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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 GlennLewis 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 page ACKNOW LEDGM ENTS ........................................ iv LIST OF TABLES ........... ................... .......... .............. vii LIST OF FIGURES ........ ...... .............. .......... ................. viii ABSTRACT................................................ xi 1 INTRODUCTION ................... .................. .............. .... ......... ....... History and Design Evolution ................. .................................4 Heat Transfer Issues and Thermal Design........................................6 Computer Simulation................. .... .......... .. ..8 Com parative Analysis Calculations.........................................................................9 ThermoPhysical 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 CounterFlow 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 BalanceofPlant ................ ...... ........................... 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 APPENDIX A EXAMPLE OUTPUT OF THE TOPPING CYCLE CODE...............................94 B LAGRANGE M ULTIPLIERS ....................................................... 96 LIST OF REFERENCES ....................... ......... ..........97 BIOGRAPHICAL SKETCH .............. ..... ......... ................100 LIST OF TABLES Table page 61 The candidate wall material candidates with selection criteria were taken from published data in D eW itt. .......................................................................... ...............56 63 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 64 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 66 The CEA equilibrium calculation of W with the BoersmaKlein et al. fission product inventories show that tungsten does not bond with any of the fission products present in the system ..................... .......... ...... ...... 60 67 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 Figure page 11 The influence of the diffusion layer on the vapor partial pressure..............................7 21 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 22 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 23 Ts Diagram depicting cycle operating characteristics....................................18 31 UF4 vapor relative enthalpy data.................................... .................. 23 32 UF4 vapor relative entropy....................................24 33 UF4 vapor thermal conductivity comparison.................... ......................................26 34 UF4 vapor dynamic viscosity .................................... ...............26 35 Temperature dependent helium mole fraction curve fit ..........................................28 36 Temperature dependent UF4 mole fraction curve fit .....................................29 37 Temperature dependent mixture enthalpy curve fit...................................29 41 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 41 Schematic of CHEX Code. The wall resistance is not shown in the algorithm because it is a constant not a variable................................................................ 41 51 Equivalent electrical circuit with the latent transferred to the wall modeled as a source term. .......................................................48 52 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 53 Thevenin equivalent circuit analysis. The node temperature difference is shown as shorted out and the current source is shown as an open circuit. ..............................49 54 Thevenin equivalent circuit with the total heat transfer to the coolant channel drawn as qload............... ............................................ .....................49 55 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 56 Axial heat flux vs. the axial dimension comparative analysis comparison between DLM and TEM. The DLM height is greater by 7%...........................................52 57 Condensing HTC for the DLM model. The HTC goes to virtually zero as vapor is condensed. ............................ ............. ......... 53 58 Comparative analysis HTC comparison between DLM and TEM calculations. .......53 61 VCR online refreshment scheme for online refueling and fission product separation .......................................................58 62 Coolant delivery flow arrangements: (a) Series flow arrangement (b) Parallel flow arrange ent ...................................... ................................ ........ 62 63 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 64 Separated helium portion of the topping cycle. The regenerator effectiveness is 0.1 and the compressor efficiency is conservatively estimated as 0.8. .......................64 65 Separated helium portion of the topping cycle. The regenerator effectiveness is 0.3. ................................................. .........65 66 Mixture portion of the topping cycle. Pressure ratio is 10. ....................................66 67 Separated helium portion of the topping cycle. The regenerator effectiveness is 0.5 and the pressure ratio is 10. .............................................. ............... 67 68 Interface freezing anomaly observed with zero coolant bypass..............................68 69 Reflector cooling allowed increasing the interface temperature. Reflector cooling is equal to 10% of reactor power. .............. ..................... ...... ....... ........ 69 610 Condensing HTC with 55% of the bottoming cycle working fluid going through the heater. The reflector cooling is 14% of the reactor power. ..................................70 611 Effect of axial enhancement on CHEX heat flux profiles and axial height. ............71 612 Heat exchanger dimensions vs. channel aspect ratio for 100 channels....................75 613 Channel geometry and aspect ratios for 100 channels ..........................................76 614 Heat exchanger dimensions vs. aspect ratio for 500 channels ..............................76 615 O ptim um aspect ratio.....................................................77 616 Hot side pressure loss using the equivalent viscosity correlation .........................79 617 Heat exchanger geometry for r 0.01 ........................................ 80 618 Hot and cold side losses for the He/Xe mixture ....................................... ...80 619 Tube channel geom etry ............................................... ............... 82 620 Channel pressure loss vs. dimension fraction.............................. ......... ......83 621 Axial height vs. dimension fraction for different number of channels .....................84 622 Channel pressure loss vs. reactor power level...................................................85 623 Lateral dimension vs. reactor power level...................................................86 624 Cold channel velocity profiles at varying pressure ..................................... 87 A1 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 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. 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 noncondensable gas in vapor greatly impedes the condensation heat and mass transfer towards the condensation interface. This noncondensable 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 onedimensional 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. CHAPTER 1 INTRODUCTION The ultrahigh 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). 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 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 ultrahigh 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 turbomachinery. 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 U235 (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 twophase 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 noncondensable gas but is inappropriate for the CHEX calculations because 95% of all the gas/vapor molecules are noncondensable. 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 noncondensable gasses has been a preoccupation of many thermal hydraulic studies throughout the 1980's and 1990's. It is well known that the presence of a noncondensable 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 noncondensable 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 noncondensable 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 noncondensable partial pressure increases through entrainment. P Pb TOb Figure 11: 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). 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 thermophysical properties of the condensing twophase 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 thermophysical property code. The CHEX code used the mole fraction data and mixing relationships to calculate thermal transport and fluid properties of the condensing mixture. Comparative Analysis Calculations Two comparative analysis studies were conducted. The first comparative analysis checked the validity of the thermophysical 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. ThermoPhysical Property Comparative Analysis The thermophysical property code used is the NASA GlennLewis 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, 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) thermophysical 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 thermophysical 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 noncondensable helium may be compressed to the reactor pressure without damaging the compressor blades from impinging liquid droplets. After separation, the condensate is recirculated back to a mixer just prior to the VCR. The pumped UF4 liquid may then be preheated to saturated vapor before mixing or it may be aspirated directly into the reactor where it is vaporized.' 2081 K Recuperator 18 bar 927 K 87 bar CHEX MHD 1379 K 2700 K 91 bar Separator SPrecooler LP Compressor Intercooler HP Compressor 1379 K "' 1379 K 87 bar Mixer 87 bar Figure 21: 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 precooler 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 twostage compressor process with intercooling is proposed to increase the overall thermodynamic cycle efficiency. Intercooler 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 points. 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) T <1600K ln(P(at)) = 38453T 1 + 74.88 7.051n(T) T> 1600K 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 mixer. 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): VT.lowF hgh 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. ,s I(3) T Tn out,a Tout,s Phigh (4) Rearranging Equation (3): Tout,a i ou t,s ]7n ]7 n T (5) 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. Bottoming Cycle The bottoming cycle consists of the coolant feed through the topping cycle condenser and precooler (now referred to as the heater stages or heater), turbomachinery and the bottoming cycle regenerator, precooler 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. Intercooler I Heater Precooler a g Regenerator Turbine LP HP Intercooler Figure 22: 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 temperature. An optimum bottoming cycle coupled with the topping cycle description in Figure 21 operates according to the following temperatureentropy diagram. 1400  1300 1200  1100  1000  900 0 o 800 C 700 E 600 500 400 o 300 200  25 26 27 28 29 30 31 32 33 Entropy (kJ/kgK) Figure 23: Ts 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. 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. 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 standardstate pressure. For a condensed or frozen species the standard state is the substance at the condensed phase at the standardstate pressure. Most recent versions of the code have used a standard state pressure of one bar (McBride & Gordon, 1994). CEA uses the minimizationoffree 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: N g = PL ni (6) J1 Where chemical potential per kilogrammole of species j is defined as: P T,P (7) Or: o nj = n+RTln +RTln(P) for gasses (8a) And: 0 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: N Ya,n, b = 0 J=1 or (9) b bO = 0 Where stoichiometric coefficients aij are the number of kilogramatoms of element i per kilogrammole of species j, bi0 is the assigned number of kilogramatoms 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 NewtonRaphson method. Using the NewtonRaphson 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): 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+3.7x102T 3200 (12) T2 And for UF6 liquid: C (kJ/kgK) = 0.448+1.86(106)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 twophase vapor curve for constant entropy and varying pressures. Dissociation and ionization phenomena were not modeled except for the large starred data in Figure 31 and Figure 32. Inspi NASA ThermoBuild CEA x INSPI* CEA* x Ion 600 500 400 S300  200  0 o A co CO 100  I S0  1 1150 1650 2150 2650 3150 100 200 300  Temperature (K) Figure 31: UF4 vapor relative enthalpy data Inspi NASA ThermoBuild CEA Ion 0.225 0.2 0.175 0.15 : 0.125  0.1 10.075 S0.05 0.025  0 1750 1950 2150 2350 2550 2750 2950 3150 Temperature (K) Figure 32: UF4 vapor relative entropy The CEA data shows very good agreement with the Anghaie and online 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: 50M46 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 J1 N =Z N (16) (MY, ++M)2 Where: 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) 1 P=YP 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. Anghaie CEA 1950 2150 2350 2550 Temperature (K) 2750 2950 3150 Figure 33: UF4 vapor thermal conductivity comparison Anghaie CEA 1.4 ,n 1.3 0 1.2 0 0 1 E S0.9 0.8 1 750 1950 2150 2350 2550 2750 2950 Temperature (K) Figure 34: UF4 vapor dynamic viscosity 0.75 2 0.7 E S0.65 E 0.6 S0.55 . 0.5 C 8 0.45 E 0.4 0.35 0.3 11 750 3150 Least Squares Data Preparation The least squares program was created for generating thermophysical 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): 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. OE = 0,i= 1,2,...m (24) da, Leads to: n Ex Ex a Lx 2x Ex3 aJ = LTxy (25) EX EX EX a2 X2y Define: 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 twophase mixture enthalpy, species mole fractions and twophase specific heats. Plotting CEA data against curve fits derived from the least squares program show reasonable conformity. He MF (Data) He MF (Fit) 1.01 1 0.99 0.9 L" 0.98 0.95 M 0.976\ 0.95  0.94 1350 1400 1450 1500 1550 1600 1650 1700 1750 Temperature (K) Figure 35: 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 Temperature (K) 1650 1700 1750 Figure 36: Temperature dependent UF4 mole fraction curve fit Mix Enth (Data) Mix Enth (Fit) 1400 1450 1500 1550 1600 Temperature (K) 1650 1700 1750 Figure 37: 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, 0.06  0.05 0.04 0.03 0.02 0.01 1350 1350 2000 2200 2400  s 2600 l 2800 3000 3000 3200 3400 1350 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 36: 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. 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 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 twophase 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 semiannular 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 gasliquid 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 onedimensional nodal analysis and calculates the total equivalent thermal resistance at each node in order to calculate the heat flux at each node. This 1D approach has been widely used for the passive cooling system design of the Westinghouse AP600 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. 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 noncondensable gas from the interface. This balance results in a logarithmic gas concentration distribution near the interface. ColbrunHougen 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"/(TbsTw), 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 noncondensable 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 41). 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 Thm0 \AA/ Tom Re Figure 41: 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. 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 noncondensable 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: XX bX X ~ ln(Xb/X,) And rewriting Equation (32) gets: V, = X (Xg, X,) (33) gave g 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 ClausiusClapeyron 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 ClausiusClapeyron equation is: P, P i v vb if (35a) sat,, sat,b T fg For our purposes the fluid specific volume is neglected so that the twophase specific volume becomes that for the vapor alone. RT 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=Deq4Af/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 ClausiusClapeyron 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 ClausiusClapeyron equation while holding Tave constant so that: P,, f (T, T) InK T2(39) vb RTe 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) (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: Sc=[ /Dabp. CounterFlow 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). 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 calculations. Channel Geometry, Mass Flow Rates, Starting Temperatures, Fit Equations Figure 41: 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. Energy Balance q, = h, (h,l ho) (45) Thermal Circuit qn = (Th TenRo, (46) Where: 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) Or: 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 twophase frictional pressure gradient is calculated using a fanning equation format. 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 twophase flow. The frictional multiplier (pgo relates the single phase pressure gradient equivalent to the true twophase gradient. Inspection of Equation (50) shows that the single phase frictional multiplier is defined such that: 2 "Pf (51) 0Pm Jgo Using the definition of flow quality and specific volume will prove that the two phase mean density is defined as: 1 x (1 x) + P(52) PM Pg A And the density ratio in the left hand side of Equation (51) is: =P x+(1x) X (53) P, P1 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 twophase viscosity. If the second option is chosen then there are three widely adopted formulas for calculating the two phase viscosity. 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 rewritten as: dP G d 1 G d x (1x)) _+ 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 twophase 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) reduces to: 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. CHAPTER 5 DIFFUSION LAYER MODEL COMPARATIVE ANALYSIS The counterflow 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 41 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 41 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 41 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 51. 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 Thm Tcm Figure 51: 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 52, 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. Thm RS Tw Tcm S.......................o o Tcm Figure 52: Thermal circuit shown with the load resistance removed. The notation Rcw stands for the series resistance of the wall conduction and coolant channel convection. 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 53 is: Req= Rs. Rs Figure 53: 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 52. q, + q = 0 Th T R q, Tw = Thm + q,x R (63) Now that the Thevenin equivalent resistance and temperature are defined the Thevenin circuit is complete: Req load Rcw TThev=ThmTcm Figure 54: 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) Req +R 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) T Th 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 condensing system. 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. DLM TEM 80 70 : 40 i30  20  10 0  1350 1400 1450 1500 1550 1600 1650 1700 1750 Temperature (K) Figure 55: 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 35 and Figure 36, 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. SDLM TEM 80 70 60 S*50 E 40 S 30  20 10  0  0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Axial Height (m) Figure 56: 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. 0.8  0.7 0.6 < 0.5 0.4  S0.3 I 0.2 0.1 0  1350 1650 1700 1750 Figure 57: 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 Temperature (K) 1650 1700 1750 Figure 58: Comparative analysis HTC comparison between DLM and TEM calculations. 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(ThmTcm). 1400 1450 1500 1550 1600 Temperature (K) 0.34  0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18  1350 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. CHAPTER 6 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 exchanger size. 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 61: The candidate wall material candidates with selection criteria were taken from published data in DeWitt. Thermal Melting Conductivity Material Point (K) (W/mK) Tungsten 3660 174 Molybdenum 2894 138 Niobium 2741 53.7 Rehnium 2453 47.9 Silicon 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 dissociated into: Table 62: 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 He 90.000% UF3 0.026% UF4 9.947% UF5 0.026% Ceramics 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 63: 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. Mole Species Fraction He 44.966% SiF2 0.037% SiF4 0.039% UF2 0.011% UF3 0.209% UF4 4.775% UF5 0.001% C 0.077% SiC 49.885% Refractory Metals 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 64: 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. Mole Species Fraction CL 0.00038 CL2 0.00001 He 0.44983 V(cr) 0.0001 VCL4 0.04988 W(cr) 0.49981 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 BoersmaKlein, Kelling and Kistemaker have developed VCR fission product inventories using the ORIGEN computer program (BoersmaKlein 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+ 3.45E3 mole/sec UF4 (30%) Actinides Reactor 1.51E3 mole/sec CF4  >Reactor  Recombine Heavy Components Separator Light Components 2.7E3 mole/sec (7%) ponent Reprocessing Purifier Fission Products 2.8E3 mole/sec (3%) 1E4 mole/sec CF4 (100%) Figure 61: VCR online refreshment scheme for online refueling and fission product separation 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 61, 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 65. Table 65: The BoersmaKlein et al. fission product inventories are for a 1200MW VCR with UF4 enriched to 30% and refreshed every 200h. Component Component 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% 4.550E07 Kr 147 83.7 12303.9 1.007% 8.758E06 Rb 137 85.5 11713.5 0.959% 8.338E06 Sr 579 87.6 50720.4 4.152% 3.611E05 Y 252 88.9 22402.8 1.834% 1.595E05 Zr 1320 91.2 120384 9.855% 8.569E05 Nb 15 92.9 1393.5 0.114% 9.920E07 Mo 880 96 84480 6.916% 6.014E05 Tc 163 99 16137 1.321% 1.149E05 Ru 664 101.7 67528.8 5.528% 4.807E05 Sn 9 118.7 1068.3 0.087% 7.605E07 Sb 8 121.8 974.4 0.080% 6.936E07 Te 240 127.6 30624 2.507% 2.180E05 1 232 126.9 29440.8 2.410% 2.096E05 Xe 1620 131.3 212706 17.413% 1.514E04 Cs 513 132.9 68177.7 5.581% 4.853E05 Ba 742 137.4 101950.8 8.346% 7.257E05 La 446 138.9 61949.4 5.071% 4.410E05 Ce 1190 140.1 166719 13.648% 1.187E04 Pr 287 140.9 40438.3 3.310% 2.879E05 Nd 767 144.3 110678.1 9.060% 7.879E05 Pm 63 145 9135 0.748% 6.503E06 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 66: The CEA equilibrium calculation ofW with the BoersmaKlein et al. fission product inventories show that tungsten does not bond with any of the fission products present in the system. Species Species Mole Mole Fraction Fraction 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 ultrahigh 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 quasiarbitrarily 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. Design Envelope 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 balanceofplant 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 backtoback counter flow heat exchangers than no mixing is required. (a) (b) Series Parallel Arrangement Arrangement mhot TMHD,o T3 Mhot cTT3 CHEX T3 CHEX m2 Ti Tcond T2z Precooler T3 Precooler Ti mi Ti Tprec,o m cold Figure 62: 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 channel wall. 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. BalanceofPlant To illustrate the balanceofplant 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 Ts plots illustrates the effect of regenerative preheating of the helium before it reaches the reactor inlet temperature. Reactor  Isentropic Expansion  Actual Expansion  CHEX/Reg 3000 2750 2500 2250 .L 2000 E 1750 1500 1250 6.8 7 7.2 7.4 7.6 7.8 Entropy (kJ/kg K) 8 8.2 8.4 8.6 Figure 63: 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 1500 1400 1300 1200 2 1100 S1000 S900 E I 800 700 600 500 Entropy (kJ/kgK) Figure 64: 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 1500 1400 1300 1200 2 1100  2 1000 900 800 700 600 500 27 28 29 30 31 32 33 34 Entropy (kJ/kgK) Figure 65: Separated helium portion of the topping cycle. The regenerator effectiveness is 0.3. 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 64 and Figure 65 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 recycled.  Reactor .Isentropic Expansion Actual Expansion  CHEX 2650 2450 2250 2050 ) 1850 E 1650  1450  1250 6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 Entropy (kJ/kgK) Figure 66: 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 67 is practically the desired bottoming cycle heat sink temperature shown if Figure 23. 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 balanceofplant3. Precooler . Low Comp Intercooler High Comp  Preheater 1300 1100 3 900 7 E 700Z 500 300 25 27 29 31 33 35 Entrpy (kJ/kgK) Figure 67: Separated helium portion of the topping cycle. The regenerator effectiveness is 0.5 and the pressure ratio is 10. Figure 63 and Figure 65 are an optimum topping cycle balanceofplant 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. 0.35 0.3 . 0.25 E 0.2 S0.15 * F, 0.1 0.05 0 1350 1400 1450 1500 1550 1600 1650 1700 1750 Temperature (K) Figure 68: Interface freezing anomaly observed with zero coolant bypass. Unfortunately, the freezeup 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 recirculated to the VCR. The freezeup 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. 0.35 0.3 0.25  E 0.2 0.15 I . 0.1 0.05 0  1350 1400 1450 1500 1550 1600 1650 1700 1750 Temperature (K) Figure 69: 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. 0.8 0.7  0.6 < 0.5 S0.3  0.2 0.1 0 1350 1400 1450 1500 1550 1600 1650 1700 1750 Temperature (K) Figure 610: 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 changed. In Figure 611 notice the change in heat flux distribution as well as the overall height in the heat exchanger between the enhanced design and the unenhanced. 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. 71 increasing the coolant exit temperature decreases the temperature difference between hot and cold channel nodes. No Enhancement Enhancement 250 200 150 E S100 0 50 0 0.5 1 1.5 2 2.5 Axial Height (m) Figure 611: 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 turbomachinery 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 exchanger. 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. GxD, Re =G (71) Where: G =hot G= Af And: 4xAf _4x(LxW) 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 Where: A, =LxW And: 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. 300.00 250.00 ^ 200.00 + Cross Sectional Length (m) 150.00 Stack Width (m) N A HTEX Height (m) ad 100.00A 50.00 0.00 0 0.2 0.4 0.6 0.8 1 Aspect Ratio: r=L/W Figure 612: Heat exchanger dimensions vs. channel aspect ratio for 100 channels As can be seen from Figure 612, 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 width curve. 76 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 flow area. Cross Sectional Length (m) . Channel Width (m) 6.00 5.00 4.00 o 3.00 5 2.00 1.00 0.00 0 0.2 0.4 0.6 0.8 1 Aspect Ratio: r=LUW Figure 613: Channel geometry and aspect ratios for 100 channels 60 50 o 40 E 30 > 20 X 10 0 0 0.2 0.4 0.6 Aspect Ratio: r=L/W +Cross Sectional Length (m) *Stack Width (m)  HTEX Height (m) 0.8 1 Figure 614: Heat exchanger dimensions vs. aspect ratio for 500 channels 77 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 evaluated. 10 9 8 SCross Sectional 6 Length (m) 5 1 Stack Width (m) 4  HTEX Height (m) 1 0 0.004 0.009 0.014 0.019 Aspect Ratio: r=L/W Figure 615: 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 twophase 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 slightly greater. 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 67: 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. 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. 0.400 0.350 0. 0.300 f 0.250 0 j 0.200 2 0.150 *0 0.100 0.050 0.000 100 200 300 400 500 600 700 800 900 1000 Number of Channels Figure 616: Hot side pressure loss using the equivalent viscosity correlation Channel Optimization 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. 80 60 50 S 40 . Cross Sectional *0 Length (m) 30 U Stack Width (m) N HTEX Height (m) o 20 10 0 0 200 400 600 800 Number of Plates Figure 617: Heat exchanger geometry for r0.01 He/Xe Influence 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) MTotal Coolant Pressure Loss (kPa) 7.00 6.00 S5.00  J 4.00 S3.00 2.00 0 > I 00^ 100 200 300 400 500 600 700 800 900 1000 Number of Channels Figure 618: 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. I.D.. I Pitch Figure 619: 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) 4xA De= f (80) e Pw 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 620 shows the hot and cold channel pressure losses for various power ratings and channel numbers. 1.0E+07 *hot loss, N=1000, 0.5m 1.0E+06 ucold loss, N=1000, 0.5m ahot 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 4hot 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 uhot loss, N=1E4, 1.5m cold loss,N=1E4, 1.5m 1.0E+03 0.35 0.45 0.55 0.65 0.75 Dimension Fraction Figure 620: 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. 12 11 10 9 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 2 0  0.35 0.45 0.55 0.65 0.75 Dimension Fraction: f=O.D.IPitch Figure 621: 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. Power Rating 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. 1.4E+07 1.2E+07 _ 9 1.0E+07 1.O  Hot Ploss, N=1E3 S8 Cold Ploss, N=1E3 S8.0E+06 Hot Ploss, N=1E4 Cold Ploss, N=1E4 3 6.OE+06 Hot Ploss, N=1 E5 Cold Ploss, N=1E5 0 4.0E+06 2.0E+06 O.OE+00 0 500 1000 1500 2000 VCR Power Level (MW) Figure 622: 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. 4 3.5 3 o 2.5 E 2 i 1.5 j 1 0.5 0  0 200 400 600 800 1000 1200 1400 1600 VCR Power Level (MW) Figure 623: Lateral dimension vs. reactor power level Observing Figure 623 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 multithousand megawatt power plants. 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, 87 increasing the fluid density decreases the fluid velocity. This can be seen from the ideal gas model and the mass flux relationships. PM (81) RT v=G (82) P Afp 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 50 45  40 35  > 25 20 2 o ............."... M  0 15 1000 1100 1200 1300 1400 1500 1600 Temperature (K) Figure 624: 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. 