are
J, (X) = (X/2) (X /16) + (X /384)
This completes the calculation of SI/(S1+S2). If the whole
shape factor is now included in the peripheral batch non-leakage
probability calculation, the expression becomes
Pnlb = 1 ((1-Pnlr) *Ei$/$b) ($1/1-$) Pc
(SI/(S1+S2))
where $¡ are the batch fluxes of the ideal reactor (following
Bessel's Jo) and $b are the actual batch fluxes, obtained from the
fuel burning calculations. Pc is the proportionality constant or
scaling factor, which value was adjusted for the present system to
0.35"and it can be reevaluated for different core configurations
using simple diffusion theory codes. Note that in the expression
above, S1/(S1+S2) is equivalent to since these were the
fluxes associated with the equivalent homogeneous bare reactor.
Simplifying assumptions are made at this point in order to
reduce data input to CRIBUR. The assumptions and approximations

118
made below are based on the observation of the results of the
neutron transport-theory pin-cell depletion code EPRI-CELL.
a).'V, which depends mainly on the core's fast diffusion
coefficient, is assumed constant since the core fast diffusion
coefficient experiences less than a 2% change through life. The
value taken for X was 40 cm.
b). The geometric buckling is calculated as that of an
infinite cylinder whose radius is the radius of the reactor plus
reflector savings. As explained before, the reactor is considered
infinite in the axial direction because leakage in that direction
affects all batches in the same proportion, and this effect is
taken care of in the assembly burnup PDQ-7 calculation. The radial
leakage, however, affects only the peripheral batch, and is
accounted for in the core modeling.
c). For the calculation of the reflector savings, the
reflector is assumed to be "thick", since the reflector thickness
is 34 cm, which is many times larger than the reflector diffusion
length. The savings is then obtained as
cf = Dc Lr / Dr
where Dc = diffusion coefficient of the core.
Lr = diffusion length of the reflector.
Dr = diffusion coefficient of the reflector.

119
d). Lr and Dr are assumed as a function of the soluble boron
concentration of the water. Since Lr can be expressed as
Lr = VDr / T. ar
where ar = macroscopic absorption coefficient of the reflector,
it is first necessary to obtain a parametric expression of Dr and
£ar in terms of the soluble boron concentration. Observation of
the results of the pin cell code (EPRI-CELL) shows that both
variables can be closely expressed as a linear function of the
soluble boron concentration of the coolant, with a regression
coefficient very close to 1. The regression lines obtained for Dr
and ar are
£ ar (an-1) = 0.0166 + (2.727 E-5) ppm.
Dr (cm) = 0.4033 (5.135 E-6) ppm.
where ppm = concentration of soluble boron in parts per million.
e). The variables Dc (used in the expression above) and L of
the core (needed for the calculation of the non-leakage probability
during the thermal diffusion period) are also approximated with a
straight line. For any particular cycle of core fuel residence,
the approximations are accurate to a few percent, but the
regression lines change from the first to the second cycle of core
residence of the fuel. Since the present study considers an
out-in-in fuel management scheme, the fuel in the periphery is in
the first cycle of core residence, and since the peripheral fuel

120
batch is the one dominating the core radial leakage, the straight
fits of Dc and Lac are made for the first residence cycle. Should
another fuel management scheme be used (e.g. an in-out-in, for a
low leakage core configuration), the regression lines would need to
be changed, obtaining then from the pin-cell burnup code
calculations. For the present case the regression lines obtained
for these variables are:
Dc (cm) = 0.4860 1.90 E-7 BU
Hac (cm-1) = 0.1912 + 6.42 E-7 BU
where BU = average burnup of the peripheral batch, in MWd/MTO.
Once the peripheral batch has its infinite multiplication
factor modified with the probability of non-leakage, it is possible
to proceed to the calculation of the core effective multiplication
factor. At this point, CRIBUR provides for one additional
consideration by allowing any batch in the core to contain burnable
poisons. The burnable poisons are handled as a reduction of the
batch's infinite multiplication factor by a value that varies
linearly as a function of burnup between the user-provided values
for BQL and EOL. If no batch is defined as containing burnable
poisons, no further variations of batches' multiplication factors
are made. In the case where the burnable poison option is to be
used and the batch-wise worth of the poisons at BQL and EOL are not
known, it is always possible to compute than with the codes
involved in the normal burnup calculation. Specifically, EPRI-CELL
is able to provide cross sections for lumped burnable poisons
(LBP's) to be used in FDQ-7 at BQL and EOL (25). If then PDQ-7

121
quarter-assembly eigenvalue cases are run with and without the
LBP's at BGL and EOL, it is possible to obtain the reactivity worth
of the poisons for use in CRIBDR.
CFIBUR obtains the core effective multiplication factor as
Kc = 1 / ( Si(Fi/Ki))
where Fi = fraction of the core power produced by the "in-th batch.
Ki = infinite multiplication factor of batch "i" modified
for leakage and/or LBP's, if any.
The Ki's are obtained from the quarter-assembly PDQ-7
calculations, with the possible modifications described above for
any possible radial leakage and/or burnable poison presence. The
Fi's are calculated as follows:
Fi = (Ki Pi)2
I,(Ki Pi)2
where Pi = non-leakage probability for batch "i".
The above formula is purely an empirical formula which tries
to assign a fraction of the core power to each batch making it as
close as possible to the real power sharing encountered in an
actual core. This formula evolved from the expression
Fi = Ki2 / Z-i Ki2
used by M.I.T.'s research team (15), but was modified by the author
to the expression shown before which adjusts better to the batch

122
power sharing encountered in a MR core with an out-in-in in-core
fuel management scheme.
With the preceding formulas. CRIBUR takes the burnup data
provided by a FDQ-7 1/4 assembly fuel-burnup calculation with a
structure similar to the one shown on Figure 3-1-6. and computes
the criticality status of the core at the different core-life
timesteps, as represented by the graph at the bottom of the figure.
If it is determined that the core is critical at all points in
its life, the PDQ-7 1/4 assembly burnup calculation is really
representative of the core life, in the sense that the core cycle
length, the batch power level and the soluble boron concentrations
are consistent- At that point, the PDQ-7 data and the
complementary CRIBUR data that will be discussed later give the
description and main characteristics of the core burnup (discharge
burnup. batch burnups. core life, isotopics, etc-)- However, in a
general case (and mainly if the FDQ-7 burnup calculation was
assigned first-guess values) it is most probable that the core does
not maintain criticality through all its cycle timesteps. The
first action to be taken is to adjust the soluble boron letdown
curve, forcing the core to be critical at all times- This action
requires previous knowledge of the soluble boron reactivity worth,
and by altering the boron letdown curve, the core life estimate is
altered (since it is estimated as the time when soluble boron
concentration must become zero)- This will force an adjustment of
core parameters, as will be explained later-

123
The way in which CRIBUR accounts for the reactivity worth of
soluble boron is also a unique feature in a core model of this
simplicity. To the best of the author's knowledge, any code of a
complexity level comparable to CRIBUR's considers soluble boron
worth as an empirical constant, hardwired in the code; frequently
this variable is not established or used, as discussed in Section
3.1. However, an analysis of EPRI-CELL calculations show that
soluble boron worth may vary as much as 25% in the range of
concentrations that may be normally encountered in a real PWR. In
addition to that, soluble boron worth was shown to depend heavily
on the fuel cell geometry (see Chapter 2). This suggested the need
to evaluate soluble boron worth for each particular reactor case,
and if at all possible, it should be evaluated for different
concentration levels.
CRIBUR will accept a soluble boron worth constant if so
indicated by the operator, or may even provide a default constant
which is representative of the average value found in a standard
PWR. However, it also provides the opportunity of calculating it
for each case and to make it a function of the boron concentration.
If a soluble boron worth calculation is desired, it is
necessary to specify some "fake-burnup" steps in one of the
fuel-burnup cycles, in a similar way as explained on Figure 3.1.5.
Since the fuel-burnup calculation performed to provide data to
CRIBUR follows the soluble boron letdown curve for each core cycle
through he whole fuel life, the calculation has seme burnup steps
in which the boron concentration is high, some in which it is near

124
average, and some in which it is low (see Figure 3.2.5). It is
then possible to specify some "fake-burnup" steps in which soluble
boron concentration is altered by about 100 ppm. This causes a new
eigenvalue calculation, but does not affect the spectrum for the
real burnup step, which is performed with the adequate boron
concentration. By repeating the "fake-burnup" step procedure at
several boron concentration levels, CRIBUR can obtain the soluble
boron worth per ppm at the different concentrations. Observation
of different cases showed that the variation of soluble boron worth
is a closely linear function of boron concentration. CRIBUR
calculates then the regression line and obtains an expression that
allows the calculation of soluble boron worth for any
concentration.
Soluble
Boron
Figure 3.2.5. Fake-burnup Steps for Soluble Boron Worth.
Obviously, if the regression line is calculated for each of
the core cycles that form the total fuel life, slightly different

125
lines are obtained because of the varying isotopic compositions of
the fuel. It is then advisable to perform the linear fit
calculation on a cycle towards the middle of the fuel life (the
second cycle in the case of a three-batch core) in order to obtain
a soluble boron worth regression line that can be representative of
the whole core.
One may be concerned about the fact that by defining the
"fake-burnup" steps for the soluble boron worth calculation, there
is one cycle in which the burnup timesteps do not match with the
other cycles. CRIBUR solves the problem by deleting the
"fake-burnup" timesteps after the regression line has been
calculated. This causes no misadjustment as far as matching
timesteps from each cycle, because the "fake-burnup" steps were
insignificant in terms of both burnup and time.
Once the soluble boron worth can be obtained for any boron
concentration, it is a simple matter to adjust the boron letdown
curve so that the reactor is critical for all timesteps in the
cycle- However, this causes a major problem in the core life
calculation, as discussed below for a reactor with a soluble boron
letdown curve such as the solid line of the top of Figure 3.2.6.
After a criticality calculation by CRIBDR. it is found that the
letdown curve was somewhat in error, and it should be more like the
one represented in a dashed line. This new letdown curve was
calculated by using the burnup timesteps marked on the old letdown
curve, and adjusting the boron concentration for criticality- But
if this new curve were indeed to be the actual letdown curve, the

126
Soluble
Boron
Concentration
Figure 3.2.6. Problem of Erroneous Cycle Length.

127
beginning of the second and third cycles would not be at A and B,
but rather at A' and B'. which have a different burnup level. The
boron concentration that was obtained for criticality by combining
points 0. A and B must be obviously excessive, since the "new"
points should be 0. A', and B'. where A1 and B' have a larger
burnup. and therefore lower reactivity. In short- this means that
when calculating a new boron letdown curve which results in a new
cycle length, the new curve is automatically in error, because it
was not calculated with burnup timesteps whose burnup levels were
consistent with the cycle length assumed by the curve.
To solve this problem, one might be tempted to run a PDQ-7
burnup calculation with the new cycle length estimate, but this
would imply large number of PDO-7 iterations and data handling back
and forth, destroying one of the main goals of this calculatonal
scheme: reduced calculational costs and data handling. To make any
further adjustments to the cycle length and the letdown curve with
the data available to CRIBUR. it is necessary to find some value or
seme characteristic that would be implicit in the data and that
would not depend on the cycle length- Such characteristic may be
found if an approximation is accepted by assuming that the initial
guesses of the cycle length and the soluble boron letdown curve are
not in large error. If this is not the case, the cycle variation
at the end of the calculation will be so large that it will force a
new PDQ-7 calculation, anyway. If those guesses are not too
erroneous, then, the fuel was burned following a spectrum which is
correct within the desired accuracy, and therefore, its isotopics

128
along the life are within an acceptable margin of accuracy. If the
soluble boron is not considered for a moment, the fuel would have a
multiplication factor which would depend on its isotopics, which in
turn are closely related to the burnup level since the actual
burnup was calculated with a reasonable neutron spectrum. This
means that under those conditions, the fuel has an invariable
characteristic: the correspondence of its "clean" (boron-free)
multiplication factor with the burnup level. If this "clean"
multiplication factor can be obtained for each known burnup level,
it would be a set of data independent of the variations of cycle
length (again, as long as the cycle length guess was not too much
in error to start with)- This is represented in the lower axis of
Figure 3-2-6-
Since CRIBUR calculates the soluble boron worth line, it is
possible to integrate the reactivity variation caused by boron from
concentration zero to the actual concentration assigned to each
burnup timestep. and therefore it is possible to calculate the
"clean" multiplication factor that can be associated with each
initial timestep. and what is more important, that can be
associated with each timestep's burnup level. This correspondence
is kept in a table within CRIBUR- and for any burnup level, the
"clean" multiplication factor can be interpolated. The actual
multiplication factor of any particular situation is then computed
by first obtaining the "clean" multiplication factor, and then
subtracting the integrated soluble boron worth from zero boron to
the boron level that is actually assigned.

129
With this new system, it is no problem if the newly calculated
boron letdown curve implies a different cycle length: the last
timestep of each cycle is adjusted in length in order to fit the
new cycle life, new burnup levels are calculated for each timestep,
and CRIBUR proceeds to a new criticality calculation which will
yield a new boron letdown curve and a new cycle length. The
iterations are continued until the point where the burnup levels,
the cycle length and the soluble boron letdown curve are consistent
within a certain margin.
As it can be realized by observation of the expressions used
by CRIEUR for the criticality calculations, this iterative process
involves a simultaneous adjustment of the relative power shared by
each batch in the core.
Once the CRIBUR calculation has converged internally. CRIBUR
outputs the new estimates of the core life, the soluble boron
letdown curve, the relative powers assigned to each timestep. the
batch burnup levels, and the discharge burnup level. All these
data have then to be compared by the user to the data that were
specified for the EDQ-7 fuel burnup calculation, and if any
significant differences are encountered (above a certain error
margin dictated by the requirements of the study) the PDQ-7 fuel
burnup calculation must be repeated with the fuel life data
indicated by CRIBUR's output. Figure 3.2.7 shows a flowchart of
the calculations performed by CRIBUR which may further clarify its
operation and its two main loops: the criticality and cycle life
calculations, and the adjustment of cycle timesteps' burnup levels.

130
SE
inEait-Option.
Data Input
From Keyboard
Data Input From
Pre-saved Disk File
1 Data input and Array Dimensioning According to Problem Size
t
Optional Modification of Problem Data
- For Error Correction
- For Modification of Disk File
Optional Storage_of__Cage-Pata _j.n-.Disk Fils
Calculation of Step-wise and Cycle-wise Burnup Evolution
o£_Fuel. According to Assemblj..Burnup Data (Input Data)
Initialization of Work-core Matrix with K's, Time Lengths,
B_imiups-,__B-Oron ..Concentrations and Relative Powers
I
Calculation of Regression Line of Boron Worth vs. Boron ppm from:
- Timesteps Specified for such Purpose
- User-defined Constant
Elimination of Fake-burnup steps Used for Boron Worth Evaluation
Restructuration of Work-core Matrix and Pointers
Calculation of Table of Unbc
¡>rated Steps K's vs. Burnup!
1
r 1
Calculation of "Present Core Status
Step-wise and Cycle-wise Burnups
Interpolatlon-Qf. "Present-Core £
p* 1
status" Clean.K's for all Steps
rom ,Cugrcn|BumuE>-aiid-ssi,giisd..BQgon Ep
lfflodify_..BatchL-Klg. for Bumabje Poisons Presence (if any?!
£alcula-tiaiL^-.£;s-g-ape-jpxQbabilitie.s fox-peripheral Batch!
MQdi£icatiorL-Pf. Peripheral Batch K for Leakage!
t
continued
=t
Figure 3.2.7. Flowchart of CRIBUR Code

131
L
¡continuation
Icalculation of Stems' Relative Powers!
¡Calculation of Core Effective Multiplication Factor
:Jf
Calculation of New Boron Concentrations for Criticality
2jl
Calculation of New Core Li^e and New Timestep Lengths
Iflpply Convergence Accelerator
¡Core K's Close Enough_.to iLBefore, Boron Adjustment ?lN0-1go to A
YES
ICalculation_of S.teMdse Jurnups With New Set of Relative Powers
New Cycle Burnups Agree With Old Iteration's ?1 YES-
NO
i.
AERly, Conv-ejcgence.. Accelerator for Relative Powersl
1
Calculation of "Present Core Status" Burnups
Storage as "Old Iteration" Burnups
J
!GQ..tP.Bl-
Erint Results
Figure 3.2.7. Continuation.

132
3.3. Benchmarking of the Burnup Calculations
3.3.1. Available References for Comparison..
In order to benchmark the burnup calculational scheme
developed for the calculations performed in this study, it is
necessary to have a quality assessment for each step in the chain
of programs which compose the total calculational scheme. The
programs used in performing the point burnup calculations are
industry standards. They already have had sufficient benchmarking
(26,27) as to make additional work in this direction by the author
unnecessary. However, it is still necessary to check the behavior
of the core model against other models of well-known
characteristics, or if possible, against an actual reactor's
behavior.
It is clear that the best benchmarking would be the comparison
with a true reactor behavior in an equilibrium cycle, since this is
the situation that the core model tries to represent. In order to
use such a benchmark it would be necessary to know all the exact
compositions of the equilibrium-cycle real core. Unfortunately, in
an equilibrium-cycle core, the compositions of the once-burned and
twice-burned batches are not exactly known from an experimental
basis, and the analytical results are not published in the open
literature, rendering this benchmarking possibility impractical.
There is, however, one case in which the model can be compared
to a real-life reactor: at B.O.L. of a first core. In such a
situation, all batches in the core are new and clean, and thus
their compositions are known, as well as the soluble boron

133
concentration that holds the core critical at hot, full power, and
all control rods out (HFP, ARO) conditions. It is then possible to
calculate the multiplication characteristics of each separate
batch, and use them as an input to CRIBUR. Obviously, this method
allows the benchmark of the criticality calculation part of CRIBUR,
but not of the power distribution and burnup assignment part.
Another way of benchmarking CRIBUR is, as said above, to
compare its results against those of some well-known and accepted
codes. 1116 code EDQ-7 is a good candidate, because it has been
widely accepted as one of the main calculational tools in the
nuclear industry, its accuracy is well known, it can explicitly
represent the geometry of a 1/4 core, and it is possible to input
in it exactly the same data that is needed in a CRIBUR case, thus
making it possible to have exactly the same case represented in
both models.
However, this 1/4 core FDQ-7 representation has a few
problems. In a real equilibrium core, the assemblies composing any
fuel batch have different characteristics depending on their
particular history, and there are differences even between the pins
that form a fuel assembly, because sane of the pins are at the
assembly periphery, facing assemblies from other batches (and
therefore with sharply different characteristics and history) while
other pins lie inside the assembly, surrounded by pins of their
same batch and very similar life conditions. These differences
cause the interfaces between batches and between different
assemblies to be "smoothed out" as burnup increases, naturally

134
limiting local power peaking effects. Unfortunately, if a PDQ-7
1/4 core model is prepared with data identical to that used in a
CRIBUR run, each batch has one single composition which is assigned
to all of its pins, and no differences between pins with their
"smoothing" effect on power distributions exist. The compositional
differences between neighboring pins from different batches are
sharper than they would really be in an actual core, causing
exagerated effects on the power peakings. This problem invalidates
the FDQ-7 1/4 core model for the purposes of benchmarking power
distributions and burnups, although it can still be used as a
redundant reference for benchmarking of the criticality
calculations of CRIBUR, since the overall core multiplication
factor is not affected by the somewhat inexact power peaking
analysis.
A PDQ-7 run for a 1/4 core configuration, even used without
PDQ's burnup features (i.e. just for criticality and power
distribution "snapshot" analysis) presents the additional
difficulty of its involved input for a complex geometry that
includes several different composition zones. This is primarily a
problem of skill and time, and can still be reasonably viewed as a
method for benchmarking purposes.
The benchmarking references described so far allow for a good
checking of the criticality calculation part of the code, since
both the PDQ-7 1/4 core batch-homogenized model and the
real-reactor B.O.L. H.F.P.. A.R.O. give good values of the
multiplication factor, and it is easy to obtain the necessary core

135
data to reproduce the situation on CRIBUR. However, it is still
necessary to obtain a reasonable source for comparison of the power
distribution among the different batches in CRIBUR. Since CRIBUR
analyzes an equilibrium-cycle core, it was considered that the best
method would be to compare its power distributions at different
points in core life with those appearing in real cores working in
an equilibrium cycle (say cyles 3 or beyond). The actual core
compositions would not be known, but the power distributions on a
batch-v/ise fashion are likely to be similar, and provided the cores
taken for comparison are similar to the one described in CRIBUR,
the code's estimates should be very close to the values seen on the
real reactors.
3..3.2,. Criticality Benchmarking.
As described in the previous section, two different references
were used for the benchmarking of CRIBUR's criticality
calculations. One reference was composed of several non-burnup
runs of PDQ-7 with a 1/4 core configuration, where each assembly
was explicitly represented, although the geometry mesh used was
somewhat coarser than an explicit pin-by-pin representation. Each
assembly was given a uniform composition equal to the one existing
in the corresponding batch in the CRIBUR calculation. Macroscopic
cross sections were used for each region instead of microscopic,
and this simplified data handling. There was no need for
microscopic cross sections, since no isotopic changes were allowed.
Two energy groups were used, as recommended by EPRI for FWR
calculations with the ARMP code package.

136
The PDQ-7 1/4 core criticality tests were performed at four
different points in the life of an equilibrium core: at B.O.L., at
approximately 1/3 of core life, at about 2/3 of core life, and
around E.O.L. In all the cases tested, the effective
multiplication factor calculated by CRIBUR differred from PDQ's by
less than 1%, and in two out of the four cases the difference was
under 0.3% .
In an attempt to check the importance of some of the
calculational features included in CRIBUR and not normally present
in other simple burnup schemes, the same four cases were
recalculated in a version of CRIBUR in which the calculation of the
leakage probability for the outer batch was bypassed. In all four
cases the error of the multiplication factor as compared to PDQ's
was over 4%, having increased from the original CRIBUR value by
about 3.5 to 4%.
Another check of the accuracy achieved by CRIBUR as compared
to other simple schemes was done by calculating the core effective
multiplication factor (K-eff.) as an average of the batches'
infinite multiplication factors (K-inf.'s), as indicated by
Graves(28). This K-inf. averaging method yielded errors between
2.6 and 3.7% with respect to PDQ's results, which represented
errors about 2.5% higher than the ones incurred by CRIBUR in the
representation of identical cases. Table 3.3.1 shows the values
obtained for all the benchmark and comparative cases described
above.

137
Table 3.3.1. CRIBUR and Batch-averaging Method Criticality Errors.
Cycle CRIBUR % Diff No-leak CRIB. % DIFF BATCH-AVG % DIFF.
Status
K-eff.
to FDQ
K-eff.
to PDQ
K-eff.
with PDQ
B-O-C.
.9694
.23
1.0094
4.36
0.9930
2.66
1/3 cycle
.9520
.15
0.9909
4.23
0.9772
2.79
2/3 cycle
.9274
.68
0.9617
4-40
0.9524
3.38
E.O.C.
.8996
.98
0-9304
4.44
0.9235
3.67
The
other
reference used for benchmarking of
CRIBUR's
criticality calculations was a real reactor at B.O.L. for the
first core. The reactor chosen was Comanche peak, Unit 1(27). The
reasons for choosing this particular core were the following:
a). The reactor was of the same 4-loop Westinghouse type with
17X17 pin fuel assembly as the one used for the CRIBUR
calculations. This made all dimensions, compositions and
temperatures common for both the real reactor and the data used for
CRIBUR's calculation.
b). Being at B.O.L. of the first core, the compositions of
all batches were known, since no burnup had occurred yet, and
therefore no fission products existed in the core.
c). The report on the real reactor indicated the HFP, ARO
soluble boron concentration needed for criticality at that time, as
well as the worth of the burnable poison rods at B.O.L.. all of
which made the comparison of the reactor with CRIBUR a rather
straight forward procedure.

138
The comparison of CRIBUR with Comanche Peak 1 B.O.L. yielded
a K-eff. of 1.013 after accounting for the reactivity worth of the
burnable poisons. CRIBUR had, thus, a 1.3% K-eff. error with
respect to the real reactor. It may be of interest to note that
B.O.L. of the first cycle is probably the worst moment for CRIBUR
to give a good estimate of the core criticality, due to the sharp
interfaces existing in the core because of the different types of
fuel and poisons present. However, even in this situation,
CRIBUR's criticality calculation was acceptably accurate.
A comparison of criticality evaluations of the CRIBUR scheme
and the scheme used by M.I.T. (15) whose main formulas were
outlined in Section 3.2 shows differences between 1.6% and 2.3% for
the core multiplication factor. The main errors of the M.I.T.
scheme appear on the batch power distribution, as will be presented
later.
Some runs were made with CRIBUR in which some of the factors
of the calculational scheme were altered, in order to observe the
sensitivity of the results to the presence or accuracy of those
different factors. In a test where the Shape Factor was given a
constant value of unity (therefore implying that for leakage
purposes the flux shape followed a Bessel function) the core
multiplication factor experienced a sharp increase of 2.76% at EOL
and 3.26% at BOL. It is logical that these forced lower-than-real
leakages caused a larger effect at BOL, when the external batch,
which is experiencing the neutron leakage, is most reactive.

139
In a series of tests where the batch multiplication factors
were altered by 1%, the core multiplication increased around .3%
when the altered batch was the oldest one, about .36% when the
second batch was altered, and approximately .32% when the
alteration affected the first batch. The sensitivity of the core
to the accuracy of the batches multiplication factors depends on
the batch power sharing. The code showed some sensitivity to the
neutronic characteristics of the reflector: if the reflector was
considered absent of soluble boron for the radial leakage
calculation, the core multiplication factor showed a variation of
.11% at BOL (when the real boron concentration is highest) and .04%
near BOL, when the level of soluble boron is low. This
demonstrates that the variations of the reflecting characteristics
of the reflector were non-negligible.
The percentage variations of core multiplication factors
associated with the different altered factors for five points
during the core life are shown on Table 3.3.2.
Table 3.3.2. Effects of Code and Data Alterations on Core K-eff.
Core Life Time:
B.O.L.
M.O.L.
E.O.L
Alteration
Percentage Change on CR1BUR K-eff.
Shape Factor = 1
3.26
3.01
2.99
2.94
2.96
+1% K, Batch 1
0.33
0.32
0.32
0.32
0.31
+1% K, Batch 2
0.36
0.37
0.36
0.36
0.36
+1% K, Batch 3
0.29
0.30
0.30
0.30
0.26
Unborated Reflec.
0.11
0.10
0.09
0.08
0.04

140
3.3.3. Power Distribution Benchmarking.
Since it is not possible to obtain actual accurate equilibrium
core compositions for the type of reactor modeled in this study
(Westinghouse's 4-loopf 17X17 pin assembly) because there is not
yet any such reactor in an equilibrium cycle, the benchmarking of
the batch-wise power distributions calculated by CRIBUR was done by
comparing its results for the standard case with the batch-wise
power distributions of several Westinghouse cores in their
equilibrium cycles.
atypical power distributions obtained by CRIBUR yield relative
powers of 1.02, 1.09 and 0.88 for the three batches in the core,
from newest to oldest respectively, with no burnable poisons
present in either of them.
The power distributions observed in seme real cores in their
equilibrium cycles are (from newest to oldest batch respectively):
0.99, 1.05, 0.94 for Zion II ; 0.96, 1.13, 0.9 for D.C. Cook ; and
0.97, 1.15, 0.88 for H.B. Robinson 2. Core-following calculations
performed by utilities show assembly-wise power level errors of a
few percent, reaching sometimes 10%. EPRI's procedures manuals
list as acceptable an error of 5%.
Obviously, when the partial power of all the assemblies are
added up into batches, the calculations come closer to the real
values, but this is achieved with pin-by-pin or nodal
multi-dimensional calculations of a degree of sophistication far
superior to CRIBUR's. The results obtained by CRIBUR tend to
assign a few percent more power than real to the fresh batch, and a

141
few percent less to the second batch. What is also important is
that the order of batches from the highest to the lowest power
level is always kept the same as in the real cores in spite of the
closeness of the power levels of batches 1 and 2. The
calculational scheme used by M.I.T. yields a too high power level
for the first batch, while reducing that of the second batch. The
variations are of such magnitude that the second batch does not
hold the maximum power as is the case in an actual core, and as is
the case with CRIBUR results, but the maximum power sharing is
assigned to the first batch. It is important to remember that
CRIBUR1 s calculations did not assign any poisons to the fresh
batch, which are some times used in actual reactors.
From these comparisons it is possible to conclude that
CRIBUR's estimates of batch-wise power distributions are very close
to the true values found in an equilibrium-cycle FWR and quite
acceptable for relative burnup studies.
The alterations mentioned in the criticality benchmarking
section affected the power distribution, the discharge burnup and
the cycle length, proving again they are non-negligible in nature.
The elimination of the Shape Factor in calculating the radial
leakage boosts the peripheral batch power and reduces the power
sharing of the second and third batches. The discharge burnup
shows a large erroneous increase of 6.3%. This is due to the core
experiencing a very reduced radial leakage.

142
The total elimination of the radial leakage shows similar
effects, but in a yet larger scale. The discharge burnup is
increased 7% over the correct calculation.
Increasing the batches' multiplication factors by 1% increases
by a few percent the power sharing of the batch being altered, and
causes an increase in discharge burnup of 2% when the outer batch
is altered, and 1.7% when one of the internal batches is altered.
The sensitivity calculations were performed using burnup data
corresponding to the standard, full power FWR.

CHAPTER IV
DETAILED SHARP BURNUP OPTIMIZATION STUDIES .
Cas.es -Chosen for.Flnal Study
Once the burnup calculational method is developed and the set
of programs to be used are tested (as described in Chapter III), it
is necessary to define the burnup cases that have to be considered
for this study. It is important to realize that although the
burnup calculational scheme developed for this work requires a
reduced level of computational effort for a quality calculation, a
considerable amount of work is required to complete each burnup
study. It is not only necessary to complete the chain of
EPRI-CELL, NUFUNCHER, PDQ-7 and CRIBUR, but it normally takes two
or three iterations of PDQ-7 and CRIBUR before all the core cycle
data converges. This represents a significant volume of data
handling, and thus, it is important to avoid burnup calculations
that are not required for the purpose of the study.
Since the main concern of this study is to determine the
effects of reduced power densities on the neutronics, fuel
utilization and general isotopic behavior of the reactor, it is
obviously necessary to examine several power density levels. Two
different plant arrangements were considered as possible in
Chapter I for the low-power cores; the main designs were the
multiple-core plant with or without preheater core, and the
143

144
single-core plant with reduced power output. As already indicated
in Chapter I, one of the key requirements for acceptance of the
SHARP concept by the industry is the minimization of significant
redesign, and this requirement is only fulfilled by the single-core
plant concept. The preheater reactor was also excluded from
consideration for the same reason. Only the single-core design is,
therefore, considered in the in-depth burnup studies.
The preliminary burnup calculations described in Chapter II
use a 25% power level core as the SHARP core; this would correspond
to a nuclear plant with only about 300 Me power output. It is not
very likely for a plant of such reduced power output to result
economically attractive. This aspect will be further discussed in
Chapter V. An adequate power level for a reduced-size plant may
start at about 500 MW, which would correspond to about 40% of the
power level of a standard plant. However, since studies of cores
with power densities around 75% that of the standard core have
already been done (2) (involving significant core redesign), the
author considered that the scoping studies will explore a power
range between 25% and 50% that of the standard reactor.
A power output of 50% that of the standard plant may be very
attractive for plants sited near medium-sized consumption centers.
Thus, the burnup study of low-power cores between 25% and 50% power
level is likely to give a good perspective of the results that may
be expected from the SHARP'S in the range more likely attractive to
industry. The power levels finally chosen for detailed study were
25%, 33% and 50% of the standard reactor power density.

145
Given a power density, there are a number of parameters that
can be allowed to vary and therefore are susceptible to
optimization, without changing much the design of the core. Some
of these parameters are: lattice pitch, fuel pin diameter, fuel
enrichment, burnable poison loading and/or management,
consideration of gadolinium-loaded fuel pins, in-core fuel
management scheme, etc. Some of these parameters are already being
considered in industry studies directed to improve fuel
utilization, and their use is discussed in greater detail in
Appendix A. It is obvious that the optimization of all these
parameters is well beyond the scope of this work. It is then
necessary to choose the best set of parameters that can be used for
core optimization within the established constraints of these
studies.
The main criteria necessary for the selection of the
parameters are the reliability and sensitivity of the calculational
methods to the parameter(s) and the acceptability of the change by
the industry.
One parameter that could be changed or modified is the in-core
fuel management scheme. The calculational method developed in
Chapter III is able to accept such change, but the alternative
mangemert schemes (v.g. in-out-in, explained in Appendix A) are
still considered problematic in the industry, and they would have
little probability of being accepted. Therefore, this study does
not consider any variational in-core fuel management scheme, and

146
the standard out-in-in scheme is used throughout the burnup
optimization calculations.
The use of burnable poisons is very common in the industry,
but although they can be represented in the calculational method,
it would be hard to reliably account for small variations in them,
and comparison with a "standard" core would be difficult, since
this is normally a particular case-to-case type of parameter.
Gadolinium loaded pins are still far from widespread consideration
for commercial PWR's, and the calculational schemes available for
this work would not be able to account for them properly, so they
are also not considered in the studies.
Enrichment is a parameter that is likely to be varied in most
studies, and several works have pointed out that slightly higher
enrichments may improve ore utilization, aside from extending core
cycles, but this forcedly requires the use of some type of burnable
poison, and it is therefore not considered here for the reasons
mentioned above. However, further research is needed in this
direction if suitable benchmark cases become available, since this
is a parameter which might result of significant effect, and which
should not present large problems of acceptance by either the
industry or the regulatory agencies.
A parameter that is simple to vary and that has shown to have
a significant effect in the preliminary calculations is the fuel
lattice pitch. Its effects on the neutron spectrum cause
significant changes on the isotopic evolution of the fuel, and
therefore on the discharge burnup levels, the ore utilization, etc.

147
Variation of this parameter is likely to be accepted without much
problem by the industry, mainly because it is not a significant
change in the design and operation of the plant, nor does it
require special types of calculations for core-follow control.
This parameter is therefore chosen for core optimization in the
burnup studies of the low-power cores. Since there are sound
guidelines on the range of pitches that are acceptable from the
preliminary calculations described on Chapter II, the starting
pitch for the parametric studies was selected at 1.20 cm and
incremented by 0.05 cm for each burnup calculation, until a net
reduction of burnup is encountered. The decrease in burnup is
likely to occur around a pitch of 1.45 cm according to the results
of the preliminary calculations.
If a 1.45 cm pitch shows the expected decrease in burnup
(after experiencing a maximum peak in a smaller pitch), this
requires a total of 19 burnup cases: six pitch values for each of
the three low-power cores (25%, 33%, and 50% power density), plus
one case for the standard core, which is only studied at its
nominal pitch, as a reference case. Thermal-hydraulic safety
considerations prevent the use of the standard power density core
with larger pitches, as explained in next chapter.
4.2. Results of Optimization
4.2.1..Neutronics and Isotopic Results.
The burnup studies performed for the different power level
cores and the changing pitches generated data sets that require

148
processing to analyze the key issues and the significant
differences between the various cases studied.
The results shown in this section are directly obtained from
the FDQ-7 CRIBUR burnup optimization calculations or frcan simple
manipulations of their output data.
The first set of data which is needed for core optimization
analysis is the U-235 and plutonium isotopes number densities at
loading and discharge. However, the number densities per se are
not a common means of comparing core performances and they are
listed in Appendix D1 for reference purposes. The figures are used
later in this chapter in order to obtain other commonly used
comparative indices. As stated before, all the burnup studies
were done under the assumption of once-through fuel management, and
thus, reprocessing was not considered. In these studies, the only
fissile material loaded into the core was U-235. However, for the
sake of completeness and because it will be necessary to reprocess
spent fuel to recycle fissile species, some figures concerning
these discharged isotopes have been calculated and are presented in
this section.
Table 4.2.1 presents the comparison of burnup achievements for
the different cores, fissile species accounting, and ore usage.
The first two columns in Table 4.2.1 define the core case studied.
Note that the two parameters varied through the study are the power
level and the fuel cell pitch. The full-power, standard pitch core
results are included as the reference case for comparison purposes.
The third column shows the effective core cycle duration as

149
Table 4.2.1. Burnup Achievements, Fissile Isotope Usage and Ore Usage.
Pite
cm
i Power
%
Cycle
F.P.H.
Dis. BU.
MWd/MTU
Dis. Fis.
MF/MWd
(a)
Rec. Fis.
% Dis/Ld.
Net Fis.
MT/MWd
(a)
Pu Dis.
MT/MWd
(a)
U308
OT/MWd
(b)
1.20
50
15588
35172
4.195
47.16
4.619
2.751
1.899
1.20
33
23751
35725
4.020
45.90
4.658
2.685
1.870
1.20
25
32058
36166
3.914
45.24
4.657
2.651
1.847
1.25
100(c)
8018
36183
3.637
42.06
4.931
2.465
1.846
1.25
50
16529
37295
3.418
40.73
4.894
2.414
1.791
1.25
33
24837
37359
3.357
40.07
4.941
2.381
1.788
1.25
25
33359
37634
3.287
39.52
4.950
2.354
1.775
1.30
50
16704
37689
3.054
36.77
5.171
2.218
1.772
1.30
33
25329
38099
2.949
35.90
5.187
2.176
1.753
1.30
25
33924
38271
2.897
35.42
5.203
2.154
1.745
1.35
50
16901
38134
2.720
33.14
5.409
2.071
1.752
1.35
33
25526
38397
2.672
32.77
5.402
2.033
1.740
1.35
25
34162
38541
2.628
32.36
5.415
2.013
1.733
1.40
50
16534
37306
2.636
31.43
5.673
1.967
1.791
1.40
33
24994
37595
2.555
30.69
5.691
1.935
1.777
1.40
25
33415
37698
2.514
30.29
5.710
1.917
1.772
(a):* 1.0 E-7 (b):* 1.0 E-4 (c): Reference Case.

150
obtained by CRIBUR calculations. The figures are, thus, full-power
hours of operation before E.O.L. is reached. Of course, in this
context "full power" refers to the total power level assigned to
each particular core. Note that for all three reduced power levels
studied, the maximum cycle length is obtained for a pitch of 1.35
cm as compared to the 1.25 cm of the standard core. It seems
reasonable to assume that the standard power core would also
present a longer cycle life with a pitch of 1.35 cm, but
thermal-hydraulics and safety limitations prevent the use of such
pitch, as explained in Chapter V. The improved burnups achievable
with the different pitches for the different fractions of full
power translate into cycle lengths higher than the factor of two-,
three- or four-fold that could be expected from the cores operating
at the 50, 33 or 25 percent power level with respect to the
standard core. Thus, the 50%, 33% and 25% power cores with the
1.35 cm pitch show a cycle length which is respectively 2.11, 3.18
and 4.26 times the cycle length of the standard core. This means
that for the 50% power core (which is the most commercially
attractive of the cases considered) the same initial fuel load
would provide not only the natural double core life as compared to
the standard plant but an additional 11%. The 33% and the 25%
power cores would provide additional cycle lengths of 18% and 26%
respectively. However, this additional cycle lengths, although
very important from the plant cycle and refueling outage schedule
viewpoints, should not be confused with the additional burnup
achieved by each core, which is discussed below. The basic

151
difference is that the burnup level is an absolute measure,
independent of the core power level, while the cycle length (even
the full-power cycle length, as represented in this third column)
is a measure that is dependent on the power level of the plant.
The fourth column of table 4.2.1, labeled Dis. BU. MWd/MTU,
shows the burnup levels achieved by each core, as calculated by
CRIBUR. This is probably the best measure of the amount of energy
provided by a given amount of initial fuel loaded in the different
cores for the case when no reprocessing is being done. This
measure of burnup does not take credit for discharged fissile
materials, which would be of obvious interest in the case of a
reprocessing policy. However, under the present U.S. policy, this
can not be accounted for, and the burnup achievable is a good
measure of the energy obtained from a certain amount of initial
fuel. The levels of burnup achieved by the best pitch cell (1.35
cm) for the 50%, 33% and 25% power level cores are 5.4%, 6.1% and
6.5% above that achieved by the standard core, respectively.
Referring these burnup increments to a common basis, the 50% power
core yields 0.11% extra burnup per percent power reduction with
respect to the standard plant; the increments for the 33% and 25%
power density cores are 0.092% and 0.087% per percent reduction
with respect to the full-power plant. These figures represent a
15% and a 20% lower value than that obtained in the power reduction
to 50%. The implicit meaning of these figures is that the
additional burnup obtained with low power density cores with
respect to the standard plant is largest for the initial power

152
reductions, but as lower power densities are experienced, the
percent increase in burnup per percent power reduction appears to
be smaller. This is a significant reason to limit the power
reduction to a "safer and optimum level" instead of going to lower
power levels, since little is gained in terms of burnup with the
further power reductions. Other than the safety-related issue, the
improvements in burnup is one of the most significant contributions
of the reduced-power cores.
The fifth column of table 4.2.1, labeled Dis. Fis. MT/MWd
shows the mass of fissile material that is discharged from each
core with the spent fuel. The remaining fissile material is
composed of U-235, Pu-239 and Pu-241.
All fresh fuel loaded in the cores consisted of uranium
enriched 3.1% in U-235. The total mass of uranium per unit volume
loaded with the fresh fuel can be easily obtained by multiplying
the number density of U-235 in the fresh fuel and dividing by the
enrichment and by Avogadro's number. In a similar manner, the
total mass of fissile materials per unit volume present in the
discharged fuel is immediately obtained from the discharge number
densities of U-235, Pu-239 and Pu-241.
With these, it is possible to obtain a ratio of the total mass
of fissile material discharged from the reactor versus the total
mass of uranium loaded with the fresh fuel, which is a constant
value for all the cores. Recalling that the burnup level of the
discharged fuel is expressed in terms of MWd per Metric Ton of
uranium loaded, it is possible to divide the above ratio by the

153
burnup value, which will yield the discharged fissile mass per MWd
produced. This is the value displayed in the fifth column of
Table 4.2.1.
The calculations show that the total mass of discharged
fissile material per MWd of energy produced decreases steadily as
the pitch is increased. This is due to the softening of the
neutron spectrum caused by the increasing amounts of moderator
present in the core, which reduces resonance absorption in U-238,
and therefore reduces the production of plutonium isotopes. For
each particular pitch, the discharged mass of fissile nuclides per
unit energy produced is reduced as the power level of the core is
reduced,, There are two factors contributing to this effect: first,
as the power level is reduced, the Doppler broadening of resonances
is also reduced, having the same effect explained above when
discussing the production of plutonium isotopes; and second, as the
lower-power cores are driven to higher burnups due to the extra
reactivities supplied by the lower xenon levels and the reduced
Doppler effect, more incremental fissile material is consumed in
energy production than is gained during the additional time for
plutonium conversion.
These observations about the reduction of discharged fissile
nuclides as pitch is increased will have further significance when
fuel reprocessing is contemplated, since the further burnup
achieved by the fuel would also reduce the worth of the discharged
fuel.

154
The sixth column on table 4.2.1, labeled Rec. Fis. % Dis/Ld
shows the percent value of the weight of discharged (i.e.
recyclable) fissile nuclides as compared to the weight of loaded
fissile material. The values obtained for this column are plotted
as the lower set of curves in Figure 4.2.1, for easier
interpretation. Since the only fissile material loaded in the core
is -235, this column can be easily obtained by adding the number
densities of the discharged fissile species multiplied by their
respective atomic weights, and dividing by the same figure,
referred to the loaded U-235.
This column shows two main facts: first, it shows more clearly
the tendencies observed in the previous column, and second it
points out that the mass of discharged fissile materials is between
one third and one half of the mass of fissile material initially
loaded into the core. This implies the large potential economical
value of the discharged fuel when incorporated into a spent fuel
reprocessing policy.
The seventh column of the table, labeled Net Fis. MT/MWd
indicates the net mass of fissile material spent per MWd thermal
produced. In this case, net mass stands for the difference between
the fissile mass loaded into the core and the fissile mass
discharged with the spent fuel. Subtracting the fissile mass
discharged from the reactor from the mass of U-235 loaded with the
fresh fuel yields the net usage of fissile mass, and dividing it by
the burnup level yields the desired value of net fissile mass used

155
Recyclable' Net Fissile

156
per unit energy generated. These values are also plotted in Figure
4.2.1, as the upper set of curves.
For any given pitch, the net fissile consumption shows very
small changes (about or less than one percent) as power level is
changed, although a slight trend can be seen of increasing net
consumption for lower power levels. As pitches are changed, the
net fissile consumption increases for more moderated pitches,
causing the once-through, burnup optimized pitch of 1.35 cm to
consume about 10% more net fissile mass as the present standard
pitch. These results are in agreement with the observations on
fissile mass discharge appearing on column 5 of this same table.
This presents again the controversial point that the 1.35 cm pitch
is an optimum point for burnup and ore utilization under the
assumption of a once-through fuel cycle. This is obviously not a
favorable pitch from the ore conservation point of view if a
reprocessing policy is implemented.
The eighth column on Table 4.2.1, labeled PU Dis. MT/MWd shows
the total amount of plutonium that is discharged from the core per
MWd thermal produced. This figure has obvious interest from the
nuclear weapon proliferation point of view. The figures have been
obtained with an expression identical to the one used for the
discharged fissile mass (shown in the fifth column of the same
table) by substituting Pu-240 for the U-235. The plutonium
discharge values are plotted in Figure 4.2.2 as the upper set of
curves.

157
From this point of view, the optimized pitch has an advantage
over the current standard reactor. For each particular pitch, the
amount of plutonium discharged is reduced a few percent as the
power level is cut down. This is due to the softer neutron
spectrum existing in the lower power cores caused by the reduced
Doppler effect and the lower xenon level. As the pitch is
increased, the discharge of plutonium is sharply reduced. Thus,
the 50% power, 1.35 cm pitch core yields almost 16% less plutonium
than the standard reactor for the same energy production. The
significant variation in the yield of plutonium associated with the
pitch change is obviously due to the spectrum shift caused by the
increased amount of moderator present in the larger pitches.
The last column on Table 4.2.1, labeled U308 fET/MWd shows the
ore use for each of the cores on a per-unit-energy basis, under the
assumptions of once-through fuel cycle, uniform enrichment to 3.1%
U-235, and a tails enrichment of 0.2% Using standard tables for
feed-to-product ratio in the enrichment cascades (28) the ore use
can be obtained with the expression
Ore usage (MT/MWd) = 6.68 / BU (MWd/MTU)
The results for this column are also plotted on Figure 4.2.2,
as the lower set of curves. As could be expected, in this case the
best burnup cell presents the best ore utilization. Thus, the 50%
power core, 1.35 cm pitch uses 5.1% less ore than the current
standard reactor, for the same generation of energy. For each
particular pitch, the lower power cores show better ore utilization

158

159
than the higher power ones by some tenths of a percent. This is
due to the extra reactivities associated with the reduced Doppler
and xenon levels. However, as pitch is varied, ore utilization
reaches a minimum at about 1.35 cm pitch, and increases for any
further pitch increase or decrease. This results from the
existence of two conflicting effects which substantially influence
the core cycle length: First, there is the moderating effect of the
coolant. Since all LWR's work on an undermoderated configuration
for safety reasons, an increase in pitch means additional
moderation, and therefore an increase in the multiplication factor
of the core. This applies, of course, until the best moderated
cell is obtained, which occurs at about 1.5 cm pitch. Second,
there is the conversion of U-238 into plutonium. Plutonium
accounts for about 50% of the core power at E.O.L., and therefore,
the amoiont of plutonium present in the core heavily affects core
life. Since plutonium is converted from the parasitic capture of
neutrons in U-238, a better moderated pitch has a softer spectrum
and reduces the resonance capture in U-238, resulting in a lower
plutonium inventory at E.O.L. (this effect could be observed in
the previous column on this table). For very undermoderated
pitches, an increase of pitch leads to a larger effect from the
additional moderation than from the reduced plutonium yield and
core life is increased. When larger pitches are used, the
reduction of plutonium inventory outweighs the effect of the
slightly softer spectrum, causing a reduction of core life.

160
4.2.2, Plant Operational Data Variations.
The previous section showed how the variations of power
density level and the changes of fuel lattice pitch affect the
neutronics characteristics and the isotopic evolution of the core.
These variations result in changes in plant operational data which
imply sharp differences from both the operations and the economic
viewpoints. This section is intended to show the main plant
operational characteristics associated with each of the studied
cores, and their underlying implications. The data shown in this
section are obtained from the burnup and cycle lengths shown in the
preceding section, which were based on the PDQ-7 CRIBUR burnup
results.
Table 4.2.2 establishes the basis in which to compare the
availability and capacity factors of the different reactors, and
their cycle lengths and refueling outages. These parameters are of
major importance since they represent large differences from the
economical viewpoint. One of the main economical incentives is
reducing plant outage time. This is a well known fact, which value
increases with time. Presently, depending on the utility location,
its dependence on oil, coal, gas, the time of the year, etc., one
outage day represents an additional expense of one to two million
dollars.
As on Table 4.2.1, the first two columns of Table 4.2.2 are
used for description of the core case studied, and refer to the
power level and the pin array pitch. The columns that follow on

161
Table 4.2.2. Plant Operational Data Comparison.
Pitch
cm
Power
%
Real Cycle
Days
Life Reful
Days
Refuel
% Life
Availa.
%
Capacity
%
Size Mod.
Cap., %
1.20
50
923
534
4.88
80.85
70.10
75.04
1.20
33
1382
357
3.26
82.23
71.30
76.40
1.20
25
1850
267
2.43
82.93
71.90
77.10
1.25
100 (
a) 496
993
9.06
77.30
67.02
64.85
1.25
50
976
505
4.61
81.08
70.30
75.26
1.25
33
1443
342
3.12
82.35
71.40
76.52
1.25
25
1923
256
2.34
83.01
71.97
77.17
1.30
50
986
500
4.57
81.12
70.33
75.30
1.30
33
1471
335
3.06
82.40
71.44
76.57
1.30
25
1955
252
2.30
83.04
72.00
77.21
1.35
50
997
495
4.52
81.16
70.37
75.34
1.35
33
1482
333
3.04
82.42
71.46
76.59
1.35
25
1969
250
2.29
83.06
72.01
77.22
1.40
50
976
505
4.61
81.08
70.30
75.26
1.40
33
1452
340
3.10
82.37
71.41
76.53
1.40
25
1926
256
2.34
83.01
71.97
77.18
(a): Reference Case

162
this table require seme background discussion about the main
factors influencing the plant cycle length and the power produced.
First, it is assumed that a standard full-power plant with one-year
cycle time (including the refueling outage time) has an
availability factor of 75%. The refueling portion of an outage is
considered to last 45 days, according to industry surveys(29),
although this number is just a "meaningful average", with actual
values oscillating around it. The capacity factor of the plant is
assumed to be 65%, according to Department of Energy reports(30).
In order to calculate the availability and capacity factors of
the reduced-power cores, it is convenient to define some new terms.
The first one is the "Active-Time Availability Factor" (referred
to from here on as ATAF). This is the availability factor of the
plant if the 45 days refueling time are not considered. The
conditions affecting the on-line and outage times of the different
plants can be considered equal for all plants, except for the
incidence of the refueling time, which depends on the length of the
core cycle. At this point it is assumed that all plants have the
same ATAF, i-e., that all plants have the same amount of
non-refueling outages. This assumption should be qualified as very
conservative for the low power cores, since they have improved
Nuclear Steam Supply System (NSSS) conditions as compared to the
standard plant; for the rest of the plant, operating conditions are
essentially identical to the standard plant. In actuality, the low
power core should have a distinct advantage in savings on unplanned
outages.

163
The MAP can be calculated from the standard plant assumptions
stated above, considering that the product of the ATAF by its
corresponding "cycle time" must equal the product of the actual
availability factor by the actual cycle time:
MAF (TCT-ROT) = AF TCT
where ATAF = Active-time availability factor.
TCT = Total cycle time. (365 days for the standard plant).
ROT = Refueling time. (45 days).
AF = Availability factor. (75% for the standard plant).
According to the standard plant assumptions, this definition
yields a value for the ATAF of 85.5%.
Another parameter defined to establish the capacity factor
variations is termed "Operating Capacity Factor". The ATAF defines
the level of on-line time of the plants when the refueling time is
not considered. However, even during the on-line time, the plants
do not deliver their full power. There are a number of reasons
that prevent the plants from delivering 100% power level at all
times while they are connected to the grid: One cause is the
de-rating of the plant due to technological causes, frequently
associated with the fuel and its thermal conditions; another is the
power runbacks, or forced temporary de-rating following a power
transient, which is normally required in order to assure the fuel
and primary coolant system integrity and performance. With these
and other similar losses of capacity factor in mind, the average
power level of the plant during the on-line time is defined here as

164
the Operating Capacity Factor. This is another factor that is
hereby considered to remain constant for all types of plants for
comparison purposes. The operating Capacity factor (which will be
referred to as OCF from here on) can be defined and calculated from
the assumptions stated above for the standard plant in. the
following way:
From the definition of ATAF, the on-line time is
On-line time = ATAF (TCT ROT)
where ATAF = Active Time Availability Factor.
TCT = Total Cycle Time.
ROT = Refueling Time.
Then, using the concepts of OCF and the conventional Capacity
Factor (herein referred to as CF), it is possible to write
OCF ATAF (TCT ROT) = CF TCT
Fran where OCF can be obtained and calculated using the
assumptions stated for the standard plant:
OCF = (CF TCT) / (ATAF (TCT ROT)) = 0.867
where OCF = Operating Capacity Factor (Capacity Factor of on-line
time).
CF = Conventional Capacity Factor (Including all time).
One significant factor that can differentiate the plants is
the length of the real cycle. Op till now, the burnup calculations

165
have described the burnup levels achievable and their associated
power generation times. However, these are full-power generation
times, and it is now necessary to turn than into actual plant
times. In a standard plant, the real time or Total Cycle Time and
the full-power time are related through the Capacity Factor, so
that
FPT = CF TCT
where FPT = Full-Power time.
In the case of the low power density cores, it is not possible
to use the same concept, since the CF is unknown because of the
changing weight of the refueling outage with the different powers
and degrees of burnup. However, the OCF defined above can be used
to overcome this problem. Using the last two equations,
FPT = CF TCT = OCF ATAF (TCT ROT)
Since OCF is considered constant, as well as ATAF and ROT,
while FPT is obtained from the burnup studies, it is possible to
obtain the total cycle time of any plant as
TCT = ( FPT / (OCF ATAF) ) + ROT
The FPT tabulated in Table 4.2.1 is in hours (from the burnup
calculations) while the TCT is more conveniently expressed in days.

166
Substituting the values obtained for OCF and ATAF allows the above
expression to be written as
TCT (days) = ((FPT (hours) / 0.74) + 1080) / 24
This total cycle time in days is listed in the third column of
Table 4.2.2 for all the cases under study, and is also plotted in
Figure 4.2.3 for easier examination and comparison purposes.
Note first that the cycle length shown in Table 4.2.2 for the
standard reactor is more than one year. The reason for this is
that the burnup levels obtained for the discharged fuel from the
calculations are higher than the actual burnup levels of the fuel
discharged from real reactors. There are several reasons for this
fact. First, the calculational model assumes a slightly high fuel
enrichment (later figures seen to point to 3.0% enrichment instead
of the 3.1% considered in this study as the equilibrium cycle fuel
enrichment). Second, the burnup study assumes uniform burnup of
the fuel; the actual fuel burnup distribution is far from being
uniform, and this causes the calculational model to sustain
criticality somewhat longer than the real core. It should also be
noted that the actual average equilibrium cycle time for FWR's in
the U.S. during the period 1973 to 1979 was somewhat greater than
one year. In any case, even with seme possible absolute error
associated with the actual values of cycle length, the relative
comparison between the different cores should be accurate, since
the bias applies to all cores in the same sense.

Cycle Length
Days
167
25% Power
1000 -
50% Power
500 -
Std. Core
T
1.20
130
1.40 Pitch
cm
Figure 4.2.3. Real Cycle Length vs. Pitch.

168
Note that the cycle length of the 50% power, 1.35 cm pitch
core is very closely twice that of the standard reactor. This
means that if a one-year cycle applies for the standard plant, a
two-year term would apply for the 50% power one. This seems
obvious, but hidden in here is the fact that for this two year
cycle, one refueling outage has been turned into power-producing
time. In other words, the refueling outage time is reduced to
approximately one-half for this configuration of what it is in a
standard plant. This will be seen better in the next column of the
table.
The fourth column on Table 4.2.2, labeled Life Reful Days,
shows the expected total number of days that the plant would spend
in refueling during a thirty year life period. The resulting data
are also displayed in Figure 4.2.4 for easier reference and
comparison. The purpose of this column is to realize the number of
days gained from refueling downtime for power-producing days. The
figure is obtained by first finding the number of cycles that the
reactor would complete in a thirty year period, and multiplying the
number by the forty-five days length assumed for refueling. Note
that the 50% power, 1.35 cm pitch reactor shows a savings of 498
days over the life time of the plant, which would represent a
considerable amount of money saved.
Another form of looking at the savings in refueling outage
time is found in column 5 of Table 4.2.2, labeled Refuel % Life.
This column shows the percentage of the total plant life that is
spent in refueling. A variation of this figure would be closely

169
Days
Figure 4.2.4. Total Life Refueling Time vs. Pitch

170
related to the variation of the capital costs part of the total
cost of energy. It is interesting to note that over 9% of the
total plant life in a standard plant is spent refueling, while this
figure is reduced to slightly over 4-5% for the optimized 50% power
core, and about 3-0% and 2.3% for the 33% and 25% power level
cores. Note that the variation of pitches does not heavily affect
the percent impact of the refueling time.
Using more conventional parameters, column 6 on Table 4.2.2
labeled Availa. % shows the availability factors that can be
expected from the different reactors assuming that they encounter
the same percentage of unexpected problems during normal operation
(which is a pessimistic assumption when comparing the low power
plants to the full power standard plant). The only variable that
is supposed to affect the availability of the plant in this
calculation is the relative effect of refueling time. The
availability factor can be obtained from the formulas defining the
Active Time Availability Factor, which was recognized above as an
invariable index- The expression for the availability factor is
AF = ATAF (TCT ROT) / TCT
As it was expected, the standard plant shows an availability
factor close to 75% (it is slightly higher due to the longer
calculated cycle time caused by the reasons stated above). The
pitch-optimized, 50% power core shows an availability factor of
about 4% over that of the standard plant, while the highest
availability, which of course is obtained by a 25% power core, is

171
just about 2% above that of the 50% power reactor. The
availability factors of all the cases considered are plotted as the
top set of curves of Figure 4.2.5, plus the point representing the
standard core, which is significantly displaced downwards.
Column 7 on Table 4.2.2, labeled Capacity %, is another
frequently used parameter for measure of the performance of a
plant. In this case, since all plants are so far assumed to have
the same amount of unplanned outages, the capacity factor is
obtained by use of the Operating Capacity Factor. From the
equation defining the OCF, it is possible to write
CF = OCF ATfiF (TCT ROT) / TCT = AF OCF
As could be expected, the standard plant shows a capacity
factor slightly above 65%, while the 50% power, pitch-optimized
core holds a capacity factor of about 3.3% above that of the
standard reactor, which means an increase of 5% energy production
in the same period of time, for equivalent installed powers. The
25% power core capacity factor is increased by only about 1.6% over
that of the 50% power core. The capacity factors of all the plants
under study are plotted as the bottom set of curves of Figure
4.2.5.
The last column of Table 4.2.2, labeled Size Mod. Cap., %,
again calculates the capacity factors that can be expected from the
different cores, but this time the calculation takes into account
the historically proven fact that smaller size plants actually have
better capacity factors than large plants for equally long cycles

172
%

173
(the standard one-year cycle). The Atomic Industrial Forum(30)
reports capacity factors of 62.3% for large plants (1000 Me and
above, such as the reference plant of this study) and 69.4 for
plants sized between 400 and 749 Me. This figure is used here for
the calculation of the capacity factor of all reduced power plants.
Obviously, the capacity factor is again affected by the cycle
length, since this causes different spacings between refueling
outages. The new, "size corrected" capacity factors are then
obtained with the following considerations:
Let TCTo be the total cycle time of the reference plant for
any particular plant size. In this case it is one year, since the
capacity factors collected from real-world data correspond in their
vast majority to yearly cycles. The plant-size dependent Capacity
Factor is called SCF, and is obtained from statistics of existing
different plant sizes.
In TCTo days, there are TCTo SCF full-power operation days.
During the same time period, the plant is available (TCTo ROT) *
ATAF days. So, in the TCTo days, there are
TCTo SCF full-power days per active day.
ATAF (TCTo ROT)
This is the concept of operating capacity factor, now applied
to any size-dependent actual capacity factor. If the plant can
operate for a known FPT full power time, just as shown before, the
total cycle time should be

174
TCT = (FPT / (OCF ATAF)) + ROT or
TCT = (FPT (TCTo ROT) / (TCTo SCF)) + ROT
and then, the capacity factor must be
CF = FPT / TCT = FPT / (FPT (TCTO ROT) / (TCTO SCF)) + ROT
Since this capacity factor takes into account the variable
capacity factor of a certain-size plant for a given standard cycle
length, it is called in Table 4-2.2 the Size-modified capacity
factor- The size-modified capacity factor is also plotted in
Figure 4.2.5, as the middle set of curves in the figure.
Note that when this size correction is taken into account, the
50% power. 1-35 cm pitch reactor shows a capacity factor over 10%
above that of a standard plant which represents a 16% increase in
the power being produced in equal amounts of time by equivalent
power installations. This is a very significant increment, and its
economical impact is bound to be of large importance because it
reduces the impact of the capital and O&M costs on the total cost
of power, and in the case of nuclear-generated power, these two
items are responsible for a large proportion of the energy cost.
Under the present assumptions, the 50% pitch-optimized power plant
would reach capacity factors of about 75%. Also, in net energy
generation, an optimized SHARP will be closer to the output of a
standard size plant than its rated power indicates-

175
4.2.3. Ore Usage and Enrichment Needs..
The last comparison used in the study is complementing the
information shown in Section 4.2.1, but establishing a common
baseline on which to compare all the different plants.
Table 4.2.3 refers to the usage of ore and enrichment units
for the different plants considered, but with all data normalized
to the power generation of 1125 Me, so that the figures on the
table can be compared on a same energy output basis. As in Tables
4.2.1 and 4.2.2, the two first columns are used for core
identification. The third column shows the total mass of plutonium
discharged yearly. It is assumed here that the total energy
production of 1125 Me over the 30 year life time is 22.4 GWy.
Then, the plutonium mass discharged yearly by each core, for an
energy production equivalent to one year of operation of the
standard plant is obtained as
Disch. Pu (Kg) = (MTPu/Md) 22400*30*365.25 / 0.33
where MTPu/Md is the total plutonium mass discharged per Md
thermal produced. This figure was calculated for each core in
Table 4.2.1.
As could be expected, the plutonium discharge is reduced as
the fuel pitch becomes larger. The cause is the softening of the
neutron spectrum due to the better moderated geometry obtained with
the increased amount of water present in the fuel cell. For each
particular pitch, the plutonium yield is also reduced as the power
level decreases, although the effect is much milder than the one

176
Table 4.2.3. Ore, Enrichment and Plutonium Normalized to 1125 MWe
Pitch
cm
Power
%
Pu Disch.
Kg/Yr
Ore Usage
MT/Life
Ore Savings
% Over Std.
Enrich. Need
SWU/Life
1.20
50
225.08
4662
-2.87
3158
1.20
33
219.69
4589
-1.28
3109
1.20
25
216.87
4533
-0.05
3071
1.25
100 (c
i) 201.72
4531
0.00
3070
1.25
50
197.51
4396
2.98
2978
1.25
33
194.83
4389
3.15
2973
1.25
25
192.61
4357
3.86
2952
1.30
50
181.48
4350
4.00
2947
1.30
33
178.07
4304
5.03
2916
1-30
25
176.21
4284
5.46
2902
1.35
50
169.45
4300
5.12
2913
1.35
33
166.30
4270
5.76
2893
1.35
25
164.66
4254
6.12
2882
1.40
50
160.95
4395
3.01
2978
1-40
33
158.33
4361
3.75
2955
1.40
25
156.88
4349
4.02
2947
(a): Reference Case

177
associated with the change of pitch. The cause of the spectral
shift associated with the power level variation is the change of
the Doppler broadening of resonances, which is decreased as the
fuel temperature decreases. These two effects add up to reduce the
plutonium discharge by 16% for the 50%, 1.35 cm pitch core, as
compared to the standard reactor. This is obviously a favorable
point for the low-power core from the non-proliferation point of
view.
The fourth column on Table 4.2.3, labeled Ore Usage WT/Life,
shows the total amount of U308 ore used by each core for the
production of 22.4 GWy (e) of energy. This is, as mentioned above,
the total energy production assumed in the lifetime of the standard
plant. An overall plant efficiency of 33% was assumed for all
plants, since the thermodynamic conditions are supposed to be
identical. Since Table 7.2.1 showed the U308 needs per MWd (th)
produced, those figures can be used in order to obtain the lifetime
ore consumption as
Ore consumption (MT) = 2.45448 E+7 (MT/MKd th)
where MT/MWd th is the specific ore consumption shown on Table
4.2.1. Note that the specific ore consumption was calculated under
the assumptions of 3.1% enrichment in U-235 of the fresh fuel and
0.2% enrichment of the enrichment plant tails. Obviously, this
column shows the ore usage under the assumption of once-through
fuel management.
Note that savings of 5.1% of ore can be achieved by the 50%
power core with the optimized pitch. The ore usage figures follow

178
a parallel trajectory to the discharge burnup figures. For each
pitch, reducing the power level reduces the ore requirements, but
as pitches are changed, a minimum ore requirement is reached for
1.35 cm pitch.
The comparison of ore usage with respect to the standard plant
is made in column 5 of Table 4-2-3- under the label Ore Savings %
Over Std. It is easier to see in this column that large savings
are achieved with the reduction of power from 100% power level to
50%, but further power reductions result in additional ore savings
of a much smaller magnitude, per unit of power reduction. This is
one of the main reasons calling for the power reduction to be
carried to no more than 50% of the standard value. Actually, an
optimum SHARP power level will be higher than 50%. and will be
affected by other economic considerations.
Column 6 of Table 4-2-3 shows the total enrichment needs of
each plant, in SWU. This column assumes also the total energy
production of all configurations to be 22.4 GWy (e), the fresh fuel
enrichment to be 3-1%, and the tails assay to be 0-2%.

CHAPTER V
THERMAL-HYDRAULICS AND ECONOMIC CONSIDERATIONS
5.1. Safety Related Thermal-hvdraulic Considerations
5.1.1. Motivation
As indicated in the previous chapters, the reduced-power cores
that would be installed in a SHARP would have very nearly the same
physical design as the standard full-power reactor, except for some
possible variation in the fuel array pitch. It has also been
stated that the balance of the plant must operate at conditions
essentially equivalent to those of the standard reactor, in order
to maintain thermodynamic efficiency, and to avoid the need for
significant redesign of components in the balance of the plant.
The above conditions require the reactor coolant to have
identical thermodynamic characteristics at the inlet and outlet of
the reduced-power core as it would have with the standard reactor.
It is obvious then, that if the core is operating at a reduced
power level, the coolant flow must be reduced accordingly in order
to obtain the same enthalpy at the outlet. There is, however, a
consideration of major importance related to reducing core flow
rate: the heat transfer conditions are affected by the different
power density and by the different coolant flow conditions. This
presents a most important issue from the standpoint of plant
safety, since it is well known that one of the critical
179

180
technological and safety aspects of the standard plant is the
assurance of an adequate heat transfer under normal, transient and
accident conditions. Mark Miller performed a detailed study(16) on
the behavior of the low-power core as compared to the standard
reactor and some of the main conclusions of this study are
presented here as illustrations of the thermal-hydraulic and safety
comparison between the two reactor concepts.
5.1.2 ,--Thermal-hvdraulic_.Studies.
There are several parameters that are considered of key
importance for thermal-hydraulic safety considerations. Some of
these are the fuel average temperature, which is an indication of
the stored heat content of the core and which is an important
parameter in determining the emergency core cooling system
characteristics; the fuel centerline temperature, of particular
importance since that is the hottest point in the pellet, the clad
surface temperature, and the departure from nucleate boiling ratio
(DNBR) which is the ratio between the surface critical heat flux
(the surface heat flux that would produce film boiling) and the
actual surface heat flux.
The DNBR varies from point to point in the reactor, since the
heat transfer conditions change from one point on the fuel to
another, as does the heat flux level. A safety margin can be
related to the minimum DNBR found in the whole core. Thus, the
minimum DNBR is one of the criteria that can be used for
thermal-hydraulic safety comparisons. The importance of the clad

181
surface temperature as an indication of core safety becomes
apparent when considering the possibility of clad failure. The
clad surface temperature depends on the heat flux level and on the
thermodynamic conditions of the coolant. Although the coolant is
kept by definition at equivalent conditions in all the cores under
study, the heat flux experiences significant differences, which may
result in changes in the clad surface temperature. Finally, the
fuel temperature depends not only on the thermodynamic coolant
conditions and on the clad-to-coolant heat transfer mode, but it
depends to a high degree on the heat generation rate at each point
of the fuel pins. The heat generation rate is different from point
to point in the core, and it is related to space-dependent nuclear
properties of the fuel and the core itself: the particular fuel
composition at each point (which depends on the power generation
history of that point), the poison concentration, the proximity to
water holes or control rods, etc. all contribute to an uneven
distribution of the heat generation rate throughout the core. This
unevenness has to be taken into account when studying the limiting
heat transfer and temperature conditions, and it is done by means
of the so-called power peaking factors. It is normally assumed
that the heat generation rate has an axial variation similar to a
cosine function; a radial variation due to different geometrical
factors is also assumed. The power peaking factors used for the
present calculation are 1.64 for the radial value, and 2.62 for the
overall core. These power peaking factors and the
thermal-hydraulic safety parameters stated above are included in

182
the TEMPRETGl) code, developed by Miller and based on a
combination of industry-accepted heat transfer correlations, which
was used for the safety-related thermal-hydraulic calculations.
Figure 5-1-1 shows the fuel centerline temperature at the most
disadvantageous point- as a function of the core power density
level and of the coolant mass flow rate. The relative mass flow
rate (on the x-axis) is referred to the standard core's nominal
mass flow rate. The marked points correspond to the flow rates
that yield outlet coolant enthalpies equivalent to those found in
the standard core. Note that for high mass flow rates (as compared
to the value required for equivalent outlet enthalpy), the heat
transfer takes place as forced convection, and is dependent on the
flow conditions. Once a certain low mass flow rate is achieved,
nucleate boiling takes place, and the heat transfer is essentially
independent of the flow conditions. The fuel centerline
temperature remains at an essentially constant level as the coolant
flow is further reduced. This situation would hold as long as the
coolant mass flow rate is not reduced below the point where film
boiling would start. Figure 5-1.2 is a plot similar to Figure
5-1-1 but it represents the volume averaged temperature of the
fuel- The same trends can be observed as in Figure 5.1.1. but they
are smoothed out by the averaging process, which for each flow rate
includes portions of fuel that are under forced convection
conditions as well as other portions which are under nucleate
boiling conditions.

£83
Temperature
deg. F
2200
2000
1800
1600
1400
1200
Figure 5.1.1. Fuel Centerline Temperature vs. Relative Mass Flow Rate

184
Figure 5-1.3 shows the fuel clad surface temperature as a
function of the the coolant flow rate for different fractions of
the standard power density, for both the maximum clad surface
temperature point, and for the point corresponding to the maximum
heat flux rate. The transition from the forced convection to the
nucleate boiling conditions is more apparent here, since the clad
surface temperature is the parameter most affected by this heat
transfer condition. The sharp break point displayed in the curves
results from changes in the type of correlation used for computing
the temperature level. Note that at the normal operating flow rate
for each particular core, all cores have practically the same
maximum clad temperature, regardless of the fraction of standard
power level considered.
Figure 5-1.4 shows the minimum DNBR for the three low power
density cores as a function of the coolant mass flow rate. Again,
the marked point on each curve represents the mass flow rates that
correspond to the standard outlet coolant enthalpy. The graph also
shows the minimum DNBR level of the standard full-power reactor-
Note that all the low power density cores have an advantage with
regard to the minimum DNBR when compared to the standard core, for
equivalent coolant conditions. However, it is important to keep in
mind that Figure 51-4 corresponds to cores with the standard fuel
lattice pitch- One should recall that the burnup studies showed
that a pitch of 1-35 cm is advantageous from the fuel utilization
point of view as compared to the standard 1-2573 cm pitch. It is
important to examine the thermal-hydraulic behavior of the cores at

185
Figure 5.1.3. Clad Surface Temperature vs. Relative Mass Flow Rate.

186

187
that burnup-optimized pitch configuration, as well as at the
standard value. Figure 5.1.5 shows the results of these
calculations.
Figure 5.1.5 is a plot of the percent savings relative to
standard core values achieved in the minimum DNBR, and in the
maximum and average fuel temperatures as a function of the core
power density level. Both the standard fuel pitch and the
burnup-optimized fuel pitch have been considered. These
calculations assume a coolant flow rate that maintains the outlet
coolant enthalpy at a the nominal standard core level.
When the standard pitch is considered, reducing the power
density causes a constant improvement in the minimum DNBR, tending
to an infinite value as the "zero-power" reactor is approached.
Obviously, the savings for the 100% power level is zero. Both the
maximum and the average fuel temperatures show an improvement, but
it is clearly bound, and tends to level off. Note that the fuel
temperatures (maximum or average) are not affected by the pitch
value.
When the burnup-optimized pitch is considered, the same
coolant mass flow requires a lower fluid velocity, which tends to
worsen the heat transfer conditions. This fact is apparent in the
corresponding DNBR curve, which appears shifted downwards from the
one corresponding to the standard pitch. This is the fact that
prevents the standard full-power reactor from operating at the
burnup-optimized fuel pitch, as was indicated in the preliminary
calculations chapter. The minimum DNBR would be 20% worse than it

188
Percent
Savings
Figure 5.1.5. Percent Savings in DNBR-minimum and Fuel Temperature

189
is in the standard reactor, which is unacceptable from a fuel
safety standpoint. Further examination of the curve shows that at
a power density level of about 75%. the minimum DNBR becomes
equivalent to that of the standard reactor with the standard pitch.
Any power density level chosen below 75% can be used with the
burnup-optimized fuel pitch value, and it still presents improved
heat transfer safety conditions as compared to the current standard
reactor-
It is important to remember that the thermal-hydraulic safety
conditions examined here correspond to the steady-state operation
of the core. The transient analysis being conducted for these
cores are still under preparation at the time of this report- Seme
of the transient conditions that should be examined are an
overpower transient, a partial and a total loss of coolant flow
with or without power reduction, and a depressurization transient.
It can be stated that the low power density cores show strong
advantages over the standard core with respect to thermal transient
behavior- The reduced fuel temperature allows for a larger amount
of heat to be stored in the fuel before critical heat transfer
conditions are reached. This fact, combined with the lower power
density level of the cores, allows for significantly longer
reaction times before any damage occurs to the fuel. In transient
situations or in loss-of-coolant flow accidents, the larger
burnup-improved pitch would seem to have a slight advantage again,
since it provides a higher inventory of water and larger coolant
channels in the core than the standard pitch. This will tend to

190
reduce the probabilities of fuel dryout and the consequential
cladding failure.
5.2. Economic Evaluation
5.2.1.Introduction.
The technical aspects surveyed so far about the SHARP clearly
show its advantages over the standard plant in the aspects of
operational safety, once-through fuel utilization, high radioactive
waste storage and disposal, reduction of refueling outages and
associated personnel radiation exposure, etc. However, it is
evident that the SHARP can be viewed as an "oversized" plant,
requiring more fuel inventory than standard power density cores for
an equivalent amount of output power, as well as having some of the
plant equipment larger than would be necessary at standard power
densities for the production of a given amount of power. All these
considerations indicate that the capital cost of the SHARP will be
larger than that of the standard plant. There is, however, the
belief that the reduction of refueling time and improved fuel
utilization will help in reducing the impact of the increased
capital cost. It thus becomes obvious that an economical
comparative analysis is needed in order to asess the final cost of
energy produced by the SHARP versus the cost of power produced by a
standard plant.
The cost of energy produced by a nuclear power plant, is
typically broken down into three main components: capital cost,
fuel cost, and operations and maintenance cost. In the case of the
SHARP, it might be possible to think that the enhanced safety could

191
result in some monetary savings due to reductions in licensing
times, in plant construction, etc. but these are rather
speculative, and they are considered out of the scope of this
report. The economical comparison presented here is based on the
costs of fuel, capital and operations and maintenance of several
different power density plants. A more detailed definition of the
sizes of the main plant components for each reduced-power density
plant is given in next section.
5.2.2. The Economic Comparison.Studies.
The economic comparison studies are presented for a set of
five different plant output powers. They correspond to 35%, 50%,
60%, 70%, and 100% of the standard plant power as defined in
Chapter II. Each of the plants considered in this study is
composed of a full size Nuclear Steam Supply System (NSSS) (as used
for the standard plant); a Balance Of Plant (BOP) of the nominal
size corresponding to the plant output power; and finally the
containment building sized for 120% of the size that would
correspond to a standard power density plant of the same power
rating. All plants are assigned a 30 years life, and startup is
assumed in 1993. Costs are levelized to and expressed in 1993
dollars,
Three possible inflation scenarios are considered for each
case studied. The low inflation scenario assumes an inflation rate
of 5% per year; it assigns an interest rate of 7% for borrowed
funds and bonds, and a 13% rate of return is assumed for common
equity (stocks). The moderate inflation scenario assumes a yearly

192
inflation rate of 9%; the assigned interest rate for borrowed funds
and bonds is 11%. and the rate of return on common equity is 17%.
Finally- the high inflation scenario assumes a yearly inflation
rate of 13%; it assigns an interest rate of 15% to borrowed funds
and bonds, and the return on common equity is placed at 21% -
The economic calculations were performed by Hersperger(l) with
the help of three industry accepted codes: CONCEPT-IV(32) for the
calculation of the capital cost associated with each plant; GEM(33)
for the calculation of the fuel cycle costs; and PCWERCCK34) which
computes the total generation costs. A brief description of the
three codes is found in Appendices B9. BIO and Bll, respectively.
Table 5-2.1 presents the capital costs of the different plant
sizes and the percentage increase with respect to the cost of the
standard plant, for the three inflation scenarios defined. It is
evident that the capital cost per KWh is a uniformly increasing
function, whose slope becomes increasingly steep as the plant power
level is reduced. Any power reduction beyond about 50% of the
standard becomes absolutely impractical since the capital cost is a
very significant proportion of the total power generation cost. The
trends of capital cost versus plant power rating for the different
inflation scenarios are plotted in Figure 5-2.1-
Table 5-2.2 presents the fuel costs for the different plant
sizes and inflation scenarios. The percentage increases of cost
with respect to the standard plant fuel cost (for each particular
inflation scenario) are also shown. These fuel costs include all
fuel-related expenses, such as mining, enrichment, fabrication.

193
Table 5.2.1
Capital Costs (Mills/KWh) and Percent Increases over Standard Plant
Power Low-inflation Moderate-inflation High-inflation
Level Cost % Inc. Cost % Inc. Cost % Inc.
35%
50.22
73.3
108.64
72.6
221.07
72.0
50%
40.89
41.1
88.59
40.8
180.55
40.5
60%
37.10
28.1
80.43
27.8
163.98
27.6
70%
34.37
18.6
74.62
18.6
152.33
18.5
100%
28.97
0.0
62.94
0.0
128.52
0.0
Fuel Costs (Mills/KWh)
Table 5.2.2
and Percent Increases over Standard Plant
Power
Level
Low-inflation
Cost % Inc.
Moderate-inflation
Cost % Inc.
High-inflation
Cost % Inc.
35%
26.52
22.4
54.95
15.9
100.32
4.7
50%
24.87
14.8
53.30
12.4
103.70
8.2
60%
23.73
9.5
51.24
8.1
101.14
5.5
70%
22.93
5.8
49.78
5.0
99.18
3.5
100%
21.67
0.0
47.41
0.0
95.85
0.0

194
Capital Cost
Mills/KWh
Figure 5.2.1. Capital Cost vs. Plant Power.
Note: Power is expressed as percent of Standard Plant.
% Power

195
etc. up to transportation and storage of spent fuel. As with the
rest of this economic evaluation, all plants are supposed to start
operation in January 1993, and the costs are expressed in 1993
dollars. Note that the fuel cost is one item that was expected to
improve the economics of the low power density plants due to
increases in burnup. However, they all show a larger fuel cost
when compared to the standard plant, because it takes much longer
for a given batch of fuel to produce its share of energy- This
means that in a standard plant, a batch of fuel has produced all
its energy (and therefore has produced all its income) in a
three-year period; in the case of a 50% power plant, it takes more
than six years for the same batch of fuel to produce all of its
energy- The income is therefore retarded, and the capital invested
in the fuel batch suffers higher interest charges, which outweigh
the relatively small increment of burnup that is obtained from the
fuel-
It is clear from the figures on Table 5-2.2 that the turnout
time for the fuel costs is important for the economy of the fuel
cycle. This strong sensitivity to time suggests that it might be
an advantage for the SHARP to consider low enrichment fuels, which
would have reduced enrichment costs and would result in shorter
core cycles. On the other hand, studies performed in the
industry(3) show that reducing the enrichment results in a worse
ore utilization. These two considerations indicate the need for a
more complete study of the economic behavior of the SHARP"s as fuel
enrichment is allowed to vary. Figure 5-2-2 is a plot of the :

196
Fuel Cycle Cost
Mills/KWh
Figure 5.2.2. Fuel Cycle Cost vs. Plant Power.
Note: Power is expressed as percent of Standard Plant.
Power

197
fuel cycle cost vs. plant size, for the three inflation scenarios
being considered.
The operations and maintenance (O&M) costs are considered
constant for all the plants studied. They are fixed as 7.18
mills/KWh for the low-inflation scenario, 11.25 mills/KWh for the
moderate-inflation scenario, and 17.34 mills/KWh for the
high-inflation scenario. These values have been calculated from
present average values for nuclear plants, and modified with an
inflationary factor. Note that the O&M costs should be dependent
on plant size, since the burnup studies showed drastically
different core cycle lives depending on the power density level.
However, the O&M part of generation cost is a very small proportion
of the total power generation cost, and the speculative differences
that could be applied would not affect the final generation cost
but by some tenths of a percent. It should be realized, however,
that the longer cycles and reduced refueling outages have been
taken into account in the evaluation of the fuel cycle costs. One
element that has not been considered is the cost of replacement
power for the different refueling downtimes. Although their cost is
presently very high due to the fact that most replacement power is
produced by oil or gas units, it is difficult to forecast their
cost at significantly later times. In any case, this could account
for an additional one or two percent reduction of power generation
cost for the low power density plants, which has not been accounted
for in the figures presented here.

198
Table 5.2.3 contains the total generation costs of the plants
under consideration, for the three inflation scenarios, as well as
their cost increment with respect to the full-power plant. Note
that like the capital cost, and unlike the fuel cost, the total
generation costs exhibit percentage increases at reduced power
which are essentially independent of the inflation level. This
table is the most significant one from the economic viewpoint,
since it shows the cost of the energy from each plant
configuration, after all parameters have been taken into account.
Power levels between 60% and 70% of the standard level show
moderate total cost increases. The transition from 60% to 50%
starts to show a sharper increase in generation cost, and it is
evident that power levels below 50% can not be considered since
their economic penalty is unreasonably large. These trends can be
better observed in the curves shown in Figure 5.2.3. which are a
plot of the total generation cost vs. the plant size, for the
three inflation scenarios.

199
Table 5.2.3
Generation Costs (Mills/KWh) and Percent Increases over Standard Plant
Power
Level
Low-inflation
Cost % Inc.
Moderate-inflation
Cost % Inc.
High-inflation
Cost % Inc.
35%
83.92
45.1
174.84
43.8
338.73
40.1
50%
72.94
26.1
153.14
25.9
301.59
24.8
60%
68.01
17.6
142.92
17.5
282.46
16.9
70%
64.48
11.5
135.65
11.5
268.85
11.2
100%
57.82
0.0
121.60
0.0
241.71
0.0

Total Generation Cost
Mills/KWh
Figure 5.2.3. Total Generation Cost vs. Plant Power.
Note: Power is expressed as percent of Standard Plant.
Power

CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH
6.1. Introduction
The primary purpose of this work is to establish the neutronic
and fuel utilization characteristics of the SHARP (Safer Highly
Available Reactor Plant), as well as to develop an accurate and
cost-effective calculational tool for the scoping studies of fuel
burnup. In addition, the primary thermal-hydraulics, safety and
economic characteristics of the concept and its advantages and
disadvantages over the present standard PWR reactor plant are
analyzed and summarized to provide a comprehensive review of the
key issues needed to judge its technical and economic merits.
The SHARP concept uses a standard, large size PWR reactor with
minimal design alterations to drive a reduced power rating electric
generation plant. The reactor is used at a reduced linear power,
while maintaining coolant thermodynamic characteristics in order to
keep the steam cycle efficiency. The immediate result of the power
density reduction is a significant improvement in technological
operational limits and an overall enhancement of reactor safety,
although it also results in a capital cost increase per unit of
installed power. The reduced power density entails differences in
the nuclear characteristics of the core, which require a thorough
study and optimization.
201

202
Standard industry neutronic codes (like PHROG, BRT, MONA and
LEOPARD) were used for the scoping studies conducted in order to
pinpoint the key differential characteristics of the SHARP core as
compared to the standard FWR core, and to determine the type and
range of parameters on which to conduct the in-depth optimization
study. One of the key issues distinguishing the SHARP concept is
the improvement of burnup and ore utilization, and this aspect
required thorough analysis. The available burnup calculational
methods were reviewed and found to be inappropiate for the type of
study required for the in-depth optimization stage of the SHARP
investigations. The problems with existing burnup calculational
schemes was that either computational requirements and expenses
were too large or accuracy was insufficient.
A new computer code called CRIBUR was developed for studying
the burnup performance of PWR's with a reduced computational cost
and a significantly increased accuracy as compared to previous
schemes of similar cost. CRIBUR is used in conjunction with
accepted industry codes (EPRI-CELL, NUFUNCHER, FDQ-7) to complete
the burnup calculational scheme. The final set of codes allows the
evaluation of isotopic composition through the fuel life, is able
to assign time-varying power levels to the different fuel batches
present in the core, and performs the burning of the fuel in a core
environment that closely simulates the criticality condition of the
reactor through life. The last feature increases the accuracy of
this type of burnup calculational scheme because it improves the
calculation of the absorption cross section of the moderator

203
medium; it therefore leads to an improved determination of the
neutron spectrum and the isotopic dynamics. The time-varying power
level capability is an unusual feature for this type of burnup
calculational tool. and has a significant impact on
spectrum-to-burnup level correspondence.
CRIBUR results were benchmarked against detailed
two-dimensional neutron diffusion theory codes and against actual
core data and were shown to consistently yield better results than
other popular calculational methods like the ones using
batch-averaging for neutron multiplication factor calculations(28),
or the ones using the linear reactivity assumptions for end-of-life
estimates(15). CRIBUR criticality calculations agree with actual
core BOL data to within 1-3%. Batchwise power distribution
calculations agree with multi-dimensional core follow calculations
to within a few percent.
The computational cost of the burnup calculational scheme is
little affected by CRIBUR. since it only takes a minimal fraction
of the computer time needed for EPRI-CELL and PDQ-7 calculations
which are required to prepare the batch burnup data for CRIBUR.
One complete core burnup calculation can be performed with one
EERI-CELL pin life calculation (requiring about three minutes of
CPU time of the IBM 370-3033 unit), one NUFUNCHER calculation
(requiring about ten seconds). three 1/4-assembly life burnup PDQ-7
calculations (requiring approximately one minute of CPU time each
on the same unit) and three CRIBUR calculations (each requiring
just a few seconds of computer time)-

204
6.2. Neutronics, Burnup and Ore Usage
The in-depth burnup studies were carried out on a series of
cores in which power density and fuel lattice pitch were used as
variable parameters. The standard Westinghouse 4-loop 17X17 fuel
assembly PWR core was used as a reference core against which the
SHARP results were compared. The reduction of power density causes
a decrease of Doppler resonance absorptions in the fuel and a
reduction of xenon concentration level, resulting in additional
reactivity being available for further burning of the fuel. The
variation of fuel lattice pitch was shown to have an even larger
effect on discharge burnup than the reduction of power density
level. These trends can be observed in Figure 6.2.1. It appears
that the initial power level reduction (from 100% to 50% of the
standard core power level) results in a higher increase of burnup
per unit power reduction than further power level reductions to 33
or 25% of the standard reactor's power. The burnup increments per
percent power reduction are 0.11% for the 100% to 50% power
reduction, 0.092% for the 50% to 33% reduction and 0.087% for the
33% to 25% power reduction. These figures clearly indicate that
the effect of power density level reductions on burnup are more
beneficial for the initial reductions. This factor, favoring
smaller reductions in power density, is further reinforced by the
fast deterioration of the economic aspects of the plants with power
density levels below 50% of the standard plant.
Examination of the effects of fuel pitch variations shows that
all power density level cores have a maximum discharge burnup level

205
Discharge Burnup
GWd/MTU
25% Power
36
34
-i
1.25
I
1.30
,
1.35
p-*-
1.40
Pitch cm
Figure 6.2.1. Discharge Burnup vs. Pitch.
Note: Data fran EPRI-CELL, NUPUNCHER, FDQ-7, CRIBUR Burnup Calculations.
Power Levels Indicated as Percent of Standard Core Power.

206
at a pitch of 1.35 cm. This burnup-optimized pitch can not be used
by the standard reactor for thermal-hydraulic safety reasons, but
presents no problem for the SHARP'S using a power density level of
70% of the standard value, or less. The additional burnups
achieved by the SHARP'S fall well within the range of acceptable
burnup level for the present fuel technology, and thus present no
need for clad structure or pin geometry redesign.
The burnup-optimized pitch is smaller than the optimum
moderated pitch (which is about 1.6 cm). This is due to the sharp
reduction of plutonium conversion associated with the softer
spectrum present in the better moderated pitches. Plutonium
discharge per unit energy produced from the burnup-optimized 50%
power density SHARP is 16% lower than from the standard PWR. This
is a positive effect from the nuclear weapons non-proliferation
viewpoint. However, in a fuel reprocessing policy environment,
such a low plutonium discharge would result in a loss of spent fuel
value.
Under the present once-through fuel policy, the SHARP improves
ore utilization as a result of the improved discharge burnup
capabilities. The 50% power core with the optimized pitch shows a
5.4% better ore utilization than the standard PWR plant. The 33%
power and 25% power SHARP'S show ore utilizations which are 6.1%
and 6.5% better than the standard plant, respectively.
With spent fuel reprocessing in mind, the net consumption of
fissile material per unit power produced was calculated. This took
into account the fissile mass loaded with the fresh fuel as U-235

207
and the discharged fissile mass, composed of U-235 and plutonium.
Fuel lattice pitch appears as a key factor influencing this
variable; the better moderated pitch configurations show a sharp
increase in ore usage per unit energy generated due to the
reduction of plutonium conversion. The 50% power, burnup optimized
pitch SHARP shows a 10% increase of net fissile species usage with
respect to the standard plant. However, the power density level
shows practically no influence in net fissile species consumption;
the 50% power level SHARP using the standard reactor pitch shows a
difference of less than 1% with respect to the standard plant.
6.3... Plant Operations Considerations
The reduction of linear power generation entails significant
changes in the plant's operational characteristics. The two key
effects are the improved safety of the reactor and the increase of
the core cycle life. The heat transfer conditions at the pin
surface are drastically ameliorated by the reduction in the linear
power. Core minimum DNBR savings of 60% with respect to the
standard core value are obtained in a 50% power density core using
the standard fuel pitch. The same core presents approximately 30%
minimum DNBR savings when using the burnup-optimized pitch. Using
the burnup optimized pitch in a standard power density reactor
would result in an unacceptable 20% worse minimum DNBR. The power
density reduction has been also shown to reduce the fuel average
temperature by 300 deg. K, resulting in improved fuel conditions,
including a reduction of gaseous fission product gas migration.

208
Another important improved operational variable is the xenon
concentration level. A 50% power level SHARP has an equilibrium
xenon concentration reduced by more than 25% with respect to the
standard plant value. This implies significant reduction of
problems associated with xenon oscillations or xenon concentration
transients caused by reactor shutdowns or large power variations.
The SHARP concept results in largely increased core cycle
lives as a result of two adding factors: the reduction of power
level itself, which entails a longer time in order to produce a
given amount of energy from the fuel, and the associated discharge
burnup level increase, which further contributes to the core life.
One implication of these longer core cycles is the reduction of
refueling outage time, and its associated high power replacement
costs. The 50% power, pitch optimized reactor shows a savings of
about 50% in total number of refueling outage days in the plant
life. A consequential effect is the increase of the plant
availability and capacity factors. The 50% power, pitch optimized
reactor shows a plant capacity factor increase of about 3.5% with
respect to the standard plant under the assumption of equal
probability of unplanned outages. When historical data concerning
reduced size plants was reviewed, it became apparent that these
plants have actually less unplanned outages than the large ones.
Evaluations of plant capacity factors based on forced outage
statistical data indicate a capacity factor increase of slightly
over 10% with respect to the standard plant for the 50% power,
burnup optimized SHARP.

209
Another consideration of importance from the operations point
of view is the likelihood of reduction of annual allowed radiation
exposure doses for plant personnel, which would have a large effect
on the operation of nuclear plants. The exposure level can be
significantly reduced by the SHARP due to the reduction of
refueling and forced outages.
The SHARP concept appears clearly as having significant
advantages over the standard plant from the operations point of
view.
6.4. Economic Effects
The SHARP concept presents both positive and negative factors
from the economic viewpoint. The reduction of refueling outage
time is an asset because of better utilization of the large capital
investments resulting in a reduction of the net energy generation
cost. The replacement power cost was not taken into account in the
present economic study because of the uncertainties in determining
it in a long term extrapolation, which is heavily influenced by the
particular characteristics of each utility. In any case, this is a
factor that could further improve the SHARP energy production cost.
The increase of plant capacity factor was taken into account in
the economic performance studies.
The increase of discharge burnup level results in direct
although small savings in ore and fuel enrichment and handling
costs

210
On the negative side, the SHARP present a much higher capital
cost per unit installed power, which ranges between 23% and 41%
above that of the standard plant, for power density levels between
65% and 50% of the standard value. Inflation level variations have
little influence on the relative capital costs when comparing
SHARP'S to standard power density level plants of equal output
rating. The fuel cycle cost analysis shows that in spite of the
reduced ore and enrichment needs, the total fuel cost is higher for
the SHARP than for the standard plant. The main reason for this
behavior is the higher interest costs associated with the long core
residence of the fuel. The fuel costs for the SHARP range between
5% and 15% above those of the standard plant's for the power
density range defined above. Inflation level has a significant
effect on fuel cycle cost, reducing the difference between the
SHARP and the standard plant as inflation rates become higher.
This is due to differences in revenue schedules on the cost of fuel
shipment and storage.
The overall economic evaluation of the SHARP indicates
generation cost increases for the SHARP between 17% and 27% with
respect to the standard plant for power levels between 65% and 50%
of the standard plant. Considering these results, a power
reduction beyond 50% of the standard value appears to be an
undesirable arrangement when cost/benefits are considered. The
combined consideration of the economic and safety aspects shows
that a power density level between 50% and 65% of the standard
plant would result in improved safety with respect to the standard

211
plant, and with a generation cost increase of about 20%, with a
slight tendency to decrease for higher inflation rate scenarios.
^-Suggestions ^for Future Research
The calculations performed in this study showed the
significant potential for operational and safety improvements in
PWR plant performance under the SHARP concept. Steady-state
thermal-hydraulics calculations showed an important side of the
improved safety of this concept. However, no transient or accident
scenario calculations have yet been carried out, and the particular
characteristics of the SHARP indicate the possibility of even more
significantly improved parameters under transient or accident
situations.
Fuel performance can be further improved with the use of new
fuel designs (like hollow fuel pins) which will see their improved
properties upgraded by the better thermal conditions of the
SHARP'S. A new parameter should be introduced in the neutronic and
economic studies of the SHARP: this parameter is fuel enrichment.
It has been shown(3) that increased fuel enrichments (up to about
5% U-235 enrichment) result in improved fuel utilization. This
fact, theoretically verified for standard power level cores, can be
synergistically potentiated with the demonstrated better burnup
capabilities of a reduced-power core. In the economic field, this
will impact in the conflicting factors of reduced refueling outages
and increased fuel interest costs, which will require new
evaluation and optimization.

212
The SHARP concept must also be tested for its advantages in a
fuel reprocessing policy environment. More involved isotopic
accounting methods will be required, but the favorable heat
transfer situation of the SHARP'S allows wide possibilities of play
with core design variables that should certainly result in
significant improvement of fuel cycle characteristics under a
reprocessing policy environment.
It is also necessary to make a thorough evaluation of the
possible monetary advantages obtainable from the improved safety
aspects of the SHARP. These monetary advantages can come from
reduced safety systems requirements, shortened licensing
procedures,etc. Although apparently more speculative in nature,
these aspects can result in significant savings, affecting the
capital cost of the plant, and thus, the highest portion of the
power generation cost.

APPENDIX A
METHODS OF IMPROVING BUENUP IN PWR'S
A.l. Introduction
A. 1.1. Motivation and Constraints.
LWR nuclear power plants were rapidly deployed in the U.S.
because of their significant advantages in known technology over
all other types of reactors when the industrial application of
nuclear energy began. The existence of the enrichment facilities
and the Nuclear Navy program were key to the selection of the LWR's
as the main U.S. reactor concept for deployment purposes. However,
LWR's are obviously not the most efficient users of uranium
resources. The "burners" were needed for rapid deployment and,
with the LMFBR and reprocessing in mind, the LWR fuel efficiency
was not a major concern. The fact that 'the discharged fuel
contains large amounts of U-235 and plutonium was important from
the point of view of resource utilization, with reprocessing and
the breeder "around the corner".
However, in the present political situation in the U.S. where
reprocessing has been indefinitely delayed and the development of
the LMFBR is questionable, the fuel utilization problem requires a
whole new view and approach. The new goal is now to obtain the
maximum energy generation with the minimum ore consumption and with
the lowest enrichment and fuel fabrication costs. Some of these
213

214
goals are at times in conflict, and a compromise (often just an
optimum economic solution) must be reached by an adequate
combination of these goals and the operational characteristics of
the reactors.
Increasing the energy obtainable from a certain amount of ore
may involve a very wide range of actions, from just operational
policy changes to some reactor redesign. Some actions which in
their conceptual sense might be in the right direction towards
reducing fuel cost and/or ore requirements may actually be
forbidden by safety or operational constraints. Some of the main
restrictions to bear in mind while searching for methods of
improving burnup are described below:
a). Linear power in the fuel pins is normally kept at a very
high level. Any attempt to increase it must insure that adequate
heat transfer at the pins' surfaces will be maintained, in order to
avoid damages due to excessive temperatures or thermal shock. In
general, any modification tending to increase the average linear
power or the power peaking factor will need very special care.
b). Fuel pin structural materials and the fuel itself suffer
physical degradation due to the radiation damage induced by the
high fluences that they must withstand. Fuel manufacturers are
trying to obtain fuel designs that will be able to withstand higher
fluences than the ones presently allowed (goals are set for about
45 GWd/MTU average burnup to be obtained in the next 10 to 16
years) (2). Hence, it is not reasonable to contemplate a solution
that would require fuel exposures much larger than the present

215
ones, if the solution is expected to be applicable in the near
future. An average burnup of about 40 GWd/MTU would look
reasonable as an upper limit in an optimistic scope for the near
future.
c). Somewhat in connection with point a) mentioned above is
the fact that if significant changes are made to the core lattice
geometry, it is indispensable to assure heat transfer conditions
consistent with operational and safety requirements of the core.
For example, a reduction of pin diameter entices a reduction of
linear power since the pin heat transfer surface would be reduced
as the diameter is reduced.
d). The fully loaded core can not have an extremely high
excess reactivity at BOL because this would force a high soluble
boron concentration in the coolant, and this might cause a positive
moderator temperature coefficient (MTC), which is unacceptable from
a safety point of view. One way of avoiding the problems caused by
a large excess reactivity at BOL is the use of burnable poisons,
which also serve the purpose of reducing power peaking.
A,1..2..^..S.ghemes-for improving Purnup.
There are a number of schemes that can be implemented to
increase the burnup obtained from a certain load of fuel. This
section reviews the main schemes involved in extending burnup,
while Section A.2 describes the possible practical ways of
implementing these principles in a PWR power plant and goes deeper
into the effects involved in the necessary modifications.

216
The core life terminates at end-of-life (EQL) because of two
basic reasons: neutron economics (neutrons are lost by leakage,
parasitic absorption, or both) and lack of fissile material (the
fissile material is burned out and any bred fissile material is
insufficient after awhile to outweigh the neutron losses). Any
design scheme aimed at extending burnup must modify these
conditions which affect the reactor, especially at EOL.
a). Reducing parasitic capture. A nuclear reactor has a
number of materials that absorb neutrons preventing them from
causing fissions in the fissile material. These are structural
materials (cladding, grids, barrel, etc.), poisons (both control
rods or residuals from burnable poisons), fission products
(produced as the fuel is burned), the fuel itself and the coolant
(either borated or unborated, the coolant presents some neutron
absorption). A basic scheme to increase burnup could involve the
reduction of the parasitic absorptions in the core. This can be
done in different ways, although not all of them may necessarily be
applicable in a real reactor, since relative burnup improvement
must be weighed against any required redesign and secondary
effects.
Parasitic absorption by fission products may be reduced,
obviously, by reducing the amount of fission products present at
EOL. This can be done by shortening the reactor cycle life and
reducing the reload batch size. This way, at EOL there is more
"low burned" fuel, and therefore less absorbing fission products.
Another way of achieving the reduction of fission products in the

217
newer batches is the use of burnable poisons that will mostly
disappear at EOL and will hold the power of the new batch down,
preventing the development of large amounts of fission products in
that batch. Burnable poisons should ideally be imbedded in the
fuel material itself rather than in poison rods. In this way they
can achieve a more thorough burn by EOL, and they don't have the
absorbing structural material associated with the poison rods.
Parasitic absorption in the fuel itself can be reduced by
reducing the power level of the reactor. This would reduce the
fuel temperature, and therefore the Doppler broadening of
absorption resonances. If some more effective means of cooling the
fuel could be devised, the same effect would be achievable without
the need for reducing the core power level.
Absorptions in the structural materials of the core could be
reduced by replacing stainless steel parts with parts manufactured
with less absorbing materials, and/or reducing the amounts of
structural material present in the core.
Absorptions in the coolant are difficult to reduce at EOL,
since no boron is present in the water. However, from the sole
point of view of absorptions in water at any point in the cycle
life, they can still be reduced by using lattices with less water
or by using the water-heavy water mixture whose other use and side
consequences will be described later in this section.
b). Reducing neutron leakage. There is a small proportion of
neutrons that leak out of the core and are lost for fissioning
purposes. There are some ways in which this leakage can be

218
reduced. One way is to place older fuel at the periphery of the
reactor, instead of the presently current practice of having the
fresh batch placed at the core periphery. 'This is further
discussed in section A.2.3. Another way would be to improve the
reflecting characteristics of the reflector, for example by
changing the material that forms the core baffle.
c). Improving moderation. For safety reasons, FWR's employ a
water/fuel ratio which is below the optimum for moderation. At
current EOL's, the core could still be critical if the water/fuel
ratio was closer to the optimum. Assuming the necessary safety
margins could be assured, alterations of fuel lattice pitch or fuel
pin diameter could improve the moderating characteristics of the
system. A similar effect can be achieved by altering the water
density, which can be done without need for fixed alterations of
the system.
d). Improving conversion ratio. Since at EOL a large part of
the power of the reactor is generated by the plutonium that has
been bred from absorptions in U-238, increasing the conversion
ratio would produce more plutonium, and therefore would tend to
extend the core life. There are two ways in which this can be
achieved in a FWK: The first is to employ the "spectral-shift
controlled reactor" (further described in section A.2.9), which
involves the mixing of light water and heavy water in a proportion
that can be optimized throughout the core life. Since heavy water
does not have the high moderating power of light water or the
higher neutron absorption, a harder spectrum is obtained in the

219
reactor when the heavy water proportion of the coolant is
increased, and a higher conversion ratio is possible. Another way
is to have a variable pitch, which may allow for different
water/fuel ratios at different times in life or in different
regions of the core.
A. 2. Techniques for Improvement of Burnup
A.2.1. Increasing Number of Batches.
In most PWR's, fuel management is based on adding or
withdrawing the fuel elements in batches, in such a way that all
the elements belonging to a certain batch have at any time in life
a similar burnup level.
Some simplified methods of calculating burnup capabilities of
PWR's use the assumption that all batches in the core share the
same fraction of the total core power (provided they have the same
number of fuel assemblies, which is normally the case) and that the
core multiplication factor can be obtained as the average of all
the batches' multiplication factors (28). This method was used,
for example, by L.E. Strawbridge of Westinghouse. Under this
assumptions it is possible to determine that the burnup at
discharge of a fuel batch, Xd, can be calculated as:
Xd = Xc ( 2*N / (N+l) )
where Xc = Average burnup of the core at EOC.
N = Number of batches present in the core.

220
An analysis of the extreme values of N yields an appreciation
of the available choices. N=1 corresponds to a reactor where the
whole core loading is renewed at each refueling, and it is obvious
that in this case the discharge burnup is the same as the core
average burnup at EOL. For N very large, Xd tends to be equal to
2*Xc, i.e., the discharge burnup can be, in the limit, twice as
large as the reactor average burnup at EOL. This would be the case
of an on-line refueling reactor such as a CANDU or a MAGNOX. The
increased burnup achievable, according to this theory, from the
same initial fuel as the number of batches increases can be
observed in Figure A.2.1.
Relative
Discharge
Burnup
2.0
1.5
1.0
Figure A.2.1. Increase in Fuel Burnup due to Partial Refueling.
Although the assumptions used for this analysis are not quite
true (i.e. the batches do not really share the same amount of
power and the core K-eff. is not really the average of the
batches K-eff.'s ) it is certainly true that increasing the number

221
of batches of a reactor will drive the fuel to higher burnups, for
the same initial enrichment. On the other hand, if the enrichment
is kept constant, having more batches in the core reduces the size
of the reload batch. Consequently, the time between refueling
shutdowns is reduced, which is generally not desirable because it
lowers plant availability.
The inmediate solution to these "very-short" cycles is to
increase fuel enrichment so that a higher core average burnup can
be achieved at EOL, and thus, longer times can elapse between
consecutive refuelings. R.L. Hellens showed (3) that in addition
to improving the discharge burnup, increasing the number of batches
and fuel enrichment also improves ore utilization for enrichments
up to about 5% U-235, as can be observed on Figure A.2.2.
In any case, a 5% enriched fuel used in a 4- or 5-batch core
would theoretically drive the fuel to burnup levels far beyond the
present technological possibilities. If fuel pins were made which
could stand average fuel burnups of about 50 GWd/MTU, potential ore
savings of up to 15% could be achieved. However, with the present
state-of-the-art fuel and the assumptions stated in Section A.l,
it is feasible to obtain 40 GWd/MTU with a 4-batch core, a 12-month
cycle and about 4% enriched fuel, with an associated improvement in
ore utilization of about 10% .
The main problem associated with such a maneuver, as described
above, is that increasing the number of batches in a core
containing a constant number of fuel assembly locations increases
the probability of placing highly burned assemblies adjacent to

222
Ore Utilization
MKd/ST U3Og

223
fresh and relatively highly enriched assemblies. This represents a
serious problem for in-core fuel management, since this induces
severe power peaking problems which are difficult to negate.
Burnable poisons utilization is then necessary, which further
complicates core management. Burnable poisons and their associated
advantages and disadvantages are described below.
A.2.2. Burnable_?pis.pns.
Burnable poisons accomplish several functions simultaneously:
reduction of core reactivity at BOL, allowing a reduced chemical
shim and thus make a positive MTC less likely; reduction of power
peaking from one assembly to another; power shaping within a fuel
assembly; and power reduction on the fresh assemblies, which
stretches the cycle length. Each of these aspects is examined
separately below.
There are two ways of implementing burnable poisons in EWR's.
The first and only one used currently in PWR's enploys the
so-called "Lumped Burnable Poisons" (LBP's) also called Burnable
Poison Rods (BPR's). They consist basically of rods having nearly
the same geometry as a fuel pin and they are loaded with a poison
(often borated Pyrex) with concentrations of boron such that it may
be completely burned in one reactor cycle life. The rods are
placed inside a thimble so that they can be easily removed at the
end of the first cycle, leaving the fuel assembly unpoisoned for
the following cycles. This type of poison does not represent a
very difficult problem for core calculations. Although the
calculations normally performed to determine the effective cross

224
sections as they change through the cycle and the isotopical burnup
evolution of the LBP involve some trial-and-error adjustments, they
can be treated rather easily. However, they present some physical
problems:
a). The absorption cross section is not close to zero at EQL
because of all the structural material associated with the BFR and
because some poison atoms are always left. This causes a residual
negative reactivity which is undesirable at EOL, since it causes
the cycle to be shortened.
b). When the BPR's are removed for later cycles, the channel
that housed them is filled with moderator and mildly absorbing
flow-control devices, forming a sort of flux trap, and causing the
neighboring pins to have a higher-than-normal power. In other
words, the assembly is left with a power distortion.
c). The array spaces used for BPR's can never be used for
fuel pins. The poison rods are removed at the end of the first
cycle of core residence, but the guide tubes remain in their
places, and there is no reasonable way of removing them and placing
fuel pins in these locations. Even if such a manipulation were
possible, severe local power peaking problems would appear because
of the drastic differences of burnup status among neighboring pins.
LBP's therefore represent a loss of energy that could have been
produced by the fuel pins that were displaced.
The second way of implementing the burnable poisons is to load
the fuel pellets with Gadolinium. This technique is used in BWR's

225
but has not been used yet in PWR's as far as the author knows,
although some tests are being conducted.
The reactivity worth of gadolinium-loaded fuel pins and its
evolution through core life are much harder to calculate accurately
than the BPR's and they also require increases in fuel fabrication
cost, because more types of fuel are needed and more complex and
careful control is required during manufacture. On the other hand,
Gadolinium-loaded pins present some substantial potential
advantages such as:
a). Complete burning of the poison by EOL is more likely
because of the poison being mixed with the fuel itself.
b). No residual absorption due to extra structural materials
is present.
c). No fuel loading is lost, since Gadolinium is imbedded in
the fuel pellets.
d). No power shape distortions due to water holes that are
present in following cycles.
Gadolinium-loaded fuel pins seem to have a promising future in
FWR applications. A review all the functions accomplished by the
burnable poisons follows.
a). Since burnable poisons are added to the most reactive
fuel elements or to the regions where power would peak, their
presence causes a strong reduction of the core reactivity. The
effect of the poisons is maximum at BOL, since they have not been
burned yet. This effectively reduces the amount of chemical shim
needed for criticality of the core, which is an interesting effect,

226
since the high soluble boron concentration needed at BOL may cause
a positive MIC, with intrinsic safety implications.
b). Since they are placed in the fuel assemblies having the
highest power levels, the burnable poisons reduce the power peaks
between assemblies and thus allow the whole-core power level to be
increased while preserving safe heat transfer conditions in the
most restrictive assemblies. The end result is a higher power
output by the reactor.
c). For assemblies having very assymetrical boundaries (e.g.
a fresh assembly having the baffle and reflector on one side and
another fresh assembly on the opposite side), burnable poisons may
be placed in such a way that they reduce the power tilt that would
appear within the assembly itself. This avoids severe power
peaking problems in subsequent cycles, when the assembly does not
have burnable poisons anymore. An example of such use of LBP's is
shown in Figure A.2.3 (37).
d). Since their reactivity and burnup are held down during
the whole first cycle, the poison-loaded fresh assemblies show a
higher reactivity at EOL, which allows the core to stretch the
burnup obtainable in each cycle, and with this, the discharge
burnup obtainable from the same initial fuel. This can be seen
rather easily using Strawbridge's rough approximation which states
that for a core having several batches of the same initial
enrichment (i.e. for an equilibrium-cycle core), EOL will be
reached when the core average burnup reaches a certain constant

227
1
1
¡
\
\
\




-

\
/
/
\
t
f
f
t
.
o 0 9
a
o 0
. -
' f
m *

Figure A.2.3. Orientations of Non-symmetric Burnable Poisons.

228
value, determined by the enrichment of the fuel and the in-core
fuel management scheme.
With this assumption, the core-average burnup at EOL can be
calculated (for a three-batch core, for example) as:
CA = CB*P1 + CB*(P1+P2) + CB*3 = CB 2*P1 + P2 + 3
3 3
where CA = Core Average Burnup at EOL.
CB = Cycle Burnup.
PI = Relative power of Batch 1 (in the core for the first
cycle).
P2 = Relative power of Batch 2 (in the core for the second
cycle).
Then, if for a given enrichment and fuel management scheme CA
is fixed, it is possible to obtain CB as
CB = CA_* 3
2*P1 + P2 + 3
Since the discharge burnup for the fuel will be three times
the cycle burnup, it is obvious that the lower PI, the higher the
discharge burnup can be, and P2 has the same effect, to a lesser
degree. These considerations indicate that the burnup of the fuel
should be shifted towards the end of its life as far as it is
possible.
It is important to bear in mind that the assumptions used for
these calculations are not exact, and it is obviously not
reasonable to rely too heavily on the precise numerical results

229
obtained from them. However, the trends shown are close to
reality, and thus it is possible to state that the use of burnable
poisons increases the discharge burnup (and therefore ore
utilization) by reducing the fresh batch's relative power.
A.2.3. Low-Leakage Fuel Management.
Most LWR's operate on a multiple-batch fuel management scheme
because of the increased burnup obtainable (as shown in Section
A.2.1) and the associated possibility of power flattening. It is
also most common in large EWR's to use a 3-batch scheme, where the
fresh batch is loaded at the core periphery, and the once-burned
and twice-burned batches are placed in the inner region of the core
in a "checkerboard" pattern. One such typical disposition is shown
in Figure A.2.4 (26).
This disposition of the "second cycle" batch being closely
mixed with the "third cycle" batch is used to drive the latter,
which in turn is used to avoid power peaking in the former; this
peaking can occur because of the relatively high reactivity of the
"second cycle" batch and its location in a high-worth area in the
core. The "first cycle" high reactivity batch at the core
periphery raises the power in that zone, which is a low worth
region; this same effect helps maintain the batch's power at a
reasonable level. However, Figure A.2.3 is an example which shows
that even in the peripheral "low worth" situation, some "first
cycle" fuel elements may need LBP's in order to prevent their power
from being too high.

230
1

1

1

1
/

1

1

i
/
/
1

1

1

1
/

I

1

1
/
/
1

1



/

1

1

/
/
1
/
1
/
/
/
/
/
/
/
1
/
Batch 1 Batch 2 Batch 3
Figure A.2.4. Batch Distribution in Core

231
It is reasonable that such an in-core fuel management scheme,
which uses the close mixing of different-burnup fuel assemblies and
the different worth of the regions in the core, obtains a flat or
near-flat power distribution without much complication, and avoids
high power peaking factors. However, this scheme is rather poor
when one is concerned with neutron economy. The "freshest" batch
lies on the periphery, surrounded by a steel barrel and a
moderately absorbing refelector (depending on the chemical shim),
and its high neutron production is offset by a disproportionate
neutron loss in these adjacent materials. This is no great problem
at BOL because there is excess reactivity just about everywhere,
but unproductive and disproportionate neutron losses at EOL cause
the reactor to go sub-critical earlier than otherwise possible.
A new "low-leakage" fuel management scheme has been developed,
where neutron leakage is minimized at all times, thus trying to
stretch cycle life by simple neutron economy. In this low-leakage
scheme; the fresh batch is placed in the inner region of the core,
"checkerboarded" with the "third cycle" batch, while the "second
cycle" batch is put on the peripheral region. Consequently, this
fuel management scheme is called "in-out-in" to describe the
succeeding positions of the batch through its life, as opposed to
"out-in-in" which describes the placement of the fuel in the
currently used scheme.
With this in-out-in scheme, the "third cycle" batch is driven
by the fresh batch. This allows the oldest batch to be driven to
higher burnup at the same time that the fresh batch is more

232
effectively used by the close vicinity of the highly burned
assemblies, and its neutrons are not lost in the baffle or
reflector, but are utilized in an old batch for further fissions or
conversion. The "second cycle" batch being in the core periphery
yields a smaller neutron leakage than a "first cycle" would
(particularly important at EOL) because its pins have already been
burned for two complete cycles by the time EOL is reached.
Obviously, this "second cycle" batch does not generally need LBP's
at all.
This in-out-in scheme is better than the out-in-in scheme as
far as neutron economy is concerned, and is able to improve burnup
by about 3% (2). However, it presents the problem of high power
peakings. This problem is aggravated by the fact that power
peaking does not improve as the cycle advances (as was the case
with the out-in-in scheme) but it gets worse. This is due to the
fact that the fresh batch burns its poisons (which are obviously
necessary) and appears relatively clean and low-burned at EOL while
facing a crisp-burned "third cycle" batch. In addition to being an
operational problem, this is also a calculational problem, because
the whole cycle burnup must be followed on a pin-by-pin basis
before the power peaking constraints may be obtined, since they
appear at EOL.
To the best knowledge of the author, there is yet no
commercial power reactor using this kind of fuel management, but it
is being given serious study and consideration. It appears as if
Gadolinium-loaded pins might be of considerable help in reducing

233
the power peaking problem associated with the in-out-in scheme.
However, this would entail, as mentioned in Section A.2.2, yet a
further calculational, technological and economic problem.
A.2.4. Alterations of Euel-rto-Water. Ratio.
Changing the relative amounts of water and fuel in the core
can have a significant impact in the nuclear characteristics of the
reactor. When more water is present, the neutron energy spectrum
is softened and the multiplication factor is raised (since PWK's
work with an undermoderated configuration for safety reasons). On
the other hand, the epithermal neutron population is then
decreased, and captures in U-238 are reduced, thus reducing the
conversion ratio and limiting the amount of plutonium contributing
to stretch the cycle length. The reverse effects appear when the
water proportion is decreased.
The two effects described are competing from the point of view
of cycle length; however, considering the present design of PWR's,
burnup can be increased by going to a more moderated configuration.
There are four possible ways of changing the fuel-to-water ratio:
a). Change the pin diameter.
b). Change the lattice pitch.
c). Change the effective fuel density.
Changing the pin diameter has the advantage of needing a very
limited redesign, since it is possible to keep the same basic fuel
assembly design and backfit the change into operating reactors. As
a matter of fact, Westinghouse Corporation has a new optimized pin
of reduced radius that will be slowly introduced into the market

234
and will eventually be installed in reactors that are now using the
larger pin design assemblies.
Changing the lattice pitch avoids the problems associated with
heat transfer, but makes backfit into current LWR's much more
difficult since it could call for major redesign of the reactor
core unless the pitch variation were very small.
Changing the fuel density can be done in several ways:
reducing the fuel density and filling the "empty" space with some
low cross section material; or just reducing the physical fuel
density; or using annular fuel pellets. In any case, the fissile
inventory can be maintained by increasing the 0-235 enrichment.
Combustion Engineering did a study (3) where fuel density was
changed, while keeping the total U-235 inventory constant. Other
alterations were also examined, both separately and in combination
with the fuel density alterations. If the fuel density is kept at
the present value, ore savings of about 6% can be achieved by going
to a larger pitch. Larger savings can be theoretically achieved
(up to about 10%) by reducing the fuel density to about 60% of the
present value while keeping the same pitch, but this would drive
the fuel to burnups far beyond the acceptable limits. However, a
combined variation in which the fuel density is reduced to about
80% of nominal, and the lattice pitch is increased could allow the
ore utilization to be improved by about 8% while keeping the
exposures within acceptable limits. From the core cycle point of
view, this would correspond to a rather high burnup combined with a

235
Enrichment W/o
Figure A.2.5. Discharge Burnup vs. Enrichment at Several Fuel Densities.

236
Ore Utilization
Enrichment Wo
Figure A.2.6. Ore Utilization vs. Enrichment at Several Fuel Densities.

237
long cycle length. Figures A.2.5 and A.2.6 show the results of the
study performed by Combustion Engineering.
A.2.5. Low Power Density.
Reducing the power density of the core is another means of
improving burnup. It also has the very desirable side effect of
greatly increasing the safety, relieving the general operating
conditions and hence reducing the problems of operating the
reactor.
Reducing the power density of the reactor does not necessarily
imply a reduction of the coolant temperature which would result in
a reduced thermodynamic plant efficiency. In the reduced power
density cores, the coolant is kept at the standard operating
temperature by altering the flow conditions, and thus, the plant
efficiency remains unaltered. The increase in safety of the
reactor is due to the reduced surface heat flux (which makes
reaching critical heat transfer conditions more unlikely) and to
the reduced heat storage and temperature of the fuel.
There are two main factors contributing to the burnup
extension of a low power density core: The reduced Doppler effect
and the reduced Xenon concentration. This modification can
contribute significantly to the relief of several thermalhydraulic
and operational problems, while presenting no new technological
obstacles. However, it has the disadvantage of not being
retrofittable (unless current plants were to under-use much of
their balance-of-plant equipment) and of requiring a higher fuel
inventory per installed power. On the other hand, it presents a

238
great improvement in plant operational flexibility, and may allow
for longer core cycles, reduced outage time and reduced personnel
radiation exposure, which are very desirable features.
Westinghouse Corporation conducted research on this subject
(2) for a reactor whose power density was reduced to 75% of the
current standard value. A possible ore savings of 3% were
obtained, although the extra costs associated with the rest of the
fuel cycle placed the option at the same fuel cycle cost level as
the standard reactors.
The author performed some calculations for reactors in which
power densities were reduced to 50, 33 and 25% of the standard
values and obtained burnup increases of 3 to 4% with respect to the
standard reactor, depending on the power density chosen. Allowing
the pitch to vary and going to better-moderated configurations
increased the burnup improvements to levels between 5.5 and 7%
above the standard core, depending again on the power density
chosen.
It appears that most of the burnup improvement achievable is
obtained when cutting the power from 100% to 50%; little
improvement is obtained for further reductions to 33 or 25% power
density levels. Table A.2.1 shows the discharge burnups obtained
for the various power levels and their percent increase over the
standard core burnup.

239
Table A.2.1. Discharge Burnups of Low power Density Cores.
Power Level
Discharge Burnup (MWd/MTU)
% burnup increase
%
Std. Pitch
Optim. Pitch
Std. Ptch. Opt. Ptch.
100
36183
0.0
50
37296
38134
3.1 5.4
33
37359
38397
3.3 6.1
25
37634
38541
4.0 6.5
A.2.6. Flattening-Axial Power Distribution.
If all the pellets in a FWR had the same composition at a
given point in time, and no special absorbing materials were
present, the reactor power distribution will approximate a cosine
in the axial direction, which means that each pin would be burning
much faster around the reactor mid-plane than towards the top or
the bottom. Since the reactor must be designed for the most
unfavorable spot to work safely, this means that most of the core
is under-employed with regard to power generation and fuel
utilization. It is, therefore, highly desirable to alter the
"natural" axial cosine shape of the reactor power distribution and
make it flatter, so that power peaking is reduced and fuel average
discharge burnup is increased by a more uniform burnup.
Such capability is available in present reactors, although it
might be possible to develop improvements. One available axial
power flattening method currently in use employs partial-length

240
neutron absorbing rods. They are distributed throughout the core
cross section and then normally placed at about mid-height in the
axial direction in order to reduce, as far as possible, the power
peak that normally appears at the core midplane. The main problem
associated with the partial-length control rods is that they take
the space of fuel pins or full-length control rods, and it is just
not possible to deploy many of them; this causes the axial power
flattening to be imperfect. Figure A.2.7 shows the effect of a
partial-length rod on the axial power distribution of the Oconee-1
reactor (38).
It is hard to devise an element that could do a similar job
without occupying fuel pin or full-length control rod spaces
because partial length rods need to be movable. This is true for
several reasons: one is that they may be used to help control
reactivity transients, like xenon oscillations; another is the fact
that if an absorbing material is placed in a fuel assembly in a
non-movable manner, it causes a reduction of the core life because
it absorbs some of the highest worth neutrons in the reactor, which
is not desirable at EOL.
A.2.7. Increasing Enrichment.
If the enrichment of the reload batch of a reactor is
increased, it seems obvious that the core should be able to achieve
higher burnup, causing the discharge burnup to be also higher. If
the fuel management scheme is kept constant and the same batch size
is used, achieving a higher cycle burnup would entail longer
cycles, which is a desirable trend from the economic point of view.

241
Normalized
Core
Position
Figure A.2.7. Power Shaping Effect of Partial Length Rod

242
However, a higher enriched fuel requires larger quantities of ore
for each core reload. Although for very low enrichments an
increased enrichment also causes a better ore utilization, a
maximum is soon reached after which further enrichment entails more
ore consumption for the same energy produced, as can be observed in
Figures A.2.2 and A.2.8. In addition to this, the combination of
higher enriched feed fuel and further burned old batches will tend
to cause more severe power peaking problems, making burnable
poisons probably necessary.
Combustion Engineering research on this topic (3) shows that
with a 3-batch out-in-in fuel management scheme (which is the most
commonly used in large PWR's), ore utilization could be
theoretically improved by about 6% by increasing fuel enrichment to
about 4.8% U-235. However, this would require the fuel to be
driven to over 50 GWd/MTU, which is beyond the present
technological limits.
According to the limits set in Section A.l, ore utilization
can be improved by about 3 to 4% by going to fuel enrichments of
about 3.8% U-235. Figure A.2.8 illustrates the ore utilization
improvement achievable for different values of fuel enrichment and
their associated discharge burnups assuming a 3-batch out-in-in
fuel management scheme.
A.2.8. End of Cycle Coastdown.
Power coastdown at EOC is an operation which is often used by
utilities to meet shutdown schedules rather than for burnup or
resource utilization improvement. There are two ways of performing

243
Ore utilization
3 4 5 6
Fuel Enrichment W/o
Figure A.2.8. Ore Utilization vs. Fuel Enrichment and Burnup.

244
the EOC coastdown: reducing the coolant temperature while keeping
the full thermal power of the reactor, or reducing both the coolant
temperature and the core power. The first system allows for
increased burnup, but practically no ore savings is obtained at all
as shown by Westinghouse's research (2) on this subject. The
second system may allow a 7% savings in ore cost and a 4% savings
in fuel cycle cost. However, the total generation cost may not be
improved at all, or at best may be improved to some limited extent;
this is a consequence of the extra costs of the replacement power
for all the period during which the plant is delivering a reduced
power level.
Coastdown must be carefully planned, because driving the core
to a limit will shorten the length of the next cycle. EOC power
coastdown appears as a feasible operational adjustment, but its
overall economy is not quite clear for a systematic use.
A.2.9. Other Possibilities of Minor Importance.
The options discussed in the previous sections appear to be
the best present candidates if a policy of increasing ore
utilization or extending burnup is to be seriously undertaken in
the near future. There are, however, many other ideas which might
in some way help increase burnup, but which either represent a very
small potential effect or their implementation would be difficult
and/or expensive. Some of these less possible ideas are discussed
below.
a). Zircaloy baffle. Present reactors use a steel baffle,
which presents a considerable absorption cross section. Zircaloy

245
could perform the same functions, presenting better nuclear
characteristics. Backfit of this idea would be really expensive,
while potential ore savings are very small.
b). Forcing a "migrating" axial power peak. By means of
partial length rods, it might be possible to force the axial power
peak to travel along the reactor axis, forcing a more leveled
burnup towards the extremes of the fuel. Although some extension
in burnup might be expected, it is possible that rather strange
power distributions would appear. The concept would require a
really sophisticated system of partial length rods.
c). Reversible-halves fuel assemblies. Since the axial power
peak is towards the center of the reactor while the top and bottom
ends have a lower power, one way of obtaining a more uniform
discharge burnup along the whole axis length would be to use fuel
assemblies cut in two halves. At the third cycle, for example, the
two halves could be turned upside down, causing the center zone to
be at the extremes, and vice versa. During the third cycle, the
assemblies would receive the maximum burn at a region that had the
lowest exposure previously. Of course, this would increase the
fuel fabrication cost in a significant proportion. It would also
increase the refueling outage time because of the higher complexity
of operations required, and probably most important of all, it
would drastically worsen the power peaking problem.
d). Another way of flattening the power distribution could be
through axial enrichment zoning of the fuel. This would reduce the

246
axial power peaking factor, but it would result in an increase of
fuel fabrication cost as well.
e). Since it is apparent that for once-through fuel cycles a
better-moderated configuration entails ore savings, this effect
could be achieved with the use of hollow fuel pins, such that the
coolant could be circulated both around the pin and inside the
hole. This would sharply improve moderation, and it would keep the
fuel colder since the thickness of fuel lumps would be reduced and
the area of fuel facing the coolant would be increased for the same
fuel loading, Extra reactivity would be available from reduced
Doppler effect due to the lower temperature of the fuel, and from
the increased water-to-fuel ratio and better-moderating geometry.
However, this idea would require major redesign of the reactor, and
it would significantly increase fuel fabrication costs.
f). Spectral-shift controlled reactor. A FWR has excess
reactivity all through its life, except at the EQL, where all
poisons are removed in order to extend the cycle as long as
possible. At the EOL much of the core power is generated by the
plutonium isotopes that have been bred during the cycle from
neutron captures in -238. If the amount of plutonium available at
the EQL could be increased, a longer cycle life could be expected.
One way of increasing the conversion ratio of the reactor is to
harden the neutron energy spectrum by reducing the moderating
capabilities of the coolant. A harder spectrum reduces the core
multiplication factor, but this is no problem until the EQL, when
multiplication has to be as high as possible.

247
One way of achieving this is to use a mixture of light and
heavy water as moderator. Heavy water has a lower moderating power
than light water, and thus the neutrons are less moderated,
suffering more resonance absorptions in the fertile isotope.
Towards EOL, heavy water can be replaced by light water in order to
obtain the maximum moderating power. This method was studied for a
mixed uranium-Thorium fueled reactor by Combustion Engineering (3)
and they predicted large potential ore savings with this method.
The obvious problem of this method is the cost associated with
the production and management of the mixture of light and heavy
water, which would place a heavy burden in the fuel cycle cost, in
addition to all the re-design needed in order to accomodate these
changes into FWR plants.

APPENDIX B1
BBT CODE DESCRIPTION
The BRT (9) code is a revised version of the THERMOS (20)code,
with several, improvements which result in a more exact handling of
cross sections and improvement of numerical techniques.
BRT solves the integral form of the transport equation for
either a slab or cylindrical geometry, with a maximum of 30
specified space points. It accepts up to 8 different material
mixtures, and the neutron cross sections are evaluated from a 30
energy groups library ranging from energy 0 to 0.683 eV The code
computes the neutron densities, fluxes and currents as a function
of ¡space point and energy group. It also calculates
energy-dependent and energy-averaged material mixture cross
sections and fluxes, and mixture dependent and total region
averaged macroscopic cross sections for each isotope considered in
the region under consideration. The code also prints and punches
on cards cell-homogenized macroscopic cross sections which may be
used as material descriptors for later problems, utilizing the
whole cell as a simple material region.
The code is structured in a way that it may be used either for
the detailed analysis of the spatial and energy dependent
characteristics of the region being considered (or of any of its
component material mixtures) or as a region homogenizer, in order
248

249
to obtain thermal group, cell-homogenized properties, for use as a
mixture description in a later case of BRT itself, or a multigroup
diffusion theory code. These two last aspects were used in the
present study for the HPWRP preliminary calculations described in
Chapter II.
Some of the improved calculational features of BRT as compared
to THERMOS are the following:
a). BRT includes a transport kernel modification for the
treatment of void regions.
b). The code has been modified to accept generalized boundary
conditions.
c). BRT may account for a perpendicular leakage factor
indicated by an input buckling value or by perpendicular
dimensions.
d). As explained above, the code can print and punch cell
smeared data for use in later cases.
e). BRT can optionally apply a linear anisotropic scattering
correction approximation.
The main characteristics of the input specified by the user
are the following:
a). Specification of number of space and energy points, and
total number of isotopes used in the calculation. The isotopes can
be from the BRT library or can be specified by the user. If they
are specified by the user, it is necessary to input their cross
sections, in addition to the concentrations for each material
mixture

250
b). Option for the anisotropic scattering correction, for an
external slowing down source, and for transversal buckling. The
latter two require specification of the slowing down source and the
buckling or the equivalent transversal dimensions respectively.
c). Description of the problem geometry: number of material
regions, number of mesh spaces assigned to each of them, thickness
of each region, type of geometry.
d!). Specification of the type of boundary conditions and
option for the cell-smeared calculations.
e). Specification of maximum iteration parameters and
convergence criteria.
f). Edit options specification.

APPENDIX B2
PHRDG CODE DESCRIPTION
PHROG (8) is a computer code to generate fast neutron spectra
and fast multigroup cross sections for use in diffusion or
transport theory codes in reactor design. PHROG uses a B1 or a Pi
approximation to the energy dependent neutron transport equation to
obtain energy dependent fluxes and currents.
The isotopic cross section data are based on an equal
lethargy, 68-group cross section library with energies ranging from
0.414 eV to 10 MeV. The code is an evolution of the GAM-1 (19)
code with a number of improvements, such as temperature dependence
of fine group resolved and unresolved resonance cross section data,
Dancoff correction factor calculation, etc.
Some other significant characteristics of the code are the
following:
a). Integration of resolved resonances may include the
effects of up to three admixed scatterer isotopes, whose atomic
weight and scattering importance (depending on their relative
isotopic abundance and their scattering cross section) are input by
the user.
b). The flux spectrum weighting functions can be generated by
the code for the particular case being studied, or may be supplied
by the user from a previous calculation.
251

252
c). The code can perform a blackness calculation for the
generation of diffusion theory constants for thin, highly absorbing
regions.
d). The code provides options for user input of cross
sectional and/or source data. It is also capable of punching the
group-collapsed data in card formats usable by neutron diffusion
theory codes such as MONA (10) or CORA (35).
The most significant data input to the code by the user are
the following:
a). Indication of the type of calculation desired: PI, Bl,
cell problem with supplied flux and current spectra, or blackness
calculation.
b). Type of source required: Input by the user, or choosing
among a selection of source spectra available from within the code.
c). Definition of number of broad groups desired and number
of isotopes in the calculation.
d). Definition of the problem geometry. It can either be a
slab, cylindrical or spherical geometry.
e). Definition of the isotopes used, their number densities,
and the properties of the related scatterer isotopes.
f). Leakage factors and buckling.
g). Isotopic information for resonance calculations.

APPENDIX B3
MONA CODE DESCRIPTION
The MONA (10) code is a multigroup, one-dimensional, diffusion
theory neutronics analysis code, developed as an enhancement of the
CORA (33) code. It is suitable for studies where space and energy
variables can not be separated, but angular flux dependence is not
of great concern, and thus there is no need for the more complex
and time consuming several-dimensional neutron transport theory
codes. MONA accepts a slab, cylindrical or spherical geometry.
The energy range can be cut into up to 50 energy groups, with full
upscattering capabilities. Up to 50 material regions can be
specified, and a variety of boundary conditions are available and
can be defined separately for each energy group.
The description of the materials in each region can be done by
means of macroscopic cross sections, or with microscopic cross
sections and isotopic number densities. The source term can be
calculated by the code or it can be specified by the user. The
input format conforms to that of the cards punched by PHROG-
The code obtains the energy- and space-dependent fluxes and
adjoint fluxes, performs a generation factor calculation and is
able to make criticality searches based on a variety of possible
parameters, such as poison concentration, region boundary position,
nuclide concentration. perpendicular buckling, etc. The

254
calculational technique includes several methods of acceleration
and extrapolation, such as Chebyshev polynomials for the fission
source iterations.
The code allows great flexibility to the user. Some of the
main user input parameters are the following:
a). Specification of number of energy groups, number of
regions, type of geometry, type of boundary conditions.
b). Selection of type of calculation: forward and/or adjoint
fluxes, search, request for energy group coalescence, etc.
c). Definition of the type and input of data (cross sections,
fluxes, neutron source, number densities, search materials, etc)
and format of data output.
d). Specification of iteration data convergence criteria,
and normalization values.
e). Geometrical definition of the problem.

APPENDIX B4
LEOPARD CODE DESCRIPTION
LEOPARD (11) is a fuel cell code intended to calculate neutron
flux spectra and isotope cross sections as they change with fuel
depletion. It is based on modified versions of the MUFT (12) and
SOFOCATE (13) codes. The code performs spectrum calculations and
fuel burnup calculations, assuming the fuel is imbedded in an
infinite array of elementary cells. The code provides the
possibility of specifying the presence of materials that cause
alterations of neutron spectrum, but which are not related to the
elementary cell itself. Such is the case of instrument thimbles,
water holes, etc.
Although the spectrum calculations take into account the
geometry of the fuel cell, the burnup routines are entirely
non-dimensional and fuel depletion is considered uniform throughout
the pellet region. The code accounts for the thermal expansions of
the different materials in the cell, according to the temperatures
supplied by the user- Input to the code is reduced and relatively
simple. However. LEOPARD has the following drawbacks:
a) The power level of the fuel can not be changed from one
timestep to the next, and therefore, the fuel is burned at a
constant power level which results in a false equivalence of time
and burnup level for each particular case.
255

256
b)- The calculation of resonance self-shielding is allowed
only for U-238. which is particularly inaccurate for high levels of
burnup where other isotopes (such as Pu-239 or Pu-240) have
significant concentrations.
On the other hand, the code is extremely flexible, allows the
variation of poison levels during life, and runs in moderate times:
A total fuel burnup calculation can be run in just a few minutes of
computer time. The most significant data input by the user are the
following:
a). Geometrical data describing the cell and the "extra"
region.
b). Composition specification for each material zone.
c). Temperatures for each region, densities, pressure, and
perpendicular buckling.
d). Flags indicating units of input data, type of cell
geometry (square or hexagonal), number of broad groups to be used
for the output cross section tables, option for buckling search,
etc.
The code prints a report for each timestep specified during
the life of the core- The main items listed on each report are the
following:
a). Cell-homogenized number densities of each isotope at the
beginning of the timestep. and conversion achieved during the
timestep.
b). Microscopic cross sections of all the isotopes present in
the cell, and macroscopic cross sections of the smeared cell. All

257
cross section tables are produced for the number of energy groups
specified by the user.
c). Fast and thermal fluxes, average neutron velocity,
group-wise and total multiplication factors, etc.

APPENDIX B5
EPRI-CELL CODE DESCRIPTION
The EPRI-CELL (19) code computes the space, energy and burnup
dependence of the neutron spectrum and isotopic cross sections
within cylindrical cells of light water reactor fuel rods. Its
primary output consists of broad-group, microscopic, exposure
dependent cross sections for subsequent use in multidimensional
neutron diffusion theory depletion analysis.
The code is an answer to the need for a well standardized and
proven code able to supply the sets of burnup and/or nuclide
concentration-dependent cross sections for use in the large
multigroup, multidimensional, diffusion theory codes used in the
nuclear industry for fuel management calculations and core
following- This requires the following characteristics of the
code:
a). The code must be based on well-proven previous
computational schemes and must take into account all the parameters
that have been shown to have a significant effect on the fuel
characteristics as it undergoes the fission process.
b). The code must be flexible enough to allow representation
of all the types of fuel situations that can be normally
encountered when performing LWR calculations, yet it has to be as
simple as possible to the user- This is solved by providing two
258

259
input options, one of which is called the "engineering" input
option, and defaults most of the commonly used data, while the
other option, called the "general" input option, requires full
specification of the problem.
c). The code must provide an output which is capable of being
easily processed for input to the core model codes.
The result is a code with an extremely simplified input, but
nonetheless, with a large amount of options available. It is based
on the very well-known codes THERMOS (21), GAM (20) and
CINDER (22). It simulates the burning of a LWR fuel pin in which
the geometry and compositions are specified by the user, as well as
the circumstancial data concerning the burnup process, such as
power level variation through time, soluble boron concentration,
etc. The output includes a printed listing of microscopic isotope
cross sections, macroscopic cell-averaged cross sections, isotopic
number densities, multiplication factor, etc. and a magnetic
device (tape or disk) data set with the same type of data in an
adequate format to be transformed into cross section tables for the
core model codes.
Some of the main characteristics of EERI-CELL are the
following:
a). The code includes the effects of Dancoff factors,
correcting also for the fuel rods that are not surrounded by other
fuel rods but that are adjacent to interassembly water gaps, to
water holes, instrument thimbles, etc.

260
b). It also includes the spectral effects of the structural
materials that are present in the reactor but that can not be
assigned to the geometry of the elementary fuel cell.
c). The code accounts for the resonance energy shielding of
all the nuclides present in the cell.
d). The thermal energy cutoff used is 1.855 eV, which is more
adequate than other codes' lower values.
e). Some calculational and convergence techniques of the
"parent" codes have been modified to improve their accuracy and/or
to optimize convergence.
f). The isotopic neutron cross sections are computed from the
microgroup libraries of GAM and THERMOS, with 62 and 35 energy
groups respectively.
g). The depletion calculations are performed for each mesh
space inside the fuel pellet, and isotope accounting is kept
separate for each mesh region. Depletion is done with a four group
spectrum and cross section data set.
The main items input by the user in the general input option
are the following:
a). Definition of the geometry of the problem: Number of
material zones, thickness of each zone and number of mesh points
assigned to each zone.
b). Specification of the nuclides present in the cell
(maximum of 25) and their number densities for each material
composition.

261
c). Volumetric fraction and density fraction of each
composition in each material zone.
d). Number of timesteps desired for the total calculation;
duration of each timestep, power level associated with each
timestep and soluble boron concentration.
e). Option flags determining buckling search, correction for
resonance overlap, presence of heavy scatterer, boundary
conditions, type of isotopic library data, number of broad groups
used for collapsing and editing (minimum of 2, maximum of 5), type
of data table output, etc.
f). Convergence criteria, temperatures for each material
zone, type of fission spectrum, optional group-dependent buckling
specification, information about nuclides for which resonance
calculations are requested, etc.
The input for the engineering option assumes default values
for many of the variables used by the code.The input is reduced to
a minimum while keeping a fair flexibility, and still allowing
representation of most of the classical problems in fuel burnup
calculations. The main items input in the engineering option are
the following:
a). Weight fractions of fissile materials in the fuel pellet
and density fraction of the pellet.
b). Volume fractions of structural materials in the buffer
extra region surrounding the fuel cell.
c). Composition of the cladding material.

262
d). Number of total timesteps used in the calculation,
soluble boron levels and power levels.
e). Geometrical data: radii of the zones of the fuel cell,
pellet and clad inner and outer diameter, pitch, and extra region
volumetric fraction.
f). Option flags concerning units used in input, buckling
search, number of collapsed groups, type of edit and output data
tables, etc.
g). Temperatures, spectrum specification and convergence
criteria

APPENDIX B6
NUPUNCHER CODE DESCRIPTION
NUPUNCHER (23) is a coupling code whose function is to prepare
the cross section tables output by EPRI-CELL to the magnetic device
into a format compatible with the HARMONY (24) tables needed for
PDQ-7 (17) calculations. This results in a substantial
simplification of the most cumbersome part of the input to PDQ-7.
Although NUPUNCHER's task is apparently simple, it contains a
considerable amount of options due to the flexibility built into
the HARMONY tables. The code generates the required macroscopic
and microscopic cross section tables using the "burnup" variable as
basic independent mask, but allows the user to specify any other
number of masks for control of any particular set of microscopic
cross sections. For example, it is common practice to assign the
absorption cross section of Pu-240 to a table controlled by the
isotope's own number density rather than assigning it to the
general table, controlled by the burnup level of the fuel.
NUPUNCHER accommodates any of these types of changes and generates
the HARMONY tables accordingly. The main items input to the code
by the user are the following:
a). Identification of the data set containing the EC-DATA
file generated by EPRI-CELL and definition of the material names to
be used in the HARMONY tables.
263

264
b). Table identification number, interpolation information,
number of energy groups and EPRI-CELL timestep number to be used
for non-interpolating data.
c). Table number for macroscopic tables, type of data to be
used in the macroscopic tables, first mask number and first
interpolating table number.
d). Definition of interpolating tables that must use a mask
other than "burnup"; definition of the mask.
e). Option to punch PDQ-7 depletion chains.

APPENDIX B7
FDQ-7 CODE DESCRIPTION
PEQ-7 (17) is a multigroup. multidimensional diffusion theory
code, used for the modelling and following of reactor cores. It is
one of the most powerful calculational tools now in use in the
nuclear industry. It has an extreme flexibility, and almost any
neutronic situation where diffusion theory can be applied can be
adequately represented with the code-
This great power and flexibility entails a long and
complicated input specification, to the point that several codes
have been developed (such as NUPUNCHER (23). described in APPENDIX
B6- and CHIMP (36)) which assist the user by taking cross section
data from codes like EERI-CELL (19) and LEOPARD (11) and translate
them into a format suitable for input to PDQ-7- These auxiliary
codes still leave a significant task to the user, but they can take
care of the most voluminous and cumbersome part of the input deck.
PDQ-7 has no cross section library of its own. Cross sections
for all the isotopes must be provided from an external source.
However. PDQ-7 has a cross section handling routine. HARMONY (24),
which allows the code to evaluate the cross sections of all
isotopes as a function of up to three independent variables, and
make interpolations in order to provide the most accurate estimate
of the cross sections for each situation of the core. The current
265

266
practice is to specify most of the cross sections as a function of
the burnup level of the fuel. It is also possible, for example, to
specify them as a function of both the burnup level and the soluble
boron concentration; the code will make the necessary
multi-dimensional interpolation for the particular situation of the
isotope at the moment of cross section evaluation. Note that the
cross sections are evaluated for each isotope and for each material
zone in the core, so it is possible to have different cross
sections for a given isotope at a given time, if the isotope is
present in two different regions. In order to make this possible,
the burnup level is computed and accounted for as a separate item
for each diffusion mesh space. This is an indication of the
extreme sophistication of the cross section handling procedures,
but it suggests as well the fact that PDQ-7 requires enormous
amounts of computer memory and time.
The code is also very generalized and flexible in the geometry
specification. It accepts one to three dimensions, with the mesh
spacing of either rectangular or hexagonal type in the X and Y
directions. The mesh and material zone specifications are done in
such a way that it is simple to build large structures with
repetitive patterns, as is the case with most reactors.
PDQ-7 requires also the specification of the radioactive and
transmutation chains with all the decay times involved in them.
This represents a further task for the user (although it is usually
taken care of by the auxilliary codes mentioned above) but it

267
enables the user to choose as simple or as complex of an isotope
accounting procedure as may be needed for each particular job.
Another involved task left to the user is the specification of
the groups of chains that must be assigned to each material region,
the interpolating tables that correspond to each set of isotopes,
and the respective masks (independent variables) that are
controlling each of the interpolation tables. PDQ-7 can also admit
"burnable" and "non-burnable" compositions, which obviously receive
a different treatment. Non-burnable materials can also have
interpolating tables, but their cross sections are stored in the
macroscopic form, since there is no need for accounting of isotopic
number densities.
Finally, the code requires a set of flag and options
specifications, indicating the type of problem to be solved, the
type of boundary conditions, the type and arrangement of the output
edits, etc. It is also necessary to specify the timesteps desired
in the case of a burnup study, as well as the power level, which
can be specified for each timestep. It is obvious that a complete
description of all the capabilities of the code escapes the context
of this work.
The items present in a normal output for a burnup study are
listed below. It is important to keep in mind that it is hard to
define a "normal" output from PDQ-7, since most of the output is
optional, and is determined by the user.
a). General definition of the problem: type of problem,
geometrical size, figure composition, timestep length, etc.

268
b). Pictures indicating the geometrical arrangement of the
problem, along with the specification of all material compositions.
c). Multiplication factor iterations.
d). Fluxes, power generation, absorption rates,
multiplication factors, macroscopic cross sections, etc. groupwise
and total, for each edit region defined, which may include separate
material regions (v.g. each particular fuel assembly) and/or any
combination of them, which can be used to obtain batch-wise or core
total values.
e). Description of the burnup step. It may be done in one
single calculation, or it may include partial timestep
renormalizations in order to correct for isotopic variations during
the timestep burning time. Isotopic concentrations for each edit
region defined by the user are also listed.
f). Information about the file handling for data transfer
between the timestep calculations.

APPENDIX B8
TEMPRET CODE DESCRIPTION
TEMPRET31) is a thermalhydraulics code developed by Mark
Miller for the prediction of two-dimensional, steady state thermal
conditions in a PWR elementary fuel cell. The code accepts a wide
range of operating characteristics of the fuel, and is able of
treating temperature dependent thermodynamic properties of the fuel
material, the cladding and the coolant. The heat generation rate
can be allowed to vary both in the axial and in the radial
directions. The code includes some special features such as the
possibility of treating either solid or central-voided fuel pins.
Some of the more significant input data to the code are the
following:
a). Geometrical description of the problem.
b). Tabular entry of materials properties.
c). Definition of thermodynamic status of coolant at core
inlet.
d). Definition of heat generation distribution.
The output from the code includes the following items:
a). Bulk coolant thermodynamic conditions at each axial node.
b). Reynolds number, Prandtl number, and other heat transfer
related data for each axial node.
269

270
c). Departure from nucleate boiling (DNB) heat rate and the
ratio for the actual heat rate at each axial node.
d). Coolant quality versus axial position.
e). Clad outer and inner temperature, fuel average and peak
temperature and specific heat content, versus axial position.

APPENDIX B9
CONCEPT-IV CODE DESCRIPTION
CCNCEPT-IV (32) is a code designed for the computation of the
capital cost of an electric generating plant using the steam cycle
for power generation. The code is able to compute time- and
size-dependent costs of the components of the plant based on a
historical data base and an escalation prediction model.
The code breaks the cost of the plant into a set of accounts,
including the reactor plant (for the case of a nuclear unit), the
turbine plant, structures and facilities, engineering and
construction, management services, etc. Each of the different
agcounts is broken down into labor, materials and equipment costs.
A large number of sub-accounts are considered, yielding a highly
detailed cost break-down.
The input needed for the code includes items such as the
following:
a), Plant type and location.
b). Net electrical capacity of the plant.
c)Date of purchase of steam generating unit (NSSS for the
case of a nuclear unit, or equivalent for the case of a
fossil-fired unit).
271

272
d). Date of reception of the construction permit.
e). Date of commercial operation.
f). Interest rate.
The output of the code is extremely flexible, and is basically
defined by the user. It can be reduced to a short one-page summary
of the major accounts, or it can include detailed listings of all
the subaccounts. The code provides also for the possibility of
printing a total cumulative cash flow curve.

APPENDIX BIO
GEM CODE DESCRIPTION
The initials GEM stand for General Economic Model. The
GEM (33) code is devised to calculate the costs of the fuel cycle
of a nuclear electric power plant. The code is able to calculate
the costs for just a batch or a set of batches of fuel, or it may
be used for the computation of the fuel costs through the entire
life of the plant.
The code takes into account all the steps of the fuel cycle,
from mining through spent fuel shipment and storage, including the
costs of extraction, enrichment, etc. and the schedules associated
with them.
The code computes both the cumulative fuel cost and the
batchwise yearly costs. The calculations are done through three
distinct types of economic analysis: Cash Flow, Allocated Costs,
and Yearly Cash Flow. The three models yield identical results,
but they present them in different forms.
The input to the code includes mainly the following items:
a). Uranium prices.
b). Fabrication and Service costs.
c). Feed losses.
d). Plutonium prices.
e). Fuel weights.
273

274
f). Number of enrichments and their magnitudes,
concentration, and other enrichment data.
g). Economic parameters and payment schedules.
h). Escalation information.
The output from the code includes the following items:
a). Batch economic analysis.
b). Yearly batch and case costs.
c). Yearly and cumulative fuel cycle costs.
d). Case cashflow.
e). Allocated cost analysis.
tails

APPENDIX Bll
POWEROO CODE DESCRIPTION
The PCWERCO (34) code is used for computing the total busbar
cost of electric power generation. This includes all the factors
that affect the cost of generation: fuel costs (including shipment
costs, possible credits for reprocessable fissile materials, ore
costs, etc.), capital costs, investment costs, plant life, capacity
factors, tax rates, etc.
The code makes use of three different methods of calculating
the power cost. The payout method is the most fundamental one, and
it requires a trial-and-error approach. The present worth method
avoids this problem by translating all costs through the plant life
to a single point in time. More popular than the present worth
method is the fixed-charge-rate method; although some
approximations are made in its application, which render it less
rigorous and more empirical.
The input stream to POWERCQ includes the following items:
a). Plant Investmebt.
b). Non-fuel working capital.
c). Project life.
d). Design capacity.
e). Depreciable life.
f). Economic parameters.
275

276
g). Cost of initial core.
h). Annual expenditures for uranium, fuel fabrication,
transportation, reprocessing (if any), spent fuel credits (if any)
etc.
The main data output by the code are the following:
a). Power cost tabulation.
b). Payout tabulation.
c). Tax-deductible expenses.
d). Annual income tax calculation.
e). Fixed charge calculations.
f). Constant annual sales income.

APPENDIX Cl
CRIBUR CODE SOURCE LISTING
10 REM *********************************************
20 REM PWR BORON LETDCWN EVALUATION, + K-EFF AND EOC BURNUP.
30 REM *********************************************
40 CLS:CLEAR1000:DEFINTI,J,K
50 REM *********************************************
60 REM DIMENSIONING AND REACTOR CONFIGURATION
70 REM *********************************************
80 PRINT "PROGRAM FOR CRITICALITY, BORON WORTH, BATCH POWER AND"
90 PRINT"CYCLE LIFE EVALUATION OF A MULTI-BATCH PWR":INFUT"DATA
INPUT FROM (K)EYBOARD OR (D)ISK";A$:IFA$="D"THEN2190
100 CLS s INPUT "TOTAL # OF TIMESTEPS AND EFECTIVE # OF TIMESTEPS PER
CYCLE";TS,CT:IFCT>TSTHEN100
110 INPUT "NUMBER OF BORON WORTH EVALUATION POINTS"; BE: IFBEO0THEN140
120 INPUT "CONSTANT BORON WORTH IN %K/100PPM, ORJUST
ENTER" ;BW:BW=BW*1. OE-4: IFBW=0THEN BW=-1.0E-4
130 GOTO210
140 INPUT"# OF FAKE STEPS TO BE KITLED AFTER BORON-WORTH
EVALUATION";CA:IFCA=0THEN16OELSEDLMBK(CA)
150 FORI=lTOCA:INPUT"STEP # TO BE KILLED";BK (I) :NEXTI
160 BE=BE*2:DIMBW(BE)
170 REM *******************************************
277

278
180 REM INPUTTING VECTOR OF T-STEPS FOR B-WORTH EVALUATION
190 PRINT"INPUT THE PAIRS OF TIME-STEPS USED FOR B-WORTH EVALUATION"
200 FORI=lTOBE/2:PRINT"PAIR # "I" ="; :INPUTBW(2*1-1) ,BW(2*I) :NEXTI
210 INPUT"# OF BATCHES IN THE CORE (MAX=5)";NB:IFNB>5 OR NBC1THEN210
220 DIMR(TS,12) ,RC(CT,8) fRR(NB,2) :INPUT"POWER LEVEL OF THE REACTOR
IN MW THERMAL";PL
230 INPUT"FUEL LOADING OF THE CORE IN MTU
(DEFAULT=94.18)";FL:IFFL=0THEN FL=94.18
240 REM ******************************************
250 REM BATCH CONFIGURATION AND OUTER BATCH FLAGGING
260 BN=0
270 FORI=1TOCT:BB=0
280 FORJ=lTONB: PRINT"STEP # THAT FORMS BATCH "J" AT CYCLE STEP
"I;:INPUT RC(If2+J):NEXTJ
290 INPUT"# OF THE STEP FORMING THE PERIPHERAL BATCH IN THIS
CYCLE-STEP";RC(1,8)
300 FORK=3TO7:IFRC(I,8)=RC(I,K)THENBB=K:NEXTK
310 IFBB< > 0THEN3 3 0
320 PRINT"THIS PERIPHERAL BATCH DOES NOT BELONG TO THIS
TIME-STEP!!":GOTO290
330 IFBN=0THEN360
340 IF(K-2)=BNTHEN360
350 PRINT" WARNING 11!PRINT "PERIPHERAL BATCH IN THIS TIME
STEP DOES NOT BELONG TO PERIPHERAL BATCH OF PREVIOUS
STEP. RECHECK.":GCTO290
360 BN=K-2:NEXTI

279
370 REM ******************************************
380 REM LBP'S WORTH AT BOC AND EOC
390 BB=0:BN=0:BP=0:INPUTnENTER # OF BATCH (1 IS THE NEWEST)
CONTAINING BURNABLE POISON. TYPE 'ENTER' IF NONE
DOES." ;;BP: IFBP=0THEN410
400 INPUT"BATCHWISE REACTIVITY WORTH OF POISON RODS (%K/K) AT BOC &
EOC";BB,BN:IFBN>BBTHEN400
410 FORI=1TOCT :RC (I r 2) =0: NEXTI
420 REM *******************************************
430 REM INPUT OF ALL DATA IN 'R' MATRIX
440 CLS: PRINT "INPUT ENDING TIMES (IN HOURS FROM BOL) OF EACH
TIMESTEP"
450 FOPJ=1TOTS:PRINT"ENDING TIME OF TIMESTEP "I; :INPUTR(I,4) :NEXTI
460 CLS:PRINT"INPUT BORON PPM FOR EACH TIMESTEP"
470 FORI=lTCTS:PRINT"BORON PPM OF TIMESTEP "I; :INPUTR(I,6):NEXTI
480 CLS:PRINT"INPUT RELATIVE POWER FOR EACH TIMESTEP"
490 FORI=lTOTS:PRINT"RELPWR OF TIMESTEP "I; :INPUTR(I,2) :NEXTI
500 CLS:PRINT"INPUT K-INF FOR EACH TIMESTEP"
510 FOR I=1TOTS
520 PRINT"TIMESTEP "I" K-INF="?:INPUTR(I,1):IFR(I,1)<3THEN540
530 PRINT"THIS K-INF LOOKS PRETTY WILD 111 l":GOTO520
540 NEXTI
550 REM ***************************************
560 REM CHECKING INPUT
570 REM ***************************************
580 CLS:PRINT"INPUT RELATIVE FLUXES FOR EACH TIMESTEP"

280
590 F0RI=1TOTS:PRINT"FLUX OP TB1ESTEP"I; :INPUTR(I,3) :NEXTI
600 CLS: PRINT "CHECK TIMESTEP ENDING TIMES IF ANY ONE TO BE CHANGED,
INPUT TIMESTEP NUMBER. OTHERWISE JUST PRESS 'ENTER11 ":PRINT" "
610 FORI=1TOTS:PRINT"TSTEP "In ENDS AT nR(I,4)n HR",:NEXTI
620 CH=0:INPUT"ANY CHANGED STEP # ";CH:IFCH=0THEN64G
630 INPUT"NEW VALUE"?R(CH,4):GOTO600
640 CLS:PRIMTnCHECK BORON PPM FOR EACH STEP. ENTER STEP TO BE
CHANGED, OR 0":PRINT" "
650 F0RI=1T0TS:PRINT"TIMESTEP "I" HAS "R(I,6)n PPM",:NEXTI
660 INPUT "ANY CHANGED STEP #";CH:IFCH=0THEN680
670 INPUT "NEW VALUE";R(CH,6) :CH=0:GOTO640
680 CLS:PRINT"CHECK K FOR EACH TIMESTEP":PRINTn
:PRINT"TSTP";TAB(6)"K-INF";TAB(30)nTSTP";TAB(36)"K-INF"
690
FORI=2TOTSSTEP2:PRINTI-1;TAB(5)R(I-1,1);TAB(30)I;TAB(35)R(I,1) :NEXTI
700 CH=0:INPUT"ANY CHANGED STEP";CH:IFCH=0THEN720
710 INPUT"NEW K,";R(CH,1) :GOTO680
720 CLS: PRINT "CHECK RELATIVE PCWER FOR EACH STEP. ENTER STEP TO BE
CHANGED, OR 0"
730 FORI=lTOTS: PRINT "TIMESTEP "I" HAS RELPOWR="R(I,2) :NEXTI
740 INPUT"ANY CHANGED STEP";CH:IFCH=0THEN760
750 INPUT"NEW VALUE";R(CH,2):CH=0:GOTO720
760 CLS:PRINT"CHECK CORE BATCH COMPOSITION IN CYCLE.":PRINT"ENTER
STEP AND COLUMN TO BE CHANGED OR 'ENTER' lIWICE.":A$="########"
770 PRINT" STEP BATCH-1 BATCH-2 BATCH-3 BATCH-4 BATCH-5
OUTER-BATCH"

281
780 FORI^ITOCT: PRINT USING
A$;I?RC(I,3)?RC(I,4);RC(I,5);RC(I,6);RC(I,7);RC(I,8):NEXTI
790 INPUT"ANY CHANGED STEP AND COLUMN";CH,CC:IFCH=0THEN810
800 COCC+2: INPUT "NEW VALUE";RC(CH,CC) :CH=0:GOT0760
810 COO :CLS: PRINT "CHECK BATCH WITH LBP AND LBP'S BATCHWISE WORTH
(%K/K) AT BOC, EOC ": PRINT "BATCH "BP, "BOC %K/K="BB,"EOC %K/K="BN
820 INPUT "NEW VALUES FOR ALL, OR HIT ENTER; CC, CH, A: IFCOOTHEN8 40
830 BP=CC:BB=CH:BN=A:GCT0810
840 CO0:CLS:PRINTnCHECK POWER LEVEL AND FUEL
LOADING":PRINT"POWER="PL"MW THERMAL","FUEL LOADING="FL"MTU"
850 INPUT "NEW VALUES FOR BOTH, OR HIT 'ENTER' ";CC,CH:IFCO0THEN870
860 PL=CC:FL=CH:GOT0840
870 CLS:PRINT"CHECK BORON WORTH EVALUATION POINTS": IFBEO0THEN900
880 PR3NT"NO BORON WORTH EVALUATION POINTS. BORON
WORTH="BW:INPUT"NEW VALUE, OR JUST 'ENTER;CH:IFCH=0THEN980
890 BW=CH:CH=0:GOT0870
900 IFCA=0THEN930ELSEPRINT"STEPS TO BE KILLED AFTER B-WORTH
EVALUATION (IN INCREASING ORDER)
910 FORI=lTOCA:PRINTBK(I),;iNEXTI
920 PRINT" ":PRINT" "
930 PRINT"TIMESTEPS FOR EVALUATION :"
940 F0RI=1T0BE/2:PRINTBW(I*2-1),BW(I*2) :NEXTI
950 INPUT"WISH TO INPUT A NEW SET (1), OR NO (0) ";CH:IFCH=0THEN980
960 FORI=lTOBE/2;PRINT"PAIR #"I:INPUTBW(I*2-1) ,BW(I*2) :NEXTI
970 CH=0:GOT0870
980 REM ****************************************

282
990 REM STORAGE IN DISK
1000 GOSUB2480
1010 REM *****************************************
1020 REM INITIALIZING WORK MATRIX
1030 KB=10:RP=2
1040 FORI=1TOTS:R(I,8)=R(I,2) :R(I,9)=R(I,4) :R(I,7)=R(I,6) :NEXTI
1050 GOSB2380
1060 FORI=1TONB:RR(I,1)=RR(I,0) :RR(I,2)=RR(I,0) tNEXTI
1070 RB=12:OC=R(RC(CT,3) ,4)
1080 FORI=lT0TS:R(If12)=R(I,10):NEXTI
1090 REM BORON WORTH VS. CICLE. INTERPOLATED TO MAKE BORON WORTH AS
A FUNCTION OF PPM
1100 IFBE>0THEN1120
1110 Y=BW:GOT01170
1120 DEF FNMP(Yl,Y2)=(R(Ylr6)+R(Y2,6))/2
1130 DEF FNW0(Y1,Y2) = (R(Y1,1)-R(Y2,1))/(R(Y1,6)-R(Y2,6))
1140 IFBE>2THEN1190
1150 Y=FNWO(BW(l)fBW(2))
1160 FORI=lTOCT:RC(I,1)=Y:NEXTI
1170 WA=Y :WB=0
1180 GOTO1250
1190 XY=0:SX=0:SY=0:XX=0:PRIOT"CALCULATING REGRESION LINE OF BORON
WORM VS. Pm"
1200 FORI=lTOBE/2:X=FNMP(EW(2*I-1),BW(2*I))
1210 Y=FNWO(BW(2*1-1),BW(2*I))
1220 XY=XY+X*Y:SX=SX+X:SY=SY+Y:XX=XX+X*X:NEXTI

283
1230 WB=(XY-(SX*SY/(BE/2)))/(XX-(SX*SX/(BE/2))) :WA=(SY-WB*SX)/(BE/2)
1240 REM ******************************************
1250 REM WORTH (DELTA K/PPM) =WAfWB*PPM
1260 IFCA=0THEN127OELSEGOSUB3170
1270 F0RI=1T0CT:CH=RC(1,3) :RC(Ifl)=WMV*R(CHr6) :NEXT1
1280 REM CALCULATING BORON-FREE K-INF FOR CORRELATION WITH BU.
1290 F0RI=1T0TS:R(I-1,11)=R(I,1)-(WA*R(I,6))-(WB*(R(I,6)[2)/2):NEXTI
1300
R(TS,ll)=R(TS-l,ll)+(R(TS,10)-R(TS-l,10))*(R(TS-lfll)-R(TS-2,ll))/(R(
TS-1,10)-R(TS-2,10))
1310 REM *******************************************
1320 REM BEGINNING OF CORE K-EFF CALCULATION
1330 RP=8:GOSUB2380
1340 GQSUB2810
1350 IFBP=0THEN1390
1360 REM CALC. OF LBP-CAUSED REACTIVITY DEFFECT
1370
FORI=1TOCT:CH=RC(I,BPf2) :CC=RC(l,BEH-2) :R(CH,0)=R(CH/0)-BN/100-(BB-BN)
* (R(CH,12) -R(CC,12)) / (100*(R(RC(CP,BIH-2) ,10) -R(CC,10))) :NEXTI
1380 REM ********************************************
1390 REM MODIFICATION OF K FOR THE EXTERNAL BATCH
1400 GOSUB2680
1410 REM *********************************************
1420 REM OBTAINING BATCH RELATIVE POWERS AND CHECKING FOR DIFFERENCE
WITH INITIALS
1430 0H=2.0

284
1440 FORI=1TQCT:SX=0
1450 F0RJ=1T0NB:CH=RC(I,J+2):IFCH=RC(I,8)THENFB=RC(I,2)ELSEPB=1
1460 SX=SX+((R(CH,0)*PB)[TH):NEXTJ
1470 FORJ=lTONB:CH=RC(I,J+2):IFCH=RC(I,8)THENPB=RC(1,2)ELSEPB=1
1480 R(CH,5)=((R(CH,0)*PB) [TH)*NB/SX:NEXTJ
1490 NEXTI
1500 MR=0
1510 FORI=1TOTS:IFR(I,5)-OTHENNEXTI
1520 CH=ABS( (R(I,5)-R(If2) )/(RMR3HENMR==CH
1530 NEXTI
1540 REM **********************************************
1550 REM OBTAINING CORE K FOR EACH REAL TIMESTEP AND CHECKING
DEVIATION
1560 MK=0
1570 FORI=1TOCT:SX=0
1580 FORJ=1TONB:CH=RC(I,J+2):SX=SX+(R(CH,5)/R(CH,0)):NEXTJ
1590 CC=NB/SX:RC(I,0)=CC:IF(ABS(CC-1))>MKTHENMK=ABS(CC-1)
1600 NEXTI
1610 GOSUB3330
1620 REM ********************************************
1630 REM GETTING BORON LETDOWN CURVE
1640 FORI=1TOCT:CC=(1-RC(I,0) )/(WAfWB*R(RC(I,3) ,7))
1650 FORJ=lTONB:CH=RC(I,J+2):R(CH,7)=R(CH,7)+CC:NEXTJ,I
1660 REM ******************************************
1670 REM OBTAINING END OF CYCLE TIME
1680 RRINT"OBTAINING EOC TIME BY INTERPOLATION FOR 0 PPM, K=l"
FOR

285
1690
I=CT:XY=0 :XX=0: SX=0: SY=0 :KK=CT~2: J=0: IFCT=20ffiINKK=lELSEIFCT<2raENl760
1700 CH=RC(I,3) :X=R(CH,7) :IFI=OTHEN1740ELSECC=RC(1-1,3) :Y=R(CC,9)
1710 IFCC<=1THEN1720ELSEIFY=R(RC(1-2,3),9)THENKK=KK-l:GOTO1730
1720 XY=XY+X*Y:SX=SX+X:SY=SY+Y:XX=XX+X*X:J=J+1
1730 1=1-1:IFI>=KKTHEN1700
1740 B=(XY-(SX*SY/J))/(XX-(SX*SX/J))
1750 A=(SY-B*SX)/J:GOTO177Q
1760 A=R(1,9)
1770 REM A IS NOW THE EOC TIME
1780 A=(AKX))/2:OC=A
1790 PRINTnEND OF CYCLE TIME="A
1800 G0TO3030
1810 REM **************************************
1820 REM ***************************************
1830 REM OUTPUT
1840 REM ***************************************
1850 INPUTINPUT A TITLE FOR THE CASE"jA$
1860 INPUT "TURN PRINTER ON, THEN ENTER" ;X
1870 LPRINTA$ jLPRINT" "iLPRINT" n:A$="#########":OK=0
1880 FORI=lTONB:LPRINTnBATCHnI:B$=n######.##":LPRINT"STEP #
n.
r
1890 F0:KJ=1T0CT:K=RC(J,I+2) :LPRINT USING A$;K;:NEXTJ
1900 LPRINT" ":LPRINTnINITIAL STEPS ENDING TIMES";
1910 FORJ=lTOCT:K=RC (J, 1+2):LPRINT USING B$;R(K,4) ; :NEXTJ:GOTO1960

286
1920 LPRINT" "¡LPRINT"NEW STEPS ENDING TIMES ";
1930 FORJ=lTOCT:K=RC(J,I+2):LPRINT USING B$;R(K,9);:NEXTJ
1940 LPRINT" LPRINT "NEW TIMESTEP DURATION ";
1950 FORJ=lTOCT:K=RC(Jf 1+2) ¡X=R(Kf9)-OK¡LPRINT USING
B$;X;¡OK=R(K,9)¡NEXTJ¡B$="###.#####"¡GOTO1980
1960 B$="###.#####"¡LPRINT" LPRINT"INITIAL STEPS K-INF.
1970 FORJ=lTOCT:K=RC(J,I+2):LPRINT USING B$;R(K,1);¡NEXTJ¡GOTO2000
1980 LPRINT" ":LPRINT"RMCTIVITY-CORRECTED K-INF";
1990 FORJ=lTOCT:K=RC(JrI+2):LPRINT USING B$?R(K,0);¡NEXTJ¡GOTO2040
2000 LPRINT" ":LPRINT"CLEAN K-INF.
2010 FORJ=lTOCT:K=RC(J,1+2)¡LPRINT USING B$;R(K-lfll);NEXTJ
2020 LPRINT" "LPRINT"INITIAL BORON PPM. ";
2030 FORJ=lTOCT:K=RC(J, 1+2) LPRINT USING A$;R(Kf6) ; NEXTJ¡G0TO2060
2040 LPRINT" ¡LPRINT"NEW BORON PPM.
2050 FORJ=1TOCTK=RC(J,I+2) ¡LPRINT USING A$;R(Kf7); ¡NEXrj¡GarO2080
2060 LPRINT" ";LPRINT"INITIAL RELATIVE POWERS
2070 FORJ=lTOCT:K=RC(J,1+2)¡LPRINT USING
B$;R(Kf2);¡NEXTJ¡B$="######.##"¡GOTO1920
2080 LPRINT" LPRINT "NEW RELATIVE POWERS ";
2090 FORJ=lTOCT¡K=RC(J, 1+2) ¡LPRINT USING B$;R(K,5) ; ¡NEXTJ
2100 LPRINT" "¡LPRINT" "¡LPRINT" "
2110 NEXTI
2120 LPRINT" "¡LPRINT" "¡X=0
2130 FORI=lTONB¡LPRINT"ESTIMATED BURNUP FOR
CYCLE"I"IS"RR(I,0) "MWD/MTU"¡X=X+RR(Ir0) ¡NEXTI
2140 K=RC(CT,3) ¡LERINT"END OF LIFE BURNUP IS ESTIMATED

287
AT"X"MWD/MTU" :LPRH7T"CYCLE LENGTH IS ESTIMATED AS"A"HOURS, WHILE
INPUT ESTIMATE WAS"R(K,4)"HOURS"
2150 LPRINT"MAXIMUM PERCENT CHANGE BETWEEN OLD AND NEW ESTIMATES OF
STEP RELATIVE PCWERS IS"MR
2160 LPRINT"BATCHWISE BORON WORTH REGRESSION LINE IS:
W="WA"+ ("WB"*PPM)"
2170 STOP
2180 REM ***************************************
2190 REM ROUTINE FOR DATA INPUT FROM DISK
2200 INFUT"FILESPEC"?A$
2210 OPEN"R", 1, A$:FIELD 1,2 AS Xl$,2 AS X2$,2 AS X3$,2 AS X4$,4 AS
X5$,4 AS X6$f2 AS X7$,4 AS X8$,4 AS X9$,229 AS N$
2220 GET
1:TS=CVI(X1$) :CT=CVI(X2$) :NB=CVI(X3$) :BE=CVI(X4$) :PL=CVS(X5$) :FL=CVS(
X6$) :A$=N$:BP=CVI(X7$) :BB=CVS(XB$) :BN=C7S(X9$)
2230
CC=201:CH=1: 1=1:K=1 :DIMR(TS, 12) ,RC(CT,8) ,RR(NB,2): IFBEO0THEN2250
2240 BW=-1. OE-4:GOTO2260
2250 DIMBW(BE)
2260 FORJ=3T08:RC(I,J)=CVS(MID$(A$,Kf4)) :K=K+4:NEXTJ
2270 1=1+1:IFKXZCTHEN2290
2280 IF K=CTTHEN2260ELSE2300
2290 GET 1:CC=223:FIELD 1,255 AS N$:K=1:A$=N$:GOT02280
2300 IFCH>TSTHEN2330
2310 F0RJ=lT06:R(CH,J)=CVS 2320 CH=CE+l:GOTO2270

288
2330 IFBE=0THEN2370
2340 F0RJ=1T0BE:EW(J)=CVI(MID$(A$,K,2)):K=K+2:NEXTJ
2350 CA=CVI (MID$ (A$fK,2)) :K=K+2: IFCA=0THEN237OELSEDIMBK(CA)
2360 F0RJ=1T0CA:BKCJ)=CVI(MID$(A$,K,2)):K=K+2:NEXTJ
2370 CLOSE :GOT0560
2380 REM OBTAINING EOC BRNUPS (CYCLE AND STEPWISE)
2390 X=0:CC=1
2400
FORI=lTOTS:X=X+PL*R(I,RP)*(R(I,9)-R(I-l,9))/(24*FL) :R(I,RB)=X:IFI=RC(
CT,CC+2)THEN2410ELSENEXTI
2410 CH=0
2420 FORJ=OTOCC-1:CH=CH+RR(J,0):NEXTJ
2430 RR(CC,0) =X-CH:COCC+1: IFKTSTHENNEXTI
2440 GOSOB3270
2450 RETURN
2460 REM THE FINAL BURNUP OF EACH CYCLE I HAS BEEN PLACED ON RR(I,0)
2470 REM ***************************************
2480 REM ROUTINE FOR STORAGE OF DATA IN DISK
2490 CLS:INPUT"DO YOU WANT TO SAVE DATA ON DISK? (Y)ES OR
(N) O" ;A$: IFLEFT$ (A$, 1) <>nY"lBENRETURN
2500 INPUTnFILESPEC";A$:OPEN"Rn,1rA$:FIELD 1,2 AS Xl$r2 AS X2$,2 AS
X3$,2 AS X4$r4 AS X5$,4 AS X6$,2 AS X7$,4 AS X8$,4 AS X9$,229 AS N$
2510 LSET X1$=MKI$(TS):LSET X2$=MKI$(CT):LSET X3$=MKI$(NB):LSET
X4$=MKI$(BE):LSET X5$=MKS$(PL):LSET X6$=MKS$(FL):LSET
X7$=MKI$(BP):LSET X8$=MKS$(BB):LSET X9$=MKS$(BN)
2520 A$="":CC=201:CH=1:I=1

289
2530 FORJ=3TCe:A$=A$+MKS$(RC(I,J)):NEXTJ
2540 1=1+1: IFLEN(A$) >CCTHEN2560
2550 IFK=CTTHEN2530ELSE2570
2560 LSETN$=A$:PUT l:A$=nn:CC=223: FIELD 1,255 AS N$:GOTO2550
2570 IFCH>TSTHEN2600
2580 F0RJ=1T06 :A$=A$+MKS$ (R(CH,J)) :NEXTJ
2590 CEKBFl :GCTO2540
2600 IFBE=0THEN2640
2610 F0RJ=1TCBE:A$=A$+MKI$(BW(J)):NEXTJ
2620 A$=A$+MKI$(CA):IFCA=0THEN2640
2630 FCRJ=lrPOCA:A$=A$+MKI$ (BK(J)) :NEXTJ
2640 IFA$=""THEN2660
2650 LSETN$=A$:PUT 1
2660 CLOSE:RETURN
2670 REM ***************************************
2680 REM ROUTINE FOR CALCULATION OF MODIFICATION OF K FOR THE
EXTERNAL BATCH
2690 FORI=1TOCT:CH=RC(I,8):TA=-20.0:BU=R(CH-1,12)
2700 L2=(0.486-(1.9E-7*BU))/(0.1912+(6.42E-7*BU))
2710
SA=(0.486-(1.9E-7*BU))/(SQR((0.4033-(5.135E-6*R(CH,7)))*(0.0166+(2.72
7E-5*R(CH,7)))))
2720 B2=(2.405/(168.53+SA))[2:UB=0
2730 REM SHAPE FACTOR CALCULATION
2740 PORJ=lTONB:UH=RC(I,J+2):UB=UB+R(UH,3):NEXTJ
2750

290
Z1=2.405*SQR( (NB-D/NB) :Zl=(Zl/2)-( (Zl[3)/16)+((Zlf5)/384) :SG=(1/(1-Z
1*1.84292*SQR( (NB-D/NB))) [2:SF=SG*R(CH,3)/UB
2760 FR=EXP(TA*B2)/(1+L2*B2)
2770 PB=1-((1-PR)*SF*0.7)
2780 R(CH,0)=R(CH,0)*PB:RC(If2)=PB:PRINTnMODIF FOR EXTERNAL BATCH
T-STEP"I"="PB
2790 NEXTI
2800 RETURN
2810 REM CALCULATING WORK K-INF FROM PRESENT-STATUS BUKNUPS AND
BORON CONCENTRATIONS.
2820 R(1,0)=R(0,11)+(WA*R(1,7))+(WB*(R(1,7) E2)/2) :IFTS<2THENRETURN
2830 FORI=2TOTS:X=R(1-1, 12):XL=0:XM=0
2840 FORJ=0TOTS:IFR(Jr10) 2850 IFR(J,10)>=XTHENXM=J
2860 IFXM=J AND JO0THENJ=TS
2870 NEXTJ
2880 IFXL=XMTHENPRIOT"WARNINGI! XMfXL III":STOP
2890 IFXM=0 AND XLOO THEN XM=TS:XL=TS-2
2900 IFABS(R(XL11)-R(XMf11))<0.001 AND XL>lTHENXL=XL-l:GOTO2900
2910
R(If 0)=R(XL,ll)+( (X-R(XL,10)) *(R(XM,ll)-R(XLfll) )/(R(XM,10)-R(XL,10))
) :R(I,0)=R(I0)+(WA*R(If7))+(WB*(R(If7) [2)/2) :NEXTI
2920 RETURN
2930 REM CHECKING FOR CONSISTENCY OF CYCLES' B
2940 CLS:PRINTTAB(15);nK CONVERGED. CHECKING BURNUPS."
2950 XX=0:RP=5:GOSUB2380

291
2960 P0RI=1T0NB:CH=ABS((RR(I,0)-FR(I,2) )/RR(I,2)) :IFCH>XXTHENXX=CH
2970 NEXTI
2980 IFKX<=0.01THEN1850
2990 FORI=LTOTS:R(I,8) = (R(I,5)+R(I,8))/2:NEXTI
3000 RP=8:GOSUB2380
3010 FORI=1TONB:RR(I,2)=RR(I,0):NEXTI
3020 GOTO1340
3030 REM ASSIGNING NEW STEP ENDING TIMES
3040 X=0:Y=R(RC(CT,3),9)
3050 FORJ=2TONB:CH=RC (CT,J+D+l:CC=RC (CT,J+2)
3060 FORI=CHTOCC:R(If9)=R(If9)+(A-Y)*(J-1) :NEXTI,J
3070 F0RI=1T0NB
3080 FORJ=lTOCT:CH=RC(J,I+2) :Y=R(CH,9) :IFY>=AfXTHEN3100
3090 IFJ=CTTHEN3100ELSEIFY 3100 R(CH,9)=X+A:NEXTJ
3110 X=X+A:NEXTI
3120 F0RI=1T0NB:CC=CT
3130 CH=RC(CC,I+2):IFR(CH-lf9)<>A*ITHEN3150
3140 R(CH,7) =0:COCC-1:IFCC=0THEN3150ELSE3130
3150 NEXTI
3160 IFMK<0.01THEN2930ELSE1330
3170 F0RI=CAT01STEP-1
3180 FORJ=BK(I)TOTS-1
3190 PQRK=0TO12:R(J,K)=R(J+1,K):NEXTK,J,I
3200 TS=TS-CA
3210 F0RI=1T0CT

292
3220 FORJ=3TONEH-2
3230 P0RK=CMK)1STEP-1 sIFRC (I, J) >BK(K) THENRC(I, J) =RC (I, J) -K:G0T0325Q
3240 NEXTK
3250 NEXTJrl
3260 RETURN
3270 CLS
3280 PRINT"CYCLES1 BURNUPS MATRIX"
3290 F0RI=1T0NB
3300 FORJ=0TO2:PRINTRR(I, J),;:NEXTJ
3310 PRINT" ":NEXTI
3320 RETURN
3330 CLS:PRINTnSTEPn,nEND TIME","WORK K"f"CORE K"
3340 FORI=1TOTS:PRINTI,R(I,9),R(I,0),;
3350 IFI<=CTTHENPRINTRC(1,0)ELSEPRINT" "
3360 NEXTI
3370 RETURN

APPENDIX C2
SAMPLE RUN OF CRIBUR
Cribur was first developed for use on a microcomputer, and for
ease of operation, it was designed for interactive use. Therefore,
all information to and from the computer is on the CRT screen,
except the final output, which is printed on hard copy. The
following listing is a reproduction of the screen messages that
appear on a sample run of the code, and of the final printout.
During execution, the code prints sets of data on the screen that
indicate the progress of the calculation and the convergence of the
iterations. These data are not presented here.
Screen 1:
PROGRAM FOR CRITICALITY, BORON WORTH, BATCH POWER
AND CYCLE LIFE EVALUATION OF A MULTI-BATCH PWR.
DATA INPUT FROM (K)EYBOARD OR (D)ISK ? D
FILESPEC ? HP50A252
Screen 2:
CHECK TIMESTEP ENDING TIMES. IF ANY ONE TO BE CHANGED, INPUT
TIMESTEP NUMBER. OTHERWISE, JUST PRESS "ENTER".
TSTEP 1 ENDS AT 200 HR TSTEP 2 ENDS AT 2000 HR
TSTEP 3 ENDS AT 6000 HR TSTEP 4 ENDS AT 12000 HR
293

294
TSTEP 5 ENDS AT 16300 HR
TSTEP 7 ENDS AT 18300 HR
TSTEP 9 ENDS AT 22301 HR
TSTEP 11 ENDS AT 28301 HR
TSTEP 6 ENDS AT 16500 HR
TSTEP 8 ENDS AT 22300 HR
TSTEP 10 ENDS AT 28300 HR
TSTEP 12 ENDS AT 32600 HR
TSTEP 13 ENDS AT 32800 HR TSTEP 14 ENDS AT 34600 HR
TSTEP 15 ENDS AT 38600 HR TSTEP 16 ENDS AT 44600 HR
TSTEP 17 ENDS AT 48900 HR ANY CHANGED STEP # ? '
Screen 3:
CHECK BORON PPM FOR EACH STEP. ENTER STEP TO BE CHANGED, OR "ENTER"
TIMESTEP 1
TIMESTEP 3
TIMESTEP 5
TIMESTEP 7
TIMESTEP 9
TIMESTEP 11
TIMESTEP 13
TIMESTEP 15
TIMESTEP 17
Screen
CHECK K FOR
HAS 1310 PPM
HAS 1040 PPM
HAS 310 PPM
HAS 1170 PPM
HAS 650 PPM
HAS 150 PPM
HAS 1310 PPM
HAS 1040 PPM
HAS 310 PPM
4:
EACH TIMESTEP
TIMESTEP 2
TIMESTEP 4
TIMESTEP 6
TIMESTEP 8
TIMESTEP 10
TIMESTEP 12
TIMESTEP 14
TIMESTEP 16
ANY CHANGED
HAS 1170 PPM
HAS 750 PPM
HAS 1310 PPM
HAS 1040 PPM
HAS 750 PPM
HAS 310 PPM
HAS 1170 PPM
HAS 750 PPM
STEP # ?
TSTP
K-INF
TSTP
K-INF
1
1.1662
2
1.1489
3
1.1478
,4
1.1415
5
1.1320
;t:'6
1.0032

295
7
1.0142
8
1.0137
9
1.0216
10
1.0127
11
1.0288
12
1.0136
13
0.8989
14
0.9103
15
0.9126
16
0.9181
17
0.9276
ANY CHANGED
Screen 5:
CHECK RELATIVE PCWER FOR EACH STEP.
ENTER STEP TO BE CHANGED, OR PRESS "ENTER".
TIMESTEP 1 HAS RELPWRf 1.176 TIMESTEP 2 HAS RELPWRf 1.14
TIMESTEP 3 HAS RELPWRf 1.14
TIMESTEP 5 HAS RELPWR= 1.11
TIMESTEP 7 HAS RELFWR= 1.03
TIMESTEP 9 HAS RELPWRf 1.03
TIMESTEP 11 HAS RELFWR= 1.03
TIMESTEP 13 HAS RELPWRf 0.81
TIMESTEP 15 HAS RELPWRf 0.83
TIMESTEP 17 HAS RELPWRf 0.86
Screen 6:
TIMESTEP 4 HAS RELPWRf 1.13
TIMESTEP 6 HAS RELPWRf 1.01
TIMESTEP 8 HAS RELPWRf 1.03
TIMESTEP 10 HAS RELPWRf 1.03
TIMESTEP 12 HAS RELPWRf 1.03
TIMESTEP 14 HAS RELPWRf 0.83
TIMESTEP 16 HAS RELPWRf 0.84
ANY CHANGED STEP ?
CHECK CORE BATCH COMPOSITION IN CYCLE.
ENTER STEP AND COLUMN TO BE CHANGED, OR JUST PRESS "ENTER"
STEP BATCH-1 BATCH-2 BATCH-3 BATCH-4 BATCH-5
11 6 13 0 0
2 2 7 14 0 0
OUTER-BATCH
1
2

296
3 3 8 15 O O 3
4 4 10 16 O O 4
5 5 12 17 0 0 5
ANY CHANGED STEP AND COLUMN ?
Screen 7:
CHECK BATCH WITH LBP AND LBP'S BATCHWISE WORTH (%K/K) AT BOC, EOC
BATCH 0 BOC %K/K= 0 EOC %K/K= 0
NEW VALUES FOR ALL OR JUST PRESS "ENTER" ?
CHECK POWER LEVEL AND FUEL LOADING
POWER= 1700 MW THERMAL FUEL LOADING- 94.18 MTU
NEW VALUES FOR BOTH OR HIT "ENTER" ?
Screen 8:
CHECK BORON WORTH EVALUATION POINTS
STEPS TO BE KILLED AFTER B-WORTH EVALUATION (IN INCREASING ORDER):
9 11
TIMESTEPS FOR EVALUATION:
6 7
9 10
11 12
WISH TO INPUT A NEW SET Y/N ? N
Screen 9:
DO YOU WANT TO STORE DATA ON DISK? Y/N ? N

Code output reproduction:
HP50A252. 50% PCWER. PITCH = 1.25
297
BATCH 1
STEP #
1
INITIAL STEPS ENDING TIME
200.00
INITIAL STEPS K-INF.
1.16620
CLEAN K-INF.
1.28232
INITIAL BORON PPM
1310
INITIAL RELATIVE POWERS
1.17600
NEW STEPS ENDING TIMES
200.00
NEW TIMESTEP DURATION
200.00
REACTmTY-CORRECTED K-INF
1.08072
NEW BORON PPM
1321
NEW RELATIVE PCWERS
1.05971
BATCH 2
STEP #
6
INITIAL STEPS ENDING TIME
16500.00
INITIAL STEPS K-INF.
1.00320
CLEAN K-INF.
1.11932
INITIAL BORON PPM
1310
INITIAL RELATIVE POWERS
1.01000
NEW STEPS ENDING TIMES
16729.40
NEW TIMESTEP DURATION
200.00
2
3
4
5
2000.00
6000.00
12000.00
16300.00
1.14890
1.14780
1.14150
1.13200
1.25395
1.24228
1.21141
1.16201
1170
1040
750
310
1.14000
1.14000
1.13000
1.11000
2000.00
6000.00
12000.00
16529.40
1800.00
4000.00
6000.00
4529.36
1.06587
1.06482
1.06278
1.05874
1200
1055
761
331
1.02967
1.02789
1.02451
1.01790
7
8
9
10
18300.00
22300.00
28300.00
32600.00
1.01420
1.01370
1.01270
1.01360
1.11925
1.10818
1.08261
1.04361
1170
1040
750
310
1.03000
1.03000
1.03000
1.03000
18529.40
22529.40
28529.40
33058.70
1800.00
4000.00
6000.00
4529.36
REACTIVIT'/-CORRECTED K-INF 1.01486 1.02252 1.02000 1.01807 1.01601

298
NEW BORON PPM
1321
1200
1055
761
331
NEW RELATIVE POWERS
1.08646
1.10146
1.09566
1.09068
1.08486
BATCH 3
STEP #
11
12
13
14
15
INITIAL STEPS ENDING TIME
32800.00
34600.00
38600.00
44600.00
48900.00
INITIAL STEPS K-INF.
0.89893
0.91031
0.91261
0.91807
0.92765
CLEAN K-INF.
1.01505
1.01536
1.00709
0.98798
0.95766
INITIAL BORON PPM
1310
1170
1040
750
310
INITIAL RELATIVE POWERS
0.81000
0.83000
0.83000
0.84000
0.86000
NEW STEPS ENDING TIMES
33258.70
35058.70
39058.70
45058.70
49588.10
NEW TIMESTEP DURATION
200.00
1800.00
4000.00
6000.00
4529.36
REACTIVITY-CORRECTED K-INF 0.89967
0.90816
0.91227
0.91697
0.92399
NEW BORON PPM
1321
1200
1055
761
331
NEW RELATIVE POWERS
0.85383
0.86886
0.87645
0.88481
0.89724
ESTIMATED BURNUP FOR CYCLE 1 IS 12736.5 MWD/MTU
ESTIMATED BURNUP FOR CYCLE 2 IS 13568.3 MWD/MTU
ESTIMATED BURNUP FOR CYCLE 3 IS 10990.7 MWD/MTU
END OF LIFE BURNUP IS ESTIMATED AT 37295.5 MWD/MTU
CYCLE LENGTH IS ESTIMATED AS 16529.4 HOURS, WHILE
INPUT ESTIMATE WAS 16300 HOURS.
MAXIMUM PERCENT CHANGE BETWEEN OLD AND NEW ESTIMATES OF
STEP RELATIVE POWERS IS 9.88877
BATCHWISE BORON WORTH REGRESSION LINE IS: W=-9.9343E-5+(1.63E-8*PPM)
\

APPENDIX D
ISOTOPIC AND SPECTRAL DATA FROM BURNUP CALCULATIONS
Chapter IV describes the burnup optimization calculations
performed for the standard and the low power density cores, and the
main neutronic and plant performance results that can be drawn from
them.
Table D.l shows the cell-homogenized number densities of
U-235, Pu-239, Pu-240 and Pu-241 at the beginning and end of life
of the fuel pins for all the core cases studied in Chapter IV. All
plutonium concentrations are assumed to be zero at BOL, since the
initial fuel composition is suposed to contain only uranium
isotopes. Note that the BOL concentration of U-235 decreases as
the pitch is increased. This is the effect of the cell
homogenization of the number density. All fuels are assumed to
start with a 3.1% enriched uranium composition. A pellet density
of 93% is assigned, which is the default assumed by EPRI-CELL. The
number densities shown include also the effect of thermal fuel
expansion, as calculated by EPRI-CELL.
Table D.2 lists the cell average neutron velocity for the
sixteen cores studied, for their BOL, MOL and EOL. As expected,
the cores with larger pitches show a lower velocity, due to the
better neutron moderation. The power level for any particular
pitch does not show a significant effect on the average velocity.
299

300
Fuel burnup affects the neutron velocity, consistently causing a
softening of the spectrum as fuel exposure increases.

301
Table D.l
Cell-Homogenized Number Densities (Atoms/Barn CM X 10b )
Pitch %
Power
BOL U-235
EOL U-235
EOL Pu-239
EOL Pu-240
EOL Pu-241
1.20
50
258.24
62.08
47.63
19.38
12.07
1.20
33
258.24
59.82
46.97
19.68
11.74
1.20
25
258.24
58.35
46.96
19.87
11.52
1.25
100
238.00
50.44
39.08
17.53
10.57
1.25
50
238.00
47.46
38.97
18.33
10.51
1.25
33
238.00
46.73
38.49
18.36
10.15
1.25
25
238.00
45.79
38.36
18.45
9.92
1.30
50
220.00
39.47
32.47
16.77
8.96
1.30
33
220.00
38.11
32.14
16.86
8.72
1.30
25
220.00
37.44
31.98
16.89
8.51
1.35
50
204.00
32.40
27.59
15.78
7.61
1.35
33
204.00
31.99
27.32
15.51
7.54
1.35
25
204.00
31.46
27.19
15.52
7.36
1.40
50
189.70
29.56
23.50
13.99
6.56
1.40
33
189.70
28.61
23.23
14.06
6.38
1.40
25
189.70
28.15
23.08
14.09
6.22

302
Table D.2
Average Cell Neutron Velocities (eV)
Pitch %
Power
BOL Speed
MOL Speed
EQL Speed
1.20
50
1.93
1.89
1.87
1.20
33
1.93
1.89
1.87
1.20
25
1.93
1.89
1.87
1.25
100
1.89
1.86
1.83
1.25
50
1.90
1.86
1.83
1.25
33
1.89
1.85
1.83
1.25
25
1.89
1.85
1.83
1.30
50
1.87
1.82
1.80
1.30
33
1.87
1.82
1.80
1.30
25
1.87
1.82
1.79
1.35
50
1.85
1.80
1.78
1.35
33
1.85
1.80
1.77
1.35
25
1.85
1.80
1.77
1.40
50
1.83
1.78
1.75
1.40
33
1.83
1.78
1.75
1.40
25
1.83
1.78
1.75

REFERENCES
1. Hersperger, E., "Economic Analysis of Low Power Density FWR
Plants," Master's Thesis, College of Engineering, University
of Florida (1981).
2. Daby, D., "Fuel Utilization Improvements in a Gnce-through PWR
Fuel Cycle," Final Report on Task 6, WCAP-9547, Westinghouse
Electric Corp. (1979).
3. Hellens, R.L., "Evaluation of Methods of Improving Fuel
Utilization for Once-through Fuel Cycles," Nuclear Power
Systems, Combustion Engineering, Inc. (1979).
4. Westinghouse Nuclear Training Operations, "Plant Information
Manual. 3400 MW Plant," Westinghouse Electric Corp. (1975).
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Engineering Science," CRC Press, Inc. (1976).
6. "Steam: Its Generation and Use," Babcock & Wilcox Co.
(1975).
7. Duderstadt, J.J. and Hamilton, L.J., "Nuclear Reactor
Analysis," John Wiley & Sons (1976).
8. Curtis, R.L., "PHROG A Fortran IV Program to Generate Fast
Neutron Spectra and Average Cross Sections," IN-1435, Aerojet
Nuclear Co. (1971).
9. Bennett, C.L. and Purcell, W.L., "BRT-I: Battelle-Revised
Thermos," BNWL 1434, Batteile Pacific Northwest Laboratory
(1970).
10. Putnam, G.E., "MONA A Multigroup, One-Dimensional Neutronics
Analysis Code," ANCR-1051, Aerojet Nuclear Co. (1972).
11. Barry, R.F., "LEOPARD A Spectrum-dependent Non-spatial
Depletion Code for the IBM 7094," WCAP-3269-26, Atomic Power
Division, Westinghouse Electric Corp. (1963).
12. Bohl, H., Gelbard, E., and Ryan, G., "MUFT-4 Fast Neutron
Spectrum Code for the IBM 704," WAPD-TM-72, Atomic Power
Division, Westinghouse Electric Corp. (1957).
303

304
13. Amster, H., and Suarez, R., "The Calculation of Thermal
Constants Averaged over a Wigner-Wilkins Flux Spectrum:
Description of the SOFOCATE Code," WAPD-TM-39, Atomic Power
Division, Westinghouse Electric Corp. (1957).
14. Macnabb, W.V., "Two Near-term Alternatives for Improved
Nuclear Fuel Utilization," Nuclear Technology, Vol. 49 (1980).
15. Driscoll, M.J., "Improved FWR Core Designs, Bimonthly Progress
Report No. 2," IPWRCD-2, MIT Nuclear Engineering Department
(1980).
16. Miller, M., "Thermalhydraulics Analysis of Alternative PWR
Core Designs," Master's Thesis, Nuclear Engineering
Department, University of Florida (1981).
17. Cadwell, W.R., "PDQ-7 Reference Manual," WAPD-TM-678, Atomic
Power Division, Westinghouse Electric Corp. (1967).
18. Ver Planck, D.M., "Manual for the Reactor Analysis Program
SIMULATE," YAEC 1158, Electric Power Research Institute
(1978).
19. Eich, W.A., Cobb, W.R., and Tivel, D.E., "EPRI-CELL Code
Description," EPRI ARMP System Documentation, Electric Power
Research Institute (1975).
20. Joanou, G.D., and Dudek, J.S., "GAM-I: A consistent P-1
Multigroup Code for the Calculation of Fast Neutron Spectra
and Multigroup Constants," GA-1850 (1961).
21. Honeck, H.C., "THERMOS, A Thermalization Transport Theory Code
for Reactor Lattice Calculations," BNL 5826 (1961).
22. England, T.R., "CINDER: A One-point Depletion and Fission
Product Program," WAPD-TM-334, Atomic Power Division,
Westinghouse Electric Corp. (1964).
23. Rothleder, B.M., "NUPUNCHER Code Description," EPRI ARMP
System Documentation, Electric Power Research Institute
(1975).
24. Breen, R.J., Marlowe, O.J., and Pfeifer, C.J., "HARMONY:
System for Nuclear Reactor Depletion Computation,"
WAPD-TM-478, Atomic Power Division, Westinghouse Electric
Corp. (1965).
25. Rothleder, B.M., Blake, R.A., Fisher, J.R., and Kendrik, E.D.,
"PWR Core Modeling Procedures for Advanced Recycle Methodology
Program," Research Project 976-1, Electric Power Research
Institute (1979).

305
26. Bell, J.L.r "Comparisons between ARMP Calculations and
Measurements from D.C. Cook Unit 2, Cycle 1," American
Electric Power Service Corp., ARMP Users' Group Meeting
(1980).
27. Chen, E., "Comanche Peak Unit 1 Core Model," Texas Utilities
Services, Inc., ARMP Users' Group Meeting (1980).
28. Graves, H.W., "Nuclear Fuel Management," John Wiley & Sons,
Inc. (1979).
29. Burns, E.T., "Refueling Outage Trends in Light Water
Reactors," EPRI NP-842, Research Project 705-1, Electric Power
Research Institute (1978).
30. Atomic Industrial Forum, "Reprint from UPDATE 'Nuclear Power
Program Information and Data'," Office of Nuclear Reactor
Programs, D.O.E. (1980).
31. Miller, M., "TEMPRET, A Computer Code for the Steady-state
Thermal Analysis of a Single PWR Fuel Pin / Coolant Channel,"
Nuclear Engineering Department, University of Florida (1981).
32. "CONCEPT-IV A Computer Code for Conceptual Cost
Estimates of Steam-Electric Power Plants," Office of Energy
Systems Analysis, U.S. ERDA (1975).
33. Hughes, J.A., and Hang, D.F., "GEM General Economic Model to
Analyze Nuclear Fuel Cycle Costs," University of Illinois
(1973).
34. Salmon, R., "POWERCO A Procedure and a Computer Code for
Calculating the Cost of Electricity Produced by Nuclear Power
Stations," ORNL (1966).
35. Anderson, E.C., and Putnam, G.E., "CORA A Few Group
Diffusion Theory Code for One-Dimensional Reactor Analysis,"
IN-1416 (1970).
36. Cacciapouti, R.J., and Sarja, A.C., "CHIMP-II A Computer
Program for Handling Input Manipulation and Preparation for
PWR Reload Core Analysis," Yankee Atomic Electric Co. (1976).
37. Impink, A.J., Jr., "Reactor Core Physics Design and Operating
Data for Cycles 1 and 2 of the Zion-2 PWR Power Plant," EPRI
NF-1232, Project 519-6, Electric Power Research Institute
(1979).
38. Flores, L., "Nuclear Reliability Program. EPRI-NODE Power
Distribution Corrparisons at Oconee," Duke Power Co. (1980).

BIOGRAPHICAL SKETCH
Cesar Molins-Bartra was born the second son of Cesar Molins
and Maria Bartra, on the twenty-second of May, 1953, in Barcelona,
Catalonia, Spain. In 1975 he received the degree of Ingeniero
Industrial Superior en Tcnicas Energticas from the Polytechnical
University of Barcelona, where he acted as an assistant professor
while finishing his military duties as an artillery officer at the
nearby city of Gerona.
In 1977 he was awarded a research assistantship at the
University of Florida, U.S.A., through the sponsorship of the
Fulbright-Hayes graduate students exchange program. He received
the Master of Engineering degree from the Nuclear Engineering
Department in 1978. In 1980 he was assigned for a nine-month stay
at O.R.N.L. where he conducted the main body of the research
towards his Ph.D. degree.
306
u..

I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
rv \
\i
k/V
Nils J,
, Diaz,\
jfchairman )
Engineering Sciences
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Edward E. Carroll
Professor of Nuclear
Engineering Sciences
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Edward T. Dugan
Visiting Assistant Pfbfessor of
Nuclear Engineering Sciences
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Marvin L. Muga
Professor of Chemist:

I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Calvin C. Oliver
Professor of Mechanical
Engineering
This dissertation was submitted to the Graduate
Faculty of the College of Engineering and to the
Graduate Council, and was accepted as partial fulfillment
of the requirements for the degree of Doctor of
Philosophy.
June, 1981
Dean, College of Engineering
Dean for Graduate Studies and
Research



180
technological and safety aspects of the standard plant is the
assurance of an adequate heat transfer under normal, transient and
accident conditions. Mark Miller performed a detailed study(16) on
the behavior of the low-power core as compared to the standard
reactor and some of the main conclusions of this study are
presented here as illustrations of the thermal-hydraulic and safety
comparison between the two reactor concepts.
5.1.2 ,--Thermal-hvdraulic_.Studies.
There are several parameters that are considered of key
importance for thermal-hydraulic safety considerations. Some of
these are the fuel average temperature, which is an indication of
the stored heat content of the core and which is an important
parameter in determining the emergency core cooling system
characteristics; the fuel centerline temperature, of particular
importance since that is the hottest point in the pellet, the clad
surface temperature, and the departure from nucleate boiling ratio
(DNBR) which is the ratio between the surface critical heat flux
(the surface heat flux that would produce film boiling) and the
actual surface heat flux.
The DNBR varies from point to point in the reactor, since the
heat transfer conditions change from one point on the fuel to
another, as does the heat flux level. A safety margin can be
related to the minimum DNBR found in the whole core. Thus, the
minimum DNBR is one of the criteria that can be used for
thermal-hydraulic safety comparisons. The importance of the clad


246
axial power peaking factor, but it would result in an increase of
fuel fabrication cost as well.
e). Since it is apparent that for once-through fuel cycles a
better-moderated configuration entails ore savings, this effect
could be achieved with the use of hollow fuel pins, such that the
coolant could be circulated both around the pin and inside the
hole. This would sharply improve moderation, and it would keep the
fuel colder since the thickness of fuel lumps would be reduced and
the area of fuel facing the coolant would be increased for the same
fuel loading, Extra reactivity would be available from reduced
Doppler effect due to the lower temperature of the fuel, and from
the increased water-to-fuel ratio and better-moderating geometry.
However, this idea would require major redesign of the reactor, and
it would significantly increase fuel fabrication costs.
f). Spectral-shift controlled reactor. A FWR has excess
reactivity all through its life, except at the EQL, where all
poisons are removed in order to extend the cycle as long as
possible. At the EOL much of the core power is generated by the
plutonium isotopes that have been bred during the cycle from
neutron captures in -238. If the amount of plutonium available at
the EQL could be increased, a longer cycle life could be expected.
One way of increasing the conversion ratio of the reactor is to
harden the neutron energy spectrum by reducing the moderating
capabilities of the coolant. A harder spectrum reduces the core
multiplication factor, but this is no problem until the EQL, when
multiplication has to be as high as possible.


228
value, determined by the enrichment of the fuel and the in-core
fuel management scheme.
With this assumption, the core-average burnup at EOL can be
calculated (for a three-batch core, for example) as:
CA = CB*P1 + CB*(P1+P2) + CB*3 = CB 2*P1 + P2 + 3
3 3
where CA = Core Average Burnup at EOL.
CB = Cycle Burnup.
PI = Relative power of Batch 1 (in the core for the first
cycle).
P2 = Relative power of Batch 2 (in the core for the second
cycle).
Then, if for a given enrichment and fuel management scheme CA
is fixed, it is possible to obtain CB as
CB = CA_* 3
2*P1 + P2 + 3
Since the discharge burnup for the fuel will be three times
the cycle burnup, it is obvious that the lower PI, the higher the
discharge burnup can be, and P2 has the same effect, to a lesser
degree. These considerations indicate that the burnup of the fuel
should be shifted towards the end of its life as far as it is
possible.
It is important to bear in mind that the assumptions used for
these calculations are not exact, and it is obviously not
reasonable to rely too heavily on the precise numerical results


125
lines are obtained because of the varying isotopic compositions of
the fuel. It is then advisable to perform the linear fit
calculation on a cycle towards the middle of the fuel life (the
second cycle in the case of a three-batch core) in order to obtain
a soluble boron worth regression line that can be representative of
the whole core.
One may be concerned about the fact that by defining the
"fake-burnup" steps for the soluble boron worth calculation, there
is one cycle in which the burnup timesteps do not match with the
other cycles. CRIBUR solves the problem by deleting the
"fake-burnup" timesteps after the regression line has been
calculated. This causes no misadjustment as far as matching
timesteps from each cycle, because the "fake-burnup" steps were
insignificant in terms of both burnup and time.
Once the soluble boron worth can be obtained for any boron
concentration, it is a simple matter to adjust the boron letdown
curve so that the reactor is critical for all timesteps in the
cycle- However, this causes a major problem in the core life
calculation, as discussed below for a reactor with a soluble boron
letdown curve such as the solid line of the top of Figure 3.2.6.
After a criticality calculation by CRIBDR. it is found that the
letdown curve was somewhat in error, and it should be more like the
one represented in a dashed line. This new letdown curve was
calculated by using the burnup timesteps marked on the old letdown
curve, and adjusting the boron concentration for criticality- But
if this new curve were indeed to be the actual letdown curve, the


172
%


106
Note that the point in Figure 3.2.1 where the soluble boron
concentration reaches a zero value and jumps back up to the value
it had at BOL signifies the core cycle duration. Once the assembly
is burned according to the power and boron concentrations assumed
for all its life, all these burnup environment data together with
the multiplication factors of the assembly at each time step are
fed into the core model code. The core model code must evaluate
the multiplication factor of the whole core, the power level of
each batch, and the core cycle life duration; it thus furnishes
data for a better estimate of the real behavior of the core through
life. If the newly calculated data from the core model code agree
closely with the power levels, boron concentrations and cycle
lengths previously input to PDQ-7, the guess is final, and burnup,
core life and isotopics can be obtained from PDQ-7 and the core
model. If the core model calculated data differ significantly from
the data input to PDQ-7, it is necessary to modify the assembly
burnup data according to the core model, and run a new PDQ-7 burnup
case, starting a new iteration.
Figure 3.2.2 shows a flow chart of the codes and data handling
necessary in order to obtain the fuel burnup data to be input to
the core model code. The core model code is described in the next
section.
3,2.2. The_CRIBUR Core Model.
The CRIBUR code was developed as a simple model for core life
calculations which would include all the points outlined in Section


161
Table 4.2.2. Plant Operational Data Comparison.
Pitch
cm
Power
%
Real Cycle
Days
Life Reful
Days
Refuel
% Life
Availa.
%
Capacity
%
Size Mod.
Cap., %
1.20
50
923
534
4.88
80.85
70.10
75.04
1.20
33
1382
357
3.26
82.23
71.30
76.40
1.20
25
1850
267
2.43
82.93
71.90
77.10
1.25
100 (
a) 496
993
9.06
77.30
67.02
64.85
1.25
50
976
505
4.61
81.08
70.30
75.26
1.25
33
1443
342
3.12
82.35
71.40
76.52
1.25
25
1923
256
2.34
83.01
71.97
77.17
1.30
50
986
500
4.57
81.12
70.33
75.30
1.30
33
1471
335
3.06
82.40
71.44
76.57
1.30
25
1955
252
2.30
83.04
72.00
77.21
1.35
50
997
495
4.52
81.16
70.37
75.34
1.35
33
1482
333
3.04
82.42
71.46
76.59
1.35
25
1969
250
2.29
83.06
72.01
77.22
1.40
50
976
505
4.61
81.08
70.30
75.26
1.40
33
1452
340
3.10
82.37
71.41
76.53
1.40
25
1926
256
2.34
83.01
71.97
77.18
(a): Reference Case


190
reduce the probabilities of fuel dryout and the consequential
cladding failure.
5.2. Economic Evaluation
5.2.1.Introduction.
The technical aspects surveyed so far about the SHARP clearly
show its advantages over the standard plant in the aspects of
operational safety, once-through fuel utilization, high radioactive
waste storage and disposal, reduction of refueling outages and
associated personnel radiation exposure, etc. However, it is
evident that the SHARP can be viewed as an "oversized" plant,
requiring more fuel inventory than standard power density cores for
an equivalent amount of output power, as well as having some of the
plant equipment larger than would be necessary at standard power
densities for the production of a given amount of power. All these
considerations indicate that the capital cost of the SHARP will be
larger than that of the standard plant. There is, however, the
belief that the reduction of refueling time and improved fuel
utilization will help in reducing the impact of the increased
capital cost. It thus becomes obvious that an economical
comparative analysis is needed in order to asess the final cost of
energy produced by the SHARP versus the cost of power produced by a
standard plant.
The cost of energy produced by a nuclear power plant, is
typically broken down into three main components: capital cost,
fuel cost, and operations and maintenance cost. In the case of the
SHARP, it might be possible to think that the enhanced safety could


254
calculational technique includes several methods of acceleration
and extrapolation, such as Chebyshev polynomials for the fission
source iterations.
The code allows great flexibility to the user. Some of the
main user input parameters are the following:
a). Specification of number of energy groups, number of
regions, type of geometry, type of boundary conditions.
b). Selection of type of calculation: forward and/or adjoint
fluxes, search, request for energy group coalescence, etc.
c). Definition of the type and input of data (cross sections,
fluxes, neutron source, number densities, search materials, etc)
and format of data output.
d). Specification of iteration data convergence criteria,
and normalization values.
e). Geometrical definition of the problem.


29
With the pellet temperature profile expression in hand, the
pellet average temperature may be obtained as
o fR
Tave = (2 / Rz) T(r) r dr = To Jo
For the 20% linear power, preheater cores, the calculational
scheme was the same except for the temperature step from coolant to
clad surface, in which case a subcooled convection heat transfer
expression was used, such as the following:
AT = q" / h
where q1' = surface heat flow rate, Btu/hr sq.ft
h = forced convection coefficient, Btu/hr sq.ft deg. F
The forced convection coefficient may be calculated as
h = 0.0023 Re0*8 Pr0*4 (K/D)
where K = coolant thermal conductivity, Btu/hr ft deg.F
D = hydraulic diameter of the coolant channel, ft
Re = Reynolds number
Pr = Prandti number
Once the thermal calculations were completed, the standard
core fuel average temperature was checked against the vendor's
literature, and found to differ by a few degrees; this difference
is probably due to slightly different conductivity coefficients
across the core (due to temperature changes), which are not taken


se
a). The fuel burned in the "basic" codes must be imbedded in
a flux spectrum which is as close as possible to the one existing
in the actual reactor for the fuel being studied.
b). Boron concentration must be kept as close as possible to
its true value throughout the core life, in order to avoid
erroneous spectra that would affect both the criticality
calculations and the isotopic evolution.
c). The model must be able to follow a "real time" evolution
of the core, as opposed to using burnup as a time measurement,
because burnup is not uniform through the core life, and some
isotope appearing or being burned earlier or later in the core life
affects the spectrum, the criticality study, the isotopics, and
eventually the cycle length.
d). In order to accomplish the previous point, the model must
be able to assign the right power to the fuel at each point in
life, i.e., if a "real time" model is sought, it is essential to
have a correct time-power-soluble boron correspondence.
e). In order to obtain the correct batch power assignment at
all times during the fuel life, it is necessary to have a fair
estimate of neutron non-leakage probability for each batch, which
if at all possible, should be made automatically adjustable through
life.
With all these ideas in mind, the model was developed as
described in the next sections, starting with a fuel pin code,
going to a fuel assembly burnup calculation, and finally feeding
the data to the core-simulation code.


290
Z1=2.405*SQR( (NB-D/NB) :Zl=(Zl/2)-( (Zl[3)/16)+((Zlf5)/384) :SG=(1/(1-Z
1*1.84292*SQR( (NB-D/NB))) [2:SF=SG*R(CH,3)/UB
2760 FR=EXP(TA*B2)/(1+L2*B2)
2770 PB=1-((1-PR)*SF*0.7)
2780 R(CH,0)=R(CH,0)*PB:RC(If2)=PB:PRINTnMODIF FOR EXTERNAL BATCH
T-STEP"I"="PB
2790 NEXTI
2800 RETURN
2810 REM CALCULATING WORK K-INF FROM PRESENT-STATUS BUKNUPS AND
BORON CONCENTRATIONS.
2820 R(1,0)=R(0,11)+(WA*R(1,7))+(WB*(R(1,7) E2)/2) :IFTS<2THENRETURN
2830 FORI=2TOTS:X=R(1-1, 12):XL=0:XM=0
2840 FORJ=0TOTS:IFR(Jr10) 2850 IFR(J,10)>=XTHENXM=J
2860 IFXM=J AND JO0THENJ=TS
2870 NEXTJ
2880 IFXL=XMTHENPRIOT"WARNINGI! XMfXL III":STOP
2890 IFXM=0 AND XLOO THEN XM=TS:XL=TS-2
2900 IFABS(R(XL11)-R(XMf11))<0.001 AND XL>lTHENXL=XL-l:GOTO2900
2910
R(If 0)=R(XL,ll)+( (X-R(XL,10)) *(R(XM,ll)-R(XLfll) )/(R(XM,10)-R(XL,10))
) :R(I,0)=R(I0)+(WA*R(If7))+(WB*(R(If7) [2)/2) :NEXTI
2920 RETURN
2930 REM CHECKING FOR CONSISTENCY OF CYCLES' B
2940 CLS:PRINTTAB(15);nK CONVERGED. CHECKING BURNUPS."
2950 XX=0:RP=5:GOSUB2380


156
per unit energy generated. These values are also plotted in Figure
4.2.1, as the upper set of curves.
For any given pitch, the net fissile consumption shows very
small changes (about or less than one percent) as power level is
changed, although a slight trend can be seen of increasing net
consumption for lower power levels. As pitches are changed, the
net fissile consumption increases for more moderated pitches,
causing the once-through, burnup optimized pitch of 1.35 cm to
consume about 10% more net fissile mass as the present standard
pitch. These results are in agreement with the observations on
fissile mass discharge appearing on column 5 of this same table.
This presents again the controversial point that the 1.35 cm pitch
is an optimum point for burnup and ore utilization under the
assumption of a once-through fuel cycle. This is obviously not a
favorable pitch from the ore conservation point of view if a
reprocessing policy is implemented.
The eighth column on Table 4.2.1, labeled PU Dis. MT/MWd shows
the total amount of plutonium that is discharged from the core per
MWd thermal produced. This figure has obvious interest from the
nuclear weapon proliferation point of view. The figures have been
obtained with an expression identical to the one used for the
discharged fissile mass (shown in the fifth column of the same
table) by substituting Pu-240 for the U-235. The plutonium
discharge values are plotted in Figure 4.2.2 as the upper set of
curves.


23
Table 2.1.4. Identification of Cases for the Scoping Study.
Cass
# Enrich.
Mod.Temp.
Fuel Temp.
Poisons
Power
K-eff(*)
1
3%
293 K
293 K
0
0
1.3532
2
3%
583 K
293 K
0
0
1.2933
3
3%
583 K
1005 K
0
100%
1.2675
3A
2.6%
583 K
1005 K
0
100%
1.233
4
3%
583 K
700 K
0
25%
1.2770
4A
2.6%
583 K
700 K
0
25%
1.242
5
2.6%
293 K
293 K
0
0
1.3184
6
3%
293 K
293 K
1300 pm
0
1.1579
7
3%
420 K
520 K
0
20%
1.3304
8
2%
420 K
520 K
0
20%
1.2260
9
3%
583 K
1005 K
1856 PPM
100%
1.0781
* Calculated with PHFOG-BRT-MONA as shown in Sections 2.2.2
and
2.2.3


209
Another consideration of importance from the operations point
of view is the likelihood of reduction of annual allowed radiation
exposure doses for plant personnel, which would have a large effect
on the operation of nuclear plants. The exposure level can be
significantly reduced by the SHARP due to the reduction of
refueling and forced outages.
The SHARP concept appears clearly as having significant
advantages over the standard plant from the operations point of
view.
6.4. Economic Effects
The SHARP concept presents both positive and negative factors
from the economic viewpoint. The reduction of refueling outage
time is an asset because of better utilization of the large capital
investments resulting in a reduction of the net energy generation
cost. The replacement power cost was not taken into account in the
present economic study because of the uncertainties in determining
it in a long term extrapolation, which is heavily influenced by the
particular characteristics of each utility. In any case, this is a
factor that could further improve the SHARP energy production cost.
The increase of plant capacity factor was taken into account in
the economic performance studies.
The increase of discharge burnup level results in direct
although small savings in ore and fuel enrichment and handling
costs


I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Calvin C. Oliver
Professor of Mechanical
Engineering
This dissertation was submitted to the Graduate
Faculty of the College of Engineering and to the
Graduate Council, and was accepted as partial fulfillment
of the requirements for the degree of Doctor of
Philosophy.
June, 1981
Dean, College of Engineering
Dean for Graduate Studies and
Research


CHAPTER IV
DETAILED SHARP BURNUP OPTIMIZATION STUDIES .
Cas.es -Chosen for.Flnal Study
Once the burnup calculational method is developed and the set
of programs to be used are tested (as described in Chapter III), it
is necessary to define the burnup cases that have to be considered
for this study. It is important to realize that although the
burnup calculational scheme developed for this work requires a
reduced level of computational effort for a quality calculation, a
considerable amount of work is required to complete each burnup
study. It is not only necessary to complete the chain of
EPRI-CELL, NUFUNCHER, PDQ-7 and CRIBUR, but it normally takes two
or three iterations of PDQ-7 and CRIBUR before all the core cycle
data converges. This represents a significant volume of data
handling, and thus, it is important to avoid burnup calculations
that are not required for the purpose of the study.
Since the main concern of this study is to determine the
effects of reduced power densities on the neutronics, fuel
utilization and general isotopic behavior of the reactor, it is
obviously necessary to examine several power density levels. Two
different plant arrangements were considered as possible in
Chapter I for the low-power cores; the main designs were the
multiple-core plant with or without preheater core, and the
143


85
Pin Cell
K oo
Figure 3.1.4. Grouping of Pin Cell Data into Core Data.


56
Figure 2.3.4. Eta and Fast Fission Factor vs. Pitch


191
result in some monetary savings due to reductions in licensing
times, in plant construction, etc. but these are rather
speculative, and they are considered out of the scope of this
report. The economical comparison presented here is based on the
costs of fuel, capital and operations and maintenance of several
different power density plants. A more detailed definition of the
sizes of the main plant components for each reduced-power density
plant is given in next section.
5.2.2. The Economic Comparison.Studies.
The economic comparison studies are presented for a set of
five different plant output powers. They correspond to 35%, 50%,
60%, 70%, and 100% of the standard plant power as defined in
Chapter II. Each of the plants considered in this study is
composed of a full size Nuclear Steam Supply System (NSSS) (as used
for the standard plant); a Balance Of Plant (BOP) of the nominal
size corresponding to the plant output power; and finally the
containment building sized for 120% of the size that would
correspond to a standard power density plant of the same power
rating. All plants are assigned a 30 years life, and startup is
assumed in 1993. Costs are levelized to and expressed in 1993
dollars,
Three possible inflation scenarios are considered for each
case studied. The low inflation scenario assumes an inflation rate
of 5% per year; it assigns an interest rate of 7% for borrowed
funds and bonds, and a 13% rate of return is assumed for common
equity (stocks). The moderate inflation scenario assumes a yearly


50
main points of view: they represent additional energy obtained from
the same initial ore, and they entail longer core cycles, which
tends to reduce costs of refueling outages.
The expected burnups shown by the 20% power, preheater core
are really surprising. The better moderation boosts reactivity in
such a way that for 3% enriched fuel, a burnup increase of 32%
above that of the standard core would be expected. If the
preheater core is loaded with low enriched fuel, e.g. an enrichment
of 2%, it would still yield 88% of the burnup expected from a
standard reactor loaded with 3% enriched fuel.
However, it should be kept in mind that the burnup results
obtained from these LEOPARD calculations do not take into account
the significantly different neutronic situation of the various
batches present in the core, which will definitely affect the
isotopic dynamics, and therefore the core life. In any case, these
figures give good hope for better fuel utilization by the SHARP'S.
Table 2.3.1. Core Reactivities at B.O.L.
Case #
ENRICH
K-eff.
Xe worth
BOL reactivity
3
3%
1.268
-3.5%
23.3%
3A
2.6%
1.233
-3.5%
19.8%
4
3%
1.277
-2.3%
25.4%
4A
2.6%
1.242
-2.3%
21.9%
7
3%
1.330
-2.1%
30.9%
8
2%
1.226
-2.1%
20.5%
9>.


6
and operational characteristics. A higher plant capital cost per
unit of installed power is expected (1) because of the losses
associated with the economics of size, i.e. because of the
additional investment in the "oversize" Nuclear Steam Supply System
(NSSS) components However, the overall cost of energy production
would be competitive because of the improved operating conditions
and safety.
Note that although the power density (and thus, the heat flux
across the fuel elements of the core) is reduced, the thermodynamic
conditions of the coolant would be maintained at the same level as
in the standard plants to avoid loss of thermodynamic efficiency of
the plant. This can be easily achieved by reducing the coolant
flow across the core, within established heat transfer limitations.
Reducing the heat flux in the fuel while maintaining standard
coolant conditions reduces the temperature of the fuel, which
improves fuel/clad thermal conditions, reduces stresses in the
fuel, and hopefully reduces fuel pin failures.
The advantages that can be obtained from the SHARP are:
a). From the heat transfer point of view:
- Lower temperatures in the fuel pellets, due to the
lower heat flux. This would imply reduced thermal-related damage
to the pellets and to the cladding, and therefore, reduced pin
failures.
- Reduced probability of reaching critical heat
transfer conditions


216
The core life terminates at end-of-life (EQL) because of two
basic reasons: neutron economics (neutrons are lost by leakage,
parasitic absorption, or both) and lack of fissile material (the
fissile material is burned out and any bred fissile material is
insufficient after awhile to outweigh the neutron losses). Any
design scheme aimed at extending burnup must modify these
conditions which affect the reactor, especially at EOL.
a). Reducing parasitic capture. A nuclear reactor has a
number of materials that absorb neutrons preventing them from
causing fissions in the fissile material. These are structural
materials (cladding, grids, barrel, etc.), poisons (both control
rods or residuals from burnable poisons), fission products
(produced as the fuel is burned), the fuel itself and the coolant
(either borated or unborated, the coolant presents some neutron
absorption). A basic scheme to increase burnup could involve the
reduction of the parasitic absorptions in the core. This can be
done in different ways, although not all of them may necessarily be
applicable in a real reactor, since relative burnup improvement
must be weighed against any required redesign and secondary
effects.
Parasitic absorption by fission products may be reduced,
obviously, by reducing the amount of fission products present at
EOL. This can be done by shortening the reactor cycle life and
reducing the reload batch size. This way, at EOL there is more
"low burned" fuel, and therefore less absorbing fission products.
Another way of achieving the reduction of fission products in the


80
The second step uses such a wide range of calculational
approaches or methods, that it is difficult to specify the general
input requirements and output data. As a general rule, the
explicit core model (in which the fuel is represented either pin by
pin or in relatively small nodes, and control materials are
specifically treated) needs the initial isotopics information, the
microscopic cross section libraries generated by the first step of
the burnup calculation, the core geometry data, the power history,
the control materials history, and general editing information,
while the output includes core isotopics in zone averages, pin by
pin, or whatever region type is used by the code versus core
history; criticality evaluations, power distributions, burnup
information in average or explicit for each unit considered in the
core, region averaged multigroup neutronics data, etc. Codes of
this type are for example FDQ-7(17), SIMULATE(18), etc.
Figure 3.1.3 shows the geometry and material zones
specifications needed for a two-dimensional calculation of a
quarter assembly of a FWR with FDQ-7. Note the extreme detail of
the geometrical description, and the relatively large number of
different compositions considered. It is easy to imagine the
extreme complexity of the calculation of a whole core in three
dimensions, with the necessity to identify different fuel batches,
control rod assemblies, burnable poison rods assemblies, assemblies
with water holes, etc. Nodal codes avoid some of the complexity
and reduce computational time by lumping each fuel assembly into a
few nodes, but they require node-interaction parameters and albedos


211
plant, and with a generation cost increase of about 20%, with a
slight tendency to decrease for higher inflation rate scenarios.
^-Suggestions ^for Future Research
The calculations performed in this study showed the
significant potential for operational and safety improvements in
PWR plant performance under the SHARP concept. Steady-state
thermal-hydraulics calculations showed an important side of the
improved safety of this concept. However, no transient or accident
scenario calculations have yet been carried out, and the particular
characteristics of the SHARP indicate the possibility of even more
significantly improved parameters under transient or accident
situations.
Fuel performance can be further improved with the use of new
fuel designs (like hollow fuel pins) which will see their improved
properties upgraded by the better thermal conditions of the
SHARP'S. A new parameter should be introduced in the neutronic and
economic studies of the SHARP: this parameter is fuel enrichment.
It has been shown(3) that increased fuel enrichments (up to about
5% U-235 enrichment) result in improved fuel utilization. This
fact, theoretically verified for standard power level cores, can be
synergistically potentiated with the demonstrated better burnup
capabilities of a reduced-power core. In the economic field, this
will impact in the conflicting factors of reduced refueling outages
and increased fuel interest costs, which will require new
evaluation and optimization.


197
fuel cycle cost vs. plant size, for the three inflation scenarios
being considered.
The operations and maintenance (O&M) costs are considered
constant for all the plants studied. They are fixed as 7.18
mills/KWh for the low-inflation scenario, 11.25 mills/KWh for the
moderate-inflation scenario, and 17.34 mills/KWh for the
high-inflation scenario. These values have been calculated from
present average values for nuclear plants, and modified with an
inflationary factor. Note that the O&M costs should be dependent
on plant size, since the burnup studies showed drastically
different core cycle lives depending on the power density level.
However, the O&M part of generation cost is a very small proportion
of the total power generation cost, and the speculative differences
that could be applied would not affect the final generation cost
but by some tenths of a percent. It should be realized, however,
that the longer cycles and reduced refueling outages have been
taken into account in the evaluation of the fuel cycle costs. One
element that has not been considered is the cost of replacement
power for the different refueling downtimes. Although their cost is
presently very high due to the fact that most replacement power is
produced by oil or gas units, it is difficult to forecast their
cost at significantly later times. In any case, this could account
for an additional one or two percent reduction of power generation
cost for the low power density plants, which has not been accounted
for in the figures presented here.


APPENDIX C2
SAMPLE RUN OF CRIBUR
Cribur was first developed for use on a microcomputer, and for
ease of operation, it was designed for interactive use. Therefore,
all information to and from the computer is on the CRT screen,
except the final output, which is printed on hard copy. The
following listing is a reproduction of the screen messages that
appear on a sample run of the code, and of the final printout.
During execution, the code prints sets of data on the screen that
indicate the progress of the calculation and the convergence of the
iterations. These data are not presented here.
Screen 1:
PROGRAM FOR CRITICALITY, BORON WORTH, BATCH POWER
AND CYCLE LIFE EVALUATION OF A MULTI-BATCH PWR.
DATA INPUT FROM (K)EYBOARD OR (D)ISK ? D
FILESPEC ? HP50A252
Screen 2:
CHECK TIMESTEP ENDING TIMES. IF ANY ONE TO BE CHANGED, INPUT
TIMESTEP NUMBER. OTHERWISE, JUST PRESS "ENTER".
TSTEP 1 ENDS AT 200 HR TSTEP 2 ENDS AT 2000 HR
TSTEP 3 ENDS AT 6000 HR TSTEP 4 ENDS AT 12000 HR
293


186


145
Given a power density, there are a number of parameters that
can be allowed to vary and therefore are susceptible to
optimization, without changing much the design of the core. Some
of these parameters are: lattice pitch, fuel pin diameter, fuel
enrichment, burnable poison loading and/or management,
consideration of gadolinium-loaded fuel pins, in-core fuel
management scheme, etc. Some of these parameters are already being
considered in industry studies directed to improve fuel
utilization, and their use is discussed in greater detail in
Appendix A. It is obvious that the optimization of all these
parameters is well beyond the scope of this work. It is then
necessary to choose the best set of parameters that can be used for
core optimization within the established constraints of these
studies.
The main criteria necessary for the selection of the
parameters are the reliability and sensitivity of the calculational
methods to the parameter(s) and the acceptability of the change by
the industry.
One parameter that could be changed or modified is the in-core
fuel management scheme. The calculational method developed in
Chapter III is able to accept such change, but the alternative
mangemert schemes (v.g. in-out-in, explained in Appendix A) are
still considered problematic in the industry, and they would have
little probability of being accepted. Therefore, this study does
not consider any variational in-core fuel management scheme, and


58


140
3.3.3. Power Distribution Benchmarking.
Since it is not possible to obtain actual accurate equilibrium
core compositions for the type of reactor modeled in this study
(Westinghouse's 4-loopf 17X17 pin assembly) because there is not
yet any such reactor in an equilibrium cycle, the benchmarking of
the batch-wise power distributions calculated by CRIBUR was done by
comparing its results for the standard case with the batch-wise
power distributions of several Westinghouse cores in their
equilibrium cycles.
atypical power distributions obtained by CRIBUR yield relative
powers of 1.02, 1.09 and 0.88 for the three batches in the core,
from newest to oldest respectively, with no burnable poisons
present in either of them.
The power distributions observed in seme real cores in their
equilibrium cycles are (from newest to oldest batch respectively):
0.99, 1.05, 0.94 for Zion II ; 0.96, 1.13, 0.9 for D.C. Cook ; and
0.97, 1.15, 0.88 for H.B. Robinson 2. Core-following calculations
performed by utilities show assembly-wise power level errors of a
few percent, reaching sometimes 10%. EPRI's procedures manuals
list as acceptable an error of 5%.
Obviously, when the partial power of all the assemblies are
added up into batches, the calculations come closer to the real
values, but this is achieved with pin-by-pin or nodal
multi-dimensional calculations of a degree of sophistication far
superior to CRIBUR's. The results obtained by CRIBUR tend to
assign a few percent more power than real to the fresh batch, and a


49
power cores relative to the standard core. One simple way of
making such estimates is to calculate the total reactivity of the
cores at B.O.L., and to assume that reactivity is lost linearly as
core burnup increases, which is a reasonable approximation for a
first-estimate calculation.
The most significant parameters altering the core reactivity
at B.O.L. when the fuel enrichment, core geometry and moderator
conditions are fixed, are the fuel temperature and the equilibrium
xenon concentration. Fuel temperature affects the core reactivity
in the amounts determined by the Doppler coefficient, explained
earlier in this section. The reactivity worth of xenon in the
different cores can not be obtained from the LEOPARD burnup
calculations performed in the scoping study, but it can be obtained
from the vendors literature (4). Combining these xenon reactivity
modifications with the core effective multiplication factors listed
in Table 2.1.4, it is possible to obtain a comparison of the
reactivity available at B.O.L. for each core, and therefore, an
expectation of their comparative discharge burnup levels. These
figures appear on Table 2.3.1 for the standard core, the 25% power
level core, and the 20% power level preheater core. Two fuel
enrichments are considered for each power level.
Using the linear reactivity assumption (15), Table 2.3.1 shows
that a 25% power SHARP may yield a discharge burnup 10.6% above
that of a standard reactor, for an average fuel enrichment of 2.6%
U-235 in both cores. The burnup increase of 9% appears for a fuel
enrichment of 3.1%. These burnup increases are desirable from two
*u.\


42
or control rods, calculated by the PHROG-BRT-MONA scheme, as
explained in sections 2.2.2 and 2.2.3.
The core effective multiplication factors obtained for the
different cores are used to calculate bulk reactivity coefficients.
Thus, the average moderator temperature reactivity coefficient
(M.T.C.) is -20.67 p.c.m./deg.K of moderator temperature variation
for the temperature range between 293 K and 583 K. This rather
large reactivity coefficient would be the main asset of a core such
as core #8, which would use very lew enriched fuel or even spent
fuel discharged from a standard plant. A reduction of average
moderator temperature of 163 K, as described on table 2.1.4, would
provide 3.37 % of extra reactivity as compared to an identical
core working with normal moderator temperature. This excess
reactivity would enable the core to work with the less reactive
fuel or burn the standard fuel beyond the current discharge burnup
levels. However, it should be kept in mind that these preheater
cores are in some way a technical speculation, and that the real
interest of this study is centered in the low-power cores with
standard moderator conditions, which enables than to drive a
standard balance of plant in a single-core arrangement.
As the average linear power produced by the fuel is varied
according to the different power densities of the SHARP cores, so
is the average fuel temperature, even though the moderator
conditions remain unchanged. This variation of fuel temperature
causes a change of the absorption resonances width, due to the
Doppler effect, and this causes a change of the core reactivity.


LIST OF TABLES
Page
2.1.1. Main Core Parameters for the Standard Reactor. ... 19
2.1.2. Basic Fuel Cell of the Standard Reactor 19
2.1.3. Basic Cell Composition. Core #1 21
2.1.4. Identification of Cases for the Scoping Study ... 23
2.1.5. Tabulation of Case Number Densities Which Differ
from Those of Case #1 24
2.2.1. Basic Cell Temperatures vs. Power Level 30
2.3.1. Core Reactivities at B.O.L 50
2.3.2. Scoping Burnup and Isotopic Results 65
3.3.1. CRIBUR and Batch-averaging Method Criticality Errors 137
3.3.2. Effects of Code and Data Alterations on Core K-eff. 139
4.2.1. Burnup Achievements, Fissile Isotope Usage and
Ore Usage ...... 149
4.2.2. Plant Operational Data Comparison 161
4.2.3. Ore, Enrichment and Plutonium Normalized to 1125 MWe 176
5.2.1. Capital Costs (Mills/KWh) and Percent Increases
over Standard Plant 193
5.2.2. Fuel Costs (Mills/KWh) and Percent Increases over
Standard Plant 193
5.2.3. Generation Costs (Mills/KWh) and Percent Increases
over Standard Plant 199
A.2.1. Discharge Burnups of Low Power Density Cores .... 239
D.l. Cell-Homogenized Number Densities
(Atoms/Barn cm X l(r) 301
D.2. Average Cell Neutron Velocities (eV) 302
viii


40
Critical
Boron
Concentration
ppn
2500
2000
1000
GWd/MT
Figure 2-2.4. Comparison of Boron Letdown frcm LEOPARD and BURNCJP.


21
Table 2.1.3. Basic Cell Composition. Core #1.
Isotope
Region
Pure # Dens.(*)
Cell Avg.
# Dens.(*)
H
4
6.7 E-2
3.6957
E-2
0
4
3.35E-2
0
1
4.4009 E-2
3.3152
E-2
U-235
1
6.6830 E-4
2.2281
E-4
U-238
1
2.1337 E-2
7.1137
E-3
Zr
3
4.2808 E-2
4.345
E-3
Ni
3
0... E-10
0...
E-10
Sn
3
4.8556 E-4
4.9285
E-5
Fe
3
1.4946 E-4
1.517
E-5
Cr
3
7.6426 E-5
7.7573
E-6
He
2
1.9 E-3
2.565
E-5
* Units are atoms per barn-cm.


78
Soluble
Boron
Concentration
Figure 3.1.2. Exanples of Burnup Conditions and Timestep
Specification for Fuel Cell Burnup Calculation.


165
have described the burnup levels achievable and their associated
power generation times. However, these are full-power generation
times, and it is now necessary to turn than into actual plant
times. In a standard plant, the real time or Total Cycle Time and
the full-power time are related through the Capacity Factor, so
that
FPT = CF TCT
where FPT = Full-Power time.
In the case of the low power density cores, it is not possible
to use the same concept, since the CF is unknown because of the
changing weight of the refueling outage with the different powers
and degrees of burnup. However, the OCF defined above can be used
to overcome this problem. Using the last two equations,
FPT = CF TCT = OCF ATAF (TCT ROT)
Since OCF is considered constant, as well as ATAF and ROT,
while FPT is obtained from the burnup studies, it is possible to
obtain the total cycle time of any plant as
TCT = ( FPT / (OCF ATAF) ) + ROT
The FPT tabulated in Table 4.2.1 is in hours (from the burnup
calculations) while the TCT is more conveniently expressed in days.


116
If n is the number of batches present in the core, recalling
that the first zero of the Jo function is at 2.405 and that all the
batches in the core are assumed to have the same volume, it is
possible to express the radius Xo which will separate the inner
zone from the peripheral batch (see Figure 3.2.4) as:
Xo = \l(7T*(2.405)2 /n) ((n-l)/7T) or
Xo = 2.405 \](n-l)/n
Figure 3.2.4. Bessel-shaped Flux Distribution.
It is now necessary to obtain the ratio of SI to S1+S2.
Recalling that the integral of the Jo Bessel function can be
-x
2 7T x Jo(X) dX = 2 It x Jx (X)
obtained as


31
into account in the thermal calculations. Since this difference
was not significant from the nuclear point of view, all the results
were modified slightly to maintain consistency with the vendor's
data in the neutronic calculations. The consistent data permit the
further benchmarking of the nuclear calculations. Table 2.2.1
shows the main temperatures involved in the calculation for all the
different power levels studied. All temperatures are in degrees
Fahrenheit, unless otherwise specified.
2.2.2. Neutron Cross Section Calculations.
Section 1 of this Chapter described the set of cores chosen
for the scoping observation of the differences between a standard
core and a low power density core. It is necessary to gather a
series of codes of known reliability which can yield an accurate
neutronic picture of a LWR core. Parametric studies can then be
conducted which will define significant factors differentiating a
SHARP from a standard plant. In these parametric studies, the
absolute accuracy of the results is not as significant as their
relative values which are used to establish figures of merit.
The best neutron cross section calculational method available
for the scoping study involved the use of several standard codes.
This method allows great flexibility in specifying isotopic number
densities, geometries, etc., but it requires a relatively large
amount of manual data handling from code to code. This section
briefly describes these codes.


225
but has not been used yet in PWR's as far as the author knows,
although some tests are being conducted.
The reactivity worth of gadolinium-loaded fuel pins and its
evolution through core life are much harder to calculate accurately
than the BPR's and they also require increases in fuel fabrication
cost, because more types of fuel are needed and more complex and
careful control is required during manufacture. On the other hand,
Gadolinium-loaded pins present some substantial potential
advantages such as:
a). Complete burning of the poison by EOL is more likely
because of the poison being mixed with the fuel itself.
b). No residual absorption due to extra structural materials
is present.
c). No fuel loading is lost, since Gadolinium is imbedded in
the fuel pellets.
d). No power shape distortions due to water holes that are
present in following cycles.
Gadolinium-loaded fuel pins seem to have a promising future in
FWR applications. A review all the functions accomplished by the
burnable poisons follows.
a). Since burnable poisons are added to the most reactive
fuel elements or to the regions where power would peak, their
presence causes a strong reduction of the core reactivity. The
effect of the poisons is maximum at BOL, since they have not been
burned yet. This effectively reduces the amount of chemical shim
needed for criticality of the core, which is an interesting effect,


134
limiting local power peaking effects. Unfortunately, if a PDQ-7
1/4 core model is prepared with data identical to that used in a
CRIBUR run, each batch has one single composition which is assigned
to all of its pins, and no differences between pins with their
"smoothing" effect on power distributions exist. The compositional
differences between neighboring pins from different batches are
sharper than they would really be in an actual core, causing
exagerated effects on the power peakings. This problem invalidates
the FDQ-7 1/4 core model for the purposes of benchmarking power
distributions and burnups, although it can still be used as a
redundant reference for benchmarking of the criticality
calculations of CRIBUR, since the overall core multiplication
factor is not affected by the somewhat inexact power peaking
analysis.
A PDQ-7 run for a 1/4 core configuration, even used without
PDQ's burnup features (i.e. just for criticality and power
distribution "snapshot" analysis) presents the additional
difficulty of its involved input for a complex geometry that
includes several different composition zones. This is primarily a
problem of skill and time, and can still be reasonably viewed as a
method for benchmarking purposes.
The benchmarking references described so far allow for a good
checking of the criticality calculation part of the code, since
both the PDQ-7 1/4 core batch-homogenized model and the
real-reactor B.O.L. H.F.P.. A.R.O. give good values of the
multiplication factor, and it is easy to obtain the necessary core


296
3 3 8 15 O O 3
4 4 10 16 O O 4
5 5 12 17 0 0 5
ANY CHANGED STEP AND COLUMN ?
Screen 7:
CHECK BATCH WITH LBP AND LBP'S BATCHWISE WORTH (%K/K) AT BOC, EOC
BATCH 0 BOC %K/K= 0 EOC %K/K= 0
NEW VALUES FOR ALL OR JUST PRESS "ENTER" ?
CHECK POWER LEVEL AND FUEL LOADING
POWER= 1700 MW THERMAL FUEL LOADING- 94.18 MTU
NEW VALUES FOR BOTH OR HIT "ENTER" ?
Screen 8:
CHECK BORON WORTH EVALUATION POINTS
STEPS TO BE KILLED AFTER B-WORTH EVALUATION (IN INCREASING ORDER):
9 11
TIMESTEPS FOR EVALUATION:
6 7
9 10
11 12
WISH TO INPUT A NEW SET Y/N ? N
Screen 9:
DO YOU WANT TO STORE DATA ON DISK? Y/N ? N


136
The PDQ-7 1/4 core criticality tests were performed at four
different points in the life of an equilibrium core: at B.O.L., at
approximately 1/3 of core life, at about 2/3 of core life, and
around E.O.L. In all the cases tested, the effective
multiplication factor calculated by CRIBUR differred from PDQ's by
less than 1%, and in two out of the four cases the difference was
under 0.3% .
In an attempt to check the importance of some of the
calculational features included in CRIBUR and not normally present
in other simple burnup schemes, the same four cases were
recalculated in a version of CRIBUR in which the calculation of the
leakage probability for the outer batch was bypassed. In all four
cases the error of the multiplication factor as compared to PDQ's
was over 4%, having increased from the original CRIBUR value by
about 3.5 to 4%.
Another check of the accuracy achieved by CRIBUR as compared
to other simple schemes was done by calculating the core effective
multiplication factor (K-eff.) as an average of the batches'
infinite multiplication factors (K-inf.'s), as indicated by
Graves(28). This K-inf. averaging method yielded errors between
2.6 and 3.7% with respect to PDQ's results, which represented
errors about 2.5% higher than the ones incurred by CRIBUR in the
representation of identical cases. Table 3.3.1 shows the values
obtained for all the benchmark and comparative cases described
above.


45
variation of soluble boron concentration would mean different total
amounts of boron being added to or retrieved from the core, which
implies that the soluble boron reactivity coefficient may also
change for different lattice geometries.
Soluble boron worth variations are analyzed as a function of
boron concentration, core burnup, core power level, moderator
density, and basic cell geometry. These calculations were based on
the LEOPARD scoping core burnup calculations.
When adjusting soluble boron concentration to keep the reactor
critical through life, the worth of boron decreases slightly from
the B.O.L. until about 2000 or 3000 MWd/MTU and increases
thereafter, with a very slowly increasing slope. This variation is
shown in Figure 2.3.1. This figure shows also that the variation
of power level, without variation of moderator conditions, causes
practically no change in the boron reactivity coefficient. On the
other hand, either a variation of pitch or a change of moderator
density (temperature) cause sharp changes in boron reactivity
worth. In both cases, as expected, boron reactivity worth
increases for the changes that imply a larger absolute amount of
boron present in the core (i.e. for an increased pitch or for an
increased moderator density).
Figure 2.3.2 shows more clearly the effect of lattice pitch on
boron worth. It is also more evident that the variation of boron
worth at critical through life is larger for larger pitches. The
cause of the monotonical increase of boron worth with life is due
to two adding effects: the self-shielding effect of boron itself


252
c). The code can perform a blackness calculation for the
generation of diffusion theory constants for thin, highly absorbing
regions.
d). The code provides options for user input of cross
sectional and/or source data. It is also capable of punching the
group-collapsed data in card formats usable by neutron diffusion
theory codes such as MONA (10) or CORA (35).
The most significant data input to the code by the user are
the following:
a). Indication of the type of calculation desired: PI, Bl,
cell problem with supplied flux and current spectra, or blackness
calculation.
b). Type of source required: Input by the user, or choosing
among a selection of source spectra available from within the code.
c). Definition of number of broad groups desired and number
of isotopes in the calculation.
d). Definition of the problem geometry. It can either be a
slab, cylindrical or spherical geometry.
e). Definition of the isotopes used, their number densities,
and the properties of the related scatterer isotopes.
f). Leakage factors and buckling.
g). Isotopic information for resonance calculations.


257
cross section tables are produced for the number of energy groups
specified by the user.
c). Fast and thermal fluxes, average neutron velocity,
group-wise and total multiplication factors, etc.


22
discharged from standard FWR plants. This core was thought of as a
possibility for further use of standard plant spent fuel.
Table 2.1.4 identifies the cores used for the scoping study
with their main distinguishing characteristics. The
characteristics of these cores were selected in order to obtain
indicative figures on reactivity coefficients and burnup
variations. Each core has a case I.D. assigned to it, which is
used for future reference. The power levels indicated are in
percent relative to the standard core's full power (6 KW/ft. or
3400 MWth. total core power).
In all cases, both the fuel and the moderator are treated as
having a uniform temperature throughout their masses. In the
reduced power density systems, the moderator temperature is always
kept at the average coolant temperature of the standard plant core,
in order to maintain the thermodynamic characteristics of the steam
cycle, and the thermal efficiency of the plant. The only cores
with different coolant temperature are the ones intended for
preheater operation. The fuel average temperatures depend
obviously on the linear power density, and the values used for each
of the scoping study cores are shown in next section.
Table 2.1.4 shows also the core effective neutron
multiplication factor (K-eff.) obtained for each of the scoping
study cores in a BOL, clean, unrodded situation. The calculational
procedures used for the scoping study are different than the ones
used for the detailed burnup study, and they are specified later in
this chapter.


164
the Operating Capacity Factor. This is another factor that is
hereby considered to remain constant for all types of plants for
comparison purposes. The operating Capacity factor (which will be
referred to as OCF from here on) can be defined and calculated from
the assumptions stated above for the standard plant in. the
following way:
From the definition of ATAF, the on-line time is
On-line time = ATAF (TCT ROT)
where ATAF = Active Time Availability Factor.
TCT = Total Cycle Time.
ROT = Refueling Time.
Then, using the concepts of OCF and the conventional Capacity
Factor (herein referred to as CF), it is possible to write
OCF ATAF (TCT ROT) = CF TCT
Fran where OCF can be obtained and calculated using the
assumptions stated for the standard plant:
OCF = (CF TCT) / (ATAF (TCT ROT)) = 0.867
where OCF = Operating Capacity Factor (Capacity Factor of on-line
time).
CF = Conventional Capacity Factor (Including all time).
One significant factor that can differentiate the plants is
the length of the real cycle. Op till now, the burnup calculations


237
long cycle length. Figures A.2.5 and A.2.6 show the results of the
study performed by Combustion Engineering.
A.2.5. Low Power Density.
Reducing the power density of the core is another means of
improving burnup. It also has the very desirable side effect of
greatly increasing the safety, relieving the general operating
conditions and hence reducing the problems of operating the
reactor.
Reducing the power density of the reactor does not necessarily
imply a reduction of the coolant temperature which would result in
a reduced thermodynamic plant efficiency. In the reduced power
density cores, the coolant is kept at the standard operating
temperature by altering the flow conditions, and thus, the plant
efficiency remains unaltered. The increase in safety of the
reactor is due to the reduced surface heat flux (which makes
reaching critical heat transfer conditions more unlikely) and to
the reduced heat storage and temperature of the fuel.
There are two main factors contributing to the burnup
extension of a low power density core: The reduced Doppler effect
and the reduced Xenon concentration. This modification can
contribute significantly to the relief of several thermalhydraulic
and operational problems, while presenting no new technological
obstacles. However, it has the disadvantage of not being
retrofittable (unless current plants were to under-use much of
their balance-of-plant equipment) and of requiring a higher fuel
inventory per installed power. On the other hand, it presents a


166
Substituting the values obtained for OCF and ATAF allows the above
expression to be written as
TCT (days) = ((FPT (hours) / 0.74) + 1080) / 24
This total cycle time in days is listed in the third column of
Table 4.2.2 for all the cases under study, and is also plotted in
Figure 4.2.3 for easier examination and comparison purposes.
Note first that the cycle length shown in Table 4.2.2 for the
standard reactor is more than one year. The reason for this is
that the burnup levels obtained for the discharged fuel from the
calculations are higher than the actual burnup levels of the fuel
discharged from real reactors. There are several reasons for this
fact. First, the calculational model assumes a slightly high fuel
enrichment (later figures seen to point to 3.0% enrichment instead
of the 3.1% considered in this study as the equilibrium cycle fuel
enrichment). Second, the burnup study assumes uniform burnup of
the fuel; the actual fuel burnup distribution is far from being
uniform, and this causes the calculational model to sustain
criticality somewhat longer than the real core. It should also be
noted that the actual average equilibrium cycle time for FWR's in
the U.S. during the period 1973 to 1979 was somewhat greater than
one year. In any case, even with seme possible absolute error
associated with the actual values of cycle length, the relative
comparison between the different cores should be accurate, since
the bias applies to all cores in the same sense.


157
From this point of view, the optimized pitch has an advantage
over the current standard reactor. For each particular pitch, the
amount of plutonium discharged is reduced a few percent as the
power level is cut down. This is due to the softer neutron
spectrum existing in the lower power cores caused by the reduced
Doppler effect and the lower xenon level. As the pitch is
increased, the discharge of plutonium is sharply reduced. Thus,
the 50% power, 1.35 cm pitch core yields almost 16% less plutonium
than the standard reactor for the same energy production. The
significant variation in the yield of plutonium associated with the
pitch change is obviously due to the spectrum shift caused by the
increased amount of moderator present in the larger pitches.
The last column on Table 4.2.1, labeled U308 fET/MWd shows the
ore use for each of the cores on a per-unit-energy basis, under the
assumptions of once-through fuel cycle, uniform enrichment to 3.1%
U-235, and a tails enrichment of 0.2% Using standard tables for
feed-to-product ratio in the enrichment cascades (28) the ore use
can be obtained with the expression
Ore usage (MT/MWd) = 6.68 / BU (MWd/MTU)
The results for this column are also plotted on Figure 4.2.2,
as the lower set of curves. As could be expected, in this case the
best burnup cell presents the best ore utilization. Thus, the 50%
power core, 1.35 cm pitch uses 5.1% less ore than the current
standard reactor, for the same generation of energy. For each
particular pitch, the lower power cores show better ore utilization


154
The sixth column on table 4.2.1, labeled Rec. Fis. % Dis/Ld
shows the percent value of the weight of discharged (i.e.
recyclable) fissile nuclides as compared to the weight of loaded
fissile material. The values obtained for this column are plotted
as the lower set of curves in Figure 4.2.1, for easier
interpretation. Since the only fissile material loaded in the core
is -235, this column can be easily obtained by adding the number
densities of the discharged fissile species multiplied by their
respective atomic weights, and dividing by the same figure,
referred to the loaded U-235.
This column shows two main facts: first, it shows more clearly
the tendencies observed in the previous column, and second it
points out that the mass of discharged fissile materials is between
one third and one half of the mass of fissile material initially
loaded into the core. This implies the large potential economical
value of the discharged fuel when incorporated into a spent fuel
reprocessing policy.
The seventh column of the table, labeled Net Fis. MT/MWd
indicates the net mass of fissile material spent per MWd thermal
produced. In this case, net mass stands for the difference between
the fissile mass loaded into the core and the fissile mass
discharged with the spent fuel. Subtracting the fissile mass
discharged from the reactor from the mass of U-235 loaded with the
fresh fuel yields the net usage of fissile mass, and dividing it by
the burnup level yields the desired value of net fissile mass used


280
590 F0RI=1TOTS:PRINT"FLUX OP TB1ESTEP"I; :INPUTR(I,3) :NEXTI
600 CLS: PRINT "CHECK TIMESTEP ENDING TIMES IF ANY ONE TO BE CHANGED,
INPUT TIMESTEP NUMBER. OTHERWISE JUST PRESS 'ENTER11 ":PRINT" "
610 FORI=1TOTS:PRINT"TSTEP "In ENDS AT nR(I,4)n HR",:NEXTI
620 CH=0:INPUT"ANY CHANGED STEP # ";CH:IFCH=0THEN64G
630 INPUT"NEW VALUE"?R(CH,4):GOTO600
640 CLS:PRIMTnCHECK BORON PPM FOR EACH STEP. ENTER STEP TO BE
CHANGED, OR 0":PRINT" "
650 F0RI=1T0TS:PRINT"TIMESTEP "I" HAS "R(I,6)n PPM",:NEXTI
660 INPUT "ANY CHANGED STEP #";CH:IFCH=0THEN680
670 INPUT "NEW VALUE";R(CH,6) :CH=0:GOTO640
680 CLS:PRINT"CHECK K FOR EACH TIMESTEP":PRINTn
:PRINT"TSTP";TAB(6)"K-INF";TAB(30)nTSTP";TAB(36)"K-INF"
690
FORI=2TOTSSTEP2:PRINTI-1;TAB(5)R(I-1,1);TAB(30)I;TAB(35)R(I,1) :NEXTI
700 CH=0:INPUT"ANY CHANGED STEP";CH:IFCH=0THEN720
710 INPUT"NEW K,";R(CH,1) :GOTO680
720 CLS: PRINT "CHECK RELATIVE PCWER FOR EACH STEP. ENTER STEP TO BE
CHANGED, OR 0"
730 FORI=lTOTS: PRINT "TIMESTEP "I" HAS RELPOWR="R(I,2) :NEXTI
740 INPUT"ANY CHANGED STEP";CH:IFCH=0THEN760
750 INPUT"NEW VALUE";R(CH,2):CH=0:GOTO720
760 CLS:PRINT"CHECK CORE BATCH COMPOSITION IN CYCLE.":PRINT"ENTER
STEP AND COLUMN TO BE CHANGED OR 'ENTER' lIWICE.":A$="########"
770 PRINT" STEP BATCH-1 BATCH-2 BATCH-3 BATCH-4 BATCH-5
OUTER-BATCH"


261
c). Volumetric fraction and density fraction of each
composition in each material zone.
d). Number of timesteps desired for the total calculation;
duration of each timestep, power level associated with each
timestep and soluble boron concentration.
e). Option flags determining buckling search, correction for
resonance overlap, presence of heavy scatterer, boundary
conditions, type of isotopic library data, number of broad groups
used for collapsing and editing (minimum of 2, maximum of 5), type
of data table output, etc.
f). Convergence criteria, temperatures for each material
zone, type of fission spectrum, optional group-dependent buckling
specification, information about nuclides for which resonance
calculations are requested, etc.
The input for the engineering option assumes default values
for many of the variables used by the code.The input is reduced to
a minimum while keeping a fair flexibility, and still allowing
representation of most of the classical problems in fuel burnup
calculations. The main items input in the engineering option are
the following:
a). Weight fractions of fissile materials in the fuel pellet
and density fraction of the pellet.
b). Volume fractions of structural materials in the buffer
extra region surrounding the fuel cell.
c). Composition of the cladding material.


APPENDIX B8
TEMPRET CODE DESCRIPTION
TEMPRET31) is a thermalhydraulics code developed by Mark
Miller for the prediction of two-dimensional, steady state thermal
conditions in a PWR elementary fuel cell. The code accepts a wide
range of operating characteristics of the fuel, and is able of
treating temperature dependent thermodynamic properties of the fuel
material, the cladding and the coolant. The heat generation rate
can be allowed to vary both in the axial and in the radial
directions. The code includes some special features such as the
possibility of treating either solid or central-voided fuel pins.
Some of the more significant input data to the code are the
following:
a). Geometrical description of the problem.
b). Tabular entry of materials properties.
c). Definition of thermodynamic status of coolant at core
inlet.
d). Definition of heat generation distribution.
The output from the code includes the following items:
a). Bulk coolant thermodynamic conditions at each axial node.
b). Reynolds number, Prandtl number, and other heat transfer
related data for each axial node.
269


181
surface temperature as an indication of core safety becomes
apparent when considering the possibility of clad failure. The
clad surface temperature depends on the heat flux level and on the
thermodynamic conditions of the coolant. Although the coolant is
kept by definition at equivalent conditions in all the cores under
study, the heat flux experiences significant differences, which may
result in changes in the clad surface temperature. Finally, the
fuel temperature depends not only on the thermodynamic coolant
conditions and on the clad-to-coolant heat transfer mode, but it
depends to a high degree on the heat generation rate at each point
of the fuel pins. The heat generation rate is different from point
to point in the core, and it is related to space-dependent nuclear
properties of the fuel and the core itself: the particular fuel
composition at each point (which depends on the power generation
history of that point), the poison concentration, the proximity to
water holes or control rods, etc. all contribute to an uneven
distribution of the heat generation rate throughout the core. This
unevenness has to be taken into account when studying the limiting
heat transfer and temperature conditions, and it is done by means
of the so-called power peaking factors. It is normally assumed
that the heat generation rate has an axial variation similar to a
cosine function; a radial variation due to different geometrical
factors is also assumed. The power peaking factors used for the
present calculation are 1.64 for the radial value, and 2.62 for the
overall core. These power peaking factors and the
thermal-hydraulic safety parameters stated above are included in


160
4.2.2, Plant Operational Data Variations.
The previous section showed how the variations of power
density level and the changes of fuel lattice pitch affect the
neutronics characteristics and the isotopic evolution of the core.
These variations result in changes in plant operational data which
imply sharp differences from both the operations and the economic
viewpoints. This section is intended to show the main plant
operational characteristics associated with each of the studied
cores, and their underlying implications. The data shown in this
section are obtained from the burnup and cycle lengths shown in the
preceding section, which were based on the PDQ-7 CRIBUR burnup
results.
Table 4.2.2 establishes the basis in which to compare the
availability and capacity factors of the different reactors, and
their cycle lengths and refueling outages. These parameters are of
major importance since they represent large differences from the
economical viewpoint. One of the main economical incentives is
reducing plant outage time. This is a well known fact, which value
increases with time. Presently, depending on the utility location,
its dependence on oil, coal, gas, the time of the year, etc., one
outage day represents an additional expense of one to two million
dollars.
As on Table 4.2.1, the first two columns of Table 4.2.2 are
used for description of the core case studied, and refer to the
power level and the pin array pitch. The columns that follow on


77
g). Some models may allow for the specification of variable
power levels for the different timesteps, the specification of
Dancoff factors or some geometric data that may allow the code to
calculate than, the specification of a perpendicular buckling, or a
buckling search in order to obtain a critical spectrum, etc.
Most of these first-step pin cell codes are self-chaining, in
the sense that the user specifies the cell geometry, isotopics and
burnup history at the beginning of the code execution, and the code
performs the burnup steps and the cross section evaluations after
each timestep without user intervention. Other schemes, like the
one used for some of the preliminary calculations in the SHARP
study, involve several codes to perform the pin cell burnup, and
they need the user to manually handle the cross section libraries
and the burnup isotopics back and forth between one code and
another. The advantages and disadvantages of each method rely
mainly on the flexibility and freedom of data handling and geometry
specification versus convenience and man time requirements. Figure
3.1.2 shows some possible schematics of the data specification for
this first step of burnup calculations.
The output of these first-step codes normally includes the
following data:
a) Microscopic cross sections for each isotope present in
the fuel cell, for the number of neutron energy groups specified in
the input (normally between two and five groups) and for each
specified timestep.


176
Table 4.2.3. Ore, Enrichment and Plutonium Normalized to 1125 MWe
Pitch
cm
Power
%
Pu Disch.
Kg/Yr
Ore Usage
MT/Life
Ore Savings
% Over Std.
Enrich. Need
SWU/Life
1.20
50
225.08
4662
-2.87
3158
1.20
33
219.69
4589
-1.28
3109
1.20
25
216.87
4533
-0.05
3071
1.25
100 (c
i) 201.72
4531
0.00
3070
1.25
50
197.51
4396
2.98
2978
1.25
33
194.83
4389
3.15
2973
1.25
25
192.61
4357
3.86
2952
1.30
50
181.48
4350
4.00
2947
1.30
33
178.07
4304
5.03
2916
1-30
25
176.21
4284
5.46
2902
1.35
50
169.45
4300
5.12
2913
1.35
33
166.30
4270
5.76
2893
1.35
25
164.66
4254
6.12
2882
1.40
50
160.95
4395
3.01
2978
1-40
33
158.33
4361
3.75
2955
1.40
25
156.88
4349
4.02
2947
(a): Reference Case


250
b). Option for the anisotropic scattering correction, for an
external slowing down source, and for transversal buckling. The
latter two require specification of the slowing down source and the
buckling or the equivalent transversal dimensions respectively.
c). Description of the problem geometry: number of material
regions, number of mesh spaces assigned to each of them, thickness
of each region, type of geometry.
d!). Specification of the type of boundary conditions and
option for the cell-smeared calculations.
e). Specification of maximum iteration parameters and
convergence criteria.
f). Edit options specification.


212
The SHARP concept must also be tested for its advantages in a
fuel reprocessing policy environment. More involved isotopic
accounting methods will be required, but the favorable heat
transfer situation of the SHARP'S allows wide possibilities of play
with core design variables that should certainly result in
significant improvement of fuel cycle characteristics under a
reprocessing policy environment.
It is also necessary to make a thorough evaluation of the
possible monetary advantages obtainable from the improved safety
aspects of the SHARP. These monetary advantages can come from
reduced safety systems requirements, shortened licensing
procedures,etc. Although apparently more speculative in nature,
these aspects can result in significant savings, affecting the
capital cost of the plant, and thus, the highest portion of the
power generation cost.


APPENDIX B3
MONA CODE DESCRIPTION
The MONA (10) code is a multigroup, one-dimensional, diffusion
theory neutronics analysis code, developed as an enhancement of the
CORA (33) code. It is suitable for studies where space and energy
variables can not be separated, but angular flux dependence is not
of great concern, and thus there is no need for the more complex
and time consuming several-dimensional neutron transport theory
codes. MONA accepts a slab, cylindrical or spherical geometry.
The energy range can be cut into up to 50 energy groups, with full
upscattering capabilities. Up to 50 material regions can be
specified, and a variety of boundary conditions are available and
can be defined separately for each energy group.
The description of the materials in each region can be done by
means of macroscopic cross sections, or with microscopic cross
sections and isotopic number densities. The source term can be
calculated by the code or it can be specified by the user. The
input format conforms to that of the cards punched by PHROG-
The code obtains the energy- and space-dependent fluxes and
adjoint fluxes, performs a generation factor calculation and is
able to make criticality searches based on a variety of possible
parameters, such as poison concentration, region boundary position,
nuclide concentration. perpendicular buckling, etc. The


267
enables the user to choose as simple or as complex of an isotope
accounting procedure as may be needed for each particular job.
Another involved task left to the user is the specification of
the groups of chains that must be assigned to each material region,
the interpolating tables that correspond to each set of isotopes,
and the respective masks (independent variables) that are
controlling each of the interpolation tables. PDQ-7 can also admit
"burnable" and "non-burnable" compositions, which obviously receive
a different treatment. Non-burnable materials can also have
interpolating tables, but their cross sections are stored in the
macroscopic form, since there is no need for accounting of isotopic
number densities.
Finally, the code requires a set of flag and options
specifications, indicating the type of problem to be solved, the
type of boundary conditions, the type and arrangement of the output
edits, etc. It is also necessary to specify the timesteps desired
in the case of a burnup study, as well as the power level, which
can be specified for each timestep. It is obvious that a complete
description of all the capabilities of the code escapes the context
of this work.
The items present in a normal output for a burnup study are
listed below. It is important to keep in mind that it is hard to
define a "normal" output from PDQ-7, since most of the output is
optional, and is determined by the user.
a). General definition of the problem: type of problem,
geometrical size, figure composition, timestep length, etc.


41
neutronic effects on the other isotopes (due to spectral effects)
could not be simulated. However, the agreement between the two
methods can still be considered good, considering the different
conceptual approaches, and the small amount of core information
included in them. The most reassuring fact is that the quoted
beginning of life (BGL) core reactivity and the quoted end of life
(EOL) burnup level of the first core for the standard reactor lie
just between the values obtained from the two models, with LEOPARD
giving a better result for the EOL burnup.
With all these considerations in mind, LEOPARD was used with
the "average fuel" enrichment for the scoping burnup calculations.
These calculations used a fuel pin with core-averaged enrichment, a
perpendicular buckling corresponding to the axial leakage of a
standard core, and an approximately critical boron letdown curve.
Calculations were performed for the standard core, a 25% power
density core using varying lattice pitches, and a 20% power density
core with reduced moderator temperature (the preheater core,
designated as core #8 in Table 2.1.4). The results of the scoping
calculations are shown in the next Section.
2.3. Results of Preliminary Work
2.3.1. Reactivity Coefficients.
Table 2.1.4 shows the main parameters characterizing the
different cores that were chosen for the scoping study of the
SHARP. The table includes the effective multiplication factors
(K-eff.) of the cores at beginning of life (B.O.L.) with no xenon


203
medium; it therefore leads to an improved determination of the
neutron spectrum and the isotopic dynamics. The time-varying power
level capability is an unusual feature for this type of burnup
calculational tool. and has a significant impact on
spectrum-to-burnup level correspondence.
CRIBUR results were benchmarked against detailed
two-dimensional neutron diffusion theory codes and against actual
core data and were shown to consistently yield better results than
other popular calculational methods like the ones using
batch-averaging for neutron multiplication factor calculations(28),
or the ones using the linear reactivity assumptions for end-of-life
estimates(15). CRIBUR criticality calculations agree with actual
core BOL data to within 1-3%. Batchwise power distribution
calculations agree with multi-dimensional core follow calculations
to within a few percent.
The computational cost of the burnup calculational scheme is
little affected by CRIBUR. since it only takes a minimal fraction
of the computer time needed for EPRI-CELL and PDQ-7 calculations
which are required to prepare the batch burnup data for CRIBUR.
One complete core burnup calculation can be performed with one
EERI-CELL pin life calculation (requiring about three minutes of
CPU time of the IBM 370-3033 unit), one NUFUNCHER calculation
(requiring about ten seconds). three 1/4-assembly life burnup PDQ-7
calculations (requiring approximately one minute of CPU time each
on the same unit) and three CRIBUR calculations (each requiring
just a few seconds of computer time)-


30
Table 2.2.1. Basic Cell Temperatures vs. Power Level.
Power level % 100 50 33 25 20 (pre)
Clad surface 593.3
Clad increment 46.4
Gap increment 120.1
Fuel surface 759.9
Fuel centerline 1846.7
Fuel average 1303.3
Fuel average (K) 979.0
Vendor normalized (K) 1005.0
592.8
592.5
592.4
304.2
23.2
15.5
11.6
9.3
60.1
40.0
30.0
24.0
676.1
648.1
634.0
337.5
1219.5
1010.3
905.7
555.0
947.8
829.2
769.9
446.3
781.9
716.0
682.0
503.2
805.0
735.0
700.0
520.0
Note: Temperatures obtained from thermal calculations as described
in Section 2.2.1.
Temperatures expressed in degrees Fahrenheit, unless
otherwise specified as Kelvin Column 20 (pre) refers to the 20% power, preheater core.


206
at a pitch of 1.35 cm. This burnup-optimized pitch can not be used
by the standard reactor for thermal-hydraulic safety reasons, but
presents no problem for the SHARP'S using a power density level of
70% of the standard value, or less. The additional burnups
achieved by the SHARP'S fall well within the range of acceptable
burnup level for the present fuel technology, and thus present no
need for clad structure or pin geometry redesign.
The burnup-optimized pitch is smaller than the optimum
moderated pitch (which is about 1.6 cm). This is due to the sharp
reduction of plutonium conversion associated with the softer
spectrum present in the better moderated pitches. Plutonium
discharge per unit energy produced from the burnup-optimized 50%
power density SHARP is 16% lower than from the standard PWR. This
is a positive effect from the nuclear weapons non-proliferation
viewpoint. However, in a fuel reprocessing policy environment,
such a low plutonium discharge would result in a loss of spent fuel
value.
Under the present once-through fuel policy, the SHARP improves
ore utilization as a result of the improved discharge burnup
capabilities. The 50% power core with the optimized pitch shows a
5.4% better ore utilization than the standard PWR plant. The 33%
power and 25% power SHARP'S show ore utilizations which are 6.1%
and 6.5% better than the standard plant, respectively.
With spent fuel reprocessing in mind, the net consumption of
fissile material per unit power produced was calculated. This took
into account the fissile mass loaded with the fresh fuel as U-235


222
Ore Utilization
MKd/ST U3Og


214
goals are at times in conflict, and a compromise (often just an
optimum economic solution) must be reached by an adequate
combination of these goals and the operational characteristics of
the reactors.
Increasing the energy obtainable from a certain amount of ore
may involve a very wide range of actions, from just operational
policy changes to some reactor redesign. Some actions which in
their conceptual sense might be in the right direction towards
reducing fuel cost and/or ore requirements may actually be
forbidden by safety or operational constraints. Some of the main
restrictions to bear in mind while searching for methods of
improving burnup are described below:
a). Linear power in the fuel pins is normally kept at a very
high level. Any attempt to increase it must insure that adequate
heat transfer at the pins' surfaces will be maintained, in order to
avoid damages due to excessive temperatures or thermal shock. In
general, any modification tending to increase the average linear
power or the power peaking factor will need very special care.
b). Fuel pin structural materials and the fuel itself suffer
physical degradation due to the radiation damage induced by the
high fluences that they must withstand. Fuel manufacturers are
trying to obtain fuel designs that will be able to withstand higher
fluences than the ones presently allowed (goals are set for about
45 GWd/MTU average burnup to be obtained in the next 10 to 16
years) (2). Hence, it is not reasonable to contemplate a solution
that would require fuel exposures much larger than the present


226
since the high soluble boron concentration needed at BOL may cause
a positive MIC, with intrinsic safety implications.
b). Since they are placed in the fuel assemblies having the
highest power levels, the burnable poisons reduce the power peaks
between assemblies and thus allow the whole-core power level to be
increased while preserving safe heat transfer conditions in the
most restrictive assemblies. The end result is a higher power
output by the reactor.
c). For assemblies having very assymetrical boundaries (e.g.
a fresh assembly having the baffle and reflector on one side and
another fresh assembly on the opposite side), burnable poisons may
be placed in such a way that they reduce the power tilt that would
appear within the assembly itself. This avoids severe power
peaking problems in subsequent cycles, when the assembly does not
have burnable poisons anymore. An example of such use of LBP's is
shown in Figure A.2.3 (37).
d). Since their reactivity and burnup are held down during
the whole first cycle, the poison-loaded fresh assemblies show a
higher reactivity at EOL, which allows the core to stretch the
burnup obtainable in each cycle, and with this, the discharge
burnup obtainable from the same initial fuel. This can be seen
rather easily using Strawbridge's rough approximation which states
that for a core having several batches of the same initial
enrichment (i.e. for an equilibrium-cycle core), EOL will be
reached when the core average burnup reaches a certain constant


230
1

1

1

1
/

1

1

i
/
/
1

1

1

1
/

I

1

1
/
/
1

1



/

1

1

/
/
1
/
1
/
/
/
/
/
/
/
1
/
Batch 1 Batch 2 Batch 3
Figure A.2.4. Batch Distribution in Core


APPENDIX BIO
GEM CODE DESCRIPTION
The initials GEM stand for General Economic Model. The
GEM (33) code is devised to calculate the costs of the fuel cycle
of a nuclear electric power plant. The code is able to calculate
the costs for just a batch or a set of batches of fuel, or it may
be used for the computation of the fuel costs through the entire
life of the plant.
The code takes into account all the steps of the fuel cycle,
from mining through spent fuel shipment and storage, including the
costs of extraction, enrichment, etc. and the schedules associated
with them.
The code computes both the cumulative fuel cost and the
batchwise yearly costs. The calculations are done through three
distinct types of economic analysis: Cash Flow, Allocated Costs,
and Yearly Cash Flow. The three models yield identical results,
but they present them in different forms.
The input to the code includes mainly the following items:
a). Uranium prices.
b). Fabrication and Service costs.
c). Feed losses.
d). Plutonium prices.
e). Fuel weights.
273


259
input options, one of which is called the "engineering" input
option, and defaults most of the commonly used data, while the
other option, called the "general" input option, requires full
specification of the problem.
c). The code must provide an output which is capable of being
easily processed for input to the core model codes.
The result is a code with an extremely simplified input, but
nonetheless, with a large amount of options available. It is based
on the very well-known codes THERMOS (21), GAM (20) and
CINDER (22). It simulates the burning of a LWR fuel pin in which
the geometry and compositions are specified by the user, as well as
the circumstancial data concerning the burnup process, such as
power level variation through time, soluble boron concentration,
etc. The output includes a printed listing of microscopic isotope
cross sections, macroscopic cell-averaged cross sections, isotopic
number densities, multiplication factor, etc. and a magnetic
device (tape or disk) data set with the same type of data in an
adequate format to be transformed into cross section tables for the
core model codes.
Some of the main characteristics of EERI-CELL are the
following:
a). The code includes the effects of Dancoff factors,
correcting also for the fuel rods that are not surrounded by other
fuel rods but that are adjacent to interassembly water gaps, to
water holes, instrument thimbles, etc.


239
Table A.2.1. Discharge Burnups of Low power Density Cores.
Power Level
Discharge Burnup (MWd/MTU)
% burnup increase
%
Std. Pitch
Optim. Pitch
Std. Ptch. Opt. Ptch.
100
36183
0.0
50
37296
38134
3.1 5.4
33
37359
38397
3.3 6.1
25
37634
38541
4.0 6.5
A.2.6. Flattening-Axial Power Distribution.
If all the pellets in a FWR had the same composition at a
given point in time, and no special absorbing materials were
present, the reactor power distribution will approximate a cosine
in the axial direction, which means that each pin would be burning
much faster around the reactor mid-plane than towards the top or
the bottom. Since the reactor must be designed for the most
unfavorable spot to work safely, this means that most of the core
is under-employed with regard to power generation and fuel
utilization. It is, therefore, highly desirable to alter the
"natural" axial cosine shape of the reactor power distribution and
make it flatter, so that power peaking is reduced and fuel average
discharge burnup is increased by a more uniform burnup.
Such capability is available in present reactors, although it
might be possible to develop improvements. One available axial
power flattening method currently in use employs partial-length


151
difference is that the burnup level is an absolute measure,
independent of the core power level, while the cycle length (even
the full-power cycle length, as represented in this third column)
is a measure that is dependent on the power level of the plant.
The fourth column of table 4.2.1, labeled Dis. BU. MWd/MTU,
shows the burnup levels achieved by each core, as calculated by
CRIBUR. This is probably the best measure of the amount of energy
provided by a given amount of initial fuel loaded in the different
cores for the case when no reprocessing is being done. This
measure of burnup does not take credit for discharged fissile
materials, which would be of obvious interest in the case of a
reprocessing policy. However, under the present U.S. policy, this
can not be accounted for, and the burnup achievable is a good
measure of the energy obtained from a certain amount of initial
fuel. The levels of burnup achieved by the best pitch cell (1.35
cm) for the 50%, 33% and 25% power level cores are 5.4%, 6.1% and
6.5% above that achieved by the standard core, respectively.
Referring these burnup increments to a common basis, the 50% power
core yields 0.11% extra burnup per percent power reduction with
respect to the standard plant; the increments for the 33% and 25%
power density cores are 0.092% and 0.087% per percent reduction
with respect to the full-power plant. These figures represent a
15% and a 20% lower value than that obtained in the power reduction
to 50%. The implicit meaning of these figures is that the
additional burnup obtained with low power density cores with
respect to the standard plant is largest for the initial power


66
means that the isotopic and spectral effects occurring during the
life of the core cause a shift of the EQL reactivity, favoring less
moderated configurations. The main reason for this effect is a
conflict between optimum moderation and enhanced conversion ratio:
Conversion ratio is enhanced when the resonance absorption in U-238
is increased, but this is obviously an effect which reduces the
immediate reactivity of the core. The trade-off appears when
realizing that at E.O.L., a large proportion of the power is
generated by the converted plutonium. Thus, if conversion ratio is
reduced in trying to optimize moderation (and thus increasing BOL
reactivity), the core reaches EQL with a reduced amount of
plutonium, and therefore becomes subcritical at a lower burnup
level.
Note on Table 2.3.2 the relatively high level of burnup
achieved by core 8, in spite of being fueled with only 2% enriched
fuel. The effect must be attributed primarily to the different
moderator characteristics, in addition to the effect of the reduced
power level.
It appears at this point that the best pitch from the burnup
viewpoint is larger than the one used in the standard reactor, but
smaller than the one yielding maximum reactivity at BOL. However,
the model used for these scoping calculations can not be assumed to
accurately represent a reactor in an equilibrium cycle since BOL
compositions with no burned fuel were employed. The question is
then whether the optimum burnup pitch obtained in these studies
will still be the best for an "equilibrium cycle" core or if it


93
Soluble
Figure 3.1.7. Effect of Time-dependent Power Level and
Soluble Boron Concentration Specification.


198
Table 5.2.3 contains the total generation costs of the plants
under consideration, for the three inflation scenarios, as well as
their cost increment with respect to the full-power plant. Note
that like the capital cost, and unlike the fuel cost, the total
generation costs exhibit percentage increases at reduced power
which are essentially independent of the inflation level. This
table is the most significant one from the economic viewpoint,
since it shows the cost of the energy from each plant
configuration, after all parameters have been taken into account.
Power levels between 60% and 70% of the standard level show
moderate total cost increases. The transition from 60% to 50%
starts to show a sharper increase in generation cost, and it is
evident that power levels below 50% can not be considered since
their economic penalty is unreasonably large. These trends can be
better observed in the curves shown in Figure 5.2.3. which are a
plot of the total generation cost vs. the plant size, for the
three inflation scenarios.


233
the power peaking problem associated with the in-out-in scheme.
However, this would entail, as mentioned in Section A.2.2, yet a
further calculational, technological and economic problem.
A.2.4. Alterations of Euel-rto-Water. Ratio.
Changing the relative amounts of water and fuel in the core
can have a significant impact in the nuclear characteristics of the
reactor. When more water is present, the neutron energy spectrum
is softened and the multiplication factor is raised (since PWK's
work with an undermoderated configuration for safety reasons). On
the other hand, the epithermal neutron population is then
decreased, and captures in U-238 are reduced, thus reducing the
conversion ratio and limiting the amount of plutonium contributing
to stretch the cycle length. The reverse effects appear when the
water proportion is decreased.
The two effects described are competing from the point of view
of cycle length; however, considering the present design of PWR's,
burnup can be increased by going to a more moderated configuration.
There are four possible ways of changing the fuel-to-water ratio:
a). Change the pin diameter.
b). Change the lattice pitch.
c). Change the effective fuel density.
Changing the pin diameter has the advantage of needing a very
limited redesign, since it is possible to keep the same basic fuel
assembly design and backfit the change into operating reactors. As
a matter of fact, Westinghouse Corporation has a new optimized pin
of reduced radius that will be slowly introduced into the market


ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation to the
members of his supervisory committee N.J. Diaz, E.E. Carroll,
E.T. Dugan, C.C. Oliver and M.L. Muga for their interest and help
in preparing this dissertation. Special thanks are given to Dr.
N.J. Diaz for originating the dissertation subject and providing
direction of the research project. The author is deeply indebted
to Dr. E.T. Dugan for providing continued technical guidance and
criticism and invaluable help in obtaining necessary calculational
tools, together with endless patience in correcting this report.
The author is indebted to J.A. Wethington, Jr., M.J. Ohanian,
G.R. Dalton, W.H. Ellis, G.S. Roessler and W.G. Vernetson for the
friendship and support given through the years of this research.
The main body of the calculations of this study were performed
at Oak Ridge National Laboratory, Tennessee, where the author met a
number of great friends who were instrumental in the execution of
this work. Special thanks are given to J.C. Cleveland for making
the necessary arrangements that made the stay at O.R.N.L. possible
and for his technical advice and assistance; to Rafael Perez and
iii


112
Plb = Nleak / Ntotal = Plr Nr / Nb
This would require knowledge of the total core and peripheral
batch's neutron populations, but since the parameter of interest is
really the ratio of these populations rather than the individual
values of any of than, it is possible to make a further
approximation that will simplify the calculation. The
approximation assumes that the neutron population in the peripheral
batch has approximately the same average speed as the whole core
population. This is not completely exact, because the different
isotopic compositions of the various batches, and the fact that
fast neutrons tend to leak in a larger proportion than thermal
neutrons, cause slight differences in the batch averaged neutron
speeds but the error incurred is of negligible importance. Once
this assumption is accepted, since the neutron flux is
(p = n v
where n = neutron density.
v = average neutron speed.
it is possible to express the ratio of neutron populations of the
core and the peripheral batch as
Nr = 2-1 $1 Vi
Nb Vb
where Vi = volume of batch 1i' in the core.
Vb = volume of the peripheral batch.


168
Note that the cycle length of the 50% power, 1.35 cm pitch
core is very closely twice that of the standard reactor. This
means that if a one-year cycle applies for the standard plant, a
two-year term would apply for the 50% power one. This seems
obvious, but hidden in here is the fact that for this two year
cycle, one refueling outage has been turned into power-producing
time. In other words, the refueling outage time is reduced to
approximately one-half for this configuration of what it is in a
standard plant. This will be seen better in the next column of the
table.
The fourth column on Table 4.2.2, labeled Life Reful Days,
shows the expected total number of days that the plant would spend
in refueling during a thirty year life period. The resulting data
are also displayed in Figure 4.2.4 for easier reference and
comparison. The purpose of this column is to realize the number of
days gained from refueling downtime for power-producing days. The
figure is obtained by first finding the number of cycles that the
reactor would complete in a thirty year period, and multiplying the
number by the forty-five days length assumed for refueling. Note
that the 50% power, 1.35 cm pitch reactor shows a savings of 498
days over the life time of the plant, which would represent a
considerable amount of money saved.
Another form of looking at the savings in refueling outage
time is found in column 5 of Table 4.2.2, labeled Refuel % Life.
This column shows the percentage of the total plant life that is
spent in refueling. A variation of this figure would be closely


17
Hie optimum fuel cycle appears to be one which obtains maximum
energy per ton of ore and which also has long core cycles.
Discharge burnups, however, must be limited, considering present
fuel technology, to somewhere between 30 and at the most 40
GWd/MTU. In addition, the size of the batches should be kept small
(i.e. the number of batches should not be reduced) in order to
maintain good ore utilization.
The reduced power density cores are expected to yield
increased burnups given a certain core management scheme, but they
are not expected to vary in extremely large proportions. This
would probably make the low-power cores able to comply with all the
desirable conditions mentioned above: long core cycles because of
the reduced power density and the associated additional burnup, and
better ore utilization because of the extended burnup obtained from
the same initial core load.
The comparison of the fuel cycle performance of the SHARP
versus a standard FWR plant can be done from many different frames
of reference, each of which would enhance the comparison of the
cores in a particular aspect. However, the two most significant
ones would probably be the following:
a). Maintaining feed enrichment and core structure, evaluate
the differences in burnup achievable, cycle length, and ore
utilization.
b), Maintaining core structure and discharge burnup, evaluate
the differences in ore requirements, enrichment needs, and cycle
lengths,,


268
b). Pictures indicating the geometrical arrangement of the
problem, along with the specification of all material compositions.
c). Multiplication factor iterations.
d). Fluxes, power generation, absorption rates,
multiplication factors, macroscopic cross sections, etc. groupwise
and total, for each edit region defined, which may include separate
material regions (v.g. each particular fuel assembly) and/or any
combination of them, which can be used to obtain batch-wise or core
total values.
e). Description of the burnup step. It may be done in one
single calculation, or it may include partial timestep
renormalizations in order to correct for isotopic variations during
the timestep burning time. Isotopic concentrations for each edit
region defined by the user are also listed.
f). Information about the file handling for data transfer
between the timestep calculations.


I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
rv \
\i
k/V
Nils J,
, Diaz,\
jfchairman )
Engineering Sciences
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Edward E. Carroll
Professor of Nuclear
Engineering Sciences
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Edward T. Dugan
Visiting Assistant Pfbfessor of
Nuclear Engineering Sciences
I certify that I have read this study and that
in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in
scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Marvin L. Muga
Professor of Chemist:


302
Table D.2
Average Cell Neutron Velocities (eV)
Pitch %
Power
BOL Speed
MOL Speed
EQL Speed
1.20
50
1.93
1.89
1.87
1.20
33
1.93
1.89
1.87
1.20
25
1.93
1.89
1.87
1.25
100
1.89
1.86
1.83
1.25
50
1.90
1.86
1.83
1.25
33
1.89
1.85
1.83
1.25
25
1.89
1.85
1.83
1.30
50
1.87
1.82
1.80
1.30
33
1.87
1.82
1.80
1.30
25
1.87
1.82
1.79
1.35
50
1.85
1.80
1.78
1.35
33
1.85
1.80
1.77
1.35
25
1.85
1.80
1.77
1.40
50
1.83
1.78
1.75
1.40
33
1.83
1.78
1.75
1.40
25
1.83
1.78
1.75


12
methodological errors that could bias the estimates towards either
system.
f). Overview of economic evaluation of the SHARP concept as
compared to the standard PWR to establish the overall economic
advantages or disadvantages that may be expected from the SHARP
concept. The economic evaluation must consider the variations in
the costs of interest and escalation, as well as the possible size
or rating variations of the plant components which depend on the
plant power rating. For example, when studying a single-core SHARP
for a power level of 50% that of a standard plant, the NSSS is
dimensioned equal to that of the standard, full-power plant, but
the BOP is dimensioned for only the new 50% power rating.
The next chapter describes the SHARP parameters and the
preliminary calculations carried out in order to establish the
basic expected performance of the low power cores.
Chapter III describes the burnup calculational methods used in
the industry; they are compared to the needs of this study, and a
new method is developed, which best suits the scope of this work
and results in reduced computational effort. The new method should
serve as a valuable industry-wide burnup calculational tool because
of its accuracy, ease of utilization and low computational cost.
Chapter IV shows the cases chosen for in-depth study and the
results obtained from the burnup calculations, plus some data
referred to the expected comparative plant performance.


242
However, a higher enriched fuel requires larger quantities of ore
for each core reload. Although for very low enrichments an
increased enrichment also causes a better ore utilization, a
maximum is soon reached after which further enrichment entails more
ore consumption for the same energy produced, as can be observed in
Figures A.2.2 and A.2.8. In addition to this, the combination of
higher enriched feed fuel and further burned old batches will tend
to cause more severe power peaking problems, making burnable
poisons probably necessary.
Combustion Engineering research on this topic (3) shows that
with a 3-batch out-in-in fuel management scheme (which is the most
commonly used in large PWR's), ore utilization could be
theoretically improved by about 6% by increasing fuel enrichment to
about 4.8% U-235. However, this would require the fuel to be
driven to over 50 GWd/MTU, which is beyond the present
technological limits.
According to the limits set in Section A.l, ore utilization
can be improved by about 3 to 4% by going to fuel enrichments of
about 3.8% U-235. Figure A.2.8 illustrates the ore utilization
improvement achievable for different values of fuel enrichment and
their associated discharge burnups assuming a 3-batch out-in-in
fuel management scheme.
A.2.8. End of Cycle Coastdown.
Power coastdown at EOC is an operation which is often used by
utilities to meet shutdown schedules rather than for burnup or
resource utilization improvement. There are two ways of performing


4
optimization study of the fuel utilization capabilities of the
Safer Highly Available Reactor Plants versus the standard LWR,
plants currently in operation or under construction, and
concurrently, to develop calculational tools that will facilitate
and enhance the accuracy of scoping-type burnup studies. The
primary study must be complemented with indications of the
comparative safety and economic performance of the SHARP with
respect to standard FWR plants.
1.2. The SHARP Concept
The main goals pursued by the "Safer Highly Available
Reactor Plant" (SHARP) are the following:
- Enhance reactor safety.
- Reduce the safety and operational problems caused by
the operating conditions prevalent in current LWR systems.
- Reduce lengthy and costly research and licensing
procedures.
- Offer a solution which utilizes current technology
and can be implemented in a relatively short time.
- Reduce economic penalties to make a commercially
viable operation.
The dominant feature of the SHARP is the use of a standard PWR
core, without core changes requiring technological modifications,
at a reduced power density to reduce safety-related and operational
problems associated with standard full power operating conditions.
The single low power density core may then be employed in an
essentially standard plant configuration, yielding a reduced power


185
Figure 5.1.3. Clad Surface Temperature vs. Relative Mass Flow Rate.


215
ones, if the solution is expected to be applicable in the near
future. An average burnup of about 40 GWd/MTU would look
reasonable as an upper limit in an optimistic scope for the near
future.
c). Somewhat in connection with point a) mentioned above is
the fact that if significant changes are made to the core lattice
geometry, it is indispensable to assure heat transfer conditions
consistent with operational and safety requirements of the core.
For example, a reduction of pin diameter entices a reduction of
linear power since the pin heat transfer surface would be reduced
as the diameter is reduced.
d). The fully loaded core can not have an extremely high
excess reactivity at BOL because this would force a high soluble
boron concentration in the coolant, and this might cause a positive
moderator temperature coefficient (MTC), which is unacceptable from
a safety point of view. One way of avoiding the problems caused by
a large excess reactivity at BOL is the use of burnable poisons,
which also serve the purpose of reducing power peaking.
A,1..2..^..S.ghemes-for improving Purnup.
There are a number of schemes that can be implemented to
increase the burnup obtained from a certain load of fuel. This
section reviews the main schemes involved in extending burnup,
while Section A.2 describes the possible practical ways of
implementing these principles in a PWR power plant and goes deeper
into the effects involved in the necessary modifications.


16
Obviously, one is concerned with obtaining the maximum energy
from each ton of ore. In addition to this, utilities are also
interested in having long core cycles. Presently there is a trend
to change from a 12-month to an 18-month cycle scheme because of
the reduction in refueling outage time and personnel radiation
exposure, which might result in attractive savings in power
generation cost.
If the currently standard core structure were maintained
(3-batch, out-in scheme) it would only take an increased fuel
enrichment and some help from burnable poisons in order to have an
18-month cycle instead of a 12-month one. By so doing, the ore
utilization is favored by the fact that the fuel is discharged at a
higher burnup. A study by Combustion Engineering(3) shows that ore
utilization is improved for increasing burnup levels up to about 50
GWd/MTU and fuel enrichments of about 4.5%. The problem, however,
is that an 18-month cycle would yield high discharge burnup levels
as compared to what is acceptable based on the present level of
fuel technology.
In order to keep the discharge burnup in an acceptable range,
given the present fuel technology, and to take advantage of the
reduced refueling outages of an 18-month scheme, utilities are
forced to switch to larger batch sizes. This enables the power
generation cost to be reduced, but with worse ore economy than that
obtained in a 12-month core cycle. This is of some concern from a
fuel resources viewpoint, mainly if reprocessing is not considered.


59
to decrease significantly as larger pitches are reached, but the
reduction of distance is insignificant, indicating again the key
role of the changed mean free path.
Finally, the thermal utilization factor, f, or ratio of
thermal neutrons absorbed in the fuel to total thermal absorptions
in the cell, shows a drop as pitch is increased. This effect is
directly related to the amount of absorbing moderator atoms present
in the fuel cell. Note that as pitch increases, the curve for core
8, which is naturally below that of cores 3A and 4A, drops faster
than that of cores 3A and 4A; the reason being simply that the same
increase of pitch adds more moderator atoms into core 8 than for
the other cores, because of the higher moderator density of core 8.
Many of the reactivity and isotopic differences between the
standard core and the low power density cores have been attributed
to neutron energy spectral effects. Figure 2.3.6 illustrates some
aspects of the spectral changes. The graph represents the ratio of
thermal flux to first fast group (of the three fast groups used for
the criticality calculations) for cores 3A, 4A and 8, and for short
burnup times, up to about 50 days of equivalent full power
operation of the standard reactor. These figures were obtained
from short burnup calculations performed with the
PHFOG-KOT-MONA-BURNUP scheme. The effect of the increased
moderator density of core 8 is much larger than all the other
effects differentiating cores 3A and 4A. However, all cores show a
parallel behavior as burnup increases: there is an immediate drop
of thermal flux right at the beginning of life. This may be


35
Figure 2.2.2. Geometry Used for MOM Criticality Calculations.


235
Enrichment W/o
Figure A.2.5. Discharge Burnup vs. Enrichment at Several Fuel Densities.


70
U-235 specific consumption begins to increase. It is thus seen
that both U-235 specific usage and plutonium relative burning
remain fairly close to the optimum values for the pitches that
allow highest burnup and therefore longest cycle times, while
worsening rapidly after the region of interest.
2.4. Scoping Work Conclusions
The scoping calculations show that there is good hope for a
significant increase of discharge burnup from the same initial fuel
by using low power density cores. The main effects contributing to
the necessary extra reactivity are the reduced level of xenon and
the reduced effect of Doppler broadening of resonances. A 9%
increase of burnup seems to be achievable by going to a 25% power
density core, while up to 16% improvement appears feasible by
varying the cell pitch. However, these calculations were performed
with a calculational tool which is not well-suited for the burnup
analysis of a core consisting of fuel of various enrichments and
exposure levels. Some changes are surely to be expected when using
an adequate core model for these burnup calculations. The burnup
calculations made in this scoping study are indicative of a
first-load core behavior rather than of an equilibrium core; the
latter is the configuration of most interest for burnup or economic
studies.
It has become evident that a maximum level of automatization
is necessary for the calculational techniques to be used in
detailed burnup calculations. The adoption of LEOPARD to
substitute the chain of PHROG BRT MONA BURNUP proved to be


64
Discharge
Burnup
GWd/MT
21.5
21.0
20.5
20.0
Figure 2.3.7. Discharge Burnup of 25% Ibwer Density Core vs. Pitch.


217
newer batches is the use of burnable poisons that will mostly
disappear at EOL and will hold the power of the new batch down,
preventing the development of large amounts of fission products in
that batch. Burnable poisons should ideally be imbedded in the
fuel material itself rather than in poison rods. In this way they
can achieve a more thorough burn by EOL, and they don't have the
absorbing structural material associated with the poison rods.
Parasitic absorption in the fuel itself can be reduced by
reducing the power level of the reactor. This would reduce the
fuel temperature, and therefore the Doppler broadening of
absorption resonances. If some more effective means of cooling the
fuel could be devised, the same effect would be achievable without
the need for reducing the core power level.
Absorptions in the structural materials of the core could be
reduced by replacing stainless steel parts with parts manufactured
with less absorbing materials, and/or reducing the amounts of
structural material present in the core.
Absorptions in the coolant are difficult to reduce at EOL,
since no boron is present in the water. However, from the sole
point of view of absorptions in water at any point in the cycle
life, they can still be reduced by using lattices with less water
or by using the water-heavy water mixture whose other use and side
consequences will be described later in this section.
b). Reducing neutron leakage. There is a small proportion of
neutrons that leak out of the core and are lost for fissioning
purposes. There are some ways in which this leakage can be


APPENDIX B4
LEOPARD CODE DESCRIPTION
LEOPARD (11) is a fuel cell code intended to calculate neutron
flux spectra and isotope cross sections as they change with fuel
depletion. It is based on modified versions of the MUFT (12) and
SOFOCATE (13) codes. The code performs spectrum calculations and
fuel burnup calculations, assuming the fuel is imbedded in an
infinite array of elementary cells. The code provides the
possibility of specifying the presence of materials that cause
alterations of neutron spectrum, but which are not related to the
elementary cell itself. Such is the case of instrument thimbles,
water holes, etc.
Although the spectrum calculations take into account the
geometry of the fuel cell, the burnup routines are entirely
non-dimensional and fuel depletion is considered uniform throughout
the pellet region. The code accounts for the thermal expansions of
the different materials in the cell, according to the temperatures
supplied by the user- Input to the code is reduced and relatively
simple. However. LEOPARD has the following drawbacks:
a) The power level of the fuel can not be changed from one
timestep to the next, and therefore, the fuel is burned at a
constant power level which results in a false equivalence of time
and burnup level for each particular case.
255


219
reactor when the heavy water proportion of the coolant is
increased, and a higher conversion ratio is possible. Another way
is to have a variable pitch, which may allow for different
water/fuel ratios at different times in life or in different
regions of the core.
A. 2. Techniques for Improvement of Burnup
A.2.1. Increasing Number of Batches.
In most PWR's, fuel management is based on adding or
withdrawing the fuel elements in batches, in such a way that all
the elements belonging to a certain batch have at any time in life
a similar burnup level.
Some simplified methods of calculating burnup capabilities of
PWR's use the assumption that all batches in the core share the
same fraction of the total core power (provided they have the same
number of fuel assemblies, which is normally the case) and that the
core multiplication factor can be obtained as the average of all
the batches' multiplication factors (28). This method was used,
for example, by L.E. Strawbridge of Westinghouse. Under this
assumptions it is possible to determine that the burnup at
discharge of a fuel batch, Xd, can be calculated as:
Xd = Xc ( 2*N / (N+l) )
where Xc = Average burnup of the core at EOC.
N = Number of batches present in the core.


no
multiplication factor of the batch in the way described below.
Since the reflector is composed of cooling water, it contains
varying amounts of soluble boron at different points in life, which
makes its reflecting characteristics variable with time. This
effect is accounted for in CRIBUR as described below.
Due to the fact that the peripheral batch has some neutron
leakage into the reflector, its multiplication factor is affected
by the probability of a neutron leaking out of the fuel zone-
Expressed in mathematical terms, this would be:
Kb = Kib Pnlb
where Kb = K of the peripheral batch, after radial leakage
modification.
Kib = K inf- of the batch, as obtained from the point-burnup
calculation.
Pnlb = Probability of non-leakage of neutrons from the
peripheral batch.
By definition of probability, it is possible to write:
Pnlb = 1 Plb
where Plb = probability of neutron leakage from the external batch.
And similarly, it is possible to write for the whole reactor:
Pnlr = 1 Plr
where Plr = Probability of radial leakage for the whole reactor.


266
practice is to specify most of the cross sections as a function of
the burnup level of the fuel. It is also possible, for example, to
specify them as a function of both the burnup level and the soluble
boron concentration; the code will make the necessary
multi-dimensional interpolation for the particular situation of the
isotope at the moment of cross section evaluation. Note that the
cross sections are evaluated for each isotope and for each material
zone in the core, so it is possible to have different cross
sections for a given isotope at a given time, if the isotope is
present in two different regions. In order to make this possible,
the burnup level is computed and accounted for as a separate item
for each diffusion mesh space. This is an indication of the
extreme sophistication of the cross section handling procedures,
but it suggests as well the fact that PDQ-7 requires enormous
amounts of computer memory and time.
The code is also very generalized and flexible in the geometry
specification. It accepts one to three dimensions, with the mesh
spacing of either rectangular or hexagonal type in the X and Y
directions. The mesh and material zone specifications are done in
such a way that it is simple to build large structures with
repetitive patterns, as is the case with most reactors.
PDQ-7 requires also the specification of the radioactive and
transmutation chains with all the decay times involved in them.
This represents a further task for the user (although it is usually
taken care of by the auxilliary codes mentioned above) but it


APPENDIX D
ISOTOPIC AND SPECTRAL DATA FROM BURNUP CALCULATIONS
Chapter IV describes the burnup optimization calculations
performed for the standard and the low power density cores, and the
main neutronic and plant performance results that can be drawn from
them.
Table D.l shows the cell-homogenized number densities of
U-235, Pu-239, Pu-240 and Pu-241 at the beginning and end of life
of the fuel pins for all the core cases studied in Chapter IV. All
plutonium concentrations are assumed to be zero at BOL, since the
initial fuel composition is suposed to contain only uranium
isotopes. Note that the BOL concentration of U-235 decreases as
the pitch is increased. This is the effect of the cell
homogenization of the number density. All fuels are assumed to
start with a 3.1% enriched uranium composition. A pellet density
of 93% is assigned, which is the default assumed by EPRI-CELL. The
number densities shown include also the effect of thermal fuel
expansion, as calculated by EPRI-CELL.
Table D.2 lists the cell average neutron velocity for the
sixteen cores studied, for their BOL, MOL and EOL. As expected,
the cores with larger pitches show a lower velocity, due to the
better neutron moderation. The power level for any particular
pitch does not show a significant effect on the average velocity.
299


13
Chapter V is a brief description of the thermal-hydraulics
safety-related aspects of the SHARP as compared to the standard
plant, and an insight into the comparative economic behavior of the
SHARP.
Chapter VI contains the main conclusions of this study and
recommendations for future research on the topic, considering
points that have appeared as unresolved and potentially
advantageous questions concerning the SHARP concept.


APPENDIX Bll
POWEROO CODE DESCRIPTION
The PCWERCO (34) code is used for computing the total busbar
cost of electric power generation. This includes all the factors
that affect the cost of generation: fuel costs (including shipment
costs, possible credits for reprocessable fissile materials, ore
costs, etc.), capital costs, investment costs, plant life, capacity
factors, tax rates, etc.
The code makes use of three different methods of calculating
the power cost. The payout method is the most fundamental one, and
it requires a trial-and-error approach. The present worth method
avoids this problem by translating all costs through the plant life
to a single point in time. More popular than the present worth
method is the fixed-charge-rate method; although some
approximations are made in its application, which render it less
rigorous and more empirical.
The input stream to POWERCQ includes the following items:
a). Plant Investmebt.
b). Non-fuel working capital.
c). Project life.
d). Design capacity.
e). Depreciable life.
f). Economic parameters.
275


204
6.2. Neutronics, Burnup and Ore Usage
The in-depth burnup studies were carried out on a series of
cores in which power density and fuel lattice pitch were used as
variable parameters. The standard Westinghouse 4-loop 17X17 fuel
assembly PWR core was used as a reference core against which the
SHARP results were compared. The reduction of power density causes
a decrease of Doppler resonance absorptions in the fuel and a
reduction of xenon concentration level, resulting in additional
reactivity being available for further burning of the fuel. The
variation of fuel lattice pitch was shown to have an even larger
effect on discharge burnup than the reduction of power density
level. These trends can be observed in Figure 6.2.1. It appears
that the initial power level reduction (from 100% to 50% of the
standard core power level) results in a higher increase of burnup
per unit power reduction than further power level reductions to 33
or 25% of the standard reactor's power. The burnup increments per
percent power reduction are 0.11% for the 100% to 50% power
reduction, 0.092% for the 50% to 33% reduction and 0.087% for the
33% to 25% power reduction. These figures clearly indicate that
the effect of power density level reductions on burnup are more
beneficial for the initial reductions. This factor, favoring
smaller reductions in power density, is further reinforced by the
fast deterioration of the economic aspects of the plants with power
density levels below 50% of the standard plant.
Examination of the effects of fuel pitch variations shows that
all power density level cores have a maximum discharge burnup level


240
neutron absorbing rods. They are distributed throughout the core
cross section and then normally placed at about mid-height in the
axial direction in order to reduce, as far as possible, the power
peak that normally appears at the core midplane. The main problem
associated with the partial-length control rods is that they take
the space of fuel pins or full-length control rods, and it is just
not possible to deploy many of them; this causes the axial power
flattening to be imperfect. Figure A.2.7 shows the effect of a
partial-length rod on the axial power distribution of the Oconee-1
reactor (38).
It is hard to devise an element that could do a similar job
without occupying fuel pin or full-length control rod spaces
because partial length rods need to be movable. This is true for
several reasons: one is that they may be used to help control
reactivity transients, like xenon oscillations; another is the fact
that if an absorbing material is placed in a fuel assembly in a
non-movable manner, it causes a reduction of the core life because
it absorbs some of the highest worth neutrons in the reactor, which
is not desirable at EOL.
A.2.7. Increasing Enrichment.
If the enrichment of the reload batch of a reactor is
increased, it seems obvious that the core should be able to achieve
higher burnup, causing the discharge burnup to be also higher. If
the fuel management scheme is kept constant and the same batch size
is used, achieving a higher cycle burnup would entail longer
cycles, which is a desirable trend from the economic point of view.


34
calculations, such as buckling searches, poison searches, etc. In
this case, a normal effective multiplication factor (K-eff.)
calculation is requested.
The code accepts a cylindrical geometry, but it is a
one-dimensional diffusion-theory code. In order to obtain accurate
results for the neutron multiplication factor in one-dimensional
cylindrical geometry, accurate values for the perpendicular
bucklings to account for perpendicular leakage are required. In
the absence of this information, it is known from previous
experience that an "equivalent" spherical system can yield accurate
results for the neutron multiplication factor. For the
"equivalent" spherical configuration, the core volume is kept equal
to that of the cylindrical reactor, while for the reflector, the
thickness is kept equal to the cylindrical reactor value. The core
region has 199 mesh points allocated, and 34 mesh points are
assigned to the reflector, thus assuring that the mesh spacing is
smaller than the neutron diffusion length. This is necessary for
these calculations if neutron diffusion theory is expected to
describe the core fluxes and currents with an acceptable level of
accuracy. An extrapolated zero-flux boundary condition is
specified for the outer boundary. Figure 2.2.2 illustrates the
geometric model used with MONA, for the criticality calculations.
Two methods were used for the scoping burnup calculations.
The first method involves the use of the calculational scheme
already used for the criticality calculations. Once the reactor
criticality status is established, soluble boron is added to the


86
until it agrees with the cycle length estimated by the timestep
collapsing procedure.
This scheme, however, involves gross errors because the
absence of soluble boron causes variations of some percent in the
cross sections of some isotopes such as U-235 and Pu-239. This
entails errors in isotope concentrations and as a result, errors in
the multiplication factor calculations.
Another cell burnup method, which partially solves this
problen, specifies a soluble boron concentration in the moderator
with a value similar to the time-averaged boron concentration of a
real core. This concentration is approximately 400 ppn. But then
the fuel cell multiplication factors can not be directly used for
the calculation of the core multiplication factor, because they
involve the soluble boron poisoning. The problem is solved by
using a very short "pseudo-burnup" timestep, with no soluble boron,
after each real burnup timestep. A "pseudo-burnup" timestep is a
very short timestep which practically adds no burnup to the fuel,
so that the isotopics are not altered, but still allows for the
eigenvalue calculation in the boron-free configuration. This way,
the fuel is always burned with a spectrum influenced by the 400 ppn
of soluble boron, but the multiplication factors are obtained from
the pseudo-burnup timesteps which contain no boron, and the "clean"
batch reactivity can still be obtained through the fuel life.
Figure 3.1.5 illustrates this method, showing the boron
specification and the multiplication factors that may be obtained
through the core life.


221
of batches of a reactor will drive the fuel to higher burnups, for
the same initial enrichment. On the other hand, if the enrichment
is kept constant, having more batches in the core reduces the size
of the reload batch. Consequently, the time between refueling
shutdowns is reduced, which is generally not desirable because it
lowers plant availability.
The inmediate solution to these "very-short" cycles is to
increase fuel enrichment so that a higher core average burnup can
be achieved at EOL, and thus, longer times can elapse between
consecutive refuelings. R.L. Hellens showed (3) that in addition
to improving the discharge burnup, increasing the number of batches
and fuel enrichment also improves ore utilization for enrichments
up to about 5% U-235, as can be observed on Figure A.2.2.
In any case, a 5% enriched fuel used in a 4- or 5-batch core
would theoretically drive the fuel to burnup levels far beyond the
present technological possibilities. If fuel pins were made which
could stand average fuel burnups of about 50 GWd/MTU, potential ore
savings of up to 15% could be achieved. However, with the present
state-of-the-art fuel and the assumptions stated in Section A.l,
it is feasible to obtain 40 GWd/MTU with a 4-batch core, a 12-month
cycle and about 4% enriched fuel, with an associated improvement in
ore utilization of about 10% .
The main problem associated with such a maneuver, as described
above, is that increasing the number of batches in a core
containing a constant number of fuel assembly locations increases
the probability of placing highly burned assemblies adjacent to


91
It would still be possible to use the "clean" pseudo-burnup steps,
but the author decided that the adjustment of the boron letdown
curve to the point where its value should be zero was best suited
for this purpose. The advantage of using the boron curve for the
EOL estimate is that it contains the constantly varying spectral
and self-shielding changes, which depend on core burnup status and
boron concentration, while other popular models neglect these
aspects.
The above paragraphs discussed the need for an accurate
determination of the soluble boron concentration level in the fuel
cell burnup code, but almost as important as the boron is the
specification of the true power density level through the fuel
life.
It is frequent to see that the simple burnup schemes do not
specify changing power density levels for the fuel as it progresses
through burnup. It is even frequent to see almost an equivalence
being used when referring to real-time core life and burnup since
they are considered to be so closely equivalent. However, it is
well known that the different batches in the core do not share
equal power levels in spite of the efforts to flatten the power
distributions. As a general rule, the batches being burned for the
first and second cycles generally hold a higher power level than
the third-cycle batch.
When assigning a boron letdown curve for the fuel cell burnup,
one is not only assigning a certain boron concentration to a core
life-time, but also to each burnup point through the fuel life.


298
NEW BORON PPM
1321
1200
1055
761
331
NEW RELATIVE POWERS
1.08646
1.10146
1.09566
1.09068
1.08486
BATCH 3
STEP #
11
12
13
14
15
INITIAL STEPS ENDING TIME
32800.00
34600.00
38600.00
44600.00
48900.00
INITIAL STEPS K-INF.
0.89893
0.91031
0.91261
0.91807
0.92765
CLEAN K-INF.
1.01505
1.01536
1.00709
0.98798
0.95766
INITIAL BORON PPM
1310
1170
1040
750
310
INITIAL RELATIVE POWERS
0.81000
0.83000
0.83000
0.84000
0.86000
NEW STEPS ENDING TIMES
33258.70
35058.70
39058.70
45058.70
49588.10
NEW TIMESTEP DURATION
200.00
1800.00
4000.00
6000.00
4529.36
REACTIVITY-CORRECTED K-INF 0.89967
0.90816
0.91227
0.91697
0.92399
NEW BORON PPM
1321
1200
1055
761
331
NEW RELATIVE POWERS
0.85383
0.86886
0.87645
0.88481
0.89724
ESTIMATED BURNUP FOR CYCLE 1 IS 12736.5 MWD/MTU
ESTIMATED BURNUP FOR CYCLE 2 IS 13568.3 MWD/MTU
ESTIMATED BURNUP FOR CYCLE 3 IS 10990.7 MWD/MTU
END OF LIFE BURNUP IS ESTIMATED AT 37295.5 MWD/MTU
CYCLE LENGTH IS ESTIMATED AS 16529.4 HOURS, WHILE
INPUT ESTIMATE WAS 16300 HOURS.
MAXIMUM PERCENT CHANGE BETWEEN OLD AND NEW ESTIMATES OF
STEP RELATIVE POWERS IS 9.88877
BATCHWISE BORON WORTH REGRESSION LINE IS: W=-9.9343E-5+(1.63E-8*PPM)
\


170
related to the variation of the capital costs part of the total
cost of energy. It is interesting to note that over 9% of the
total plant life in a standard plant is spent refueling, while this
figure is reduced to slightly over 4-5% for the optimized 50% power
core, and about 3-0% and 2.3% for the 33% and 25% power level
cores. Note that the variation of pitches does not heavily affect
the percent impact of the refueling time.
Using more conventional parameters, column 6 on Table 4.2.2
labeled Availa. % shows the availability factors that can be
expected from the different reactors assuming that they encounter
the same percentage of unexpected problems during normal operation
(which is a pessimistic assumption when comparing the low power
plants to the full power standard plant). The only variable that
is supposed to affect the availability of the plant in this
calculation is the relative effect of refueling time. The
availability factor can be obtained from the formulas defining the
Active Time Availability Factor, which was recognized above as an
invariable index- The expression for the availability factor is
AF = ATAF (TCT ROT) / TCT
As it was expected, the standard plant shows an availability
factor close to 75% (it is slightly higher due to the longer
calculated cycle time caused by the reasons stated above). The
pitch-optimized, 50% power core shows an availability factor of
about 4% over that of the standard plant, while the highest
availability, which of course is obtained by a 25% power core, is


Code output reproduction:
HP50A252. 50% PCWER. PITCH = 1.25
297
BATCH 1
STEP #
1
INITIAL STEPS ENDING TIME
200.00
INITIAL STEPS K-INF.
1.16620
CLEAN K-INF.
1.28232
INITIAL BORON PPM
1310
INITIAL RELATIVE POWERS
1.17600
NEW STEPS ENDING TIMES
200.00
NEW TIMESTEP DURATION
200.00
REACTmTY-CORRECTED K-INF
1.08072
NEW BORON PPM
1321
NEW RELATIVE PCWERS
1.05971
BATCH 2
STEP #
6
INITIAL STEPS ENDING TIME
16500.00
INITIAL STEPS K-INF.
1.00320
CLEAN K-INF.
1.11932
INITIAL BORON PPM
1310
INITIAL RELATIVE POWERS
1.01000
NEW STEPS ENDING TIMES
16729.40
NEW TIMESTEP DURATION
200.00
2
3
4
5
2000.00
6000.00
12000.00
16300.00
1.14890
1.14780
1.14150
1.13200
1.25395
1.24228
1.21141
1.16201
1170
1040
750
310
1.14000
1.14000
1.13000
1.11000
2000.00
6000.00
12000.00
16529.40
1800.00
4000.00
6000.00
4529.36
1.06587
1.06482
1.06278
1.05874
1200
1055
761
331
1.02967
1.02789
1.02451
1.01790
7
8
9
10
18300.00
22300.00
28300.00
32600.00
1.01420
1.01370
1.01270
1.01360
1.11925
1.10818
1.08261
1.04361
1170
1040
750
310
1.03000
1.03000
1.03000
1.03000
18529.40
22529.40
28529.40
33058.70
1800.00
4000.00
6000.00
4529.36
REACTIVIT'/-CORRECTED K-INF 1.01486 1.02252 1.02000 1.01807 1.01601


TABLE OF CONTENTS
gage.
ACKNOWLEDGEMENTS iii
LIST OF TABLES viii
LIST OF FIGURES ix
ABSTRACT xii
I.INTRODUCTION 1
1.1. Background 1
1.2. The SHARP Concept 4
1.3. Previous Studies of Low Power Density Cores . 9
1.4. Purpose of this Study 10
II.SCOPING WORK 14
2.1. Problem Framing 14
2.2. Calculational Methods for the Scoping Study . 25
2.2.1. Heat Transfer Calculations 25
2.2.2. Neutron Cross Section Calculations . 31
2.2.3. Criticality and Burnup Methods .... 33
2.3. Results of Preliminary Work 41
2.3.1. Reactivity Coefficients 41
2.3.2. Neutronics and Safety Considerations 51
2.3.3. Burnup Achievements and Isotopic
Inventories 63
2.4. Scoping Work Conclusions 70
III.BURNUP CALCULATION METHODS 72
3.1. Burnup Calculational Methods and Present Needs 72
3.2. Method Developed for this Study 99
3.2.1. The Fuel-burning Codes 99
3.2.2. The CRIBUR Core Model 106
3.3. Benchmarking of the Burnup Calculations .... 132
3.3.1. Available References for Comparison . 132
3.3.2. Criticality Benchmarking 135
3.3.3. Power Distribution Benchmarking .... 140
v


62
One of the main parameters of concern when analyzing the heat
transfer conditions in a EWR is the minimum DNBR (Departure from
Nucleate Boiling Ratio) existing in the core. The minimum DNBR
indicates how far the core is from reaching a film boiling heat
transfer condition, which would result in heavy overheating and
damage of the fuel.
Mark Miller performed calculations(16) on the subject of the
DNBR for several power densities and several flow rates and
compared the results to the DNBR actually found in the standard
reference core. A more detailed analysis of the thermal-hydraulic
performance of the low-power cores is presented in Chapter V. It
is, however, important to point out here that all the reduced-power
cores show a higher DNBR than the standard core, under equal outlet
enthalpy condition, and using the same fuel lattice geometry. For
larger pitches (which may be of interest from the discharge burnup
viewpoint) the low-power cores can accept some pitch increase and
maintain the outlet enthalpy and still stay at a better DNBR than
the standard reactor. However, the standard core can not use
larger pitches without seriously reducing its thermal-hydraulic
safety margins, due to the reduction of coolant velocity associated
with the increase of fuel lattice pitch. This is an unequivocal
and key point proving the substantially higher safety level of the
low power reactors as compared to the standard ones.


128
along the life are within an acceptable margin of accuracy. If the
soluble boron is not considered for a moment, the fuel would have a
multiplication factor which would depend on its isotopics, which in
turn are closely related to the burnup level since the actual
burnup was calculated with a reasonable neutron spectrum. This
means that under those conditions, the fuel has an invariable
characteristic: the correspondence of its "clean" (boron-free)
multiplication factor with the burnup level. If this "clean"
multiplication factor can be obtained for each known burnup level,
it would be a set of data independent of the variations of cycle
length (again, as long as the cycle length guess was not too much
in error to start with)- This is represented in the lower axis of
Figure 3-2-6-
Since CRIBUR calculates the soluble boron worth line, it is
possible to integrate the reactivity variation caused by boron from
concentration zero to the actual concentration assigned to each
burnup timestep. and therefore it is possible to calculate the
"clean" multiplication factor that can be associated with each
initial timestep. and what is more important, that can be
associated with each timestep's burnup level. This correspondence
is kept in a table within CRIBUR- and for any burnup level, the
"clean" multiplication factor can be interpolated. The actual
multiplication factor of any particular situation is then computed
by first obtaining the "clean" multiplication factor, and then
subtracting the integrated soluble boron worth from zero boron to
the boron level that is actually assigned.


Ill
The problem is now to obtain a reasonable value for Pnlb.
Considering that the zone occupied by the peripheral batch is very
thick in terns of the neutron diffusion length, it is possible to
assume that none of the neutrons leaking from the core into the
reflector come, from the inner region batches, and therefore all
net losses are from the peripheral batch population. Since in the
present large FWR cores the radial power distribution is maintained
fairly flat, it is also reasonable to assume that there is no
significant net neutron current between the outer and the inner
core zones. In this case, the probability of leakage of a neutron
belonging to the peripheral batch can be expressed as the
probability of leakage of a neutron from the whole reactor, scaled
up by the ratio of population of the whole core to that of the
peripheral batch. This can be mathematically expressed as follows:
Plr Nr = Plb Nb
where Nr = reactor neutron population.
Nb = peripheral batch neutron population.
Recalling that the probability of a neutron leaking can be
expressed as
Plb = Nleak / Ntotal
where Nleak = number of neutrons leaked.
Ntotal = neutron population.
It is possible to rewrite the peripheral batch neutron leakage
probability as


279
370 REM ******************************************
380 REM LBP'S WORTH AT BOC AND EOC
390 BB=0:BN=0:BP=0:INPUTnENTER # OF BATCH (1 IS THE NEWEST)
CONTAINING BURNABLE POISON. TYPE 'ENTER' IF NONE
DOES." ;;BP: IFBP=0THEN410
400 INPUT"BATCHWISE REACTIVITY WORTH OF POISON RODS (%K/K) AT BOC &
EOC";BB,BN:IFBN>BBTHEN400
410 FORI=1TOCT :RC (I r 2) =0: NEXTI
420 REM *******************************************
430 REM INPUT OF ALL DATA IN 'R' MATRIX
440 CLS: PRINT "INPUT ENDING TIMES (IN HOURS FROM BOL) OF EACH
TIMESTEP"
450 FOPJ=1TOTS:PRINT"ENDING TIME OF TIMESTEP "I; :INPUTR(I,4) :NEXTI
460 CLS:PRINT"INPUT BORON PPM FOR EACH TIMESTEP"
470 FORI=lTCTS:PRINT"BORON PPM OF TIMESTEP "I; :INPUTR(I,6):NEXTI
480 CLS:PRINT"INPUT RELATIVE POWER FOR EACH TIMESTEP"
490 FORI=lTOTS:PRINT"RELPWR OF TIMESTEP "I; :INPUTR(I,2) :NEXTI
500 CLS:PRINT"INPUT K-INF FOR EACH TIMESTEP"
510 FOR I=1TOTS
520 PRINT"TIMESTEP "I" K-INF="?:INPUTR(I,1):IFR(I,1)<3THEN540
530 PRINT"THIS K-INF LOOKS PRETTY WILD 111 l":GOTO520
540 NEXTI
550 REM ***************************************
560 REM CHECKING INPUT
570 REM ***************************************
580 CLS:PRINT"INPUT RELATIVE FLUXES FOR EACH TIMESTEP"


289
2530 FORJ=3TCe:A$=A$+MKS$(RC(I,J)):NEXTJ
2540 1=1+1: IFLEN(A$) >CCTHEN2560
2550 IFK=CTTHEN2530ELSE2570
2560 LSETN$=A$:PUT l:A$=nn:CC=223: FIELD 1,255 AS N$:GOTO2550
2570 IFCH>TSTHEN2600
2580 F0RJ=1T06 :A$=A$+MKS$ (R(CH,J)) :NEXTJ
2590 CEKBFl :GCTO2540
2600 IFBE=0THEN2640
2610 F0RJ=1TCBE:A$=A$+MKI$(BW(J)):NEXTJ
2620 A$=A$+MKI$(CA):IFCA=0THEN2640
2630 FCRJ=lrPOCA:A$=A$+MKI$ (BK(J)) :NEXTJ
2640 IFA$=""THEN2660
2650 LSETN$=A$:PUT 1
2660 CLOSE:RETURN
2670 REM ***************************************
2680 REM ROUTINE FOR CALCULATION OF MODIFICATION OF K FOR THE
EXTERNAL BATCH
2690 FORI=1TOCT:CH=RC(I,8):TA=-20.0:BU=R(CH-1,12)
2700 L2=(0.486-(1.9E-7*BU))/(0.1912+(6.42E-7*BU))
2710
SA=(0.486-(1.9E-7*BU))/(SQR((0.4033-(5.135E-6*R(CH,7)))*(0.0166+(2.72
7E-5*R(CH,7)))))
2720 B2=(2.405/(168.53+SA))[2:UB=0
2730 REM SHAPE FACTOR CALCULATION
2740 PORJ=lTONB:UH=RC(I,J+2):UB=UB+R(UH,3):NEXTJ
2750


55
and it is one of the main factors responsible for the high
reactivity found for core 8 in spite of its low fuel enrichment
(see Table 2.3.1). However, it is also important to note that this
same curve shift prevents considering a different pitch for core 8,
because it is already working at a point much closer to the optimum
moderation ratio, and any further change of pitch would bring the
core too close to a possibly overmoderated configuration, which as
explained above, is not acceptable from a safety point of view.
Figures 2.3.4 and 2.3.5 show the changes in the four factors
of the "four-factor formula" for cores 3A, 4A and 8, as pitches are
varied. These are the results of the criticality calculations
performed for BOL with PHROG-BRT-MONA. It is apparent that the
neutronic effect of the higher moderator density of core 8 is far
larger than the change caused by the Doppler effect difference
between cores 3A and 4A. None of the four factors exhibit a
difference of more than one percent in their corresponding values
between cores 3A and 4A. However, the largest differences are
observed in the resonance escape probability for very
undermoderated lattices. This could be expected because this is
the configuration and the factor that give the maximum enhancement
to the Doppler effect variation.
The change of moderator density causes a sharp difference of
spectra which is reflected in all the four factors. The fast
fission factor, epsilon, is consistently lower for core 8 than for
the other two. Note that the fast fission factor experiences a
sharp rise when the lattice pitch reaches very small values and


247
One way of achieving this is to use a mixture of light and
heavy water as moderator. Heavy water has a lower moderating power
than light water, and thus the neutrons are less moderated,
suffering more resonance absorptions in the fertile isotope.
Towards EOL, heavy water can be replaced by light water in order to
obtain the maximum moderating power. This method was studied for a
mixed uranium-Thorium fueled reactor by Combustion Engineering (3)
and they predicted large potential ore savings with this method.
The obvious problem of this method is the cost associated with
the production and management of the mixture of light and heavy
water, which would place a heavy burden in the fuel cycle cost, in
addition to all the re-design needed in order to accomodate these
changes into FWR plants.


'
89
The above reasons induced the author to consider that the
specification of the boron level that would keep the reactor
critical at all times is of significant importance for an accurate
determination of the core life and should not be neglected, as is
commonly done in simplified burnup calculations.
Following the soluble boron concentration through the life of
the fuel implies a fuel cell burnup boron specificaton such as the
one shown on Figure 3.1.6. This solves the problen of the
erroneous neutron spectrum, but creates the new problem of the
boron letdown curve needing careful adjustment, mainly concerning
the core life duration. This is needed because specifying too
short or too long of a cycle time on the boron letdown curve could
cause the period of fuel burning to be done with a very low boron
concentration when the highest concentration would be required, or
vice versa, and this would bring back the problems of erroneous
isotopics and eigenvalue estimates because of inexact spectra.
This boron determination requirement is solved by the calculational
method described later in this chapter. This new method contains
important differences with the schemes described above. The
previous schemes relied on an extrapolation or interpolation of the
core multiplication factor (or sometimes on the core reactivity,
which is supposed to follow a more linear behavior) for the
determination of the core life. The scheme used in this work,
where soluble boron is constantly adjusted to its real value, can
not rely on these eigenvalues since the core eigenvalue should by
definition be kept at a value of unity throughout the core life.


146
the standard out-in-in scheme is used throughout the burnup
optimization calculations.
The use of burnable poisons is very common in the industry,
but although they can be represented in the calculational method,
it would be hard to reliably account for small variations in them,
and comparison with a "standard" core would be difficult, since
this is normally a particular case-to-case type of parameter.
Gadolinium loaded pins are still far from widespread consideration
for commercial PWR's, and the calculational schemes available for
this work would not be able to account for them properly, so they
are also not considered in the studies.
Enrichment is a parameter that is likely to be varied in most
studies, and several works have pointed out that slightly higher
enrichments may improve ore utilization, aside from extending core
cycles, but this forcedly requires the use of some type of burnable
poison, and it is therefore not considered here for the reasons
mentioned above. However, further research is needed in this
direction if suitable benchmark cases become available, since this
is a parameter which might result of significant effect, and which
should not present large problems of acceptance by either the
industry or the regulatory agencies.
A parameter that is simple to vary and that has shown to have
a significant effect in the preliminary calculations is the fuel
lattice pitch. Its effects on the neutron spectrum cause
significant changes on the isotopic evolution of the fuel, and
therefore on the discharge burnup levels, the ore utilization, etc.


163
The MAP can be calculated from the standard plant assumptions
stated above, considering that the product of the ATAF by its
corresponding "cycle time" must equal the product of the actual
availability factor by the actual cycle time:
MAF (TCT-ROT) = AF TCT
where ATAF = Active-time availability factor.
TCT = Total cycle time. (365 days for the standard plant).
ROT = Refueling time. (45 days).
AF = Availability factor. (75% for the standard plant).
According to the standard plant assumptions, this definition
yields a value for the ATAF of 85.5%.
Another parameter defined to establish the capacity factor
variations is termed "Operating Capacity Factor". The ATAF defines
the level of on-line time of the plants when the refueling time is
not considered. However, even during the on-line time, the plants
do not deliver their full power. There are a number of reasons
that prevent the plants from delivering 100% power level at all
times while they are connected to the grid: One cause is the
de-rating of the plant due to technological causes, frequently
associated with the fuel and its thermal conditions; another is the
power runbacks, or forced temporary de-rating following a power
transient, which is normally required in order to assure the fuel
and primary coolant system integrity and performance. With these
and other similar losses of capacity factor in mind, the average
power level of the plant during the on-line time is defined here as


2
The many operational transients? safety-related incidents and
the rare but significant accidents resulting in extended reactor
shutdown, e.g. Three Mile Island and others, clearly substantiate
this assertion.
A key dilemma appears immediately: a very high power density
reactor has intrinsic serious safety problems; however, economic
considerations have resulted in larger, high power density reactors
working closer to their technological limits and requiring improved
engineering safeguards. The industry has naturally worked towards
extracting the maximum power from a given core, and also towards
extending the reactor fuel cycle time in an attempt to reduce the
economic impact of refueling outages and fuel carrying charges.
Both these goals tend to demand performance from the reactors in a
manner that is not always compatible with strict safety
considerations.
Several questions need to be asked at this point. First: Is
there a way of relieving the LWR's cores from these limiting
situations without incurring an unacceptable economic penalty?
Second: Will the new concept or approach need extensive (and thus
costly and long) research and development and therefore have no
impact in the short term? Third: Will the new concept be easily
licensable, or will it require completely new licensing regulations
and processes that would take many years to develop and create
significant uncertainties for its deployment?
The first part of the first question has a relatively obvious
answer from the technical point of view: reducing the power density


147
Variation of this parameter is likely to be accepted without much
problem by the industry, mainly because it is not a significant
change in the design and operation of the plant, nor does it
require special types of calculations for core-follow control.
This parameter is therefore chosen for core optimization in the
burnup studies of the low-power cores. Since there are sound
guidelines on the range of pitches that are acceptable from the
preliminary calculations described on Chapter II, the starting
pitch for the parametric studies was selected at 1.20 cm and
incremented by 0.05 cm for each burnup calculation, until a net
reduction of burnup is encountered. The decrease in burnup is
likely to occur around a pitch of 1.45 cm according to the results
of the preliminary calculations.
If a 1.45 cm pitch shows the expected decrease in burnup
(after experiencing a maximum peak in a smaller pitch), this
requires a total of 19 burnup cases: six pitch values for each of
the three low-power cores (25%, 33%, and 50% power density), plus
one case for the standard core, which is only studied at its
nominal pitch, as a reference case. Thermal-hydraulic safety
considerations prevent the use of the standard power density core
with larger pitches, as explained in next chapter.
4.2. Results of Optimization
4.2.1..Neutronics and Isotopic Results.
The burnup studies performed for the different power level
cores and the changing pitches generated data sets that require


44
which implies higher moderator density. This rather large
reactivity coefficient associated with the fuel enrichment could be
used in order to stretch burnup of the cores by increasing the fuel
enrichment, which up to a certain point will result in a better ore
utilization (14).
Higher fuel enrichment would, however, bring problems
associated with higher power peaking requiring the permanent use of
burnable poisons and the technological problem of loss of clad
integrity at very high burnup levels.
The reactivity coefficient of soluble boron in the moderator
is given an in-depth study for two main reasons: First, the soluble
boron concentration that makes a reactor critical at the different
points through the core's life is a very useful and realistic way
of estimating the cycle life of the core. However, for this to be
feasible, it is important to know the reactivity worth of soluble
boron at all times through the core's life. This coefficient may
be expected to vary, depending mainly on boron concentration and
core burnup status; these two variables are somewhat related, but
they certainly don't follow any exact relationship. Second, just
reducing the power density may cause the standard reactor lattice
geometry not to be the optimal one from the burnup point of view,
since the nuclear characteristics of the fuel are somewhat
perturbed. Some changes in lattice geometry, such as fuel pitch may
be required then in order to optimize the core for the new
low-power situation. If this is the case, the ratio of
moderator-to-fuel is likely to be changed, and then, a given


208
Another important improved operational variable is the xenon
concentration level. A 50% power level SHARP has an equilibrium
xenon concentration reduced by more than 25% with respect to the
standard plant value. This implies significant reduction of
problems associated with xenon oscillations or xenon concentration
transients caused by reactor shutdowns or large power variations.
The SHARP concept results in largely increased core cycle
lives as a result of two adding factors: the reduction of power
level itself, which entails a longer time in order to produce a
given amount of energy from the fuel, and the associated discharge
burnup level increase, which further contributes to the core life.
One implication of these longer core cycles is the reduction of
refueling outage time, and its associated high power replacement
costs. The 50% power, pitch optimized reactor shows a savings of
about 50% in total number of refueling outage days in the plant
life. A consequential effect is the increase of the plant
availability and capacity factors. The 50% power, pitch optimized
reactor shows a plant capacity factor increase of about 3.5% with
respect to the standard plant under the assumption of equal
probability of unplanned outages. When historical data concerning
reduced size plants was reviewed, it became apparent that these
plants have actually less unplanned outages than the large ones.
Evaluations of plant capacity factors based on forced outage
statistical data indicate a capacity factor increase of slightly
over 10% with respect to the standard plant for the 50% power,
burnup optimized SHARP.


CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH
6.1. Introduction
The primary purpose of this work is to establish the neutronic
and fuel utilization characteristics of the SHARP (Safer Highly
Available Reactor Plant), as well as to develop an accurate and
cost-effective calculational tool for the scoping studies of fuel
burnup. In addition, the primary thermal-hydraulics, safety and
economic characteristics of the concept and its advantages and
disadvantages over the present standard PWR reactor plant are
analyzed and summarized to provide a comprehensive review of the
key issues needed to judge its technical and economic merits.
The SHARP concept uses a standard, large size PWR reactor with
minimal design alterations to drive a reduced power rating electric
generation plant. The reactor is used at a reduced linear power,
while maintaining coolant thermodynamic characteristics in order to
keep the steam cycle efficiency. The immediate result of the power
density reduction is a significant improvement in technological
operational limits and an overall enhancement of reactor safety,
although it also results in a capital cost increase per unit of
installed power. The reduced power density entails differences in
the nuclear characteristics of the core, which require a thorough
study and optimization.
201


20
The basic reference core used in the scoping study is
described in Table 2.1.1; its fuel cell characteristics are
described in Table 2.1.2. The fuel used for the reference core is
uranium dioxide, enriched to 3% in U-235 isotope; the moderator is
considered at standard operating pressure (2250 psia.), but at rocm
temperature (293 K). No soluble or lumped poisons are considered,
nor are any fission products present for all the scoping beginning
of life (BOL) calculations, unless otherwise specified.
Under these conditions, the composition of the basic fuel cell
is as shown in Table 2.1.3. Region 1 corresponds, as in Table
2.1.2, to the fuel pellet; Region 2 is the gap between pellet and
clad; Region 3 is the Zircaloy-4 clad and Region 4 is the light
water moderator-coolant. This reference core will be named
core #1.
The pin-average linear power for the standard reactor is
6 KW/ft. A pin-average linear power of 1.5 KW/ft. is used for the
scoping studies of the SHARP. This is one fourth the linear power
of the standard Westinghouse core.
In an attempt to cover a wider scope and range of possible
applications of low-power reactors, the scoping study includes some
exploration of a very low power, low enrichment core which could be
used as a preheater in a multi-core configuration similar to the
one illustrated in Figure 1.2.2. This core would have lower than
standard moderator temperature and a very low power (20% that of
the standard core). Such a low power density, low moderator
temperature core might possibly be fueled with spent fuel


231
It is reasonable that such an in-core fuel management scheme,
which uses the close mixing of different-burnup fuel assemblies and
the different worth of the regions in the core, obtains a flat or
near-flat power distribution without much complication, and avoids
high power peaking factors. However, this scheme is rather poor
when one is concerned with neutron economy. The "freshest" batch
lies on the periphery, surrounded by a steel barrel and a
moderately absorbing refelector (depending on the chemical shim),
and its high neutron production is offset by a disproportionate
neutron loss in these adjacent materials. This is no great problem
at BOL because there is excess reactivity just about everywhere,
but unproductive and disproportionate neutron losses at EOL cause
the reactor to go sub-critical earlier than otherwise possible.
A new "low-leakage" fuel management scheme has been developed,
where neutron leakage is minimized at all times, thus trying to
stretch cycle life by simple neutron economy. In this low-leakage
scheme; the fresh batch is placed in the inner region of the core,
"checkerboarded" with the "third cycle" batch, while the "second
cycle" batch is put on the peripheral region. Consequently, this
fuel management scheme is called "in-out-in" to describe the
succeeding positions of the batch through its life, as opposed to
"out-in-in" which describes the placement of the fuel in the
currently used scheme.
With this in-out-in scheme, the "third cycle" batch is driven
by the fresh batch. This allows the oldest batch to be driven to
higher burnup at the same time that the fresh batch is more


142
The total elimination of the radial leakage shows similar
effects, but in a yet larger scale. The discharge burnup is
increased 7% over the correct calculation.
Increasing the batches' multiplication factors by 1% increases
by a few percent the power sharing of the batch being altered, and
causes an increase in discharge burnup of 2% when the outer batch
is altered, and 1.7% when one of the internal batches is altered.
The sensitivity calculations were performed using burnup data
corresponding to the standard, full power FWR.


243
Ore utilization
3 4 5 6
Fuel Enrichment W/o
Figure A.2.8. Ore Utilization vs. Fuel Enrichment and Burnup.


28
With the previous expressions, it is possible to obtain the
pellet surface temperature. Since the temperature of interest for
nuclear calculations is the average temperature of the pellet, it
is necessary to know the temperature profile in the pellet, as a
function of the distance to the centerline, so that volumetric
weighting of the temperature can be done.
The temperature profile inside the pellet (assuming uniform
heat generation) is given by
T(r) = To o r2
7T R* 4 Kf
where T(r) = temperature of the pellet at radius r deg. F
To = temperature of the pellet centerline, deg. F
q' = linear heat flow rate, Btu/hr ft
R = pellet outer radius, ft
Kf = pellet thermal conductivity, Btu/hr ft deg. F
r = distance from centerline, ft
One more quantity is needed at this point: the pellet
centerline temperature, which is given by
To = Ts + g1
4 TT Kf
where To = pellet centerline temperature, deg. F
Ts = pellet surface temperature, deg. F


APPENDIX B5
EPRI-CELL CODE DESCRIPTION
The EPRI-CELL (19) code computes the space, energy and burnup
dependence of the neutron spectrum and isotopic cross sections
within cylindrical cells of light water reactor fuel rods. Its
primary output consists of broad-group, microscopic, exposure
dependent cross sections for subsequent use in multidimensional
neutron diffusion theory depletion analysis.
The code is an answer to the need for a well standardized and
proven code able to supply the sets of burnup and/or nuclide
concentration-dependent cross sections for use in the large
multigroup, multidimensional, diffusion theory codes used in the
nuclear industry for fuel management calculations and core
following- This requires the following characteristics of the
code:
a). The code must be based on well-proven previous
computational schemes and must take into account all the parameters
that have been shown to have a significant effect on the fuel
characteristics as it undergoes the fission process.
b). The code must be flexible enough to allow representation
of all the types of fuel situations that can be normally
encountered when performing LWR calculations, yet it has to be as
simple as possible to the user- This is solved by providing two
258


238
great improvement in plant operational flexibility, and may allow
for longer core cycles, reduced outage time and reduced personnel
radiation exposure, which are very desirable features.
Westinghouse Corporation conducted research on this subject
(2) for a reactor whose power density was reduced to 75% of the
current standard value. A possible ore savings of 3% were
obtained, although the extra costs associated with the rest of the
fuel cycle placed the option at the same fuel cycle cost level as
the standard reactors.
The author performed some calculations for reactors in which
power densities were reduced to 50, 33 and 25% of the standard
values and obtained burnup increases of 3 to 4% with respect to the
standard reactor, depending on the power density chosen. Allowing
the pitch to vary and going to better-moderated configurations
increased the burnup improvements to levels between 5.5 and 7%
above the standard core, depending again on the power density
chosen.
It appears that most of the burnup improvement achievable is
obtained when cutting the power from 100% to 50%; little
improvement is obtained for further reductions to 33 or 25% power
density levels. Table A.2.1 shows the discharge burnups obtained
for the various power levels and their percent increase over the
standard core burnup.


90
Soluble
Boron
First Cycle T Second Cycle T Third Cycle
Relative
Power
Figure 3.1.6. Pin Cell Burning with Soluble Boron Letdown Curve
and Power Level Following.


113
Since normally all the batches in the core have the same
number of assemblies, all Vi's are equal, and can be factored out
of the summation and cancelled with the Vb of the denominator. The
probability of non-leakage from the peripheral batch becomes
Pnlb = 1 ((1-Pnlr) *£i /$b)
It is now necessary to find some way of obtaining the
probability of neutron non-leakage from the reactor and the
batches' average fluxes with the minimum data handling possible,
since all these data have to be input to the core model.
Hiere are well known expressions that determine, as a good
approximation, the probability of neutron non-leakage from an
homogeneous reactor, the reflector effect being accounted for with
the use of the "reflector savings". The batches' fluxes can be
obtained from the fuel burnup calculations. Considering that
neutrons can leak during slowing down or after becoming thermal,
the probability of non-leakage can be expressed as
Pnlr = (1 + L2* B2) expi-TT* b2)
where L = diffusion length of the core.
B = geometrical buckling.
V = neutron age.
However, it is important to note that the above expression
used for determining Pnlr is correct when applied to a bare
homogeneous reactor. The reflector effect is mocked up by adding


124
average, and some in which it is low (see Figure 3.2.5). It is
then possible to specify some "fake-burnup" steps in which soluble
boron concentration is altered by about 100 ppm. This causes a new
eigenvalue calculation, but does not affect the spectrum for the
real burnup step, which is performed with the adequate boron
concentration. By repeating the "fake-burnup" step procedure at
several boron concentration levels, CRIBUR can obtain the soluble
boron worth per ppm at the different concentrations. Observation
of different cases showed that the variation of soluble boron worth
is a closely linear function of boron concentration. CRIBUR
calculates then the regression line and obtains an expression that
allows the calculation of soluble boron worth for any
concentration.
Soluble
Boron
Figure 3.2.5. Fake-burnup Steps for Soluble Boron Worth.
Obviously, if the regression line is calculated for each of
the core cycles that form the total fuel life, slightly different


193
Table 5.2.1
Capital Costs (Mills/KWh) and Percent Increases over Standard Plant
Power Low-inflation Moderate-inflation High-inflation
Level Cost % Inc. Cost % Inc. Cost % Inc.
35%
50.22
73.3
108.64
72.6
221.07
72.0
50%
40.89
41.1
88.59
40.8
180.55
40.5
60%
37.10
28.1
80.43
27.8
163.98
27.6
70%
34.37
18.6
74.62
18.6
152.33
18.5
100%
28.97
0.0
62.94
0.0
128.52
0.0
Fuel Costs (Mills/KWh)
Table 5.2.2
and Percent Increases over Standard Plant
Power
Level
Low-inflation
Cost % Inc.
Moderate-inflation
Cost % Inc.
High-inflation
Cost % Inc.
35%
26.52
22.4
54.95
15.9
100.32
4.7
50%
24.87
14.8
53.30
12.4
103.70
8.2
60%
23.73
9.5
51.24
8.1
101.14
5.5
70%
22.93
5.8
49.78
5.0
99.18
3.5
100%
21.67
0.0
47.41
0.0
95.85
0.0


195
etc. up to transportation and storage of spent fuel. As with the
rest of this economic evaluation, all plants are supposed to start
operation in January 1993, and the costs are expressed in 1993
dollars. Note that the fuel cost is one item that was expected to
improve the economics of the low power density plants due to
increases in burnup. However, they all show a larger fuel cost
when compared to the standard plant, because it takes much longer
for a given batch of fuel to produce its share of energy- This
means that in a standard plant, a batch of fuel has produced all
its energy (and therefore has produced all its income) in a
three-year period; in the case of a 50% power plant, it takes more
than six years for the same batch of fuel to produce all of its
energy- The income is therefore retarded, and the capital invested
in the fuel batch suffers higher interest charges, which outweigh
the relatively small increment of burnup that is obtained from the
fuel-
It is clear from the figures on Table 5-2.2 that the turnout
time for the fuel costs is important for the economy of the fuel
cycle. This strong sensitivity to time suggests that it might be
an advantage for the SHARP to consider low enrichment fuels, which
would have reduced enrichment costs and would result in shorter
core cycles. On the other hand, studies performed in the
industry(3) show that reducing the enrichment results in a worse
ore utilization. These two considerations indicate the need for a
more complete study of the economic behavior of the SHARP"s as fuel
enrichment is allowed to vary. Figure 5-2-2 is a plot of the :


295
7
1.0142
8
1.0137
9
1.0216
10
1.0127
11
1.0288
12
1.0136
13
0.8989
14
0.9103
15
0.9126
16
0.9181
17
0.9276
ANY CHANGED
Screen 5:
CHECK RELATIVE PCWER FOR EACH STEP.
ENTER STEP TO BE CHANGED, OR PRESS "ENTER".
TIMESTEP 1 HAS RELPWRf 1.176 TIMESTEP 2 HAS RELPWRf 1.14
TIMESTEP 3 HAS RELPWRf 1.14
TIMESTEP 5 HAS RELPWR= 1.11
TIMESTEP 7 HAS RELFWR= 1.03
TIMESTEP 9 HAS RELPWRf 1.03
TIMESTEP 11 HAS RELFWR= 1.03
TIMESTEP 13 HAS RELPWRf 0.81
TIMESTEP 15 HAS RELPWRf 0.83
TIMESTEP 17 HAS RELPWRf 0.86
Screen 6:
TIMESTEP 4 HAS RELPWRf 1.13
TIMESTEP 6 HAS RELPWRf 1.01
TIMESTEP 8 HAS RELPWRf 1.03
TIMESTEP 10 HAS RELPWRf 1.03
TIMESTEP 12 HAS RELPWRf 1.03
TIMESTEP 14 HAS RELPWRf 0.83
TIMESTEP 16 HAS RELPWRf 0.84
ANY CHANGED STEP ?
CHECK CORE BATCH COMPOSITION IN CYCLE.
ENTER STEP AND COLUMN TO BE CHANGED, OR JUST PRESS "ENTER"
STEP BATCH-1 BATCH-2 BATCH-3 BATCH-4 BATCH-5
11 6 13 0 0
2 2 7 14 0 0
OUTER-BATCH
1
2


46
Boron Worth
pcm/pptn
Figure 2.3.1. Soluble Boron Worth at Critical vs. Burnup.


Cycle Length
Days
167
25% Power
1000 -
50% Power
500 -
Std. Core
T
1.20
130
1.40 Pitch
cm
Figure 4.2.3. Real Cycle Length vs. Pitch.


73
coupled with thermo-hydraulics models. These type of studies
require large computational efforts and highly specialized
personnel, which makes them extremely costly.
b). Burnup studies performed by utilities, either for core
following or for licensing. These also have to be detailed
studies, because they require strong economical decisions affecting
the core cycle, or have significant safety considerations affecting
the licensing procedure. However, there is presently a trend by
utilities to use more empirical codes (such as nodal codes) which
run at lower costs than the fully detailed pin-by-pin studies.
Nodal codes generally need base detailed diffusion theory
calculations for normalization purposes, but still result in an
overall lower cost than the fine mesh neutron diffusion theory
codes, while being able to maintain the information necessary for
adequate core management.
c). Studies performed by engineering companies or government
agencies for assessment of energy policies or new reactor concepts
or fuel cycles. These studies fall in a whole new class. They are
not directly concerned with the behavior of each particular pin,
not even of assembly-wise details. They look at overall results of
burnup, time scales, and monetary volumes. Although they need to
yield reasonable results, the numerical results themselves are
normally less critical, and emphasis is placed on the comparative
results of two or more different concepts. These studies generally
use some type of empirical formula or simplified core model which


141
few percent less to the second batch. What is also important is
that the order of batches from the highest to the lowest power
level is always kept the same as in the real cores in spite of the
closeness of the power levels of batches 1 and 2. The
calculational scheme used by M.I.T. yields a too high power level
for the first batch, while reducing that of the second batch. The
variations are of such magnitude that the second batch does not
hold the maximum power as is the case in an actual core, and as is
the case with CRIBUR results, but the maximum power sharing is
assigned to the first batch. It is important to remember that
CRIBUR1 s calculations did not assign any poisons to the fresh
batch, which are some times used in actual reactors.
From these comparisons it is possible to conclude that
CRIBUR's estimates of batch-wise power distributions are very close
to the true values found in an equilibrium-cycle FWR and quite
acceptable for relative burnup studies.
The alterations mentioned in the criticality benchmarking
section affected the power distribution, the discharge burnup and
the cycle length, proving again they are non-negligible in nature.
The elimination of the Shape Factor in calculating the radial
leakage boosts the peripheral batch power and reduces the power
sharing of the second and third batches. The discharge burnup
shows a large erroneous increase of 6.3%. This is due to the core
experiencing a very reduced radial leakage.


92
This means that a certain boron poisoning is assigned for each
isotopic status of the fuel. Figure 3.1.7 illustrates the
consequences of not specifying the different power levels of the
fuel as it advances through its life.
If the power level is maintained constant, as is the case with
most of the simple burnup calculational schemes, the end of core
cycles correspond to 1/3 of the discharge burnup, 2/3 of the
discharge burnup, and the total discharge burnup, in a three-batch
core. Thus, the burnup levels slightly after 1/3 and 2/3 of the
discharge burnup are burned with a very high boron level, which
corresponds to the beginning of the core life. However, the real
core situation has a higher burnup at the beginning of the second
and third cycles, because of the higher power level of the fuel
during the first two cycles of core residence. This burnup-time
mismatch of the flat-powered model as compared to the real core
burnup situation may cause a sustained error in the correspondence
of soluble boron to burnup level, as shown on Figure 3.1.7, with
the resulting errors in batch multiplication factors and neutron
spectrum.
Figure 3.1.7 shows a hypothetical boron letdown curve for a
core cycle, and using the same time axis, the fuel burnup is
represented on the ordinates for both a case where the power level
is kept constant and for a case where cycles 1 and 2 have a
somewhat larger power level than cycle 3, as is the case in a real
core. It is possible to see the sustained boron concentration
mismatch, which becomes most accentuated in the vicinity of the


244
the EOC coastdown: reducing the coolant temperature while keeping
the full thermal power of the reactor, or reducing both the coolant
temperature and the core power. The first system allows for
increased burnup, but practically no ore savings is obtained at all
as shown by Westinghouse's research (2) on this subject. The
second system may allow a 7% savings in ore cost and a 4% savings
in fuel cycle cost. However, the total generation cost may not be
improved at all, or at best may be improved to some limited extent;
this is a consequence of the extra costs of the replacement power
for all the period during which the plant is delivering a reduced
power level.
Coastdown must be carefully planned, because driving the core
to a limit will shorten the length of the next cycle. EOC power
coastdown appears as a feasible operational adjustment, but its
overall economy is not quite clear for a systematic use.
A.2.9. Other Possibilities of Minor Importance.
The options discussed in the previous sections appear to be
the best present candidates if a policy of increasing ore
utilization or extending burnup is to be seriously undertaken in
the near future. There are, however, many other ideas which might
in some way help increase burnup, but which either represent a very
small potential effect or their implementation would be difficult
and/or expensive. Some of these less possible ideas are discussed
below.
a). Zircaloy baffle. Present reactors use a steel baffle,
which presents a considerable absorption cross section. Zircaloy


68
considered when comparing results of burnup or uranium resources
utilization.
Since the present study assumes all fresh fuel to have the
same characteristics, increasing the discharge burnup implies
improving ore utilization. In this aspect, core 4A has a better
fuel utilization than core 3A. Although core 8 is able to achieve
a high burnup for a low-enriched fuel, the total use of U-235 per
unit of energy produced appears to be somewhat worse than that of
cores 3A and 4A. However, core 8 is not starting from the same
type of fuel, and therefore this result should not be considered as
a negative point for core 8. In addition, core 8 could be used to
further burn the fuel that is ordinarily discharged from standard
PWR's which would in fact represent an improvement of overall ore
utilization if fuel reprocessing is still not being considered.
Figure 2.3.8 shows the use of U-235 per unit of energy
produced, and the relative amounts of plutonium-to-U-235 burned for
the different cores, and for some varying pitches. Note that the
amount of plutonium burned is maximum for the tightest pitches, due
to the better conversion ratio associated with the hard-flux,
undermoderated configurations. As a logical result, U-235
consumption per unit energy produced tends to increase with pitch.
It is interesting to realize that for a slight increase of pitch
above the standard (from 1.25 to 1.3 cm), the U-235 specific
consumption actually decreases slightly. This is due to the
reactivity enhancement which occurs in going to larger pitches.
The poorer plutonium production quickly offsets this factor and the


133
concentration that holds the core critical at hot, full power, and
all control rods out (HFP, ARO) conditions. It is then possible to
calculate the multiplication characteristics of each separate
batch, and use them as an input to CRIBUR. Obviously, this method
allows the benchmark of the criticality calculation part of CRIBUR,
but not of the power distribution and burnup assignment part.
Another way of benchmarking CRIBUR is, as said above, to
compare its results against those of some well-known and accepted
codes. 1116 code EDQ-7 is a good candidate, because it has been
widely accepted as one of the main calculational tools in the
nuclear industry, its accuracy is well known, it can explicitly
represent the geometry of a 1/4 core, and it is possible to input
in it exactly the same data that is needed in a CRIBUR case, thus
making it possible to have exactly the same case represented in
both models.
However, this 1/4 core FDQ-7 representation has a few
problems. In a real equilibrium core, the assemblies composing any
fuel batch have different characteristics depending on their
particular history, and there are differences even between the pins
that form a fuel assembly, because sane of the pins are at the
assembly periphery, facing assemblies from other batches (and
therefore with sharply different characteristics and history) while
other pins lie inside the assembly, surrounded by pins of their
same batch and very similar life conditions. These differences
cause the interfaces between batches and between different
assemblies to be "smoothed out" as burnup increases, naturally


199
Table 5.2.3
Generation Costs (Mills/KWh) and Percent Increases over Standard Plant
Power
Level
Low-inflation
Cost % Inc.
Moderate-inflation
Cost % Inc.
High-inflation
Cost % Inc.
35%
83.92
45.1
174.84
43.8
338.73
40.1
50%
72.94
26.1
153.14
25.9
301.59
24.8
60%
68.01
17.6
142.92
17.5
282.46
16.9
70%
64.48
11.5
135.65
11.5
268.85
11.2
100%
57.82
0.0
121.60
0.0
241.71
0.0


19
Table 2.1.1. Main Core Parameters for the Standard Reactor.
Core Shape
Cylindrical
Radius
168.53 cm
Active Height
365.00 cm
Reflector Thickness
34.00 cm
Active Volume
3.2568 E+7 cm'
Heavy Metal Loading
94.418 WTO
Array Geometry
Rectangular
Pitch
1.2573 cm
Coolant Pressure
2250 psia
Avg. Coolant Temperature
583 K
Thermal Power
3400 MWth
Pin Average Linear Power
6 KW/ft
Table 2.1.2. Basic Fuel Cell of the Standard Reactor.
Region
Material
Radius(cm)
Thick.(cm)
Vol. Fraction
1
Fuel
0.4096
0.4096
0.3334
2
Gap
0.4178
0.0082
0.0135
3
Clad
0.4750
0.0572
0.1015
4
Water
0.7094
0.2344
0.5516


65
Table 2.3.2. Scoping Burnup and Isotopic Results.
Case
Pitch
Energy
Cycle
U-235
(Kq/GWd)
U-238 Fiss
Pu
(Kg/GWd)
I.D.
(cm)
(GWd)
Life
Fiss.
Disch.
(Kg/GWd)
Fiss.
DISCH.
3A
1.25
1737
ly 316d
0.6343
0.6319
0.0805
0.3291
0.4064
3A
1,60
1849
ly 360d
0.7073
0.4779
0.0495
0.2686
0.2726
4A
1.25
1893
8y 049d
0.6218
0.5293
0.0854
0.3519
0.3895
4A
1.30
2022
8y 252d
0.6210
0.4519
0.0787
0.3512
0.3474
4A
1.35
2033
8y 269d
0.6370
0.4307
0.0708
0.3355
0.3169
4A
1.40
2028
8y 261d
0.6518
0.4195
0.0647
0.3207
0.2969
4A
1.45
2031
8y 266d
0.6587
0.4112
0.0613
0.3127
0.2844
4A
1.60
1881
8y 031d
0.7033
0.4604
0.0499
0.2676
0.2610
8
1.25
1511
8y 043d
0.6387
0.4785
0.0707
0.3227
0.3795
8
1.40
1362
7y 116d
0.7092
0.5386
0.0545
0.2688
0.3448
Note: Ave. fuel enrichment is 2.6% for all cores except for core 8
it is 2%.
Total energy generated assumes a core with 94.42 MTU loading.
Cycle life is calculated assuming 75% availability.
For Core 3A, pitch = 1.2573 which is the Westinghouse value
for a standard core using 17X17 assemblies.
Data obtained from LEOPARD burnup calculations.


218
reduced. One way is to place older fuel at the periphery of the
reactor, instead of the presently current practice of having the
fresh batch placed at the core periphery. 'This is further
discussed in section A.2.3. Another way would be to improve the
reflecting characteristics of the reflector, for example by
changing the material that forms the core baffle.
c). Improving moderation. For safety reasons, FWR's employ a
water/fuel ratio which is below the optimum for moderation. At
current EOL's, the core could still be critical if the water/fuel
ratio was closer to the optimum. Assuming the necessary safety
margins could be assured, alterations of fuel lattice pitch or fuel
pin diameter could improve the moderating characteristics of the
system. A similar effect can be achieved by altering the water
density, which can be done without need for fixed alterations of
the system.
d). Improving conversion ratio. Since at EOL a large part of
the power of the reactor is generated by the plutonium that has
been bred from absorptions in U-238, increasing the conversion
ratio would produce more plutonium, and therefore would tend to
extend the core life. There are two ways in which this can be
achieved in a FWK: The first is to employ the "spectral-shift
controlled reactor" (further described in section A.2.9), which
involves the mixing of light water and heavy water in a proportion
that can be optimized throughout the core life. Since heavy water
does not have the high moderating power of light water or the
higher neutron absorption, a harder spectrum is obtained in the


305
26. Bell, J.L.r "Comparisons between ARMP Calculations and
Measurements from D.C. Cook Unit 2, Cycle 1," American
Electric Power Service Corp., ARMP Users' Group Meeting
(1980).
27. Chen, E., "Comanche Peak Unit 1 Core Model," Texas Utilities
Services, Inc., ARMP Users' Group Meeting (1980).
28. Graves, H.W., "Nuclear Fuel Management," John Wiley & Sons,
Inc. (1979).
29. Burns, E.T., "Refueling Outage Trends in Light Water
Reactors," EPRI NP-842, Research Project 705-1, Electric Power
Research Institute (1978).
30. Atomic Industrial Forum, "Reprint from UPDATE 'Nuclear Power
Program Information and Data'," Office of Nuclear Reactor
Programs, D.O.E. (1980).
31. Miller, M., "TEMPRET, A Computer Code for the Steady-state
Thermal Analysis of a Single PWR Fuel Pin / Coolant Channel,"
Nuclear Engineering Department, University of Florida (1981).
32. "CONCEPT-IV A Computer Code for Conceptual Cost
Estimates of Steam-Electric Power Plants," Office of Energy
Systems Analysis, U.S. ERDA (1975).
33. Hughes, J.A., and Hang, D.F., "GEM General Economic Model to
Analyze Nuclear Fuel Cycle Costs," University of Illinois
(1973).
34. Salmon, R., "POWERCO A Procedure and a Computer Code for
Calculating the Cost of Electricity Produced by Nuclear Power
Stations," ORNL (1966).
35. Anderson, E.C., and Putnam, G.E., "CORA A Few Group
Diffusion Theory Code for One-Dimensional Reactor Analysis,"
IN-1416 (1970).
36. Cacciapouti, R.J., and Sarja, A.C., "CHIMP-II A Computer
Program for Handling Input Manipulation and Preparation for
PWR Reload Core Analysis," Yankee Atomic Electric Co. (1976).
37. Impink, A.J., Jr., "Reactor Core Physics Design and Operating
Data for Cycles 1 and 2 of the Zion-2 PWR Power Plant," EPRI
NF-1232, Project 519-6, Electric Power Research Institute
(1979).
38. Flores, L., "Nuclear Reliability Program. EPRI-NODE Power
Distribution Corrparisons at Oconee," Duke Power Co. (1980).


301
Table D.l
Cell-Homogenized Number Densities (Atoms/Barn CM X 10b )
Pitch %
Power
BOL U-235
EOL U-235
EOL Pu-239
EOL Pu-240
EOL Pu-241
1.20
50
258.24
62.08
47.63
19.38
12.07
1.20
33
258.24
59.82
46.97
19.68
11.74
1.20
25
258.24
58.35
46.96
19.87
11.52
1.25
100
238.00
50.44
39.08
17.53
10.57
1.25
50
238.00
47.46
38.97
18.33
10.51
1.25
33
238.00
46.73
38.49
18.36
10.15
1.25
25
238.00
45.79
38.36
18.45
9.92
1.30
50
220.00
39.47
32.47
16.77
8.96
1.30
33
220.00
38.11
32.14
16.86
8.72
1.30
25
220.00
37.44
31.98
16.89
8.51
1.35
50
204.00
32.40
27.59
15.78
7.61
1.35
33
204.00
31.99
27.32
15.51
7.54
1.35
25
204.00
31.46
27.19
15.52
7.36
1.40
50
189.70
29.56
23.50
13.99
6.56
1.40
33
189.70
28.61
23.23
14.06
6.38
1.40
25
189.70
28.15
23.08
14.09
6.22


196
Fuel Cycle Cost
Mills/KWh
Figure 5.2.2. Fuel Cycle Cost vs. Plant Power.
Note: Power is expressed as percent of Standard Plant.
Power


127
beginning of the second and third cycles would not be at A and B,
but rather at A' and B'. which have a different burnup level. The
boron concentration that was obtained for criticality by combining
points 0. A and B must be obviously excessive, since the "new"
points should be 0. A', and B'. where A1 and B' have a larger
burnup. and therefore lower reactivity. In short- this means that
when calculating a new boron letdown curve which results in a new
cycle length, the new curve is automatically in error, because it
was not calculated with burnup timesteps whose burnup levels were
consistent with the cycle length assumed by the curve.
To solve this problem, one might be tempted to run a PDQ-7
burnup calculation with the new cycle length estimate, but this
would imply large number of PDO-7 iterations and data handling back
and forth, destroying one of the main goals of this calculatonal
scheme: reduced calculational costs and data handling. To make any
further adjustments to the cycle length and the letdown curve with
the data available to CRIBUR. it is necessary to find some value or
seme characteristic that would be implicit in the data and that
would not depend on the cycle length- Such characteristic may be
found if an approximation is accepted by assuming that the initial
guesses of the cycle length and the soluble boron letdown curve are
not in large error. If this is not the case, the cycle variation
at the end of the calculation will be so large that it will force a
new PDQ-7 calculation, anyway. If those guesses are not too
erroneous, then, the fuel was burned following a spectrum which is
correct within the desired accuracy, and therefore, its isotopics


169
Days
Figure 4.2.4. Total Life Refueling Time vs. Pitch


121
quarter-assembly eigenvalue cases are run with and without the
LBP's at BGL and EOL, it is possible to obtain the reactivity worth
of the poisons for use in CRIBDR.
CFIBUR obtains the core effective multiplication factor as
Kc = 1 / ( Si(Fi/Ki))
where Fi = fraction of the core power produced by the "in-th batch.
Ki = infinite multiplication factor of batch "i" modified
for leakage and/or LBP's, if any.
The Ki's are obtained from the quarter-assembly PDQ-7
calculations, with the possible modifications described above for
any possible radial leakage and/or burnable poison presence. The
Fi's are calculated as follows:
Fi = (Ki Pi)2
I,(Ki Pi)2
where Pi = non-leakage probability for batch "i".
The above formula is purely an empirical formula which tries
to assign a fraction of the core power to each batch making it as
close as possible to the real power sharing encountered in an
actual core. This formula evolved from the expression
Fi = Ki2 / Z-i Ki2
used by M.I.T.'s research team (15), but was modified by the author
to the expression shown before which adjusts better to the batch


194
Capital Cost
Mills/KWh
Figure 5.2.1. Capital Cost vs. Plant Power.
Note: Power is expressed as percent of Standard Plant.
% Power


Page.
B5. EERI-CELL CODE DESCRIPTION ............. 258
B6. NPUNCHER CODE DESCRIPTION 263
B7. PDQ-7 CODE DESCRIPTION 265
B8. TEMPRET CODE DESCRIPTION .............. 269
B9. CONCBFT-IV CODE DESCRIPTION 271
BIO. GEM CODE DESCRIPTION 273
Bll. FQWERCO CODE DESCRIPTION 275
Cl. CRIBDR CODE SOURCE LISTING 277
C2. SAMPLE RUN OF CRIBUR 293
D. ISOTOPIC AND SPECTRAL DATA FROM BURNUP CALCULATIONS. 299
REFERENCES 303
BIOGRAPHICAL SKETCH 306
vii


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
BURMJP AND FEASIBILITY STUDY OF
LOW POWER DENSITY PWR'S
By
Cesar Molins-Bartra
June 1981
Chairman: Nils J- Diaz
Major Department: Nuclear Engineering Sciences
Operational and safety problems of current Pressurized Water
Reactors are often associated with the high power density level of
the cores. An alternate use of current-design cores is proposed by
reducing the power density. The effects should be improved safety,
improved ore utilization, and improved operational characteristics.
A scoping study is performed in order to define core
parameters suitable for optimization under the low power density
characteristics, while minimizing redesign requirements.
A neutronic optimization study of the reactor cores is
performed by systematic changes in the fuel lattice pitch. A new
core burnup computational model (CRIBUR) is developed, which allows
calculation of the burnup and isotopic analysis of a multi-batch
core in its equilibrium cycle with a moderate computational and
human effort. CRIBUR provides better accuracy and sensitivity than
other known existing models of comparable scope, with a moderate
xii


88
The advantage of this method is that the core isotopics do not
contain the heavy errors due to errors in the neutron energy
spectrum, and that the method is still rather simple. On the other
hand, the pin cell burnup calculations require a much larger number
of timesteps, which increases the computational costs, and
although the final isotopics may not be grossly wrong, there is no
guarantee that the chosen boron average concentration is really the
average for the real reactor life.
PDQ-7 assembly burnup calculations conducted have shown that
an average error of 50 ppm in boron concentration (even in
real-time concentration following calculations) causes isotopic
errors of 1.2% in Pu-239 and 1.27% in Pu-241 for a burnup level of
34000 MWd in a fuel cell configuration of a tight 1.20 cm pitch
(containing very small amounts of moderator and its associated
soluble boron). For a large pitch cell of 1.60 cm, isotopic
differences of 3% in Pu-239 were found after only 9000 MWd burnup,
for an average error in boron concentration of 100 ppm. These
facts require a very precise specification of soluble boron
concentration if accurate isotopics are needed, and this is the
case if one is concerned about core life determination. The system
described on Figure 3.1.5 has the further disadvantage that the
fuel is burned without the critical spectrum, and although the
discharge isotopics may not be grossly in error, the errors
existing at any particular time during the core life are unknown,
and so are the errors incurred in calculating the core
multiplication factors, which in turn will determine the core life.


48
and the buildup of Pu-239. As core life advances, the boron
concentration is steadily reduced; the boron self-shielding also
decreases, causing an increase in boron worth. The contribution of
Pu-239 buildup to the increase of the boron worth increases with
core life. It should be noted that by the end of the core life,
about 40 % of the core power is generated by Pu-239 that has been
bred through parasitic captures in U-238. Pu-239 has a higher
thermal fission cross section than U-235, and it has a resonance
peak in the thermal energy region (tenths of an eV.). This causes
the reactor to increase its sensitivity to the presence of a
thermal poison (such as boron) that can remove neutrons from the
high fission cross section energy region.
The rate of increase of boron worth with core life is faster
for larger lattices, because of the effect of increased boron
inventory in larger lattices.
Thus, iron the point of view of the SHARP, it is possible to
conclude that the reduced power density will not affect boron worth
to a significant extent as long as the moderator conditions and
cell geometry characteristics remain unchanged. However, if
geometry changes are performed in order to optimize the cell
behavior for burnup or fuel utilization, boron worth may be
expected to show a significant variation, tending to increase for
larger pitches and tending to increase towards E.O.L., mainly if
the burnup levels achieved are high.
An important goal of the scoping study, is to make a first
estimate of the burnup advantages that can be expected from the low


276
g). Cost of initial core.
h). Annual expenditures for uranium, fuel fabrication,
transportation, reprocessing (if any), spent fuel credits (if any)
etc.
The main data output by the code are the following:
a). Power cost tabulation.
b). Payout tabulation.
c). Tax-deductible expenses.
d). Annual income tax calculation.
e). Fixed charge calculations.
f). Constant annual sales income.


BUKNUP AND FEASIBILITY' STUDY OF
LOW PCWER DENSITY FWRS
BY
CESAR MOLINS-BARTRA
A DISSERTATION PRESENTED TO IRE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1981


r
k


5
rating plant. Alternatively, it could be employed in parallel with
several other low power density cores powering a single standard
secondary side so as to yield a plant of normal or full power
rating. This multi-core arrangement was first considered
attractive to maintain the overall capital cost down and maintain
large plant output. However, from the commercial point of view, it
is obviously more reasonable to study the single-reactor reduced
power rating plant, whose design and construction are well known.
It is important to consider also that the economy of scale afforded
by the present large plants lies in the reduced number of critical,
expensive components used for the production of a large amount of
power. The low power density concept breaks away from this
constraint and attempts to demonstrate that there is no large
advantage in designing large power output plants as compared to
smaller sizes, when all factors influencing plant economics are
considered. Furthermore, it is frequent to find electrical demand
areas where the large power rating plants are oversized and a
smaller plant is better suited for such places.
A single-core SHARP, which is the main concept analyzed in
this work, is a plant using a standard, full-sized reactor vessel
operating at a reduced power level, with the balance of plant
dimensioned according to the desired rated output power (50% of
standard, 33% of standard, etc.). The use of a single full-sized
core for the low power density system accomplishes the dual goal of
avoiding extensive plant redesign, and operating a core at a power
level clearly within its technological limits with enhanced safety


100
b). It has known restrictions concerning the self-shielding
calculations. These restrictions might be detrimental in the SHARP
study since the somewhat high burnups expected entail
higher-than-normal concentration of highly absorbing nuclides such
as fission products and plutonium.
The EPRI-CELL(19) code was also considered. Although
EPRI-CELL is not a "classical" code yet because it is fairly new
(released to industry users around 1977), the calculational scheme
it uses is based on well-proven calculational methods, and the
benchmarking done to date proves its high accuracy. EPRI-CELL uses
a GAM-THERMOS-CINDER(20-22) scheme for cross section evaluation and
fuel burning isotopic chains. It presents basically all the
advantages of LEOPARD (namely, it is a pin-cell code of simple
input and automatically chains spectrum and burnup calculations)
and it does not present the two limitations of LEOPARD, thus
allowing a higher flexibility and accuracy in the pin-burn
calculation. The only disadvantage of EPRI-CELL when compared to
LEOPARD is its running cost, which is about one order of magnitude
higher. Despite this drawback, EPRI-CELL was chosen as the
pin-burn code, with the initial intention of using it as the only
calculational tool external to the core model.
However, when examining the documentation of the EPRI ARMP
code package (of which EPRI-CELL is a part), it is evident that the
purpose of EPRI-CELL is to generate sets of cross-sections through
the life of the fuel for use in generating the HARMONY tables for
PDQ-7. PDQ-7 is then expected to be used for the actual core-life


26
deg. K but at the same 2250 psia pressure. For all the other
cores, subcooled nucleate boiling has been assumed at the pin
surface, while cores #7 and #8 were assumed to have subcooled
convection, due to the much lower bulk coolant temperature and to
the very low heat flux rate. Figure 2.2.1 shows the geometry
assumed in the pin for the heat transfer calculations.
Figure 2.2.1. Pin Cell Geometry.
Given the heat transfer conditions found at the pin surface,
the temperature increment between the bulk coolant and the clad
surface may be obtained by the Jens & Lottes expression:


207
and the discharged fissile mass, composed of U-235 and plutonium.
Fuel lattice pitch appears as a key factor influencing this
variable; the better moderated pitch configurations show a sharp
increase in ore usage per unit energy generated due to the
reduction of plutonium conversion. The 50% power, burnup optimized
pitch SHARP shows a 10% increase of net fissile species usage with
respect to the standard plant. However, the power density level
shows practically no influence in net fissile species consumption;
the 50% power level SHARP using the standard reactor pitch shows a
difference of less than 1% with respect to the standard plant.
6.3... Plant Operations Considerations
The reduction of linear power generation entails significant
changes in the plant's operational characteristics. The two key
effects are the improved safety of the reactor and the increase of
the core cycle life. The heat transfer conditions at the pin
surface are drastically ameliorated by the reduction in the linear
power. Core minimum DNBR savings of 60% with respect to the
standard core value are obtained in a 50% power density core using
the standard fuel pitch. The same core presents approximately 30%
minimum DNBR savings when using the burnup-optimized pitch. Using
the burnup optimized pitch in a standard power density reactor
would result in an unacceptable 20% worse minimum DNBR. The power
density reduction has been also shown to reduce the fuel average
temperature by 300 deg. K, resulting in improved fuel conditions,
including a reduction of gaseous fission product gas migration.


APPENDIX B9
CONCEPT-IV CODE DESCRIPTION
CCNCEPT-IV (32) is a code designed for the computation of the
capital cost of an electric generating plant using the steam cycle
for power generation. The code is able to compute time- and
size-dependent costs of the components of the plant based on a
historical data base and an escalation prediction model.
The code breaks the cost of the plant into a set of accounts,
including the reactor plant (for the case of a nuclear unit), the
turbine plant, structures and facilities, engineering and
construction, management services, etc. Each of the different
agcounts is broken down into labor, materials and equipment costs.
A large number of sub-accounts are considered, yielding a highly
detailed cost break-down.
The input needed for the code includes items such as the
following:
a), Plant type and location.
b). Net electrical capacity of the plant.
c)Date of purchase of steam generating unit (NSSS for the
case of a nuclear unit, or equivalent for the case of a
fossil-fired unit).
271


129
With this new system, it is no problem if the newly calculated
boron letdown curve implies a different cycle length: the last
timestep of each cycle is adjusted in length in order to fit the
new cycle life, new burnup levels are calculated for each timestep,
and CRIBUR proceeds to a new criticality calculation which will
yield a new boron letdown curve and a new cycle length. The
iterations are continued until the point where the burnup levels,
the cycle length and the soluble boron letdown curve are consistent
within a certain margin.
As it can be realized by observation of the expressions used
by CRIEUR for the criticality calculations, this iterative process
involves a simultaneous adjustment of the relative power shared by
each batch in the core.
Once the CRIBUR calculation has converged internally. CRIBUR
outputs the new estimates of the core life, the soluble boron
letdown curve, the relative powers assigned to each timestep. the
batch burnup levels, and the discharge burnup level. All these
data have then to be compared by the user to the data that were
specified for the EDQ-7 fuel burnup calculation, and if any
significant differences are encountered (above a certain error
margin dictated by the requirements of the study) the PDQ-7 fuel
burnup calculation must be repeated with the fuel life data
indicated by CRIBUR's output. Figure 3.2.7 shows a flowchart of
the calculations performed by CRIBUR which may further clarify its
operation and its two main loops: the criticality and cycle life
calculations, and the adjustment of cycle timesteps' burnup levels.


171
just about 2% above that of the 50% power reactor. The
availability factors of all the cases considered are plotted as the
top set of curves of Figure 4.2.5, plus the point representing the
standard core, which is significantly displaced downwards.
Column 7 on Table 4.2.2, labeled Capacity %, is another
frequently used parameter for measure of the performance of a
plant. In this case, since all plants are so far assumed to have
the same amount of unplanned outages, the capacity factor is
obtained by use of the Operating Capacity Factor. From the
equation defining the OCF, it is possible to write
CF = OCF ATfiF (TCT ROT) / TCT = AF OCF
As could be expected, the standard plant shows a capacity
factor slightly above 65%, while the 50% power, pitch-optimized
core holds a capacity factor of about 3.3% above that of the
standard reactor, which means an increase of 5% energy production
in the same period of time, for equivalent installed powers. The
25% power core capacity factor is increased by only about 1.6% over
that of the 50% power core. The capacity factors of all the plants
under study are plotted as the bottom set of curves of Figure
4.2.5.
The last column of Table 4.2.2, labeled Size Mod. Cap., %,
again calculates the capacity factors that can be expected from the
different cores, but this time the calculation takes into account
the historically proven fact that smaller size plants actually have
better capacity factors than large plants for equally long cycles


118
made below are based on the observation of the results of the
neutron transport-theory pin-cell depletion code EPRI-CELL.
a).'V, which depends mainly on the core's fast diffusion
coefficient, is assumed constant since the core fast diffusion
coefficient experiences less than a 2% change through life. The
value taken for X was 40 cm.
b). The geometric buckling is calculated as that of an
infinite cylinder whose radius is the radius of the reactor plus
reflector savings. As explained before, the reactor is considered
infinite in the axial direction because leakage in that direction
affects all batches in the same proportion, and this effect is
taken care of in the assembly burnup PDQ-7 calculation. The radial
leakage, however, affects only the peripheral batch, and is
accounted for in the core modeling.
c). For the calculation of the reflector savings, the
reflector is assumed to be "thick", since the reflector thickness
is 34 cm, which is many times larger than the reflector diffusion
length. The savings is then obtained as
cf = Dc Lr / Dr
where Dc = diffusion coefficient of the core.
Lr = diffusion length of the reflector.
Dr = diffusion coefficient of the reflector.


computational effort- The code is benchmarked against actual core
data and against multi-dimensional diffusion theory core
calculations, and its sensitivity to several of the calculational
parameters is also tested-
The thermal-hydraulic behavior of the low-power cores is
compared to that of the standard reactors, and their enhanced
safety margins are clearly demonstrated.
The low-power cores yield higher burnup levels than the
standard reactors. Ore utilization is also improved in a
once-through fuel management policy- Isotopic comparisons are
presented. Core cycles are drastically increased- Plant
availability and capacity factors are also increased as a result of
both the reduced impact of refueling downtimes and the reduced
forced outage time resulting from the improvement of operational
characteristics.
An economic comparison of the low-power cores is presented as
a function of the core power level and the level of inflation.
Low-power cores are at an economic disadvantage when compared to
the standard reactor primarily because of the impact of the
increased capital cost- The fuel cycle cost is also higher because
of the long fuel core residence time- This economic disadvantage
needs to be weighed against the iirproved safety and operational
reliability to determine the commercial feasibility of the concept.
xiii


32
Two codes are used for the calculation of cross sections.
PHRQG8) is used for calculation of the fast-group neutron cross
sections, and the original library, consisting of a 68 energy group
cross section data set is collapsed into three fast groups which
are used in the core model code. A more detailed explanation of
PHROG can be found in Appendix B2. As is shown in the next
section, the core criticality calculations are done with a model
which defines two distinct regions: fuel and reflector. Fast cross
section calculations need to be run for both of these regions. The
fuel region is run first, and the group-wise fluxes and currents
existing in the fuel are used as weighting functions for the
-calculation of the reflector constants.
The thermal-group cross sections are calculated with the
Battelle-Revised Thermos (BRT) code(9) using a 30-group cross
section library. Thermal cross sections are collapsed into a
single thermal group. It is also necessary to obtain separate
cross sections for the fuel region and for the reflector. The fuel
region does not present any problem, since it can be well
represented by the calculation of a unit fuel cell. However, BRT
does not allow an intrinsic representation of the reflector region
without an adjacent core. The reflector region is then calculated
from a two-region slab reactor configuration, where one of the
regions has the average core region characteristics, while the
other represents the reflector. The geometry chosen was that of a
slab. Because of limitations in the number of mesh spaces allowed
by BRT, inaccuracies at the core-reflector boundary arise when a


122
power sharing encountered in a MR core with an out-in-in in-core
fuel management scheme.
With the preceding formulas. CRIBUR takes the burnup data
provided by a FDQ-7 1/4 assembly fuel-burnup calculation with a
structure similar to the one shown on Figure 3-1-6. and computes
the criticality status of the core at the different core-life
timesteps, as represented by the graph at the bottom of the figure.
If it is determined that the core is critical at all points in
its life, the PDQ-7 1/4 assembly burnup calculation is really
representative of the core life, in the sense that the core cycle
length, the batch power level and the soluble boron concentrations
are consistent- At that point, the PDQ-7 data and the
complementary CRIBUR data that will be discussed later give the
description and main characteristics of the core burnup (discharge
burnup. batch burnups. core life, isotopics, etc-)- However, in a
general case (and mainly if the FDQ-7 burnup calculation was
assigned first-guess values) it is most probable that the core does
not maintain criticality through all its cycle timesteps. The
first action to be taken is to adjust the soluble boron letdown
curve, forcing the core to be critical at all times- This action
requires previous knowledge of the soluble boron reactivity worth,
and by altering the boron letdown curve, the core life estimate is
altered (since it is estimated as the time when soluble boron
concentration must become zero)- This will force an adjustment of
core parameters, as will be explained later-


262
d). Number of total timesteps used in the calculation,
soluble boron levels and power levels.
e). Geometrical data: radii of the zones of the fuel cell,
pellet and clad inner and outer diameter, pitch, and extra region
volumetric fraction.
f). Option flags concerning units used in input, buckling
search, number of collapsed groups, type of edit and output data
tables, etc.
g). Temperatures, spectrum specification and convergence
criteria


245
could perform the same functions, presenting better nuclear
characteristics. Backfit of this idea would be really expensive,
while potential ore savings are very small.
b). Forcing a "migrating" axial power peak. By means of
partial length rods, it might be possible to force the axial power
peak to travel along the reactor axis, forcing a more leveled
burnup towards the extremes of the fuel. Although some extension
in burnup might be expected, it is possible that rather strange
power distributions would appear. The concept would require a
really sophisticated system of partial length rods.
c). Reversible-halves fuel assemblies. Since the axial power
peak is towards the center of the reactor while the top and bottom
ends have a lower power, one way of obtaining a more uniform
discharge burnup along the whole axis length would be to use fuel
assemblies cut in two halves. At the third cycle, for example, the
two halves could be turned upside down, causing the center zone to
be at the extremes, and vice versa. During the third cycle, the
assemblies would receive the maximum burn at a region that had the
lowest exposure previously. Of course, this would increase the
fuel fabrication cost in a significant proportion. It would also
increase the refueling outage time because of the higher complexity
of operations required, and probably most important of all, it
would drastically worsen the power peaking problem.
d). Another way of flattening the power distribution could be
through axial enrichment zoning of the fuel. This would reduce the


115
25% of the total population present in the outer batch. The
proportionality constant can be easily obtained with a few simple
calculations using a diffusion theory code. This procedure gives a
simple way of scaling the leakage according to the population or
flux distribution. Another aproximation needed (which was also
used before) is that of considering the average speed of neutrons
being constant across the reactor; this allows the use of
volumetric flux averages instead of neutron populations for the
leakage calculations.
With these assumptions in mind, the shape factor can be
expressed as
SF = ($b/£i)
(SI / (S1+S2))
where b= actual flux average of the peripheral batch.
<|)i= actual flux average of batch "i" in the core.
Sl= neutron population of the peripheral batch in the
Jo-shaped core.
S2= neutron population of the internal core zone in the
Jo-shaped core.
For any given number of batches, SI and S2 would be constant
values, but since CRIBUR was designed to handle cores with any
number of batches up to five, it is necessary to express SI and S2
as a function of the number of batches present in the core, since
the peripheral batch will correspond to a varying thickness of the
peripheral shell.


APPENDIX B1
BBT CODE DESCRIPTION
The BRT (9) code is a revised version of the THERMOS (20)code,
with several, improvements which result in a more exact handling of
cross sections and improvement of numerical techniques.
BRT solves the integral form of the transport equation for
either a slab or cylindrical geometry, with a maximum of 30
specified space points. It accepts up to 8 different material
mixtures, and the neutron cross sections are evaluated from a 30
energy groups library ranging from energy 0 to 0.683 eV The code
computes the neutron densities, fluxes and currents as a function
of ¡space point and energy group. It also calculates
energy-dependent and energy-averaged material mixture cross
sections and fluxes, and mixture dependent and total region
averaged macroscopic cross sections for each isotope considered in
the region under consideration. The code also prints and punches
on cards cell-homogenized macroscopic cross sections which may be
used as material descriptors for later problems, utilizing the
whole cell as a simple material region.
The code is structured in a way that it may be used either for
the detailed analysis of the spatial and energy dependent
characteristics of the region being considered (or of any of its
component material mixtures) or as a region homogenizer, in order
248


71
most advantageous, and a similarly automated scheme should be used
for the detailed calculations. However, a new burnup calculational
scheme must be developed which allows the adequate representation
of a several-batch reactor core in its equilibrium cycle.
After these scoping studies are completed, the in-depth study
should be centered on the cores that can best conduce to
short-term, commercially feasible low power density systems. The
in-depth study is, therefore, centered on a single-core concept
having a reduced power density in the 25 to 60% of the standard
core. The cores selected for the in-depth study will be discussed
in Chapter IV.
The safety aspects of the low power cores are extremely
favorable. The fuel average temperature is drastically reduced,
indicating a reduced risk of pin damage. The heat transfer
conditions have their safety margins, e.g. the DNBR, significantly
improved, which makes the probability of critical heat transfer
conditions much lower than it is in the standard reactor.


210
On the negative side, the SHARP present a much higher capital
cost per unit installed power, which ranges between 23% and 41%
above that of the standard plant, for power density levels between
65% and 50% of the standard value. Inflation level variations have
little influence on the relative capital costs when comparing
SHARP'S to standard power density level plants of equal output
rating. The fuel cycle cost analysis shows that in spite of the
reduced ore and enrichment needs, the total fuel cost is higher for
the SHARP than for the standard plant. The main reason for this
behavior is the higher interest costs associated with the long core
residence of the fuel. The fuel costs for the SHARP range between
5% and 15% above those of the standard plant's for the power
density range defined above. Inflation level has a significant
effect on fuel cycle cost, reducing the difference between the
SHARP and the standard plant as inflation rates become higher.
This is due to differences in revenue schedules on the cost of fuel
shipment and storage.
The overall economic evaluation of the SHARP indicates
generation cost increases for the SHARP between 17% and 27% with
respect to the standard plant for power levels between 65% and 50%
of the standard plant. Considering these results, a power
reduction beyond 50% of the standard value appears to be an
undesirable arrangement when cost/benefits are considered. The
combined consideration of the economic and safety aspects shows
that a power density level between 50% and 65% of the standard
plant would result in improved safety with respect to the standard


94
cycle ends, due to the fact of having the two burnup schemes start
their second and third cycles with different burnups.
This boron mismatch causes yet a further problem: at the BOL,
the constant-power scheme forms the core with batches having
burnups of 0, 1/3 discharge and 2/3 discharge. Meanwhile, the
varying-power scheme starts the cycle with burnup of 0 for the
first batch, somewhat more than 1/3 of discharge burnup for the
second batch, and somewhat more than 2/3 of the discharge burnup
for the third batch. These increments are of the order of 5% of
the burnup level for the first batch, and of the order of 8% of the
burnup level for the second batch, which represent non-negligible
differences in the batches' reactivities. Having a higher level of
burnup in two out of the three batches, the varying-power scheme
presents a lower reactivity than the constant-power scheme, thus
requiring a lower level of soluble boron in order to achieve
criticality. This situation is maintained, with different sizes of
mismatch, through the whole core cycle, thus causing evidently a
different critical boron letdown curve for each of the schemes, and
yielding a necessarily different core cycle length. This, together
with the spectral effects of soluble boron are the two main causes
that require maintaining the real power levels during the fuel cell
burnup calculations.
Once it is clear how the simplified burnup models can use a
fuel cell burnup scheme with its chained timesteps and all the
different adjustable data in order to obtain the core behavior
through life, it is necessary to look at the different methods


189
is in the standard reactor, which is unacceptable from a fuel
safety standpoint. Further examination of the curve shows that at
a power density level of about 75%. the minimum DNBR becomes
equivalent to that of the standard reactor with the standard pitch.
Any power density level chosen below 75% can be used with the
burnup-optimized fuel pitch value, and it still presents improved
heat transfer safety conditions as compared to the current standard
reactor-
It is important to remember that the thermal-hydraulic safety
conditions examined here correspond to the steady-state operation
of the core. The transient analysis being conducted for these
cores are still under preparation at the time of this report- Seme
of the transient conditions that should be examined are an
overpower transient, a partial and a total loss of coolant flow
with or without power reduction, and a depressurization transient.
It can be stated that the low power density cores show strong
advantages over the standard core with respect to thermal transient
behavior- The reduced fuel temperature allows for a larger amount
of heat to be stored in the fuel before critical heat transfer
conditions are reached. This fact, combined with the lower power
density level of the cores, allows for significantly longer
reaction times before any damage occurs to the fuel. In transient
situations or in loss-of-coolant flow accidents, the larger
burnup-improved pitch would seem to have a slight advantage again,
since it provides a higher inventory of water and larger coolant
channels in the core than the standard pitch. This will tend to


132
3.3. Benchmarking of the Burnup Calculations
3.3.1. Available References for Comparison..
In order to benchmark the burnup calculational scheme
developed for the calculations performed in this study, it is
necessary to have a quality assessment for each step in the chain
of programs which compose the total calculational scheme. The
programs used in performing the point burnup calculations are
industry standards. They already have had sufficient benchmarking
(26,27) as to make additional work in this direction by the author
unnecessary. However, it is still necessary to check the behavior
of the core model against other models of well-known
characteristics, or if possible, against an actual reactor's
behavior.
It is clear that the best benchmarking would be the comparison
with a true reactor behavior in an equilibrium cycle, since this is
the situation that the core model tries to represent. In order to
use such a benchmark it would be necessary to know all the exact
compositions of the equilibrium-cycle real core. Unfortunately, in
an equilibrium-cycle core, the compositions of the once-burned and
twice-burned batches are not exactly known from an experimental
basis, and the analytical results are not published in the open
literature, rendering this benchmarking possibility impractical.
There is, however, one case in which the model can be compared
to a real-life reactor: at B.O.L. of a first core. In such a
situation, all batches in the core are new and clean, and thus
their compositions are known, as well as the soluble boron


CHAPTER III
BURNUP CALCULATION METHODS
3.1. Burnup Calculational Methods and Present Needs
Once the philosophy behind the development of the SHARP is
established and a power density or a range of power densities are
chosen, the key work to be done in the area of fuel utilization is
the optimization of the pin-cell of the SHARP. The fuel
utilization and performance (burnup and isotopic results) are then
to be compared with those obtained from a standard power density
core to determine its relative merit. This optimization and
comparison require a large number of reactor life calculations,
where items such as batch-wise burnup, discharge burnup and
physical cycle life, etc. are of interest among others.
Many different organizations are interested in developing and
performing reactor fuel cycle and burnup calculations. The wide
range of organizations and specific interests has caused the
development of different calculational schemes, in order to
accomodate the different cost/effort/accuracy ratios desired for
each particular type of study. Some of the more important types of
burnup studies are the following:
a). Vendor burnup studies and core optimization, performed as
design work. These studies are most complete, and they generally
include three-dimensional, pin-by-pin studies which are closely
72


82
which must be normalized with a detailed core calculation, as well
as requiring adjustment of a large number of empirical factors.
There are other computational models which run at lower costs
and may not require the technical expertise needed for running full
blown PDQ-7 models or three-dimensional nodal codes. These models
normally accept a lower degree of geometrical sophistication, or
they may do just a part of the burnup calculation. For example,
they may perform the criticality and eigenvalue calculations, but
not be able to perform the actual core burning, which then has to
be done by a separate code. This is the case of MONA, which was
used in some of the preliminary studies of the SHARP. This type of
codes are useful for criticality studies or spectral effects
analysis, but they are of little use for a real life burnup study
because of the enormous manual data handling required. However,
they have the advantage of the low cost and the relatively simple
input.
Other types of calculational models are often used for scoping
or medium accuracy burnup calculations. They normally are
simplified codes which use part of the data obtained from the first
step of the burnup. calculations, and then introduce an approximate
core model, using empirical factors or formulas whenever the
available data is insufficient for a rigorous treatment of the
items needed for the calculation. However, these types of codes
cover a wide range of sophistication and thus their accuracy and
the information that can be obtained from them is largely variable.
The core model used for the SHARP can be included in this


APPENDIX B6
NUPUNCHER CODE DESCRIPTION
NUPUNCHER (23) is a coupling code whose function is to prepare
the cross section tables output by EPRI-CELL to the magnetic device
into a format compatible with the HARMONY (24) tables needed for
PDQ-7 (17) calculations. This results in a substantial
simplification of the most cumbersome part of the input to PDQ-7.
Although NUPUNCHER's task is apparently simple, it contains a
considerable amount of options due to the flexibility built into
the HARMONY tables. The code generates the required macroscopic
and microscopic cross section tables using the "burnup" variable as
basic independent mask, but allows the user to specify any other
number of masks for control of any particular set of microscopic
cross sections. For example, it is common practice to assign the
absorption cross section of Pu-240 to a table controlled by the
isotope's own number density rather than assigning it to the
general table, controlled by the burnup level of the fuel.
NUPUNCHER accommodates any of these types of changes and generates
the HARMONY tables accordingly. The main items input to the code
by the user are the following:
a). Identification of the data set containing the EC-DATA
file generated by EPRI-CELL and definition of the material names to
be used in the HARMONY tables.
263


43
The Doppler coefficient of reactivity obtained from Table 2.1.4 is
-3.99 p.c.m./deg.K of fuel temperature change, for a variation
between 293 K and 700 K (which correspond to 0 power and 25% power
respectively). The coefficient drops to -3.12 p.c.m./deg.K for the
temperature range between 700 K and 1005 K (25% power to full power
range). This shows the well known behavior of saturation of the
Doppler effect for increasing temperatures of the fuel. The
average coefficient that applies for the total range from 0 power
to full power is -3.62 p.c.m./deg.K The reduced fuel temperature
is one of the main effects contributing to an increase in the
reactivity of a low-power core with respect to the standard core.
As was mentioned before, the moderator conditions would remain
unchanged for the different power level cores for thermodynamic
reasons, and only the fuel temperature would experience a variation
due to the change of the heat production rate.
Fuel enrichment obviously has an effect on core reactivity,
and therefore, it is also possible to define a reactivity
coefficient for it. For enrichments between 2.6% and 3% in U-235,
the reactivity coefficient oscillates between 8.705% / 1%
enrichment variation for a cold, clean reactor and 8.643% / 1%
enrichment variation for a hot, clean, full power reactor. For the
enrichment step between 2% and 3% of 0-235, the 1/5 power,
preheater core shows a reactivity coefficient of 10.44% / 1%
enrichment variation. This larger magnitude may be due to the
double effect of bracketing a lower enrichment end, and having a
better moderated configuration due to the low moderator temperature


108
3.1 as being of significant importance for the determination of the
core life. Consequently. CRiBUR needs the input of a string of
infinite multiplication factors of a fuel batch burned with a
scheme in which the soluble boron letdown curve has been specified,
as well as the power levels of the batch through its life. CRIBUR
calculations include the following:
a). The criticality status of the core at each timestep
through its life, by combining the multiplication factors of the
batches in the corresponding time.
b). The necessary adjustments to the soluble boron letdown
curve in order to maintain core criticality.
c). A new estimate of the core life by interpolation or
extrapolation of the soluble boron letdown curve.
d). The power level of each batch during its core residence.
The fact that the core model emphasizes the core criticality
through life gave it the name of CRIBUR (CRItical BURning).
CRIBUR considers the core as an infinite cylinder surrounded
by a reflector- Radial leakage is obviously not the same for the
peripheral batch as for the inner batches, and thus it requires a
special treatment; on the other hand, the axial leakage is common
to all the batches, and is directly accounted for in the
2-dimensional PDQ-7 assembly calculation. thus making it
unnecessary to keep track of it in the core model.
In this study, the core model represents a Westinghouse 3400
MWth. four-loop system with a core radius of 168.53 cm. This


174
TCT = (FPT / (OCF ATAF)) + ROT or
TCT = (FPT (TCTo ROT) / (TCTo SCF)) + ROT
and then, the capacity factor must be
CF = FPT / TCT = FPT / (FPT (TCTO ROT) / (TCTO SCF)) + ROT
Since this capacity factor takes into account the variable
capacity factor of a certain-size plant for a given standard cycle
length, it is called in Table 4-2.2 the Size-modified capacity
factor- The size-modified capacity factor is also plotted in
Figure 4.2.5, as the middle set of curves in the figure.
Note that when this size correction is taken into account, the
50% power. 1-35 cm pitch reactor shows a capacity factor over 10%
above that of a standard plant which represents a 16% increase in
the power being produced in equal amounts of time by equivalent
power installations. This is a very significant increment, and its
economical impact is bound to be of large importance because it
reduces the impact of the capital and O&M costs on the total cost
of power, and in the case of nuclear-generated power, these two
items are responsible for a large proportion of the energy cost.
Under the present assumptions, the 50% pitch-optimized power plant
would reach capacity factors of about 75%. Also, in net energy
generation, an optimized SHARP will be closer to the output of a
standard size plant than its rated power indicates-


138
The comparison of CRIBUR with Comanche Peak 1 B.O.L. yielded
a K-eff. of 1.013 after accounting for the reactivity worth of the
burnable poisons. CRIBUR had, thus, a 1.3% K-eff. error with
respect to the real reactor. It may be of interest to note that
B.O.L. of the first cycle is probably the worst moment for CRIBUR
to give a good estimate of the core criticality, due to the sharp
interfaces existing in the core because of the different types of
fuel and poisons present. However, even in this situation,
CRIBUR's criticality calculation was acceptably accurate.
A comparison of criticality evaluations of the CRIBUR scheme
and the scheme used by M.I.T. (15) whose main formulas were
outlined in Section 3.2 shows differences between 1.6% and 2.3% for
the core multiplication factor. The main errors of the M.I.T.
scheme appear on the batch power distribution, as will be presented
later.
Some runs were made with CRIBUR in which some of the factors
of the calculational scheme were altered, in order to observe the
sensitivity of the results to the presence or accuracy of those
different factors. In a test where the Shape Factor was given a
constant value of unity (therefore implying that for leakage
purposes the flux shape followed a Bessel function) the core
multiplication factor experienced a sharp increase of 2.76% at EOL
and 3.26% at BOL. It is logical that these forced lower-than-real
leakages caused a larger effect at BOL, when the external batch,
which is experiencing the neutron leakage, is most reactive.


107
USER-PROVIDED DATA
CODES
LIBRARIES
Figure 3.2.2. Flowchart of Data for a Fuel Burnup Calculation Using
EDQ-7 1/4-assembly Geometry.


BIOGRAPHICAL SKETCH
Cesar Molins-Bartra was born the second son of Cesar Molins
and Maria Bartra, on the twenty-second of May, 1953, in Barcelona,
Catalonia, Spain. In 1975 he received the degree of Ingeniero
Industrial Superior en Tcnicas Energticas from the Polytechnical
University of Barcelona, where he acted as an assistant professor
while finishing his military duties as an artillery officer at the
nearby city of Gerona.
In 1977 he was awarded a research assistantship at the
University of Florida, U.S.A., through the sponsorship of the
Fulbright-Hayes graduate students exchange program. He received
the Master of Engineering degree from the Nuclear Engineering
Department in 1978. In 1980 he was assigned for a nine-month stay
at O.R.N.L. where he conducted the main body of the research
towards his Ph.D. degree.
306
u..


287
AT"X"MWD/MTU" :LPRH7T"CYCLE LENGTH IS ESTIMATED AS"A"HOURS, WHILE
INPUT ESTIMATE WAS"R(K,4)"HOURS"
2150 LPRINT"MAXIMUM PERCENT CHANGE BETWEEN OLD AND NEW ESTIMATES OF
STEP RELATIVE PCWERS IS"MR
2160 LPRINT"BATCHWISE BORON WORTH REGRESSION LINE IS:
W="WA"+ ("WB"*PPM)"
2170 STOP
2180 REM ***************************************
2190 REM ROUTINE FOR DATA INPUT FROM DISK
2200 INFUT"FILESPEC"?A$
2210 OPEN"R", 1, A$:FIELD 1,2 AS Xl$,2 AS X2$,2 AS X3$,2 AS X4$,4 AS
X5$,4 AS X6$f2 AS X7$,4 AS X8$,4 AS X9$,229 AS N$
2220 GET
1:TS=CVI(X1$) :CT=CVI(X2$) :NB=CVI(X3$) :BE=CVI(X4$) :PL=CVS(X5$) :FL=CVS(
X6$) :A$=N$:BP=CVI(X7$) :BB=CVS(XB$) :BN=C7S(X9$)
2230
CC=201:CH=1: 1=1:K=1 :DIMR(TS, 12) ,RC(CT,8) ,RR(NB,2): IFBEO0THEN2250
2240 BW=-1. OE-4:GOTO2260
2250 DIMBW(BE)
2260 FORJ=3T08:RC(I,J)=CVS(MID$(A$,Kf4)) :K=K+4:NEXTJ
2270 1=1+1:IFKXZCTHEN2290
2280 IF K=CTTHEN2260ELSE2300
2290 GET 1:CC=223:FIELD 1,255 AS N$:K=1:A$=N$:GOT02280
2300 IFCH>TSTHEN2330
2310 F0RJ=lT06:R(CH,J)=CVS 2320 CH=CE+l:GOTO2270


8
fuel in the core, which would allow more of the mid-life fission
products to decay while in the reactor.
- Reduced ore requirements due to the larger burnups
achievable from the same initial cores.
- Reduced enrichment needs.
d). From the operations point of view:
- Increased plant availability and capacity factor due
to the longer inherent fuel cycle.
- Operations well within technological limits with
reduced failure of components and reduced forced outage periods.
- Decreased operational transients and constraints.
- Better load following capability because of the
large technological margins available for maneuvering.
- Reduced overall personnel radiation exposure.
- Reduced fuel handling cost because of less material
being handled and because of its lower specific (per MWd generated)
radioactivity (which results in reduced personnel radiation
ejqoosure).
- Possibility of reduced spent fuel storage and
transportation requirements.
- Easier inventory control and reduced risk of
proliferation because of the reduction of fuel handling operations.
All these advantages appear to be qualitatively obvious, but
the question remains whether or not they can outweigh the economic
disadvantage mentioned before.


£83
Temperature
deg. F
2200
2000
1800
1600
1400
1200
Figure 5.1.1. Fuel Centerline Temperature vs. Relative Mass Flow Rate


67
will be a different one. In order to answer this question it is
necessary to perform burnup calculations with a model that can
reasonably represent a core in an equilibrium cycle, i.e. there
must be some burned fuel present at BOL for each reload core.
These scoping studies, however, can serve as guidelines for
"equilibrium cycle" calculations.
There are a number of criteria that can be considered
important when examining the results of a burnup analysis from an
isotopic or fuel utilization point of view:
a). Search for the maximum energy output from the same load
of fuel.
b). Search for the minimum discharge of plutonium, for
non-proliferation purposes.
c). Search for the minimum amount of net fissile material
being consumed per unit of energy produced. This would account for
the fissile species being discharged with the spent fuel, and
therefore would assume a spent fuel reprocessing policy.
d). In the same way as the criterion above looked for the
best resource utilization, it might also be important to look for
the best economical combination of resource utilization and cost of
fuel cycle (including fuel enrichment, fabrication, storage,
reprocessing, etc.).
Although the criteria that involve reprocessing appear more
meaningful from the scientific or technological point of view than
those which do not include it, the present political situation in
the U.S. prevents commercial reprocessing, and this must be


33
cylindrical shape is considered. The basic features of ERT are
described in Appendix Bl.
Four sets of cross sections are obtained with the calculations
described above: one set with three-group fast cross sections for
the core region; another set with three-group fast cross sections
for the reflector; a third set with thermal cross sections for the
core, and finally a set with thermal cross sections for a slab
core-reflector configuration, from which only the reflector
constants are used. These cross section sets are then organized
for input to the core models, which are discussed in the next
section.
2.2.3. Criticality and Burnup Methods.
The first objective of the scoping study is to obtain the
neutron multiplication factors of the cores described in Table
2.1.4. These multiplication factors are used to estimate the main
reactivity coefficients involved in the neutronic aspects that
differentiate the SHARP core from the standard core. The second
goal of the scoping study is to obtain a first estimate of the
burnup levels achievable by each particular core, as well as the
isotopics associated with them. Following is the description of
the methods used for achieving both of these purposes.
The core modeling for the criticality calculations is done
with the HONA(IO) code which is described in more detail in
Appendix B3. MONA accepts the cross section input prepared by
PHROG and BRT, and information about the core geometry and material
region compositions. The code can perform a number of


119
d). Lr and Dr are assumed as a function of the soluble boron
concentration of the water. Since Lr can be expressed as
Lr = VDr / T. ar
where ar = macroscopic absorption coefficient of the reflector,
it is first necessary to obtain a parametric expression of Dr and
£ar in terms of the soluble boron concentration. Observation of
the results of the pin cell code (EPRI-CELL) shows that both
variables can be closely expressed as a linear function of the
soluble boron concentration of the coolant, with a regression
coefficient very close to 1. The regression lines obtained for Dr
and ar are
£ ar (an-1) = 0.0166 + (2.727 E-5) ppm.
Dr (cm) = 0.4033 (5.135 E-6) ppm.
where ppm = concentration of soluble boron in parts per million.
e). The variables Dc (used in the expression above) and L of
the core (needed for the calculation of the non-leakage probability
during the thermal diffusion period) are also approximated with a
straight line. For any particular cycle of core fuel residence,
the approximations are accurate to a few percent, but the
regression lines change from the first to the second cycle of core
residence of the fuel. Since the present study considers an
out-in-in fuel management scheme, the fuel in the periphery is in
the first cycle of core residence, and since the peripheral fuel


236
Ore Utilization
Enrichment Wo
Figure A.2.6. Ore Utilization vs. Enrichment at Several Fuel Densities.


135
data to reproduce the situation on CRIBUR. However, it is still
necessary to obtain a reasonable source for comparison of the power
distribution among the different batches in CRIBUR. Since CRIBUR
analyzes an equilibrium-cycle core, it was considered that the best
method would be to compare its power distributions at different
points in core life with those appearing in real cores working in
an equilibrium cycle (say cyles 3 or beyond). The actual core
compositions would not be known, but the power distributions on a
batch-v/ise fashion are likely to be similar, and provided the cores
taken for comparison are similar to the one described in CRIBUR,
the code's estimates should be very close to the values seen on the
real reactors.
3..3.2,. Criticality Benchmarking.
As described in the previous section, two different references
were used for the benchmarking of CRIBUR's criticality
calculations. One reference was composed of several non-burnup
runs of PDQ-7 with a 1/4 core configuration, where each assembly
was explicitly represented, although the geometry mesh used was
somewhat coarser than an explicit pin-by-pin representation. Each
assembly was given a uniform composition equal to the one existing
in the corresponding batch in the CRIBUR calculation. Macroscopic
cross sections were used for each region instead of microscopic,
and this simplified data handling. There was no need for
microscopic cross sections, since no isotopic changes were allowed.
Two energy groups were used, as recommended by EPRI for FWR
calculations with the ARMP code package.


47
Boron Worth
Figure 2.3.2. Soluble Boron Worth at Critical. Changing Pitch.


229
obtained from them. However, the trends shown are close to
reality, and thus it is possible to state that the use of burnable
poisons increases the discharge burnup (and therefore ore
utilization) by reducing the fresh batch's relative power.
A.2.3. Low-Leakage Fuel Management.
Most LWR's operate on a multiple-batch fuel management scheme
because of the increased burnup obtainable (as shown in Section
A.2.1) and the associated possibility of power flattening. It is
also most common in large EWR's to use a 3-batch scheme, where the
fresh batch is loaded at the core periphery, and the once-burned
and twice-burned batches are placed in the inner region of the core
in a "checkerboard" pattern. One such typical disposition is shown
in Figure A.2.4 (26).
This disposition of the "second cycle" batch being closely
mixed with the "third cycle" batch is used to drive the latter,
which in turn is used to avoid power peaking in the former; this
peaking can occur because of the relatively high reactivity of the
"second cycle" batch and its location in a high-worth area in the
core. The "first cycle" high reactivity batch at the core
periphery raises the power in that zone, which is a low worth
region; this same effect helps maintain the batch's power at a
reasonable level. However, Figure A.2.3 is an example which shows
that even in the peripheral "low worth" situation, some "first
cycle" fuel elements may need LBP's in order to prevent their power
from being too high.


256
b)- The calculation of resonance self-shielding is allowed
only for U-238. which is particularly inaccurate for high levels of
burnup where other isotopes (such as Pu-239 or Pu-240) have
significant concentrations.
On the other hand, the code is extremely flexible, allows the
variation of poison levels during life, and runs in moderate times:
A total fuel burnup calculation can be run in just a few minutes of
computer time. The most significant data input by the user are the
following:
a). Geometrical data describing the cell and the "extra"
region.
b). Composition specification for each material zone.
c). Temperatures for each region, densities, pressure, and
perpendicular buckling.
d). Flags indicating units of input data, type of cell
geometry (square or hexagonal), number of broad groups to be used
for the output cross section tables, option for buckling search,
etc.
The code prints a report for each timestep specified during
the life of the core- The main items listed on each report are the
following:
a). Cell-homogenized number densities of each isotope at the
beginning of the timestep. and conversion achieved during the
timestep.
b). Microscopic cross sections of all the isotopes present in
the cell, and macroscopic cross sections of the smeared cell. All


52
over-moderated configuration, the moderator temperature coefficient
(WTC) becomes zero or positive, making the core inherently
unstable. Stability and safety require a negative MTC, which means
a pitch must be selected which yields an undermoderated
configuration.
The scoping study included an exploration of the reactivity
changes occurring in the fuel cell as pitch is varied, for seme of
the more representative core cases being considered. Figure 2.3.3
shows the plots of the infinite multiplication factor versus pitch
for the full power standard core (core 3A), the 25% power core
(core 4A) and the 20% power core with lower moderator temperature
(core 8) as obtained from PHROG-BRT-MONA. The plots correspond to
BOL, with no poisons, control rods or xenon present in the core,
but with the temperatures corresponding to the assigned core power
levels.
The curves show a uniform increase of reactivity between core
3A and core 4A, due to the reduced Doppler effect. Remember that
no xenon effect is taken into account in these curves, since the
cores are clean and with zero burnup. Curves 3A and 4A do not
show, however, any visible change of shape; there is only a
vertical shift due to the gained reactivity. This means that the
best moderated pitch is the same independently of the power density
(under equal moderator conditions). With this in mind, the
overmoderation / undermoderation safety criteria for the low-power
cores would admit some relaxation as compared to the standard core,
due to the inherently safer operating conditions of the SHARP.


177
associated with the change of pitch. The cause of the spectral
shift associated with the power level variation is the change of
the Doppler broadening of resonances, which is decreased as the
fuel temperature decreases. These two effects add up to reduce the
plutonium discharge by 16% for the 50%, 1.35 cm pitch core, as
compared to the standard reactor. This is obviously a favorable
point for the low-power core from the non-proliferation point of
view.
The fourth column on Table 4.2.3, labeled Ore Usage WT/Life,
shows the total amount of U308 ore used by each core for the
production of 22.4 GWy (e) of energy. This is, as mentioned above,
the total energy production assumed in the lifetime of the standard
plant. An overall plant efficiency of 33% was assumed for all
plants, since the thermodynamic conditions are supposed to be
identical. Since Table 7.2.1 showed the U308 needs per MWd (th)
produced, those figures can be used in order to obtain the lifetime
ore consumption as
Ore consumption (MT) = 2.45448 E+7 (MT/MKd th)
where MT/MWd th is the specific ore consumption shown on Table
4.2.1. Note that the specific ore consumption was calculated under
the assumptions of 3.1% enrichment in U-235 of the fresh fuel and
0.2% enrichment of the enrichment plant tails. Obviously, this
column shows the ore usage under the assumption of once-through
fuel management.
Note that savings of 5.1% of ore can be achieved by the 50%
power core with the optimized pitch. The ore usage figures follow


264
b). Table identification number, interpolation information,
number of energy groups and EPRI-CELL timestep number to be used
for non-interpolating data.
c). Table number for macroscopic tables, type of data to be
used in the macroscopic tables, first mask number and first
interpolating table number.
d). Definition of interpolating tables that must use a mask
other than "burnup"; definition of the mask.
e). Option to punch PDQ-7 depletion chains.


REFERENCES
1. Hersperger, E., "Economic Analysis of Low Power Density FWR
Plants," Master's Thesis, College of Engineering, University
of Florida (1981).
2. Daby, D., "Fuel Utilization Improvements in a Gnce-through PWR
Fuel Cycle," Final Report on Task 6, WCAP-9547, Westinghouse
Electric Corp. (1979).
3. Hellens, R.L., "Evaluation of Methods of Improving Fuel
Utilization for Once-through Fuel Cycles," Nuclear Power
Systems, Combustion Engineering, Inc. (1979).
4. Westinghouse Nuclear Training Operations, "Plant Information
Manual. 3400 MW Plant," Westinghouse Electric Corp. (1975).
5. Bolz, R.E. and Tuve, G.L., "CRC Handbook of Tables for Applied
Engineering Science," CRC Press, Inc. (1976).
6. "Steam: Its Generation and Use," Babcock & Wilcox Co.
(1975).
7. Duderstadt, J.J. and Hamilton, L.J., "Nuclear Reactor
Analysis," John Wiley & Sons (1976).
8. Curtis, R.L., "PHROG A Fortran IV Program to Generate Fast
Neutron Spectra and Average Cross Sections," IN-1435, Aerojet
Nuclear Co. (1971).
9. Bennett, C.L. and Purcell, W.L., "BRT-I: Battelle-Revised
Thermos," BNWL 1434, Batteile Pacific Northwest Laboratory
(1970).
10. Putnam, G.E., "MONA A Multigroup, One-Dimensional Neutronics
Analysis Code," ANCR-1051, Aerojet Nuclear Co. (1972).
11. Barry, R.F., "LEOPARD A Spectrum-dependent Non-spatial
Depletion Code for the IBM 7094," WCAP-3269-26, Atomic Power
Division, Westinghouse Electric Corp. (1963).
12. Bohl, H., Gelbard, E., and Ryan, G., "MUFT-4 Fast Neutron
Spectrum Code for the IBM 704," WAPD-TM-72, Atomic Power
Division, Westinghouse Electric Corp. (1957).
303


27
Ts Te + 60 ( a" / IQ6 )25
exp ( p / 900 )
where Ts = pin surface temperature, deg. F
Tc = bulk coolant temperature, deg. F
q'' = surface heat flow rate, Btu/hr sq.ft
p = coolant pressure, psia
The temperature increment across the clad is given by the
expression:
A Tel = q' do In ( do / di )
2 K S
where ATcl = increment of temperature across the clad, deg. F
q' = linear heat flow rate, Btu/hr ft
do = clad outer diameter, ft
di = clad inner diameter, ft
K = clad thermal conductivity, Btu/hr ft deg.F
S = clad surface per unit pin length, ft
Since the thickness of the gap is extremely small, the
temperature change across the gap may be obtained as
A Tg =
K S
where ATg = increment of temperature across the gap, deg. F
q1 = linear heat flow rate, Btu/hr ft
K = thermal gap conductance, Btu/hr sq.ft deg.F
= surface of heat transfer per unit pin length, ft.
S


15
Parameters of interest from the safety point of view would be:
a). The moderator temperature reactivity coefficient (MTC).
b). The Doppler reactivity coefficient.
c). The fuel average temperature.
d). The power peaking factors.
e). The soluble boron reactivity worth (at the concentrations
needed throughout a cycle life for each particular reactor system).
f). Power density and linear power (KW/'ft).
Fi:om the neutronics point of view, some of the more important
parameters to be observed are:
a). The changes of Doppler reactivity coefficient at the
different power density levels.
b). Xenon concentration.
c). Core effective multiplication factor at beginning of
life.
c). Evolution of isotopics.
d). Neutron energy spectrum variations.
e). Effects of varying enrichments if enrichments different
from those normally used for present FWR's can be considered.
Pnom the fuel cycle point of view, there are two main
variables that have to be optimized with regard to energy
production, but which are bound by conditions like the maximum
power peaking factor, fuel enrichment costs, thermal-hydraulic
safety aspects, etc. These two main variables are the following:
a). The total energy obtainable per ton of uranium ore.
b]. The core cycle length.


79
b). Macroscopic cross sections of the fuel and the whole
cell, weighted with the neutron spectrum present at each timestep,
and with the number of energy groups requested.
c). Infinite multiplication factor of the fuel cell, and
effective multiplication factor if a perpendicular buckling was
specified.
d). Average neutron speed.
e). Concentration of each isotope present in the cell,
regionwise and cell homogenized.
f). Cumulative burnup level since the beginning of the burnup
calculation.
g). Other data depending on each particular model, such as
the calculated Dancoff factors, fraction of the total core power
produced by each fissile isotope, cumulative fission densities,
conversion factors, etc.
The second step of the burnup calculations involves taking the
data generated by the first code and using it in a core model which
accepts overall core information regarding geometry and core
operation in order to simulate the actual life of the reactor.
This step is the one showing the widest variations from one type of
burnup study to another. The most sophisticated models are able to
follow the core life without further iterations, while others need
to feed their data back to step one of the burnup calculation, for
a new iteration. The flow of data from step one to two and back
depends on the sophistication of each of the models and the burnup
data sought from the overall study.


202
Standard industry neutronic codes (like PHROG, BRT, MONA and
LEOPARD) were used for the scoping studies conducted in order to
pinpoint the key differential characteristics of the SHARP core as
compared to the standard FWR core, and to determine the type and
range of parameters on which to conduct the in-depth optimization
study. One of the key issues distinguishing the SHARP concept is
the improvement of burnup and ore utilization, and this aspect
required thorough analysis. The available burnup calculational
methods were reviewed and found to be inappropiate for the type of
study required for the in-depth optimization stage of the SHARP
investigations. The problems with existing burnup calculational
schemes was that either computational requirements and expenses
were too large or accuracy was insufficient.
A new computer code called CRIBUR was developed for studying
the burnup performance of PWR's with a reduced computational cost
and a significantly increased accuracy as compared to previous
schemes of similar cost. CRIBUR is used in conjunction with
accepted industry codes (EPRI-CELL, NUFUNCHER, FDQ-7) to complete
the burnup calculational scheme. The final set of codes allows the
evaluation of isotopic composition through the fuel life, is able
to assign time-varying power levels to the different fuel batches
present in the core, and performs the burning of the fuel in a core
environment that closely simulates the criticality condition of the
reactor through life. The last feature increases the accuracy of
this type of burnup calculational scheme because it improves the
calculation of the absorption cross section of the moderator


*
Page
6.2.1. Discharge Burnup vs. Pitch 205
A.2.1. Increase in Fuel Burnup due to Partial Refueling . 220
A.2.2. Ore Utilization vs. Enrichment and Cycle Length . 222
A.2.3. Orientations of Non-symmetric Burnable Poisons . 227
A.2.4. Batch Distribution in Core 230
A.2.5. Discharge Burnup vs. Enrichment at Several
Fuel Densities .... ...... 235
A.2.6. Ore Utilization vs. Enrichment at Several
Fuel Densities 236
A.2.7. Power Shaping Effect of Partial Length Rod .... 241
A.2.8. Ore Utilization vs. Fuel Enrichment and Burnup . 243
xi


162
this table require seme background discussion about the main
factors influencing the plant cycle length and the power produced.
First, it is assumed that a standard full-power plant with one-year
cycle time (including the refueling outage time) has an
availability factor of 75%. The refueling portion of an outage is
considered to last 45 days, according to industry surveys(29),
although this number is just a "meaningful average", with actual
values oscillating around it. The capacity factor of the plant is
assumed to be 65%, according to Department of Energy reports(30).
In order to calculate the availability and capacity factors of
the reduced-power cores, it is convenient to define some new terms.
The first one is the "Active-Time Availability Factor" (referred
to from here on as ATAF). This is the availability factor of the
plant if the 45 days refueling time are not considered. The
conditions affecting the on-line and outage times of the different
plants can be considered equal for all plants, except for the
incidence of the refueling time, which depends on the length of the
core cycle. At this point it is assumed that all plants have the
same ATAF, i-e., that all plants have the same amount of
non-refueling outages. This assumption should be qualified as very
conservative for the low power cores, since they have improved
Nuclear Steam Supply System (NSSS) conditions as compared to the
standard plant; for the rest of the plant, operating conditions are
essentially identical to the standard plant. In actuality, the low
power core should have a distinct advantage in savings on unplanned
outages.


54
Figure 2.3.3 shows the pitch used by the standard reactor. Note
that it complies with the safety criterion of working in an
undemoderated configuration. However, it seems that the pitch
could still be allowed to increase somewhat without compromising
the safety, since the operation point of the standard reactor is
rather far from the peak of the curve. Later in this section it is
explained why the standard reactor can not take advantage of the
extra reactivity achievable from a slight increase of pitch, while
the low-power core can.
When examining the curve for core #8 on Figure 2.3.3, a sharp
change of shape can be noticed as compared to the other curves.
Reactor 8's curve has a larger curvature; its reactivity is much
more sensitive to the variations of pitch. This fact is easily
explainable: Reactor 8 works with reduced moderator temperature,
but at the same pressure as reactors 3A and 4A. The moderator has,
therefore, a higher density. This can be observed on Table 2.1.5.
The higher moderator density causes a given change of pitch to
represent a larger increase in moderation, i.e. there are more mean
free paths or more interactions taking place in the moderator
region between pins at higher moderator density. It is as if the
x-axis of the curve for core #8 has been compressed as compared to
the standard moderator temperature ones.
One of the consequences of the higher moderator density is a
reduction of the distance between the standard reactor working
pitch and the optimum moderation point. This effectively raises
the multiplication factor of core 8 when using the standard pitch.


175
4.2.3. Ore Usage and Enrichment Needs..
The last comparison used in the study is complementing the
information shown in Section 4.2.1, but establishing a common
baseline on which to compare all the different plants.
Table 4.2.3 refers to the usage of ore and enrichment units
for the different plants considered, but with all data normalized
to the power generation of 1125 Me, so that the figures on the
table can be compared on a same energy output basis. As in Tables
4.2.1 and 4.2.2, the two first columns are used for core
identification. The third column shows the total mass of plutonium
discharged yearly. It is assumed here that the total energy
production of 1125 Me over the 30 year life time is 22.4 GWy.
Then, the plutonium mass discharged yearly by each core, for an
energy production equivalent to one year of operation of the
standard plant is obtained as
Disch. Pu (Kg) = (MTPu/Md) 22400*30*365.25 / 0.33
where MTPu/Md is the total plutonium mass discharged per Md
thermal produced. This figure was calculated for each core in
Table 4.2.1.
As could be expected, the plutonium discharge is reduced as
the fuel pitch becomes larger. The cause is the softening of the
neutron spectrum due to the better moderated geometry obtained with
the increased amount of water present in the fuel cell. For each
particular pitch, the plutonium yield is also reduced as the power
level decreases, although the effect is much milder than the one


159
than the higher power ones by some tenths of a percent. This is
due to the extra reactivities associated with the reduced Doppler
and xenon levels. However, as pitch is varied, ore utilization
reaches a minimum at about 1.35 cm pitch, and increases for any
further pitch increase or decrease. This results from the
existence of two conflicting effects which substantially influence
the core cycle length: First, there is the moderating effect of the
coolant. Since all LWR's work on an undermoderated configuration
for safety reasons, an increase in pitch means additional
moderation, and therefore an increase in the multiplication factor
of the core. This applies, of course, until the best moderated
cell is obtained, which occurs at about 1.5 cm pitch. Second,
there is the conversion of U-238 into plutonium. Plutonium
accounts for about 50% of the core power at E.O.L., and therefore,
the amoiont of plutonium present in the core heavily affects core
life. Since plutonium is converted from the parasitic capture of
neutrons in U-238, a better moderated pitch has a softer spectrum
and reduces the resonance capture in U-238, resulting in a lower
plutonium inventory at E.O.L. (this effect could be observed in
the previous column on this table). For very undermoderated
pitches, an increase of pitch leads to a larger effect from the
additional moderation than from the reduced plutonium yield and
core life is increased. When larger pitches are used, the
reduction of plutonium inventory outweighs the effect of the
slightly softer spectrum, causing a reduction of core life.


9
The present study conducted on the low power density
single-core, reduced power rating plant using essentially the
standard plant configuration is also timely because of the present
worldwide interest towards building new, small size plants. There
are many situations where a 1000+ Me plant is just too large, and
so is the capital investment associated with it.
1.3. Previous Studies of Low Power Density Cores
The general trend followed by industry since the application
of nuclear reactors for commercial production of energy has been to
increase plant size and power densities in an effort to reduce the
capital cost per KW installed. The trend of higher power densities
reduced both the materials involved in the reactor construction and
the fuel inventory necessary for a given plant power rating.
A few commercial jreactor concepts having significantly lower
power densities than the LWR's are in operation, but they present
higher capital cost and reduced operating experience. Such is the
case, for example, of the Magnox reactors.
Lower power densities for LWR's were, thus, not considered
seriously until recently under the NSAP(2) project
(Nonproliferation Alternative Systems Assessment Program) where low
power density cores were investigated primarily as a means for
increasing the fuel utilization in a once-through fuel management
scheme. This would reduce fuel handling operations, and therefore
proliferation risks, while also providing additional safety
margins.
fc*.


39
calculation. Obviously, LEOPARD does not model the core and the
reflector since it is just a pin cell, one-dimensional code, but
neutron leakage effects are considered by means of a perpendicular
buckling, which enables the code to perform rough estimates of
criticality.
In an effort to compare the two burnup calculational methods,
parallel calculations were run for one core representing a standard
reactor's first load. Since no fuel regions were considered in the
MONA core model, the fuel enrichment was specified as an average of
the concentrations of the three real batches. This was also the
only v/ay of simulating the situation in LEOPARD, because it
obviously can not accept several enrichments simultaneously. Note,
however, that this is a rather crude way of representing a core,
because it is very different to have three distinct regions with
various enrichments or burnup levels than to have a large region
with averaged characteristics. However, the procedure was deemed
adequate for a scoping comparative study.
The results from the LEOPARD and MGNA-et-seq. comparison
showed a discharge burnup discrepancy of about 13%. Figure 2.2.4
shows the estimated soluble boron letdown curve obtained from each
calculational scheme. The agreement is very good up to about two
thirds of the core life, but the separation increases thereafter.
This is most probably due to the absence of proper treatment of the
fission products in the MONA method, because they could not be
included in the fuel pellet region, and therefore were only
accounted for as a reduction of core reactivity,
but their


96
The above estimate of the core multiplication factor, although
frequently used for rough criticality estimates, is rather
simple-minded, since it does not take into account the fact that
there are batches which for their combination of power sharing and
geometrical situation represent a heavier weight in the core
multiplication factor. It does not take into account either the
fact that the geometrical disposition of the batches in the core
causes some batches to experience a much larger radial neutron
leakage than others.
More complex type of "simple" burnup calculation core models
use weighting factors from the batches' power sharing in the
calculation of the core multiplication factor. For example,
VK = Zj (Fi/Ki)
where K = Core effective multiplication factor.
Fi = Fraction of the core power produced by batch "i".
' Ki = Multiplication factor of batch "i".
In this case, the power fraction generated by each batch in
the core may be obtained as
Fi = Ki"
( ZiKin)
This calculational scheme is significantly more complete than
the simple K-averaging one described before, but still does not
take into account the very important fact that one of the batches


97
lies on the core periphery, thus experiencing a radial leakage
which is quite different than that of the internal batches. There
is a model which corrects for this effect by multiplying the
peripheral batch multiplication factor by an empirical non-leakage
constant, which is again an improvement of the core model.
However, the radial leakage of the peripheral batch is mainly
affected by the reflector characteristics, and since the
reflector's main component is cooling water, the neutronic changes
experienced by the coolant during the core life must heavily affect
the reflector's performance. Indeed, the changes in boron
concentration in the water from B.O.L. through E.O.L. cause the
thermal absorption cross section to vary by a factor of about
three, and even the transport cross section is affected by some
percent. This suggests that if soluble boron concentration is
important for the spectral effects in the core, it is also
important in its effect on the reflector characteristics which
directly affect the radial leakage, and this effect should be also
taken into account if at all possible. It appears that this effect
has not been taken into account in known simplified burnup models;
it has been included in the model used in this work.
Whenever a simplified model for the representation of a
complex system is used, it is necessary to reach some compromise
and take into account only those factors which are considered of
first importance, while bypassing others of lower importance. The
factors considered of importance in the model used in this work are
summarized in the following specifications:


130
SE
inEait-Option.
Data Input
From Keyboard
Data Input From
Pre-saved Disk File
1 Data input and Array Dimensioning According to Problem Size
t
Optional Modification of Problem Data
- For Error Correction
- For Modification of Disk File
Optional Storage_of__Cage-Pata _j.n-.Disk Fils
Calculation of Step-wise and Cycle-wise Burnup Evolution
o£_Fuel. According to Assemblj..Burnup Data (Input Data)
Initialization of Work-core Matrix with K's, Time Lengths,
B_imiups-,__B-Oron ..Concentrations and Relative Powers
I
Calculation of Regression Line of Boron Worth vs. Boron ppm from:
- Timesteps Specified for such Purpose
- User-defined Constant
Elimination of Fake-burnup steps Used for Boron Worth Evaluation
Restructuration of Work-core Matrix and Pointers
Calculation of Table of Unbc
¡>rated Steps K's vs. Burnup!
1
r 1
Calculation of "Present Core Status
Step-wise and Cycle-wise Burnups
Interpolatlon-Qf. "Present-Core £
p* 1
status" Clean.K's for all Steps
rom ,Cugrcn|BumuE>-aiid-ssi,giisd..BQgon Ep
lfflodify_..BatchL-Klg. for Bumabje Poisons Presence (if any?!
£alcula-tiaiL^-.£;s-g-ape-jpxQbabilitie.s fox-peripheral Batch!
MQdi£icatiorL-Pf. Peripheral Batch K for Leakage!
t
continued
=t
Figure 3.2.7. Flowchart of CRIBUR Code


To Him who cares for all
and who gave us this world,
as an insignificant sample of my deep appreciation


104
burnup-dependent cross sections generated by EPRI-CELL into a card
deck with an adequate format for the HARMONY(24) part of PDQ-7.
The deck includes the description of the isotopic chains and their
constants. It also allows for the cross sections to be fitted
against any variable that can be deemed significant on any given
cross section, for any particular isotope (for example, Pu-240
thermal absorption cross section is generally not expressed as a
function of the pin burnup, but rather as a function of Pu-240
concentration itself). NUPUNCHER greatly simplifies the task of
preparing input to PDQ-7.
Once the quarter-assembly geometry is set in PDQ-7, it is
necessary to burn the fuel following the circumstances encountered
in an actual reactor core as closely as possible. One of the main
parameters that influences the evolution of the fuel is the neutron
energy spectrum, as was explained above. The neutron energy
spectrum existing in the reactor at any given time depends on many
factors, some of which are related to the physical design of the
core, and therefore are implicit in the geometric and compositional
description of the quarter-assembly, while others depend on factors
that may vary during the core life. Two of the main variable
parameters influencing the neutron spectrum are the soluble boron
concentration and the power level. The soluble boron concentration
directly affects the neutron spectrum because it is a strong
thermal-neutron absorber. The power level affects the neutron
spectrum in an indirect way, because it determines the
concentration of xenon present in the fuel, and xenon is also a


51
2.3.2. Neutronics and Safety Considerations.
The reduction of power density in the cores of the SHARP
causes, among other effects, a reduction of the Doppler effect, a
reduction of xenon concentration, and a reduction of fuel
temperature. These changes entail unavoidable differences in the
neutronic characteristics of the SHARP as compared to the standard
reactor. Thus, it is necessary to check the neutronic changes
associated with the power reduction, because some of their aspects
are closely related to plant safety, while others are related to
fuel performance or isotopic evolution.
As was pointed out in the preceding section, the neutronic
differences existing between the standard reactor and the low-power
reactors may result in a difference in the optimum fuel arrangement
in the core. In this study, the variation of fuel arrangement is
limited to the variation of fuel lattice pitch. This variation
implies a change of the thermal-hydraulic characteristics of the
fuel cell, as well as a change of the fuel-to-moderator ratio,
whose heavy effect on the core neutronics is well known. The pin
geometry is kept constant and equal to the standard core design.
The immediate effect of changing the pitch or
fuel-to-moderator ratio is a change in the neutron energy spectrum,
which in turn causes a change of cell reactivity, and as fuel
burnup progresses, it affects the isotopics evolution. There is an
optimum pitch which yields the maximum reactivity for a given fuel
pin. However, such a configuration is not safe from the
operational point of view. At the optimum point or in an


227
1
1
¡
\
\
\




-

\
/
/
\
t
f
f
t
.
o 0 9
a
o 0
. -
' f
m *

Figure A.2.3. Orientations of Non-symmetric Burnable Poisons.


285
1690
I=CT:XY=0 :XX=0: SX=0: SY=0 :KK=CT~2: J=0: IFCT=20ffiINKK=lELSEIFCT<2raENl760
1700 CH=RC(I,3) :X=R(CH,7) :IFI=OTHEN1740ELSECC=RC(1-1,3) :Y=R(CC,9)
1710 IFCC<=1THEN1720ELSEIFY=R(RC(1-2,3),9)THENKK=KK-l:GOTO1730
1720 XY=XY+X*Y:SX=SX+X:SY=SY+Y:XX=XX+X*X:J=J+1
1730 1=1-1:IFI>=KKTHEN1700
1740 B=(XY-(SX*SY/J))/(XX-(SX*SX/J))
1750 A=(SY-B*SX)/J:GOTO177Q
1760 A=R(1,9)
1770 REM A IS NOW THE EOC TIME
1780 A=(AKX))/2:OC=A
1790 PRINTnEND OF CYCLE TIME="A
1800 G0TO3030
1810 REM **************************************
1820 REM ***************************************
1830 REM OUTPUT
1840 REM ***************************************
1850 INPUTINPUT A TITLE FOR THE CASE"jA$
1860 INPUT "TURN PRINTER ON, THEN ENTER" ;X
1870 LPRINTA$ jLPRINT" "iLPRINT" n:A$="#########":OK=0
1880 FORI=lTONB:LPRINTnBATCHnI:B$=n######.##":LPRINT"STEP #
n.
r
1890 F0:KJ=1T0CT:K=RC(J,I+2) :LPRINT USING A$;K;:NEXTJ
1900 LPRINT" ":LPRINTnINITIAL STEPS ENDING TIMES";
1910 FORJ=lTOCT:K=RC (J, 1+2):LPRINT USING B$;R(K,4) ; :NEXTJ:GOTO1960


288
2330 IFBE=0THEN2370
2340 F0RJ=1T0BE:EW(J)=CVI(MID$(A$,K,2)):K=K+2:NEXTJ
2350 CA=CVI (MID$ (A$fK,2)) :K=K+2: IFCA=0THEN237OELSEDIMBK(CA)
2360 F0RJ=1T0CA:BKCJ)=CVI(MID$(A$,K,2)):K=K+2:NEXTJ
2370 CLOSE :GOT0560
2380 REM OBTAINING EOC BRNUPS (CYCLE AND STEPWISE)
2390 X=0:CC=1
2400
FORI=lTOTS:X=X+PL*R(I,RP)*(R(I,9)-R(I-l,9))/(24*FL) :R(I,RB)=X:IFI=RC(
CT,CC+2)THEN2410ELSENEXTI
2410 CH=0
2420 FORJ=OTOCC-1:CH=CH+RR(J,0):NEXTJ
2430 RR(CC,0) =X-CH:COCC+1: IFKTSTHENNEXTI
2440 GOSOB3270
2450 RETURN
2460 REM THE FINAL BURNUP OF EACH CYCLE I HAS BEEN PLACED ON RR(I,0)
2470 REM ***************************************
2480 REM ROUTINE FOR STORAGE OF DATA IN DISK
2490 CLS:INPUT"DO YOU WANT TO SAVE DATA ON DISK? (Y)ES OR
(N) O" ;A$: IFLEFT$ (A$, 1) <>nY"lBENRETURN
2500 INPUTnFILESPEC";A$:OPEN"Rn,1rA$:FIELD 1,2 AS Xl$r2 AS X2$,2 AS
X3$,2 AS X4$r4 AS X5$,4 AS X6$,2 AS X7$,4 AS X8$,4 AS X9$,229 AS N$
2510 LSET X1$=MKI$(TS):LSET X2$=MKI$(CT):LSET X3$=MKI$(NB):LSET
X4$=MKI$(BE):LSET X5$=MKS$(PL):LSET X6$=MKS$(FL):LSET
X7$=MKI$(BP):LSET X8$=MKS$(BB):LSET X9$=MKS$(BN)
2520 A$="":CC=201:CH=1:I=1


109
dimension is used for radial leakage calculations, and although it
is hardwired into the program, it may be easily changed if a
different core radius is considered. Figure 3.2.3 shows a
schematic drawing of the geometry used in the core model.
Reflector
Outer Batch
Inner Batch
Figure 3.2.3. Core Geometry Used in CRIBUR
Note that the core is considered as being composed of two fuel
regions: an inner zone containing all the fuel batches except the
one placed at the core periphery, and a peripheral region
containing only the peripheral batch. The reflector then surrounds
the fuel region. This configuration is used in order to account
for the different radial leakage effects on the different batches.
In a large FWR core such as the one being studied, it is reasonable
to assume that no significant net radial leakage occurs from the
non-peripheral batches, and therefore, all net neutron leakage in
the radial direction is associated with the peripheral batch. This
neutron leakage of the peripheral batch is accounted for in the
core criticality calculations by modifying the infinite


184
Figure 5-1.3 shows the fuel clad surface temperature as a
function of the the coolant flow rate for different fractions of
the standard power density, for both the maximum clad surface
temperature point, and for the point corresponding to the maximum
heat flux rate. The transition from the forced convection to the
nucleate boiling conditions is more apparent here, since the clad
surface temperature is the parameter most affected by this heat
transfer condition. The sharp break point displayed in the curves
results from changes in the type of correlation used for computing
the temperature level. Note that at the normal operating flow rate
for each particular core, all cores have practically the same
maximum clad temperature, regardless of the fraction of standard
power level considered.
Figure 5-1.4 shows the minimum DNBR for the three low power
density cores as a function of the coolant mass flow rate. Again,
the marked point on each curve represents the mass flow rates that
correspond to the standard outlet coolant enthalpy. The graph also
shows the minimum DNBR level of the standard full-power reactor-
Note that all the low power density cores have an advantage with
regard to the minimum DNBR when compared to the standard core, for
equivalent coolant conditions. However, it is important to keep in
mind that Figure 51-4 corresponds to cores with the standard fuel
lattice pitch- One should recall that the burnup studies showed
that a pitch of 1-35 cm is advantageous from the fuel utilization
point of view as compared to the standard 1-2573 cm pitch. It is
important to examine the thermal-hydraulic behavior of the cores at


153
burnup value, which will yield the discharged fissile mass per MWd
produced. This is the value displayed in the fifth column of
Table 4.2.1.
The calculations show that the total mass of discharged
fissile material per MWd of energy produced decreases steadily as
the pitch is increased. This is due to the softening of the
neutron spectrum caused by the increasing amounts of moderator
present in the core, which reduces resonance absorption in U-238,
and therefore reduces the production of plutonium isotopes. For
each particular pitch, the discharged mass of fissile nuclides per
unit energy produced is reduced as the power level of the core is
reduced,, There are two factors contributing to this effect: first,
as the power level is reduced, the Doppler broadening of resonances
is also reduced, having the same effect explained above when
discussing the production of plutonium isotopes; and second, as the
lower-power cores are driven to higher burnups due to the extra
reactivities supplied by the lower xenon levels and the reduced
Doppler effect, more incremental fissile material is consumed in
energy production than is gained during the additional time for
plutonium conversion.
These observations about the reduction of discharged fissile
nuclides as pitch is increased will have further significance when
fuel reprocessing is contemplated, since the further burnup
achieved by the fuel would also reduce the worth of the discharged
fuel.


192
inflation rate of 9%; the assigned interest rate for borrowed funds
and bonds is 11%. and the rate of return on common equity is 17%.
Finally- the high inflation scenario assumes a yearly inflation
rate of 13%; it assigns an interest rate of 15% to borrowed funds
and bonds, and the return on common equity is placed at 21% -
The economic calculations were performed by Hersperger(l) with
the help of three industry accepted codes: CONCEPT-IV(32) for the
calculation of the capital cost associated with each plant; GEM(33)
for the calculation of the fuel cycle costs; and PCWERCCK34) which
computes the total generation costs. A brief description of the
three codes is found in Appendices B9. BIO and Bll, respectively.
Table 5-2.1 presents the capital costs of the different plant
sizes and the percentage increase with respect to the cost of the
standard plant, for the three inflation scenarios defined. It is
evident that the capital cost per KWh is a uniformly increasing
function, whose slope becomes increasingly steep as the plant power
level is reduced. Any power reduction beyond about 50% of the
standard becomes absolutely impractical since the capital cost is a
very significant proportion of the total power generation cost. The
trends of capital cost versus plant power rating for the different
inflation scenarios are plotted in Figure 5-2.1-
Table 5-2.2 presents the fuel costs for the different plant
sizes and inflation scenarios. The percentage increases of cost
with respect to the standard plant fuel cost (for each particular
inflation scenario) are also shown. These fuel costs include all
fuel-related expenses, such as mining, enrichment, fabrication.


282
990 REM STORAGE IN DISK
1000 GOSUB2480
1010 REM *****************************************
1020 REM INITIALIZING WORK MATRIX
1030 KB=10:RP=2
1040 FORI=1TOTS:R(I,8)=R(I,2) :R(I,9)=R(I,4) :R(I,7)=R(I,6) :NEXTI
1050 GOSB2380
1060 FORI=1TONB:RR(I,1)=RR(I,0) :RR(I,2)=RR(I,0) tNEXTI
1070 RB=12:OC=R(RC(CT,3) ,4)
1080 FORI=lT0TS:R(If12)=R(I,10):NEXTI
1090 REM BORON WORTH VS. CICLE. INTERPOLATED TO MAKE BORON WORTH AS
A FUNCTION OF PPM
1100 IFBE>0THEN1120
1110 Y=BW:GOT01170
1120 DEF FNMP(Yl,Y2)=(R(Ylr6)+R(Y2,6))/2
1130 DEF FNW0(Y1,Y2) = (R(Y1,1)-R(Y2,1))/(R(Y1,6)-R(Y2,6))
1140 IFBE>2THEN1190
1150 Y=FNWO(BW(l)fBW(2))
1160 FORI=lTOCT:RC(I,1)=Y:NEXTI
1170 WA=Y :WB=0
1180 GOTO1250
1190 XY=0:SX=0:SY=0:XX=0:PRIOT"CALCULATING REGRESION LINE OF BORON
WORM VS. Pm"
1200 FORI=lTOBE/2:X=FNMP(EW(2*I-1),BW(2*I))
1210 Y=FNWO(BW(2*1-1),BW(2*I))
1220 XY=XY+X*Y:SX=SX+X:SY=SY+Y:XX=XX+X*X:NEXTI


11
c). Preliminary neutronic and fuel cycle study of the defined
cores with known and reliable calculational tools in order to have
a good estimate of the performance characteristics of each core.
This would include obtaining first estimates of basic core
parameters (such as reactivity worth of boron, coolant temperature,
Doppler reactivity coefficient, etc.) and of the expected variation
of core and fuel cycle parameters (burnup, cycle length, etc.).
d). Exploration and evaluation of currently available
calculational tools that may be suitable for this type of study,
and development of new calculational schemes that may accomplish
the established goals with the optimum cost/results ratio and serve
the nuclear industry as an accurate scoping tool.
For example, this study needs a good set of calculational
tools for the neutronics calculations, where power levels, fuel
temperatures, etc. can be easily specified and changed, since
these are some of the main parameters that will differentiate the
low-power core from the standard core. It also requires reliable
and economical means of calculating core burnup distribution, core
life time, etc.
e). Detailed neutronic and fuel cycle studies of the cores
selected from the parametric variations performed in the
preliminary study. This study must result in an optimization of
the SHARP cores from fuel-cycle and plant operation points of view.
The results of the SHARP study must be compared to those of
standard FWR's analyzed with the same calculational tools, in order
to obtain relative figures of merit with a minimum of


APPENDIX B7
FDQ-7 CODE DESCRIPTION
PEQ-7 (17) is a multigroup. multidimensional diffusion theory
code, used for the modelling and following of reactor cores. It is
one of the most powerful calculational tools now in use in the
nuclear industry. It has an extreme flexibility, and almost any
neutronic situation where diffusion theory can be applied can be
adequately represented with the code-
This great power and flexibility entails a long and
complicated input specification, to the point that several codes
have been developed (such as NUPUNCHER (23). described in APPENDIX
B6- and CHIMP (36)) which assist the user by taking cross section
data from codes like EERI-CELL (19) and LEOPARD (11) and translate
them into a format suitable for input to PDQ-7- These auxiliary
codes still leave a significant task to the user, but they can take
care of the most voluminous and cumbersome part of the input deck.
PDQ-7 has no cross section library of its own. Cross sections
for all the isotopes must be provided from an external source.
However. PDQ-7 has a cross section handling routine. HARMONY (24),
which allows the code to evaluate the cross sections of all
isotopes as a function of up to three independent variables, and
make interpolations in order to provide the most accurate estimate
of the cross sections for each situation of the core. The current
265


120
batch is the one dominating the core radial leakage, the straight
fits of Dc and Lac are made for the first residence cycle. Should
another fuel management scheme be used (e.g. an in-out-in, for a
low leakage core configuration), the regression lines would need to
be changed, obtaining then from the pin-cell burnup code
calculations. For the present case the regression lines obtained
for these variables are:
Dc (cm) = 0.4860 1.90 E-7 BU
Hac (cm-1) = 0.1912 + 6.42 E-7 BU
where BU = average burnup of the peripheral batch, in MWd/MTO.
Once the peripheral batch has its infinite multiplication
factor modified with the probability of non-leakage, it is possible
to proceed to the calculation of the core effective multiplication
factor. At this point, CRIBUR provides for one additional
consideration by allowing any batch in the core to contain burnable
poisons. The burnable poisons are handled as a reduction of the
batch's infinite multiplication factor by a value that varies
linearly as a function of burnup between the user-provided values
for BQL and EOL. If no batch is defined as containing burnable
poisons, no further variations of batches' multiplication factors
are made. In the case where the burnable poison option is to be
used and the batch-wise worth of the poisons at BQL and EOL are not
known, it is always possible to compute than with the codes
involved in the normal burnup calculation. Specifically, EPRI-CELL
is able to provide cross sections for lumped burnable poisons
(LBP's) to be used in FDQ-7 at BQL and EOL (25). If then PDQ-7


188
Percent
Savings
Figure 5.1.5. Percent Savings in DNBR-minimum and Fuel Temperature


182
the TEMPRETGl) code, developed by Miller and based on a
combination of industry-accepted heat transfer correlations, which
was used for the safety-related thermal-hydraulic calculations.
Figure 5-1-1 shows the fuel centerline temperature at the most
disadvantageous point- as a function of the core power density
level and of the coolant mass flow rate. The relative mass flow
rate (on the x-axis) is referred to the standard core's nominal
mass flow rate. The marked points correspond to the flow rates
that yield outlet coolant enthalpies equivalent to those found in
the standard core. Note that for high mass flow rates (as compared
to the value required for equivalent outlet enthalpy), the heat
transfer takes place as forced convection, and is dependent on the
flow conditions. Once a certain low mass flow rate is achieved,
nucleate boiling takes place, and the heat transfer is essentially
independent of the flow conditions. The fuel centerline
temperature remains at an essentially constant level as the coolant
flow is further reduced. This situation would hold as long as the
coolant mass flow rate is not reduced below the point where film
boiling would start. Figure 5-1.2 is a plot similar to Figure
5-1-1 but it represents the volume averaged temperature of the
fuel- The same trends can be observed as in Figure 5.1.1. but they
are smoothed out by the averaging process, which for each flow rate
includes portions of fuel that are under forced convection
conditions as well as other portions which are under nucleate
boiling conditions.


178
a parallel trajectory to the discharge burnup figures. For each
pitch, reducing the power level reduces the ore requirements, but
as pitches are changed, a minimum ore requirement is reached for
1.35 cm pitch.
The comparison of ore usage with respect to the standard plant
is made in column 5 of Table 4-2-3- under the label Ore Savings %
Over Std. It is easier to see in this column that large savings
are achieved with the reduction of power from 100% power level to
50%, but further power reductions result in additional ore savings
of a much smaller magnitude, per unit of power reduction. This is
one of the main reasons calling for the power reduction to be
carried to no more than 50% of the standard value. Actually, an
optimum SHARP power level will be higher than 50%. and will be
affected by other economic considerations.
Column 6 of Table 4-2-3 shows the total enrichment needs of
each plant, in SWU. This column assumes also the total energy
production of all configurations to be 22.4 GWy (e), the fresh fuel
enrichment to be 3-1%, and the tails assay to be 0-2%.


260
b). It also includes the spectral effects of the structural
materials that are present in the reactor but that can not be
assigned to the geometry of the elementary fuel cell.
c). The code accounts for the resonance energy shielding of
all the nuclides present in the cell.
d). The thermal energy cutoff used is 1.855 eV, which is more
adequate than other codes' lower values.
e). Some calculational and convergence techniques of the
"parent" codes have been modified to improve their accuracy and/or
to optimize convergence.
f). The isotopic neutron cross sections are computed from the
microgroup libraries of GAM and THERMOS, with 62 and 35 energy
groups respectively.
g). The depletion calculations are performed for each mesh
space inside the fuel pellet, and isotope accounting is kept
separate for each mesh region. Depletion is done with a four group
spectrum and cross section data set.
The main items input by the user in the general input option
are the following:
a). Definition of the geometry of the problem: Number of
material zones, thickness of each zone and number of mesh points
assigned to each zone.
b). Specification of the nuclides present in the cell
(maximum of 25) and their number densities for each material
composition.


76
The first step involves using a code in which a
representative unit cell of the reactor is burned. Macroscopic
core geometry is avoided; the emphasis is placed in burning a unit
fuel cell and surrounding it with neutronic conditions as close as
possible to those to be encountered by the "average" fuel element
during the real core life. The purpose of this calculational step
is to obtain neutronic characteristics of the fuel as it undergoes
burnup, which will then be used in the second step of the burnup
calculations.
Input to these first-step codes normally includes the
following:
a). Fuel cell geometry and initial isotopics.
b). Temperatures of fuel and moderator.
c). Power density (normally linear power density).
d). Definition of a buffer region surrounding the fuel cell
in order to modify the neutron spectrum for the presence of
structural materials (guide thimbles, fuel assembly cans, spacer
grids, water holes, etc).
e). Specification of a series of burnup timesteps that will
determine the points where the code has to recalculate the
neutronic status of the fuel cell, which has been modified by the
burnup process.
f). Specification of controllable poisons present in the fuel
cell. This is normally specified as soluble boron concentration in
the moderator, and may be specified for each of the burnup
timesteps defined for the calculation.


24
Table 2.1.5. Tabulation of Case Number Densities Which Differ fresa
Those of Case #1.
Case # Isotope
Region
Pure #Dens(*)
Cell Hem.
#Dens(*)
2
0
4
2.346 E-2
2.76149
E-2
2
H
4
4.6926 E-2
2.58845
E-2
3
Same as
core #2
4
Same as
core #2
5
U-235
1
5.7924 E-4
1.9312
E-4
5
U-238
1
2.1426 E-2
7.14335
E-3
6
B
4
7.258 E-5
4.0037
E-5
7
0
4
3.0982 E-2
3.17624
E-2
7
H
4
6.1964 E-2
3.41795
E-2
8
U-235
1
4.4561 E-4
1.48565
E-4
8
U-238
1
2.1559 E-2
7.18790
E-3
8
0
4
3.0982 E-2
3.17624
E-2
8
H
4
6.1964 E-2
3.41795
E-2
9
B
4
7.258 E-5
4.0037
E-5
9
0
4
2.346 E-2
2.76149
E-2
9
H
4
4.6926 E-2
2.58845
E-2
3A
U-235
1
5.7924 E-4
1.9312
E-4
3A
U-238
1
2.1426 E-2
7.14335
E-3
3A
0
4
2.346 E-2
2.76149
E-2
3A
H
4
4.6926 E-2
2.58845
E-2
4A
Same as
core #3A
* Units are atoms per barn-cm.


300
Fuel burnup affects the neutron velocity, consistently causing a
softening of the spectrum as fuel exposure increases.


137
Table 3.3.1. CRIBUR and Batch-averaging Method Criticality Errors.
Cycle CRIBUR % Diff No-leak CRIB. % DIFF BATCH-AVG % DIFF.
Status
K-eff.
to FDQ
K-eff.
to PDQ
K-eff.
with PDQ
B-O-C.
.9694
.23
1.0094
4.36
0.9930
2.66
1/3 cycle
.9520
.15
0.9909
4.23
0.9772
2.79
2/3 cycle
.9274
.68
0.9617
4-40
0.9524
3.38
E.O.C.
.8996
.98
0-9304
4.44
0.9235
3.67
The
other
reference used for benchmarking of
CRIBUR's
criticality calculations was a real reactor at B.O.L. for the
first core. The reactor chosen was Comanche peak, Unit 1(27). The
reasons for choosing this particular core were the following:
a). The reactor was of the same 4-loop Westinghouse type with
17X17 pin fuel assembly as the one used for the CRIBUR
calculations. This made all dimensions, compositions and
temperatures common for both the real reactor and the data used for
CRIBUR's calculation.
b). Being at B.O.L. of the first core, the compositions of
all batches were known, since no burnup had occurred yet, and
therefore no fission products existed in the core.
c). The report on the real reactor indicated the HFP, ARO
soluble boron concentration needed for criticality at that time, as
well as the worth of the burnable poison rods at B.O.L.. all of
which made the comparison of the reactor with CRIBUR a rather
straight forward procedure.


284
1440 FORI=1TQCT:SX=0
1450 F0RJ=1T0NB:CH=RC(I,J+2):IFCH=RC(I,8)THENFB=RC(I,2)ELSEPB=1
1460 SX=SX+((R(CH,0)*PB)[TH):NEXTJ
1470 FORJ=lTONB:CH=RC(I,J+2):IFCH=RC(I,8)THENPB=RC(1,2)ELSEPB=1
1480 R(CH,5)=((R(CH,0)*PB) [TH)*NB/SX:NEXTJ
1490 NEXTI
1500 MR=0
1510 FORI=1TOTS:IFR(I,5)-OTHENNEXTI
1520 CH=ABS( (R(I,5)-R(If2) )/(RMR3HENMR==CH
1530 NEXTI
1540 REM **********************************************
1550 REM OBTAINING CORE K FOR EACH REAL TIMESTEP AND CHECKING
DEVIATION
1560 MK=0
1570 FORI=1TOCT:SX=0
1580 FORJ=1TONB:CH=RC(I,J+2):SX=SX+(R(CH,5)/R(CH,0)):NEXTJ
1590 CC=NB/SX:RC(I,0)=CC:IF(ABS(CC-1))>MKTHENMK=ABS(CC-1)
1600 NEXTI
1610 GOSUB3330
1620 REM ********************************************
1630 REM GETTING BORON LETDOWN CURVE
1640 FORI=1TOCT:CC=(1-RC(I,0) )/(WAfWB*R(RC(I,3) ,7))
1650 FORJ=lTONB:CH=RC(I,J+2):R(CH,7)=R(CH,7)+CC:NEXTJ,I
1660 REM ******************************************
1670 REM OBTAINING END OF CYCLE TIME
1680 RRINT"OBTAINING EOC TIME BY INTERPOLATION FOR 0 PPM, K=l"
FOR


101
calculations, since it allows for multi-dimensional and complex
geometry specifications. PDQ-7 calculations are long and
cumbersome if one is trying to model a two or three-dimensional
large grid, but it may be used with reasonable ease for a small
geometry (a 1/4 fuel assembly in two dimensions, for example).
This last fact suggested that PDQ-7 could be used instead of
EFRI-CELL in order to perform the fuel burnup calculations. This
may appear awkward since it first takes an EPRI-CELL run to feed
the cross section tables to PDQ-7. The system requires more codes
in order to obtain apparently the same data for the core model, and
manual data handling with PDQ-7 is far more cumbersome than with
EPRI-CELL, even for a simple geometry case. However, there are a
number of advantages in using PDQ-7 instead of EPRI-CELL for the
fuel-burning calculations. Some of these advantages are given
below:
a). Given the type of core model that must be used for the
burnup calculations of the SHARP study, it normally takes more than
one iteration to adjust all the core life parameters described in
the previous section (soluble boron letdown curve, batch relative
powers, etc.) and therefore, each case studied requires several
fuel burnup calculations. If PDQ-7 is used, the iteration only
requires repeating the PDQ-7 calculation and not the EPRI-CELL one,
if the fuel cell characteristics are not altered. Each 1/4
assembly EDQ-7 calculation costs only one fourth as much as an
EPRI-CELL one, because PDQ-7 is a diffusion-theory code working
with pre-tabulated few-group cross section libraries, as opposed to


37
Figure 2.2.3. Code and Data Flow for Preliminary Burnup Calculation.


APPENDIX Cl
CRIBUR CODE SOURCE LISTING
10 REM *********************************************
20 REM PWR BORON LETDCWN EVALUATION, + K-EFF AND EOC BURNUP.
30 REM *********************************************
40 CLS:CLEAR1000:DEFINTI,J,K
50 REM *********************************************
60 REM DIMENSIONING AND REACTOR CONFIGURATION
70 REM *********************************************
80 PRINT "PROGRAM FOR CRITICALITY, BORON WORTH, BATCH POWER AND"
90 PRINT"CYCLE LIFE EVALUATION OF A MULTI-BATCH PWR":INFUT"DATA
INPUT FROM (K)EYBOARD OR (D)ISK";A$:IFA$="D"THEN2190
100 CLS s INPUT "TOTAL # OF TIMESTEPS AND EFECTIVE # OF TIMESTEPS PER
CYCLE";TS,CT:IFCT>TSTHEN100
110 INPUT "NUMBER OF BORON WORTH EVALUATION POINTS"; BE: IFBEO0THEN140
120 INPUT "CONSTANT BORON WORTH IN %K/100PPM, ORJUST
ENTER" ;BW:BW=BW*1. OE-4: IFBW=0THEN BW=-1.0E-4
130 GOTO210
140 INPUT"# OF FAKE STEPS TO BE KITLED AFTER BORON-WORTH
EVALUATION";CA:IFCA=0THEN16OELSEDLMBK(CA)
150 FORI=lTOCA:INPUT"STEP # TO BE KILLED";BK (I) :NEXTI
160 BE=BE*2:DIMBW(BE)
170 REM *******************************************
277


53
Figure 2.3.3. Infinite Multiplication Factor vs. Pitch.


158


224
sections as they change through the cycle and the isotopical burnup
evolution of the LBP involve some trial-and-error adjustments, they
can be treated rather easily. However, they present some physical
problems:
a). The absorption cross section is not close to zero at EQL
because of all the structural material associated with the BFR and
because some poison atoms are always left. This causes a residual
negative reactivity which is undesirable at EOL, since it causes
the cycle to be shortened.
b). When the BPR's are removed for later cycles, the channel
that housed them is filled with moderator and mildly absorbing
flow-control devices, forming a sort of flux trap, and causing the
neighboring pins to have a higher-than-normal power. In other
words, the assembly is left with a power distortion.
c). The array spaces used for BPR's can never be used for
fuel pins. The poison rods are removed at the end of the first
cycle of core residence, but the guide tubes remain in their
places, and there is no reasonable way of removing them and placing
fuel pins in these locations. Even if such a manipulation were
possible, severe local power peaking problems would appear because
of the drastic differences of burnup status among neighboring pins.
LBP's therefore represent a loss of energy that could have been
produced by the fuel pins that were displaced.
The second way of implementing the burnable poisons is to load
the fuel pellets with Gadolinium. This technique is used in BWR's


CHAPTER V
THERMAL-HYDRAULICS AND ECONOMIC CONSIDERATIONS
5.1. Safety Related Thermal-hvdraulic Considerations
5.1.1. Motivation
As indicated in the previous chapters, the reduced-power cores
that would be installed in a SHARP would have very nearly the same
physical design as the standard full-power reactor, except for some
possible variation in the fuel array pitch. It has also been
stated that the balance of the plant must operate at conditions
essentially equivalent to those of the standard reactor, in order
to maintain thermodynamic efficiency, and to avoid the need for
significant redesign of components in the balance of the plant.
The above conditions require the reactor coolant to have
identical thermodynamic characteristics at the inlet and outlet of
the reduced-power core as it would have with the standard reactor.
It is obvious then, that if the core is operating at a reduced
power level, the coolant flow must be reduced accordingly in order
to obtain the same enthalpy at the outlet. There is, however, a
consideration of major importance related to reducing core flow
rate: the heat transfer conditions are affected by the different
power density and by the different coolant flow conditions. This
presents a most important issue from the standpoint of plant
safety, since it is well known that one of the critical
179


232
effectively used by the close vicinity of the highly burned
assemblies, and its neutrons are not lost in the baffle or
reflector, but are utilized in an old batch for further fissions or
conversion. The "second cycle" batch being in the core periphery
yields a smaller neutron leakage than a "first cycle" would
(particularly important at EOL) because its pins have already been
burned for two complete cycles by the time EOL is reached.
Obviously, this "second cycle" batch does not generally need LBP's
at all.
This in-out-in scheme is better than the out-in-in scheme as
far as neutron economy is concerned, and is able to improve burnup
by about 3% (2). However, it presents the problem of high power
peakings. This problem is aggravated by the fact that power
peaking does not improve as the cycle advances (as was the case
with the out-in-in scheme) but it gets worse. This is due to the
fact that the fresh batch burns its poisons (which are obviously
necessary) and appears relatively clean and low-burned at EOL while
facing a crisp-burned "third cycle" batch. In addition to being an
operational problem, this is also a calculational problem, because
the whole cycle burnup must be followed on a pin-by-pin basis
before the power peaking constraints may be obtined, since they
appear at EOL.
To the best knowledge of the author, there is yet no
commercial power reactor using this kind of fuel management, but it
is being given serious study and consideration. It appears as if
Gadolinium-loaded pins might be of considerable help in reducing


173
(the standard one-year cycle). The Atomic Industrial Forum(30)
reports capacity factors of 62.3% for large plants (1000 Me and
above, such as the reference plant of this study) and 69.4 for
plants sized between 400 and 749 Me. This figure is used here for
the calculation of the capacity factor of all reduced power plants.
Obviously, the capacity factor is again affected by the cycle
length, since this causes different spacings between refueling
outages. The new, "size corrected" capacity factors are then
obtained with the following considerations:
Let TCTo be the total cycle time of the reference plant for
any particular plant size. In this case it is one year, since the
capacity factors collected from real-world data correspond in their
vast majority to yearly cycles. The plant-size dependent Capacity
Factor is called SCF, and is obtained from statistics of existing
different plant sizes.
In TCTo days, there are TCTo SCF full-power operation days.
During the same time period, the plant is available (TCTo ROT) *
ATAF days. So, in the TCTo days, there are
TCTo SCF full-power days per active day.
ATAF (TCTo ROT)
This is the concept of operating capacity factor, now applied
to any size-dependent actual capacity factor. If the plant can
operate for a known FPT full power time, just as shown before, the
total cycle time should be


152
reductions, but as lower power densities are experienced, the
percent increase in burnup per percent power reduction appears to
be smaller. This is a significant reason to limit the power
reduction to a "safer and optimum level" instead of going to lower
power levels, since little is gained in terms of burnup with the
further power reductions. Other than the safety-related issue, the
improvements in burnup is one of the most significant contributions
of the reduced-power cores.
The fifth column of table 4.2.1, labeled Dis. Fis. MT/MWd
shows the mass of fissile material that is discharged from each
core with the spent fuel. The remaining fissile material is
composed of U-235, Pu-239 and Pu-241.
All fresh fuel loaded in the cores consisted of uranium
enriched 3.1% in U-235. The total mass of uranium per unit volume
loaded with the fresh fuel can be easily obtained by multiplying
the number density of U-235 in the fresh fuel and dividing by the
enrichment and by Avogadro's number. In a similar manner, the
total mass of fissile materials per unit volume present in the
discharged fuel is immediately obtained from the discharge number
densities of U-235, Pu-239 and Pu-241.
With these, it is possible to obtain a ratio of the total mass
of fissile material discharged from the reactor versus the total
mass of uranium loaded with the fresh fuel, which is a constant
value for all the cores. Recalling that the burnup level of the
discharged fuel is expressed in terms of MWd per Metric Ton of
uranium loaded, it is possible to divide the above ratio by the


283
1230 WB=(XY-(SX*SY/(BE/2)))/(XX-(SX*SX/(BE/2))) :WA=(SY-WB*SX)/(BE/2)
1240 REM ******************************************
1250 REM WORTH (DELTA K/PPM) =WAfWB*PPM
1260 IFCA=0THEN127OELSEGOSUB3170
1270 F0RI=1T0CT:CH=RC(1,3) :RC(Ifl)=WMV*R(CHr6) :NEXT1
1280 REM CALCULATING BORON-FREE K-INF FOR CORRELATION WITH BU.
1290 F0RI=1T0TS:R(I-1,11)=R(I,1)-(WA*R(I,6))-(WB*(R(I,6)[2)/2):NEXTI
1300
R(TS,ll)=R(TS-l,ll)+(R(TS,10)-R(TS-l,10))*(R(TS-lfll)-R(TS-2,ll))/(R(
TS-1,10)-R(TS-2,10))
1310 REM *******************************************
1320 REM BEGINNING OF CORE K-EFF CALCULATION
1330 RP=8:GOSUB2380
1340 GQSUB2810
1350 IFBP=0THEN1390
1360 REM CALC. OF LBP-CAUSED REACTIVITY DEFFECT
1370
FORI=1TOCT:CH=RC(I,BPf2) :CC=RC(l,BEH-2) :R(CH,0)=R(CH/0)-BN/100-(BB-BN)
* (R(CH,12) -R(CC,12)) / (100*(R(RC(CP,BIH-2) ,10) -R(CC,10))) :NEXTI
1380 REM ********************************************
1390 REM MODIFICATION OF K FOR THE EXTERNAL BATCH
1400 GOSUB2680
1410 REM *********************************************
1420 REM OBTAINING BATCH RELATIVE POWERS AND CHECKING FOR DIFFERENCE
WITH INITIALS
1430 0H=2.0


Total Generation Cost
Mills/KWh
Figure 5.2.3. Total Generation Cost vs. Plant Power.
Note: Power is expressed as percent of Standard Plant.
Power


18
Mother of the other possible scenarios for comparison could
be to maintain the discharge burnup constant and allow fuel
enrichment, cycle length and core structure to vary. However, it
is the feeling of the author that these cases would not aid
significantly in demonstrating the differences of a SHARP as
compared to a standard plant.
Once the main parameters of interest have been defined, it is
necessary to define a reference core that represents a standard
reactor plant, and whose characteristics and performance is known
and usable as a frame of reference for comparison of the SHARP
characteristics and as a benchmark of the calculational methods
used in the study.
After the reference core is defined, it is necessary to define
a basic SHARP core and a series of "variational" cores that will be
used in order to obtain the coefficients, parameters, and
comparative results defined above. After the main SHARP
characteristics are obtained, another set of cores will be selected
for the in-depth study and comparison with the standard plant. The
calculational tools needed for the in-depth study will be defined
based on the requirements and restrictions observed in the scoping
calculations.
The core chosen for the standard reference plant is a
Westinghouse four-loop, 3400 MWth. FWR, with 17x17 pin fuel
assemblies(4) whose main parameters appear on table 2.1.1. Table
2.1.2 describes the main characteristics of the basic fuel cell of
this standard reactor.


APPENDIX B2
PHRDG CODE DESCRIPTION
PHROG (8) is a computer code to generate fast neutron spectra
and fast multigroup cross sections for use in diffusion or
transport theory codes in reactor design. PHROG uses a B1 or a Pi
approximation to the energy dependent neutron transport equation to
obtain energy dependent fluxes and currents.
The isotopic cross section data are based on an equal
lethargy, 68-group cross section library with energies ranging from
0.414 eV to 10 MeV. The code is an evolution of the GAM-1 (19)
code with a number of improvements, such as temperature dependence
of fine group resolved and unresolved resonance cross section data,
Dancoff correction factor calculation, etc.
Some other significant characteristics of the code are the
following:
a). Integration of resolved resonances may include the
effects of up to three admixed scatterer isotopes, whose atomic
weight and scattering importance (depending on their relative
isotopic abundance and their scattering cross section) are input by
the user.
b). The flux spectrum weighting functions can be generated by
the code for the particular case being studied, or may be supplied
by the user from a previous calculation.
251


234
and will eventually be installed in reactors that are now using the
larger pin design assemblies.
Changing the lattice pitch avoids the problems associated with
heat transfer, but makes backfit into current LWR's much more
difficult since it could call for major redesign of the reactor
core unless the pitch variation were very small.
Changing the fuel density can be done in several ways:
reducing the fuel density and filling the "empty" space with some
low cross section material; or just reducing the physical fuel
density; or using annular fuel pellets. In any case, the fissile
inventory can be maintained by increasing the 0-235 enrichment.
Combustion Engineering did a study (3) where fuel density was
changed, while keeping the total U-235 inventory constant. Other
alterations were also examined, both separately and in combination
with the fuel density alterations. If the fuel density is kept at
the present value, ore savings of about 6% can be achieved by going
to a larger pitch. Larger savings can be theoretically achieved
(up to about 10%) by reducing the fuel density to about 60% of the
present value while keeping the same pitch, but this would drive
the fuel to burnups far beyond the acceptable limits. However, a
combined variation in which the fuel density is reduced to about
80% of nominal, and the lattice pitch is increased could allow the
ore utilization to be improved by about 8% while keeping the
exposures within acceptable limits. From the core cycle point of
view, this would correspond to a rather high burnup combined with a


81
Diffusion Calculation Mesh
l~" I Fuel Pins
i'x V\ \ i Interassembly Water Gap
fyx;::~i Water Holes (or LBP's)
Instrument Thimble
Figure 3.1.3. Geometry and Composition Specification for a FDQ-7
1/4-assembly Burnup Calculation.


10
This low power density study for the NASAP project was carried
out by Westinghouse Corp., and their approach to low power density
was that of increasing somewhat the overall core dimensions, while
maintaining or even augmenting the output power. This approach
meant only a moderate reduction in power density (about 23% as
compared to the 50 to 75% reductions considered in the SHARP
concept). This system was found to be advantageous from the points
of view of safety and fuel utilization when compared to the
standard high power plants. On the other hand, the need for
redesign of the core, the pressure vessel, fuel handling
mechanisms, etc., was found to require large investments both in
time and in capital, and the idea was not considered practical by
the private industry unless governmental support was provided.
1^4. Purpose of this Study
The main purpose of this study is to research and develop the
neutronic and fuel utilization characteristics of the SHARP
concept, and for purposes of completeness, to assess its overall
safety and economic capabilities in comparison with standard PWR
plants.
Several steps are necessary in order to accomplish this task:
a). Definition of what constitues a SHARP for the purpose of
this study (ranges of power densities, safety considerations and/or
goals, etc.).
b). Definition of a set of possible reactor cores on which
parametric studies can be effectively conducted.


102
EPRI-CELL, which is a neutron transport theory code, working with
raw fine-group libraries. However, PDQ-7's calculations may be as
accurate or even more accurate than EPRI-CELL's as will be
explained below.
b). For a well-versed user, PDQ-7 has almost no restrictions.
The code allows for extraordinary flexibility in the specification
of the parameters for simulation of the core environment for the
fuel burning process, and it has very powerful editing
capabilities.
c). EPRI-CELL simulates the reactor environment affecting the
fuel pin by placing a buffer region around the moderator ring.
This buffer region is composed of coolant and a certain proportion
of structural materials which modify the fuel cell neutron
spectrum. This system yields reasonably accurate average isotopics
for the fuel batches, but there are really very few pins in any
assembly which would behave as a "batch average" pin. PDQ-7, on
the other hand, can specify the geometry of a 1/4 fuel assembly as
described on Figure 3.1.3 in the previous section. That
description treats each fuel pin in the assembly as a separate
unit, and it is obvious that the pins neighboring water holes or
facing the interassembly water gap do not see the same spectrum as
a pin surrounded by identical pins. These spectral differences
among fuel pins may translate into slight differences in the
calculation of batch-wise multiplication factors due to the
combination of different spectra and different isotopic composition
of each pin in the fuel assembly. If this is the case, PDQ-7's


220
An analysis of the extreme values of N yields an appreciation
of the available choices. N=1 corresponds to a reactor where the
whole core loading is renewed at each refueling, and it is obvious
that in this case the discharge burnup is the same as the core
average burnup at EOL. For N very large, Xd tends to be equal to
2*Xc, i.e., the discharge burnup can be, in the limit, twice as
large as the reactor average burnup at EOL. This would be the case
of an on-line refueling reactor such as a CANDU or a MAGNOX. The
increased burnup achievable, according to this theory, from the
same initial fuel as the number of batches increases can be
observed in Figure A.2.1.
Relative
Discharge
Burnup
2.0
1.5
1.0
Figure A.2.1. Increase in Fuel Burnup due to Partial Refueling.
Although the assumptions used for this analysis are not quite
true (i.e. the batches do not really share the same amount of
power and the core K-eff. is not really the average of the
batches K-eff.'s ) it is certainly true that increasing the number


241
Normalized
Core
Position
Figure A.2.7. Power Shaping Effect of Partial Length Rod


274
f). Number of enrichments and their magnitudes,
concentration, and other enrichment data.
g). Economic parameters and payment schedules.
h). Escalation information.
The output from the code includes the following items:
a). Batch economic analysis.
b). Yearly batch and case costs.
c). Yearly and cumulative fuel cycle costs.
d). Case cashflow.
e). Allocated cost analysis.
tails


CHAPTER II
SCOPING WORK
2.1. Problem Framing
The purpose of this chapter is to study a sequence of
different low-power cores in order to calculate first estimates of
the low-power cores characteristics. A reference standard core
must be defined against which one may compare the characteristics
of the low-power cores. Finally, the results of the comparisons of
the different reduced power density cores versus the standard one
will be used to define the characteristics of selected cores that
will undergo an in-depth study. The possible need for new
calculational tools to perform such in depth study will also be
examined. This chapter will, therefore, study a series of
low-power cores. The results of this study will be used to select
the better candidate cores, and a range of their expected
characteristics, for further analysis.
The study of the SHARP requires an initial definition of the
parameters of the reactor and/or of the fuel cycle that may be used
as a means of comparing the SHARP with the standard PWR plant.
These parameters may be classified into three main categories of
interest: Safety, Neutronics, and Fuel Cycle. These are key areas
of investigation which have both independent and interrelated
problems.
14


84
timesteps would span exactly the expected duration of one core
cycle. This would allow the data from the corresponding timesteps
in each cycle to be grouped into a core timestep. Figure 3.1.4
illustrates this "timestep synthesis" from a hypothetical cell
burnup scheme in which five timesteps were allowed for each of the
three cycles of core residence of the fuel. The data sought is the
infinite multiplication factor of the core, based on the unpoisoned
multiplication factor of the pin through its life. The figure
shows the conceptual grouping of each timestep's data, but not the
calculations involved in it, which will be dicussed later. Note
that the timestep pattern is repeated in each cycle of the cell
burnup scheme, in order to make the timesteps correspond to the
same time of core life.
However, there are several ways of specifying the pin cell
burnup and of doing the calculations for collapsing the cell
timesteps into core life timesteps. Some of the systems used for
pin cell burnup specification will now be examined.
The simplest scheme, as mentioned before, burns the fuel cell
without any poisoning and at the nominal power, for the length of
time that is estimated that the fuel will remain in the core. Then
the multiplication factors of the cell timesteps are collapsed in
order to obtain the core multiplication factor through the core
life. An interpolation or extrapolation of the curve of core
multiplication factor as it reaches a value of unity determines the
end of the core cycle. The core cycle length is then optionally
used to perform new iterations of the fuel cell burnup calculation.


CHAPTER I
INTRCOUCTXON
1.1. Background
The most widely used reactor system for present and near
future commercial production of nuclear energy is the Light Water
Reactor (LWR). The LWR's were developed originally as compact,
light-weight, high-power units suitable for ship propulsion and
power, and they indeed performed as intended. The large research
and development efforts of the U.S. military establishment for the
LWR systems made them attractive for commercial power generation,
and thus, they were deployed in land-based electric power plants.
LWR's have been generating commercial electricity for a
considerable amount of time, totaling over 500 reactor-years of
operation in the United States alone. However, the safety systems
and engineering safeguards intended to prevent or to contain and
reduce the impact of accidents have become extremely sophisticated
and costly. This is due to several cumulative reasons: 1) the fact
that LWR's work under conditions that are close to technological
limits, 2) the high power density typical of these compact
reactors, 3) the fuel and coolant conditions, and 4) the inherent
"core-cooling" problems in case of accidents or severe transients.
1


155
Recyclable' Net Fissile


36
moderator in the amount necessary to force the core to be exactly
critical.
With the reactor critical, the fuel-region homogenized
composition that was input to MONA and the flux spectrum generated
by MONA for the fuel region are input into a BURNUP code, which
performs the isotopic burning of the mixture to the burnup degree
specified, and at the power level desired. The BURNUP code used in
this case is geometrically non-dimensional, but it allows for the
specification of a number of time-steps after which the homogenized
composition of the fuel region is recalculated. The flux spectrum
is assumed to remain constant throughout the burnup calculation.
After the burnup calculation, the homogenized fuel-region
composition is used again for a new core criticality evaluation, in
the way explained above. Figure 2.2.3 shows a schematic diagram of
the flow of data involved in this type of burnup calculation.
This burnup calculation scheme has some very obvious
drawbacks:
a). Each burnup timestep requires an inordinate amount of
effort.
b). There is a large amount of punched card handling and
typing, which largely increases the probability of human error in
the calculations.
c). The BURNUP code used has a rather low degree of
sophistication, resulting in a reduced number and complexity of
radioactive chains.


75
Figure 3.1.1. Basic Steps in a Burnup Calculation


61
attributed to the appearance of xenon. Note that the cores are
always burned in the critical condition, which means that at BOL
there is a high concentration of soluble boron keeping the core at
critical. This boron concentration is rapidly reduced as xenon
builds up to an equilibrium level. It would seem then, that the
core is changing one thermal poison for another, and therefore no
significant change of neutron spectrum would occur, but this is not
the case. The fact that the thermal poison is in the fuel (in the
case of xenon) as opposed to being in the moderator (in the case of
boron) has a significant effect on the neutron spectrum. This is
due to the different relative changes of thermal absorption cross
section occurring in the fuel and in the moderator. Thus, as
burnup increases, the thermal fraction of the neutron spectrum
continues to drop slowly (the spectrum hardens) due to the
generation of fission products, which are thermal poisons located
within the fuel, as is the case with xenon.
A key point in the safety of operation of a new reactor is its
thermal-hydraulic behavior. It has been stated before that the low
power reactors should deliver the outgoing coolant in the same
thermodynamic conditions as the standard reactor, because this is a
basic factor in the thermal efficiency of the plant. It is obvious
that if a reactor delivers a reduced amount of power, it is also
forced to deliver a reduced coolant mass flow in order to keep the
thermodynamic conditions at the outlet unchanged, and this flow
reduction encompasses different characteristics
transfer conditions.
in the heat


270
c). Departure from nucleate boiling (DNB) heat rate and the
ratio for the actual heat rate at each axial node.
d). Coolant quality versus axial position.
e). Clad outer and inner temperature, fuel average and peak
temperature and specific heat content, versus axial position.


69


Page
3.1.6. Pin Cell Burning with Soluble Boron Letdown Curve
and Power Level Following ... 90
3.1.7. Effect of Time-dependent Power Level and Soluble
Boron Concentration Specification . 93
3.2.1. Estimates of Boron Letdown and Relative Power . 105
3.2.2. Flowchart of Data for a Fuel Burnup Calculation
Using PDQ-7 1/4-Assembly Geometry 107
3.2.3. Core Geometry Used in CRIBUR 109
3.2.4. Bessel-shaped Flux Distribution 116
3.2.5. Fake-burnup Steps for Soluble Boron Worth 124
3.2.6. Problem of Erroneous Cycle Length 126
3.2.7. Flowchart of CRIBUR Code 130
4.2.1. Recyclable Fissile and Net Fissile Usage vs. Pitch 155
4.2.2. Plutonium Discharge and U^DgUse vs. Pitch 158
4.2.3. Real Cycle Length vs. Pitch 167
4.2.4. Total Life Refueling Time vs. Pitch 169
4.2.5. Availability and Capacity Factors vs. Pitch .... 172
5.1.1. Fuel Centerline Temperature vs. Relative
Mass Flow Rate 183
5.1.2. Fuel Average Temperature vs. Relative Mass Flow Rate 185
5.1.3. Clad Surface Temperature vs. Relative Mass Flow Rate 185
5.1.4. DNBR-minimum vs. Relative Mass Flow Rate 186
5.1.5. Percent Savings in DNBR-minimum and Fuel Temperature 188
5.2.1. Capital Cost vs. Plant Power 194
5.2.2. Fuel Cycle Cost vs. Plant Power . 196
5.2.3. Total Generation Cost vs. Plant Power 200
x


57
there is almost no moderator in the cell. In the range of pitches
that can be reasonably considered for practical purposes (between
1.2 and 1.4 cm) the fast fission factor slowly decreases as more
moderator is brought into the cell.
Eta, the number of neutrons produced per thermal neutron
absorbed in the fuel, is the least sensitive of the factors, as it
remains almost flat through the whole range of lattice pitches; it
only shows a fall for very tight configurations. However, it is
important to note that there is still a difference between core 8
and cores 3A and 4A. This shows that more important than the
amount of moderator present in the cell is the variation of mean
free path associated with the change of moderator density.
The resonance escape probability p, shows a very predictable
pattern, increasing as the amount of moderator increases, but
showing a very clear saturation effect as large pitches are
reached, However, in the range of practically reasonable pitches,
p shows a rather steep positive slope; this factor is the one which
is primarily responsible for the increase of reactivity achieved by
increases of pitch in this range. Note that core 8 stays
consistently above cores 3A and 4A throughout the range of pitches
examined in spite of the larger relative number of resonance
absorber atoms (mainly U-238) present in core 8. This is a direct
effect of the different mean free path caused by the increased
moderator density. If only the better moderation effect and
saturation effect of p for larger pitches existed, the distance
between the curves for core 8 and cores 3A and 4A could be expected


278
180 REM INPUTTING VECTOR OF T-STEPS FOR B-WORTH EVALUATION
190 PRINT"INPUT THE PAIRS OF TIME-STEPS USED FOR B-WORTH EVALUATION"
200 FORI=lTOBE/2:PRINT"PAIR # "I" ="; :INPUTBW(2*1-1) ,BW(2*I) :NEXTI
210 INPUT"# OF BATCHES IN THE CORE (MAX=5)";NB:IFNB>5 OR NBC1THEN210
220 DIMR(TS,12) ,RC(CT,8) fRR(NB,2) :INPUT"POWER LEVEL OF THE REACTOR
IN MW THERMAL";PL
230 INPUT"FUEL LOADING OF THE CORE IN MTU
(DEFAULT=94.18)";FL:IFFL=0THEN FL=94.18
240 REM ******************************************
250 REM BATCH CONFIGURATION AND OUTER BATCH FLAGGING
260 BN=0
270 FORI=1TOCT:BB=0
280 FORJ=lTONB: PRINT"STEP # THAT FORMS BATCH "J" AT CYCLE STEP
"I;:INPUT RC(If2+J):NEXTJ
290 INPUT"# OF THE STEP FORMING THE PERIPHERAL BATCH IN THIS
CYCLE-STEP";RC(1,8)
300 FORK=3TO7:IFRC(I,8)=RC(I,K)THENBB=K:NEXTK
310 IFBB< > 0THEN3 3 0
320 PRINT"THIS PERIPHERAL BATCH DOES NOT BELONG TO THIS
TIME-STEP!!":GOTO290
330 IFBN=0THEN360
340 IF(K-2)=BNTHEN360
350 PRINT" WARNING 11!PRINT "PERIPHERAL BATCH IN THIS TIME
STEP DOES NOT BELONG TO PERIPHERAL BATCH OF PREVIOUS
STEP. RECHECK.":GCTO290
360 BN=K-2:NEXTI


APPENDIX A
METHODS OF IMPROVING BUENUP IN PWR'S
A.l. Introduction
A. 1.1. Motivation and Constraints.
LWR nuclear power plants were rapidly deployed in the U.S.
because of their significant advantages in known technology over
all other types of reactors when the industrial application of
nuclear energy began. The existence of the enrichment facilities
and the Nuclear Navy program were key to the selection of the LWR's
as the main U.S. reactor concept for deployment purposes. However,
LWR's are obviously not the most efficient users of uranium
resources. The "burners" were needed for rapid deployment and,
with the LMFBR and reprocessing in mind, the LWR fuel efficiency
was not a major concern. The fact that 'the discharged fuel
contains large amounts of U-235 and plutonium was important from
the point of view of resource utilization, with reprocessing and
the breeder "around the corner".
However, in the present political situation in the U.S. where
reprocessing has been indefinitely delayed and the development of
the LMFBR is questionable, the fuel utilization problem requires a
whole new view and approach. The new goal is now to obtain the
maximum energy generation with the minimum ore consumption and with
the lowest enrichment and fuel fabrication costs. Some of these
213


249
to obtain thermal group, cell-homogenized properties, for use as a
mixture description in a later case of BRT itself, or a multigroup
diffusion theory code. These two last aspects were used in the
present study for the HPWRP preliminary calculations described in
Chapter II.
Some of the improved calculational features of BRT as compared
to THERMOS are the following:
a). BRT includes a transport kernel modification for the
treatment of void regions.
b). The code has been modified to accept generalized boundary
conditions.
c). BRT may account for a perpendicular leakage factor
indicated by an input buckling value or by perpendicular
dimensions.
d). As explained above, the code can print and punch cell
smeared data for use in later cases.
e). BRT can optionally apply a linear anisotropic scattering
correction approximation.
The main characteristics of the input specified by the user
are the following:
a). Specification of number of space and energy points, and
total number of isotopes used in the calculation. The isotopes can
be from the BRT library or can be specified by the user. If they
are specified by the user, it is necessary to input their cross
sections, in addition to the concentrations for each material
mixture


205
Discharge Burnup
GWd/MTU
25% Power
36
34
-i
1.25
I
1.30
,
1.35
p-*-
1.40
Pitch cm
Figure 6.2.1. Discharge Burnup vs. Pitch.
Note: Data fran EPRI-CELL, NUPUNCHER, FDQ-7, CRIBUR Burnup Calculations.
Power Levels Indicated as Percent of Standard Core Power.


95
available for combining the several fuel cell timesteps that form
the complete core at each particular time.
The core is composed, at any timestep of its life, of a
certain number of fuel batches. Each of these fuel batches is
represented in the simplified burnup core model by one timestep
from each of the cycles specified in the fuel cell burnup
calculation, as shown on Figure 3.1.4. The problem is, at this
point, to determine the core multiplication factor from the
multiplication factors of each batch. This can also be done in a
number of ways, with varying degrees of complexity. The simplest
model gathers the batches' infinity multiplication factors and
averages them. The result is considered to be the multiplication
factor of the core. That is,
Kc =
n
where Kc = Core effective multiplication factor.
Ki = Infinite multiplication factor of batch "i".
n = number of batches in the core.
Note that the fuel cell burnup schemes often include a
perpendicular leakage factor, so that what is referred here as
"batch infinite multiplication factor" may well be already
corrected for axial leakage, which makes the core multiplication
factor appear much more realistic when compared to the more
sophisticated core criticality calculations.


87
Soluble
Boron
Pin Cell
First Cycle
Second Cycle
Third Cycle
Figure 3.1.5. Constant-boron Pin Cell Burning with Boron-free
Pseudo-burnup Steps.


60
Figure 2.3.6. Thermal-to-Fast Flux Ratio.


139
In a series of tests where the batch multiplication factors
were altered by 1%, the core multiplication increased around .3%
when the altered batch was the oldest one, about .36% when the
second batch was altered, and approximately .32% when the
alteration affected the first batch. The sensitivity of the core
to the accuracy of the batches multiplication factors depends on
the batch power sharing. The code showed some sensitivity to the
neutronic characteristics of the reflector: if the reflector was
considered absent of soluble boron for the radial leakage
calculation, the core multiplication factor showed a variation of
.11% at BOL (when the real boron concentration is highest) and .04%
near BOL, when the level of soluble boron is low. This
demonstrates that the variations of the reflecting characteristics
of the reflector were non-negligible.
The percentage variations of core multiplication factors
associated with the different altered factors for five points
during the core life are shown on Table 3.3.2.
Table 3.3.2. Effects of Code and Data Alterations on Core K-eff.
Core Life Time:
B.O.L.
M.O.L.
E.O.L
Alteration
Percentage Change on CR1BUR K-eff.
Shape Factor = 1
3.26
3.01
2.99
2.94
2.96
+1% K, Batch 1
0.33
0.32
0.32
0.32
0.31
+1% K, Batch 2
0.36
0.37
0.36
0.36
0.36
+1% K, Batch 3
0.29
0.30
0.30
0.30
0.26
Unborated Reflec.
0.11
0.10
0.09
0.08
0.04


3
of the cores and making other technological fixes while maintaining
the basic reactor design and technology will increase safety
margins in the critical operational variables. However, the plant
economics could suffer a significant penalty since a reduced power
density requires a higher capital investment per unit of installed
power, and this is the major part of the cost of nuclear-produced
energy. On the other hand, the reduction of power density might
allow a better fuel economy, reduce refueling times, reduce
personnel radiation exposures and waste handling and allow a better
overall plant capacity, thus utilizing better the invested capital.
Favorable answers to the second and third questions strongly
dictate that presently known and proven technology be used if the
solution is to be regarded as a viable alternative by the electric
power generating industry.
The Safer Highly Available Reactor Plant (SHARP) concept
described below appears to be a solution meeting these overall
demands., It would definitely enhance safety by improving the core
power distribution, thermal-hydraulics conditions, heat removal
requirements and overall plant operations. It would not require
significant new engineering and technical innovations, since
minimal variations would be applied to the currently used reactor
designs. These new plants will probably be easier to license than
the current plants. The economic aspect is the only phase on which
a conclusion is not easily forthcoming, and it requires in-depth
analysis before reasonable cost comparisons can be established.
The purpose of this work is to conduct an in-depth comparative and


Pedro Otaduy (the Godfather and the executive coordinator of the
Spanish colony at O.R.N.L., respectively) and to the rest of the
Spanish group for their continued moral support and friendship; and
to F. Clark and T. Priiran for their administrative and technical
advice.
The author acknowledges that during his tenure at the
University of Florida he was financially supported by the National
Science Foundation, the Hauck fund, Oak Ridge National Laboratory
and two consecutive University of Florida Graduate School
Fellowships.
The author keeps a special thought for Keith Johnson for his
untiring help in computing matters, and to Pedro Salas for the many
hours of friendship and sharing in happiness and in distress.
iv


38
This method was, however, used for some cases, but a faster
and at least equally accurate method had to be obtained. The
second burnup method employed the LEOPARD program(11). This
program, which is discussed in more detail in Appendix B4, makes an
automated chain of calculations involving cross sections, spectra
and burr'tp for an elementary fuel cell. The cross sections
calculations are based on the MUETQ2) and SOFOCATE(13) codes. The
geometry is always that of a fuel pin, which may be surrounded by a
buffer zone simulating the structural materials present in the core
but not associated with the elementary fuel cell.
LEOPARD requires an initial pin composition, a power level, a
soluble boron concentration history, and a burnup timestep
structure. The code calculates each isotope's cross sections
collapsed to a specified number of broad groups, performs a
multiplication factor calculation, burns the fuel, recalculates the
pin composition, and resumes the loop until the completion of all
the burnup timesteps.
Although the cross section generation in LEOPARD is not quite
as accurate as the one used for the criticality calculations, the
burnup calculational structure is far superior to the one in the
BURNUP code, and the automatic coupling of the calculations allows
for the effortless and error free handling of a much larger number
of isotopes, which definitely adds to the accuracy of the overall
calculation. It is important to note the drastic reduction of
effort required for similar results between the first
(PHRDG-BRT-MCNA-BUFNUP) and the second (LEOPARD) method of burnup


99
1.2. Method Developed for this Study
3.2.1. The Fuel-burning Codes.
As it was explained in the preceding section, any burnup
calculation must start with a fuel-burning scheme which provides
the necessary data for the subsequent core model. In the case of
the SHARP study, the number of initial calculations needed was
rather large, due to the significant number of different power
levels and pitches that had to be studied. This required a rather
automatized calculational procedure which should be of a relatively
low cost. In addition, since the differences between the several
cores to be studied might not be too large, calculations need to
retain maximum accuracy while again not incurring in excessive
costs. The first task was to choose the code or codes to be used
for the fuel-burning step.
One of the more classic pin-burn codes is LEOPARD(11), which
is described in more detail in Appendix B.5. LEOPARD uses a
MUFT-SGFOCATE(12,13) scheme for the calculation of cross sections
for a pin-cell geometry, and it has been and is being widely used
in industry for many of the burnup study types described on the
preceding section. LEOPARD offers flexibility, low cost and simple
input requirements; however, LEOPARD presents sane problems for the
SHARP study:
a). It does not allow for a change of the pin power during
the burnup process, thus making specification 4 of the model
requirements hard to keep.


126
Soluble
Boron
Concentration
Figure 3.2.6. Problem of Erroneous Cycle Length.


294
TSTEP 5 ENDS AT 16300 HR
TSTEP 7 ENDS AT 18300 HR
TSTEP 9 ENDS AT 22301 HR
TSTEP 11 ENDS AT 28301 HR
TSTEP 6 ENDS AT 16500 HR
TSTEP 8 ENDS AT 22300 HR
TSTEP 10 ENDS AT 28300 HR
TSTEP 12 ENDS AT 32600 HR
TSTEP 13 ENDS AT 32800 HR TSTEP 14 ENDS AT 34600 HR
TSTEP 15 ENDS AT 38600 HR TSTEP 16 ENDS AT 44600 HR
TSTEP 17 ENDS AT 48900 HR ANY CHANGED STEP # ? '
Screen 3:
CHECK BORON PPM FOR EACH STEP. ENTER STEP TO BE CHANGED, OR "ENTER"
TIMESTEP 1
TIMESTEP 3
TIMESTEP 5
TIMESTEP 7
TIMESTEP 9
TIMESTEP 11
TIMESTEP 13
TIMESTEP 15
TIMESTEP 17
Screen
CHECK K FOR
HAS 1310 PPM
HAS 1040 PPM
HAS 310 PPM
HAS 1170 PPM
HAS 650 PPM
HAS 150 PPM
HAS 1310 PPM
HAS 1040 PPM
HAS 310 PPM
4:
EACH TIMESTEP
TIMESTEP 2
TIMESTEP 4
TIMESTEP 6
TIMESTEP 8
TIMESTEP 10
TIMESTEP 12
TIMESTEP 14
TIMESTEP 16
ANY CHANGED
HAS 1170 PPM
HAS 750 PPM
HAS 1310 PPM
HAS 1040 PPM
HAS 750 PPM
HAS 310 PPM
HAS 1170 PPM
HAS 750 PPM
STEP # ?
TSTP
K-INF
TSTP
K-INF
1
1.1662
2
1.1489
3
1.1478
,4
1.1415
5
1.1320
;t:'6
1.0032


223
fresh and relatively highly enriched assemblies. This represents a
serious problem for in-core fuel management, since this induces
severe power peaking problems which are difficult to negate.
Burnable poisons utilization is then necessary, which further
complicates core management. Burnable poisons and their associated
advantages and disadvantages are described below.
A.2.2. Burnable_?pis.pns.
Burnable poisons accomplish several functions simultaneously:
reduction of core reactivity at BOL, allowing a reduced chemical
shim and thus make a positive MTC less likely; reduction of power
peaking from one assembly to another; power shaping within a fuel
assembly; and power reduction on the fresh assemblies, which
stretches the cycle length. Each of these aspects is examined
separately below.
There are two ways of implementing burnable poisons in EWR's.
The first and only one used currently in PWR's enploys the
so-called "Lumped Burnable Poisons" (LBP's) also called Burnable
Poison Rods (BPR's). They consist basically of rods having nearly
the same geometry as a fuel pin and they are loaded with a poison
(often borated Pyrex) with concentrations of boron such that it may
be completely burned in one reactor cycle life. The rods are
placed inside a thimble so that they can be easily removed at the
end of the first cycle, leaving the fuel assembly unpoisoned for
the following cycles. This type of poison does not represent a
very difficult problem for core calculations. Although the
calculations normally performed to determine the effective cross


25
Sane of the cores identified in Table 2.1.4 show variations in
moderator temperature or fuel enrichment that imply changes in the
cell number densities as compared to the reference cell. The
coolant and fuel densities for the different temperature situations
were calculated with the help of standard tables for each
materials,6). Table 2.1.5 shows the changed number densities for
the affacted isotopes and applicable cases. For the rest of the
isotopes and/or cases, the reference core number densities apply.
2.2. Calculational Methods for the Scoping Study.
2.2.1. Heat Transfer Calculations.
Some of the main differences between a standard reactor and a
SHARP are related to the differences in the fuel temperature. It
is therefore necessary to know the fuel temperatures that
correspond to several different power-density cores before their
neutronic study can be started. However, for the purpose of the
present study, it is not necessary to obtain extremely accurate
results, since the data will be used for obtaining varying nuclear
properties of the fuel, which are not drastically affected by a few
degrees of uncertainty in the temperature.
The calculational scheme used may be found in any complete
Nuclear Heat Transfer or Reactor Analysis text(7) It is assumed
that a uniform volumetric heat source exists in the fuel region;
that the bulk moderator operating conditions are kept constant for
all the different cores (583 K, 2250 psia.) except for the
preheater cores (cases #7 and #8) which have their moderator at 420


144
single-core plant with reduced power output. As already indicated
in Chapter I, one of the key requirements for acceptance of the
SHARP concept by the industry is the minimization of significant
redesign, and this requirement is only fulfilled by the single-core
plant concept. The preheater reactor was also excluded from
consideration for the same reason. Only the single-core design is,
therefore, considered in the in-depth burnup studies.
The preliminary burnup calculations described in Chapter II
use a 25% power level core as the SHARP core; this would correspond
to a nuclear plant with only about 300 Me power output. It is not
very likely for a plant of such reduced power output to result
economically attractive. This aspect will be further discussed in
Chapter V. An adequate power level for a reduced-size plant may
start at about 500 MW, which would correspond to about 40% of the
power level of a standard plant. However, since studies of cores
with power densities around 75% that of the standard core have
already been done (2) (involving significant core redesign), the
author considered that the scoping studies will explore a power
range between 25% and 50% that of the standard reactor.
A power output of 50% that of the standard plant may be very
attractive for plants sited near medium-sized consumption centers.
Thus, the burnup study of low-power cores between 25% and 50% power
level is likely to give a good perspective of the results that may
be expected from the SHARP'S in the range more likely attractive to
industry. The power levels finally chosen for detailed study were
25%, 33% and 50% of the standard reactor power density.