analysis and optimization of a jet-pumped...
TRANSCRIPT
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 FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2006
Copyright 2006
by
Sherif M. 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.
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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.
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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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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 DISCUSSION: 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: COOLING 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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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 refrigeration cycle efficiency definitions....27
4-1 Optimization variables and their limits ....................................................................67
4-2 Constraints used in the optimization ........................................................................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, 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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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 geometry showing the different state points. .............10
2-2 Three-dimensional ejector operating surface 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 constant 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 showing 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 characteristics from Addy and Dutton [2]. .............................71
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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 Ap1/Am3=0.25. .....................................................................................................72
5-5 Break-off compression and mass flow characteristics 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 stagnation pressure ratio on the compression ratio...........................................................................................................................77
6-3 Effect of jet-pump geometry and stagnation pressure ratio on the System Mass Ratio (SMR). ............................................................................................................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 temperature...............................................................................................................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 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 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
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6-16 Effect of secondary superheat on Qsc/Qcool...............................................................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 SITMAP system. ...........................................................................90
6-19 Effect of the turbine inlet pressure on the amount of the SMR and overall efficiency of the SITMAP 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 mixed regime............................................................................................................93
6-22 Effect of the entrainment ratio on secondary nozzle exit conditions in the mixed regime.......................................................................................................................93
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 MPa. ......................................................................96
6-26 Effect of the evaporator temperature on ξT, 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 superheat 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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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 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 thermodynamic states in the jet-pump turbo-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-pump inlet conditions. ............................................................................................126
7-7 MSMR and SMR are equal for Wext = 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 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 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
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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). .................136
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). .................137
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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
Sherif M. 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
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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.
1
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.
Figure 1-1. Schematic of the Solar Integrated Thermal Management and Power
(SITMAP) cycle
2
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
3
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.
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 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
4
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 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 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
5
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
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 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.
8
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
9
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,
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 secondary mass flow rate
for a given secondary pressure and a given primary pressure.
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.
11
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 (entrainment ratio), s pW W , is
independent of the backpressure ratio, 3m osP P , so that these two surfaces are
perpendicular to the s pW W - 3m osP P 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, 1 1p sP 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, 1 1s pP 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
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 stagnation pressure ratio,
op osP P , in Figure 2-2. As 3m osP P is increased from zero, s pW 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 3m osP P
causes a significant drop in s pW W . Hence, the points along break-off curve a-b-c are of
particular importance since they represent the highest values of 3m osP P for which
s pW 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.
13
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 φ 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
14
performance variables, entrainment ratio,φ, 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
15
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
( )2
rate of momentumrate of kinetic energy 2m
mVKm V
= =&
& (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 exchange effectiveness expressions. It was found
that in almost all evaluations, the design point value of SMEE ranged between 0.1-0.3.
16
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,φ, 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
17
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 φ is low. As φ
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 φ. For low values of φ, the highest
compression ratio occurs at the lowest area ratio. For the higher values of φ, 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 (concentrator)
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
18
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].
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 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
r
ACRA
= (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
20
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.
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
21
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 (α 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 (α) in the visible and
22
near infrared spectrum (0.3-2.5 μm) and low emittance (ε) 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
23
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.
Material Short-wave Absorptivity
Long-waveemissivity
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
24
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
25
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,
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 = 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.
outexergy
in
EE
η Σ=
Σ (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,
27
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 compressionFirst Law I net HW Qη = c inCOP Q W= Exergy exergy net inW Eη = exergy c inE Wη =
Second law II revη η η= II revCOP COPη = reinjection has to be accounted for. Therefore, a modified definition of the form
outR
hs
EE
η Σ=
Σ (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],
usefulexergy
in useful
Ei
E EΣ
=Σ − Σ
(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.
II revη η η= (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].
28
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
( )1net
exergyh o h
WQ T T
η =−
(2.8)
while the second law efficiency is
( )1net
exergyh c h
WQ T T
η =−
(2.9)
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.
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
Th
Tr
Wnet Cyclic device
29
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
net cI
h
W QQ
η += (2.10)
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
30
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
net cI
h
W EQ
η += (2.11)
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.
( ), , , ,c cf in cf out o cf in cf outE m h h T 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.
31
, ,
net cexergy
hs in hs out
W EE E
η +=
− (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 “reversible cycle” for use with variable
temperature heat input and rejection. A T-s diagram of the cycle is shown in Figure 2-5.
34
12
1LorenzQQ
η = − (2.14)
If the heat input and rejection were written in terms of the heat source and heat rejection
fluids, the efficiency would be given as:
( )( )
, ,
, ,
1 hr hr out r inLorenz
hs hs in hs out
m h hm h h
η−
= −−
(2.15)
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
( )( )
, ,
, ,
hr out r inhs
hr hs in hs out
s smm s s
−=
− (2.16)
The efficiency expression for the Lorenz cycle then reduces to
( ) ( )( ) ( )
, , , ,
, , , ,
/1
/hr out r in hr out r in
Lorenzhs in hs out hs in hs out
h h s sh h s s
η− −
= −− −
(2.17)
This can also be written as
32
( )( )1 s hr
Lorenzs hs
T
Tη = − (2.18)
Here, the temperatures in the expression above are entropic average temperatures, of the
form
2 1
2 1s
h hTs s
−=
− (2.19)
Figure 2-5. The T-S diagram for a Lorenz cycle
For constant specific heat fluids, the entropic average temperature can be reduced to
( )2 1
2 1lnsT TT
T T−
= (2.20)
The Lorenz efficiency can therefore be written in terms of temperatures as
( ) ( )( ) ( )
, , , ,
, , , ,
/ ln /1
/ ln /hr out hr in hr out r in
Lorenzhs in hs out hs in hs out
T T T TT T T T
η−
= −−
(2.21)
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
T
s
1
2
4
3
33
( )( ) ( )
s cfLorenz
s shr cf
TCOP
T T=
− (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 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
,out c c
I sysh
W W fQQ
η − += (2.23)
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
,net c
I sysh
W fQQ
η += (2.24)
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 cascaded cycle can also be related using the
efficiencies of the cascaded devices
out h HEW Q η= (2.25)
c cW Q COP= (2.26)
By specifying identical refrigeration to work ratios (r) in the combined cycle and the
corresponding reversible cascaded cycle as
c netr Q W= (2.27)
and using Equation(2.23) and Equations(2.25-2.26), one can arrive at the efficiency of the
cascaded system as
( ),
11
1I sys HE
r f COPr COP
η η⎡ ⎤−⎢ ⎥= +⎢ ⎥+⎢ ⎥⎣ ⎦
2.28
assuming the cascaded cycle to be reversible, the efficiency expression reduces to
Th
WnetCyclic device Tr
Tc
Th
Wnet HE
Tc
Tr
REF
Wout
Qh
Qc
35
,
11
1Lorenz
I rev LorenzLorenz
r f COPr COP
η η
⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎝ ⎠= +⎢ ⎥+⎢ ⎥
⎢ ⎥⎣ ⎦
(2.29)
Here Lorenzη 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
,II I I revη η η= (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
, , ,
net cIII
I rev net rev c rev
W fQW fQ
ηηη
+= =
+ (2.31)
This reduces further to
( )( ), ,
11
netIII
I rev net rev
W frW fr
ηηη
+= =
+ (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 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,
,1
1I rev LorenzLorenz
rr COP
η η⎡ ⎤+
= ⎢ ⎥+⎣ ⎦ (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
36
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
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 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.
,net c practical
I effh
W Q COPQ
η+
= (2.34)
,,
net c II refI eff
h
W EQ
ηη
+= (2.35)
,, ,
net c practicalexergy eff
hs in hs out
W Q COPE E
η+
=−
(2.36)
,,
, ,
net c II refexergy eff
hs in hs out
W EE E
ηη
+=
− (2.37)
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 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, me ne seA 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
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 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, φ, 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.
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,
VAA
Vpint
pi
nt
pint=
ρρ
(3.1)
The velocity at the nozzle throat is calculated using conservation of energy,
pi nt ne
si se
se si
deme
40
Vh h
AA
ntpi nt
nt
pi
nt
pi
=−
−⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
12
12
12
ρρ
(3.2)
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
as
Pρ
⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠
(3.3)
M Va
= (3.4)
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
V h V hne nt nt ne= + −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥2 1
22
12 (3.5)
Ant/Ane is calculated using the continuity equation
AA
VV
nt
ne
ne
nt
ne
nt=
ρρ
(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 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-
41
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, boφ . The value
of (Ppo/Pso)bo is represented by line “bd” in Figure 2-2, and boφ is represented by the
curve “abc”. (Ppo/Pso)bo affect the jet-pump operation as follows:
po po
so so bo
P PP P
⎛ ⎞< ⎜ ⎟
⎝ ⎠ ⇔ bo inletchokeφ = φ
po po
so so bo
P PP P
⎛ ⎞> ⎜ ⎟
⎝ ⎠ ⇔ bo fabriφ = φ
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=1.
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 φinlet 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 φinlet choke is then calculated
from continuity as
se se seinletchoke
ne ne ne
V AV A
ρφρ
= (3.7)
Figure 3-2. Schematic for the jet-pump with constant area mixing, showing the Fabri
choked state s2.
pi nt ne
si se
se si
s2
s2
n2 deme
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.
• Ms2 = 1.
• The primary inlet static pressure is always larger than secondary inlet static pressure, Pne > Pse.
The following analysis is used to calculate φfabri 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
2 2 2 2 2 2se se ne ne s s n n p n s s p ne s seP A P A P A P A m V m V m V m V+ − − = + − −& & & & (3.8)
dividing by pm& yields
( ) ( ) ( )2 2 2 2 2 21
se se ne ne s s n n n ne Fabri s sep
P A P A P A P A V V V Vm
φ+ − − = − + −&
(3.9)
( )( )
( )( )
2 2 2 2 2
2 2
se se ne ne s s n n n neFabri
p s se s se
P A P A P A P A V Vm V V V V
φ+ − − −
∴ = −− −&
(3.10)
( )( )( )
2 22 2
2
22
ne s n nese ne s n
n nese se ne seFabri
ne s sene ne s se
se
A A A AP P P PV VA A A A
A V VV V VA
φρ
⎛ ⎞+ − −⎜ ⎟ −⎝ ⎠∴ = −
−− (3.11)
The iteration scheme starts by guessing a value for seP , knowing that se sis s= , that
defines the state (se). From the energy equation
( )1
22se si seV h h⎡ ⎤= −⎣ ⎦ (3.12)
44
Fabriφ can then be calculated as,
se se seFabri
ne ne ne
V AV A
ρφρ
= (3.13)
It should be noted that the area ratio ne seA A is an input to the SITMAP code. Then a
guess is made for 2sP , and 2s ses s= , which defines state (s2). The velocity 2sV can be
obtained from the energy equation between (se) and (s2)
122
2 222se
s se sVV h h
⎡ ⎤⎛ ⎞= − +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦ (3.14)
Ps2 is iterated on till Ms2=1. The area ratio 2s seA A is calculated from the continuity
equation between (se) and (s2),
2
2 2
s se se
se s s
A VA V
ρρ
= (3.15)
For constant-area mixing 2 2ne se s nA A A A+ = + , then
2 21n se s se
ne ne se ne
A A A AA A A A
⎛ ⎞= + − ⎜ ⎟
⎝ ⎠ (3.16)
seP is iterated on till the values for Fabriφ 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.
45
Secondary Flow
When the jet-pump is operating in the mixed regime ( boφ < φ ), 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.
V h V hse si si se= + −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥2 1
22
12 (3.17)
AA
VV
ne
se
se
ne
se
ne=
1φ
ρρ
(3.18)
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
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
46
( ) ( ) 1me ne se ne ne se se p ne p se p meP A A P A P A m V m V m Vφ φ− + + + = − − + +& & & (3.19)
given &m A Vp ne ne ne= ρ , and the constant area mixing process, me ne seA A A= + , Equation 19
can be rearranged as
( ) ( )
( )
2 2
1
ne nene me se me ne ne se se
se seme
nene ne
se
A AP P P P V VA AV AV
A
ρ ρ
φ ρ
− − − + +=
+ (3.20)
Then the enthalpy hme is calculated from the energy equation for the mixing chamber
2 2 21 1 1 11 2 2 2me ne ne se se meh h V h V Vφ
φ⎡ ⎤⎛ ⎞ ⎛ ⎞= + + + −⎜ ⎟ ⎜ ⎟⎢ ⎥+ ⎝ ⎠ ⎝ ⎠⎣ ⎦
(3.21)
Then from continuity
1me me me
ne ne ne
A VA V
ρφρ
= − (3.22)
Pme is iterated on till the value of φ 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.
If Mme > 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.
ρ ρme me ss ssV V= (3.23)
47
P V P Vme me me ss ss ss+ = +ρ ρ2 2 (3.24)
h V h Vme me ss ss+ = +12
12
2 2 (3.25)
( )ρ ρss ss ssP h= , (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
VAA
AA Vde
me
de
me
ne
ne
deme=
ρρ
(3.27)
V h V hde me me de= + −⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥2
12
2
12
(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.
48
Boiler
Jet-pump
(re)
Pump
Turbine(ti)
(pi)(si)
Recuperator
Radiator(jpe)
(bi)(pe)
Evaporator
(te)
(ei)
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,
h hei re= , 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
49
same state, sti (or any other independent property like the enthalpy). Then the pump work
can be calculated as
( )&&
Wm
P Ppumpp
pump repe re= −
η ρ (3.29)
Energy balance across the pump yields,
pe re pumph h W= + & (3.30)
Now state (pe) is defined. The recuperator efficiency is assumed to be 0.7 and is
defined as
max
= H C
recupQ or Q
Qη (3.31a)
where,
maxQ ( , )⎡ ⎤= −⎣ ⎦& p te te pem h h P T (3.31b)
Equation 3.31a, and 3.31b are combined yielding,
( , )−
=−
te pirecup
te te pe
h hh h P T
η (3.32)
( , )1
−∴ =
−pi recup te pe
terecup
h h P Th
ηη
(3.33)
The specific enthalpy from Equation (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,
ti tets ti
t
h hh hη−
= − (3.34)
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
50
( )t p ti teW m h h= −& & (3.35)
Pti is iterated on (repeat the entire SITMAP analysis) until the net work, T pW W−& & , is
positive. In other words the analysis stops when it finds the minimum turbine inlet
pressure that yields positive net work, i.e. 0− ≥& &t pW W .
A converged solution has now been obtained for the SITMAP cycle. The following
equations complete the analysis:
( )& &Q m h hevap p si ei= −φ (3.36)
( )( )& &Q m h hrad p de re= + −1 φ (3.37)
( )& &Q m h hsc p ti pe= − (3.38)
( ) ( )recup p bi pe p te piQ m h h m h 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, satT , 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
51
region, and is denoted satrA , and the second part operates in the superheated region and is
denoted SHrA . 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
( ) ( )4 41 ( )sat satsat i r r sat sm h x h G CR A A T Tα εσ= − = − −⎡ ⎤⎣ ⎦& (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 forsatrA .
( )( )4 4
1
( )sat
sat ir
sat s
m h x hA
G CR T Tα εσ
= −⎡ ⎤⎣ ⎦=− −
& (3.41)
Superheated region analysis
4 4( ) ( ( ) )SHp r SmC dT G CR A W T x T dxα εσ= − −& (3.42)
52
4 4( ) ( ( ) )p SdTmC G CR W W T x Tdx
α εσ= − −& (3.43)
Let 1 1
SH SH SH
x dT dTx dx dxL L dx L dx
= ⇒∴ = ⇒∴ =% %%
(3.44)
4 4( ) ( ( ) )pS
SH
mC dT G CR W W T x TL dx
α εσ= − −&
%%
(3.45)
Multiply through by SHL
4 4( ) ( ( ) )SH SHp r r S
dTmC G CR A A T x Tdx
α εσ= − −& %%
(3.46)
Now we non-dimensionlize the dependent variable T dividing it by the evaporator
temperature, we let
*
e
TTT
= ⇒ * 1
e
dT dTT
∴ = ⇒ *
edT dTTdx dx
∴ =% %
4 4*
4 * *( ) ( ( ) )SH SHp e r r e s
dTmC T G CR A A T T x Tdx
α εσ= − −& %%
(3.47)
If we divide both sides by SHr p eA mC T& and rearrange
4 43*
* *1 ( ) ( ( ) )SH
es
r p e p
TdT G CR T x TA dx mC T mC
εσα= − −%
% & &
4 4
*
3* *( ) ( ( ) )
SHre
sp e p
dT A dxTG CR T x T
mC T mCεσα
=− −
%
%& &
(3.48)
This separable ordinary differential equation can be written in the form
4
*
* ( ) SHrdT A dx
a bT x=
−%
% (3.49)
Where
53
43
*( ) es
p e p
TG CRa TmC T mC
εσα= +
& &; and
3e
p
TbmCεσ
=&
Integrating equation 3.49 for the limits
* * *sat oT T T≤ ≤ ; and 0 1x≤ ≤%
yields the expression below for the area of the superheated region of the collector,SHrA .
*
*
1 *4 1 1 1 11 * *4 4 4 41
4
314 4
2 tan ln ln
4
o
SH
sat
T
r
T
b T a b T a b Ta
Ab a
−⎧ ⎫⎡ ⎤
⎡ ⎤ ⎡ ⎤⎪ ⎪− − + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪⎢ ⎥⎪ ⎪⎣ ⎦= ⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭
(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.
SH satr r rA A A= + (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:
* * *satT T T≤ ≤ ; and 0 x x≤ ≤% %
Figure 3-5 shows a typical temperature profile in the solar collector.
54
0 0.25 0.5 0.75 1x/L
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
T*Pso = 128 kPaTso = 79.4 KTs = 78.4G = 1300 W/m2
CR = 100α = 0.95ε = 0.1Ar = 1.31 m2
T*in = 3.43
T*out = 5.19
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.
( )4 4( ) r r eff sm h G CR A A T Tα εσΔ = − −& (3.52)
In the above equation the enthalpy difference, hΔ , is the overall enthalpy
difference between the inlet and outlet of the solar collector. Solving the above equation
for effT yields
14
4 ( ) reff s
r
G CR A m hT TA
αεσ
⎡ ⎤− Δ= +⎢ ⎥
⎣ ⎦
& (3.53)
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 available is the product of
the solar irradiation, G[W/m2], and the aperture area of the concentrator, Aa [m2].
a
m hGA
η Δ=&
(3.54)
The aperture area can be calculated from the concentration ratio expression.
a rA CR 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, ηrad,
dAm
T dhradp
radrad rad=
−−
&
η σ4 (3.56)
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
56
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.
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
orad
sysradcol
mmmm
m,
~ ++= (3.57)
Equation (3.57) can be separated and rewritten in terms of collector and radiator areas
W
Qs
Ts
Power Cycle Refrigeration Cycle
TH Te,res
QH Q’e
57
orad
sys
orad
radcolrad
col
mm
A
AAm
,,
~ ++
=λλ
(3.58)
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.
( )4 4s rad rad sQ A T Tεσ= − (3.59)
The idealized passive radiator model operates perfectly (ε = 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
( )4 4,passive rad o e s eQ A T T Qσ= − = (3.60)
Defining a new non-dimensional parameter, α, as rad
col
λλα = , performing an overall
energy balance on the active system yielding H e sQ Q Q+ = , and substituting Equations
(3.54), (3.59), and (3.60) into Equation (3.58), yields
4
4 4 44
4 4,
11 11
1 1
s
syses e e eH
e e col sun rad rad rad os s
rad rad
TmTT T T TQm
Q T G T T mT TT T
αεσε η
⎡ ⎤ ⎫⎛ ⎞ ⎛ ⎞⎛ ⎞⎢ ⎥ ⎪⎜ ⎟ ⎜ ⎟− ⎜ ⎟⎧ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎪⎪ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟= − + + +⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎛ ⎞ ⎛ ⎞⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎢ ⎥⎝ ⎠ ⎜ ⎟ ⎜ ⎟⎩ − −⎜ ⎟ ⎜ ⎟ ⎪⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎭
%
(3.61)
58
Substituting the following definitions
H
WQ
η = ; eQCOPW
= ; e
H
QCOPQ
η = (3.62)
CP η
ηξ = ; C
R COPCOP
=ξ ; RPT ξξξ = (3.63)
col
radC T
T−=1η ;
erad
eC TT
TCOP
−= ;
suncol
e
GT
ηαεσ
ζ4
= (3.64)
e
colcol
TTT =* ;
e
radrad
TT
T =* ; e
ss
TTT =* (3.65)
into Equation (3.61) yields
( )( )( ) orad
sys
srad
s
sradradcolT
sradcol
mm
TTT
TTTTTTTm
,4*4*
4*
4*4***
4*** 11111~ +⎪⎭
⎪⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−
+⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
−−−
= ζξε
(3.66)
Non-dimensionalizing the third term on the right hand side of Equation (3.66) yields
mmm
mm
mm
actt
sys
orad
actt
actt
sys ~,,
,
,⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ (3.67)
But syscolradactt mmmm ++=, , therefore
( )( )( ) m
mm
TTT
TTTTTTTm
actt
sys
srad
s
sradradcolT
sradcol ~11111~,
4*4*
4*
4*4***
4***
+⎪⎭
⎪⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−
+⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
−−−
= ζξε
(3.68)
Defining actt
sys
mm
,
=μ yields
( )( )( )
( ) ⎪⎭
⎪⎬⎫
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
⎩⎨⎧
−−−
−= 4*4*
4*
4*4***
4*** 111111~
srad
s
sradradcolT
sradcol
TTT
TTTTTTTm ζ
ξμε (3.69)
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 Tcol* , radiator
temperature *radT , and the overall percentage Carnot efficiency ξT.
60
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
61
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: )(1 Xgm+
Subject to equality and inequality type constraints as given below
0)( =Xgi neqi ,...,1= (4. 1)
0 ( ) ( )ig X ub n i≤ ≤ + mneqi ,.....,1+= (4. 2)
The variables are constrained by an upper and lower bounds.
62
)()( iubXilb i ≤≤ ni ,...,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, mnn XX ++ ,...,1
The optimization program then becomes
Minimize: )(1 Xgm+
Subject to:
0)( =− +ini XXg , mi ,...,1= (4. 4)
)()( iubXilb i ≤≤ , mni += ,...,1 (4. 5)
0)()( == iubilb , neqnni ++= ,...,1 (4. 6)
0)( =ilb , mnneqni +++= ,...,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 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
63
0),( =xyg (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.
)()),((1 xFxxygm =+
This equation is reasonably valid in the neighborhood of the current point to a simpler
reduced problem.
Minimize )(xF
Subject to the variable limits for the components of the vector x.
uxl ≤≤ (4. 9)
The gradient of the reduced objective )(xF , )(xF∇ 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
ii yg ∂∂ 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
64
35]. It can be mathematically explained in terms of the gradients of the objective
functions and inequality constraints as:
∑=
=+ =∇+∇
mj
jjjm XguXg
11 0)()( (4. 10)
0≥ju , 0)()( =− jubXgu jj (4. 11)
)()( jubXg j ≤ , mj ,....,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 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.
)( jubVu j ∂
∂−= (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 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
65
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
66
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.
67
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 ratioAnt/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
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 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
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 < ηcol < 0.99 Collector efficiency has to be lower than 0.99 0 0.99 Δhrecup > 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
69
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, φ, 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-
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 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
γs 1.405
γp 1.405
MWs / MWp 1
Tso / Tpo 1
p1 m3 se neA A 1/(1 A A )= + 0.25,0.333
Mp1 = Mne 4
p sm / m 1/= φ& & 2 - 20
71
0 100 200 300Ppi/Psi
0
2
4
6
8
10
12
14
16
18
201/
φ
Mp1 = 4, Ap1/Am3 = 0.33333Mp1 = 4, Ap1/Am3 = 0.25
Figure 5-1. Break-off mass flow characteristics from the JETSIT simulation code.
Figure 5-2. Break-off mass flow characteristics from Addy and Dutton [2].
72
0 100 200 300Ppi/Psi
0
2
4
6
8
10
12P
me/P
si
0
5
10
15
20
25
1/φ
Mp1 = 4, Ap1/Am3 = 0.33333Mp1 = 4, Ap1/Am3 = 0.25Mp1 = 4, Ap1/Am3 = 0.33333Mp1 = 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.
73
Figure 5-5. Break-off compression and mass flow characteristics from Addy and Dutton
[2], for Ap1/Am3=0.333.
74
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
75
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 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 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.
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 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.
5 10 15 20 25 30Ppi/Psi
0
1
2
3
4
5
6
7
8
9
φ
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/Ane,Ane/Ase = 0.35,0.3
Figure 6-1. Effect of jet-pump geometry and stagnation pressure ratio on the breakoff
entrainment ratio.
77
5 10 15 20 25 30Ppi/Psi
1
2
3
4
Pde
/Psi
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/Ane,Ane/Ase = 0.35,0.3
Figure 6-2. Effect of jet-pump geometry and stagnation pressure ratio on the
compression ratio.
5 10 15 20 25 30Ppi/Psi
3
4
5
6
7
8
9
10
11
SM
R
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/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).
78
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 Pc. 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 (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 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
79
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
80
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.
81
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 Qsc) 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 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.
5 10 15 20 25 30Ppi/Psi
200000
400000
600000
800000
1E+06
1.2E+06
1.4E+06
1.6E+06
1.8E+06
Qra
d
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/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.
82
5 10 15 20 25 30Ppi/Psi
81
82
83
84
85
86
87
88
89
90
91
92
93
T rade
ff
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/Ane,Ane/Ase = 0.35,0.3
Figure 6-6. Effect of jet-pump geometry and stagnation pressure ratio on the radiator
temperature.
5 10 15 20 25 30Ppi/Psi
185000
190000
195000
200000
205000
210000
215000
220000
Qsc
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/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.
83
5 10 15 20 25 30Ppi/Psi
0
200000
400000
600000
800000
1E+06
1.2E+06
1.4E+06
1.6E+06
1.8E+06
Qco
ol
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/Ane,Ane/Ase = 0.35,0.3
Figure 6-8. Effect of jet-pump geometry and stagnation pressure ratio on the specific
cooling capacity.
5 10 15 20 25 30Ppi/Psi
2
4
6
8
10
12
Qra
d/Qco
ol
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/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.
84
5 10 15 20 25 30Ppi/Psi
2
4
6
8
10
12
14
Qsc
/Qco
ol
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/Ane,Ane/Ase = 0.35,0.3
Figure 6-10. Effect of jet-pump geometry and stagnation pressure ratio on the cooling
specific heat input.
5 10 15 20 25 30Ppi/Psi
0
2
4
6
8
ηCO
P
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/Ane,Ane/Ase = 0.35,0.3
Figure 6-11. Effect of jet-pump geometry and stagnation pressure ratio on the overall
cycle efficiency.
85
Figure 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 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, ξT. It can be seen that there is a maximum for ξT 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.
5 10 15 20 25 30Ppi/Psi
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
ξ T
Ant/Ane,Ane/Ase = 0.25,0.1Ant/Ane,Ane/Ase = 0.25,0.2Ant/Ane,Ane/Ase = 0.25,0.3Ant/Ane,Ane/Ase = 0.35,0.1Ant/Ane,Ane/Ase = 0.35,0.2Ant/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 the overall Carnot efficiency.
86
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=1) 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
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 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 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 secondary flow superheat on the breakoff
entrainment ratio. It can be seen that φ 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
if xsi<1, but once the secondary flow is saturated vapor, the amount of superheat does not
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
SM
R
xsi = 0.5xsi = 1.05 degrees superheat10 degrees superheat15 degrees superheat
Figure 6-13. Effect of secondary superheat 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 = 0.5xsi = 1.05 degrees superheat10 degrees superheat15 degrees superheat
Figure 6-14. Effect of secondary superheat on the break-off compression ratio.
88
1.35
1.4
1.45
1.5
1.55
1.6
1.65Q
rad/Q
cool
xsi = 0.5xsi = 1.05 degrees superheat10 degrees superheat15 degrees superheat
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
/Qco
ol
xsi = 0.5xsi = 1.05 degrees superheat10 degrees superheat15 degrees superheat
Figure 6-16. Effect of secondary superheat on Qsc/Qcool.
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
φ
xsi = 0.5xsi = 1.05 degrees superheat10 degrees superheat15 degrees superheat
Figure 6-17. Effect of secondary superheat on the break-off mass flow 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 increases 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 cooling capacity it is better from the SMR and the
90
overall cycle efficiency standpoint to operate at the lowest possible turbine inlet pressure
that yields minimum amount of positive net work.
0 2 4 6 8 10 12Pti/Pte
0
100000
200000
300000
400000
500000
600000
Wne
t
0
100000
200000
300000
400000
500000
600000
700000
Qsc
WnetQsc
Figure 6-18. Effect of the turbine inlet pressure on the amount of net work rate and
specific heat input to the SITMAP system.
0 2 4 6 8 10 12Pti/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
SM
R
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
ηCO
P
SMRηCOP
Figure 6-19. Effect of the turbine inlet pressure on the amount of the SMR and overall
efficiency of the SITMAP system.
Even though it is useful to investigate the 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
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 cooling capacity; therefore,
the SMR definition is accurate only when the only output of the SITMAP system is
cooling. When there is net work output (as well as cooling), the SITMAP system mass
should be compared to that of an ideal passive radiator plus the mass 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 this 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 entrainment ratio is determined to be the maximum
possible (break-off entrainment ratio) for a given jet-pump geometry and inlet states. To
investigate the jet-pump performance in the mixed regime, the entrainment ratio was
varied in the range breakoff0 < φ < φ . 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 the mixed regime as the entrainment ratio decreases, the
compression ratio increases as well as the SMR because it is favorable to operate the jet-
pump at the maximum entrainment ratio possible, 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 SMR 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
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 jet-
pump 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 6-
22. This causes the velocity of the mixed stream, Vme, to increase as well. Eventually,
the diffuser exit total pressure starts to increase 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.
2 2.5 3 3.5 4 4.5φ
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
SM
R
1.246
1.248
1.25
1.252
1.254
1.256
1.258
1.26
1.262
1.264
1.266P
de/P
si
Figure 6-20. SMR and Compression ratio behavior in the mixed regime.
93
2 2.5 3 3.5 4 4.5φ
120000
125000
130000
135000
140000
145000
150000
Pm
e
60
70
80
90
100
110
Vm
e
PmeVme
Figure 6-21. Effect of the entrainment ratio on the mixed chamber exit conditions in the
mixed regime.
2 2.5 3 3.5 4 4.5φ
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 ratio on secondary nozzle exit conditions in the
mixed regime.
94
2 2.5 3 3.5 4 4.5φ
0.975
1
1.025
1.05
1.075
1.1
1.125
1.15
Pm
e/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.
2 2.5 3 3.5 4 4.5φ
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Qra
d/Qco
ol
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Qsc
/Qco
ol
Figure 6-24. Effect of entrainment ratio on specific heat transfer ratios in the mixed
regime.
Figure 6-20 shows that from the SMR standpoint it is better to operate the jet-pump
near the break-off conditions because this yields the lowest SMR. The reason for that is
shown in Figure 6-24. It can be seen that if the value of φ drops below its break-off
value, this causes the ratio of specific heat input to cooling load and the ratio of the
specific heat rejected to the specific cooling load to increase which in turn increases the
95
SMR. Therefore, in this study the jet-pump 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 temperature corresponding to a given mission.
However, it is also important to understand the effect of varying the evaporator
temperature on the SITMAP cycle performance. For that purpose, all the cycle
parameters are kept fixed and the evaporator temperature is varied in the range
70K<Tevap<90 K. The primary stagnation pressure ratio is kept fixed at Ppo = 3.3 MPa
with 10 degrees superheat (the effect of different degrees of primary superheat is
presented later in this study). Table 6-4 summarizes the cycle configuration.
Table 6-4. SITMAP cycle configuration to study the effect of the evaporator temperature, Tevap
Variable name Description Ppo 3.3 MPa
xpo 10 degrees superheat
Pso Variable Tevap = 70,75,80,85,89 K
xso 1.0
Ant/Ane 0.25
Ane/Ase 0.1
Ts 0
Figure 6-25 through Figure 6-28 show the results for this configuration. The first
trend that can be noticed is that the compression ratio is decreasing with increasing Tevap
(which also means higher Pso, since xso=1). This is expected, since the primary stagnation
96
pressure is fixed, higher secondary pressures 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.
75 80 85 90Tevap
1
1.5
2
2.5
3
3.5
φ
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
φPde/Psi
Figure 6-25. Effect of the evaporator temperature 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 ξT, 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.
97
75 80 85 90Tevap
0.08
0.09
0.1
0.11
0.12
0.13
0.14
ξ 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
SM
R
ξTSMR
Figure 6-26. Effect of the evaporator temperature on ξT, and SMR, for Ppo = 3.3 MPa.
75 80 85 90Tevap
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
Qra
d/Qco
ol
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
/Qco
ol
Qrad/QcoolQsc/Qcool
Figure 6-27. Effect of the evaporator temperature 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 helps lower the SMR. As the radiator
temperature increases, this helps decrease the required radiator size and thus lowers the
overall system mass.
98
75 80 85 90Tevap
83
84
85
86
87
88
89
90
91
T rade
ff
Figure 6-28. Effect of the evaporator temperature on the effective radiator temperature,
for Ppo = 3.3 MPa.
Primary Flow Superheat Heat Effect
To study the effect of the degree of superheat of the primary flow going into the jet-
pump 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 configuration 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 flow quality increases towards saturated
vapor and into the superheated regime, the SMR 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
99
rejected per unit cooling load increases which in turn causes the SMR to increase. This
effect is expected since increasing the primary 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 primary flow did not seem to have any
significant effect on the compression ratio, as can be seen in Figure 6-32.
1 2 3 4 5 6 7 8 9 10 11 12 13 140.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
SM
R
Ppo/Pso = 10Ppo/Pso = 20Ppo/Pso = 25
POINT xpo
1 0.12 0.23 0.34 0.45 0.56 0.67 0.78 0.89 0.910 5 degrees superheat11 10 degrees superheat12 15 degrees superheat13 25 degrees superheat14 30 degrees superheat
Figure 6-29. Effect of primary flow superheat on the SMR.
100
1 2 3 4 5 6 7 8 9 10 11 12 13 141.2
1.3
1.4
1.5
1.6
1.7
1.8
Qra
d/Qco
ol
Ppo/Pso = 10Ppo/Pso = 20Ppo/Pso = 25
Figure 6-30. Effect of primary flow superheat on the Qrad/Qcool.
1 2 3 4 5 6 7 8 9 10 11 12 13 140.2
0.3
0.4
0.5
0.6
0.7
0.8
Qsc
/Qco
ol
Ppo/Pso = 10Ppo/Pso = 20Ppo/Pso = 25
Figure 6-31. Effect of primary flow superheat on the Qsc/Qcool.
101
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Pde
/Psi
Ppo/Pso = 10Ppo/Pso = 20Ppo/Pso = 25
Figure 6-32. Effect of primary flow 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. Since the SMR values are still well above unity,
this means that, for the range of operation considered, 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 environmental 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, (ΔTload
= Tevap-Ts), gives the advantage to the ideal passive radiator. However, if ΔTload
decreases, the size of a passive radiator needed to achieve a given cooling load increases.
This gives the advantage from a mass standpoint to active systems over their passive
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 evaporator 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 range of system parameters considered, ΔTload
has to be very small (maximum of 2 degrees) in order for the active system to have the
advantage (SMR<1). Further analysis at higher compression ratios should be considered
to see if the active system could be an attractive option over a wider range of Ts/Tevap.
0 0.25 0.5 0.75 1Ts/Tevap
0
1
2
3
4
5
6
7
8
9
10
11
12
SM
R
Tsi=97.7K,Pde/Psi=1.0504,φ=11.9145Tsi=79.4K,Pde/Psi=1.1667,φ=6.6673Tsi=79.4K,Pde/Psi=3.702,φ=0.14
Figure 6-33. Sink temperature effect on SMR.
Zoom in shown in next figure.
103
0.99 0.9925 0.995Ts/Tevap
0.98
0.99
1
1.01
1.02
1.03
SM
R
Tsi=97.7K,Pde/Psi=1.0504,φ=11.9145Tsi=79.4K,Pde/Psi=1.1667,φ=6.6673Tsi=79.4K,Pde/Psi=3.702,φ=0.14
Figure 6-34. Compression ratio effect on the SMR < 1 regime.
System Optimization
An obvious question that should be asked now is whether or not the active system
is going to continue gaining more advantage with increasing compression ratio, or is
there is an optimum system configuration at which the break-off environmental sink
temperature, bosT , has a minimum value? (N.B. bo
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 ( bosT ) with changing jet-
pump 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 bosT < Ts < Tevap ⇔ SMR < 1; if Ts < bo
sT ⇔ SMR > 1
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 keep increasing with increasing compression
ratios. Figure 6-36 shows that the compression ratio increases (and φ decreases) with
increasing values of both area ratios.
These results show that when the difference between the sink and evaporator
temperature, ΔTload, is small enough there are competing effects that act in such a way so
that increasing the compression ratio does not necessarily give 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 ratio increases, and the entrainment ratio
decreases and this lead to an increase in the SMR. 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.
0 0.2 0.4 0.6 0.8 1Ane/Ase
77.3
77.4
77.5
77.6
77.7
77.8
77.9
78
78.1
78.2
T sbo
Ant/Ane = 0.1Ant/Ane = 0.2
Figure 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.
105
0 0.2 0.4 0.6 0.8 1Ane/Ase
1.5
2
2.5
3
3.5
4
Pde
/Psi
0.5
1
1.5
2
2.5
3
3.5
4
φ
Pde/Psi, (Ant/Ane = 0.1)Pde/Psi, (Ant/Ane = 0.2)φ, (Ant/Ane = 0.1)φ, (Ant/Ane = 0.2)
Figure 6-36. Compression ratio and entrainment ratio variation 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. However, there is another 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 operating 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 stagnation pressure will always lead to
increasing compression ratios for a given jet-pump geometry. However, as shown above,
increasing the compression ratio does not always give the system added advantage from a
mass point of view. Therefore, the effect of Ppo on the system behavior should be
106
investigated. Figure 6-37 shows that actually the stagnation pressure ratio is a parameter
that needs to be optimized for any given jet-pump configuration.
10 20 30 40 50 60Ppi/Psi
77
77.2
77.4
77.6
77.8
78
78.2
78.4
78.6
78.8T Sbo
Ant/Ane = 0.1Ane/Ase = 0.3Tpo = 150 KPso = 128 kPaTso = 79.4 K
SMR < 1
SMR > 1
Figure 6-37. Effect of stagnation pressure ratio on the break-off sink temperature (77.1).
So far the jet-pump has always been constrained to work at break-off conditions,
where the entrainment ratio is maximum, and the compression ratio is minimum. These
are the best conditions because they yield the 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 difference 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 temperature 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 the mixed regime increases the value of the break-
off entrainment ratio, in turn lowering the system performance from a mass standpoint.
107
Therefore, the rest of the system optimization analysis will be restricted to operation at
the break-off entrainment ratio.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9φ
77.25
77.5
77.75
78
78.25
78.5
78.75
79
T sbo
Ant/Ane = 0.1Ane/Ase = 0.3Pso = 128 kPaTso = 79.4 KPpo/Pso = 40
SMR > 1
SMR < 1
Figure 6-38. Break-off sink temperature behavior 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 seen 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.
108
0 0.2 0.4 0.6 0.8Ane/Ase
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
SM
R
Ant/Ane = 0.1Ant/Ane = 0.2Ant/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.
10 15 20 25 30 35 40 45 50 55 60 65Ppi/Psi
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
SM
R
Ant/Ane = 0.1Ane/Ase = 0.3Pso = 128 kPaTso = 79.4 KTs = 78.4 KTpo = 150 K
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 environmental 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
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]
φ 0.66
Pde/Psi 2.549
Ts 78.4 [K]
Qcool 117 [kW]
ηjet-pump 29.1 %
bosT 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 output 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
110
and the SITMAP system will also be optimized for different missions, with the MSMR as
the new objective function to be minimized.
111
CHAPTER 7 RESULTS AND DISCUSSION: COOLING AND WORK OUTPUTS
The System Mass Ratio (SMR) presented earlier 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 system 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 previous definition of SMR is not adequate
because it doesn’t 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; radiator, collector, and a
general system mass comprising the turbo-machinery and piping present in an active
system. This is shown mathematically by
,
, ,
col rad sys SITMAP
rad o Carnot Rankine
m m mMSMR
m m+ +
=+
(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
112
, , ,Carnot Rankine col CR rad CRm m m= + (7.2)
The modified SMR expression can be separated as follows
,
, , , ,
sys SITMAPcol rad
rad o Carnot Rankine rad o Carnot Rankine
mm mMSMRm m m m
+= +
+ + (7.3)
I II
The first term on the RHS of the equation above can be expanded into
, , ,
col rad
rad o col CR rad CR
m mIm m m
+=
+ + (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
, , ,
col col rad rad
rad rad o rad rad CR col col CR
A AIA A A
λ λλ λ λ
+=
+ + (7.5)
Where massarea
λ = (7.6)
If we divide by radλ
, , ,
colcol rad
rad
colrad o rad CR col CR
rad
A AI
A A A
λλ
λλ
+=
+ + (7.7)
rad
col
λλα = (7.8)
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).
113
,
Hcol
a col
QGA
η = (7.9)
Where Aa,col is the solar collector aperture area, which will be referred to as Acol for
simplicity. The reason this is the aperture area and not the collector receiver area is that
λcol is defined as the mass per unit aperture area in this analysis. If λcol was defined as
mass per unit receiver area, then the denominator 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-planetary or solar missions this may not be
the case.
( )4 4rad
radrad s
QA
T Tεσ=
− (7.10)
The idealized passive radiator model operates perfectly (ε = 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
( ), 4 4e
rad oe s
QAT Tσ
=−
(7.11)
In a similar sense the Carnot Rankine cycle 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 the Carnot Rankine cycle is given by
( ),
, 4 4rad CR
rad CRrad s
QA
T Tσ=
− (7.12)
114
Substituting Equations (7.8)-( 7.12) into equation (7.7)
1 2
4 4
, ,4 4 4 4
( )
( ) ( )
radH
col rad s
rad CR H CRe
e s rad s col
QQG T TI Q QQ
T T T T G
αη εσ
ασ σ η
+−
=+ +
− −
(7.13)
3 4 5
The overall SITMAP system can be divided into a power subsystem and a
refrigeration subsystem. The two subsystems 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 e ext radQ Q W Q+ = + (7.14)
Where
, ,rad L ref L powerQ Q Q= + (7.15)
ext T PW W W= − (7.16)
Substituting Equation (7.14) in Equation (7.13) yields
4 4
, ,4 4 4 4
( )
( ) ( )
H e extH
col rad s
rad CR H CRe
e s rad s col
Q Q WQG T TI Q QQ
T T T T G
αη εσ
ασ σ η
+ −+
−=
+ +− −
(7.17)
Qe
Wext Power
Sub-system RefrigerationSub-system
Wint
QL,refQL,power
QH
115
If we divide the numerator and denominator by Qe,
4 4
, ,
4 4 4 4
1
( )
1( ) ( )
extH
e eH
col e rad s
rad CR H CR
e e
e s rad s col
WQQ QQ
G Q T TI Q QQ Q
T T T T G
αη εσ
ασ σ η
+ −+
−=
+ +− −
(7.18)
4 4
, ,
4 4 4 4
1
( )
1( ) ( )
extH H
e H eH
col e rad s
rad CR H CR
e e
e s rad s col
WQ QQ Q QQ
G Q T TI Q QQ Q
T T T T G
αη εσ
ασ σ η
+ −+
−=
+ +− −
(7.19)
4 4
, ,
4 4 4 4
1 1
( )
1( ) ( )
extH
e HH
col e rad s
rad CR H CR
e e
e s rad s col
WQQ QQ
G Q T TI Q QQ Q
T T T T G
αη εσ
ασ σ η
⎛ ⎞− +⎜ ⎟
⎝ ⎠+−
=
+ +− −
(7.20)
4 4 4 4
, ,
4 4 4 4
11
( ) ( )
1( ) ( )
ext
H H
col rad s e rad s
rad CR H CR
e e
e s rad s col
WQ Q
G T T Q T T
I Q QQ Q
T T T T G
αη εσ εσ
ασ σ η
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠⎜ ⎟+ +⎜ ⎟− −⎜ ⎟⎜ ⎟⎝ ⎠=
+ +− −
(7.21)
which can be written as
116
4 4 4 4
, , ,
,4 4 4 4
11
( ) ( )
1( ) ( )
ext
H H
col rad s e rad s
rad CR H CR H CR
H CR e e
e s rad s col
WQ Q
G T T Q T T
I Q Q QQ Q Q
T T T T G
αη εσ εσ
ασ σ η
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟
⎝ ⎠⎜ ⎟+ +⎜ ⎟− −⎜ ⎟⎜ ⎟⎝ ⎠=
+ +− −
(7.22)
The first law efficiency of the power subsystem is given by
ext in
H
W WQ
η += (7.23)
∴ ext in
H H
W WQ Q
η= − (7.24)
∴ ext in ext
H ext H
W W WQ W Q
η= − (7.25)
∴ 1ext in
H ext
W WQ W
η⎛ ⎞
+ =⎜ ⎟⎝ ⎠
(7.26)
If we define the work ratio,WR as follows
ext
in
WWRW
= (7.27)
If Equation (7.27) is plugged into Equation (7.26)
∴( )
( )1ext
H
WRWQ WR
η=
+ (7.28)
The coefficient of performance of the refrigeration subsystem is given by
e
in
QCOPW
= (7.29)
Therefore,
117
e ext in
in H
Q W WCOPW Q
η⎛ ⎞⎛ ⎞+
= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
(7.30)
1e ext
H in
Q WCOPQ W
η⎛ ⎞⎛ ⎞
= +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
(7.31)
Substituting Equation (7.27) into Equation (7.31)
1H
e
Q WRQ COPη
+= (7.32)
The Carnot efficiencies of the power and refrigeration subsystems are defined as
,
1ext radc
H CR col
W TQ T
η = = − (7.33)
,
e ec
in c rad e
Q TCOPW T T
= =−
(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.
, ,
ext ec c
H CR in c
W QCOPQ W
η⎛ ⎞⎛ ⎞
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
(7.35)
, ,
e ext inc c
H CR in in c
Q W WCOPQ W W
η⎛ ⎞ ⎛ ⎞⎛ ⎞
= ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ (7.36)
But since the ratio of internal work to the carnot internal work can be written as
,
in c
in c
W COPW COP
= (7.37)
Substituting Equations (7.27) and (7.37) in Equation (7.36) yields
( ),
e cc c
H CR
Q COPCOP WRQ COP
η⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
(7.38)
If we define
118
( ) cc
COPWR WRCOP
⎛ ⎞= ⎜ ⎟⎝ ⎠
(7.39)
then Equation (7.38) can be written as
,H CR c
e c c
Q WRQ COPη
= (7.40)
The overall energy conservation equation for the Carnot Rankine cycle is given by
, ,ext rad CR H RCW Q Q+ = (7.41)
Dividing both sides by ,H RCQ yields
,
, ,
1rad CRext
H RC H RC
QWQ Q
+ = (7.42)
Plugging Equation (7.33) into Equation (7.42)
,
,
1rad CRc
H RC
η= − (7.43)
Substituting Equations (7.28), (7.32), (7.40), (7.43) into Equation (7.22)
( )
4 4 4 4
4 4 4 4
1 1 11( ) ( )
11( ) ( )
col rad s rad s
c c
e s c c rad s col
WRWRWR
G T T COP T T
IWR
T T COP T T G
ηα
η εσ η εσ
η ασ η σ η
⎛ ⎞−⎜ ⎟ +++ +⎜ ⎟
− −⎜ ⎟⎜ ⎟⎝ ⎠=
⎛ ⎞−+ +⎜ ⎟− −⎝ ⎠
(7.44)
If we define
Tc c
COPCOP
ηξη
= (7.45)
Plugging Equation (7.45) into Equation (7.44)
119
( )
4 4 4 4
4 4 4 4
1 1 11( ) ( )
11( ) ( )
col rad s T c c rad s
c c
e s c c rad s col
WRWRWR
G T T COP T T
IWR
T T COP T T G
ηα
η εσ ξ η εσ
η ασ η σ η
⎛ ⎞−⎜ ⎟ +++ +⎜ ⎟
− −⎜ ⎟⎜ ⎟⎝ ⎠=
⎛ ⎞−+ +⎜ ⎟− −⎝ ⎠
(7.46)
If we substitute the definitions of the Carnot efficiencies from Equations (7.33), and
(7.34) into Equation (7.46)
( )
4 4 4 4
4 4 4 4
1 1 11( ) ( )
1
1 11
( ) ( )1
col rad s rad srad eT
col rad e
rad
colc
e s rad s colrad e
col rad e
WRWRWR
G T T T TT TT T T
ITTWR
T T T T GT TT T T
ηα
η εσ εσξ
ασ σ η
⎛ ⎞−⎜ ⎟ +++ +⎜ ⎟
− −⎛ ⎞⎛ ⎞⎜ ⎟ −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ −⎝ ⎠⎝ ⎠=⎛ ⎞⎛ ⎞
− −⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟+ +⎜ ⎟− −⎛ ⎞⎛ ⎞
− ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠
(7.47)
If we normalize all the temperatures by the evaporator temperature, Te
( )
( ) ( )
( )
4 4 4 4
4
*4 * * 4 * *
* *
*4 *
* *
1 1 1111
1
111 1
1
col rade rad s e rad sT
col rad
c
crade s
col rad
WRWRWR
G TT T T T T TT T
I
WRTT TT T
ηα
η εσ εσξ
αησ
⎛ ⎞⎛ ⎞⎜ ⎟ ⎛ ⎞−⎜ ⎟ ++ ⎜ ⎟ ⎜ ⎟+ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎛ ⎞ ⎜ ⎟− −⎜ ⎟ −⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ −⎝ ⎠⎝ ⎠⎝ ⎠=⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟ + ⎜ ⎟⎜ ⎟ ⎛ ⎞⎛ ⎞⎜ ⎟− −⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠
( )4 4
*
*
4 * *
rad
col
ol e rad s
TT
G T T Tσ
⎛ ⎞⎜ ⎟⎜ ⎟+⎜ ⎟−⎜ ⎟⎝ ⎠
(7.48)
If we multiply the numerator and denominator of Equation (7.48) by 4eTεσ and define
4e
col
TG
αεσζη
= (7.49)
120
( )
( ) ( )
( )
4 4 4 4
4 4
** * * *
* *
*
*
** *
* *
1 1 1111
1
11 11
radrad s rad sT
col rad
rad
c col
rads rad s
col rad
WRWRWR
TT T T TT T
IT
WR TTT T TT T
η
ζξ
εε ζ
⎛ ⎞⎛ ⎞⎜ ⎟ ⎛ ⎞−⎜ ⎟ ++ ⎜ ⎟ ⎜ ⎟+ +⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎛ ⎞ ⎜ ⎟− −⎜ ⎟ −⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ −⎝ ⎠⎝ ⎠⎝ ⎠=⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟ + +⎜ ⎟⎜ ⎟ ⎛ ⎞⎛ ⎞⎜ ⎟− −−⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠
( )4*
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
(7.50)
Equation (43) can be rearranged as
( )
( )( )( )
( ) ( )
( )( )( )( ) ( )
4 4 4 4
4 4 4
* *
* ** * * *
* * *
* ** * * *
1 1 1 11
1
111
col rad
T col radrad s rad s
col c rad rad
col rads col rad s
WRT WR TWR
T TT T T T
IT WR T T
T TT T T T
η
ζξ
ε ζε
⎛ ⎞ ⎛ ⎞− ⎛ ⎞⎜ ⎟ + −+ ⎜ ⎟⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎜ ⎟− −⎜ ⎟⎝ ⎠⎜ ⎟ ⎝ ⎠⎝ ⎠=⎛ ⎞ ⎛ ⎞⎛ ⎞−⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠
(7.51)
The only term left is term II in Equation (7.3) which is given by
,
, ,
sys SITMAP
rad o Carnot Rankine
mTermII
m m=
+ (7.52)
Which can be written as
, ,
, , ,
sys SITMAP total SITMAP
total SITMAP rad o Carnot Rankine
m mTermII
m m m⎛ ⎞⎛ ⎞
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠ (7.53)
where the second term on the right hand side is the MSMR. If we define μ is the ratio of
the general system mass comprising the turbo-machinery and piping present in the
SITMAP system to its total mass. It is given mathematically by,
,
,
sys SITMAP
total SITMAP
mm
μ = (7.54)
Then Equation (7.52) become
121
( ),
, ,
sys SITMAP
rad o Carnot Rankine
mTermII MSMR
m mμ= =
+ (7.55)
Therefore, the MSMR expression given in Equation (7.3) can be written as
1TermIMSMR TermI TermII
μ= + =
− (7.56)
Substituting Equation (44) into Equation (49) yields the final expression for the MSMR
( )
( )
( )( )( )
( ) ( )
( )( )( )( ) ( )
4 4 4 4
4 4 4
* *
* ** * * *
* * *
* ** * * *
1 1 1 11
11 11
1
col rad
T col radrad s rad s
col c rad rad
col rads col rad s
WRT WR TWR
T TT T T T
MSMRT WR T T
T TT T T T
η
ζξ
ε μ ζε
⎛ ⎞ ⎛ ⎞− ⎛ ⎞⎜ ⎟ + −+ ⎜ ⎟⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎜ ⎟− −⎜ ⎟⎝ ⎠⎜ ⎟ ⎝ ⎠⎡ ⎤ ⎝ ⎠= ⎢ ⎥− ⎛ ⎞ ⎛ ⎞⎛ ⎞⎣ ⎦ −⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠
(7.57)
It can be seen that if the external work output, Wext is 0, thus WR = 0, η = 0, and WRc =
0, the MSMR expression boils down to the original SMR expression presented earlier in
Equation (3.69), as shown below.
( )( )
( )( ) ( )
( )
4 4 4 4
4
* *
* ** * * *
*
11 1
11 1
1
col rad
T col radrad s rad s
s
T T
T TT T T TMSMR
T
ζξ
ε μ
⎛ ⎞ ⎛ ⎞⎛ ⎞−⎜ ⎟ ⎜ ⎟⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠=
− ⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
(7.58)
( ) ( )( )( )
( )( )
( )
4 4
4 4 4 4
* * * *
* ** * * *
1 1 11 11
col rad s s
T col radrad s rad s
T T T TMSMR SMR
T TT T T Tζ
ε μ ξ
⎛ ⎞⎛ ⎞ ⎛ ⎞− − −⎜ ⎟⎜ ⎟ ⎜ ⎟= = + +⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠
(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.
122
Jet-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 through two mechanisms. The first is the shear
stresses at the tangential interfaces 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 secondary fluid, a turbo-machinery analog of the
ejector is used, shown in Figure 7-2. In this analogy, the primary fluid expands through a
turbine, from state 2 to state 5 which drives a compressor through 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 the 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
23
4
5
123
Figure 7-3. T-s diagram illustrating the thermodynamic states in the jet-pump turbo-
machinery analog.
The isentropic efficiencies of the compressor and turbine are given by,
' 11
4 1compressor
h hh h
η−
=−
(7.60)
'
2 5
2 2turbine
h hh h
η −=
− (7.61)
The work balance between the turbine and compressor can be expressed as
compressor turbineW W= (7.62)
( ) ( )1 4 1 2 2 5m h h m h h− = −& & (7.63)
Plugging the turbo-machinery efficiencies, Equations (7.60) and (7.61) into Equation
(7.63) yields
( )'
'11
2 2turbinecompressor
h hh hφ η
η⎛ ⎞−
= −⎜ ⎟⎜ ⎟⎝ ⎠
(7.64)
Equation (7.64) can be rearranged as
'
'
11
2 2turbine compressor
h hh h
η η φ⎛ ⎞−
= ⎜ ⎟⎜ ⎟−⎝ ⎠ (7.65)
T
s
31’
1
2
52’
4
124
The product of efficiencies in Equation (7.65) is referred to in the 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 pressure at the evaporator exit to the total
pressure of the mixed flow, to the isentropic 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 discharge from the compressor at point 4 and the
discharge from the turbine at point (5) combine and form the mixed state at point 3.
Point 3 corresponds to the discharge from an equivalent jet-pump. Therefore, the overall
jet-pump efficiency can be written as
'
'
11
2 2jet pump turbine compressor
h hh h
η η η φ−
⎛ ⎞−= = ⎜ ⎟⎜ ⎟−⎝ ⎠
(7.66)
It can be seen that there are multiple possible combinations of ,turbine compressorη η that
would yield the same overall jet-pump efficiency, jet pumpη − . As the compression
efficiency, compressorη increase the amount of internal work, Wint decrease and consequently
the COP of refrigeration subsystem increase, as shown in Figure 7-4. However, while
the COP increases, the efficiency, η (Equation (7.23)) decreases. Since the change in
compressorη does not affect the overall cycle parameters such as the specific cooling load,
specific heat input, or even the external work rate output, changing compressorη 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 compressorη , so compressorη can assume any value between 0 and 1 as long as the
product compressor turbineη η satisfies Equation (7.66).
125
The value of compressorη depends on the technology of the jet-pump. Research in
still ongoing trying to improve the work transfer between the primary and secondary
streams [24]. An interesting deduction from this analysis is the value of compressorη that
needs to be achieved in order for a jet-pump refrigeration system to compete against
commercially available refrigeration systems. It can be seen from Figure 7-4 that for a
COP of 3, compressorη needs to be about 45%.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ηcompressor
20000
25000
30000
35000
40000
45000
50000
Win
t
2.5
3
3.5
4
4.5
5
5.5
6
6.5
CO
P
WintCOP
Figure 7-4. Effect of compression efficiency on jet-pump characteristics.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ηcompressor
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
MS
MR
126
Figure 7-5. Effect of compression efficiency on MSMSR.
For a given jet-pump geometry and inlet states 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 higher jet-pump efficiencies yield higher COP and
thus less heat transfer rates (specific heat input and specific heat rejected) per unit cooling
load leading to smaller and lighter heat exchangers.
Figure 7-7 verifies that the MSMR and SMR expressions are equal at zero
external work output.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ηjet-pump
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
Pjp
e/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
MS
MR
Pjpe/Psi
MSMR
Figure 7-6. Jet-pump efficiency effect on the compression ratio and MSMR for given jet-pump inlet conditions.
127
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5Pjpe/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
SM
R
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
MS
MR
SMRMSMR
Figure 7-7. MSMR and SMR are equal for Wext = 0.
Since it is not possible to specify the value of compressorη for a given set of cycle
parameters, and since the MSMR is not affected 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
'
'
11
2 2compressor turbine
h hh h
η η φ⎛ ⎞−
= = ⎜ ⎟⎜ ⎟−⎝ ⎠ (7.67)
The abovementioned assumption brings closure 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 jet-
pump 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 increase. However, as the amount of work
increases the size of the ideal Carnot system that would produce the same amount of
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 parameters 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
φ 0.663
Ts 78.4 [K]
Qcool 117 [kW]
ηjet-pump 29 %
ηcompressor = ηturbine 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 increasing the work output has almost no effect on the
MSMR. Therefore, it can be concluded that the competing effects discussed earlier
balance each other out not giving any significant advantage to increasing the work output
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+06 9E+06 1E+07 1.1E+07 1.2E+07 1.3E+07 1.4E+07Pti
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
/Qco
ol
300000
310000
320000
330000
340000
350000
360000
370000
380000
390000
400000
410000
420000
430000
Wex
t
Qsc/QcoolWext
Figure 7-8. High pressure effect on the cooling specific heat input and external work output for a given jet-pump inlet conditions.
8E+06 9E+06 1E+07 1.1E+07 1.2E+07 1.3E+07 1.4E+07Pti
0.2
0.21
0.22
0.23
0.24
0.25
0.26
0.27
0.28
0.29
0.3
MS
MR
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
ξ T
MSMRξT
Figure 7-9. High pressure effect on the MSMR and efficiency for a given jet-pump inlet conditions.
130
System Optimization for MSMR
The SITMAP cycle performance with net external work output can be studied
using another way of constraining 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 was increased by increasing the high pressure
and keeping all the other cycle parameter fixed. It was shown that there is no significant
advantage from the MSMR standpoint in increasing 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 stagnation 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 pressure (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 primary nozzle geometry on the MSMR. It
can be seen that there is an optimum primary nozzle area ratio that yields the minimum
MSMR. Similar trends are noticed 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 specify the optimum jet-pump geometry and
stagnation pressure ratio for the mission parameters mentioned above and Pti = 14.2MPa.
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 further supports the conclusion that Pti should
not be considered as an optimization variable. However, Pti can be used a design
parameter to adjust the amount of work output for a given optimum cycle configuration.
0.1 0.2 0.3Ant/Ane
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
MS
MR
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ane/Ase = 0.4Ppo/Pso = 26Pso = 128 MPa
Figure 7-10. Primary nozzle geometry effect on MSMR at Ane/Ase = 0.4, Ppo/Pso = 26, Pso =128 kPa.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ane/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
MS
MR
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ant/Ane = 0.2Ppo/Pso = 26Pso = 128 kPa
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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5 10 15 20 25 30 35 40 45Ppo/Pso
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
MS
MR
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ant/Ane = 0.2Ane/Ase = 0.4Pso = 128 kPa
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 independent 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 variables 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 competing effects lead to the presence of an
optimum configuration that leads to the minimum MSMR for the specific mission under
investigation.
133
0 0.1 0.2 0.3 0.4Ant/Ane
1
1.5
2
2.5
3
3.5
4
4.5
Pde
/Psi
0
0.5
1
1.5
2
2.5
φ
Pde/Psi (Pti = 8 MPa)φ (Pti = 8 MPa)
Ane/Ase = 0.4Ppo/Pso = 26Pso = 128 MPa
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.
0 0.1 0.2 0.3 0.4Ant/Ane
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Qra
d/Qco
ol
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ane/Ase = 0.4Ppo/Pso = 26Pso = 128 MPa
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.
134
0 0.1 0.2 0.3 0.4Ant/Ane
0
5
10
15
20
25
30
Qsc
/Qco
ol
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ane/Ase = 0.4Ppo/Pso = 26Pso = 128 MPa
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.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ane/Ase
1
1.5
2
2.5
3
3.5
4
4.5
Pde
/Psi
0
5
10
15
20φ
Pde/Psi (Pti = 8 MPa)φ (Pti = 8 MPa)
Ant/Ane = 0.2Ppo/Pso = 26Pso = 128 kPa
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.
135
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ane/Ase
2
3
4
5
6
7
8
Qra
d/Qco
ol
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ant/Ane = 0.2Ppo/Pso = 26Pso = 128 kPa
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.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ane/Ase
0123456789
10111213141516
Qsc
/Qco
ol
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ant/Ane = 0.2Ppo/Pso = 26Pso = 128 kPa
Figure 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.
136
5 10 15 20 25 30 35 40 45Ppo/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
φ
Pde/Psi (Pti = 8 MPa)φ (Pti = 8 MPa)
Ant/Ane = 0.2Ane/Ase = 0.4Pso = 128 kPa
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.
5 10 15 20 25 30 35 40 45Ppo/Pso
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Qra
d/Qco
ol
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ant/Ane = 0.2Ane/Ase = 0.4Pso = 128 kPa
Figure 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), and Pso = 128 MPa.
137
5 10 15 20 25 30 35 40 45Ppo/Pso
0
5
10
15
20
25
Qsc
/Qco
ol
Pti = 8 MPaPti = 9 MPaPti = 10 MPaPti = 11 MPaPti = 12 MPaPti = 13 MPaPti = 14 MPa
Ant/Ane = 0.2Ane/Ase = 0.4Pso = 128 kPa
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 the 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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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] φ 0.66 Pde/Psi 2.549 Ts 78.4 [K] Qcool 117 [kW] Wext 433 [kW] ηjet-pump 29.1 %
bosT 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] φ 0.51 Pde/Psi 2.66 Ts 78.4 [K] Qcool 88.4 [kW] Wext 460 [kW] ηjet-pump 25.7%
bosT 75.02 [K]
MSMR 0.45
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CHAPTER 8 CONCLUSIONS
The combined power and cooling SITMAP cycle has been analyzed and optimized.
The cycle is comprised of a refrigeration cycle 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 subsystem with the low pressure stream from the
refrigeration subsystem, thus providing the necessary compression in the refrigeration
subsystem. The methodology followed for the 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, then developing the optimization techniques
for the SITMAP cycle, and finally applying 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 irradiance which is a function of the distance
from the sun.
The results from the jet-pump model were in very good agreement with results
available in the literature for perfect gases. This gives the confidence in the accuracy of
the flow model as well as its implementation.
The jet-pump performance followed the expected 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.
140
• Higher entrainment ratios led to lower compression ratios. • Secondary flow superheat did not have significant effect on the entrainment ratio.
However, entrainment increases if the secondary 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 exchangers (radiator and solar collector) to be larger and heavier, leading to higher SMR.
• Two-phase jet-pumps usually have significantly higher entrainment ratios than single-phase ejectors. This is attributed 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) expression 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 expression 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 radiator with the same cooling
capacity plus the mass of an ideal Rankine 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 larger 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 the 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 produces any work output, the optimum cycle configuration for a given set of mission parameters is the same. Said differently, the increase in the overall system mass due to the addition of the work output is
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
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CHAPTER 9 RECOMMENDATIONS
More work is needed to better understand 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 and a discussion elaborating on some of these
topics follows.
• Include frictional losses in the jet-pump model. • Investigate the use of different working fluids 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 amplifiers or unsteady ejectors can be a good starting point.
• An economically-based objective function can be used when optimizing for terrestrial aplications to determine the feasibility of using this system.
The jet-pump model used in this study assumed frictionless flow throughout the jet-
pump as an approximation. A quasi one-dimensional frictional model needs to be
included for more realistic results.
In this study, the SITMAP system is optimized 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 recognize that the ultimate determining factor in
cycle design for most applications is the cost. Some case studies 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 these applications has different
143
constraints, which must be considered in an optimization study. For example, there is a
limit in geothermal applications on the temperature of the geofluid leaving the cycle, to
prevent silica precipitation. The cycle designs developed can be compared to standard
cycles in use now and to the most promising alternative determined, particularly through
an economic analysis.
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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 famous for the Pyramids of Giza. He received
his bachelor’s degree in mechanical 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 master’s degree in mechanical engineering. During
his master’s Sherif worked on Computational Fluid Dynamics (CFD) modeling of
multiphase flows. In fall 2001, Sherif joined the University of Florida Mechanical and
Aerospace Engineering Department and started working on the combined power and
refrigeration system as his dissertation topic.