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Analysis and optimization of a jet-pumped combined power/refrigeration cycle

University of Florida Institutional Repository
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PAGE 1

ANALYSIS AND OPTIMIZATION OF A JET-PUMPED COMBINED POWER/REFRIGERATION CYCLE By SHERIF M. KANDIL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Sherif M. Kandil

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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.

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iv 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 st udents to be self-starters, w ho work independently on their projects. I appreciate his pa tience and mentorship in area s within and beyond the realm of research and graduate sc hool. Dr. Sherif Ahmed Sherif was a terrific source of discussion, advice, encouragement, support a nd hard to find journal proceedings. Dr. Sherifs support made my year s 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 wo rk, for which I am grateful. Dr. Bruce Carroll has to be thanked fo r his advice on jet-pumps. Dr. Leon Lasdon from the University of Texas sent me the FORTRAN version of the GRG code and answered my questions very promptly. Mr s. Becky Hoover and Pam Simon have to be thanked for their help with all my administ rative problems during my time here and their constant reminders to finish up. I would like to particularly thank my fa mily 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 a nd new, too many to name individually, who have been great pals and c onfidants over the years.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 2 LITERATURE REVIEW.............................................................................................8 Related Work................................................................................................................8 Jet-pumps and Fabri Choking.......................................................................................9 Solar Collectors..........................................................................................................17 Solar Irradiance...................................................................................................18 Concentration Ratio.............................................................................................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 Cascaded Cycle Analogy.....................................................................................33 Use of the Different Efficiency Definitions........................................................36 3 MATHEMATICAL MODEL.....................................................................................38 Jet-pump Analysis......................................................................................................38 Primary Nozzle....................................................................................................39

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vi Flow Choking Analysis.......................................................................................40 Secondary Flow...................................................................................................45 Mixing Chamber..................................................................................................45 Diffuser................................................................................................................46 SITMAP Cycle Analysis............................................................................................47 Overall Analysis..................................................................................................48 Solar Collector Model.........................................................................................50 Two-phase region analysis...........................................................................51 Superheated region analysis.........................................................................51 Solar collector efficiency.............................................................................55 Radiator Model....................................................................................................55 System Mass Ratio.....................................................................................................55 4 CYCLE OPTIMIZATION..........................................................................................60 Optimization Method Background.............................................................................60 Search Termination.....................................................................................................63 Sensitivity Analysis....................................................................................................64 Application Notes.......................................................................................................64 Variable Limits...........................................................................................................66 Constraint Equations...................................................................................................67 5 CODE VALIDATION................................................................................................69 6 RESULTS AND DISCUSSI ON: COOLING AS THE ONLY OUTPUT.................74 Jet-pump Geometry Effects........................................................................................75 Stagnation Pressure Ratio Effect................................................................................78 Secondary Flow Superheat Effect..............................................................................86 Turbine Pressure Effect..............................................................................................89 Mixed Regime Analysis.............................................................................................91 Evaporator Temperature Effect..................................................................................95 Primary Flow Superheat Heat Effect..........................................................................98 Environmental Sink Temperature Effect..................................................................101 System Optimization................................................................................................103 7 RESULTS AND DISCUSSION: C OOLING AND WORK OUTPUTS.................111 Jet-pump Turbo-machinery Analogy........................................................................122 System Optimization for MSMR..............................................................................130 8 CONCLUSIONS......................................................................................................139 9 RECOMMENDATIONS..........................................................................................142 LIST OF REFERENCES.................................................................................................144 BIOGRAPHICAL SKETCH...........................................................................................148

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vii LIST OF TABLES Table page 2-1 Effect of the distance from the sun on solar irradiance............................................19 2-2 Properties of some selective surfaces.......................................................................23 2-3 Rankine cycle and vapor compression refr igeration cycle efficiency definitions....27 4-1 Optimization variab les and their limits....................................................................67 4-2 Constraints used in the optimization........................................................................68 5-1 Representative constant-a rea ejector configuration.................................................70 6-1 Input parameters to the JETSIT cycle simulation code............................................75 6-2 SITMAP cycle parameters input to the JETSIT simulation code............................79 6-3 SITMAP cycle configuration to study th e effect of secondary flow superheat.......86 6-4 SITMAP cycle configuration to study th e effect of the evaporator temperature, Tevap..........................................................................................................................95 6-5 SITMAP cycle configuration to study the primary flow superheat.........................98 6-6 Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K).............................109 7-1 Base case cycle parameters to study the MSMR behavior.....................................128 7-2 Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K), Pti = 14.2 MPa...138 7-3 Optimum Cycle parameters for Pso = 140 kPa (Tevap = 80.2 K), Pti = 14.2 MPa...138

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viii LIST OF FIGURES Figure page 1-1 Schematic of the Solar Integrated Thermal Management and Power (SITMAP) cycle.......................................................................................................................... .1 1-2 Schematic of the Solar Integrated Thermal Management and Power (SITMAP) cycle with regeneration..............................................................................................3 2-1 A schematic of the jet-pump geomet ry showing the different state points..............10 2-2 Three-dimensional ejector operating surf ace depicting the different flow regimes [2]............................................................................................................................ .13 2-3 Relationship between concentration ratio and temperature of the receiver [11]......20 2-4 A cyclic heat engine working between a hot and cold reservoir..............................28 2-5 The T-S diagram for a Lorenz cycle........................................................................32 2-6 Thermodynamic representation of (a) combined power/cooling cycle and (b) cascaded cycle..........................................................................................................34 3-1 Schematic for the jet-pump with constant area mixing............................................39 3-2 Schematic for the jet-pump with cons tant area mixing, showing the Fabri choked state s2......................................................................................................................42 3-3 Jet-pump schematic showing the control volume for the mixing chamber analysis.....................................................................................................................45 3-4 A schematic of the SITMAP cycle show ing the notation for the different state points........................................................................................................................48 3-5 Typical solar collector temperature profile..............................................................54 3-6 Overall system schematic for SMR analysis............................................................56 5-1 Break-off mass flow characteristics from the JETSIT simulation code...................71 5-2 Break-off mass flow characteris tics from Addy and Dutton [2]..............................71

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ix 5-3 Break-off compression and mass flow characteristics.............................................72 5-4 Break-off compression and mass flow ch aracteristics from Addy and Dutton [2], for Ap1/Am3=0.25......................................................................................................72 5-5 Break-off compression and mass flow ch aracteristics from Addy and Dutton [2], for Ap1/Am3=0.333....................................................................................................73 6-1 Effect of jet-pump geometry and stagnation pressure ratio on the breakoff entrainment ratio......................................................................................................76 6-2 Effect of jet-pump geometry and st agnation pressure ratio on the compression ratio.......................................................................................................................... .77 6-3 Effect of jet-pump geometry and st agnation pressure ratio on the System Mass Ratio (SMR).............................................................................................................77 6-4 T-s diagram for the refrigerat ion part of the SITMAP cycle...................................78 6-5 Effect of jet-pump geometry and st agnation pressure ratio on the amount of specific heat rejected................................................................................................81 6-6 Effect of jet-pump geometry and st agnation pressure ratio on the radiator temperature...............................................................................................................82 6-7 Effect of jet-pump geometry and st agnation pressure ratio on the amount of specific heat input.....................................................................................................82 6-8 Effect of jet-pump geometry and st agnation pressure ratio on the specific cooling capacity........................................................................................................83 6-9 Effect of jet-pump geometry and stagnation pressure ratio on the cooling specific rejected heat................................................................................................83 6-10 Effect of jet-pump geometry and stagnation pressure ratio on the cooling specific heat input.....................................................................................................84 6-11 Effect of jet-pump geometry and st agnation pressure ratio on the overall cycle efficiency..................................................................................................................84 6-12 Effect of jet-pump geometry and stag nation pressure ratio on the ratio of the overall cycle efficiency to th e overall Carnot efficiency.........................................85 6-13 Effect of secondary superheat on the overall system mass ratio (SMR)..................87 6-14 Effect of secondary superheat on the break-off compression ratio..........................87 6-15 Effect of secondary superheat on Qrad/Qcool.............................................................88

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x 6-16 Effect of secondary superheat on Qsc/Qcool...............................................................88 6-17 Effect of secondary superheat on th e break-off mass flow characteristics..............89 6-18 Effect of the turbine inlet pressure on the amount of net work rate and specific heat input to the SITMAP system............................................................................90 6-19 Effect of the turbine inlet pressu re on the amount of the SMR and overall efficiency of the SITMAP system............................................................................90 6-20 SMR and Compression ratio beha vior in the mixed regime....................................92 6-21 Effect of the entrainment ratio on th e mixed chamber exit conditions in the mixed regime............................................................................................................93 6-22 Effect of the entrainment ratio on sec ondary nozzle exit conditions in the mixed regime.......................................................................................................................93 6-23 Jet-pump compression behavi or in the mixed regime..............................................94 6-24 Effect of entrainment ratio on specific heat transfer ratios in the mixed regime.....94 6-25 Effect of the evaporator temperature on the break-off entrainment ratio and the compression ratio, for Ppo = 3.3 MPa.......................................................................96 6-26 Effect of the evaporator temperature on T, and SMR, for Ppo = 3.3 MPa...............97 6-27 Effect of the evaporator temperature on the cooling specific rejected specific heat, Qrad/Qcool, and the cooling specific heat input, Qsc/Qcool..................................97 6-28 Effect of the evaporator temperature on the effective radiator temperature, for Ppo = 3.3 MPa..........................................................................................................98 6-29 Effect of primary flow superheat on the SMR.........................................................99 6-30 Effect of primary flow superheat on the Qrad/Qcool.................................................100 6-31 Effect of primary flow superheat on the Qsc/Qcool..................................................100 6-32 Effect of primary flow s uperheat on the compression ratio...................................101 6-33 Sink temperature effect on SMR............................................................................102 6-34 Compression ratio effect on the SMR < 1 regime..................................................103 6-35 Effect of jet-pump geometry on the break-off sink temperature, for PpoPso=25, Pso=128 kPa, Tevap=79.4 K, 10 degrees primary superheat....................................104

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xi 6-36 Compression ratio and entrainment ratio variation with jetpump geometry, for Ppo/Pso=25...............................................................................................................105 6-37 Effect of stagnation pressure ratio on the break-off sink temperature (77.1).........106 6-38 Break-off sink temperature behavi or in the mixed regime (77.1)..........................107 6-39 Effect of jet-pump geometry on the SMR for Ppo/Pso=40, Tpo=150 K, Pso=128 kPa, Tevap=79.4 K, Ts = 78.4...................................................................................108 6-40 Effect of stagnation pressure ratio on the SMR.....................................................108 7-1 Schematic of a cooling and power combined cycle...............................................114 7-2 A schematic of the turbo-machinery analog of the jet-pump.................................122 7-3 T-s diagram illustrating the thermodyna mic states in the jet-pump turbomachinery analog...................................................................................................123 7-4 Effect of compression efficien cy on jet-pump characteristics...............................125 7-5 Effect of compression efficiency on MSMSR.......................................................126 7-6 Jet-pump efficiency effect on the co mpression ratio and MSMR for given jetpump inlet conditions.............................................................................................126 7-7 MSMR and SMR are equal for Wext = 0................................................................127 7-8 High pressure effect on the cooling speci fic heat input and ex ternal work output for a given jet-pump inlet conditions.....................................................................129 7-9 High pressure effect on the MSMR a nd efficiency for a given jet-pump inlet conditions...............................................................................................................129 7-10 Primary nozzle geometry effect on MSMR at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa.................................................................................................................131 7-11 Jet-pump geometry effect on MSMR at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa..........................................................................................................................131 7-12 Stagnation pressure ratio eff ect on MSMR at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa......................................132 7-13 Primary nozzle geometry effect on the compression ratio and the entrainment ratio at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa...................................................133 7-14 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

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xii 7-15 Primary nozzle geometry effect on th e specific heat input per unit specific cooling load at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa.......................................134 7-16 Jet-pump geometry effect on the compre ssion ratio and the entrainment ratio at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa.........................................................134 7-17 Jet-pump 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 7-18 Jet-pump geometry effect on the specif ic heat input per unit specific cooling load at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa.............................................135 7-19 Stagnation pressure ratio effect on the compression ratio and the entrainment ratio at a fixed jet-pump geometry (Ant/Ane=0.2, Ane/Ase=0.4)..............................136 7-20 Stagnation pressure ratio effect on the specific heat rejected per unit specific cooling load at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4)..................136 7-21 Stagnation pressure ratio effect on th e specific heat input per unit specific cooling load at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4)..................137

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xiii Abstract of Dissertation Pres ented 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 JET-PUMPED COMBINED POWER/REFRIGERATION CYCLE By Sherif M. Kandil May 2006 Chair: William Lear Cochair: S. A. Sherif Major Department: Mechanic al and Aerospace Engineering The objectives of this study were to an alyze and optimize a jet-pumped combined refrigeration/power system, and assess its f easibility, as a thermal-management system, for various space missions. A mission is herein defined by the cooling load temperature, environmental sink temperature, and solar irra diance which is a function of the distance and orientation relative to the sun. The cycl e is referred to as the Solar Integrated Thermal Management and Power (SITMAP) cycl e. The SITMAP cycle is essentially an integrated vapor compression cycle and a Ra nkine cycle with the compression device being a jet-pump instead of the conventional compressor. This study presents a detailed component analysis of the jet-pump, allowing for two-phase subsonic or supersonic flow, as we ll as an overall cycle analysis. The jetpump analysis is a comprehensive one-dime nsional flow model where conservation laws are applied and the various Fabri choking re gimes are taken into account. The objective of the overall cycle analysis is to calculate the various th ermodynamic state points within the cycle using appropriate conservation laws Optimization techni ques were developed

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xiv and applied to the overall cycle, with the overa ll system mass as the objective function to be minimized. The optimization technique utilizes a generalized reduced gradient algorithm. The overall system mass is evaluated fo r 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 sec ond is when the system is pr oducing 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 cooli ng only or both cooling and work). It was found that the active SITMAP syst em would only have an advantage over its passive counterpart when there is a small difference between the evaporator and sink temperatures. Typically, the minimum temp erature 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 jet-pump 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 mean s 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 ope rating cost per payload kilogram. It was also found that the work output did not ha ve a significant effect on the system performance from a mass point of view, becau se 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.

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1 CHAPTER 1 INTRODUCTION The increased interest in space exploration and the importance of a human presence in space motivate space power and thermal mana gement improvements. One of the most important aspects of the desi red enhancements is to have lightweight space power generation and thermal management capabilitie s. Onboard power generation adds weight to the space platform not only due to its inhe rent weight, but also due to the increased weight of the required therma l 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 co sts. The system is referred to as the Solar Integrated Thermal Management and Power system (SITMAP) [33]. Figure 1-1 shows the standard SITMAP system. Figure 1-1. Schematic of the Solar Inte grated Thermal Management and Power (SITMAP) cycle

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2 Figure 1-2 illustrates the operation of the SI TMAP system considered in this study. The cycle is essentially a combined vapor compression cycle and Rankine cycle with the compression device being a jet-pump instead of the conventional compressor. The jetpump has several advantages for space appl ications, as it involves no moving parts, which decreases the weight and vibration le vel while increasing the reliability. The power part of the SITMAP cycle is a Rankine cycle, which drives the system. The jetpump 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 pre ssure increase in the refrigeration cycle. High pressure superheated vapor is genera ted 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 recupera tor where it exchanges heat with the cold flow going into the solar colle ctor, thereby reducing the coll ector size and weight. After the recuperator the flow goes into the jetpump providing the high pr essure primary (or motive) stream. The primary stream draws low pressure secondary flow from the evaporator. The two streams mix in th e jet-pump where the secondary flow is compressed by mixing with the primary flow a nd the combined flow is ejected to the radiator where heat is rejected from th e 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 reduc tion in the expansion device, and the other stream is pressurized through the pump and th en goes into the rec uperator where it is

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3 heated up by exchanging heat w ith 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. Heat from: Solar Collector, Radioisotope Waste Heat, and/or Electronics Heat Rejection Expansion Valve Jet Pump Pump/Capillary Pump High-Pressure Vapor Liquid/Vapor Liquid/Vapor Radiator Liquid Liquid Liquid Liquid Turbine Recuperator Figure 1-2. Schematic of the Solar Inte grated Thermal Management and Power (SITMAP) cycle with regeneration The jet-pump, also referred to as an ejector in the literature, is the simplest flow induction device [24]. It exchanges energy and moment um by direct contact between a high-pressure, high-energy primary fluid a nd a relatively low-energy low-pressure secondary fluid to produce a discharge of inte rmediate pressure and energy level. The high-pressure 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 th e jet-pump to function as a compressor. In steady ejectors, momentum can be imparted from the primary fluid to the secondary fluid

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4 by the shear stresses at the tangential in terface between the primary and secondary streams as a result of turbulence and viscosity [24]. Ejector refrigeration has c ontinued to draw considerab le 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 characteristic s there are many applications where ejector refrigeration is used, such as cooling of buildings, automotive air-conditioning, solar powered ejector air-conditioning, and industrial process cooling. However, despite the abovementioned st rong points, conventional steady-flow 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 appl ications where low cost energy from steam, solar energy, or waste heat sources is av ailable, and where large condensers can be accommodated. However, if major improvements in the jet-pump (ejector) efficiency can be attained, significant improvement in the COPs of such systems will be realized and jet-pumped refrigeration systems will pr esent strong competition to conventional vapor compression systems. Alternatives to the SITMAP system for space applications can be either other active systems such as cryo-coolers or passive systems such as a radiator. Conventional

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5 cryo-coolers are generally bulky, heavy, and induce high vibration levels. Passive radiators have to operate at a temperature lo wer than the cooling load temperature which causes the radiator to be larg er and thus heavier. The pr oposed system eliminates some moving parts, which decreases the vibrat ion level and enhances reliability. A major contribution of this study is the detailed analysis of the two-phase jetpump. All previous work in th e literature is limited to jet-pu mps with a perfect gas as the working fluid. Flow choking phenomena are al so accounted for, as discussed in Fabri and Siestrunk [18], Dutton and Carroll [12], and Addy et al. [2]. The SITMAP cycle performance is evaluate d 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 th e 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 coo ling capacity. In the second case the system performance is evaluated using a more gene ral form of the aforementioned figure of merit, referred to as the Modified Syst em 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 techni ques developed in this study are general and applicable for any working fluid. Ho wever, 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 cryogeni c nitrogen is that it can be used as a

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6 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 optimi ze 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 Jet-pump and SITMAP). The code includ es the jet-pump two-phase one-dimensional flow model, as well as the SITMAP cy cle, and SMR analyses. A thermodynamic properties subroutine was incorporated in the code to dynamically calc ulate the properties of the working fluid instead of using a data file which can limit the range of simulation parameters. The thermodynamic properties so ftware used is called REFPROP and is developed by the National Institute fo r Standards and Technology (NIST). A commercially available optimization program was incorporated in the JetSit cycle simulation code. The optimization routin e 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 non-lin ear function. The cycle optimi zation method chosen for the analysis of this cycle is a search method. S earch methods are used to refer to a general class of optimization methods that search within a domain to arrive at the optimum solution.

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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 me thod is presented later in this study.

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8 CHAPTER 2 LITERATURE REVIEW Related Work The work presented in this study is a con tinuation 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 investigat ed in this study fo r onboard spacecraft applications. Nord et al. [33] used Refrigerant 134-a 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 a pplications. This allows the SITMAP cycle to be solely driven by solar thermal input. They did not consider the choked regimes in their jet-pump analysis, because their analysis only involved constant-pressure mixing in the jet-pump. 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. Fre udenberg et al. estimated the range of each parameter for a typical thermally actuated co oling system operating in space. They

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9 investigated the effect of varying each of the parameters within the estimated range, comparing their analysis to a base model ba sed on the average value of each of the ranges. Many systems dealing with power a nd thermal management have been proposed for which this analysis can be used, in cluding absorption cooling systems and solarpowered vapor jet refrigeration systems. Jet-pumps and Fabri Choking The Fabri choking phenomenon was first analyzed by Fabri and Siestrunk [18] in the study of supersonic air ejectors. They di vided the operation of the supersonic ejector into three regimes, namely, the supersonic re gime (SR), the saturated supersonic regime (SSR), and the mixed regime (MR). The s upersonic 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 indica ted by the dotted line in Figure 2-1. 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 ei ther at se or s2, the entrainment ratio becomes independent of the backpressure dow nstream. 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 investigati on to verify the various flow regimes. They generated various performance curv es relating the entrainment ratio, the compression ratio, and the ratio of the primar y flow stagnation pressure to the exit pressure (Ppi / Pde). Fabri and Paulon went on to di scuss the optimum jet ejector design,

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10 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 sec ondary mass flow rate for a given secondary pressure and a given primary pressure. Figure 2-1. A schematic of the jet-pump ge ometry showing the different state points. Addy et al. [2] studied supersonic ejectors a nd the regimes defined by Fabri and Siestrunk [18]. They wrote computer codes an alyzing constant-area and constantpressure ejectors. Their flow model wa s one-dimensional and assumed perfect gas behavior. They also conducted an experi mental study to which they compared their analytical results. Addy et al concluded that the constant-a rea ejector model predicts the operational characteristics of ejector systems mo re realistically than the constant-pressure model. They introduced a three-dimens ional 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 2-2. Figure 2-2 depicts a three-dimensional ejector solution surface. It should be noted that in Figure 2-2 the ejector geometry, and the primary to secondary stagnation temperature ratio are fixed. The surfaces show all the different flow regimes. Addy et al.

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11 also presented the details of the break-o ff conditions for transition from one operating regime to another. The po ssible transitions are between: The saturated supersonic and supersonic regimes, break-off curve b-d. The saturated supersonic and mixed regimes, break-ff curve b-c. The supersonic and mixed regimes, break-off curve a-b. In both the SR and SSR regimes the ma ss flow ratio (entrainment ratio), s pWW, is independent of the backpressure ratio, 3 mosPP so that these two surfaces are perpendicular to the s pWW 3 mosPP 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, 11psPP, the primary stream will expand fo rming 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, 11 s pPP, 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 ca se 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, al so independent of the backpressure. In the MR regime, however, the backpressure is high enough that the secondary flow remains

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12 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 stagna tion pressure ratio, oposPP in Figure 2-2. As 3mosPP is increased from zero, s pWW remains constant until break-off curve a-b-c, which separates the backpressure-independent from the backpressure-dependent regimes, is reache d. From here, a slight increase in the 3mosPP causes a significant drop in s pWW Hence, the points along break-off curve a-b-c are of particular importance since they represent the highest values of 3mosPP for which s pWW remains fixed. For this reason, it is adva ntageous to design ejectors to operate in a back-pressure independent regime at or near this break-off curve. The criterion for determining each tran sition was based on the pressure ratio Pse/Pne, and the Mach number at the minimum throat ar ea, either at se, or s2. If the Mach number at the minimum throat area was unit y, 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 break-off 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.

PAGE 27

13 Figure 2-2. Three-dimensional ejector opera ting surface depicting the different flow regimes [2]. Dutton and Carroll [12] discussed another importa nt 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 entrainm ent ratio to be independent of the backpressure. In their analysis they coul d 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 till they obtained a solution and that was at Mme = 1. This led them to the conclusi on 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 one-dimensional c onstant area flow model for optimizing a large class of supe rsonic ejectors utilizing perfec t gases as a working fluid. Given the primary and secondary gases and th eir temperatures, the scheme determines the values of the design parameters Mne, and Ane/Ame, which optimize one of the

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14 performance variables, entrainment ratio,, compression ratio, Pme/Psi, or Ppi/Psi given the value of the other two. Al-Ansary and Jeter [3] conducted a computational fluid dynamics (CFD) study of single phase ejectors utilizing an ideal gas as a working fluid. Th eir work studied the complex flow patterns within an ejector. CFD analysis was used to explain the changes in secondary flow rate with the primary in let 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 turbulen ce model used. Al-Ansary and Jeter [3] also showed that the mechanism by which the mi xed flow compresses at the exit of the mixing chamber, me is not the widely us ed one-dimensional normal shock. They found that compression occurs through a seri es of oblique shocks induced by boundary layer separation in the diffuser. Al-Ansary and Jeter [3] also conducted an experime ntal study to investigate the effect of injecting fine droplets of a nonvol atile 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 two-phase concept needs further exploration. Eames [14] conducted a theoretical study into a new method for designing jetpumps used in jet-pump cycle refrigerators. The method assumes a constant rate of momentum change (CRMC) within the mi xing section, which in this case is a converging-diverging diffuser. The temper ature and pressure were calculated as a function of the axial distance in the diffuse r, and then a function was derived for the geometry of the diffuser that removes the thermodynamic shock process by allowing the

PAGE 29

15 momentum of the flow to change at a c onstant 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 associat ed with the shock process encountered in conventional diffusers. They concluded that diffusers designed using the CRM C method yield a 50% increase in the compression ratio than a c onventional jet-pump 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 quantif y 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 moment um transfer devices rather than fluid moving devices. Since the mome ntum transfer mechan ism in ejectors is inherently dissipative in nature (shear forces instead of pres sure forces), there is no ideal process to compare the ejec tor performance to. Unlike turbomachinery, which can perform ideally in an isentropic process. Ro an developed a correcti on factor defined as 2rate of momentum rate of kinetic energy 2mmV K mV (2.1) 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 excha nge effectiveness expressions. It was found that in almost all evaluations, the design point value of SMEE ranged between 0.1-0.3.

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16 However, SMEE was not found constant for a wide range of off-design performance, especially for large change s in the secondary flow. Earlier work done on two-phase ejectors in the University of Florida includes Lear et al. [29], and Sherif et al. [38]. These two studies deve loped a one-dimensional model for two-phase ejectors with constant-pre ssure mixing. The primary and secondary streams had the same chemical composition, while the primary str eam was in the twophase regime and the secondary flow was either saturated or sub-cooled 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 Rankine-Hugoniot relations for two-phase 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 fi xed geometry. Qualitative agreement was found with single-phase ejector performance. Parker et al. [34] work is considered the most rele vant work in the literature to the ejector work presented in this study. They an alyzed the flow in two-phase ejectors with constant-area mixing. They confined their analysis to the mixed regime where the entrainment ratio,, 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 in let conditions and the geometry of the ejector, and varying the entrainment ratio versus th e compression ratio showed the first trend. Since all the data are in the mixed regime. The expected trend of decreasing compression

PAGE 31

17 ratio with increasing entrainment ratio was obs erved. They investigated this trend for various primary to secondary nozzles exit area ratios (Ase/Ane, see Figure 2-2). An interesting observation was found; that low Ase/Ane is desired when is low. As 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 For low values of the highest compression ratio occurs at the lowest ar ea ratio. For the higher values of there are maximum compression ratios. When the va lue of the optimum compression ratio was plotted against the entrainment ratio, the re lationship was found to be linear, which simplifies the design proce dure. Parker et al. [34] did not mention the working fluid used in their study. Solar Collectors For many applications it is desirable to de liver energy at temperatures possible with flat-plate collectors. Ener gy delivery temperatures can be increased by decreasing the area from which heat losses occur. This is done by using an optical device (concentrator) between the source of radiation and the ener gy-absorbing surface. The smaller absorber will have smaller heat losses compared to a flat-plate 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 ra tios (concentration ratio definition is presented later in this section) from low valu es close to unity to high values of the order of 105. Increasing concentration ratios mean in creasing temperatures at which energy can

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18 be delivered and increasing requirements for precision in optical quality and positioning of the optical system. Thus cost of deliver ed energy from a concentrating collector is a function of the temperature at which it is available. At th e 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 concentr ating solar collector are functions of the temperature the heat is transferred at, it is im portant to come up with a simple model that relates the solar collector efficiency to its te mperature profile. Such a model is presented in details later in this section. The mode l assumes an uncovered cylindrical absorbing tube used as a receiver with a linear con centrator. Since the SITMAP cycle is primarily for space applications, the only form of heat tran sfer considered in the model is radiation. The model assumes one-dimensional temperature gradient along the fl ow direction (i.e. no temperature gradients around the circumferen ce 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 2-1shows 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 sola r irradiance, as expected. The distance from the sun is in Astronomical Units AU. One AU is the average dist ance between the earth and the sun, and it is about 150 million Km or 93 million miles [11].

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19 Table 2-1. 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 th is 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 rA CR A (2.2) The concentration ratio has an uppe r limit that depends on whether the concentration is a three-dimensional (circula r) concentrator or tw o-dimensional (linear) concentrators. Concentrators can be divided into two categories: non-imaging and imaging. Nonimaging concentrators do not produce clearly de fined images of the sun on the absorber. However, they distribute the ra diation from all parts of the so lar disc onto all parts of the absorber. The concentration ratios of linea r non-imaging 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 2-3 from Duffe and Beckman [11], shows practical

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20 ranges of concentration ratios and types of optical systems needed to deliver energy at various temperatures. The lower limit curve re presents 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 co rresponds to collection efficiencies of 40-60% and represents a probable range of operation. Figure 2-3 also shows approximate ranges in which several types of reflectors might be used. Figure 2-3. Relationship between concentrati on ratio and temperature of the receiver [11]. It should noted that Figure 2-3 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 con centrator technologies. These technologies

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21 included inflatable structures, light weight primary concentrators, and high efficiency secondary concentrators. Secondary concen trators 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 e fficiency. Mason [32] also indicated that the use of s econdary concentrators also eas es the pointing and surface accuracy requirements of the primary concentrator, making the inflatable structure a more feasible option. Typical seconda ry concentrators are hollow, reflective parabolic cones. Recent studies at Glenn Research Center ha ve also investigated the use of a solid, crystalline refractive secondary concentrat or for solar thermal propulsion which may provide considerable improvement in effici ency by eliminating reflective losses. Mason [32] reported that the Earth Concentra tion ratio of the parabolic, thin-film inflatable primary concentrator is 1600. The Earth Concentration ratio is defined as the concentration ratio as required at 1 Astronom ical Unit (AU). An Astronomical Unit is approximately the mean distance between the Ea rth 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 c onversion device depends on the absorbing surface and its optical and thermal characterist ics. The efficiency can be increased by increasing the absorbed solar energy ( close to unity) and by decreasing the thermal losses. Surfaces/coatings having selective response to the solar spectrum are called selective surfaces/coatings. Su ch surfaces offer a cost effective way to increase the efficiency of solar collectors by providing high solar absorptance () in the visible and

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22 near infrared spectrum (0.3-2.5 m) and low emittance () in the infrared spectrum at higher wavelengths, to reduce therma l losses due to radiation. Materials that behave optimally for solar heat conversion do not exist in nature. Virtually all black materials have high sola r absorptance and also have high infrared emittance. Thus it is necessary to manufact ure 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 th e coating must remain stable under longterm 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 semi conductors and reflectors or dielectrics and metals are made in order to combine two disc rete layers to obtain the desired optical effect. Another method is the use of wa velength discriminating materials by physical surface roughness to produce the desired in th e visible and infrared. This could be by deliberately making a surface rough, which is a mi rror for the infrared (high reflectivity). Such surfaces (example: CuO) are deposit ed on metal substrates to enhance the

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23 selectivity. Table 2-2 gives the properties of few selective surfaces [8]. Effective selective surfaces have solar absorptiviti es around 0.95 and emissivities at about 0.1. Table 2-2. Properties of some selective surfaces. Material Short-wave Absorptivity Long-wave emissivity Black Nickel on Nickel-plated steel 0.95 0.07 Black Chrome on Nickel-plated steel0.95 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 low-pressure lowtemperature cooling cycle for comfort air-conditioning. The cy cle is driven solely by sola r energy, and it utilizes a jetpump as the compression device, with steam as the working fluid. The cycle has no mechanical moving parts as it utilizes potenti al energy to create th e pressure difference between the solar collector pressure and the condenser pressure, by elevating the condenser above the solar collector. In their steam-jet ejector analysis, Khattab et al. [25] used a primary convergingdiverging nozzle to expand the motive steam (primary flow) a nd accelerate it to supersonic speed, which then entrains the va por coming from the evaporator. Constant pressure mixing was assumed in the mixing regi on. They also neglected the velocity of the entrained secondary flow in their mome ntum 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 co mpression shock wave. Khattab et al. wrote a simulation program that studied the performance of the steam-jet cooling cycle under different design and opera ting conditions, and constructed a set of

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24 design charts for the cycle as well as the ej ector geometry. The i nputs to the simulation program were the solar gene rator and evaporator temperatures and the condenser saturation temperature. Dorantes and Estrada [10] presented a mathematical simulation for the a solar ejector-compression refrigeration system, used as an ice make r, with a capacity of 100 kg of ice/day. They took into c onsideration the varia tion of the solar co llector efficiency with climate, which in turn affects the system efficiency. Freon R142-b 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 varia tion 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 syst em 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 single-phase flow (superheated refrigerant) going into the ejector from both streams. Tamm et al. [41,42] performed theoretical and ex perimental studies, respectively, on a combined absorption refrigeration/Ranki ne power cycle. A binary ammonia-water 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 poten tial 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

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25 expected trends for vapor generation and ab sorption condensation processes, as well as potential for combined turbine work and refr igeration output. Furt her 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 ex ergy 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 highe r exergy efficiency, as is the case with 1st law efficiency. The largest exergy destructi on occurs in the absorb er, 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 effi ciency as an objective function for a given sensible heat source and a fixed ambient te mperature. The objective function depended on eight free variables, namely, the absorber temperature, boiler te mperature, rectifier temperature, super-heater temperature, inle t 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 su ch 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 f act that there are two different simultaneous outputs, namely power and refrigeration. An efficiency expression has to appropriately weigh the cooling component in order to al low comparison of this cycle with other cycles. This section presents several expres sions from the literature for the first law,

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26 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 = Usef ul 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 si mply as efficiency, in the case of power cycles, and as a coefficient of pe rformance for refrigeration cycles. Table 2-3 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 irre versibilities in devices. Two examples are given in Table 2-3 where Ec is the change in exergy of the cooled medium. out exergy inE E (2.4) Resource utilization efficiency [9] is a special case of th e exergy efficiency that is more suitable for use in some cases. Consider for instance a geothermal power cycle,

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27 where the geofluid is reinjected into th e ground after transferri ng heat to the cycle working fluid. In this case, the unextracted avai lability of the geofluid that is lost on Table 2-3. Rankine cycle and vapor comp ression refrigeration cycle efficiency definitions. Cycle type Rankine Vapor compression First Law InetHWQ cinCOPQW Exergy exergynetinWE exergycinEW Second law IIrev IIrevCOPCOP reinjection has to be accounted for. Theref ore, a modified definition of the form out R hsE E (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 ra tio of useful exergy to exergy loss in the process [1], useful exergy inusefulE i EE (2.6) Second law efficiency Second law efficiency is defined as the rati o of the efficiency of the cycle to the efficiency of a reversible cycle operating between the same thermodynamic conditions. IIrev (2.7) The reversible cycle efficiency is the fi rst law efficiency or COP depending on the cycle being considered. The s econd law efficiency of a refr igeration cycle (defined in terms of a COP ratio) is also called the thermal efficiency of refrigeration [5]. For constant temperature heat addition and rejec tion conditions, the reversible cycle is the Carnot cycle. On the other hand for sensible heat addition and rejec tion, the Lorenz cycle is the applicable reversible cycle [30].

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28 The exergy efficiency and second law efficien cy are often similar or even identical. For example, in a cycle operating between a hot and a cold reservoir (see Figure 2-4), the exergy efficiency is 1net exergy hohW QTT (2.8) while the second law efficiency is 1net exergy hchW QTT (2.9) Where To is the ambient or the ground state temp erature. 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. Figure 2-4. A cyclic heat engine working between a hot and cold reservoir The Choice of Efficiency Definition The first law, exergy and second law effi ciency 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 temperat ure heat source (that has a much higher Th Tr Wne t Cyclic device

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29 availability), the first law efficiency will not distinguish between the performances of the two plants. Using an exergy or second law effi ciency though will show that one of these plants has higher losses than the other. Th e first law efficiency, though, is still a very useful measure of plant performance. For ex ample, 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 e fficiency or second law efficiency is an excellent choice when comparing energy c onversion 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 cy cle performance. The other goal is to compare this cycle with other en ergy conversion options. First law efficiency Following the pattern of first law efficien cy definitions given in the previous section, a simple definition for the first law efficiency would be netc I hWQ Q (2.10) Equation (2.10) overestimates the efficien cy of the cycle, by not attributing a quality to the refrigeration output. Using th is definition, in some cases, the first law efficiency of the novel cycle approaches Car not values or even exceeds them. Such a situation appears to violate the fact that the Carnot efficien cy specifies the upper limit of first law conversion efficien cies (the Carnot cycle is not the reversible cycle corresponding to the combined cycle; this is discussed la ter in this chapter). The confusion arises due to the addition of work and refrigeration in the output. Refrigeration

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30 output cannot be considered in an effici ency expression without accounting for its quality. To avoid this confusion, it may be be tter to use the definition of the first law efficiency given as netc I hWE Q (2.11) The term Ec represents the exergy associated w ith 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. A lternately, to account for irreversibilities of heat transfer in the refrigeration heat exch anger, the exergy change of the chilled fluid would be considered. ,,,,ccfincfoutocfincfoutEmhhTss (2.12) Rosen and Le [37] studied efficiency expressi ons for processes integrating combined heat and power and district cool ing. 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 wo rk needed to produce the cooling output. This results in refrigerati on output being weighted very poorly in relation to work. Exergy efficiency Following the definition of exergy effici ency described previously in Equation (2.13), the appropriate equation fo r 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 th e heat source, which is equivalent to the exergy input into the cycle.

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31 ,, netc exergy hsinhsoutWE EE (2.13) 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 r eversible cycle for use with variable temperature heat input and rejection. A T-s diagram of the cycle is shown in Figure 2-5. 34 121LorenzQ Q (2.14) If the heat input and rejection were written in terms of the heat s ource and heat rejection fluids, the efficiency would be given as: ,, ,,1hrhroutrin Lorenz hshsinhsoutmhh mhh (2.15) Knowing that processes 4-1 a nd 2-3 are isentropic, it is eas ily shown that in terms of specific entropies of the heat source and heat rejection fluids that ,, ,, hroutrin hs hr hsinhsoutss m m ss (2.16) The efficiency expression for the Lorenz cycle then reduces to ,,,, ,,,,/ 1 /hroutrinhroutrin Lorenz hsinhsouthsinhsouthhss hhss (2.17) This can also be written as

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32 1s hr Lorenz s hsT T (2.18) Here, the temperatures in the expression above are entropic average temperatures, of the form 21 21 shh T ss (2.19) Figure 2-5. The T-S diag ram for a Lorenz cycle For constant specific heat fluids, the entrop ic average temperature can be reduced to 21 21lnsTT T TT (2.20) The Lorenz efficiency can therefore be written in terms of temperatures as ,,,, ,,,,/ln/ 1 /ln/hrouthrinhroutrin Lorenz hsinhsouthsinhsoutTTTT TTTT (2.21) It is easily seen that if the heat transfer processes were isothermal, like in the Carnot cycle, the entropic average temperatures w ould reduce to the temperature of the heat reservoir, yielding the Carnot efficiency. Sim ilarly the COP of a Lorenz refrigerator can be shown to be T s 1 2 4 3

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33 s cf Lorenz ss hrcfT COP TT (2.22) 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 togeth er 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 combin ed power/refrigeration cycle. Figure 2-6 shows the analogy, with a dotted line around the compone nts in the cascaded cy cle representing a black box. One way to look at an ideal combined cycl e would be as two Lorenz cycle engines cascaded together (Figure 2-6b). 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 ha ve identical inlet and exit temperatures in both cases. The first law efficiency of the cascaded system, using a weight factor f for refrigeration is ,outcc Isys hWWfQ Q (2.23) The weight factor, f is a function of the th ermal boundary conditions. Therefore, the first law efficiency of the combined cycle can also be written as ,netc Isys hWfQ Q (2.24)

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34 (a) (b) Figure 2-6. Thermodynamic representation of (a) combined power/cooling cycle and (b) cascaded cycle The work and heat quantities in the cascad ed cycle can also be related using the efficiencies of the cascaded devices outhHEWQ (2.25) ccWQCOP (2.26) By specifying identical refrigeration to wo rk ratios (r) in the combined cycle and the corresponding reversible cascaded cycle as cnetrQW (2.27) and using Equation(2.23) and Equations(2.25-2.26), one can arrive at th e efficiency of the cascaded system as ,1 1 1IsysHErf COP rCOP 2.28 assuming the cascaded cycle to be reversible the efficiency expression reduces to Th Wne t Cyclic device T r Tc Th Wne t HE Tc Tr REF Wou t Qh Qc

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35 ,1 1 1Lorenz IrevLorenz Lorenzrf COP rCOP (2.29) Here Lorenz 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 IIIIrev (2.30) If the new cycle and its equivalent reversib le cascaded cycle have identical heat input (Qh), the second law efficiency can also be written as ,,, netc I II IrevnetrevcrevWfQ WfQ (2.31) This reduces further to ,,1 1net I II IrevnetrevWfr Wfr (2.32) 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 versi on. 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 corr esponding reversible cycle efficiency would be, ,1 1IrevLorenz Lorenzr rCOP (2.33) 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

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36 respect to the work output. Being a second la w efficiency, the expres sion 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 a nd second law efficiencies have been recommended for the combined power and co oling cycle in Equations (2.11, 2.13 and 2.31) respectively. These definitions give thermodynamically consistent evaluations of cycle performance, but they ar e 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 di scussed 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 th en would be to compare the thermodynamic performance of the novel cycle wi th other options designed to meet the same load. If the performance of both cycles were evaluated us ing 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

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37 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 di vide the exergy of cooling by a reasonable second law efficiency of refriger ation (also called thermal efficiency of refrigeration). Such efficiencies are named effective efficiencies in this study. netcpractical Ieff hWQCOP Q (2.34) , netcIIref Ieff hWE Q (2.35) ,, netcpractical exergyeff hsinhsoutWQCOP EE (2.36) , ,, netcIIref exergyeff hsinhsoutWE EE (2.37)

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38 CHAPTER 3 MATHEMATICAL MODEL Jet-pump Analysis First, it should be noted that the inputs to the jet-pump model are: Fully defined stagnation state at the jet-pump primary inlet. Fully defined stagnation state at the jet-pump secondary inlet. Primary nozzle area ratio Ant/Ane. Secondary to primary area ratio, Ane/Ase. The outputs of the jet-pump model are: Break-off entrainment ratio. Mixed flow conditions at the jet-pump exit. The following general assumptions ar e made for the jet-pump analysis: Steady flow at all state points. Uniform flows at all state points. One-dimensional flow throughout the jet-pump. Negligible shear stresses at the jet-pump walls. Constant-area mixing,meneseAAA 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 3-1 shows a schematic of the jetpump. The high-pressure primary flow from the power part of the cycle (State pi) is expanded in a converging-diverging

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39 supersonic nozzle to supersonic speed. Due to viscous interaction secondary flow is entrained into the jet-pump. Constant-area mixing of the high velocity primary and the lower velocity secondary streams takes plac e in the mixing chamber. The mixed flow enters the diffuser where it is slowed down n early to stagnation conditions. The method for calculating the diffuser exit st ate and the entrainment ratio, given the jet-pump geometry and the primary and secondary stag nation states is presented next. For each region of the jet-pump flow-field conservation laws and process assumptions are used to develop a well posed mathematical model of the flow physics. Figure 3-1. Schematic for the je t-pump 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, V A A Vpi nt pi nt pi nt (3.1) The velocity at the nozzle throat is ca lculated using conservation of energy, pi nt ne si se se si de me

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40 V hh A Ant pint nt pi nt pi 1 2 12 1 2 (3.2) Mach number at the nozzle throat is calculate d using Equations (3.3), and (3.4). The s in Equation (3.3) signifies an isentropic process a s P (3.3) M V a (3.4) Pnt is iterated on until the Mach number is eq ual to unity at the primary nozzle throat. The properties at the nozzle exit are obt ained by assuming isentropic flow, sne=snt, and iterating on Pne. Conservation of energy is used to calculate the primary nozzle exit velocity as VhVhnentntne 2 1 22 1 2 (3.5) Ant/Ane is calculated using the continuity equation A A V Vnt ne ne nt ne nt (3.6) 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 jetpump. Either one of these mechanisms dictates the break-off value for the entrainment ratio for a given jet-pump configuration. Each mechanism correspond s to a different jet-

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41 pump operating regime. The first choking m echanism is referred to as inlet choking and it takes place when the jet-pump 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 c hoking and it takes plac e when the jet-pump is operating in the supersonic regime. In th is regime the secondary flow chokes at an aerodynamic throat inside the mixing chamber. For a given jet-pump geometry, there is a break-off 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 brea k-off (maximum) entrainment ratio, b o. The value of (Ppo/Pso)bo is represented by line bd in Figure 2-2, and b o is represented by the curve abc. (Ppo/Pso)bo affect the jet-pump operation as follows: popo soso b oPP PP b oinletchoke popo soso b oPP PP b ofabri The break-off 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 jet-pump geometry and stagnati on conditions at the primary inlet, the state (ne) at the primary nozzle exit can be defined using the proce dure presented in the previous section. Then (Pso)bo is the stagnation pressure corresponding to the conditions: Pse=Pne, and Mse=1.

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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 r eaches 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 seconda ry, therefore, the only place where the secondary can choke is at the in let to the mixing chamber. Figure 3-2 shows a schematic of the jet-pump. To calculate inlet choke corresponding to the saturated supers onic 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 give n stagnation pressure, Psi. Then inlet choke is then calculated from continuity as s esese inletchoke neneneVA VA (3.7) Figure 3-2. Schematic for the jet-pump w ith constant area mixing, showing the Fabri choked state s2. pi nt ne si se se si s2 s2 n2de me

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43 The following is a list of the general assumptions made in the Fabri choking analysis: The primary and secondary flows stay dis tinct and don not mix till sections (n2), and (s2), respectively. The primary and secondary flows are isen tropic between (se)-(s2), and (ne)-(n2), respectively. Ms2 = 1. The primary inlet static pressure is always larger than secondary inlet static pressure, Pne > Pse. The following analysis is used to calculate fabri corresponding to the supersonic regime. The momentum equation for the c ontrol volume shown by the dotted line in Figure 3-2 can be written as 222222 s esenenessnnpnsspnessePAPAPAPAmVmVmVmV (3.8) dividing by pm yields 2222221 s esenenessnnnneFabrisse pPAPAPAPAVVVV m (3.9) 22222 22 s esenenessnnnne Fabri pssessePAPAPAPAVV mVVVV (3.10) 22 22 2 2 2 nesnne senesn nne sesenese Fabri ne s se nenesse seAAAA PPPP VV AAAA A VV VVV A (3.11) The iteration scheme starts by guessing a value for s eP, knowing that s esiss that defines the state (se). From the energy equation 1 22sesiseVhh (3.12)

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44 Fabri can then be calculated as, s esese Fabri neneneVA VA (3.13) It should be noted that the area ratio neseAA is an input to the SITMAP code. Then a guess is made for 2 s P, and 2 s sess which defines state (s2). The velocity 2 s V can be obtained from the energy equa tion between (se) and (s2) 1 2 2 222 2se ssesV Vhh (3.14) Ps2 is iterated on till Ms2=1. The area ratio 2 s seAA is calculated from the continuity equation between (se) and (s2), 2 22 s sese s essAV AV (3.15) For constant-area mixing 22nesesnAAAA then 221nsesse neneseneAAAA AAAA (3.16) s eP is iterated on till the values for Fabri 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 encount ered in the analysis performed for this study.

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45 Secondary Flow When the jet-pump is operating in the mixed regime ( b o ), the following secondary flow analysis is used to calculate the Pse for the given conditions. Pse is iterated on assuming isentropic fl ow in the secondary nozzle (sse = ssi) till the following conservation equations are satisfied. VhVhsesisise 2 1 22 1 2 (3.17) A A V Vne se se ne se ne 1 (3.18) Pse iteration stops when Ane/Ase matches its input value. Th en the Mach number at the secondary exit is calculated us ing Equations (3.3) and (3.4). Mixing Chamber Figure 3-3. Jet-pump schematic showing th e 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 3-3 can be written as

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46 1menesenenesesepnepsepmePAAPAPAmVmVmV (3.19) given mAVpnenene, and the constant area mixing process,meneseAAA Equation 19 can be rearranged as 221nene nemesemenenesese sese me ne nene seAA PPPPVV AA V A V A (3.20) Then the enthalpy hme is calculated from the energy equation for the mixing chamber 2221111 1222menenesesemehhVhVV (3.21) Then from continuity 1mememe neneneAV AV (3.22) Pme is iterated on till the value of from Equation 3.22 matches its input value. Then the mixing chamber exit Mach number is calcul ated 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 shoc k occurs at the diffuse r inlet where the Mach number is closest to unity and, thus, the stagnation pressure loss over the shock is minimized. If Mme > 1, The pressure downstream of the shock, Pss, is iterated on till the following set of conservation equations acro ss the shock between (me) and (ss) is satisfied. memessssVV (3.23)

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47 PVPVmememessssss22 (3.24) hVhVmemessss1 2 1 222 (3.25) ssssssPh (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 V A A A A Vde me de me ne ne de me (3.27) VhVhdememede 2 1 22 1 2 (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 jet-pump analysis needed for the SITMAP cycle analysis is the jet-pump exit pressure, which correspond s to the radiator pre ssure in the SITMAP cycle. Figure 3-4 shows a schematic of the cycle wi th 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.

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48 Boiler Jetp ump ( re ) Pum p Turbine ( ti ) (p i ) ( si ) Recuperator Radiator ( jp e ) ( b i ) ( p e ) Eva p orator ( te ) ( ei ) Figure 3-4. A schematic of the SITMAP cy cle showing the notation for the different state points. The method used to achieve a converged so lution for the SITMAP cycle given the jet-pump inlet and exit states and entrainment ratio follows. Overall Analysis Knowing the pressure and assu ming that the condenser ex it state is saturated liquid (xre=0), this defines the radiator ex it state. Also the pressure at the evaporator inlet is the same as the jet-pump secondary inlet pressu re, and assuming iso-enthalpic expansion, hheire, this defines the evaporator inlet state (ei). So straight out of the jet-pump analysis all the states in the refrigerati on part of the SITMAP cycle are defined. System convergence requires a double-itera tive solution. The first step requires guessing the high pressure in the cycle, turbine inlet pressure, Pti, and the entropy at the

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49 same state, sti (or any other independent property lik e the enthalpy). Then the pump work can be calculated as W m PPpump p pumpre pere (3.29) Energy balance across the pump yields, perepumphhW (3.30) Now state (pe) is defined. The recuperator efficiency is assumed to be 0.7 and is defined as max H C recupQorQ Q (3.31a) where, maxQ(,) ptetepemhhPT (3.31b) Equation 3.31a, and 3.31b are combined yielding, (,) tepi recup tetepehh hhPT (3.32) (,) 1 pirecuptepe te recuphhPT h (3.33) The specific enthalpy from Equati on (3.33) and the fact that Pts = r Ppi can then be used to calculate an isentropic turbine exit state. From the definition of turbine efficiency, tite tsti thh hh (3.34) The entropy at the turbine inlet, sti is iterated on until the entropy at the turbine inlet state matches that of the is entropic turbine exit state. The turbine work is calculated as

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50 tptiteWmhh (3.35) Pti is iterated on (repeat the entire SITMAP analysis) until the net work,TpWW is positive. In other words the analysis stops when it finds the minimum turbine inlet pressure that yields positive net work, i.e. 0 tpWW. A converged solution has now been obtained for the SITMAP cycle. The following equations complete the analysis: Qmhhevappsiei (3.36) Qmhhradpdere 1 (3.37) Qmhhscptipe (3.38) recuppbipeptepiQmhhmhh (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 wo rk values are per unit primary flow rate and their units are [J/kg]. These values will be refe rred to during this study as heat rate or work rate. Solar Collector Model If the working fluid comes into the solar co llector as a two-phase mixture, part of the heat exchange in the collector will take place at a constant temperature equal to the saturation temperature, s atT at the collector pressure. The re st of the heat exchange in the collector will be in the superheated region wh ere 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. Th e first is the part operating in the two-phase

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51 region, and is denoted s atrA and the second part operates in the superheated region and is denoted SHrA The working fluid will always be a ssumed to be either in the two-phase region or in the superheated region coming into the solar co llector and never in the subcooled region. This assumption was found to be always true within the range of cycle parameters investigated in this study. The ma in 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 fl uid temperature at any given location in the solar collector. This assumption neglects th e 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. Two-phase region analysis An energy balance can be written for the po rtion of the solar collector operating in the two-phase region as follows 441()satsat s atirrsatsmhxhGCRAATT (3.40) The specific enthalpy difference in the ab ove equation is between the enthalpy of saturated vapor at the collector pressure a nd the enthalpy of the working fluid coming into the solar collector. The above equation can be solved for s atrA 441 ()satsati r satsmhxh A GCRTT (3.41) Superheated region analysis 44()(())SHprSmCdTGCRAWTxTdx (3.42)

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52 44()(())pSdT mCGCRWWTxT dx (3.43) Let 11SHSHSH x dTdT xdxdx LLdxLdx (3.44) 44()(())p S SHmC dT GCRWWTxT Ldx (3.45) Multiply through by SHL 44()(())SHSHprrSdT mCGCRAATxT dx (3.46) Now we non-dimensionlize the dependent va riable T dividing it by the evaporator temperature, we let eT T T *1edTdT T edTdT T dxdx 44* 4**()(())SHSHperresdT mCTGCRAATTxT dx (3.47) If we divide both sides by SHrpe A mCT and rearrange 443 **1() (())SHe s rpepT dTGCR TxT AdxmCTmC 44* 3 **() (())SHr e s pepdT A dx T GCR TxT mCTmC (3.48) This separable ordinary differential equation can be written in the form 4* *()SHrdT A dx abTx (3.49) Where

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53 43 *()e s pepT GCR aT mCTmC ; and 3 e pT b mC Integrating equation 3.49 for the limits *** s atoTTT ; and 01 x yields the expression below for the area of the superheated region of the collector,SHr A *1 4 1111 1** 4444 1 4 3 1 442tanlnln 4o SH s atT r TbT abTabT a A ba (3.50) This expression is obtained using the symbolic integration feature of Mathematica. The total solar collector area, rA is equal to the summation of the areas of the superheated region and that of the saturated (two-phase) region. SHsatrrrAAA (3.51) The ODE shown in Equation 3.48 can be so lved a second time for the temperature profile in the solar as function of the axia l distance for the calculated solar collector receiver area. To obtain the temperature profile the ODE is integrated between the following limits: *** satTTT ; and 0 x x Figure 3-5 shows a typical temperature profile in the solar collector.

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54 00.250.50.751 x/L 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 T* Pso=128kPa Tso=79.4K Ts=78.4 G=1300W/m2CR=100 =0.95 =0.1 Ar=1.31m2T* in=3.43 T* out=5.19 Figure 3-5. Typical solar collector temperature profile. To calculate an effective collector temper ature, an energy balance is performed on the solar collector as a whole similar to the energy balances performed on the two-phase and superheated regions of the solar collector. 44()rreffsmhGCRAATT (3.52) In the above equation the enthalpy difference, h is the overall enthalpy difference between the inlet and outlet of the solar collector. Solving the above equation for effT yields 1 4 4()r effs rGCRAmh TT A (3.53)

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55 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 availa ble is the product of the solar irradiation, G[W/m2], and the aperture area of the concentrator, Aa [m2]. amh GA (3.54) The aperture area can be calculated fro m the concentration ratio expression. arACRA (3.55) In this model the value of the concentration ratio will be assumed based on typical values for current technologies av ailable for deep space applications. Radiator Model Equation (3.56) represents the energy ba lance between the fluid and the radiator; the emissivity has been lumped in to an overall radiator efficiency, rad, dA m Tdhrad p rad radrad 4 (3.56) If superheat exists at the radiator inle t, Equation (3.56) must be numerically integrated to account for the changing temperat ure in the superheated region. For the rare case of either mixed or saturated vapor conditions at the jet-pump exit, Equation (3.56) can be analytically integrated, using the cons tant value of the saturation temperature at the radiator pressure. System Mass Ratio Figure 3-6 shows a schematic for the ther mally actuated heat pump system being considered. The power subsystem accepts heat from a high-temperature source and supplies the power needed by the refrigeration subsystem. Both systems reject heat via a

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56 radiator to a common heat si nk. The power cycle supplie s just enough power internally to maintain and operate the refrigeration loop However, in principle, the power cycle could provide power for other onboard syst ems 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 compressi on cycles for the cooling subsystem. Figure 3-6. 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, collect or, and a general syst em mass comprising the turbomachinery and piping present in an active system. This is shown mathematically by o rad sys rad colm m m m m,~ (3.57) Equation (3.57) can be separated and rewritte n in terms of collector and radiator areas W Qs Ts Power Cycle Refrigeration Cycle TH Te,res QH Qe

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57 o rad sys o rad rad col rad colm m A A A m, ,~ (3.58) The solar collector is modeled by exam ining the solar energy incident on its surface. This energy is proportional to the co llector efficiency, the cross-sectional 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 th e radiator and the environment is given below. For deep space applications, the en vironmental reservoir temperature may be neglected, but for near-planetary or so lar missions this may not be the case. 44 sradradsQATT (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 44 passiveradoeseQATTQ (3.60) Defining a new non-dimensional parameter, as rad col performing an overall energy balance on the active system yielding H esQQQ and substituting Equations (3.54), (3.59), and (3.60) into Equation (3.58), yields 4 444 4 44 ,1 11 1 11s s ys e seee H eecolsunradradrado ss radradT m T TTTT Q m QTGTTm TT TT (3.61)

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58 Substituting the following definitions HW Q; eQ COP W ; e H Q COP Q (3.62) C P ; C RCOP COP ; R P T (3.63) col rad CT T 1; e rad e CT T T COP ; sun col eG T 4 (3.64) e col colT T T *; e rad radT T T *; e s sT T T (3.65) into Equation (3.61) yields o rad sys s rad s s rad rad col T s rad colm m T T T T T T T T T T m, 4 4 4 4 4 * 4 * *1 1 1 1 1 ~ (3.66) Non-dimensionalizing the third term on the ri ght hand side of E quation (3.66) yields m m m m m m mact t sys o rad act t act t sys~, , (3.67) But sys col rad act tm m m m ,, therefore m m m T T T T T T T T T T mact t sys s rad s s rad rad col T s rad col~ 1 1 1 1 1 ~, 4 4 4 4 4 * 4 * (3.68) Defining act t sysm m, yields 4 4 4 4 4 * 4 * *1 1 1 1 1 1 ~s rad s s rad rad col T s rad colT T T T T T T T T T m (3.69) Equation (3.69) represents the SMR in term s 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 ar e quantities that can be computed for a given application. It should be noted that thr ee of the SMR parameters are dictated by the

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59 SITMAP cycle analysis; those parame ters are the collector temperature Tcol *, radiator temperature radT and the overall percentage Carnot efficiency T.

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60 CHAPTER 4 CYCLE OPTIMIZATION The combined cycle has been studied by a simple simulation model coupled to an optimization algorithm. The simulation model pr esented in the previous chapter is based on simple mass, energy, and momentum balan ces. The properties of the working fluid are dynamically calculated using a software called REFPROP made by the National Institute for Standards and Technology (N IST). The source code for REFPROP was integrated within the simulation code to al low 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 certa in 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 non-linear functions. Th e cycle optimization method chosen for the analysis of this cycle is a search method. S earch 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 init ial starting point in search schemes. The optimization algorithm picks a new point in th e neighborhood of th e initial point such that the objective function (the function being optimized) value improves without violating any constraints. A si mple method of determining the direction of change is to

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61 calculate the gradient of the objective function at the current point [38]. Such methods are also classified as steepest ascent (or des cent) methods, since the algorithm looks for the direction of maximum change. By repeating th ese steps until a term ination 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 sear ch might have to be modified such that the bounds are not violated. The Generalized Redu ced 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 GR G algorithm. A commercially available program called the LSGRG2 was used for SITM AP cycle optimization. LSGRG2 is able to handle more variables and constraints th an 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: ) (1X gm Subject to equality and inequality type constraints as given below 0 ) ( X gi neq i ,..., 1 (4. 1) 0()()igXubni m neq i ,....., 1 (4. 2) The variables are constrained by an upper and lower bounds.

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62 ) ( ) ( i ub X i lbi n i ,..., 1 (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, m n nX X ,...,1 The optimization program then becomes Minimize: ) (1X gm Subject to: 0 ) ( i n iX X g m i ,..., 1 (4. 4) ) ( ) ( i ub X i lbi m n i ,..., 1 (4. 5) 0 ) ( ) ( i ub i lb neq n n i ,..., 1 (4. 6) 0 ) ( i lb m n neq n i ,..., 1 (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 e quality constraints, while the variables are positive for the inequality constraints. The va riables 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 constr aints 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 n-nb natural variables and the nb sl ack variables associated with the binding constraints. These n variables are called the non-basic variables. The binding constraints can be written as

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63 0 ) ( x y g (4. 8) Here y and x are vectors of the nb basic and n non-basic variables respectively and g is a vector of the binding constraint functions. Th e binding constraints Equation (4.8) can be solved for y in terms of x, reducing the objective to a f unction of x only. ) ( ) ), ( (1x F x x y gm This equation is reasonably valid in the neighborhood of the current point to a simpler reduced problem. Minimize ) ( x F Subject to the variable limits for the components of the vector x. u x l (4. 9) The gradient of the reduced objective) ( x F ) ( x F is called the reduced gradient. Now the search direction can be determin ed 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 i iy g should be nonsingular at the current point. A more detailed descripti on 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 improve d feasible point cannot be found in a particular iteration. A well known test for optimality is by checking if the Kuhn-Tucker conditions are satisfied. Th e Kuhn-Tucker conditions ar e explained in detail in [15, and

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6435]. It can be mathematically explained in terms of the gradients of the objective functions and inequality constraints as: m j j j j mX g u X g1 10 ) ( ) ( (4. 10) 0ju, 0 ) ( ) ( j ub X g uj j (4. 11) ) ( ) (j ub X gj, m j,...., 1 (4. 12) Here, uj is a Lagrange Multiplier for the inequality constraints. Unfortunately, the Kuhn-Tucker conditions are valid only for strictly convex problems, a definition that most optimization problems do not satisf y. 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 gl obal optimum [15]. The procedure is to run the optimization program starting from several in itial 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 activ e constraints can be determined using the corresponding Lagrange multipliers. ) (j ub V uj (4. 13) 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 sche me, 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

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65 of the optimum returned by the program. Prio r 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 fi xed at any value close to their optimum. The relative scaling of the va riables affects the accuracy of the differentiation and the actual value of th e components in the grad ient, 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 proces s a lot more stable. This can be achieved by keeping all the variables in the optimizati on 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 criter ion, particularly for the Newton-Raphson method used during the one-dimensional search can cause premature termination of the optimization program. The accuracy of the num erical 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 accu racy of the objective function calculation and constraints were accurate enough. Same results were obtained using both forward and central difference gradient calculations. Special atten tion should be paid to make sure that the convergence criterion for the op timization process is no t more stringent than

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66 that of the objective function and constrai nts 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, pa rticularly 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 poin t 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 th e resulting point is indeed a global optimum. The optimization proc ess 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, li mits 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 LSGR G2 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 4-1 shows the upper and lower bounds of the variables used in the cycle optimization. Some of the bounds are ar bitrarily specified when a clear value was not available.

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67 Table 4-1. Optimization variables and their limits Variable Lower Limit Upper LimitName 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 th at 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 defi ning constraint func tions and setting an upper and lower bound for the function. Table 4-2 summarizes the constraint equations used for simulation of the basic cycle. If th e 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 4-2 follows. A constraint was used to make sure that the jet-pump co mpression ratio is greater than one to ensure that there will be cooling produced. The radi ator temperature has to be higher than the environmental sink temperature to ensure that heat can be rejected in the radiator. The evaporator temperature also ha s to be higher than the environmental sink temperature; otherwise the SMR cannot be used as the figur e of merit. The reason is that if the evaporator temperature is lower than the sink temperature then a passi ve 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

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68 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 th e other solar collector paramete rs such as the concentration ratio. Another constraint is used to ensure the right direction of heat transfer in the recuperator. The next constr aint ensures that there is positive work output from the turbine. The last constraint ensures th at the objective function (SMR) is positive. Table 4-2. Constraints us ed in the optimization Constraint Description Lower Limit Upper Limit Pjpe/Psi > 1 Jet-pump compression ratio has to be higher than unity. 1 1E+30 Trad/Ts > 1 Radiator temperature must be higher than the sink temperature. 1 1E+30 Tevap/Ts > 1 Evaporator temperature has to be higher than the sink temperature. 1 1E+30 0 < col < 0.99 Collector efficiency has to be lower than 0.99 0 0.99 hrecup > 0 Recuperator has to have positive heat gain 0 1E+30 0 < Pte/Pti < 1 Pressure ratio across the turbine has to be lower than unity. 0 1 Objective System Mass ratio 0 1E+30

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69 CHAPTER 5 CODE VALIDATION Jet-pump Results In order to validate the JETSIT simula tion code, results are compared to the literature using single-phase models. Addy and Dutton [2] studied constant-area ejectors assuming ideal gas behavior of the working fluid. Changes were made to the working fluid properties subroutine in the JETSIT si mulation code to include an ideal gas model instead of using REFPROP subroutines. The ejector configuration that Addy and Dutton studied and for which the compar ison was made is presented in Table 5-1. Figure 5-1 and Figure 5-2 show the results from the JETS IT code and those of Addy and Dutton, respectively. It should be not ed 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, used in this study. Comparing results shown in Figure 5-1 and Figure 5-2 it can be seen that the JETSIT code gave the exact same break-off mass flow results presented by Addy and Dutton. Figure 5-3 shows the compression characte ristics at break-off conditions. The region above the break-off curves represents the mixed regime where the entrainment ratio is dependent on the back pressure, while the region below the break-off curves represent the supersonic and saturated supersonic regimes where the mass flow is independent of the back-pressure. The bold lines in Figure 5-3 show the same entrainment ratio values at break-off conditions shown in Figure 5-1, but were included in Figure 5-3 for ease of comparison with the Addy and Dutton results shown in Figure 5-

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70 4, and Figure 5-5. Addy and Dutton show the break-off mass flow rates in Figure 5-4, and Figure 5-5 below the vertical lines which match the valu es shown by the bold curves in Figure 5-3. The vertical lines under the break-off curves in Addy and Dutton results are used to demonstrate the fact that the ma ss flow stays constant in the supersonic and saturated supersonic regimes, even if th e back-pressure drops. Comparing the results shown in Figure 5-3 to those in Figure 5-4, and Figure 5-5 it can be seen that the JETSIT code was able to duplicate the compressi on ratio results obtained by Addy and Dutton [2]. This gives confidence in the accuracy of the results ge nerated in this study for the two-phase ejector. It should also be noted that the jet-pump results presented in this study will not be in perfect agreement with the real-life performance of such device because of the simplifying assumptions made in the model, such as the isentropic flow assumption in the all the jet-pump nozzles. Al so the accuracy of the results will be bound by the precision of the thermodynamic pr operties routines us ed (REFPROP 7). Table 5-1. Representative cons tant-area ejector configuration Variable Value s 1.405 p 1.405 MWs / MWp 1 Tso / Tpo 1 p 1m3seneAA1/(1AA) 0.25,0.333 Mp1 = Mne 4 psm/m1/ 2 20

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71 0100200300 Ppi/Psi 0 2 4 6 8 10 12 14 16 18 20 1/ Mp1=4,Ap1/Am3=0.33333 Mp1=4,Ap1/Am3=0.25 Figure 5-1. Break-off mass fl ow characteristics from the JETSIT simulation code. Figure 5-2. Break-off mass flow char acteristics from Addy and Dutton [2].

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72 0100200300 Ppi/Psi 0 2 4 6 8 10 12 Pme/Psi 0 5 10 15 20 25 1/ Mp1=4,Ap1/Am3=0.33333 Mp1=4,Ap1/Am3=0.25 Mp1=4,Ap1/Am3=0.33333 Mp1=4,Ap1/Am3=0.25 Figure 5-3. Break-off compression and mass flow characteristics. Figure 5-4. Break-off compression and mass flow characteristics from Addy and Dutton [2], for Ap1/Am3=0.25.

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73 Figure 5-5. Break-off compression and mass flow characteristics from Addy and Dutton [2], for Ap1/Am3=0.333.

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74 CHAPTER 6 RESULTS AND DISCUSSION: COOL ING AS THE ONLY OUTPUT A computer code was deve loped to exercise the thermodynamic simulation and optimization techniques developed in the chap ters 3 and 4 for the SITMAP cycle. The code is called JetSit (short for Jet-pump and SITMAP). The input parameters to the JETSIT simulation code are summarized in Table 6-1. The primary and secondary stagnation states can be define d 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 jet-pump performance at a given cooling load temperature. Parametric analysis was performed to study the effect of differ ent parameters on the jet-pump and SITMAP cycle performance. Th ese parameters are the jet-pump 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 en tering the jet-pump, evaporator temperature, quality of primary flow enteri ng the jet-pump, work rate produced (work rate is the amount of power produced per unit pr imary mass flow rate, in J/kg), as well as the environmental sink temperature, Ts. Following the parametric study, system-lev el 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

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75 temperature (evaporator temperature), Tevap or Tso, the environmental sink temperature, Ts, and the solar irradiance, G. The solar irradiance is fixed th roughout 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 chapte r optimization results for the Modified System Mass Ratio (MSMR) will be presented where there will be both cooling and work output. Table 6-1. Input parameters to the JETSIT cycle simulation code Variable name Description Ppo Jet-pump primary inlet stagnation pressure xpo Jet-pump primary inlet quality Pso Jet-pump secondary inlet stagnation pressure xso Jet-pump secondary inlet quality Ant/Ane Primary nozzle area ratio Ane/Ase Ratio of primary nozzle exit area to the secondary nozzle exit area. Ts Environmental sink temperature. Jet-pump Geometry Effects Figure 6-1 illustrates the effect of the jet-pump geometry on the break-off entrainment ratio. The jet-pump 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 6-1 shows the variation of the break-off entrainment ratio versus the stagnation pressure ratio for different jet-pump geometries. It can be seen that lower primary nozzle area ratio, Ant/Ane, (i.e. higher Mne) allow more secondary flow entrai nment. This is expected, since the entrainment mechanism is by viscous inte raction between the secondary and primary streams. Therefore, faster primary flow s hould be able to entrain more secondary flow.

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76 The effect of the second area ratio, Ane/Ase is also illustrated in Figure 6-1. 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 rati o means that more area is available for the secondary flow relative to that available fo r the primary flow and thus more secondary flow can be entrained before choking takes place. It can be seen from Figure 6-3 that the jet-pump geometry yielding the maximum entrainment ratio, also corresponds to the mini mum SMR. The reason for that is that the maximum entrainment ratio corresponds to the minimum compression ratio, as can be seen in Figure 6-2, which in turn correspond to the minimum Qrad/Qcool, and Qsc/Qcool. 51015202530 Ppi/Psi 0 1 2 3 4 5 6 7 8 9 Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-1. Effect of jet-pump geometry and stagnation pressure ratio on the breakoff entrainment ratio.

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77 51015202530 Ppi/Psi 1 2 3 4 Pde/Psi Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-2. Effect of jet-pump geomet ry and stagnation pressure ratio on the compression ratio. 51015202530 Ppi/Psi 3 4 5 6 7 8 9 10 11 SMR Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-3. Effect of jet-pump geometry and stagnation pressure ratio on the System Mass Ratio (SMR).

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78 The reason why these specific heat tran sfer ratios decrease with decreasing compression ratio can be explained using the T-s diagram in Figure 6-4. 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 Pc. If we let Pa be the evaporator pre ssure and consider two cases. The first case is when Pb is the radiator pressure (1-2-4-5-1), the second is when the compression ratio is higher and Pc is the radiator pressure (1 -3-4-5-1). Because of the fact that state 4 is always c onstrained to be saturated liquid, it can be seen that as the condenser pressure increases, the amount of h eat rejected in the ra diator also decreases (Q3-4 < Q2-4), however, the amount of coo ling decreases even faster (Q1-5 << Q1-5). This causes the specific heat transfer ratios Qrad/Qcool, and Qsc/Qcool to go down, leading to lower values of the SMR. Figure 6-4. T-s diagram for the refrig eration part of the SITMAP cycle. Stagnation Pressure Ratio Effect The SITMAP cycle parameters used to st udy the effect of the stagnation pressure ratio as well as the jet-pump geometry effect s on the cycle performance are presented in

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79Table 6-2. As mentioned before the stagnati on pressure ratio is varied by changing the primary inlet stagnation pressure, Ppo. The secondary stagnation pr essure 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 jet-pump 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 jet-pump secondary inlet flow is always restricted to saturated vapor. The secondary flow parameters correspond to Tevap = 79.4 K. The jet-pump 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 primar y to secondary flow areas going into the mixing chamber. The environmental sink te mperature, 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 6-2. 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 6-1 showed the effect of the je t-pump geometry and stagnation pressure ratio on the break-off entrainment ratio. It can be seen that the break-off value of the

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80 entrainment ratio decreases with increasing st agnation pressure ratio. This should be expected because, since the secondary stagnati on inlet pressure is fi xed, a higher primary stagnation pressure corresponds to a higher ba ckpressure. The higher backpressure has an adverse effect on the entrainment proce ss allowing less secondary flow entrainment before choking occurs. Figure 6-2 and Figure 6-3 show the variation of the compression ratio and the SMR, respectively, with Ppi/Psi, for different jet-pump geom etries. The compression ratio and SMR are calculated at the break-off entrai nment ratio. Therefor e all of these data points correspond to points on the a-b-c (break-off) curve in Figure 2-2. It can be seen in Figure 6-2 that as the ratio Ppi/Psi increases, the compression ratio increases as well, which is expected. However, the SMR incr eases with increasing compression ratios. Therefore, it is not advantageous from a mass standpoint to in crease the stagnation pressure ratio. This can be explained by cons idering the other parameters that affect the SMR. Such parameters are shown in Figure 6-5 through Figure 6-8. Figure 6-5 through Figure 6-8 show the effect of stagnation pressure ratio and jetpump geometry on the following quantities: am ount of specific heat rejected, radiator temperature, amount of speci fic heat input, and cooling capacity. As the stagnation pressure ratio increases all of the aforementioned quantities ch ange in a way that should lead to a decrease in the value of SMR. A ll the heat exchange quantities decrease which leads to smaller heat exchangers, which in tu rn should lead to lowe r SMR. The radiator temperature, shown in Figure 6-6, increases with increas ing 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 6-3, the SMR behavior cont radicts this expected trend.

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81 SMR increases with increasing Ppi/Psi. This is because of the f act 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 Qsc) is not of relevance. The parameters that actually affect the SMR are the specific heat transfer rate s normalized by the specific cooling capacity. Thus, even though Qrad and Qsc decrease, which causes Arad, and Asc to decrease as well, SMR still increases because the cooling capacity, Qcool, 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 6-9, and Figure 6-10 that show an increase in the values of Qrad/Qcool, and Qsc/Qcool, respectively, with increasing stagnation pressure ratio, Ppi/Psi. 51015202530 Ppi/Psi 200000 400000 600000 800000 1E+06 1.2E+06 1.4E+06 1.6E+06 1.8E+06 Qrad Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-5. Effect of jet-pump geometry and stagnation pressure ratio on the amount of specific heat rejected.

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82 51015202530 Ppi/Psi 81 82 83 84 85 86 87 88 89 90 91 92 93 Tradeff Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-6. Effect of jet-pump geometry and stagnation pressure ratio on the radiator temperature. 51015202530 Ppi/Psi 185000 190000 195000 200000 205000 210000 215000 220000 Qsc Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-7. Effect of jet-pump geometry and stagnation pressure ratio on the amount of specific heat input.

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83 51015202530 Ppi/Psi 0 200000 400000 600000 800000 1E+06 1.2E+06 1.4E+06 1.6E+06 1.8E+06 Qcool Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-8. Effect of jet-pump geometry and stagnation pressure ratio on the specific cooling capacity. 51015202530 Ppi/Psi 2 4 6 8 10 12 Qrad/Qcool Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-9. Effect of jet-pump geometry and stagnation pressure ratio on the cooling specific rejected heat.

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84 51015202530 Ppi/Psi 2 4 6 8 10 12 14 Qsc/Qcool Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-10. Effect of jet-pump geometry a nd stagnation pressure ratio on the cooling specific heat input. 51015202530 Ppi/Psi 0 2 4 6 8 COP Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-11. Effect of jet-pump geometry a nd stagnation pressure ratio on the overall cycle efficiency.

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85Figure 6-11 shows the overall efficiency of the SITMAP system. The overall efficiency is the ratio of specific cooling produced, Qcool, to the required specific heat input, Qsc, which is the inverse of the ratio presented in Figure 6-10. Thus it is expected that the overall efficiency would decrease with increasing stagnation pressure ratio. It should be noted that this definition of th e overall efficiency assumes a work balance between the mechanical pump and the turbine. Figure 6-12 show an interesting trend for the ratio of overall cycle efficiency to that of a Carnot cycle, T. 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 ma ximized. However, in this study overall system mass is the objective since the SI TMAP cycle is studied specifically for space applications. 51015202530 Ppi/Psi 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 T Ant/Ane,Ane/Ase=0.25,0.1 Ant/Ane,Ane/Ase=0.25,0.2 Ant/Ane,Ane/Ase=0.25,0.3 Ant/Ane,Ane/Ase=0.35,0.1 Ant/Ane,Ane/Ase=0.35,0.2 Ant/Ane,Ane/Ase=0.35,0.3 Figure 6-12. Effect of jet-pump geometry and stagnation pressure ratio on the ratio of the overall cycle efficiency to th e overall Carnot efficiency.

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86Secondary Flow Superheat Effect In all the results pr esented so far the jet-pump seconda ry inlet (evaporator exit) is constrained to be saturated vapor (xsi=1) at the corresponding ev aporator 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 jet-pump inlet wi th all the other parameters fixed. The complete configuration is presented in Table 6-3. Table 6-3. SITMAP cycle conf iguration 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 xso 0.5,1.0 (Tevap = 79.4 K) 5, 10,and 15 degrees superheat Ant/Ane 0.25 Ane/Ase 0.1 Ts 0 Figure 6-14 show that the degr ee of superheat does not ha ve a significant effect on the compression characteristics of the jet-pum p. However, increasing the degree of superheat increases the cooli ng capacity of the SITMAP cycle and improves the SITMAP cycle performance in terms of decreasing the amount of Qrad and Qsc per unit cooling load, as shown in Figure 6-15, and Figure 6-16, respectively. This causes the SMR to drop, as shown in Figure 6-13. Figure 6-17 shows the effect of the s econdary flow superheat on the breakoff entrainment ratio. It can be seen that decreases with incr easing 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 superhea t has more influence if xsi<1, but once the secondary flow is satura ted vapor, the amount of superheat does not

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87 have a strong effect on SMR. Thus, it can be concluded that it is advantageous to operate the jet-pump with the secondary flow either as a saturated vapor or in the superheated regime. 2.7 2.8 2.9 3 3.1 3.2 3.3 SMR xsi=0.5 xsi=1.0 5degreessuperheat 10degreessuperheat 15degreessuperheat Figure 6-13. Effect of secondary superh eat on the overall system mass ratio (SMR). 1.415 1.42 1.425 1.43 1.435 1.44 1.445 1.45 Pde/Psi xsi=0.5 xsi=1.0 5degreessuperheat 10degreessuperheat 15degreessuperheat Figure 6-14. Effect of secondary supe rheat on the break-off compression ratio.

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88 1.35 1.4 1.45 1.5 1.55 1.6 1.65 Qrad/Qcool xsi=0.5 xsi=1.0 5degreessuperheat 10degreessuperheat 15degreessuperheat Figure 6-15. Effect of secondary superheat on Qrad/Qcool. 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Qsc/Qcool xsi=0.5 xsi=1.0 5degreessuperheat 10degreessuperheat 15degreessuperheat Figure 6-16. Effect of secondary superheat on Qsc/Qcool.

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89 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 xsi=0.5 xsi=1.0 5degreessuperheat 10degreessuperheat 15degreessuperheat Figure 6-17. Effect of secondary superhea t on the break-off mass fl ow characteristics. Turbine Pressure Effect The effect of the turbine inlet pressure was studied by fixing the jet-pump geometry and inlet states and allowing the turbine pressure (high pressure in the Rankine part of the SITMAP cycle) to increase beyond the value that yields minimum positive net work, as discussed earlier. Increasing the turbine pressure increases the work input to the pump, the work output from the turbine, and the amount of heat input to the SITMAP system. Figure 6-18 shows that even though the net work increases at higher turbine inlet pressures, the increase in the amount of specific heat input is still higher. This leads to a decrease in the overall cycle efficiency, as shown in Figure 6-19. Since the amount of specific heat input incr eases with increasing Pti, and the specific cooling capacity is fixed, this causes the SMR to increase, as shown in Figure 6-19. Therefore, it can be concluded that for a given jet-pump geometry and cooli ng capacity it is better from the SMR and the

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90 overall cycle efficiency standpoi nt to operate at the lowest possible turbine inlet pressure that yields minimum amount of positive net work. 024681012 Pti/Pte 0 100000 200000 300000 400000 500000 600000 Wnet 0 100000 200000 300000 400000 500000 600000 700000 Qsc WnetQsc Figure 6-18. Effect of the tu rbine inlet pressure on the amount of net work rate and specific heat input to the SITMAP system. 024681012 Pti/Pte 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 SMR 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 COP SMR COP Figure 6-19. Effect of the tu rbine inlet pressure on the amount of the SMR and overall efficiency of the SITMAP system. Even though it is useful to investigate th e effect of having nonzero net work output on the SMR, it has to be kept in mind that using the SMR as a figure of merit for the SITMAP system when there is a nonzero net work output is not the most accurate

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91 representation for the system performance. The SMR compares the mass of the SITMAP system to that of an ideal passive radiator that has the same coo ling capacity; therefore, the SMR definition is accurate only when the only output of the SITMAP system is cooling. When there is net work output (a s well as cooling), the SITMAP system mass should be compared to that of an ideal passive radiator plus the ma ss of an ideal Carnot Rankine cycle that has the same net work output. This new figure of merit will be derived and studied in later sections of th is study. The new figure of merit will be referred to as the Modified System Mass Ratio, or MSMR. Mixed Regime Analysis So far in this study the value of entrainmen t ratio is determined to be the maximum possible (break-off entrainment ratio) for a gi ven jet-pump geometry and inlet states. To investigate the jet-pump performance in th e mixed regime, the entrainment ratio was varied in the range b reakoff0. The results presented below are for Psi=128 kPa, xsi=1, Ppi/Psi=10, Ant/Ane=0.25, and Ane/Ase=0.1. For these conditions the break-off entrainment ratio is 4.28. It is expected that in th e mixed regime as the entrai nment ratio decreases, the compression ratio increases as well as the SM R because it is favorable to operate the jetpump at the maximum entrainment ratio possibl e, as shown before. The expected trend for SMR was indeed observed, as shown in Figure 6-20, however, the compression ratio behavior was different. Figure 6-20 shows the variation of SM R and the compression ratio with the entrainment ratio. It can be seen that as the entrainment ratio increases, the compression ratio decreases at first as expected, but then it starts to increase again. This behavior is

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92 due to the effect of the kinetic energy of the secondary stream It should be noted that the compression ratio here is defined as the ratio of the total or stagnation pressure at the jetpump diffuser exit, Pde, to that at the secondary inlet, Psi, and hence the kinetic energy effect should be taken into consideration. Figure 6-21 shows that even though it is true that the static backpressure, Pme, always decreases with increasing entrainment ratio, the velocity at the secondary nozzle exit, Vse, increases significantly, as shown in Figure 622. This causes the velocity of the mixed stream, Vme, to increase as well. Eventually, the diffuser exit total pressure starts to incr ease due to the kinetic energy effect, even though the static backpressure is still decreasing. Figure 6-23 shows the effect of entrainment ratio on the static backpressure, Pme, and the total pressure at the diffuser exit, Pde. 22.533.544.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 SMR 1.246 1.248 1.25 1.252 1.254 1.256 1.258 1.26 1.262 1.264 1.266 Pde/Psi Figure 6-20. SMR and Compression rati o behavior in the mixed regime.

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93 22.533.544.5 120000 125000 130000 135000 140000 145000 150000 Pme 60 70 80 90 100 110 Vme PmeVme Figure 6-21. Effect of the entrainment ratio on the mixed chamber exit conditions in the mixed regime. 22.533.544.5 75000 80000 85000 90000 95000 100000 105000 110000 115000 120000 Pse 50 60 70 80 90 100 110 120 130 140 150 Vse PseVse Figure 6-22. Effect of the entrainment ra tio on secondary nozzle exit conditions in the mixed regime.

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94 22.533.544.5 0.975 1 1.025 1.05 1.075 1.1 1.125 1.15 Pme/Psi 1.248 1.25 1.252 1.254 1.256 1.258 1.26 1.262 1.264 1.266 Pde/Psi Pme/PsiPde/Psi Figure 6-23. Jet-pump compression behavior in the mixed regime. 22.533.544.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Qrad/Qcool 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Qsc/Qcool Figure 6-24. Effect of entrainment ratio on specific heat transfer ratios in the mixed regime. Figure 6-20 shows that from the SMR standpoi nt it is better to operate the jet-pump near the break-off conditions because this yiel ds the lowest SMR. The reason for that is shown in Figure 6-24. It can be se en that if the value of drops below its break-off value, this causes the ratio of specific heat input to cool ing load and the ratio of the specific heat rejected to the sp ecific cooling load to increase which in turn increases the

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95 SMR. Therefore, in this study the jetpump and the SITMAP cycle operation will be studied at break-off conditions. Evaporator Temperature Effect All the data sets presented so far in this study have the evaporator pressure fixed at 128 kPa, which corresponds to an evaporator temperature, Tevap=79.4 K. The reason Tevap has been fixed throughout; is that it is more realistic to study the SITMAP cycle performance at a given cooling load temp erature corresponding to a given mission. However, it is also import ant to understand the effect of varying the evaporator temperature on the SITMAP cycle performan ce. For that purpose, all the cycle parameters are kept fixed and the evaporat or temperature is varied in the range 70K
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96 pressure is fixed, higher secondary pressu res will lead to lower compression ratios. Higher compression ratios cause the entrainment ratio to decrease, since the backpressure is increasing, as can be seen in Figure 6-25. 75808590 Tevap 1 1.5 2 2.5 3 3.5 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 Pde/Psi Pde/Psi Figure 6-25. Effect of the evaporator temp erature on the break-off entrainment ratio and the compression ratio, for Ppo = 3.3 MPa. Figure 6-26 shows that increasing Tevap has a favorable effect on both T, and the SMR. The reason for this is shown in Figure 6-27. The ratios Qrad/Qcool and Qsc/Qcool decrease significantly with increasing Tevap (or Tso). This positively impacts the overall efficiency as well as the SMR of the cycle.

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97 75808590 Tevap 0.08 0.09 0.1 0.11 0.12 0.13 0.14 T 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 SMR TSMR Figure 6-26. Effect of th e evaporator temperature on T, and SMR, for Ppo = 3.3 MPa. 75808590 Tevap 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Qrad/Qcool 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Qsc/Qcool Qrad/QcoolQsc/Qcool Figure 6-27. Effect of the evaporator te mperature on the cooling specific rejected specific heat, Qrad/Qcool, and the cooling specific heat input, Qsc/Qcool, for Ppo =3.3MPa. Figure 6-28 shows another trend that help s lower the SMR. As the radiator temperature increases, this helps decrease the required radiator size and thus lowers the overall system mass.

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98 75808590 Tevap 83 84 85 86 87 88 89 90 91 Tradeff Figure 6-28. Effect of the evaporator temper ature on the effective ra diator temperature, for Ppo = 3.3 MPa. Primary Flow Superheat Heat Effect To study the effect of the degree of superh eat of the primary fl ow going into the jetpump on the cycle performance, the quality of the primary flow is varied between xpi=0.1 to 30 degrees of superheat with all the other cycle parameters are kept fixed at the values summarized in Table 6-5. Table 6-5. SITMAP cycle configurati on to study the primary flow superheat. Variable name Description Ppo/Pso 10,20,25 xpo variable Pso 128 kPa xso 1.0 Ant/Ane 0.25 Ane/Ase 0.1 Ts 0 Figure 6-29 shows that as the primary fl ow quality increases towards saturated vapor and into the superheated regime, the SM R increases. The reason for this is shown in Figure 6-30, and Figure 6-31. It can be seen that as the primary flow quality moves towards the superheated regime, the amount of specific heat input and specific heat

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99 rejected per unit cooling load increases which in turn causes the SMR to increase. This effect is expected since increasing the pr imary flow quality (or temperature in the superheated regime) requires more heat input to achieve and thus more heat to reject. Meanwhile, increasing the quality of the pr imary flow did not seem to have any significant effect on the compression ratio, as can be seen in Figure 6-32. 1234567891011121314 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 SMR Ppo/Pso=10 Ppo/Pso=20 Ppo/Pso=25 POINTxpo10.1 20.2 30.3 40.4 50.5 60.6 70.7 80.8 90.9 105degreessuperheat 1110degreessuperheat 1215degreessuperheat 1325degreessuperheat 1430degreessuperheat Figure 6-29. Effect of primar y flow superheat on the SMR.

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100 1234567891011121314 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Qrad/Qcool Ppo/Pso=10 Ppo/Pso=20 Ppo/Pso=25 Figure 6-30. Effect of prim ary flow superheat on the Qrad/Qcool. 1234567891011121314 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Qsc/Qcool Ppo/Pso=10 Ppo/Pso=20 Ppo/Pso=25 Figure 6-31. Effect of prim ary flow superheat on the Qsc/Qcool.

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101 1234567891011121314 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Pde/Psi Ppo/Pso=10 Ppo/Pso=20 Ppo/Pso=25 Figure 6-32. Effect of primary fl ow superheat on the compression ratio. Environmental Sink Temperature Effect A very important trend should be noticed in all the results presented so far regarding the SITMAP cycle operation. From the SMR standpoint, it has always been advantageous to operate the SITMAP system at the highest possible entrainment ratio, hence the lowest possible compression ratio. This means that the cycle is being driven towards operating as a passive radiator. Sin ce the SMR values are still well above unity, this means that, for the range of operation co nsidered, the SITMAP system does not have an advantage over a passive radiator from a system mass point of view. The reason for this is that the environm ental sink temperature, Ts, assumed so far in this study (Ts = 0) is a significantly lower than the evaporator temperatures (cooling load temperatures) considered. The large difference between the evaporator temperature and Ts=0 K, ( Tload = Tevap-Ts), gives the advantage to the idea l passive radiator. However, if Tload decreases, the size of a passive radiator needed to achieve a given cooling load increases. This gives the advantage from a mass sta ndpoint to activ e systems over their passive

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102 counterparts, yielding SMR values below unity. This is shown in Figure 6-33. It can be seen that as the sink temperature, Ts, gets closer to the evapor ator temperature, the SMR starts dropping to values below unity. This proves that in this regime the active systems have the advantage over the passive systems. Figure 6-34 is a zoom in on the right end of the curve in Figure 6-33. It can be seen that at higher compression ratios (i.e. low entrainment ratios) the SMR starts dropping towards unity at lower Ts/Tevap values; proving that active systems have more of an advantage at higher compression ratios. It should also be noted that for the ra nge of system parameters considered, Tload has to be very small (maximum of 2 degrees) in order for th e active system to have the advantage (SMR<1). Further analysis at hi gher compression ratios should be considered to see if the active system could be an attractive option over a wider range of Ts/Tevap. 00.250.50.751 Ts/Tevap 0 1 2 3 4 5 6 7 8 9 10 11 12 SMR Tsi=97.7K,Pde/Psi=1.0504, =11.9145 Tsi=79.4K,Pde/Psi=1.1667, =6.6673 Tsi=79.4K,Pde/Psi=3.702, =0.14 Figure 6-33. Sink temperature effect on SMR. Zoom in shown in next figure.

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103 0.990.99250.995 Ts/Tevap 0.98 0.99 1 1.01 1.02 1.03 SMR Tsi=97.7K,Pde/Psi=1.0504, =11.9145 Tsi=79.4K,Pde/Psi=1.1667, =6.6673 Tsi=79.4K,Pde/Psi=3.702, =0.14 Figure 6-34. Compression ratio effect on the SMR < 1 regime. System Optimization An obvious question that shoul d be asked now is whether or not the active system is going to continue gaining more advantag e with increasing compression ratio, or is there is an optimum system configuration at which the break-off environmental sink temperature, b o sT, has a minimum value? (N.B. b o sT is the value of Ts at which SMR = 1 for a given cycle configuration). Figure 6-35 presents the answer to this question. Figure 6-35 shows the variation of Ts at which SMR=1 ( b o sT) with changing jetpump geometry, for an evaporator temperature, Tevap=79.4 K, and stagnation ratio, Ppo/Pso=25. The significance of this parameter is that it represents the value of the sink temperature below which the active system loses its mass advantage over its passive counterpart. The following relations can further illustrate this concept, if b o sT < Ts < Tevap SMR < 1; if Ts < b o sT SMR > 1

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104 It can be seen that for a given primary nozzle geometry, the value of Ts has an optimum (minimum) value and it does not k eep increasing with increasing compression ratios. Figure 6-36 shows that the co mpression ratio increases (and decreases) with increasing values of both area ratios. These results show that when the difference between the sink and evaporator temperature, Tload, is small enough there are competing e ffects that act in such a way so that increasing the compression ratio does not necessarily gi ve more advantage to the active system (i.e. lowering SMR). As shown earlier in Figure 6-1 through Figure 6-3, as both area ratios increase the compression ra tio increases, and the entrainment ratio decreases and this lead to an increase in the SM R. This is due to the increase in the ratio of specific heat input and the specific heat rejected per unit specific cooling load, as shown previously in Figure 6-9 and Figure 6-10. 00.20.40.60.81 Ane/Ase 77.3 77.4 77.5 77.6 77.7 77.8 77.9 78 78.1 78.2 Ts bo Ant/Ane=0.1 Ant/Ane=0.2 Figure 6-35. Effect of jet-pump geomet ry on the break-off sink temperature, for PpoPso=25, Pso=128 kPa, Tevap=79.4 K, 10 degrees primary superheat.

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105 00.20.40.60.81 Ane/Ase 1.5 2 2.5 3 3.5 4 Pde/Psi 0.5 1 1.5 2 2.5 3 3.5 4 Pde/Psi,(Ant/Ane=0.1) Pde/Psi,(Ant/Ane=0.2) ,(Ant/Ane=0.1) ,(Ant/Ane=0.2) Figure 6-36. Compression ratio and entrainment ratio variati on with jet-pump geometry, for Ppo/Pso=25. This effect was dominant when the system was operating with a big difference between the evaporator and the sink temperatures. Ho wever, there is anothe r effect that higher compression ratios have on the SMR that competes with the previous effect. As the compression ratio increases the active systems gain advantage over passive systems at lower sink temperatures, an effect that helps lower the value of SMR. This later effect has more influence when the system is ope rating with a small difference between the evaporator and sink temperatures. These two competing effects cause the system to behave in the fashion shown in Figure 6-35. Another parameter that affects the SITMAP system operation is the primary stagnation pressure, Ppo. Increasing the primary stagnati on pressure will always lead to increasing compression ratios for a given jetpump geometry. However, as shown above, increasing the compression ratio does not always give the system added advantage from a mass point of view. Ther efore, the effect of Ppo on the system behavior should be

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106 investigated. Figure 6-37 shows that actually the sta gnation pressure ratio is a parameter that needs to be optimized for any given jet-pump configuration. 102030405060 Ppi/Psi 77 77.2 77.4 77.6 77.8 78 78.2 78.4 78.6 78.8 TS bo Ant/Ane=0.1 Ane/Ase=0.3 Tpo=150K Pso=128kPa Tso=79.4KSMR<1 SMR>1 Figure 6-37. Effect of stagnation pressure ra tio on the break-off sink temperature (77.1). So far the jet-pump has always been cons trained to work at break-off conditions, where the entrainment ratio is maximum, a nd the compression ratio is minimum. These are the best conditions because they yield th e lowest SMR when the system is operating at a high difference between the evaporator and the sink temperatures. However, when the system is operating at low temperature di fference between the evaporator and the sink temperatures, increasing the compression can have a favorable effect on the SMR. Therefore, the system operation in the mixed regime might yield lower SMR, and thus it should be investigated. Figure 6-38 shows the break-off sink temp erature behaves when the jet-pump operates in the mixed regime with lower entrainment ratios (thus higher compression ratios). It can be seen that operating in th e mixed regime increases the value of the breakoff entrainment ratio, in turn lowering the system performance from a mass standpoint.

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107 Therefore, the rest of the system optimizati on analysis will be rest ricted to operation at the break-off entrainment ratio. 00.10.20.30.40.50.60.70.80.9 77.25 77.5 77.75 78 78.25 78.5 78.75 79 Ts bo Ant/Ane=0.1 Ane/Ase=0.3 Pso=128kPa Tso=79.4K Ppo/Pso=40 SMR>1 SMR<1 Figure 6-38. Break-off sink temperature be havior in the mixed regime (77.1). Figure 6-39 shows the effect of the jet-pump geometry on the SMR. It can be seen that the same competing effects discussed earlier cause the SMR to have an optimum (minimum) at certain jet-pump geometry. Figure 6-40 shows the effect of the stagnation pressure ratio on the SMR. It can also be s een that the stagnation pressure ratio has an optimum value. The values of the optimum jet-pump design and stagnation pressure ratio will be evaluated using the optimization technique discussed in chapter 4.

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108 00.20.40.60.8 Ane/Ase 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 SMR Ant/Ane=0.1 Ant/Ane=0.2 Ant/Ane=0.3 Figure 6-39. Effect of jet-pump geometry on the SMR for Ppo/Pso=40, Tpo=150 K, Pso=128 kPa, Tevap=79.4 K, Ts = 78.4. 101520253035404550556065 Ppi/Psi 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 SMR Ant/Ane=0.1 Ane/Ase=0.3 Pso=128kPa Tso=79.4K Ts=78.4K Tpo=150K Figure 6-40. Effect of stagnation pressure ratio on the SMR. The optimization process presented in chapter 4 is used to find the optimum values for Ant/Ane, Ane/Ase, and Ppo/Pso for a given mission. Again, a mission is defined by the evaporator temperature and the envi ronmental sink temperature. During the optimization process the variables were allowed to vary in the following ranges 0.01 < Ant/Ane < 0.99 0.01 < Ane/Ase < 1.0

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109 2 < Ppo/Pso < 65 The optimum cycle configuration for Pso=128 kPa, and Ts = 78.4 K is listed in Table 6-6. Table 6-6. Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K). Variable name Value Ppo/Pso 15.6 Tpo 150 [K] Pso 128 [kPa] xso 1.0 (Tevap = 79.4 K) Ant/Ane 0.29 Ane/Ase 0.41 Pti 2.04 [MPa] 0.66 Pde/Psi 2.549 Ts 78.4 [K] Qcool 117 [kW] jet-pump 29.1 % bo s T 74.3 [K] SMR 0.27 In the following chapter of this study the SITMAP cycle performance will be studied for cases where there is a net work out put as well as cooling. For that purpose a Modified System Mass Ratio (MSMR) will be derived as a new figure of merit. MSMR is the ratio of the mass of the SITMAP system to the mass of two different ideal systems. The first is an ideal passive radiator with the same cooling capacity as the SITMAP system, and the second is an ideal Rankine cycle with the same work output as the SITMAP system. The effect of the different cycle parameters on MSMR will be studied

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110 and the SITMAP system will also be optimized for different missions, with the MSMR as the new objective function to be minimized.

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111 CHAPTER 7 RESULTS AND DISCUSSION: C OOLING AND WORK OUTPUTS The System Mass Ratio (SMR) presented ea rlier was defined as the ratio between the mass of the overall SITMAP system and that of an idealized passive radiator. This definition assumes a work balance between the turbine and the pump in the SITMAP system. Thus the only useful output of the sy stem was just its cooling capacity. That was the reason why the SMR definition compared the SITMAP system to an ideal passive radiator with the same cooling capacity However, if we want to allow for a net work output from the SITMAP system, the prev ious definition of SMR is not adequate because it doesnt take into account the work output. In this section a modified SMR definition is presented. The modified definition compares the mass of the overall SITMAP system to that of an ideal passive radiator with the same cooling capacity, plus the mass of a Carnot Rankine cycle with the same net work output. The overall system mass is divided into three terms; radiat or, collector, and a general system mass comprising the turbo-m achinery and piping present in an active system. This is shown mathematically by ,, colradsysSITMAP radoCarnotRankinemmm MSMR mm (7.1) Where the mass of the Carnot cycle can be broken down in the same manner the ideal passive radiator mass was. It is assumed that the passive radiator and the Carnot systems are ideal and hence no general system mass is accounted for. Therefore, the Carnot system mass is given by

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112,,, CarnotRankinecolCRradCRmmm (7.2) The modified SMR expression can be separated as follows ,,,, sysSITMAP colrad radoCarnotRankineradoCarnotRankinem mm MSMR mmmm (7.3) I II The first term on the RHS of the equation above can be expanded into ,,, colrad radocolCRradCRmm I mmm (7.4) Where the subscript CR is short for Carnot Rankine. Term II in Equation (7.3) will be dealt with later on in the analysis. Equation (7.4) can be separated and rewritten in terms of collector and radiator areas ,,, colcolradrad radradoradradCRcolcolCRAA I AAA (7.5) Where mass area (7.6) If we divide by rad ,,, col colrad rad col radoradCRcolCR radAA I AAA (7.7) rad col (7.8) The solar collector is modeled by exam ining the solar energy incident on its surface. This energy is proportional to the co llector efficiency, the cross-sectional area that is absorbing the flux, and the local radiant solar heat flux, G, same as Equation (3.54).

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113 H col acolQ GA (7.9) Where Aa,col is the solar collector aperture ar ea, which will be referred to as Acol for simplicity. The reason this is the aperture ar ea and not the collector receiver area is that col is defined as the mass per unit aperture area in this analysis. If col was defined as mass per unit receiver area, then the denomin ator of Equation (7.9) would have been multiplied by the concentration ratio of the solar collector. The radiant energy transfer rate between the radiator of the SITMAP system and the environment is given below. For deep space applications, the environmental reservoir temperature may be neglected, but for near-pla netary or solar missions this may not be the case. 44 rad rad radsQ A TT (7.10) 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 44 e rado esQ A TT (7.11) In a similar sense the Carnot Rankine cycl e is assumed to have an ideal passive radiator operating at the same radiator temperature in the SITMAP system. Therefore, the area of the ideal radiator of th e Carnot Rankine cycle is given by , 44 radCR radCR radsQ A TT (7.12)

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114 Substituting Equations (7.8)-( 7.12) into equation (7.7) 1 2 44 ,, 4444() ()()rad H colrads radCRHCR e esradscolQ Q GTT I QQ Q TTTTG (7.13) 3 4 5 The overall SITMAP system can be di vided into a power subsystem and a refrigeration subsystem. The two subsyste ms and their interactions are shown in Figure 7-1. Figure 7-1. Schematic of a cooling and power combined cycle The overall energy conservation of the combined cycle can be written as H eextradQQWQ (7.14) Where ,, radLrefLpowerQQQ (7.15) extTPWWW (7.16) Substituting Equation (7.14) in Equation (7.13) yields 44 ,, 4444() ()()Heext H colrads radCRHCR e esradscolQQW Q GTT I QQ Q TTTTG (7.17) Qe Wext Power Sub-system Refrigeration Sub-system Wint QL,re f QL,power QH

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115 If we divide the numerator and denominator by Qe, 44 ,, 44441 () 1 ()()ext H ee H colerads radCRHCR ee esradscolW Q QQ Q GQTT I QQ QQ TTTTG (7.18) 44 ,, 44441 () 1 ()()ext HH eHe H colerads radCRHCR ee esradscolW QQ QQQ Q GQTT I QQ QQ TTTTG (7.19) 44 ,, 444411 () 1 ()()ext H eH H colerads radCRHCR ee esradscolW Q QQ Q GQTT I QQ QQ TTTTG (7.20) 4444 ,, 44441 1 ()() 1 ()()ext H H colradserads radCRHCR ee esradscolW Q Q GTTQTT I QQ QQ TTTTG (7.21) which can be written as

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116 4444 ,, , 44441 1 ()() 1 ()()ext H H colradserads radCRHCR HCR HCRe e esradscolW Q Q GTTQTT I QQ Q QQ Q TTTTG (7.22) The first law efficiency of the power subsystem is given by extin HWW Q (7.23) extin H HWW QQ (7.24) extinext H extHWWW QWQ (7.25) 1extin HextWW QW (7.26) If we define the work ratio,WR as follows ext inW WR W (7.27) If Equation (7.27) is plugged into Equation (7.26) 1ext HWR W QWR (7.28) The coefficient of performance of the refrigeration subsystem is given by e inQ COP W (7.29) Therefore,

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117 eextin inHQWW COP WQ (7.30) 1eext HinQW COP QW (7.31) Substituting Equation (7.27) into Equation (7.31) 1H eQ WR QCOP (7.32) The Carnot efficiencies of the power and refrigeration subsystems are defined as ,1extrad c H CRcolWT QT (7.33) ee c incradeQT COP WTT (7.34) In the above expressions it is assumed that the Carnot system does the same amount of work as the external work of the SITMAP system, Wext. ,, exte cc H CRincWQ COP QW (7.35) ,, eextin cc H CRinincQWW COP QWW (7.36) But since the ratio of internal work to the carnot internal work can be written as inc incWCOP WCOP (7.37) Substituting Equations (7.27) and ( 7.37) in Equation (7.36) yields ec cc HCRQCOP COPWR QCOP (7.38) If we define

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118 c cCOP WRWR COP (7.39) then Equation (7.38) can be written as HCR c eccQ WR QCOP (7.40) The overall energy conservation equation for the Carnot Rankine cycle is given by ,, extradCRHRCWQQ (7.41) Dividing both sides by H RCQ yields ,,1radCR ext HRCHRCQ W QQ (7.42) Plugging Equation (7.33) in to Equation (7.42) ,1radCR c HRCQ Q (7.43) Substituting Equations (7.28), (7.32), (7.40), (7.43) into Equation (7.22) 4444 44441 11 1 ()() 1 1 ()()colradsrads cc esccradscolWR WR WR GTTCOPTT I WR TTCOPTTG (7.44) If we define T ccCOP COP (7.45) Plugging Equation (7.45) into Equation (7.44)

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119 4444 44441 11 1 ()() 1 1 ()()colradsTccrads cc esccradscolWR WR WR GTTCOPTT I WR TTCOPTTG (7.46) If we substitute the definitions of the Car not efficiencies from Equations (7.33), and (7.34) into Equation (7.46) 4444 44441 11 1 ()() 1 11 1 ()() 1colradsrads rade T colrade rad col c esradscol rade colradeWR WR WR GTTTT TT TTT I T T WR TTTTG TT TTT (7.47) If we normalize all the temperatures by the evaporator temperature, Te 44 44 4* 4**4** ** 4* **1 11 1 1 1 1 1 1 1 1 1col rad eradserads T colrad c c rad es colradWR WR WR G T TTTTTT TT I WR T TT TT 44* 4** rad col ol eradsT T G TTT (7.48) If we multiply the numerator and denominator of Equation (7.48) by 4 eT and define 4 e colT G (7.49)

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120 44 44 4 4* **** ** * ** **1 11 1 1 1 1 1 1 1 1rad radsrads T colrad rad ccol rad srads colradWR WR WR T TTTT TT I T WRT T TTT TT 4* (7.50) Equation (43) can be rearranged as 4444 4 44** ** **** ** ** ****1 11 1 1 1 1 1 1colrad Tcolrad radsrads colcrad rad colrad scolradsWR TWRT WR TT TTTT I TWRT T TT TTTT (7.51) The only term left is term II in Equation (7.3) which is given by ,, sysSITMAP radoCarnotRankinem TermII mm (7.52) Which can be written as , ,,, sysSITMAP totalSITMAP totalSITMAPradoCarnotRankinem m TermII mmm (7.53) where the second term on the right hand side is the MSMR. If we define is the ratio of the general system mass comprising the tu rbo-machinery and piping present in the SITMAP system to its total mass. It is given mathematically by, , s ysSITMAP totalSITMAPm m (7.54) Then Equation (7.52) become

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121 ,, sysSITMAP radoCarnotRankinem TermIIMSMR mm (7.55) Therefore, the MSMR expression given in Equation (7.3) can be written as 1 TermI MSMRTermITermII (7.56) Substituting Equation (44) into Equation (49) yields the final expression for the MSMR 4444 4 44** ** **** ** ** ****1 11 1 1 1 1 1 1 1colrad Tcolrad radsrads colcrad rad colrad scolradsWR TWRT WR TT TTTT MSMR TWRT T TT TTTT (7.57) It can be seen that if the external work output, Wext is 0, thus WR = 0, = 0, and WRc = 0, the MSMR expression boils down to the or iginal SMR expression presented earlier in Equation (3.69), as shown below. 4444 4** ** **** *1 11 1 1 1 1colrad Tcolrad radsrads sTT TT TTTT MSMR T (7.58) 44 4444**** ** ****111 11 1colradss Tcolrad radsradsTTTT MSMRSMR TT TTTT (7.59) Therefore, the SMR can be considered as a special case of the MSMR for cases when the SITMAP system has no net work output.

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122Jet-pump Turbo-machinery Analogy The only remaining challenge to complete the MSMR derivation is the calculation of the internal work, Wint, which is necessary to calculate the COP of the refrigeration subsystem, and the ratios WR, and WRc. The challenge lies in the nature of work interaction between the primary and secondary streams in the jet-pump. Momentum transfer occurs between the two streams thr ough two mechanisms. The first is the shear stresses at the tangential inte rfaces between primary and secondary fluids as a result of turbulence and viscosity. The second mechanism is the work of pressure forces acting normal to the interface, and is called pressure exchange. In order to evaluate the amount of work that goes into compressing the seconda ry fluid, a turbo-machinery analog of the ejector is used, shown in Figure 7-2. In this analogy, th e primary fluid expands through a turbine, from state 2 to state 5 which dr ives a compressor thr ough which the secondary flow is compressed from state 1 to state 4. The two streams then mix and get to state 3. This process is illustrated on the T-s diagram in Figure 7-3. In this analogy it is assumed that all the work transfer takes place before th e two streams mix. It is also assumed that pressures at 4 and 5 are the same. Figure 7-2. A schematic of the turbo-machinery analog of the jet-pump Compressor Turbine 1 2 3 4 5

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123 Figure 7-3. T-s diagram illustrating the th ermodynamic states in the jet-pump turbomachinery analog. The isentropic efficiencies of th e compressor and turbine are given by, '1 1 41 compressorhh hh (7.60) '25 2 2 turbinehh hh (7.61) The work balance between the turbine and compressor can be expressed as compressorturbineWW (7.62) 141225mhhmhh (7.63) Plugging the turbo-machinery efficiencies, E quations (7.60) and (7.61) into Equation (7.63) yields '1 1 2 2turbine compressorhh hh (7.64) Equation (7.64) can be rearranged as '1 1 2 2turbinecompressorhh hh (7.65) T s 3 1 1 2 5 2 4

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124 The product of efficiencies in Equation (7.65) is referred to in th e literature as the overall jet-pump efficiency. Equation (7.65) is a ratio of the isentropic work done in compressing the secondary fluid from the total pr essure at the evaporator exit to the total pressure of the mixed flow, to the isentr opic work done by a turbine in expanding the primary flow from the total pressure at the primary inlet to the total pressure of the mixed flow. It should also be noted that the di scharge from the compressor at point 4 and the discharge from the turbine at point (5) combin e and form the mixed state at point 3. Point 3 corresponds to the discharge from an e quivalent jet-pump. Therefore, the overall jet-pump efficiency can be written as '1 1 2 2jetpumpturbinecompressorhh hh (7.66) It can be seen that there are mu ltiple possible combinations of ,turbinecompressor that would yield the same overa ll jet-pump efficiency, jetpump As the compression efficiency, compressor increase the amount of internal work, Wint decrease and consequently the COP of refrigeration subs ystem increase, as shown in Figure 7-4. However, while the COP increases, the efficiency, (Equation (7.23)) decrease s. Since the change in compressor does not affect the overall cycle parameters such as the specific cooling load, specific heat input, or even the ex ternal work rate output, changing compressor has no effect on the MSMR. This is shown in Figure 7-5. It can be seen that MSMR is not affected by the value of compressor so compressor can assume any value between 0 and 1 as long as the product compressorturbine satisfies Equation (7.66).

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125 The value of compressor depends on the technology of the jet-pump. Research in still ongoing trying to improve the work tr ansfer between the primary and secondary streams [24]. An interesting deduction from this analysis is the value of compressor that needs to be achieved in order for a jet-pum p refrigeration system to compete against commercially available refrigeration systems. It can be seen from Figure 7-4 that for a COP of 3, compressor needs to be about 45%. 0.30.40.50.60.70.80.91 compressor 20000 25000 30000 35000 40000 45000 50000 Wint 2.5 3 3.5 4 4.5 5 5.5 6 6.5 COP WintCOP Figure 7-4. Effect of compression efficiency on je t-pump characteristics. 0.30.40.50.60.70.80.91 compressor 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 MSMR

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126 Figure 7-5. Effect of comp ression efficiency on MSMSR. For a given jet-pump geometry and inlet st ates it is useful to know what the maximum achievable compression ratio can be if the jet-pump is assumed to be ideal, and to see how the compression ratio changes with jet-pump efficiency. Figure 7-6 shows that for the jet-pump configuration defined later in Table 7-1, the maximum compression ratio is 7.5 if the jet-pump is assumed ideal. As expected Figure 7-6 also shows that as the jet-pump efficiency is improved the MSMR decreases for a given jet-pump configuration. This is expected since highe r jet-pump efficiencies yield higher COP and thus less heat transfer rates (specific heat input and specific heat re jected) per unit cooling load leading to smaller a nd lighter heat exchangers. Figure 7-7 verifies that the MSMR a nd SMR expressions are equal at zero external work output. 00.10.20.30.40.50.60.70.80.91 jet-pump 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Pjpe/Psi 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 MSMR Pjpe/PsiMSMR Figure 7-6. Jet-pump efficiency effect on the compression ratio and MSMR for given jetpump inlet conditions.

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127 11.522.533.544.555.566.577.5 Pjpe/Psi 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 SMR 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 MSMR SMR MSMR Figure 7-7. MSMR and SMR are equal for Wext = 0. Since it is not possible to specify the value of compressor for a given set of cycle parameters, and since the MSMR is not aff ected by it, it will be assumed that the compressor and turbine efficiencies in the jet-pump turbo-machinery analogy are equal, and are given by '1 1 2 2compressorturbinehh hh (7.67) The abovementioned assumption brings closur e to the MSMR analysis. Following, the MSMR behavior is investigated for the cycle parameters summarized in Table 7-1. For this set of cycle parameters, the cooling load, Qcool, is about 117 kW, and the overall jetpump efficiency is 29%. It is expected that as the amount of work produced by the SITMAP system is increased the amount of heat input and heat rejection will also increase which will cause the size of the heat exchangers to increase. This will cause the overall system mass and hence the MSMR to in crease. However, as the amount of work increases the size of the ideal Carnot system that would produce the same amount of

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128 work also increases which causes the MSMR to decrease. Therefore, there are competing effects that makes it important to investigate the effect of increasing external work output on the MSMR. Table 7-1. Base case cycle parame ters to study the MSMR behavior. Variable name Value Ppo/Pso 15.5 Tpo 150 [K] Pso 128 kPa xso 1.0 (Tevap = 79.4 K) Ant/Ane 0.29 Ane/Ase 0.4 Pti Variable 0.663 Ts 78.4 [K] Qcool 117 [kW] jet-pump 29 % compressor = turbine 53.9 % For a given set of cycle parameters, the amount of net external work output, Wext, can be increased by increasing the high pressure, Pti (turbine inlet pressure). Figure 7-8 shows that as the high pressure is increased the amount of net work output increases, so does the cycle efficiency (Equation (7.23)). It can also be seen that for the same set of cycle parameters, the amount of specific heat input increases by the same amount as the work rate output. This is the reason the two graph lines coincide in Figure 7-8. Figure 7-9 shows that incr easing the work output has almost no effect on the MSMR. Therefore, it can be concluded th at the competing effects discussed earlier balance each other out not givi ng any significant advantage to increasing the work output

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129 by increasing the cycle high pressure. In other words the increase in the mass of the Carnot system with increasing work output is offset by the increase in the actual SITMAP system size. 8E+069E+061E+071.1E+071.2E+071.3E+071.4E+07 Pti 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Qsc/Qcool 300000 310000 320000 330000 340000 350000 360000 370000 380000 390000 400000 410000 420000 430000 Wext Qsc/QcoolWext Figure 7-8. High pressure effect on the coo ling specific heat input and external work output for a given jet-pump inlet conditions. 8E+069E+061E+071.1E+071.2E+071.3E+071.4E+07 Pti 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 MSMR 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3 1.31 1.32 T MSMRT Figure 7-9. High pressure effect on the MSMR and efficiency for a given jet-pump inlet conditions.

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130System Optimization for MSMR The SITMAP cycle performance with net external work output can be studied using another way of constraini ng the problem. This can be done by fixing the cycle high pressure (the turbine inlet pressure, Pti) and allowing the other cycle parameters to change. Previously, the net work output wa s increased by increasing the high pressure and keeping all the other cycle parameter fixed. It was show n that there is no significant advantage from the MSMR standpoint in increa sing the cycle high pressure, as shown in Figure 7-9 above. Five variables will be considered in the following optimization analysis. These variables are the primary nozzle area ratio, Ant/Ane, primary to secondary area ratio at the mixing chamber inlet, Ane/Ase, the primary to secondary sta gnation pressure ratio at the jet-pump inlet, Ppo/Pso, evaporator pressure (cooling load temperature), Pso, and the cycle high pressure, Pti. The effect of the evaporator pr essure (cooling load temperature), Pso will be studied separately at the end of this section. Figure 7-10 through Figure 7-12 show the effect of each of the first four variables mentioned above on the MSMR. The mission parameters for these figures are Pso = 128 MPa, which corresponds to an evaporator pressure Tso = 79.4 K, and the environmental sink temperature, Ts = 78.4 K. Figure 7-10 shows the effect of the prim ary nozzle geometry on the MSMR. It can be seen that there is an optimum prim ary nozzle area ratio that yields the minimum MSMR. Similar trends are not iced for the effect of Ane/Ase, and Ppo/Pso on MSMR, shown in Figure 7-11 and Figure 7-12, respectively. The optimization techniques presented in chapter 4 will be used to sp ecify the optimum jet-pump geometry and stagnation pressure ratio for the mission para meters mentioned above and Pti = 14.2MPa.

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131 It should be noted that in Figure 7-10 through Figure 7-12 the turbine inlet pressure, Pti had no significant effect on MSMR, which fu rther supports the conclusion that Pti should not be considered as an optim ization variable. However, Pti can be used a design parameter to adjust the amount of work output for a given optimum cycle configuration. 0.10.20.3 Ant/Ane 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 MSMR Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ane/Ase=0.4 Ppo/Pso=26 Pso=128MPa Figure 7-10. Primary nozzle geometry effect on MSMR at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa. 00.10.20.30.40.50.60.70.80.91 Ane/Ase 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 MSMR Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ant/Ane=0.2 Ppo/Pso=26 Pso=128kPa Figure 7-11. Jet-pump geometry effect on MSMR at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa.

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132 51015202530354045 Ppo/Pso 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 MSMR Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ant/Ane=0.2 Ane/Ase=0.4 Pso=128kPa Figure 7-12. Stagnation pressure ratio effect on MSMR at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa. Figure 7-13 through Figure 7-21 show the competing effects that cause the MSMR trends discussed above It can be seen from Figure 7-13, Figure 7-16, and Figure 7-19 that as each of the respective independe nt variables increase, the compression ratio increase which gives the SITMAP system the advantage over the passive system. However, it can also be seen that as the va riables increase the amount of specific heat transfers per unit specific cooling load (Qsc/Qcool, and Qrad/Qcool) increase as well, which has an adverse effect on the MSMR. These comp eting effects lead to the presence of an optimum configuration that leads to the minimum MSMR for the specific mission under investigation.

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133 00.10.20.30.4 Ant/Ane 1 1.5 2 2.5 3 3.5 4 4.5 Pde/Psi 0 0.5 1 1.5 2 2.5 Pde/Psi(Pti=8MPa)(Pti=8MPa) Ane/Ase=0.4 Ppo/Pso=26 Pso=128MPa Figure 7-13. Primary nozzle geometry effect on the compression ratio and the entrainment ratio at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa. 00.10.20.30.4 Ant/Ane 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Qrad/Qcool Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ane/Ase=0.4 Ppo/Pso=26 Pso=128MPa Figure 7-14. 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.

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134 00.10.20.30.4 Ant/Ane 0 5 10 15 20 25 30 Qsc/Qcool Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ane/Ase=0.4 Ppo/Pso=26 Pso=128MPa Figure 7-15. 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. 00.10.20.30.40.50.60.70.80.91 Ane/Ase 1 1.5 2 2.5 3 3.5 4 4.5 Pde/Psi 0 5 10 15 20 Pde/Psi(Pti=8MPa)(Pti=8MPa) Ant/Ane=0.2 Ppo/Pso=26 Pso=128kPa Figure 7-16. Jet-pump geometry effect on the compression ratio and the entrainment ratio at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa.

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135 00.10.20.30.40.50.60.70.80.91 Ane/Ase 2 3 4 5 6 7 8 Qrad/Qcool Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ant/Ane=0.2 Ppo/Pso=26 Pso=128kPa Figure 7-17. Jet-pump geometry effect on the specific heat rejected per unit specific cooling load at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa. 00.10.20.30.40.50.60.70.80.91 Ane/Ase 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Qsc/Qcool Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ant/Ane=0.2 Ppo/Pso=26 Pso=128kPa Figure 7-18. Jet-pump geometry effect on the specific heat input pe r unit specific cooling load at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa.

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136 51015202530354045 Ppo/Pso 1 1.5 2 2.5 3 3.5 4 4.5 5 Pde/Psi 0.5 1 1.5 2 2.5 3 Pde/Psi(Pti=8MPa)(Pti=8MPa) Ant/Ane=0.2 Ane/Ase=0.4 Pso=128kPa Figure 7-19. Stagnation pressure ratio effect on the compression ratio and the entrainment ratio at a fixed jet-pump geometry (Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa. 51015202530354045 Ppo/Pso 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Qrad/Qcool Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ant/Ane=0.2 Ane/Ase=0.4 Pso=128kPa Figure 7-20. Stagnation pressu re ratio effect on the specific heat rejected per unit specific cooling load at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa.

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137 51015202530354045 Ppo/Pso 0 5 10 15 20 25 Qsc/Qcool Pti=8MPa Pti=9MPa Pti=10MPa Pti=11MPa Pti=12MPa Pti=13MPa Pti=14MPa Ant/Ane=0.2 Ane/Ase=0.4 Pso=128kPa Figure 7-21. Stagnation pressure ratio effect on the specific heat input per unit specific cooling load at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa. The optimization process presented in chapter 4 is used to find the optimum values for Ant/Ane, Ane/Ase, and Ppo/Pso. During the optimization process the variables were allowed to vary in the following ranges 0.01 < Ant/Ane < 0.99; 0.01 < Ane/Ase < 1.0; 2 < Ppo/Pso < 65 The optimum cycle configuration for Pso = 128 kPa, and Pso= 140 kPa are listed in Table 7-2, and Table 7-3, respectively. Comparing the results for the two evaporator pressures, it can be seen that the higher evaporator pressure (i.e. higher evaporator temperature) yield a higher optimum MSMR. This is due to the difference between the evaporator and sink temperatures. For th e same environmental sink temperature, Ts = 78.4 K, the higher evaporator pressure corresponds to a higher temperature difference between the evaporator and the environment. The higher temperature difference gives more advantage from a mass standpoint to the passive system, which in turn increases the MSMR. The MSMR is fairly sensitive to this temperature difference. An increase from

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138 one degree difference to 1.8 degrees caused the optimum MSMR to increase by about 70%. Table 7-2. Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K), Pti = 14.2 MPa. Variable name Value Ppo/Pso 15.6 Tpo 150 [K] Pso 128 [kPa] xso 1.0 (Tevap = 79.4 K) Ant/Ane 0.29 Ane/Ase 0.41 Pti 14.2 [MPa] 0.66 Pde/Psi 2.549 Ts 78.4 [K] Qcool 117 [kW] Wext 433 [kW] jet-pump 29.1 % bo s T 74.05 [K] MSMR 0.27 Table 7-3. Optimum Cycle parameters for Pso = 140 kPa (Tevap = 80.2 K), Pti = 14.2 MPa. Variable name Value Ppo/Pso 12.86 Tpo 150 [K] Pso 140 [kPa] xso 1.0 (Tevap = 80.2 K) Ant/Ane 0.33 Ane/Ase 0.54 Pti 14.2 [MPa] 0.51 Pde/Psi 2.66 Ts 78.4 [K] Qcool 88.4 [kW] Wext 460 [kW] jet-pump 25.7% bo s T 75.02 [K] MSMR 0.45

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139 CHAPTER 8 CONCLUSIONS The combined power and cooling SITMAP cycle has been analyzed and optimized. The cycle is comprised of a refrigeration cy cle combined with a Rankine cycle with the jet-pump acting as the joining device between the two subsystems. The jet-pump mixes the high pressure stream from the power subsys tem with the low pressure stream from the refrigeration subsystem, thus providing the necessary compression in the refrigeration subsystem. The methodology followed for th e study consisted of developing a robust one-dimensional model for the two-phase jet-pump (ejector) to capture the details the physics of the different choking phenomena, th en developing the optimization techniques for the SITMAP cycle, and finally applyi ng the jet-pump flow model and optimization techniques to specific missions. A mission is defined by the cooling load temperature, environmental sink temperature, and solar irra diance which is a function of the distance from the sun. The results from the jet-pump model we re in very good agreement with results available in the literature for perfect gases. This gives th e confidence in the accuracy of the flow model as well as its implementation. The jet-pump performance followed the expect ed trends. Following is a list of the general trends noticed in jet-pump performance: Low primary nozzle area ratio, Ant/Ane (i.e. higher primary flow Mach number), caused more entrainment since entrainment is caused primarily by viscous interaction between the primary and secondary streams. Low primary to secondary area ratio, Ane/Ase, led to higher entrainment ratio due to the larger area available for secondary flow at lower Ane/Ase.

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140 Higher entrainment ratios led to lower compression ratios. Secondary flow superheat did not have signi ficant effect on the entrainment ratio. However, entrainment increases if the sec ondary flow is in the two-phase region. It is not desirable, from a SMR point of view, to operate the jet-pump in the mixed regime, because the SMR is lowest at the break-off entrainment ratio. Higher compression ratios led to higher ratio of heat input and heat rejected per unit cooling load. This caused the heat ex changers (radiator and solar collector) to be larger and heavier, leading to higher SMR. Two-phase jet-pumps usually have signi ficantly higher entrainment ratios than single-phase ejectors. This is attribut ed to the lower specific volume of the working fluid in the two-phase regime. The overall SITMAP cycle performance with cooling as the only output was evaluated using the System Mass Ratio (SMR) e xpression derived in the literature. This expression is a ratio of the mass of the combined cycle to that of an ideal passive radiator with the same cooling capacity. However, to evaluate the cycle performance with both work and cooling outputs, a more general expr ession was derived. The new expression is referred to as the Modified System Mass Ratio (MSMR). The MSMR is the ratio of the mass of the combined cycle to that of an ideal passive radi ator with the same cooling capacity plus the mass of an ideal Ranki ne cycle with the same work output. The SITMAP system optimization process led to the following conclusions: SMR and MSMR values are lowest at the break-off entrainment ratio Higher compression ratios lead to larg er and heavier heat exchangers. The most significant parameter on SMR and MSMR is Tload ( Tload = Tevap-Ts). For large values of Tload, SMR is significantly greater than unity and the system is driven to operate as a passive radiator. For Tload around 5 degrees, active systems start to gain advantage and SMR and MSMR drop below unity. This is due to th e large passive systems required at this small temperature difference to achieve a given cooling load. At low Tload competing effects change the way high compression ratios influence the system mass. Whether or not the SITMAP system produ ces any work output, the optimum cycle configuration for a given set of mission para meters is the same. Said differently, the increase in the overall system mass due to the addition of the work output is

PAGE 155

141 offset by the mass of the Carnot Rankine system in the MSMR expression, leading to the MSMR being practically the same no matter what the work output is. The SITMAP cycle maximum pressure (pressure at the turbine inlet) does not have significant effect on the overall system mass

PAGE 156

142 CHAPTER 9 RECOMMENDATIONS More work is needed to better understa nd and improve the performance of the proposed SITMAP system. Some recommendations to extend the results of this work are presented below. The suggestions are listed a nd a discussion elaborating on some of these topics follows. Include frictional losses in the jet-pump model. Investigate the use of different working fl uids in the SITMAP system, which might improve the system performance. Optimizing the system for other objective functions, such as the cooling load, the work output, or the overall cycle efficiency, can lead to further insight especially if this cycle is considered for applications other than space applications where mass is of paramount significance. More efficient jet-pumps will be needed for the SITMAP system to be competent in other applications. Hydrokinetic amplif iers or unsteady ejectors can be a good starting point. An economically-based objective functi on can be used when optimizing for terrestrial aplications to determine th e feasibility of using this system. The jet-pump model used in this study a ssumed frictionless flow throughout the jetpump as an approximation. A quasi one-dimensional frictional model needs to be included for more realistic results. In this study, the SITMAP system is op timized for space applications, therefore, minimum system mass is of the most importance. However, if this system was to be used for other applications, it is important to rec ognize that the ultimate determining factor in cycle design for most applicati ons is the cost. Some case st udies where a cycle design is developed for specific applications might be useful. For instance, geothermal, and waste heat conversion application can be considered. Each of thes e applications has different

PAGE 157

143 constraints, which must be considered in an optimization study. For example, there is a limit in geothermal applications on the temperat ure of the geofluid leaving the cycle, to prevent silica precipitation. The cycle desi gns developed can be compared to standard cycles in use now and to the most promisi ng alternative determined, particularly through an economic analysis.

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144 LIST OF REFERENCES 1. Abrahamsson, K., Jernqvist, A., Aly, G., 1994, Thermodynamic Analysis of Absorption Heat Cycles, Proceedings of the International Absorption Heat Pump Conference, AES-Vol. 31, ASME, New York, pp. 375-383. 2. Addy, A. L., Dutton, J. C., and Mikkelsen, C.D., Supersonic Ejector-Diffuser Theory and Experiments, Report No UILU-ENG-82-4001, Department of Mechanical and Industrial Engineering, University of Illinois at UrbanaChampaign, Urbana, Illinois, August 1981. 3. Al-Ansary, H. A., and Jeter, S. M., Numerical and Experimental Analysis of Single-Phase and Two-Phase Flow in Ej ectors, HVAC&R Research, Vol. 10, No. 4, pp. 521-538, October 2004. 4. Anderson, H., Assessment of Solar Powered Vapor Jet Air-conditioning System, International Solar Energy Congress and Exposition (ISES), Los Angeles, California, pp. 408, 1975. 5. American Society of Heating, Re frigerating and Air-C onditioning Engineers (ASHRAE) Handbook of Fundamentals, 1997, Inc., Atlanta, SI Edition. 6. Chai, V.W. and Lansing, F.L., A Ther modynamic Analysis of a Solar-powered Jet Refrigeration System, DSN Progress Repor t 41-42, Jet Propulsion Laboratory, pp. 209-217, 1977. 7. Chen, L.T, Solar Powered Vapor-Compressive Refrigeration System Using Ejector as the Thermal Compressors, Proceedings of the National Science Council, No. 10, Part 3, pp. 115-132, 1977. 8. Choudhury, G. M., Selective Surface fo r Efficient Solar Thermal Conversion, Bangladesh Renewable Energy Newsletter Vol. 1, No. 2, pp. 1-3, 2000. 9. DiPippo, R., 1980, Geothermal Energy as a Source of Electri city: A Worldwide Survey of Design and Operation of Geothe rmal Power Plants, U.S. Department of Energy, Washington, D.C., 388p. 10. Dorantes, R., Estrada, C. A., and Pilatowsky, I., Mathematical Simulation of a Solar Ejector-Compression Refrigeration System, Applied Thermal Engineering, Vol. 16, pp. 669-675, 1996.

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145 11. Duffe, J. A., and Beckman, W. A., So lar Engineering of Thermal Processes, John Wiley and Sons, New York. 12. Dutton, J. C., and Carroll, B. F., Limitations of Ejector Performance Due to Exit Choking, Journal of Fluids Engineer ing, Vol. 110, pp. 91-93, March 1988. 13. Dutton, J. C., and Carroll, B. F., Optimal Supersonic Ejector Designs, ASME ASME Journal of Fluids Engineering, Vol. 108,No. 4, pp. 414-420, 1986. 14. Eames, I. W., A New Prescription for the Design of Supersonic Jet-Pumps: The Constant Rate of Momentum Change Met hod, Applied Thermal Engineering, Vol. 22, No. 2, pp. 121-131, February 2002. 15. Edgar, T. F., Himmelblau, D. M., and Lasdon, L.S., 2001, Optimization of Chemical Processes, McGraw-Hill, New York, 650p. 16. Elger, D.F., McLam, E.T., and Taylor S.J., A New Way to Represent Jet Pump Performance, ASME Journal of Fluids Engineering, Vol. 113, No. 3, pp. 439-444, 1991. 17. Fabri, J. and Paulon, J., Theory and E xperiments on Air-to-Air Supersonic Ejectors, NACA-TM-1410, September 1958. 18. Fabri, J. and Siestrunk, R., Supers onic Air Ejectors, Advances in Applied Mechanics, Vol. V, H.L. Dryden and Th. von Karman (editors), Academic Press, New York, pp. 1-33, 1958. 19. Fairuzov, Y.V. and Bredikhin, V.V., Two Phase Cooling System with a Jet Pump for Spacecraft, AIAA Journal of Thermophysic s and Heat Transfer, Vol. 9, No. 2, pp. 285-291, April-June 1995. 20. Freudenberg, K., Lear, W. E., and Sher if, S. A., Parametric Analysis of a Thermally Actuated Cooling System for Space Applications, Proceedings of the ASME Advanced Energy Systems Division, AES-Vol. 40, pp. 499-510, November 2000. 21. Garg, H. P., and Prakash, J., Solar En ergy, Fundamentals and Applications, Tata Mc-Graw Hill, New Delhi. 22. Hasan, A. A., Goswami, D. Y., and Vijayaraghavan, S., First and Second Law Analysis of a New Power and Refrigeration Thermodynamic Cycle Using a Solar Heat Source, Solar Energy, Vol. 73, No. 5, pp. 385-393, 2002. 23. Hasan, A. A., Goswami, D. Y., Exergy Analysis of a Combined Power and Refrigeration Thermodynamic Cycle Dr iven by a Solar Heat Source, ASME Journal of Solar Energy Engineering, Vol. 125, pp. 55-60, February 2003.

PAGE 160

146 24. Hong, J. H., Alhussan, K., Zhang, H., a nd Garris, A. G., A Novel Thermally Driven rotor-vane/pressure-exchange Ejector Refr igeration System with Environmental Benefits and Energy Efficiency, Journal of Energy, Vol. 29, pp. 2331-2345, 2004. 25. Khattab, N. M., and Barakat, M. H., Modeling the Desi gn and Performance Characteristics of Solar Steam-Jet Cooling for Comfort Air Conditioning, Solar Energy, 4, pp. 257-267, 2002. 26. Lansing, F.L. and Chai, V.W., Performance of Solar-Powered Vapor-Jet Refrigeration Systems with Selected Wo rking Fluids, DSN Progress Report 42-44, Jet Propulsion Laboratory, pp. 245-248, 1978. 27. Lasdon, L., Waren, A. D., Jain, A., and Ratner, M., Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming, ACM Transactions on Mathematical Software, 4, pp. 34-50, 1978. 28. Lasdon, L. S., and Warren, A. D., 1997, GRG2 Users Guide, Unpublished Manuscript, University of Texas, Aus tin, Dept. of General Business, 26p. 29. Lear, W.E., Sherif, S.A., Steadham, J.M ., Design Considerations of Jet Pumps with Supersonic Two-Phase Flow and Shoc ks for Refrigeration and Thermal Management Applications, International J ournal of Energy Research, Vol. 24, pp. 1373-1389, 2000. 30. Lorenz, V. H., 1894, Die Ausntzung der Brennstoffe in den Khlmaschinen, Zeitschrift fr die gesammte Klte-Industrie, 1, pp. 10-15. 31. Lu, S., and Goswami, Y., 2003, Optimization of a Novel Combined Power/Refrigeration Thermodynamic Cycl e, ASME Journal of Solar Energy Engineering, Vol. 125, pp. 212-217, May 2003. 32. Mason, L. S., Solar Stirling for Deep Space Applications, Space Technology and Applications Internationa l Forum, American Institute of physics, Albuquerque, New Mexico, January 30-Februaury 3, 2000. 33. Nord, J.W., Lear, W.E., and Sherif, S. A., Design Analysis of a Heat-Driven Jet Pumped Cooling System for Space Ther mal Management Applications, AIAA Journal of Propulsion and Power, Vol. 17, No. 3, pp. 566-570, 2001. 34. Parker, G. M., Lear, W. E., and Sherif, S. A., On an Optimum Geometry of a TwoPhase Ejector, ASME Proceedings of the 36th Intersociety Energy Conversion Engineering Conference, Savannah, Georgia, paper no. IECEC2001-TM-14, Vol. 2, pp. 1187-1194, July 29-August 2, 2001. 35. Reklaitis, G. V., Ravindran, A., and Ragsdell, K. M., 1983, Engineering Optimization: Methods and Applications John Wiley and Sons, New York, 684p.

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147 36. Roan, V. P., An Ejector Perf ormance Correlation Factor, AIAA 27th Joint Propulsion Conference, Sacramento, CA, AIAA-91-2545, pp. 1-4, June 24-26 1991. 37. Rosen, M.A., and Le, M., 1995, Efficien cy Measures for Pr ocesses Integrating Combined Heat and Power and District Cooling, Thermodynamics and the Design, Analysis, and Improvement of Energy Systems, AES-Vol. 35, pp. 423434. 38. Sherif, S.A., Lear, W.E., Steadham, J.M., Hunt, P.L., and Holladay, J.B., Analysis and Modeling of a Two-Phase Jet Pump of a Thermal Management System for Aerospace Applications, International Jour nal of Mechanical Sciences, Vol. 42, No. 2, pp. 185-198, February 2000. 39. Stoecker, W. F., 1989, Design of Thermal Systems, McGraw-Hill, New York, 565p. 40. Szargut, J., Morris, D. R., and Steward, F. R., 1988, Exergy Analysis of Thermal, Chemical and Metallurgi cal Processes, Hemisphere, New York, 332p. 41. Tamm, G., Goswami, D. Y., Lu, S., a nd Hasan, A.A., Novel Combined Power and Cooling Thermodynamic Cycle for Low Te mperature heat Sources, Part I: Theoretical Investigation, ASME Journal of Solar Energy Engineering, Vol. 125, pp. 218-222, May 2003. 42. Tamm, G., Goswami, D. Y., Lu, S., a nd Hasan, A.A., Novel Combined Power and Cooling Thermodynamic Cycle for Low Te mperature heat Sources, Part II: Experimental Investigation, ASME Journa l of Solar Energy Engineering, Vol. 125, pp. 218-222, May 2003. 43. Vijayaraghavan, S., 2003, Thermodynamic Study on Alternate Binary Working Fluid Combinations and Configurations for a Combined Power and Cooling Cycle, Ph.D. Dissertation, University of Florida, Gainesville, FL.

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148 BIOGRAPHICAL SKETCH Sherif Kandil was born on November 29th,1976, in Cairo, Egypt. Cairo is the capital city of Egypt and its largest, most fa mous for the Pyramids of Giza. He received his bachelors degree in mech anical engineering from the American University in Cairo in 1998. For a year after that he worked as a research and development engineer in the largest air-conditioning company in Egypt called Miraco-Carrier He then joined West Virginia University where he got his masters degree in mechanical engineering. During his masters Sherif worked on Computat ional Fluid Dynamics (CFD) modeling of multiphase flows. In fall 2001, Sherif joined the University of Florida Mechanical and Aerospace Engineering Department and star ted working on the combined power and refrigeration system as his dissertation topic.


Permanent Link: http://ufdc.ufl.edu/UFE0013733/00001

Material Information

Title: Analysis and optimization of a jet-pumped combined power/refrigeration cycle
Physical Description: Mixed Material
Language: English
Creator: Kandil, Sherif M. ( Dissertant )
Lear, William E. ( Thesis advisor )
Sherif, Sherif Ahmed ( Thesis advisor )
Hahn, David ( Reviewer )
Ingley, Skip ( Reviewer )
Carroll, Bruce ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2006
Copyright Date: 2006

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, Ph.D
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering

Notes

Abstract: The objectives of this study were to analyze and optimize a jet-pumped combined refrigeration/power system, and assess its feasibility, as a thermal-management 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 jet-pump instead of the conventional compressor. This study presents a detailed component analysis of the jet-pump, allowing for two-phase subsonic or supersonic flow, as well as an overall cycle analysis. The jet-pump analysis is a comprehensive one-dimensional 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 jet-pump primary nozzle area ratio, A sub nt/A sub ne, the primary to secondary area ratio, A sub ne/A sub se, and the primary to secondary stagnation pressure ratio, P sub po/P sub so. 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.
Subject: ejector, jet, optimization, space, thermal, two
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 162 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2006.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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Permanent Link: http://ufdc.ufl.edu/UFE0013733/00001

Material Information

Title: Analysis and optimization of a jet-pumped combined power/refrigeration cycle
Physical Description: Mixed Material
Language: English
Creator: Kandil, Sherif M. ( Dissertant )
Lear, William E. ( Thesis advisor )
Sherif, Sherif Ahmed ( Thesis advisor )
Hahn, David ( Reviewer )
Ingley, Skip ( Reviewer )
Carroll, Bruce ( Reviewer )
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2006
Copyright Date: 2006

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering thesis, Ph.D
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering

Notes

Abstract: The objectives of this study were to analyze and optimize a jet-pumped combined refrigeration/power system, and assess its feasibility, as a thermal-management 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 jet-pump instead of the conventional compressor. This study presents a detailed component analysis of the jet-pump, allowing for two-phase subsonic or supersonic flow, as well as an overall cycle analysis. The jet-pump analysis is a comprehensive one-dimensional 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 jet-pump primary nozzle area ratio, A sub nt/A sub ne, the primary to secondary area ratio, A sub ne/A sub se, and the primary to secondary stagnation pressure ratio, P sub po/P sub so. 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.
Subject: ejector, jet, optimization, space, thermal, two
General Note: Title from title page of source document.
General Note: Document formatted into pages; contains 162 pages.
General Note: Includes vita.
Thesis: Thesis (Ph. D.)--University of Florida, 2006.
Bibliography: Includes bibliographical references.
General Note: Text (Electronic thesis) in PDF format.

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 003589390
System ID: UFE0013733:00001


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ANALYSIS AND OPTIMIZATION OF A JET-PUMPED 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 self-starters, 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 jet-pumps. 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
Jet-pum 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

Jet-p 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 o-phase 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

Jet-pum 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

Jet-pum p Turbo-m 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

2-1 Effect of the distance from the sun on solar irradiance ......................................19

2-2 Properties of som e selective surfaces......................................... ......... ............... 23

2-3 Rankine cycle and vapor compression refrigeration cycle efficiency definitions....27

4-1 Optim ization variables and their lim its ........................................ ............... 67

4-2 Constraints used in the optim ization ............................................. ............... 68

5-1 Representative constant-area ejector configuration ..............................................70

6-1 Input parameters to the JETSIT cycle simulation code................ .................75

6-2 SITMAP cycle parameters input to the JETSIT simulation code..........................79

6-3 SITMAP cycle configuration to study the effect of secondary flow superheat .......86

6-4 SITMAP cycle configuration to study the effect of the evaporator temperature,
T evap ......................................................................................... ....9 5

6-5 SITMAP cycle configuration to study the primary flow superheat .......................98

6-6 Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K) ..........................109

7-1 Base case cycle parameters to study the MSMR behavior................................... 128

7-2 Optimum Cycle parameters for Pso = 128 kPa (Tevap = 79.4 K), Pti = 14.2 MPa. ..138

7-3 Optimum Cycle parameters for Pso = 140 kPa (Tevap = 80.2 K), Pti = 14.2 MPa. ..138
















LIST OF FIGURES


Figure pge

1-1 Schematic of the Solar Integrated Thermal Management and Power (SITMAP)
cy c le ....................................................... ..................... . 1

1-2 Schematic of the Solar Integrated Thermal Management and Power (SITMAP)
cycle w ith regeneration ........................ ...................... ... .. ....... ................ .3

2-1 A schematic of the j et-pump geometry showing the different state points .............10

2-2 Three-dimensional ejector operating surface depicting the different flow regimes
[2 ] ........................................................ ............................ . 1 3

2-3 Relationship between concentration ratio and temperature of the receiver [11]......20

2-4 A cyclic heat engine working between a hot and cold reservoir............................28

2-5 The T-S diagram for a Lorenz cycle ............................................. ............... 32

2-6 Thermodynamic representation of (a) combined power/cooling cycle and (b)
cascaded cycle ..................................................................... ..........34

3-1 Schematic for the jet-pump with constant area mixing .........................................39

3-2 Schematic for the jet-pump with constant area mixing, showing the Fabri choked
state s2 ...............................................................................42

3-3 Jet-pump schematic showing the control volume for the mixing chamber
an aly sis. ............................................................................. 4 5

3-4 A schematic of the SITMAP cycle showing the notation for the different state
p o in ts. ............................................................................ 4 8

3-5 Typical solar collector temperature profile. .................................. .................54

3-6 Overall system schematic for SMR analysis................................. .................56

5-1 Break-off mass flow characteristics from the JETSIT simulation code ..................71

5-2 Break-off mass flow characteristics from Addy and Dutton [2]............................71









5-3 Break-off compression and mass flow characteristics. .........................................72

5-4 Break-off compression and mass flow characteristics from Addy and Dutton [2],
for Api/Am3=0.25. ........................................ ............ ................ 72

5-5 Break-off compression and mass flow characteristics from Addy and Dutton [2],
for Api/Am3=0.333 ................................................... ..............73

6-1 Effect of jet-pump geometry and stagnation pressure ratio on the breakoff
entrainm ent ratio. ................................................... ................. 76

6-2 Effect of jet-pump geometry and stagnation pressure ratio on the compression
ra tio ........................................................................... 7 7

6-3 Effect of jet-pump geometry and stagnation pressure ratio on the System Mass
R atio (SM R ). .........................................................................77

6-4 T-s diagram for the refrigeration part of the SITMAP cycle. .............................78

6-5 Effect of jet-pump geometry and stagnation pressure ratio on the amount of
specific heat rejected. .......... ................................... .......... .. ........ .... 81

6-6 Effect of jet-pump geometry and stagnation pressure ratio on the radiator
tem perature ..................... ... ......... ..... ....... ..... .................. ......... 82

6-7 Effect of jet-pump geometry and stagnation pressure ratio on the amount of
specific heat input ..... ........ ................................... .................... 82

6-8 Effect of jet-pump geometry and stagnation pressure ratio on the specific
co olin g cap city ............................. .................................................. ............... 83

6-9 Effect of jet-pump geometry and stagnation pressure ratio on the cooling
specific rejected heat. ........ ..... ............................... ....... ...... ............ 83

6-10 Effect of jet-pump geometry and stagnation pressure ratio on the cooling
specific heat input ............ ..... .............................. ........ .......... ............ 84

6-11 Effect of jet-pump geometry and stagnation pressure ratio on the overall cycle
efficiency ......... ...... ............ ..................................... ........................... 84

6-12 Effect of jet-pump geometry and stagnation pressure ratio on the ratio of the
overall cycle efficiency to the overall Carnot efficiency. ......................................85

6-13 Effect of secondary superheat on the overall system mass ratio (SMR) ..................87

6-14 Effect of secondary superheat on the break-off compression ratio ..........................87

6-15 Effect of secondary superheat on Qrad/Qcool. ............... ..................................88









6-16 Effect of secondary superheat on Qs/Q oo.................................... ..................... 88

6-17 Effect of secondary superheat on the break-off mass flow characteristics ............89

6-18 Effect of the turbine inlet pressure on the amount of net work rate and specific
heat input to the SITM AP system ........................................ ....................... 90

6-19 Effect of the turbine inlet pressure on the amount of the SMR and overall
efficiency of the SITM AP system .................. ...................... ....................... 90

6-20 SMR and Compression ratio behavior in the mixed regime. ..................................92

6-21 Effect of the entrainment ratio on the mixed chamber exit conditions in the
m ixed regime e ............. .... ..... .. .................................. ... ....... .. 93

6-22 Effect of the entrainment ratio on secondary nozzle exit conditions in the mixed
re g im e ......... .. ..................................... ................................................... 9 3

6-23 Jet-pump compression behavior in the mixed regime.............. ................. 94

6-24 Effect of entrainment ratio on specific heat transfer ratios in the mixed regime.....94

6-25 Effect of the evaporator temperature on the break-off entrainment ratio and the
compression ratio, for Ppo = 3.3 M Pa. ............................. .... ....................... 96

6-26 Effect of the evaporator temperature on T, and SMR, for Ppo = 3.3 MPa ...............97

6-27 Effect of the evaporator temperature on the cooling specific rejected specific
heat, Qrad/Qcool, and the cooling specific heat input, Qsc/Qcool..............................97

6-28 Effect of the evaporator temperature on the effective radiator temperature, for
P p o = 3 .3 M P a ..................................................................... 9 8

6-29 Effect of primary flow superheat on the SMR. ................................... ..................99

6-30 Effect of primary flow superheat on the Qrad/Qcool............... .... ..................... 100

6-31 Effect of primary flow superheat on the Qsc/Qoo............................. ..............100

6-32 Effect of primary flow superheat on the compression ratio.............. ................101

6-33 Sink temperature effect on SM R. .............................................. ........ ....... 102

6-34 Compression ratio effect on the SMR < 1 regime.................................. .....103

6-35 Effect of jet-pump geometry on the break-off sink temperature, for PpoPso=25,
Pso=128 kPa, Tevap=79.4 K, 10 degrees primary superheat ..............................104









6-36 Compression ratio and entrainment ratio variation with jet-pump geometry, for
Ppo/Pso=25.. ................................................... ............ 105

6-37 Effect of stagnation pressure ratio on the break-off sink temperature (77.1).........106

6-38 Break-off sink temperature behavior in the mixed regime (77.1).........................107

6-39 Effect of jet-pump geometry on the SMR for Ppo/Pso=40, Tpo=150 K, Pso=128
kP a, T evap= 79.4 K T = 78.4.................................................................. 108

6-40 Effect of stagnation pressure ratio on the SMR. .........................................108

7-1 Schematic of a cooling and power combined cycle ........................ ............114

7-2 A schematic of the turbo-machinery analog of the jet-pump..............................122

7-3 T-s diagram illustrating the thermodynamic states in the jet-pump turbo-
m machinery analog ........................ .............................. .. ........ .... ....... 123

7-4 Effect of compression efficiency on jet-pump characteristics.............................125

7-5 Effect of compression efficiency on MSMSR. ....................................................126

7-6 Jet-pump efficiency effect on the compression ratio and MSMR for given jet-
pum p inlet conditions. ............................................... ................. ............. 126

7-7 M SM R and SM R are equal for W ext = 0. .................................... .................127

7-8 High pressure effect on the cooling specific heat input and external work output
for a given jet-pump inlet conditions. .............. .............................................. 129

7-9 High pressure effect on the MSMR and efficiency for a given jet-pump inlet
con edition s. ....................................................................... 12 9

7-10 Primary nozzle geometry effect on MSMR at Ane/Ase = 0.4, Ppo/Pso = 26, Pso
= 12 8 kP a ...... ............... ...... ................................................. ............................ 13 1

7-11 Jet-pump geometry effect on MSMR at Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128
kP a ......... ................... ............... .......................... ........ ....... 13 1

7-12 Stagnation pressure ratio effect on MSMR at a fixed jet-pump
geometry(Ant/Ane=0.2, Ane/Ase=0.4), and Pso = 128 MPa. .................................... 132

7-13 Primary nozzle geometry effect on the compression ratio and the entrainment
ratio at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa. .............................................133

7-14 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









7-15 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

7-16 Jet-pump geometry effect on the compression ratio and the entrainment ratio at
Ant/Ane = 0.2, Ppo/Pso = 26, and Pso =128 kPa...................................................... 134

7-17 Jet-pump 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

7-18 Jet-pump 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

7-19 Stagnation pressure ratio effect on the compression ratio and the entrainment
ratio at a fixed jet-pump geometry (Ant/Ane=0.2, Ane/Ase=0.4) ..............................136

7-20 Stagnation pressure ratio effect on the specific heat rejected per unit specific
cooling load at a fixed jet-pump geometry(Ant/Ane=0.2, Ane/Ase=0.4). ...............1.36

7-21 Stagnation pressure ratio effect on the specific heat input per unit specific
cooling load at a fixed jet-pump 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 JET-PUMPED 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 jet-pumped combined

refrigeration/power system, and assess its feasibility, as a thermal-management 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 jet-pump instead of the conventional compressor.

This study presents a detailed component analysis of the jet-pump, allowing for

two-phase subsonic or supersonic flow, as well as an overall cycle analysis. The jet-

pump analysis is a comprehensive one-dimensional 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 jet-pump 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 1-1

shows the standard SITMAP system.





Radiator
Soling Jet Pump
Expansion CLoad
Valve 4


Evaporator

Pump ) Waste Heat Solaor
Collector

-Power

Waste CQsolar


Figure 1-1. Schematic of the Solar Integrated Thermal Management and Power
(SITMAP) cycle









Figure 1-2 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 jet-pump 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 jet-pump 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 jet-pump 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:
High-Pressure 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 1-2. Schematic of the Solar Integrated Thermal Management and Power
(SITMAP) cycle with regeneration

The jet-pump, 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

high-pressure, high-energy primary fluid and a relatively low-energy low-pressure

secondary fluid to produce a discharge of intermediate pressure and energy level. The

high-pressure stream goes through a converging-diverging 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 jet-pump 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 low-grade 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 air-conditioning, solar

powered ejector air-conditioning, and industrial process cooling.

However, despite the abovementioned strong points, conventional steady-flow

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 jet-pump (ejector) efficiency

can be attained, significant improvement in the COPs of such systems will be realized

and jet-pumped 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 cryo-coolers or passive systems such as a radiator. Conventional









cryo-coolers 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 two-phase jet-

pump. All previous work in the literature is limited to jet-pumps 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 Jet-pump and SITMAP). The code includes the jet-pump two-phase one-dimensional

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 non-linear 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 134-a 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 jet-pump analysis, because their analysis only involved constant-pressure mixing in

the jet-pump. 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.

Jet-pumps 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 2-1. 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 2-1. A schematic of the jet-pump 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 constant-area and constant-

pressure ejectors. Their flow model was one-dimensional and assumed perfect gas

behavior. They also conducted an experimental study to which they compared their

analytical results. Addy et al. concluded that the constant-area ejector model predicts the

operational characteristics of ejector systems more realistically than the constant-pressure

model. They introduced a three-dimensional 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 2-2.

Figure 2-2 depicts a three-dimensional ejector solution surface. It should be noted

that in Figure 2-2 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 break-off conditions for transition from one operating

regime to another. The possible transitions are between:

* The "saturated supersonic" and "supersonic" regimes, break-off curve b-d.
* The "saturated supersonic" and "mixed" regimes, break-ff curve b-c.
* The "supersonic" and "mixed" regimes, break-off curve a-b.

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 2-2. As Pm3,/Po is increased from zero, W, W, remains constant

until break-off curve a-b-c, which separates the backpressure-independent from the

backpressure-dependent regimes, is reached. From here, a slight increase in the Pm3 /P

causes a significant drop in FWF Hence, the points along break-off curve a-b-c 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 back-pressure independent regime at or near this break-off 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 break-off 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 2-2. Three-dimensional 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 one-dimensional 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.

Al-Ansary 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. Al-Ansary 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 one-dimensional normal shock. They

found that compression occurs through a series of oblique shocks induced by boundary

layer separation in the diffuser.

Al-Ansary 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

two-phase concept needs further exploration.

Eames [14] conducted a theoretical study into a new method for designing jet-

pumps used in jet-pump cycle refrigerators. The method assumes a constant rate of

momentum change (CRMC) within the mixing section, which in this case is a

converging-diverging 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 jet-pump 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 0.1-0.3.









However, SMEE was not found constant for a wide range of off-design performance,

especially for large changes in the secondary flow.

Earlier work done on two-phase ejectors in the University of Florida includes Lear

et al. [29], and Sherif et al. [38]. These two studies developed a one-dimensional model

for two-phase ejectors with constant-pressure 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 sub-cooled 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 Rankine-Hugoniot relations for two-phase 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 single-phase 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 two-phase ejectors with

constant-area 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 2-2). 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

flat-plate 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 energy-absorbing surface. The smaller absorber

will have smaller heat losses compared to a flat-plate 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 one-dimensional 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 2-1shows 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 2-1. 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 three-dimensional (circular) concentrator or two-dimensional (linear)

concentrators.

Concentrators can be divided into two categories: non-imaging 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 non-imaging 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 2-3 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 40-60%

and represents a probable range of operation. Figure 2-3 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 2-3. Relationship between concentration ratio and temperature of the receiver
[11].

It should noted that Figure 2-3 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, thin-film

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 (0.3-2.5 [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 2-2 gives the properties of few selective surfaces [8]. Effective

selective surfaces have solar absorptivities around 0.95 and emissivities at about 0.1.

Table 2-2. Properties of some selective surfaces.
.rial Short-wave Long-wave
Material
Absorptivity emissivity
Black Nickel on Nickel-plated steel 0.95 0.07
Black Chrome on Nickel-plated 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 low-pressure low-temperature cooling cycle for

comfort air-conditioning. 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 steam-jet 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 steam-jet

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

ejector-compression 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 R142-b 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 single-phase 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 ammonia-water

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, super-heater 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 2-3 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 2-3 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 2-3. 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 2-4), 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 2-4. 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 T-s diagram of the cycle is shown in Figure 2-5.


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 4-1 and 2-3 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 2-5. The T-S 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 2-6 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 2-6b). 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 2-6. 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(2.25-2.26), 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

Jet-pump Analysis

First, it should be noted that the inputs to the jet-pump model are:

* Fully defined stagnation state at the jet-pump primary inlet.
* Fully defined stagnation state at the jet-pump secondary inlet.
* Primary nozzle area ratio Ant/Ane.
* Secondary to primary area ratio, Ane/Ase.

The outputs of the jet-pump model are:

* Break-off entrainment ratio.
* Mixed flow conditions at the jet-pump exit.

The following general assumptions are made for the jet-pump analysis:

* Steady flow at all state points.

* Uniform flows at all state points.

* One-dimensional flow throughout the jet-pump.

* Negligible shear stresses at the jet-pump walls.

* Constant-area 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 3-1 shows a schematic of the jet-pump. The high-pressure primary flow

from the power part of the cycle (State pi) is expanded in a converging-diverging










supersonic nozzle to supersonic speed. Due to viscous interaction secondary flow is

entrained into the jet-pump. Constant-area 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 jet-pump

geometry and the primary and secondary stagnation states is presented next. For each

region of the jet-pump flow-field 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 3-1. Schematic for the jet-pump 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 break-off value for the entrainment

ratio for a given jet-pump 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 jet-pump 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 jet-pump

is operating in the "supersonic" regime. In this regime the secondary flow chokes at an

aerodynamic throat inside the mixing chamber.

For a given jet-pump geometry, there is a break-off 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 break-off (maximum) entrainment ratio, 4bo. The value

of (Ppo/Pso)bo is represented by line "bd" in Figure 2-2, and 4bo is represented by the

curve "abc". (Ppo/Pso)bo affect the jet-pump operation as follows:


PP < PP bo = lnietchoke
Pso Pso bo


P- > P o o fabri
P P
so so bo

The break-off 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 jet-pump 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 3-2 shows a schematic of the jet-pump. 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



I--------I .
Si
Se I


me
n2 de
pi nt ne m



Se I
"I 2
i.s2



Figure 3-2. Schematic for the jet-pump 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 3-2 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 constant-area 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 jet-pump 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 3-3. Jet-pump 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 3-3 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 jet-pump analysis needed for the SITMAP cycle analysis

is the jet-pump exit pressure, which corresponds to the radiator pressure in the SITMAP

cycle.

Figure 3-4 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.















Jet-pump




(ei) (si) (pi)
Evaporator




Pump
(te)
I Boiler
(pe) (bi) (t) Turbine
Recuperator


Figure 3-4. 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

jet-pump 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 jet-pump secondary inlet pressure, and assuming iso-enthalpic expansion,

he, = hre, this defines the evaporator inlet state (ei). So straight out of the jet-pump

analysis all the states in the refrigeration part of the SITMAP cycle are defined.

System convergence requires a double-iterative 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 two-phase 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 two-phase









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 two-phase

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.

Two-phase region analysis

An energy balance can be written for the portion of the solar collector operating in

the two-phase region as follows

m [h, (x = 1) -h ]= G(CR)caA uo-A (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, o-W(T4(x)- T4)dx (3.42)









dT
1hC, = G(CR)aW- co-W(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 o-W(T4 (,() T4) (3.45)
LSH dx

Multiply through by LsH

dT
ihCP d = G(CR)aA, -scA, (T4() -T4) (3.46)

Now we non-dimensionlize 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, -co-A T (T*)-T ') (3.47)


If we divide both sides by A mCHCT and rearrange

1 dT G(CR)a co-T
A, d jhCpT hCC ( C

dT*
dT = A di (3.48)
G(CR)a co-T3 (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
2tan-1 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 (two-phase) 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 3-5 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 3-5. 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 two-phase

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 jet-pump exit, Equation (3.56)

can be analytically integrated, using the constant value of the saturation temperature at

the radiator pressure.

System Mass Ratio

Figure 3-6 shows a schematic for the thermally actuated heat pump system being

considered. The power subsystem accepts heat from a high-temperature 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 3-6. 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 cross-sectional 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 near-planetary or solar missions this may not be the case.

Qs = eArdad T-d4 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 non-dimensional 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)
1-i ,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

Non-dimensionalizing 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 non-linear 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 n-nb natural variables and the nb slack variables associated with the binding

constraints. These n variables are called the non-basic 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 non-basic 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 Kuhn-Tucker

conditions are satisfied. The Kuhn-Tucker 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)
J-1

uJ > 0, uJ g (X) -ub(j) =0 (4. 11)

g, (X)
Here, uj is a Lagrange Multiplier for the inequality constraints.

Unfortunately, the Kuhn-Tucker 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 Newton-Raphson

method used during the one-dimensional 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 4-1 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 4-1. 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 4-2 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 4-2 follows. A constraint

was used to make sure that the jet-pump 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 4-2. Constraints used in the optimization
Lower Upper
Constraint DescriptionLower Upper
escripioLimit Limit
Pipe/Psi > I Jet-pump 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

Jet-pump Results

In order to validate the JETSIT simulation code, results are compared to the

literature using single-phase models. Addy and Dutton [2] studied constant-area 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 5-1. Figure 5-1

and Figure 5-2 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 5-1 and

Figure 5-2 it can be seen that the JETSIT code gave the exact same break-off mass flow

results presented by Addy and Dutton.

Figure 5-3 shows the compression characteristics at break-off conditions. The

region above the break-off curves represents the "mixed regime" where the entrainment

ratio is dependent on the back pressure, while the region below the break-off curves

represent the "supersonic" and "saturated supersonic" regimes where the mass flow is

independent of the back-pressure. The bold lines in Figure 5-3 show the same

entrainment ratio values at break-off conditions shown in Figure 5-1, but were included

in Figure 5-3 for ease of comparison with the Addy and Dutton results shown in Figure 5-









4, and Figure 5-5. Addy and Dutton show the break-off mass flow rates in Figure 5-4,

and Figure 5-5 below the vertical lines which match the values shown by the bold curves

in Figure 5-3. The vertical lines under the break-off 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 back-pressure drops. Comparing the results

shown in Figure 5-3 to those in Figure 5-4, and Figure 5-5 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

two-phase ejector. It should also be noted that the jet-pump results presented in this

study will not be in perfect agreement with the real-life performance of such device

because of the simplifying assumptions made in the model, such as the isentropic flow

assumption in the all the jet-pump nozzles. Also the accuracy of the results will be bound

by the precision of the thermodynamic properties routines used (REFPROP 7).

Table 5-1. Representative constant-area 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 = / 2-20














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 5-1. Break-off mass flow characteristics from the JETSIT simulation code.


P /Pso
Pc F SO


Figure 5-2. Break-off 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 5-3. Break-off compression and mass flow characteristics.


0o To 200 300
Ppo/Pso

Figure 5-4. Break-off 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 break-off 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 5-5. Break-off 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 Jet-pump and SITMAP). The input parameters to the

JETSIT simulation code are summarized in Table 6-1. 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 jet-pump performance at a given cooling

load temperature.

Parametric analysis was performed to study the effect of different parameters on the

jet-pump and SITMAP cycle performance. These parameters are the jet-pump 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 jet-pump, evaporator

temperature, quality of primary flow entering the jet-pump, 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, system-level 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 6-1. Input )arameters to the JETSIT cycle simulation code
Variable name Description
Pp Jet-pump primary inlet stagnation
pressure
Xpo Jet-pump primary inlet quality

Ps Jet-pump secondary inlet stagnation
pressure

Xso Jet-pump 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.


Jet-pump Geometry Effects

Figure 6-1 illustrates the effect of the jet-pump geometry on the break-off

entrainment ratio. The jet-pump 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 6-1 shows the

variation of the break-off entrainment ratio versus the stagnation pressure ratio for

different jet-pump 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 6-1. 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 6-3 that the jet-pump 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 6-2, 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 6-1. Effect of jet-pump 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 6-2. Effect of jet-pump 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 6-3. Effect of jet-pump 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 T-s diagram in Figure 6-4. 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 (1-2-4'-5'-1), the second is when

the compression ratio is higher and Pc is the radiator pressure (1-3-4-5-1). 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

(Q3-4 < Q2-4'), however, the amount of cooling decreases even faster (Q-5 << Q-5'). 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 6-4. T-s 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 jet-pump geometry effects on the cycle performance are presented in









Table 6-2. 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 jet-pump 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 jet-pump

secondary inlet flow is always restricted to saturated vapor. The secondary flow

parameters correspond to Tevap = 79.4 K. The jet-pump 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 6-2. 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 6-1 showed the effect of the jet-pump geometry and stagnation pressure

ratio on the break-off entrainment ratio. It can be seen that the break-off 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 6-2 and Figure 6-3 show the variation of the compression ratio and the

SMR, respectively, with Ppi/Psi, for different jet-pump geometries. The compression ratio

and SMR are calculated at the break-off entrainment ratio. Therefore all of these data

points correspond to points on the a-b-c (break-off) curve in Figure 2-2. It can be seen in

Figure 6-2 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 6-5 through Figure 6-8.

Figure 6-5 through Figure 6-8 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 6-6, 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 6-3, 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 6-9, and Figure 6-10 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 6-5. Effect of jet-pump geometry and stagnation pressure ratio on the amount of
specific heat rejected.


























85

84 -

83 -

82 -,

81

5 10 15
piPsi
Figure 6-6. Effect of jet-pump geometry and
temperature.


stagnation pressure ratio on the radiator


220000

215000



210000



205000
S -
a
200000


195000


190000


185000
-


piPsi
Figure 6-7. Effect of jet-pump 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 6-8. Effect of jet-pump 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 6-9. Effect of jet-pump 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 6-10. Effect of jet-pump 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 6-11. Effect of jet-pump geometry and stagnation pressure ratio on the overall
cycle efficiency.










Figure 6-11 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 6-10. 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 6-12 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 6-12. Effect of jet-pump 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 jet-pump 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 jet-pump inlet with all the other parameters fixed. The

complete configuration is presented in Table 6-3.

Table 6-3. 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 6-14 show that the degree of superheat does not have a significant effect on

the compression characteristics of the jet-pump. 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 6-15, and Figure 6-16, respectively. This causes the SMR to

drop, as shown in Figure 6-13.

Figure 6-17 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