Scaling of a Space Molten Salt Reactor Concept THESIS

Transcription

1 Scaling of a Space Molten Salt Reactor Concept THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Robert K Palmer Graduate Program in Mechanical Engineering The Ohio State University 2015 Master's Examination Committee: Dr. Thomas Blue, Advisor Dr. Xiaodong Sun

2 Copyright by Robert K Palmer 2015

3 Abstract The Space Molten Salt Reactor Concept considered for this study is a design for a 500 kw electric, 2 MW thermal liquid fueled molten salt reactor coupled to a Brayton power cycle to be used for Lunar or Martian surface power or for nuclear electric propulsion. To validate the concept, a scale terrestrial model would be built and tested. The goal of this work is to produce a design for a scale model or models that could be used to simulate the behavior of an actual reactor. Two scaling approaches were used. In the first approach, a laboratory-scale model was designed by focusing on power, cost, and safety. In the second approach, the model was scaled in order to achieve nearer similarity with the prototype. ii

4 Dedication This document is dedicated to my family and friends who have had faith in me and have been patient with me through many trials. iii

5 Acknowledgments I would like to thank Dr. Thomas Blue, Dr. Xiaodong Sun, and all the students working on the Steckler project for their guidance and help. iv

6 Vita May Macon County High School, TN B.A. Mathematics, Western Kentucky University M.S. Mathematics, Western Kentucky University M.S. Aeronautics and Astronautics, Purdue University Fields of Study Major Field: Mechanical Engineering v

7 Table of Contents ABSTRACT... II DEDICATION... III ACKNOWLEDGMENTS... IV VITA... V TABLE OF CONTENTS... VI LIST OF TABLES... VIII LIST OF FIGURES... IX NOMENCLATURE... X CHAPTER 1: INTRODUCTION AND BACKGROUND PROBLEM STATEMENT PROJECT OBJECTIVES STECKLER MOLTEN SALT SPACE REACTOR... 2 CHAPTER 2: SCALING APPROACH GOVERNING EQUATIONS NON DIMENSIONAL GOVERNING EQUATIONS GENERAL SIMILARITY LAWS CHAPTER 3: SCALING PROCESS vi

8 3.1 SCALING THE PRIMARY HEAT EXCHANGER AND LOOP SCALING THE SECONDARY HEAT EXCHANGER AND LOOP SCALING THE TERTIARY HEAT EXCHANGER AND LOOP CHAPTER 4: SCALING RESULTS LABORATORY-SCALE MODEL SIMILARITY-MAXIMIZED MODEL CHAPTER 5: EXACT SOLUTION TO THE INTEGRAL MOMENTUM EQUATION CHAPTER 6: RADIATOR AND ELECTRICAL HEATING ELEMENT SIZING RADIATOR SIZING HEATING ELEMENT SIZING OF THE WATER MODEL CHAPTER 7: CONCLUSIONS CHAPTER 8: FUTURE WORK REFERENCES APPENDIX: FLUID PROPERTIES vii

9 List of Tables Table 1: Property Ratios of Substitute Liquods for Laboratory-Scale Model Table 2: Property Ratios of Substitute Material Table 3: PHX Scaling Values for Laboratory-Scale Model Table 4: SHX Scaling Values for Laboratory-Scale Model Table 5: THX Scaling Values for Laboratory-Scale Model Table 6: Property Ratios for Maximized Model Table 7: PHX Scaling Values for the Maximized Model Table 8: SHX Scaling Values for the Maximized Model Table 9: THX Scaling Values for the Maximized Values Table 10: Required Radiator Surface Areas for Prototype and Models Table 11: Properties of Fuel Salt at 1450 K [1] Table 12: Properties of Liquid Lithium at 1344 K [1] Table 13: Properties fo Helium at 15 Bar [1] Table 14: Properties of NaK at 556 K [1] viii

10 List of Figures Figure 1: Power Conversion Scheme of the Reactor [1]... 2 Figure 2: Primary Heat Exchanger [3] Figure 3: SHX Fin Geometry [8] Figure 4: Geometry of Unit Cell [1] Figure 5: Radiation Heat Transfer Geometry Figure 6: Radiator Surface Area as a Function of Heat Sink Temperature ix

11 Nomenclature a cross sectional area A s solid cross sectional area Ar gas flow cross-sectional area in SHX/THX c 1 coolant 1 c 2 coolant 2 c p specific heat C d drag coefficient d diameter d i tube inner diameter d o tube outer diameter D hydraulic diameter f friction factor F heat exchanger effectiveness F v radiation view factor h heat transfer coefficient h g gas channel height h l liquid metal channel height I current k thermal conductivity K orifice or expansion/contraction coefficient l length L total length m mass n number N tube number Nu Nusselt number p pump pump pressure P power Pr Prandtl number p h total heated perimeter R thermal resistance Re Reynolds number Pr Prandtl number q thermal power per unit volume q heat flux Q total total thermal power Q thermal power r p compressor pressure ratio R thermal resistance s SHX/THX fin width x

12 t T u U V w y z α β γ δ ε ξ η ηc ηt ρ σ SHX/THX fin thickness temperature velocity total heat transfer coefficient voltage work spatial variable into solid perpendicular to fluid flow axial spatial variable thermal diffusivity turbine pressure loss term ratio of specific heats conduction thickness emissivity wetted perimeter thermal efficiency compressor efficiency turbine efficiency density Stefan Boltzmann constant Subscripts i g l m o p r R s ith section gas side liquid side model reference section prototype representative variable ratio solid xi

13 Chapter 1: Introduction and Background 1.1 Problem Statement Research at The Ohio State University conducted under the Ralph Steckler Space Grant Colonization Research and Technology Development Opportunity has resulted in the preliminary design of a liquid-fueled molten salt reactor coupled to a Brayton power cycle [1]. The ultimate goal of the reactor would be to provide power for a Lunar or Martian colony or to provide power for nuclear electric propulsion. The reactor as designed would produce 500 kilowatts electrical power from two megawatts of thermal power. In order to demonstrate this technology, a terrestrial model would need to be built to simulate a prototype reactor. The goal of this project is to size a scaled model of the Steckler Molten Salt Space Reactor. If built, this model could be tested and used to predict the behavior of the prototype reactor. 1.2 Project Objectives The goals of this project are to 1) utilize appropriate mass, momentum, and energy equations, in both dimensional and non dimensional form, to model each fluid loop. 1

14 2) develop similarity relationships to relate values of the model and the prototype. 3) use these similarity relationships to create a scaled model that if built could be tested and used to predict the prototype reactor. 1.3 Steckler Molten Salt Space Reactor The Steckler Molten Salt Space Reactor is a liquid-fueled molten salt nuclear reactor coupled to a Brayton power cycle [1]. Figure 1 displays the power conversion scheme of the reactor system. Figure 1: Power Conversion Scheme of the Reactor [1] 2

15 The reactor is designed to produce approximately 2 MW thermal power and 500 kw electrical power delivered to the turbo alternator. The primary heat exchanger (PHX) is a conventional baffled-counterflow shell and tube heat exchanger, which transfers heat from the fuel salt, a mixture of lithium and uranium fluorides, to lithium coolant. The secondary (SHX) and tertiary (THX) heat exchangers are both offset-strip fin liquid-gas heat exchangers. Lithium transfers heat to helium in the SHX, then after the turbine, helium transfers residual heat to a sodium-potassium eutectic fluid (NaK) in the THX. These heat exchangers will be described in more detail in following sections. Finally, waste heat is rejected into space by a large radiator. 3

16 Chapter 2: Scaling Approach 2.1 Governing Equations For the Steckler reactor prototype and model, the following set of one-dimensional equations were used. These are modified from the equations used by Ishii and Kataoka which were derived for a reactor cooling system using natural circulation [2]. Nomenclature and variables used by Ishii and Kataoka were also employed in this analysis. Mass: ρ i u i a i = ρ 0 u r a 0 (1) Integral Momentum: Fluid Energy: ρ du r a 0 l dt a i i i = Δp pump ρu r 2 i + K) d i (a 0 (2) 2 (fl a i ) 2 ρc p ( T i + u T i t i ) = 4h (T z d si T i ) + q (3) Solid Energy: ρ s c ps T si t = k s 2 T si + q s (4) 4

17 Boundary Condition: k s T si y = h(t i T si ) (5) These represent a generalized set of equations that can be simplified using the following assumptions. 1) The liquid fuels and coolants are assumed to be incompressible. This implies that the momentum equation can be solved independently of the energy equations. 2) The gaseous coolant is also assumed to be ideal and incompressible except in the compressor and turbine. This assumption is reasonable provided that the fluid velocity is less than 30% of the local speed of sound of the gas. 3) Heat transfer to the environment (except through the radiator) is negligible. 4) Heat generation occurs only in the core. 5) Pump power is negligible. 6) Axial heat flow in the solid is negligible. 7) The temperature change through heat exchanger pipes and fins is assumed to have a linear profile. 8) The inaccuracies of using temperature-averaged thermodynamic properties of working Mass: fluids are negligible. Using these assumptions, these equations can be reduced to the following relationships. Integral Momentum: u i a i = u r a 0 (6) 5

18 ρ du r a 0 l dt a i i i = ΔP pump ρu r 2 i + K) d i (a 0 (7) 2 (fl a i ) 2 Fluid Energy: ρc p ( T i + u T i t i ) = δ 4h z i1 (T d si T i ) + δ i0 q (8) Solid Energy: ρ s c ps T si t = δ i1 k s 2 T si (9) Boundary Condition: k s T si y = δ i1h(t i T si ) (10) Here, the Kronecker delta notation is used to indicate in which each energy source or sink term is relevant in which part of the loop (0 indicates the reactor core and 1 indicates a heat exchanger). 2.2 Non Dimensional Governing Equations These governing equations are non-dimensionalized, using the following nondimensional variables [2]. U i = u i u 0, U r = u r u 0 6

19 L i = l i l 0, Z = z l 0, Y = y δ τ = tu 0 l 0 θ = T T 0 A i = a i a 0 2 = δ 2 2 These variables relate values at any point to their corresponding values in the reference section, which is the reactor core. The variables produce the following non-dimensional governing equations. Mass: U i A i = U r (11) Momentum: 7

20 du r L i dτ A i i = 1 2 ΔP i U r 2 2 F i A i 2 i (12) Fluid Energy: θ i τ + U r A i θ i Z = δ i1st(θ si θ i ) + δ i0 Q 1 (13) Solid Energy: θ si τ = δ i1t 2 θ si (14) Boundary Condition: θ si Y = δ i1bi(θ i θ si ) (15) This variable substitution also leads to the following non dimensional numbers [2]. Pump pressure number: 8

21 Δp = p pump pump driving head = 1 2 ρu 0 2 dynamic pressure loss Friction number: F i = ( fl d + K) friction force = i inertia force Stanton number: St = 4hl 0 wall convection = ρc p u 0 d axial convection Fuel heat source number: Q = q l 0 heat source = ρc p u 0 T 0 axial energy change Time ratio number: T = α sl 0 transport time δ 2 = u 0 conduction time Biot number: 9

22 Bi = hδ wall convection = k s conduction Also, the reference scale for the steady state temperature change in the core can be determined from a one dimensional analysis of the core, yielding T 0 = Q total ρc p u 0 a 0 (16) The reference scale for the velocity can be obtained from solving Equation 7 in steady state, yielding u 0 = 2 p pump ρ ( fl 2 d +K) i (a 0 i ) a i (17) Finally, the conduction thickness δ can be determined by noting that the hydraulic diameter is defined by D h = 4a ξ (18) where ξ is the wetted perimeter, and the conduction thickness is [2] δ = a s ξ 10 (19)

23 where a s is the solid cross-sectional area. Equations 18 and 19 lead to δ = D 4 (a s a ) (20) 2.3 General Similarity Laws by For any property or value, similarity ratio between the model and the prototype is given ψ R = ψ model ψ prototype (21) For each of the preceding non dimensional numbers, a similarity ratio can be defined. It is desirable for all similarity ratios to be one, or as close to one as possible, so that the model behavior will most closely approximate the prototype behavior Geometric Similarity Geometric similarity requires that A ir = (a i a 0 ) m = 1 (22) (a i a 0 ) p which implies that A ir = A 0R d 2 ir = d 2 0R d 0R = d ir (23) 11

24 From this relationship, the continuity equation requires that u 0R = u ir (24) Dynamic Similarity From the momentum equation, dynamic similarity is achieved if ( L i i ) = 1 (25) A i R and and ΔP R = P pumpr = 1 (26) ρ Ru 0R ( F i i A2) = 1 (27) i R A stronger condition on Equation 25 is complete axial geometrical similarity, which is given by (l i l 0 ) m (l i l 0 ) p = 1 (28) which implies that L ir = L 0R (29) 12

25 It can be shown that Equation 28 implies Equation 25. Also, Equation 27 can generally be achieved by properly chosen orifices to produce the correct friction coefficient numbers. Returning to Equation 17 and using Equations 24 and 27, the similarity ratio for the velocity which satisfies Equation 26 is u ir = u 0R = p pumpr 0.5 ρ R 0.5 (30) Energy Similarity Complete energy similarity requires that St R = Q 1R = T R = Bi R = 1 (31) However, this is nearly impossible to achieve if substitute fuels or materials are used. If a similarity relationship for Equation 16 is used, then Here T 0R = Q totalr ρ R C pr u 0R A 0R (32) Q totalr = q Rl 0R A 0R (33) It can be shown that 13

26 Q 1R = 1 (34) The remaining energy ratios can be made equal to one depending upon the choice of model fluids and materials and requirements of the model. 14

27 Chapter 3: Scaling Process Two general scaling approaches were taken. In the first, a small, cost effective model for a laboratory is scaled. In the second approach, a liquid metal and liquid metal salt model that achieves a high level of similarity is sized. For the laboratory-scale model, cost, convenience, available power, and safety are the primary considerations. Cost and power considerations require that the model produces a small fraction of the total power of the prototype, but produces enough power so that the core and heat exchanger temperature differences are measurable with conventional instruments. Also, the model must be large enough so that the use of specialized fabrication is limited. Finally, common liquid and gaseous coolants should be used to reduce costs and safety concerns. Once the substitute fluids and materials have been chosen, the independent ratios that can be chosen are d R, l R, Q totalr, and, u R (or p pumpr ). These ratios are chosen to create a model that could be operated in a laboratory. The following sections describe how each loop and heat exchanger are scaled, and then the results of this scaling approach. 15

28 3.1 Scaling the Primary Heat Exchanger and Loop 1 The Primary Heat Exchanger (PHX) is a baffled, counter-flow shell and tube heat exchanger depicted in Figure 2. The fuel flows through the tubes, and the coolant flows on the shell side. The thermodynamic and pressure characteristics of the heat exchanger must be scaled. Figure 2: Primary Heat Exchanger [3] Once d R, l R, Q totalr, and u R (or P pumpr ) are chosen, the heat transfer coefficient ratio for the fuel, h fr, is determined. The tube-side heat transfer coefficient in the prototype is given by h tube = Nu tube k d tube (35) 16

29 The Nusselt number for fully turbulent flow is given by [4] Nu tube = 0.023Re d 0.8 Pr (36) Therefore, using algebra, the tube side heat transfer coefficient ratio is h tuber = k R ρ R ur PrR d 0.17 tr μ0.83 (37) R For the shell side, the heat transfer coefficient for the prototype is [5] h shell = jcp shell m shell A shell Pr (38) Here, A shell is the cross-sectional cross-flow area of the coolant at the center baffle position, and j is a coefficient determined by corrective factors for flow in the shell. The heat transfer coefficient ratio is [6] h shellr = j R Cp shellr m shellr A shellr Pr R (39) It can be shown that A shellr = l R d R (40) 17

30 If it is assumed that j R = 1 (41) and with m shellr = ρ R u R d R 2 (42) then, the ratio is h shellr = Cp shellr ρ R u R d R l R Pr R (43) Equations 37 and 43 are used to determine the tube-side and shell-side heat transfer coefficients of the model, or the respective velocity ratios can be determined by setting each heat transfer coefficient equal to one. To size the model heat exchanger, the heat transfer surface area Am must be determined from the equation A m = ( Q FU i T lm )m (44) where F is an effectiveness factor, and the overall heat transfer coefficient is given by U i m = [ d i + d i ln(d 0 d i ) d 0 h shell 2k s h tube ]m (45) 18

31 The model power Q is determined from the total power ratio and the tube-side temperature change is determined from Equation 32. Then, the model shell-side temperature change is given by T shell = ( Q m shellcp shell )m (46) Here, the shell-side mass flow rate is calculated from the previously determined shell-side velocity and the diameter ratio. Then the logarithmic mean temperature difference for the counter-flow model is T lm = (T f,in T c,out ) (T f,out T c,in ) ln[(t f,in T c,out )/(T f,out T c,in )] (47) The heat transfer surface area is also related to the length and number of tubes by A m = π(d i l tube N) m (48) where N is the number of tubes. Once the heat transfer area is known, the inner tube diameter and the tube length are both determined from the chosen diameter and tube ratios, the number of tubes needed can be determined, which completes the thermodynamic scaling of the PHX. The pressure losses for the heat exchanger tubes, core, and piping are determined using 19

32 P = f l ρ u2 2d i (49) where the Darcy-Weisbach friction factor for fluid in the tubes is approximated by [7] 1 f = 2 log(re f) (50) which must be solved iteratively or approximated by a polynomial curve fit. Care must be taken in choosing the velocity and diameter ratios so that the tube-side pressure losses are not too large, as the pressure loss is proportional to u 2. These same relationships can also be used to determine the pressure losses through the piping and core. d i 3.2 Scaling the Secondary Heat Exchanger and Loop 2 The Secondary Heat Exchanger (SHX) is a liquid to gas offset-strip fin counter-flow heat exchanger. Figure 3 displays the basic fin structure, and Figure 4 displays the geometry of a two unit cells. The smaller channels shown in Figure 4 are for the lithium coolant, and as the figure shows, there are four lithium channels for every unit helium cell. For a given heat exchanger, there are r rows containing n cells, and these arrangements are m deep. For example, Figure 3 below displays r = 1 and m = 3. Each set of lithium channels is bordered by helium cells on both sides, so there are r rows of lithium channels and r + 1 rows of helium channels, with helium and lithium flowing in opposite directions. 20

33 Figure 3: SHX Fin Geometry [8] Figure 4: Geometry of Unit Cell [1] 21

34 To scale secondary loop and the SHX, the PHX shell-side mass flow rate must be conserved for the entire loop, and the shell-side temperature change determined for the PHX by Equation 47 must also be used for the liquid-side of the SHX. For the lithium side, the heat transfer coefficient for a liquid metal in a non circular duct is [9] h c1 = k ln l D l = k l D l ( Re D 0.8 Pr 0.8 ) (51) where the hydraulic diameter of the lithium channel is given by [1] where D l = 2s lh l s l +h l (52) s l = s 3t l 4 (53) It can be shown algebraically that D lr = D R (54) So, the heat transfer coefficient ratio for the liquid side of the THX is h c1r = k lrnu lr D lr (55) 22

35 In this ratio, Equation 52 may be used for the model fluid if the fluid is a liquid metal; if not, Equation 37 must be used for the model fluid Nusselt number. Equation 55 may be used to determine the model liquid-side heat transfer coefficient, or may be used to find the model liquid-channel Reynolds number if it has not been previously chosen. Once the Reynolds number is known, then the mass flow rate per cell for the model can be determined from the following expression, m cell,l = (ρ l u l h l s 3t 4 ) m (56) Once the cell mass flow rate is known, then from [1] m cell,l = m l,total 4nr (57) the product nr can be determined. To determine n, r, and m, the gaseous side of the heat exchanger must be scaled. For the gaseous side, the heat transfer coefficient is [5] [9] h c2 = k gnu g D g = 0.005Re l 0.5 Re D Pr k g D g (58) where Re l is the Reynolds number based upon the length of the fin, and the hydraulic diameter of the helium channel is [1] 23

36 D g = 2sh gl sl+h g l+th g (59) Due to the presence of the fin length in this relationship, this hydraulic diameter does not scale directly as the diameter ratio d R. However, if the following scaling ratios are used, s R = h gr = t R = d R (60) and L R = L 0R (61) then, the correctly scaled gas channel hydraulic diameter can be determined. Next, the heat transfer coefficient ratio h c2r = k grnu gr D gr (62) can be used to determine the model gaseous-side heat transfer ratio, or can be used to determine the gaseous-side cell velocity. Once the velocity is known, the Reynolds number can be calculated, and the gaseous cell mass flow rate can be determined from m cell,g = (ρ g u g h g s) m (63) 24

37 Next, the total model gaseous mass flow rate must be determined. First, the total gas-flow cross sectional area for the prototype must be calculated from Ar total,p = π 4 [D g 2 n(r + 1)] p (64) Once this value has been calculated, then the total gaseous mass flow rate for the model is calculated using m g,total = (ρ g u g ) m A 0R Ar total,p (65) Then, from [1] m cell,g = m g,total n(r+1) (66) the product n(r + 1) can be calculated. From Equations 57 and 66, n and r can be determined. Solving for both yields, n = m l,total 4m cell,l m g,total (67) 4m cell,g and 25

38 r = m l,total m l,total 4m cell,l 4m cell,l m 1 g,total 4m cell,g (68) Next, the thermal resistance per cell of the model is calculated using [1] R m = 1 h c1 A l + 1 2s l k + ( h c2 A g a m +t + 2s l k a m ) 1 (69) where k is the thermal conductivity of the material, A l and A g are the cross-sectional areas of the liquid and gaseous channels respectively, and a m is the width of the material between the gaseous and liquid channels. If a m scales with the diameter ratio d R, then the thermal resistance per cell of the model can be calculated. The liquid-side temperature change has already been determined from the previous loop, and the gaseous-side temperature change of the model can be calculated from T g,m = ( Q ) m g,totalcp g m (70) Then, the logarithmic mean temperature difference can be calculated from Equation 47. The number of cells needed to transfer the energy through the model heat exchanger is given by N cells = Q R m T lm = mnr (71) 26

39 and from this equation, m can be determined. This completes the thermodynamic scaling of the model SHX. Next, the pressure losses through the model SHX must be calculated. The friction factor and pressure drop though the liquid channels can be determined by Equations 50 and 51 used for the model PHX. For the pressure drop through the gaseous channels, the pressure drop can be calculated using Equation 50, and the friction factor for Reynolds numbers less than is [8] f = 9.62Re 0.74 α 0.19 δ 0.31 γ 0.27 ( Re 4.4 α 0.92 δ 3.8 γ 0.24 ) 0.1 (72) where the ratios α, δ, and γ are defined in Figure 3. For Reynolds numbers greater than 20000, the friction factor is [5] f = C dt 2l Re l 0.5 (73) where C d = 0.88 [5]. 3.3 Scaling the Tertiary Heat Exchanger and Loop 3 The Tertiary Heat Exchanger (THX) is a gas to liquid offset-strip fin heat exchanger identical in design to the SHX. The total mass flow rate for the gaseous fluid determined from the scaling of the SHX must be conserved, but the temperature change of the gas though the THX is different since the compressor and turbine powers must be taken into account. 27

40 The turbine power is given by [10] w T = η T m g,totalcp g T T,in (1 β loss ) = η T m g,totalcp g (T T,in β loss T T,out ) (74) γ 1 r γ p where T T,in is the turbine inlet temperature, and is equal to the temperature exiting the gasside of the SHX. The terms η T, Cp g, γ are the turbine efficiency, gas specific heat, and, gas ratio of specific heats, respectively, and T T,out is the turbine outlet temperature. The term β loss is a pressure loss term, and [10] β loss = ( P T,outP C,out P T,in P C,in ) γ 1 γ (75) The same value for the prototype is used for the model. Equation 74 can be solved for the pressure ratio r p of the model, yielding γ r p,m = β γ 1 loss [1 ( w T ) η T m g,totalcp g T T,in m ] γ 1 γ (76) where it is assumed that the turbine power ratio is equal to the total power ratio w TR = Q totalr (77) 28

41 The model turbine outlet temperature T T,out, which is equal to the THX inlet temperature, can also be calculated from Equation 74 [10]. T T,out = 1 w T (T β T,in ) (78) loss η T m g,totalcp g The turbine power is delivered to the turbo alternator and to the compressor, or w T = w C + w Alt (79) With efficiency η for both the prototype and model, w Alt,m = ηq totalr(q total) (80) So, Equations 79 and 80 can be used to determine the compressor power for the model. Then, knowing that the entrance temperature to the gaseous side of the SHX which was determined previously is equal to the exit temperature of the compressor, the gaseous exit temperature of the model THX, which is equal to the compressor entrance temperature, can be determined from the following equation for compressor power. w C = η C m g,totalcp g (T C,out T C,in ) (81) 29

42 where η C is the compressor efficiency, and T C,out and T C,in are the compressor outlet and inlet temperatures, respectively. Finally, the power rejected by the THX can be calculated by Q THX = m g,totalcp g (T T,out T C,in ) (82) From Equations 74 81, the inlet and outlet temperatures are determined. The rest of the scaling process for the THX is identical to that of the SHX, and the scaling of the THX is completed. 30

43 Chapter 4: Scaling Results The following sections describe the results of the scaling for the laboratory-scale model and the model that is scaled specifically to achieve a high level of similarity. 4.1 Laboratory-Scale Model For the laboratory-scale model, a simple, safe, and cost effective model is needed. A total thermal power in the range of 10 kw to 20 kw was desired. Stainless steel 316 was selected as the substitute material for the core, heat exchangers, and piping, with water chosen to replace all the liquids and helium selected for the gas loop. The prototype delivers 71.2% of its thermal energy to the turbine, with 25.6% of the total thermal power delivered to the turbo-alternator. These percentages were used in the model as well. In the model, complete axial and geometric similarities were preserved, and the diameter ratio, length ratio, velocity ratio, and power ratio were independently chosen. The model was then tuned to produce a model with suitable pressure drops and temperature changes within the liquid range of water. One issue with a model using water is the relatively small liquid range of water at atmospheric pressure. If waste heat is to be radiated to either the inside of a laboratory or 31

44 to the outside environment, then the difference between the radiator and the ambient temperature should be as large as possible. However, this requires the model reactor exit temperature to be close to the boiling point of water so that the logarithmic mean temperature difference of each heat exchanger to be as large as possible to minimize the size and mass of each heat exchanger. This limited liquid temperature prevented the thermal power of the water model from exceeding 10 kw. Therefore, a liquid with a larger liquid temperature range was sought. Propylene glycol, C3H8O2, is a double alcohol and is a commonly used in automotive antifreeze [11]. It has more than twice the liquid range of water, and was chosen as a model fluid for a propylene glycol and helium system. The following tables display the values of the prototype compared to the water/helium and propylene glycol/helium systems. 32

45 Fuel Salt Water Propylene Glycol Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Lithium Water Propylene Glycol Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio NaK Water Propylene Glycol Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Helium at 15 bar (1040 K) Helium at 5 bar (345 K) Helium at 5 bar (345 K) Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Helium at 4.2 bar (729 K) Helium at 3 bar (345 K) Helium at 3 bar (345 K) Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Table 1: Property Ratios of Substitute Liquods for Laboratory-Scale Model Molybdenum Stainless Steel 316 Density ratio Specific heat ratio Thermal conductivity ratio Thermal diffusivity ratio Table 2: Property Ratios of Substitute Material 33

46 Prototype Water Model Propylene Glycol Model Diameter ratio Length ratio Tube velocity ratio Shell velocity ratio Power ratio Power (W) 2,000,000 10,000 20,000 Tube inner diameter (m) Tube outer diameter (m) PHX length (m) Tube number Fuel mass flow rate (kg/s) Tube velocity (m/s) Tube Reynolds number Shell mass flow rate (kg/s) Shell velocity (m/s) Shell Reynolds number Tube heat transfer coefficient (W/m 2 -K) Shell heat transfer coefficient (W/m 2 -K) Fuel temperature change (K) Fuel inlet temperature (K) Fuel outlet temperature (K) Coolant temperature change (K) Coolant inlet temperature (K) Coolant outlet temperature (K) Logarithmic mean temperature difference (K) Tube pressure drop (Pa) Tube Stanton number ratio Tube time number ratio Tube Biot number ratio Shell Stanton number ratio Shell time number ratio Shell Biot number ratio Table 3: PHX Scaling Values for Laboratory-Scale Model 34

47 Prototype Water Model Propylene Glycol Model Diameter ratio Length ratio Liquid side velocity ratio Gas side velocity ratio t (m) s (m) l (m) Gas channel height (m) Liquid channel height (m) Gas channel hydraulic diameter (m) Liquid channel hydraulic diameter (m) n r m Gas cell mass flow rate (kg/s) Gas channel velocity (m/s) Gas channel Reynolds number Gas channel heat transfer coefficient (W/m 2 -K) Liquid cell mass flow rate (kg/s) Liquid channel velocity (m/s) Liquid channel Reynolds number Liquid channel heat transfer coefficient (W/m 2 -K) Gas side temperature change (K) Gas side inlet temperature (K) Gas side outlet temperature (K) Liquid side temperature change (K) Liquid side inlet temperature (K) Liquid side outlet temperature (K) Logarithmic mean temperature difference (K) Gas side Pressure drop (Pa) Liquid side pressure drop (Pa) Liquid Stanton number ratio Liquid time number ratio Liquid Biot number ratio Gas Stanton number ratio Gas time number ratio Gas Biot number ratio Table 4: SHX Scaling Values for Laboratory-Scale Model 35

48 Prototype Water Model Propylene Glycol Model Diameter ratio Length ratio Liquid side velocity ratio Gas side velocity ratio t (m) s (m) L (m) Gas channel height (m) Liquid channel height (m) Gas channel hydraulic diameter (m) Liquid channel hydraulic diameter (m) n r m Gas cell mass flow rate (kg/s) Gas channel velocity (m/s) Gas channel Reynolds number Gas channel heat transfer coefficient (W/m 2 -K) Liquid cell mass flow rate (kg/s) Liquid channel velocity (m/s) Liquid channel Reynolds number Liquid channel heat transfer coefficient (W/m 2 -K) Gas side temperature change (K) Gas side inlet temperature (K) Gas side outlet temperature (K) Liquid side temperature change (K) Liquid side inlet temperature (K) Liquid side outlet temperature (K) Logarithmic mean temperature difference (K) Gas side Pressure drop (Pa) Liquid side pressure drop (Pa) Rejection power (W) Compression ratio Gas Stanton number ratio Gas time number ratio Gas Biot number ratio Liquid Stanton number ratio Liquid time number ratio Liquid Biot number ratio Table 5: THX Scaling Values for Laboratory-Scale Model 36

49 As the data indicate, neither water/helium or propylene glycol/helium achieve a high level of similarity with the chosen parameters. Thermal power was kept low for each model so that such a model could be run in a relatively small research lab. The water/helium model power of 10 kw was used because higher powers caused the liquid exit temperature of the THX to fall below the freezing point of water. The need to keep the power low combined with the differences in the density and specific heat ratios between the prototype fuel and the water result in temperature changes of a few Kelvin or less in the liquids in the model heat exchangers. The fluid temperature changes calculated would be difficult to measure and maintain. These temperature changes could be increased by decreasing the diameter ratio, but the result would be larger pressure drops through the heat exchangers. The THX liquid exit temperatures for both models are close to 300 K, indicating that a large radiator would be needed to reject the waste heat if the ambient temperature is close to a typical room temperature or close to average daily temperatures. The propylene glycol model was scaled at a larger thermal power of 20 kw, so these liquid heat exchanger temperatures changes were improved. However, propylene glycol is much more viscous than water, and this resulted in larger pressure losses than the corresponding water system. As expected, both water and propylene glycol were poor substitutes for the liquid metals due to the much lower thermal conductivities of water and propylene glycol. In the PHX, low heat transfer coefficients resulted in an increased number of tubes. Improving heat transport by significantly increasing the velocity was prohibited due to the higher pressure drop which would result. 37

50 As the Stanton number, time ratio number, and Biot number ratios indicate, neither model system produced acceptable similarity to the prototype. Although prototype behavior could be inferred from these models through the similarity ratios, a better model would require more compatible fluids and materials. 4.2 Similarity-Maximized Model Next, a model that maximizes the similarity between itself and the prototype is scaled. For this model, the same 95%molybdenum-5% rhenium alloy used in the prototype is used, and similar temperature ranges are employed. Therefore, ρ sr = Cp sr = k sr = 1 (83) assuming these properties do not change significantly since similar temperature ranges are used. Due to the need for energy similarity, Lithium and NaK are used in the model. Helium is also used, but at different pressures. Due to the smaller compressor ratio that will result at lower power, helium at 9 bar and 4 bar pressures are used. Operating temperatures for the model are close to the average temperatures through the respective prototype heat exchangers. Finally, both FLiBe and FKiNaK are used as substitute fuels [11]. FLiBe is a mixture of lithium fluoride (LiF) and Beryllium Fluoride (BeF2), while FLiNaK is a mixture of lithium fluoride (LiF), sodium fluoride(nf), and potassium fluoride (KF). Both have been 38

51 studied extensively for use as coolants in nuclear reactors. These are considered as model fuels because their properties more resemble the fuel salt than water or propylene glycol. A thermal power of 100 kw for the model was chosen, resulting in a model suitable for a larger laboratory or dedicated facility. Table 6 below displays the property ratios for the substitute fluids. Fuel Salt FLiBe FliNaK Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Helium at 15 bar (1040 K) Helium at 9 bar (840 K) Helium at 9 bar (840 K) Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Helium at 4.2 bar (729 K) Helium at 4 bar (840 K) Helium at 4 bar (840 K) Density ratio Specific heat ratio Viscosity ratio Thermal conductivity ratio Prandtl number ratio Table 6: Property Ratios for Maximized Model To maximize similarity, the Biot number ratio is set equal to one. And with Equation 83 above, for the same diameter and length ratios Bi R = h R δ R = 1 δ R = 1 (84) h R Substituting this result into the time ratio number with the same solid materials and setting it equal to one yields 39

52 T R = l R = h R 2 l R δ 2 R u R u R = 1 h R = ( u R l R ) 0.5 (85) Finally, substituting this result into the Stanton number ratio and setting it equal to one yields St R = h R l R ρ R Cp R u R d R = l R 0.5 ρ R Cp R u R 0.5 d R = 1 u R = l R (ρ R Cp R d R ) 2 (86) And, with the same fluids, this becomes u R = l R d R 2 (87) This result also implies that for the same fluids, h R = 1 d R 2 (88) These relationships were used to create scaled models. As with the small model, the prototype delivers 71.2% of its thermal energy to the turbine, with 25.6% of the total thermal power delivered to the turbo-alternator. Complete axial and geometric similarities were also preserved. The following tables list the results of the scaling for the FLiBe and FLiNaK systems. 40

53 Prototype FLiBe Model FLiNaK Model Diameter ratio Length ratio Tube velocity ratio Shell velocity ratio Power ratio 1 Power (W) Tube inner diameter (m) Tube outer diameter (m) PHX length (m) Tube number Fuel mass flow rate (kg/s) Tube velocity (m/s) Tube Reynolds number Shell mass flow rate (kg/s) Shell velocity (m/s) Shell Reynolds number Tube heat transfer coefficient (W/m 2 -K) Shell heat transfer coefficient (W/m 2 -K) Fuel temperature change (K) Fuel inlet temperature (K) Fuel outlet temperature (K) Coolant temperature change (K) Coolant inlet temperature (K) Coolant outlet temperature (K) Logarithmic mean temperature difference (K) Tube pressure drop (Pa) Tube Stanton number ratio Tube time number ratio Tube Biot number ratio Shell Stanton number ratio Shell time number ratio Shell Biot number ratio Table 7: PHX Scaling Values for the Maximized Model 41

54 Prototype FLiBe Model FLiNaK Model Diameter ratio Length ratio Liquid side velocity ratio Gas side velocity ratio t (m) s (m) l (m) Gas channel height (m) Liquid channel height (m) Gas channel hydraulic diameter (m) Liquid channel hydraulic diameter (m) n r m Gas cell mass flow rate (kg/s) Gas channel velocity (m/s) Gas channel Reynolds number Gas channel heat transfer coefficient (W/m 2 -K) Liquid cell mass flow rate (kg/s) Liquid channel velocity (m/s) Liquid channel Reynolds number Liquid channel heat transfer coefficient (W/m 2 -K) Gas side temperature change (K) Gas side inlet temperature (K) Gas side outlet temperature (K) Liquid side temperature change (K) Liquid side inlet temperature (K) Liquid side outlet temperature (K) Logarithmic mean temperature difference (K) Gas side Pressure drop (Pa) Liquid side pressure drop (Pa) Liquid Stanton number ratio Liquid time number ratio Liquid Biot number ratio Gas Stanton number ratio Gas time number ratio Gas Biot number ratio Table 8: SHX Scaling Values for the Maximized Model 42

55 Prototype FLiBe Model FLiNaK Model Diameter ratio Length ratio Liquid side velocity ratio Gas side velocity ratio t (m) s (m) L (m) Gas channel height (m) Liquid channel height (m) Gas channel hydraulic diameter (m) Liquid channel hydraulic diameter (m) n r m Gas cell mass flow rate (kg/s) Gas channel velocity (m/s) Gas channel Reynolds number Gas channel heat transfer coefficient (W/m 2 -K) Liquid cell mass flow rate (kg/s) Liquid channel velocity (m/s) Liquid channel Reynolds number Liquid channel heat transfer coefficient (W/m 2 -K) Gas side temperature change (K) Gas side inlet temperature (K) Gas side outlet temperature (K) Liquid side temperature change (K) Liquid side inlet temperature (K) Liquid side outlet temperature (K) Logarithmic mean temperature difference (K) Gas side Pressure drop (Pa) Liquid side pressure drop (Pa) Rejection power (W) Compression ratio Gas Stanton number ratio Gas time number ratio Gas Biot number ratio Liquid Stanton number ratio Liquid time number ratio Liquid Biot number ratio Table 9: THX Scaling Values for the Maximized Values 43

56 The data from the previous tables show that a scaled model can produce thermal similarity of the prototype if similar fluids and materials are used, except for a substitute fuel salt. The one half diameter scale was chosen so that manufacturing of the heat exchangers would not be extremely difficult and costly. Both the FLiBe and FLiNaK systems produce nearly identical results when identical diameter and length ratios are chosen, so the choice between the FLiBe and FLiNaK can be based upon cost and availability. The pressure losses in all three model heat exchangers are consistent with those in the prototype. Another characteristic of the similarity-maximized model which is both an advantage and disadvantage is that it operates at high temperatures with fluids which are solid at room temperature. This fact makes the operation of the model more realistic since the startup and shutdown of the reactor system must be taken into consideration. Corrosion and safety concerns must also be incorporated into the operation. But, this same trait makes this model more difficult and costly to operate. 44

57 Chapter 5: Exact Solution to the Integral Momentum Equation Equation 2, the integral momentum equation, can be solved exactly. Separating variables yields ρ a 0 i l a i du r i Δp pump ρu r (fl d +K) i (a 0 i ) a i = dt (89) The left side of this equation is of the form, C 1 C 2 C 3 x 2 (90) Integrating yields = C 1 tanh 1[ C3 C2 x] C 2 C 3 x 2 = C 1dx C 2 C 3 + C (91) So, 45

58 u r 0 ρ a 0 i l a i dx i 2 ) a i Δp pump ρx2 2 (fl d +K) i (a 0 i t = dt 0 (92) becomes ρ a 0 i l a i tanh 1 [(Δp pump ) 1 2 ( ρ i 2 (fl 2 i d +K) i (a 0) a i ) (Δp pump ) 1 2 ( ρ i (fl d +K) i (a 0) a ) i 1 2 u r ] = t (93) Solving for u r yields u r = [ Δp pump 2 ρ 2 (fl d +K) i (a 0 i ) a i ] 1 2 tanh [ (Δp ρ pump 2 i (fl d +K) i (a 0) a ) i ρ a 0 t] (94) i l a i i Due to the domain of the inverse hyperbolic function, the following condition is required Δp pump ρ 2 i (fl + K) d i (a 0 u 2 r (95) a i ) 2 This condition is satisfied by the steady state solution. 46

59 Chapter 6: Radiator and Electrical Heating Element Sizing 6.1 Radiator Sizing The radiator discharging the reactor waste heat consists of a series of heat pipes which must be sized based upon the THX coolant mass flow rate, inlet and outlet temperatures, and the temperature of the radiating environment. Figure 5 displays the heat transfer geometry. 47

60 T s da R m THX dq1 da THX T THX Figure 5: Radiation Heat Transfer Geometry The differential heat dq 1 transferred from the THX coolant to a differential area da THX is dq 1 = m THXc p,thx dt THX (96) Assuming that the heat pipe is the same temperature as the fluid in which it is in contact, the differential heat dq 2 transferred from a differential radiator area da R to the environment at temperature T s is 48

61 4 dq 2 = εσf v da R (T THX T 4 s ) (97) At steady state, these must be equal, so 4 m THXc p,thx dt THX = εσf v da R (T THX T 4 s ) (98) Separating variable and integrating A R εσf v ds 0 m THXc p,thx T THX,out dt T 4 T4 s = (99) T THX,in yields A R = m THXc p,thx εσf v 2[tan 1 ( T THX,in Ts ) tan 1 ( T THX,out )]+ln[ (T THX,out T s)(t THX,in +Ts) Ts ] (T THX,out +Ts)(T THX,in Ts) 4T s 3 (100) Here, the leading negative sign is need since T THX,out < T THX,in. For the same parameters, this expression yields the same total radiator surface area to within 1% as a similar relationship developed by Juhasz [13]. Table 10 provides the needed radiator surface areas for the small-scale models and the high-similarity models. The FLiBe and FLiNaK models yield the same results. Here, the rejection temperature is 50 F (283.2 K), 49

62 the view factor parameter F v is assumed to equal one, and perfect black body radiation is assumed, implying that the emissivity equals one. Rejection Power (W) Radiator Surface Area (m 2 ) Prototype Water Model Propylene Glycol Model FLiBe/FLiNaK Models Table 10: Required Radiator Surface Areas for Prototype and Models Since the temperature of the environmental heat sink varies throughout the year, for a constant power, the required radiator surface area would also vary. Figure 6 provides displays the variation the radiator surface area as the radiation sink temperature varies from 32 F (273.1 K) to 70 F (294.3 K) for the water model. Figure 6: Radiator Surface Area as a Function of Heat Sink Temperature 50

63 6.2 Heating Element Sizing of the Water Model The power needed by the water model can be supplied by electric pencil bar cartridge heaters. To size the heating elements, the steady state model core is considered. At steady state, the energy balance of a core cross-section of with dx heated by a heat flux q is m c p dt = q da heat (101) where A heat is the surface from which the heat flux is generated, and da heat = p h dx (102) where p h is the combined heated perimeter of one or more pencil bar elements. Since these elements are cylindrical, then p h = πnd n (103) where this the total heated perimeter of n cylindrical elements each of diameter d n. Inserting this relationship into the above equation, rearranging, and integrating yields T out T in dt = πnd nq m c p L 0 dx (104) assuming that the heating elements run the length of the core. So, T out T in = πnd nq m c p L (105) 51

64 Choosing a surface loading of 2 watts per cubic centimeter, which is well within the capabilities of such heating elements [14], implies that q = W/m 2. Solving for the product nd i using the dimensions and parameters of the water model gives nd n = m c p (T out T in ) πq L = m (106) Therefore, four one-centimeter diameter pencil bar heating elements can be used to generate the required power of 10 kw. Finally, the power dissipated from the resistance heating is related to the required voltage and current by P dissipation = V I (107) If a common voltage of 240 V AC is desired, then the needed current is approximately 42 A. Using Equations 49 and 50, the core pressure drop with the inclusion of the heaters in the core is 2.02 Pa, which greater than the calculated pressure drop of 0.91 Pa in an open core. Both of these values are negligible compared to the primary loop pressure drop of over 6000 Pa. 52