COMPACT HEAT EXCHANGER DESIGN FOR TRANSFERRING HEAT FROM A
VAPOR CORE REACTOR INTO A GAS TURBINE POWER PLANT
By
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
2004
Copyright 2004
by
Samuel E. Bays
This document is dedicated to my loving wife, Nikki.
ACKNOWLEDGMENTS
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.
iv
TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................. iv
LIST OF TABLES............................................................................................................ vii
LIST OF FIGURES ......................................................................................................... viii
ABSTRACT....................................................................................................................... xi
1 INTRODUCTION ........................................................................................................1
History and Design Evolution ......................................................................................4 Heat Transfer Issues and Thermal Design....................................................................6 Computer Simulation....................................................................................................8 Comparative Analysis Calculations..............................................................................9
Thermo-Physical Property Comparative Analysis ................................................9 Diffusion Layer Theory Comparative Analysis ..................................................10
2 THERMODYNAMIC ANALYSIS METHOD .........................................................11
Topping Cycle ............................................................................................................11 Intercooler...................................................................................................................13 Topping Cycle Code Description ...............................................................................13 Bottoming Cycle.........................................................................................................17
3 CONDENSATION PROPERTY MAPPING.............................................................19
The CEA Code............................................................................................................20 Thermodynamic Properties .................................................................................22 Thermal Transport Properties..............................................................................24
Least Squares Data Preparation ..................................................................................27 4 THERMAL HYDRAULIC MODEL DEVELOPMENT...........................................31
The Heat Transfer Model............................................................................................34 Diffusion Layer Theory Development ................................................................36 Counter-Flow Nodal Analysis.............................................................................40
Pressure Loss Model...................................................................................................43 Frictional Pressure Loss ......................................................................................44
v
Accelerational Pressure Loss...............................................................................45 5 DIFFUSION LAYER MODEL COMPARATIVE ANALYSIS ...............................47
Comparative Analysis Method ...................................................................................47 Impact of DLM...........................................................................................................51
6 COMPACT HEAT EXCHANGER DESIGN............................................................55
Wall Material Selection ..............................................................................................56 Ceramics ..............................................................................................................57 Refractory Metals ................................................................................................57 Fission Product Test ............................................................................................58
Design Envelope.........................................................................................................61 Coolant Temperature Selection ...........................................................................61 Balance-of-Plant ..................................................................................................63 Interface Freezing Phenomenon ..........................................................................67
Rating and Sizing........................................................................................................71 Plate CHEX Rating and Sizing............................................................................73
Plate heat exchanger pressure losses ............................................................77 Channel optimization ...................................................................................79 He/Xe 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 Thermodynamic Tools................................................................................................90 Pressure Loss ..............................................................................................................91 Interface Freezing Phenomenon .................................................................................92 Channel Velocity Considerations ...............................................................................92 Remarks ......................................................................................................................92
APPENDIX A EXAMPLE OUTPUT OF THE TOPPING CYCLE CODE......................................94
B LAGRANGE MULTIPLIERS ...................................................................................96
LIST OF REFERENCES...................................................................................................97
BIOGRAPHICAL SKETCH ...........................................................................................100
vi
LIST OF TABLES
Table page 6-1 The candidate wall material candidates with selection criteria were taken from
published data in DeWitt..........................................................................................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 VCl4. ................................................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-1 Data index description ...............................................................................................95
vii
LIST OF FIGURES
Figure page 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, MHD 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 Rcw stands for the series resistance of the wall conduction and coolant channel convection..........48
viii
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 modeled. .........................................................................................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 separation .................................................................................................................58
6-2 Coolant delivery flow arrangements: (a) Series flow arrangement (b) Parallel flow arrangement ..............................................................................................................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
ix
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 Optimum 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 geometry ............................................................................................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
x
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
By
Samuel E. Bays
August 2004
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.
xi
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.
xii
CHAPTER 1 INTRODUCTION
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
technology.
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).
Modern 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
1
2
vapor core removes these limitations allowing the fuel and coolant to be at the same
temperature.
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
3
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
4
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 modern VCR concepts.
5
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.
6
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
7
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.
P
Pvb
Pgb
Tvb
Figure 1-1: The influence of the diffusion layer on the vapor partial pressure. Depiction
adapted from Collier (Collier, 1972).
8
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).
Computer Simulation
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
condensing mixture.
9
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
10
data and the mole fraction data which has proven to be accurate (Anghaie, 1992)(NIST,
5/2/2004).
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
not modeled.
CHAPTER 2 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.
Topping Cycle
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.
11
12
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.1
Recuperator 2081 K 18 bar
927 K 87 bar
CHEX MHD
1379 K
2700 K Separator 91 bar
Precooler VCR
LP Compressor
Intercooler
HP Compressor
1379 K 1379 K Mixer 87 bar 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 or 1GW.
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.
13
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
effici han at
hese
ooler
There is a side benefit to incorpo oling in the design. Because
interc HEX)
be the
The CEA code prove ination of thermodynamic
state
evaluate dissociation and ionization mole fractions in the reactor and MHD throughput.
ency of the compressor stage dictates a lower compressor inlet temperature t
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. T
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.
Interc
rating interco
ooling occurs in a separate heat exchanger device than the UF4 condenser (C
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
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
d as a valuable tool for rapid determ
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
14
A file writing code was created using the C++ language for rapid generation o
CEA input decks for each state point in the primary cycle. C++ was used because of its
f
wide s
nows
l all topping cycle
calcu
ture and pressure. After the output file is read, the user is asked for the system
pressu e
es
variety of features offered in its file input/output system. Another C++ code read
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 k
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 unti
lations 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
points.
The first state calculated is the reactor output. The user is asked for the reactor
tempera
re ratio. The file writing code then calculates the MHD outlet pressure; reads th
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 calculat
the actual MHD exit condition. In the next calculation, the file writing code calculates
15
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).
( )
(TTP atm ln)7.0(0.7)3.0(217.74)1000(37977ln 1)( ±−±+±−= − )
( ) ( )KT
TTPKT
atm
1600ln05.788.7438453ln
16001)(
>−+−=
≤−
(1)
After this point, the batch program is then thrown into an infinite loop. Each loop
comp
es
lium or preheater side of the regenerator is evaluated first in order to
calcu n
g
l
e
letion 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 seri
of calculation steps focuses on the UF4, helium and mixture properties before and after
the mixer.
The he
late the compressor inlet and exit states. The user is asked for the regeneratio
effectiveness. This is defined as the ratio of the MHD exhaust heat used for preheatin
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 contro
volume around the reactor. The helium mass flow rate is backed out using the mixtur
mole fractions set at: UF4 (5%) and He (95%).
reactorQm =&
& ( )inoutmix hh − (2)
( )mix
mixHeHeHelium M
mMym && ××= (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
16
is used, an optimum pressure ratio must be determined to maximize cycle efficienc
optimum intercooler pressure is given by (Todreas & Kazimi, 1993):
highlowi PPP =
This equation assumes that the high and low compressors’ inlet
y. The
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.
inaout
insout
TTTT
−
−
,
,c =η
(4)
(3)
( ) γγ 1,
−
=
i
high
in
sout
PP
TT
Rearranging Equation (3):
cin
soutc
in
aout
soutc
aout
TT
TT
TT
ηη
η
+−=
−=
−
1
11
,,
,,
n
inin TT
And substituting Equatio (4):
( )
i
caoutin
P
TT
ηγγ
×= −1
,
state is defined as the low pressure compressor exit and
chighP η+−
1
(5)
The intercooler entrance
the intercooler exit is determined as the high pressure compressor inlet. Therefore, the
interc ooler 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.
17
Bottoming Cycle
The bottoming cycle consists of the coolant feed through the topping cycle ant feed through the topping cycle
condenser and precooler (now referred to as the heater stages or heater), turbo-machinery
and th ols the
and therefore eliminating unnecessary pressure loss penalty in the CHEX.
Thou
heater inlet
tempe
ine
thus
condenser and precooler (now referred to as the heater stages or heater), turbo-machinery
and th ols the
and therefore eliminating unnecessary pressure loss penalty in the CHEX.
Thou
heater inlet
tempe
ine
thus
e bottoming cycle regenerator, pre-cooler and intercooler. A split stream co
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.
e bottoming cycle regenerator, pre-cooler and intercooler. A split stream co
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.
Topping
Figure 2-2: Bottoming cycle schematic showing the split stream configuration to
date topping cycle intercooling. Later this separate cooling streamwill prove advantageous for reducing the mass flow rate through the heater
Figure 2-2: Bottoming cycle schematic showing the split stream configuration to
date topping cycle intercooling. Later this separate cooling streamwill prove advantageous for reducing the mass flow rate through the heater
Intercooler
Heater
Regenerator Precooler
Turbine HP LP
Intercooler
accommoaccommo
gh intercooling reduces the compressor work required, it also reduces the
temperature from what it would be without intercooling. This lower
gh intercooling reduces the compressor work required, it also reduces the
temperature from what it would be without intercooling. This lower
rature would require more heat transfer to achieve the desired turbine inlet
temperature. To supply this additional heat transfer, the regenerator recovers turb
waste heat and recycles it by preheating the He/Xe to the heater inlet temperature,
rature would require more heat transfer to achieve the desired turbine inlet
temperature. To supply this additional heat transfer, the regenerator recovers turb
waste heat and recycles it by preheating the He/Xe to the heater inlet temperature,
18
providing the necessary power required to heat the fluid to the maximum bottoming cy
temperature.
An optimum bottoming cycle coupled with the topping cycle description in Figure
2-1 operates a
cle
ccording to the following temperature-entropy diagram.
200
300
400
500
600
700
800
900
1000
1100
1200
1400
25 26 27 28 29 30 31 32 33
Entropy (kJ/kgK)
Tem
pera
ture
(K)
1300
Figure 2-3: T-s Diagram depicting cycle operating characteristics
Here, the minimum cycle temperature is about 300 K and the maximum heater
t law cycle efficiency
is 40%
temperature is about 1300K. The pressure ratio is 6.24 and the firs
. 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.
CHAPTER 3 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
• Density
• Sonic Velocity
• Viscosity
• 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
code.
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.
19
20
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
processing plants.
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:
21
∑=
=N
jjj ng
1µ (6)
Where chemical potential per kilogram-mole of species j is defined as:
n
g
jiPTj
j n≠
∂∂
=,,
µ (7)
Or:
)ln(ln0
PRTnn
RT jjj
++= µµ for gasses (8a)
And:
µµ 0jj = for condensed phases (8b)
Note: the superscript 0 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:
0
0
0
1
0
=−
=−∑=
iu
N
jijij
bbor
bna
(9)
Where stoichiometric coefficients aij are the number of kilogram-atoms of element i
per kilogram-mole of species j, bio 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
22
( ) 01
0
1 1=∂−+∂
+=∂ ∑∑ ∑
== =
N
jiii
N
jj
l
iijij bbnaG λλµ (10)
Where: G ∑ = −+=l
i iibbg
10 )(λ
Equation (10) is the requirement for equilibrium. Minimization is obtained
iteratively by updating nj, λj, 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.
Thermodynamic Properties
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 &
Shapiro, 2000):
)()( 12122
1
TTCdTTTT
Chh pp −==− ∫ (11a)
==− ∫ TTCdTT
T TTC
ss pp 2
12 ln)(
2
1
(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.
23
Liquid enthalpies and entropies were calculated using the same integrations from
Equation (11) but with temperature dependent specific heats. Where the specific heat for
UF4 liquid is given by:
22 3200107.33.136)/(
TTKmolJCp −×+=−
− (12)
And for UF6 liquid:
236 )10(71.7)10(86.1448.0)/( −− −+=− TTKkgkJC p (13)
CEA thermodynamic properties are taken from Gurvich (1982) (McBride et al.,
2002). Thermodynamic data is presented standardized to a fixed temperature reference
datum (h1 and s1). 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.
-300
-200
-100
0
100
200
300
400
500
600
1150 1650 2150 2650 3150
Temperature (K)
H(T
)-H(1
800)
(kJ/
kg)
Inspi NASA ThermoBuild CEA INSPI* CEA* Ion
Figure 3-1: UF4 vapor relative enthalpy data
24
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
1750 1950 2150 2350 2550 2750 2950 3150
Temperature (K)
S(T)
-S(1
800)
(kJ/
kg-K
)
Inspi NASA ThermoBuild CEA Ion
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:
=Ω
4.1
6.42,2
,50
lnT
M jji (14)
This lack of accurate data made it necessary to compare with an alternative source.
25
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).
∑∑=≠=
+=
N
iN
ijj
ijii
iimix
yy
y
1
1φ
ηη (15)
∑∑=≠=
+=
N
iN
ijj
ijii
iimix
yy
y
1
1ψ
λλ (16)
Where:
21241212
141
+
+=
ji
j
j
i
j
iij MM
MMM
ηη
φ (17)
+
−−+=
2)(
)142.0)((41.21
ji
jijiijij MM
MMMMφψ (18)
Here the symbol ηmix and λmix 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
iii MyM
1
(19) ∑=
=
=N
ii
totii
pp
pyp
1
(20)
∑=
=N
iiii uMyM
u1
1 , ∑=
=N
iiii hMyM
h1
1 , ∑=
=N
iiii sMyM
s1
1 (21)
26
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.
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
1750 1950 2150 2350 2550 2750 2950 3150Temperature (K)
Ther
mal
Con
duct
ivity
(mW
/cm
K)
Anghaie CEA
Figure 3-3: UF4 vapor thermal conductivity comparison
0.8
0.9
1
1.1
1.2
1.3
1.4
1750 1950 2150 2350 2550 2750 2950 3150Temperature (K)
Dyn
amic
Vis
cosi
ty (m
illiP
oise
)
Anghaie CEA
Figure 3-4: UF4 vapor dynamic viscosity
27
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,
1999):
2210)( TaTaaTy ++= (22)
The error function is given by:
([∑=
++−=n
ii TaTaayE
1
22210 )] (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.
miaE
i
,...2,1,0 ==∂∂ (24)
Leads to:
∑∑∑
=
∑∑∑∑∑∑∑∑
yxxyy
aaa
xxxxxxxxn
22
1
0
432
32
2
(25)
Define:
[ ]
∑∑∑∑∑∑∑∑
=432
32
2
xxxxxxxxn
X (26) [ ]
=
2
1
0
aaa
A [ ]
∑∑∑
=yx
xyy
Y2
28
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.
[ ] [ ][ ] 1−= XYA (27)
Where: [X]-1 is the inverse matrix of [X]
[ ] [ ]XX TransposeCofactorsX =−1 (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.
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1350 1400 1450 1500 1550 1600 1650 1700 1750
Temperature (K)
Hel
ium
Mol
e Fr
actio
n
He MF (Data) He MF (Fit)
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
29
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.
0
0.01
0.02
0.03
0.04
0.05
0.06
1350 1400 1450 1500 1550 1600 1650 1700 1750
Temperature (K)
UF4
Mol
e Fr
actio
n
UF4 MF (Data) UF4 MF (Fit)
Figure 3-6: Temperature dependent UF4 mole fraction curve fit
-3400
-3200
-3000
-2800
-2600
-2400
-2200
-2000
1350 1400 1450 1500 1550 1600 1650 1700 1750
Temperature (K)
Mix
ture
Ent
halp
y (k
J/kg
)
Mix Enth (Data) Mix Enth (Fit)
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,
30
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/φn. The heat transfer per node is calculated
based on the enthalpy change across the node from Figure 3-6: qn=mdot(∆H)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.
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
exchanger are:
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
31
32
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 churn 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
33
channels. Thermodynamic equilibrium may also be assumed to evaluate the changing
gas-liquid thermodynamic state as it is being cooled through the channel.1
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 regime.
34
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
35
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.
Thm Rs Rw Rc Tcm Thm
Rc
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
concentration.
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.
36
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”l) heat flux:
ivivfglstiw y
TkVcMiqqqTTh
∂∂
+−=+==− ∞ """)( (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, Mv is the
molecular weight of the vapor species, kv 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:
∂∂
−=y
XgcDVcXcV igigi (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:
( )i
g
i
g
gi Xy
Dy
XX
DV
∂∂
=
∂
∂= ln1 (31)
Considering a diffusion layer thickness δg, the condensation velocity is redefined.
37
( )ln()ln( gigbg
i XXDV −∂
= ) (32)
At this point it is convenient to define the log mean mole fraction so that:
( )ibib
ave XXXX
Xln
−=
And rewriting Equation (32) gets: ( )gigbgaveg
i XXXD
−∂
=,
V (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.
( vbvigavegt
i PPXPDV −
∂=
,
) (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:
fg
fg
bsatisat
vbvi
Tvi
TTPP
=−−
,,
(35a)
For our purposes the fluid specific volume is neglected so that the two-phase
specific volume becomes that for the vapor alone.
38
tavevv
avefg PXM
RTv
,
= (35b)
The condensation velocity in terms of temperature difference is now:
( bigavegave
avevvfgi TTXRT
XMDiV −
∂=
,2
, ) (36)
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 δg. 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.
−
=∂
=DMPi
TRL
TTqLSh
vtfg
ave
bi
l
g22
32''
φ (37)
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:
= 2
2
221
o
ovofg
avec TR
DMPiT
kφ
(38)
Where: )/ln(
))1/()1ln((
gigb
gigb
XXXX −−
−=φ
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=hlL/kc or where the characteristic length
in a closed channel is L=Deq≡4Af/Pwet: Sh=hl Deq /kc.
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
39
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 Φ in
the condensation conductivity. This can be done by integrating the Clausius-Clapeyron
equation while holding Tave constant so that:
2
)(ln
ave
bifg
vb
vi
RT
TTiPP −
=
(39)
The ideal gas equation is used to equate the saturation partial pressure ratios to the
mole fraction ratio Xvi/Xvb:
avevvb
avevvi
avevvb
avevvi
vb
vi
RTMXRTMX
RTMcRTMc
PP
== (40)
Where cv 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: hl = 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
40
( ) ( )( ) ( ) 3.08.0
3.08.0
Re023.0
PrRe023.0
ScSh
Nu
=
= (41)
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:
Sc=µ/Dabρ.
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
calculated.
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).
41
)()(
