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ANALYSIS AND OPTIMIZATION OF A JETPUMPED COMBINED POWER/REFRIGERATION CYCLE By SHERIFF M. KANDIL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by SherifM. Kandil I would like to dedicate this work to my family Mohamed Kandil, Nayera Elsedfy, and my sister Nihal M Kandil. I would like them to know that their support has been invaluable. ACKNOWLEDGMENTS The work presented in this dissertation was completed with the encouragement and support of many wonderful people. Working with Dr. Bill Lear has been a tremendous experience. He expects his students to be selfstarters, who work independently on their projects. I appreciate his patience and mentorship in areas within and beyond the realm of research and graduate school. Dr. Sherif Ahmed Sherif was a terrific source of discussion, advice, encouragement, support and hard to find journal proceedings. Dr. Sherif s support made my years here a lot easier and made me feel home. Dr. David Hahn, Dr. Skip Ingley, and Dr. Bruce Carroll agreed to be on my committee and took the time to read and critique my work, for which I am grateful. Dr. Bruce Carroll has to be thanked for his advice on jetpumps. Dr. Leon Lasdon from the University of Texas sent me the FORTRAN version of the GRG code and answered my questions very promptly. Mrs. Becky Hoover and Pam Simon have to be thanked for their help with all my administrative problems during my time here and their constant reminders to finish up. I would like to particularly thank my family for putting up with me being so far away from home, and for their love, support and eternal optimism. This section is not complete without mentioning friends, old and new, too many to name individually, who have been great pals and confidants over the years. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ............. ......... .. .................. ............... ............ vii LIST OF FIGURES ............. ............................ .................. viii A B S T R A C T .......................................... ..................................................x iii CHAPTER 1 INTRODUCTION ............... ................. ........... ................. ... .... 1 2 L IT E R A TU R E R E V IE W .................................................................. .....................8 R e late d W o rk .................................................................................. 8 Jetpum ps and Fabri Choking......................................................... .............. 9 S o lar C o llecto rs ................................................................17 Solar Irradiance ........................ ...... ........ ................... ......... 18 C on centration R atio ....................................................................................... 19 Selective Surfaces ............................................... .. ...... ................. 21 Combined Power/Refrigeration Cycles ........................................... ............... 23 Efficiency Definitions for the Combined Cycle..................................................25 Conventional Efficiency Definitions ............... ........................................... 26 First law efficiency .......................................... ... ...... ... .. ........ .. 26 Exergy efficiency .................................... ................. ..... ..... 26 Second law efficiency ............................................................................ 27 The Choice of Efficiency Definition .................. ......................................... 28 Efficiency Expressions for the Combined Cycle....................................... 29 First law efficiency .......................................... ... ...... ... .. ........ .. 29 Exergy efficiency .................................... ................. ..... ..... 30 Second law efficiency ............................................................................ 31 Lorenz cycle ................................... ......... ................. 31 C ascaded Cycle A nalogy................................................................ .......... ..... 33 Use of the Different Efficiency Definitions .................................................36 3 M A THEM A TICAL M ODEL.......................................................... ............... 38 Jetp u m p A n aly sis ............................................................................ ................ .. 3 8 P rim ary N ozzle ............. .............................................................. ........ .. .. .. 39 v Flow Choking A analysis ......................................................... .............. 40 Secondary Flow ................................................... .... .. ........ .... 45 M ix ing C h am b er........... ............................................ .............. .. .... .... .. ..4 5 D iffuser.................................................. 46 SITM A P Cycle A analysis .................................................. .............................. 47 O v erall A n aly sis .............................................................4 8 Solar Collector M odel ...................... ..... ........ ................ 50 Tw ophase region analysis ................................... .............. ............... 51 Superheated region analysis ................................ ...................................... 51 Solar collector efficiency ........................................ ......... ............... 55 Radiator M odel ................................... ................ ..................... 55 Sy stem M ass R atio ................. .................................. ...... ........ .. ............ 55 4 C Y C LE O PTIM IZA TIO N ........................................................................... ... ..... 60 Optim ization M ethod Background ........................................ ......... ............... 60 Search T erm nation .......... .............................................................. ........... ...... 63 Sensitivity A analysis .......................................... .. .. .... ........ .... .. ... 64 A application N otes ........................................................... .. ........ .... 64 V a riab le L im its ..................................................................................................... 6 6 C on straint E qu ation s........... ............................................................ ...... .... ... ..67 5 C O D E V A L ID A T IO N ..................................................................... .....................69 6 RESULTS AND DISCUSSION: COOLING AS THE ONLY OUTPUT.................74 Jetpum p G eom etry E effects ............................................................. .....................75 Stagnation Pressure R atio Effect ........................................ .......................... 78 Secondary Flow Superheat Effect ........................................ ......................... 86 Turbine Pressure E effect ........................................... .................. ............... 89 M ixed R egim e A naly sis ..................................................................... ..................9 1 Evaporator Tem perature Effect ............................................................................ 95 Prim ary Flow Superheat H eat Effect................................... .................................... 98 Environm ental Sink Tem perature Effect............................................................... 101 System O ptim ization .......................................... .................... ......... 103 7 RESULTS AND DISCUSSION: COOLING AND WORK OUTPUTS................. 11 Jetpum p Turbom achinery A nalogy............................................. .....................122 System Optimization for MSMR .. .. ......................... ...................130 8 CONCLUSIONS ....................................... .........................139 9 RECOMMENDATIONS ........... .. ......... ........................ 142 LIST OF REFERENCES ......... ...................................... ........ .. ............... 144 BIO GR A PH ICA L SK ETCH .................................... ........... ......................................148 LIST OF TABLES Table page 21 Effect of the distance from the sun on solar irradiance ......................................19 22 Properties of som e selective surfaces......................................... ......... ............... 23 23 Rankine cycle and vapor compression refrigeration cycle efficiency definitions....27 41 Optim ization variables and their lim its ........................................ ............... 67 42 Constraints used in the optim ization ............................................. ............... 68 51 Representative constantarea ejector configuration ..............................................70 61 Input parameters to the JETSIT cycle simulation code................ .................75 62 SITMAP cycle parameters input to the JETSIT simulation code..........................79 63 SITMAP cycle configuration to study the effect of secondary flow superheat .......86 64 SITMAP cycle configuration to study the effect of the evaporator temperature, T evap ......................................................................................... ....9 5 65 SITMAP cycle configuration to study the primary flow superheat .......................98 66 Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K) ..........................109 71 Base case cycle parameters to study the MSMR behavior................................... 128 72 Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K), Pti = 14.2 MPa. ..138 73 Optimum Cycle parameters for Pso = 140 kPa (Tevap = 80.2 K), Pti = 14.2 MPa. ..138 LIST OF FIGURES Figure pge 11 Schematic of the Solar Integrated Thermal Management and Power (SITMAP) cy c le ....................................................... ..................... . 1 12 Schematic of the Solar Integrated Thermal Management and Power (SITMAP) cycle w ith regeneration ........................ ...................... ... .. ....... ................ .3 21 A schematic of the j etpump geometry showing the different state points .............10 22 Threedimensional ejector operating surface depicting the different flow regimes [2 ] ........................................................ ............................ . 1 3 23 Relationship between concentration ratio and temperature of the receiver [11]......20 24 A cyclic heat engine working between a hot and cold reservoir............................28 25 The TS diagram for a Lorenz cycle ............................................. ............... 32 26 Thermodynamic representation of (a) combined power/cooling cycle and (b) cascaded cycle ..................................................................... ..........34 31 Schematic for the jetpump with constant area mixing .........................................39 32 Schematic for the jetpump with constant area mixing, showing the Fabri choked state s2 ...............................................................................42 33 Jetpump schematic showing the control volume for the mixing chamber an aly sis. ............................................................................. 4 5 34 A schematic of the SITMAP cycle showing the notation for the different state p o in ts. ............................................................................ 4 8 35 Typical solar collector temperature profile. .................................. .................54 36 Overall system schematic for SMR analysis................................. .................56 51 Breakoff mass flow characteristics from the JETSIT simulation code ..................71 52 Breakoff mass flow characteristics from Addy and Dutton [2]............................71 53 Breakoff compression and mass flow characteristics. .........................................72 54 Breakoff compression and mass flow characteristics from Addy and Dutton [2], for Api/Am3=0.25. ........................................ ............ ................ 72 55 Breakoff compression and mass flow characteristics from Addy and Dutton [2], for Api/Am3=0.333 ................................................... ..............73 61 Effect of jetpump geometry and stagnation pressure ratio on the breakoff entrainm ent ratio. ................................................... ................. 76 62 Effect of jetpump geometry and stagnation pressure ratio on the compression ra tio ........................................................................... 7 7 63 Effect of jetpump geometry and stagnation pressure ratio on the System Mass R atio (SM R ). .........................................................................77 64 Ts diagram for the refrigeration part of the SITMAP cycle. .............................78 65 Effect of jetpump geometry and stagnation pressure ratio on the amount of specific heat rejected. .......... ................................... .......... .. ........ .... 81 66 Effect of jetpump geometry and stagnation pressure ratio on the radiator tem perature ..................... ... ......... ..... ....... ..... .................. ......... 82 67 Effect of jetpump geometry and stagnation pressure ratio on the amount of specific heat input ..... ........ ................................... .................... 82 68 Effect of jetpump geometry and stagnation pressure ratio on the specific co olin g cap city ............................. .................................................. ............... 83 69 Effect of jetpump geometry and stagnation pressure ratio on the cooling specific rejected heat. ........ ..... ............................... ....... ...... ............ 83 610 Effect of jetpump geometry and stagnation pressure ratio on the cooling specific heat input ............ ..... .............................. ........ .......... ............ 84 611 Effect of jetpump geometry and stagnation pressure ratio on the overall cycle efficiency ......... ...... ............ ..................................... ........................... 84 612 Effect of jetpump geometry and stagnation pressure ratio on the ratio of the overall cycle efficiency to the overall Carnot efficiency. ......................................85 613 Effect of secondary superheat on the overall system mass ratio (SMR) ..................87 614 Effect of secondary superheat on the breakoff compression ratio ..........................87 615 Effect of secondary superheat on Qrad/Qcool. ............... ..................................88 616 Effect of secondary superheat on Qs/Q oo.................................... ..................... 88 617 Effect of secondary superheat on the breakoff mass flow characteristics ............89 618 Effect of the turbine inlet pressure on the amount of net work rate and specific heat input to the SITM AP system ........................................ ....................... 90 619 Effect of the turbine inlet pressure on the amount of the SMR and overall efficiency of the SITM AP system .................. ...................... ....................... 90 620 SMR and Compression ratio behavior in the mixed regime. ..................................92 621 Effect of the entrainment ratio on the mixed chamber exit conditions in the m ixed regime e ............. .... ..... .. .................................. ... ....... .. 93 622 Effect of the entrainment ratio on secondary nozzle exit conditions in the mixed re g im e ......... .. ..................................... ................................................... 9 3 623 Jetpump compression behavior in the mixed regime.............. ................. 94 624 Effect of entrainment ratio on specific heat transfer ratios in the mixed regime.....94 625 Effect of the evaporator temperature on the breakoff entrainment ratio and the compression ratio, for Ppo = 3.3 M Pa. ............................. .... ....................... 96 626 Effect of the evaporator temperature on T, and SMR, for Ppo = 3.3 MPa ...............97 627 Effect of the evaporator temperature on the cooling specific rejected specific heat, Qrad/Qcool, and the cooling specific heat input, Qsc/Qcool..............................97 628 Effect of the evaporator temperature on the effective radiator temperature, for P p o = 3 .3 M P a ..................................................................... 9 8 629 Effect of primary flow superheat on the SMR. ................................... ..................99 630 Effect of primary flow superheat on the Qrad/Qcool............... .... ..................... 100 631 Effect of primary flow superheat on the Qsc/Qoo............................. ..............100 632 Effect of primary flow superheat on the compression ratio.............. ................101 633 Sink temperature effect on SM R. .............................................. ........ ....... 102 634 Compression ratio effect on the SMR < 1 regime.................................. .....103 635 Effect of jetpump geometry on the breakoff sink temperature, for PpoPso=25, Pso=128 kPa, Tevap=79.4 K, 10 degrees primary superheat ..............................104 636 Compression ratio and entrainment ratio variation with jetpump geometry, for Ppo/Pso=25.. ................................................... ............ 105 637 Effect of stagnation pressure ratio on the breakoff sink temperature (77.1).........106 638 Breakoff sink temperature behavior in the mixed regime (77.1).........................107 639 Effect of jetpump geometry on the SMR for Ppo/Pso=40, Tpo=150 K, Pso=128 kP a, T evap= 79.4 K T = 78.4.................................................................. 108 640 Effect of stagnation pressure ratio on the SMR. .........................................108 71 Schematic of a cooling and power combined cycle ........................ ............114 72 A schematic of the turbomachinery analog of the jetpump..............................122 73 Ts diagram illustrating the thermodynamic states in the jetpump turbo m machinery analog ........................ .............................. .. ........ .... ....... 123 74 Effect of compression efficiency on jetpump characteristics.............................125 75 Effect of compression efficiency on MSMSR. ....................................................126 76 Jetpump efficiency effect on the compression ratio and MSMR for given jet pum p inlet conditions. ............................................... ................. ............. 126 77 M SM R and SM R are equal for W ext = 0. .................................... .................127 78 High pressure effect on the cooling specific heat input and external work output for a given jetpump inlet conditions. .............. .............................................. 129 79 High pressure effect on the MSMR and efficiency for a given jetpump inlet con edition s. ....................................................................... 12 9 710 Primary nozzle geometry effect on MSMR at Ane/Ase = 0.4, Ppo/Pso = 26, Pso = 12 8 kP a ...... ............... ...... ................................................. ............................ 13 1 711 Jetpump geometry effect on MSMR at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kP a ......... ................... ............... .......................... ........ ....... 13 1 712 Stagnation pressure ratio effect on MSMR at a fixed jetpump geometry(Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa. .................................... 132 713 Primary nozzle geometry effect on the compression ratio and the entrainment ratio at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa. .............................................133 714 Primary nozzle geometry effect on the specific heat rejected per unit specific cooling load at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa.......................................133 715 Primary nozzle geometry effect on the specific heat input per unit specific cooling load at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa..................................... 134 716 Jetpump geometry effect on the compression ratio and the entrainment ratio at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa...................................................... 134 717 Jetpump geometry effect on the specific heat rejected per unit specific cooling load at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa........................ ...............135 718 Jetpump geometry effect on the specific heat input per unit specific cooling load at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa.........................................135 719 Stagnation pressure ratio effect on the compression ratio and the entrainment ratio at a fixed jetpump geometry (Ant/Ane=0.2, Ane/Ase=0.4) ..............................136 720 Stagnation pressure ratio effect on the specific heat rejected per unit specific cooling load at a fixed jetpump geometry(Ant/Ane=0.2, Ane/Ase=0.4). ...............1.36 721 Stagnation pressure ratio effect on the specific heat input per unit specific cooling load at a fixed jetpump geometry(Ant/Ane=0.2, Ane/Ase=0.4). ...............1.37 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ANALYSIS AND OPTIMIZATION OF A JETPUMPED COMBINED POWER/REFRIGERATION CYCLE By SherifM. Kandil May 2006 Chair: William Lear Cochair: S. A. Sherif Major Department: Mechanical and Aerospace Engineering The objectives of this study were to analyze and optimize a jetpumped combined refrigeration/power system, and assess its feasibility, as a thermalmanagement system, for various space missions. A mission is herein defined by the cooling load temperature, environmental sink temperature, and solar irradiance which is a function of the distance and orientation relative to the sun. The cycle is referred to as the Solar Integrated Thermal Management and Power (SITMAP) cycle. The SITMAP cycle is essentially an integrated vapor compression cycle and a Rankine cycle with the compression device being a jetpump instead of the conventional compressor. This study presents a detailed component analysis of the jetpump, allowing for twophase subsonic or supersonic flow, as well as an overall cycle analysis. The jet pump analysis is a comprehensive onedimensional flow model where conservation laws are applied and the various Fabri choking regimes are taken into account. The objective of the overall cycle analysis is to calculate the various thermodynamic state points within the cycle using appropriate conservation laws. Optimization techniques were developed and applied to the overall cycle, with the overall system mass as the objective function to be minimized. The optimization technique utilizes a generalized reduced gradient algorithm. The overall system mass is evaluated for two cases using a mass based figure of merit called the Modified System Mass Ratio (MSMR). The first case is when the only output is cooling and the second is when the system is producing both cooling and work. The MSMR compares the mass of the system to the mass of an ideal system with the same useful output (either cooling only or both cooling and work). It was found that the active SITMAP system would only have an advantage over its passive counterpart when there is a small difference between the evaporator and sink temperatures. Typically, the minimum temperature difference was found to be about 5 degrees for the missions considered. Three optimization variables proved to have the greatest effect on the overall system mass, namely, the jetpump primary nozzle area ratio, Ant/Ane, the primary to secondary area ratio, Ane/Ase, and the primary to secondary stagnation pressure ratio, Ppo/Pso. SMR and MSMR as low as 0.27 was realized for the mission parameters investigated. This means that for the given mission parameters the overall SITMAP system mass can be as low as 27% of the mass of an ideal system, which presents significant reduction in the operating cost per payload kilogram. It was also found that the work output did not have a significant effect on the system performance from a mass point of view, because the increase in the system mass due to the additional work output is offset by the increase in the mass of the Carnot power system that produces the same amount of work. CHAPTER 1 INTRODUCTION The increased interest in space exploration and the importance of a human presence in space motivate space power and thermal management improvements. One of the most important aspects of the desired enhancements is to have lightweight space power generation and thermal management capabilities. Onboard power generation adds weight to the space platform not only due to its inherent weight, but also due to the increased weight of the required thermal management systems. This study presents a novel thermal management and power system as an effort to decrease the mass of thermal management systems onboard spacecraft, thereby lowering costs. The system is referred to as the Solar Integrated Thermal Management and Power system (SITMAP) [33]. Figure 11 shows the standard SITMAP system. Radiator Soling Jet Pump Expansion CLoad Valve 4 Evaporator Pump ) Waste Heat Solaor Collector Power Waste CQsolar Figure 11. Schematic of the Solar Integrated Thermal Management and Power (SITMAP) cycle Figure 12 illustrates the operation of the SITMAP system considered in this study. The cycle is essentially a combined vapor compression cycle and Rankine cycle with the compression device being a jetpump instead of the conventional compressor. The jet pump has several advantages for space applications, as it involves no moving parts, which decreases the weight and vibration level while increasing the reliability. The power part of the SITMAP cycle is a Rankine cycle, which drives the system. The jet pump acts as the joining device between the thermal and power parts of the system, by mixing the high pressure flow from the power cycle with the low pressure flow from the refrigeration part of the system providing a pressure increase in the refrigeration cycle. High pressure superheated vapor is generated in the solar collector, which then passes through the turbine extracting work from the flow. The mechanical power produced by the turbine can be used to drive the mechanical pump as well as other onboard applications. This allows the SITMAP cycle to be solely driven by solar thermal input. The flow then goes through the recuperator where it exchanges heat with the cold flow going into the solar collector, thereby reducing the collector size and weight. After the recuperator the flow goes into the jetpump providing the high pressure primary (or motive) stream. The primary stream draws low pressure secondary flow from the evaporator. The two streams mix in the jetpump where the secondary flow is compressed by mixing with the primary flow and the combined flow is ejected to the radiator where heat is rejected from the fluid to the surroundings, resulting in a condensate at the exit of the radiator. Flow is then divided into two streams; one stream enters the evaporator after a pressure reduction in the expansion device, and the other stream is pressurized through the pump and then goes into the recuperator where it is heated up by exchanging heat with the hot stream coming out of the turbine. The flow then goes into the solar collector where it is vaporized again and the cycle repeats itself. Jet Pump Turbine Heat from: HighPressure Solar Waste Heat, Sand/or Liquid apo Electronics Heat Rejection Recuperator Liquid/Vapo Liquid Rcprt Expansion Valve Liquid Liquid Liquid Lq Pump/Capillary Pump Figure 12. Schematic of the Solar Integrated Thermal Management and Power (SITMAP) cycle with regeneration The jetpump, also referred to as an "ejector" in the literature, is the simplest flow induction device [24]. It exchanges energy and momentum by direct contact between a highpressure, highenergy primary fluid and a relatively lowenergy lowpressure secondary fluid to produce a discharge of intermediate pressure and energy level. The highpressure stream goes through a convergingdiverging nozzle where it is accelerated to supersonic speed. By viscous interaction the high velocity stream entrains secondary flow. More secondary flow is entrained until the secondary flow is choked whether at the inlet to the mixing compartment or at an aerodynamic throat inside the mixing compartment. Conditions for both choking mechanisms are described in detail in later sections of this study. The two streams mix in a constant area mixing chamber. The transfer of momentum between the two streams gives rise to an increase in the stagnation pressure of the secondary fluid and enables the jetpump to function as a compressor. In steady ejectors, momentum can be imparted from the primary fluid to the secondary fluid by the shear stresses at the tangential interface between the primary and secondary streams as a result of turbulence and viscosity [24]. Ejector refrigeration has continued to draw considerable attention due to its potential for low cost, its utilization of lowgrade energy for refrigeration, its simplicity, its versatility in the type of refrigerant, and its low maintenance due to the absence of moving parts. Another important advantage of ejector refrigeration is that high specific volume vaporized refrigerants can easily be compressed with an ejector of reasonable size and cost. This allows a wide variety of environmentally friendly refrigerants to be used. As a result of these characteristics there are many applications where ejector refrigeration is used, such as cooling of buildings, automotive airconditioning, solar powered ejector airconditioning, and industrial process cooling. However, despite the abovementioned strong points, conventional steadyflow ejectors suffer low COPs. Therefore, more energized primary flow must be provided in order to attain a given cooling requirement. The thermal energy contained in this driving fluid must also be rejected in the condenser (or radiator). Hence, the use of ejector refrigeration systems has been limited to applications where low cost energy from steam, solar energy, or waste heat sources is available, and where large condensers can be accommodated. However, if maj or improvements in the jetpump (ejector) efficiency can be attained, significant improvement in the COPs of such systems will be realized and jetpumped refrigeration systems will present strong competition to conventional vapor compression systems. Alternatives to the SITMAP system for space applications can be either other active systems such as cryocoolers or passive systems such as a radiator. Conventional cryocoolers are generally bulky, heavy, and induce high vibration levels. Passive radiators have to operate at a temperature lower than the cooling load temperature which causes the radiator to be larger and thus heavier. The proposed system eliminates some moving parts, which decreases the vibration level and enhances reliability. A major contribution of this study is the detailed analysis of the twophase jet pump. All previous work in the literature is limited to jetpumps with a perfect gas as the working fluid. Flow choking phenomena are also accounted for, as discussed in Fabri and Siestrunk [18], Dutton and Carroll [12], and Addy et al. [2]. The SITMAP cycle performance is evaluated in this study for two cases. The first case is when the only output is cooling and the second is when the system is producing both cooling and work. In the first case the system performance is evaluated using a mass based figure of merit, called the System Mass Ratio (SMR). The SMR, first presented by Freudenberg et al. [20], is the ratio of the overall system mass to the mass of an ideal passive radiator with the same cooling capacity. In the second case the system performance is evaluated using a more general form of the aforementioned figure of merit, referred to as the Modified System Mass Ratio (MSMR). The MSMR compares the mass of the overall system to that of a passive radiator with the same cooling capacity plus the mass of a Carnot Rankine system with the same work output. The MSMR and SMR are equal when the system is only producing cooling. The cycle analysis and optimization techniques developed in this study are general and applicable for any working fluid. However, in this study, cryogenic nitrogen was used as an example working fluid since it is readily present onboard many spacecraft for other purposes. Another advantage of cryogenic nitrogen is that it can be used as a working fluid in a conventional evaporator, or the nitrogen tank can be used as the evaporator, in this case the nitrogen is used to cool itself which eliminates the need for the evaporator heat exchanger; adding further mass advantage to the system. The final stage of this study is to optimize the recuperated SITMAP cycle, with the SMR (or MSMR) as an objective function to find out the optimum cycle configuration for different missions. To achieve this, a computer code was developed for the thermodynamic simulation and optimization of the cycle. The code is called JetSit (short for Jetpump and SITMAP). The code includes the jetpump twophase onedimensional flow model, as well as the SITMAP cycle, and SMR analyses. A thermodynamic properties subroutine was incorporated in the code to dynamically calculate the properties of the working fluid instead of using a data file which can limit the range of simulation parameters. The thermodynamic properties software used is called REFPROP and is developed by the National Institute for Standards and Technology (NIST). A commercially available optimization program was incorporated in the JetSit cycle simulation code. The optimization routine is written by Dr. Leon Lasdon of the University of Texas in Austin and it utilizes a Generalized Reduced Gradient algorithm, and is called LSGRG2. The optimization of the working of the cycle is a non linear programming (NLP) problem. A NLP problem is one in which either the objective function or at least one of the constraints is a nonlinear function. The cycle optimization method chosen for the analysis of this cycle is a search method. Search methods are used to refer to a general class of optimization methods that search within a domain to arrive at the optimum solution. 7 When implementing steepest ascent type of search methods for constrained optimization problems, the constraints pose some limits on the search algorithm. If a constraint function is at its bound, the direction of search might have to be modified such that the bounds are not violated [28]. The Generalized Reduced Gradient (GRG) method was used to optimize the cycle. GRG is one of the most popular NLP methods in use today [39]. A detailed description of the GRG method is presented later in this study. CHAPTER 2 LITERATURE REVIEW Related Work The work presented in this study is a continuation of the work done by Nord et al. [33] and Freudenberg et al. [20]. Nord et al. [33] investigated the same combined power and thermal management cycle investigated in this study for onboard spacecraft applications. Nord et al. [33] used Refrigerant 134a as the working fluid in their analysis. The mechanical power produced by the turbine can be used to drive the mechanical pump as well as other onboard applications. This allows the SITMAP cycle to be solely driven by solar thermal input. They did not consider the choked regimes in their jetpump analysis, because their analysis only involved constantpressure mixing in the jetpump. The different Fabri choking regimes will be defined in detail later in this section. Freudenberg et al. [20], motivated by the novel SITMAP cycle developed by Nord et al. [33], developed an expression for a system mass ratio (SMR) as a mass based figure of merit for any thermally actuated heat pump with power and thermal management subsystems. SMR is a ratio between the overall mass of the SITMAP system to the mass of an ideal passive radiator, where there is no refrigeration subsystem, in which the ideal radiator operates at a temperature lower than the cooling load temperature. SMR depends on several dimensionless parameters including three temperature parameters as well as structural and efficiency parameters. Freudenberg et al. estimated the range of each parameter for a typical thermally actuated cooling system operating in space. They investigated the effect of varying each of the parameters within the estimated range, comparing their analysis to a base model based on the average value of each of the ranges. Many systems dealing with power and thermal management have been proposed for which this analysis can be used, including absorption cooling systems and solar powered vapor jet refrigeration systems. Jetpumps and Fabri Choking The Fabri choking phenomenon was first analyzed by Fabri and Siestrunk [18] in the study of supersonic air ejectors. They divided the operation of the supersonic ejector into three regimes, namely, the supersonic regime (SR), the saturated supersonic regime (SSR), and the mixed regime (MR). The supersonic regime refers to the operating conditions when the primary flow pressure at the inlet of the mixing section is larger than the secondary flow pressure (Pne > Pse) which causes the primary flow to expand into the secondary flow, as indicated by the dotted line in Figure 21. This causes the secondary flow to choke in an aerodynamic throat (Ms2 = 1) in the mixing chamber. The saturated supersonic regime is a limiting case of the supersonic flow regime, where Psi increases and the secondary flow chokes at the inlet to the mixing chamber (Mse = 1). In both of these flow regimes, once the flow is choked either at "se" or "s2," the entrainment ratio becomes independent of the backpressure downstream. The third regime is the regime encompassing flow conditions before choking occurs. In the mixed flow regime, the entrainment ratio is dependent on the upstream and downstream conditions. Fabri and Paulon [17] performed an experimental investigation to verify the various flow regimes. They generated various performance curves relating the entrainment ratio, the compression ratio, and the ratio of the primary flow stagnation pressure to the exit pressure (Ppi / Pde). Fabri and Paulon went on to discuss the optimum jet ejector design, concluding that it corresponds to the lowest secondary pressure for a fixed primary pressure and a given secondary mass flow rate; or to the highest secondary mass flow rate for a given secondary pressure and a given primary pressure. p i ,2 1 Figure 21. A schematic of the jetpump geometry showing the different state points. Addy et al. [2] studied supersonic ejectors and the regimes defined by Fabri and Siestrunk [18]. They wrote computer codes analyzing constantarea and constant pressure ejectors. Their flow model was onedimensional and assumed perfect gas behavior. They also conducted an experimental study to which they compared their analytical results. Addy et al. concluded that the constantarea ejector model predicts the operational characteristics of ejector systems more realistically than the constantpressure model. They introduced a threedimensional performance curve, which has the entrainment ratio, the ratio of the secondary stagnation pressure to the primary stagnation pressure (Psi/Ppi), and the compression ratio as the three axes, see Figure 22. Figure 22 depicts a threedimensional ejector solution surface. It should be noted that in Figure 22 the ejector geometry, and the primary to secondary stagnation temperature ratio are fixed. The surfaces show all the different flow regimes. Addy et al. also presented the details of the breakoff conditions for transition from one operating regime to another. The possible transitions are between: * The "saturated supersonic" and "supersonic" regimes, breakoff curve bd. * The "saturated supersonic" and "mixed" regimes, breakff curve bc. * The "supersonic" and "mixed" regimes, breakoff curve ab. In both the SR and SSR regimes the mass flow ratio entrainmentt ratio), W,/W, is independent of the backpressure ratio, Pm /Ps so that these two surfaces are perpendicular to the W,/W, Pm3/Po plane. This independence of backpressure is due to the previously mentioned secondary choking phenomenon. For a short distance downstream from the mixing duct inlet, the primary and secondary streams remain distinct. If the primary static pressure at the mixing duct inlet exceeds that of the secondary, P, > P,, the primary stream will expand forming an "aerodynamic nozzle" in the secondary stream which causes the secondary stream to accelerate. For a low enough backpressure the secondary stream will choke at this aerodynamic throat, so that its mass flow rate becomes independent of the backpressure. These are the conditions encountered in the SR regime. In the SSR regime, on the other hand, the secondary inlet static pressure exceeds that of the primary, PI, > P%, so that the secondary stream expands against the primary stream inside the mixing tube. Thus, the minimum area encountered by the secondary stream in this case occurs at the mixing tube inlet and for a low enough backpressure the secondary stream will choke there. The secondary mass flow rate in the SSR regime is, therefore, also independent of the backpressure. In the MR regime, however, the backpressure is high enough that the secondary flow remains subsonic throughout the mixing duct and its mass flow rate is therefore dependent (in fact, strongly dependent) on the backpressure. Consider a plane of constant primary to secondary stagnation pressure ratio, Po /Pos in Figure 22. As Pm3,/Po is increased from zero, W, W, remains constant until breakoff curve abc, which separates the backpressureindependent from the backpressuredependent regimes, is reached. From here, a slight increase in the Pm3 /P causes a significant drop in FWF Hence, the points along breakoff curve abc are of particular importance since they represent the highest values of Pm,3Pos for which W,/W, remains fixed. For this reason, it is advantageous to design ejectors to operate in a backpressure independent regime at or near this breakoff curve. The criterion for determining each transition was based on the pressure ratio Pse/Pne, and the Mach number at the minimum throat area, either at "se," or "s2." If the Mach number at the minimum throat area was unity, the ejector operates in the either the "saturated supersonic" or the "supersonic" regime, while if the Mach number was less than unity, the ejector operates in the mixed regime. The breakoff conditions for each of the transitions mentioned above are 1. Mse = 1, and Pse/Pne = 1; 2. Mse = 1, and Pse/Pne > 1; and 3. Mse < 1, and Pse/Pne < 1, and Ms2 1. Breakoff Curves Figure 22. Threedimensional ejector operating surface depicting the different flow regimes [2]. Dutton and Carroll [12] discussed another important limitation on the maximum entrainment ratio due to exit choking. This is the case when the flow chokes at the mixing chamber exit, causing the entrainment ratio to be independent of the backpressure. In their analysis they could not find a mixed flow solution because the entrainment ratios considered were higher than the value that would cause the mixing chamber exit flow to choke. They lowered the value of 4 till they obtained a solution and that was at Mme = 1. This led them to the conclusion that mixed flow choking at the exit is a different limitation for these cases, not the usual Fabri inlet choking phenomenon. Dutton and Carroll [13] developed a onedimensional constant area flow model for optimizing a large class of supersonic ejectors utilizing perfect gases as a working fluid. Given the primary and secondary gases and their temperatures, the scheme determines the values of the design parameters Mne, and Ane/Ame, which optimize one of the performance variables, entrainment ratio,4, compression ratio, Pme/Psi, or Ppi/Psi given the value of the other two. AlAnsary and Jeter [3] conducted a computational fluid dynamics (CFD) study of single phase ejectors utilizing an ideal gas as a working fluid. Their work studied the complex flow patterns within an ejector. CFD analysis was used to explain the changes in secondary flow rate with the primary inlet pressure, as well as how and when choking of the secondary flow happens. It was found that the CFD results are strongly dependent on the grid resolution and the turbulence model used. AlAnsary and Jeter [3] also showed that the mechanism by which the mixed flow compresses at the exit of the mixing chamber, "me" is not the widely used onedimensional normal shock. They found that compression occurs through a series of oblique shocks induced by boundary layer separation in the diffuser. AlAnsary and Jeter [3] also conducted an experimental study to investigate the effect of injecting fine droplets of a nonvolatile liquid into the primary flow to reduce irreversibilities in the mixing chamber. The results showed that this could be advantageous when the secondary flow is not choked. However, they mentioned that the twophase concept needs further exploration. Eames [14] conducted a theoretical study into a new method for designing jet pumps used in jetpump cycle refrigerators. The method assumes a constant rate of momentum change (CRMC) within the mixing section, which in this case is a convergingdiverging diffuser. The temperature and pressure were calculated as a function of the axial distance in the diffuser, and then a function was derived for the geometry of the diffuser that removes the thermodynamic shock process by allowing the momentum of the flow to change at a constant rate as it passes through the mixing diffuser, which allows the static pressure to rise gradually from entry to exit avoiding the total pressure loss associated with the shock process encountered in conventional diffusers. They concluded that diffusers designed using the CRMC method yield a 50% increase in the compression ratio than a conventional jetpump for the same entrainment ratio. Motivated by the fact that there is no universally accepted definition for ejector efficiency, Roan [36] derived an expression to quantify the ejector performance based on its ability to exchange momentum, between the primary and secondary streams, rather than energy. The effectiveness term is called the Stagnation Momentum Exchange Effectiveness (SMEE). Roan [36] viewed ejectors as momentum transfer devices rather than fluid moving devices. Since the momentum transfer mechanism in ejectors is inherently dissipative in nature (shear forces instead of pressure forces), there is no ideal process to compare the ejector performance to. Unlike turbomachinery, which can perform ideally in an isentropic process. Roan developed a correction factor defined as rate of momentum h V K= (2.1) Rate of kinetic energy h(V2/2) for the primary stream and multiplied it by the work potential from the primary flow (energy effectiveness) yielding a new expression for the momentum exchange effectiveness. A similar correction factor was developed for the secondary stream and applied to the compression work performed on the secondary stream yielding a momentum exchange effectiveness expression for the secondary stream. SMEE was then defined as the ratio of the momentum exchange effectiveness expressions. It was found that in almost all evaluations, the design point value of SMEE ranged between 0.10.3. However, SMEE was not found constant for a wide range of offdesign performance, especially for large changes in the secondary flow. Earlier work done on twophase ejectors in the University of Florida includes Lear et al. [29], and Sherif et al. [38]. These two studies developed a onedimensional model for twophase ejectors with constantpressure mixing. The primary and secondary streams had the same chemical composition, while the primary stream was in the two phase regime and the secondary flow was either saturated or subcooled liquid. Since the mixing process occurred at constant pressure, they did not consider the secondary flow choking regimes in the mixing chamber, but their model allowed for supersonic flow entering the diffuser inducing the formation of a normal shock wave, which was modeled using the RankineHugoniot relations for twophase flow. Their results showed geometric area ratios as well as system state point information as a function of the inlet states and entrainment ratio. These results are considered a series of design points as opposed to an analysis of an ejector of fixed geometry. Qualitative agreement was found with singlephase ejector performance. Parker et al. [34] work is considered the most relevant work in the literature to the ejector work presented in this study. They analyzed the flow in twophase ejectors with constantarea mixing. They confined their analysis to the mixed regime where the entrainment ratio,4, is dependent on the backpressure, and vice versa. This is why they did not consider the Fabri choking phenomenon in their study. Their results showed two trends in ejector performance. Fixing the inlet conditions and the geometry of the ejector, and varying the entrainment ratio versus the compression ratio showed the first trend. Since all the data are in the mixed regime. The expected trend of decreasing compression ratio with increasing entrainment ratio was observed. They investigated this trend for various primary to secondary nozzles exit area ratios (Ase/Ane, see Figure 22). An interesting observation was found; that low Ase/Ane is desired when 4 is low. As 4 increases past a certain threshold, a larger Ase/Ane is required for higher compression ratios. The second trend that Parker et al. [34] investigated was the compression ratio as a function of the area ratio Ase/Ane, for constant 4. For low values of 4, the highest compression ratio occurs at the lowest area ratio. For the higher values of 4, there are maximum compression ratios. When the value of the optimum compression ratio was plotted against the entrainment ratio, the relationship was found to be linear, which simplifies the design procedure. Parker et al. [34] did not mention the working fluid used in their study. Solar Collectors For many applications it is desirable to deliver energy at temperatures possible with flatplate collectors. Energy delivery temperatures can be increased by decreasing the area from which heat losses occur. This is done by using an optical device concentratorr) between the source of radiation and the energyabsorbing surface. The smaller absorber will have smaller heat losses compared to a flatplate collector at the same absorber temperature [11]. For that reason a concentrating solar collector will be used in this study since weight and size are of profound importance in space applications. Concentrators can have concentration ratios (concentration ratio definition is presented later in this section) from low values close to unity to high values of the order of 105. Increasing concentration ratios mean increasing temperatures at which energy can be delivered and increasing requirements for precision in optical quality and positioning of the optical system. Thus cost of delivered energy from a concentrating collector is a function of the temperature at which it is available. At the highest range of concentration, concentrating collectors are called solar furnaces. Solar furnaces are laboratory tools for studying material properties at high temperatures and other high temperature processes. Since the cost and efficiency of a concentrating solar collector are functions of the temperature the heat is transferred at, it is important to come up with a simple model that relates the solar collector efficiency to its temperature profile. Such a model is presented in details later in this section. The model assumes an uncovered cylindrical absorbing tube used as a receiver with a linear concentrator. Since the SITMAP cycle is primarily for space applications, the only form of heat transfer considered in the model is radiation. The model assumes onedimensional temperature gradient along the flow direction (i.e. no temperature gradients around the circumference of the receiver tube). Before getting into the details of the solar collector model, it would be useful to define few concepts that will be used throughout the model. Solar Irradiance Solar irradiance is defined as the rate at which energy is incident on a surface, per unit area of the surface. The symbol G is used for solar irradiance. The value of the solar irradiance is a function of the distance from the sun. Table 21shows typical values of the solar irradiance for the different planets in our solar system. It can be seen that the planets closer to the sun have stronger solar irradiance, as expected. The distance from the sun is in Astronomical Units, AU. One AU is the average distance between the earth and the sun, and it is about 150 million Km or 93 million miles [11]. Table 21. Effect of the distance from the sun on solar irradiance. Planet Distance from Sun [AU] Solar Irradiance, G [W/m2] Mercury 0.4 9126.6 Venus 0.7 2613.9 Earth 1 1367.6 Mars 1.5 589.2 Jupiter 5.2 50.5 Saturn 9.5 14.9 Uranus 19.2 3.71 Neptune 30.1 1.51 Pluto 39.4 0.89 Concentration Ratio The concentration ratio definition used in this study is an area concentration ratio, CR, the ratio of the area of the concentrator aperture to the area of the solar collector receiver. A CR = A(2.2) The concentration ratio has an upper limit that depends on whether the concentration is a threedimensional (circular) concentrator or twodimensional (linear) concentrators. Concentrators can be divided into two categories: nonimaging and imaging. Non imaging concentrators do not produce clearly defined images of the sun on the absorber. However, they distribute the radiation from all parts of the solar disc onto all parts of the absorber. The concentration ratios of linear nonimaging concentrators are in the low range and are generally below 10 [11]. Imaging concentrators are analogous to camera lenses. They form images on the absorber. The higher the temperature at which energy is to be delivered, the higher must be the concentration ratio and the more precise must be the optics of both the concentrator and the orientation system. Figure 23 from Duffe and Beckman [11], shows practical ranges of concentration ratios and types of optical systems needed to deliver energy at various temperatures. The lower limit curve represents concentration ratios at which the thermal losses will equal the absorbed energy. Concentration ratios above that curve will result in useful gain. The shaded region corresponds to collection efficiencies of 4060% and represents a probable range of operation. Figure 23 also shows approximate ranges in which several types of reflectors might be used. 104 " io2 1 0 '  i' ; 102 III I 0 500 1000 1500 Receiver Temp, C Figure 23. Relationship between concentration ratio and temperature of the receiver [11]. It should noted that Figure 23 is from Duffe and Beckman [11] and is included just for illustration, and does not correspond to any conditions simulated in this study. Mason [32], from NASA Glenn research center studied the performance of solar thermal power systems for deep space planetary missions. In his study, Mason incorporated projected advances in solar concentrator technologies. These technologies included inflatable structures, light weight primary concentrators, and high efficiency secondary concentrators. Secondary concentrators provide an increase in the overall concentration ratio as compared to primary concentrators alone. This reduces the diameter of the receiver aperture thus improving overall efficiency. Mason [32] also indicated that the use of secondary concentrators also eases the pointing and surface accuracy requirements of the primary concentrator, making the inflatable structure a more feasible option. Typical secondary concentrators are hollow, reflective parabolic cones. Recent studies at Glenn Research Center have also investigated the use of a solid, crystalline refractive secondary concentrator for solar thermal propulsion which may provide considerable improvement in efficiency by eliminating reflective losses. Mason [32] reported that the Earth Concentration ratio of the parabolic, thinfilm inflatable primary concentrator is 1600. The Earth Concentration ratio is defined as the concentration ratio as required at 1 Astronomical Unit (AU). An Astronomical Unit is approximately the mean distance between the Earth and the Sun. It is a derived constant and used to indicate distances within the solar system. Selective Surfaces The efficiency of any solar thermal conversion device depends on the absorbing surface and its optical and thermal characteristics. The efficiency can be increased by increasing the absorbed solar energy (a close to unity) and by decreasing the thermal losses. Surfaces/coatings having selective response to the solar spectrum are called selective surfaces/coatings. Such surfaces offer a cost effective way to increase the efficiency of solar collectors by providing high solar absorptance (c) in the visible and near infrared spectrum (0.32.5 [tm) and low emittance (s) in the infrared spectrum at higher wavelengths, to reduce thermal losses due to radiation. Materials that behave optimally for solar heat conversion do not exist in nature. Virtually all black materials have high solar absorptance and also have high infrared emittance. Thus it is necessary to manufacture selective materials with the required optical properties. The selective surface and/or coating should have the following physical properties [21]. 1. High absorptance for the ultraviolet solar spectrum range and low emittance in the infrared spectrum. 2. Spectral transition between the region of high absorptance and low emittance be as sharp as possible. 3. The optical and physical properties of the coating must remain stable under long term operation at elevated temperatures, thermal cycling, air exposure, and ultraviolet radiation. 4. Adherence of coating to substrate must be good. 5. Coating should be easily applicable and economical for the corresponding application. Selectivity can be obtained by many ways. For example, there are certain intrinsic materials, which naturally possess the desired selectivity. Hafnium carbide and tin oxide are examples of this type. Stacks of semiconductors and reflectors or dielectrics and metals are made in order to combine two discrete layers to obtain the desired optical effect. Another method is the use of wavelength discriminating materials by physical surface roughness to produce the desired in the visible and infrared. This could be by deliberately making a surface rough, which is a mirror for the infrared (high reflectivity). Such surfaces (example: CuO) are deposited on metal substrates to enhance the selectivity. Table 22 gives the properties of few selective surfaces [8]. Effective selective surfaces have solar absorptivities around 0.95 and emissivities at about 0.1. Table 22. Properties of some selective surfaces. .rial Shortwave Longwave Material Absorptivity emissivity Black Nickel on Nickelplated steel 0.95 0.07 Black Chrome on Nickelplated steel 0.95 0.09 Black Chrome on galvanized steel 0.95 0.16 Black Chrome on Copper 0.95 0.14 Black Copper on Copper 0.88 0.15 CuO on Nickel 0.81 0.17 CuO on Aluminum 0.93 0.11 PbS crystals on Aluminum 0.89 0.20 Combined Power/Refrigeration Cycles Khattab et al. [25] studied a lowpressure lowtemperature cooling cycle for comfort airconditioning. The cycle is driven solely by solar energy, and it utilizes a jet pump as the compression device, with steam as the working fluid. The cycle has no mechanical moving parts as it utilizes potential energy to create the pressure difference between the solar collector pressure and the condenser pressure, by elevating the condenser above the solar collector. In their steamjet ejector analysis, Khattab et al. [25] used a primary converging diverging nozzle to expand the motive steam (primary flow) and accelerate it to supersonic speed, which then entrains the vapor coming from the evaporator. Constant pressure mixing was assumed in the mixing region. They also neglected the velocity of the entrained secondary flow in their momentum equation. The compression takes place in the diffuser that follows the mixing chamber by making sure that the flow at the supersonic diffuser throat is supersonic to get the necessary compression shock wave. Khattab et al. wrote a simulation program that studied the performance of the steamjet cooling cycle under different design and operating conditions, and constructed a set of design charts for the cycle as well as the ejector geometry. The inputs to the simulation program were the solar generator and evaporator temperatures and the condenser saturation temperature. Dorantes and Estrada [10] presented a mathematical simulation for the a solar ejectorcompression refrigeration system, used as an ice maker, with a capacity of 100 kg of ice/day. They took into consideration the variation of the solar collector efficiency with climate, which in turn affects the system efficiency. Freon R142b was used as the working fluid. They fixed the geometry of the ejector for a base design case. Then they studied the effect of the annual variation of the condenser temperature, Tc, and the generator temperature, TG on the heat transfer rate at the generator and the evaporator as well as the overall COP of the cycle. They presented graphs of the monthly average ice production, COP, as well as collector and system efficiencies. They found that the average COP, collector efficiency, and system efficiency were 0.21, 0.52, 0.11, respectively. In their analysis, Dorantes et al. [10] always assumed singlephase flow (superheated refrigerant) going into the ejector from both streams. Tamm et al. [41,42] performed theoretical and experimental studies, respectively, on a combined absorption refrigeration/Rankine power cycle. A binary ammoniawater system was used as the working fluid. The cycle can be used as a bottoming cycle using waste heat from a conventional power cycle, or as an independent cycle using low temperature sources as geothermal and solar energy. Tamm et al. [41] performed initial parametric study of the cycle showing the potential of the cycle to be optimized for 1st or 2nd law efficiencies, as well as work or cooling output. Tamm et al. [42] performed a preliminary experimental study to compare to the theoretical results. Results showed the expected trends for vapor generation and absorption condensation processes, as well as potential for combined turbine work and refrigeration output. Further theoretical work was done on the same cycle by Hasan et al. [22, 23]. They performed detailed 1st and 2nd law analyses on the cycle, as well as exergy analysis to find out where the most irreversibilities occur in the cycle. It was found that increasing the heat source temperature does not necessarily produce higher exergy efficiency, as is the case with 1st law efficiency. The largest exergy destruction occurs in the absorber, while little exergy destruction occurs in the boiler. Lu and Goswami [31] used the Generalized Reduced Gradient algorithm developed by Lasdon et al. [27] to optimize the same combined power and absorption refrigeration cycle discussed in references [22, 23, 41, 42]. The cycle was optimized for thermal performance with the second law thermal efficiency as an objective function for a given sensible heat source and a fixed ambient temperature. The objective function depended on eight free variables, namely, the absorber temperature, boiler temperature, rectifier temperature, superheater temperature, inlet temperature of the heat source, outlet temperature of the heat source, and the high and low pressures. Two typical heat source temperatures, 360 K and 440 K, were studied. Lu et al. also presented some optimization results for other objective functions such as power and refrigeration outputs. Efficiency Definitions for the Combined Cycle The SITMAP cycle is combined power and cooling cycle. Evaluating the efficiency of combined cycles is made difficult by the fact that there are two different simultaneous outputs, namely power and refrigeration. An efficiency expression has to appropriately weigh the cooling component in order to allow comparison of this cycle with other cycles. This section presents several expressions from the literature for the first law, second law and exergy efficiencies for the combined cycle. Some of the developed equations have been recommended for use over others, depending on the comparison being made. Conventional Efficiency Definitions Performance of a thermodynamic cycle is conventionally evaluated using an efficiency or a coefficient of performance (COP). These measures of performance are generally of the form Measure of performance = Useful output / Input (2.3) First law efficiency The first law measure of efficiency is simply a ratio of useful output energy to input energy. This quantity is normally referred to simply as efficiency, in the case of power cycles, and as a coefficient of performance for refrigeration cycles. Table 23 gives two typical first law efficiency definitions. Exergy efficiency The first law fails to account for the quality of heat. Therefore, a first law efficiency does not reflect all the losses due to irreversibilities in a cycle. Exergy efficiency measures the fraction of the exergy going into the cycle that comes out as useful output [40]. The remaining exergy is lost due to irreversibilities in devices. Two examples are given in Table 23 where Ec is the change in exergy of the cooled medium. EE 7 exergy (2.4) Resource utilization efficiency [9] is a special case of the exergy efficiency that is more suitable for use in some cases. Consider for instance a geothermal power cycle, where the geofluid is reinjected into the ground after transferring heat to the cycle working fluid. In this case, the unextracted availability of the geofluid that is lost on Table 23. Rankine cycle and vapor compression refrigeration cycle efficiency definitions. Cycle type Rankine Vapor compression First Law r, = Wt /QH COP = Qc /W Exergy 7exerg = Wnet /En rexergy = Ec/Wn Second law 7, = 7/lrev 1r7 = COP/COPW reinjection has to be accounted for. Therefore, a modified definition of the form EE 7R o (2.5) is used, where the Ehs is the exergy of the heat source. Another measure of exergy efficiency found in the literature is what is called the exergy index defined as the ratio of useful exergy to exergy loss in the process [1], YE exegy F (2.6) Second law efficiency Second law efficiency is defined as the ratio of the efficiency of the cycle to the efficiency of a reversible cycle operating between the same thermodynamic conditions. r7 = /177re, (2.7) The reversible cycle efficiency is the first law efficiency or COP depending on the cycle being considered. The second law efficiency of a refrigeration cycle (defined in terms of a COP ratio) is also called the thermal efficiency of refrigeration [5]. For constant temperature heat addition and rejection conditions, the reversible cycle is the Carnot cycle. On the other hand for sensible heat addition and rejection, the Lorenz cycle is the applicable reversible cycle [30]. The exergy efficiency and second law efficiency are often similar or even identical. For example, in a cycle operating between a hot and a cold reservoir (see Figure 24), the exergy efficiency is 7exergy et (2.8) Qh (1 To /h) while the second law efficiency is g net (2.9) e" Q (1 lTT, ) Where To is the ambient or the ground state temperature. For the special case where the cold reservoir temperature Tr is the same as the ground state temperature To, the exergy efficiency is identical to the second law efficiency. Th Cyclic Wnet device Tr Figure 24. A cyclic heat engine working between a hot and cold reservoir The Choice of Efficiency Definition The first law, exergy and second law efficiency definitions can be applied under different situations [43]. The first law efficiency has been the most commonly used measure of efficiency. The first law does not account for the quality of heat input or output. Consider two power plants with identical first law efficiencies. Even if one of these power plants uses a higher temperature heat source (that has a much higher availability), the first law efficiency will not distinguish between the performances of the two plants. Using an exergy or second law efficiency though will show that one of these plants has higher losses than the other. The first law efficiency, though, is still a very useful measure of plant performance. For example, a power plant with a 40% first law efficiency rejects less heat than one of the same capacity with a 30% efficiency; and so would have a smaller condenser. An exergy efficiency or second law efficiency is an excellent choice when comparing energy conversion options for the same resource. Ultimately, the choice of conversion method is based on economic considerations. Efficiency Expressions for the Combined Cycle When evaluating the performance of a cycle, there are normally two goals. One is to pick parameters that result in the best cycle performance. The other goal is to compare this cycle with other energy conversion options. First law efficiency Following the pattern of first law efficiency definitions given in the previous section, a simple definition for the first law efficiency would be Whet +Q r, =net (2.10) Qh Equation (2.10) overestimates the efficiency of the cycle, by not attributing a quality to the refrigeration output. Using this definition, in some cases, the first law efficiency of the novel cycle approaches Carnot values or even exceeds them. Such a situation appears to violate the fact that the Carnot efficiency specifies the upper limit of first law conversion efficiencies (the Carnot cycle is not the reversible cycle corresponding to the combined cycle; this is discussed later in this chapter). The confusion arises due to the addition of work and refrigeration in the output. Refrigeration output cannot be considered in an efficiency expression without accounting for its quality. To avoid this confusion, it may be better to use the definition of the first law efficiency given as Wi +E ri U (2.11) Qh The term Ec represents the exergy associated with the refrigeration output. In other words, this refers to the exergy transfer in the refrigeration heat exchanger. Depending on the way the cycle is modeled, this could refer to the change in the exergy of the working fluid in the refrigeration heat exchanger. Alternately, to account for irreversibilities of heat transfer in the refrigeration heat exchanger, the exergy change of the chilled fluid would be considered. S= h [ ,,, h, To (s s ,)] (2.12) Rosen and Le [37] studied efficiency expressions for processes integrating combined heat and power and district cooling. They recommended the use of an exergy efficiency in which the cooling was weighted using a Carnot COP. However, the Carnot COP is based on the minimum reversible work needed to produce the cooling output. This results in refrigeration output being weighted very poorly in relation to work. Exergy efficiency Following the definition of exergy efficiency described previously in Equation (2.13), the appropriate equation for exergy efficiency to be used for the combined cycle is given below. Since a sensible heat source provides the heat for this cycle, the denominator is the change in the exergy of the heat source, which is equivalent to the exergy input into the cycle. W +E texergy net E (2.13) Ehs,in Ehs,out Second law efficiency The second law efficiency of the combined cycle needs a suitable reversible cycle to be defined. Once that is accomplished, the definition of a second law efficiency is a simple process. Lorenz cycle The Lorenz cycle is the appropriate "reversible cycle" for use with variable temperature heat input and rejection. A Ts diagram of the cycle is shown in Figure 25. Lorenz = 1 34 (2.14) Q12 If the heat input and rejection were written in terms of the heat source and heat rejection fluids, the efficiency would be given as: mh (hhrout rhn) 1 ILorenz 1 hr h ) (2.15) mhs hhs,n hhs,out ) Knowing that processes 41 and 23 are isentropic, it is easily shown that in terms of specific entropies of the heat source and heat rejection fluids that mhs hrout S) (2.16) mhr Shsin Shs,out) The efficiency expression for the Lorenz cycle then reduces to ULorenz (hhr,ro rn) / (hrout r,n) (2.17) Thhhn hsout )n / Sh, hs, outw This can also be written as Loren= (2.18) Here, the temperatures in the expression above are entropic average temperatures, of the form Ts= (2.19) S2 S 2 1 4 s Figure 25. The TS diagram for a Lorenz cycle For constant specific heat fluids, the entropic average temperature can be reduced to T 2 = (2.20) In (T2~1) The Lorenz efficiency can therefore be written in terms of temperatures as Lorenz 1(T, out hrn) l mn(Thout n (2.21) (Ths,in Thseout ) / I (Ths,,n / Thsiout) It is easily seen that if the heat transfer processes were isothermal, like in the Carnot cycle, the entropic average temperatures would reduce to the temperature of the heat reservoir, yielding the Carnot efficiency. Similarly the COP of a Lorenz refrigerator can be shown to be COPL, ='f) (2.22) Lorenz ( )Cf Cascaded Cycle Analogy An analogy to the combined cycle is a cascaded power and refrigeration cycle, where part of the work output is directed into a refrigeration machine to obtain cooling. If the heat engine and refrigeration machine were to be treated together as a black box, the input to the entire system is heat, while output consists of work and refrigeration. This looks exactly like the new combined power/refrigeration cycle. Figure 26 shows the analogy, with a dotted line around the components in the cascaded cycle representing a black box. One way to look at an ideal combined cycle would be as two Lorenz cycle engines cascaded together (Figure 26b). Assume that the combined cycle and the cascaded arrangement both have the same thermal boundary conditions. This assumption implies that the heat source fluid, chilled fluid and heat rejection fluid have identical inlet and exit temperatures in both cases. The first law efficiency of the cascaded system, using a weight factor f for refrigeration is WO.' W, + f (2.23) ris= ou W (2.23) Qh The weight factor, f is a function of the thermal boundary conditions. Therefore, the first law efficiency of the combined cycle can also be written as Ws net+ fQ (2.24) Qh (a) (b) Figure 26. Thermodynamic representation of (a) combined power/cooling cycle and (b) cascaded cycle The work and heat quantities in the cascaded cycle can also be related using the efficiencies of the cascaded devices W., = QhrHE (2.25) W,= QcCOP (2.26) By specifying identical refrigeration to work ratios (r) in the combined cycle and the corresponding reversible cascaded cycle as r = Qc Wne (2.27) and using Equation(2.23) and Equations(2.252.26), one can arrive at the efficiency of the cascaded system as r,=i+ = rnH 1+ 2.28 s1 + rICOP assuming the cascaded cycle to be reversible, the efficiency expression reduces to r f Ycop 17,rev Loren + 1+r Lore (2.29) Here Loren, is the first law efficiency of the Lorenz heat engine and COPLorenz is the COP of the Lorenz refrigerator. A second law efficiency would then be written as 17 = 71/17/ ,rv (2.30) If the new cycle and its equivalent reversible cascaded cycle have identical heat input (Qh), the second law efficiency can also be written as r, = Wn, + fc (2.31) rIrev Wnet,rev + fc,rev This reduces further to 17, t W, (1 + frt) r71 r, (2.32) .II ,rev W (1+f) Evidently, the refrigeration weight factor (f) does not affect the value of the second law efficiency. This is true as long as f is a factor defined such that it is identical for both the combined cycle and the analogous cascaded version. This follows if f is a function of the thermal boundary conditions. Assuming a value of unity for f simplifies the second law efficiency expression even further. The corresponding reversible cycle efficiency would be, 17,_ = rLren 1 +r O(2.33) 1 + rCOPLoren. The resulting second law efficiency equation is a good choice for second law analysis. The expression does not have the drawback of trying to weight the refrigeration with respect to the work output. Being a second law efficiency, the expression also reflects the irreversibility present in the cycle, just like the exergy efficiency. Use of the Different Efficiency Definitions Expressions for the first law, exergy and second law efficiencies have been recommended for the combined power and cooling cycle in Equations (2.11, 2.13 and 2.31) respectively. These definitions give thermodynamically consistent evaluations of cycle performance, but they are not entirely suitable for comparing the cycle to other energy conversion options. Substituting for refrigeration with the equivalent exergy is equivalent to replacing it with the minimum work required to produce that cooling. This would be valid if in the equivalent cascaded arrangement, the refrigeration machine were reversible. Therefore, when comparing the combined cycle with other options, such a substitution is debatable. This is where the difficulty arises in arriving at a reasonable definition of efficiency. Two cases are discussed here to illustrate the point. Case 1: Comparing this Cycle to Other Combined Cooling and Power Generation Options Consider the situation where the novel cycle is being designed to meet a certain power and refrigeration load. The goal then would be to compare the thermodynamic performance of the novel cycle with other options designed to meet the same load. If the performance of both cycles were evaluated using Equations (2.11, 2.13 and 2.31), such a comparison would be perfectly valid. Case 2: Comparing a Combined Cycle to a Power Cycle In some instances, a combined cycle would have to be compared to a power cycle. For example, this cycle could be configured so as to operate as a power cycle. In this situation, the refrigeration would have to be weighted differently, so as to get a valid comparison. One way of doing this would be to use a practically achievable value of refrigeration COP to weight the cooling output. Another option is to divide the exergy of cooling by a reasonable second law efficiency of refrigeration (also called thermal efficiency of refrigeration). Such efficiencies are named "effective" efficiencies in this study. Wet + Qc COPrctc (2.34) eQh net + /='ref (2.35) eQh nexet Qryc CO' praccal (2.36) exeeff E E/ rf (2.37) ehs,xn Ehs,out CHAPTER 3 MATHEMATICAL MODEL Jetpump Analysis First, it should be noted that the inputs to the jetpump model are: * Fully defined stagnation state at the jetpump primary inlet. * Fully defined stagnation state at the jetpump secondary inlet. * Primary nozzle area ratio Ant/Ane. * Secondary to primary area ratio, Ane/Ase. The outputs of the jetpump model are: * Breakoff entrainment ratio. * Mixed flow conditions at the jetpump exit. The following general assumptions are made for the jetpump analysis: * Steady flow at all state points. * Uniform flows at all state points. * Onedimensional flow throughout the jetpump. * Negligible shear stresses at the jetpump walls. * Constantarea mixing, Ame = A, + A,. * Spacing between the primary nozzle exit and the mixing section entrance is zero. * Adiabatic mixing process. * Negligible change in potential energy. * The primary and secondary flows are assumed to be isentropic from their respective stagnation states to the entrance of the mixing section. Figure 31 shows a schematic of the jetpump. The highpressure primary flow from the power part of the cycle (State pi) is expanded in a convergingdiverging supersonic nozzle to supersonic speed. Due to viscous interaction secondary flow is entrained into the jetpump. Constantarea mixing of the high velocity primary and the lower velocity secondary streams takes place in the mixing chamber. The mixed flow enters the diffuser where it is slowed down nearly to stagnation conditions. The method for calculating the diffuser exit state and the entrainment ratio, 4, given the jetpump geometry and the primary and secondary stagnation states is presented next. For each region of the jetpump flowfield conservation laws and process assumptions are used to develop a well posed mathematical model of the flow physics. si se me pi nt ne de se si Figure 31. Schematic for the jetpump with constant area mixing. Primary Nozzle To obtain the properties at the nozzle throat, Pnt is guessed and, since isentropic flow is assumed, Snt = Spi. The primary nozzle inlet velocity can be calculated using the continuity equation, vp, Pt Ant vt (3.1) ppT APt The velocity at the nozzle throat is calculated using conservation of energy, hp ,, h (3.2) S1_ Pnt A4t 2 PP, AP, Mach number at the nozzle throat is calculated using Equations (3.3), and (3.4). The 's' in Equation (3.3) signifies an isentropic process a = (3.3) M = (3.4) a Pnt is iterated on until the Mach number is equal to unity at the primary nozzle throat. The properties at the nozzle exit are obtained by assuming isentropic flow, Sne=Snt, and iterating on Pne. Conservation of energy is used to calculate the primary nozzle exit velocity as Ve = [2 hn, +i, hne (3.5) Ant/Ane is calculated using the continuity equation Ant Pne Vne Ane Pnt nt The Mach number at the primary nozzle exit is calculated using Equations (3.3) and (3.4). Pne is iterated on till Ant/Ane matches its input value. Flow Choking Analysis There are two different choking mechanisms that can take place inside the jet pump. Either one of these mechanisms dictates the breakoff value for the entrainment ratio for a given jetpump configuration. Each mechanism corresponds to a different jet pump operating regime. The first choking mechanism is referred to as inlet choking and it takes place when the jetpump is operating in the "saturated supersonic" regime. In this regime the secondary flow chokes at the inlet to the mixing chamber. The second choking mechanism is referred to as Fabri choking and it takes place when the jetpump is operating in the "supersonic" regime. In this regime the secondary flow chokes at an aerodynamic throat inside the mixing chamber. For a given jetpump geometry, there is a breakoff value for the stagnation pressure ratio, (Ppo/Pso)bo, that determines which of the two choking mechanisms will take place and dictate the value of the breakoff (maximum) entrainment ratio, 4bo. The value of (Ppo/Pso)bo is represented by line "bd" in Figure 22, and 4bo is represented by the curve "abc". (Ppo/Pso)bo affect the jetpump operation as follows: PP < PP bo = lnietchoke Pso Pso bo P > P o o fabri P P so so bo The breakoff conditions for transition from one operating regime to another are: 1. Mse = 1, and Pse/Pne = 1 (for transition from "saturated supersonic" to "supersonic) 2. Mse = 1, and Pse/Pne > 1 (for transition from "mixed" to "saturated supersonic") 3. Mse < 1, and Pse/Pne < 1, and Ms2 = 1 (for transition from "mixed" to "supersonic"). For a given jetpump geometry and stagnation conditions at the primary inlet, the state (ne) at the primary nozzle exit can be defined using the procedure presented in the previous section. Then (Pso)bo is the stagnation pressure corresponding to the conditions: Pse=Pne, and Mse=l. 42 For Fabri choking to occur Pse has to be less than Pne. In this case the primary flow expands in the mixing chamber constricting the available flow area for the secondary stream causing it to accelerate. Then the secondary stream reaches sonic velocity at an aerodynamic throat in the mixing chamber, causing the secondary mass flow rate to become independent of downstream conditions. However, when Pse is greater than Pne the primary cannot expand into the secondary, therefore, the only place where the secondary can choke is at the inlet to the mixing chamber. Figure 32 shows a schematic of the jetpump. To calculate inlet choke corresponding to the "saturated supersonic" regime, iterations are done on Pse till it reaches the critical pressure (pressure at which the Mach number is equal to unity) corresponding to the given stagnation pressure, Psi. Then inlet choke is then calculated from continuity as PseV' (3.7) Anletchoke p eV (37) Pnene Ane II . Si Se I me n2 de pi nt ne m Se I "I 2 i.s2 Figure 32. Schematic for the jetpump with constant area mixing, showing the Fabri choked state s2. 43 The following is a list of the general assumptions made in the Fabri choking analysis: * The primary and secondary flows stay distinct and don not mix till sections (n2), and (s2), respectively. * The primary and secondary flows are isentropic between (se)(s2), and (ne)(n2), respectively. S Ms2 = 1. * The primary inlet static pressure is always larger than secondary inlet static pressure, Pne > Pse. The following analysis is used to calculate 4fabri corresponding to the "supersonic" regime. The momentum equation for the control volume shown by the dotted line in Figure 32 can be written as ,eA,, + PA,, P2A,2 4zA,2 = 2Vn2 + ,h2,, ,pVn ,yjsV (3.8) dividing by Mi yields 1(P,,A, +Pne An,,e A,2 PA, )= (n2 )+ Fbn (Vz2 e) (3.9) P (P A + [,,A,,, AA2 A 4,2A,2) (Vz2 3V ) Spe + Ane A P 2 A2 Ane SFabrn = e A A A A ) (V, ie) (3.11) PneVe An e As e (j _e) The iteration scheme starts by guessing a value for Pe, knowing that s = s,, that defines the state (se). From the energy equation V = [2(h, h )Y2 (3.12) Fabn can then be calculated as, hFabn PseV,, (3.13) pneVneAne It should be noted that the area ratio A,,/A,, is an input to the SITMAP code. Then a guess is made for P2, and s2 = ss, which defines state (s2). The velocity Vs2 can be obtained from the energy equation between (se) and (s2) Vs= 2 h, hs2 + (3.14) Ps2 is iterated on till Ms2=l. The area ratio As2/As, is calculated from the continuity equation between (se) and (s2), As2 PseVe (3.15) As, ps2Vs2 For constantarea mixing An + As = A2 + A2, then Ae2 A As 2 An2 = Ae AA (3.16) A A A A ne ne se ne PI is iterated on till the values for Fabn, from equations (3.11) and (3.13) match. There is another limit on the maximum entrainment ratio referred to, only in one source in the literature, as exit choking and was first addressed by Dutton et al. 11. It refers to conditions when the flow chokes at the mixing chamber exit, state (me). However, such conditions were never encountered in the analysis performed for this study. Secondary Flow When the jetpump is operating in the mixed regime ()< < bo), the following secondary flow analysis is used to calculate the Pse for the given conditions. Pse is iterated on assuming isentropic flow in the secondary nozzle (sse = ssi) till the following conservation equations are satisfied. Ve = 2 hs, + V2 hse 2 (3.17) Ane 1PseVe (3.18) Ase 0 Pne Vne Pse iteration stops when Ane/Ase matches its input value. Then the Mach number at the secondary exit is calculated using Equations (3.3) and (3.4). Mixing Chamber n, ori  l or r sm pi, or po   Figure 33. Jetpump schematic showing the control volume for the mixing chamber analysis. In the beginning it should be noted that at this point, the state points (se) and (ne) are fully defined. The entrainment ratio is also known from the previous choking analysis. The mixed pressure, Pme is iterated on till the following set of equations is satisfied. The momentum equation for the control volume shown by the dotted line in Figure 33 can be written as Pme (Ane + Ae)+ eAne + P A e = pVe pe +(1 + ) pVme (3.19) given izp = pneAnVne, and the constant area mixing process, Am = An + A e, Equation 19 can be rearranged as (P P) Ane (P _)+pnPV A +pAVe e meA e ne A Vm  (3.20) (1+ ) PneVnA Ase Then the enthalpy hme is calculated from the energy equation for the mixing chamber 1 1 ( 12 1V2 he = he + ) +V2 V+e (3.21) Then from continuity = PeAmeVme 1 (3.22) PneAneVne Pme is iterated on till the value of q from Equation 3.22 matches its input value. Then the mixing chamber exit Mach number is calculated using Equations (3.3) and (3.4). Diffuser If the mixing chamber exit flow is supersonic. In such a case, a shock exists in the diffuser. This analysis assumes that the shock occurs at the diffuser inlet where the Mach number is closest to unity and, thus, the stagnation pressure loss over the shock is minimized. IfMme > 1, The pressure downstream of the shock, Pss, is iterated on till the following set of conservation equations across the shock between (me) and (ss) is satisfied. PmeVme = Ps,V,Y (3.23) Pme +PmeVme = Pss + PssVs (3.24) 1 1 (3 hme + 2 Vm = hss + 2 V (3.25) Pss = (Pss hss) (3.26) To obtain the diffuser exit state (de) for the case of Mme > 1, follow the following procedure for Mme less than or equal to 1, replacing the subscript 'me' with 'ss.' If Mme < 1, then to obtain the properties at the diffuser exit, Pde is iterated on assuming isentropic flow in the diffuser (sde = Sme) till the following continuity and energy conservation equations are satisfied de e Ane Ade e =[hme +Ve hde (3.28) Then the Mach number at the diffuser exit is calculated using Equations (3.3) and (3.4). SITMAP Cycle Analysis The only output from the jetpump analysis needed for the SITMAP cycle analysis is the jetpump exit pressure, which corresponds to the radiator pressure in the SITMAP cycle. Figure 34 shows a schematic of the cycle with all state point notations. The pump, and turbine, efficiencies were estimated to be 95%. Frictional pressure losses in the system were lumped into an estimated pressure ratio over the various heat exchangers of r = 0.97. Jetpump (ei) (si) (pi) Evaporator Pump (te) I Boiler (pe) (bi) (t) Turbine Recuperator Figure 34. A schematic of the SITMAP cycle showing the notation for the different state points. The method used to achieve a converged solution for the SITMAP cycle given the jetpump inlet and exit states and entrainment ratio follows. Overall Analysis Knowing the pressure and assuming that the condenser exit state is saturated liquid (xre=0), this defines the radiator exit state. Also the pressure at the evaporator inlet is the same as the jetpump secondary inlet pressure, and assuming isoenthalpic expansion, he, = hre, this defines the evaporator inlet state (ei). So straight out of the jetpump analysis all the states in the refrigeration part of the SITMAP cycle are defined. System convergence requires a doubleiterative solution. The first step requires guessing the high pressure in the cycle, turbine inlet pressure, Pti, and the entropy at the same state, sti (or any other independent property like the enthalpy). Then the pump work can be calculated as pp Ppe Pre) (3.29) plpumpPre Energy balance across the pump yields, hp = + Wpu (3.30) pe re pump Now state (pe) is defined. The recuperator efficiency is assumed to be 0.7 and is defined as 7rep or (3.31a) Qmax where, Qmax = p h, t h(Pe, 'pe)] (3.31b) Equation 3.3 la, and 3.31b are combined yielding, he hi 7recup he h(P (3.32) hte f h(eI Tpe) ht, = p recuph( eTpe) (3.33) 1 7recup The specific enthalpy from Equation (3.33) and the fact that Pts = rxPpi can then be used to calculate an isentropic turbine exit state. From the definition of turbine efficiency, h, = h h, h't (3.34) 7t The entropy at the turbine inlet, sti, is iterated on until the entropy at the turbine inlet state matches that of the isentropic turbine exit state. The turbine work is calculated as W m (h, h,,) (3.35) Pti is iterated on (repeat the entire SITMAP analysis) until the net work, W, W is positive. In other words the analysis stops when it finds the minimum turbine inlet pressure that yields positive net work, i.e. W W, > 0. A converged solution has now been obtained for the SITMAP cycle. The following equations complete the analysis: Qevap =0 lp (h, her) (3.36) Qrad =mp(1+i)(hd hre) (3.37) s = ,p (ht hpe) (3.38) ecup = (, hpe) = hp (hT h,) (3.39) It should be noted that the primary mass flow rate in this analysis is assumed to be unity, therefore, all the heat transfer and work values are per unit primary flow rate and their units are [J/kg]. These values will be referred to during this study as heat rate or work rate. Solar Collector Model If the working fluid comes into the solar collector as a twophase mixture, part of the heat exchange in the collector will take place at a constant temperature equal to the saturation temperature, Ts,, at the collector pressure. The rest of the heat exchange in the collector will be in the superheated region where the temperature of the working fluid is a function of the position in the solar collector. Therefore, in this analysis the solar collector area is divided into two parts. The first is the part operating in the twophase region, and is denoted A, and the second part operates in the superheated region and is denoted A r The working fluid will always be assumed to be either in the twophase region or in the superheated region coming into the solar collector and never in the sub cooled region. This assumption was found to be always true within the range of cycle parameters investigated in this study. The main reason is the presence of the recuperator which heats up the working fluid prior to the solar collector. It should also be noted that it is always assumed that the temperature of the solar collector receiver is equal to the working fluid temperature at any given location in the solar collector. This assumption neglects the thermal resistance of the receiver wall. Since in this study the SITMAP cycle is assumed to operate in outer space; the only form of heat transfer considered in the solar collector analysis is radiation. Twophase region analysis An energy balance can be written for the portion of the solar collector operating in the twophase region as follows m [h, (x = 1) h ]= G(CR)caA uoA (T7 T4) (3.40) The specific enthalpy difference in the above equation is between the enthalpy of saturated vapor at the collector pressure and the enthalpy of the working fluid coming into the solar collector. The above equation can be solved for A, . m ha (x =1)h A hh ( [ ) ]_ (3.41) G(CR)a o NT, T4) Superheated region analysis hlCpdT = G(CR)aA, oW(T4(x) T4)dx (3.42) dT 1hC, = G(CR)aW coW(T4 (x) T4) (3.43) dx x 1 dT 1 dT Let x = >.di~= dx>.. (3.44) LSH LSH dx LSH da rizC dT = G(CR)aW oW(T4 (,() T4) (3.45) LSH dx Multiply through by LsH dT ihCP d = G(CR)aA, scA, (T4() T4) (3.46) Now we nondimensionlize the dependent variable T dividing it by the evaporator temperature, we let T 1 dT dT* T= drT dT =Tr Te T di dx dT* CT, = G(CR)a, coA T (T*)T ') (3.47) If we divide both sides by A mCHCT and rearrange 1 dT G(CR)a coT A, d jhCpT hCC ( C dT* dT = A di (3.48) G(CR)a coT3 (4 ( 6tC,T, #C, This separable ordinary differential equation can be written in the form dT* dT Ar dx (3.49) a bT () Where G(CR)a coT T3 sT3 a= + eT and b= mCT, mCp "' mC Integrating equation 3.49 for the limits Tt <:T yields the expression below for the area of the superheated region of the collector, A.SH LT 2tan1 b In a4 T + In a 4+b 4T SH b (3.50) W4b4a4 This expression is obtained using the symbolic integration feature of Mathematica. The total solar collector area, Ar, is equal to the summation of the areas of the superheated region and that of the saturated (twophase) region. A = ASH + Ar, (3.51) The ODE shown in Equation 3.48 can be solved a second time for the temperature profile in the solar as function of the axial distance for the calculated solar collector receiver area. To obtain the temperature profile the ODE is integrated between the following limits: T,:t T* T*; and 0 ! Figure 35 shows a typical temperature profile in the solar collector. 5.2 P. = 128 kPa T = 3.43 STo = 79.4K T =519 5 T = 78.4 out G = 1300 W/m2 4.8 CR = 100 a= 0.95 4.6 = 0.1 A,= 1.31 m2 4.4 I 4.2 4 3.8 3.6 3.4 I I I 0 0.25 0.5 0.75 1 x/L Figure 35. Typical solar collector temperature profile. To calculate an effective collector temperature, an energy balance is performed on the solar collector as a whole similar to the energy balances performed on the twophase and superheated regions of the solar collector. mAh = G(CR)aA, EaA(T, T4) (3.52) In the above equation the enthalpy difference, Ah, is the overall enthalpy difference between the inlet and outlet of the solar collector. Solving the above equation for Tff yields T~, =T4+ J(3.53) Tf +G(CR)aA, thz ]Y (353 E,04 Solar collector efficiency The efficiency of the solar collector can be calculated as the ratio of the useful gain to the total amount of available solar energy. The total energy available is the product of the solar irradiation, G[W/m2], and the aperture area of the concentrator, Aa [m2]. iiAh '7= (3.54) GA, The aperture area can be calculated from the concentration ratio expression. A, = CR x A (3.55) In this model the value of the concentration ratio will be assumed based on typical values for current technologies available for deep space applications. Radiator Model Equation (3.56) represents the energy balance between the fluid and the radiator; the emissivity has been lumped into an overall radiator efficiency, rlrad, mi dArd = Trd dhrd (3.56) 7rad' If superheat exists at the radiator inlet, Equation (3.56) must be numerically integrated to account for the changing temperature in the superheated region. For the rare case of either mixed or saturated vapor conditions at the jetpump exit, Equation (3.56) can be analytically integrated, using the constant value of the saturation temperature at the radiator pressure. System Mass Ratio Figure 36 shows a schematic for the thermally actuated heat pump system being considered. The power subsystem accepts heat from a hightemperature source and supplies the power needed by the refrigeration subsystem. Both systems reject heat via a radiator to a common heat sink. The power cycle supplies just enough power internally to maintain and operate the refrigeration loop. However, in principle, the power cycle could provide power for other onboard systems if needed. Both the power and refrigeration systems are considered generic and can be modeled by any specific type of heat engine such as the Rankine, Sterling, and Brayton cycles for the power subsystem and gas refrigeration or vapor compression cycles for the cooling subsystem. TH Q' e W Power Cycle Refrigeration Cycle \ Figure 36. Overall system schematic for SMR analysis. The System Mass Ratio (SMR) is defined as the ratio between the mass of the overall system and that of an idealized passive system. The overall system mass is divided into three terms; radiator, collector, and a general system mass comprising the turbomachinery and piping present in an active system. This is shown mathematically by m = moo d +m ,, (3.57) rad ,o Equation (3.57) can be separated and rewritten in terms of collector and radiator areas col Acol +Arad = ad s+ys (3.58) rad,o mrad,o The solar collector is modeled by examining the solar energy incident on its surface. This energy is proportional to the collector efficiency, the crosssectional area that is absorbing the flux, and the local radiant solar heat flux, G, same as Equation (3.54). The radiant energy transfer rate between the radiator and the environment is given below. For deep space applications, the environmental reservoir temperature may be neglected, but for nearplanetary or solar missions this may not be the case. Qs = eArdad Td4 4T) (3.59) The idealized passive radiator model operates perfectly (; = 1) at the temperature of the evaporator, i.e. the load temperature. Since there is no additional thermal input, the heat transferred to the radiator is equal to that transferred from the evaporator. The ideal passive area for a radiator is consistent with Op a ve = uArd,o(T T4)= Qe (3.60) P col Defining a new nondimensional parameter, a, as a = performing an overall Prad energy balance on the active system yielding QH + Qe = Q, and substituting Equations (3.54), (3.59), and (3.60) into Equation (3.58), yields {4 a1 + 4 sys m = 1 +r + ad ,\ rad I (3.61) Substituting the following definitions W (3.62) 1I = ; COP = Q ; 7COP (3.62) QH W QH ~P ; R_ COP = PR (3.63) PiC COPc c =1 COPc Te eTe (3.64) 1i ,COpc ; c (3.64) To Trd Te 7col suG T T. = (3.65) Too = ; Trad= ; T T7 Te T into Equation (3.61) yields S 1 +T*co (T*rad , 1 radr 1 4 (3.66) 4 T coi T rad rad T s T rad iT s radio Nondimensionalizing the third term on the right hand side of Equation (3.66) yields mss m,, act msys m (3.67) t,act J rad,o t,act But m,at = mrad + mco + m therefore m. + s + m4 , E I ( Tcoi Trad [ T rad T *s4) T rad T s t,act Defining p/ = m yields M t,act 1 T* ,(T* d I T 4 1 4 1 4 4 e( )co T*rad Trad s T*rad4 T*4 Equation (3.69) represents the SMR in terms of seven system parameters. Three of these parameters are based on temperature ratios and the remaining four are based on system properties. All of the parameters are quantities that can be computed for a given application. It should be noted that three of the SMR parameters are dictated by the 59 SITMAP cycle analysis; those parameters are the collector temperature T*,, radiator temperature Trd and the overall percentage Carnot efficiency rT. CHAPTER 4 CYCLE OPTIMIZATION The combined cycle has been studied by a simple simulation model coupled to an optimization algorithm. The simulation model presented in the previous chapter is based on simple mass, energy, and momentum balances. The properties of the working fluid are dynamically calculated using a software called REFPROP made by the National Institute for Standards and Technology (NIST). The source code for REFPROP was integrated within the simulation code to allow for dynamic properties calculation. The optimization is performed by a search method. Search methods require an initial point to be specified. From there the algorithm searches for a "better" point in the feasible domain of parameters. This process goes on until certain criteria that indicate that the current point is optimum are satisfied. Optimization Method Background The optimization of the working of the cycle is a non linear programming (NLP) problem. A NLP problem is one in which either the objective function or at least one of the constraints are nonlinear functions. The cycle optimization method chosen for the analysis of this cycle is a search method. Search methods are used to refer to a general class of optimization methods that search within a domain to arrive at the optimum solution. It is necessary to specify an initial starting point in search schemes. The optimization algorithm picks a new point in the neighborhood of the initial point such that the objective function (the function being optimized) value improves without violating any constraints. A simple method of determining the direction of change is to calculate the gradient of the objective function at the current point [38]. Such methods are also classified as steepest ascent (or descent) methods, since the algorithm looks for the direction of maximum change. By repeating these steps until a termination condition is satisfied, the algorithm is able to arrive at an optimized value of the objective. When implementing steepest ascent type methods for constrained optimization problems, the constraints pose some limits on the search algorithm. If a constraint function is at its bound, the direction of search might have to be modified such that the bounds are not violated. The Generalized Reduced Gradient (GRG) method was used to optimize the cycle. GRG is one of the most popular NLP methods in use today. A description of the GRG method can be found in several sources [15, 35, and 39]. There are several variations of the GRG algorithm. A commercially available program called the LSGRG2 was used for SITMAP cycle optimization. LSGRG2 is able to handle more variables and constraints than the GRG2 code, and is based on a sparse matrix representation of the problem Jacobian (matrix of first partial derivatives). The method used in the software has been discussed by Edgar et al. [15] and Lasdon et al. [27]. A brief description of the concept of the algorithm is presented below: Consider the optimization problem defined as: Minimize objective function: g,,, (X) Subject to equality and inequality type constraints as given below g, (X) = 0 i = 1,...,neq (4. 1) 0 < g (X)< ub(n + i) i= neq + 1,.....,m (4.2) The variables are constrained by an upper and lower bounds. lb(i) < X, < ub(i) i = ,...,n (4.3) Here is the variable vector consisting of n variables. As in many optimization algorithms, the inequality constraints are set to equality form by adding slack variables, Xnl,..., X+m The optimization program then becomes Minimize: gm,, (X) Subject to: g,(X) X, = 0, i ,...,m (4.4) lb(i) lb(i)= ub(i) = 0, i = n + 1,...,n + neq (4.6) lb(i)= 0, i = n + neq + 1,...,n + m (4.7) The last two equations specify the bounds for the slack variables. Equation (4.6) specifies that the slack variables are zero for the equality constraints, while the variables are positive for the inequality constraints. The variables are called the natural variables. Consider any feasible point (satisfies all constraints), which could be a starting point, or any other point after each successful search iteration. Assume that 'nb' of the constraints are binding, or in other words, hold as equality constraints at a bound. In the GRG algorithm used in the LSGRG2 software, using the nb binding constraint equations, nb of the natural variables (called basic variables) are solved for in terms of the remaining nnb natural variables and the nb slack variables associated with the binding constraints. These n variables are called the nonbasic variables. The binding constraints can be written as g(y,x) = 0 (4.8) Here y and x are vectors of the nb basic and n nonbasic variables respectively and g is a vector of the binding constraint functions. The binding constraints Equation (4.8) can be solved for y in terms of x, reducing the objective to a function of x only. g,,, (y(x), x) = F(x) This equation is reasonably valid in the neighborhood of the current point to a simpler reduced problem. Minimize F(x) Subject to the variable limits for the components of the vector x. < x < u (4.9) The gradient of the reduced objectiveF(x), VF(x) is called the reduced gradient. Now the search direction can be determined from the reduced gradient. A basic descent algorithm can now be used to determine an improved point from here. The choice of basic variables is determined by the fact that the nb by nb basis matrix consisting of ag, /1y, should be nonsingular at the current point. A more detailed description of the theory and the implementation of the GRG algorithm and the optimization program can be found in the literature [15, 27, and 28]. This algorithm is a robust method that appears to work well for the purposes of optimizing this cycle, the way it has been implemented in our study. Search Termination The search will terminate if an improved feasible point cannot be found in a particular iteration. A well known test for optimality is by checking if the KuhnTucker conditions are satisfied. The KuhnTucker conditions are explained in detail in [15, and 35]. It can be mathematically explained in terms of the gradients of the objective functions and inequality constraints as: Vg+, (X)+ u Vg,(X) = 0 (4. 10) J1 uJ > 0, uJ g (X) ub(j) =0 (4. 11) g, (X) Here, uj is a Lagrange Multiplier for the inequality constraints. Unfortunately, the KuhnTucker conditions are valid only for strictly convex problems, a definition that most optimization problems do not satisfy. A disadvantage of using a search method, such as the GRG algorithm that has been used in this study, is that the program can terminate at a local optimum. There is no way to conclusively determine if the point of termination is a local or global optimum [15]. The procedure is to run the optimization program starting from several initial points to verify whether or not the optimum point is actually the optimum in the domain investigated. Sensitivity Analysis The sensitivity of the results to the active constraints can be determined using the corresponding Lagrange multipliers. V uJ (4. 13) O Bub(j) where, V is the value of the objective at the optimum. Application Notes There are some factors in the optimization of the cycle studied using LSGRG2 that are interesting to mention. In a search scheme, it is possible that the termination point could be a local optimum or not an optimum at all. It is necessary to determine the nature of the "optimum" returned by the program. Prior to the optimization, during setup, close attention should be paid to: * Scaling of the variables * Limits set for different convergence criteria * Method used to numerically calculate the gradient * Variables that the objective function is not very sensitive to in the vicinity of the optimum. These variables cause convergence problems at times. They should be taken out of the optimization process and fixed at any value close to their optimum. The relative scaling of the variables affects the accuracy of the differentiation and the actual value of the components in the gradient, which determines the search direction. From experience, it is very useful to keep all the optimization variables at same order of magnitude. This makes the optimization process a lot more stable. This can be achieved by keeping all the variables in the optimization subroutines at same order of magnitude and then multiply them by the necessary constants when they are passed to the subroutine that calculates the objective function and the constraints. Another very important parameter in the optimization process is the convergence criterion. Too small a convergence criterion, particularly for the NewtonRaphson method used during the onedimensional search can cause premature termination of the optimization program. The accuracy of the numerical gradient can affect the search process. However, in this study forward differencing scheme was accurate enough for the search to proceed forward as long as the accuracy of the objective function calculation and constraints were accurate enough. Same results were obtained using both forward and central difference gradient calculations. Special attention should be paid to make sure that the convergence criterion for the optimization process is not more stringent than that of the objective function and constraints calculation. This can cause convergence problems. Once the program was setup, the following methods were used in the process in order to obtain a global optimum: * For each case, several runs were performed, from multiple starting points. * The results were perturbed and optimized, particularly with respect to what would be expected to be very sensitive variables, to see if a better point could be obtained and to make sure that the optimum point obtained is an actual global optimum within the range of variables investigated. * Another method is to change the scaling of variables that appear to be insensitive to check if better points can be obtained. At the end of this process, it is assumed with confidence that the resulting point is indeed a global optimum. The optimization process using GRG is to a certain extent an "art" not "science". Unfortunately, this is a problem with almost all NLP methods currently in use. Variable Limits In any constrained optimization problem, limits of variable values have to be specified. The purpose of specifying limits is to ensure that the values at optimum conditions are achievable, meaningful, and desirable in practice. An upper and lower bound is specified for the variables in the LSGRG2 optimization program. If the variable is to be held fixed, the upper bound is set to be equal to the lower bound, both of which are set equal to the value of the parameter. Unbounded variables are specified by setting a very large limit. Table 41 shows the upper and lower bounds of the variables used in the cycle optimization. Some of the bounds are arbitrarily specified when a clear value was not available. Table 41. Optimization variables and their limits Variable Lower Limit Upper Limit Name and Units Ppo/Pso 2 65 Primary to secondary stagnation pressure ratio Ant/Ane 0.01 0.99 Primary nozzle throat to exit area ratio Ane/Ase 0.01 1.0 Primary to secondary nozzle exit area ratio The actual domain in which these variables may vary is further restricted by additional constraints that are specified. Constraint Equations To ensure that cycle parameters stay within limits that are practical and physically achievable, it is necessary to specify limits in the form of constraint equations. Constraints are implemented in GRG2 by defining constraint functions and setting an upper and lower bound for the function. Table 42 summarizes the constraint equations used for simulation of the basic cycle. If the constraint is unbounded in one direction, a value of the order of 1030 is specified. In GRG2, the objective function is also specified among the constraint functions. The program treats the objective function as unbounded. A brief discussion of the constraints specified in Table 42 follows. A constraint was used to make sure that the jetpump compression ratio is greater than one to ensure that there will be cooling produced. The radiator temperature has to be higher than the environmental sink temperature to ensure that heat can be rejected in the radiator. The evaporator temperature also has to be higher than the environmental sink temperature; otherwise the SMR cannot be used as the figure of merit. The reason is that if the evaporator temperature is lower than the sink temperature then a passive radiator cannot be used for cooling, and since the SMR is the ratio of the overall SITMAP system mass to that of an ideal passive radiator with the same cooling capacity, then if a passive radiator is not a viable option for cooling then SMR cannot be a viable expression for measuring the cycle performance from a mass standpoint. The solar collector efficiency has to be between 0 and 1, this constraint is just to ensure that there are no unrealistic values for the heat input or the other solar collector parameters such as the concentration ratio. Another constraint is used to ensure the right direction of heat transfer in the recuperator. The next constraint ensures that there is positive work output from the turbine. The last constraint ensures that the objective function (SMR) is positive. Table 42. Constraints used in the optimization Lower Upper Constraint DescriptionLower Upper escripioLimit Limit Pipe/Psi > I Jetpump compression ratio has to be higher than 1 1E+30 Pjpe/Psi > 1 1 1E+30 unity. Trad/Ts> 1 Radiator temperature must be higher than the sink 1 1E+30 Trad/Ts > 1 1 1E+30 temperature. STevap/T > I Evaporator temperature has to be higher than the Tvap/T sink temperature. 1 1 0 < col < 0.99 Collector efficiency has to be lower than 0.99 0 0.99 Ahrecup > 0 Recuperator has to have positive heat gain 0 1E+30 St/ 1 Pressure ratio across the turbine has to be lower 0 < Pte/Pti < 1 0 1 than unity. Objective System Mass ratio 0 1E+30 CHAPTER 5 CODE VALIDATION Jetpump Results In order to validate the JETSIT simulation code, results are compared to the literature using singlephase models. Addy and Dutton [2] studied constantarea ejectors assuming ideal gas behavior of the working fluid. Changes were made to the working fluid properties subroutine in the JETSIT simulation code to include an ideal gas model instead of using REFPROP subroutines. The ejector configuration that Addy and Dutton studied and for which the comparison was made is presented in Table 51. Figure 51 and Figure 52 show the results from the JETSIT code and those of Addy and Dutton, respectively. It should be noted that Addy and Dutton define the entrainment ratio as the ratio of the primary mass flow rate to that of the secondary, which is the inverse of the entrainment ratio, 4, used in this study. Comparing results shown in Figure 51 and Figure 52 it can be seen that the JETSIT code gave the exact same breakoff mass flow results presented by Addy and Dutton. Figure 53 shows the compression characteristics at breakoff conditions. The region above the breakoff curves represents the "mixed regime" where the entrainment ratio is dependent on the back pressure, while the region below the breakoff curves represent the "supersonic" and "saturated supersonic" regimes where the mass flow is independent of the backpressure. The bold lines in Figure 53 show the same entrainment ratio values at breakoff conditions shown in Figure 51, but were included in Figure 53 for ease of comparison with the Addy and Dutton results shown in Figure 5 4, and Figure 55. Addy and Dutton show the breakoff mass flow rates in Figure 54, and Figure 55 below the vertical lines which match the values shown by the bold curves in Figure 53. The vertical lines under the breakoff curves in Addy and Dutton results are used to demonstrate the fact that the mass flow stays constant in the "supersonic" and "saturated supersonic" regimes, even if the backpressure drops. Comparing the results shown in Figure 53 to those in Figure 54, and Figure 55 it can be seen that the JETSIT code was able to duplicate the compression ratio results obtained by Addy and Dutton [2]. This gives confidence in the accuracy of the results generated in this study for the twophase ejector. It should also be noted that the jetpump results presented in this study will not be in perfect agreement with the reallife performance of such device because of the simplifying assumptions made in the model, such as the isentropic flow assumption in the all the jetpump nozzles. Also the accuracy of the results will be bound by the precision of the thermodynamic properties routines used (REFPROP 7). Table 51. Representative constantarea ejector configuration Variable Value Ys 1.405 yp 1.405 MWs / MWp 1 Tso / Tpo 1 Api/Am3 =1/(1+Ase/Ane) 0.25,0.333 Mpl = Mne 4 ilmp / mils = / 220 20 18 16 14 12 10 8 6 4 2 0 Mp = 4, Apl/A,3 = 0.33333 Mp = 4, Apl/Am3 = 0.25 I I I I I I I I P, /Pi PI SI Figure 51. Breakoff mass flow characteristics from the JETSIT simulation code. P /Pso Pc F SO Figure 52. Breakoff mass flow characteristics from Addy and Dutton [2]. 12  Mpi = 4, Ap/Am3 = 0.33333 Mpi =4, Ap/Am3 = 0.25 Mp = 4, Apl/Am3 = 0.33333  M =4, ApIAm3 = 0.25 100 pi 200 *I i SI rI I // 0 300 Figure 53. Breakoff compression and mass flow characteristics. 0o To 200 300 Ppo/Pso Figure 54. Breakoff compression and mass flow characteristics from Addy and Dutton [2], for Apl/Am3=0.25. Mp =4 APi/A3 = 0.25 YP = ,s = 1.4 Mws/MWP = 1.0 T /T 1 "MR", locus of breakoff points Tso/Tpo = 1.0 \ I I I I I I I I I MP = 4 Ap/AM3 = 0.333 7, = s =1.4 Mws/Mw, = 1 Tso/Tpo = 1 /,V SSSR   "SR" , t i I I I I 1  I i P,/Pso Figure 55. Breakoff compression and mass flow characteristics from Addy and Dutton [2], for Api/Am3=0.333. 10 6 0 4 2 i q CHAPTER 6 RESULTS AND DISCUSSION: COOLING AS THE ONLY OUTPUT A computer code was developed to exercise the thermodynamic simulation and optimization techniques developed in the chapters 3 and 4 for the SITMAP cycle. The code is called JetSit (short for Jetpump and SITMAP). The input parameters to the JETSIT simulation code are summarized in Table 61. The primary and secondary stagnation states can be defined by any two independent properties (P, x, h, s). For any given set of data presented in this study, the stagnation pressure ratio Ppo/Pso is varied by changing Ppo and not Pso. The reason is that for a given set of data the evaporator temperature needs to be fixed to simulate the jetpump performance at a given cooling load temperature. Parametric analysis was performed to study the effect of different parameters on the jetpump and SITMAP cycle performance. These parameters are the jetpump geometry given by two area ratios, the primary nozzle area ratio, Ant/Ane, and the primary to secondary area ratio at the mixing duct inlet, Ane/Ase, the primary to secondary stagnation pressure ratio, Ppo/Pso, quality of secondary flow entering the jetpump, evaporator temperature, quality of primary flow entering the jetpump, work rate produced (work rate is the amount of power produced per unit primary mass flow rate, in J/kg), as well as the environmental sink temperature, Ts. Following the parametric study, systemlevel optimization was performed, where the SITMAP system is optimized for given missions with the SMR as an objective function to be minimized. A specific system mission is defined by the cooling load temperature (evaporator temperature), Tevap or Tso, the environmental sink temperature, Ts, and the solar irradiance, G. The solar irradiance is fixed throughout this study at 1367.6 W/m2. Results in this chapter are confined to the case where the only output from the system is cooling. In the next chapter optimization results for the Modified System Mass Ratio (MSMR) will be presented where there will be both cooling and work output. Table 61. Input )arameters to the JETSIT cycle simulation code Variable name Description Pp Jetpump primary inlet stagnation pressure Xpo Jetpump primary inlet quality Ps Jetpump secondary inlet stagnation pressure Xso Jetpump secondary inlet quality Ant/Ane Primary nozzle area ratio Ratio of primary nozzle exit area to Ane/Ase the secondary nozzle exit area. Ts Environmental sink temperature. Jetpump Geometry Effects Figure 61 illustrates the effect of the jetpump geometry on the breakoff entrainment ratio. The jetpump geometry is defined by two area ratios. The first ratio is the primary nozzle throat to exit area ratio, Ant/Ane, and the second is the primary to secondary area ratio at the mixing chamber entrance, Ane/Ase. Figure 61 shows the variation of the breakoff entrainment ratio versus the stagnation pressure ratio for different jetpump geometries. It can be seen that lower primary nozzle area ratio, Ant/Ane, (i.e. higher Mne) allow more secondary flow entrainment. This is expected, since the entrainment mechanism is by viscous interaction between the secondary and primary streams. Therefore, faster primary flow should be able to entrain more secondary flow. The effect of the second area ratio, Ane/Ase is also illustrated in Figure 61. It can be seen that lower primary to secondary area ratios, Ane/Ase, allows for more entrainment. This trend is expected since a lower area ratio means that more area is available for the secondary flow relative to that available for the primary flow and thus more secondary flow can be entrained before choking takes place. It can be seen from Figure 63 that the jetpump geometry yielding the maximum entrainment ratio, also corresponds to the minimum SMR. The reason for that is that the maximum entrainment ratio corresponds to the minimum compression ratio, as can be seen in Figure 62, which in turn correspond to the minimum Qrad/Qcool, and Qsc/Qcool. 9 SAn/Ane,Ane/Ase = 0.25,0.1 An/Ane,Ane/Ase = 0.25,0.2 8 An/Ane,Ane/Ase = 0.25,0.3 An/An,An/A = 0.35,0.1 An/An,An/A = 0.35,0.2 7 An/Ane,Ane/As = 0.35,0.3 6 5  4 3 \ 2 5 10 15 20 25 30 pi/Psi Figure 61. Effect of jetpump geometry and stagnation pressure ratio on the breakoff entrainment ratio. An/Ane,Ane/Ase = 0.25,0.1 An/Ane,Ane/Ase = 0.25,0.2 An/Ane,Ane/Ase = 0.25,0.3  AAn,Ane/Ase = 0.35,0.1  AAn,Ane/Ae = 0.35,0.2  AAn,Ane/Ase = 0.35,0.3 5 10 15 20 25 30 pi/Psi Figure 62. Effect of jetpump geometry and stagnation pressure ratio on the compression ratio. An/Ane,Ane/Ase = 0.25,0.1 11 An/AneAne/Ase = 0.25,0.2 An/Ane,Ane/Ase = 0.25,0.3 AnA,Ane/Ase = 0.35,0.1 10 An/Ane,Ane/Ase = 0.35,0.2 An/Ane,Ane/Ase = 0.35,0.3 5 10 15 20 25 30 pi/Psi Figure 63. Effect of jetpump geometry and stagnation pressure ratio on the System Mass Ratio (SMR). The reason why these specific heat transfer ratios decrease with decreasing compression ratio can be explained using the Ts diagram in Figure 64. It should be noted that all the heat transfer are per unit primary flow rate and that is the reason why they are referred to as specific heat transfer. This figure shows three different constant pressure lines, Pa, Pb, and P,. If we let Pa be the evaporator pressure and consider two cases. The first case is when Pb is the radiator pressure (124'5'1), the second is when the compression ratio is higher and Pc is the radiator pressure (13451). Because of the fact that state 4 is always constrained to be saturated liquid, it can be seen that as the condenser pressure increases, the amount of heat rejected in the radiator also decreases (Q34 < Q24'), however, the amount of cooling decreases even faster (Q5 << Q5'). This causes the specific heat transfer ratios Qrad/Qcool, and Qsc/Qcool to go down, leading to lower values of the SMR. T c I 4  s Figure 64. Ts diagram for the refrigeration part of the SITMAP cycle. Stagnation Pressure Ratio Effect The SITMAP cycle parameters used to study the effect of the stagnation pressure ratio as well as the jetpump geometry effects on the cycle performance are presented in Table 62. As mentioned before the stagnation pressure ratio is varied by changing the primary inlet stagnation pressure, Ppo. The secondary stagnation pressure is kept fixed to simulate cycle performance at a fixed cooling load temperature. The stagnation pressure ratio was varied within the range 5 < Ppo/Pso < 25. The jetpump primary inlet thermodynamic state is fully defined by the degree of superheat as well as the pressure. The primary inlet superheat is fixed at 10 degrees for this simulation. The jetpump secondary inlet flow is always restricted to saturated vapor. The secondary flow parameters correspond to Tevap = 79.4 K. The jetpump geometry is defined by two area ratios, the first is Ant/Ane which is the primary nozzle throat to exit area ratio. The second area ratio is Ane/Ase, which is the ratio of the primary to secondary flow areas going into the mixing chamber. The environmental sink temperature, Ts, is kept at 0 K for this simulation. This is a typical value for deep space missions. The parameters that are fixed in this simulation will be varied later on to study their individual effect on the overall cycle performance. Table 62. SITMAP cycle parameters input to the JETSIT simulation code Variable name Description Ppo/Pso 5 < Ppo/Pso < 25 Xpo 10 degrees superheat Pso 128 kPa Xso 1.0 (Tevap = 79.4 K) Ant/Ane 0.25, 0.35 Ane/Ase 0.1, 0.2, 0.3 Ts 0 Figure 61 showed the effect of the jetpump geometry and stagnation pressure ratio on the breakoff entrainment ratio. It can be seen that the breakoff value of the entrainment ratio decreases with increasing stagnation pressure ratio. This should be expected because, since the secondary stagnation inlet pressure is fixed, a higher primary stagnation pressure corresponds to a higher backpressure. The higher backpressure has an adverse effect on the entrainment process allowing less secondary flow entrainment before choking occurs. Figure 62 and Figure 63 show the variation of the compression ratio and the SMR, respectively, with Ppi/Psi, for different jetpump geometries. The compression ratio and SMR are calculated at the breakoff entrainment ratio. Therefore all of these data points correspond to points on the abc (breakoff) curve in Figure 22. It can be seen in Figure 62 that as the ratio Ppi/Psi increases, the compression ratio increases as well, which is expected. However, the SMR increases with increasing compression ratios. Therefore, it is not advantageous from a mass standpoint to increase the stagnation pressure ratio. This can be explained by considering the other parameters that affect the SMR. Such parameters are shown in Figure 65 through Figure 68. Figure 65 through Figure 68 show the effect of stagnation pressure ratio and jet pump geometry on the following quantities: amount of specific heat rejected, radiator temperature, amount of specific heat input, and cooling capacity. As the stagnation pressure ratio increases all of the aforementioned quantities change in a way that should lead to a decrease in the value of SMR. All the heat exchange quantities decrease which leads to smaller heat exchangers, which in turn should lead to lower SMR. The radiator temperature, shown in Figure 66, increases with increasing stagnation pressure ratio as well, and this also leads to smaller radiator size that should also lead to lower SMR. However, as can be seen in Figure 63, the SMR behavior contradicts this expected trend. SMR increases with increasing Ppi/Psi. This is because of the fact that the SMR is a ratio of the mass of the SITMAP system to that of a passive radiator producing the same amount of cooling. Therefore, the amount of heat exchanged between the SITMAP system and its environment (Qrad, and Qs,) is not of relevance. The parameters that actually affect the SMR are the specific heat transfer rates normalized by the specific cooling capacity. Thus, even though Qrad and Qs, decrease, which causes Arad, and As, to decrease as well, SMR still increases because the cooling capacity, Qool, decreases faster which causes the size of the corresponding passive radiator to decrease at the same rate, yielding a lower SMR. This argument is evident in Figure 69, and Figure 610 that show an increase in the values of Qrad/Qcool, and Qsc/Qcool, respectively, with increasing stagnation pressure ratio, Ppi/Psi. An/AneAneA/Ase = 0.25,0.1 1.8E+06 An/AneAne/Ase = 0.25,0.2 An/Ane,Ane/Ase = 0.25,0.3 1.6E+06 An/AneAne/Ase = 0.35,0.1 1.6E+06 An/Ane,Ane/Ase = 0.35,0.2 An/AneAne/Ase = 0.35,0.3 1.4E+06 1.2E+06 21E+06 800000  600000 400000  200000  I I I I ' 5 10 15 20 25 30 PJPI pi /Psi Figure 65. Effect of jetpump geometry and stagnation pressure ratio on the amount of specific heat rejected. 85 84  83  82 , 81 5 10 15 piPsi Figure 66. Effect of jetpump geometry and temperature. stagnation pressure ratio on the radiator 220000 215000 210000 205000 S  a 200000 195000 190000 185000  piPsi Figure 67. Effect of jetpump geometry and stagnation pressure ratio on the amount of specific heat input. 10 15 20 25 30 1.8E+06 An/An,Ane/Ase = 0.25,0.1 An/Ane,Ane/Ase = 0.25,0.2 1 6E+06 An/AneAne/Ase = 0.25,0.3 S _ An/An,Ane/Ase = 0.35,0.1 AAn,Ane/Ae = 0.35,0.2 1.4E+06 AAne,Ane/Ase = 0.35,0.3 1.2E+06  S1E+06 0 0  U 0800000 \ 600000 \ 400000 200000 5 10 15 Pp/Psi Figure 68. Effect of jetpump geometry and cooling capacity. 12  20 25 30 stagnation pressure ratio on the specific An/Ane,Ane/Ase = 0.25,0.1 An/Ane,Ane/Ase = 0.25,0.2 An/Ane,Ane/Ae = 0.25,0.3 An/Ane,Ane/Ase = 0.35,0.1 An/Ane,Ane/Ae = 0.35,0.2 An/Ane,Ane/Ase = 0.35,0.3 pi/Psi Figure 69. Effect of jetpump geometry and stagnation pressure ratio on the cooling specific rejected heat. SAn/An,Ane/Ase = 0.25,0.1 14 Afn/An,Ane/Ase = 0.25,0.2 An/Ane,Ane/Ase = 0.25,0.3 AnAn,An/As = 0.35,0.1 An/AnA,Ane/Ase = 0.35,0.2 12 An/Ane,Ane/Ase = 0.35,0.3 pi /Psi Figure 610. Effect of jetpump geometry and stagnation pressure ratio on the cooling specific heat input. 8 8 An/Ane,Ane/Ase = 0.25,0.1 An/Ane,Ane/Ase = 0.25,0.2 S An/Ane,Ane/As = 0.25,0.3 \ An/An,Ane/Ase = 0.35,0.1 An/An,Ane/Ase = 0.35,0.2 An/Ane,Ane/Ase = 0.35,0.3 6 0 4 \ O 4 2 5 10 15 20 25 30 pi/Psi Figure 611. Effect of jetpump geometry and stagnation pressure ratio on the overall cycle efficiency. Figure 611 shows the overall efficiency of the SITMAP system. The overall efficiency is the ratio of specific cooling produced, Qool, to the required specific heat input, Qs, which is the inverse of the ratio presented in Figure 610. Thus it is expected that the overall efficiency would decrease with increasing stagnation pressure ratio. It should be noted that this definition of the overall efficiency assumes a work balance between the mechanical pump and the turbine. Figure 612 show an interesting trend for the ratio of overall cycle efficiency to that of a Carnot cycle, rT. It can be seen that there is a maximum for T at a given stagnation pressure ratio. This trend lends itself to optimization analysis if the overall cycle efficiency is the objective function to be maximized. However, in this study overall system mass is the objective since the SITMAP cycle is studied specifically for space applications. 0.11 0.1 0.09  0.08 / " o 0.07 0.06  0.05  An/Ane,Ane/Ase = 0.25,0.1 An/Anenene/Ase = 0.25,0.2 0.04 A nAneAne/Ase = 0.25,0.3 AAnAne/Ase = 0.35,0.1 An/Ane,Ane/Ase = 0.35,0.2 0.03  nAn,An,/Ase 0.35,0.3 I . 5 10 15 20 25 30 Ppi/Psi Figure 612. Effect of jetpump geometry and stagnation pressure ratio on the ratio of the overall cycle efficiency to the overall Camot efficiency. Secondary Flow Superheat Effect In all the results presented so far the jetpump secondary inlet (evaporator exit) is constrained to be saturated vapor (xsi=l) at the corresponding evaporator pressure. To study the effect of the degree of superheat of the secondary flow on the performance of the SITMAP cycle, the JETSIT simulation code was ran for different degrees of superheat in the secondary jetpump inlet with all the other parameters fixed. The complete configuration is presented in Table 63. Table 63. SITMAP cycle configuration to study the effect of secondary flow superheat Variable name Description Ppo 1.28 MPa (Ppo/Pso = 10) Xpo 10 degrees superheat Pso 128 kPa 0.5,1.0 (Tevap= 79.4 K) Xso 5, 10,and 15 degrees superheat Ant/Ane 0.25 Ane/Ase 0.1 Ts 0 Figure 614 show that the degree of superheat does not have a significant effect on the compression characteristics of the jetpump. However, increasing the degree of superheat increases the cooling capacity of the SITMAP cycle and improves the SITMAP cycle performance in terms of decreasing the amount of Qrad and Qs, per unit cooling load, as shown in Figure 615, and Figure 616, respectively. This causes the SMR to drop, as shown in Figure 613. Figure 617 shows the effect of the secondary flow superheat on the breakoff entrainment ratio. It can be seen that 4 decreases with increasing secondary flow superheat. This is due to the decrease in the secondary flow density at higher degrees of superheat. It should be noted that the amount of secondary superheat has more influence ifxsi 