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COMPACT HEAT EXCHANGER DESIGN FOR TRANSFERRING HEAT FROM A
VAPOR CORE REACTOR INTO A GAS TURBINE POWER PLANT
SAMUEL E. BAYS
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING
UNIVERSITY OF FLORIDA
Samuel E. Bays
This document is dedicated to my loving wife, Nikki.
I would like to thank my parents for raising me well and teaching me patience and
that hard work is a virtue. I thank my wife for standing by me and encouraging me in my
work. I give special thanks to my faculty advisor, Dr. Samim Anghaie, for his receptive
and insightful suggestions. I would also like to thank the other members of my advisory
panel, Dr. Edward Dugan and Dr. Wei Shyy, for their sensible recommendations. I thank
my friend and colleague Dr. Blair Smith for his thoughtful questions and an attentive ear
for my ideas. Special thanks go to Ms. Bonnie McBride of NASA Glenn-Lewis
Laboratory for her invaluable assistance with the Chemical Equilibrium with
Applications Code. I thank my department chairman, Dr. Alireza Haghighat, for asking
me how I was doing.
TABLE OF CONTENTS
ACKNOW LEDGM ENTS ........................................ iv
LIST OF TABLES ........... ................... .......... .............. vii
LIST OF FIGURES ........ ...... .............. .......... ................. viii
1 INTRODUCTION ................... .................. .............. .... ......... .......
History and Design Evolution ................. .................................4
Heat Transfer Issues and Thermal Design........................................6
Computer Simulation................. .... .......... .. ..8
Com parative Analysis Calculations.........................................................................9
Thermo-Physical Property Comparative Analysis.........................................9
Diffusion Layer Theory Comparative Analysis ................................................10
2 THERMODYNAMIC ANALYSIS METHOD ....................................................11
T popping C y cle ...................................... ............................................ 11
Intercooler......................... ... ................. 13
Topping Cycle Code D description ....................................................... 13
Bottoming Cycle..................... ......................... 17
3 CONDENSATION PROPERTY MAPPING....................................................19
The CEA Code.................................................. 20
Therm odynam ic Properties ........................................ ................. 22
Therm al Transport Properties.............................. .................... 24
Least Squares D ata Preparation ......................................................... .... ...........27
4 THERMAL HYDRAULIC MODEL DEVELOPMENT................ ...... .....31
T he H eat T ran sfer M odel ....................................................................................... 34
Diffusion Layer Theory Development ................................. ..........36
Counter-Flow N odal A nalysis..................................................................... 40
Pressure Loss M odel ................... .............................. ... ...... ...... .. ................. 43
Frictional Pressure L oss .............................................. ............... 44
A ccelerational Pressure L oss ...................... .......................... ............... 45
5 DIFFUSION LAYER MODEL COMPARATIVE ANALYSIS .............................47
Com parative Analysis M ethod ............................................ ............... 47
Impact of DLM ............. .................. ................ ..............51
6 COMPACT HEAT EXCHANGER DESIGN.......................................................55
W all M material Selection ...................... ................ ............................................ 56
Ceramics............... ............. ..................57
Refractory M etals ............................................... ........ 57
Fission Product Test ........................................................... 58
Design Envelope.............................. ......... ...... ...........61
Coolant Temperature Selection ............... .......................... ......... 61
Balance-of-Plant ................ ...... ........................... 63
Interface Freezing Phenomenon ............... ....................... .......... 67
Rating and Sizing................................................................... ...............7
Plate CHEX Rating and Sizing............... ..............73
Plate heat exchanger pressure losses.......... ........................................77
Channel optim ization ............................................ ............... 79
H e/X e influence.................. .......... ................. 80
Tube CHEX Rating and Sizing ............................. ............... 81
Optimum dimension fraction .......................................... 83
Power rating ......................... ... ............... ........ 85
Coolant Channel Pressure and Velocity .......................................86
7 SUMMARY AND CONCLUSIONS................................................................... 89
Thermodynamic Performance ................ ................................89
Computational Tools ............................................... ........ 89
T herm odynam ic T ools..................................................................90
Pressure Loss .......................................................91
Interface Freezing Phenom enon ........................................... ............... 92
Channel V elocity Considerations ........................................................ 92
Rem arks ...................................................... ........ 92
A EXAMPLE OUTPUT OF THE TOPPING CYCLE CODE...............................94
B LAGRANGE M ULTIPLIERS ....................................................... 96
LIST OF REFERENCES ....................... ......... ..........97
BIOGRAPHICAL SKETCH .............. ..... ......... ................100
LIST OF TABLES
6-1 The candidate wall material candidates with selection criteria were taken from
published data in D eW itt. .......................................................................... ...............56
6-3 The dissociation mole fractions shown are for a starting mixture containing one mole
of SiC, 0.9 moles of helium and 0.1 moles of UF4............... ...............57
6-4 The dissociation mole fractions generated from the CEA code are for one mole of W
reacted with 0.9 mole of He and 0.1 Mole of VC14. ........................................58
6-6 The CEA equilibrium calculation of W with the Boersma-Klein et al. fission product
inventories show that tungsten does not bond with any of the fission products
present in the system ..................... .......... ...... ...... 60
6-7 The empirical correlations compared below are given are for frictional and
accelerational loss only. Because of their small contribution to the total pressure
head, including gravitational head would give a negative pressure loss...............78
A D ata index description .............................................................................................95
LIST OF FIGURES
1-1 The influence of the diffusion layer on the vapor partial pressure..............................7
2-1 Schematic Diagram showing optimum topping cycle operating conditions. The
regeneration effectiveness: 0.25, VMHD isentropic efficiency: 0.7, Compressor
isentropic efficiency: 0.8. The reactor power could be 100MW or 1GW...........12
2-2 Bottoming cycle schematic showing the split stream configuration to accommodate
topping cycle intercooling. Later this separate cooling stream will prove
advantageous for reducing the mass flow rate through the heater and therefore
eliminating unnecessary pressure loss penalty in the CHEX............... ...............17
2-3 T-s Diagram depicting cycle operating characteristics....................................18
3-1 UF4 vapor relative enthalpy data.................................... .................. 23
3-2 UF4 vapor relative entropy....................................24
3-3 UF4 vapor thermal conductivity comparison.................... ......................................26
3-4 UF4 vapor dynamic viscosity .................................... ...............26
3-5 Temperature dependent helium mole fraction curve fit ..........................................28
3-6 Temperature dependent UF4 mole fraction curve fit .....................................29
3-7 Temperature dependent mixture enthalpy curve fit...................................29
4-1 Thermal circuit showing the parallel latent and sensible thermal resistances in series
with the wall and coolant channel convective thermal resistances. The figure
nomenclature represents thermal resistances instead of HTC's..............................35
4-1 Schematic of CHEX Code. The wall resistance is not shown in the algorithm
because it is a constant not a variable................................................................ 41
5-1 Equivalent electrical circuit with the latent transferred to the wall modeled as a
source term. .......................................................48
5-2 Thermal circuit shown with the load resistance removed. The notation Rw stands for
the series resistance of the wall conduction and coolant channel convection..........48
5-3 Thevenin equivalent circuit analysis. The node temperature difference is shown as
shorted out and the current source is shown as an open circuit. ..............................49
5-4 Thevenin equivalent circuit with the total heat transfer to the coolant channel drawn
as qload............... ............................................ .....................49
5-5 Axial heat flux vs. temperature comparative analysis comparison between DLM and
TEM. The TEM model shows higher mass flux because there is nod diffusion
layer resistance m odeled. ............................................................. 51
5-6 Axial heat flux vs. the axial dimension comparative analysis comparison between
DLM and TEM. The DLM height is greater by 7%...........................................52
5-7 Condensing HTC for the DLM model. The HTC goes to virtually zero as vapor is
condensed. ............................ ............. ......... 53
5-8 Comparative analysis HTC comparison between DLM and TEM calculations. .......53
6-1 VCR online refreshment scheme for online refueling and fission product
6-2 Coolant delivery flow arrangements: (a) Series flow arrangement (b) Parallel flow
arrange ent ...................................... ................................ ........ 62
6-3 Mixed He/UF4 portion of topping cycle. The cycle pressure ratio of 5 and the MHD
isentropic efficiency is 0.7. The portion of the CHEX/Reg between 1700K and
1950K is the superheated portion of the MHD output.........................64
6-4 Separated helium portion of the topping cycle. The regenerator effectiveness is 0.1
and the compressor efficiency is conservatively estimated as 0.8. .......................64
6-5 Separated helium portion of the topping cycle. The regenerator effectiveness
is 0.3. ................................................. .........65
6-6 Mixture portion of the topping cycle. Pressure ratio is 10. ....................................66
6-7 Separated helium portion of the topping cycle. The regenerator effectiveness is 0.5
and the pressure ratio is 10. .............................................. ............... 67
6-8 Interface freezing anomaly observed with zero coolant bypass..............................68
6-9 Reflector cooling allowed increasing the interface temperature. Reflector cooling is
equal to 10% of reactor power. .............. ..................... ...... ....... ........ 69
6-10 Condensing HTC with 55% of the bottoming cycle working fluid going through the
heater. The reflector cooling is 14% of the reactor power. ..................................70
6-11 Effect of axial enhancement on CHEX heat flux profiles and axial height. ............71
6-12 Heat exchanger dimensions vs. channel aspect ratio for 100 channels....................75
6-13 Channel geometry and aspect ratios for 100 channels ..........................................76
6-14 Heat exchanger dimensions vs. aspect ratio for 500 channels ..............................76
6-15 O ptim um aspect ratio.....................................................77
6-16 Hot side pressure loss using the equivalent viscosity correlation .........................79
6-17 Heat exchanger geometry for r 0.01 ........................................ 80
6-18 Hot and cold side losses for the He/Xe mixture ....................................... ...80
6-19 Tube channel geom etry ............................................... ............... 82
6-20 Channel pressure loss vs. dimension fraction.............................. ......... ......83
6-21 Axial height vs. dimension fraction for different number of channels .....................84
6-22 Channel pressure loss vs. reactor power level...................................................85
6-23 Lateral dimension vs. reactor power level...................................................86
6-24 Cold channel velocity profiles at varying pressure ..................................... 87
A-1 Sample output of the thermal design code package...............................................94
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 Engineering
COMPACT HEAT EXCHANGER DESIGN FOR TRANSFERRING HEAT FROM A
VAPOR CORE REACTOR INTO A GAS TURBINE POWER PLANT
Samuel E. Bays
Chair: Samim Anghaie
Major Department: Nuclear and Radiological Engineering
The very high temperature vapor core nuclear reactor offers so many advantages in
terms of fuel management, plant efficiency, fuel cycle economics and waste minimization
that it is the subject of interest for 21st century nuclear power technology. The vapor core
design has always been blocked from prototype development by engineering problems
related to containment of high temperature fluoride gasses and their effect on plant
components such as the heat exchanger. In a vapor core, the gas/vapor phase nuclear fuel
is uniformly mixed with the topping cycle working fluid. Heat is generated
homogeneously throughout the working fluid, thus extending the metallurgical heat
source temperature restriction. Because of the high temperature, magnetohydrodynamic
generation is employed for topping cycle power extraction. Since magnetohydrodynamic
generators only work in high temperature partial plasma domains, they are ineffective for
deriving power from lower temperature hot gas.
The usable heat energy in the magnetic generator exhaust is recovered in a heat
recovery Brayton power cycle to be converted into electricity. The heat is transferred
into this bottoming cycle via a compact heat exchanger. This work addresses the design
issues pertaining to balance of plant and optimizing the compact heat exchanger design.
A series of computer codes was written to define the design envelope as well as rate and
size the heat exchanger itself. Various issues regarding pressure loss, channel velocity,
pressure gradient across channel walls and the high vapor freezing point guide a natural
design evolution. The working fluid of the topping cycle is helium and uranium
tetrafluoride vapor. It is well known that the presence of a non-condensable gas in vapor
greatly impedes the condensation heat and mass transfer towards the condensation
interface. This non-condensable gas entrainment or diffusion layer problem was
addressed in the heat exchanger calculation. A novel diffusion layer theory algorithm
was adopted to calculate a condensing heat transfer coefficient that was used to model the
sensible and latent heat transfer as parallel processes. The heat exchanger computer code
models these parallel processes in a one-dimensional nodal analysis scheme with the hot
condensing channel in counter flow with a coolant channel. The independent variable
separating each node is temperature change, thus allowing channel heat transfer area to
be calculated as output along with other thermal hydraulic deliverables such as heat flux,
pressure loss, channel velocity and Mach number.
The ultra-high temperature vapor core reactor (VCR) has been a common
conceptual side note for advanced nuclear power generation because of its novel
approach to the nuclear fuel cycle and design simplicity. Simply stated, a VCR is
essentially a hollow drum surrounded by an external beryllium oxide (BeO)
reflector/moderator. A mixture of fluidized uranium fuel and gas coolant passes through
the core where reflected neutrons returned from the BeO force a chain reaction.
Historically this concept has always illuminated a definite potential for futuristic
application of nuclear power technology. However, VCR power plant designs have never
successfully been taken from the drawing board and scaled laboratory experiments into
prototype design. The VCR potential in improved fuel economy, high level waste
minimization and plant efficiency have preserved interest in further developing the
The primary advantage of using a vapor core is that the uranium fuel is in a
fluidized state and homogeneously mixed with the reactor coolant (Diaz et al., 1993).
Modem reactor cores operate at temperatures dictated by the fuel and cladding melting
points. Because of the high thermal resistances in the fuel and cladding the coolant
temperature has to be much lower than the peak temperature of the fuel. This is
thermodynamically disadvantageous because the thermal power could be used more
efficiently if the working fluid temperature better matched that of the heat source. A
vapor core removes these limitations allowing the fuel and coolant to be at the same
Very high reactor coolant temperatures become inviting for application of
magnetohydrodynamic generation (MHD) (Clement & Williams, 1970). MHD uses
Lorentz force to create electromotive field (EMF) by applying a perpendicular magnetic
field to the high velocity ionized gas in the reactor output. MHD generators perform the
same function as conventional turbines but can only operate efficiently at temperatures
above 1800K. At these ultra-high temperatures the dissociation of the uranium
tetrafluoride fuel (UF4), fission products and electrical conductivity enhancing seed
gasses becomes pronounced. MHD allows high volumetric electric conversion at
temperatures beyond conventional turbo machinery metallurgical limitations and fully
utilizes the VCR's high volumetric power generation ability.
Since MHD only works in high temperature partial plasma domains it is ineffective
for deriving power from lower temperature hot gas. Therefore, the usable heat in the
MHD exhaust must be recovered in a heat recovery Brayton power cycle to be converted
into electricity. Previous studies at the university level have focused on the VCR and
MHD components. The largely unexplored avenue of the VCR/MHD plant design is the
thermal hydraulic performance of the heat transfer system (HTS). In order to minimize
construction cost associated with nuclear plant containment structures, the HTS physical
size must be minimized. At the same time pressure drops in the topping and bottoming
cycle fluids must also be minimized in order to maximize plant performance.
Waste heat recovery from the MHD using a compact heat exchanger (CHEX)
allows further valuable thermal power to be extracted from the hot topping cycle rejected
heat at temperatures within operating conditions of conventional Brayton power cycle
turbo-machinery. This is where a combined cycle becomes useful for thermodynamic
gain. The freezing point of uranium tetrafluoride is 1309 K (McBride et al., 2002).
Therefore, heat rejection from a topping cycle containing the VCR/MHD primary loop
must operate above this temperature. Coincidentally, this temperature roughly
approximates the maximum allowable turbine inlet temperature of most modern Brayton
power cycle engines (General Atomics, 5/2/2004). This fact becomes extremely useful
because the CHEX coolant must be kept at a temperature high enough to keep the heat
exchanger wall temperature above the UF4 freezing point.
This work details the thermal hydraulic performance of the compact heat exchanger
and the selection of thermodynamic state points for the topping and heat recovery
bottoming cycles. The thermodynamic analysis establishes a design envelope for the
CHEX design. This design envelope supports evaluating heat exchanger input and
coolant temperatures as well as mass flow rates.
The deliverables of this study entail a balance of plant methodology and its impact
on CHEX rating and sizing. However, it should be stressed that though simple control
volume relationships have been used to analyze topping and bottoming side operating
conditions the detailed balance of plant design and components other than the CHEX are
outside the scope of this work.
The rating and sizing problem takes into account the expected pressure losses,
maximum fluid velocities and problems encountered with the UF4 freezing point. The
work investigates the affect of plant power rating and bottoming cycle working fluid
composition on CHEX mass flow rates and presents a natural design evolution of the
CHEX channel geometry based on those two variables. Special effort is given to
addressing CHEX materials feasibility for hot gas containment and safety considerations.
This work is intended as a preliminary design discussion that makes an attempt to
characterize the thermal hydraulic performance of the heat exchanger. Though
quantitative assessments are made, the lack of experimental data limits the accuracy of
the design calculations to within the assumptions provided in the discussions to follow.
This study frames the potential ability of the compact heat exchanger within its
applicability to transferring usable heat energy into the gas turbine cycle.
History and Design Evolution
Analytical studies on the VCR began in the United States at Los Alamos National
Laboratory in 1955 (Bell, 1955). The reactors considered were fueled by uranium
hexafluoride (UF6) gas and surrounded by a spherical moderating reflector composed of
heavy water, beryllium or graphite. These reactors were coined "cavity reactors" (Diaz,
1985) because VCR power density is dependant on the molecular density of the fuel gas
at less than atmospheric pressure for those systems. The early history of VCR theoretical
and analytical studies is outlined by Diaz (Diaz, 1985).
Early studies. A closed system reactor separated by an optically transparent
containment vessel from the rocket propellant known as the nuclear "light bulb" was
theorized by Latham (Latham, 1966). Also, a coaxial flow open system gas core was
considered in 1971. A problem encountered with these systems was materials
incompatibility with the UF6 gas at extreme high temperatures.1
1 Today refractory metals and ceramics research is increasing our understanding in
materials technology. However, metallurgical limitations and containment are still the
dominating technological issues in modem VCR concepts.
The first comparison of theoretical predictions and experimental data were carried
out and reported in Moscow in 1959. The core had internal moderation (Beryllium) and a
graphite reflector. It went critical with 3340 g of uranium at 90% enriched in U-235
(Diaz, 1985). The first such study in the United States was conducted in 1962.
Criticality studies for the gas core were continued throughout the 1970's.
Much of the earlier investigations centered on developing the gas core concept for
rocket engines in which nuclear energy was converted into thrust by expulsion of heated
gasses through a rocket nozzle in the plasma state. However, in 1969 the Rand
Corporation reported a study on a theoretical 4000MW thermal spherical plasma core
with a five foot radius at 11 atmosphere pressure. The central cavity was surrounded by a
moderating reflector region and banks of energy conversion devices. A significant series
of theoretical and experimental investigations were carried out at The Georgia Institute of
Technology from 1968 through 1975. These studies focused on plasma cores, breeder
reactor power plants and advanced energy conversion systems with extensive work in
MHD power extraction.
Present work. In 1985, the University of Florida Innovative Space Power and
Propulsion Institute (INSPI) was chartered and sponsored by the Strategic Defense
Initiative's Innovative Science and Technology Office as a consortium of university and
industrial research on advanced nuclear power concepts. The primary concept initially
considered was a uranium tetrafluoride mixture in alkaline metal working fluid in a
closed loop Rankine cycle for electrical power generation using an MHD. UF4 was
chosen instead of UF6 because of its chemical stability with the temperature resistant
refractory metals and their carbides necessary to contain the high temperature plasma.
Questions pertaining to the containment of alkaline metals challenged the
feasibility of a terrestrial application of the concept. Therefore, the liquid metal topping
cycle was replaced with a UF4/helium Brayton cycle. This Brayton system will be the
choice topping cycle evaluated in this work (FAS, 5/2/2004).
Also, topping cycle pressures much higher than previous studies are being
considered. The higher pressure allows for a higher power density and can also be used
to accelerate the fissioning plasma through a nozzle into the MHD duct. This
acceleration increases the plasma velocity enabling Lorentz force to produce large EMF
(Anghaie et al., 2001).
Heat Transfer Issues and Thermal Design
Because of its very high dew point the UF4 vapor must be fully condensed and
separated from the helium coolant to get the maximum temperature change for the
topping cycle working fluid. This creates a complicated design issue because sensible
and latent heat transfer from the MHD exhaust must be considered.
Latent heat transfer is the heat required for phase change of vapor molecules into
liquid. This phase change occurs at the temperature of the channel wall. Normal
condensing two-phase heat transfer calculations assume that the channel wall temperature
is at the same temperature as that of the bulk fluid. The vapor heat transfer coefficient is
modified by a gas entrainment correction factor. This approximation is acceptable when
considering only small amounts of non-condensable gas but is inappropriate for the
CHEX calculations because 95% of all the gas/vapor molecules are non-condensable.
According to Newton's law of cooling the convective or sensible heat transfer of the
gas/vapor phase dictates that the bulk fluid temperature must be higher than the channel
wall. Therefore a better heat transfer model must be devised for modeling this parallel
heat transfer process.
The modeling of vapor condensation in the presence of non-condensable gasses has
been a pre-occupation of many thermal hydraulic studies throughout the 1980's and
1990's. It is well known that the presence of a non-condensable gas in a condensing
vapor greatly impedes the vapor heat and mass transfer process towards the channel
walls. This is because as the vapor cools and becomes saturated it moves toward
condensation sites at the wall to form droplets and eventually a condensate film. The
non-condensable gas becomes entrained in the vapor as it moves to the condensation
interface and blocks the cooling vapor from reaching the wall. An equilibrium balance
ensues as the entrainment process towards the wall matches non-condensable gas
diffusion away from the interface. In order for the channel total pressure to remain
constant, the vapor partial pressure decreases towards the wall as the non-condensable
partial pressure increases through entrainment.
Figure 1-1: The influence of the diffusion layer on the vapor partial pressure. Depiction
adapted from Collier (Collier, 1972).
There are three well adopted methods for calculating the true condensation heat and
mass transfer coefficients that account for this entrainment obstruction. These are the
degradation factor method initially proposed by Vierow and Schrock, a diffusion layer
theory initially proposed by Peterson, and a third a fundamental mass transfer
conductance model (Kuhn et al., 1997). The diffusion layer model is the most
mechanistic approach for engineering computer model simulation and will be discussed
in detail as it is used in the analysis of the CHEX (Peterson et al., 1992).
A series of computer codes were created for modeling the topping and bottoming
cycle thermodynamic performance. Another set of computer codes prepared curve fits to
map the changing thermo-physical properties of the condensing two-phase mixture. This
data was then used in the CHEX heat transfer calculations.
The CHEX design process progressed through a natural evolution due to increasing
design constraints from a plate type heat exchanger to a tube bank class of heat
exchanger. Both models were designed using a one dimensional nodal analysis code that
assessed the temperature and mass flow rate inputs from the design envelope and
calculated performance criterion such as the pressure drop heat exchanger rating and size.
The code also received tabulated and polynomial fits of enthalpy, entropy and mole
fractions generated in the topping cycle analysis. These properties were generated using
a NASA thermo-physical property code. The CHEX code used the mole fraction data
and mixing relationships to calculate thermal transport and fluid properties of the
Comparative Analysis Calculations
Two comparative analysis studies were conducted. The first comparative analysis
checked the validity of the thermo-physical property code against published data on
uranium fluoride and helium data. The second comparative analysis analyzed the
difference between the novel diffusion layer theory used in a nodal analysis technique for
the heat transfer calculations and a more rudimentary heat conductance model.
Thermo-Physical Property Comparative Analysis
The thermo-physical property code used is the NASA Glenn-Lewis Chemical
Equilibrium with Applications package (CEA). This code uses minimization of free
energy to determine mixture chemical equilibrium while considering the various possible
fluoride species and their dissociation reactions. The code was found to be quite reliable
for thermodynamic properties such as equilibrium mole fraction, enthalpy and entropy. It
did show problems calculating density for pure liquid such as when the UF4 condensate
is being pumped to the reactor inlet pressure. It also gave poorly accurate data on
thermal transport properties such as viscosity and thermal conductivity. However, it did
provide good mixture specific heat data. The reason for the thermal transport property
discrepancy is that though the NASA code uses well adopted thermodynamic data
provided by Gurvich; it has zero thermal transport data such as thermal conductivity and
viscosity on uranium tetrafluoride (McBride & Gordon, 1996).
If the code does not have data on a particular species it derives an estimated value
of the property based on a fundamental collision integral for approximating molecular
interactions. Therefore, the CHEX fluid properties: thermal conductivity, viscosity,
density and specific heat are calculated using mixing relationships with well published
data and the mole fraction data which has proven to be accurate (Anghaie, 1992)(NIST,
Diffusion Layer Theory Comparative Analysis
The heat transfer results were compared with an alternate theory during the plate
heat exchanger design. Diffusion layer theory (DLT) allows for the sensible and latent
heat transfer to be modeled as parallel thermal resistances. The comparative analysis
calculation used the same nodal scheme as the counter flow heat exchanger model but
modeled the latent heat transfer as a heat current source. A similar thermal circuit to the
DLT model was constructed. The comparative analysis modeled the convective
resistance in parallel with the current source. The similarities between the thermal circuit
analysis and electrical circuit analysis made possible the application of Kirchov's current
law. A Thevenin equivalent circuit (TEC) could then be constructed using the wall and
coolant channel resistances as the load.
This circuit analysis methodology made possible the construction of heat flux plot
comparisons between DLT and TEC. The calculations showed that modeling the
diffusion layer with DLT gave a larger surface area required to transport the same amount
of heat to the cold channel than TEC. This is because diffusion layer heat and mass
transfer resistances decreases the heat flux across the length of the counter flow CHEX.
The TEC heat fluxes were overall greater because these diffusion layer resistances were
THERMODYNAMIC ANALYSIS METHOD
The CHEX thermal design consists of four steps: (1) thermodynamic analysis of
topping and bottoming cycle performance, (2) thermo-physical property data base
development, (3) plate heat exchanger scoping calculations and comparative analysis and
(4) design considerations for tube heat exchanger design evolution. The thermodynamic
calculations are used to set the CHEX inlet and outlet conditions. During the
thermodynamic analysis the appropriate thermo-physical properties are data based and set
to fit equations with respect to temperature. A least squares algorithm is used to generate
these fits on the fly. These set points and fit equations are then fed into the scoping
calculation where they are used to generate temperature and axial dependent heat flux
plots for the counter flow CHEX. This code also calculates the frictional, accelerational
and gravitational head for the condensing fluid and the coolant channel frictional head.
The topping cycle is essentially a VCR/MHD Brayton power cycle with a mixture
of helium and UF4 vapor as the working fluid. The vapor must be condensed into a
liquid and separated so that the non-condensable helium may be compressed to the
reactor pressure without damaging the compressor blades from impinging liquid droplets.
After separation, the condensate is re-circulated back to a mixer just prior to the VCR.
The pumped UF4 liquid may then be pre-heated to saturated vapor before mixing or it
may be aspirated directly into the reactor where it is vaporized.'
2081 K Recuperator
91 bar Separator
1379 K "-' 1379 K
87 bar Mixer 87 bar
Figure 2-1: Schematic Diagram showing optimum topping cycle operating conditions.
The regeneration effectiveness: 0.25, MHD isentropic efficiency: 0.7,
Compressor isentropic efficiency: 0.8. The reactor power could be 100MW
The separated helium exchanges heat in a pre-cooler before it is compressed back
to the reactor pressure where it rejoins the UF4. Finally, the mixture enters the BeO
moderator/reflector and is heated to the reactor outlet temperature where it travels
1 This second scheme may have a neutronic advantage because the average fuel density
because the average fuel density near the reactor inlet will be enhanced by the liquid
droplets so that the lower part of the VCR could be considered a liquid drop reactor.
through the MHD duct. The MHD duct expands and cools the now partial plasma until it
is at the heat exchanger inlet temperature and pressure thus completing the loop.
Precooling is generally required for the helium feed because the isentropic
efficiency of the compressor stage dictates a lower compressor inlet temperature than at
separation in order to arrive at the proper mixer temperature and pressure. Thus the
amount of precooling is an indication of thermodynamic losses in the compressor. These
losses degrade the overall topping cycle performance. Therefore, regenerative heating of
the helium and a two-stage compressor process with intercooling is proposed to increase
the overall thermodynamic cycle efficiency.
There is a side benefit to incorporating intercooling in the design. Because
intercooling occurs in a separate heat exchanger device than the UF4 condenser (CHEX)
and precooler, the selection of coolant is independent of the main bottoming cycle
working fluid. This means that the mass flow rate through the intercooler may not be the
same as that through the condenser and precooler which is the primary heat source for the
bottoming cycle. The reduction in mass flow rate through the heater will have a positive
effect on the CHEX coolant channel pressure loss and fluid velocities.
Topping Cycle Code Description
The CEA code proved as a valuable tool for rapid determination of thermodynamic
state points in the topping cycle parametric evaluation. CEA input decks are short and
simple to create. They require two properties to set a state. The code uses an extensive
library of well adopted references on uranium and uranium Fluoride thermophysical
properties. Not only does it determine condensed species but it can also be used to
evaluate dissociation and ionization mole fractions in the reactor and MHD throughput.
A file writing code was created using the C++ language for rapid generation of
CEA input decks for each state point in the primary cycle. C++ was used because of its
wide variety of features offered in its file input/output system. Another C++ code reads
output files from CEA and gleaned the thermodynamic properties for storage in a
separate text file that summarized all the CEA runs. A batch file may be used to run the
file writing code, CEA and the file reading code in tandem. The file writing code knows
what state point calculation to perform by reading a separate tracking file that records
which calculation was previously performed. The file reading routine reads this file also
and advances the number stored in the file to the next calculation.
Using the tracking file and batch program allows for the writing, CEA and reading
routines to be performed as though they were a single package until all topping cycle
calculations are performed. It also allows for the CEA code to be used in its original
release form from the Glenn Lewis Lab without being altered. A table of state point
calculations is given in Appendix A to show an example output of topping cycle state
The first state calculated is the reactor output. The user is asked for the reactor
temperature and pressure. After the output file is read, the user is asked for the system
pressure ratio. The file writing code then calculates the MHD outlet pressure; reads the
entropy from the previous run and generates an entropy/pressure CEA input file for
isentropic expansion in the MHD. The user is then asked for the isentropic efficiency of
the MHD usually taken to be about 0.6 or 0.7(INSPI). The file writing code calculates
the actual MHD exit condition. In the next calculation, the file writing code calculates
the UF4 dew point at the MHD outlet by matching the left hand side and right hand side
of the following saturation vapor pressure curve (Anghaie, 1992).
In(P(a)) -37977(1000)T 1 + 74.217(0.3) -7.0(0.7) n(T)
ln(P(at)) = -38453T 1 + 74.88 -7.051n(T)
After this point, the batch program is then thrown into an infinite loop. Each loop
completion decreases the mixture entropy until the file reading program reads a UF4
mole fraction less than 0.1%. This is the criterion for full condensation. The next series
of calculation steps focuses on the UF4, helium and mixture properties before and after
The helium or preheater side of the regenerator is evaluated first in order to
calculate the compressor inlet and exit states. The user is asked for the regeneration
effectiveness. This is defined as the ratio of the MHD exhaust heat used for preheating
the compressed helium just prior to mixing. Knowledge of this and the helium mass flow
rate is used to calculate the preheater inlet or compressor chain exit condition. The
mixture mass flow rate is calculated from an energy balance calculation for a control
volume around the reactor. The helium mass flow rate is backed out using the mixture
mole fractions set at: UF4 (5%) and He (95%).
Mt 'reactor h,,,) (2)
Hehum -(YHeX MHe nXxm,x (2a)
The user is asked for the compressors' isentropic efficiency. This is used to
calculate the intercooler state points and the precooler entrance state. Since intercooling
is used, an optimum pressure ratio must be determined to maximize cycle efficiency. The
optimum intercooler pressure is given by (Todreas & Kazimi, 1993):
This equation assumes that the high and low compressors' inlet temperatures are
the same and the helium is modeled as an ideal gas. Knowing this and the compressor
chain outlet temperature, e.g. the preheater inlet temperature, the inlet temperature for
both compressors is determined.
Tout,s Phigh (4)
Rearranging Equation (3):
Tout,a i ou t,s
]7n ]7 n
The intercooler entrance state is defined as the low pressure compressor exit and
the intercooler exit is determined as the high pressure compressor inlet. Therefore, the
intercooler exit is already calculated and the isentropic efficiency definition of Equation
(3) for the first stage compressor may be used to determine the intercooler inlet.
The bottoming cycle consists of the coolant feed through the topping cycle
condenser and precooler (now referred to as the heater stages or heater), turbo-machinery
and the bottoming cycle regenerator, pre-cooler and intercooler. A split stream cools the
topping cycle intercooler and rejoins the heater flow before entering the turbine. The
application of the bottoming cycle heat transfer devices is very similar to that of the
topping cycle. The intercooler reduces the work requirement to compress the low
pressure helium/xenon working fluid up to the heater pressure.
Precooler a g Regenerator
Figure 2-2: Bottoming cycle schematic showing the split stream configuration to
accommodate topping cycle intercooling. Later this separate cooling stream
will prove advantageous for reducing the mass flow rate through the heater
and therefore eliminating unnecessary pressure loss penalty in the CHEX.
Though intercooling reduces the compressor work required, it also reduces the
heater inlet temperature from what it would be without intercooling. This lower
temperature would require more heat transfer to achieve the desired turbine inlet
temperature. To supply this additional heat transfer, the regenerator recovers turbine
waste heat and recycles it by preheating the He/Xe to the heater inlet temperature, thus
providing the necessary power required to heat the fluid to the maximum bottoming cycle
An optimum bottoming cycle coupled with the topping cycle description in Figure
2-1 operates according to the following temperature-entropy diagram.
25 26 27 28 29 30 31 32 33
Figure 2-3: T-s Diagram depicting cycle operating characteristics
Here, the minimum cycle temperature is about 300 K and the maximum heater
temperature is about 1300K. The pressure ratio is 6.24 and the first law cycle efficiency
is 40%. It should be noted that the topping cycle condenser pressure for this model is
about 20 bars and the heater side pressure is 50 bars.
CONDENSATION PROPERTY MAPPING
The condensation thermodynamic and thermal transport properties were mapped
with CEA between the UF4 dew point and the CHEX separation condition. Property
tables and fit equations were generated for:1
* Sonic Velocity
* Specific Heat
* Thermal Conductivity
* Prandtl Number
* helium Mole Fraction
* UF4 Mole Fraction
* Gas/Vapor Phase Enthalpy
* Mixture Enthalpy
Most important of these were the fits for mixture enthalpies, specific heats, gas
and liquid phase mole fractions because those curve fits were directly used in the CEA
It is understood that there is thermodynamic pressure loss caused by removal of the
UF4 vapor phase from the bulk gas mixture. However, the condensing mixture pressure
was assumed constant for simplification of the fit equations as a single dimensional fit.
1 These properties were fit to a second order polynomial.
In the heat transfer calculations to follow, the loss of vapor atoms in the condensing
mixture is compensated by addition of equal moles of helium gas molecules. This has the
effect of maintaining the pressure constant while negligibly increasing the mass flow rate
in the CHEX.
The CEA Code
CEA is a thermodynamic and thermal transport property evaluation code. It is
commonly used for finding chemical equilibrium of reaction products, rocket
performance calculations, detonation problems and modeling thermodynamic systems
with complex species compositions. Some applications include the design and analysis
of compressors, turbines, nozzles, engines, shock tubes, heat exchangers, and chemical
All thermodynamic properties are orientated to a reference standard state. For a
gas, the standard state is the hypothetical ideal gas at the standard-state pressure. For a
condensed or frozen species the standard state is the substance at the condensed phase at
the standard-state pressure. Most recent versions of the code have used a standard state
pressure of one bar (McBride & Gordon, 1994).
CEA uses the minimization-of-free energy formulation for finding chemical
equilibrium between reactant species. This can be accomplished using two different
methodologies: (1) Minimization of the system Gibbs Energy or (2) Minimization of the
system Helmholtz energy. Gibbs energy is used when pressure is specified as one of the
thermodynamic states. The Helmholtz method is used when specific volume or density is
given. (McBride & Gordon, 1994) For N species, the Gibbs energy per kilogram of
mixture is defined as:
g = PL ni (6)
Where chemical potential per kilogram-mole of species j is defined as:
P T,P (7)
= n+RTln -+RTln(P) for gasses (8a)
PL = p for condensed phases (8b)
Note: the superscript o stands for the chemical potential in the standard state.
The minimization process is performed by making use of Lagrangian multipliers2
and subjecting the minimization process to certain constraints such as the mass balance:
Ya,n, b = 0
b- bO = 0
Where stoichiometric coefficients aij are the number of kilogram-atoms of element i
per kilogram-mole of species j, bi0 is the assigned number of kilogram-atoms of element i
per kilogram of total reactants.
In order to find the minimum extremum of (6) using the constraints of (9) we must
first observe the first derivative test of (6) and (9). Then, multiplying the derivative (9)
by the Lagrangian multiplier and adding to derivative of (6) produces:
2 See Appendix B for a description of Lagrangian Multipliers
=G =1 + /1,a)n ( -b hoA,= o (10)
J=1 1=1 J=1
Where: G=g+ _= A(b, bI )
Equation (10) is the requirement for equilibrium. Minimization is obtained
iteratively by updating nj, 2j, moles of gas components and when required temperature.
This is done by using a Newton-Raphson method. Using the Newton-Raphson method
and the extensive property relationships for gas mixtures, CEA calculates thermodynamic
properties of the system at equilibrium.
The reliability of the CEA program for the CHEX calculation lies in the agreement
between CEA and accepted literature on uranium Fluoride thermophyscial property
equations and data. Therefore, CEA thermodynamic data was tabulated at various
temperatures for UF4 and UF6 and benchmarked with enthalpy and entropy data derived
from specific heat relationships from Anghaie (1992) for UF4 and Dugan and Oliver
(1984) for UF6 (Anghaie, 1992)(Dugan & Oliver, 1984).
Enthalpy and entropy were calculated for the comparison with CEA using the
incompressible perfect gas model with constant specific heat for UF4 gas (Moran &
h2- h = T2 C(T)dT C( T1) (11a)
s2 -S SI 2- T =C C In (11b)
It is understood that the mixture may not perform exactly as a perfect gas at high
temperatures but the assumption is appropriate to ascertain accurate general behavior.
Liquid enthalpies and entropies were calculated using the same integration from
Equation (11) but with temperature dependent specific heats. Where the specific heat for
UF4 liquid is given by:
C,(J/mol -K)=136.3+3.7x10-2T 3200 (12)
And for UF6 liquid:
C (kJ/kg-K) = 0.448+1.86(10-6)T -7.71(103)T -2 (13)
CEA thermodynamic properties are taken from Gurvich (1982) (McBride et al.,
2002). Thermodynamic data is presented standardized to a fixed temperature reference
datum (hi and si). All CEA runs were generated with a pressure of one atmosphere or
one bar with the exception of properties noted with an which notes that these data for
vapor were generated inside the UF4 two-phase vapor curve for constant entropy and
varying pressures. Dissociation and ionization phenomena were not modeled except for
the large starred data in Figure 3-1 and Figure 3-2.
Inspi NASA ThermoBuild CEA x INSPI* CEA* x Ion
CO 100 -
1 1150 1650 2150 2650 3150
Figure 3-1: UF4 vapor relative enthalpy data
Inspi NASA ThermoBuild CEA Ion
: 0.125 -
1750 1950 2150 2350 2550 2750 2950 3150
Figure 3-2: UF4 vapor relative entropy
The CEA data shows very good agreement with the Anghaie and on-line
Thermobuild library. As expected the Thermobuild data is almost identical to the CEA
data because both sources are produced by the same institution with the same data
library. The Anghaie data agreed very strongly even up to high temperatures. The small
discrepancy at temperatures greater than 2000K should not affect the CHEX design
calculations because they are outside the design envelope.
Thermal Transport Properties
Transport properties mapped with CEA were also compared with reference
properties. CEA does not have thermal transport properties for all species in the
thermodynamic database. In other words, it has data on UF6 but not for UF4. Therefore,
CEA estimates thermal transport data for UF4 using the collision integral:
Q, = In 14(14)
This lack of accurate data made it necessary to compare with an alternative source.
There is a discrepancy between CEA data and reference data taken from Anghaie
as high as 20%. Therefore, CEA thermal transport data fits were not used in the CHEX
heat transfer calculations. Gas/vapor thermal transport properties used in the CHEX code
were derived using Anghaie's data and the CEA equilibrium mole fractions in the
following equations (Watanabe & Anghaie, 1993).
7 =1 Ny, (15)
Y + (1 8
N =Z N (16)
y 4 77PJ / Po+ A
M [ 2.4 1(1, -(9) (, -0.)142 (18)
yM, y (KM, M ()
Here the symbol lmix and Xmix refer to the mixture viscosity and thermal
conductivity respectively (McBride & Gordon, 1994).
Other CEA mixture properties may be found using the following mixing rules for
thermodynamic properties (Moran & Shapiro, 2000).
N A =yptot
M=/I y,M, (19) N (20)
u ZyAMuy,, h ZyMlh, s Zy^M^s^ (21)
1- 1 y A/I, Ms 121)
Where M is the apparent molecular weight of the mixture, Mi is the molecular
weight of component i, yi is the mole fraction of component i, pi is the partial pressure of
component i and ui, hi, si, are the specific internal energy, enthalpy and entropy of
mixture component i at the system temperature and partial pressure pi.
1950 2150 2350 2550
2750 2950 3150
Figure 3-3: UF4 vapor thermal conductivity comparison
1950 2150 2350 2550 2750 2950
Figure 3-4: UF4 vapor dynamic viscosity
Least Squares Data Preparation
The least squares program was created for generating thermo-physical curve fit
equations on the fly without having to enter the CEA data into an external spreadsheet or
other software for analysis. The least squares program reads data from a file and assigns
the data to arrays.
A second order polynomial is well suited for fitting most of the data. A set of
linear equations is developed to minimize the error function corresponding to (Echoff,
y(T) =a,+aT+a,T2 (22)
The error function is given by:
E= [y,- (a ,+aT+a2T2)2 (23)
The error function can be minimized by differentiating E with respect to each
coefficient and setting them equal to zero. This forms the set of algebraic equations
which are solved simultaneously.
= 0,i= 1,2,...m (24)
n Ex Ex a
Lx 2x Ex3 aJ = LTxy (25)
EX EX EX a2 X2y
n Ex 2 a y
[X]= Lx x x3 [A]= a [Y]= xy (26)
Ex2 Ex3 x4 a2 X2Y
We can solve for [A] using Cramer's Rule. The drawback for using Cramer's rule
is dimensionality. It only works if there are an equal number of rows as there are
columns in the [X] matrix, the determinant of the coefficients is nonzero and the size of
[X] must be small for computational time reasons. However, for our application
Cramer's rule is efficient enough to give us quick reliable results.
[A]= [Y][x] (27)
Where: [X]-1 is the inverse matrix of [X]
[X] [- -p (28)
The primary use for the fit equations is for plotting the two-phase mixture
enthalpy, species mole fractions and two-phase specific heats. Plotting CEA data against
curve fits derived from the least squares program show reasonable conformity.
He MF (Data) He MF (Fit)
1350 1400 1450 1500 1550 1600 1650 1700 1750
Figure 3-5: Temperature dependent helium mole fraction curve fit
Notice an increasing disagreement for the enthalpy curve fit for increasing
temperature. This disagreement probably stems from number error or some other data
processing anomaly and only appears for enthalpy and only at irregular applications of
the software. Therefore, it is recommended to check the validity of the curve fits before
applying the heat exchanger design code.
UF4 MF (Data) -- UF4 MF (Fit)
1400 1450 1500 1550 1600
1650 1700 1750
Figure 3-6: Temperature dependent UF4 mole fraction curve fit
Mix Enth (Data) Mix Enth (Fit)
1400 1450 1500 1550 1600
1650 1700 1750
Figure 3-7: Temperature dependent mixture enthalpy curve fit
If the fit coefficients need to be modified they can be done manually by changing
the values directly in the global output text file where all fit coefficients are stored. Also,
if this error goes unchecked it can only make the heat exchanger sizing calculation more
conservative because the total heat transfer and coolant mass flow rate for the CHEX are
calculated based off of the thermodynamic output (See Appendix A). The heat transfer
area of the heat exchanger is calculated in the nodal analysis by dividing the heat transfer
per node by the calculated heat flux3: An=qn/(pn. The heat transfer per node is calculated
based on the enthalpy change across the node from Figure 3-6: qn=mdot(AH)n If the
decrease in enthalpy for each node is over predicted then the heat transfer for that node is
over predicted. Hence, it can only increase the heat transfer area.
3 The nodal analysis algorithm will be discussed in detail in Chapter 4.
THERMAL HYDRAULIC MODEL DEVELOPMENT
The most important bottoming cycle device is the main heat exchanger, the heater.
The heater is the main bottoming cycle heat source and takes the place of the burner in
normal Brayton power engines. The two main governing design parameters of the heat
1. Complete phase change of the uranium fuel
2. Low heat exchanger pressure loss with respect to cycle pressure ratio
The compact heat exchanger (CHEX) design was selected because it is commonly
used in many industrial applications where heat has to be transferred between two gas
streams. Compact heat exchangers are desirable when high heat transfer rates are
required but heat transfer coefficients in at least one of the fluids are low. Since gas
convective heat transfer coefficients are low, compact heat exchangers are the choice
device. The CHEX creates a large surface area per volume of heat exchanger by dividing
the flow into many channels separated by plates or tube bundles (Kuppan, 2000).
It is desirable to minimize the size of the heat exchanger because of the price
associated with building large containment structures to house reactor components.
Therefore, an optimum heat exchanger design must be found that does not compromise
pressure loss for space allowance in the plant.
Many times compact heat exchangers employ extended surfaces such as fins and
tubes to enhance the heat transfer surfaces in the heat exchanger volume. The CHEX
design codes do not employ extended surfaces because of the fouling risk due to plating
out of fluorides and fission products. Instead the CHEX design is kept as simple as
possible to make CHEX maintenance as easy as possible. This way if a plate or tube
(depending on channel geometry, discussed later) is damaged or has suffered chemical
deposition; the plate or plates can simply be replaced during routine maintenance without
having to scrap the entire unit.
A computer code was created that models the latent and sensible heat transfer
process in an unmixed counter flow compact plate heat exchanger. The code reads in the
output generated by the topping and bottoming cycle thermodynamic analysis and
generates the total surface area required for heat transfer. It also calculates the total
pressure drop in the hot and cold side fluids.
It was found that the two-phase void fraction of UF4 liquid in the primary stream
was very near one throughout the condensation process. This is due to the very high
density of UF4 liquid compared to the bulk gas/vapor density. Observation of the gas
and liquid phase mass flux and densities on a flow pattern map indicates that the
condensing mixture is in a state of chum flow throughout the entire CHEX.
Churn flow is sometimes referred to as semi-annular flow indicating that the flow is
a homogeneous solution of vapor and liquid though the liquid coalesces near the channel
walls (Collier, 1972). Because of the homogeneous nature of the flow regime it is
assumed that there is no stable condensate film on the CHEX channel walls and the
homogeneous fog flow model may be used for evaluating pressure drop in the primary
channels. Thermodynamic equilibrium may also be assumed to evaluate the changing
gas-liquid thermodynamic state as it is being cooled through the channel.'
The mixture enthalpy and species mole fractions were calculated using the CEA
code for decreasing temperatures throughout the condensing channel. The enthalpy and
mole fraction data from CEA was tabulated and a least squares fit was found to
characterize the thermodynamic states. The coefficients from these curve fits were then
uploaded into the heat exchanger code for easy determination of changes in temperature
dependent bulk fluid thermodynamic and thermal transport properties.
A computer model of latent and sensible heat transfer from a hot channel passing a
condensing mixture of helium and UF4 to an adjacent cooling channel passing a helium
or helium/xenon mixture is analyzed. The diffusion layer model was used to solve the
heat and mass transport problem. The utility of using the diffusion layer model is that a
condensation heat transfer coefficient is formulated allowing for the sensible and latent
heat transfer resistances be modeled in parallel as a single thermal resistance (Herranz et
al., 2001). This parallel equivalent resistance is then modeled in series with the coolant
convective and wall conductive thermal resistances to complete the total thermal circuit.
The code constructs a one-dimensional nodal analysis and calculates the total equivalent
thermal resistance at each node in order to calculate the heat flux at each node. This 1-D
approach has been widely used for the passive cooling system design of the
Westinghouse AP-600 Reactor and the General Electric Simplified Boiling Water
Reactor plant concepts (Herranz et al., 1997, 1998).
1 This work does not address the possible affect of partial or unstable films being
developed and the potential liquid subcooling before the condensate leaves the wall
interface. Further experimental data is required to study UF4 condensate in this flow
Using the hot channel mass flow rate and enthalpy curve fits, the code calculates
the heat removal for a given temperature drop across each node. With knowledge of both
heat transfer and heat flux, the area required to remove heat from each node is calculated.
The code marches node by node until the UF4 vapor is completely condensed. Thus the
total heat transfer area is calculated.
Because of the large temperature change across the heat exchanger, axial heat
conduction between nodes along the channel walls may become an important issue for a
final design analysis. It is not considered for this work to keep the calculations simple
and limited to the thermal hydraulic issues.
The Heat Transfer Model
Sensible and latent heat transfer calculations must describe the heat and mass
transport problem from the bulk mixture to the condensation interface where the vapor is
making phase change. These methods require that the physical conditions at the interface
be known in order to calculate an appropriate mass transfer coefficient. The history of
these types of calculations is outlined by Peterson et al. (Peterson et al., 1992). In 1934
Colburn and Hougen proposed that a balance exists between convective mass transfer and
diffusion of non-condensable gas from the interface. This balance results in a logarithmic
gas concentration distribution near the interface. Colbrun-Hougen type film models can
be cumbersome in practice because they require extensive iterations to match the
condensation mass flux with the heat transport through the condensate film and external
heat removal thermal resistances. Traditionally for vertical surfaces in nuclear
applications an empirical curve fit of total heat transfer coefficient data versus gas to
steam weight ratio measured by Uchida et al. (1965) has been applied. Other researchers
(Henderson and Marchello, 1969 and Vierow and Schrock, 1991) have correlated
condensation data as the ratio of experimental heat transfer coefficient, defined as
qt"/(Tbs-Tw), to the Nusselt solution for the vapor alone.
With lack of experimental data, a very mechanistic approach to heat transfer
degradation may be applied using thermodynamics and a fundamental solution to mass
transport in diffusion layers with the non-condensable gas (Peterson, 2000). Then a
condensation thermal conductivity and heat transfer coefficient are formulated based on
the heat and mass transfer analogy (Herranz et al., 1997) 2
This heat transfer coefficient (HTC) is then modeled in parallel with the
convective HTC for the bulk mixture to calculate an equivalent thermal resistance (See
Figure 4-1). The convective HTC represents the sensible heat input to the wall while the
condensation HTC represents the latent heat transfer to the wall.
Thm0 \AA/ Tom
Figure 4-1: Thermal circuit showing the parallel latent and sensible thermal resistances
in series with the wall and coolant channel convective thermal resistances.
The figure nomenclature represents thermal resistances instead of HTC's.
If the concentration of vapor decreases, the latent heat transfer goes to zero. This
can be seen as the condensation resistance going to infinity as the condensation HTC
goes to zero. This occurs when the bulk gas concentration matches the interface gas
2 It is understood that the discussion to follow describes the diffusion layer in terms of
concentration and entrainment. Because this is only a preliminary conceptual analysis,
other factors such as radial temperature gradient related diffusion and radial property
variations are neglected.
Diffusion Layer Theory Development
The derivation of the Diffusion Layer Model (DLM) is outlined by Peterson, 1992
(Peterson, 2000). We need to develop an energy balance that equates total heat
transmitted through the wall from the hot side to the total heat received by the cold side.
The total (q"t) heat flux through the coolant channel wall must equal the sensible (q"s)
and latent (q"i) heat flux:
h(T, T) = q", = q", +q" = -icM,V, +k, (29)
Where hw represents the combined thermal resistances of the condensate, film and
coolant, ifg is the average heat of formation, c is the total molar density, My is the
molecular weight of the vapor species, k, is the gas/vapor thermal conductivity, and y is
the coordinate normal to the interface.
To calculate the mass transport to the wall we need to calculate the average molar
velocity. The average molar velocity away from the interface, Vi, is related to the non-
condensable gas mole fraction Xg by Fick's law:
c V, = cX cD (30)
Where D is the mass diffusion coefficient determined using the Wilke and Lee
Correlation (Poling, et al., 2001).
The interface is impermeable to non-condensable gas, so the absolute gas velocity
at the interface is zero, thus the condensation velocity is:
C dV, = D lr D ln(Xg,) (31)
Considering a diffusion layer thickness 6g, the condensation velocity is redefined.
V f= (ln(Xgb)-ln(X,)) (32)
At this point it is convenient to define the log mean mole fraction so that:
X ~ ln(Xb/X,)
And rewriting Equation (32) gets: V, = X (Xg, X,) (33)
This will become important later as the condensation velocity becomes dependent
on the change in saturation pressure in the bulk fluid and at the interface. Assuming ideal
gas behavior, the mole fractions can be expressed in terms of the species partial pressure.
V,= D (P, Pvb (34)
The partial pressures of the vapor at the interface, the bulk fluid and the total
pressure are Pvi, Pvb and Pt respectively. Note that Pt= Pvb + Pgb and Pt= Pvi + Pgi.
Notice that the condensation velocity is now dependent upon the difference in
partial pressure in the bulk fluid and at the interface. The Clausius-Clapeyron equation
can be used to relate the partial pressure difference to a difference in saturation
temperature in the bulk fluid and at the interface. This assumes that the bulk fluid vapor
is saturated. Using the equation in the derivation requires that heat of vaporization (ifg)
and relative specific volume (vfg) do not change drastically between the bulk and
interface temperatures. As an approximation the Clausius-Clapeyron equation is:
P, P i
v vb -if (35a)
sat,, s-at,b T fg
For our purposes the fluid specific volume is neglected so that the two-phase
specific volume becomes that for the vapor alone.
vf = (35b)
f9 MX^v' P,
The condensation velocity in terms of temperature difference is now:
RV = vave(v -Tb) (36)
RTve g, ave g
The Sherwood number defines the unitless concentration gradient of vapor at the
interface and can be defined as the characteristic length divided by the diffusion layer
thickness 6g. Combining the latent heat term from Equation (29) with Equations (30) and
(36) we define the Sherwood number in terms of the bulk temperature difference.
L q R2 T3
SA = L- = T L ave (37)
_Tb 2PM2f MD
Upon inspection of Equation (37), the first term on the right hand side is defined as
the condensation HTC. The terms to the right of the characteristic length make up the
inverse of the effective condensation conductivity, defined as:
1 i 2, PMvD,
k T= e fRK 2-- (38)
ln((1 Xgb ) /(1 Xg,))
Where: # =-
In(Xgb / X, )
The foregoing definitions have been made such that the Sherwood number can
describe the latent heat flux in terms of the diffusion layer mass transfer problem.
Equation (37) now takes the familiar form: Sh=hiL/kc or where the characteristic length
in a closed channel is L=Deq-4Af/Pwet: Sh=hi Deq /k,.
Thinking back to the derivation of Xave it becomes clear how the Clausius-
Clapeyron equation is used to calculate the difference in saturation partial pressures or
concentrations for the bulk and the interface temperatures. Earlier this was done to
simplify the form of the condensation HTC. However, the Clausius-Clapeyron equation
is also necessary to attain the mole fractions of vapor at the interface for calculating P in
the condensation conductivity. This can be done by integrating the Clausius-Clapeyron
equation while holding Tave constant so that:
P,, f (T, T)
The ideal gas equation is used to equate the saturation partial pressure ratios to the
mole fraction ratio Xvi/Xvb:
P, c,2M,,RTa XV2M,,RT^
e RTmMvRTe (40)
Pb cCbMRTe XMIRTe
Where c, represents the molar concentration of vapor molecules and R is the mass
specific gas constant. Note that Equation (40) neglects the expansion of gas with respect
to temperature by using an average temperature just as with Equation (39). This average
temperature is taken as the arithmetic mean of the bulk and interface temperatures.
The bulk fluid mole fractions are already predetermined and presented as a function
of the bulk mean temperature distribution from the thermodynamic analysis. The
calculation of the mole fraction at the interface is simply the bulk mole fraction
multiplied by Xvi/Xvb calculated in Equations (39) and (40) (Lock, 1994).
Once the condensation conductivity is defined, we can calculate the condensation
heat transfer coefficient by: hi = Sh kc/Deq. The Sherwood number is calculated using the
heat and mass transfer analogy such that for turbulent flows (Incropera & DeWitt, 1996)3:
3 These correlations are used to give a general idea of the flow behavior. More advanced
Nussult relationships may be required or even developed experimentally to give the
Nu= 0.023(Re)O8 (Pr) (41)
Sh= 0.023(Re)O" (Sc)0
Where Nu is the local Nusselt Number, Pr is the local Prandtl Number, and Sc is
the local Schmidt Number by definition the ratio of momentum and mass diffusivity:
Counter-Flow Nodal Analysis
The code constructs a one dimensional nodal analysis and calculates the total
equivalent thermal resistance in order to calculate the heat flux at each node. Using the
enthalpy curve fits generated by CEA and mass flow rates from the thermodynamic
analysis, the code calculates the heat removal for a given temperature drop across each
node. With knowledge of both heat transfer and heat flux, the area required to remove
heat from each node is calculated. The heat transfer relations are used to march node by
node until the UF4 vapor is completely condensed. Thus the total heat transfer area is
The length of the hot channel is segmented into N nodes. Each node has an inlet
temperature and an exit temperature. The exit temperature of node n becomes the inlet
temperature for node n+1. The bulk mean temperature used in the heat transfer analysis
is the arithmetic mean of the inlet and exit temperatures.
The cold channel is also broken into N nodes. The inlet and exit temperature for
the coolant loop are governed by the thermodynamic cycle evaluations. The coolant
mass flow rate is determined using an energy balance for a control volume around the
entire heat exchanger.
highest accuracy for a final design calculation. Errors for Equation (41) may be as high
as 25% (Incropera & DeWitt, 1996).
1h h h(hh, hho (42)
m =) (42)
cP (T -T )
Where the heat exchanger inlet enthalpy hhi is taken as the dew point temperature
and the outlet temperature hho is the fully condensed state from the topping cycle
Mass Flow Rates,
Figure 4-1: Schematic of CHEX Code. The wall resistance is not shown in the algorithm
because it is a constant not a variable.
Each coolant node has an inlet and exit temperature. An energy balance is applied
between each hot and cold node to calculate the change in coolant temperature at each
node. The inlet temperature to node n is calculated based off of the energy balance. The
exit temperature of node n+1 is the inlet temperature of node n.
T = T, -mth(hh -ho ,)/hmCp (43)
The cold mean temperature is calculated based upon the calculated inlet and exit
temperature and the hot mean temperature. The arithmetic mean is not used for coolant
nodes because each hot channel node is treated as an infinite heat source at temperature
Thm surrounding each cold node. This assumption is appropriate because the side plenum
area of each rectangular channel is only a small fraction of the total plate area where
virtually all of the heat transfer is taking place. It is also appropriate because the hot
channel mass flow rate and specific heat are greater than in the cold channel. Hence, if
the length of the heat exchanger went to infinity the cold channel would be heated to the
hot channel inlet temperature.
Therefore, a log mean bulk temperature is determined. For the sake of discussion
let us infer that the n subscript is implied.
Tp = Th (Th, T~, ) (T T~o ) (44)
SIn((T, T,)/(Th, To ))
The heat flux between nodes of the hot and cold channel are determined by
combining the node energy balance with a thermal circuit analysis for each node.
q, = h, (h,l ho) (45)
qn = (Th TenRo, (46)
R -I + + (47)
to Ah, A +h, + h k hj
Combining Equations (45) and (46) and keeping in mind Equations (43), (44) and
(47) gives a relationship for temperature changes across hot channel nodes and the wall
surface areas required to transport the equivalent amount of heat for that temperature
change into the coolant channel.
A, 2h (hm,,,x (Th ) h,,, (Tho )(48)
h, (Thm Tm)
n jhh (h,,, (Th) hm, (Tho )49)
P x h (Thm Tcm)
Where P is the channel heated perimeter and Zn is the axial length of the node n
portion of the heat exchanger.
The nodal calculation starts at the top of the heat exchanger where the hot fluid is
entering and the coolant is exiting. It then decrements the hot node inlet temperature by
an amount AT to get the node exit and computes the corresponding enthalpy change. It
then calculates the cold node inlet temperature based on Equation (43). Next it calculates
the bulk mean temperatures, thermal properties taken and HTC's taken at the bulk mean
temperatures and HTC's. Finally, it calculates the total equivalent HTC and An and zn.
As output, it sums up the Zn's to get the total length of the CHEX.
Pressure Loss Model
The differential pressure loss is a function of temperature dependent thermal
transport properties, mass quality and node geometry. After the heat transfer calculations
are complete for each node the computer code calculates the differential pressure loss and
multiplies this number by the calculated length of the node. The code then sums the total
pressure loss across the heat exchanger as it marches through all nodes.
Frictional Pressure Loss
The Homogeneous Equilibrium Model (HEM) treats the flow velocity of the liquid
and gas phases as equal to each other (Todreas & Kazimi, 1993). This makes sense
because the void fraction is very nearly one throughout the channel. The process may be
visualized as this. Vapor is cooled to its saturation point and moves toward the cooled
wall where condensation through droplet growth and eventually flooding creates an
unstable liquid film there. Due to the bulk gas/vapor velocity, this film breaks away and
coalesces in the channel to form liquid filaments. These filaments are entrained in the
gas/vapor stream and move with the velocity of the gas/vapor.
The two-phase frictional pressure gradient is calculated using a fanning equation
dPT = 2fp G 02 2fgo G, =02 2 dP go
go -] g (50)
dz D p go D P go dz D L Yod
The subscript go stand for the single gas/vapor phase flowing with the same mass
flux in the channel as the two-phase flow. The frictional multiplier (pgo relates the single
phase pressure gradient equivalent to the true two-phase gradient.
Inspection of Equation (50) shows that the single phase frictional multiplier is
defined such that:
2 "Pf (51)
Using the definition of flow quality and specific volume will prove that the two-
phase mean density is defined as:
1 x (1 x)
PM Pg A
And the density ratio in the left hand side of Equation (51) is:
=P x+(1-x) X (53)
The friction factor ratio in the right hand side may be expressed as the ratio of:
fTP C/Re"P= p (54)
fgo CRe0 / Ug
Where C and n are empirical constants taken for turbulent flow conditions to be:
C=0.316 and n=0.25 or C=0.184 and n=0.2.
There are two methods of defining Equation (54). The first is that the two phase
viscosity and the single phase viscosity are assumed equivalent. The second is that an
empirical model must be assumed for calculating the two-phase viscosity. If the second
option is chosen then there are three widely adopted formulas for calculating the two-
McAdams et al.: [ x 0 x) (54)
Cichiti eta al.: u, = x/, + (1- x)#,u (55)
Dukler et al.: np = pm[xpg/Pg +(1 -x)p/p,] (56)
The code employs all three formulas as well as the equal viscosity assumption and
allows the user to pick which assumption is preferred.
Accelerational Pressure Loss
The differential accelerational pressure loss is the change of kinetic energy in the
fluid as a result of the flow area of each fluid varying with respect to phase change. The
flow area occupied by the gas/vapor component becomes less as more vapor is
condensed. This change in flow area causes the gas velocity to increase thus increasing
the accelerational pressure loss. The change in accelerational pressure loss across the
node may be described as (Todreas):
dp .V2 1,2 G Gm
PI 1 Pv2 mV2
dz ace 2 ot 2 2pm out 2pm (57)
Since the HEM mass flux is constant across the channel, Equation (57) may be
dP G d 1 G d x (1-x))
dz acc 2 dz Pm 2 dz p, I (58)
Equation (58) was written in terms of mass quality (dx/dz) so that it may be
converted into a simpler form in which the phase change across the node is accounted but
also assumes that the two-phase mean density does not change across the length of the
node and may be evaluated at bulk fluid temperature Thm for the node.
dP G, [dvt, dvg dv, v ,dx~
dz 2 dz dz dz) dz
Assuming that the fluid is incompressible across the node than Equation (59)
dP G 1 [dx G mx (60)
dz 2 L dz= 2 pg p, dz
The differential quality is calculated by using the thermodynamic mole fraction fits
at the node inlet and outlet temperatures and divided by the calculated node length.
DIFFUSION LAYER MODEL COMPARATIVE ANALYSIS
The counter-flow heat exchanger model was developed without DLM to contrast
the effect of the diffusion layer on the heat exchanger size. A simplified model of latent
heat transfer was devised for the node n. It can be shown that the heat input into the wall
in Figure 4-1 is roughly equivalent to the sum of a constant latent heat current source and
a sensible heat resistance input into the wall.
Comparative Analysis Method
Just as in Figure 4-1 the sensible heat is modeled as a temperature drop multiplied
by the convective resistance. Latent heat, however, is modeled as a current source
equivalent to the heat of formation at the bulk mean temperature multiplied by the change
in vapor mass across n.
qs = hcoun X (Thm T) (61)
q, = [h,, (T,, ) lh,, (To)] x if, (T,, ) (62)
Once simple definitions of latent and sensible heat currents are defined, Kirchoff s
Current Law is applied for analyzing the thermal current and temperature differences.
The figure shown below is equivalent to Figure 4-1 with the exception that the latent heat
is modeled as a thermal current source.
The Thevenin equivalent circuit methodology commonly applied to electrical
circuit analysis is then applied to Figure 5-1. In the Thevenin equivalent circuit
methodology, one resistance in the circuit is defined as the load and removed from the
circuit (Rizzoni, 2000).
R, Thw Rw
Figure 5-1: Equivalent electrical circuit with the latent transferred to the wall modeled as
a source term.
The Thevenin equivalent voltage then becomes the voltage drop across the
removed load resistance.1 In the thermal circuit of Figure 5-2, the coolant channel
resistance Re and wall resistance Rw is defined as the load because heat transfer there is
independent of the physical processes in the hot duct. The notation Rcw stands for the
series resistance of the wall conduction and coolant channel convection.
Figure 5-2: Thermal circuit shown with the load resistance removed. The notation Rcw
stands for the series resistance of the wall conduction and coolant channel
1 It should be noted that for this discussion when analyzing thermal circuits, voltage
changes across portions of the circuit are replaced by temperature changes.
The next step is determining the equivalent resistance. This is done by shorting out
the voltage sources and short circuiting the current sources.
The equivalent resistance in Figure 5-3 is: Req= Rs.
Figure 5-3: Thevenin equivalent circuit analysis. The node temperature difference is
shown as shorted out and the current source is shown as an open circuit.
Finally, we must determine the temperature drop across the load. The open circuit
nodal analysis method is applied at Tw of Figure 5-2.
q, + q = 0
Tw = Thm + q,x R (63)
Now that the Thevenin equivalent resistance and temperature are defined the
Thevenin circuit is complete:
Figure 5-4: Thevenin equivalent circuit with the total heat transfer to the coolant channel
drawn as qload.
The heat current seen by the load may be expressed in terms of the Thevenin
temperature and equivalent resistance.
qload T (64)
Substituting Equation (63) for T, in Equation (65):
qjR, +T,,, -T = (R, +R, ,)q.,1 (65)
Or: q,R, + T,, T,, = + I (66)
A,, h, h, k
Solving for An gives:
q, /1 1 L
qh h_ kI I
A,, = h h, hc (67)
Where qload is defined as the total heat transported across the node n found from
thermodynamics in Equation (68).
qload = n = h (hh, (Th,) hho (Tho) (68)
The enthalpy relationships are acquired from the CEA curve fits. Since the CEA
data reflects the changing mixture properties while accounting for the degree of
condensation, these curve fits reflect temperature as well as phase change in the
An important characteristic of Equation (64) is that as latent heat goes to zero the
heat transfer model becomes identical to that of a simple composite wall heat resistance
with only convective heat transfer on each side of the wall.
T Tm Th T(69)
q = "' (69)
SR, R, + R, + R
This can be observed by allowing qi in Equation (67) to go to zero and rearranging
the equation to be in the form of Equation (69).
Impact of DLM
The impact of modeling the diffusion layer proves that a larger surface area is
required than a model without a diffusion layer resistance. This can be gauged by
comparing axial heat flux profiles using DLM with the Thevenin equivalent TEM model.
1350 1400 1450 1500 1550 1600 1650 1700 1750
Figure 5-5: Axial heat flux vs. temperature comparative analysis comparison between
DLM and TEM. The TEM model shows higher mass flux because there is
nod diffusion layer resistance modeled.
The heat flux profiles show that the heat flux for the TEM model is generally
greater than DLM. The heat flux for both models converges towards the end of the
channel when virtually all the vapor is condensed. This point could be considered the
effective fully condensed point. Because of the quadratic nature of the mole fraction
curve fits seen in Figure 3-5 and Figure 3-6, the vapor mole fraction may increase
unnaturally at temperatures below the local minimum of the fit. The CHEX code will not
allow this and retain the vapor minimum value. If this happens the comparative analysis
version of the code will assume full condensation and continue to the last node assuming
there is no change in vapor mass flow rate forcing Equation (62) to zero. This artificial
correction is allowed for because the vapor concentration beyond this effective point is
very near zero.
The reduced DLM heat fluxes have a negative heat flux on CHEX size. As
expected, the length of the heat exchanger for DLM is greater than TEM by about 7%.
Therefore, DLM is the more accurate and most conservative method as well.
S 30 -
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Axial Height (m)
Figure 5-6: Axial heat flux vs. the axial dimension comparative analysis comparison
between DLM and TEM. The DLM height is greater by 7%.
Also, as expected the condensation heat transfer coefficient goes to zero as the
concentration of vapor vanishes.
Plotting the total HTC for the channel shows that the HTC assumes a minimum
baseline after total condensation has occurred. This common base line consists of the
wall conductive and the hot and cold convective resistances.
1650 1700 1750
Figure 5-7: Condensing HTC for the DLM model. The HTC goes to virtually zero as
vapor is condensed.
DLM HTC TEM HTC
1400 1450 1500 1550 1600
1650 1700 1750
Figure 5-8: Comparative analysis HTC comparison between DLM and TEM
The TEM HTC was calculated by dividing the node heat transfer from Equation
(68) by the node heat transfer area calculated by Equation (67). This was then divided by
the difference between Thm and Tem: HTC=qn/An(Thm-Tcm).
1400 1450 1500 1550 1600
The above plot comparisons demonstrate the validity of the DLM method by
comparing it with a rudimentary brute force calculation of the latent heat transfer process.
Since DLM models a physical process that has a negative impact on the heat exchanger
size, it is the most conservative calculation to use as well as the most accurate. The
CHEX code ensures a physical answer by preventing numerical anomalies from the curve
fit process to be introduced into the heat transfer calculation. The resulting effective full
condensation point demonstrates convergence of the DLM and TEM methods when the
vapor mole fraction is virtually nonexistent at low temperatures near the CHEX exit.
COMPACT HEAT EXCHANGER DESIGN
The VCR/MHD combined cycle plant feasibility is dependent upon the size and the
complexity of the heat transfer system. Remember, if the HTS size can be marginalized
to be proportional to that of the VCR/MHD than requirement of a massive containment
structure can be argued as mute. This can only be possible by minimizing the CHEX
physical size. The purpose for selecting a compact heat exchanger was to maximize the
heat transfer surface area per volume of heat exchanger. This has the benefit of reducing
heat exchanger size but it also raises the question of pressure loss penalty.
The design methodology shall optimize the heat exchanger size while minimizing
pressure loss in both the hot and cold channels. Other design aspects such as structural
integrity related to wall pressure gradients and channel velocity must also be considered.
It is well known that the channel velocity can be reduced by increasing the channel
pressure. If the pressure difference between the hot and cold channel is great than the
selection of channel geometry regarding plate or tube arrays will be influenced. Large
pressure differences across the separating channel wall could lead to plate bowing or
deflection in the middle. Therefore, tube channel geometry is desirable when attempting
to minimize channel velocity. This is the case for larger mass flow rates through the
CHEX that are necessary at higher power ratings while trying to maintain a small heat
Wall Material Selection
The first obvious solution to minimizing heat exchanger size and pressure loss is to
pick a wall material with a high thermal conductivity. However, considering the ultra-
high temperatures in the VCR/MHD output a material with a very high melting point
must also be selected. This constrains the problem to primarily refractory metals and
some ceramics. The selection is further complicated by the fact that the material chosen
must be resistant to chemical fouling from ionized uranium and fission product fluorides
present in the VCR/MHD working fluid. A list of possible materials is provided.
Table 6-1: The candidate wall material candidates with selection criteria were taken from
published data in DeWitt.
Material Point (K) (W/mK)
Tungsten 3660 174
Molybdenum 2894 138
Niobium 2741 53.7
Rehnium 2453 47.9
Carbide 3100 490
Tantalum 3269 57.5
The CEA code was used to perform chemical equilibrium assessment on these few
choice materials. The condition set was one mole of the wall material and one mole of
coolant at 18 bars and over 2000 Kelvin temperatures. A mixture of 0.10 moles of UF4
and 0.90 moles of helium was selected as the hot fluid. At these conditions the UF4
Table 6-2: The initial UF4 mole fraction in this CEA calculation was twice that used in
the actual CHEX design to get a detectable level of uranium fluorides in the CEA output.
Species Mole Fraction
SiC has the highest thermal conductivity and has a melting point over 3000K.
However, SiC had equilibrium reactions with the uranium fluoride gasses. The
equilibrium mole fractions of SiC with the above species are listed below:
Table 6-3: The dissociation mole fractions shown are for a starting mixture containing
one mole of SiC, 0.9 moles of helium and 0.1 moles of UF4.
Unlike their' ceramic competitor refractory metals had no equilibrium dissociation
phenomenon. Therefore, they are the primary candidate material for CHEX walls. It
should be noted that CEA has no data on refractory metal fluorides. It does have data for
tungsten hexachloride (WC16). However, CEA does not have data for uranium chlorides.
Therefore, tungsten was tested with the metallic chloride vanadium tetrachloride (VC14)
in lieu of UF4 to see if dissociation reactions would create the product WC16. At
270KJ/mole and 222KJ/mole respectively, vanadium and uranium have similar bond
enthalpies (Winter, 5/2/2004). Also, since fluorine and chlorine are in the same column
of the periodic table they should have similar bond characteristics with vanadium and
uranium. Thus, pure vanadium tetrachlorides were tested at the VCR temperature and
pressure to get an idea of the bulk dissociation reactions. These bulk fluid dissociation
products were used in the next simulation with fission products present.
Table 6-4: The dissociation mole fractions generated from the CEA code are for one
mole of W reacted with 0.9 mole of He and 0.1 Mole of VC14.
Fission Product Test
Now that tungsten has proven resistant to chemical attack, it must be tested against
the entire range of fission product compounds existing in the topping cycle loop.
Previous studies by Boersma-Klein, Kelling and Kistemaker have developed VCR fission
product inventories using the ORIGEN computer program (Boersma-Klein et al., 1984).
These inventories were based on a 1200 MW thermal VCR with a uranium holdup of 0.6
tons enriched to 30% and a refreshment time of 200 hours.1
UF4+UF6+CF4+ 3.45E-3 mole/sec UF4 (30%)
Actinides Reactor 1.51E-3 mole/sec CF4
------ >Reactor --
Recombine Heavy Components Separator
2.7E-3 mole/sec (7%) ponent
2.8E-3 mole/sec (3%) 1E-4 mole/sec
Figure 6-1: VCR online refreshment scheme for online refueling and fission product
1 The VCR working fluid was 70% UF4 and 30% CF4. The fission product inventories
reported were normalized to the amount of UF4 in the working fluid. These normalized
inventories were multiplied by the UF4 mole fraction for the He/UF4 mixture to get the
relevant fission product inventory for this system.
From Figure 6-1, it can be seen that the fission products amount to be
approximately 1% of the uranium concentration. The molar concentration of total
uranium in the test is 10%. The helium accounted for 89.9% and fission products
accounted for 0.1% of all moles in the fluid. The fission product inventory given in the
study is given in Table 6-5.
Table 6-5: The Boersma-Klein et al. fission product inventories are for a 1200MW VCR
with UF4 enriched to 30% and refreshed every 200h.
Percent of Mole
Atomic Total Fraction in
Quantity Wt Quantity Fission UF4/He
Element (g) (g/mol) (mol) Prod System
Br 8 79.9 639.2 0.052% 4.550E-07
Kr 147 83.7 12303.9 1.007% 8.758E-06
Rb 137 85.5 11713.5 0.959% 8.338E-06
Sr 579 87.6 50720.4 4.152% 3.611E-05
Y 252 88.9 22402.8 1.834% 1.595E-05
Zr 1320 91.2 120384 9.855% 8.569E-05
Nb 15 92.9 1393.5 0.114% 9.920E-07
Mo 880 96 84480 6.916% 6.014E-05
Tc 163 99 16137 1.321% 1.149E-05
Ru 664 101.7 67528.8 5.528% 4.807E-05
Sn 9 118.7 1068.3 0.087% 7.605E-07
Sb 8 121.8 974.4 0.080% 6.936E-07
Te 240 127.6 30624 2.507% 2.180E-05
1 232 126.9 29440.8 2.410% 2.096E-05
Xe 1620 131.3 212706 17.413% 1.514E-04
Cs 513 132.9 68177.7 5.581% 4.853E-05
Ba 742 137.4 101950.8 8.346% 7.257E-05
La 446 138.9 61949.4 5.071% 4.410E-05
Ce 1190 140.1 166719 13.648% 1.187E-04
Pr 287 140.9 40438.3 3.310% 2.879E-05
Nd 767 144.3 110678.1 9.060% 7.879E-05
Pm 63 145 9135 0.748% 6.503E-06
CEA does not have thermodynamic data on: Y, Tc, Ru, Sb, Te, La, Ce, Pr, Nd and
Pm. Therefore, equilibrium reactions of tungsten with these elements could not be
studied. However, all elements except for antimony and tellurium are metals or
lanthanides and antimony does not have a sizeable concentration compared with the rest
of the inventory. Tellurium only makes up 2.5% of the total inventory. Therefore,
chemical attack by these fission products may not be a major concern. Equilibrium
analysis of these fission products with UF4 and helium yields the following fluorides. It
should be noted that CEA does not report concentrations below one hundred thousandths
therefore many species in the inventory were automatically dropped from the calculation.
Introducing one mole of tungsten did not change the equilibrium concentration
and had no chemical interaction with the other species in the system. Notice that the
species mole fractions in the tungsten run are half of the mole fractions of the
comparative analysis run. This was done to normalize all concentrations to one.
Table 6-6: The CEA equilibrium calculation ofW with the Boersma-Klein et al. fission
product inventories show that tungsten does not bond with any of the fission
products present in the system.
in Fluid Including
Species Alone Tungsten
BaCL2 0.00007 0.00004
CL 0.00075 0.00038
CL2 0.00001 0.00001
CsCL 0.00005 0.00002
He 0.89889 0.44972
SrCL2 0.00004 0.00002
VCL4 0.09961 0.04984
Xe 0.00015 0.00008
Mo(cr) 0.00006 0.00003
V(cr) 0.00026 0.00013
Zr(b) 0.00009 0.00004
W(cr) n/a 0.49969
This data strongly indicates that tungsten and other refractory metals make a good
candidate for constructing an ultra-high temperature CHEX. Molybdenum was selected
for the CHEX wall material because of its ability to be rolled into long sheets. Tungsten
was not selected because of its very high hardness. This hardness is why tungsten is only
used in small proportions for high temperature applications such as arc welding, etc.
A thickness of about 1mm was quasi-arbitrarily selected for the wall thickness
because it is the dimension commonly used in conventional modern compact heat
exchangers. If a tube geometry is used the tube thickness was chosen to be 5% of the
tube outer diameter. This is also a common rule of thumb for tube bundle design.
The design envelope is an important aspect of the CHEX optimization. It
determines the CHEX operating characteristics such as mass flow rate and the inlet and
outlet temperature range. This is important to rating and sizing the heat exchanger
because the mass flow rate affects the channel pressure drop and velocity. The coolant
temperature domain affects materials selection and the condensation interface.
Therefore, it is necessary to visualize the balance-of-plant for a VCR/MHD in a
combined cycle with a gas turbine plant.
Coolant Temperature Selection
One of the major issues that affect cycle efficiency is the temperature of the heat
source and that of the heat sink. The heat source for the vapor core is set between 2500
and 3000 K because even the refractory metals will melt at temperatures much higher
than this. The heat sink temperature is essentially dictated by volatility. The boiling
point of the UF4 is directly proportional to the partial pressure of UF4 in the mixture.
The lower this partial pressure the more volatile the UF4 vapor will be and hence the
lower the temperature of complete condensation and the coolant inlet temperature.
The necessity of a topping cycle precooler also impacts the coolant temperature.
The precooler temperature range is of course below the CHEX minimum temperature.
This crates two possibilities for providing coolant to the CHEX and precooler. If the two
devices are run in parallel then they will have the same coolant inlet temperature and will
require recombining the coolant before entering the bottoming cycle turbine. If they are
operated in series such as two back-to-back counter flow heat exchangers than no mixing
TMHD,o T3 Mhot
CHEX T3 CHEX
Precooler T3 Precooler
Ti mi Ti
Tprec,o m cold
Figure 6-2: Coolant delivery flow arrangements: (a) Series flow arrangement (b)
Parallel flow arrangement
The series flow arrangement creates minor difficulty in getting a precooler coolant
outlet temperature that is equal to or less than the CHEX hot side exit temperature.
Obviously, this temperature difference is necessary in order to ensure the positive flow of
heat from the hot fluid to the cold fluid. This situation arises from simultaneously
solving the energy balance for both heat exchangers in a way that the coolant exit
temperature of the precooler becomes the coolant inlet temperature of the CHEX.
t HneCp,He Tcond Tprec,o)
T2 = T, +H ,,CP H (T-, (70a)
2 1 rhboCP bot
TT T op (hdewpt -hcond )
T2 3 IbotCnPbot (70b)
This situation is possible but difficult to do when considering the maximum CHEX
coolant temperature should be kept low2 due to turbine inlet temperature metallurgical
considerations. However, this situation becomes increasingly important in the later
discussion of condensation interface freezing because methods for increasing the coolant
temperature in the CHEX must be investigated to prevent UF4 vapor freezing at the
The parallel configuration guaranties that the CHEX coolant inlet temperature is
below Tcond. However, because of the possibility of interface freezing the parallel
configuration becomes more desirable because the coolant entering the CHEX is greater
than that entering the precooler.
To illustrate the balance-of-plant it is necessary to discuss a set of test cases
highlighting the thermodynamic design. The first case applies regeneration with the
helium preheater. The second case demonstrates the pitfalls of going to a higher pressure
ratio without changing the UF4 mole fraction.
Topping cycle regeneration has a profound impact on the bottoming cycle heater
inlet temperature. Because the full condensation temperature basically sets the reactor
inlet temperature, the preheater inlet temperature is solely dependent on the amount of
recycled heat from the CHEX. The more heat recycled the greater the temperature
change will be on the preheater thus lowering the compressor temperatures and hence the
2 The turbine inlet temperature is considered somewhere in the projected range of the Gas
Turbine helium Modular Reactor system proposed by General Atomics which is
between 1000K and 1200K.
minimum temperature for the cycle. The following T-s plots illustrates the effect of
regenerative preheating of the helium before it reaches the reactor inlet temperature.
-Reactor -- Isentropic Expansion -- Actual Expansion -- CHEX/Reg
6.8 7 7.2 7.4 7.6 7.8
Entropy (kJ/kg K)
8 8.2 8.4 8.6
Figure 6-3: Mixed He/UF4 portion of topping cycle. The cycle pressure ratio of 5 and
the MHD isentropic efficiency is 0.7. The portion of the CHEX/Reg between
1700K and 1950K is the superheated portion of the MHD output.
- Precooler -.- Low Comp Intercooler -u- High Comp -- Preheater
Figure 6-4: Separated helium portion of the topping cycle. The regenerator effectiveness
is 0.1 and the compressor efficiency is conservatively estimated as 0.8.
Precooler -.- Low Comp Intercooler -u- High Comp -- Preheater
2 1100 --
27 28 29 30 31 32 33 34
Figure 6-5: Separated helium portion of the topping cycle. The regenerator effectiveness
As can be seen from the graphs changing the amount of regeneration does not
change the mixture portion of the reactor because the state points for the mixed portion of
the cycle are set by the reactor temperature and pressure, the cycle pressure ratio and the
fully condensed temperature. It can be seen from Figure 6-4 and Figure 6-5 that the
preheater exit and precooler inlet temperature coincide with the fully condensed point but
the cycle minimum temperature is controlled by the regenerator effectiveness.
Increasing the pressure ratio decreases the compressor inlet temperature even
further and hence also lowers the topping cycle minimum temperature. This is because it
decreases the volatility of the vapor thus lowering the fully condensed point. It also
increases the level of precooling considerably. Bearing in mind that the heater inlet
temperature must be lower than this exit temperature for positive heat flow, the amount
of regeneration and pressure ratio dictate the heater inlet temperature.
The heater inlet temperature becomes important because it affects the amount of
allowable compression in the bottoming cycle. If the heater inlet temperature is too low,
the bottoming cycle compressor chain inlet temperature will be too low to allow heat
rejection from the bottoming cycle. In order for the turbine plant to have the same
efficiency as the earlier example it will have to have a temperature below that of the
ambient temperature of the environment outside the plant. This can be seen in this next
graph where the topping cycle pressure ratio is ten and 50% of the MHD exhaust heat is
-- Reactor -.-Isentropic Expansion --Actual Expansion --- CHEX
6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8
Figure 6-6: Mixture portion of the topping cycle. Pressure ratio is 10.
The topping cycle efficiency is around 15% which is at least 2% higher than the
previous example. It also has a lower reactor temperature which is attractive from a
materials aspect. However, the minimum topping cycle temperature in Figure 6-7 is
practically the desired bottoming cycle heat sink temperature shown if Figure 2-3. This
means that whatever turbine plant design possible, the minimum bottoming cycle
temperature is going to be ridiculously low. Therefore, modest levels of topping cycle
regeneration and pressure ratio are necessary to assure a realistic balance-of-plant3.
Precooler -.- Low Comp Intercooler High Comp -- Preheater
3 900 7
25 27 29 31 33 35
Figure 6-7: Separated helium portion of the topping cycle. The regenerator effectiveness
is 0.5 and the pressure ratio is 10.
Figure 6-3 and Figure 6-5 are an optimum topping cycle balance-of-plant because it
has a high efficiency of about 13% and has a minimum precooler temperature of 700K.
This inlet temperature is comparable to the reactor inlet temperature for present Modular
helium Reactor designs (General Atomics, 5/2/2004) 4.
Interface Freezing Phenomenon
The CHEX coolant outlet temperature is relatively arbitrary compared to the inlet
temperature. Because of the high temperature freezing point of UF4, condensing vapor
traveling to the cooled wall may experience freezing before breaking away from the
3 A regenerative effectiveness of 0.25 was found to reduce the VCR/MHD output
temperature to the UF4 dew point. This is helpful for the realistic plant configuration
because all condensation happens in only the CHEX device.
4 The cycle efficiency is highly dependent upon the estimation of compressor isentropic
efficiency. This efficiency is estimated at 80% but higher compressor efficiency can
yield topping cycle efficiency as high as 17%.
condensation interface to rejoin the churn flow liquid filaments as discussed in Chapter 4.
Considering the metallurgical constrained turbine inlet temperature is at best
optimistically around 1300K, the CHEX coolant temperatures must be increased by some
special means. To accomplish this, a certain level of coolant bypass is required to
decrease the mass flow rate through the heater thus increasing the temperature change
across the heated portion of the bottoming cycle working fluid. The unheated portion
will have to be mixed with the heated portion prior to entering the turbine. Another
strategy employed was reflector cooling by the bottoming cycle working fluid between
the precooler and CHEX.
The interface freezing phenomenon occurs because the interface temperature is
below the freezing point of the UF4 which is around 1300K. This is seen in the heat
exchanger condensing heat transfer coefficient by the type of heat of formation used in
the diffusion layer calculation. A sharp discontinuity is observed in the condensing heat
transfer coefficient if the wall temperature drops below the vapor freezing point.
1350 1400 1450 1500 1550 1600 1650 1700 1750
Figure 6-8: Interface freezing anomaly observed with zero coolant bypass.
Unfortunately, the freeze-up phenomenon is completely natural. If this were to
happen in a real heat exchanger, condensing vapor would becomes super saturated vapor
as it traveled towards the interface and freeze immediately after changing into the liquid
state. This situation is undesirable because the frozen UF4 could not be removed from
the system and re-circulated to the VCR.
The freeze-up phenomenon can be avoided by using the series heat exchanger
arrangement and allowing heat to be added to the bottoming cycle working fluid through
BeO reflector cooling. Coolant exiting the precooler is heated in the reflector before
entering the CHEX. Trial and error experimentation proved that reflector cooling up to
13% of the reactor power had virtually no impact on reducing the interface temperature.
The discontinuity is moved by only one calculation node toward the coolant inlet.
1350 1400 1450 1500 1550 1600 1650 1700 1750
Figure 6-9: Reflector cooling allowed increasing the interface temperature. Reflector
cooling is equal to 10% of reactor power.
The next strategy employed was bypassing some of the coolant away from the
heater. The bypass method combined with reflector cooling allowed for the right
combination of coolant reduction to have a large enough temperature increase in the
precooler and reflector to prevent interface freezing. However, a delicate balance ensues
between the maximum coolant outlet temperature that ensures positive heat flow from the
hot channel into the cold channel and the mass flow rate that will keep the interface
temperature above freezing5.
1350 1400 1450 1500 1550 1600 1650 1700 1750
Figure 6-10: Condensing HTC with 55% of the bottoming cycle working fluid going
through the heater. The reflector cooling is 14% of the reactor power.
Changing the coolant mass flow rates and temperatures has an affect on the CHEX
design envelope. For example, with reduced mass flow rates the mass flux in the coolant
channel will be less thus reducing pressure loss. The heat flux distribution is also
In Figure 6-11 notice the change in heat flux distribution as well as the overall
height in the heat exchanger between the enhanced design and the un-enhanced. This
change is the result of an order of magnitude heat flux reduction. That is because
5 This truly is a design envelope definition because too much reflector cooling makes the
coolant inlet temperature so high that positive heat flow impossible near the CHEX
coolant entrance. Too high a mass flow rate does not prevent interface freezing.
increasing the coolant exit temperature decreases the temperature difference between hot
and cold channel nodes.
No Enhancement Enhancement
0 0.5 1 1.5 2 2.5
Axial Height (m)
Figure 6-11: Effect of axial enhancement on CHEX heat flux profiles and axial height.
It has been shown that a coolant bypass of 50% to 55% through the heater coupled
with reflector cooling of approximately 10% is sufficient to ensure positive heat flow and
prevent interface freezing. These are design guidelines and are considered part of the
design envelope for the CHEX rating and sizing discussion in the following sections.
Rating and Sizing
Now that a design envelope has been established the CHEX design must be
optimized. This discussion needs to address the following issues:
* Pressure Loss vs. Size
* Coolant choice impact on pressure drop and channel velocity
* Selection of CHEX channel geometry (Plate vs. Tube)
Varying concentrations of helium and xenon were tested in the bottoming cycle
design. Helium is the coolant of choice because of its extremely high specific heat and it
thermal transport properties. Xenon has a high molecular weight which has the ability to
increase the density of the working fluid when mixed with helium. This is advantageous
for the bottoming cycle turbo-machinery design because it increases the mixture density
thus reducing compression demand.
Xenon does not affect the thermodynamic performance of the bottoming cycle
because it does not appreciably impact the specific heat ratio. However, mixing the two
species together dramatically reduces the mixture specific heat. A large reduction in
specific heat drastically increases the coolant mass flow rate and thus mass flux in the
CHEX coolant channels. This changes the operating characteristics of the heat
For a 100MW VCR, using helium in the plate channel geometry does not even
create a mentionable amount of coolant pressure loss. Adding xenon has an order of
magnitude effect on the coolant pressure loss. As will be seen later, increasing the power
rating while trying to maintain a small CHEX size using He/Xe as the coolant has such a
profound impact on pressure loss and fluid velocity that an alternate channel geometry
must be assumed to mitigate the high mass flux penalties. Higher coolant channel
pressures may be necessary to increase the density in the coolant channel thus
minimizing the coolant velocity. If this is the case than a different channel geometry may
be desired to handle the large pressure differential experienced across the channel wall.
A tube bank passing the condensing mixture cooled by a counter flow stream of the
coolant is the preferred geometry because tubes are less susceptible to shear stresses
when placed in compression with the higher pressure fluid outside the tube. The CHEX
code allows the user to pick the channel geometry.
Plate CHEX Rating and Sizing
Heat exchanger rating and sizing is the design specification optimizations such as
flow rate, number of flow passages, physical dimensions etc. The rating and size
attributes of the CHEX are solved for the specified topping and bottoming cycle mass
flow and temperature requirements. The channel geometry is controlled by a user
defined aspect ratio equal to the channel length divided by its width. The code also
requires a user defined starting hot channel inlet Reynolds number. A mass flux is then
calculated that meets the following equations.
Re =G (71)
D,, = (72)
S P 2L + 2W
Where length (L) and width (W) are defined by the aspect ration: r=L/W. The nhot term
above is the total topping cycle mass flow rate divided by the number of plates or
channels in the heat exchanger.
It can be seen from the relations given above and plots of the heat exchanger
physical dimensions that the size and shape of the heat exchanger are governed by two
factors: (1) the channel aspect ratio, (2) the number of plates used to divide the flow.
But first, a few descriptive terms must be defined. The channel length corresponds to the
heat exchanger cross sectional length while the channel width multiplied by the number
of plates gives the heat exchanger stacked width. The flow area and hence length and
width is directly proportional to the channel mass flow. Therefore, a large heat
exchanger length is indicative of a large mass flow rate per channel caused by too few
plates in the heat exchanger.
Heat exchanger length and pressure drop are closely related to channel geometry.
The channel flow area is related to the heated perimeter through the equivalent diameter.
The aspect ratio governs the heated perimeter so that the flow area may be written as:
Rearranging Equation (72):
P,=2L+2W=2L+2L/r= 2 rA1(1+1/r)
P,= 2 ( r +1/ j) (73)
Rearranging Equation (71) and substituting Equation (73):
( th 4
c2 -X -
A f ('I/ 2 (K 1r + l/)r ) (7 4 )
Once the channel geometry is found, the code launches the nodal analysis and
pressure drop subroutines. These subroutines determine the axial height of the heat
exchanger and corresponding pressure drop. Since the equivalent diameter and HTC are
directly proportional through h=NuxDeq/k, an excessively tall heat exchanger may be the
result of very small HTC's caused by a low equivalent diameter. The solution is to
decrease the aspect ratio thus making the equivalent diameter larger.
P, = 2 A(r+l1/- )
4xAf 2 xA (75)
D = i
," PJ + 1 J
r = L/W
A test case was performed on a 100MW example similar to that used in the
preceding sections. The heat exchanger length, width and height are plotted as functions
of the channel aspect ratio and the number of plates used to divide the flow.
^ 200.00 +- Cross Sectional
150.00 --Stack Width (m)
N -A- HTEX Height (m)
0 0.2 0.4 0.6 0.8 1
Aspect Ratio: r=L/W
Figure 6-12: Heat exchanger dimensions vs. channel aspect ratio for 100 channels
As can be seen from Figure 6-12, the cross sectional length is weekly dependant on
the aspect ratio while the stack width and heat exchanger height are greatly dependant.
This can be explained by the fact that the channel width is the direct inverse of the
channel length and the stack height curve is a factor of 100 multiplied by the channel
Though variable on aspect ratio, plotting a case with 500 channels shows that the
magnitudes of the heat exchanger length and stack width are directly proportional to the
Cross Sectional Length (m) -.- Channel Width (m)
0 0.2 0.4 0.6 0.8 1
Aspect Ratio: r=LUW
Figure 6-13: Channel geometry and aspect ratios for 100 channels
0 0.2 0.4 0.6
Aspect Ratio: r=L/W
-*-Stack Width (m)
-- HTEX Height (m)
Figure 6-14: Heat exchanger dimensions vs. aspect ratio for 500 channels
Since the number of channels is five times greater, the flow area is five times less.
This requirement stems from a Reynolds number and topping cycle mass flow rate
requirement. Thus, for the same aspect ratio the 500 channel case will give a heat
exchanger height five times that of the 100 channel case.
Comparing the two cases show that an aspect ratio of 0.01 is the optimum value to
give a desirable compact heat exchanger shape. Given a realistic perspective on
geometric specifications the pressure loss penalty associated with those specifications is
6 Length (m)
5 1 -Stack Width (m)
4 -- HTEX Height (m)
0.004 0.009 0.014 0.019
Aspect Ratio: r=L/W
Figure 6-15: Optimum aspect ratio
Plate Heat Exchanger Pressure Losses
If the coolant fluid is chosen as pure helium the pressure loss is more pronounced
in the hot than in the cold side flow because the presence of liquid increases the average
viscosity and density of the two-phase flow. Of course the impact of liquid on viscosity
is dependent upon the empirical correlation used. Observation of pressure loss on three
different base cases show that the McAdams and Dukler viscosity correlations gave
similar results and slightly over predicted the viscosity equivalence correlation. The
Cichiti correlation was generally within an order of magnitude of the other three but
Accelerational pressure loss calculated in Equation (60) is not dependant on
viscosity and hence is independent of the viscosity correlation used. Therefore, the
change in total pressure drop associated with the viscosity method is only due to the
frictional loss contribution. From the previous discussions an aspect ratio of 0.01 is the
optimum value to give a desirable compact heat exchanger shape. Therefore, the three
test scenarios will be for varying channel numbers and an aspect ratio of 0.01.
Table 6-7: The empirical correlations compared below are given are for frictional and
accelerational loss only. Because of their small contribution to the total
pressure head, including gravitational head would give a negative pressure
Number of Channels
Correlation 100 250 500
fTP/fgo=l1 3.55867 22.2417 88.9668
McAdams 3.67906 22.9941 91.9764
Cichiti 4.63106 28.9441 115.776
Dukler 3.56351 22.2719 89.0877
Notice the disparity in pressure loss for different numbers of channels. There is a
definite influence of channel number on pressure loss associated with channel geometry
or flow area. Inspection of Equations (51) and (60) will show that the pressure loss is
proportional to the square of the mass flux and hence increases by a factor equal to the
square root of the number of flow channels for increasing channels.
100 200 300 400 500 600 700 800 900 1000
Number of Channels
Figure 6-16: Hot side pressure loss using the equivalent viscosity correlation
For an aspect ratio of 0.01 the pressure loss for any number of plates up to 1000
plates is negligible compared to the difference in topping cycle high and low pressures.
At this point we can be critical on the size of the heat exchanger and not be concerned
with pressure loss penalty. As it turns out, there is a minimum cut off for number of
channels or plates that give a symmetric CHEX shape.
More plates will reduce the length and height of the heat exchanger but will do
virtually nothing for the stack width. This is because more plates will simply reduce the
flow area per channel but also reduce the mass flow rate per channel by the same amount.
Thus, the mass flux does not change. Considering the asymptotic increase in pressure
loss with increasing channels it is best to select a lower number of plates.
S 40 -. Cross Sectional
*0 Length (m)
30 -U Stack Width (m)
N --HTEX Height (m)
0 200 400 600 800
Number of Plates
Figure 6-17: Heat exchanger geometry for r-0.01
The higher mass flow rates due to reduced specific heat in a He/Xe coolant do have
a considerable impact on CHEX sizing. Large concentrations of xenon can make coolant
pressure loss comparable or even in excess of the hot channel losses. This is
demonstrated in figure 1 for a 50/50% He/Xe mixture.
-*-Total Primary Pressure Loss (kPa) -M-Total Coolant Pressure Loss (kPa)
100 200 300 400 500 600 700 800 900 1000
Number of Channels
Figure 6-18: Hot and cold side losses for the He/Xe mixture
As can be seen the cold pressure loss far exceeds the hot pressure loss for the
He/Xe case for increasing channels. The hot pressure loss trend is slightly greater than
the pure helium case because the heat exchanger becomes 36% longer than the helium
case. This is due to the lower thermal conductivity of the He/Xe mixture.
The disparity in pressure loss between channels may give reason for changing to
alternative channel geometry. Alternate channel geometry could be found that equalizes
the mass flux and hence pressure loss in both channels thus forcing both hot and cold
flows to share the pressure loss penalty.
This could be done by simply making the cold channel width greater than the hot
channel width but this would increase the overall stacked width dimension of the heat
exchanger. A more desirable solution may be found by switching to tube type geometry.
The tubes would pass the condensing hot mixture while coolant would pass outside of the
tubes in a counter flow fashion.
Tube CHEX Rating and Sizing
The tube geometry allows for more flexibility in mass flow and pressure
conditions. In the plate design the hot and cold channels have the same geometry. The
tube design decouples the flow area and equivalent diameter calculation by allowing the
user to tailor the geometry of both channels. The channel flow area for both channels is
now controlled through the tube diameter and the square lattice pitch. Tubes are
increasingly convenient at higher power levels and higher pressures because of their
structural strength when in compression by pressure differences across the wall.
The CHEX code offers a channel geometry option to facilitate the tube design.
Once selected the code asks for a dimension fraction defined as: f=O.D./Pitch. The outer
diameter (O.D.) is the inner diameter (I.D.) minus the wall thickness. The user is also
asked for a specified lateral or side dimension for the CHEX or if it is desired for the
code to find this based off of the number of channels and the hot inlet Reynolds number
in a similar fashion that it found the channel geometry for the plate CHEX.
Figure 6-19: Tube channel geometry
If the lateral dimension is specified the code can calculate the pitch based by
dividing this length by the square root of the number of channels.
P = L/N (76)
It then calculates the O.D. by using the dimension fraction.
O.D. =Pxf (77)
The tube I.D. is assumed to be 90% that of the O.D. With that in mind the hot
channel flow area and heated perimeter is calculated. The cold channel equivalent
diameter, flow area and heated perimeter are calculated as such:
A = P2- (78)
P, = P,. = z(O.D.) (79)
De= f (80)
Selecting the same Reynolds number and adjusting the number of channels to keep
the mass flow rate per channel the same ensures that the diameter and pitch of the tubes
will remain constant by satisfying Equation (76) and Equation (77). Then the lateral
dimension can be calculated using Equation (79).
Optimum Dimension Fraction
The 100MW case was analyzed using the tube channel geometry and varying the
dimension fraction, lateral dimension and number of channels. The objective was to
determine the optimum dimension fraction that would equate pressure losses in both
channels for a range of operating conditions. The lateral dimensions used were 0.5m,
1.0m, and 1.5m. The number of channels was also varied between 1000 and 10,000
channels. Figure 6-20 shows the hot and cold channel pressure losses for various power
ratings and channel numbers.
-*-hot loss, N=1000, 0.5m
1.0E+06 -u-cold loss, N=1000, 0.5m
-a-hot loss, N=1E4, 0.5m
0 cold loss,N=1E4,0.5m
U n hot loss, N=1000, 1.0m
S .E --cold loss, N=1000, 1.0m
0 1.e E6 -4-hot loss, N=10E4, 1.5m
3, -cold loss,N=1E4, 1.0m
U) hot loss, N=1000, 1.5m
cold loss, N=1000 1.5m
1.0E+04 -u-hot loss, N=1E4, 1.5m
cold loss,N=1E4, 1.5m
0.35 0.45 0.55 0.65 0.75
Figure 6-20: Channel pressure loss vs. dimension fraction
Notice, the pressure loss does not change for varying the number of channels in the
geometry because the mass flow rates and flow areas are reduced proportionally. The
pressure loss does change with varying the lateral dimension because the flow area is
changing. It becomes apparent that a dimension fraction of about 0.6 becomes the break
even point for both channels sharing the pressure loss penalty.
A heat exchanger with equally proportioned sides with marginal pressure loss can
be found with a lateral dimension of one meter, a dimension fraction of 0.6 and 10,000
channels. This is readily observed on the following plot.
E --- N=1 E3, 0.5m
S7 --- N=1 E4, 0.5m
6 N=1E3, 1.0m
E N=1E4, 1.0m
5 N=1E3, 1.5m
S4 -- N=1E4, 1.5m
0.35 0.45 0.55 0.65 0.75
Dimension Fraction: f=O.D.IPitch
Figure 6-21: Axial height vs. dimension fraction for different number of channels
The hot and cold pressure loss for this geometry is roughly 0.5 bars. The maximum
velocity using He/Xe is 45m/s. This occurs at the maximum coolant temperature because
the coolant density is the lowest.
Though pressure loss may not be a concern for the 100MW, increasing the power
rating of the heat exchanger for 1000MW does have a profound pressure loss penalty for
the same heat exchanger. Increasing the reactor power by 10 increases the mass flow rate
to the CHEX proportionally, thus increasing the channels' mass fluxes and pressure
losses by the same factor. Since frictional and accelerational pressure loss is directly
proportional to the square of the mass flux the pressure loss can easily go up two orders
of magnitude from that of the 100MW case.
1.O -- Hot Ploss, N=1E3
S8 --Cold Ploss, N=1E3
Hot Ploss, N=1E4
Cold Ploss, N=1E4
Hot Ploss, N=1 E5
-Cold Ploss, N=1E5
0 500 1000 1500 2000
VCR Power Level (MW)
Figure 6-22: Channel pressure loss vs. reactor power level
The heat exchanger height is also affected by the increased rating. As expected,
increasing the power rating while keeping the lateral dimension constant increases the
height. This makes sense because the added surface area is required to get the heat out.
Also, increasing the number of channels decreases the length because it adds more
surface area per volume as is the nature of the CHEX.
0 200 400 600 800 1000 1200 1400 1600
VCR Power Level (MW)
Figure 6-23: Lateral dimension vs. reactor power level
Observing Figure 6-23 the plot makes it clear that the lateral dimension increase is
directly proportional to the square root of the VCR power level. This is expected because
the lateral dimension is directly proportional to the number of channels per side which in
turn is inversely proportional to the number of channels used.
The 1000MW CHEX has only roughly three times the lateral dimensions that the
100MW CHEX. Because of the square root relationship, even a 10GW plant with100
times the mass flow rate to the heat exchanger would only increase the lateral dimension
by a factor of 10. This revelation makes it inviting for large multi-thousand megawatt
Coolant Channel Pressure and Velocity
The coolant channel pressure determines the mean fluid velocity because it controls
the density of the fluid. For a given mass flux or mass flow rate with a fixed flow area,
increasing the fluid density decreases the fluid velocity. This can be seen from the ideal
gas model and the mass flux relationships.
Therefore, increasing the mass flow rate as a result of changing the He/Xe
composition plays a hand in determining the coolant pressure. The figure below shows
the affect of increasing channel pressure on the He/Xe coolant velocity.
18 bar 36 bar 50 bar 70 bar 100 bar
1000 1100 1200 1300 1400 1500 1600
Figure 6-24: Cold channel velocity profiles at varying pressure
Velocity and pressure play an important role in the heat exchanger design that has
not previously been addressed. Conventional closed loop gas turbine engines usually
operate at a low pressure of around 20 bars (Walsh, 1998). The resulting high density
fluid entering the engine enables very high power output for a given size of plant. This
means the heater pressure is established over 50 bars using the pressure ratio in the
bottoming cycle design. The higher pressure creates a problem with the plate heat
exchanger design in the form of pressure differential across the thin membrane like walls.
A large pressure differential could lead to the plate bowing in the middle towards the
lower pressure side of the channel walls. Switching to the tube channel geometry adds
structural strength through the theory of shells. Since the tube is placed under
compression the curved surface is less likely to bow. The situation is analogous to the
use of arches instead of strait beams to support heavy loads (Kelkar, 1987).
In addition, one benefit of using the coolant bypass that was used to prevent
interface freezing is that it reduces the mass flow rates in the channel. This reduction
decreases the coolant channel mass flux. Equation (82) and knowing that only 55% of
the total bottoming cycle working fluid goes through the CHEX proves that without the
bypass the mass flow rate and hence the coolant channel velocities would increase by:
100%/55%=181%. Therefore, due to the nature of the CHEX design envelope the
coolant channel velocity must be low.