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THERMODYNAMIC STUDIES ON ALTERNATE BINARY WORKING FLUID COMBINATIONS AND CONFIGURATIONS FOR A COMBINED POWER AND COOLING CYCLE By SANJAY VIJAYARAGHAVAN 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 2003 ACKNOWLEDGMENTS The work presented in this dissertation was completed with the encouragement and support of many wonderful people. Working with Dr. Yogi Goswami has been a tremendous experience. He expects his students to be selfstarters, who work independently on their projects. I appreciate his patience and mentorship in areas within and beyond the realm of research and graduate school. Dr. Sherif Sherif was a terrific source of discussion, advice, encouragement and hard to find AES proceedings. Dr. James Klausner, Dr. David Hahn, and Dr. Ulrich Kurzweg agreed to be on my committee and took the time to read and critique my work, for which I am grateful. Dr. Hahn was also on my master's committee and supervised me during my assignment as a TA for two terms. Dr. Bill Lear has to be thanked for advice on jet pumps and Dr. Skip Ingley for his interest in the cycle. Dr. Leon Lasdon from the University of Texas sent me the FORTRAN version of the GRG code and answered my questions very promptly. Although my particular project did not require much of his marvelous skills, the senior engineering technician at the Solar Park, Chuck Garretson, was very supportive. Watching him and working with him have taught me many things. He is a wonderful resource for any student at the lab. Barbara Graham over the years and now recently Vitrell McNair at the solar office have cheerfully helped me in many ways in the course of my stay here. Mrs. Becky Hoover will have to be thanked for her help and constant reminders to finish up. My fellow students at the solar lab have been stimulating company. Some of them particularly have to be acknowledged. Gunnar Tamm, Chris Martin and Nitin Goel took interest in my work and provided constructive feedback. Gunnar has also been at the lab almost as long as I have, and he has been fun to work with. Former students such as Viktoria and Andrew Martin, and Adrienne Cooper have stayed in touch and encouraged me. I would like to particularly thank my family for putting up with my being so far away from home, and for their love, support and eternal optimism. This section is not complete without mentioning friends, old and new, too many to name individually, who have been great pals and confidants over the years. TABLE OF CONTENTS Page A C K N O W L E D G M E N T S ......... .. ................................................................. ............... ii LIST OF TABLES ......... .......... ............................. .......... .............. viii L IST O F FIG U R E S .......... .. .......... .................................. .. ........ ........ ..... ix N O M EN CL A TU R E ........... ........................................................................ xiii A B ST R A C T ............................................................................. ........................................ xvi CHAPTER 1 B A C K G R O U N D ............ ................................................. .. ...... ............... .. 1 In tro d u ctio n .................................................................. 1 Background ................................. .......................... ..... ..... ........ 2 R ankine C y cle ................. ...... ............................. 3 Power Cycles for Low Temperature Heat Sources ......................................... 5 O organic R ankine cycles (O R C) .......................................................... ..... 5 Supercritical cycles...................................... 6 M alone and R obertson cycle ................................................... ....... 8 Kalina cycle ................ .................... .......... ....... 9 Other am m onia based cycles ........................... .. ... ...................... 13 Use of Mixtures as Working Fluids in Thermodynamic Cycles ...................... 14 The Combined Power and Cooling Cycle ................................. .............. 16 Introdu action ............................................................ ....... .. 16 A summary of past research on the cycle ...................................................... 18 Specific motivation for work in this dissertation................... 21 2 EFFICIENCY DEFINITIONS FOR THE COMBINED CYCLE............... 24 C conventional Efficiency D definitions ............................... ........................................ 24 First Law Efficiency ................. ................. ..... ...................... 25 E x ergy E fficien cy ................................................. .............. ...... .............. 2 5 Second Law Efficiency .............. ......... ............................... .............. 26 The Choice of Efficiency Definition ......... ........................... .. ............ 28 Efficiency Expressions for the Combined Cycle ............................................... 28 F first L aw E fficien cy ............................................................. .......................... 2 8 Exergy Efficiency .......... ................ ........................................ ............. 30 Second L aw Efficiency ............................................... .............................. 30 Lorenz cycle ........................................... 30 Cascaded cycle analogy .................................................... 32 V validity of E xpressions......................... ........... ....... .... ..... .......... 35 Case 1: Comparing this Cycle to Other Combined Cooling and Power G generation O options ............................................................... .. 36 Case 2: Comparing a Combined Cycle to a Power Cycle ......................... 36 Some Examples of Application to the Combined Cycle ......................................... 37 Conclusions ................ .......... .............. .............................. 41 3 CYCLE SIMULATION AND OPTIMIZATION .............................................. 42 O ptim ization M ethod B ackground........................................................ .............. 42 Search T erm nation ........................................ ...................... .. ................ 46 Sensitivity Analysis .............. ...................................... ...... .............. 46 A application N otes ......... ........................ ................ ..... .. .......... 47 Cycle Modeling ............... ......... .................. 48 State P oint C alculation ................................................... ........... ........... 49 State point 1 ...................................... .............. 49 P o in t 2 ................. .............. .............. ...... 5 1 Points 5 and 11 ............. ........................... 52 Points 6 and 7 .. ... ........ ................... .............. 53 P point 8.. ................ ............................ 53 P point 9.. ................ ............................ 54 P point 10 ............................................................................................. 54 Points 12 and 3 .............. ................... .............. 55 Point 13 .................. ......... ............ 56 P point 14 ............................................................................................. 56 P point 15 ............................................................................................. 57 P point 4.............................................. 57 V variable L im its ...................................................... .............. 57 Constraint Equations ........ .......... ......... ................ .............. 59 M odel Limitations................... ... ................... .............. 61 4 OPTIMIZATION OF BASIC CYCLE CONFIGURATION USING AMMONIA WATER MIXTURE AS THE WORKING FLUID .................................... 62 Sim ulated C condition s ....................................................................... 63 Optimization results ................................... ......... ................. 65 R source U tilization Efficiency .................................................................... 65 Exergy Efficiency ..........................................67 First Law Efficiency ............. .................. .............. 69 Exergy Analysis ............................................... ........ ...... ......... 70 Optim ization Considering Losses ................................................................... 74 D iscu ssion ........................................................................................................ 79 C conclusions ............................................................................................................ 84 v 5 RESULTS AND DISCUSSION: ORGANIC WORKING FLUIDS ........................ 86 W working F lu id S election ............................................................................................ 86 Prelim inary Simulation ....................... ................................................................ 89 Simulation of Higher Boiling Components .................................................... 94 D discussion ............................................................................................................. 99 Factors Affecting Cycle Performance .............................................. ........... 100 Rectification ............................ .............. 100 B oiler C conditions .............................................................. .. 10 1 Basic Solution Concentration .............................. 104 Pressure and Tem perature Ratio............................................... ................. 104 Volatility Ratio .. ....................... ....... .. ............ .......... 108 Liquid Formation During Expansion....................... ..... .............. 109 Conclusions ..................................... ................................ .......... 113 6 RESULTS AND DISCUSSION: IMPROVED CONFIGURATIONS................. 115 M o tiv atio n ............................................................................. 1 1 5 Reflux Mixed with Boiler Inlet................................. .............. 115 A addition of a Preheater .......................................................... ... .............. 118 Jet Pum p A assisted C ycle ....................................................................... 120 Jet Pum p B background ......................................................... .............. 122 Jet Pum p A nalysis............................................. .................... .. ................ 123 Primary nozzle ...... ........ ...... ................ ..1............... 126 Secondary nozzle ....... ........ ...................... ..1.... .... .... 128 M ixing section ........ ... .. ........... .... ......... ......... ... ......... ..... 128 D iffuser ....................................................... ... .. ... ............ .............. 130 Simplified M odel U sed in Simulation .................................... .............. 131 R esu lts ................................................................................... .............. 13 2 Conclusions................. ............... ..... .............. 135 Distillation (Thermal Compression) Methods ............................................... 135 Using Heat Source ... ..... ............................................................. 136 Using Absorber Heat Recovery ................. ..... ............. .. .............. 142 Conclusions ......................................... 149 7 CONCLUSIONS ............ ............. ............ .............. 150 Summary of Results ......................................... ................... 150 Future W ork ............................................................. ........ 153 APPENDIX A WORKING FLUID PROPERTY CALCULATION .......................................... 157 P property P reduction M ethods ................................................................................... 157 M ethods Used for Properties in this Study .................................... .............. 159 Ammonia Water Properties Prediction................... ....................... 159 Organic Fluid Mixture Properties ........................................ ............. 160 B OPTIM IZED STATE POINTS..................................................... 164 Conditions ............................................................... .... ...... ........ 164 Basic Cycle Configuration........................... .......... .... 165 Fluid: A m m onia W ater.................................................... 165 Fluid: PropanenUndecane ............... ........ ........... .............. 168 Fluid: IsobutanenUndecane....................... ........... 169 Configuration with Preheater............................. .............. 170 Configuration with Jet Pum p .............. .............. ............................ ...... 170 Configuration with Kalina Thermal Compression....................................... 171 U sing heat source for distillation.............. ........................ ....... ....... 171 Using absorber heat recovery for distillation ............ ...... ............ .. 173 C SIMULATION SOURCE CODE .................... .. ................. .............. 176 B asic C ycle C configuration ...................... .. .. .................... .................... .............. 176 H eader file............................................ 176 Cycle Sim ulation .................. ...................... ..... .... ...... .. ......... ..... 177 Interface to Supertrapp .......................................................... ... .............. 186 LIST OF REFEREN CES ................................................... ................................. 202 BIOGRAPHICAL SKETCH .............................................................. ...208 LIST OF TABLES Table p 1.1 Efficiency definitions used in various papers published on the cycle................. 22 2.1 Some examples of efficiencies applied to the Rankine cycle and the vapor com pression refrigeration cycle..................................................... .............. 26 2.2 Cycle parameters that yield optimum exergy and second law efficiencies ............ 38 2.3 Efficiency and cycle parameters optimized for effective exergy efficiency .......... 39 2.4 Efficiency and cycle parameters optimized for effective first law efficiency......... 40 3.1 Independent cycle parameters and their limits ............................................ .. 58 3.2 Constraints used in the optimization .............. ...... ......................... ........... 60 5.1 List of working fluids Considered Initially .............. ..... .... .......... .... 88 5.2 Higher boiling components considered ............. ...................... .......... ........ 95 5.3 Calculated R/cp values for certain gases in the temperature range of the cycle b ein g stu d ied ................................................... ................ 10 6 5.4 V olatility ratio of selected pairs.................................................... 109 5.5 Ivalues of certain pure components at 300 K and corresponding saturation p re ssu re ...................... .. .. ......... .. .. ............................................... 1 10 LIST OF FIGURES Figure pge 1.1 A schematic diagram of a simple Rankine cycle................................................. 4 1.2 Diagram showing a supercritical Rankine cycle on the Ts diagram for isobutane.. 7 1.3 Arrangem ent of a sim ple K alina cycle ........................................... .............. 9 1.4 Schematic diagram of Kalina System 12................. .......................... .............. 12 1.5 Illustration of the difference in temperature profiles for a pure fluid being boiled and a multicomponent mixture being boiled ....... ...... ........ ......................... 15 1.6 The basic configuration of the combined power and cooling cycle..................... 17 1.7 Effect of turbine inlet pressure on the thermal efficiency (%) of the cycle ........... 20 1.8 Effect of turbine inlet pressure on the cooling capacity (kJ/kg) of the cycle ......... 20 2.1 A cyclic heat engine working between a hot and cold reservoir ............................. 27 2.2 The TS diagram for a L orenz cycle.............................................. ... ................. 31 2.3 Thermodynamic representation of (a) combined power/cooling cycle and (b) cascaded cycle ..... ...................................... ............... 33 3.1 The basic cycle configuration with the variables shown...................................... 50 4.1 Block diagram showing the basic scheme of the combined power and cooling cycle. Same as Fig. 1.6 ...... .... ........................................ .... ............ .. 64 4.2 Optimized resource utilization efficiencies for the basic cycle configuration ........ 66 4.3 Optimized exergy efficiencies for the basic cycle configuration ........................ 68 4.4 Optimized First Law Efficiencies for the Cycle Configuration in Fig. 4.1 ............ 70 4.5 Exergy destruction in the cycle represented as a percentage of exergy of the heat source, for optimized RUE corresponding to 280 K turbine exit temperature........ 71 4.6 Exergy destruction in the cycle represented as a percentage of exergy of the heat source, for optimized RUE corresponding to pure work output............................ 71 4.7 Exergy destruction in the cycle for optimized exergy efficiency corresponding to 280 K turbine exit tem perature...................................................... .............. 73 4.8 Exergy destruction in the cycle for optimized exergy efficiency corresponding to pure w ork output.................... .................................... .. .. .... ...... .............. 73 4.9 Optimized RUE for the basic cycle configuration using a non isentropic turbine and pump ...... ........... ............................ 75 4.10 Effect of recovery heat exchanger effectiveness on RUE at optimum RUE conditions, with cooling output ............................................. ............. ........ 75 4.11 Optimized exergy efficiencies for the basic cycle configuration using a non isentropic turbine and pum p .............. .................................................................. 77 4.12 Effect of recovery heat exchanger effectiveness on effective efficiency .............. 77 4.13 Exergy destruction in the optimized RUE case.............. ............ ......... ..... 78 4.14 Exergy destruction in the optimized exergy efficiency case ................................. 78 4.15 Working fluid temperatures and vapor fractions in boiler: at maximum RUE and w ith co o lin g ..................................................................................... 8 0 4.16 Working fluid temperatures and vapor fractions in boiler: at maximum RUE with pure w ork output.................... .............. .................... .. .... ............ .. .............. 80 4.17 Cycle pressure ratios and the influence of solution heat exchanger effectiveness.. 82 4.18 Cooling to work output ratios at optimum conditions ....................................... 82 4.19 First law efficiency at optimized exergy efficiency conditions............................... 83 4.20 First law efficiency at optimum RUE conditions ........................................ 84 5.1 Basic configuration simulated. In this configuration, the condensate from the rectifier is redirected to the absorber. ............. ...... .......................... .......... 90 5.2 Optimized second law efficiencies of the combined power and cooling cycle using organic w working fluid pairs..................... ................................. .......................... 92 5.3 Alternate configuration, where the condensate from the rectifier is mixed with the strong solution inlet stream to the boiler. ............. ............................... ....... ....... 93 5.4 Comparison of the second law efficiency performance of the two configurations in Fig. 5.1 and 5.3, using a propanehexane mixture as the working fluid................ 94 5.5 Optimized exergy efficiency using higher boiling non volatile mixtures ............ 95 5.6 Optimized resource utilization efficiency using higher boiling non volatile mixtures ........ ..................................... ............... 96 5.7 Pressure ratios at optimum exergy efficiency.............................. .................... 98 5.8 Pressure ratios at optim um RU E ........................................ .................. ...... 98 5.9 Maximum (limiting) pressure ratio using some working fluid mixtures at various basic solution concentrations in the absorber and using a 360 K heat source....... 100 5.10 Phase diagram of a part of the cycle using isobutanendecane at 400 K, optim ized for exergy efficiency.................................... ............................. ...... 102 5.11 Close up of a portion of Fig. 5.10............... .................................... ................. 102 5.12 Effect of low pressure (propanehexane mixture as working fluid)................. 105 5.13 Calculated isentropic temperature ratio as a function of pressure ratio using perfect gas assume options .................................. .............. ................................... 107 5.14 Ts diagram for concentrated propanenundecane mixtures ............................. 111 5.15 Ts diagram for concentrated isobutanenundecane mixtures............................. 112 5.16 Ts diagram for concentrated propanenhexane mixtures.............................. 112 5.17 Ts diagram for concentrated ammoniawater mixtures..................................... 113 6.1 Modification of the basic cycle configuration which has the liquid condensate from the rectifier mixed with the strong solution at the boiler inlet.................... 116 6.2 Optimum exergy efficiencies obtained with the modified configuration shown in F ig 6 .1 ........................................................ ................. 1 1 7 6.3 Major exergy destruction categories for the modified cycle, at optimum exergy efficiency conditions........................................ ........................... ...... .... .. 118 6.4 Modified cycle configuration with part of the basic solution being preheated by the heat source fluid............ ................. ......................... ... .. .......... 119 6.5 Optimized RUE for the modified configuration in Fig. 6.4 compared to the base con figu ration in F ig 4 .1 ...................................... ............................................ 12 1 6.6 Major exergy losses in different parts of the modified cycle, optimized for RUE 121 6.7 Schematic drawing of a jet pump showing the different sections and the flow th rou g h it .................. .................................. ...... .......... ..... 12 4 6.8 Cycle configuration incorporating the jet pump..................... .............. 132 6.9 Improvement in resource utilization efficiency with the addition of a jet pump... 134 6.10 Influence of the choice of turbine exit temperatures on the improvement achievable with ajet pump using an isentropic turbine and pump....................... 134 6.11 Cycle configuration using the heat source to produce vapor in the distiller ......... 137 6.12 Maximum RUE of the cycle configuration modified with heat source fluid powered thermal compression modification....................... ... ........... ... 139 6.13 First law efficiency at maximum RUE conditions for configuration with heat source powered thermal compression modification ......................... ................. 139 6.14 Exergy destruction in the cycle with heat source powered thermal compression modification, when operated to provide power and cooling ............................... 140 6.15 Exergy destruction in the cycle with heat source powered thermal compression modification, when operated to provide only power output................................ 140 6.16 Some parameters for optimized RUE conditions in the cooling domain ............. 141 6.17 Some parameters for optimized RUE in the work domain........................... .. 142 6.18 Cycle configuration using heat of condensation to produce vapor in the distiller 144 6.19 Maximum RUE of the cycle configuration modified with a condensing mixture providing heat of distillation......................................................... .............. 145 6.20 First law efficiency at maximum RUE conditions for configuration using heat of condensation to produce vapor in the distiller ............................................... 145 6.21 Exergy destruction in cooling domain in modified cycle with a condensing mixture providing heat of distillation ..... ........... .. ................................. 146 6.22 Exergy destruction in work domain with a condensing mixture providing heat o f d istillatio n .................................................................... .............. 14 7 6.23 Some parameters for optimized RUE (strict definition) in the cooling domain for the modified cycle with a condensing mixture providing heat of distillation ....... 148 6.24 Some parameters for optimized RUE in the work domain for the modified cycle with a condensing mixture providing heat of distillation............................. 148 NOMENCLATURE COP Coefficient of Performance cp Specific heat E Exergy f refrigeration weight factor h specific enthalpy i Exergy Index Imf Liquid mass fraction m mass NTU Number of Transfer Units Obj Objective function p Pressure Q Heat Interaction T Temperature S Entropy s specific entropy R Universal gas constant r Ratio of cooling to work output v Specific volume vmf Vapor mass fraction W Work Interaction x Mass fraction of ammonia (volatile component) in a binary mixture y Mole fraction of volatile component in a binary mixture Greek Symbols a21 Volatility of component 2 w.r.t component 1 r7 Efficiency Subscripts 0 Reference state b Boiling basic Referring to basic solution stream, from absorber Carnot Carnot Cycle c Cooling cf Chilled Fluid crit Critical eff Effective exergy Exergy exit At Exit of a Device fg Liquid to gas (representing phase change from liquid to gas) HE Heat engine h Heat Source high High hr Heat rejection Fluid hs Heat Source Fluid in Input Lorenz Lorenz Cycle low Low m Melting max maximum net Net out Output p Pump R Resource Utilization Efficiency r Reduced rev Reversible ref Refrigeration sys System useful Useful I First Law II Second Law 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 THERMODYNAMIC STUDIES ON ALTERNATE BINARY WORKING FLUID COMBINATIONS AND CONFIGURATIONS FOR A COMBINED POWER AND COOLING CYCLE By Sanjay Vijayaraghavan August 2003 Chair: D. Y. Goswami Major Department: Mechanical and Aerospace Engineering A combined power and cooling cycle was investigated. The cycle is a combination of the Rankine cycle and an absorption refrigeration cycle. A binary mixture of ammonia and water is partially boiled to produce a vapor rich in ammonia. This vapor is further enriched in a rectifier/condenser and after superheating, expanded through a turbine. The vapor exiting the turbine in this cycle is cold enough to extract refrigeration output. By suitable selection of operational parameters for the cycle, the useful output can have a large range of refrigeration to work ratios. This combined cycle is being proposed for applications with lower temperature heat sources, with the primary objective of producing power. Some examples of energy sources include solar, geothermal, or industrial waste heat. Evaluating the efficiency of this cycle is made difficult by the fact that there are two different outputs, namely power and refrigeration. An efficiency expression has to suitably weight the cooling component in order to be able to compare this cycle with other cycles. Several expressions are proposed for the first law and second law efficiencies for the combined cycle based on existing definitions in the literature. Some of the developed equations have been recommended for use over others, depending on the comparison being made. This study extended the application of the cycle to working fluids other than ammoniawater mixtures, specifically to organic fluid mixtures. It was found that very low temperatures (well below ambient) are not achievable using organic fluid mixtures, while with an ammoniawater mixture; temperatures that were substantially below ambient were obtained under similar conditions. Thermodynamic efficiencies obtained with hydrocarbon mixtures are lower than those seen with an ammoniawater mixture as the working fluid. Based on the exergy analysis, the cycle configuration has been modified to improve its second law efficiency. A significant improvement in the resource utilization efficiency of more than 25% was achieved with the best among the improved schemes. Increased efficiencies can also be obtained for the cases where only work output is desired. CHAPTER 1 BACKGROUND Introduction The goal of this study is to investigate the use of organic fluid binary mixtures as possible alternate working fluids for a new thermodynamic power cycle (henceforth referred to as the cycle) proposed by Goswami [1]. The power cycle under study is unique in that the working fluid exiting the turbine could be cold enough to extract refrigeration. This cycle is a combination of the Rankine and absorption refrigeration cycles. The performance of the cycle has been studied extensively and optimized in earlier studies [28]. A mixture of ammonia and water has been considered so far for the working fluid. An experimental setup was built and a preliminary verification of ammonia boiling and condensation behavior performed [8]. The cycle has been found to be more suited to low temperature heat sources, based on earlier simulation results. The use of a volatile component such as ammonia allows vapor to be formed at high enough pressures that are useful for power generation. It is, therefore, anticipated that this cycle will be used in low temperature applications such as those involving solar, geothermal, and waste heat sources. Traditionally, the trend in thermal energy conversion in general, is to use higher temperatures in thermal energy conversion plants. Heat associated with high temperatures has a larger availability, and therefore systems operating at higher temperatures normally achieve greater first law efficiencies of conversion. Some of the best efficiencies (around 58% first law efficiency) achieved today are with combined cycle power plants. Attempts are being made to break the 60% efficiency barrier in some combined cycle plants being designed currently [9]. In combined cycle plants being built today, a gas turbine power cycle is the first stage, and the exhaust from the turbine is used to run a steam power plant. In the case of solar power generation, to operate at high temperatures, considerable concentration of the solar radiation is required. Solar energy being a dilute resource, the collection system is a large part of the cost of a solar power plant. The optics required for achieving high concentration increases the cost of the collection system. Some of the biggest disadvantages of solar energy as an energy source for electricity generation are a) the dilute nature of the resource, b) the intermittent nature of the energy and c) the high cost of producing power. It is hoped that by using cheaper, low concentration collection systems and a cycle that utilizes a large percentage of the exergy in that temperature range, the cost of solar power production can be brought down. The combined power and cooling cycle is being developed to efficiently utilize the exergy of low temperature heat sources. Since the cycle can operate at low temperatures, it could be applied to other low temperature heat sources such as geothermal and waste heat. Background Thermal power cycles can be classified on the basis of the working fluid used, as vapor power cycles and gas power cycles. In a vapor power cycle, the gas that spins the turbine is obtained from vaporizing a liquid. An example of such a cycle is the classic Rankine cycle. In a gas power cycle, such as the Brayton cycle, the working fluid is in a gaseous state throughout the cycle. Rankine Cycle The most commonly used vapor power cycle is the Rankine cycle. Even though a description of the Rankine cycle can be found in any engineering thermodynamics textbook [10], it is briefly covered here. The simple Rankine cycle, shown in Fig. 1.1 consists of four steps. The working fluid is pumped to a high pressure and circulated through the boiler. The fluid is boiled at a constant pressure in the boiler after which the highpressure vapor produced is expanded through a turbine, thus extracting work from it. The vapor exiting the turbine is condensed in a condenser by rejecting heat to a cooling fluid. There are several modifications to the Rankine cycle that are used to achieve better efficiencies. These include superheating, reheating and regeneration. Water (Steam) is the working fluid of choice for most vapor power cycles. Water works over a broad range of temperatures and pressures, has a large heat capacity, and is stable, safe and very environmentally friendly. The energy sources used to generate steam include gas, coal, oil, and nuclear sources. A small percentage of steam power plants use geothermal and solar energy sources. The first commercially successful steam engine was the one patented by Thomas Savery in 1698. The betterknown Newcomen's engine eventually displaced this engine in the early 18th century [11]. The steam power plant has come a long way in the 100 odd years since central steam generating plants started being built. The first central station steam turbine in the United States operated (in 1900) with steam conditions of IMPa and 483 K. In contrast, a typical central steam power station today operates at high pressure turbine inlet pressures and temperatures of 16.4 MPa and 800 K respectively [11]. Fig. 1.1 A schematic diagram of a simple Rankine cycle While steam is the working fluid of choice today, there are certain situations in which it does not work efficiently, particularly with low temperature resources. For example, consider the case of a binary1 geothermal power plant. In such a plant, a fluid such as isobutane is boiled using a relatively low temperature geothermal fluid, and used to spin a turbine working on a Rankine cycle. If steam were to be used in such applications (or if the plant is designed as a flash type plant), very low pressures and large vacuums at turbine exit would result. 1 It needs to be pointed out that in a binary geothermal power plant, a high pressure fluid is heated by a hot geofluid, boiled and expanded through a turbine. This is not the same as a plant using a binary fluid mixture for a working fluid. Power Cycles for Low Temperature Heat Sources The cycle being studied here appears to work well with low temperature heat sources, based on past studies. Low temperature heat sources are usually those deriving their energy from geothermal, solar or some waste heat sources. Solar energy can be used at various temperatures, depending on the collection method used. Low temperature heat sources have low availability and are not usually the energy sources of choice. Such sources would be considered useful only if some economic advantage is found in their utilization. For instance, a geothermal power plant could prove to be economically feasible to supply an area in the vicinity of a geothermal steam field. Several power cycles that are suitable for use with low temperature heat sources have been proposed in the literature and have been used in practice. The Rankine cycle has been adapted for use with low temperature heat sources by using low boiling working fluids such as organic fluids. Organic Rankine cycles (ORC) have been extensively used in binary geothermal power plants and low temperature solar power conversion. Several ammoniawater mixture based cycles have also been proposed for low temperature applications. Organic Rankine cycles (ORC) Since the first Rankine cyclebased thermal power plant was built, there have been several improvements in the configuration, components, and materials used, but the working fluid of choice has almost always been water. While water has several properties that make it a very good choice as a working fluid, in low temperature applications, better choices are available. Organic working fluids are a popular choice for such applications. Despite the fact that these fluids have lower heats of vaporization than water, which requires larger flow rates, smaller turbine sizes are obtained due to the higher density at the turbine exit conditions. A variety of fluids, both pure components and binary and ternary mixtures, have been considered for use in ORCs. These include saturated hydrocarbons such as propane, isobutane, pentane, hexane and heptane; aromatics such as benzene and toluene; refrigerants such as R11, R113, R114; and some other synthetic compounds such as Dowtherm A. Organic Rankine cycles have been proposed and used in a variety of applications * Binary geothermal power plants: A flash type geothermal power plant is unsuitable for low temperature, liquid dominated, geothermal resources (nominally below 180 C). Instead, the geofluid is used to boil an organic working fluid that is subsequently expanded in a turbine. Isobutane is an example of a common working fluid in such applications. * Solar thermal power: ORC plants can also be applied to the conversion of low and medium temperature (say up to 300 C) solar heat. The Coolidge plant [12] that was a 200 kW plant built near the town of Coolidge, AZ, is a good example. For higher temperature heat sources, toluene seems to be a common choice of working fluid. * Bottoming cycle applications: ORCs have been proposed for the bottoming cycle in some applications Mixtures of organic fluids have also been studied for use in ORCs. The advantages of the use of mixtures will be discussed later in this chapter. Binary mixtures have been found to have a better performance compared to pure fluid ORCs [1316]. Organic fluid turbines have been found to be very reliable and to have a relatively high efficiency even at small sizes. Due to their reliability, some people have even suggested their use in space applications, with the ORC as the bottoming stage of a Brayton or Rankine cycle power plant. Supercritical cycles The use of mixtures is one way to obtain good thermal matching with sensible heat sources. Other methods that have been proposed include multipressure boiling and supercritical operation. Multi pressure boiling has not been very popular in the industry because of the costs involved. The other option is to use supercritical cycles. The cycle high pressure and temperature exceed that at the working fluid critical point in a supercritical cycle. The boiling process does not pass through a distinct twophase region, and a better thermal match is obtained in the boiler. Figure 1.2 shows the boiling process in a supercritical cycle on a Ts diagram. Notice the better thermal match in a boiling heat exchanger. Supercritical cycles have to operate at a higher pressure, since the boiler pressure has to exceed the critical pressure of the working fluid. This is a disadvantage. Equipment costs go up at higher pressures, although there is an improvement in performance. 420 heat source 400  380  360  q) 2 3 / S340  1) E w 320 I 300  280  260 Fig. 1.2 Diagram showing a supercritical Rankine cycle on the Ts diagram for isobutane Supercritical cycles have been studied in the United States for geothermal applications as a part of the DOE Heat Cycle Research Program [17] and have been found to improve geofluid effectiveness (power output per unit mass of geothermal fluid consumed, usually expressed as kWh/lb). Boretz [18] studied the use of supercritical operation in space applications. He suggested the use of a supercritical cycle in order to avoid the twophase boiling encountered at subcritical conditions. Several power cycles utilizing ammoniawater mixtures as the working fluid have been proposed in the literature. A common characteristic of most of these cycles is that all or part of the heat rejection occurs in an absorber condenser. These range from the simple Maloney and Robertson cycle to the relatively sophisticated Kalina cycle. Some of these cycles are discussed below. Maloney and Robertson cycle Although ammonia based cycles have been proposed earlier in the literature, in this chapter the oldest cycle considered is one by Maloney and Robertson [19]. Maloney and Robertson studied the ammoniawater binary mixture as a candidate for a simple absorption based power cycle. They studied the cycle within the range of properties available to them at that time and concluded that in all cases a steam Rankine cycle was more efficient. The range of temperatures, pressures, and compositions for which property data were available at the time of the study was limited. This is a serious shortcoming of Maloney and Robertson's results. Further, the cycle did not have a super heater included in the loop, which could have changed somewhat the efficiencies obtained. Kalina cycle Another ammoniawater mixture based cycle of more recent origin that shows considerable promise is the Kalina cycle [20,21]. The first Kalina cycle was first developed in the late 1970's and early 1980's. This cycle, developed by Alexander Kalina, is essentially a Rankine cycle that utilizes a binary mixture of ammonia and water as the working fluid. This results in better thermal matching in the boiler, as will be discussed later. Another feature of the Kalina cycle is the extensive internal heat recovery and exchange arrangement that minimizes irreversibilities in the heat transfer processes in the cycle. Many of the Kalina cycle configurations proposed also have an arrangement that uses the heat in the vapor leaving the turbine in a distiller to concentrate the liquid stream that is boiled and expanded in the turbine. 10 0 Turbine Boiler 96 11 9r 5 Distiller Recovery t7 Hx. Fig. 1.3 Arrangement of a simple Kalina cycle [20] The basic version of the Kalina cycle is shown in Fig. 1.3. The composition of the working fluid passing through the boiler (or the working solution) is different from the basic solution in the absorber. This is accomplished through the distillation method that has also been called a "thermal compression" arrangement in some papers. The hot vapor leaving the turbine is used to distil a fraction of the pressurized basic solution (state 2) to produce vapor (state 6) rich in ammonia. The remaining liquid is combined with the cooled turbine exhaust vapor in an absorber (state 13). The turbine exhaust is normally at too high a concentration to be fully condensed at the corresponding pressure. Combining with the weak liquid enables condensation. The vapor from the distiller is combined with the rest of the basic solution and condensed (state 8) to form the working fluid. The working fluid is vaporized, and superheated (state 10) in the boiler and expanded in the turbine. The advantages of a Kalina cycle plant are the following * The use of a mixture results in a better thermal match in the boiler due to variable temperature boiling * Better internal heat recovery is also possible due to use of mixtures * The distillation arrangement along with absorption condensation allows a lower turbine exit pressure despite using a high concentration vapor. Higher work output is therefore obtained Several studies have shown that the Kalina cycle performs substantially better than a steam Rankine cycle system [2022]. ElSayed and Tribus [23] showed that the Kalina cycle has 1020% higher second law efficiency than a simple Rankine cycle operating between the same thermal boundary conditions. A second law analysis of various Kalina systems shows that using a binary fluid and the resulting reduced irreversibility generation in the boiler is one source of the improved efficiency of the cycle [2425]. Marston [26] verified some of the results of ElSayed, Tribus, and Kalina and through a parametric analysis concluded that composition at turbine inlet and temperature at the separator of a simple Kalina cycle are the key parameters in optimizing cycle performance. Several Kalina cycle variations have been proposed, each optimized for different applications. A common feature of each of these "systems" as the developers of this cycle prefer to call the versions, is the improved and sometimes counterintuitive application of internal heat recovery in the cycle to minimize exergy losses. For instance, in a bottoming cycle for utility applications, (Kalina System 6) [22], the vapor is actually cooled between the IP and LP turbines and used to evaporate part of the working fluid stream. The Kalina Cycle System 12 [27] is a variation of the Kalina cycle that was proposed for geothermal applications. This version does not have the trademark distillation arrangement. Instead, a more complicated network of recovery heat exchangers is used to improve efficiency. Figure 1. 4 shows the Kalina 12 system. Bliem [28] studied the application of the Kalina cycle to hydrothermal (liquid dominated geothermal) energy conversion. The supercritical, recuperated Rankine cycle technology studied as part of the heat cycle research program proved to have a slightly higher second law efficiency compared to the Kalina System 12. The Heber binary geothermal plant summer design conditions were used as a reference. Both these technologies were superior to the supercritical isobutane based Rankine cycle being used at Heber. STurbine HP TurbineLP   7 Superheater Reheater Geofluid  8 in 5 I In Evaporator 2 3 Recovery Hx. 1 Evaporator 1 10 Geofluid out Recovery Hx. 2 Condenser Fig. 1.4 Schematic diagram of Kalina System 12 The Kalina type distillationcondensation subsystem was also evaluated in supercritical organic cycles, but the improvement was found to be minimal. Bliem and Mines [29] also showed that the Kalina System 12 and the supercritical cycle approach a realistic upper limit of conversion, for the relatively high temperature resources (460 K and 500 K) studied. Ibrahim and Klein [30] compared the thermodynamic performance of the Maloney and Robertson cycle and the Kalina cycle based on a second law analysis. The second law efficiency was defined as the ratio of the first law efficiency of the cycle to that of a "maximum power" cycle operating between the same thermodynamic conditions. The maximum power cycle used is essentially the Lorenz cycle. These concepts are explained in greater detail in chapter 2. Based on the results from that study, the Kalina cycle outperforms the Maloney and Robertson cycle at large heat exchanger sizes (large NTU). The Kalina cycle has been criticized for the fact that obtaining the output predicted by calculations for the cycle requires 100% vaporization of the working fluid in a single pass countercurrent boiler. The heat exchanger surfaces would dry out at high vapor fractions, resulting in lower overall heat transfer coefficients and a larger heat exchange area. The first Kalina geothermal plant (of 2 MWe capacity) [31] built in Husavik, Iceland, does not use the system 12 configuration. Instead, about 70 % of the flow is vaporized and the remaining liquid throttled back through a series of recovery heat exchangers. Over the last twenty years, the Kalina cycle has caught a lot of attention from the engineering community, but that has translated to very few plants actually being built. The first bottoming cycle demonstration of the plant was at the Energy Technology and Engineering Center, a DOE facility near Canoga Park, California [32]. A 3 MW demonstration plant was constructed that started operation in 1992. Tests were conducted till the end of 1996. Other ammonia based cycles Rogdakis and Antonopoulos [33] proposed an ammoniawater absorption power cycle. This cycle is somewhat similar to the Kalina cycle, with the major difference being that the distillationabsorption condensation process is replaced with a simple absorption condensation process. The authors calculated a substantially higher (greater than 20%) first law efficiency as compared to a steam Rankine cycle, while keeping similar superheating and condensation temperatures. The improvement is particularly good at low heat source temperatures. An application of this cycle for recovering waste heat from gas turbine exhaust is discussed by Kouremenos et al. [34]. Use of Mixtures as Working Fluids in Thermodynamic Cycles In a conventional Rankine cycle in which a pure fluid is used, boiling occurs at the saturation temperature corresponding to its pressure. This results in a mismatch between the temperature profile of the heat source fluid (fuel combustion products, geothermal fluids etc.) and the working fluid in the boiler. Consequently, there is significant exergy destruction during heat transfer in the boiler. One of the methods around this limitation, practiced in industry, is to boil at different pressures, resulting in a moderate increase in efficiency. The other option is to use a mixture in the working fluid. The use of mixtures as working fluids in Rankine cycles is not a new idea. As a mixture vaporizes, the changing composition of the liquid results in boiling occurring over a range of temperatures. This gives what is called a "temperature glide" between the temperatures at which the mixture starts boiling to those at which the fluid is completely vaporized. Figure 1.5 qualitatively shows an example of the difference in boiling behavior just described. With a multicomponent mixture, the fluid boils throughout in the boiler, achieving a good temperature match with the heat source fluid. While the pure fluid boils at one temperature and is then subsequently superheated, the mixture boils at varying temperatures and achieves a better thermal match in the boiler [35]. The condensation of mixtures is also a variable temperature process. This results in a better thermal match with the heat rejection fluid. The advantages of better thermal matching in the condenser are debatable. In a power plant, there is normally very little cost associated with the heat rejection fluid itself. The size of the condenser is the Pure Fluid mixtures Heat S rce," Q pin point 2 'Working Fluid E I // Q (Heat) Fig. 1.5 Illustration of the difference in temperature profiles for a pure fluid being boiled and a multicomponent mixture being boiled. limitation. Also, for complete condensation of a fluid, the bubble point at the cycle low pressure has to be a little above the heat rejection fluid temperature. The use of mixtures has another advantage, particularly in geothermal power plants. Any degradation in the quality of the heat resource with time can be countered to a certain extent by changing the composition of the working fluid. Several cycles have been developed in the literature to take advantage of mixtures. Binary mixtures have been recommended as working fluids in binary geothermal power plants. It is found that a geothermal resource is utilized more efficiently by the use of these mixtures. The Combined Power and Cooling Cycle Introduction A unique feature of the combined power and cooling cycle proposed by Prof Goswami is the simultaneous production of power and refrigeration (cooling) in the same loop. Other combined power and cooling cycles utilize the waste heat rejected from the power cycle to run a coupled heat fired cooling cycle, such as an absorption refrigeration cycle. The cycle can be viewed as a combination of the Rankine cycle and an absorption refrigeration cycle. A binary mixture of ammonia and water has been used as the working fluid in all studies performed so far. Using that as an example, consider the schematic (basic configuration) of the cycle shown in Fig. 1.6. A mixture of ammonia and water (borrowing from absorption refrigeration literature, this mixture is called the strong solution, or basic solution) is pumped to a high pressure. This stream is preheated and pumped to the boiler, where it is partially boiled. Being the component with a lower boiling point, the vapor generated is rich in ammonia. A "rectifier" is used to increase the concentration of ammonia in the vapor by condensing some of the water out. The rectified vapor is superheated and expanded to low temperatures in an expander such as a turbine. This is possible since ammonia is a volatile component that does not condense at the temperatures and pressures at the turbine exit. The cold ammonia is used to produce refrigeration. The remaining hot liquid in the boiler, called the weak solution, is used to preheat the working fluid in the recovery or solution heat exchanger. This high pressure liquid is then throttled back to the absorber. The vapor is absorbed into the weak solution and 17 condensed in the absorber. The absorber takes the place of a regular condenser in the cycle. Superheater 9 Fig. 1.6 The basic configuration of the combined power and cooling cycle A portion of the strong solution stream is used to recover heat from the condensing vapor in the rectifier. The word rectification is used in the absorption cooling literature, by some authors, to refer to a particular configuration that employs a rectification column to purify the ammonia vapor. We use the word rectification to refer to a process used to purify vapor leaving the boiler in all the papers and reports associated with the study of this cycle. Some of the key features of the cycle are listed below. * Output consists of both power and cooling * Uses a mixture as the working fluid, which makes it suitable for sensible heat sources * Absorption condensation is used to condense the vapor * Works best using low temperature heat sources A summary of past research on the cycle The combined power and cooling cycle has been the subject of a sustained research effort at the University of Florida for the past eight years. A summary of the work done to date and the status of the development of the cycle is discussed below. The results of a parametric analysis of the cycle using low temperature sensible heat sources are reported in Goswami and Xu [2] and Xu et al.[36]. The analysis established that theoretically, both power and cooling could be obtained from the proposed configuration. Ideal turbine and pump were assumed, heat losses and pressure drops were neglected. The following range of parameters were studied 1. Boiler Temperature 400 K 2. Condenser Temperature 350 K 400 K 3. Turbine Inlet Temperature 410 K 500 K 4. Turbine Inlet Pressure 18 bar 32 bar 5. Ammonia Concentration 0.20 0.55 6. Absorber Temperature 280 K 7. Recovery HX. Exit Temp. 350 K The parametric study results indicate the general behavior of the cycle. Figure 1.7 shows the effect of turbine inlet pressure on the thermal efficiency2 of the cycle. For a set absorber pressure (as in a parametric analysis), the pressure ratio increases with a higher pressure, but the vapor generation rate drops until the pressure becomes too large to generate any vapor at all. A similar effect is seen with the cooling capacity. A larger pressure ratio results in a larger temperature drop in the turbine, but the drop in vapor generation rate limits cooling at higher pressures (see Fig. 1.8). The boiler temperature strongly influences the output, as the vapor generated goes up. The heat input also goes up with increased vapor generation; therefore the thermal efficiency reaches a limit after initially increasing. The cooling capacity of the cycle is strongly influenced by the superheater and condenser temperatures. A higher superheater temperature raises the turbine exit temperature, until no refrigeration is produced at all. The parametric study also clearly shows that there is an optimum value of parameters for the best operation of the cycle. Based on these results, Lu and Goswami [3] performed a mathematical optimization of the efficiency of the cycle using an optimization program that uses the Generalized Reduced Gradient (GRG) algorithm. The optimization program has since then been used extensively to maximize the thermodynamic output of the cycle in several studies [5,6]. Lu and Goswami [4] also used the program to determine the lowest temperatures that could be achieved using the cycle. They concluded that temperatures as low as 205 K are achievable, however, the vapor generation drops substantially at very low temperatures since lower absorber pressures and purer ammonia vapor generation is required. Hasan and Goswami [5] 2 See Table 1.1 for the definition of efficiency 22 u 20 E S18 16  x=0.47 x=0.5 x=0.53 16 18 20 22 24 26 28 30 32 34 turbine inlet pressure (bar) Fig. 1.7 Effect of turbine inlet pressure on the thermal efficiency (%) of the cycle [2] 30 x=0.47 x=0.5 25 Ax=0.53 20 i 15 rC 0 0 10 5 16 18 20 22 24 26 28 30 32 34 turbine inlet pressure (bar) Fig. 1.8 Effect of turbine inlet pressure on the cooling capacity (kJ/kg) of the cycle [2] performed an exergy analysis of results optimized for second law efficiency and looked at various aspects of the optimized results such as the refrigeration to work output ratio, pressure ratio etc. An experimental system was built to verify the actual performance of the combined power and cooling cycle. The initial system was a simplified version of the cycle without an actual turbine present. A throttling valve and a heat exchanger performed the functions of expanding and cooling the vapor, thus simulating the turbine in the loop. The experimental system also lacked a rectification arrangement that would have been useful to purify the vapor. Operating this limited version of the cycle [8] verified the boiling and absorption processes. The results indicated that both power and cooling can be obtained simultaneously and that efficiencies close to the predicted values should be achievable. Specific motivation for work in this dissertation Efficiency definition. While a significant amount of work has been done on the theoretical analysis of the cycle, there are some shortcomings that need to be addressed. The first question deals with the right efficiency definitions for the cycle. Since the output consists of both power and cooling, the questions that arises is how the two components would be added so as to arrive at a meaningful efficiency definition. Table 1.1 summarizes the efficiency definitions used in different papers written on the cycle. Initial papers simply added the work and cooling output to generate an efficiency, which really is an "energy" efficiency. Later papers divided the cooling output by a Carnot COP (evaluated at the average of the inlet and exit temperatures of the refrigeration heat exchanger). Such a definition gives a very small weight to the cooling output. The subject of efficiency definitions is discussed further in chapter 2. Table 1.1 Efficiency Definitions used in Various Papers Published on the Cycle Reference First Law Definition Second Law Definition Goswami and Xu [2] (Wet + Q,)/Qh N.A. Xu et al. [36] Lu and Goswami [3] (We + QO)/Qh (,, + Q)/E,,, Hasan and Goswami [5] (Wt + Q )/Qh [Wne + (QOCOP E a o)]/Eh,,,. Tamm et al. [7] Cycle simulation. The cycle simulation program written by Lu and Hasan had some shortcomings that were discovered when early simulations were being performed to compare results. These are listed below: * A pinch point check was not implemented in the recovery heat exchanger. Sometimes the strong solution starts boiling there resulting in the pinch point condition being violated * The fraction of the strong solution stream entering the rectifier was calculated by setting the temperature after the rectifier and using an energy balance. This temperature need not be set; it can actually be optimized. * The liquid condensate in the rectifier was assumed to return to the boiler and boil as a separate stream. Yet this was not considered when calculating pinch point in the boiler. The result was that the pinch point was improperly calculated. In addition, the design of such a heat exchanger with three streams is also complicated. Working fluid. All the work performed so far has been based on using an ammoniawater mixture as the working fluid. It is logical to assume that other mixtures could also be used in this new thermodynamic cycle to provide simultaneous power and cooling output. Organic fluids have been used in Rankine cycles designed for low temperature heat sources. Therefore, these fluids seem like natural choice for use in this cycle. Based on these motivations, the work presented in the subsequent chapters follows an outline as follows. First, there is a discussion of the various efficiency definitions that can be used for this cycle. That is followed by a description of the simulation and optimization methods. The basic cycle configuration is optimized using the proposed efficiency definitions, the results of which are discussed in chapter 4. Chapter 5 contains results of simulations performed using organic fluid mixtures and an analysis of the results. Some modified and improved configurations of the cycle are proposed and discussed in chapter 6. The final chapter suggests some directions for the development of the cycle in the light of this study. CHAPTER 2 EFFICIENCY DEFINITIONS FOR THE COMBINED CYCLE The combined power and cooling cycle is a combination of the Rankine cycle and an absorption refrigeration cycle. Evaluating the efficiency of this cycle 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 chapter develops several expressions for the first law, second law and exergy efficiencies for the combined cycle based on existing definitions in the literature. Some of the developed equations have been recommended for use over others, depending on the comparison being made. Finally, some of these definitions are applied to the cycle and the performance of the cycle optimized for maximum efficiency. A Generalized Reduced Gradient (GRG) method was used to perform the optimization. This method is described in detail in the following chapter. 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 ofperformance = useful output / input (2.1) 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 2.1 gives two typical first law efficiency definitions. For the case of an absorption refrigeration cycle, the input is in the form of heat and pump work. Therefore, its COP is expressed in terms of refrigeration output, heat input, and pump work as [37] COP =Qc/(Qh +w) (2.2) 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 [38]. The remaining exergy is lost due to irreversibilitites in devices. *Rexergy = lE,,u /IE,' (2.3) Two examples are given in table 2.1. For the absorption refrigeration cycle, the corresponding exergy efficiency expression is given as rexergy = Ec / (E + W) (2.4) Here Ec is the change in exergy of the cooled medium. A resource utilization efficiency [39] is a special case of the exergy efficiency that is more suitable for use in some cases. Consider for instance a binary geothermal power cycle, where the geofluid is reinjected into the ground after transferring heat to the cycle working fluid. In this case, the unextracted availability of the geofluid that is lost on Table 2.1 Some examples of efficiencies applied to the Rankine cycle and the vapor compression refrigeration cycle Cycle Type Rankine Vapor Compression First Law 1I = Wne/Qh (2.5) COP= Qc/W, (2.8) Exergy re, = Wn,/E, (2.6) qer = Ec/W, (2.9) Second Law rH = r/rrev, (2.7) qH = COP/COP_ (2.10) reinjection has to be accounted for. Therefore, a modified definition of the form 1R = ZEu, /Eh (2.11) 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 [40], IE er = useful T 'Ein Euseful (2.12) 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. 7, = 717/rev (2.13) 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 [37]. 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 [41]. 27 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 Fig. 2.1), the exergy efficiency is W Snexr et (2.14) Qh 10 .Th) while the second law efficiency is Wet nr ne (2.15) Q h (1 h Where To is the ambient or the ground state temperature. For the special case where the cold reservoir temperature Tr is the same as the ground state temperature To, the exergy efficiency is identical to the second law efficiency. Th Qh Cyclic W nfav/ira net Fig. 2.1 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. The first law efficiency has been the most commonly used measure of efficiency. The first law does not account for the quality of heat input or output. Consider two power plants with identical first law efficiencies. Even if one of these power plants uses a higher temperature heat source (that has a much higher availability), the first law efficiency will not distinguish between the performance 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 probably have a smaller condensor. 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 The performance evaluation functions discussed above will be applied to the combined power and cooling 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. For instance, to compare the novel cycle with a power cycle that uses waste heat to run an absorption refrigeration system. 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 i = (Wn,, + Qc)/ (2.16) Equation (2.16) 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 Carnot3 efficiency specifies the upper limit of first law conversion efficiencies. The confusion arises due to the addition of work and refrigeration in the output. Refrigeration output cannot be considered in an efficiency expression without accounting for its quality. To avoid this confusion, it may be better to use the definition of the first law efficiency given as 7 = (We, + Ec)/Qh (2.17) 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. Ec =m [h cf,,n hf,out To (s f,, scf,out) (2.18) Rosen and Le [42] 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 3 The Carnot cycle is not the reversible cycle corresponding to the combined cycle. This is discussed later in this chapter. 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 Eq.(2.3), 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. rexer, = (Wne + Ec )/(Eh ,,, Eh o,,t) (2.19) Second Law Efficiency The second law efficiency of the combined cycle needs a suitable reversible cycle to be defined. Once that is accomplished, the definition of a second law efficiency is a simple process. Lorenz cycle The Lorenz cycle is the appropriate "reversible cycle" for use with variable temperature heat input and rejection. A TS diagram of the cycle is shown in Fig. 2.2. S1 34 12Loren Q, (2.20) If the heat input and rejection were written in terms of the heat source and heat rejection fluids, the efficiency would be given as: mhr(hhr,.,,o hh,..,,) 1Lorenz mh (hr, ,n h~,o, (2.21) Knowing that processes 41 and 23 are isentropic, it is easily shown that in terms of specific entropies of the heat source and heat rejection fluids that mhs _(Shr out Shr,n ) mhr (Shin Sh.o.ut ) (2.22) The efficiency expression for the Lorenz cycle then reduces to lLorenz 1 (hhrout hhr, )/ (Shr out Shr, ,ln) sn s,out hszn hs,out (2.23) This can also be written as 7lLoren. 1 TVshs (2.24) Here, the temperatures in the expression above are entropic average temperatures, of the form Fig. 2.2 The TS diagram for a Lorenz cycle (2.25) T = (h2 hi) (S2 S) For constant specific heat fluids, the entropic average temperature can be reduced to (T T,) T = T2 (2.26) In(T2 Ti ) The Lorenz efficiency can therefore be written in terms of temperatures as L (T routt Thrn )/ln(Thr,out /Thr, n ) = 1 (T Th )Iln(Th Tht ) (2.27) (Thsn hsout ln( n hs,out (2.27) 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 COPLo = (2.28) )hr ),f 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 2.3 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 (Fig 2.3b). Assume that the combined cycle and the cascaded arrangement both have the same thermal boundary conditions. This assumption implies a b Fig. 2.3 Thermodynamic representation of (a) combined power/cooling cycle and (b) cascaded cycle 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 for refrigeration is w, w, + feo Sws t + fQ (2.29) Qh The weight factor,f is a function of the thermal boundary conditions. Therefore, the first law efficiency of the combined cycle can also be written as Whet + fQ rI = Q (2.30) Qh The work and heat quantities in the cascaded cycle can also be related using the efficiencies of the cascaded devices Wot = QhrHE (2.31) W, = Q /COP (2.32) By specifying identical refrigeration to work ratios (r) in the combined cycle and the corresponding reversible cascaded cycle as r = Q /Wet (2.33) and using Eq.(2.29) and Eqs.(2.31 2.33), one can arrive at the efficiency of the cascaded system as Ss 7HE r(f l COPOP) (2.34) lIys = rHE 1+ +rCOP ] assuming the cascaded cycle to be reversible, the efficiency expression reduces to l, rev = 'Lorenz f COPLoren (2.35) 1 + rCc7Lorenz Here 7Lorenz 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 rhI = r, /1l, rev (2.36) If the new cycle and its equivalent reversible cascaded cycle have identical heat input (Qh), the second law efficiency can also be written as n, W + fQ Wnet + fQ (2.37) r1I,rev Wnetrev + fc, rev This reduces further to net 1 + f net (2.38) Wnet.rev (1 + fr) W.netrev 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 iffis a function of the thermal boundary conditions. Assuming a value of unity forfsimplifies the second law efficiency expression even further. The corresponding reversible cycle efficiency would be, l+r r7,rev = Lorenz 1+ r/COP 1 (2.39) 1 / 1 + _Lorenz The resulting second law efficiency equation is a good choice for second law analysis. The expression does not have the drawback of trying to weight the refrigeration with respect to the work output. Being a second law efficiency, the expression also reflects the irreversibility present in the cycle, just like the exergy efficiency. Validity of Expressions Expressions for the first law, exergy and second law efficiencies have been recommended for the combined power and cooling cycle in Eqs. (2.17), (2.19) and (2.37) 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 Eqs.(2.17), (2.19) and (2.35), such a comparison would be perfectly valid. Case 2: Comparing a Combined Cycle to a Power Cycle In some instances, a combined cycle would have to be compared to a power cycle. For example, this cycle could be configured so as to operate as a power cycle. In this situation, the refrigeration would have to be weighted differently, so as to get a valid comparison. One way of doing this would be to use a practically achievable value of refrigeration COP to weight the cooling output. Another option is to divide the exergy of cooling by a reasonable second law efficiency of refrigeration (also called thermal efficiency of refrigeration). Such efficiencies are named "effective" efficiencies in this study. (Wnet + Qc COPpractcal) r1,ef Qh (2.40) (WUnet + E, GIH,ref ) 7i,eff = h (2.41) Rexergy,eff = (Wnet + Q COpractical )/(Ehs,n Ehs,out (2.42) lexergy,eff = (net + Ec /I, ref )/ (E,,in Ehsout ) (2.43) Some Examples of Application to the Combined Cycle An optimization program was used to optimize the performance of the Goswami cycle. A description of the methodology adopted is given in the following chapter. Optimization results using different efficiency definitions as objective functions are given in this section. The cycle simulations were performed using simple energy and mass balances. Approach temperature limits of 5 K were imposed on all heat exchangers. Boiler pinch points were assumed to be at least 5 K. It was assumed that there were no pressure losses in the devices and that the turbine and pump operated isentropically. The vapor at the turbine exit is constrained to be at least 90% dry. A turbine exit temperature of 270 K or lower was required to generate refrigeration. The optimization was also constrained so that refrigeration was always generated. The heat source fluid was assumed to be hot water at 400 K at the saturation pressure of water at that temperature. The absorber temperature was 5 K above the ambient which was assumed to be 298 K. Saturation conditions were assumed in the absorber to fix the concentration of ammonia in the binary mixture. The schematic of the cycle in Fig. 1.6 differs slightly from previously published versions in that the liquid reflux from the rectifier is sent back to the absorber. The optimization results of the cycle evaluated using the exergy efficiency and second law efficiency Eqs.(2.19) and (2.36) are both identical. That is not surprising, since exergy efficiency and second law efficiency are very similar. The cycle parameters are presented in Table 2.2. It was assumed that the chilled fluid was liquid water for these simulations. Table 2.2 Cycle parameters that yield optimum exergy and second law efficiencies. Parameters Optimized Case exery (%) 61.35 ( (%) 61.35 Tboder,exit (K) 395 Tectifierexit (K) 365.15 superheater, exit (K) 365.15 Tturbne,exit 269.96 Phgh (bars) 10.84 pow (bars) 1.00 basic, Ammonia mass fraction in absorber 0.288 1h hs, Mass flow rate of heat source (kg/s) 10.93 basic Mass flow rate of basic solution (kg/s) 3 Wnet, Net work output (kW) 79.15 Q,, Cooling output (kW) 25.99 Tables 2.3 and 2.4 give results optimized for effective efficiencies. Different second law efficiencies of refrigeration were assumed. The exergy of cooling was calculated on the working fluid side. It is seen that assuming different second law efficiencies of refrigeration has a significant effect on the equivalent exergy efficiency. Table 2.3 Efficiency and cycle parameters optimized for effective exergy efficiency S, o Case Case Case laorcmro rTlOfo T (%,ref (%) 1exergy,eff (%) h,,eff (%) Tboder, exit (K) fWer, ^ (K) Trectifier,ext (K) Tuperheater, exit (K) tu rbineexit Phgh (bars) Pow (bars) xbasc Ammonia mass fraction in absorber ih,, Mass flow rate of heat source basic Mass flow rate of basic solution Wet, Net Work Output Qc, Cooling Output I LLILLIII~L~IJ ~LLJ~I 0.374 0.360 0.390 0.388 10.82 10.79 10.92 10.95 69.2 59.1 72.2 58.1 97.4 34.2 87.63 30.55 30 72.52 17.70 395 342.85 342.85 253.68 16.7 1.8 II 50 65.20 15.89 395 343.88 343.88 252.10 15.3 1.64 III 70 62.74 15.11 395 368.75 368.75 270 17.4 2.0 IV 100 62.10 15.08 395 370.67 370.67 270 18.0 1.96 40 Table 2.4 Efficiency and cycle parameters optimized for effective first law efficiency SCase Case Case lroirntro r Of T r1nu,ref (%) 7h,eff (%) Tbodlerexit (K) Trecfier,exit (K) Tsuperheater,exit (K) turbine,exit Phgh (bars) Piow (bars) xbsic, Ammonia mass fraction in absorber basic Mass flow rate of basic solution Wet, Net work output Qc, Cooling output I LLILLIII~L~IJ ~LLJ~I 0.369 0.355 0.327 0.388 64.5 55.1 66.83 53.80 68.26 48.18 87.63 30.55 II 50 15.92 395 344.92 344.92 251.49 15.46 1.59 III 70 15.21 395 348.99 348.99 249.72 13.37 1.31 IV 100 15.08 395 370.67 370.67 270 18.0 1.96 30 17.73 395 343.69 343.69 253.08 16.79 1.74 To get a feel for second law efficiencies, a typical 10 EER vapor compression air conditioning system operates at a COP of around 3 at the standard rated conditions. The corresponding Carnot COP, assuming 280 K cold temperature and an ambient temperature of 308 K (selected based on standard rating conditions) is around 10. This implies a second law efficiency of 30%. Refrigeration cycles are inherently irreversible since they include a throttling process. The last case in tables 2.3 and 2.4 is the case where a second law efficiency of refrigeration is not considered. It is interesting to note that the optimization results in table 2.3, case IV are different from those in table 2.2. The reason for the difference is that irreversibilites in the refrigeration heat exchanger are considered in the optimization in table 2.2. The exergy of cooling Ec, is calculated on the working fluid side in both tables 2.3 and 2.4. Conclusions In defining efficiencies of a combined power and cooling cycle, it is necessary to weight the refrigeration output to obtain meaningful values. Definitions of first law, exergy and second law efficiencies have been developed in this chapter. From the basis of a strict thermodynamic analysis, the definitions given in Eqs. (2.17) and (2.19) are the correct efficiency definitions of the cycle. However, these definitions do not realistically weight the cooling content. When comparing the combined power and cooling cycle to one producing work alone, effective efficiencies defined in Eqs. (2.41) and (2.43) are recommended. It has also been shown in this paper that the weight assigned to refrigeration output has an impact on the optimum parameters for the cycle. CHAPTER 3 CYCLE SIMULATION AND OPTIMIZATION The combined cycle has been studied by a simple simulation model coupled to an optimization algorithm. The simulation model is simplistic, and is based on simple mass and energy balances. Ammonia water properties are calculated using a Gibbs free energy based method [43] while organic fluid properties were estimated using a NIST program that is based on a corresponding states method (see Appendix A). 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 is one in which either the objective function or one of the constraints are nonlinear functions. The cycle optimization method chosen for the analysis of this cycle is a search method. Search methods are used to refer to a general class of optimization methods that search within a domain to arrive at the optimum solution. It is necessary to specify an initial starting point in search schemes. The optimization algorithm picks a new point in the neighborhood of the initial point such that the objective function (the function being optimized) value improves without violating any constraints. A simple method of determining the direction of change is to calculate the gradient of the objective function at the current point [44]. 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 [4446]. There are several variations of the GRG algorithm. A commercially available program called the GRG2 was used for ammonia water optimization, and a more recent version called the LSGRG2 was used for alternate fluid work. 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 Himmelblau et al. [46] and Lasdon et al. [47]. A brief description of the concept of the algorithm, heavily adapted [46, 47] is presented below: Consider the optimization problem defined as: Minimize objective function: g,, (X) Subject to equality and inequality type constraints as given below g,(X)= 0, i=,...,neq (3.1) 0< g, (X) < ub(n +i), i = neq + ,...,m (3.2) The variables are constrained by an upper and lower bound. lb(i) < X, < ub(i), i = 1,...,n (3.3) Here X 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, Xl,...,Xn+m The optimization program then becomes Minimize: g, (X) Subject to: g, (X)X,+ =0, i= ,...,m (3.4) lb(i) < X, < ub(i), i = 1,...,n+m (3.5) lb(i) = ub(i) = 0, i = n + 1,..., n + neq (3.6) lb(i) = O, i = n+neq+l,...,n+m (3.7) The last two equations specify the bounds for the slack variables. Eq. (3.6) specifies that the slack variables are zero for the equality constraints, while the variables are positive for the inequality constraints. The variables X,,..., X, 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 GRG2 and LSGRG2 software, using the nb binding constraint equations, nb of the natural variables (called basic variables) are solved for in terms of the remaining nnb natural variables and the nb slack variables associated with the binding constraints. These n variables are called the nonbasic variables. The binding constraints can be written as g(y,x)=O (3.8) Here y and x are vectors of the nb basic and n nonbasic variables respectively and g is a vector of the binding constraint functions. The binding constraints Eq. (3.8) can be solved fory in terms of x, reducing the objective to a function of x only. gl (y(x), x) = F(x) This equation is reasonably valid in the neighborhood of the current point to a simpler reduced problem. Minimize F(x) Subject to the variable limits for the components of the vector x. l x u (3.9) The gradient of the reduced objective F(x), VF(x) is called the reduced gradient. Now the search direction can be determined from the reduced gradient. A basic descent algorithm can now be used to determine an improved point from here. The choice of basic variables is determined by the fact that the nb by nb basis matrix consisting of (g, /ly, )should be nonsingular terms 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 elsewhere [4648]. This algorithm is a robust method that appears to work well for the purposes of optimizing this cycle, the way it has been implemented in our study. Search Termination The search will terminate if an improved feasible point cannot be found in a particular iteration. A well known test for optimality is by checking if the KuhnTucker conditions are satisfied. The KuhnTucker conditions are explained in detail in [45,46]. It can be mathematically explained in terms of the gradients of the objective functions and inequality constraints as: Vg +, (X)+ uVg, (X) 0 (3.10) uJ >0, u,[g (X)ub(j)]=0 (3.11) g, (X)< ubj), j= 1,...,m (3.12) Here, u, is a Lagrange Multiplier for the inequality constraints. Unfortunately, the KuhnTucker conditions are valid only for strictly convex problems, a definition that most optimization problems do not satisfy. A disadvantage of using a search method, such as the GRG algorithm that has been used in this study, is that the program can terminate at a local optimum. There is no way to conclusively determine if the point of termination is a local or global optimum [46]. The procedure is to run the optimization program starting from several initial points. Sensitivity Analysis The sensitivity of the results to the active constraints can be determined using the corresponding Lagrange multipliers. u= (3.13) Saub(j) Where, Vis the value of the objective at the optimum. Application Notes There are some factors in the optimization of the cycle studied using GRG2 and LSGRG2 that are interesting to mention. In a search scheme, it is possible that the termination point could be a local optimum, or not an optimum at all. It is necessary to determine the nature of the "optimum" returned by the program. Prior to the optimization, during setup, close attention should be paid to: * Scaling of the variables * Limits set for different convergence criteria * Method used to numerically calculate the gradient 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. Experimenting with the numerical calculation of the gradient is useful during scaling. Too small a convergence criterion, particularly for the NewtonRaphson method used during the onedimensional search can cause premature termination of the optimization program. The accuracy of the numerical gradient affects the search process. This was clearly seen when setting up the program with the SUPERTRAPP program. A forward differencing scheme was not accurate enough for the search to proceed forward. 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. * Optimized results using different objective functions were useful as starting points for other cases. * Examining the constraints indicated if the point was truly an optimum. For instance, at maximum second law efficiencies, the pinch point in the boiler was expected to be at its lower bound. * The results were perturbed and optimized, particularly with respect to what would be expected to very sensitive variables, to see if a better point could be obtained. For example, in RUE optimization, the heat source flow rate is a very sensitive variable. Perturbing the optimum results w.r.t the heat source flow rate was very useful to get better points. * Another method is to change the scaling of variables that appear to be insensitive to check if better points can be obtained. * For each configuration, comparing the results for different heat source and turbine exit temperatures was used to identify nonoptimum and local optimum points. At the end of this exhaustive process, it is assumed with reasonable confidence that the resulting point is indeed a global optimum. The optimization process using GRG is to a certain extent "art" not "science". Unfortunately, this is a problem with almost all NLP methods currently in use. Cycle Modeling Several cycle configurations have been studied in this dissertation. The model used for the basic cycle configuration shown in Fig. 3.1 is first discussed. The modeling of additional features in modified configurations is discussed, if necessary, when the alternate configuration is introduced. Consider the basic cycle configuration shown in Fig. 3.1 Eight independent variables are sufficient to determine the operation of the cycle. Two more variables are required to determine heat source conditions. Pressure drops and heat losses are neglected. * Cycle high pressure * Cycle low pressure * Absorber temperature * Temperature at boiler exit * Temperature at rectifier / condenser exit * Superheater exit temperature * Effectiveness of the heat recovery heat exchanger, R1 * Ratio of the mass flow rates of the strong solution in the heat recovery heat exchanger and the rectifier, R2 * Heat source temperature * Mass flow rate of heat source The absorber temperature is set a little above the ambient temperature. The following general assumptions were used in the simulation * Pressure losses in piping and equipment are neglected. * An isentropic efficiency is assumed for the pump and turbine * All processes are assumed to end in equilibrium thermodynamic states. Using the variables and assumptions listed above, the properties at all state points in the cycle can be determined. For a binary mixture, two intensive properties and the composition of the mixture are sufficient to establish all the other properties. Some combinations include pressure, temperature and composition (p, T,x); pressure, specific enthalpy and composition (p,h,x); or pressure, specific entropy and composition (p,s,x). Property calculation methods are discussed in Appendix B. State Point Calculation The calculation of state points for each state in Fig. 3.1 is described below. State point 1 The mixture in the absorber is assumed to be at saturation conditions. The low pressure of the cycle and the temperature of the absorber allow the calculation of ammonia concentration of the basic solution. Absorber temperature is assumed to be 5 K above ambient temperature. A reference flow rate is assumed and all other flow rates in the (1R m, m = 1 kg/s reference flow rate Fig. 3.1 The basic cycle configuration with the variables shown Superheater system are scaled to this value. The choice of reference flow rate is to an extent decided by the optimization program. The value is determined after a few trials so that other parameters (particularly heat source mass flow rate) are reasonably large. This is important so that error in gradient calculation is limited. Relative magnitudes of the optimized variables are important to ensure reasonable values in gradient calculations. In equation form: T, = Tabsorbev (3.14a) P = Plo (3.14b) x, = satcon(p, T,) (3.14c) ih, =3 (3.14d) Satcon is a function in the ammoniawater properties program to calculate saturated liquid concentration given the pressure and temperature. Knowing three properties, the remaining properties of the binary mixture can be determined. This is implicit in the rest of this section. Point 2 The pump pressurizes the strong solution to the high pressure in the cycle. Knowing the isentropic efficiency of the pump, the pressure and enthalpy at state 2 is easily computed. The composition of the solution remains the same as in state 1. P2 = Phgh (3.15a) s.sen = S1 (3.15b) (3.15c) X2 = X1 Calculate enthalpy at pump exit, h,,,,en, for isentropic compression. Then, knowing the pump efficiency, the actual enthalpy can be calculated using Eq. (3.15d) below. h2 h1 77 2 hI (3.15d) h.se,, h, Points 5 and 11 Since the boiler exit temperature and pressure are known, vaporliquid equilibrium correlations yield the vapor and liquid compositions leaving the boiler. A mass balance is used to determine the vapor and liquid mass fractions. T5 = T, = Tboler (3.16a) P5 = P11 = Phigh (3.16b) Use VLE data to get equilibrium compositions of vapor and liquid. The vapor and liquid mass fractions in the boiler can be computed from a mass balance as shown below. vmfboler 4 1 (3.16c) X5 11 lmfbod er _5 x 4 (3.16d) X5 X I x5 "11 Therefore, the mass flow rates would be Ih5 = vmfboler1 4 (3.16e) 1111 =lmfboilerim4 (3.16f) 53 Points 6 and 7 The rectifier exit temperature and pressure are known, as well as the boiler vapor mass fraction and composition. This allows the calculation of vapor and liquid compositions and mass fractions leaving the rectifier. T6 T7 = Tectfer (3.17a) P6 P7 = Phgh (3.17b) VLE data is used to get equilibrium compositions of vapor and liquid. Once again, a mass balance gives the vapor and liquid mass fraction leaving the rectifier as Vmfrecfer = X5 (3.17c) x7 x Infrecr = X 5 (3.17d) x7 6 Therefore, the mass flow rates would be Mi7 = vmfretfier j5 (3.17e) h6 = mfrechfier1j5 (3.17f) Point 8 The superheater exit temperature is specified for each simulation, and the pressures and composition of the vapor are known. T8 superheater (3.18a) P = Phgh (3.18b) (3.18c) X8 = X7 54 Point 9 The turbine isentropic efficiency is assumed, the turbine exit pressure is the system low pressure and the composition of the vapor is known. Therefore state points at the turbine exit can be calculated. P9 = Plow (3.19a) 8sen = s, (3.19b) x9 = X, (3.19c) Calculate an enthalpy corresponding to isentropic expansion and use in equation below to calculate enthalpy at turbine exit. hs h9 t = 9 (3.19d) h8 hsen Point 10 If the vapor is cold enough, it is assumed that the vapor leaves the refrigeration heat exchanger at 5 K below ambient. If the vapor leaving the turbine is not cold enough to yield refrigeration, state 10 is identical to state 9. The pressure and composition are already known. If T9 < Tm T,0 = To 5 (3.20a) Po = P9 (3.20b) 10 = x9 (3.20c) Else, point 10 is identical to state point 9. Points 12 and 3 The maximum heat recovered from the weak solution is decided by the approach temperature limits. A fraction of this heat (variable R,) is recovered. x12 = x,1 (3.21a) PA2 = 11 (3.21b) A 5 K approach temperature difference is assumed in the recovery heat exchanger, i.e., TI = T2 + 5. Other properties at the state point are calculated using this temperature to obtain maximum possible heat transfer. This assumption eliminates the use of a constraint at that point. QHRHX,max,hot = ml(hl, hl ) (3.21c) On the other side of the heat exchanger, P3 =Phgh (3.21d) 3 = X (3.21e) Ih3 = Rh2, (3.21f) Set the maximum temperature at the heat exchanger outlet T'= T, 5 (3.21g) Now a cold side maximum heat transfer can be calculated QHRHXmaxcold = l3 (h h3) (3.21h) The smaller of the values from Eq. (3.21c) and Eq. (3.21h) represents the maximum heat transfer possible. Now, using the ratio of the actual heat transfer to the maximum possible heat transfer, RI, enthalpy h12 is calculated as h12 = hll QHR x (3.21i) / 11 h3 =h2+(RlQHRHXmx /1 '/3) (3.21j) This allows the calculation of other state points. Note that R1 is a pseudo heat exchanger effectiveness. The advantage of this assumption is that approach temperature constraints can be eliminated for the heat exchanger. Point 13 The liquid reflux stream from the rectifier, and that from the boiler mix to form the fluid at state 13. The mixing is modeled as a constant total enthalpy process. P13 =Phgh (3.9a) h13 6h6 12h(3.9b) m13 /m6 X6 +/12 X12 x"13 = (3.9c) m13 Point 14 The weak solution at high pressure is throttled to a low pressure. The throttling is modeled as a constant enthalpy process. h14 = h13 (3.10a) (3.10b) P14 = Plow 57 X14 = 13 (3.10c) Point 15 The heat lost by the condensing fluid in the rectifier is recovered by the part of the strong solution stream flowing through the rectifier. A simple energy balance allows the determination of the state of the fluid at point 15. P15 =Phgh (3.12a) 15 = X2 (3.12b) h15 = (1 R2 )2 (3.12c) Qrectier = lhh5 lh6h6 l7h7 (3.12d) h15 = h2 +Qrecfer /5 (3.12e) Point 4 The two strong solution streams mix to form the fluid at state 4. This is also modeled as a constant total enthalpy process, as in statel3. The composition of the two streams are the same, and hence no mass balance is needed. h4 = h3 + h1 (3.13a) X4 =2 (3.13b) P4= Phgh (3.13c) 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 GRG 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 3.1 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 Table 3.1 Independent cycle parameters and their limits Variable Lower Limit Upper Limit Name and Units Tabsorber T + 5 To +5 Absorber Temperature (K) Tboder T + 5 Theatsource 5 Boiler Exit Temperature (K) Tecfi T + 5 That.ou 5 Rectifier Exit Temperature (K) Tsuperheater T + 5 Theaturce 5 Superheater Exit Temperature (K) Ph/gh 2 100 Cycle High Pressure (bars) ow 1 100 Cycle Low Pressure (bars) , T, .T, Heat Source Temperature (K) Theatsource Theatsource Theatsource Heat Source Temperature (K) hheatsource 0 18 Mass Flow Rate of Heat Source Fluid (kg/s) R 0 1 Pseudo Heat Exchanger Effectiveness R2 0.01 0.99 Flow Split Ratio not available. For instance, although the rectifier exit temperature can reach its upper limit, the value always has to be below the boiler exit temperature. The actual domain in which these variables may vary is further restricted by additional constraints that are specified. The lower limit for the low pressure is set at 1 bar, to avoid vacuum pressures anywhere in the system. At vacuum pressures, noncondensable gases enter the system. Additional equipment is required to remove these gases. The highest pressures in the optimization are set arbitrarily at around 100 bars. 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 3.2 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 3.2 follows. The rectifier/condenser exit temperature should be below the boiler temperature and superheater temperature. A minimum approach temperature of 5K is assumed for all devices. A pinch point temperature difference of 5K is also assumed in the boiler. For the contingency that boiling could begin in the rectifier or recovery heat exchanger, a pinch point calculation is implemented in those two devices also. To ensure that the reflux flow from the boiler is in the right direction, a constraint is used to maintain that value positive. Additional constraints are used to keep the vapor mass fraction at the turbine exit (dryness of vapor) greater than 90%. Two additional constraints (that might 60 Table 3.2 Constraints used in the optimization Constraint Description Lower Limit Upper Limit T Toler Measure of condensation in the rectifier 1E+30 0 rectifier boiler Trecqfier Tu erheater Measure of Superheating 1E+30 0 T6 T Rectifier exit approach AT 5 1E+30 5 T5 Rectifier inlet approach AT 5 1E+30 Ths,l T7 Boiler exit approach AT 5 1E+30 Ths, T4 Boiler inlet approach AT 5 1E+30 Th,, Pinch Point AT in Boiler 5 1E+30 pinch,boiler Tpnh,HRHX Pinch AT in HRHX (if applicable) 5 1E+30 Tplnch,ectiie Pinch AT in Rectifier (if applicable) 5 1E+30 ,hl x100 Boiler reflux mass flow rate 0 1E+30 vmf.rbne x100 Vapor mass fraction at turbine exit 90 100 vmfboler x100 Vapor mass fraction at boiler exit 1E05 100 Imfrecer x 100 Liquid mass fraction 1E05 100 QC Cycle Cooling Output 0.1 1E+30 Objective Efficiency Percentage 0 100 be redundant) are used to ensure that some vapor is generated in the boiler and that there is some condensation in the rectifier. These ensure that there is no heat addition to the vapor in the rectifier. A final constraint is used so that there is always some minimum positive cooling achieved in the cycle. This is to make sure that the cycle behavior does not go into a mode where only power is produced. Model Limitations The model used to simulate the cycle is a simple one with the advantage of being computationally fast. The results are expected to give a good idea of the trends that would be seen in real equipment. Such simple models are regularly found in the literature for preliminary analysis of thermodynamic cycles. The efficiencies obtained from such models would be higher than actually achievable efficiencies. While the results of the optimization study will indicate an approximate value for the maximum efficiency of the cycle, the cycle parameters are not necessarily practical. The ultimate predictor of the usefulness of the cycle is in the economics of using it. CHAPTER 4 OPTIMIZATION OF BASIC CYCLE CONFIGURATION USING AMMONIA WATER MIXTURE AS THE WORKING FLUID Initial parametric studies of the combined power and cooling cycle using ammonia water mixtures as the working fluid suggested that some optimum conditions of operation exist for the cycle performance [2,36]. An optimization scheme was implemented to determine the best conditions of operation (from a thermodynamic efficiency perspective) for various applications [3], such as for utilizing geothermal and solar resources, and for achieving very low temperature cooling. Work done in the past has used a variety of different efficiency definitions. In this chapter, some of the past optimization work has been repeated with the efficiency definitions discussed in chapter 2. Additional results are presented, that use isentropic efficiencies for the turbine and pump. The modeling of the cycle in earlier work was improved upon for the optimization work discussed in this chapter. Two additional variables were added to vary the effectiveness of the Heat Recovery Heat eXchanger (HRHX) and to control the ratio in which the strong solution stream was split between the HRHX and the rectifier. Earlier modeling had the liquid reflux from the rectifier going back into the boiler and boiling as a separate stream. Such a model makes the pinch point calculation in the boiler difficult. The stream was diverted to a point after the recovery heat exchanger. Additional pinch point checks were introduced in the rectifier and HRHX to account for the possibility of the start of boiling of the strong solution before entering the boiling heat exchanger. Simulated Conditions The modeling of the cycle in the optimization study carried out here has been discussed in the previous chapter. One of the questions that remained unanswered in earlier optimization work is the influence of the turbine exit temperature (the low temperature in the cycle) on the optimized efficiencies. Turbine exit temperatures were varied from 255 K to 280 K in 5 K increments. Optimizing for work output maximization, one ends up working with a cycle similar to the Maloney and Robertson cycle. A low pressure limit of 5 bars was used in the work optimization in order that the cycle remained an absorption type cycle. This limit was arbitrarily set so that there is a reasonable amount of absorbent in the basic solution at the absorber temperature. To study the effect of heat source temperatures, four values, 360 K, 400 K, 440 K and 480 K were picked for simulation. This covers a range of low to medium temperature solar heat. The heat source was assumed to be water at the saturation pressure corresponding to the heat source temperature. The ambient temperature was assumed to be 298 K for all simulations. The objective functions corresponding to efficiency definitions developed in chapter 2 are shown in Eqs. (4.14.5) ObjR ,,R =(We,+E)/Eim, (4.1) Obex = exerg = (Wnet + E )/(EhsLn EL ut ) (4.2) Obj = r = (Wnet + E )/h (4.3) ObjRk, =R,eff = (Wet +E c/l iref )/E,sn (4.4) Objexergy eff = exergyeff = (Wnet +E c ref r(E ,n Ehsout ) (4.5) i Superheater 11 Heat :Source Outlet 8 Heat Source Inlet Recovery Tur Heat Turbine L Exchanger 12 1 9 13 Refrigeration 2 14 Heat Exchanger 1 Absorber 10 Pump Fig. 4.1 Block diagram showing the basic scheme of the combined power and cooling cycle. Same as Fig. 1.6 Exergy losses in each component were calculated in the optimized results using the entropy generation in each device. The turbine and pump were the only devices assumed without losses in the model. Results discussed later in this chapter assume a value for isentropic efficiencies for these devices. The exergy loss in each process is calculated using the GuoyStodola theorem, from an entropy balance on each device, as in Eq. (4.6). Eos = ToSg,, = To(ISo,, S,,) (4.6) Here, To is the ground state temperature with respect to which exergy is defined, set at 298 K for all cases. The ground state pressure was set at 1.013 bar (1 atmosphere). Optimization results Resource Utilization Efficiency The resource utilization efficiency is the recommended choice in evaluating an energy conversion device for resources that are "discarded" after use in the cycle. A good example is the case of a geothermal application, where, the heating fluid is reinjected into the ground after extraction of energy from it in the power plant. The availability of the geofluid at the point of reinjection is wasted exergy that the power plant is incapable of utilizing. This efficiency can also be applied to other applications. In a coal power plant, the resource utilization efficiency would consider either the exergy of the coal and the air used for combustion, or the exergy of the products of combustion (hot gases and any solid residue such as ash). Unutilized exergy leaving the power plant through hot combustion products leaving the smokestack and in hot ash dumped into an ash pit is wasted. Optimizing for resource utilization efficiency is a good choice to 100 80 60 3 work 40 40 280 K 255 K 20  20 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.2 Optimized resource utilization efficiencies for the basic cycle configuration ensure that maximum use is made of the energy source or fuel. The resource utilization efficiency is written as (also see Eqs. (4.3) and (2.11)) (wet +Ec) i h... h To (sh so) (4.7) The specific enthalpy ho and specific entropy So are calculated at the ground state. To maximize the efficiency of Eq. (4.7), the sum of the net work output and the exergy of cooling has to be increased, while the mass flow rate of the heat source is decreased. The objective function, therefore, is very sensitive to the flow rate of the heat source. The optimum resource utilization efficiencies (RUE) are shown in Fig. 4.2. Three different cases have been plotted. Maximum efficiency obtained using two different turbine exit temperature limits are shown. There is a slight drop off in RUE with lower turbine exit temperatures. Optimizing purely for work output gives a better RUE value. An upper limit of 5 bars (0.5 MPa) was used for the work maximization simulations. Without this constraint, the optimum conditions were seen to be at conditions where the cycle operates using nearly pure ammonia as a Rankine cycle. The absorber model used is not valid at those conditions and the results are also not valid. Exergy Efficiency In solar thermal power plants, normally solar energy is collected and stored in a heat storage medium. This medium acts as the heat source for the power plant. The exergy of the heat source is not lost in the unextracted heat leaving in the fluid exiting the cycle because it is recirculated through the collector system to the storage tank. In such a case, the exergy efficiency is a reasonable choice for the objective function. This assumes though that the collection system efficiency is unrelated to the inlet and outlet temperatures of the cycle, which might not be the case. Even for solar resources, ultimately, resource utilization efficiency can be calculated, if the solar radiation were to be treated as the input. The definition of exergy of solar radiation is still being debated, but the values of different definitions in the literature give values that are fairly close to each other. Exergy efficiency is defined in terms of the exergy transferred from the heat source fluid to the working fluid in the boiler. This function is relatively less sensitive to mass flow rate of the heat source. Changing the mass flow rate would be compensated by a change in specific exergy change in the heat source fluid, in the denominator of the exergy efficiency equation, Eq. (4.6). A large heat source flow rate results in a small temperature drop of the heat source and a bad thermal match in the boiler. 68 100 80 work S60 280 K C: 255 K LU >, 40 w 20  20 0 ii i  i i 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig.4.3 Optimized exergy efficiencies for the basic cycle configuration (W,,,et +Ec) 1exergvy [ (, E s (4.6) m' lh,, hsout o (Shsin hrs,out For the exergy efficiency case, as seen in Fig. 4.3, efficiency is maximum at 400 K (of the four temperatures considered here), which is still lower than the efficiency in the work output optimized case, with a 5 bar upper limit on pressure. The optimized results are not bound at the 5 bar pressure limit. The reason for the drop in efficiency at higher temperatures is primarily due to the boiler temperature not increasing corresponding to the increase in heat source temperatures. There are limits to the temperatures in the boiler and condenser at which refrigeration can be obtained efficiently, since it becomes difficult to achieve sufficient temperature drop in the turbine across a given pressure ratio. A similar effect can also be seen in first law efficiency optimization (see Fig. 4.4) where the efficiency curve plateaus at higher heat source temperatures. The effect of lower turbine exit temperatures is to lower optimum efficiencies slightly. First Law Efficiency The first law efficiency is not sensitive to mass flow rate of the heat source. Once the optimum operating conditions of the cycle are determined, as long as the pinch point conditions are not violated, changing the mass flow rate does not affect the first law efficiency at all, since the heat input to the cycle remains the same. An advantage of using a binary mixture is the improved thermal matching in the boiler. Since the first law efficiency does not account for the heat source behavior, the exergy efficiency is a better choice for evaluating binary mixtures. It might be a better choice to use the first law efficiency corresponding to the optimum exergy efficiency. A first law efficiency is a useful measure of the cycle's performance. It is a direct measure of the heat transfer requirements in the boiler and condenser. A cycle with a high first law efficiency would have a much smaller boiler heat transfer area requirement per unit work output. Similarly, the condenser load being smaller, would use a much smaller condenser for the same boiler size. Note that this is a simplistic statement. Several other factors such as the heat transfer coefficients and pressures play a role in the size of the equipment. Optimized first law efficiencies in the refrigeration domain are shown in Fig.4.4 Once again, the work optimization results show a higher value of efficiency. The effect of lower temperatures in the cycle is a small drop off in efficiency. It is also seen from the results that the cycle optimized in the refrigeration domain fails to take advantage of higher heat source temperatures. The optimized parameter values for heat source 70 30 25 I work 220 S20 280 K S 255 K 4 15 LL J S10  340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.4 Optimized First Law Efficiencies for the Cycle Configuration in Fig. 4.1 temperatures of 440 and 480 K are almost identical. Optimizing in the work output domain, one sees an increase in efficiency going from 440 K to 480 K. This implies that while the first law efficiency improves, a corresponding increase in exergy efficiency is not seen. Exergy Analysis An exergy analysis was performed to determine the various pathways of losses in the cycle. For the optimized RUE results, the major exergy losses in different parts of the cycle are plotted in Figs. 4.5 and 4.6. Figure 4.5 shows the losses occurring for the case where cooling is produced and the turbine exit temperature limit was set to 280 K. A large fraction of the exergy of the heat source is lost in the fluid leaving the cycle. Approximately 30 % of the exergy available in the heat source is lost in this fashion. The 0 03 40 o 4', EP a) r xo &s 20 (0 0 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.5 Exergy destruction in the cycle represented as a percentage of exergy of the heat source, for optimized RUE corresponding to 280 K turbine exit temperature heat addition v internal Hx I heat rejection + throttling   unrecovered from heat source 4* 0, 40 360 380 400 420 440 460 480 40 30 30 40 420 440 460 480 Heat Source Temperature (K) Fig. 4.6 Exergy destruction in the cycle represented as a percentage of exergy of the heat source, for optimized RUE corresponding to pure work output heat addition v internal Hx  heat rejection + throttling 0  unrecovered from heat source  , V 0 20 40 (0 3 > (0 y) 3 losses in the heat addition process (boiler and superheater combined) are also high due to the poor thermal matching in the boiler. At the optimized condition, the mass flow rate of the heat source is very small. At these conditions, the strong solution enters the boiler at a temperature slightly above the bubble point. This removes the pinch point constraint in the boiler and allows a small heat source flowrate at the expense of thermal matching. In the work domain, there is no rectification at optimum conditions. The process of rectification is a source of exergy loss and is unnecessary if refrigeration is not desired. At higher heat source temperatures (440 K and 480 K), there is no heat recovered in the solution heat exchanger under optimum conditions. The hot weak solution is throttled and then cooled in the absorber, with considerable losses. The heat source is used to preheat the working fluid and therefore leaves the cycle at low temperatures. As a result, the exergy losses in heat rejection and throttling go up, while the loss through unrecovered exergy in the heat source drops (Fig. 4.6). Similar plots are shown for exergy destruction in the optimized exergy efficiency cases in Figs. 4.7 and 4.8. In the case with cooling output, the losses in the boiler are small. There is good thermal matching in the boiler, except at the 480 K heat source temperature. The exergy loss during heat rejection, in the absorber, is a major source of irreversibility in the system. That is surprising because the vapor has been expanded to very low temperatures and a large fraction of the heat recovered from the weak solution. Apparently, the mixing and absorption losses are high. Once again the optimum parameters in the work domain do not show any rectification. The internal heat exchange category in Fig. 4.8 is the sum of exergy destruction in the solution heat exchanger and rectifier. It is seen that exergy destruction 50  40 00 2. 30 (b 2 20 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.7 Exergy destruction in the cycle for optimized exergy efficiency corresponding to 280 K turbine exit temperature 50 S* heat addition internal Hx S40  heat rejection + throttling 3X =S 2)Gd3 20 0 >, x U (010 . 10 I I i i r 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.8 Exergy destruction in the cycle for optimized exergy efficiency corresponding to pure work output heat addition I internal Hx E heat rejection + throttling a^ v  U _ in the boiler and superheater is small. The thermal matching in the boiler is reasonably good in the exergy efficiency optimized cases, but it appears to be poorer than in the cooling domain. In the work domain, the requirement for work output does not limit the boiler exit temperature. However, the superheating seen at optimum conditions is larger and therefore, the thermal match in the boiler is poorer. In addition, the vapor fraction in the boiler is large enough that there is not enough weak solution to preheat the strong solution sufficiently to eliminate the pinch point in the boiler at 440 K and 480 K. This accounts for the odd looking inflections in the curves in Fig. 4.8 and contributes to increased exergy losses in the boiler. Optimization Considering Losses The optimization discussed so far assumes isentropic turbine and pump. Only approach temperature constraints are placed on the solution heat exchanger. The resulting effectiveness is close to 95%. Additional simulations were performed considering the turbine and pump as non ideal devices and using a lower effectiveness for the recovery heat exchanger. Optimization calculations were also performed using effective efficiency definitions developed in chapter 2. An isentropic efficiency of 0.85 was assumed for the turbine, while a value of 0.8 was assumed for the pump. The values assumed are relatively optimistic, actual values would depend on the type of expander used and the size and scale of the equipment. Although the effect is predictable, the basic cycle configuration was also optimized using heat exchangers with lower effectiveness of 70 % and 80%. A turbine exit temperature limit of 270 K was set for all simulations. The optimized RUE is plotted in Fig. 4.9. The effect of including an isentropic efficiency for the turbine and pump is to lower the RUE (see Fig. 4.2). In effective 10  0  340 U  S qRi, Eq. (4.1)  'R with irer=0.4, Eq. 14 4 '1R.t with ,llr,=0.3, Eq. :4 4  Only work output 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.9 Optimized RUE for the basic cycle configuration using a non isentropic turbine and pump 40 30   UL 20 REq. (4.1), =0.8 10  R.ef with nir,=0.4, Eq. 14.4i ,.=0.8 lR, Eq. (4.1 ..=0.7  lRef with nr ,,=0.3, Eq. (4.4), e=0.7 0  340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.10 Effect of recovery heat exchanger effectiveness on RUE at optimum RUE conditions, with cooling output efficiency definitions, as discussed earlier, the cooling component is given a larger weight. Therefore, better efficiencies are obtained and a larger amount of refrigeration is produced at optimum conditions. The cycle was also optimized so as to obtain maximum work output with an upper limit on low pressure set at 5 bars, as in the previous optimizations. As seen in the earlier results, the efficiency is higher when optimizing for work output. It can be seen that with sufficient weight given to the refrigeration output, the cycle could have a higher effective efficiency in the refrigeration mode. Figure 4.10 represents the effect of solution heat exchanger effectiveness on the RUE. As expected, a lower effectiveness results in a lower efficiency. Figures 4.11 and 4.12 show optimized exergy efficiencies for the cycle with an irreversible turbine and pump. It is seen that effective exergy efficiencies are comparable to the optimum efficiencies obtained for pure work output at lower heat source temperatures. At higher source temperatures, the optimum conditions in the pure work output case are still superior. The best efficiency in the refrigeration domain is seen at the heat source temperature of 400 K. At higher source temperatures, the boiler exit temperature does not get very close to the heat source inlet temperature because it is difficult to drop to cold temperatures through the turbine if the inlet temperature is very high. The effect of lower recovery heat exchanger effectiveness is a drop in efficiency. The exergy destruction in some of the major processes in the cycle (appropriately normalized), when optimized for RUE and exergy efficiency in the cooling domain, is plotted in Figs. 4.13 and 4.14. It is seen that the effect of turbine irreversibility is to lower the RUE by about 5% (absolute) and the exergy efficiency by about 10%. Once again the largest source of losses in the RUE optimization is the 77 S 0  lexergy, Eq. (4.2) i 1i..,,,,w~ with ,i,,,,,=0.4, Eq. (4.5) l egy.enwith, ll.re,=0.3, Eq. (4.5) _  Only work output 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.11 Optimized exergy efficiencies for the basic cycle configuration using a non isentropic turbine and pump .. ., Eq. (4.2), s=0.8 60  O  ,1. _.. _n with ,lll~=0.4, Eq. (4.5), E=0.8  m E, Eq. (4.2), e=0.7  i with r =0.4, Eq. (4.5), c=0.7 X C) v  L 20 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.12 Effect of recovery heat exchanger effectiveness on effective efficiency 60 .) c 40 w 0) x LU 20 0 78  Heat addition 60 Internal Hx.  Heat rejection + throttling 2  Turbine I 0 ....... A ....... Unrecovered from heat source 2 5 40  A. ... .. . . . 20 CU  ._  _    0 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.13 Exergy destruction in the optimized RUE case 50  Heat addition S v Internal Hx. 40  heat rejection + throttling (3 0 Turbine ._ o 30  ._c Q >D^ 1' 20 " ,  V 0  340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.14 Exergy destruction in the optimized exergy efficiency case unrecovered exergy lost from the heat source. The bad thermal match in the boiler is reflected in the large exergy destruction during heat addition. The losses in the absorber are also quite big. The dominant losses when maximizing exergy efficiency (Fig. 4.14) are in the absorber and the internal heat recovery process. Discussion The basic configuration of the cycle has been optimized in order to gauge its thermodynamic performance. An exergy analysis is performed to determine the potential areas of improvement in the cycle. The results of the exergy analysis are applied to improving the performance of the cycle, which is discussed in a later chapter. It is seen that RUE values obtained with the cycle are quite low. The optimized cycle state points show that the strong solution starts boiling before entering the boiler (in the HRHX and rectifier), thereby eliminating the "pinch point" in the boiler. Figures 4.15 and 4.16 show the boiler inlet and exit conditions of the working fluid at the heat source temperatures studied. The cycle performance is maximized by reduction of mass flow rate of the heat source until the approach temperature constraint at the boiler entrance (state 4 in Fig. 4.1) is at the bound. High exergy destruction occurs in the boiler, when the parameters are set to yield maximum RUE while providing both power and cooling (see Figs. 4.5 and 4.13). The reason for the observed trend is the low vapor fraction in the cycle (boiler) when operated to produce power and refrigeration (see Fig. 4.15). It is seen that the vapor fraction is between 9 and 15 % when producing work and cooling at high efficiency, while in the work domain it is between 20 and 28 % (Fig. 4.16). In order to expand the vapor to low temperatures, relatively high concentration vapor is required. Other requirements include a high pressure ratio, and lower pressures at the turbine exit such that the temperature drop in the turbine is high. All these factors 0.18 0.16  0.14 0.12 0.10 f o.o 0.08 0.06 m  0.04 S0.02  0.00 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.15 Working fluid temperatures and vapor fractions in boiler: at maximum RUE and with cooling 420 400  380  a E 360 aI Boiler exit vapor mass fraction Boiler Boiler inlet 0.30 0.25 0.20 c o 0.15 0.10 .  0.05 320 i i I.i 0.00 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.16 Working fluid temperatures and vapor fractions in boiler: at maximum RUE with pure work output A . Boiler exit vapor mass fraction 4 /J Boiler Exit Bubble Point I I promote low vapor fraction in the boiler. Consequently, there is a large temperature difference at the boiler exit (between heat source state II and vapor at state 5) and the result is poor thermal matching in the boiler. A large temperature glide would be needed for good thermal matching in the boiler, which only occurs at large vapor fractions. From Fig. 4.1, it can be seen that the exergy in the heat source fluid leaving the boiler is not utilized elsewhere. In order to remove as much of the exergy out of the heat source as possible, the temperature of the heat source should be dropped to the maximum extent possible in the boiler. Since the fluid enters the boiler at a condition where it is beginning to boil, the bubble point of the strong solution should be low. This is achieved by having a high basic (strong) solution concentration and lower boiler pressures. However when operating at these conditions, the pressure ratios are lower resulting in a lower temperature drop in the turbine. Figure 4.17 clearly shows that the pressure ratios are lower in the RUE optimized results compared to the exergy efficiency optimized results. Interestingly, it is seen that setting a lower limit for recovery heat exchanger effectiveness results in a substantial reduction in pressure ratio in the optimized exergy efficiency results. With ammonia water mixtures, lower absorber pressures result in higher pressure ratios. However, the basic (strong) solution concentration then becomes smaller limiting the vapor fractions in the boiler. With the resulting increase in the weak solution flow, the load on the recovery heat exchanger goes up. A reduction in the effectiveness of the recovery unit pushes the optimum towards higher absorber pressures and lower pressure ratios. 22 20  18  16 .* 14 12 uI 10 ,l 6 4 2 0  exergy effrcrency optimized   RUE optimized 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig.4.17 Cycle pressure ratios and the influence of solution heat exchanger effectiveness 1.2  energy efficiency optimized :  RUE optimized qlnsef=0.3 O 0.6  S llrere=11 Ir 0 0.4 0.2  i  i  i  i   i i 0.2 . 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig.4.18 Cooling to work output ratios at optimum conditions 83 Figure 4.18 shows the cooling to work output ratio at optimum conditions using the basic cycle configuration. Clearly, assigning a larger weight to the refrigeration output results in a larger cooling output at optimum conditions in the cooling domain. Unlike classic refrigeration cycles, the cold fluid is in the form of dry or slightly wet vapor. The cooling is by sensible heating of the vapor, and not by the vaporization of a twophase mixture as in a typical evaporator. When weight assigned to cooling is larger, in some cases, the turbine output temperature drops well below the set limit of 270 K in the simulations. The corresponding first law efficiency of the cycle at optimum exergy efficiency and optimum RUE conditions is plotted in Fig. 4.19 and 4.20 respectively. Since the heat source temperature is low, the corresponding first law efficiency is generally low. 30 a* nEq, (2.17) 25 re, Eq. (2.41), rllref=0.4 Onlywork output 3 20 r 15  5  *r 10 A LL 0 I I 340 360 380 400 420 440 460 480 500 Heat Source Temperature (K) Fig. 4.19 First law efficiency at optimized exergy efficiency conditions 