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Tandem Use of Monte Carlo and Deterministic Methods for Analysis of Large Scale Heterogeneous Radiation Systems

Permanent Link: http://ufdc.ufl.edu/UFE0021460/00001

Material Information

Title: Tandem Use of Monte Carlo and Deterministic Methods for Analysis of Large Scale Heterogeneous Radiation Systems
Physical Description: 1 online resource (86 p.)
Language: english
Creator: Mock, Travis Owings
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: boltzmann, carlo, criticality, differencing, eigenvalue, fission, fixed, mcnp, monte, parallel, penburn, pentran, source, transport
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Monte Carlo stochastic methods of radiation system analysis are among the most popular of computation techniques. Alternatively, deterministic analysis often remains a minority calculation method among nuclear engineers, since large memory requirements inhibit abilities to accomplish 3-D modeling. As a result, deterministic codes are often limited to diffusion type solvers with transport corrections, or limited geometry capabilities such as 1-D or 2-D geometry approximations. However, there are some 3-D deterministic codes with parallel capabilities, used in this work, to abate such issues. The future of radiation systems analysis is undoubtedly evaluation through parallel computation. Large scale heterogeneous systems are especially difficult to model on one machine due to large memory demands, and it becomes advantageous not only to split computational requirements through individual particle interactions, (as is the method for stochastic parallelization), but also to split the geometry of the problem across machines in parallel. In this effort, first presented is a method for multigroup cross section generation for deterministic code use, followed by radiation system analysis performed using parallel 3-D MCNP5 (Monte Carlo) and parallel 3-D PENTRAN (Sn deterministic) such that these two independent calculation methods were used in order to boost confidence of the final results. Two different radiation systems were modeled: an eigenvalue/criticality problem, and a fixed source shielding problem. The work shows that in some systems, stochastic methods are not easily converged, and that tandem use of deterministic calculations provides, at the very least, another means by which the evaluator can increase problem solving efficiency and accuracy. Following this, lessons learned are presented, followed by conclusions and future work.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Travis Owings Mock.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Sjoden, Glenn E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021460:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021460/00001

Material Information

Title: Tandem Use of Monte Carlo and Deterministic Methods for Analysis of Large Scale Heterogeneous Radiation Systems
Physical Description: 1 online resource (86 p.)
Language: english
Creator: Mock, Travis Owings
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: boltzmann, carlo, criticality, differencing, eigenvalue, fission, fixed, mcnp, monte, parallel, penburn, pentran, source, transport
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Monte Carlo stochastic methods of radiation system analysis are among the most popular of computation techniques. Alternatively, deterministic analysis often remains a minority calculation method among nuclear engineers, since large memory requirements inhibit abilities to accomplish 3-D modeling. As a result, deterministic codes are often limited to diffusion type solvers with transport corrections, or limited geometry capabilities such as 1-D or 2-D geometry approximations. However, there are some 3-D deterministic codes with parallel capabilities, used in this work, to abate such issues. The future of radiation systems analysis is undoubtedly evaluation through parallel computation. Large scale heterogeneous systems are especially difficult to model on one machine due to large memory demands, and it becomes advantageous not only to split computational requirements through individual particle interactions, (as is the method for stochastic parallelization), but also to split the geometry of the problem across machines in parallel. In this effort, first presented is a method for multigroup cross section generation for deterministic code use, followed by radiation system analysis performed using parallel 3-D MCNP5 (Monte Carlo) and parallel 3-D PENTRAN (Sn deterministic) such that these two independent calculation methods were used in order to boost confidence of the final results. Two different radiation systems were modeled: an eigenvalue/criticality problem, and a fixed source shielding problem. The work shows that in some systems, stochastic methods are not easily converged, and that tandem use of deterministic calculations provides, at the very least, another means by which the evaluator can increase problem solving efficiency and accuracy. Following this, lessons learned are presented, followed by conclusions and future work.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Travis Owings Mock.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Sjoden, Glenn E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021460:00001


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2756421fb98eb380868209bc57902fac
ba6875d8df8444cbbb06be25dd66f3d78ee509fb







TANDEM USE OF MONTE CARLO AND DETERMINISTIC METHODS FOR ANALYSIS
OF LARGE SCALE HETEROGENEOUS RADIATION SYSTEMS





















By

TRAVIS OWINGS MOCK


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENT S FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007




























O 2007 Travis Owings Mock





























To the beautiful Lily Boynton Kaye, and my magnificent family, Todd, Elaine, and Ron Mock-
without their support, someone else would have completed this work.









ACKNOWLEDGMENT S

I must acknowledge the technical advice and help provided by Dr. Glenn Sj oden. At this

stage of life, it is difficult to define a role model, but Dr. Sj oden has become mine with a

seemingly endless wealth of knowledge, and dedication for work and family. Also, I must

acknowledge Kevin Manalo for his help with Fortran coding, and Monte Carlo Neutral Particle

(MCNP5) modeling techniques.












TABLE OF CONTENTS


page

ACKNOWLEDGMENT S ............ ..... ._ .............. 4.....


LIST OF TABLES................ ...............7.


LIST OF FIGURES ............ ..... ._ ...............8....


AB STRACT ................. ................. 11..............


CHAPTER


1 INTRODUCTION ................. ................. 13..............


Radiation Transport and Diffusion Theory .............. .................... 13
Transport Solution............... ................ 17

2 PREVIOUS WORK ................. ................. 19......... ....


Deterministic Codes ................. .. ............ ... ......... ..... .......... 1
Oak Ridge's Reactor Physics and Shielding Package ................. ...........................19
Tetrahedral Mesh Transport ................. ................. 20......... ....
Oak Rid ge' sTran sport Package ................. ........... ................. 20...
Parallel Environment Neutral Particle Transport............... ...............2
Monte Carlo Stochastic Method of Solution ................. ...............21...............


3 MULTIGROUP CROSS SECTION GENERATION ................. .............. ......... .....23


Candidate Design Fuel Pin Data ................. ...............23.......... ....
Process for Cross Section Collapse ................. ...............24........... ...
Candidate Design Muligroup Binning ........._._........___ .....__ ............2
Codes for Cross Section Generation ..........._. ..........__ ... ...............25.

Single Pin Mesh Modeling in Cartesian 3 -D Sn Transport. .........._. .... ......_. ........26
Group Albedo Treatment in the 3-D Deterministic Lattice Cell ................... ...............30
Single Pin Monte Carlo and Determini sti c Eigenvalue Compari sons ................... ...........3 2
Monte Carlo and Deterministic Flux Comparison............... ...............3
Di scu ssi on ................. ................. 39.............


4 RE ACT OR MODELING ................. ................. 40......... ....


Candidate Design ................. ........... ...............40.......
Candidate Design Parameters ................. ...............40................
Candidate Design Reactor Core Design ................. ...............42........... ...
Reactor Phy si cs Analy si s M ethod s ............... ... ..... ......... ................. 45...
Full Core Candidate Design Monte Carlo Model ................. ...............45........... ..
Radial Power Profile e................ ................. 48......... ...












Axial Power Profile .........__........_. ...............50...
Power Peaking Summary .........__........_. .............. 50....
Deterministic Analysis of Candidate Design .........__........_. ....__ ............5
1/8th Core Candidate Design Model .............. .. ... ...... ... ..._. ............5
Deterministic Criticality Calculation, 1/8th Core Candidate Design .........._... ..............52
Power Peaking in the 1/8th Core Candidate Design ....__ ......_____ ...... ....__........5
Full Core Power Density Comparisons ............ .....___ ...............56..
Further Refinement of the Deterministic Model ............__......__ ....___ ...........5
Di scu ssi on ................. ...............62................


5 FIXED SOURCE SIMULATION ................. ...............63.......... .....


3 -D Gamma Transport Methods .........._. ......... ...............63...
Forward and Ad joint Operators ..........._._ ........_. ...............64...
Forward and Adj oint D etector Response ........._._.. .....__.. .....__. ..........6
Responses Aliased to Dose Rate ........._._.. .....__.. ...............66....
Solution Methodology ........._._.... ...............68..___.........
Modeling Procedure ........._._.. .....__.. ...............69.....
Source Spectra ........._._.... ...............69..___.........
R results ........._.. .... .... ....._._. .. ...... ... ._ ... .. .... ... .......7
Monte Carlo Simulation of the Container Assembly ........._._.. .....__.. ........._......70
Deterministic (Sn) Forward Transport Simulation ...._.__... ..... ..___.. ......._...... ....72
Di scu ssi on ........._._.... ...............79..___.. .....


6 CONCLUSIONS ........._._.... ...............82..___.........


7 FUTURE WORK ........._._.... ................ 83...__.. ....


Candidate Design Deterministi c/Monte Carlo Agreement ....._____ .........__ ............... .83
Parallel Environment Burnup ........._._.._......_.. ...............84....

LIST OF REFERENCES ........._._.._......_.. ...............85....


BIOGRAPHICAL SKETCH ........._._.._......_.. ...............86.....











LIST OF TABLES

Table page

3-1 8 group collapsed energy bin bounds ................. ...............25........... ..

3-2 6 group collapsed energy bin bounds ................. ...............25........... ..

3-3 Candidate Design single pin geometry parameters ................. ...............27........... ..

3-4 Excess mass balance and percent error for a 35x34 mesh full pin ................. ................. 28

3-5 Excess mass balance and percent error for a 21xl9 mesh full pin ................. ................. 29

3-6 Albedo boundary distance parameters ................. ......... ......... ............3

3-7 8 group albedo factors ................. ...............32........... ...

3-8 6 group albedo factors ................. ...............32........... ...

3-9 Monte Carlo and Deterministic single pin eigenvalue results for 8 groups .......................33

3-10 Monte Carlo and deterministic single pin eigenvalue results for 6 groups ........................34

4-1 Monte Carlo Candidate Design k-code full core results ................. .......... ...............48

4-2 Power density per pin (maximum error < 4.2%) ........._.._.. ....._.. .......__. .......4

4-3 Computed power density per axial zone (errors < 4.2%) .......__. .......... .... ........._...50

4-4 Power Peaking Factor Summary ........._.._........___ ...............50..

4-5 Final values for 6-group convergence comparison ....._._.__ ........_._ ......._._........5

4-6 Axial zone boundaries, zone and hottest channel power densities computed using the
determine sti c model ................. ...............55.......... .....

4-7 Hot channel power peaking factors compared to Monte Carlo ................. ............. .....55

4-8 Average power density by zone compared to Monte Carlo ........_.. ........ ...._............56

5-1 Interpolated Flux to Dose Conversion Factors, 1 cSv = 1 reml2............. ...................66

5-2 Deterministic and Monte Carlo dose comparisons of dose rate (rem/hr) at the Region
of Interest ................. ................. 8......... 0....










LIST OF FIGURES


Fiu~re page

3-1 Fuel element detail for Candidate Design. ........._ ...... __.. ......__ .........2

3-2 Whole-core U-23 5 fission rate (average relative error 3%) and selected 8 -group
structure. ................. ................. 24..............

3-3 Flow chart of the codes and used in the cross section generation process, where
codes are represented in blue, files are represented in yellow. ................. ............... .....26

3-4 35x34 coarse mesh cell for deterministic model. ..........._..._ ..............__... 28....._ .

3-5 21xl9 coarse mesh cell for deterministic model. .....__.___ .... ... ._._ ... .._._ .......2

3-6 Single pin Candidate Design material description used in the Monte Carlo model. .........33

3-7 21xl9 mesh tally scheme for Monte Carlo single pin model ..........._..._. .....................34

3-8 Group 1 flux comparison (6 group model). .............. ...............35....

3-9 Group 2 flux comparison (6 group model). .............. ...............36....

3-10 Group 3 flux comparison (6 group model). .............. ...............36....

3-11 Group 4 flux comparison (6 group model). .............. ...............37....

3-12 Group 5 flux comparison (6 group model). .............. ...............37....

3-13 Group 6 flux comparison (6 group model). .............. ...............38....

4-1 Plant Overview of original Candidate Design.9 ..........._... ............_ ..... 40...._ ..

4-2 Candidate Design reactor core views.9 A) North-South Candidate Design core view.
B) East-West Candidate Design core view. ................. ...............41...............

4-3 Enlarged East-West Core cutaway.9 ................ ......... ...............42.....

4-4 Horizontal slice of core and surrounding beam ports.9 ................ ................. ........ 44

4-5 X-Z (y=0 slice) plane view of full core Candidate Design Monte Carlo Model ...............45

4-6 X-Y (z=0 slice) plane view of full core Candidate Design Monte Carlo Model. .............46

4-7 Monte Carlo model full core pin numbering scheme. ............_... ........._..._... .........47

4-8 Radial (averaged over axial length) total power pin power densities based on relative
errors of less than 4.2% for each value. ........._..._........... ...............48..










4-9 1/8th COre Candidate Design. .........._.... ...............51....__.. ..

4-10 Convergence of the 1/8th COre Candidate Design for deterministic models. ................ .....52

4-11 Relative pin power for pins contained in 1/8th Candidate Design (W/cc). Monte Carlo
values are converged to a standard deviation of less than 4.2% (1-sigma) ................... .....54

4-12 Monte Carlo and Deterministic 6 group computed power densities (W/cc). Monte
Carlo values are converged to a standard deviation of less than 4.2% (1-sigma). .............56

4-13 Ratio of Monte Carlo power density to Deterministic power density. Monte Carlo
values are converged to a standard deviation of less than 4.2% (1-sigma). ................... ....57

4-14 Monte Carlo 6 group fluxes ................. ................. 58............

4-15 Deterministic 6 group fluxes ................. ...............59........... ...

4-16 Monte Carlo to Deterministic power ratios for S10 refined mesh (left) and S10
coarser mesh (right). Monte Carlo values are converged to a standard deviation ............59

4-17 PENTRAN differencing scheme map, blue = Directional Theta Weighted, red =
Exponential Directional Iterative. ................. ...............60................

4-18 Monte Carlo to Deterministic power ratios for S10 refined mesh with updated
differencing. Monte Carlo values are converged to a standard deviation. ........................61

5-1 Standard fuel pellet gamma source spectrum normalized from specified upper MeV
bound s. ................. ...............69.......... .....

5-2 Cobalt-60 source spectrum normalized from specified MeV upper bounds ................... ...70

5-3 Monte Carlo Neutral Particle version 5 generated picture of region of interest
detector box position and assumed container assembly immersed in water. .....................71

5-4 Gamma flux at 0.65 MeV (average), proj ected on SS3 04 materials immersed in
water ................. ...............73.................

5-5 Gamma flux, 0.08 MeV (average), proj ected on stainless steel 304 materials. .................74

5-6 Gamma group adj oint function, aliased to Importance at 0.08 MeV (average)
proj ected on SS304 container materials immersed in water. ................ ................. ... 75

5-7 Dose for a single Fuel unit (mean burnup) in the Region of Interest as a function of
source height at the x-y center of nearest container (Adj oint Deterministic. ................. ....77

5-8 Dose from single fuel unit sources coupled for various burnups (low, mean, and
high) in the Region of Interest as a function of source height at x-y center. ................... ...77










5-9 Dose caused by 1 gram alloyed Co-60 (3-cycle irradiation) in the "ROI Detector
Box" region as a function of source height at x-y center of nearest container. .................. 78

5-10 Dose caused by 1g alloyed Co-60 (5-cycle irradiation) as a function of source height
at x-y center of nearest container. ................. ...............79..............









Abstract of Thesi s Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

TANDEM USE OF MONTE CARLO AND DETERMINISTIC METHODS FOR ANALYSIS
OF LARGE SCALE HETEROGENEOUS RADIATION SYSTEMS

By

Travis Owings Mock

August 2007

Chair: Glenn Sjoden
Major: Nuclear Engineering Sciences

Monte Carlo stochastic methods of radiation system analysis are among the most popular

of computation techniques. Alternatively, deterministic analysis often remains a minority

calculation method among nuclear engineers, since large memory requirements inhibit abilities

to accomplish 3 -D modeling. As a result, deterministic codes are often limited to diffusion type

solvers with transport corrections, or limited geometry capabilities such as 1-D or 2-D geometry

approximations. However, there are some 3 -D deterministic codes with parallel capabilities,

used in this work, to abate such issues. The future of radiation systems analysis is undoubtedly

evaluation through parallel computation. Large scale heterogeneous systems are especially

difficult to model on one machine due to large memory demands, and it becomes advantageous

not only to split computational requirements through individual particle interactions, (as is the

method for stochastic parallelization), but also to split the geometry of the problem across

machines in parallel.

In this effort, first presented is a method for multigroup cross section generation for

deterministic code use, followed by radiation system analysis performed using parallel 3 -D

MCNP5 (Monte Carlo) and parallel 3 -D PENTRAN (Sn deterministic) such that these two

independent calculation methods were used in order to boost confidence of the final results. T wo









different radiation systems were modeled: an eigenvalue/criti cality problem, and a fixed source

shielding problem. The work shows that in some systems, stochastic methods are not easily

converged, and that tandem use of deterministic calculations provides, at the very least, another

means by which the evaluator can increase problem solving efficiency and accuracy. Following

this, lessons learned are presented, followed by conclusions and future work.









CHAPTER 1
INTRODUCTION

Analysis of reactor and radiation systems aims at one central problem, the resolution of the

neutron or photon distribution within a radiation system. Several different methods have been

devised in order to obtain neutron densities, neutron flux, gamma flux, and subsequently reaction

rates within radiation systems. The most notable of simulation methods include transport theory,

diffusion theory (for neutrons), and Monte Carlo methods. Diffusion theory is an approximation

to transport theory, and Monte Carlo methods are a statistical sampling approach. However,

deterministic transport theory is subj ect to its own set of constraints due to phase space

discretization, most notably multigroup cross section generation. Therefore, the most accurate

modeling approach for radiation system analysis is one that utilizes a combination of techniques.

Radiation Transport and Diffusion Theory

The transport equation was developed by Ludwig Boltzmann in 1872 to describe the

kinetic theory of gases.' For neutral particle transport, the linear Boltzmann equation (LBE) is

used to accurately model neutron and photon behavior in radiation systems. The integro-

differential form for neutron multiplying systems is shown in Equation 1-1:


-+ 0z V y(F~, E, 02, t) + o-(F, E)W(F~, E, 02, t) = q, (F~, E, 02, t)
v at
(E) (1-1)




Where;

* v = neutral particle speed
* S" = particle angular flux
* 0Z = unit vector in direction of particle motion
* P = particle coordinate location in space
* E = particle energy
* t = time









* 0- = total macroscopic cross section
* q,, = external independent particle source
* o, = differential scattering cross section
* X = fission spectrum
* o-, = fission production cross section
* v = average number of fission neutrons produced per fission
Most students of nuclear engineering live in fear of the neutron transport equation. The

neutron transport equation, although relatively simple to understand, is difficult to solve in 3 -D

systems without immense computing power and parallel computation capabilities. Diffusion

theory is often used as a substitute for the neutron transport equation in these cases because it is

limitati ons.

To arrive at the diffusion equation from Equation 1-1, we integrate angular flux over 4 Fr

steradians such that the following is realized:


di7(, E,6, ) -4F, E t)(1-2)

If we integrate the LBE over all angles, we obtain the zeroth angular moment. Without showing

all the steps of the integration, the zeroth moment balance equation takes the form of the

follow ng:

1 84(F, E, t)
+ V J(F, E, t) + 0-(F, E) (F, E, t) = q,, (F, E, t)
v 8 t (1-3)



Where a neutron current term, V J(F, E, t) becomes a second unknown, leaving two unknowns

and only one equation. This divergence of current is the net leakage from the system. To

resolve the equation, a relationship must be developed between ~(F, E, t) and J(F, E, t) The

relationship is established by introducing jirst angular moment:


di i(LBE)(1-4)









This new relation, after some manipulation, becomes the following:

1~~ V E~tdn nsy(F, E, 02, t) + cr(F, E)J(F, E, t) = q,, (F, E, t)
v dt 4x(1-5)
+ SdE'a, (F, E'~ -E)J(F, E, t)

Equation 1-5 may be of further use assuming that angular flux weakly depends on angle,

so that, using first moments,2 we assume:


y (F, E, 0, t) = 1 Q(F, E, t) + 302 J(F, E, t) (1-6)
47r

Substitution of Equation 1 -6 into Equation 1-5 with subsequent integration and simplification

yields the P1 balance equations, as:

1 8 (F, E, t)
+ V J(F, E, t) + a(F, E)#(F, E, t) = q,, (F, E, t)
v 8 t (1-7)
+ %dE'o, (F, E'~ ->) (F, E,~ t) + X(E) RdE'v e (F, E) (r, E' t)

And

1 8J(F, E, t) 1
+ -V #(F, E, t) + (F, E)J(F, E, t) = q,, (F, E, t)
v dt 3 (1-8)



Taking the one speed form of the P1 equations, (i.e. with integration over all possible energies)

yields the one speed balance equations as:

1 84(F, t)
+ V J(F, t) + o(F, E) (F, t) = q,, (F, t)
v dt (1-9)
+ o, (F)#(F, t) + va, (F)#(F, t)

And

1 8J(F, t) 1
+-V4(F,t)+o(F)J(F,t) =q,,,(F,t)
v dt 3 (1-10)
+ o,, (F) J(F, t)









dJ(F, t)
Further assumptions can be made about .It is acceptable to assume the relative


variation in current will not dominate the collision term, (e.g. unless there is a severe accident


situation), and therefore the M(F~t) term is assumed to be zero.2 This is particularly convenient


because it yields an approximation for J(F, t) itself from Equation (1-10) to give:


J(F, t) = ( o (F, I~lt) (1-11)


Where the diffusion constant can be defined as:


D! = ( o (1-12)

Substitution into Equation (1-9) gives the one speed diffusion equation as:

1 8 (F, t)
-D on2(F, t) + (F, E)#(F, t) = q,t(F, t)
v 8t (1-13)
+ o, (F) (F, t) + vo-, (F)#(F, t)

Diffusion theory for neutrons is based on the idea that neutrons migrate from an area of

high concentration to an area of low concentration, much like diffusion of gases and heat.

However, in most diffusive processes, diffusing particles are characterized by what can be

described as frequent scattering. The problem is that the cross section for neutron scattering is

often on the order of 10-24 CM12 2 As a result, with typical atom densities, neutrons may travel for

several centimeters before a collision or scattering event takes place, depending on neutron

energy. What' s more, the dimensions of fuel rods, control rods, and other absorbers in a reactor

or radiation system are of the same order as the mean free path (mfp) of a neutron.









Diffusion theory fails to describe neutron behavior and becomes inaccurate in regions

where the flux can change radically.2 The regions in a reactor or radiation system where the flux

changes very rapidly include:

* Material interfaces, where two different materials meet

* Strong absorbers (especially, control rods)

* At the boundary of a system

Therefore, it' s not hard to imagine that these assumptions break down in a heterogeneous

radiation system. To give reasonable solutions, diffusion theory is often specifically "tuned" by

correction factors in order to produce transpor~rtrt~t~r~t-crtrected~fluxes and reaction rates. Needless to

say, that the desired solution, often termed the "true solution, is only obtained through the

transport theory, by solving the LBE. In most new reactor systems, every "tuned" diffusion

theory is proving to be inadequate as reactor margins and efficiency are pressed.

Transport Solution

Today, 3 -D transport reactor modeling is done primarily through stochastic Monte Carlo

simulation, perhaps most popularly utilized by Los Alamos National Laboratory' s, (LANL)

MCNP series. This is in part due to the fact that solving the LBE deterministically creates

difficulties and generating multigroup cross section parameters can be difficult, and proper

inning of energy groups, and therefore modeling of resonances, can become treacherous.

MCNP uses "continuous energy" bins at a variety of temperatures, and many concerns about

accurate representation of cross section resonances and energy dependence are lifted.

As a result, nuclear engineers and physicists have learned to rely heavily on Monte Carlo

methods, sometimes taking for granted the issues related to the statistics involved in obtaining a

converged Monte Carlo solution. Radiation systems analysis with stochastic Monte Carlo

methods alone can never be exact, but rather, are bounded by statistics and tests for statistical









convergence. Thus, while the Monte Carlo method is quite useful, it may lead to difficulties in

achieving a converged solution. Transport theory renders a scientist with a near "exact" global

solution, providing the languid evaluator has the right tools and discretization of the phase space

in Equation 1-1, including reasonably accurate geometric modeling and Doppler broadened, self-

shielded multigroup cross section generation tools. This brings us to a discussion involving the

advantages and disadvantages of both Monte Carlo, and deterministic transport solutions.

The following chapters describe a multigroup cross section generation and modeling

approach in an effort to analyze two radiation systems using Sn and Monte Carlo simulation. In

Chapter 2, previous work and different analytical methods are outlined. Chapter 3 highlights

cross section generation, and presents a study involving a procedure for generating Doppler

broadened energy weighted material cross sections for a 3 -D Candidate Design (CD) pin cell.

Chapter 4 presents a 3 -D reactor model and simulation of the neutron flux behavior in a

pressurized heavy water, natural uranium reactor using 3 -D Sn Parallel Environment Transport

theory, (PENTRAN) in tandem with Monte Carlo Neutral Particle version 5 (MCNP5), while

utilizing the cross section generation methods outlined in Chapter 3. In Chapter 5, a fixed source

shielding problem is introduced and solved using Monte Carlo in tandem with transport theory

that includes demonstration of the advantages of forward and adj oint equation. This is followed

by a discussion of conclusions, future work, and references.









CHAPTER 2
PREVIOUS WORK

Deterministic Codes

Although deterministic codes exist for evaluation of heterogeneous radiation systems,

many fall short of geometry, memory, or parallel processing requirements. There are few

parallel deterministic transport codes available and therefore, previous work involving analysis

of large scale heterogeneous radiation systems is limited. Analysis and validation of

deterministic code packages, through modeling of increasingly detailed reactor and fixed source

systems, remains the method of quality control among the use of different deterministic radiation

system simulation techniques.

Oak Ridge's Reactor Physics and Shielding Package

Oak Ridge National Lab (ORNL) has developed many useful codes for the public (U. S.

citizens only) and their radiation systems analysis needs. The multitude of codes includes

criticality safety codes using both stochastic and deterministic methods, as well as sophisticated

reactor fuel burnup modules for certain standarddddd~~~~~~dddddd reactor systems (pressurized water reactor,

boiling water reactor). However, deterministic 3 -D transport remains a beast yet dominated in

nuclear science. NEW Transport code (NEWT), X-Section Dynamics for Reactor Nucleonics with

Petrie Modifications (XSDRNPM) and the Discrete Ordinates Oak Ridge System (DOORS)

system are the only ORNL codes available for transport analysis.

XSDRNPM is a one dimensional (1-D) Sn transport algorithm and is used in Scale version

5.1 (Scale5.1) for eigenvalue (k-effective) determination, cross-section collapsing, production of

bias factors for use in Monte Carlo shielding calculations, and shielding analysis.3 Diffusion

theory can also be executed through use of XSDRNPM. However, 1 -D models are less useful

for large heterogeneous systems.









NEWT (although it is not the newest addition to SCALES), was developed as a multigroup

discrete ordinates radiation transport code with non orthogonal meshing to allow for complex

two-dimensional (2-D) neutron transport models.4 NEWT is commonly spoken of in tandem

with Triton (a transport and depletion module for nuclear fuel) and used for infinite lattice

calculations and comparisons. It is especially useful for multigroup cross section generation and

Doppler broadening. Unfortunately, NEWT has no parallel capabilities, and does not account for

three dimensional (3-D) effects.

Tetrahedral Mesh Transport

Attila solves the first order steady state form of the Linear Boltzmann Equation (LBE)

using the discrete ordinates method, and does so using tetrahedral mesh geometries and angular

discretization.5 Tetrahedral mesh geometry provides for impressive accuracy in material

representation, but requires a commercial mesh generator. However, the complex tetrahedral

mesh limits the differencing scheme versatility in the code, and therefore Attila is tied to linear

discontinuous finite-element spatial differencing. A notable advantage to this scheme is that

since Attila allows for discontinuities between element faces, and the Linear Discontinuous

Finite Element Method (LDFEM) is able to capture sharp gradients with large meshes allowing

for lower Sn angular discretization.5 However, this can pose challenges to yielding accurate

solutions in thick shielding problems. To date, Attila is not parallel, and memory issues are

significant in cases where complex geometry couples with high angular discretization

requirements.

Oak Ridge's Transport Package

Three-Dimensional Oak Ridge Transport (TORT), available through the Oak Ridge

National Lab (ORNL) Discrete Ordinates Oak Ridge System (DOORS), is another ORNL code

that solves the Linear Boltzmann Equation (LBE) using the method of discrete ordinates, but in









3-D. TORT's limitations apply specifically to fixed source problems where penetration of

particles through large non-scattering media produce errors in TORT's calculations because of

non-physical "ray effects". Further limitations memory limitations and no scalable parallel.

Parallel Environment Neutral Particle Transport

Parallel Environment Transport (PENTRAN) is a scalable parallel code that solves the

steady state form of the LBE in 3 -D while using Cartesian mesh geometries and Sn angular

discretization.6 The package, as mentioned previously, operates in a distributed parallel

environment, solving the issue of memory availability by further distributing the phase space

across 3 dimensions of domain decomposition, including space (Cartesian mesh spatial

discretization), angle (angular discretization), and energy group (energy group discretization).

PENTRAN is very efficient with adaptive differencing schemes, and has schemes for coarse

mesh/partial current rebalance and Taylor proj section methods for linking grid discontinuities and

mitigating ray effects. The very flexible differencing schemes allow for a solution that i s tailored

to the experience of the user. Also, PENTRAN is in continuous development.

Monte Carlo Stochastic Method of Solution

As mentioned in the previous section, Oak Ridge National Lab (ORNL) uses a variety of

codes, of which employ stochastic Monte Carlo methods to assist in problem simulations. The

Keno criticality module Monte Carlo is an incredibly useful tool for criticality calculations, and

much study has been directed towards obtaining solutions in loosely coupled systems. However,

the code is not meant to model fixed source transport problems. Further applications ofKeno

include its use in depletion calculations with the use of Triton.

Los Alamos National Lab's (LANL) Monte Carlo Neutral Particle code (MCNP) has

achieved high notoriety in the field of nuclear engineering and simulation, but there are

situations in which loosely coupled systems provide waning confidence in source convergence.









Such elusive resolutions require a keen eye for the statistical nature of the method of solution.

Although statistical tests such as "drift-in-mean" and "1/N" tests can detect non-convergence of

a source, they cannot provide completely reliable indications of non-convergence.7 Previous

work in Monte Carlo solution methods include loosely coupled system simulations such as

criticality safety geometries, eigenvalue, and fixed source shielding problems, and in particular

address solutions to such problems.










CHAPTER
MULTIGROUP CROSS SECTION GENERATION

In order to specifically address the problem involving the generation of multigroup

Doppler broadened/self shielded material cross sections, a procedure was developed using Scale

version 5.1 (Scale5.1) with the NEW Transport along with Triton (NEWT-Triton) transport tool.

In NEWT, a general purpose 23 8 group library can be collapsed to obtain a useful cross section

library for use in 3 -D Sn transport in Parallel Environment Transport (PENTRAN). This method

was developed in the process of building a Candidate Design (CD) fuel pin, performing a 3 -D

unit cell or lattice eigenvalue calculation, and comparing the results against Monte Carlo Neutral

Particle version 5 (MCNP5). An example of the procedure is highlighted in subsequent sections.

Candidate Design Fuel Pin Data

The Candidate Design (CD) fuel rod, depicted in Figure 3 -1, modeled for the study was

taken from the full core CD with the following properties;

* Uranium-Mo Alloy Annular Fuel, 0.72 w%/ U-235 enriched; 98.5 w% U, 1.5 w% Mo
* Inner Fuel Radius: 0 cm (reduced for natural uranium fuel design)
* Outer Fuel Radius: 1.78 cm
* Fuel Rod Length: 250 cm
* Fuel Rod Pitch: 13.5 cm
* Cladding, inner: 0.1 cm, Alloy 6061T6
* Cladding, outer: 0.1 cm, Alloy 6061T6
* Inner Pressure Tube Radius: 2.0 cm, Pressure Tube thickness: 0.15 cm





Figre3-. ue eemntdeaifoCadatDegn
Fo.:rth simulation, the inr imte f h nnlr yefuldeitd nFiue3- a
considered to be zero, impDlyigasldfuleeet










Process for Cross Section Collapse

Although the single pin infinite lattice calculation performed in order to validate the cross

section collapse procedure using the Scale version 5.1 (Scale5.1) package only models 2-D

geometries, the process was shown to be accurate for 3 -D calculations, as will be shown by the

following sections.

Candidate Design Muligroup Binning

To select a proper group structure most applicable to the core physics Candidate Design

(CD) problem, the entire CD reactor core, to be described in the following chapter, was modeled

in a continuous energy Monte Carlo Neutral Particle version 5 (MCNP5) for approximately 10

hours on 16 64-bit Opteron processors. Results of this Monte Carlo simulation yielded

approximately 3% tally statistics throughout all 238 energy groups. MCNP5 continuous energy

groups were binned to be identical with the general Scale version 5.1 (Scale5.1) library.

Following analysis of this data, eight (8) total energy groups were collapsed from the original

238 groups that could best represent problem physics. Note these groups were identified through

use of the global integrated U-23 5 fission rate tally, with contributions from all fueled core

regions portrayed in Figure 3 -2. Table 3-1 shows each collapsed group's upper energy limits.


CD Full Core, 1.35% Enriched Annular Fuel, Fission Rate in U-235

G8 G7 G6 G5 G4 G3 321 G1









Energy In MeV

Figure 3-2. Whole-core U-235 fission rate (average relative error 3%) and selected 8-group
structure.










Table 3-1. 8 group collapsed energy bin bounds
Group Upper Group
Bounds(MeV)
1 20
2 1.36
3 4.40E-01
4 5.20E-02
5 3.90E-03
6 6.83E-04
7 1.22E-04
8 3.00E-06

The study was also collapsed to six groups for in order to reduce memory requirements.

The upper bounds for the six group collapse are shown in Table 3 -2.

Table 3-2. 6 group collapsed energy bin bounds
Group Upper Group Bounds
(MeV)
1 20
2 1.36
3 4.40E-01
4 3.90E-03
5 1.22E-04
6 3.00E-06

Codes for Cross Section Generation

The 8 and 6 group libraries presented in the previous section were collapsed from the 238

group Scale version 5.1 (Scale5.1) general library for the pin cell materials and rendered into

deterministic cross section libraries using tools that were developed or significantly expanded

and enhanced in support of this proj ect: the Scale Form (SCALFORM) tool renders Scale5.1

microscopic cross sections in a readable text format following the use of the Scale5.1 Alpo tool.

Mixing the rendered text based microscopic cross sections from SCALFORM was then

accomplished using an updated group cross section mixer called Group Independent Mixer

(GMIX), which provides material macroscopic cross sections, and fuel mixture group fission









source "chi" values. For this work, a mean average integrated chi fission spectrum that accounts

for the effects of multiple fissile nuclides was developed and incorporated into GMIX. A

flowchart outlining the cross section generation process, termed the single uranium Pin (suPin)

process, is shown in Figure 3 -3.


Output~ suPin.out (data file)


Output s uPinXS


suPin.inp


Input


NEWT-TRITON


.xsa (log file) Oupu


Output suPin.out (log file)


suPin.grp

su Pin.xrf


GMIX


Output


suPinMAC.xs


Input


su Pin.gmx Input Output suPinMAC.chi


Figure 3 -3. Flow chart of the codes and used in the cross section generation proce ss, where
codes are represented in blue, files are represented in yellow.


Single Pin Mesh Modeling in Cartesian 3-D Sn Transport

Following the generation of macroscopic cross sections, from the 2 -D NEW Transport

(NEWT) infinite lattice calculation, a 3 -D calculation with Parallel Environment Transport

(PENTRAN) was modeled. Again, the purpose of the PENTRAN calculation was to validate the










cross section generation procedure in a 3 -D lattice calculation for further use on the 1/8th core

Candidate Design (CD) model. For the simulation, the full pin model was used with the

parameters shown in Table 3 -3.

Table 3 -3. Candidate Design single pin geometry parameters

Description length (cm)

active fuel length 250


Length of D20 stacked above/below active fuel 79

rfyne-ortere 1.780

1st clad rozere; 1.992

Pressure tube rnner 2.430

Pressure tube rozere; 2.530
fuel pitch 12.563

Note that the fuel pitch in the single pin model is 12.563 cm, which is an adjusted

equivalent square pitch to the triangular pitch of 13.5 cm in the full core model. The single pin

model considers an infinite square lattice, whereas the full core model has fuel elements in a

triangular array.

PENTRAN is a 3 -D Cartesian discrete ordinates code, and therefore mesh refinement must

be implemented to represent the geometry specification of a desired design. In some cases, the

design can be modeled to exact geometry, in cases with curved shapes, some approximations

must be used. For the single lattice cell study, two geometries for the coarse mesh cell

containing cylindrical fuel elements were considered. These two geometries will be referred to

as a 35x34 coarse mesh cell and a 21xl9 coarse mesh cell; more specifically, these numbers refer

to the fine mesh structure of the coarse mesh containing the fuel element.










The 35x34 coarse mesh cell was intended to represent the cylindrical geometry of the fuel

element to near exact design geometry specifications. The 35x34 coarse mesh cell is shown

below in Figure 3-4.






















Figure 3 -4. 35x34 coarse mesh cell for deterministic model.


The 35x34 (x-y) coarse mesh scheme in Figure 3 -4 yields the approximations for masses in the

model as shown in Table 3-4.

Table 3-4. Excess mass balance and percent error for a 35x34 mesh full pin


Model Excess
Mass(g)

139.00

-98.20

-28.40

197.00

-19.90


Material

U-Mo

Al-inner

D20-inner

Al -outer

D20-outer


Model Mass (g)

47,000.00

1,597.60

1,642.40

1,249.00

65,200.00


Target Mass (g)

47,139.00

1,499.40

1,614.00

1,446.20

65,180.00


Percent Error

0.29%

6.15%

1.73%

15.79%

0.03%










This model worked well for representation of a unit cell calculation; however, due to the

anticipated possible memory limitations encountered in developing the 1/8 Symmetry core

model of the Candidate Design deterministically, it became necessary to perform a single lattice

cell parameter study in an effort to limit the number of fine mesh cells defining the geometry

within coarse mesh cells containing fuel pins and in outer coarse meshes in a triangular pitch

assembly arrangement for the CD. After a brief study, a practical Cartesian meshing scheme was

chosen. This scheme involved a blend of minimal meshing to achieve a balance of fissile mass,

and minimize memory requirements in PENTRAN; this is depicted in Figure 3 -5.





U-natural

SAl-inner

SD20-Inner

SAl-outer

M D20-outer






Figure 3 -5. 21xl9 coarse mesh cell for deterministic model.


The excess mass for the 21xl9 mesh scheme is shown in Table 3 -5.

Table 3-5. Excess mass balance and percent error for a 21xl9 mesh full pin
Model Excess Model Target Mass Percent
Material Mass(g) Mass (g) (g) Error
U-natural -35.6 46820.0 46784.0 0.08%
Al-inner -136.0 1559.4 1423.0 8.75%
D20-inner 91.2 1762.0O 1853.2 5.18%
Al-outer -12.3 1039.6 1027.3 1.18%









From inspection of Table 3 -4, using Cartesian "voxels" as building blocks for meshed

representation of the fuel assembly, there will inherently be a material approximation in

representing curved boundary surfaces, so that approximately 9% of the aluminum cladding is

underrepresented in the 21xl9 mesh model, and the D20 within the flow channel is

overrepresented by ~5%. However, for the purposes of the study, this is acceptable, since it is

most important to properly represent the targeted mass of fissile material present as closely as

possible, along with reasonably close conservation of the moderator mass.

Group Albedo Treatment in the 3-D Deterministic Lattice Cell

Due to the large size of the 1/8 core Candidate Design (CD) model, full representation of

the almost meter of water above and below the fuel elements (79cm thick), the core was not

tractable for the deterministic model to fit on the 8-node, 16 processor Einstein cluster.

Therefore, the model size may be reduced by implementing group dependent albedo factors,

computed using a 3 -D lattice cell calculation.

To reduce the amount of heavy water atop the pins in the CD 1/8 core model, group

albedos were calculated by using the multigroup solutions from a single pin lattice cell

calculation and mapping albedo factors for each of 15 fine z-levels within the 79 cm of heavy

water directly above the fuel pin location (every 5.267cm). To determine an appropriate location

in the 79cm of heavy water reflector to truncate or chop" the D20 and implement a group-

dependent albedo factor boundary condition, a calculation was made using flux-weighted cross

sections to truncate the model at a distance of 6 mean free paths above the fuel. The highly

resolved 35x34 mesh scheme pin cell model, described in the previous section, was used to

obtain group albedo factors for the full D20 reflected model, and with additional investigation

using both a 35x34 and a 21xl9 mesh albedo based model.

The formulations used for determining an average mean free path follows:














g=1 I=1

With N,, = o,Mx Where hx=thickness of the medium (3 -2)


F = exp(-N,f ) (3-3)


Where ~, is the it fine mesh group g flux above the fuel element, my, represents the total

macroscopic cross section in group g, and y, is the fine mesh volume. The fraction F of

uncollided neutrons at Nmean free paths is given by Equation 3 -3. A 6-mean free path (6 mfp)

length of reflector is often used as a thickness to apply a group albedo boundary. This was

applied here; applicable data are presented in Table 3 -6.

Table 3-6. Albedo boundary distance parameters

Parameter Value units

"t 0.44464 cml

3*mfp 6.7470 cm
6*mfp 13.494 cm

Here we should note that value of 6 mean free paths for the heavy water using the 3 -D unit

cell data yielded a reflector distance which also happens to be equivalent to the pitch of the fuel

pins, 13.5 cm implying the fuel pins are decoupledd." Further, flux -weighted group albedo

factors were found using the following equation:






1=1 (3-4)










The group dependent albedo factors determined from Equation 3 -4 are easily recognized as

the fraction of returned current computed for the mesh cells in the heavy water above the pin in

mesh i, and of energy group g, where the desired heavy water cut is to be made. Due to the z-

fine mesh partitions of the problem throughout the heavy water above the fuel, a value of 1 0.53

cm, or 4.68 mean free paths above the fuel element. Albedo factors for the 8 group and 6 group

model are provided in Table 3 -7 and 3 -8 respectively.

Table 3-7. 8 group albedo factors
Group Upper Group Bounds(MeV) ag
1 20.0 0.0823

2 1.36 0.2714
3 4.4E-1 0.4089
4 5.2E-2 0.4536
5 3.9E-3 0.5063
6 6.83E-4 0.5501
7 1.22E-4 0.6239
8 3.0E-6 0.9956


Table 3-8. 6 group albedo factors
Group Upper Group Bounds (MeV) ag
1 20 0.0824

2 1.36 0.2715

3 4.40E-01 0.4811

4 3.90E-03 0.5451

5 1.22E-04 0.6189

6 3.00E-06 0.9949


Single Pin Monte Carlo and Deterministic Eigenvalue Comparisons

The pin that was modeled in Monte Carlo Neutral Particle version 5 (MCNP5) is shown in

Figure 3-6. It is easy to see the advantages of MCNP5 when compared to the discretized

geometry required for Parallel Environment Transport (PENTRAN).












Al 6061
-- Clad








a i U-Mot Fuel I







D20



Figure 3-6. Single pin Candidate Design material description used in the Monte Carlo model.


No material approximations or albedo calculations were needed for the MCNP5 model,

and geometries were modeled exactly as shown in Table 3-2. The results and comparisons for

all single pin eigenvalue calculations compared to MCNP5's continuous energy calculation are

shown in Table 3 -9 and 3-10.

Table 3-9. Monte Carlo and Deterministic single pin eigenvalue results for 8 groups
Percent difference from
Model Final kgf Standard Deviation
MCNP

MCNP continuous energy 1.091 4.20E-04 N/A
PENTRAN 35x34 Full
1.086 4.85E-05 0.46%
D20 Reflector
PENTRAN 35x34 Albedo
1.084 1.23E-04 0.64%
Treated Reflector
PENTRAN 21xl9 Albedo
1.092 1.94E-04 0.09%
Treated Reflector










Table 3-10. Monte Carlo and deterministic single pin eigenvalue results for 6 groups
Percent difference from
Model Final kerr Standard Deviation MCNP



MCNP continuous
energy 1.091 4.20E-04 N/A

PENTRAN 35x34 Full
D20 Reflector 1.086 4.95E-05 0.46%

PENTRAN 21xl9
Albedo Treated
Refl ector 1.090 1.75E-05 0.09%

The 6 group 21xl9 fuel pin provided the most desirable results with the least amount of

memory requirements. Therefore, the 21xl9 fuel pin was selected as the pin design of the 1/8th

core Sn model.

Monte Carlo and Deterministic Flux Comparison

In order to collect volumetric fluxes, a mesh tally in Monte Carlo Neutral Particle version

5 (MCNP5) was incorporated Figure 3 -7 shows a 21xl9 mesh overlaying the pin region, which

was selected to provide adequate aliasing with the single pin Deterministic model. Radial flux

profiles were gathered with only one z-mesh spanning the active fuel length.


I II


21xl9 mesh tally scheme for Monte Carlo single pin model.


Figure 3-7.












Figures 3 -8 through 3-13 compare radial flux profiles for both MCNP5 and Parallel


Environment Transport (PENTRAN) for a single pin unit cell, allowing direct spectral


comparison of group fluxes. The figures display relative flux through the centerline of the single


pin model for both MCNP5 and PENTRAN, where tallies are aliased to the 6 group energy bins.


The relative percent of total neutrons in each group (determined from integration over neutron


density) are provided along with maximum and minimum flux values in both Monte Carlo and


deterministic models, and maximum relative error between the two models.





Group 1 Flux 1.36 MeV to 20 MeV

~Max % Oifference between MCNP/PENTRAN = 1.131%
II IIII I MCNP Results
0 Max Flux


S1 37E-02
Mmn RE
0 66%
a, It Lk I I ~ \UI II Al Clad 0 00217% of total
o I i I I l ull II I neutrons


D20 --M N

-= PENTRAN

Al Clad III1II ID20
Nat U
X I l~l I I PENTRAN Results
0 / Max

of total
Mneurn

a000




0 1 2 3 4 5 6 7 8 9 10 11 12 13
x (cm)



Figure 3-8. Group 1 flux comparison (6 group model).























N ~Max % Difference between MCNP/PENTRAN = 1 208%



o
0








o










0 13579 1 1 1 1
x cm





Fiue39 ru lxcoprsn( ru oe)




Gru lx-3. e o4 e


Max % Difference between MCNP/PENTRAN = 1 482%


Group 2 Flux 440 keV to 1.36 MeV


IVCNP Results
Ma Flux
0 001065
Mmn Flux
0000207
MaxRE
1 34%
Mmn RE
0 69%

0 00700% of total
neutrons





-*-CNP

- -PENTRANj




PENTRAN Results







of total
neutrons


~
8
o






m
6
o
o






co
f8
iii,
g





8
8
o







8
8


IVCNP Results
Max Flux
0 001755







ein 6ns f oa







-* MCNP
-- PENTRAN



PENTRAIN
Results
Max
0 001736
Mmn
0001220

0 271%
of total neutrons


00 1 2 3 4 5 6 7 8 9 10 11 12 13

x (cm)






Figure 3-10. Group 3 flux comparison (6 group model).











































ii;


Group 4 Flux 122 eV to 3.9 keV

Max % Difference between MCNP/PENTRAN = 1 375%


0 12 3 4


O
o
o
o
8
o

8
8
o

8
8
o
x
EB
f8
lo

B
8
o

6
8
o

6
8
o

8
8
o


IVINP Results
Ma Flux
0 000943
Mmn Flux
0 00 772



0 467% of total
neutrons



-*MCNP
-=-PNTRAN

PENTRAIN
Results
0 000953
Mmn
0 000804
3 615%
of total neutrons


5 6 7 8 9 10 11 12 13
x (cm)


Figure 3-11. Group 4 flux comparison (6 group model).






Group 5 Flux 3 eV to 122 eV


IVCNP Results
Max Flux
0 000928
Mmn Flux
0 000632
Max RE
0 80%
MmnRE
0 53%
2 558% of total
neutrons



-* MCNP
-* PENTRAN

PENTRAIN
Results
Max
0 000928
Mmn
0 000633
13 382%
of total neutrons


012


3456
x (cm)


8 9 10 11 12 13


Figure 3-12. Group 5 flux comparison (6 group model).













Group 6 Flux 0 eV to 3 eV

o Max % Difference between MCNP/PENTRAN = 4 671%
II IIII I I MCNP Results
Max Flux
0 0055


0 32%
MmnRE
0 23%

~II I i/I Iof ~totalneurons



-* MCNP
---PENTAN
iij PENTRAN


0 004714
0 002370
8 1 I I I I II 1 82 645%
o ~of total neutrons



0013457810 11 1213
x (cm)



Figure 3-13. Group 6 flux comparison (6 group model).



Differences between the MCNP and PENTRAN models for the single pin can be attributed


to the multigroup energy group treatment and structure between the Monte Carlo and


Deterministic computations, remembering that MCNP uses continuous energy bins, whereas


PENTRAN uses discrete energy group bins mixed using the SCALES code with the SUPIN


procedure developed for this proj ect. Multigroup cross sections represent averaged behavior in


regions with strong resonances, particularly at high energies in U-23 5, and especially U-23 8.'


In addition, the mean nzaxinsun difference between the MCNP5 continuous energy pin cell


fluxes and the 6-group model was 1.71%. However, based on the assessment of total neutrons in


the system (computed using group neutron densities (equal to group flux/ velocity) integrated


over the system volume for each energy group), 82.6% of all neutrons appear in group 6 (most


thermal group) according to PENTRAN results, and 96.9% of the neutrons according to MCNP5,









with a maximum flux difference in group 6 of 4.67% between deterministic and Monte Carlo

values, outside of the standard error of the MCNP results.

Discussion

The single uranium Pin (suPin) process extracts Doppler broadened, self shielded cross

sections after a user specified collapse from the Scale version 5.1 (Scale5.1) multigroup library.

Results for eigenvalue calculations in both 3-D Sn transport with Parallel Environment Transport

(PENTRAN) and Monte Carlo Neutral Particle version 5 (MCNP5) were consistent, and

therefore are applicable for development of a 6 group full core model.









CHAPTER 4
REACTOR MODELING

This chapter describes the problem and evaluation of the Candidate Design (CD), with

information on results, and limitations of the analysis. The structures of the CD are depicted in

the plant overview shown in Figure 4-1


Figure 4-1. Plant Overview of original Candidate Design.9


Candidate Design

Candidate Design Parameters

The Candidate Design (CD) is based on a high flux pressurized heavy water reactor

(PHWR), located inside a 6061-T6 aluminum cylindrical reactor tank vessel (boron, lithium, and

cadmium free aluminum) with walls 1.3 cm thick, and a bottom that is 3.5 cm thick. The

"reactor tank" design is designed for a power level of between 10 and 35 MWt with certain










limitations; it is heavy water moderated and cooled. Secondary heat exchangers match heat

transfer from the heavy water cooling circuit to a light water circuit. The initial design of the

reactor was modified for fueling with naturally enriched uranium. Average fuel burnup for the

core was specified to be 2400 MWD/MT, peak is 4000 MWD/MT for the 15 MWt design (note a

peaking of 1.67 is attributed to these values); burnups for the natural uranium core are strongly

dependent upon power density and irradiation history. A North-South slice and East-West slice

of the CD reactor component essentials are provided in Figure 4-2, with an expanded view in

Figure 4-3.














A -. B .

Figre4-2 Cndiat Deig ractr oreviws. A NrthSuhCniae eincr iw
B) EatWs Caddt Deig cor view.


Altoug Fgur 41, igre -2 an Fgur 43 ae ot ssntil or hecomue
moelngofth C, t sneesar t ginanaprcitin orth sz o te eato ad h

potntalmeor rqureens ha my e ncunerd.Aloitisimoran t ntetht an









of the structures labeled in Figures 4-1 and Figure 4-2 were not modeled in deterministic Sn

Parallel Environment Transport (PENTRAN), or stochastic Monte Carlo Neutral Particle version

5 (MCNP5). Boundary approximations, including vacuum boundaries and albedo boundary

conditions at the edge of the actual reactor core give satisfactory results.


Figure 4-3. Enlarged East-West Core cutaway.9


Candidate Design Reactor Core Design

The Candidate Design (CD) core holds a total of solid fuel slugs in fuel assemblies

immersed in heavy water. Remaining positions house up to six B4C COntrol rods. Each fuel

assembly is placed in the core using a 13.5 cm hexagonal pitch, and is composed of four









"stacked" naturally enriched fuel rods clad in aluminum. Thermal hydraulic analysis and heat

transfer parameter limits restrict the power increase, so that a ramped overpower operation in this

design is bounded to < 35 MW. The CD reactor and fuel parameter are summarized by the

follow ng;

* 26 MWth Design Power

* 166 fuel assembly positions

* 5 fuel rods per assembly

* Solid natural uranium

* Up to 6 shim rods

* Central channel

* D20 around, above, and below core

* Aluminum reactor wall

* Aluminum support plate

* Graphite around outer core

The core contains the fuel inside individual pressure tubes spanning over a radius of ~90

cm in a 13.5 cm hexagonal lattice, which is then surrounded by 40 cm of heavy water as a

primary reflector, made up from the outer edge of fuel channel sub-assemblies up to the physical

tank wall. Immediately outside the tank (separated by a 0.8 cm air gap for air cooling) sits a

large mass of graphite fashioned as a 14-sided prism at an approximate inner radius of 130O cm

and a thickness extending out at least 62.8 cm from the inner radius. At the edge of the graphite

is another air gap of 2 cm, followed by a 100 MT cast steel thermal shield 19 cm thick. The

graphite is stacked on top of a bismuth support plate (to avoid n-gamma reactions since natural

bismuth-209 has only a 20 mb neutron capture cross section) to support the graphite. The

graphite is purified nuclear grade graphite (< 3 ppm natural boron) composed of20 cm x 20 cm









interlocking stringer blocks configured to be very resistant to shifting, even when heated. The

air gaps support filtered atmospheric air flow used for a secondary cooling mechanism. The

reactor tank is under pressure with helium cover gas, and based on the data available for the CD,

tank seals can withstand ~2 psi of heavy water overpressure during an excursion. Beyond the

graphite is a 200 cm thick concrete biological shield. Figure 4-4 shows a horizontal slice of the

core and the surrounding beam ports.






































Figure 4-4. Horizontal slice of core and surrounding beam ports.9









Reactor Physics Analysis Methods

Full Core Candidate Design Monte Carlo Model

A full core model for the Candidate Design (CD) was generated with complete detail for

the reactor system components using Monte Carlo Neutral Particle version 5 (MCNP5).

Figure4-5 and Figure 4-6 rendered using the MCNP5 viewer, and provide the details of the

schematic for the full core Monte Carlo model. In Figure 4-5, individual fuel pins with five fuel

slugs are labeled, in addition to the aluminum reactor vessel and graphite reflector surrounding

the core. The geometry of the pins is the same as in Table 4-4, a fuel pitch of 1 3.5 cm (triangular

lattice of 166 pins).


Figure 4-5. X-Z (y=0 slice) plane view of full core Candidate Design Monte Carlo Model





























J ~~ +++ & 0a6 *A&
eassemenessel
*
*+ + 1 499* *
meemseema weee
*
+ ** ** **

















Figure 4-6. X-Y(z=0 slice) plane view of full core Candidate Design Monte Carlo Model.






With a full core model in MCNP5, calculations to determine pin power, and peaking
factors were performed. To assist in a core map reference, a pin numbering index was
implemented. This numbering index is depicted in a simplified diagram of the full core model,
with pins sequentially labeled 1 thru 166 for the CD as shown in Figure 4-7.


0@@@@@@@
@@@@@@@@@
@@@@@@@@@@
@@@@@@@@@@@@
@@@@@@@@@@@@
@@@@@@@@@@@@>
@@@ @ @ @@@@i
@@@@@@@@@@@@@








Figure 4-7. Monte Carlo model full core pin numbering scheme.
The full core model was simulated on the Einstein cluster using a "k-code" simulation
sequence using MCNP5 to determine the overall criticality of the system. Table 4-1 provides a
summary of the computed keff summary.








Table 4-1. Monte Carlo Candidate Design k-code full core results
Parameter Value Units


kef
Eag Ofa n COMSing fission


1.04231 +/- 0.00048
218. 3
2.464


keV
neutrons/fission


prompt removal lifetime 8.1916E-04 +/- 1.89E-06 Seconds
CPU Time, Einstein, 16 Procs 5.09 Hours (+/- 5% pin power)

Radial Power Profile

A radial pin power profile (relative to a total reactor power of 26 MWr) is depicted in

Figure 4-8 and Table 4-2 with power densities given in W/cc. This was established using F7

tallies to yield total fission power in each pin. As anticipated, the reactor power is the highest

near the core center, in the immediate region which surrounds the largest experimental cavity;

specifically, this is at pin number 76, with 0.901% of the total power generated, or 94.53 W/cc.
The core average radial pin power was 63.6 W/cc.


Pows m ce



@B S @@***@ @@@ ~

888 88 888
9999****989**~



888 8 O 88 **6) G
999898990~~
***888*


Figure 4-8. Radial (averaged over axial length) total power pin power densities based on relative
errors of less than 4.2% for each value.










Table 4-2. Power density per pin (maximum error < 4.2%)
pin number (W/cc) pin number (W/cc) pin number (W/cc)


pin number
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
Average


(W/cc)
73.892
70.729
66.285
60.786
52.727
50.391
55.137
54.497
59.581
64.665
70.428
72.876
67.113
64.100
57.397
53.141
50.730
51.446
50.354
57.171
63.121
65.833
59.430
55.476
49.902
48.998
47.529
46.362
52.312
57.962
58.677
54.610
49.827
45.985
46.588
45.006
45.947
48.697
53.141
52.990
51.446
45.232
42.897
63.195


51.484
53.555
54.798
54.572
55.401
48.734
47.379
54.723
54.271
57.585
55.739
55.288
51.823
50.015
50.843
57.660
57.020
60.033
61.539
63.347
61.125
56.832
53.555
53.291
60.711
59.656
63.799
68.356
71.143
68.469
66.473
63.686
55.589
54.723
60.033
59.807
67.151
72.424
74.985
77.583
72.085


70.503
63.724
57.133
55.288
58.828
59.129
66.774
71.633
78.299
79.353
76.830
77.546
71.030
64.402
54.534
53.932
60.598
58.865
65.155
72.725
78.901
80.333
81.011
80.634
77.018
70.277
62.519
54.120
54.836
58.338
63.347
73.892
82.027
87.338
94.531
90.426
85.379
80.672
69.109
58.752
53.103


83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123


61.878
59.167
72.461
89.823
93.853
75.889
64.552
54.647
52.049
59.694
61.991
76.114
80.408
86.697
93.439
87.865
86.057
78.600
67.226
58.564
48.659
47.680
57.660
57.510
63.422
72.650
75.625
80.220
78.751
78.148
74.834
67.528
59.543
52.049
47.567
54.685
55.626
62.632
68.168
70.767
74.118









Axial Power Profile

Only five axial zones were modeled to obtain power density. Using five zones was

convenient since five slugs compose a single pin. Since each fuel slug (~9.450 kg or ~500 cc) is

the same volume and defined as a cell in the Monte Carlo Neutral Particle version 5 (MCNP5)

model, fission power tallies are conveniently obtained. The results of the axial power densities

are in Table 4-3.

Table 4-3. Computed power density per axial zone (errors < 4.2%)
Axial Zone Power Hot Channel Power
Axial Zone (W/cc) (W/cc)
1, (1.05 cm to 51.05 cm) 47.03 69.11
2, (53.15 cm to 103.15 cm) 72.69 112.23
3, (105.25 cm to 155.25 cm) 80.76 122.21
4, (157.35 cm to 207.35 cm) 70.79 103.95
5, (209.45 cm to 259.45 cm) 44.70 65.16
Average 63.20 94.53

Power Peaking Summary

Calculated power peaking factor (PPF) values representing peak to average assembly

power densities are given in Table 4-4. Using Table 4-3, the radial PPF was calculated by

locating the maximum power density (pin 76) of the 166 pins and dividing by the average power

density. The axial PPF in the hottest pin was calculated by segmenting pin 76 into five (5)

individual zones and calculating a local PPF for that pin alone. The PPF in the hottest

pin/channel is calculated by multiplying the radial PPF with the axial PPF in the hottest pin.

Table 4-4. Power Peaking Factor Summary

Power Peaking Factor Summary

radial PPF 1.496

axial PPF in the hottest pin 1.293

PPF in the hottest pin/channel 1.934

Therefore, 1.934 was the calculated power peaking factor using the Monte Carlo full core model.










Deterministic Analysis of Candidate Design

1/8th COre Candidate Design Model

The Candidate Design (CD)was generated using Latte triangular lattice generator. Latte

was developed in an effort to ease construction of the model. Previously, the user would have to

construct each individual pin. Latte was used to create the whole core assembly from the 21xl9

single pin coarse mesh described in the previous chapter. Memory requirements for the Einstein

8-node, 16 processor cluster dictated that at most 1/8th of the core could be modeled, resulting in

a modified core design without the empty channels of the original core design. A 1/8th COTO

symmetry was obtained by modeling half the core in the z-direction and '/ core in the x-y plane.

The 1/8th CD is shown in Figure 4-9.



100











0 20 40 60 80 100


Figure 4-9. 1/8th COre Candidate Design.


It should be noted that the graphite reflector was omitted due to the approximation that the

D20 already serves as an infinite reflector under the specified operating conditions. The 1/8th

Core Parallel Environment Transport (PENTRAN) model used a block-adaptive mesh grid,

where high resolution meshing is applied in zones where needed (in fuel pin zones). Meshing

can be relaxed in the peripheral heavy water region. The final 1/8 core model used contained





361,584 mesh cells, and as many as 260 million multigroup simultaneous equations in a

power/source iteration for determination of the criticality eigenvalue and pin power distribution.

Heavy water moderated systems are some of the most difficult systems to converge, since

neutrons scatter quickly, and remain in thermal energies (as evidenced by the thermal group

albedos of 0.995 determined previously).

Deterministic Criticality Calculation, 1/8th COre Candidate Design

The Parallel Environment Transport (PENTRAN) deterministic criticality calculation

presented several challenges due to limited cluster memory. Three calculations were performed

using 21xl9 x-y meshes for each fuel pin, studying convergence effects of changing the Sn

ordinate parameters and number of energy groups. kgf convergence for 8 group models using S4

and S8 angular quadrature, and a 6 group model using S10 angular quadrature were studied; all

cases were run with P1 scattering anisotropy. All calculations were initially performed using the

Directional Theta Weighted (DTW) differencing scheme locked in all 1710 coarse meshes. The

behavior of the criticality eigenvalue kgf and the convergence comparisons are provided in

Figure 4-10 and Table 4-5, respectively.


Figure 4-10. Convergence of the 1/8th COre Candidate Design for deterministic models.









Table 4-5. Final values for 6-group convergence comparison
CPU
Standard Final k- Procs Procs Procs Time
Model Final k Deviation Iteration tol erance Space Group Angle MB/Proc (hrs)

S4 8
groups 1.00237 8. 10E-0 5 84 1.00E-04 4 3 1 2714.3 54.7

S8 8
groups 0.99741 5.85E-05 75 1.00E-04 10 1 1 3994.8 87.3

S10 6
groups 0.99807 5.15E-05 75 1.00E-04 5 3 1 3270.3 99.2

Power Peaking in the 1/8th COre Candidate Design

Parallel Environment Transport (PENTRAN) is capable of delivering fission source

information directly to the user via PENTRAN Data extractor (PENDATA), and can be used to

yield volumetric fission neutron source values in the fuel, sorted by energy group. The

volumetric fission source data corresponds to the fission neutron source in the fuel in user-

specified fine meshes within each coarse mesh volume. Therefore, group sorted fission sources

(# of neutrons/cc) in fuel values were integrated over energy group and fuel zone volume, and

divided by total zone fuel volume and the average number of neutrons per fission to obtain the

mean fission reaction density per zone. The source density per zone was then scaled to yi eld a

computed core thermal power of 26 MWt in accordance with Candidate Design (CD)

specifications.


Powr~nsi~ =l I= 190M~eV 1.602E -13J 1 W 41
zone "ia fiss M~eV J/s



Radial pin power peaking was calculated by summing over zones of fuel pin power

volumes axially, and determining peak to average ratios between radial pin power densities. The

results of the pin mean power densities are shown in Figure 4-1 1.










P ow


density (W~lcc)
97T

82
66
51
46

41

26





~888





88888


9888


**8~888

Figure 4-11. Relative pin power for pins contained in 1/8th Candidate Design (W/cc). Monte
Carlo values are converged to a standard deviation of less than 4.2% (1-sigma)










Axial power peaking was calculated in a similar manner, using 6 z-axial zones, where the

pin power was summed over radial volumes in each axial zone. The hottest half pin was split

over 6 z levels. Axial zone boundaries and associated zone averaged power densities are in

Table 4-6.

Table 4-6. Axial zone boundaries, zone and hottest channel power densities computed using the
deterministic model
Axial Zone Hot Channel
No. Boundaries in cm
Power (W/cc) Power (W/cc)
1 0-20 69.191 113.220
2 20-40 68.099 111.324
3 40-60 65.887 107.359
4 60-80 62.626 101.718
5 80-100 58.218 94.298
6 100-125 53.196 86.244
Average 62.484 101.716

Power peaking values for the 1/8th COre deterministic computation are shown in Table 4-7.

Average power density for each zone is shown in Table 4-8.

Table 4-7. Hot channel power peaking factors compared to Monte Carlo


Percent difference from
Type Peak to Average Power Ratio MCNP5

PENTRAN Radial 1.628 8.82%

MCNP Radial 1.496 N/A

PENTRAN Axial 1.113 13.92%

MCNP Axial 1.293 N/A



PENTRAN Power Peaking
Factor 1.934 6.73%

MCNP Power Peaking Factor 1.812 N/A









Table 4-8. Average power density by zone compared to Monte Carlo

Average Power Density Percent difference from
Zone (W/cc) MCNP5

MCNP5 Radial 63.6 0.01%

PENTRAN Radial Full Core 63.605 N/A


MCNP5 Axial (Hottest Pin) 94.53 7.65%
PENTRAN Axial (Hottest
Pin) 102.361 N/A

Full Core Power Density Comparisons

The pin powers for reactor power scaled to 26 MWt were computed as noted in previous

sections, and are reproduced here for direct comparison. A side-by-side pin power comparison is

provided in Figure 4-12, and the ratio of power density per pin is given in Figure 4-13.





< : MCNP


~~~~~~


Figure 4-12. Monte Carlo and Deterministic 6 group computed power densities (W/cc). Monte
Carlo values are converged to a standard deviation of less than 4.2% (1-sigma).

















O 9 809 989

0.8 O .0 Q @
Fiue41.Rto fMneCropwe est oDtemnsi oe eniy ot al





diuevito (-1-sigma) of 4.2%e orless Power density rtios divergiitioefro unity ionthe pinso


primarily along the outer periphery, attributed largely to the use of inadequate mesh densities in
outer core regions of the Parallel Environment Transport (PENTRAN) model. Higher mesh










densities are needed there to resolve the neutron leakage streaming off of the core, along with an

investigation of adaptive differencing scheme behavior. The inadequate local mesh resolution at

the edge of the core causes the leakage to be overestimated there, so that this directly explains

why a lower system criticality eigenvalue (kegf= 1.00) resulted from the deterministic models

(compared to the kegf=1.04 in the Monte Carlo model). In addition, since the PENTRAN model

was a 1/8 core symmetry model, some differences near the center of the core can be attributed to

the empty channels not modeled in PENTRAN model which also affect k. Given these facts, the

models are comparable, although the differences could be narrowed further by refinement of the

deterministic model.

The following plots show that there was a severe flux drop at the outer edges of the core.

The flux drop may not have been accurately modeled in the PENTRAN calculation due to

limited memory capabilities. Further study was provided after expanding models on a new

cluster named Bohr.



MCNP 6 Group
Group 60Group 54rloup 4-M**Group 3-M**roup 2- -Group 1








0 1 2 3 0 0 0 70 80 9

0.0y cm)


Fiue -4.Mnt aro6 rupfuxs










PENTRAIN 6 Group
Group 6-E-roup5-Mroup 4--Group3iCroup 2- -Group 1

0.00025








0.00


0 20 40 60 80 100 120
y (cm)


Figure 4-15. Deterministic 6 group fluxes.


Further Refinement of the Deterministic Model

The Bohr cluster supplied more speed of calculation and memory for the Parallel

Environment Transport (PENTRAN) simulation. To a limit, the meshing surrounding fuel

elements on the perimeter of the model could be refined and the results are shown in Figure 4 -16.









G @QQ0990






O OO se


Figure 4-16. Monte Carlo to Deterministic power ratios for S10 refined mesh (left) and S10
coarser mesh (right). Monte Carlo values are converged to a standard deviation of
less than 4.2% (1-sigma).









Mesh densities in Figure 4-16 were increased by a factor of 2 coarse meshes surrounding

periphery fuel pins.

Further study was employed in an effort to determine more accurate differencing schemes

for the PENTRAN model. The Exponential Directional Iterative (EDI) differencing scheme has

shown promising results in test problems and was implemented on the perimeter pins as

indicated by Figure 4-17.10 Figure 4-18 presents the ratios, Monte Carlo to deterministic power

densities, for the calculation.


Figure 4-17. PENTRAN differencing scheme map, blue = Directional Theta Weighted, red
Exponential Directional Iterative.














































Figure 4-18. Monte Carlo to Deterministic power ratios for S10 refined mesh with updated
differencing. Monte Carlo values are converged to a standard deviation of less than
4.2% (1-sigma).










The power ratios are closer to unity in the previous problem areas located at the top of the

figure. This demonstrates the need for a differencing scheme like EDI to account for streaming

and edge effects in core physics calculations. Also, agreement for the pin on the upper left

corner of Figure 4-18 still remains an issue. This could be due to higher stochastic error in the

fission source inherent in the pins.

Discussion

Tandem methods for the analysis of the reactor model provided two independent solutions

that, for the most part, were in good agreement. Monte Carlo Neutral Particle version 5

(MCNP5) yielded a converged solution with no approximations for geometric parameters, while

the Parallel Environment Transport (PENTRAN) calculation had to be carefully constructed so

that a converged solution could be obtained. MCNP5 and PENTRAN were in reasonable

agreement for power density calculations on all but the periphery fuel pins. For peripheral fuel,

one must note whether or not the Monte Carlo calculations truly converged, or if the Sn solutions

are accurate for the discretization used. Further investigative techniques were employed in

PENTRAN in order to access problem regions, which were prominent in the periphery of the

model, with and improved with discretization and higher order differencing algorithm. The

analysis identifies areas of investigation for both deterministic and Monte Carlo solution

approaches.









CHAPTER 5
FIXED SOURCE SIMULATION

In the course of handling packaged radionuclides, it is often necessary to assay closed

containers of radioactive materials for inventory and accountability, and to validate gamma ray

dose rates and source spectra. Often, materials in sealed containers can be well characterized

using gamma ray spectrometry and dosimetry when the container is readily accessible. With

regard to emphasis on material accountability and minimizing personnel doses, it is useful to

understand the magnitude and the spectrum of a containerized source without requiring the

container to be opened. However, some packaged or containerized sources are often stored in

locations not completely accessible for repeated or convenient survey via spectrometry, such as

in spent fuel pools, fuel handling areas, etc, so that inventory, accountability, and control must be

more generally verified through routine dose rate measurement audits. These measurements may

be carried out in spent fuel pools at water depths of one to two atmospheres of pressure, and only

a small probe ion chamber measurement for dose assessment is possible, given operational,

environmental, or geometry constraints. Therefore, a study was commissioned at the FINDS

Institute at the University of Florida to employ 3 -D gamma radiation transport calculations to

explore the extent of potential variations in measured dose rate from hypothetical containerized

sources in a simulated spent fuel pool using standard stainless steel storage containers (S SC).

These 3 -D transport calculations were compared to and validated by Monte Carlo calculations

under the use of MCNP5 code.

3-D Gamma Transport Methods

Performing the gamma ray transport computations necessary for this study involved

obtaining either a deterministic solution (achieved in our studies via the Sn method), or

statistically (achieved using the Monte Carlo method). Both solution methods were applied to









provide consistent results in characterizing dose rates in this application. The steady-state multi-

group form of the transport equation operating on the forward group angular flux y, (F) using

standard notation with total group cross section o,(F'), differential group scattering cross section


es,, ,, (r, 6'-O2), and source terms q (F', ^) g, is:11



g'=1 4a

Forward and Adjoint Operators

Principally, scattering from all other energy groups g' into group g is dominated by

downscattering from higher energies to lower energies. 1 The adj oint transport operator H' can

be derived using the adj oint identity for real valued functions and the forward multi -group

transport operator, where the angled brackets represent integration over all independent

variables, in this case phase space: "




Using Equation 4-1, it can be seen that the forward operator is



g'=1 4K (5-3)

The angular adj oint (importance) function is, lyg. Applying the adj oint boundary condition

that particles leaving a bounded system have an importance of zero in all groups (converse of the

forward vacuum boundary condition) with the above equations, and requiring a continuous

importance function mathematically leads to the multi -group adj oint transport operator:"

Hi 8+~(;C dRXb.(;A~~G
g'=1 4n (5-4)









Note the minus sign on the streaming term indicates that adjoint particles travel along a

reverse direction (relative to the forward transport formulation), where scattering progresses

from group g back to other groups g (those groups formerly contributing to group g in the

forward equation).

Forward and Adjoint Detector Response

To render a traditional detector response R (e.g., absorptions per unit time) attributed to a

specific reaction using an adj oint transport solution, the desired R must be linked to the adj oint

"importance" by aliasing the adj oint source to the cross section causing a response R in the

detector. To illustrate this, consider a fixed forward source-detector problem, where the

radiation flux between the source and the detector must satisfy the transport equation: 1


Hy, = q#(5-5)

Where q g is the gamma source density, typically with units of y/cm3/S. Alternatively, the

adjoint transport equation should also be satisfied using an adj oint source aliased to the group

detector response cross section o-d, typically with units of l/cm:


H' a~s (5-6)

Applying Equations 5 -2, 5 -5, and 5-6, and integrating over all variables results in the very useful

expression for detector response R:


R = yr o-d g g g (5-7)

Therefore, a traditional detector response can be obtained from the adj oint transport solution by

integration of the product of the source distribution with the adjoint function computed over the

problem phase space, for any arbitrary source distribution. Thus, detector response R caused by

several sources may be computed in two ways:











* Directly from the left side of Equation 5 -7 from several individual forward transport
computations for each radiation source, or

* From the coupling of a single adj oint transport computation with the set of different source
termsll


Responses Aliased to Dose Rate

Dose rate response (D) computations in our study were rendered using energy group


dependent flux-to-dose conversion factors (FDCFs) for gamma radiation (Chilton et. all, 1984).

The units of these factors are in rem-cm2 (cSv-cm2), So that when a (forward) gamma flux is

determined in a region in y/cm2/S, the product of the group gamma fluxes with each group FDCF


was computed, summed over all groups, and this yields the gamma dose rate in rem/s at that


point. FDCFs interpolated onto BUGLE-96 multigroup energy bins are presented in Table 5-1.


Table 5-1. Interpolated Flux to Dose Conversion Factors, 1
Group MeV (hi) MeV (avg)
1 1.40E+01 1.20E+01
2 1.00E+01 9.00E+00
3 8.00E+00 7.50E+00
4 7.00E+00 6.50E+00
5 6.00E+00 5.50E+00
6 5.00E+00 4.50E+00
7 4.00E+00 3.50E+00
8 3.00E+00 2.50E+00
9 2.00E+00 1.75E+00
10 1.50E+00 1.25E+00
11 1.00E+00 9.00E-01
12 8.00E-01 7.50E-01
13 7.00E-01 6.50E-01
14 6.00E-01 5.00E-01
15 4.00E-01 3.00E-01
16 2.00E-01 1.50E-01
17 1.00E-01 8.00E-02
18 6.00E-02 4.50E-02
19 3.00E-02 2.50E-02
20 2.00E-02 1.00E-02


cSv = 1 reml2
FDCF (cSv cm2)
3.10E-09
2.44E-09
2.13E-09
1.93E+09
1.72E-09
1.50E-09
1.28E-09
1.03E-09
8.11E-10
6.40E-10
5.09E-10
4.45E-10
4.00E-10
3.26E-10
2.07E-10
1.09E-10
6.80E-11
4.96E-11
3.90E-11
3.50E-11









The desired dose rate can also be rendered using an adj oint transport computation if the

adjoint source is properly aliased to yield D ." To accomplish this for the adj oint transport

calculation, the group adj oint source must be set equal to the group FDCFs with units of cSv-

cm .

Once the adj oint function is determined at the location of interest, the dose can be rendered

using Equation 5-7, where the forward source, being aliased to the FDCFs, must be a gamma ray


flux with units of '.Since placement of a small localized source is defined as a source rate
cm2S

(7 /s), this must be properly converted to an estimated flux in the problem for proper

computation of the dose rate D (assuming volumetric variable dimensions cancel):


D = 7 FDC~j=wvg = (5-8)

Therefore, to convert a localized source rate to a source flux for coupling to the adjoint

function (to then yield a dose rate), an equivalent radius (req) must be determined for

computation of an effective surface area (1/(4 nreq2)), which is used to divide into the source

rate to yield an effective source flux, 41- Since forward and adjoint calculations must each


independently render the same D, this equality was used in this work to determine the

equivalent radius (req) for each source rate. Therefore, use of the equivalent surface area in

coupling a localized source with the adj oint aliased to the FDCFs is necessary to yield the correct

D, within the practical limits of trncation error in the two deterministic transport computations.

Flux to dose conversions are assumed to be applicable in this case without the specific

need to consider specific electron transport effects in the transport computations, since the range

of ~1 to~2 MeV (the dominant source term) photoelectrons in water is <0.5 to <1cm; lower

energy electrons caused by interactions with lower energy photons have even lower ranges in









water. Also, charged particle equilibrium can be assumed. As a result, dose rates computed

using FDCFs coupled to transport-determined photon fluxes are accurate.

Solution Methodology

Two source terms were selected to be simulated in closed containers for comparison: a

single "average burnup" power reactor fuel pellet, and a Co-60 source. Transport computational

models were constructed to determine the variation in the expected dose rate near the bottom of

"typical" spent fuel pool using a gamma ray ion chamber placed a short distance from a standard

stainless steel container in a location designated as the "Region of Interest Detector Box," or ROI

Detector Box position. In performing 3 -D computations to determine the effect of dose rate

attributed to the sources adjacent to the storage container, the following computations were

performed:

* Forward 3 -D Monte Carlo photon transport computations using the MCNP5 code.
Continuous energy gamma ray treatments of the source are to compute a dose in the ROI
Detector Box. Tallies are set up in MCNP using group bins aliased to the group structure
of the BUGLE-96 gamma library to enable direct comparison to deterministic results;
fluxes are rendered into a dose rate by coupling with ANSI flux-to-dose conversions
interpolated onto BUGLE-96 group energy bins.

* Forward deterministic 3-D Discrete Ordinates (Sn) computations solving the 3 -D radiation
transport equation using the PENTRAN 3-D Sn code to compute dose in the ROI Detector
Box using the BUGLE-96 multigroup gamma library. Resulting fluxes are rendered as a
dose rate by coupling with flux-to-dose conversions.

* Adj oint 3 -D Di screte Ordinates computations solving the adj oint transport equation with
an adjoint source aliased to the ANSI standard flux-to-dose conversion factors, with this
adjoint source placed in the ROI Detector Box location. The adjoint transport result yields
the relative contribution to the dose rate for any photon source placed anywhere in the
model .

Each of these computational approaches amounted to an independent method of

determining a dose, and when converged, the result from each model for equivalent source terms

should yield the same solution (within the statistical uncertainty for the Monte Carlo, and some

small variation between the continuous energy and multigroup energy treatments, etc).










Modeling Procedure


In constructing the models for this effort, the following procedures were used:

* Evaluation of gamma source terms using the Scale version 5.1 (Scale5.1) package
considering a single "standard" boiling water reactor (BWR) fuel pellet (assuming a
19,800 MWD/MTU in a mean flux of 2.6E+13 n/cm2/S OVer a 3 cycle burnup history), and
a Co-60 alloy source

* Use of the BUGLE-96 multigroup gamma cross section library (Sn) and continuous energy
(Monte Carlo) cross sections

* Construction of 3 -D Monte Carlo Neutral Particle version 5 (MCNP5) and 3 -D Sn Parallel
Environment Transport (PENTRAN) models

* Execution of models on computer systems (parallel clusters), post processing and analysis
of results

Source Spectra

The simulated fuel pellet gamma source, rendered using Scale version 5.1 (Scale5.1),

yielded a total of 1.8441E+10 photons per second, and is characterized by the normalized

spectrum data in Figure 5-1.


2.0 E02
3.00E-02
6.00E-02
1.00E-01
2.00E-01
4.00E-01
6.00E-01
7.00E-01
8.00E-01
1.00E40
1.50E-00
2.00E-00
3.00E-00
4.00E-00
5.00E40
6.00E-00
7.00500 0
8.00E-00
1.00E-t1
1.40E-t1


N o and rd

7.00E-02
1.28E-01
5.97E-02
5.16E-02
2.61E-02
B.OBE-03
5.03E-01
3.32E-03
3.72 E-03
7.21E-03
3.07 E-04
1.34E-OG
9.27E-10
3.11E-10
1.05E-10
3.52E-11
1.18E-11
5.17E-12
2.45E-13


1.0E+13

1.0E+11

1.0E+09

1.OE+07

1.0E+05

1.0E+03

1.0E+01

1 .OE-01

1 .0E-03
1.0E-02


1.0E-01 1.0E+00
MeV


1.0E+01 1.0E+02


Figure 5-1. Standard fuel pellet gamma source spectrum normalized from specified upper MeV
bounds.


Average Fuel Pellet Gamma Source










This data is partitioned into upper group energy bins compatible with a BUGLE-96 group

structure. The cobalt-60 alloy source is given in Figure 5 -2, yielding a total of 1.99E+12 photons

per second.



anny Normalized
CO*60 Alloy Source 2.00E-02 7.47E-03
1.0OE+14 3.00E-02 3.50E-03
6.00E-02 3.4HE-03
1.0E+12 1.00E-01 1.25E-03
2.00E-01 0.00E+00
> 1.0E+10 Q-00E-01 0.00E+00
6.00E-01 0.00E+00
'311.0E+08 7.00E-01 0.00E+00
I H.00E-01 0.00E+00
S1.0E+06 1.00E+00 0.00E+00
1.50E+00 B.H4QE-01
1.E0 2.00E+00 0.00E+00
3.0E 0 0.00E+0
4.00E+00 0.00E+00
1 .0E+ 02
5.00E+00 0.00E+00
6.00E+00 0.00E+00
1.0E+ 00 7.00E+00 0.00E+00
1.0E-02 1.OE-01 1 .0E+00 1.0E+01 1 E+2 H .00E+00 0.00E+00
1.00E+01 0.00E+00
MeV
1.40E+01 0.00E+00


Figure 5-2. Cobalt-60 source spectrum normalized from specified MeV upper bounds


Results

Monte Carlo, deterministic forward, and deterministic adj oint results are presented here to

profile dose variation as a function of source position in the sealed stainless steel containers

submersed in a spent fuel pool.

Monte Carlo Simulation of the Container Assembly

A graphical rendering of the geometry used in the Monte Carlo simulation of a stainless

steel container assembly resting in a spent fuel pool given in Figure 5 -3. The simulated spent

fuel pool bottom was made of 1 cm of Type 3 04 stainless steel (SS3 04) liner assumed to be

resting on top of 15 cm of standard concrete, which in turn was assumed to be resting on O'Brien









dirt. For this model, it was also assumed that the container assembly was submerged, resting

directly on the bottom of a pool as shown (on the top surface of the SS304 liner). The water-

filled "ROI Detector Box" region, discussed previously, was located at the bottom of the can

(shown in dark red in Figure 5 -3). The Region of Interest (ROI) Detector Box geometry was

selected for the detector region for ready comparison with deterministic Sn computations

discussed in later sections.


Figure 5 -3. Monte Carlo Neutral Particle version 5 generated picture of region of interest
detector box position and assumed container assembly immersed in water.


The fuel pellet and Co-60 sources were modeled individually using Monte Carlo as point

sources inside the left-most container in Figure 5 -3 (adjacent to the ROI Detector Box), initially

at a height of 4 cm above the bottom of the stainless container, and at higher z-axis elevations in

the container. Doses were computed using Flux to Dose Conversion Factors (FDCFs) following

simulation of photon transport with volumetric flux tallies. With a consideration of the percent









contribution to the total dose, the main contribution to the dose from the fuel pellet was in energy

groups 18 to 13, corresponding to gamma energies between 0.06 MeV to 0.70 MeV respectively;

this accounts for over 97 percent of the total dose at the detector due to the fuel pellet. Based on

the Monte Carlo results, the fuel pellet would yield a dose of 5.1 (+/- 0.01) rem/hour at the ROI

Detector Box location. For the Co-60 source, considering the contribution to the total dose

(72.21 rem/hr) and, group 10, with energies ranging from 1.0 to 1.5 MeV, accounts for 56.8 % of

the total dose at the detector due to the Co-60 source.

Deterministic (Sn) Forward Transport Simulation

One forward calculation was performed using Sn methods using the Parallel Environment

Transport (PENTRAN) code in order to validate dose results for comparison with Monte Carlo

Neutral Particle version 5 (MCNP5), and establish a reference dose for the more useful adj oint

calculation (for assessment of the variation of the dose depending upon source placement,

detailed later). For the Sn computation, a mean fuel unit source was placed 4 cm above the

bottom of the outer container (equivalent to the source placement in the Monte Carlo models

discussed previously). Recall that the deterministic solution of the problem requires that the

geometry be discretized into a number of fine mesh "voxels", characterized by 252 coarse

meshes (containing fine meshes) proj ected on 6 z-levels. A total of 775,184 fine meshes were

defined to describe the 3 -D geometry of the container and surrounding water and components.

These models used an S14 quadrature with an angular splitting technique available in

PENTRAN (splitting S14 into additional directions for octant ordinates 12 and 27 with 2x2

segments, 18 added directions/octant) in an effort to mitigate ray effects in the detector region

with P1 scattering. Therefore, the biased quadrature yielded an Sn equivalent quadrature

comparable to S18. Model dimensions were 60 cm x 104 cm x 70 cm, and gamma ray water

albedos, defined in a manner similar to that in Chapter 3, were used on all sides. This problem











required the solution of 5.457 billion simultaneous equations in the PENTRAN code, and was

solved using the high performance computing clusters at the University of Florida, with infinity


norm convergence to at least 10-3 Or better.


Figure 5-4 depicts the 0.65 MeV (average) gamma flux results with all water regions

hidden for demonstration of the flux behavior. Unphysical ray effects are evident in the


calculation at the outer perimeter of the problem away from the fuel unit source term placed


inside the nearby container. Because the dose "probe" at the Region of Interest (ROI) Detector


Box location is adj acent to the container in a region where ray effects were less of a concern;


note that 35% of the ROI Detector Box dose is attributed to the 0.65 MeV averaged group.







SS304 Pipes IC X

Phi0
S3E+07
1.50949E+07
r-7.59516E+06
3.82159E+06
1.92288E+06
967520
S486819
244949
123249
S62014.2
S31203.2
15700.3
7899.77
3974.86
--- 2000

SS304 Pool Liner
(Bottom of Pool)
Probe Location


Figure 5-4. Gamma flux at 0.65 MeV (average), projected on SS304 materials immersed in
water.



Figure 5-5 represents a 0.08 MeV (average) gamma transport calculation where the ray effects

are reduced due to photon scattering interactions.











Y-1


Figure 5-5. Gamma flux, 0.08 MeV (average), projected on stainless steel 304 materials.

The PENTRAN Sn-rendered dose computed by folding group fluxes with Flux to Dose

Conversion Factors (FDCFs) at the ROI Detector Box location was 5.17 rem/hr, and compares

extremely well with the continuous energy MCNP Monte Carlo rendered dose of 5.1 (+/- 0.01)

rem/hr for the same source term.

An adj oint transport computation and resulting adj oint function becomes particularly

useful in analysis of different source locations in the geometry, and the predicted detector

response. The Sn model constructed for the forward case (detailed in the previous section) was

also used to perform an adj oint computation to analyze the effects of moving the source up

through the container assembly. The adj oint source spectrum located at the ROI Detector Box is











set to the FDCFs, and yields results that produce a dose caused by any source when the adj oint

function values are folded together with a corresponding source flux at the desired location,


anywhere in the model (reference Equation 4 -7). Figure 5-6 portrays the scalar adj oint

importance function for the 0.08 MeV (average) energy group, representing the relative

importance (in the same units of the FDCF) of the gamma rays in each group as they contribute

to the total dose. Ray effects are of smaller consequence in the adj oint solutions like this, partly

because the adj oint source i s aliased with the FDCFs di stributed over the volume of the ROI

Detector Box and is not a small localized source, and therefore the problem is much less prone to


ray effects.











I m port anceerdo s e)
3 5E-10
7 49768E-11
1.60615E-11
3 44068E-12
7 37061E-13
1 57893E-13
3.38237E-14
7 24569E-15
1 55217E-15
3 32504E-16
7.12289E-17
z. 1.52586E-17
3 26869E-18
7 00217E-19
1 5E-19






Figure 5-6. Gamma group adj oint function, aliased to Importance at 0.08 MeV (average)
proj ected on SS304 container materials immersed in water.











The adj oint source used here is the FDCF as a function of energy in the ROI Detector Box.

To compute the dose ob served in the ROI Detector B ox, the resulting adj oint at any location in

the problem (where any true source could be placed) must be coupled to an effective source flux

as noted previously. Since forward Sn and Monte Carlo results confirmed that a dose of ~5.1

rem/hr would be obtained for a single mean fuel unit located at z=4 cm in the container, and by

Equation 5-7 and Equation 5-8, the adj oint result must match this result. The equivalent radius

was calculated conservatively so that the adj oint model would yield 5.17 rem/hr at the original

forward source location; this equivalent radius was calculated to be 5.01 cm (with a

corresponding surface area equivalent of 3 15.4 cm2) for consistency among the results.

Once determined, the group source flux (e.g. for fuel or alloyed Co-60) was then multiplied

by the scalar adj oint importance function for each energy group (aliased to a dose in the ROI

Detector Box). Summing thi s then yields the integrated dose at the ROI Detector B ox caused by

the source at the location selected.

To analyze the overall effect of the source, it was then moved from the first position at 4

cm above the bottom of the outermost container along the z-axis, to 14 cm and 27 cm above the

container bottom; detailed adj oint results are shown in Figure 5 -7 for the mean fuel unit case

mapped over the 3 source locations.

Here, it is important to note that the time of calculation for the forward MCNP5 run was on

the order of 12 hours. Considering that three forward runs were computed to yield doses at three

locations, and that 1 PENTRAN calculation can provide a contiguous solution, it was shown that

PENTRAN, in this particular instance using the adj oint, saved hours of computing. Depending

on the number of points of interest, PENTRAN could save weeks of evaluation and computer

hours.































































6.66


5.18





- 2.53

- 2.19

0.73
0.57
0.49


600 -


10 50 60 70


Figure 5-7.


Dose for a single Fuel unit (mean burnup) in the Region of Interest as a function of
source height at the x-y center of nearest container (Adj oint Deterministic dose vs.
Monte Carlo dose).


Figure 5-8 exclusively shows the results for all fuel unit cases at the 3 source point locations.



Dose (z) (PENTRAN Adjoint) from Fuel Unit
Source


7 00

6 00

5 00

~400
O o
2 o
Too


0 5 10 15
z (cm)


20 25 30


Figure 5-8. Dose from single fuel unit sources coupled for various burnups (low, mean, and
high) in the Region of Interest as a function of source height at x-y center of nearest
container.


Dose as a Function of Sou rce Height Along Z

at ROI Detector Box


z(cm)










Obtaining adj oint function values using an adj oint source spectrum with flux-to-dose

conversion factors is convenient, so that other source spectra can be applied. Assuming a 1g

alloyed Co-60 source spectrum (3 -cycles) was applied to obtain a dose of 74.07 rem/hr at the

container nearest to the detector at 4 cm above the bottom of the container shown in Figure 5 -9.



Dose (z) (PENTRAN Adjoint) From I gram 3 Cycle
Alloyed Co-60 source



80.00
74.07
70.00

60.00

50.00

:E 40.oo352

30.00

20.00

10.00



0 5 10 15 20 25 30

z(c m)


Figure 5-9. Dose caused by 1 gram alloyed Co-60 (3-cycle irradiation) in the "ROI Detector
Box" region as a function of source height at x-y center of nearest container.

This compares extremely well with the Monte Carlo dose for 1 g of alloyed Co-60 of 72.2

rem/hr. Similarly, the assumed 1g alloyed Co-60 source spectrum (5-cycle) was applied and

yielded a dose of about 1668.3 rem/hr at the same location shown in Figure 5-10.









Since the doses matched so well for the fuel pellet model, there was no need to run an

MCNP5 calculation and compare results. Here the adj oint equation could save even more time.



Dose (z) (PENTRAN Adjoint) From 1 gram 5 Cycle
Alloyed Co-60 source




1,668.3






1,00.0




MO.M






0 5 10 15 20 25 30

z~lcm)


Figure 5-10. Dose caused by 1g alloyed Co-60 (5-cycle irradiation) as a function of source
height at x-y center of nearest container.


Discussion

In determining the dose at the Region of Interest (ROI) Detector Box (Ref Figure 5 -6),

Monte Carlo and deterministic radiation transport computations based on low, average, and high

burnup single segmented fuel unit sources, and also calculations of expected dose rate on a "per

gram" basis from irradiated Co-60 materials were performed. The fuel sources were determined










by simulating radiation histories of 8x8 boiling water reactor (BWR) fuel using the Scale version

5.1 (Scale5.1) package, for typical fuel cycle data. A summary of the results for the mean fuel

unit (1 fuel segment with clad) and the single gram of 3 -cycle irradiated Co-60 (assuming 100%

irradiation) is given in Table 5-2.

Table 5-2. Deterministic and Monte Carlo dose comparisons of dose rate (rem/hr) at the Region
of Interest
Source Sn Forward Monte Monte Carlo 1- Sn Adj oint Rel %
(PENTRAN) Carlo sigma rel (MCNP5) (PENTRAN) Difference
(MCNP5) *


Fuel Pellet at 5.17 5.1 <0.01 5.18 0.0153
z=4cm
Fuel Pellet at -- 2.55 <0.01 2.53 0.0072
z=14cm
Fuel Pellet at -- 0.55 <0.01 0.57 0.0291
z=27cm
Co-60 at z=4cm -- 72.2 0.04 74.07 0.0256


Co-60 at z=14cm -- 37.2 0.03 35.23 0.0544


Co-60 at z=27cm -- 9.1 0.01 8.29 0.0932


*Aliased to a Phantom Photon Dose Response in Detector Box ROI using a source flux computed
from an effective radius of 5.01 cm
Relative difference of the mean of the two
values

Because the method of calculation between the Monte Carlo (with continuous energy) and

the deterministic Sn (with multigroup energy) are completely independent, the results exude very

high confidence, where the forward Monte Carlo, Forward Sn, and Adj oint Sn results yielded

very consistent results. It is clear that each "mean fuel unit" located 4 cm off of the bottom of

the container rack will yield a dose of ~5 rem/hr when measured at the ROI Detector Box.









In addition, the adjoint results permit coupling of any source inside the container rack

assembly to predict an effective dose measured in the ROI Detector Box, useful in estimating

doses attributed to new source materials. Adj oint-derived dose results at elevated z-axi s values

were fully consistent with Monte Carlo predicted doses as indicated in Table 5-2.









CHAPTER 6
CONCLUSIONS

The advantages and disadvantages of each Monte Carlo stochastic methods, and

deterministic methods of radiation systems analysis can be listed side by side with no other

conclusion than that the two methods must be used side by side. Given the proper tools,

deterministic Parallel Environment Transport (PENTRAN) yields results that are accurate,

speedy, and compare well with Monte Carlo Neutral Particle version 5 (MCNP5) while

highlighting problem areas and designs that require further study.

The single uranium Pin (suPin) cross section generation process delivered accurate results

when compared with 3 -D models in MCNP5 and PENTRAN, while allowing for some flexibility

in geometry approximations. The suPin procedure also proved to be accurate and the 1/8th COTO

model, while comparisons with MCNP5 revealed that periphery pins need to be locked into an

EDI differencing scheme.

The adj oint equation proved to be very powerful for fixed source problems. Multigroup

adjoint study with PENTRAN reduced ray effects and matched MCNP5 dose calculations at the

Region of Interest (ROI) detector box location. Also calculations utilizing the adj oint equation

give dose importance in all regions with one calculation, rather than just one location for every

source position as does MCNP5.

With parallel machine clusters, parallel 3 -D deterministic transport calculations can be

performed on heterogeneous radiation systems under time constraints comparable to MCNP5.

Further, with understanding of adj oint transport methods, stochastic solutions can verify

parameters of the deterministic solution and reduce analysis time from weeks to days.









CHAPTER 7
FUTURE WORK

Integration of deterministic methods into the highly stochastic-saturated nuclear

reaction/eigenvalue, and flux calculation arena naturally leads to the need for deterministic

techniques in other areas of nuclear evaluation. The advantage of using two fundamentally

different methods of solution reach out to obtain a system of checks and balances on each

method. The problems of a nonconverged system are readily seen by a drastically different

deterministic solution and raises new questions about the nature of the radiation system.

Candidate Design Deterministic/Monte Carlo Agreement

Further work includes evaluation of differencing schemes for reactor physics, and methods

for obtaining multigroup cross section sets for different reactor cores. Advanced study could

provide a sleek, easy to use manual on differencing schemes for set geometries and spectrum

specific cross section sets.

Cross section sets generated by SCALES are limited to certain typical reactors and

geometry restraints apply. Future work will involve seeking other means to obtain cross section

sets for any geometry in a thermal or fast reactor system. Flexibility of reactor system modeling

should give new insight to experimental reactor design.

It is apparent from this study that different meshing, and differencing schemes are

desirable for different radiation system configurations. More work must be completed in order

to set up preferred meshing/differencing schemes on the periphery of different reactor systems.

Since PENTRAN is capable of a multitude of coarse mesh differencing schemes, it is in a prime

position to be used for further study on reactor modeling and tandem use with MCNP.









Parallel Environment Burnup

The obvious next step from using stochastically determined radiation flux values for

reaction rate calculations in a burnup solver, is to use deterministically solved flux values for

reaction rate calculations in a burnup solver. Some preliminary studies have been completed

with Parallel Environment Transport (PENTRAN) on various reactor cores to develop a parallel

3-D deterministic burnup solver. The Parallel Environment Burnup solver (PENBURN), will

allow for further evaluation and comparisons between deterministic type solvers and Monte

Carlo solvers. PENBURN will be produced independent of PENTRAN, and will be able to take

reaction rate information from Monte Carlo Neutral Particle version 5 (MCNP5) as well as

PENTRAN to solve for isotopic compositions for user specified burnsteps.









LIST OF REFERENCES


1. L. BOLTZMANN, "Lectures on Gas Theory," (English translation from original German
manuscript (1872)), University of California Press, (1964).

2. J. DUDERSTADT and L. HAMILTON, Nuclear Reactor Analysis, Wiley, New York
(1976).

3. N. GREENE and L. PETRIE, "XSDRNPM: A One-Dimensional Discrete-Ordinates Code
for Transport Analysis," Oak Ridge National Laboratory, Oak Ridge, Tennessee (2006).

4. M. DEHART, "NEWT: A New Transport Algorithm for Two-Dimensional Discrete
Ordinates Analysis in Non-Orthogonal Geometries," Oak Ridge National Laboratory, Oak
Ridge, Tennessee (2006).

5. D. LUCAS, et al., "Applications of the 3 -D Deterministic Transport Attila for Core Safety
Analysis," Americas Nuclear Energy Symposium, Miami Beach, Florida, (2004).

6. G. SJODEN and A. HAGHIGHAT, "PENTRAN A 3 -D Cartesian Parallel SN Code with
Angular, Energy, and Spatial Decomposition," Proc of the Joint Int '1Conf on
Ma'~thematical M~eth ods and' Supercomputing for Nuclear Applications, Vol I, p. 553,
Saratoga Springs, New York (1997).

7. R. BLOMQUIST and E. GELBARD, "Monte Carlo Criticality Source Convergence in a
Loosely Coupled Fuel Storage System," The 7th International Conference on
NVuclear Cri ticali ty Safe ty, Argonne National Laboratory, Argonne, IL, (2003).

8. H. FISHER, "A Nuclear Cross Section Databook," LA-1 1711l-M, Los Alamos National
Laboratory, Los Alamos, New Mexico (1989).

9. Atomic World, Vol 10, (1959).

10. G. SJODEN, "An Efficient Exponential Directional Iterative Differencing Scheme for
Three-Dimensional Sn Computations in XYZ Geometry," Nuclear Science and'
Engineering, 155, 179-189 (2007).

11. E. LEWIS and W. F. MILLER, "Computational Methods of Neutron Transport,"
American Nuclear Society Publishing Co., LaGrange Park, Illinois, (1993).

12. A. CHILTON, J. SHULTIS, and R. FAW, Principles of ~~~~~~~~~RRRRRRRRRRadato .\/ue/Jin, Prentice Hall,
Englewood Cliffs, New Jersey, (1984).









BIOGRAPHICAL SKETCH

Travis Owings Mock was born in 1981, on a cold day in the Mohave Desert, California,

the first of two children. He spent most of his early years in Florida, participating in organized

sports and water sports. After graduating high school in 2000, he began further study at the

University of Florida. He spent 2 years studying physics before committing to nuclear

engineering, and in all spent 7 years at the University of Florida.





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1 TANDEM USE OF MONTE CARLO AND DETERMINISTIC METHODS FOR ANALYSIS OF LARGE SCALE HETEROGENEOUS RADIATION SYSTEMS By TRAVIS OWINGS MOCK A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFIL LMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007

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2 2007 Travis Owings Mock

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3 To the beautiful Lily Boynton Kaye, and my magnificent family, Todd, Elaine, and Ron Mock without their support, someone else would have completed this work

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4 ACKNOWLEDGMENTS I must acknowledge the technical advice and help provided by Dr. Glenn Sjoden. At this stage of life, it is difficult to define a role model, but Dr. Sjoden has become mine with a seemingly endless wealth of knowledge, and dedication for w ork and family. Also, I must acknowledge Kevin Manalo for his help with Fortran coding, and Monte Carlo Neutral Particle (MCNP5) modeling techniques.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .................... 4 LIST OF TABLES ................................ ................................ ................................ ................................ 7 LIST OF FIGURES ................................ ................................ ................................ .............................. 8 ABSTRACT ................................ ................................ ................................ ................................ ........ 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ....................... 13 Radiation Transport and Diffusion Theory ................................ ................................ ............... 13 Transport Solution ................................ ................................ ................................ ....................... 17 2 PREVIOUS WORK ................................ ................................ ................................ .................... 19 Deterministic Codes ................................ ................................ ................................ .................... 19 ................................ ....................... 19 Tetrahedral Mesh Transport ................................ ................................ ................................ 20 ................................ ................................ ......................... 20 Parallel Environment Neutral Particle Transport ................................ ............................... 21 Monte Carlo Stochastic M ethod of Solution ................................ ................................ ............. 21 3 MULTIGROUP CROSS SECTION GENERATION ................................ .............................. 23 Candidate Design Fuel Pin Data ................................ ................................ ................................ 23 Process for Cross Section Collapse ................................ ................................ ............................ 24 Candidate Design Muligroup Binning ................................ ................................ ............... 24 Codes for Cross Sect ion Generation ................................ ................................ ................... 2 5 Single Pin Mesh Modeling in Cartesian 3 D Sn Transport ................................ ............... 26 Group Albedo Treatment in the 3 D Deterministic Latt ice Cell ................................ ...... 30 Single Pin Monte Carlo and Deterministic Eigenvalue Comparisons .............................. 32 Monte Carlo and Deterministic Flux Comparison ................................ ............................. 34 Discussion ................................ ................................ ................................ ................................ .... 39 4 REACTOR MODELING ................................ ................................ ................................ ........... 40 Candidate Design ................................ ................................ ................................ ........................ 40 Candidate Design Parameters ................................ ................................ ............................. 40 Candidate Design Reactor Core Design ................................ ................................ ............. 42 R eactor Physics Analysis Methods ................................ ................................ ............................ 45 Full Core Candidate Design Monte Carlo Model ................................ .............................. 45 Radial Power Profile ................................ ................................ ................................ ............ 48

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6 Axial Power Profile ................................ ................................ ................................ ............. 50 Power Peaking Summary ................................ ................................ ................................ .... 50 Deterministic Analysis of Candidate D esign ................................ ................................ ............ 51 1/8 th Core Candidate Design Model ................................ ................................ ................... 51 Deterministic Criticality Calculation, 1/8 th Core Candidate Design ................................ 52 Power Peaking in the 1/8 th Core Candidate Design ................................ ................................ .. 53 Full Core Power Density Comparisons ................................ ................................ ..................... 56 Further Refinement of the Deterministic Model ................................ ................................ ....... 59 Discussion ................................ ................................ ................................ ................................ .... 62 5 FIXED SOURCE SIMULATION ................................ ................................ ............................. 63 3 D Gamma Transport Methods ................................ ................................ ................................ 63 Forward and Adjoint Operators ................................ ................................ .......................... 64 Forward and Adjoint D etector Response ................................ ................................ ........... 65 Responses Aliased to Dose Rate ................................ ................................ ......................... 66 Solution Methodology ................................ ................................ ................................ ......... 68 Modeling Procedure ................................ ................................ ................................ ............ 69 Source Spectra ................................ ................................ ................................ ..................... 69 Results ................................ ................................ ................................ ................................ .......... 70 Monte Carlo Simulation of the Container Assembly ................................ ........................ 70 Deterministic (Sn) Forward Transport Simulation ................................ ............................ 72 Discussion ................................ ................................ ................................ ................................ .... 79 6 CONCLUSIONS ................................ ................................ ................................ ......................... 82 7 FUTURE WORK ................................ ................................ ................................ ........................ 83 Candidate Design Deterministic/Monte Car lo Agreement ................................ ....................... 83 Parallel Environment Burnup ................................ ................................ ................................ ..... 84 LIST OF REFERENCES ................................ ................................ ................................ ................... 85 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ............. 86

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7 LIST OF TABLES Table page 3 1 8 group collapsed energy bin bounds ................................ ................................ .................... 25 3 2 6 group collapsed energy bin bounds ................................ ................................ .................... 25 3 3 Candidate Design single pin geometry parameters ................................ .............................. 27 3 4 Excess mass balance and percent error for a 35x34 mesh full pin ................................ ...... 28 3 5 Excess mass balance and percent error for a 21x19 mesh full pin ................................ ...... 29 3 6 Albedo boundary distance parameters ................................ ................................ .................. 31 3 7 8 group albedo factors ................................ ................................ ................................ ............ 32 3 8 6 group albedo factors ................................ ................................ ................................ ............ 32 3 9 Monte Carlo and Determi nistic single pin eigenvalue results for 8 groups ....................... 33 3 10 Monte Carlo and deterministic single pin eigenvalue results for 6 groups ........................ 34 4 1 Monte Carlo Candidate Design k code full core results ................................ ...................... 48 4 2 Power density per pin (maximum error < 4.2%) ................................ ................................ .. 49 4 3 Compu ted power density per axial zone (errors < 4.2%) ................................ .................... 50 4 4 Power Peaking Factor Summary ................................ ................................ ........................... 50 4 5 Final values for 6 group convergence comparison ................................ .............................. 53 4 6 Axial zone boundaries, zone and hottest channel power densities computed using the deterministic model ................................ ................................ ................................ ................ 55 4 7 Hot channel power peaking factors compared to Monte Carlo ................................ ........... 55 4 8 Average power density by zone compared to Monte Carlo ................................ ................ 56 5 1 I nterpolated Flux to Dose Conversion Factors, 1 cSv = 1 rem 12 ................................ ......... 66 5 2 Deterministic and Monte Carlo dose comparisons of dose rate (rem/hr) at the Region of Interest ................................ ................................ ................................ ................................ 80

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8 LIST OF FIGURES Figure page 3 1 Fuel element detail for Candidate Design. ................................ ................................ ........... 23 3 2 Whole core U 235 fission rate (averag e relative error 3%) and selected 8 group structure. ................................ ................................ ................................ ................................ 24 3 3 Flow chart of the codes and used in the cross section generation process, where codes are represented in blue, files are represented in yellow. ................................ ........... 26 3 4 35x34 coarse mesh cell for deterministic model. ................................ ................................ 28 3 5 21x19 coarse mesh cell for deterministic model. ................................ ................................ 29 3 6 Single pin Candidate Design material description used in the Monte Carlo model. ......... 33 3 7 21x19 mesh tally scheme for Monte Carlo single pin model. ................................ ............. 34 3 8 Group 1 flux comparison (6 group model). ................................ ................................ .......... 35 3 9 Group 2 flux comparison (6 group model). ................................ ................................ .......... 36 3 10 Group 3 flux comparison (6 group model). ................................ ................................ .......... 36 3 11 Group 4 flux comparison (6 group model). ................................ ................................ .......... 37 3 12 Group 5 flux comparison (6 group model). ................................ ................................ .......... 37 3 13 Group 6 flux comparison (6 group model). ................................ ................................ .......... 38 4 1 Pla nt Overview of original Candidate Design. 9 ................................ ................................ ... 40 4 2 Candidate Design reactor core views. 9 A) North South Candidate Design core view. B) East West Candidate Design core view. ................................ ................................ .......... 41 4 3 Enlarged East West Core cutaway. 9 ................................ ................................ ..................... 42 4 4 Horizontal slice of core and surrounding beam ports. 9 ................................ ........................ 44 4 5 X Z (y=0 slice) plane view of full core Candidate Design Monte Carlo Model ............... 45 4 6 X Y (z=0 slice) plane view of full core Candidate Design Monte Carlo Model. .............. 46 4 7 Monte Carlo model full core pin numbering scheme. ................................ ......................... 47 4 8 Radial (averaged over axial length) total power pin power densities bas ed on relative errors of less than 4.2% for each value. ................................ ................................ ................ 48

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9 4 9 1/8 th core Candidate Design. ................................ ................................ ................................ .. 51 4 10 Convergence of the 1/8 th cor e Candidate Design for deterministic models. ...................... 52 4 11 Relative pin power for pins contained in 1/8 th Candidate Design (W/cc). Monte Carlo values are converged to a standard deviation of less th an 4.2% (1 sigma) ........................ 54 4 12 Monte Carlo and Deterministic 6 group computed power densities (W/cc). Monte Carlo values are converged to a standard deviation of less than 4.2% (1 sigma). ............. 56 4 13 Ratio of Monte Carlo power density to Deterministic power density. Monte Carlo values are converged to a standard deviation of less than 4.2% (1 sigma). ....................... 57 4 14 Monte Carlo 6 group fluxes. ................................ ................................ ................................ .. 58 4 15 Deterministic 6 group fluxes. ................................ ................................ ................................ 59 4 16 Monte Carlo to Deter ministic power ratios for S10 refined mesh (left) and S10 coarser mesh (right). Monte Carlo values are converged to a standard deviat ion ............ 59 4 17 PENTRAN differenc ing scheme map, blue = Directional Theta Weighted, red = Exponential Directional Iterative. ................................ ................................ ......................... 60 4 18 Monte Carlo to Deterministic power ratios for S10 refined mesh with updated differencing. M onte Carlo values are converged to a standard devia tion ........................ 61 5 1 Standard fuel pellet gamma source spectrum normalized from specified upper MeV bounds. ................................ ................................ ................................ ................................ .... 69 5 2 Cobalt 60 source spectrum normalized from specified MeV upper bounds ...................... 70 5 3 Monte Carlo Neutral Particle version 5 generated picture o f region of interest detector box position and assumed container assembly immersed in water. ..................... 71 5 4 Gamma flux at 0.65 MeV (average), projected on SS304 materials immersed in water. ................................ ................................ ................................ ................................ ....... 73 5 5 Gamma flux, 0.08 MeV (average), projected on stainless steel 304 materials. ................. 74 5 6 Gamma group adjoint function, aliased to Importa nce at 0.08 MeV (average) projected on SS304 container materials immersed in water. ................................ .............. 75 5 7 Dose for a single Fuel unit (mean burnup) in the Region of Interest as a function of source height at the x y center of nearest container (Adjoint Determin istic ..................... 77 5 8 Dose from single fuel unit sources coupled for various burnups (low, mean, and high) in the Regio n of Interest as a function of source height at x y center ...................... 77

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10 5 9 Dose caused by 1 gram alloyed Co 60 (3 of source height at x y center of nearest container. .................. 78 5 10 Dose caused by 1g alloyed Co 60 (5 cycle irradiation) as a function of source height at x y center of nearest container. ................................ ................................ ......................... 79

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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science TANDEM USE OF MONTE CARLO AND DETERMINISTIC METHODS FOR ANALYSIS OF LARGE SCALE HETEROGENEOUS RADIATION SYSTEMS By Travis Owings Mock August 2007 Chair: Glenn Sjoden Major: Nuclear Engineering Sciences Monte Carlo stochastic methods of radiation system analysis are among the most popular of computation techniques. Alternatively, d eterministic analysis often remains a minority calculation method among nuclear engineers, since large memory requirements inhibit abilities to accomplish 3 D modeling. As a result, deterministic codes are often limited to diffusion type solvers with tran sport corrections, or limited geometry capabilities such as 1 D or 2 D geometry approximations. However, there are some 3 D deterministic codes with parallel capabilities, used in this work, to abate such issues. The future of radiation systems analysis is undoubtedly evaluation through parallel computation. Large scale heterogeneous systems are especially difficult to model on one machine due to large memory demands, and it becomes advantageous not only to split computational requirements through indivi dual particle interactions, (as is the method for stochastic parallelization), but also to split the geometry of the problem across machines in parallel. In this effort, first presented is a method for multigroup cross section generation for deterministic code use, followed by radiation system analysis performed using parallel 3 D MCNP5 (Monte Carlo) and parallel 3 D PENTRAN (Sn deterministic) such that these two independent calculation methods were used in order to boost confidence of the final results. T wo

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12 different radiation systems were modeled: an eigenvalue/criticality problem, and a fixed source shielding problem. The work shows that in some systems, stochastic methods are not easily converged, and that tandem use of deterministic calculations provi des, at the very least, another means by which the evaluator can increase problem solving efficiency and accuracy. Following this, lessons learned are presented, followed by conclusions and future work.

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13 CHAPTER 1 INTRODUCTION Analysis of reactor and radiation systems aims at one central problem, the resolution of the neutron or photon distribution within a radiation system. Several different methods have been devised in order to obtain neutron densities, neutron flux, gamma flux, and subsequently reaction rates within radiation systems. The most notable of simulation methods include transport theory, diffusion theory (for neutrons), and Monte Carlo methods. Diffusion theory is an approximation to transp ort theory, and Monte Carlo methods are a statistical sampling approach. However, deterministic transport theory is subject to its own set of constraints due to phase space discretization, most notably multigroup cross section generation. Therefore, the most accurate modeling approach for radiation system analysis is one that utilizes a combination of techniques. Radiation Transport and Diffusion Theory The transport equation was developed by Ludwig Boltzmann in 1872 to describe the kinetic theory of gase s. 1 For neutral particle transport, the linear Boltzmann equation (LBE) is used to accurately model neutron and photon behavior in radiation systems. The integro differential form for neutron multiplying systems is shown in Equation 1 1: (1 1) Where; v = neutral particle speed = particle angular flux = unit vector in direction of particle motion = particle coordinate location in space = particle energy = time

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14 = total macroscopic cross section = external independent particle source = differential scattering cross section = fission spectrum = fission production cross section = average number of fission neutrons produced per fission Most students of nuclear engineering live in fear of the neutron transport equation. The neutro n transport equation, although relatively simple to understand, is difficult to solve in 3 D systems without immense computing power and parallel computation capabilities. Diffusion theory is often used as a substitute for the neutron transport equation i n these cases because it is limitations. To arrive at the diffusion equation from Equation 1 1, we integrate angular flux over 4 steradians such that the following is realized: (1 2) If we integrate the LBE ov er all angles, we obtain the zeroth angular moment Without showing all the steps of the integration, the zeroth moment balance equation takes the form of the following: (1 3) Where a neutron current term, b ecomes a second unknown, leaving two unknowns and only one equation. This divergence of current is the net leakage from the system. To resolve the equation, a relationship must be developed between and T he relationship is established by introducing first angular moment : (1 4)

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15 This new relation, after some manipulation, becomes the following: (1 5) Equation 1 5 may be of further use assuming that angular flux weakly depends on angle, so that, using first moments, 2 we assume: (1 6) Substitution of Equation 1 6 into Equation 1 5 with subsequent integration and simplification yields the P 1 balance equations, as: (1 7) And (1 8) Taking the one speed form of the P 1 equations, (i.e. with integration over all possible energies) yields the one speed balance equations as: (1 9) And (1 10)

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16 Further assum ptions can be made about It is acceptable to assume the relative variation in current will not dominate the collision term, (e.g. unless there is a severe accident situation), and therefore the term is assu med to be zero. 2 This is particularly convenient because it yields an approximation for itself from Equation (1 10) to give: (1 11) Where the diffusion constant can be defined as: (1 12) Substitution into Equation (1 9) gives the one speed diffusion equation as: (1 13) Diffusion theory for neutrons is based on the idea that neutrons migrate from an area of high concentration to an area of low concentration, much like diffusion of gases and heat. However, in most diffusive processes, diffusing particles are characterized by what can be described as frequent scattering. The problem is that the cross section for neutron scattering is often on the order of 10 24 cm 2 2 As a result, with typical atom densities, neutrons may travel for several centimeters before a collision or scattering event takes place, depending on neutron actor or radiation system are of the same order as the mean free path (mfp) of a neutron.

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17 Diffusion theory fails to describe neutron behavior and becomes inaccurate in regions where the flux can change radically. 2 The regions in a reactor or radiation sys tem where the flux changes very rapidly include: Material interfaces, where two different materials meet Strong absorbers (especially, control rods) At the boundary of a system rogeneous radiation system. To give reasonable solutions, diffusion theory is often specifically by correction factors in order to produce transport corrected fluxes and reaction rates. Needless to say, that the desired solution, often termed the solution, is only obtained through the tuned theory is proving to be inadequate as reactor margins and efficiency are pressed. Transport Solution Today, 3 D trans port reactor modeling is done primarily through stochastic Monte Carlo MCNP series. This is in part due to the fact that solving the LBE deterministically creates diff iculties and generating multigroup cross section parameters can be difficult, and proper binning of energy groups, and therefore modeling of resonances, can become treacherous. erns about accurate representation of cross section resonances and energy dependence are lifted. As a result, nuclear engineers and physicists have learned to rely heavily on Monte Carlo methods, sometimes taking for granted the issues related to the stati stics involved in obtaining a converged Monte Carlo solution Radiation systems analysis with stochastic Monte Carlo methods alone can never be exact, but rather, are bounded by statistics and tests for statistical

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18 convergence. Thus, while the Monte Carl o method is quite useful, it may lead to difficulties in solution, providing the languid evaluator has the right tools and discretization of the phase space in Equation 1 1, including reasonably accurate geometric modeling and Doppler broadened, self shielded multigroup cross section generation tools. This brings us to a discussion involving the advantages and disadvantages of both Monte Carlo, and deterministi c transport solutions. The following chapters describe a multigroup cross section generation and modeling approach in an effort to analyze two radiation systems using Sn and Monte Carlo simulation. In Chapter 2, previous work and different analytical meth ods are outlined. Chapter 3 highlights cross section generation, and presents a study involving a procedure for generating Doppler broadened energy weighted material cross sections for a 3 D Candidate Design (CD) pin cell. Chapter 4 presents a 3 D reacto r model and simulation of the neutron flux behavior in a pressurized heavy water, natural uranium reactor using 3 D Sn Parallel Environment Transport theory, (PENTRAN) in tandem with Monte Carlo Neutral Particle version 5 (MCNP5), while utilizing the cross section generation methods outlined in Chapter 3. In Chapter 5, a fixed source shielding problem is introduced and solved using Monte Carlo in tandem with transport theory that includes demonstration of the advantages of forward and adjoint equation. Th is is followed by a discussion of conclusions, future work, and references.

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19 CHAPTER 2 PREVIOUS WORK Deterministic Codes Although deterministic codes exist for evaluation of heterogeneous radiation systems, many fall short of geometry, memory, or parallel processing requirements. There are few parallel deterministic transport code s available and therefore, previous work involving analysis of large scale heterogeneous radiation systems is limited. Analysis and validation of deterministic code packages, through modeling of increasingly detailed reactor and fixed source systems, rema ins the method of quality control among the use of different deterministic radiation system simulation techniques. Oak Ridge National Lab (ORNL) has developed many useful codes for the public (U.S. citizens only) and their radiation systems analysis needs. The multitude of codes includes criticality safety codes using both stochastic and deterministic methods, as well as sophisticated reactor fuel burnup modules for certain standard reactor systems (pressur ized water reactor, boiling water reactor). However, deterministic 3 D transport remains a beast yet dominated in nuclear science. NEW Transport code (NEWT), X S ection D ynamics for R eactor N ucleonics with P etrie M odifications ( XSDRNPM ) and the Discrete O rdinates Oak Ridge System ( DOORS ) system are the only ORNL codes a vailable for transport analysis. XSDRNPM is a one dimensional (1 D) Sn transport algorithm and is used in Scale version 5.1 (Scale5.1) for eigenvalue (k effective) determination, cross secti on collapsing, production of bias factors for use in Monte Carlo shielding calculations, and shielding analysis. 3 Diffusion theory can also be executed through use of XSDRNPM. However, 1 D models are less useful for large heterogeneous systems.

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20 NEWT (alt hough it is not the newest addition to SCALE5), was developed as a multigroup discrete ordinates radiation transport code with non orthogonal meshing to allow for complex two dimensional (2 D) neutron transport models. 4 NEWT is commonly spoken of in tande m with Triton (a transport and depletion module for nuclear fuel) and used for infinite lattice calculations and comparisons. It is especially useful for multigroup cross section generation and Doppler broadening. Unfortunately, NEWT has no parallel capa bilities, and does not account for three dimensional (3 D) effects. Tetrahedral Mesh Transport Attila solves the first order steady state form of the Linear Boltzmann Equation (LBE) using the discrete ordinates method, and does so using tetrahedral mesh ge ometries and angular discretization. 5 Tetrahedral mesh geometry provides for impressive accuracy in material representation, but requires a commercial mesh generator. However, the complex tetrahedral mesh limits the differencing scheme versatility in the code, and therefore Attila is tied to linear discontinuous finite element spatial differencing. A notable advantage to this scheme is that since Attila allows for discontinuities between element faces, and the Linear Discontinuous Finite Element Method ( LDFEM) is able to capture sharp gradients with large meshes allowing for lower Sn angular discretization. 5 However, this can pose challenges to yielding accurate solutions in thick shielding problems. To date, Attila is not parallel, and memory issues ar e significant in cases where complex geometry couples with high angular discretization requirements. Three Dimensional Oak Ridge Transport (TORT), available through the Oak Ridge National Lab (ORNL) Discrete Ordinates Oak Ridg e System (DOORS), is another ORNL code that solves the Linear Boltzmann Equation (LBE) using the method of discrete ordinates, but in

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21 3 particles through large non scat non Parallel Environment Neutral Particle Transport Parallel Environment Transport (PENTRAN) is a scalab le parallel code that solves the steady state form of the LBE in 3 D while using Cartesian mesh geometries and Sn angular discretization. 6 The package, as mentioned previously, operates in a distributed parallel environment, solving the issue of memory av ailability by further distributing the phase space across 3 dimensions of domain decomposition, including space (Cartesian mesh spatial discretization), angle (angular discretization), and energy group (energy group discretization). PENTRAN is very effici ent with adaptive differencing schemes, and has schemes for coarse mesh/partial current rebalance and Taylor projection methods for linking grid discontinuities and mitigating ray effects. The very flexible differencing schemes allow for a solution that i s tailored to the experience of the user. Also, PENTRAN is in continuous development. Monte Carlo Stochastic Method of Solution As mentioned in the previous section, Oak Ridge National Lab (ORNL) uses a variety of codes, of which employ stochastic Monte C arlo methods to assist in problem simulations. The Keno criticality module Monte Carlo is an incredibly useful tool for criticality calculations, and much study has been directed towards obtaining solutions in loosely coupled systems. However, the code i s not meant to model fixed source transport problems. Further applications of Keno include its use in depletion calculations with the use of Triton. achieved high notoriety in t he field of nuclear engineering and simulation, but there are situations in which loosely coupled systems provide waning confidence in source convergence. 7

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22 Such elusive resolutions require a keen eye for the statistical nature of the method of solution. in convergence of a source, they cannot provide completely reliable indications of non convergence. 7 Previous work in Monte Carlo solution methods include loosely coupled syste m simulations such as criticality safety geometries, eigenvalue, and fixed source shielding problems, and in particular address solutions to such problems.

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23 CHAPTER3 MULTIGROUP CROSS SEC TION GENERATION In order to specifically address the problem involving the generation of multigroup Doppler broadened/self shielded material cross sections, a procedure was developed using Scale version 5.1 (Scale5.1) with the NEW Transport along with Triton (NEWT Triton) transport tool. In NEWT, a general purpose 238 group library can be collapsed to obtain a useful cross section library for use in 3 D Sn transport in Parallel Environment Transport (PENTRAN). This method was developed in the process of building a Candidate Design (CD) fuel pin, performing a 3 D unit cell or lattice eigenvalue calculation, and comparing the results against Monte Carlo Neutral Particle version 5 (MCNP5). An example of the procedure is highlight ed in subsequent sections. Candidate Design Fuel Pin Data Th e Candidate Design (CD) fuel rod, depicted in Figure 3 1, modeled for the study was taken from the full core CD with the following properties; Uranium Mo Alloy Annular Fuel, 0.72 w% U 235 enriched ; 98.5 w% U, 1.5 w% Mo Inner Fuel Radius: 0 cm (reduced for natural uranium fuel design) Outer Fuel Radius: 1.78 cm Fuel Rod Length: 250 cm Fuel Rod Pitch: 13.5 cm Cladding, inner: 0.1 cm, Alloy 6061T6 Cladding, outer: 0.l cm, Alloy 6061T6 Inner Pres sure Tube Radius: 2.0 cm, Pressure Tube thickness: 0.15 cm Figure 3 1 Fuel element detail for C andidate D esign. For the simulation, the inner diameter of the annular type fuel depicted in Figure 3 1 was considered to be zero, implying a solid fuel e lement.

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24 Process for Cross Section Collapse Although the single pin infinite lattice calculation performed in order to validate the cross section collapse procedure using the Scale version 5.1 (Scale5.1) package only models 2 D geometries, the process was s hown to be accurate for 3 D calculations, as will be shown by the following sections. Candidate Design Muligroup Binning To select a proper group structure most applicable to the core physics Candidate Design (CD) problem, the entire CD reactor core, to be described in the following chapter, was modeled in a continuous energy Monte Carlo Neutral Particle version 5 (MCNP5) for approximately 10 hours on 16 64 bit Opteron processors. Results of this Monte Carlo simulation yielded approximately 3% tally statis tics throughout all 238 energy groups. MCNP5 continuous energy groups were binned to be identical with the general Scale version 5.1 (Scale5.1) library. Following analysis of this data, eight (8) total energy groups were collapsed from the original 238 g roups that could best represent problem physics. Note these groups were identified through use of the global integrated U 235 fission rate tally, with contributions from all fueled core regions portrayed in Figure 3 2. Table 3 1 shows each collapsed group Figure 3 2. Whole core U 235 fission rate (average relative error 3%) and selected 8 group structure.

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25 Table 3 1. 8 group collapsed energy bin bounds Group Upper Group Bounds(MeV) 1 20 2 1.36 3 4.40E 01 4 5.20E 02 5 3.90E 03 6 6.83E 04 7 1.22E 04 8 3.00E 06 The study was also collapsed to six groups for in order to reduce memory requirements. The upper bounds for the six group collapse are shown in Table 3 2. Table 3 2. 6 group collapsed energy bin bounds Group Upper Group Bounds (MeV) 1 20 2 1.36 3 4.40E 01 4 3.90E 03 5 1.22E 04 6 3.00E 06 Codes for Cross Section Generation T he 8 and 6 group libraries prese nted in the previous section were collapsed from the 238 group Scale version 5.1 (Scale5.1) general libr ary for the pin cell materials and rendered into deterministic cross section libraries using tools that were developed or significantly expanded and enhanced in support of this project: the Scale Form ( SCALFORM ) tool renders S cale 5 .1 microscopic cross sect ions in a readable text format following the use of the S cale 5 .1 A lpo tool. Mixing the rendered text based microscopic cross sections from SCALFORM was then accomplished using an updated group cross section mixer called Group Independent Mixer ( GMIX ) whi ch provides material macroscopic cross sections, and fuel mixture group fission

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26 chi val ues. For this work, a mean average integrated chi fission spectrum that accounts for the effects of multiple fissile nuclides was developed and incorporated in to GMIX. A flowchart outlining the cross section generation process, termed the single uranium Pin (suPin) process, is shown in Figure 3 3. Figure 3 3. Flow chart of the codes and used in the cross section generation proce ss, where codes are represented in blue, files are represented in yellow. Single Pin Mesh Modeling in Cartesian 3 D Sn Transport Following the generation of macroscopic cross sections, from the 2 D NEW Transport (NEWT) infinite lattice calculation, a 3 D calculation with Parallel Environment Transport (PENTRAN) was modeled. Again, the purpose of the PENTRAN calculation was to validate the

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27 cross section generation procedure in a 3 D lattice calculation for further use on the 1/8 th core Candidate Design (CD ) model. For the simulation, the full pin model was used with the parameters shown in Table 3 3. Table 3 3. Candidate Design single pin geometry parameters Description length (cm) active fuel length 250 L ength of D 2 O stacked above/below active fuel 79 r fuel outer 1.780 1st clad r outer 1.992 Pressure tube r inner 2.430 Pressure tube r outer 2.530 fuel pitch 12.563 Note that the fuel pitch in the single pin model is 12.563 cm, which is an adjusted equivalent square pitch to the triangular pitch of 13 .5 cm in the full core model. The single pin model considers an infinite square lattice, whereas the full core model has fuel elements in a triangular array. PENTRAN is a 3 D Cartesian discrete ordinates code, and therefore mesh refinement must be impleme nted to represent the geometry specification of a desired design. In some cases, the desig n can be modeled to exact geometry, in cases with curved shapes some approximations must be used For the single lattice cell study, two geometries for the coarse mesh cell containing cylindrical fuel elements were considered. These two geometries will be referred to as a 35x34 coarse mesh cell and a 21x19 coarse mesh cell ; m ore specifically these numbers refer to the fine mesh structure of the coarse mesh contain ing the fuel element

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28 The 35x34 coarse mesh cell was intended to represent the cylindrical geometry of the fuel element to near exact design geometry specifications. The 35x34 coa rse mesh cell is shown below in F igure 3 4 Figure 3 4. 35x34 coarse me sh cell for deterministic model. The 35x34 ( x y ) coarse mesh scheme in Figure 3 4 yields the approximations for masses in the mod el as shown in Table 3 4. Table 3 4. Excess mass balance and percent error for a 35x34 mesh full pin Material Model Excess Ma ss(g) Model Mass (g) Target Mass (g) Percent Error U Mo 139.00 47,000.00 47,139.00 0.29% Al inner 98.20 1,597.60 1,499.40 6.15% D 2 O inner 28.40 1,642.40 1,614.00 1.73% Al outer 197.00 1,249.00 1,446.20 15.79% D 2 O outer 19.90 65,200.00 65,180.00 0.0 3%

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29 This model worked well for representation of a unit cell calculation; however, due to the anticipated possible memory limitations encountered in developing the 1/8 Symmetry core model of the Candidate Design deterministically, it became necessary to p erform a single lattice cell parameter study in an effort to limit the number of fine mesh cells defining the geometry within coarse mesh cells containing fuel pins and in outer coarse meshes in a triangular pitc h assembly arrangement for the CD. After a brief study, a practical Cartesian meshing scheme was chosen. This scheme involved a blend of minimal meshing to achieve a bal ance of fissile mass, and minimize memory requirements i n PENTRAN; this is depicted in Figure 3 5 Figure 3 5. 21x19 coarse mesh cell for deterministic model. The excess mass for the 21x19 mes h scheme is shown in Table 3 5 Table 3 5. Excess mass balance and percent error for a 21x19 mesh full pin Material Model Excess Mass(g) Model Mass (g) Target Mass (g) Percent Error U n atural 35.6 46820.0 46784.0 0.08% Al inner 136.0 1559.4 1423.0 8.75% D 2 O inner 91.2 1762.0 1853.2 5.18% Al outer 12.3 1039.6 1027.3 1.18%

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30 From inspec tion of Table 3 4 representation of the fue l assembly, there will inherently be a material approximation in representing curved boundary surfaces, so that approximately 9% of the aluminum cladding is underrepresented in the 21x19 mesh model, and the D 2 O within the flow channel is overrepresented by ~5%. However, for the purposes of the study this is acceptable, since it is most important to properly represent the targeted mass of fissile material present as closely as possible, along with reasonably close conservation of the moderator mass. Group Albedo Treatment in the 3 D Deterministic Lattice Cell Due to the large size of the 1/8 core Candidate Design (CD) model, full representation of the almost meter of water above and below the fuel elements (79cm thick), the core was not tractable for the de terministic model to fit on the 8 node, 16 processor Einstein cluster. Therefore, the model size may be reduced by implementing group dependent albedo factors computed using a 3 D lattice cell calculation. To reduce the amount of heavy water atop the pin s in the CD 1/8 core model, group albedos were calculated by using the multigroup solutions from a single pin lattice cell calculation and mapping albedo factors for each of 15 fine z levels within the 79 cm of heavy water directly above the fuel pin locat ion (every 5.267cm). To determine an appropriate location 2 O and implement a group dependent albedo factor boundary condition, a calculation was made using flux weighted cross sections to tru ncate the model at a distance of 6 mean free paths above the fuel. The highly resolved 35x34 mesh scheme pin cell model, described in the previous section, was used to obtain group albedo factors for the full D 2 O reflected model, and with additional inves tigation using both a 35x34 and a 21x19 mesh albedo based model. The formulations used for determining an average mean free path follows:

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31 (3 1) With Where =thickness of the medium (3 2) (3 3) Where is the i t fine mesh group g flux above the tg represents the total macroscopic cross section in group g and V i is the fine mesh volume. The fraction F of uncollided neutrons at N mean free paths is given by Equation 3 3. A 6 mean free path (6 mfp) length of reflector is often use d as a thickness to apply a group albedo boundary. This was applied here; applicable data are presented in Table 3 6. Table 3 6. Albedo boundary distance parameters Parameter Value units 0.44464 cm 1 3*mfp 6.7470 cm 6*mfp 13.49 4 cm Here we should note that value of 6 mean free paths for the heavy water using the 3 D unit cell data yielded a reflector distance which also happens to be equivalent to the pitch of the fuel ther, flux weighted group albedo factors were found using the following equation: (3 4)

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32 The group dependent albedo factors determined from Equation 3 4 are easily recognized as the fraction of returned current computed for the mes h cells in the heavy water above the pin in mesh i, and of energy group g where the desired heavy water cut is to be made. D ue to the z fine mesh partitions of the problem throughout the he avy water above the fuel, a value of 10.53 cm, or 4.68 mean free paths above the fuel element. Albedo factors for the 8 group and 6 group model are provided in Table 3 7 and 3 8 respectively. Table 3 7. 8 group albedo factors Group Upper Group Bounds(MeV) g 1 20.0 0.0823 2 1.36 0.2714 3 4.4E 1 0.408 9 4 5.2E 2 0.4 53 6 5 3.9E 3 0.506 3 6 6.83E 4 0.550 1 7 1.22E 4 0.623 9 8 3.0E 6 0.995 6 Table 3 8. 6 group albedo factors Group Upper Group Bounds (MeV) g 1 20 0.0824 2 1.36 0.2715 3 4.40E 01 0.4811 4 3.90E 03 0.5451 5 1.22E 04 0.6189 6 3.00E 06 0.9949 Singl e Pin Monte Carlo and Deterministic Eigenvalue Comparisons The pin that was modeled in Monte Carlo Neutral Particle version 5 (MCNP5) is shown in Figure 3 6. It is easy to see the advantages of MCNP5 when compared to the discretized geometry required for Parallel Environment Transport (PENTRAN).

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33 Figure 3 6. Single pin Candidate Design material description used in the Monte Carlo model. No material approximations or albedo calculations were needed for the MCNP5 model, and geometries were modeled exactl y as shown in Table 3 2. The results and comparisons for shown in Table 3 9 and 3 10. Table 3 9. Monte Carlo and Deterministic single pin eigenvalue results for 8 groups Model Final k eff Standard Deviation Percent difference from MCNP MCNP continuous energy 1.091 4.20E 04 N/A PENTRAN 35x34 Full D 2 O Reflector 1.086 4.85E 05 0.46% PENTRAN 35x34 Albedo Treated Reflector 1.084 1.23E 04 0.64% PENTRAN 21x19 Albedo T reated Reflector 1.092 1.94E 04 0.09%

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34 Table 3 10. Monte Carlo and deterministic single pin eigenvalue results for 6 groups The 6 group 21x19 fuel pin provided the most desirable results with the least amount of memory requirements. Therefore, the 21x19 fuel pin was selected as the p in design of the 1/8 th core Sn model. Monte Carlo and Deterministic Flux Comparison In order t o collect volumetric fluxes, a mesh tally in Monte Carlo Neutral Particle version 5 ( MCNP5 ) was incorporated Figure 3 7 shows a 21x19 mesh overlaying the pin regi on, which was selected to provide adequate aliasing with the single pin Deterministic model. Radial flux profiles were gathered with only one z mesh spanning the active fuel length. Figure 3 7. 21x19 mesh tally scheme for Monte Carlo single pin model Model Final k eff Standard Deviation Percent difference from MCNP MCNP continuous energy 1.091 4.20E 04 N/A PENTRAN 35x34 Full D 2 O Reflector 1.086 4.95E 05 0.46% PENTRAN 21x19 Albedo Treated Reflector 1.090 1.75E 05 0.09%

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35 Figures 3 8 through 3 13 compare radial flux profiles for both MCNP 5 and Parallel Environment Transport ( PENTRAN ) for a single pin unit cell, allowing direct spectral comparison of group fluxes. The figures display relative flux through the centerline o f the single pin model for both MCNP5 and PENTRAN, where tallies are aliased to the 6 group energy bins. The relative percent of total neutrons in each group (determined from integration over neutron density) are provided along with maximum and minimum fl ux values in both Monte Carlo and deterministic models, and maximum relative error between the two models. Figure 3 8. Group 1 flux comparison (6 group model).

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36 Figure 3 9. Group 2 flux comparison (6 group model). Figure 3 10. Group 3 flux com parison (6 group model).

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37 Figure 3 11. Group 4 flux comparison (6 group model). Figure 3 12. Group 5 flux comparison (6 group model).

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38 Figure 3 13. Group 6 flux comparison (6 group model). Differences between the MCNP and PENTRAN models for the single pin can be attributed to the multigroup energy group treatment and structure between the Monte Carlo and Deterministic computations, remembering that MCNP uses continuous energy bins, whereas PENTRAN uses discrete energy group bins mixed using the SCALE5 code with the SUPIN procedure developed for this project. Multigroup cross sections represent averaged behavior in regions with strong resonances, particularly at high energies in U 235, and especially U 238. 7 In addition, the mean maximum differen ce between the MCNP5 continuous energy pin cell fluxes and the 6 group model was 1.71%. However, based on the assessment of total neutrons in the system (computed using group neutron densities (equal to group flux/ velocity) integrated over the system vol ume for each energy group), 82.6% of all neutrons appear in group 6 (most thermal group) according to PENTRAN results, and 96.9% of the neutrons according to MCNP5,

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39 with a maximum flux difference in group 6 of 4.67% between deterministic and Monte Carlo values, outside of the standard error of the MCNP results. Discussion The single uranium Pin (suPin) pr ocess extracts Doppler broadened, self shielded cross sections after a user specified collapse from the Scale version 5.1 (Scale5.1) multigroup library. Results for eigenvalue calculations in both 3 D Sn transport with Parallel Environment Transport (PENT RAN) and Monte Carlo Neutral Particle version 5 (MCNP5) were consistent, and therefore are applicable for development of a 6 group full core model.

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40 CHAPTER 4 REACTOR MODELING This chapter describes the problem and evaluation of the Candidate Design (CD), with information on results, and limitations of the analysis The structures of the CD are depicted in the plant overview shown in Figure 4 1 Fi gure 4 1. Plant Overview of original Candidate Design. 9 Candidate Design C andidate D esign Parameters The Candidate Design (CD) is based on a high flux pressurized heavy water reactor (PHWR), located inside a 6061 T6 aluminum cylindrical reactor tank vess el (boron, lithium, and cadmium free aluminum) with walls 1.3 cm thick, and a bottom that is 3.5 cm thick. The

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41 limitations; it is heavy water moderated and cooled. Secondary heat exchangers match heat transfer from the heavy water cooling circuit to a light water circuit. The initial design of the reactor was modified for fueling with naturally enriched uranium. Average fuel burnup for the core was specified to be 2400 MWD/MT, peak is 4000 MWD/MT for the 15 MWt design (note a peaking of 1.67 is attributed to these values); burnups for the natural uranium core are strongly dependent upon power density and irradiation history. A North South slice and East West slice of the CD reactor component esse ntials are provided in Figure 4 2, with an expanded view in F igure 4 3. A B Fig ure 4 2. Candidate Design reactor core views. 9 A) North South Candidate Design core v iew. B) East West Candidate Design core view. Altho ugh Figure 4 1, Figure 4 2, and Figure 4 3 are not essential for the computer modeling of the CD, it is necessary to gain an appreciation for the size of the reactor and the potential memory requirements that may be encountered. Also, it is important to n ote that many

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42 of the structures labeled in Figures 4 1 and Figure 4 2 were not modeled in deterministic Sn Parallel Environment Transport (PENTRAN), or stochastic Monte Carlo Neutral Particle version 5 (MCNP5). Boundary approximations, including vacuum bo undaries and albedo boundary conditions at the edge of the actual reactor core give satisfactory results. Figure 4 3. Enlarged East West Core cutaway. 9 C andidate D esign Reactor Core Design The Candidate Design (CD) core holds a total of solid fuel sl ugs in fuel assemblies immersed in heavy water. Remaining positions house up to six B 4 C control rods. Each fuel assembly is placed in the core using a 13.5 cm hexagonal pitch, and is composed of four

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43 m. Thermal hydraulic analysis and heat transfer parameter limits restrict the power increase, so that a ramped overpower operation in this design is bounded to < 35 MW. The CD reactor and fuel parameter are summarized by the following; 26 MWth Design Pow er 166 fuel assembly positions 5 fuel rods per assembly Solid natural uranium Up to 6 shim rods Central channel D 2 O around, above, and below core Aluminum reactor wall Aluminum support plate Graphite around outer core The core contains the fuel inside i ndividual pressure tubes spanning over a radius of ~90 cm in a 1 3 .5 cm hexagonal lattice, which is then surrounded by 40 cm of heavy water as a primary reflector, made up from the outer edge of fuel channel sub assemblies up to the physical tank wall. Imm ediately outside the tank (separated by a 0.8 cm air gap for air cooling) sits a large mass of graphite fashioned as a 14 sided prism at an approximate inner radius of 130 cm and a thickness extending out at least 62.8 cm from the inner radius. At the edg e of the graphite is another air gap of 2 cm, followed by a 100 MT cast steel thermal shield 19 cm thick. The graphite is stacked on top of a bismuth support plate (to avoid n gamma reactions since natural bismuth 209 has only a 20 mb neutron capture cros s section) to support the graphite. The graphite is purified nuclear g rade graphite (< 3 ppm natural boron ) composed of 20 cm x 20 cm

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44 interlocking stringer blocks configured to be very resistant to shifting, even when heated. The air gaps support filtere d atmospheric air flow used for a secondary cooling mechanism. The reactor tank is under pressure with helium cover gas, and based on the data available for the CD, tank seals can withstand ~2 psi of heavy water overpressure during an excursion. Beyond th e graphite is a 200 cm thick concrete biological shield. Figure 4 4 shows a horizontal slice of the core and the surrounding beam ports. Figure 4 4. Horizontal slice of core and surrounding beam ports. 9

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45 Reactor Physics Analysis Methods Full Core C and idate D esign Monte Carlo Model A full core model for the Candidate Design (CD) was generated with complete detail for the rea ctor system components using Monte Carlo Neutral Particle version 5 (MCNP5). Figure4 5 and Figure 4 6 rendered using the MCNP5 vie wer, and provide the details of the schematic for the fu ll core Monte Carlo model. In F igure 4 5 individual fuel pins with five fuel slugs are labeled, in addition to the aluminum reactor vessel and graphite reflector surrounding the core. The geometry of the pins is the same as in T able 4 4 a fuel pitch of 13.5 cm (t riangular lattice of 166 pins). Figure 4 5. X Z (y=0 slice) plane view of full core Candidate Design Monte Carlo Model

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46 Figure 4 6. X Y (z=0 slice) plane view of full core Candidate Design Monte Carlo Model.

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47 With a full core model in MCNP5, calculations to determine pin power and peaking factors were performed. To assist in a core map reference, a pin numbering index was implemented. This numbering index is depicted in a simplifi ed diagram of the full core model, with pins sequentially labeled 1 thru 166 for the CD as shown in Figure 4 7 Figure 4 7. Monte Carlo model full core pin numbering scheme. The full core model was simulated on code imulation sequence using MCNP5 to determine the overall criticality of the s ystem. Table 4 1 provides a summar y of the computed keff summary.

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48 Table 4 1. Monte Carlo Candidate Design k code full core results Parameter Value Units k eff 1.04231 + / 0.0004 8 E avg of a n causing fission 218.3 keV avg 2.464 neutrons/ fission prompt removal lifetime 8.1916E 04 +/ 1.89 E 06 Seconds CPU Time, Einstein 16 Procs 5.09 Hours (+/ 5% pin power) Radial Power Profile A radial pin power pro file (relative to a to tal reactor power of 26 MW t ) is depicted in Figure 4 8 and Table 4 2 with power densities given in W/cc This was established using F7 tallies to yield total fission power in each pin As anticipated, t he reactor power is the highest near the core center in the immediate region which surrounds the largest experimental cavity; specifically, this is at pin number 76 with 0.901% of the total power generated, or 94.53 W/cc. The core average radial pin power was 63.6 W/cc. Figure 4 8. Radial (averaged over axial length) total power pin power densities based on relative errors of less than 4.2% for each value.

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49 Table 4 2. Power density per pin (maximum error < 4.2%) pin number (W/cc) pin number (W/cc) pin number (W/cc) pin number (W/cc) 1 51.484 42 70.5 03 83 61.878 124 73.892 2 53.555 43 63.724 84 59.167 125 70.729 3 54.798 44 57.133 85 72.461 126 66.285 4 54.572 45 55.288 86 89.823 127 60.786 5 55.401 46 58.828 87 93.853 128 52.727 6 48.734 47 59.129 88 75.889 129 50.391 7 47.379 48 66.774 89 64.5 52 130 55.137 8 54.723 49 71.633 90 54.647 131 54.497 9 54.271 50 78.299 91 52.049 132 59.581 10 57.585 51 79.353 92 59.694 133 64.665 11 55.739 52 76.830 93 61.991 134 70.428 12 55.288 53 77.546 94 76.114 135 72.876 13 51.823 54 71.030 95 80.408 136 67.113 14 50.015 55 64.402 96 86.697 137 64.100 15 50.843 56 54.534 97 93.439 138 57.397 16 57.660 57 53.932 98 87.865 139 53.141 17 57.020 58 60.598 99 86.057 140 50.730 18 60.033 59 58.865 100 78.600 141 51.446 19 61.539 60 65.155 101 67.226 142 5 0.354 20 63.347 61 72.725 102 58.564 143 57.171 21 61.125 62 78.901 103 48.659 144 63.121 22 56.832 63 80.333 104 47.680 145 65.833 23 53.555 64 81.011 105 57.660 146 59.430 24 53.291 65 80.634 106 57.510 147 55.476 25 60.711 66 77.018 107 63.422 148 49.902 26 59.656 67 70.277 108 72.650 149 48.998 27 63.799 68 62.519 109 75.625 150 47.529 28 68.356 69 54.120 110 80.220 151 46.362 29 71.143 70 54.836 111 78.751 152 52.312 30 68.469 71 58.338 112 78.148 153 57.962 31 66.473 72 63.347 113 74.834 1 54 58.677 32 63.686 73 73.892 114 67.528 155 54.610 33 55.589 74 82.027 115 59.543 156 49.827 34 54.723 75 87.338 116 52.049 157 45.985 35 60.033 76 94.531 117 47.567 158 46.588 36 59.807 77 90.426 118 54.685 159 45.006 37 67.151 78 85.379 119 55.626 160 45.947 38 72.424 79 80.672 120 62.632 161 48.697 39 74.985 80 69.109 121 68.168 162 53.141 40 77.583 81 58.752 122 70.767 163 52.990 41 72.085 82 53.103 123 74.118 164 51.446 165 45.232 166 42.897 Average 63.195

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50 Axial Power P rofile Only five axial zones were modeled to obtain power density. Using five zones was convenient since five slugs compose a single pin. Since each fuel slug (~9.450 kg or ~500 cc) is the same volume and defined as a cell in the Monte Carlo Neutral Part icle version 5 (MCNP5) model, fission power tallies are conveniently obtained. The results of the axial power densities are in Table 4 3. Table 4 3. Computed power density per axial zone (errors < 4.2%) Axial Zone Axial Zone Power (W/cc) Hot Channel Powe r (W/cc) 1, (1.05 cm to 51.05 cm) 47.03 69.11 2, (53.15 cm to 103.15 cm) 72.69 112.23 3, (105.25 cm to 155.25 cm) 80.76 122.21 4, (157.35 cm to 207.35 cm) 70.79 103.95 5, (209.45 cm to 259.45 cm) 44.70 65.16 Average 63.20 94.53 Power Peaking Summar y Calculated power peaking factor (PPF) values representing peak to average assembly power d ensities are given in Table 4 4. Using Table 4 3 the radial PPF was calculated by locating the maximum power density (pin 76) of the 166 pins and dividing by the average power density. The axial PPF in the hottest pin was calculated by segmenting pin 76 into five (5) individual zones and calculating a local PPF for that pin alone. The PPF in the hottest pin/channel is calculated by multiplying the radial PPF with the axial PPF in the hottest pin. Table 4 4. Power Peaking Factor Summary Power Peaking Factor Summary radial PPF 1.496 axial PPF in the hottest pin 1.293 PPF in the hottest pin/channel 1.934 Therefore, 1.934 was the calculated power peaking factor using the Monte Carlo full core model.

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51 Deterministic Analysis of Candidate Design 1 /8 th Core Candidate Design Model The Candidate Design (CD)was generated using L atte triangular lattice generator. Latte was developed in an effort to ease construction of t he model. Previously, the user would have to construct each individual pin. Latte was used to create the whole core assembly from the 21x19 single pin coarse mesh described in the previous chapter. Memory requirements for the Einstein 8 node, 16 process or cluster dictated that at most 1/8 th of the core could be modeled, resulting in a modified core design without the empty channels of the original core design. A 1/8 th core symmetry was obtained by modeling half the core in the z direction and core in the x y plane. The 1/8 th CD is shown in Figure 4 9 Figure 4 9. 1/8 th core Candidate Design. It should be noted that the graphite reflector was omitted due to the approximation that the D 2 O already serves as an infinite reflec tor under the specified operating conditions. The 1/8 th Core Parallel Environment Transport (PENTRAN) model used a block adaptive mesh grid, where high resolution meshing is applied in zones where needed (in fuel pin zones). Meshing can be relaxed in the peripheral heavy water region. The final 1/8 core model used contained

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52 361,584 mesh cells, and as many as 260 million multigroup simultaneous equations in a power/source iteration for determination of the criticality eigenvalue and pin power distribution Heavy water moderated systems are some of the most difficult systems to converge, since neutrons scatter quickly, and remain in thermal energies (as evidenced by the thermal group albedos of 0.995 determined previously). Deterministic Criticality Calcul ation 1/8 th Core Candidate Design The Parallel Environment Transport (PENTRAN) deterministic criticality calculation presented several challenges due to limited cluster memory. Three calculations were performed using 21x19 x y meshes for each fuel pin, studying convergence effects of changing the Sn ordinate parameters and number of energy groups. k eff convergence for 8 group models using S4 and S8 angular quadrature, and a 6 group model using S10 angular quadrature were studied; all cases were run with P 1 scattering anisotropy. All calculations were initially performed using the Directional Theta Weighted (DTW) differencing scheme locked in all 1710 coarse meshes. The behavior of the criticality eigenvalue k eff and the convergence comparisons are prov ided in Figure 4 10 and Table 4 5, respectively. Figure 4 10. Convergence of the 1/8 th core Candidate Design for deterministic models.

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53 Table 4 5. Final values for 6 group convergence comparison Model Final k Standard Deviation Final Iteration k toler ance Procs Space Procs Group Procs Angle MB/Proc CPU Time (hrs) S4 8 groups 1.00237 8.10E 05 84 1.00E 04 4 3 1 2714.3 54.7 S8 8 groups 0.99741 5.85E 05 75 1.00E 04 10 1 1 3994.8 87.3 S10 6 groups 0.99807 5.15E 05 75 1.00E 04 5 3 1 3270.3 99.2 Power Pe aking in the 1/8 th Core Candidate Design Parallel Environment Transport (PENTRAN) is capable of delivering fission source information directly to the user via PENTRAN Data extractor (PENDATA), and can be used to yield volumetric fission neutron source valu es in the fuel, sorted by energy group. The volumetric fission source data corresponds to the fission neutron source in the fuel in user specified fine meshes within each coarse mesh volume. Therefore, group sorted fission sources (# of neutrons/cc) in f uel values were integrated over energy group and fuel zone volume, and divided by total zone fuel volume and the average number of neutrons per fission to obtain the mean fission reaction density per zone. The source density per zone was then scaled to yi eld a computed core thermal power of 26 MWt in accordance with Candidate Design (CD) specifications. (4 1) Radial pin power peaking was calculated by summing over zones of fuel pin power volumes axially, and determining peak to aver age ratios between radial pin power densities. The results of the pin mean power densities are shown in Figure 4 11.

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54 Figure 4 11. Relative pin power for pins contained in 1/8 th Candidate Design (W/cc). Monte Carlo values are converged to a standard de viation of less than 4.2% (1 sigma)

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55 Axial power peaking was calculated in a similar manner, using 6 z axial zones, where the pin power was summed over radial volumes in each axial zone. The hottest half pin was split over 6 z levels. Axial zone boundari es and associated zone averaged power densities are in Table 4 6. Table 4 6. Axial zone boundaries, zone and hottest channel power densities computed using the deterministic model No. Boundaries in cm Axial Zone Power (W/cc) Hot Channel Power (W/cc) 1 0 20 69.191 113.220 2 20 40 68.099 111.324 3 40 60 65.887 107.359 4 60 80 62.626 101.718 5 80 100 58.218 94.298 6 100 125 53.196 86.244 Average 62.484 101.716 Power peaking values for the 1/8 th core deterministic computation are shown in Table 4 7. Average power density for each zone is shown in Table 4 8. Table 4 7. Hot channel power peaking factors compared to Monte Carlo Type Peak to Average Power Ratio Percent difference from MCNP 5 PENTRAN Radial 1.628 8.82% MCNP Radial 1.496 N/A PENTRAN Axi al 1.113 13.92% MCNP Axial 1.293 N/A PENTRAN Power Peaking Factor 1.934 6.73% MCNP Power Peaking Factor 1.812 N/A

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56 Table 4 8. Average power density by zone compared to Monte Carlo Zone Average Power Density (W/cc) Percent difference from MCNP 5 MCNP5 Radial 63.6 0.01% PENTRAN Radial Full Core 63.605 N/A MCNP5 Axial (Hottest Pin) 94.53 7.65% PENTRAN Axial (Hottest Pin) 102.361 N/A Full Core Power Density Comparisons The pin powers for reactor power scaled to 26 MWt were computed as noted in previou s sections, and are reproduced here for direct comparison. A side by side pin power comparison is provided in Figure 4 12, and the ratio of power density per pin is given in Figure 4 13. Figure 4 12. Monte Carlo and Deterministic 6 group computed pow er densities (W/cc). Monte Carlo values are converged to a standard deviation of less than 4.2% (1 sigma).

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57 Figure 4 13. Ratio of Monte Carlo power density to Deterministic power density. Monte Carlo values are converged to a standard deviation of less than 4.2% (1 sigma). Ideally, all of the values in Figure 4 13 should be unity. In comparing these ratios, recall that Monte Carlo Neutral Particle version 5 (MCNP5) values are converged with a standard deviation (1 sigma) of 4.2% or less. Power density ratios diverge from unity in the pins primarily along the outer periphery, attributed largely to the use of inadequate mesh densities in outer core regions of the Parallel Environment Transport (PENTRAN) model. Higher mesh

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58 densities are needed there to r esolve the neutron leakage streaming off of the core, along with an investigation of adaptive differencing scheme behavior. The inadequate local mesh resolution at the edge of the core causes the leakage to be overestimated there, so that this directly ex plains why a lower system criticality eigenvalue ( k eff = 1.00) resulted from the deterministic models (compared to the k eff =1.04 in the Monte Carlo model). In addition, since the PENTRAN model was a 1/8 core symmetry model, some differences near the cent er of the core can be attributed to the empty channels not modeled in PENTRAN model which also affect k. Given these facts, the models are comparable, although the differences could be narrowed further by refinement of the deterministic model. The followi ng plots show that there was a severe flux drop at the outer edges of the core. The flux drop may not have been accurately modeled in the PENTRAN calculation due to limited memory capabilities. Further study was provided after expanding models on a new c luster named Bohr. Figure 4 14. Monte Carlo 6 group fluxes.

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59 Figure 4 15. Deterministic 6 group fluxes. Further Refinement of the Deterministic Model The Bohr cluster supplied more speed of calculation and memory for the Parallel Environment Trans port (PENTRAN) simulation. To a limit, the meshing surrounding fuel elements on the perimeter of the model could be refined and the results are shown in Figure 4 16. Figure 4 16. Monte Carlo to Deterministic power ratios for S10 refined mesh (left) and S10 coarser mesh (right). Monte Carlo values are converged to a standard deviation of less than 4.2% (1 sigma).

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60 Mesh densities in Figure 4 16 were increased by a factor of 2 coarse meshes surrounding periphery fuel pins. Further study was employed in an effort to determine more accurate differencing schemes for the PENTRAN model. The Exponential Directional Iterative (EDI) differencing scheme has shown promising results in test problems and was implemented on the perimeter pins as indicated by Figure 4 1 7. 10 Figure 4 18 presents the ratios, Monte Carlo to deterministic power densities, for the calculation. Figure 4 17. PENTRAN differencing scheme map, blue = Directional Theta Weighted, red = Exponential Directional Iterative.

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61 Figure 4 18. Monte Car lo to Deterministic power ratios for S10 refined mesh with updated differencing. Monte Carlo values are converged to a standard deviation of less than 4.2% (1 sigma).

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62 The power ratios are closer to unity in the previous problem areas located at the top o f the figure. This demonstrates the need for a differencing scheme like EDI to account for streaming and edge effects in core physics calculations. Also, agreement for the pin on the upper left corner of Figure 4 18 still remains an issue. This could be due to higher stochastic error in the fission source inherent in the pins. Discussion Tandem methods for the analysis of the reactor model provided two independent solutions that, for the most part, were in good agreement. Monte Carlo Neutral Particle ve rsion 5 (MCNP5) yielded a converged solution with no approximations for geometric parameters, while the Parallel Environment Transport (PENTRAN) calculation had to be carefully constructed so that a converged solution could be obtained. MCNP5 and PENTRAN were in reasonable agreement for power density calculations on all but the periphery fuel pins. For peripheral fuel, one must note whether or not the Monte Carlo calculations truly converged, or if the Sn solutions are accurate for the discretization used Further investigative techniques were employed in PENTRAN in order to access problem regions, which were prominent in the periphery of the model, with and improved with discretization and higher order differencing algorithm. The analysis identifie s areas of investigation for both deterministic and Monte Carlo solution approaches.

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63 CHAPTER 5 FIXED SOURCE SIMULATION In the course of handling packaged radionuclides, it is often necessary to assay closed containers of radioactive materials for inventory and accountability, and to validate gamma ray dose rates and source spectra. Often, materials in sealed containers can be well characterized using gamma ray spectrometry and dosimetry when the container is readily accessible. With regard to emphasis on material accountability and minimizing personnel doses, it is useful to understand th e magnitude and the spectrum of a containerized source without requiring the container to be opened. However, some packaged or containerized sources are often stored in locations not completely accessible for repeated or convenient survey via spectrometry such as in spent fuel pools, fuel handling areas, etc, so that inventory, accountability, and control must be more generally verified through routine dose rate measurement audits. These measurements may be carried out in spent fuel pools at water depths of one to two atmospheres of pressure, and only a small probe ion chamber measurement for dose assessment is possible, given operational, environmental, or geometry constraints. Therefore, a study was commissioned at the FINDS Institute at the University of Florida to employ 3 D gamma radiation transport calculations to explore the extent of potential variations in measured dose rate from hypothetical containerized sources in a simulated spent fuel pool using standard stainless steel storage containers (S SC). These 3 D transport calculations were compared to and validated by Monte Carlo calculations under the use of MCNP5 code. 3 D Gamma Transport Methods Performing the gamma ray transport computations necessary for this study involved obtaining either a deterministic solution (achieved in our studies via the Sn method), or statistically (achieved using the Monte Carlo method). Both solution methods were applied to

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64 provide consistent results in characterizing dose rates in this application. The steady st ate multi group form of the transport equation operating on the forward group angular flux using standard notation with total group cross section differential group scattering cross section and source terms q ( ) g is: 11 (5 1) Forward and Adjoint Operators Principally, scattering from all other energy groups into group g is dominated by downscattering from higher energies to lower energies. 11 The adjoint transport operator can be derived using the adjoint identity for real valued functions and the forward multi group transport operator, where the angled brackets r epresent integration over all independent variables, in this case phase space: 11 (5 2) Using Equation 4 1, it can be seen that the forward operator is (5 3) The angular adjoint (importance) function is, Applying the adjoint boundary condition that particles leaving a bounded system have an importance of zero in all groups (converse of the forward vacuum boundary condition) with the above equations, and requiring a continuous importanc e function mathematically leads to the multi group adjoint transport operator: 11 (5 4)

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65 Note the minus sign on the streaming term indicates that adjoint particles travel along a reverse direction (relative to the forward transport fo rmulation), where scattering progresses from group g back to other groups (those groups formerly contributing to group g in the forward equation). Forward and Adjoint Detector Response To render a traditional detector response R (e.g., absorptions per u nit time) attributed to a specific reaction using an adjoint transport solution, the desired R must be linked to the adjoint R in the detector. To illustrate this, conside r a fixed forward source detector problem, where the radiation flux between the source and the detector must satisfy the transport equation: 11 ( 5 5) Where q g is the gamma source density, typically with units of /cm 3 /s. Alternativ ely, the adjoint transport equation should also be satisfied using an adjoint source aliased to the group detector response cross section typically with units of 1/cm: ( 5 6) Applying Equations 5 2, 5 5, and 5 6, and integrating over all variables results in the very useful expression for detector response R : ( 5 7) Therefore, a traditional detector response can be obtained from the adjoint transport solution by integration of the produc t of the source distribution with the adjoint function computed over the problem phase space, for any arbitrary source distribution. Thus, detector response R caused by several sources may be computed in two ways:

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66 Directly from the left side of Equation 5 7 from several individual forward transport computations for each radiation source, or From the coupling of a single adjoint transport computation with the set of different source terms 11 Responses Aliased to Dose Rate Dose rate response ( ) computations in our study were rendered using energy group dependent flux to dose conversion factors (FDCFs) for gamma radiation (Chilton et. all, 1984). The units of these factors are in rem cm 2 (cSv cm 2 ), so that when a (forward) gamma flux is determined in a region in /cm 2 /s, the product of the group gamma fluxes with each group FDCF was computed, summed over all groups, and this yields the gamma dose rate in rem/s at that point. FDCFs interpolated onto BUGLE 96 multigroup energy bins are presented in Table 5 1. Table 5 1. Interpolated Flux to Dose Conversion Factors 1 cSv = 1 rem 12 Group MeV (hi) MeV (avg) FDCF ( cSv cm2 ) 1 1.40E+01 1.20E+01 3.10E 09 2 1.00E+01 9.00E+00 2.44E 09 3 8.00E+00 7.50E+00 2.13E 09 4 7.00E+00 6.50E+00 1.93E+09 5 6.00E+00 5.50E+00 1.72E 09 6 5.00E+00 4.50E+00 1.50E 09 7 4.00E+00 3.50E+00 1.28E 09 8 3.00E+00 2.50E+00 1.03E 09 9 2.00E+00 1.75E+00 8.11E 10 10 1.50E+00 1.25E+00 6.40E 10 11 1.00E+00 9.00E 01 5.09E 10 12 8.00E 01 7.50E 01 4.45E 10 13 7.00E 01 6.50E 01 4.00E 10 14 6.00E 01 5.00E 01 3.26E 10 15 4.00E 01 3.00E 01 2.07E 10 16 2.00E 01 1.50E 01 1.09E 10 17 1.00E 01 8.00E 02 6.80E 11 18 6.00E 02 4.50E 02 4.96E 11 19 3.00E 02 2.50E 02 3.90E 11 20 2.00E 02 1.00E 02 3.50E 11

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67 The desired dose rate can also be rendered usi ng an adjoint transport computation if the adjoint source is properly aliased to yield 11 To accomplish this for the adjoint transport calculation, the group adjoint source must be set equal to the group FDCFs with units of cSv cm 2 Once the adjoint function is determined at the location of interest, the dose can be rendered using Equation 5 7, where the forward source, being aliased to the FDCFs, must be a gamma ray flux with units of Since placement of a small localized source is defined as a source rate ( /s ) this must be properly converted to an estimated flux in the problem for proper computation of the dose rate (assuming volumetric variable dimensions canc el) : (5 8) Therefore, to convert a localized source rate to a source flux for coupling to the adjoint function (to then yield a dose rate), an equivalent radius ( r eq ) must be determined for computation of an effective surface area ( 1/( 4 r eq 2 )), which is used to divide into the source rate to yield an effective source flux, Since forward and adjoint calculations must each independently render the same this equality was used in this work to determine the equivalent radius ( r eq ) for each source rate. Therefore, use of the equivalent surface area in coupling a localized source with the adjoint aliased to the FDCFs is necessary to yield the correct within the practic al limits of truncation error in the two deterministic transport computations. Flux to dose conversions are assumed to be applicable in this case without the specific need to consider specific electron transport effects in the transport computations, since the range of ~1 to~2 MeV (the dominant source term) photoelectrons in water is <0.5 to <1cm; lower energy electrons caused by interactions with lower energy photons have even lower ranges in

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68 water. Also, charged particle equilibrium can be assumed. As a result, dose rates computed using FDCFs coupled to transport determined photon fluxes are accurate. Solution Methodology Two source terms were selected to be simulated in closed containers for comparison: a t, and a Co 60 source. Transport computational models were constructed to determine the variation in the expected dose rate near the bottom of stainless steel ROI Detector Box position. In performing 3 D computations to determine the effect of dose rate attributed to the sources adjacent to the storage container, the following compu tations were performed: Forward 3 D Monte Carlo photon transport computations using the MCNP5 code. Continuous energy gamma ray treatments of the source are to compute a dose in the ROI Detector Box. Tallies are set up in MCNP using group bins aliased to the group structure of the BUGLE 96 gamma library to enable direct comparison to deterministic results; fluxes are rendered into a dose rate by coupling with ANSI flux to dose conversions interpolated onto BUGLE 96 group energy bins. Forward deterministic 3 D Discrete Ordinates (Sn) computations solving the 3 D radiation transport equation using the PENTRAN 3 D Sn code to compute dose in the ROI Detector Box using the BUGLE 96 multigroup gamma library. Resulting fluxes are rendered as a dose rate by coupl ing with flux to dose conversions. Adjoint 3 D Discrete Ordinates computations solving the adjoint transport equation with an adjoint source aliased to the ANSI standard flux to dose conversion factors, with this adjoint source placed in the ROI Detector B ox location. The adjoint transport result yields the relative contribution to the dose rate for any photon source placed anywhere in the model. Each of these computational approaches amounted to an independent method of determining a dose, and when conver ged, the result from each model for equivalent source terms should yield the same solution (within the statistical uncertainty for the Monte Carlo, and some small variation between the continuous energy and multigroup energy treatments, etc).

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69 Modeling Proc edure In constructing the models for this effort, the following procedures were used: Evaluation of gamma source terms using the Scale version 5.1 (Scale5.1) package 19,800 MWD/MTU in a mean flux of 2.6E+13 n/cm 2 /s over a 3 cycle burnup history), and a Co 60 alloy source Use of the BUGLE 96 multigroup gamma cross section library (Sn) and continuous energy (Monte Carlo) cross sections Construction of 3 D Monte Carlo Neutral P article version 5 (MCNP5) and 3 D Sn Parallel Environment Transport (PENTRAN) models Execution of models on computer systems (parallel clusters), post processing and analysis of results Source Spectra The simulated fuel pellet gamma source, rendered using Scale version 5.1 (Scale5.1), yielded a total of 1.8441E+10 photons per second, and is characterized by the normalized spectrum data in Figure 5 1. Figure 5 1. Standard f uel pellet gamma source spectrum normalized from specified upper MeV bounds.

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70 This d ata is partitioned into upper group energy bins compatible with a BUGLE 96 group structure. The cobalt 60 alloy source is given in Figure 5 2, yielding a total of 1.99E+12 photons per second. Figure 5 2. Co balt 60 source spectrum normalized from spec ified MeV upper bounds Results Monte Carlo, deterministic forward, and deterministic adjoint results are presented here to profile dose variation as a function of source position in the sealed stainless steel containers submersed in a spent fuel pool. Mon te Carlo Simulation of the Container Assembly A graphical rendering of the geometry used in the Monte Carlo simulation of a stainless steel container assembly resting in a spent fuel pool given in Figure 5 3. The simulated spent fuel pool bottom was made of 1 cm of Type 304 stainless steel (SS304) liner assumed to be

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71 dirt. For this model, it was also assumed that the container assembly was submerged, resting d irectly on the bottom of a pool as shown (on the top surface of the SS304 liner). The water (shown in dark red in Figure 5 3). The Region of Interest (ROI) Detec tor Box geometry was selected for the detector region for ready comparison with deterministic Sn computations discussed in later sections. Figure 5 3. Monte Carlo Neutral Particle version 5 generated picture of region of interest detector box position and assumed container assembly immersed in water. The fuel pellet and Co 60 sources were modeled individually using Monte Carlo as point sources inside the left most container in Figure 5 3 (adjacent to the ROI Detector Box), initially at a height of 4 c m above the bottom of the stainless container, and at higher z axis elevations in the container. Doses were computed using Flux to Dose Conversion Factors (FDCFs) following simulation of photon transport with volumetric flux tallies. With a consideration of the percent

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72 contribution to the total dose, the main contribution to the dose from the fuel pellet was in energy groups 18 to 13, corresponding to gamma energies between 0.06 MeV to 0.70 MeV respectively; this accounts for over 97 percent of the total dose at the detector due to the fuel pellet. Based on the Monte Carlo results, the fuel pellet would yield a dose of 5.1 (+/ 0.01) rem/hour at the ROI Detector Box location. For the Co 60 source, considering the contribution to the total dose (72.21 rem /hr) and, group 10, with energies ranging from 1.0 to 1.5 MeV, accounts for 56.8 % of the total dose at the detector due to the Co 60 source. Deterministic (Sn) Forward Transport Simulation One forward calculation was performed using Sn methods using the P arallel Environment Transport (PENTRAN) code in order to validate dose results for comparison with Monte Carlo Neutral Particle version 5 (MCNP5), and establish a reference dose for the more useful adjoint calculation (for assessment of the variation of th e dose depending upon source placement, detailed later). For the Sn computation, a mean fuel unit source was placed 4 cm above the bottom of the outer container (equivalent to the source placement in the Monte Carlo models discussed previously). Recall t hat the deterministic solution of the problem requires that the meshes (containing fine meshes) projected on 6 z levels. A total of 775,184 fine meshes were defined t o describe the 3 D geometry of the container and surrounding water and components. These models used an S14 quadrature with an angular splitting technique available in PENTRAN (splitting S14 into additional directions for octant ordinates 12 and 27 with 2 x2 segments, 18 added directions/octant) in an effort to mitigate ray effects in the detector region with P 1 scattering. Therefore, the biased quadrature yielded an Sn equivalent quadrature comparable to S18. Model dimensions were 60 cm x 104 cm x 70 cm, and gamma ray water albedos, defined in a manner similar to that in Chapter 3, were used on all sides. This problem

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73 required the solution of 5.457 billion simultaneous equations in the PENTRAN code, and was solved using the high performance computing clu sters at the University of Florida, with infinity norm convergence to at least 10 3 or better. Figure 5 4 depicts the 0.65 MeV (average) gamma flux results with all water regions hidden for demonstration of the flux behavior. Unphysical ray effects are ev ident in the calculation at the outer perimeter of the problem away from the fuel unit source term placed Box location is adjacent to the container in a region where ray effects were less of a concern; note that 35% of the ROI Detector Box dose is attributed to the 0.65 MeV averaged group. Figure 5 4. Gamma flux at 0.65 MeV (average), projected on SS304 materials immersed in water. Figure 5 5 represents a 0 .08 MeV (average) gamma transport calculation where the ray effects are reduced due to photon scattering interactions.

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74 Figure 5 5. Gamma flux, 0.08 MeV (average) projected on stainless steel 304 materials The PENTRAN Sn rendered dose computed by fol ding group fluxes with Flux to Dose Conversion Factors ( FDCFs ) at the ROI Detector Box location was 5.17 rem/hr, and compares extremely well with the continuous energy MCNP Monte Carlo rendered dose of 5.1 (+/ 0.01) rem/hr for the same source term. An adj oint transport computation and resulting adjoint function becomes particularly useful in analysis of different source locations in the geometry, and the predicted detector response. The Sn model constructed for the forward case (detailed in the previous s ection) was also used to perform an adjoint computation to analyze the effects of moving the source up through the container assembly. The adjoint source spectrum located at the ROI Detector Box is

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75 set to the FDCFs, and yields results that produce a dose caused by any source when the adjoint function values are folded together with a corresponding source flux at the desired location, anywhere in the model (reference Equation 4 7). Figure 5 6 portrays the scalar adjoint importance function for the 0.08 MeV (average) energy group, representing the relative importance (in the same units of the FDCF) of the gamma rays in each group as they contribute to the total dose. Ray effects are of smaller consequence in the adjoint solutions like this, partly because t he adjoint source is aliased with the FDCFs distributed over the volume of the ROI Detector Box and is not a small localized source, and therefore the problem is much less prone to ray effects. Figure 5 6. Gamma group adjoint function al iased to Impo rtance at 0.08 MeV (average) projected on SS304 container materials immersed in water.

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76 The adjoint source used here is the FDCF as a function of energy in the ROI Detector Box. To compute the dose observed in the ROI Detector Box, the resulting adjoint a t any location in the problem (where any true source could be placed) must be coupled to an effective source flux as noted previously. Since forward Sn and Monte Carlo results confirmed that a dose of ~5.1 rem/hr would be obtained for a single mean fuel u nit located at z =4 cm in the container, and by Equation 5 7 and Equation 5 8, the adjoint result must match this result. The equivalent radius was calculated conservatively so that the adjoint model would yield 5.17 rem/hr at the original forward source l ocation; this equivalent radius was calculated to be 5.01 cm (with a corresponding surface area equivalent of 315.4 cm 2 ) for consistency among the results. Once determined, the group source flux (e.g. for fuel or alloyed Co 60) was then multiplied by the s calar adjoint importance function for each energy group (aliased to a dose in the ROI Detector Box). Summing this then yields the integrated dose at the ROI Detector Box caused by the source at the location selected. To analyze the overall effect of the s ource, it was then moved from the first position at 4 cm above the bottom of the outermost container along the z axis, to 14 cm and 27 cm above the container bottom; detailed adjoint results are shown in Figure 5 7 for the mean fuel unit case mapped over t he 3 source locations. Here, it is important to note that the time of calculation for the forward MCNP5 run was on the order of 12 hours. Considering that three forward runs were computed to yield doses at three locations, and that 1 PENTRAN calculation c an provide a contiguous solution, it was shown that PENTRAN, in this particular instance using the adjoint, saved hours of computing. Depending on the number of points of interest, PENTRAN could save weeks of evaluation and computer hours.

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77 Figure 5 7. Dose for a single Fuel unit (mean burnup) in the Region of Interest as a function of source height at the x y center of nearest container (Adjoint Deterministic dose vs. Monte Carlo dose). Figure 5 8 exclusively shows the results for all fuel unit cases at the 3 source point locations. Figure 5 8. Dose from single fuel unit sources coupled for various burnups (low, mean, and high) in the Region of Interest as a function of source height at x y center of nearest container.

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7 8 Obtaining adjoint function va lues using an adjoint source spectrum with flux to dose conversion factors is convenient, so that other source spectra can be applied. Assuming a 1g alloyed Co 60 source spectrum (3 cycles) was applied to obtain a dose of 74.07 rem/hr at the container nea rest to the detector at 4 cm above the bottom of the container shown in Figure 5 9. Figure 5 9. Dose caused by 1 gram alloyed Co 60 (3 y center of nearest containe r. This compares extremely well with the Monte Carlo dose for 1 g of alloyed Co 60 of 72.2 rem/hr. Similarly, the assumed 1g alloyed Co 60 source spectrum (5 cycle) was applied and yielded a dose of about 1668.3 rem/hr at the same location shown in Figure 5 10.

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79 Since the doses matched so well for the fuel pellet model, there was no need to run an MCNP5 calculation and compare results. Here the adjoint equation could save even more time. Figure 5 10. Dose caused by 1g alloyed Co 60 (5 cycle irradiatio n) as a function of source height at x y center of nearest container. Discussion In determining the dose at the Region of Interest ( ROI ) Detector Box (Ref Figure 5 6), Monte Carlo and deterministic radiation transport computations based on low, average, a nd high burnup single segmented fuel unit sources, and also calculations of expected dose rate 60 materials were performed The fuel sources were determined

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80 by simulating radiation histories of 8x8 boiling water re actor ( BWR ) fuel using the Scale version 5.1 (Scale5.1) package for typical fuel cycle data A summary of the results for the mean fuel unit (1 fuel segment with clad) and the single gram of 3 cycl e irradiated Co 60 (assuming 100% irrad iation) is given i n Table 5 2. Table 5 2. Deterministic and M onte Carlo dose c omparisons of d ose r ate (rem/hr) at the Region of Interest Source Sn Forward (PENTRAN) Monte Carlo (MCNP5) Monte Carlo 1 sigma rel (MCNP5) Sn Adjoint (PENTRAN) Rel % Difference + Fuel Pellet at z=4cm 5.17 5.1 <0.01 5.18 0.0153 Fuel Pellet at z=14cm -2.55 <0.01 2.53 0.0072 Fuel Pellet at z=27cm -0.55 <0.01 0.57 0.0291 Co 60 at z=4cm -72.2 0.04 74.07 0.0256 Co 60 at z=14cm -37.2 0.03 35.23 0.0544 Co 60 at z=27cm -9.1 0.01 8.29 0.0932 *Aliased to a Phantom Photon Dose Response in Detector Box ROI using a source flux computed from an effective radius of 5.01 cm + Relative difference of the mean of the two values Because the method of calculation between the Monte Carlo (with conti nuous energy) and the deterministic Sn (with multigroup energy) are completely independent, the results exude very high confidence, where the forward Monte Carlo, Forward Sn, and Adjoint Sn results yielded of the container rack will yield a dose of ~5 rem/hr when measured at the ROI Detector Box.

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81 In addition, the adjoint results permit coupling of any source inside the container rack assembly to predict an effect ive dose measured in the ROI Detector Box, useful in estimating doses attributed to new source materials. A djoint derived dose results at elevated z axis values were fully consistent with Monte Carlo predicted doses as indicated in Table 5 2

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82 CHAPTER 6 CONCLUSIONS The advantages and disadvantages of each Monte Carlo stochastic methods, and deterministic methods of radiation systems analysis can be listed side by side with no other conclusion than that the two methods must be used side by side. Given the proper tools, deterministic Parallel Environment Transport (PENTRAN) yields results that are accurate, speedy, and compare well with Monte Carlo Neutral Particle version 5 (MCNP5) while highlighting problem areas and designs that require further study. The single uranium Pin (suPin) cross section generation process delivered accurate results when compared with 3 D models in MCNP5 and PENTRAN, while allowing for some flexibility in geometry approximations. The suPin procedure also proved to be ac curate and the 1/8 th core model, while comparisons with MCNP5 revealed that periphery pins need to be locked into an EDI differencing scheme. The adjoint equation proved to be very powerful for fixed source problems. Multigroup adjoint study with PENTRAN reduced ray effects and matched MCNP5 dose calculations at the Region of Interest (ROI) detector box location. Also calculations utilizing the adjoint equation give dose importance in all regions with one calculation, rather than just one location for eve ry source position as does MCNP5. With parallel machine clusters, parallel 3 D deterministic transport calculations can be performed on heterogeneous radiation systems under time constraints comparable to MCNP5. Further, with understanding of adjoint tran sport methods, stochastic solutions can verify parameters of the deterministic solution and reduce analysis time from weeks to days.

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83 CHAPTER 7 FUTURE WORK Integration of deterministic methods into the highly stochastic saturated nuclear reaction/eigenvalue, and flux calculation arena naturally leads to the need for deterministic techniques in other areas of nuclear evaluation. The adva ntage of using two fundamentally different methods of solution reach out to obtain a system of checks and balances on each method. The problems of a nonconverged system are readily seen by a drastically different deterministic solution and raises new ques tions about the nature of the radiation system. Candidate Design Deterministic/Monte Carlo Agreement Further work includes evaluation of differencing schemes for reactor physics, and methods for obtaining multigroup cross section sets for different reactor cores. Advanced study could provide a sleek, easy to use manual on differencing schemes for set geometries and spectrum specific cross section sets. Cross section sets generated by SCALE5 are limited to certain typical reactors and geometry restraints ap ply. Future work will involve seeking other means to obtain cross section sets for any geometry in a thermal or fast reactor system. Flexibility of reactor system modeling should give new insight to experimental reactor design. It is apparent from this s tudy that different meshing, and differencing schemes are desirable for different radiation system configurations. More work must be completed in order to set up preferred meshing/differencing schemes on the periphery of different reactor systems. Since PENTRAN is capable of a multitude of coarse mesh differencing schemes, it is in a prime position to be used for further study on reactor modeling and tandem use with MCNP.

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84 Parallel Environment Burnup The obvious next step from using stochastically determin ed radiation flux values for reaction rate calculations in a burnup solver, is to use deterministically solved flux values for reaction rate calculations in a burnup solver. Some preliminary studies have been completed with Parallel Environment Transport (PENTRAN) on various reactor cores to develop a parallel 3 D deterministic burnup solver. The Parallel Environment Burnup solver (PENBURN), will allow for further evaluation and comparisons between deterministic type solvers and Monte Carlo solvers. PENB URN will be produced independent of PENTRAN, and will be able to take reaction rate information from Monte Carlo Neutral Particle version 5 (MCNP5) as well as PENTRAN to solve for isotopic compositions for user specified burnsteps.

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85 LIST OF REFERENCES 1. L. BOLTZMANN (English translation from original German manuscript (1872)), University of California Press, ( 1964 ) 2. J. DUDERSTADT and L. HAMILTON, Nuclear Reactor Analysis Wiley, New York ( 1976 ) 3. N. GREENE and L. PETRIE Dimensional Discrete Ordinat es Code Oak Ridge National Laboratory, Oak Ridge, Tennessee (2006). 4. Dimensional Discrete Ordinates Analysis in Non Orthogon al Geometries Ridge, T ennessee (2006). 5. D. LUCAS et al. D Deterministic Transport At tila for Core Safety Americas Nuclear E nergy Symposium, Miami Beach, Florida (2004). 6. G. SJODEN and A. HAGHIGHAT, "PENTRAN A 3 D Cartesian Parallel SN Code with Angular, Energy, and Spatial Decomposition Mathematical Methods and Supercomputing for Nuclear Applications Vol I, p. 553, Saratoga Springs, New York (1997). 7. R. BLOMQUIST Loose The 7th International Conference on Nuclear Criticality Safety, Argonne National Laboratory, Argonne, IL, (2003). 8. H. FISHER s Section Databook 11711 M, Los Alamos National Laboratory, Los Alamos, New Mexico (1989). 9. Atomic World, Vol 10, (1959). 10. Three Dimensional S n Computations in XYZ Geomet Nuclear Science and Engineering 155 179 189 (2007). 11. E. LEWIS American Nuclear Society Publishing Co., LaGrange Park, Illinois, (1993). 12. A. C HILTON J. SHULTIS, and R. FAW Principles of Radiation Shielding Prentice Hall, Englewood Cliffs, New Jersey, (1984).

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86 BIOGRAPHICAL SKETCH Travis Owings Mock was born in 1981, on a cold day in the Mohave Desert, California, the first of two children. He spent most of his early years in Florida, participating in organized sports and water sports. After graduating high sch ool in 2000, he began further study at the University of Florida. He spent 2 years studying physics before committing to nuclear engineering, and in all spent 7 years at the University of Florida.