Single_Pressure_Absorption_Heat_Pump_Analysis

Single Pressure Absorption Heat Pump Analysis

A DissertationPresented to

The Academic Faculty

by

Laura Atkinson Schaefer

In Partial Fulfillmentof the Requirements for the Degree

Doctor of Philosophy in Mechanical Engineering

Georgia Institute of TechnologyMay 2000

ii

Single Pressure Absorption Heat Pump Analysis

Approved:

Samuel V. Shelton, Chairman

Richard Barke

Alan Larson

Amyn Teja

William Wepfer

Date Approved

iii

DEDICATION

To Andrew James Schaefer,

my wonderful husband

iv

ACKNOWLEDGEMENTS

During my time at Georgia Tech, I have received support from too many people

to list them all. I am deeply grateful to everyone who has encouraged and aided me.

However, there are certain professors, colleagues, friends, and family members to

whom I would like to individually express thanks.

First, I would like to give thanks to God. Without You, Lord, none of this

would have been possible.

Dr. Sam Shelton has been my advisor in not just the typical but also the

traditional sense of the word. He has provided me with direction and encouragement

in my research, my graduate studies, and my academic job search. He has had faith

in my abilities even when I doubted them, and I truly appreciate the intellectual and

moral support that he has given me. I will miss our discussions on topics ranging

from thermodynamics to environmental concerns. I would also like to thank his wife,

Mrs. Sharon Shelton, for her encouragement and advice.

Dr. William Wepfer has also been an integral part of my graduate career

at Georgia Tech. Dr. Wepfer helped me to publish my first paper in the field of

mechanical engineering, and has called numerous scholarship and career opportunities

to my attention. I appreciate his help during my job search and the time that he

devoted to serving on my reading committee.

I am also grateful to Dr. Alan Larson for his support of my job search and for

his contributions to my dissertation. Dr. Larson has the unique ability to combine a

deep, overall understanding of a work with a meticulous attention to detail.

v

I am very thankful to Dr. Amyn Teja and Dr. Richard Barke for serving

on my reading committee. Their backgrounds and perspectives have enriched my

doctoral work.

My office mates and friends at Georgia Tech have been an invaluable source of

help. I would like to thank Andy Delano for his technical support and Comas Haynes

for his prayer support. I would also like to recognize my friends Claudia Zettner, Bill

Anderson, and Jason Brown.

Finally, I would like to thank the most important people in my life: my family.

My dear husband, Andrew Schaefer, has been my rock and my foundation during

the occasionally stormy years of graduate school. His love, kindness, and humor have

sustained me. The past seven years with Andrew have been a blessing, and I look

forward to the exciting future that lies ahead of us.

My mother and father, Michael and Sherry Atkinson, my brother and sister,

Seth and Terra Atkinson, and all of my grandparents have been an unfailing source of

support. Thank you for your love, faith, and encouragement. You were there for me

even during the darkest times. I am also thankful for the love and support provided

by my husband’s parents, sister, and grandmothers. I feel that you have taken me

into your hearts, and I am so grateful for that.

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TABLE OF CONTENTS

DEDICATION iii

ACKNOWLEDGEMENTS iv

LIST OF TABLES x

LIST OF ILLUSTRATIONS xi

NOMENCLATURE xiv

SUMMARY xvii

CHAPTER

I INTRODUCTION 1Heat Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Heat Sources and Sinks . . . . . . . . . . . . . . . . . . . . . 5Heat Pump Cycles . . . . . . . . . . . . . . . . . . . . . . . . 7

The Einstein Cycle . . . . . . . . . . . . . . . . . . . . . . . . . 13Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 15

Current Research . . . . . . . . . . . . . . . . . . . . . . . . . . 16Modeling the Einstein Cycle . . . . . . . . . . . . . . . . . . 16Alterations of the Cycle Configuration . . . . . . . . . . . . . 17Gas Water Heating: An Application for the Einstein Cycle . 18

II THERMODYNAMIC MODEL OF THE CYCLE 20Overall Description of the Cycle . . . . . . . . . . . . . . . . . . 20Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . 22Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 22The Evaporator . . . . . . . . . . . . . . . . . . . . . . . . . 23The Precooler . . . . . . . . . . . . . . . . . . . . . . . . . . 26The Condenser/Absorber . . . . . . . . . . . . . . . . . . . . 28The Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 29The Bubble Pump . . . . . . . . . . . . . . . . . . . . . . . . 33

Second Law Analysis . . . . . . . . . . . . . . . . . . . . . . . . 41

vii

The Coefficient of Performance . . . . . . . . . . . . . . . . . 41An Illustrative Case: The Internal Generator Heat Exchanger 45

III THE PRINCIPLE OF CORRESPONDINGSTATES MODEL 48The van der Waals Equation of State . . . . . . . . . . . . . . . 48Two-Parameter Principle of Corresponding States . . . . . . . . 49Three-Parameter Principle of Corresponding States . . . . . . . 50Calculation of Compressibility . . . . . . . . . . . . . . . . . . . 51Calculation of the Enthalpy Departure Function . . . . . . . . . 56Ideal Gas Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 57Incompressible Liquid Theory . . . . . . . . . . . . . . . . . . . 59Saturated Liquid and Vapor Behavior: Enthalpy Versus Temper-

ature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Implementation of the Ideal Gas and Incompressible Solution

Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Determining Enthalpy Differences . . . . . . . . . . . . . . . . . 65

IV THE IDEAL SOLUTION MODEL 67Vapor-Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . 67Mole and Mass Fractions . . . . . . . . . . . . . . . . . . . . 68Mixing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

V IDEAL SOLUTION PARAMETER RESULTSAND FLUID SELECTION 73System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Searching for Optimum Properties . . . . . . . . . . . . . . . . . 73Matching the Ideal Properties to Actual Fluids . . . . . . . . . . 81Dimensions of the Search . . . . . . . . . . . . . . . . . . . . 81Absorbing Fluid . . . . . . . . . . . . . . . . . . . . . . . . . 81Pressure-Equalizing Fluid . . . . . . . . . . . . . . . . . . . . 82Refrigerant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Selection of Fluid Triplets . . . . . . . . . . . . . . . . . . . . . 83

VI THE PATEL-TEJA MODEL 87The Patel-Teja Equation of State . . . . . . . . . . . . . . . . . 87Single Substances . . . . . . . . . . . . . . . . . . . . . . . . . . 88Determination of the Patel-Teja Coefficients . . . . . . . . . . 88Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . 89Enthalpy and Entropy Departure Functions . . . . . . . . . . 90

Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Mixing Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . 92

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Enthalpy and Entropy . . . . . . . . . . . . . . . . . . . . . . 93

VII THE PATEL-TEJA MODEL RESULTS 96System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Binary Interaction Parameters . . . . . . . . . . . . . . . . . . . 96Equation of State Coefficients . . . . . . . . . . . . . . . . . . 96Fitting the Experimental Data . . . . . . . . . . . . . . . . . 97Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . 98

Actual Fluid Behavior . . . . . . . . . . . . . . . . . . . . . . . 99Ammonia-Water-Butane Results . . . . . . . . . . . . . . . . . . 101Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Effect of State Point Properties . . . . . . . . . . . . . . . . . 103Correspoding States/Ideal Solution Model Parameters . . . . 105

VIII GAS WATER HEATING: AN APPLICATIONAND ITS IMPLICATIONS 106Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Water Heating Efficiency . . . . . . . . . . . . . . . . . . . . . . 107Electric versus Gas Water Heating . . . . . . . . . . . . . . . 107Einstein Cycle Gas Water Heating . . . . . . . . . . . . . . . 109

Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Economic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Environmental . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Configuration of the Einstein Cycle as a Gas Water Heater . 114Market Potential . . . . . . . . . . . . . . . . . . . . . . . . . 114Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

IX SUMMARY, CONCLUSIONS ANDRECOMMENDATIONS 118Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 118Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 123

APPENDIX

A GENERATOR PROGRAMS 125Generator Configuration 1 . . . . . . . . . . . . . . . . . . . . . 125Generator Configuration 2 . . . . . . . . . . . . . . . . . . . . . 127Generator Configuration 3 . . . . . . . . . . . . . . . . . . . . . 132Generator Libraries . . . . . . . . . . . . . . . . . . . . . . . . . 134

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B IDEAL SOLUTION MODEL PROGRAMS 137Ideal Solution Model EES Program . . . . . . . . . . . . . . . . 137Ideal Solution Model Libraries . . . . . . . . . . . . . . . . . . . 143

C PATEL-TEJA MODEL PROGRAMS 145EES Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Program Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . 148

BIBLIOGRAPHY 170

VITA 176

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LIST OF TABLES

Table Page

3-1 Constants for Calculating Compressibility . . . . . . . . . . . . . . . 53

5-1 Parameter Variation for Increased Efficiency . . . . . . . . . . . . . . 81

5-2 Pressure Equalizing Fluids . . . . . . . . . . . . . . . . . . . . . . . 82

5-3 Refrigerants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7-1 Patel-Teja Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7-2 Binary Interaction Parameters . . . . . . . . . . . . . . . . . . . . . 98

7-3 COP for Tca = 325 K, Tevap = 295 K . . . . . . . . . . . . . . . . . 101



7-6 Ammonia-Water-Butane Results . . . . . . . . . . . . . . . . . . . . 102

7-7 Comparison of Refrigerant Performance . . . . . . . . . . . . . . . . 103

7-8 Comparison of Pressure-Equalizing Fluid Performance . . . . . . . . 104

7-9 Theoretical Fluids and the Effect on the COP . . . . . . . . . . . . . 105

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LIST OF ILLUSTRATIONS

Figure Page

1-1 A Vapor Compression Cycle (Anderson, 1972) . . . . . . . . . . . . . 2

1-2 An Ammonia Absorption Cycle (Anderson, 1972) . . . . . . . . . . . 3

1-3 An Ice Plant (Anderson, 1972) . . . . . . . . . . . . . . . . . . . . . 3

1-4 Lord Kelvin’s Warming Machine . . . . . . . . . . . . . . . . . . . . 4

1-5 The Zurich Town Hall Heat Pump . . . . . . . . . . . . . . . . . . . 6

1-6 A Vapor Compression Cycle . . . . . . . . . . . . . . . . . . . . . . . 8

1-7 A Dual Pressure Absorption Cycle . . . . . . . . . . . . . . . . . . . 10

1-8 A Single Pressure Absorption Cycle . . . . . . . . . . . . . . . . . . 12

1-9 The Einstein Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2-1 The Einstein Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2-2 The Evaporator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-3 The Ideal Solution Evaporator Model . . . . . . . . . . . . . . . . . 25

2-4 The Precooler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2-5 The Condenser/Absorber . . . . . . . . . . . . . . . . . . . . . . . . 29

2-6 The Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2-7 The Ideal Solution Model Generator . . . . . . . . . . . . . . . . . . 32

2-8 A Bubble Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2-9 Vertical Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2-10 Collier Flow Pattern Map . . . . . . . . . . . . . . . . . . . . . . . . 38

2-11 Baker Flow Pattern Map . . . . . . . . . . . . . . . . . . . . . . . . 38

xii

2-12 Effect of Diameter on Liquid and Vapor Flow Rates (Submergence

Ratio = 0.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-13 Effect of Diameter on Liquid Flow Rate and Heat Transferred to the

Bubble Pump (Submergence Ratio = 0.2) . . . . . . . . . . . . . . . 40

2-14 Pumping Capacity for h/L = 0.2, D = 8.6 mm . . . . . . . . . . . . 42

2-15 Heat Transfer To And From The Einstein Cycle . . . . . . . . . . . . 42

2-16 Alternate Generator Configurations . . . . . . . . . . . . . . . . . . . 46

3-1 Intersection of Equations 3.12 and 3.13 . . . . . . . . . . . . . . . . . 55

3-2 Saturated Liquid and Vapor Enthalpy . . . . . . . . . . . . . . . . . 60

3-3 Saturated Liquid and Vapor Enthalpies . . . . . . . . . . . . . . . . 61



5-1 The Effect of Generator Temperature on COP . . . . . . . . . . . . . 74

5-2 COP versus cp,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5-3 COP versus Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5-4 COP versus Pc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5-5 COP versus ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5-6 COP versus M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5-7 COP versus Absorbing Fluid Critical Temperature for Varying Gen-

erator Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5-8 Alternative Fluid Triplets . . . . . . . . . . . . . . . . . . . . . . . . 85

7-1 Methyl Amine-Butane, T = 288 K . . . . . . . . . . . . . . . . . . . 99

7-2 Hydrogen Chloride-Butane, P = 4 bar . . . . . . . . . . . . . . . . . 100

7-3 Ammonia-Propane, T = 293 K . . . . . . . . . . . . . . . . . . . . . 100

8-1 An Einstein Cycle Gas Heat Pump Water Heater . . . . . . . . . . . 106

8-2 Water Heater Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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8-3 Residential and Commercial Natural Gas Water Heating . . . . . . . 113

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NOMENCLATURE

Symbols

a Attractive Force Correction Factor (N ·m4

mol2)

b Volume Correction Factor ( m3

mol)

c Specific Heat ( Jkg·K )

COP Coefficient of Performance (-)

D Diameter (m)

f Fugacity (kPa, bar)

G Gibbs Energy ( Jmol)

g Gravity (ms2)

H Total Enthalpy ( Jmol)

h Specific Enthalpy ( Jkg)

M Molecular Weight ( kgkg·mol

)

m Mass Flow Rate (kgs)

P Pressure (kPa, bar)

Q Heat Transfer Rate (kW )

R Gas Constant (m3·Pa

mol·K ,m3·Pakg·K )

S Total Entropy ( Jmol·K )

s Specific Entropy ( Jkg·K )

T Temperature (C, K)

u Specific Internal Energy ( Jkg)

V Total Volume ( m3

mol)

v Specific Volume (m3

kg)

xv

x Liquid Mass or Mole Fraction (-)

y Vapor Mass or Mole Fraction (-)

Z Compressibility (-)

Greek Characters

ζc Patel-Teja Coefficient (-)

ρ Density ( kgm3 )

σ Surface Tension (Nm)

ω Acentric Factor (-)

Subscripts

abs Absorbing Fluid

c Critical Point

f Liquid

g Vapor

m Mixture

p Constant Pressure

pe Pressure-Equalizing Fluid

r Reduced Property

refrg Refrigerant

s Entropic Average

sv Superheated Vapor

u Universal

v Constant Volume

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Superscripts

∗ Ideal State

0 Simple Fluid

L Liquid

r Reference Fluid

sat Saturated State

V Vapor

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SUMMARY

Thermally-driven absorption heat pump cycles have several advantages over

conventional vapor compression cycles. They do not require a costly electric power

plant and they use environmentally-benign natural fluids. Current absorption sys-

tems are dominated by dual-pressure cycles using a solution pump (which still re-

quires a small electrical power source). Single-pressure cycles remove the need for

a pump and any electrical power. This makes them portable, inexpensive, reliable,

and silent. An entirely different approach to a single-pressure absorption cycle was

taken by Einstein and Szilard in the 1930s. Their cycle decouples the temperature

lift from the generator temperature, which allows the selection of a refrigerant that

matches the application temperature requirements.

The configuration of the Einstein cycle was examined, and changes were made

to increase the coefficient of performance. These changes were primarily implemented

on the generator side. The bubble pump performance was increased through selection

of optimum operating parameters. An external heat exchanger was added between

the generator and the condenser/absorber to improve heat recovery, and the par-

tial internal heat exchanger in the generator was expanded to a full internal heat

exchanger in order to minimize entropy generation.

The Einstein Cycle has been modeled using two separate property models: 1)

a corresponding states/ideal solution property model, and 2) a Patel-Teja/Panagio-

topoulos and Reid property model. The first model was used to predict which pa-

rameters would increase the COP and the second model was used to more accurately

predict the behavior the cycle

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For this study, three temperature conditions were evaluated: 1) Tcondens−absorb

= 325 K and Tevaporator = 295 K, 2) Tca = 325 K and Tevap = 306 K, and 3) Tca =

316 K and Tevap = 295 K. As outlined in the introduction, each of these temperature

levels are suitable for using the Einstein cycle for gas heat pump water heating. The

highest COP for the first set of temperatures, which can produce 125F water from

ambient conditions (flue gases could be used to help to maintain the evaporator at

72F), is 1.51. The best COP for the second set of temperatures, which require a

hotter evaporator, is 1.88. Finally, the COP for the third set of temperatures, where

the Einstein cycle would be used as a preheater, is 1.76.

With a COP of 1.5, an Einstein cycle gas heat pump water heater would cut

the operating costs of a conventional gas water heater by 33%. This could result in

large economic and environmental savings.

1

CHAPTER I

INTRODUCTION

Heat Pumps

Definition

Using input energy, heat pump cycles pump thermal energy from low tempera-

tures to high temperatures. Heat pumps can transfer heat from natural or man-made

thermal sources to a building or an application. Heat pumps can also be used for

cooling by transferring heat from the application to be cooled to higher-temperature

surroundings. Heat pumps are used in refrigeration, air-conditioning, space heating,

and other applications.

History

Initially, refrigeration was the primary thrust of heat pump development. The

first attempt to produce refrigeration mechanically occurred in 1755, when William

Cullen created ice with an air pump (Anderson, 1972). Cullen, however, was unable

to consistently maintain refrigeration temperatures. The first reliable refrigeration

machine was constructed by Perkins in 1834, and by the 1850s, there was an explosion

of interest in mechanical refrigeration.

Vapor compression (Figure 1-1) and ammonia absorption (Figure 1-2) cycles

were both vigorously studied in the 1860s and 1870s. The focus at that time, how-

ever, was not to provide direct refrigeration. Instead, the cycles were used to produce

2

Figure 1-1: A Vapor Compression Cycle (Anderson, 1972)

large quantities of ice (Figure 1-3) that could then be used in commercial or residen-

tial applications (such as an icebox). Breweries were the first to implement direct

refrigeration on a large scale, followed by the food-processing industries.

By the end of World War I, public utilities started to offer incentives to residen-

tial customers to purchase home refrigerators. The public slowly began to abandon

the icebox: in 1921, only five thousand domestic refrigerators were sold, but by 1930,

that number had grown to 850,000 units (Anderson, 1972). Simultaneously, heat

pumps used for air-conditioning moved from industrial to commercial and then to

residential applications by the 1950s.

In 1852, Lord Kelvin demonstrated that a heat pump (or heat engine) could

be used for heating rather than cooling when he published a paper that described

a “warming machine” driven by an engine. Kelvin’s machine, shown in Figure 1-4,

would produce heat for a building that was greater than the heat of combustion of

the coal used to drive the engine. Kelvin outlined the heating of air from 50 F to

3

Figure 1-2: An Ammonia Absorption Cycle (Anderson, 1972)

Figure 1-3: An Ice Plant (Anderson, 1972)

4

EgressCylinder

IngressCylinder

ExternalAtmosphere

ReceiverT2

ToBuilding

T3

Steam Engine Drive

T1

Figure 1-4: Lord Kelvin’s Warming Machine

80 F as an example. He stated that an ideal machine could deliver one pound of air

per second at those temperatures from a driving power of 0.28 hp, which is equivalent

to 0.2 BTU. To directly heat one pound of air from 50 F to 80 F requires 7 BTU.

Therefore, under ideal conditions, direct heating required approximately thirty-five

times the heat equivalent of the work (Morley, 1922).

Despite Kelvin’s early work, the use of heat pumps for heating water or air

was not seriously investigated until the 1920s and 30s. Morley revived Kelvin’s work

in 1922, and laid out the configuration previously shown in Figure 1-4. Haldane

(1938) analyzed experimental data from a refrigerating plant from 1891 to 1926 to

demonstrate the heating potential of a heat pump. He also constructed an exper-

imental heat pump for his own home, and tested various heat sinks and sources

(Heap, 1983). Homkes also conducted heat pump research in the early 1930s, with a

focus on producing hot water.

5

The worldwide economic depression of the 1930s greatly stimulated heat pump

development. For heating applications, large-scale implementations proved to be

more financially favorable than smaller domestic appliances. The Zurich Town Hall

used a heat pump to provide a portion of its peak heating requirements as early as

1939 (Figure 1-5). The Town Hall heat pump worked so successfully that a second

heat pump was installed in the Zurich Public Baths (Fearon, 1978).

The use of heat pumps for domestic heating applications began to be explored

in the 1950s. The Ferranti Fridge-Heater, which was marketed in Britain in 1954, was

a hybrid refrigerator-water heater with a heating output of 1.2 kW in the summer

and 0.7 kW in the winter. Unfortunately, the fridge-heater failed to establish a

substantial consumer base, and soon disappeared from the market. Since that time,

both ground and air-source heat pumps have been investigated for residential use.

Currently, electric-driven heat pumps are available for air and water heating.

Heat Sources and Sinks

Both the Town Hall and the Public Baths in Zurich had a large supply of water

that could be used as a source of low-grade energy for the evaporator. Obviously,

though, not all heat pump installations can rely on a water source. Two other

potential sources of energy that have been investigated through the years are the

earth and the air.

Air source heat pumps have a number of obvious advantages. Air is abun-

dantly available, and the evaporator can be easily incorporated into a number of

possible configurations. However, there are problems with air source heat pumps.

The heat exchangers may be quite large, and noise is produced from the large vol-

ume of air flow. Additionally, when air source heat pumps are used to supply space

heating or cooling, difficulties can arise at the extremes. For instance, in certain

6

Figure 1-5: The Zurich Town Hall Heat Pump

7

climates, frost can form on the evaporator, which blocks the airways.

The earth is a less commonly used source of low-grade heat. The vast majority

of the heat in the earth comes from the stored energy of the sun, while only a

fraction is from geothermal energy. The main problem with this large source of

energy is finding the means to extract it. Coils must be buried in the ground, limiting

the flexibility of ground source heat pump implementations. Where this can be

overcome, though, ground source heat pumps can be an appropriate choice. Various

coil configurations have been investigated since the 1940s, and it has been found that

consistent extraction rates of 30 BTU/hr/ft of pipe can be achieved.

Water and air can not only serve as good sources of free low-grade energy, they

can also be used as heat sinks for the condenser and/or absorber. One configuration

is the use of air as a heat source and water as a sink. For that case, the evaporator

can be used to provide space cooling while the condenser provides hot water for a

number of household applications.

Heat Pump Cycles

The cycles that drive the heat pumps described in preceding history can fall

into two categories: vapor compression and absorption. Additionally, absorption

cycles can be further delineated by the number of operating pressure levels. In this

study, dual pressure and single pressure absorption cycles are of particular interest.

Vapor Compression Cycles

Vapor compression cycles are the most common type of heat pump cycle.

The most basic vapor compression cycle consists of a compressor, an evaporator,

a condenser, and a throttling valve, as shown in Figure 1-6. Work is supplied to

the compressor, which increases the pressure of the working fluid. This increase in

8

Condenser

eQ

cpW

Compressor

cdQ

Evaporator

ThrottlingValve

Figure 1-6: A Vapor Compression Cycle

pressure raises the temperature of the refrigerant, and causes the vapor to become

superheated. The refrigerant then passes through the condenser, where it rejects heat

to its lower-temperature surroundings (which can be water or air) and condenses into

a saturated or subcooled liquid.

Next, the refrigerant passes through a throttling valve, which expands the

refrigerant down to the lower, original pressure. This process also reduces the tem-

perature of the refrigerant, so that it can absorb low-temperature heat. This heat is

absorbed in the evaporator, where the refrigerant is boiled back into a vapor state.

9

Absorption Cycles

When examined as an isolated system, a vapor compression cycle can be

highly efficient. However, large losses can occur in creating the work that drives

the compressor. For example, coal may be combusted to produce electricity in a

power plant, a process that has an average efficiency of only thirty percent. By using

direct thermal energy, absorption cycles can lessen or completely avoid the need to

convert heat to work and then back to heat.

Dual Pressure Absorption Cycles

The dual pressure absorption cycle shown in Figure 1-7 replaces the compres-

sor with an absorption-generation process. Dual pressure absorption cycles use two

fluids rather than a single refrigerant. The changes in pressure, temperature, and

state that occur in the condenser, throttling valve, and evaporator are the same as

in the vapor compression cycle.

When the refrigerant leaves the evaporator, however, it is not mechanically

compressed, but is instead absorbed by a second working fluid. This absorption

process causes heat to be rejected. The liquid mixture of absorbent and refrigerant is

pumped to a higher pressure and into the generator. The pumping process is driven

by a work input, but the amount of work that is required is an order of magnitude less

than in a vapor compression cycle. Direct thermal energy from a high-temperature

source is transferred to the refrigerant-absorbent mixture in the generator. This

heating process generates refrigerant vapor, which flows into the condenser. The

remaining liquid absorbent is then throttled back to the lower system pressure as it

falls from the generator into the absorber.

10

Condenser

eQ

cdQ

Evaporator

ThrottlingValve

Generator

Absorber

PumpcpW

genQ

absQ

Figure 1-7: A Dual Pressure Absorption Cycle

11

Single Pressure Absorption Cycles

Single pressure absorption cycles have a number of advantages over vapor

compression cycles. They do not require a costly electric power plant, use environ-

mentally benign natural fluids, and have waste heat available at the user site rather

than at the central power plant. Since they have no moving parts, they are also

inexpensive, reliable, and quiet.

In order to eliminate the need for any work input, single pressure absorption

cycles add a third working fluid and replace the mechanical pump with a bubble

pump. A bubble pump can move fluids across a difference in height simply by using

a heat input. One single pressure absorption cycle is illustrated in Figure 1-8. This

is the Platen and Munters diffusion-absorption cycle, which was patented in 1928.

The Platen and Munters cycle uses ammonia as a refrigerant, hydrogen as an inert

gas, and water as an absorbent. The diffusion of hydrogen in the cycle is the kinetic

limiting process.

Although some of the components in a single pressure absorption cycle resem-

ble those of a vapor compression or dual pressure absorption cycle, there are impor-

tant differences in the way that they operate. In the generator, high-temperature

heat is applied to a liquid mixture of the refrigerant and the absorbent. This gener-

ates refrigerant vapor, which is first used to “pump” the liquid absorbent through the

bubble pump into the absorber. The refrigerant vapor then flows into the condenser,

where it rejects its thermal energy to the surroundings and condenses into a liquid.

Next, the liquid refrigerant flows into the evaporator and mixes with an inert

gas. This mixing process lowers the partial pressure of the refrigerant so that it

evaporates at a lower temperature. The vapor mixture of refrigerant and inert gas

flows into the absorber, where the liquid absorbent absorbs the refrigerant. The

12

Condenser

eQ

cdQ

Evaporator

Generator

Absorber

genQ

absQ

bubblepumpQ

Figure 1-8: A Single Pressure Absorption Cycle

13

inert gas then returns to the evaporator, and the liquid mixture of absorbent and

refrigerant flows into the generator.

The Einstein Cycle

Description

In 1928, Albert Einstein and Leo Szilard also patented a single pressure ab-

sorption cycle. Unlike the Platen and Munters cycle, however, the Einstein cycle uses

a pressure-equalizing absorbate fluid rather than an inert gas. In the Einstein cycle,

butane is the refrigerant, water remains the absorbent, and ammonia becomes the

pressure-equalizing fluid.

A basic diagram of the Einstein cycle is given in Figure 1-9. The generator,

bubble pump, and evaporator remain from the Platen and Munters cycle, but the

condenser and absorber are combined into a single unit. In the evaporator, gaseous

ammonia is bubbled into liquid butane. This reduces the partial pressure of the

butane to below the system pressure, which causes to butane to evaporate at a lower

temperature. The resultant vapor mixture of butane and ammonia flows into the

condenser/absorber.

A liquid film of water falls down the side of the condenser/absorber, which ab-

sorbs the ammonia from the ammonia-butane vapor. This causes the partial pressure

of the butane to rise to the system pressure, so that the butane condenses at a higher

temperature than in the evaporator. The liquid mixture of ammonia and water is

immiscible with the butane, so it separates and flows out of the condenser/absorber

into the generator while the liquid butane flows back into the evaporator.

In the generator, the ammonia-water mixture is heated to the highest temper-

ature in the system. This high temperature causes the ammonia to evaporate from

14

Condenser/Absorber

eQ

abs/cdQ

Evaporator

Generator

genQ

bubblepumpQ

Figure 1-9: The Einstein Cycle

15

the liquid water. The liquid water is pumped back into the condenser/absorber by

the bubble pump, and the vapor ammonia flows into the evaporator, completing the

cycle.

Literature Review

Very little has been published on the Einstein cycle. In 1980, Alefeld briefly

outlined the cycle in a Physics Today guest comment. That same year, the Johns

Hopkins University Applied Physics Laboratory produced an internal report on the

Einstein cycle (Follin and Yu, 1980). The Hopkins study investigated possible alter-

native power sources for an Einstein cycle (such as solar or geothermal energy), and

calculated a maximum cooling coefficient of performance (COP) for the cycle of 0.25.

In 1984, a patent was issued to Rojey for a variation on the Einstein cycle.

While the physical cycle configuration is nearly the same as in Einstein and Szilard’s

patent, Rojey used carbon dioxide rather than ammonia for the inert fluid. He also

suggested possible alternatives for the Einstein absorbing fluid (water) and refrigerant

(butane).

One decade later, Razi et al. (1994) examined the result of powering the

Einstein cycle with solar energy for use in Morocco. They used both n-butane and

iso-butane as the refrigerant in a simple model, and found possible cooling COPs

of 0.13 to 0.19. A recent article in Scientific American also mentioned the Einstein

cycle (Dannen, 1997). However, the article focused on the partnership of Einstein

and Szilard rather than on the properties of the cycle.

The most recent and comprehensive work on the Einstein cycle was performed

by Delano for his M.S. thesis (1997) and Ph.D. dissertation (1998) at Georgia Tech.

Delano developed an Engineering Equation Solver (EES) model of the cycle using

the first and second laws. The primary emphasis of Delano’s work was to gain an un-

16

derstanding of the performance of the cycle through an analysis of the processes and

the properties of various working fluids. To model the fluids’ properties, the Patel-

Teja equation of state was fitted to experimental data using the Panagiotopoulos and

Reid mixing rules. Simplifying assumptions for both the fluid and the component

behavior were made. Delano also developed a bubble pump performance model, but

no optimization of its operating parameters was carried out.

With the EES model, Delano was able to test various refrigerant fluids, such

as i-butane, neopentane, and i-pentane, and monitor their effect on the COP and

the temperature lift from the evaporator to the condenser/absorber. He also ran

parametric studies for varying pressures, temperatures, and pinch points. While the

maximum reversible cooling COP was found to be 0.86, component irreversibilities

reduced the calculated COP to 0.15.

In addition to the computational model, Delano built a conceptual demon-

stration prototype from welded stainless steel tubing. The prototype was charged

with iso-butane, ammonia, and water. By supplying the generator with between 150

and 250 watts and the bubble pump with 50 to 70 watts, and by cooling the con-

denser/absorber with 21 C tap water, the evaporator was able to maintain tempera-

tures as low as -2 C. This prototype operated silently and reliably, and demonstrated

the viability of the Einstein cycle.

Current Research

Modeling the Einstein Cycle

The examination of the Einstein cycle performed by Delano (1998) increased

the understanding of the limitations and potential of the cycle. This study builds

on Delano’s work by using a methodical approach to increase the efficiency of the

17

Einstein cycle.

Both the cycle configuration and the fluids that are used are examined. An

ideal gas/incompressible solution model of the Einstein cycle has been completed.

This model uses the three parameter theory of corresponding states to model in-

dividual fluid behavior and Raoult’s Law to model fluid mixing, and so gives the

performance of the cycle entirely in terms of the least number of fluid properties

(such as the specific heats, masses, critical temperatures, etc.). Using this model,

an optimization of the properties has been performed in order to find the optimal

hypothetical fluids that maximize the efficiency.

Once the optimal properties were found, a number of actual fluids were chosen

that most closely matched those properties. The various fluid combinations were

then evaluated using a complex cycle model that could more accurately represent

the behavior of polar substances. The complex model uses the Patel-Teja equation

of state and Panagiotopoulos and Reid mixing rules to model fluid behavior.

Alterations of the Cycle Configuration

Changing the physical configuration of the cycle can also improve its efficiency.

A second law analysis of each cycle component can reveal sources of entropy gener-

ation. Entropy generation degrades the overall efficiency of the cycle, and so should

be minimized. The entropy generated in each component has been examined, and

appropriate improvements have been implemented. In addition, the relationship of

the diameter, submergence ratio, mass flow rate, and heat input have been analyzed

for the bubble pump, and its physical geometry has been altered to maximize its

performance.

18

Gas Water Heating: An Application for the Einstein Cycle

Previously, single (and dual) pressure absorption cycles have been primarily

used for refrigeration applications. The Einstein cycle, however, is able to produce

temperatures that are suitable for water heating. Three temperature configurations

will be explored in this study. The first will produce hot water at 52C (125F) from

ambient air temperatures of 22C (72F). The second configuration will use flue gas

exhaust to heat the evaporator, and will produce 52C hot water from an evaporator

temperature of 33C (91F). In the final configuration, the Einstein cycle will be used

as a preheater that will raise the water temperature to 43C (109F) from ambient air

temperatures of 22C (72F). The cycle performance for each of these configurations

is detailed in Chapter VII.

These temperatures were selected based on current water heater regulations.

Due to safety considerations, water heater temperatures have been reduced so that

140F is the upper limit on the tank thermostat. Furthermore, scalding temperatures

are at 120-125F, and 105F is the maximum temperature need for most applications.

These temperature levels are ideal for domestic water heating and service (commer-

cial) water heating, resulting in an increase in efficiency. The efficiency, which is

defined as the amount of heating provided to the water divided by the amount of

heating provided by the gas flame, can be increased from the current levels of 66%

to over 100%.

The higher coefficient of performance means that an Einstein cycle natural gas

heat pump water heater is a feasible alternative to traditional gas water heating. Gas

water heaters comprise over fifty-five percent of the United States’ residential water

heating market, and consume 1.3 quads of energy each year. In the commercial sector,

gas water heating accounts for 13% of the total gas usage, and consumes 0.35 quads

19

per year. An alternative gas water heater that uses less energy has the potential to

significantly reduce carbon emissions, and provide savings for the consumer. These

economic and environmental savings are analyzed, and implementation strategies and

market barriers are also discussed.

20

CHAPTER II

THERMODYNAMIC MODEL OF THE CYCLE

Overall Description of the Cycle

The Einstein cycle is a single-pressure, thermally-driven refrigeration cycle

that uses three fluids. One fluid acts as a refrigerant, one as a pressure-equalizing

fluid, and one as an absorbing fluid. Because the Einstein cycle operates at a single

pressure, no moving parts, such as a pump, are required. In order to facilitate fluid

motion, there are some slight pressure variations due to height differences in the

cycle, but they are considered to be insignificant for the purposes of this evaluation.

Figure 2-1 shows a general diagram of the Einstein cycle. The cycle shown

in this figure has been modified from Einstein’s original configuration. Various heat

exchangers, such as the internal generator solution heat exchanger, have been added

in order to create a cycle with a higher efficiency.

At point 1, liquid refrigerant leaves the condenser/absorber, passes through

a precooler, and arrives in the evaporator at point 2. The pressure-equalizing fluid

enters the evaporator as a vapor at point 4. The refrigerant and pressure-equalizing

fluids combine at a low temperature. The refrigerant evaporates and leaves the evap-

orator at point 3. The evaporation process requires heat transfer from the surround-

ings. The refrigerant and pressure-equalizing vapor mixture becomes superheated in

the precooler, and is bubbled into the condenser/absorber at point 6.

Liquid absorbing fluid from the generator is sprayed into the condenser/-

absorber (point 11). The absorbing fluid absorbs the pressure-equalizing vapor from

21

5

Qgenerator

Bub

ble

Pum

p

Reservoir

Gen

erat

or

2

Qevaporator

3

4b

7

Qcondenser/absorber

Condenser/Absorber

Evaporator

Pre-Cooler

6

1

ExternalHeat Exchanger

9

8

9g

9f

10

114

Butane (Refrigerant)

Water (Absorbing Fluid)

Ammonia (Pressure-Equalizing Fluid)

Qbub pump

Figure2-1:

TheEinstein

Cycle

22

the mixture entering at point 6. The refrigerant vapor and the remaining pressure-

equalizing vapor are then cooled until they condense. Heat of absorption and con-

densation is rejected to the environment. The refrigerant is then siphoned into the

evaporator (as noted above), and the pressure-equalizing/absorbing fluid mixture

leaves the condenser/evaporator at point 7.

The mixture stream then exchanges heat with the liquid absorbing fluid that

is to be sprayed into the condenser/absorber, and enters the generator at point 8. In

the generator, the fluid mixture is heated to separate the absorbing fluid from the

pressure-equalizing fluid. The pressure-equalizing vapor leaves the generator at point

5, and mixes with additional pressure-equalizing vapor from the bubble pump. The

two vapor streams combine at point 4b, and then pass through the precooler to point

4. To remove most of the pressure-equalizing fluid that remains in the liquid mixture,

that mixture is pumped through a bubble pump to a reservoir (point 9). The residual

pressure-equalizing vapor (point 9g) is combined with the pressure-equalizing vapor

leaving the generator, as noted above. The liquid absorbing fluid then exchanges

heat with the entering mixture stream, and leaves the generator at point 10.

Component Analysis

Introduction

In order to model the behavior of the Einstein cycle, a first and second law

analysis is applied to each component. The entropy that is generated by each process

is evaluated, and energy, mass, and species conservation equations have been devel-

oped. Two separate property models are used in order to both improve the efficiency

through optimization and accurately describe the cycle. The ideal solution model

presents simplifications for analysis of the cycle processes, and utilizes the three-

23

2Qevaporator

34

Figure 2-2: The Evaporator

parameter theory of corresponding states, as outlined in Chapters III and IV. The

more accurate property model uses the Patel-Teja equation of state and Panagiotopo-

lus and Reid mixing rules, and is described in Chapter VI. Differences between the

two models in their evaluation of the various components are noted in the following

sections.

The Evaporator

The evaporator is presented in Figure 2-2 . In the evaporator, saturated liquid

refrigerant enters at point 2, while a mixture of saturated vapor pressure-equalizing

fluid from the generator and the bubble pump is bubbled in (point 4) after passing

through the precooler. The partial pressure of the refrigerant is reduced, causing it to

evaporate and leave the evaporator as a vapor mixture with the pressure-equalizing

fluid at point 3.

The conservation of energy for the evaporator is:

24

Qevaporator = m3h3 − m2h2 − m4h4 (2.1)

Mass and species conservation equations can also be written:

m3 = m2 + m4 (2.2)

ype,3m3 = xpe,2m2 + ype,4m4 (2.3)

where ype,3 is the vapor mass fraction of the pressure equalizing fluid at point 3 and

x is the liquid mass fraction. Because there are only two fluids, the mixture mass

conservation equation and a single species conservation equation completely specify

the states.

In order to accurately model the vapor-liquid-liquid equilibrium that can occur

in the evaporator, the Patel-Teja equation of state must be used. Since the ideal

solution model cannot accurately represent the evaporator’s internal processes, an

approximation must be formulated instead. This approximate model is shown in

Figure 2-3.

For the ideal solution model, it is assumed that the control volume in the

lower section of the evaporator is in vapor-liquid equilibrium and is at a constant

temperature. It is also assumed that all of the heat transferred to the evaporator is

transferred at that temperature. Given these assumptions, the conservation equations

become:

m2 + m3p + m4 = m2p + m3 + m4p (2.4)

m2xrefr,2 + m3pyrefr,3p + m4yrefr,4 = m2pxrefr,2p + m3yrefr,3 + m4pyrefr,4p (2.5)

25

Qevaporator

4p

2

34

3p

2p

Figure 2-3: The Ideal Solution Evaporator Model

m3p = m2p + m4p (2.6)

m3pype,3p = m2pxpe,2p + m4pype,4p (2.7)

The conservation of energy for the evaporator is still equation 2.1, but a further

energy conservation statement is also needed:

Qevaporator = m3ph3p − m2ph2p − m4ph4p (2.8)

Returning to a more realistic model of the evaporator, in addition to these

energy, species, and mass conservation equations, a second law analysis should also

be performed. For the evaporator, the entropy generation is:

Sevaporator = m3s3 − m2s2 − m4s4 −Qevaporator

T evaporator,s

(2.9)

The temperature in equation 2.9, T evaporator,s, is the entropic average temper-

26

ature. When heat is transferred over a temperature difference, entropy is generated.

The entropic average temperature is a means of determining the average temperature

at which this generation occurs. Mathematically, it is defined as:

Ts =Q∫ δQT

(2.10)

The Precooler

Prior to entering the precooler, the pressure-equalizing vapor from the genera-

tor mixes with pressure-equalizing vapor from the bubble pump. (It should be noted

that the ideal solution model does not include a bubble pump.) This combination of

the pressure-equalizing streams can be quantified through mass, species, and energy

conservation equations:

m4b = m4 + m9g (2.11)

ype,4bm4b = ype,4m4 + ype,9gm9g (2.12)

m4bh4b = m4h4 + m9gh9g (2.13)

Within the precooler, as presented in Figure 2-4, no mixing occurs between

the three streams, and no external heat is added or released. Two pinch points

(minimum temperature differences) are required, since three streams are exchanging

heat. These pinch points will be specified to occur between points 2 and 4 and points

4b and 6.

T2 = T4 + pinch24 (2.14)

T4b = T6 + pinch4b6 (2.15)

27

2

6

1

344b

Figure 2-4: The Precooler

The ideal solution model does not include the bubble pump, so equation 2.15

becomes:

T5 = T6 + pinch56 (2.16)

where state point 5 is equivalent to state point 4b.

Because the streams do not mix, neither the species concentrations nor the

mass flow rates change (i.e. m1 = m2), and the conservation of mass is satisfied

through the evaporator and condenser/absorber equations. Only a single equation is

then needed to specify the states of the precooler:

m1h1 + m3h3 + m4bh4b = m2h2 + m4h4 + m6h6 (2.17)

The precooler transfers heat across finite temperature differences, generating

entropy:

Sprecooler = m2s2 + m4s4 + m6s6 − m1s1 − m3s3 − m5s5 − m9gs9g (2.18)

This equation also includes the entropy generated by mixing the pressure-

equalizing vapor streams from the generator and the bubble pump.

28

The Condenser/Absorber

Rather than utilizing a separate condenser and absorber, these units are com-

bined in the Einstein cycle (Figure 2-5). A weak liquid mixture of absorbing fluid

and pressure-equalizing fluid falls in a film down the side of the unit. A vapor mix-

ture of refrigerant and pressure-equalizing fluid is bubbled into the film, and the

pressure-equalizing fluid is absorbed, generating useful heat. The entering liquid

mixture increases its pressure-equalizing fluid concentration as it falls down the con-

denser/absorber, while the partial pressure of the pressure-equalizing vapor decreases

as it rises, thereby maximizing the absorption potential of the liquid throughout the

unit. A liquid mixture of absorbing fluid and pressure-equalizing fluid is immiscible

with the refrigerant, and is siphoned from the bottom of the condenser/absorber into

the generator. As the pressure-equalizing fluid is absorbed, the refrigerant’s partial

pressure returns to the system pressure, so it condenses at a higher temperature than

in the evaporator. Some pressure-equalizing fluid condenses as well, and that liquid

mixture then enters the precooler.

The amount of heat transferred from the condenser/absorber is found from

the conservation of energy:

Qcond/abs = m1h1 + m7h7 − m6h6 − m11h11 (2.19)

The species and mass conservation equations state that:

m1 + m7 = m6 + m11 (2.20)

xi,1m1 + xi,7m7 = yi,6m6 + xi,11m11 (2.21)

where i represents one of the fluids. Since there are three fluids present in the

29

7

6

1

11

Qcondenser/absorber

Figure 2-5: The Condenser/Absorber

condenser/absorber, the species equation must be stated for any two to completely

specify this control volume.

The entropy generation in the condenser/absorber is:

Scond/abs = m1s1 + m7s7 − m6s6 − m11s11 −Qcond/abs

T cond/abs,s

(2.22)

The Generator

The generator control volume will be defined as including the external solution

heat exchanger, the internal solution heat exchanger, the bubble pump, the reser-

voir, and the generator unit, as shown in Figure 2-6. The external heat exchanger

and a full internal solution heat exchanger are departures from Einstein’s original

patent. Combined, these elements produce pressure-equalizing fluid vapor for both

the evaporator and the condenser/absorber and liquid absorbing fluid for the con-

30

denser/absorber from a liquid pressure-equalizing/absorbing fluid mixture and a heat

input.

The conservation of energy for the entire control volume states that

Qgenerator = m5h5 + m9gh9g + m10h10 − m8h8 (2.23)

Again, since only two fluids are present, only a single species mass conservation

equation is needed in addition to the mixture conservation of mass:

m5 + m9g + m11 = m7 (2.24)

ype,5m5 + ype,9gm9g + xpe,11m11 = xpe,7m7 (2.25)

The processes in the external and internal solution heat exchangers must also

be accounted for. No mixing occurs in either the external or the internal heat ex-

changer, so the mass flow rates remain constant in each stream:

m7 = m8 (2.26)

m9f = m10 = m11 (2.27)

To account for the heat transferred in the internal and external solution heat

exchangers, additional mass, species, and energy equations are required:

m8 = m5 + m9 (2.28)

xpe,8m8 = ype,5m5 + xpe,9m9 (2.29)

m7h7 + m11h11 = m8h8 + m10h10 (2.30)

31

5

Qgenerator

Reservoir

9

7

9g

9f

10

Qbub pump

8

11

Figure 2-6: The Generator

32

5

Qgenerator

9

7

10

8

11

Figure 2-7: The Ideal Solution Model Generator

Finally, a pinch point must also be defined:

T11 = T7 + pinch7,11 (2.31)

As noted previously, for simplicity, the ideal solution model does not include

the bubble pump, which results in the configuration shown in Figure 2-7. The mass,

species, and energy conservation equations must be modified to reflect this change.

Equations 2.28-2.31 remain the same, but equation 2.23 becomes:

Qgenerator = m5h5 + m10h10 − m8h8 (2.32)

Returning to a model that includes the bubble pump, the entropy generated

by the generator is:

33

Sgenerator = m5s5 + m9gs9g + m11s11 − m7s7 −Qgenerator

T generator,s

(2.33)

The entropy generated by the internal and external solution heat exchangers

is also accounted for in equation 2.33.

The Bubble Pump

Although the bubble pump is part of the generator subsystem, it merits a

separate review. The performance of the bubble pump (or vapor lift pump) is a

function of both its physical geometry and the properties of the fluid mixture that

it carries. A bubble pump is a heated tube that lifts fluid from a lower reservoir

to a higher reservoir, as shown in Figure 2-8. Heat applied at the bottom of the

tube causes vapor bubbles to form and to rise. This creates a balance between the

buoyancy and the friction forces, which “pumps” the liquid to the upper reservoir.

Two-phase flow in a vertical pipe falls into one of five flow regimes: bubbly,

slug, churn, wispy-annular, or annular flow. These flow regimes are illustrated in

Figure 2-9 (Collier, 1981). As the heat that is added to the pipe increases, the two-

phase flow moves sequentially through these flow patterns. The first stage is bubbly

flow, where the vapor exists as discrete bubbles within the liquid phase. In the second

stage, the bubbles expand until they are bullet-shaped and nearly span the diameter

of the tube. These bubbles are separated by “slugs” of liquid, which may contain

smaller gas bubbles. The next stage is churn flow, where the large gas bubbles begin

to break down in an oscillatory fashion, so that the center of the tube is alternately

filled with the liquid and the vapor phases. The fourth stage, wispy-annular flow, can

be difficult to differentiate from annular flow. In wispy-annular flow, the liquid phase

is present both as a film on the wall of the pipe and as long filaments of agglomerated

34

L

d

H

Heat

Figure 2-8: A Bubble Pump

35

droplets dispersed throughout the central core of the flow. The final stage is annular

flow, which is characterized by a liquid film surrounding a vapor core (Collier, 1981;

Azbel, 1981).

Delano has shown that a bubble pump operates most efficiently in the slug

flow regime (1998). There are numerous methods which can be used to determine

the flow regime. Tutu (1984) recommends using a dimensionless approach that em-

ploys the liquid Froude number, which represents a ratio of the inertial force to the

gravitational force. His derivation provides the boundary between the bubbly and

slug flow regimes:

V—g

V—f= Fr−0.075

f − 0.84Fr0.075f (2.34)

where V— is the volume flow rate of the liquid (f) or the vapor (g). Taitel et al.

(1980) provide an alternate correlation for the same transition from bubbly to slug

flow:

V—f

A= 3.0

V—g

A− 1.15

[g(ρf − ρg)

ρ2fσ

]0.25(2.35)

where g is gravity, σ is the surface tension, ρ is the density, and A is the cross-

sectional area of the pipe. An upper bound for slug flow is provided by Chisholm

(1983), which predicts a transition from slug to churn flow at:

D = 19

(σ/ρf

g(1− ρg/ρf)

)(2.36)

Flow maps can also be useful in predicting the nature of the flow. For example,

Collier (1981) plots the flow regimes on axes of the vapor momentum flux versus the

liquid momentum flux, as shown in Figure 2-10. A commonly used flow map was

36

Figure 2-9: Vertical Flow Regimes

37

developed by Baker in 1954. He plotted

G

λ= 1.0212× 105 V—g ρg

πD2√ρg ρf(2.37)

versus

L λ ψ

G=0.1844V—f

σV—g

(ρ3fρg

) 12(µfρ2f

) 13

(2.38)

where G is the mass velocity of the gas phase, L is the mass velocity of the liquid

phase, and λ and ψ are functions of the fluid densities and viscosities.

Baker developed his transition lines from experimental data. Some similiarities

can be noted between the Baker flow map of Figure 2-11 and the Collier flow map

of Figure 2-10. It should be noted that all of the flow regime correlations and flow

maps described above have been developed using a homogeneous flow model. In

the homogeneous flow model, two-phase flow is treated as a single-phase flow with

properties that are a weighted average of the individual phases.

A complete derivation has been given by Delano (1998) for the dependence

of the submergence ratio (which is the ratio of the reservoir height, H , to the tube

length, L) on the fluid properties of the bubble pump:

H

L=

1

1 + sR/s+

V 2f

2gLA2(1 +K +KsR + 2sR) (2.39)

s =VgVf

sR =V—g

V—f

Griffith (1961) has given a correlation to calculate s for slug flow:

38

WispyAnnularAnnular

Churn

Bubbly

Slug Bubbly-Slug

Liquid Momentum Flux

Vap

or M

omen

tum

Flu

x

Figure 2-10: Collier Flow Pattern Map

Figure 2-11: Baker Flow Pattern Map

39

s = 1.2 + 0.2V—g

V—f

+0.35A

√gD

V—f

(2.40)

Furthermore, the friction coefficient (K) is a function of the pipe losses, including

those due to friction. Friction losses are derived from the friction factor (Reinemann

et al., 1990):

f =0.316

(Re)0.25(2.41)

where

Re =4ρ(V—g + V—f)

πµD

Since boiling produces the vapor bubbles, the amount of heat transferred to

the bubble pump determines V—g:

V—g =Qbubblepump

ρg · hfg(2.42)

In order to minimize the heat transfer that needs to be added to the bubble

pump, an optimization must be performed. The relationship between the bubble

pump tube diameter and the liquid and vapor flow rates can be seen in Figures 2-12.

A similar relationship is seen for the diameter, the liquid flow rate, and the amount

of heat transferred to the bubble pump.

Returning to the use of the bubble pump within the Einstein cycle, the goal

is to minimize the amount of heat transfer needed to pump the desired amount of

liquid and vapor. This can be expressed as the pumping efficiency:

40

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

m v ap (kg /m in)

mli

q (

kg

/s)

d = 8.6 mm

d = 7.6 mm

d = 6.6 mm

Figure 2-12: Effect of Diameter on Liquid and Vapor Flow Rates (Submergence Ratio= 0.2)

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Q bu b p ump

mli

q (

kg

/s)

d = 8.6 mm

d = 7.6 mm

d = 6.6 mm

Figure 2-13: Effect of Diameter on Liquid Flow Rate and Heat Transferred to theBubble Pump (Submergence Ratio = 0.2)

41

ηpump =m

Qbubblepump

Fixing the submergence ratio as a reasonable 0.2, the diameter with the opti-

mal slope that will still fall within the slug flow regime is 8.6 mm. For these values,

the optimum pump capacity per amount of heat transferred will obviously fall to

the left of point A on the curve of Figure 2-14. This section of the curve can be

approximated as a straight line so that

mliq = 1.22x10−2 · Qbp (2.43)

This is an improvement of 7.9% over Delano’s bubble pump efficiency. The liquid

and vapor flow rates for this section of the curve can be approximated as:

mliq = 28.6 · mvap (2.44)

These two equations describe the bubble pump’s performance within the cycle.

Second Law Analysis

The Coefficient of Performance

The heating efficiency of the cycle, or coefficient of performance (COPh), is

directly related to the cycle’s entropy generation. This can be demonstrated by

combining a first and second law analysis. By defining a control volume to include

the entire cycle, as in Figure 2-15, conservation of energy states that

42

0 20 40 60 80 100 120 1400.2

0.3

0.4

0.5

0.6

0.7

0.8

Q bubpump

mli

q

d = 8.6 mm

h/L = 0.2A

Figure 2-14: Pumping Capacity for h/L = 0.2, D = 8.6 mm

Generator Evaporator

Condenser/Absorber

Qca

Qgen Qevap

Figure 2-15: Heat Transfer To And From The Einstein Cycle

43

Qevap + Qgen − Qca = 0 (2.45)

This assumes that the sign of the rate of heat transfer is always positive in the

direction indicated in Figure 2-15.

A second law balance on a control volume that includes the entire cycle is:

∑Si =

Qca

Tc/a,s− Qevap

Tevap,s− Qgen

Tgen,s(2.46)

Si is the entropy that is generated in component i, and Ts is defined as the entropic

average temperature, as outlined in the description of the evaporator.

Equation 2.45 can be rearranged as

Qevap = −Qgen + Qca

and substituted into equation 2.46 to yield

∑Si =

Qca

Tca,s− (−Qgen + Qca)

Tevap,s− Qgen

Tgen,s(2.47)

This can then be restated as

Tevap,s∑

Si = Qca(Tevap,s − Tca,s

Tca,s) + Qgen(

Tgen,s − Tevap,sTgen,s

) (2.48)

The heating coefficient of performance is defined as the useful heating that is

desired divided by the high-temperature heating input that must be provided or pur-

chased. For the Einstein cycle, this is the heating provided by the condenser/absorber

divided by the heating required by the generator:

44

COPh =Qca

Qgen

(2.49)

Dividing equation 2.48 by Qgen therefore produces

Tevap,s

∑Si

Qgen

= COP h(Tevap,s − Tca,s

Tca,s) + (

Tgen,s − Tevap,sTgen,s

) (2.50)

Furthermore, the reversible, or ideal, heating coefficient of performance for a

system operating at three temperatures is:

COP revh =

TM(TH − TL)

TH(TM − TL)(2.51)

where TH , TM , TL are the high, middle, and low temperatures, respectively (Delano,

1998). For the Einstein cycle, this is equivalent to:

COP revh =

Tca,s(Tgen,s − Tevap,s)

Tgen,s(Tca,s − Tevap,s)(2.52)

Given this definition, manipulating equation 2.50 yields:

COP h = COP revh − Tevap,s · Tca,s

Tca,s − Tevap,s·∑

Si

Qgen

(2.53)

Therefore, the ideal COP is reduced to the actual COP by:

∑COP degradation

h,i =Tevap,s · Tca,sTca,s − Tevap,s

·∑

Si

Qgen

(2.54)

This relation illustrates that if the amount of entropy generated by a component (Si)

can be reduced, the degradation of the COP can also be lessened, and the performance

of the cycle can be improved.

45

An Illustrative Case: The Internal Generator Heat Exchanger

Examining the causes of entropy generation in each component is obviously of

interest. Entropy generation can be caused by friction, by transferring heat across a

finite temperature difference, or by mixing. The generator is presented as an example

of how a component second law analysis can provide insight on how to improve the

efficiency of the cycle.

Without an internal solution heat exchanger, and by assuming that the exter-

nal heat is added at the temperature of the mixture at each point in the generator,

the processes that occur there can be treated as fully reversible. In Delano’s study, a

partial internal solution heat exchanger was used in the generator to heat the pres-

sure equalizing/absorbing fluid mixture entering from the condenser (1998). This

heat exchanger improved the internal heat recovery, and increased the COP. How-

ever, the heat exchanger also caused the internal heat transfer to occur at quite large

temperature differences, resulting in entropy generation. By reducing the magnitude

of the temperature differences, the amount of entropy that is generated can also be

reduced.

When the partial internal solution heat exchanger is replaced by a full inter-

nal solution heat exchanger, as illustrated in Figure 2-16, the temperature differences

across which heat is transferred are lessened. The temperature differences are inher-

ent and cannot simply be eliminated through the creation of a larger heat exchanger.

The temperature profiles of the fluids inside and outside the tube are not parallel,

and a pinch point must occur at one end. An external heat exchanger was also added

between the condenser/absorber and the generator in order to preheat the entering

ammonia-water liquid mixture to thermal and vapor-liquid equilibrium conditions.

For given inlet and outlet conditions, three configurations were examined: 1)

46

5

Qgenerator

9

10

8

5

Qgenerator

9

10

8

Partial InternalHeat Exchanger

Full InternalHeat Exchanger

Figure 2-16: Alternate Generator Configurations

47

No internal or external heat exchange, 2) Partial internal and no external heat ex-

change, and 3) Full internal and external heat exchange. The amount of supplied heat

was approximately 522 kW for configuration 1, 311 kW for configuration 2, and 248

kW for configuration 3. The heating coefficient of performance is an inverse function

of the amount of heat transferred to the generator, so it can be seen that the COPh

is indeed the highest for configuration 3. (Since the inlet and outlet conditions are

fixed, the heat transferred to and from the evaporator and the condenser/absorber are

constant.) EES programs for each of the configurations are included in Appendix A.

48

CHAPTER III

THE PRINCIPLE OF CORRESPONDING

STATES MODEL

The van der Waals Equation of State

The ideal gas equation of state (PV = RuT ) treats all substances as if they

behave as a gas at pressures approaching zero. (A capital letter designates a molar

property and a lower-case letter designates a mass-based property.) In 1873, van

der Waals formulated an equation of state that could model a substance in both the

liquid and the vapor regions. His cubic equation of state

P =RuT

V − b− a

V 2(3.1)

was developed from molecular principles, and accounts for both intermolecular forces,

a, and the volume occupied by a molecule, b (van der Waals, 1873).

In order to match actual fluid behavior, the isotherms generated from the van

der Waals equation of state must have an inflection point at the critical point. This

is defined mathematically as

(∂P

∂V

)T

= 0

(∂2P

∂V 2

)T

= 0 (3.2)

at T = Tc, P = Pc, and V = Vc.

Simultaneously solving these equations yields

49

a =9

8RuTcVc b =

1

3Vc (3.3)

Substituting a and b into equation 3.1 at the critical point shows that

PcVcRuTc

=3

8(3.4)

Combining equations 3.1, 3.3, and 3.4:

P =RuT

V − Vc

3

−98RuTcVcV 2

P =RuTcVc

[T/Tc

(V/Vc)− 13

−98

(V/Vc)2

]

Pr =8Tr3Vr − 1

− 3

Vr2 (3.5)

where

Tr ≡T

TcPr ≡

P

PcVr ≡

V

Vc

are defined as the reduced temperature, pressure, and volume (Abbott and Van

Ness, 1989).

Two-Parameter Principle of Corresponding States

The significance of equation 3.5 is that it asserts that the reduced pressure

of any substance is solely a function of that substance’s reduced temperature and

50

volume. Furthermore, reduced pressure, temperature, and volume are characterized

by the same relationship for all substances.

Unfortunately, this two-parameter corresponding states model can be highly

inaccurate. Hougen and Watson (1948) calculated that errors of up to 35% can

result from this van der Waals-based model. The reason for this high level of error

is that the van der Waals equation of state does not take into account the molecular

variability that can occur in actual substances.

One way of measuring this variability is to examine the compressibility factor

(Z) of a substance. The compressibility factor is defined as

Z =Pv

RT(3.6)

where Z = 1 for an ideal gas. The critical compressibility factor is

Zc =PcvcRTc

(3.7)

Therefore, from equation 3.4, it can be deduced that the two-parameter principle of

corresponding states assumes that the critical compressibility is equal to 0.375 for all

substances.

Three-Parameter Principle of Corresponding States

In actuality, the critical compressibility can vary greatly between substances.

This would suggest that utilizing Zc as a third parameter could increase the accuracy

of the principle of corresponding states. To calculate Zc, vc must be known. Unfor-

tunately, since the differential compressibility, (∂v/∂P )T , approaches infinity at the

critical point, the critical volume cannot be accurately measured.

51

Pitzer et al. (1955) suggested an alternate third parameter that they termed

the acentric factor:

ω = log10(Prsat)Simple Fluid at Tr=0.7 − log10(Prsat)Tr=0.7 (3.8)

The acentric factor measures the deviation of intermolecular potential from a

simple fluid. A simple fluid is defined as one with a spherical shape and an inverse

sixth power potential, such as the heavier rare gases (Pitzer, 1955). Simple fluids all

exhibit a reduced saturation pressure of approximately 0.1 at a reduced temperature

of 0.7, so equation 3.8 becomes

ω = −1− log10(Prsat)Tr=0.7 (3.9)

The acentric factor can therefore be determined from the critical temperature and

pressure and from one saturation pressure measurement.

With three parameters (Pc, Tc, and ω), the principle of corresponding states

becomes highly accurate for subcooled and superheated nonpolar and slightly polar

substances. While the accuracy is lessened in the saturated region and near the

critical point, the three-parameter model still provides a reasonable approximation.

Calculation of Compressibility

Pitzer and Curl (1957) used the three-parameter principle of corresponding

states to demonstrate that the compressibility factor can be expressed as a function

of the acentric factor for a given reduced temperature and pressure:

Z(Tr, Pr, ω) = Z(0)(Tr, Pr) + ωZ(1)(Tr, Pr) + · · · (3.10)

52

This equation is based on an expansion of the compressibility factor as a simple

power series in the acentric factor. Pitzer et al. (1955) determined that the first two

terms are sufficient to accurately describe the compressibility factor in almost all

regions.

Lee and Kesler (1975) expanded this work to provide increased accuracy and to

include a wider range of temperatures (Tr ≥ 0.3). They found that the compressibility

factor of any fluid is a function of the compressibility of a simple fluid (Z(0)), the

compressibility of a reference fluid (Z(r)), and the acentric factor, where Z(0) and Z(r)

are functions of Tr and Pr:

Z = Z(0) +ω

ω(r)(Z(r) − Z(0)) (3.11)

Lee and Kesler chose n-octane as the reference fluid.

Through a combination of experimental data and the modified Benedict-

Webb-Rubin equation of state, the reduced compressibilities (Zr = Z/Zc) for both

the simple and reference cases were found to conform to:

Zr = 1 +B

Vr+

C

Vr2 +

D

Vr5 +

c4Tr

3Vr2

(β +

γ

Vr2

)exp

(− γ

Vr2

)(3.12)

where

B = b1 −b2Tr

− b3Tr

2 −b4Tr

3

C = c1 −c2Tr+

c3Tr

3

53

Table 3-1: Constants for Calculating CompressibilityConstant Simple Fluid (0) Reference Fluid (r)b1 0.1181193 0.2026579b2 0.265729 0.331511b3 0.15479 0.027655b4 0.030323 0.203488c1 0.0236744 0.0313385c2 0.0186984 0.0503618c3 0 0.016901c4 0.042724 0.041577

d1 x 104 0.155488 0.48736

d2 x 104 0.623689 0.0740336

β 0.65392 1.226γ 0.060167 0.03754

D = d1 +d2Tr

A set of constants was determined for the simple fluid and the reference fluid, as

shown in Table 3-1.

Recalling equations 3.6 and 3.7, the reduced compressibility is also

Zr =PrVrTr

(3.13)

To find Z(0), the simple fluid coefficients listed in Table 3-1 are used in equation 3.12.

That equation is then set equal to equation 3.13, a reduced temperature and pressure

are chosen, and the reduced volume is found. This is not an actual reduced volume,

but rather a pseudo-reduced volume. Once Vr(0) is calculated, equation 3.13 is used

to find Z(0). This process is then repeated using the reference fluid coefficients to

find Vr(r) and Z(r).

Due to the highly nonlinear nature of equation 3.12, solving for Vr is not a

54

simple calculation. For every reduced temperature and pressure, there are either one

or three real roots which satisfy the equations. If there is only one root, that point

corresponds to either a liquid state (Figure 3-1a) or a vapor state (Figure 3-1b). If

three real roots exist (Figure 3-1c), one root is at a liquid state (point 1), one root is

at a vapor state (point 3), and one root (point 2) has no physical significance because

it does not represent a stable state.

For superheated or subcooled substances, only one root is present, so Z(0) and

Z(r) can be calculated solely from equations 3.12 and 3.13. In the saturated region,

however, the state of the substance must be known in order to determine which root

to utilize. The definition of the acentric factor and the principle of corresponding

states dictates that the reduced saturation pressure of a substance is a function of

Tr and ω (Wong et al., 1990):

ln(Prsat) = f (0) + ωf (1) (3.14)

where

f (0) = 5.92714− 6.09648/Tr − 1.28862 ln(Tr) + 0.16934Tr6

f (1) = 15.2518− 15.6875/Tr − 13.4721 ln(Tr) + 0.43577Tr6

Therefore, when Pr > Prsat, the root that corresponds to the vapor state should be

used, and when Pr < Prsat, the root that corresponds to the liquid state should be

chosen.

Once Z(0) and Z(r) have been found, they can then be substituted into equation

3.11 to determine the fluid’s compressibility at the given reduced temperature and

55

Reduced Volume

0 0.5 1 1.5 2 2.5 3

Red

uced

Pre

ssur

e

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

Reduced Volume

Red

uced

Pre

ssur

e

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25

Red

uced

Pre

ssur

e

Reduced Volume

Figure 3-1: Intersection of Equations 3.12 and 3.13

56

pressure. The reference acentric factor (ω(r)) is a known constant, and is equal to

0.3978.

Calculation of the Enthalpy Departure Function

Once the compressibility is known, many other properties can be calculated

by using the three-parameter corresponding states model. These properties include

departure functions for enthalpy, entropy, isochoric specific heat, and isobaric specific

heat. A departure function is the difference between an actual property and the value

of that property if it behaved as an ideal gas at the same temperature and pressure.

This analysis is primarily concerned with enthalpy, so the derivation of the enthalpy

departure function will be presented.

Lee and Kesler found that a fluid’s enthalpy departure is a function of the

enthalpy departures of the simple fluid and the reference fluid:

H −H∗

RuTc=[H −H∗

RuTc

](0)+(

ω

ω(r)

)[H −H∗

RuTc

](r)−[H −H∗

RuTc

](0)(3.15)

where H∗ is the ideal gas enthalpy and H is the actual enthalpy on a mole basis.

Using fundamental property relations, the enthalpy departure function can be

related to the Gibbs energy:

H −H∗

RT= −T

[∂((G−G∗)/RuT )

∂T

]P

(3.16)

The Gibbs energy is also related to the compressibility:

G−G∗

RuT=∫ P

0(Z − 1)dP

P(constant T) (3.17)

57

so equation 3.16 becomes:

H −H∗

RuT= −T

∫ P

0

(∂Z

∂T

)P

dP

P(constant T) (3.18)

Replacing Z with equation 3.12 yields

H −H∗

RuTc= Tr

[Z − 1− b2 + 2b3/Tr + 3b4/Tr

2

TrVr

− c2 − 3c3/Tr2

2TrVr2 +

d25TrVr

5 + 3E

](3.19)

where

E =c42Tr

3γ

[β + 1−

(β + 1 +

γ

Vr2

)exp

(− γ

Vr2

)]

As with the compressibility, the simple fluid constants and reduced volume

are used to find ((H −H∗)/Ru Tc)(0), and ((H − H∗)/Ru Tc)

(r) is calculated using

the reference fluid constants and reduced volume. The usefulness of these enthalpy

departure function and saturation pressure calculations in determining ideal fluid

properties is demonstrated in the next sections.

Ideal Gas Theory

Since the theory of corresponding states produces functions that are depar-

tures from the ideal gas state, the principles of that state should be examined. Ideal

gas internal energy (u) can be represented as a function of temperature and volume:

du =

(∂u

∂T

)v

dT +

(∂u

∂v

)T

dv (3.20)

58

Constant-volume specific heat is defined as:

cv ≡(δq

dT

)v

≡(∂u

∂T

)v

(3.21)

so, substituting this into equation 3.20 results in

du = cvdT +

(∂u

∂v

)T

dv (3.22)

Recalling that Pv = RT for an ideal gas,

(∂u

∂v

)T

= 0

so this equation can be further simplified to

du = cvdT (3.23)

From equation 3.23, it is then evident that the internal energy of an ideal gas

is solely a function of temperature (Black and Hartley, 1991).

Enthalpy (H) is defined as a function of internal energy, pressure and volume:

dh ≡ du+ d(Pv) (3.24)

Returning again to the ideal gas equation of state, this equation becomes

dh = du+ d(RT ) = du+RdT (3.25)

and the enthalpy of an ideal gas is also shown to be only a function of temperature.

By defining constant-pressure specific heat:

59

cp ≡(δq

dT

)p

≡(∂h

∂T

)p

(3.26)

the differential enthalpy can then be stated as

dh = cpdT (3.27)

Incompressible Liquid Theory

For the ideal solution model, all liquid states will be assumed to behave as

an incompressible liquid at saturated liquid conditions. This is not an unreasonable

assumption, since the majority of processes in the Einstein cycle depend on operation

in the saturation region. The enthalpy at any point can be determined in a manner

similar to that employed in the previous ideal gas analysis.

Saturated Liquid and Vapor Behavior: Enthalpy Versus Temperature

For a given substance, the enthalpies of the saturated vapor (hg) and the

saturated liquid (hf ) vary with temperature in the manner illustrated by Figure 3-

2. While the slope and curvature of the lines are dependent on the substance, this

general behavior is universal. While there is a great deal of curvature demonstrated

close to the critical point, hg and, particularly, hf are primarily linear in the non-

critical region.

At a given temperature, the difference in the saturated vapor enthalpy and

the saturated liquid enthalpy is the heat of vaporization (hfg) for that temperature.

The slope of the saturated vapor line is equal to the gas specific heat at constant

pressure (cp,g), and the slope of the saturated liquid line is the fluid specific heat

60

400 450 500 550 600 650500

1000

1500

2000

2500

3000

Temperature (K)

En

tha

lpy

(k

J/k

g K

)

h g

h f

W ater

Figure 3-2: Saturated Liquid and Vapor Enthalpy

61

3 0 0 3 1 6 3 3 1 3 4 7 3 6 3 3 7 9 3 9 4 4 1 02 5 0

3 0 0

3 5 0

4 0 0

4 5 0

5 0 0

5 5 0

6 0 0

6 5 0

7 0 0

T em p er at u r e (K )

En

tha

lpy

(k

J/k

g K

)

h f Is o b u ta n eh f Is o b u ta n e

h g Is o b u ta n eh g Is o b u ta n e

3 0 0 3 1 5 3 3 0 3 4 5 3 6 0 3 7 5 3 9 02 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

1 4 0 0

1 6 0 0

T e m p e r a tu r e ( K )

En

tha

lpy

(k

J/k

g K

)

h f A m m o n iah f A m m o n ia

h g A m m o n iah g A m m o n ia

Figure 3-3: Saturated Liquid and Vapor Enthalpies

62

Ent

halp

y

Temperature

1

Tref

href

2

3

4


at constant pressure (cp,f). In deciding which specific heat to use as an additional

parameter, the shape of the saturation lines should be considered. By observing a

variety of saturation curves away from the critical point, as in Figure 3-3, it can be

seen that the saturated liquid line tends to behave in a more linear fashion than the

saturated vapor line. Therefore, the fluid specific heat at constant pressure will be

selected as the specified parameter, and will be assumed to be constant.

Implementation of the Ideal Gas and Incompressible Solution Theories

Returning to the ideal gas analysis, the enthalpy of a vapor at any point can

be found by integrating equation 3.27 from a reference state (point 0) to the point

of interest:

63

∫ h

h0

dh =∫ T

T0

cPdT (3.28)

It is helpful to assign the value of the enthalpy at a reference state, and to use

this to calculate the enthalpy at any other state. Therefore, reference values (h0, P0

and T0) are assigned for each fluid. The reference pressure for all three fluids should be

equal, and has been set to the atmospheric pressure (101 kPa) for this analysis. The

reference temperature for each fluid is the saturation temperature at the universal

reference pressure, and the reference enthalpy is the saturated liquid enthalpy at that

temperature and pressure, which is set to zero. The reference temperature therefore

varies between fluids.

Equation 3.28 can be represented graphically on the enthalpy versus tempera-

ture plots of the previous section. For example, the enthalpy of a saturated vapor can

be traced as path 1-2-3 in Figure 3-4. (This figure is a simplification of the saturation

curves from Figures 3-2 and 3-3.) Therefore, the saturated vapor enthalpy at any

temperature is:

hg = h0 + cp,f(T − T0) + hfg(T ) (3.29)

For a superheated vapor, the path of integration is 1-2-3-4, and the enthalpy

is:

hshv = h0 + cp,f(T − T0) + hfg(T ) + hsv−g(T ) (3.30)

where hsv−g is the difference in enthalpy between the superheated and saturated

vapor at T .

For an incompressible liquid, determining the enthalpy is even more basic.

64

Ent

halp

y

Temperature

1

Tref

href 2

3

4

5

6

7

hfg,1

hfg,2

hfg,3


Enthalpy is a function of both temperature and pressure, but it is only a weak

function of pressure when it is a liquid and close to the reference pressure. Since all

liquid states are assumed to be the saturated liquid state, determining the enthalpy

of a liquid is represented by path 1-2, so that

hf = h0 + cp,f(T − T0) (3.31)

It should be noted that while this method of determining the enthalpy assumes

that the fluid specific heat is a constant, it does allow for variation in the gas specific

heat curve. For example, even if the saturated vapor line flattens out or develops a

negative slope, the magnitude of the heat of vaporization will account for this. This

is illustrated through paths 1-2-3, 1-4-5, and 1-6-7 in Figure 3-5.

65

Determining Enthalpy Differences

While the fluid specific heats and the reference states are given, the differences

in enthalpy (hfg and hsv−g) used in the above equations can be accurately calculated

using corresponding states. Recall that if Tc, Pc, and ω are known, the enthalpy

departure function can be calculated in conjunction with the compressibility and

the pseudo-reduced volume. Furthermore, P sat can also be determined from equa-

tion 3.14, so the saturated liquid and vapor roots can also be determined. Therefore,

at a given temperature (T1),

(h− h∗

RTc

)T=T1x=0

and

(h− h∗

RTc

)T=T1x=1

(3.32)

are known.

The enthalpy of vaporization (hfg) at T1 is defined as:

hfg(T1) = hg(T1)− hf (T1) = hT=T1x=1

− hT=T1x=0

and the enthalpies at the saturation conditions can be further expanded to:

hT=T1x=1

= RTc

(h− h∗

RTc

)T=T1x=1

+ h∗T=T1x=1

and

hT=T1x=0

= RTc

(h− h∗

RTc

)T=T1x=0

+ h∗T=T1x=0

So the enthalpy of vaporization becomes

66

hfg(T1) = RTc

(h− h∗

RTc

)T=T1x=1

−(h− h∗

RTc

)T=T1x=0

+ (h∗

T=T1x=1

− h∗T=T1x=0

) (3.33)

However, recalling that the change in enthalpy for an ideal gas is only a func-

tion of temperature, the second term in equation 3.33 is equal to zero:

h∗T=T1x=1

− h∗T=T1x=0

= cp(T1 − T1) = cp(0) = 0

Therefore, the enthalpy of vaporization can be represented as a function of

the critical temperature, the molecular mass, and two calculated enthalpy departure

functions:

hfg(T1) =Ru

MTc

(h− h∗

RTc

)T=T1x=1

−(h− h∗

RTc

)T=T1x=0

(3.34)

The difference in enthalpy between a superheated vapor and a saturated vapor

at the same temperature (hsv−g) can be derived in a similar manner, such that:

hsv−g(T1) = RTc

(h− h∗

RTc

)T=T1

P=Pactual

−(h− h∗

RTc

)T=T1

P=Psat

(3.35)

where the departure functions are calculated at the same temperature but different

pressures.

67

CHAPTER IV

THE IDEAL SOLUTION MODEL

In the previous chapter, the enthalpies of the individual fluids are calculated

for saturated liquid, saturated vapor, and superheated vapor states. Next, the indi-

vidual fluids must be combined using equilibrium conditions and mixing rules.

Vapor-Liquid Equilibrium

The most basic statement of equilibrium for a mixture in a vapor and a liquid

phase is that the fugacities of the individual substances must be equal:

fVi = fLi (4.1)

However, this form of the fugacity relationship does not present an explicit represen-

tation of the mole fractions, temperature, and pressure. To provide a more useful

form of this equation, some simplifying assumptions can be made.

1) At constant temperature and pressure, the fugacity of the liquid for substance

i is equal to the liquid fugacity of that substance multiplied by the liquid mole

fraction of that substance:

fLi = xifLpure i

2) At moderate pressures, pressure has no effect on the fugacity of a liquid.

68

Additionally, the vapor phase is assumed to be an ideal gas:

fLpure i = P sati

3) As in assumption 1, at constant temperature and pressure, the fugacity of the

vapor for substance i is equal to the vapor fugacity of that substance multiplied

by the vapor mole fraction of that substance:

fVi = yifVpure i

4) Finally, as previously stated in assumption 2, the vapor phase is assumed to

be an ideal gas, so that the vapor fugacity of the pure component is equal to

the total pressure:

fVpure i = P

Given these assumptions, equation 4.1 is reduced to:

yiP = xiPsati (4.2)

This is Raoult’s Law (Prausnitz et al., 1986).

Mole and Mass Fractions

The total number of moles (Nm) in a vapor or a liquid is equal to the sum of

the number of moles of each component (Ni):

Nm =n∑i

Ni (4.3)

69

The vapor mole fraction is defined as the number of moles of a component of

the vapor divided by the total number of moles of the vapor:

yi ≡Ni

Nm, vapor(4.4)

and the liquid mole fraction is similarly defined:

xi ≡Ni

Nm, liquid(4.5)

The sum of the vapor mole fractions can be demonstrated to be equal to one:

n∑i

yi =n∑i

Ni

Nm, vapor=

∑ni Ni

Nm=

Nm

Nm= 1 (4.6)

Therefore, it also holds that the sum of the liquid mole fractions is equal to one

(Black and Hartley, 1991):

n∑i

xi = 1 (4.7)

At each point in the Einstein cycle, only a two-component mixture need be

considered, so equations 4.6 and 4.7 reduce to y1 + y2 = 1 and x1 + x2 = 1. In

combination with equation 4.2, the liquid and vapor mole fractions can then be

stated solely as functions of the absolute and saturations pressures:

x2 =P−P sat

1

P sat2 −P sat

1y2 = x2

P sat2

P

x1 = 1− x2 y1 = 1− y2

(4.8)

The partial pressures and the mass fractions can also now be determined:

70

pi = yiP (4.9)

xm,2 = (x2m2)/(x1m1 + x2m2) ym,2 = (y2m2)/(y1m1 + y2m2)

xm,1 = 1− xm,2 ym,1 = 1− ym,2

(4.10)

Mixing Rules

Once the mass fractions and the individual component enthalpies are known,

the enthalpy of a vapor or a liquid mixture can be calculated. To determine the

mixing rules which govern this calculation, the Gibbs energy must be examined. The

Gibbs energy of an ideal gas mixture is a function of its chemical potential (µ) and

vapor mole fractions:

Gig =n∑i

yiµi (4.11)

The component chemical potential is in turn a function of temperature, vapor mole

fraction, and individual component Gibbs energy:

µi = Gigi +RuT ln(yi) (4.12)

Substituting equation 4.12 into equation 4.11 results in:

Gig =n∑i

yiGigi +RuT

n∑i

yi ln(yi) (4.13)

Enthalpy is a function of the Gibbs energy, entropy, and temperature (H =

71

G+ TS). The molar mixture entropy is further related to the Gibbs energy as:

S = −(∂G

∂T

)P,y

Combining these relations with equation 4.13 reveals that the enthalpy of an ideal

gas mixture is equal to:

H ig =n∑i

yiHigi (4.14)

where H igi is the molar ideal gas enthalpy of component i.

Similar to equation 4.12, the chemical potential for an ideal liquid solution is

defined as following:

µi ≡ Gi +RuT ln(xi) (4.15)

It can clearly be seen that the derivation of the mixture enthalpy for an ideal liquid

solution will follow a similar path as the derivation for an ideal gas. The ideal liquid

solution mixture enthalpy is therefore analogous to the ideal gas mixture enthalpy

(Abbott and Van Ness, 1989):

H ideal soln =n∑i

xiHi (4.16)

Equations 4.14 and 4.16 can also be restated in terms of specific properties

and mass fractions such that:

hig =n∑i

ym,ihigi (4.17)

and

72

hideal soln =n∑i

xm,ihi (4.18)

The enthalpy at any point in the cycle can now be expressed by combining

these expressions for enthalpy with the relations for individual component enthalpies

and mass fractions that were derived earlier in this chapter.

73

CHAPTER V

IDEAL SOLUTION PARAMETER RESULTS

AND FLUID SELECTION

System Model

Using the corresponding states and ideal solution thermodynamic property

model outlined in Chapter II and the mixture property relations of Chapters III

and IV, an EES program was created that describes the system behavior. This

program and its libraries can be found in Appendix B.

In order to produce conditions suitable for domestic and commercial water

heating, the temperature of the condenser was set at 325 K (125F) and the evapora-

tor temperature was specified to be 295 K (72F). For the base case of the ammonia-

water-butane mixture, the generator temperature that produced the highest heating

coefficient of performance was found to be 495 K. However, the COP was only slightly

degraded by lowering the generator temperature to 425 K, as shown in Figure 5-1.

The lower generator temperature provided more stability to the solver when the fluid

parameters were varied.

Searching for Optimum Properties

Initially, a full-scale nonlinear optimization of the cycle model was attempted,

in which all the parameters were varied simultaneously. Unfortunately, this technique

caused the solver to become unstable and crash before a solution could be found. To

74

3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 01 .68

1 .70

1 .72

1 .73

1 .75

Tge ne ra tor (K)

CO

Ph

T = 425

T = 495

Figure 5-1: The Effect of Generator Temperature on COP

avoid this problem, each fluid property was instead individually varied, and the effect

on the COP was noted.

The starting point for each parameter variation was the default value from

the ammonia-water-butane triplet. Figure 5-2 shows the effect of varying the liquid

specific heat at constant pressure for the refrigerant (()refrg), pressure-equalizing fluid

(()pe), and absorbing fluid(()abs). In each graph, the dashed line denotes the base case

coefficient of performance. Figures 5-3 and 5-4 show the effect of varying the critical

temperature and critical pressure, respectively, and Figure 5-5 examines the acentric

factor’s effect. Finally, Figure 5-6 plots the changing COP versus the molecular mass.

In addition to testing the effects of parameter deviation, the contribution of

the generator temperature was also examined. As the generator temperature was

lowered from its optimum, the trend for each parameter remained the same, but the

magnitude of the COP improvement or degradation increased. Identical behavior

75

1.0 1.5 2.0 2.5 3.0 3.5 4.01.65

1.67

1.69

1.71

1.73

1.75

c pl,re frg

CO

Ph

4.8 5.2 5.7 6.1 6.6 7.07.01.65

1.67

1.69

1.71

1.73

1.75

c pl,pe

CO

Ph

3.1 3.5 3.9 4.4 4.8 5.21.65

1.67

1.69

1.71

1.73

1.75

cpl,abs

CO

Ph

Figure 5-2: COP versus cp,l

76

415 418 421 423 426 429 4324321.65

1.67

1.69

1.71

1.73

1.75

T c,re frg

CO

Ph

401 406 411 416 421 425 430 435 4401.65

1.67

1.69

1.71

1.73

1.75

T c,pe

CO

Ph

632 641 649 658 666 675 683 692 7001.65

1.67

1.69

1.71

1.73

1.75

T c,abs

CO

Ph

Figure 5-3: COP versus Tc

77

35 39 43 46 501.65

1.67

1.69

1.71

1.73

1.75

P c,re frg

CO

Ph

105 110 115 120 1251.65

1.67

1.69

1.71

1.73

1.75

P c,pe

CO

Ph

130 173 215 258 3001.65

1.67

1.69

1.71

1.73

1.75

P c,abs

CO

Ph

Figure 5-4: COP versus Pc

78

0.05 0.08 0.10 0.13 0.15 0.17 0.20 0.22 0.250.251.65

1.70

1.75

1.80

1.85

ω re frg

CO

Ph

0.25 0.30 0.35 0.40 0.45 0.50 0.550.551.65

1.70

1.75

1.80

1.85

ω pe

CO

Ph

0.22 0.29 0.36 0.44 0.51 0.58 0.651.65

1.70

1.75

1.80

1.85

ω abs

CO

Ph

Figure 5-5: COP versus ω

79

51 53 55 57 59 60 62 64 661.65

1.67

1.69

1.71

1.73

1.75

M re frg

CO

Ph

16 19 22 25 28 311.65

1.67

1.69

1.71

1.73

1.75

M pe

CO

Ph

16 17 18 19 20 211.65

1.67

1.69

1.71

1.73

1.75

M abs

CO

Ph

Figure 5-6: COP versus M

80

630 640 650 660 670 680 690 7001.65

1.70

1.75

1.80

T c,ab so rb

CO

Ph

Tgen < Tgen ,op tim um

Tgen = Tgen ,op tim um

Tgen > Tgen,optim um

Figure 5-7: COP versus Absorbing Fluid Critical Temperature for Varying GeneratorTemperature

occurred when the generator temperature was raised above the optimum. This is

illustrated for the critical temperature of the absorbing fluid in Figure 5-7.

Table 5-1 summarizes the manner in which each of the parameters must be

changed in order to increase the COP. The magnitudes of the variations are illustrated

in Figures 5-2 through 5-6, and are denoted by “+” signs in the table.

One potential difficulty with this method of optimization, however, was that

the effects of the parameters might be interrelated. In fact, once the parameters

were matched to actual fluids, this was indeed found to be the case. Constraints that

specify the maximum and minimum temperature differences between the evaporator

and the condenser/absorber must be met, and the effect of varying the refrigerant

81

Table 5-1: Parameter Variation for Increased Efficiency

Pressure-Equalizing Fluid Refrigerant Absorbing Fluid

cp,l Increase (+) Decrease (+) Decrease (++)M Increase (+) Increase (+) Decrease (+)Pc Decrease (+) Increase (++) Decrease (+)Tc Increase (++) Decrease (++) Increase (++)ω Increase (++) Decrease (+++) Increase (++)

parameters on the pressure-equalizing fluid parameters (and vice-versa) should be

explored. These difficulties are explained in more detail in the final section of this

chapter.

Matching the Ideal Properties to Actual Fluids

Dimensions of the Search

Three databases were used to gather property information on a total of 612

fluids (Cranium, 2000; PhysProp, 2000; Reid et al., 1987). For each fluid type (re-

frigerant, pressure-equalizing fluid, and absorbing fluid), the parameters were ranked

in order of their greatest effect on the COP, and the best matches were determined.

Absorbing Fluid

Einstein and Szilard used water as their absorbing fluid. The only viable alter-

native to water that was found in the fluid search was hydrazine (H4N2). Hydrazine

has the potential to raise the system’s COP since it has a higher critical temperature

and a lower critical pressure and liquid heat capacity than water. Unfortunately,

hydrazine also has a higher molecular weight and a lower acentric factor. These

variations combine to actually lower the coefficient of performance. Since water has

82

Table 5-2: Pressure Equalizing Fluids

Component M Tc (K) Pc (bar) w cp,l

(kJ

kg·K

)Ammonia 17.031 405.69 112.8 0.2526 5.272Formaldehyde 30.0263 408.04 65.9 0.2818 2.382Methyl Amine 31.058 430 74.5752 0.275 1.774Hydrogen Chloride 36.4606 324.69 83.1 0.1315 2.112Sulfur Dioxide 64.0648 430.79 78.84 0.2454 1.369Nitrogen Dioxide 46.0055 431.19 101.33 0.8511 3.25Methyl Bromide 94.9388 467.04 80 0.1922 0.857Sulfur Trioxide 80.0642 490.89 82.1 0.424 3.224Methanol 32.0422 512.68 80.97 0.564 2.842Ethanol 46.069 513.96 61.48 0.6452 2.861

unique properties, water was kept as the absorbing fluid for alternative mixtures.

Pressure-Equalizing Fluid

In contrast to the previous case, a number of alternatives were found for am-

monia. These are listed in Table 5-2, where the values of the parameters for ammonia

are also given. However, in addition to satisfying the parameter requirements, the

toxicity and environmental impact of the alternatives were also examined (Environ-

mental Defense Scorecard, 2000). For example, nitrogen dioxide initially appears to

be a viable alternative, but it is actually a pollutant that is extremely harmful to

humans and the environment. Including this sort of consideration, the list of alter-

natives was narrowed so that only formaldehyde, hydrogen chloride, methyl amine,

methanol, and ethanol were considered.

Refrigerant

As with the pressure-equalizing fluid, there were a number of alternate fluids

found for butane. Table 5-3 contains a list of these, where, as before, the values

83

Table 5-3: Refrigerants

Component M Tc (K) Pc (bar) w cp,l

(kJ

kg·K

)Butane 58.1234 425.22 37.97 0.1993 2.631Vinyl Fluoride 46.0442 327.84 52.38 0.1889 2.411Hydrogen Bromide 80.9119 363.19 85.52 0.0693 0.759Vinylacetylene 52.0758 454.04 48.6 0.092 2.059Hydrogen Sulfide 34.0819 373.57 89.63 0.0942 2.192-Butyne 54.0916 488.19 50.8 0.1301 2.387Propylene 42.0806 365.61 46.65 0.1398 2.63Propane 44.0965 369.86 42.49 0.1531 3.159Methyl Chloride 50.4875 416.29 66.8 0.1531 1.716Pentane 72.15 469.8 33.75 0.251 2.314Nitrous Oxide 44.0129 309.61 72.45 0.1409 1.943Acetylene 26.0379 308.36 61.39 0.1873 1.774Ethyl Chloride 64.5144 460.39 52.7 0.1906 1.762Dimethyl Ether 46.069 400.14 53.7 0.2036 2.805Ketene 42.0373 370.04 58.1 0.1257 2.413Isobutane 58.1234 408.2 36.5 0.183 2.219Chlorodifluoromethane 86.4684 369.34 49.71 0.2192 1.375Methyl Acetylene 40.0648 402.43 56.28 0.2161 2.448Ethyl Fluoride 48.0601 375.35 50.28 0.222 2.2

of the parameters for butane (the base case fluid) are also given. As a secondary

screen, toxicity was again considered, and certain choices, such as methyl chloride,

were eliminated.

Selection of Fluid Triplets

The Patel-Teja equation of state and Panagiotopoulos and Reid mixing rules

were selected to model the actual fluid behavior. In order to accurately do so, bi-

nary interaction parameters needed to be found for each absorbing fluid/pressure-

equalizing fluid and refrigerant/pressure-equalizing fluid pair. As outlined in the next

84

chapter, these interaction parameters are found through correlations with experimen-

tal equilibrium data.

The availability of this experimental data further narrowed the choices as to

which alternatives could be considered. For example, while there were data avail-

able for water-formaldehyde vapor-liquid equilibrium, no data could be found for

formaldehyde and any of the refrigerant fluid alternatives. Therefore, formaldehyde

was eliminated as a pressure-equalizing fluid alternative.

The various possible matches were extensively explored, which resulted in four-

teen alternative fluid triplets. These are outlined in Figure 5-8, which also includes

the base case triplet of ammonia-water-butane.

Unfortunately, once the vapor-liquid equilibrium behavior of the fluid pairs

was modeled, some of the alternatives had to be eliminated. The optimum system

pressure has been found to be that which allows the refrigerant to begin to condense

at its partial pressure and the condenser/absorber temperature. At that pressure,

the minimum evaporator temperature is then either the saturation temperature of

the pressure-equalizing fluid or the minimum boiling azeotrope of the fluid pair,

whichever is lower. For a water heating application, the temperature of the con-

denser/absorber should be 325 K and the maximum temperature of the evaporator

should be approximately 295 K.

Although the ideal model optimization indicated that using either methanol

or ethanol as the pressure-equalizing fluid would increase the COP, it was found

that the triplets containing those fluids could not produce a suitably low evaporator

temperature for a condenser/absorber temperature of 325 K. In addition to matching

actual fluids to parameter trends, parameter boundaries must also be specified. A

constraint must be used that states that the saturation temperature of the pressure-

equalizing fluid must be less than or equal to the condenser/absorber temperature

85

Water

Methanol

Ammonia

MethylAmine

Propane

Propylene

Butane

Butane

Ethanol

Propane

Propylene

Isobutane

Propane

Propylene

Butane

Pentane

Pentane

HydrogenChloride

Propane

Propylene

Butane

Figure 5-8: Alternative Fluid Triplets

86

minus the lift at the system pressure.

Equation 3.14 from the corresponding states theory relates the saturation

temperature and pressure to the acentric factor. Using that equation, the above

constraint can be stated mathematically as:

Pc,refrg exp[f0

refrg + wrefrg f 1refrg ] = Pc,pe exp[f

0pe + wpe f 1

pe ] (5.1)

where

f 0refrg , f

1refrg = Fn

(Tcond−absTc,refrg

)

f 0pe , f

1pe = Fn

(TevapTc,pe

)

and

Tcond−abs − Tevap ≥ Tlift

Since methanol and ethanol do not meet this criterion, they were eliminated

from consideration. The effect of each of the remaining fluid triplets on the system

performance was examined using the complex cycle model. As stated previously, the

following chapter explains the bases for this model: the Patel-Teja equation of state

and Panagiotopoulos and Reid mixing rules.

87

CHAPTER VI

THE PATEL-TEJA MODEL

The Patel-Teja Equation of State

As with the principle of corresponding states, the development of the Patel-

Teja equation of state begins with the cubic van der Waals equation, which was

described in Chapter III:

P =RT

v − b− a

v2

The van der Waals equation incorporates an intermolecular force parameter (a) and

a volume correction parameter (b). Furthermore, there is an inflection point at the

critical point, which is expressed as equation 3.2.

The Patel-Teja equation of state:

P =RT

v − b− a

v(v + b) + c(v − b)(6.1)

is clearly similar in form to the van der Waals equation. Patel-Teja adds a third

parameter (c). Furthermore, in addition to the constraints that (∂P/∂V )T = 0 and

(∂2P/∂V 2)T = 0 at the critical point, they added:

ζc =PcVcRuTc

(6.2)

where ζc is empirically determined. It should be noted that ζc is not equal to the

88

experimental value of the critical compressibility, but instead corresponds to the

minimum deviation in saturated liquid densities when the fugacities are equal.

Single Substances

Determination of the Patel-Teja Coefficients

In equation 6.1, b and c are constants, and a is a function of temperature.

Their values can be determined by applying the constraints of equations 3.2 and 6.2

to the Patel-Teja equation of state. The value of c is straightforward:

c = Ωc

(RuTcPc

)(6.3)

where

Ωc = 1− 3ζc (6.4)

Determining b is slightly more complicated:

b = Ωb

(RuTcPc

)(6.5)

The value of Ωb is the smallest positive root of

Ω3b + (2− 3ζc)Ω2

b + 3ζ2cΩb − ζ3c = 0 (6.6)

Finally, as mentioned previously, a is dependent on the temperature:

a = Ωa

(R2uT

2c

Pc

)α (6.7)

While Ωa is a constant:

89

Ωa = 3ζ2c + 3(1− 2ζc)Ωb + Ω2

b + 1− 3ζc (6.8)

α is a function of the reduced temperature:

α = [1 + F (1−√TR)]

2 (6.9)

Like ζc, F is an empirically-determined constant. Therefore, there are four

parameters that are required by the Patel-Teja equation of state: Tc, Pc, ζc, and F .

In their original paper on the equation of state, Patel and Teja calculated the values

of ζc and F for thirty-eight substances (1982). Furthermore, they determined that ζc

and F can be expressed as functions of the acentric factor for nonpolar fluids:

ζc = 0.329032− 0.076799ω + 0.0211947ω2 (6.10)

F = 0.452413 + 1.30982ω − 0.295937ω2 (6.11)

Equilibrium Conditions

The emperical values of ζc and F previously discussed were determined by ex-

amining the saturation curve at the equilibrium condition. Vapor-liquid equilibrium

occurs when the fugacity of the vapor phase is equal to the fugacity of the liquid

phase (as discussed in Chapter IV):

fL = fV

The fugacity can be expressed as a function of temperature, volume, and

90

pressure:

ln

(f

P

)=∫ P

0

(v

RuT− 1

P

)dP (6.12)

The Patel-Teja equation of state can then be substituted into the above inte-

gral to result in the following fugacity relationship (Patel and Teja, 1982):

ln

(f

P

)= Z − 1− ln

(Z − bP

RuT

)+

a

2RuTNln

(Z +M

Z +Q

)(6.13)

where

N =

(bc +

(b+ c)2

2

)−1/2

(6.14)

M =P

RuT

(b+ c

2−N

)(6.15)

Q =P

RuT

(b+ c

2+N

)(6.16)

In equations 6.13 through 6.16, Z is the compressibility and a, b, and c are the

parameters that were defined in the previous subsection.

Enthalpy and Entropy Departure Functions

In addition to providing the fugacity and the pressure, volume, and temper-

ature of the state of a substance, the Patel-Teja equation of state can also be used

to determine the enthalpy and entropy. As discussed in Chapter III, a departure

function is the difference between an actual property of a substance and the value

of that property if the substance behaved as an ideal gas at the same temperature

91

and pressure. For the Patel-Teja equation of state (1982), the enthalpy and entropy

departure functions are:

H −H∗

RuT= Z − 1− 1

2RuTN

(T∂a

∂T− a

)ln

Z +M

Z +Q(6.17)

S − S∗

Ru= − ln P

Z − (bP/RuT )− 1

2RuN

∂a

∂Tln

Z +M

Z +Q(6.18)

The values of N , M , and Q are given in equations 6.14 through 6.16. The differential

∂a/∂T is found by taking the derivative with respect to temperature of equation 6.7,

which results in:

∂a

∂T= −R2

uTcPcΩaF

√α

TR(6.19)

Mixtures

Mixing Rules

In addition to predicting the behavior of individual substances, the Patel-Teja

equation of state can also be used to model mixtures. To do so, mixture parameters

(am, bm, and cm) must be used in equation 6.1 rather than the individual substance

parameters. The mixture parameters are a function of the component parameters

and mole fractions:

am =∑i

∑j

xixjaij (6.20)

bm =∑i

xibi (6.21)

92

cm =∑i

xici (6.22)

The values of bi and ci for each component can be found using equations 6.5 and 6.3,

respectively.

Determing aij is more complicated. That term is a “cross-interaction parame-

ter,” and is a function of the individual component parameters (ai and aj , calculated

using equation 6.7) and one or more binary interaction coefficients. In this study, the

Panagiotopoulos and Reid rule was used to determine aij :

aij = [1− kij + xi(kij − kji)]√aiaj (6.23)

The parameters kij and kji are binary interaction coefficients (1985). They are de-

termined by minimizing the difference between experimentally and analytically de-

termined equilibrium points.

Equilibrium Conditions

The computationally determined equilibrium points are found by using the

fugacity. For mixtures, in addition to vapor-liquid equilibrium, liquid-liquid equi-

librium (fL1i = fL2

i ) and vapor-liquid-liquid equilibrium (fL1i = fL2

i = fVi ) can also

exist. The expression for the fugacity of a component in a mixture is more compli-

cated than the expression for the fugacity of a single substance. For component i,

the fugacity is equal to (Patel and Teja, 1982; Smith, 1995):

ln

(fiyiP

)= ln

(v

v − b

)+

b+ b′

v − b′+

a′D′

RT+

a

RT

[v

QD(b2 + c2 + 6bc

93

+ b′b+ 3b′c + 3bc′ + c′c) +1

QD(b′c2 + b2c′ − b′bc− c′cb) (6.24)

+D′

Q(−b2 − c2 − 6bc+ b′b+ 3b′c+ 3bc′ + c′c)

]− ln (Z)

where:

Q = −b2 − 6bc− c2

D = v2 + vb+ vc− bc

D′ = (−Q)−12 ln

(2v + b+ c−

√−Q

2v + b+ c+√−Q

)

a′ = −2∑j

∑k

yjyk(1− kjk)√ajak − 3

∑j

∑k

y2j yk(kjk − kkj)√akaj

+ 2∑j

yiyj(kij − kji)√aiaj +

∑j

yj[2− kij − kji + yj(kij − kji)]√aiaj

b′ = bi − b

c′ = ci − c

Enthalpy and Entropy

The enthalpy and entropy departure functions for each component in the mix-

ture are given by equations 6.17 and 6.18. However, since a, b, and c are now mixture

94

parameters, to determine ∂am/∂T an expression must be found for ∂aij/∂T from

equations 6.7 and 6.23:

∂aij∂T

=

√√√√Ωa,iR2T 2c,i

Pc,i

Ωa,jR2T 2c,j

Pc,j

FiTc,j

√αjTR,j − FjTc,i

√αiTR,i

2Tc,iTc,j√TR,iTR,j

(6.25)

The departure functions can be used to determine the enthalpy and entropy

of the mixture at any point. For each substance, the reference pressure is the at-

mospheric pressure, and the reference temperature is the saturation temperature of

that substance at the reference pressure. At the reference point, the enthalpy and

entropy of the saturated liquid for each substance are defined to be zero.

First, each component is evaluated as if it acted like an ideal gas at the ref-

erence temperature and pressure. The ideal gas enthalpy and entropy of each com-

ponent at this point are the negatives of the departure functions. The state of each

component is then moved from the reference temperature to the actual temperature

(the pressure remains at the reference condition). For an ideal gas, the change in

enthalpy for that process is equal to the specific heat multiplied by the difference in

temperature. Ideal gas specific heats were found in the DIPPR database (1999) and

in Assael et al. (1996). The change in entropy is the specific heat multiplied by the

natural logarithm of the actual temperature divided by the reference temperature.

Up to this point, the components have been evaluated individually; now, they are

mixed.

Once the substances are mixed, the liquid or vapor composition can be deter-

mined from the fugacities. The mixture is still treated as if it is an ideal gas at the

reference pressure and system temperature. The enthalpy of the mixture is:

95

hmix = zihi + zjhj (6.26)

where zi and zj represent either the liquid or the vapor mole fractions of components

i and j. The entropy, which incorporates an additional term that accounts for the

increase from mixing, is:

smix = zisi + zjsj −R[zi ln (zi) + zj ln (zj)] (6.27)

Finally, the mixture must move from the reference pressure to the actual pres-

sure. For this transition, the enthalpy and entropy departure functions are calculated

for the mixture at the actual temperature and pressure. In summary, the enthalpy

and entropy of a mixture at any point are:

h = zi[−hdep,liqi (Tref,i, Pref) + cid.g.p,i (Tsys − Tref,i)]

+ zj [−hdep,liqj (Tref,j, Pref) + cid.g.p,j (Tsys − Tref,j)]

+ hdepmixture(Tsys, Psys, zi, zj) (6.28)

s = zi[−sdep,liqi (Tref,i, Pref) + cid.g.p,i ln (Tsys/Tref,i)− R ln (zi)

]

+ zj[−sdep,liqj (Tref,j, Pref) + cid.g.p,j ln (Tsys/Tref,j)− R ln (zj)

]

+ sdepmixture(Tsys, Psys, zi, zj) (6.29)

These two equations can be used to find the enthalpy and entropy for any subcooled,

saturated, or superheated state.

96

CHAPTER VII

THE PATEL-TEJA MODEL RESULTS

System Model

Using the equation of state relations of Chapter VI and the thermodynamic

model of Chapter II, a highly accurate cycle model was created. The EES program

that describes the model can be found in Appendix C.

Binary Interaction Parameters

Equation of State Coefficients

The first step in using the Patel-Teja equation of state to model the behavior

of the alternate fluid triplets is to determine their binary interaction parameters

(kij and kji). Binary interaction parameters are found by minimizing the differences

between experimentally and analytically determined vapor-liquid (or vapor-liquid-

liquid) equilibrium points.

Experimental data were gathered from a number of sources for the various

fluid pairs. An effort was made to use data at temperatures and pressures similar

to the desired system operating conditions. The Patel-Teja fugacity relationship was

used to find the calculated equilibrium data. In order to use this expression for

the fugacity, the Patel-Teja equation of state coefficients were calculated from the

expressions given in the previous chapter. Some of the coefficients were previously

97

Table 7-1: Patel-Teja Coefficients

Fluid F ζc Ωa Ωb Ωc b c

Water 0.6898 0.269 0.50455 0.06510 0.193 0.01588 0.04078Ammonia 0.6271 0.282 0.48808 0.06934 0.154 0.02073 0.04604H Chloride 0.6214 0.319 0.44349 0.08179 0.042 0.02657 0.05236Methyl Amine 0.7902 0.310 0.45474 0.07851 0.071 0.03764 0.03425Butane 0.6784 0.309 0.44882 0.08023 0.056 0.0747 0.05236Propane 0.6481 0.317 0.44602 0.08105 0.049 0.0587 0.03546Propylene 0.6613 0.324 0.43799 0.083 0.028 0.05437 0.01824Pentane 0.7465 0.308 0.45653 0.078 0.076 0.09038 0.08807

known from Patel and Teja’s paper on their equation of state (1982). Table 7-1 lists

the constant coefficients for each fluid.

Fitting the Experimental Data

One of two types of equilibrium data was available for each fluid pair. Either

the temperature was fixed and the pressure varied with concentration, or the pressure

was fixed and the temperature varied with concentration. For the first case, to

reduce the error between the experimental and computation values, the following

relationship was minimized:

Pdeviation(%) =100

N

N∑i=1

∣∣∣∣∣1− Pi,calculatedPi,experimental

∣∣∣∣∣ (7.1)

where N is the number of data points, and i is each data point. For the second case,

temperature deviation was minimized:

Tdeviation(%) =100

N

N∑i=1

∣∣∣∣∣1− Ti,calculatedTi,experimental

∣∣∣∣∣ (7.2)

When more than one data set was available, the sum of the deviation of each data

98

Table 7-2: Binary Interaction Parameters

Pair kij kji Reference

Ammonia-Water -0.264 -0.294 (Gmehling et al., 1977)Ammonia-Butane 0.283 0.128 (Wilding et al., 1996)Ammonia-Pentane 0.283 0.128 Butane: (Wilding et al., 1996)Ammonia-Propylene 0.205 0.1 Propane: (Wilding et al., 1996)Ammonia-Propane 0.29 0.13 (Wilding et al., 1996)Methyl Amine-Water -0.205 -0.423 (Stumm et al., 1993)Methyl Amine-Butane 0.123 0.057 (Gmehling et al., 1977)Methyl Amine-Pentane 0.123 0.057 (Gmehling et al., 1977)Hydrogen Chloride-Water -1.4 -0.18 (Brandani et al., 1994)HCl-Propane 0.15 0.058 (Ashley and Brown, 1972)HCl-Propylene 0.142 0.025 Propane: (Ashley and Brown, 1972)HCl-Butane 0.15 0.058 (Ottenweller et al., 1943)

set was minimized.

Unfortunately, due to solver instabilities, EES could not automatically find

the minimum point, so the values of kij and kji were manually varied. For each guess

value of the binary interaction parameters, the fugacity of the liquid and vapor phase

were set equal to each other, and the pressure and temperature were calculated at

the experimental concentrations. The values of kij and kji were then changed, and

the effect on the average deviation was noted. Furthermore, in addition to math-

ematically minimizing the error, the correlation to experimental data was visually

checked.

Parameter Values

The kij and kji values for each fluid pair are given in Table 7-2. The source for

each pair’s experimental data is listed in the Reference column (a refrigerant listed in

that column indicates that the interaction parameters were extrapolated from that

data).

99

0 .0 0 0 .2 0 0 .4 0 0 .6 0 0 .8 0 1 .0 01 .7 5

2 .0 5

2 .3 5

2 .6 5

2 .9 5

3 .2 5

M ole Frac tion M ethyl Am ine

Pre

ss

ure

(b

ar)

Experimental DataExperimental Data

Calculated V LECalculated V LE

Figure 7-1: Methyl Amine-Butane, T = 288 K

For most cases, visual inspection shows that the computational results are

fairly accurate. Azeotropic pairs (Figure 7-1), zeotropic pairs (Figure 7-2), and vapor-

liquid-liquid equilibrium points (Figure 7-3) can all be modeled.

Actual Fluid Behavior

Once the binary interaction parameters are found, the equation of state co-

efficients and entropy and enthalpy departure functions can be used to find the cy-

cle COP for the alternate fluid triplets. As explained earlier, the system pressure

for each triplet is set such that the partial pressure of the refrigerant in the con-

denser/absorber is equal to the refrigerant’s saturation pressure at the temperature

of the condenser/absorber. The desired evaporator temperature ranges from 295 K

to 306 K, which may or may not be the minimum possible temperature. For each

triplet, the generator temperature was varied to find the maximum COP.

100

0 0.2 0.4 0.6 0.8 1210

232

254

276

298

320

M ole Fraction HC l

Te

mp

era

ture

(K

)

Calculated VLE

Experimental Data

Figure 7-2: Hydrogen Chloride-Butane, P = 4 bar

0 .0 0 .2 0 .4 0 .6 0 .8 1 .08

9

10

11

12

13

14

15

16

17

M ole Fraction Ammonia

Pre

ss

ure

(b

ar)

E x perim ental DataE x perim ental Data

Calc ulated V LLECalc ulated V LLE

Figure 7-3: Ammonia-Propane, T = 293 K

101

Table 7-3: COP for Tca = 325 K, Tevap = 295 K

Triplet COP Psystem

Ammonia-Butane-Water 1.51 5.25Ammonia-Propane-Water 1.49 18.3Methyl Amine-Pentane-Water 1.44 1.7Hydrogen Chloride-Propane-Water 1.41 18Hydrogen Chloride-Propylene-Water 1.39 21.7Hydrogen Chloride-Butane-Water 1.39 5.25Ammonia-Pentane-Water 1.38 1.78


Triplet COP Psystem

Ammonia-Butane-Water 1.88 5.25Ammonia-Propane-Water 1.87 18.3Methyl Amine-Pentane-Water 1.75 1.7Hydrogen Chloride-Propane-Water 1.68 18Hydrogen Chloride-Propylene-Water 1.66 21.7Hydrogen Chloride-Butane-Water 1.66 5.25Ammonia-Pentane-Water 1.63 1.78Methyl Amine-Butane-Water 1.61 5.2Ammonia-Propylene-Water 1.52 21.7

As outlined in the introduction, three temperature configurations were exam-

ined. The results for the first configuration, a condenser/absorber temperature of

325 K and and evaporator temperature of 295 K, are given in Table 7-3. The second

configuration’s (Tca = 325 K, Tevap = 306 K) results are in Tables 7-4, and the results

for the third configuration (Tca = 316 K, Tevap = 295 K) are in Table 7-5.

Ammonia-Water-Butane Results

The behavior of the cycle for the ammonia-water-butane case and the first

operating configuration is summarized in Table 7-6. The pressure is 5.25 bar, which

102


Triplet COP Psystem

Ammonia-Propane-Water 1.76 15.3Ammonia-Butane-Water 1.75 4.2Methyl Amine-Pentane-Water 1.7 1.3Hydrogen Chloride-Propylene-Water 1.62 17.8Hydrogen Chloride-Propane-Water 1.61 15Ammonia-Propylene-Water 1.61 17.6Ammonia-Pentane-Water 1.59 1.4Hydrogen Chloride-Butane-Water 1.56 4.2Methyl Amine-Butane-Water 1.53 4.2

Table 7-6: Ammonia-Water-Butane Results

COPact = 1.51 COPid = 3.46 Qevap = 439.2 kW Qca = 1307 kWQg,tot = 867.6 kW m1 = 1.24 kg/s m3 = 1.76 kg/s m5 = 0.47 kg/sm7 = 1.47 kg/s m10 = 0.96 kg/s mbp,in = 1.0 kg/s h1 = 134.2 kJ/kgh2 = 58.15 kJ/kg h3 = 694.1 kJ/kg h4 = 1519 kJ/kg h5 = 1529 kJ/kgh6 = 752.7 kJ/kg h7 = -170 kJ/kg h9 = 220.1 kJ/kg h11 = -198 kJ/kgxmb1 = 0.9957 yma6 = 0.2645 xma7 = 0.3279 xma9 = 0.0128

provides a maximum temperature lift of 53.1 K. With an evaporator temperature of

295 K and a condenser/absorber temperature of 325 K, 54% of this lift is used.

Analysis

Unfortunately, none of the alternate fluid triplets provide a better coefficient

of performance. For all three temperature cases, the water-ammonia-propane COP is

close to the water-ammonia-butane COP, with differences so small that they can be

neglected. To analyze this result, the relationship of the COP to the various states

throughout the cycle will be explored. The validity of the corresponding states/ideal

solution model will also be reexamined.

103

Table 7-7: Comparison of Refrigerant Performance

Pair COP P Qbubpump Qgen,actual

Ammonia-Butane 1.51 5.25 88.5 779.1Ammonia-Pentane 1.38 1.78 258.2 912.8

Pair mbubp mpe,5 m7 mrefrg,3

Ammonia-Butane 1 0.465 1.465 1.293Ammonia-Pentane 2.931 0.509 3.44 1.264

Effect of State Point Properties

For the first set of operating temperatures, the ammonia-water-butane and

ammonia-water-pentane triplets display the most disparate COPs. There are various

reasons for this degradation in the COP, as can be seen in Table 7-7, where the

evaporator cooling capacity has been set to 439.2 kW. The pentane case requires

a much larger mass flow rate of ammonia leaving the generator and entering the

evaporator. As a result of this, more ammonia must be absorbed by the water

stream in the condenser/absorber, and thereby desorbed in the generator.

Furthermore, the decreased pressure in the pentane case means that the con-

centration of ammonia is much higher in the liquid stream entering the bubble pump.

For that case, the bubble pump must receive nearly three times as much heat transfer

in order to remove the remaining ammonia and circulate the liquid water. In fact,

twenty-five percent of the heat that is applied to the generator goes to the bubble

pump. Previously, the bubble pump heat input has been treated as a minor factor

in the efficiency, but this is obviously no longer true. The higher bubble pump heat

transfer rate can be traced to the ammonia concentration differences between the

ammonia-butane and ammonia-pentane cases. For the ammonia-pentane mixture,

104

Table 7-8: Comparison of Pressure-Equalizing Fluid Performance

Pair COP P Qbubpump Qgen,actual

Methyl Amine-Pentane 1.44 1.7 140.4 863.6Ammonia-Pentane 1.38 1.78 258.2 912.8

Pair mbubp mpe,5 m7 mrefrg,3

Methyl Amine-Pentane 2.035 1.003 3.04 1.265Ammonia-Pentane 2.931 0.509 3.44 1.264

the relative difference in concentration between the ammonia-water mixture enter-

ing the generator and the ammonia-water mixture entering the bubble pump dictates

that more water must be pumped to provide the given ammonia requirement. Similar

results occur in both alternative temperature configurations.

The effect of changing the pressure-equalizing fluid can be seen by examining

the water-methyl amine-pentane and water-ammonia-pentane triplets for the first set

of operating temperatures (Table 7-8). Again, the evaporator load is set to 439.2 kW.

In this instance, the pressures and generator temperatures are comparable, but the

methyl amine case requires nearly twice as much pressure-equalizing fluid vapor to be

produced in the generator to meet the refrigerant’s partial pressure requirements in

the evaporator. This stems from the difference in molecular weight between ammonia

and methyl amine. The provide the same evaporator partial pressure, more mass of

pressure-equalizing fluid is required since the molecular weight of methyl amine is

nearly twice that of ammonia.

Once again, though, a driving force in lowering the COP is the amount of heat

added to the bubble pump. The bubble pump for the ammonia-pentane mixture

requires approximately twice as much heat transfer as for the methyl amine-pentane

mixture. This “parasitic pumping power” has a large influence on the COP.

105

Table 7-9: Theoretical Fluids and the Effect on the COP

Base Fluid Parameter Alteration COP COPincrease

Butane Tc to 415 from 425.2 1.52648 0.02028Butane Pc to 50 from 37.9 1.52824 0.02204Butane ω to 0.10 from 0.199 1.5278 0.0216Ammonia M to 30 from 17 1.50905 0.00285

Correspoding States/Ideal Solution Model Parameters

To test the validity of the parameter trends indicated by the corresponding

states/ideal solution model, four theoretical fluids have been created as examples.

Three of the fluids are a hypothetical alkane with properties similar to butane. One

parameter has been changed for each of the fluids to a value that was predicted

to increase the COP. The fourth fluid is an ammonia alternative with a different

molecular mass. Appropriate changes were made to each cycle model to reflect the

new parameter values, and the COP was calculated for a condenser temperature of

325 K and an evaporator temperature of 295 K. Each of the changes and the effect

on the COP is summarized in Table 7-9.

In each case, the COP was increased as predicted in Chapter V. Unfortunately,

these properties cannot be matched to actual fluids.

106

CHAPTER VIII

GAS WATER HEATING: AN APPLICATION

AND ITS IMPLICATIONS

Introduction

The temperatures that were chosen for the condenser/absorber and the evap-

orator in this study were selected for gas water heating. Configuring the Einstein

cycle as a gas heat pump water heater would produce hot water from ambient air

conditions and a gas flame, and would thereby provide an increase over conventional

gas water heating efficiency. The basic principle behind this configuration is shown

in Figure 8-1.

Residential and small commercial water heating is an energy use that is often

overlooked. Fifty-five percent of U.S. households utilize a gas water heater, and,

among those households, water heating accounts for nearly 25% of their gas usage.

In the commercial sector, natural gas water heating requires approximately 0.35

Generator Evaporator

Cond/Abs

Heat FromGas Flame

Heat FromAmbient Air

Heat From Condenser/Absorber to Tank

Einstein Cycle

Figure 8-1: An Einstein Cycle Gas Heat Pump Water Heater

107

quads per year, which is 13% of the total commercial gas usage. For an average

residential gas cost of $6.10 per thousand cubic feet and an average commercial gas

cost of $5.10 per thousand cubic feet, this translates into a cost of approximately $10

billion annually.

In addition to the fuel operating costs, gas water heaters contribute to global

warming emissions from the carbon dioxide formed during combustion. Each year,

approximately twenty-four million metric tons of carbon are emitted due to gas water

heating. An improvement in the efficiency of these gas water heaters would reduce

fossil fuel depletion, provide economic savings for the consumer, and reduce green-

house gas emissions.

Water Heating Efficiency

Electric versus Gas Water Heating

Residential water heating is a major energy use. The DOE estimates daily

hot water demand to be 64 gallons at 135F with an inlet city water temperature

of 58F, and requires testing and labeling of gas and electric water heaters at these

conditions (DOE 10CFR430, 1998). This test procedure measures the efficiency of

the water heater by comparing the energy supplied in heated water to the total daily

consumption of the water heater.

For an electric water heater using electric resistance heaters immersed in the

tank, the only losses are the stand-by losses from the stored hot water to the sur-

roundings. This is shown in Figure 2. The minimum efficiency rating for electric

water heating is 0.86, with the best available being 0.95. For discussion here, the

average will be taken to be the DOE recommended rating of 0.92. This means 92%

of the energy added to water is delivered from the tank, with the remaining 8% lost

108

Electrical Energy100%

StandbyLoss

8%

Deliveredot Water

92%

Gas CombustionEnergy - 100%

StandbyLoss

5%

DeliveredHot Water

61%

Flue Loss34%

Electric Water Heater Gas Water Heater

Figure 8-2: Water Heater Losses

to the air in stand-by losses from the tank.

In a gas water heater, in addition to the stand-by losses there is a flue loss due

to the hot combustion gases exiting the flue as shown in Figure 8-2. The minimum

efficiency rating for gas water heating is 0.53, and the best available is 0.66. For

discussion here, it is assumed that the average is the DOE recommended efficiency

level of 0.61, and that the absolute stand-by heat losses are same as for electric water

heaters. This results in a gas flue efficiency (percentage of gas combustion energy

added to the water in the tank) of (0.61/0.92) = 0.66, so that the flue efficiency

(percentage of combustion energy being transferred to the water storage tank) is:

Effflue =Qtank

Qgas

= 66% = Effwater heater (8.1)

Therefore, 34% of the gas combustion energy is going up the flue and 5% is

lost to the air in stand-by losses from the storage tank. The stand-by losses as a

percent of the gas input energy are less than the electric water heater (5% versus

8%) due to the gas input being greater for the same delivered hot water.

109

Einstein Cycle Gas Water Heating

Recall that for the Einstein cycle, the heat transfer available from the con-

denser/absorber for water heating relative to the heat transfer received from the gas

combustion is given by:

Qtank

Qgen= COPheating =

Qgen +Qevap

Qgen≥ 1 (8.2)

Heat transfer to the evaporator is at the relatively high temperature of 72F, so it

can be taken from the ambient air and is free to the consumer.

Using the typical 66% flue efficiency for gas water heaters for the heat pump,

which assumes that 66% of the gas combustion energy is transferred to the heat pump

generator:

EffHP,water heater =Qtank

QHP,gas

=EffflueQtank

Qgen

(8.3)

Combining equations 8.1, 8.2, and 8.3:

EffHP,water heater =Qtank

QHP,gas= Effflue × COPh = Effwater heater × COPh (8.4)

This shows that the heat pump water heater will have an efficiency higher

than the conventional water heater by the value of the heat pump cycle COP, which

is always greater than one.

Savings

Energy

U.S. residential and commercial water heating consumes approximately three

quadrillion Btu of primary energy each year. In the residential sector, fifty-percent

110

of households use natural gas water heaters. This translates into 1.3 quads of energy

use each year, which accounts for nearly 25% of residential natural gas usage. In

the commercial sector, gas water heaters consume 0.35 quads of primary energy each

year, which is 70% of the primary energy used for water heating. The residential and

commercial gas water heater installed base is estimated at 48 million and 1 million.

From equation 8.4, it can be seen that an Einstein Heat Pump Water Heater

with a heating COP of 1.5 would result in a 50% improvement over current natural

gas water heating technologies. This improvement would reduce gas consumption by

33%. In the residential sector, this means that 0.54 quads of energy could be saved

each year, and in the commercial sector, 0.12 quads could be saved. The reduced

energy consumption provides both economic savings and environmental benefits.

Economic

Reducing the energy consumed by water heating would provide economic sav-

ings for the consumer. Gas water heaters are already more beneficial for consumers

than electric resistance water heaters. Operation of an electric resistance water heater

for a typical household would cost approximately $300 each year. Electric heat pump

water heaters provide some improvement, but due to the required electrically-driven

compressor, their initial cost is two to three times higher than the cost of an electric

water heater, so the market penetration is limited.

However, just as electric heat pump water heaters improve upon resistance

water heaters, an affordably-priced gas heat pump water heater would be an im-

provement on existing gas water heating technology. As previously discussed, a gas

heat pump water heater that uses the Einstein cycle would have no machined or

moving parts, so it could be constructed out of inexpensive materials and would re-

quire no additional maintenance. Therefore, while the purchase cost of a natural gas

111

Einstein Heat Pump Water Heater would be slightly greater than that of a conven-

tional gas water heater, lower operating costs should compensate for the extra initial

purchase price.

The current average retail cost of a gas water heater is between $250 and $300.

It appears reasonable that the Einstein cycle gas heat pump could be manufactured

for about $120 above the production cost of a convention gas water heater. Once

the manufacturer’s gross margin of 30% and the distribution costs are included, the

total cost to the consumer would be approximately $550.

For a residential consumer, the average cost for natural gas is $6.10 per thou-

sand cubic feet. A family of four uses approximately 320 therms per year for natural

gas water heating, resulting in an average annual cost of $190. Overall, the residential

sector spends over $8 billion each year on gas water heating. With a 25% penetration

of the residential gas water heating market, $660 million could be saved each year.

A typical household would reduce their gas bill by $70 annually, and larger families

would be able to save even more.

The internal rate of return (IRR) is a useful quantity in evaluating the eco-

nomic value of purchasing an Einstein gas heat pump water heater rather than a

conventional gas water heater. IRR is calculated from:

0 = I0 +12∑n=1

In(1 + IRR)n

(8.5)

The initial incremental cost of an Einstein gas water heater (I0) is -$275, and the

yearly energy savings (In) are $70. Given that a gas water heater can be expected

to last for 12 years (n), equation 8.5 returns an IRR of 24%. Compared to a bank

account (IRR ≈ 4%) or an investment in the stock market (IRR ≈ 8%), it can be

seen that this would be a very attractive return.

112

In the commercial sector, the average cost for natural gas is $5.10 per thousand

cubic feet. Annually, the commercial sector spends $1.8 billion for natural gas water

heating. Due to the larger daily hot water usage, this sector should have better

economics than the residential sector, so gas heat pump water heaters would thereby

be expected to reach a higher market penetration of 33%, and $600 million could be

saved each year.

Environmental

In addition to the operating costs detailed above, combustion of natural gas

results in carbon emissions. With the Kyoto treaty of 1997, global warming resulting

from carbon emissions became an issue of great interest to the public. As a result,

numerous studies have been published that detail the costs associated with reducing

the United States’ greenhouse gas emissions to the treaty target levels. By using top-

down economic models, which include numerous difficult and debatable assumptions,

economists have calculated the cost for reducing annual carbon emissions to be $100

to $200 per ton of carbon. This is approximately equal to the current cost of fossil

fuels, meaning the effective cost of energy would double. Economists have calculated

that this economic cost would lead to a reduction in gross domestic production, a

societal cost that is both heavy and politically unacceptable (Passel, 1997).

Conversely, a bottom-up analysis that looks at high efficiency technology

shows that the use of such technology can give a positive economic payoff (Jacard et

al., 1996). Furthermore, by utilizing higher efficiency systems, the benefits enjoyed

by the end-user are not affected. An Einstein cycle gas heat pump water heater is

an example of this type of technology.

To provide hot water for a year for a family of four, 0.46 metric tons of carbon

are released from a natural gas water heater. In the residential sector, nearly 19

113

1.240

1.335

1.430

1.525

1.620

1.715

1985 1987 1989 1991 1993 1995 1997

Year

Pri

mar

y E

ner

gy

Co

nsu

mp

tio

n

(Qu

ads)

18.0

21.5

25.0

Car

bo

n D

ioxi

de

Em

issi

on

s (M

illio

n M

etri

c T

on

s C

arb

on

)

Figure 8-3: Residential and Commercial Natural Gas Water Heating

million metric tons of carbon are released each year due to gas water heating, while

over 5 million metric tons of carbon are released due to commercial gas water heating.

The total annual energy usage and emissions for gas water heaters can be seen in

Figure 8-3.

A 25% penetration of the residential gas water heater market by an Einstein

gas heat pump water heater would reduce carbon emissions by 1.6 million metric

tons each year. Annual carbon emissions from the commercial sector may also be

reduced by up to 1.7 million metric tons.

Under the Kyoto agreement, the U.S. must be at or below 93% of its 1990

greenhouse gas emission levels by 2008-2012. In 1990, the U.S. emitted 477 mil-

lion metric tons of carbon. By 1998, that number had grown to 517 million tons

(DOE/EIA-0383(98)). The annual carbon savings of 3.3 million metric tons outlined

above would result in meeting nearly 5% of that goal. If residential penetration

increased to approach 100%, 11% of the Kyoto target could be reached.

114

Once again, it should be emphasized that this carbon savings comes with

economic benefits (as detailed in the previous subsection) and not costs.

Implementation

Configuration of the Einstein Cycle as a Gas Water Heater

An Einstein cycle heat pump water heater could be built as an integral part

of the tank, with an external natural convection ambient heat exchanger. Within

this general framework, there are two ways that the cycle can be configured. In a

series configuration, the Einstein cycle condenser/absorber would heat the water to

an intermediate temperature, and direct heating would then increase the temperature

of the water to the final desired level.

In a quick recovery configuration, the Einstein cycle heat pump would heat

the water to the required temperature level. If the hot water level dropped below the

minimum set point, the gas flame would over-fire in a direct heat transfer mode, just

as a conventional water heater. In a configuration similar to an electric water heater,

two thermostats would provide temperature feedback to determine if the heat pump

alone should be operated, or if over-firing should be instigated.

For either configuration, the hot flue gases could be used in heat recovery. A

two-stage evaporator could be constructed that would take advantage of the heat

transfer available from both ambient temperatures and the higher-temperature vent

gases.

Market Potential

According to the Gas Appliance Manufacturers Association, over 4 million

residential and 95,000 commercial gas water heaters were shipped in 1998. These

115

numbers include both installations in new buildings and replacements for existing

units. The installed base of residential gas water heaters is estimated to be about 48

million. The short payback from the after-tax energy cost savings on the increased

Einstein heat pump water heater installed costs would be expected to yield a signif-

icant market penetration that would rise over a 10 year period to 25 percent. With

25% of the 4 million annual units, at an estimated installed cost of $550, the total

retail sales would be $350 million. A 3% penetration would yield annual sales of $42

million annually.

Barriers

An analogous case study that was performed for electric heat pump water

heaters provides insight into the barriers to implementation that might be encoun-

tered (Shelton and Schaefer, 1998). Essentially, all electric water heaters used in U.S.

residences are electric resistance water heaters. Assuming ideal thermal insulation,

one unit of electricity will generate one unit of hot water in an electric resistance

heater. However, electric heat pump water heaters pump free thermal energy from

the atmosphere rather than create it, and thereby use fifty to seventy percent less

electricity than resistance heaters. There are currently forty million electric resistance

water heaters in U.S. residential use (DOE/EE-0009(93)). If each electric resistance

water heater were replaced by a heat pump water heater, twenty-four million metric

tons of carbon would be saved per year.

In addition to reducing carbon emissions, electric heat pump water heaters

can also help the consumer to save money. While heat pumps are more expensive to

purchase and install, the reduced electricity costs are significant. At the end of an

electric heat pump water heater’s average twelve-year lifespan, the present value of

the consumer’s total lifetime expenditures will be only $1960, compared to $3120 for

116

an electric resistance water heater, a savings of nearly $1200 (DOE/EE-0009(93)).

Based on numbers similar to those given above, the DOE formally suggested

in 1994 that the minimum efficiency standard for electric water heaters be raised from

0.86 to 1.89. In homes with electric-powered water heating, this standard would ef-

fectively mandate that heat pump water heaters be installed in all new homes and

that all replacement water heaters in existing houses utilize heat pumps. Further-

more, the DOE study found that not only would consumers benefit from the raised

standard, but utilities and manufacturers would as well. The manufacturing sector

would incur some short-term retooling costs, but would then increase its profitability

(DOE/EE-0009(93); Proposed Rules, 1994).

Unfortunately, the response from utilities and manufacturers was strongly

negative. Electric utilities claimed that there was an inadequate infrastructure of

technicians who could install and service electric heat pump water heaters. They

also stated that the initial financial burden was too high for many consumers, and

that increasing the efficiency of appliances in the home, such as water heaters, would

reduce the incentive to improve power plant efficiency (Pacific Northwest National

Laboratory, 1998). Based on the public comment session, the DOE decided not to

raise the minimum efficiency standard.

A DOE-mandated higher minimum efficiency standard would greatly aid in

the commercialization of Einstein cycle gas heat pump water heaters. As in the above

case, though, obstacles from manufacturers and gas utilities would probably arise.

However, new considerations, such as the need to reduce greenhouse gas emissions,

may outweigh the concerns expressed in 1994.

Furthermore, an Einstein gas heat pump water has a number of qualities that

make it more attractive than an electric heat pump water heater. The simplicity of

the unit’s construction would greatly reduce the incremental initial cost, so that it

117

would be more affordable to a larger percentage of consumers. The relatively high

temperature of the evaporator also means that the Einstein cycle can successfully

operate in a variety of climates, so marketing efforts would not be limited to only

certain regions.

Additionally, one of the inventors of the cycle also increases its appeal. The

Einstein name is widely recognized, and is associated with genius and unique insight.

Recent presentations on the cycle at a number of universities have been strongly

attended, not only by professionals familiar with the energy field, but also by students

and other lay people. Audience members at these presentations have expressed a

great deal of enthusiasm for an Einstein cycle gas heat pump water heater, and have

been eager to learn when it might be available commercially (Schaefer, 2000a-2000e).

118

CHAPTER IX

SUMMARY, CONCLUSIONS AND

RECOMMENDATIONS

Summary and Conclusions

Previously, the Einstein cycle has been investigated primarily for refrigera-

tion applications. The cycle can also be used for heating, however, and has been

investigated at temperature levels that are suitable for natural gas water heating.

The configuration of the Einstein cycle was examined, and changes were made

to increase the heating coefficient of performance. These changes were primarily im-

plemented on the generator side. An external heat exchanger was added between the

generator and the condenser/absorber to improve heat recovery. The partial internal

heat exchanger in the generator was expanded to a full internal heat exchanger in

order to minimize entropy generation.

Additionally, the bubble pump performance was increased through selection

of optimum operating parameters. The submergence ratio of the bubble pump was

set to 0.2, and an optimum diameter was found to be 8.6 mm. These values resulted

in a 7.9% improvement in bubble pump performance over previously used operating

parameters.

The Einstein Cycle has been modeled using two separate property models: 1)

a corresponding states/ideal solution property model, and 2) a Patel-Teja/Panagioto-

poulos and Reid property model. The first model was used to predict fluid charac-

teristic parameters which would increase the heating COP and the second model was

119

used to more accurately predict the behavior of the cycle using the alternative fluids

selected from the results of the first model.

The first property model uses departure functions calculated from the three-

parameter theory of corresponding states to find the properties of individual fluids.

Corresponding states is based on the cubic equation of state first formulated by van

der Waals. In addition to van der Waals’ dependence on critical temperature and

pressure, however, corresponding states incorporates Pitzer’s acentric factor to model

fluid behavior. Corresponding states has been found to provide accurate results for

individual fluid properties at pressures and temperatures away from the critical point.

Once the properties of the individual fluids are calculated, the properties of the fluid

mixtures were found by utilizing Raoult’s Law and ideal solution mixing rules.

The corresponding states/ideal solution model requires only five characteristic

fluid parameters per fluid to model the system behavior. These are the critical

temperature, critical pressure, acentric factor, molecular weight, and the specific

heat at constant pressure. By varying these parameters, trends were observed that

could increase the COP.

Differences in the molecular weight of each of the fluids causes only small to

negligible changes in the COP. Varying the acentric factor and the critical temper-

ature results in the largest increase in the COP, with the refrigerant acentric factor

producing the largest effect. Lowering the refrigerant acentric factor from 0.20 to

0.08 should increase the COP by 6%.

For the base case of ammonia-water-butane, the generator temperature that

produced the highest heating COP was 495 K. Generator temperatures both above

(up to 575 K) and below (down to 425 K) this level lowered the COP, but not

significantly. The non-optimum generator temperatures also altered the magnitude

of the changes in the COP for the parameter variations, but did not alter the observed

120

trends.

Varying the critical temperature, critical pressure, and acentric factor of the

refrigerant affects the optimum system pressure. These three parameters determine

the saturation pressure of the refrigerant, which is needed to find the optimum system

pressure at the condenser/absorber temperature. Increasing the refrigerant critical

pressure increases the optimum system pressure, while increasing the critical tem-

perature or acentric factor lowers the system pressure.

Additionally, a constraint was developed to limit the corresponding states/-

ideal solution search. In order for the cycle to operate, the minimum available tem-

perature lift must be greater than or equal to the difference in the desired con-

denser/absorber and evaporator temperatures. At the optimum system pressure,

the temperature lift is the saturation temperature of the refrigerant at its partial

pressure in the condenser/absorber minus either the saturation temperature of the

pressure-equalizing fluid or the minimum boiling azeotrope, whichever is lower.

Unfortunately, due to computational limitations, the corresponding states/-

ideal solution model could not be optimized by varying all of the parameters si-

multaneously. This limits the corresponding states parameter optimization to single

parameter variations. Based on the results of the corresponding states/ideal solution

model, alternative fluids were sought for the three Einstein cycle working fluids.

No viable alternative was found for water, the absorbing fluid. For the pressure-

equalizing fluid, methyl amine and hydrogen chloride were selected for study as re-

placements for ammonia. Propane, propylene, and pentane were chosen for study as

refrigerant alternatives. Experimental equilibrium data were found in the literature

for each of these alternative mixtures. To model the behavior of these alternatives,

the Patel-Teja equation of state and Panagiotopoulos and Reid mixing rules were

used.

121

Like van der Waals and corresponding states, Patel-Teja is a cubic equation

of state. Using fluid parameters such as critical temperature and critical pressure,

the Patel-Teja equation of state can predict the behavior of individual fluids. In

conjunction with mixing rules, such as those developed by Panagiotopoulos and Reid,

the Patel-Teja equation of state can also accurately model fluid mixtures.

The Patel-Teja fugacity relations and the Panagiotopoulos and Reid mix-

ing rules were used to find the binary interaction parameters (kij and kji) for the

refrigerant/pressure-equalizing fluid pairs and the water/pressure-equalizing fluid

pairs. The binary interaction parameters were varied to match the calculated equilib-

rium states with experimental data. Excellent agreement was achieved for zeotropes,

vapor-liquid equilibrium azeotropes, and vapor-liquid-liquid equilibrium azeotropes.

Using the binary interaction parameters and the other characteristic fluid pa-

rameters, the effect of using the alternative fluid triplets on the cycle performance

was evaluated. Three operating conditions were examined. For the first case, the con-

denser/absorber temperature is 325 K and the evaporator temperature is 295 K. This

configuration can produce 125F water from ambient conditions (flue gases could be

used to help to maintain the evaporator at 72F). The condenser/absorber temper-

ature remains the same for the second operating configuration, but the evaporator

temperature is increased to 306 K, which would again utilize the higher temperature

thermal energy available from the flue vent gases. In the third operating configu-

ration, the Einstein cycle would be used as a water preheater, with an evaporator

temperature of 295 K and a condenser/absorber temperature of 316 K.

The Patel-Teja/Panagiotopoulos and Reid property model demonstrated that

none of the alternative refrigerants or pressure-equalizing fluids improved the COP.

One alternate refrigerant, propane, produced COPs comparable to those of the origi-

nal ammonia-water-butane mixture, but the remaining alternatives actually degraded

122

the COP. Using either ammonia-water-butane or ammonia-water-propane as the

working fluids, the Einstein cycle has a COP of 1.5 for the first operating config-

uration, 1.88 for the second configuration, and 1.76 for the third.

To test the validity of the indicated trends of the corresponding states model,

three hypothetical refrigerants and one hypothetical pressure-equalizing fluid were

also modeled using the Patel-Teja equation of state. The properties of those fluids

were altered to exactly match the improved parameters of the corresponding states

model. The hypothetical fluids each increased the COP, but to a lesser degree than

that predicted by the corresponding states/ideal solution model.

For a set evaporator load, the cycle conditions were compared for fluid com-

binations that resulted in disparate COPs. It was found that the pressure of the

system affects the absolute pressure-equalizing fluid concentration in the water. This

affects the required water mass flow per unit of ammonia circulated. When more

water must be pumped by the bubble pump, the heat input to the bubble pump is

raised to a level that can seriously degrade the COP.

Across all alternative cases, it was found that the maximum temperature lift

does not provide the highest COP. Furthermore, matching the temperature lift to the

temperature requirements also does not improve the COP. When the maximum lift

is greater than the temperature difference between the condenser/absorber and the

evaporator, the best system performance occurs at a pressure such that the partial

pressure in the condenser/absorber is the saturation pressure of the refrigerant.

It appears possible that ammonia-water-butane may be the ideal fluid triplet

for the Einstein cycle. However, ammonia-water-propane and methyl amine-water-

pentane are acceptable alternatives that do not significantly lower the COP.

With a heating COP of 1.5 from the first operating configuration, an Einstein

cycle gas heat pump water heater would cut the operating costs of a conventional gas

123

water heater by 33%. That efficiency level would result in yearly economic savings of

$70 and environmental savings of 0.15 metric tons of carbon for an average family. At

a hypothetical 100% market penetration, the national savings would be $2.6 billion

dollars and 6.4 million metric tons of carbon per year. In the commercial sector, $600

million and 1.7 million metric tons of carbon could be saved annually. The Kyoto

agreement states that the U.S. must be at or below 93% of its 1990 greenhouse

gas emission levels by 2008-20012. Currently, that means that 75 million metric

tons of carbon emissions must be eliminated. The carbon emission reductions from

widespread implementation of Einstein cycle gas heat pump water heaters would

meet nearly 11% of that goal.

While an Einstein cycle gas heat pump water heater appears to have a large

market potential, there are barriers to implementation. These might be overcome in

three ways: 1) through DOE’s minimum efficiency regulatory program, 2) through

implementation of a strategic national policy to reduce global warming emissions, and

3) through the enhanced marketing potential of a cycle named after Albert Einstein.

Recommendations

A more robust corresponding states model computer program may be able to

overcome the property optimization limitations encountered in this study. Nondi-

mensionalizing the parameters and application of nonlinear optimization solver tech-

niques may also prove beneficial. The level of potential COP improvement, however,

appears to be relatively small. Additionally, the interrelation of the various processes

in the Einstein cycle may prove to be too complicated to be accurately represented

by a corresponding states/ideal solution simplification.

When used as a gas heat pump water heater, alternate configurations for the

124

evaporator should be examined. A higher evaporator temperature raises the COP

and increases the number of potential fluid triplets. A two-stage evaporator may be

able to take advantage of both ambient-temperature heat transfer and heat transfer

from the higher-temperature flue gases.

Multistage Einstein cycles could also be investigated. The heat transferred

from the condenser/absorber of one cycle could be used to power the generator of

another cycle. This would increase the generator heat input temperature and improve

the ideal COP.

Finally, the viability of creating a mesoscopic or microscopic Einstein refrig-

eration cycle could be studied. Recently, microscopic vapor-compression cycles have

been developed for a variety of applications. Since it is a completely sealed system

with no moving parts, the Einstein cycle is a natural candidate for miniaturization.

The Einstein cycle eliminates the need for a mechanical compressor, and a compres-

sor’s machined moving parts increase the failure rate. On the microscale, however,

the continuum assumption is discarded, so the importance of boundary properties

must be explored. Capillary-driven flow will also be of importance on a microscopic

scale.

125

APPENDIX A

GENERATOR PROGRAMS

Generator Configuration 1

$INCLUDE amwatbut.txt

"Specified Points"

m_dot_1=1

T_1=315 "K"

T_2=T_1

T_3=375

P_conv=convert(bar,kPa)

P=4.5*P_conv

"Finding the Pressure of the Stream Entering

the Generator (P_1)"

T_e=289.9

x_ae=0.45

Pea|s=pressure(AMMONIA,T=T_e,x=0)

Peb|s=pressure(R600,T=T_e,x=0)

P_e=x_ae*(Pea|s-Peb|s) + Peb|s P_1=Pea|s*x_ae

"Point 1 - Entering the Generator"

P1a|s=pressure(AMMONIA,T=T_1,x=0)

P1w|s=pressure(WATER,T=T_1,x=0)

"Raoult’s law and fugacities"

x_w1=(P_1-P1a|s)/(P1w|s-P1a|s)

y_w1=(P1w|s/P_1)*x_w1

x_a1=1-x_w1 y_a1=1-y_w1

P_a1=y_a1*P_1 P_w1=y_w1*P_1

xm_w1=(x_w1*m_w)/(x_w1*m_w+x_a1*m_a)

ym_w1=(y_w1*m_w)/(y_w1*m_w+y_a1*m_a)

xm_a1=(x_a1*m_a)/(x_w1*m_w+x_a1*m_a)

126

ym_a1=(y_a1*m_a)/(y_w1*m_w+y_a1*m_a)

"Properties"

h_a1=P_1*v_f_a + u_0_a + c_v_f_a*(T_1 - T_ref_a)

h_w1=P_1*v_f_w + u_0_w + c_v_f_w*(T_1 - T_ref_w)

s_a1=s_0_a + c_p_f_a*ln(T_1/T_ref_a)

s_w1=s_0_w + c_p_f_w*ln(T_1/T_ref_w)

h_1=xm_a1*h_a1+xm_w1*h_w1

s_1=xm_a1*s_a1+xm_w1*s_w1-(xm_a1*R_a*ln(xm_a1)

+xm_w1*R_w*ln(xm_w1))

"Point 2 - Vapor Leaving"




x_w2=(P-P2a|s)/(P2w|s-P2a|s)

y_w2=(P2w|s/P)*x_w2

x_a2=1-x_w2

y_a2=1-y_w2

P_a2=y_a2*P

P_w2=y_w2*P





"Properties"

h_a2=R_a*T_2 + u_0_a + c_v_g_a*(T_2 - T_ref_a) + u_fg_a

h_w2=R_w*T_2 + u_0_w + c_v_g_w*(T_2 - T_ref_w) + u_fg_w

s_a2=s_0_a + s_fg_a + c_p_f_a*ln(T_2/T_ref_a)

- R_a*ln(P_a2/P_ref_a)

s_w2=s_0_w + s_fg_w + c_p_f_w*ln(T_2/T_ref_w)

- R_w*ln(P_w2/P_ref_w)


s_2=xm_a2*s_a2+xm_w2*s_w2

"Point3 - Solution Leaving"




127

x_w3=(P-P3a|s)/(P3w|s-P3a|s)

y_w3=(P3w|s/P)*x_w3

x_a3=1-x_w3

y_a3=1-y_w3

P_a3=y_a3*P

P_w3=y_w3*P





"Properties"

h_a3=P*v_f_a + u_0_a + c_v_f_a*(T_3 - T_ref_a)

h_w3=P*v_f_w + u_0_w + c_v_f_w*(T_3 - T_ref_w)






"Generator Control Volume"

m_dot_1=m_dot_2+m_dot_3

m_dot_1*xm_a1=m_dot_2*ym_a2+m_dot_3*xm_a3

Q_dot_g=m_dot_2*h_2+m_dot_3*h_3-m_dot_1*h_1

T_bar_g_s=Q_dot_g/(m_dot_2*s_2+m_dot_3*s_3-m_dot_1*s_1)



"Specified Points"

m_dot_1=1


P=4.5*P_conv

"Finding the Pressure of the Stream Entering the

Generator (P_1)"

T_e=289.9

x_ae=0.45

128



P_e=x_ae*(Pea|s-Peb|s) + Peb|s

P_1=Pea|s*x_ae

T_1=315 "K"

"Point 1 - Entering the Generator"





y_w1=(P1w|s/P_1)*x_w1

x_a1=1-x_w1

y_a1=1-y_w1

P_a1=y_a1*P_1

P_w1=y_w1*P_1





"Properties"









T_2=T_1

P_2=P





y_w2=(P2w|s/P_2)*x_w2

x_a2=1-x_w2

y_a2=1-y_w2

129

P_a2=y_a2*P_2

P_w2=y_w2*P_2





"Properties"









"Point3 - Solution Leaving Lower CV/Entering SHX"

T_3=375

P_3=P





y_w3=(P3w|s/P_3)*x_w3

x_a3=1-x_w3

y_a3=1-y_w3

P_a3=y_a3*P_3

P_w3=y_w3*P_3





"Properties"






130



"Point4 - Solution Leaving SHX"

T_4=T_1

P_4=P

xm_a4=xm_a3

xm_w4=xm_w3

"Properties"








"Point 5 - Solution Leaving SHX internally"

P_5=P





y_w5=(P5w|s/P_5)*x_w5

x_a5=1-x_w5

y_a5=1-y_w5

P_a5=y_a5*P_5

P_w5=y_w5*P_5





"Properties"






131



"Point 6 - Vapor Entering the SHX"

T_6=T_5

P_6=P





y_w6=(P6w|s/P_6)*x_w6

x_a6=1-x_w6

y_a6=1-y_w6

P_a6=y_a6*P_6

P_w6=y_w6*P_6





"Properties"









"Generator Control Volumes"

"Internal SHX"

m_dot_3*h_3+m_dot_1*h_1+m_dot_6*h_6=

m_dot_2*h_2+m_dot_4*h_4+m_dot_5*h_5

m_dot_4=m_dot_3

m_dot_1+m_dot_6=m_dot_2+m_dot_5

m_dot_1*xm_a1+m_dot_6*ym_a6=m_dot_2*ym_a2+m_dot_5*xm_a5

"Lower Section of the Generator"



132


T_bar_g_s=Q_dot_g/(m_dot_6*s_6+m_dot_3*s_3-m_dot_5*s_5)



"Specified Points"

m_dot_1=1

m_dot_2=m_dot_1

m_dot_4=m_dot_3

m_dot_5=m_dot_4

T_1=315 "K"

T_2=324.52

T_3=375

T_5=T_1

T_6=T_2


P=4.5*P_conv

"Finding the Pressure of the Stream Entering the Generator (P_1)"

T_e=289.9

x_ae=0.45



P_e=x_ae*(Pea|s-Peb|s) + Peb|s

P_1=Pea|s*x_ae

"Point 1 - Solution Entering External Hxgr"

CALL raoul(T_1,P_1:x_a1,x_w1,xm_a1,xm_w1,y_a1,y_w1,ym_a1,

ym_w1,P_a1,P_w1)

CALL fluprop(P_1,T_1,T_ref_a,v_f_a,c_v_f_a,c_p_f_a,u_0_a,

s_0_a:h_a1,s_a1)

CALL fluprop(P_1,T_1,T_ref_w,v_f_w,c_v_f_w,c_p_f_w,u_0_w,

s_0_w:h_w1,s_w1)

CALL fluidmix(xm_a1,xm_w1,h_a1,h_w1,s_a1,s_w1,R_a,R_w:h_1,s_1)

"Point 2 - Solution Leaving External Hxgr/Entering Generator"

CALL raoul(T_2,P:x_a2,x_w2,xm_a2,xm_w2,y_a2,y_w2,ym_a2,

ym_w2,P_a2,P_w2)

133

xm_a2=xm_a1

***NEED TO ACHIEVE THIS BY CHOICE OF T_2***

xm_w2=xm_w1

CALL fluprop(P,T_2,T_ref_a,v_f_a,c_v_f_a,c_p_f_a,u_0_a,

s_0_a:h_a2,s_a2)

CALL fluprop(P,T_2,T_ref_w,v_f_w,c_v_f_w,c_p_f_w,u_0_w,

s_0_w:h_w2,s_w2)


"Point3 - Solution Leaving Generator/Entering Internal Hxgr"


ym_w3,P_a3,P_w3)


s_0_a:h_a3,s_a3)


s_0_w:h_w3,s_w3)


"Point 4 - Solution Leaving Internal/Entering External Hxgr"

xm_a4=xm_a3

xm_w4=xm_w3


s_0_a:h_a4,s_a4)


s_0_w:h_w4,s_w4)


"Point 5 - Solution Leaving External Hxgr"

xm_a5=xm_a3

xm_w5=xm_w3


s_0_a:h_a5,s_a5)


s_0_w:h_w5,s_w5)




ym_w6,P_a6,P_w6)

CALL gasprop(P_a6,P_ref_a,T_6,T_ref_a,c_v_g_a,c_p_g_a,

u_0_a,u_fg_a,s_0_a,s_fg_a,R_a:h_a6,s_a6)

CALL gasprop(P_w6,P_ref_w,T_6,T_ref_w,c_v_g_w,c_p_g_w,

134

u_0_w,u_fg_w,s_0_w,s_fg_w,R_w:h_w6,s_w6)

CALL gasmix(xm_a6,xm_w6,h_a6,h_w6,s_a6,s_w6:h_6,s_6)

Mass, Species, and Energy Balances



0=m_dot_5*h_5-m_dot_4*h_4+m_dot_1*h_1-m_dot_2*h_2


FoM = m_dot_6*ym_a6/Q_dot_g

MassAmm=m_dot_6*ym_a6

Generator Libraries

amwatbut.txt

c p f a=4.7259088 c p f b=2.4305380 c p f w=4.6397774

c p g a=2.0942909 c p g b=1.6581138 c p g w=1.9486486

c v f a=4.2377115 c v f b=2.2874990 c v f w=4.1782731

c v g a=1.6060936 c v g b=1.5150747 c v g w=1.4871443

m a=17.030 m b=58.124 m w=18.015

P ref a=101.32 P ref b=101.32 P ref w=101.32

R a=0.4881973 R b=0.1430390 R w=0.4615043

T ref a=239.8458447 T ref b=272.4120042 T ref w=373.1676851

u 0 a=28.5827757 u 0 b=88.3527653 u 0 w=418.846656

u fg a=1292.2284431 u fg b=348.8636657 u fg w=2087.4236029

v f a=0.0014677 v f b=0.0016631 v f w=0.0010433

s 0 a=0.1212995 s 0 b=0.3498070 s 0 w=1.3063770

s fg a=5.7236373 s fg b=1.4167022 s fg w=6.0509762

raoul.lib

PROCEDURE raoul (T,P:x1,x2,xm1,xm2,y1,y2,ym1,ym2,P1,P2)

135

mw1=17.03

mw2=18.015

P1|s=pressure(AMMONIA,T=T,x=0)

P2|s=pressure(WATER,T=T,x=0)

x2=(P-P1|s)/(P2|s-P1|s)

y2=(P2|s/P)*x2

x1=1-x2

y1=1-y2

P1=y1*P

P2=y2*P

xm2=(x2*mw2)/(x2*mw2+x1*mw1)

ym2=(y2*mw2)/(y2*mw2+y1*mw1)

xm1=(x1*mw1)/(x2*mw2+x1*mw1)

ym1=(y1*mw1)/(y2*mw2+y1*mw1)

END

fluprop.lib

PROCEDURE fluprop(P,T,Tref,v,cv,cp,u0,s0:h,s)

h=P*v + u0 + cv*(T - Tref)

s=s0 + cp*ln(T/Tref)

END

gasprop.lib

PROCEDURE gasprop(P,Pref,T,Tref,cv,cp,u0,ufg,s0,sfg,R1:h,s)

h=R1*T + u0 + cv*(T - Tref) + ufg

s=s0 + sfg + cp*ln(T/Tref)- R1*ln(P/Pref)

END

fluidmix.lib

PROCEDURE fluidmix(xm1,xm2,h1,h2,s1,s2,R1,R2:h,s)

136

h=xm1*h1+xm2*h2

s=xm1*s1+xm2*s2-(xm1*R1*ln(xm1)+xm2*R2*ln(xm2))

END

gasmix.lib

PROCEDURE gasmix(xm1,xm2,h1,h2,s1,s2:h,s)

h=xm1*h1+xm2*h2

s=xm1*s1+xm2*s2

END

137

APPENDIX B

IDEAL SOLUTION MODEL PROGRAMS

Ideal Solution Model EES Program

$INCLUDE cst_sat.LIB

$INCLUDE cst_h_vap.LIB

$INCLUDE cst_h_liq.LIB

$INCLUDE fluidmix.lib

PROCEDURE fluprop2(T,Tc,Tref_r,cpl,h0:h1)

h1=h0+cpl*(T-Tref_r*Tc)

END

PROCEDURE gasprop2(T,Tc,Tref_r,P,Pc,w,M,h0,cpl:h)

$Common R

Pr=P/Pc

Tr=T/Tc

w=w

Call cst_sat(Tr,w:Prg)

CALL cst_h_vap(Tr,Pr,w:hdepvap1)

CALL cst_h_liq(Tr,Prg,w:hdepliq)

Call cst_h_vap(Tr,Prg,w:hdepvapg)

hdiff=(R/M)*(Tc)*(hdepvapg-hdepvap1)

hfg2=(R/M)*(Tc)*(hdepliq-hdepvapg)

h = h0 + cpl*(T - Tref_r*Tc) + hfg2 + hdiff

END

PROCEDURE raoul2(T,Tc1,Tc2,P,Pc1,Pc2,M1,M2,w1,w2:x1,x2,

xm1,xm2,y1,y2,ym1,ym2,P1,P2)

138

mw1=M1

mw2=M2

Tr1=T/Tc1

Tr2=T/Tc2

CALL cst_sat(Tr1,w1:Pr1|s)

CALL cst_sat(Tr2,w2:Pr2|s)

P1|s=Pr1|s*Pc1

P2|s=Pr2|s*Pc2

x2=(P-P1|s)/(P2|s-P1|s)

y2=(P2|s/P)*x2

x1=1-x2

y1=1-y2

P1=y1*P

P2=y2*P

xm2=(x2*mw2)/(x2*mw2+x1*mw1)

ym2=(y2*mw2)/(y2*mw2+y1*mw1)

xm1=(x1*mw1)/(x2*mw2+x1*mw1)

ym1=(y1*mw1)/(y2*mw2+y1*mw1)

END

Tcw=647.3

Tca=405.6

Tcb=425.25

ww=0.344

wa=0.25

wb=0.199

h0w=0

h0a=0

h0b=0

Mw=18.015

Ma=17.03

Mb=58.124

Pcw=221.2

Pca=112.8

Pcb=37.92

139

cplw=4.5

cpla=5.32

cplb=2.64

R=8.31434

water=1

ammonia=2

butane=3

TconF=125

TconF=converttemp(’K’, ’F’, Tcon)

T7=Tcon

TevapF=72

TevapF=converttemp(’K’, ’F’, Tevap)

Tgen=425

TgenF=converttemp(’K’, ’F’, Tgen)

T9=Tgen

Tcontrol=(Tcon+0.1)/Tcb

CALL cst_sat(Tcontrol,wb:P/Pcb) "Sets System Pressure"

m9=1

CALL cst_sat(Tref_rw,ww:1/Pcw)

CALL cst_sat(Tref_ra,wa:1/Pca)

CALL cst_sat(Tref_rb,wb:1/Pcb)

CALL raoul2(T7,Tca,Tcw,Pa3,Pca,Pcw,Ma,Mw,wa,ww:xa7,xw7,

xma7,xmw7,ya7,yw7,yma7,ymw7,Pa7,Pw7)

CALL fluprop2(T7,Tca,Tref_ra,cpla,h0a:ha7)

CALL fluprop2(T7,Tcw,Tref_rw,cplw,h0w:hw7)

CALL fluidmix(xma7,xmw7,ha7,hw7:h7)

xma8=xma7

xmw8=xmw7

T8=T10-3

m8=m7




140

Qext1=m7*(h8-h7)

CALL raoul2(T9,Tca,Tcw,P,Pca,Pcw,Ma,Mw,wa,ww:xa9,xw9,





T5=T8



CALL gasprop2(T5,Tca,Tref_ra,Pa5,Pca,wa,Ma,h0a,cpla:ha5)

CALL gasprop2(T5,Tcw,Tref_rw,Pw5,Pcw,ww,Mw,h0w,cplw:hw5)

CALL fluidmix(yma5,ymw5,ha5,hw5:h5)

T11=Tcon

m11=m9






Qext2=m9*(h10-h11)

Qext1=Qext2

xma10=xma9

xmw10=xmw9

m10=m9




Qint=m9*(h9-h10)

Qg = m10*h10 + m5*h5 -m8*h8

m8*xma8 = m5*yma5 + m9*xma9

m8=m9+m5

CALL raoul2(Tcon,Tca,Tcb,P,Pca,Pcb,Ma,Mb,wa,wb:xaca,xbca,

xmaca,xmbca,yaca,ybca,ymaca,ymbca,Paca,Pbca)

xmb1=ymbca

CALL raoul2(T1ca,Tca,Tcb,P,Pca,Pcb,Ma,Mb,wa,wb:xa1,xb1,

141

xma1,xmb1,ya1,yb1,yma1,ymb1,Pa1,Pb1)

T1=Tcon


CALL fluprop2(T1,Tcb,Tref_rb,cplb,h0b:hb1)

CALL fluidmix(xma1,xmb1,ha1,hb1:h1)

T2p=Tevap

T3p=T2p

T4p=T2p

CALL raoul2(T2p,Tca,Tcb,P,Pca,Pcb,Ma,Mb,wa,wb:xa2p,xb2p,

xma2p,xmb2p,ya2p,yb2p,yma2p,ymb2p,Pa2p,Pb2p)

CALL fluprop2(T2p,Tca,Tref_ra,cpla,h0a:ha2p)

CALL fluprop2(T2p,Tcb,Tref_rb,cplb,h0b:hb2p)

CALL fluidmix(xma2p,xmb2p,ha2p,hb2p:h2p)

CALL gasprop2(T4p,Tca,Tref_ra,P,Pca,wa,Ma,h0a,cpla:ha4p)

h4p=ha4p

CALL raoul2(T3p,Tca,Tcb,P,Pca,Pcb,Ma,Mb,wa,wb:xa3p,xb3p,

xma3p,xmb3p,ya3p,yb3p,yma3p,ymb3p,Pa3p,Pb3p)

CALL gasprop2(T3p,Tca,Tref_ra,Pa3p,Pca,wa,Ma,h0a,cpla:ha3p)

CALL gasprop2(T3p,Tcb,Tref_rb,Pb3p,Pcb,wb,Mb,h0b,cplb:hb3p)

CALL fluidmix(yma3p,ymb3p,ha3p,hb3p:h3p)

xma2=xma1

xmb2=xmb1

m2=m1


CALL fluprop2(T2,Tcb,Tref_rb,cplb,h0b:hb2)

CALL fluidmix(xma2,xmb2,ha2,hb2:h2)

T4=T2



CALL fluprop2(T4,Tca,Tref_ra,cpla,h0a:ha4l)

CALL fluprop2(T4,Tcw,Tref_rw,cplw,h0w:hw4l)

CALL fluidmix(xma4,xmw4,ha4l,hw4l:h4l)

CALL gasprop2(T4,Tca,Tref_ra,Pa4,Pca,wa,Ma,h0a,cpla:ha4v)

CALL gasprop2(T4,Tcw,Tref_rw,Pw4,Pcw,ww,Mw,h0w,cplw:hw4v)

CALL fluidmix(yma4,ymw4,ha4v,hw4v:h4v)

142

VP=vapor percentage

yma5 = VP*yma4 + (1-VP)*xma4

h4=VP*h4v + (1-VP)*h4l

m4=VP*m5

m4v=m4

m4l=m5-m4v

CALL raoul2(T3,Tca,Tcb,P,Pca,Pcb,Ma,Mb,wa,wb:xa3,xb3,

xma3,xmb3,ya3,yb3,yma3,ymb3,Pa3,Pb3)


CALL gasprop2(T3,Tcb,Tref_rb,Pb3,Pcb,wb,Mb,h0b,cplb:hb3)

CALL fluidmix(yma3,ymb3,ha3,hb3:h3)

m2+m3p=m2p+m3

m2*xmb2+m3p*ymb3p=m2p*xmb2p+m3*ymb3

m3p=m2p+m4p

m3p*yma3p=m2p*xma2p+m4p

m3*h3 - m2*h2 - m4*h4v = m3p*h3p - m4p*h4p - m2p*h2p

Qe = m3*h3 - m2*h2 - m4*h4v

m6=m3

yma6=yma3

ymb6=ymb3

T6=T5


CALL gasprop2(T6,Tcb,Tref_rb,Pb3,Pcb,wb,Mb,h0b,cplb:hb6)

CALL fluidmix(yma6,ymb6,ha6,hb6:h6)

Qpc1=m2*h2 + m4v*h4v + m6*h6 + m4l*h4l

Qpc2=m1*h1 + m3*h3 + m5*h5

Qpc1=Qpc2

m7+m1=m11+m6+m4l

Qca=m7*h7+m1*h1-m4l*h4l-m6*h6-m11*h11

Qcheck=Qca+Qg+Qe

COP_h = -Qca/(Qg)

COP_h_ideal = ((Tgen-Tevap)/(Tgen))/((Tcon-Tevap)/Tcon)

143

Ideal Solution Model Libraries

cst sat.lib

PROCEDURE cst_sat(Tr,w:Pr)

Pr := exp(5.92714 - (6.09648/Tr) - 1.28862*ln(Tr)

+ 0.169347*(Tr^6) + w*(15.2518 - (15.6875/Tr)

- 13.4721*ln(Tr) + 0.43577*(Tr^6)))

END

cst h liq.lib

PROCEDURE cst_h_liq(Tr,Pr,w:hdepliq)

col_num := Pr*100

row_num := 100*(Tr-0.29)

IF (col_num<1) THEN

h_0_liq := lookup(’hzeroliq’,row_num,1) -

- ((Pr-0.01)/0.01)*(lookup(’hzeroliq’,row_num,1)

- lookup(’hzeroliq’,row_num,2))

h_1_liq := lookup(’honeliq’,row_num,1)

- ((Pr-0.01)/0.01)*(lookup(’honeliq’,row_num,1)

- lookup(’honeliq’,row_num,2))

hdepliq := h_0_liq + w*h_1_liq

ELSE

h_0_liq := lookup(’hzeroliq’,row_num,col_num)

h_1_liq := lookup(’honeliq’,row_num,col_num)

hdepliq := h_0_liq + w*h_1_liq

ENDIF

END

cst h vap.lib

PROCEDURE cst_h_vap(Tr,Pr,w:hdepvap)

col_num := Pr*100

row_num := 100*(Tr-0.29)

IF (col_num<1) THEN

h_0_vap := (Pr/0.01)*(lookup(’hzerovap’,row_num,1))

h_1_vap := (Pr/0.01)*(lookup(’honevap’,row_num,1))

hdepvap := h_0_vap + w*h_1_vap

ELSE

144

h_0_vap := lookup(’hzerovap’,row_num,col_num)

h_1_vap := lookup(’honevap’,row_num,col_num)

hdepvap := h_0_vap + w*h_1_vap

ENDIF

END

fluidmix.lib

PROCEDURE fluidmix(xm1,xm2,h1,h2:h)

h=xm1*h1+xm2*h2

END

145

APPENDIX C

PATEL-TEJA MODEL PROGRAMS

EES Program

$include Pteosambtl.LIB

$include Pteosambtv.LIB

$include Pteosawl.lib

$include Pteosawv.lib

$include ptabvlehsv.LIB

$include ptawvlehsv.LIB

$include ptamwtsc.LIB

$include ptamhsv.LIB

P=5.25

State 1: Ammonia-Butane saturated liquid leaving condenser

This sets T1

xa1=0.001

CALL ptabvlehsv(T1,P,xa1,ya1:xam1,yam1,xbm1,ybm1,hv1,hl1,sv1,sl1,

vl1,vv1,fal1,fav1,fbl1,fbv1)

fal1=fav1

fbl1=fbv1

States 3: Ammonia-Butane sat. liquid/vapor in evaporator

T3=295



fal3=fav3

fbl3=fbv3

State 5: Superheated Ammonia entering PC

T5=T1

CALL ptamhsv(T5,P:hl5,hv5,sl5,sv5,vl5,vv5)

146

State 2: Butane subcooled liquid leaving pre-cooler

T2=T3+pinch23

pinch23=0

xa2=xa1

ya2=ya1



State 6: Super-Heated Ammonia-Butane leaving PC

T6=T5-pinch65

pinch65=0

ya6=ya3

xa6=xa3



State 7: Ammonia-Water sat. liquid leaving condenser

T7=T1

Pa7=P*ya3

CALL ptawvlehsv(T7,Pa7,xa7,ya7:xam7,yam7,xwm7,ywm7,hv7,hl7,sv7,sl7,

vl7,vv7,fal7,fav7,fwl7,fwv7)

fal7=fav7

fwl7=fwv7

State 9: Ammonia-Water sat. liquid Generator heat input

T9=424

CALL ptawvlehsv(T9,P,xa9,ya9:xam9,yam9,xwm9,ywm9,hv9,hl9,sv9,sl9,

vl9,vv9,fal9,fav9,fwl9,fwv9)

fal9=fav9

fwl9=fwv9

State 11: Ammonia-Water sub-cooled liquid leaving

Generator Ext Hxgr

T11=T7+pinch97

pinch97=0

xa11=xa9

ya11=ya9

xam11=xam9

CALL ptamwtsc(T11,P,xa11:h11,s11)

Bubble Pump

mbp=1

147

m9v=mbp/28.6

m9=mbp-m9v

m11=m9

Evaporator: conservation of mass and energy

m2=m1

m6=m3

m2+m4=m3

ammonia

yam4=1

m5=m4

xam2*m2+m4*yam4=m3*yam3

Condenser/Absorber overall

m1+m7=m11+m6+m9v

m1*xam1+m7*xam7=m11*xam11+m6*yam6+m9v*yam9

Pre-cooler

m3*hv3+m1*hl1+m5*hv5=m6*hv6+m2*hl2+m4*hv4

Qc=m1*hl1+m7*hl7-m11*h11-m6*hv6-m9v*hv9

Tc=T1

Qe=m3*hv3-hl2*m2-hv4*m4

Tg=T9

Te=T3

merr1=m4+m11+m9v-m7 "Error Checking"

merr2=m4*yam4+xam11*m11+yam9*m9v-m7*xam7

Intermediate State 5a

y5a=1

m5a=m5

xa8=xa7

m8=m7

m10=m11

xa10=xa9

T5a=348.8 "Determined from an external program"

T8=T5a

CALL ptamhsv(T5a,P:hl5a,hv5a,sl5a,sv5a,vl5a,vv5a)


Qdump1=m5*(hv5a-hv5)

148

Qget1=m8*(h8 - hl7)

Qdump2=m10*(h10-h11)

Qdump1+Qdump2=Qget1

Qg=m10*h10+m9v*hv9+m5a*hv5a-m8*h8


Q_bp=m9v*(hv9-hl9)

Qgact=Qg-Q_bp

Qerr=Qg+Qe+Qc "Error checking"

COP=-Qc/(Qg)

COPi=((Tg-Te)/(Tg))/((Tc-Te)/Tc)

Program Libraries

pteosambtl.lib

PROCEDURE pteosambtl(T,P,xa,k_ab,k_ba:f_ial,f_ibl,v_ml,xb,xa_mass,

xb_mass,h_depl,s_depl)

k_aa=0

k_bb=0

R = 8.314/100000 m^3*bar/mol K

ammonia

omega_a=.25

Tc_a=405.6

Pc_a=112.8

Tr_a=T/Tc_a

F_a=.627090

Zeta_a=.282

Omega_2a=.06934

Omega_1a=3*Zeta_a^2+3*(1-2*Zeta_a)*Omega_2a+Omega_2a^2+1-3*Zeta_a

Omega_3a=1-3*Zeta_a

a_Tc_a=(R^2*Tc_a^2/Pc_a)

b_Tc_a=(R*Tc_a/Pc_a)

alpha_a=(1+F_a*(1-Tr_a^.5))^2

a_a=a_Tc_a*alpha_a*Omega_1a

149

b_a=Omega_2a*b_Tc_a

c_a=Omega_3a*b_Tc_a

Aa=(a_a*P)/(R^2*T^2)

Ba=b_a*P/(R*T)

Ca=c_a*P/(R*T)

Na=(b_a*c_a+((b_a+c_a)/2)^2)^.5

Ma=((b_a+c_a)/2-Na)*P/(R*T)

Qa=((b_a+c_a)/2+Na)*P/(R*T)

butane

omega_b=.199

Tc_b=425.2

Pc_b=38

Tr_b=T/Tc_b

F_b=.678389

Zeta_b=.309

Omega_2b=0.07834

Omega_1b=3*Zeta_b^2+3*(1-2*Zeta_b)*Omega_2b+Omega_2b^2+1-3*Zeta_b

Omega_3b=1-3*Zeta_b

a_Tc_b=(R^2*Tc_b^2/Pc_b)

b_Tc_b=(R*Tc_b/Pc_b)

alpha_b=(1+F_b*(1-Tr_b^.5))^2

a_b=a_Tc_b*alpha_b*Omega_1b

b_b=b_Tc_b*Omega_2b

c_b=b_Tc_b*Omega_3b

Ab=(a_b*P)/(R^2*T^2)

Bb=b_b*P/(R*T)

Cb=c_b*P/(R*T)

Nb=(b_b*c_b+((b_b+c_b)/2)^2)^.5

Mb=((b_b+c_b)/2-Nb)*P/(R*T)

Qb=((b_b+c_b)/2+Nb)*P/(R*T)

xb=1-xa

a_aa=a_a

a_ab=(a_a*a_b)^.5*(1-k_ab+(k_ab-k_ba)*xa)

a_bb=a_b

a_ba=(a_b*a_a)^.5*(1-k_ba+(k_ba-k_ab)*xb)

a_mx=xa*xa*a_aa+xb*xb*a_bb+xa*xb*a_ab+xb*xa*a_ba

b_mx=xa*b_a+xb*b_b

c_mx=xa*c_a+xb*c_b

A_m=(a_mx*P)/(R^2*T^2)

B_m=b_mx*P/(R*T)

150

C_m=c_mx*P/(R*T)

N_m=(b_mx*c_mx+((b_mx+c_mx)/2)^2)^.5

M_m=((b_mx+c_mx)/2-N_m)*P/(R*T)

Q_m=((b_mx+c_mx)/2+N_m)*P/(R*T)

ap_1=-2*(xa*xa*a_a+xa*xb*(a_a*a_b)^.5*(1-k_ab)+xb*xb*a_b

+xb*xa*(a_b*a_a)^.5*(1-k_ba))

ap_2=-3*(xa^2*xb*(a_a*a_b)^.5*(k_ab-k_ba)

+xb^2*xa*(a_b*a_a)^.5*(k_ba-k_ab))

ap_3a=2*xa*xb*(a_a*a_b)^.5*(k_ab-k_ba)

ap_3b=2*xb*xa*(a_b*a_a)^.5*(k_ba-k_ab)

ap_4a=2*xa*a_a+xb*(a_a*a_b)^.5*(1-k_ab+1-k_ba+(k_ba-k_ab)*xb)

ap_4b=2*xb*a_b+xa*(a_b*a_a)^.5*(1-k_ba+1-k_ab+(k_ab-k_ba)*xa)

ap_a=ap_1+ap_2+ap_3a+ap_4a

ap_b=ap_1+ap_2+ap_3b+ap_4b

bp_a=b_a-b_mx

bp_b=b_b-b_mx

cp_a=c_a-c_mx

cp_b=c_b-c_mx

c_1=1

c_2=(C_m-1)

c_3=(-2*B_m*C_m-B_m^2-B_m-C_m+A_m)

c_4=(B_m^2*C_m+B_m*C_m-A_m*B_m)

pp=c_2/c_1

qq=c_3/c_1

rr=c_4/c_1

A_cubic=1/3*(3*qq-pp^2)

B_cubic=1/27*(2*pp^3-9*pp*qq+27*rr)

D_cubic=(A_cubic)^3/27+(B_cubic)^2/4

IF D_cubic<0 THEN

phi=arccos(((B_cubic^2/4)/(-A_cubic^3/27))^.5)

yr1=2*(-A_cubic/3)^.5*cos(phi/3)

yr3=2*(-A_cubic/3)^.5*cos(phi/3+4.18879020479)

yr2=2*(-A_cubic/3)^.5*cos(phi/3+2.09439510239)

Z_mv=yr1-pp/3

Z_ml=yr2-pp/3

Zr3=yr3-pp/3

ENDif

IF D_cubic>0 THEN

M_cubic=(-B_cubic/2+(D_cubic)^.5)^(1/3)

N_cubic=(-B_cubic/2-(D_cubic)^.5)^(1/3)

151

Z_mv=M_cubic+N_cubic-pp/3

Z_ml=-(M_cubic+N_cubic)/2-pp/3

endif

v_ml=Z_ml*R*T/P

Q=-(b_mx^2+6*b_mx*c_mx+c_mx^2)

D_l=v_ml^2+v_ml*(b_mx+c_mx)-b_mx*c_mx

D1_l=(1/(-Q)^.5)*ln((2*v_ml+(b_mx+c_mx)-(-Q)^.5)/(2*v_ml

+(b_mx+c_mx)+(-Q)^.5))

F_1al=v_ml/(Q*D_l)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_a

+3*bp_a*c_mx+3*b_mx*cp_a+c_mx*cp_a)

+1/(Q*D_l)*(bp_a*c_mx^2+b_mx^2*cp_a

-b_mx*bp_a*c_mx-b_mx*c_mx*cp_a)+D1_l/Q*(b_mx*bp_a

+3*b_mx*cp_a-b_mx^2+3*bp_a*c_mx+c_mx*cp_a-c_mx^2

-6*b_mx*c_mx)

phi_ial=(ln(v_ml/(v_ml-b_mx))+(b_mx+bp_a)/(v_ml-b_mx)

+(ap_a*D1_l)/(R*T)+a_mx/(R*T)*(F_1al)-ln(Z_ml))

f_ial1=exp(ln(v_ml/(v_ml-b_mx))+(b_mx+bp_a)/(v_ml-b_mx)


f_ial2=xa*P

f_ial=f_ial1*f_ial2

F_1bl=v_ml/(Q*D_l)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_b

+3*bp_b*c_mx+3*b_mx*cp_b+c_mx*cp_b)

+1/(Q*D_l)*(bp_b*c_mx^2+b_mx^2*cp_b

-b_mx*bp_b*c_mx-b_mx*c_mx*cp_b)+D1_l/Q*(b_mx*bp_b

+3*b_mx*cp_b-b_mx^2+3*bp_b*c_mx+c_mx*cp_b-c_mx^2

-6*b_mx*c_mx)

phi_ibl=(ln(v_ml/(v_ml-b_mx))+(b_mx+bp_b)/(v_ml-b_mx)

+(ap_b*D1_l)/(R*T)+a_mx/(R*T)*(F_1bl)-ln(Z_ml))

f_ibl1=exp(ln(v_ml/(v_ml-b_mx))+(b_mx+bp_b)/(v_ml-b_mx)

+(ap_b*D1_l)/(R*T)+a_mx/(R*T)*(F_1bl)-ln(Z_ml))

f_ibl2=xb*P

f_ibl=f_ibl1*f_ibl2

mwa=17.03 ammonia kg/kmol

mwb=58.124 butane kg/kmol

mwml=mwa*xa+mwb*xb

xa_mass=xa*mwa/mwml

xb_mass=1-xa_mass

aTcab=((a_Tc_a*Omega_1a)*(a_Tc_b*Omega_1b))^.5

152

daabdT=-(aTcab*F_a*(1+F_b*(1-Tr_b^.5)))/(2*(Tr_a)^.5*Tc_a)

-(aTcab*F_b*(1+F_a*(1-Tr_a^.5)))/(2*(Tr_b)^.5*Tc_b)

daadT=(Omega_1a*R^2*Tc_a/Pc_a)*(-F_a*alpha_a^.5*Tr_a^(-.5))

dabdT=(Omega_1b*R^2*Tc_b/Pc_b)*(-F_b*alpha_b^.5*Tr_b^(-.5))

daasrdT=-(Omega_1a*R^2/Pc_a)^.5*F_a/(2*(Tr_a)^.5)

dabsrdT=-(Omega_1b*R^2/Pc_b)^.5*F_b/(2*(Tr_b)^.5)

damxdT2=xa*xa*daadT*(1-k_aa)+xa*xb*(daabdT)*(1-k_ab

+(k_ab-k_ba)*xa)+xb*xb*dabdT*(1-k_bb)

+xb*xa*(daabdT)*(1-k_ba+(k_ba-k_ab)*xb)

damxdT=xa*xa*daadT*(1-k_aa)+(daasrdT*a_b^.5

+dabsrdT*a_a^.5)*xa*xb*(1-k_ab+(k_ab-k_ba)*xa)

+xb*xb*dabdT*(1-k_bb)+(daasrdT*a_b^.5

+dabsrdT*a_a^.5)*xb*xa*(1-k_ba+(k_ba-k_ab)*xb)

h_depl=100000*R*T*(Z_ml-1)-100000*(T*damxdT-a_mx)

*(1/(2*N_m))*ln((Z_ml+M_m)/(Z_ml+Q_m))

s_depl=100000*(-R*ln(P/(Z_ml-B_m))-damxdT*((1/(2*N_m))

*ln((Z_ml+M_m)/(Z_ml+Q_m))))

END

pteosambtv.lib

PROCEDURE pteosambtv(T,P,ya,k_ab,k_ba:f_iav,f_ibv,v_mv,yb,

ya_mass,yb_mass,h_depv,s_depv)

k_aa=0

k_bb=0

R = 8.314/100000 m^3*bar/mol K

ammonia

omega_a=.25

Tc_a=405.6

Pc_a=112.8

Tr_a=T/Tc_a

F_a=.627090

Zeta_a=.282

Omega_2a=.06934


Omega_3a=1-3*Zeta_a



153

alpha_a=(1+F_a*(1-Tr_a^.5))^2


b_a=Omega_2a*b_Tc_a

c_a=Omega_3a*b_Tc_a

Aa=(a_a*P)/(R^2*T^2)

Ba=b_a*P/(R*T)

Ca=c_a*P/(R*T)

Na=(b_a*c_a+((b_a+c_a)/2)^2)^.5



butane

omega_b=.199

Tc_b=425.2

Pc_b=38

Tr_b=T/Tc_b

F_b=.678389

Zeta_b=.309

Omega_2b=0.07834

Omega_1b=3*Zeta_b^2+3*(1-2*Zeta_b)*Omega_2b+Omega_2b^2+1-3*Zeta_b

Omega_3b=1-3*Zeta_b

a_Tc_b=(R^2*Tc_b^2/Pc_b)

b_Tc_b=(R*Tc_b/Pc_b)

alpha_b=(1+F_b*(1-Tr_b^.5))^2

a_b=a_Tc_b*alpha_b*Omega_1b

b_b=b_Tc_b*Omega_2b

c_b=b_Tc_b*Omega_3b

Ab=(a_b*P)/(R^2*T^2)

Bb=b_b*P/(R*T)

Cb=c_b*P/(R*T)

Nb=(b_b*c_b+((b_b+c_b)/2)^2)^.5

Mb=((b_b+c_b)/2-Nb)*P/(R*T)

Qb=((b_b+c_b)/2+Nb)*P/(R*T)

yb=1-ya

a_aa=a_a*(1-k_aa)

a_ab=(a_a*a_b)^.5*(1-k_ab+(k_ab-k_ba)*ya)

a_bb=a_b*(1-k_bb)

a_ba=(a_b*a_a)^.5*(1-k_ba+(k_ba-k_ab)*yb)

a_mx=ya*ya*a_aa+yb*yb*a_bb+ya*yb*a_ab+yb*ya*a_ba

b_mx=ya*b_a+yb*b_b

c_mx=ya*c_a+yb*c_b

154

A_m=(a_mx*P)/(R^2*T^2)

B_m=b_mx*P/(R*T)

C_m=c_mx*P/(R*T)

N_m=(b_mx*c_mx+((b_mx+c_mx)/2)^2)^.5



ap_1=-2*(ya*ya*a_a*(1-k_aa)+ya*yb*(a_a*a_b)^.5*(1-k_ab)

+yb*yb*a_b*(1-k_bb)+yb*ya*(a_b*a_a)^.5*(1-k_ba))

ap_2=-3*(ya^2*yb*(a_a*a_b)^.5*(k_ab-k_ba)

+yb^2*ya*(a_b*a_a)^.5*(k_ba-k_ab))

ap_3a=2*ya*yb*(a_a*a_b)^.5*(k_ab-k_ba)

ap_3b=2*yb*ya*(a_b*a_a)^.5*(k_ba-k_ab)

ap_4a=2*ya*a_a*(1-k_aa)+yb*(a_a*a_b)^.5*(1-k_ab+1

-k_ba+(k_ba-k_ab)*yb)

ap_4b=2*yb*a_b*(1-k_bb)+ya*(a_b*a_a)^.5*(1-k_ba+1

-k_ab+(k_ab-k_ba)*ya)


ap_b=ap_1+ap_2+ap_3b+ap_4b

bp_a=b_a-b_mx

bp_b=b_b-b_mx

cp_a=c_a-c_mx

cp_b=c_b-c_mx

c_1=1

c_2=(C_m-1)

c_3=(-2*B_m*C_m-B_m^2-B_m-C_m+A_m)


pp=c_2/c_1

qq=c_3/c_1

rr=c_4/c_1




IF D_cubic<0 THEN

phi=arccos(((B_cubic^2/4)/(-A_cubic^3/27))^.5)

yr1=2*(-A_cubic/3)^.5*cos(phi/3)

yr3=2*(-A_cubic/3)^.5*cos(phi/3+4.18879020479)

yr2=2*(-A_cubic/3)^.5*cos(phi/3+2.09439510239)

Z_mv=yr1-pp/3

Z_ml=yr2-pp/3

Zr3=yr3-pp/3

155

ENDIF

IF D_cubic>0 THEN

M_cubic=(-B_cubic/2+(D_cubic)^.5)^(1/3)

N_cubic=(-B_cubic/2-(D_cubic)^.5)^(1/3)

Z_mv=M_cubic+N_cubic-pp/3

Z_ml=-(M_cubic+N_cubic)/2-pp/3

ENDIF

v_ml=Z_ml*R*T/P

v_mv=Z_mv*R*T/P


D_v=v_mv^2+v_mv*(b_mx+c_mx)-b_mx*c_mx

D1_v=(1/(-Q)^.5)*ln((2*v_mv+(b_mx+c_mx)-(-Q)^.5)/(2*v_mv

+(b_mx+c_mx)+(-Q)^.5))

F_1av=v_mv/(Q*D_v)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_a


+1/(Q*D_v)*(bp_a*c_mx^2+b_mx^2*cp_a-b_mx*bp_a*c_mx

-b_mx*c_mx*cp_a)+D1_v/Q*(b_mx*bp_a+3*b_mx*cp_a-b_mx^2

+3*bp_a*c_mx+c_mx*cp_a-c_mx^2-6*b_mx*c_mx)

f_iav=ya*P*exp(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_a)/(v_mv-b_mx)

+(ap_a*D1_v)/(R*T)+a_mx/(R*T)*(F_1av)-ln(Z_mv))

f_iav1=exp(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_a)/(v_mv-b_mx)


f_iav2=ya*P

phi_iav=(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_a)/(v_mv-b_mx)


F_1bv=v_mv/(Q*D_v)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_b

+3*bp_b*c_mx+3*b_mx*cp_b+c_mx*cp_b)

+1/(Q*D_v)*(bp_b*c_mx^2+b_mx^2*cp_b-b_mx*bp_b*c_mx

-b_mx*c_mx*cp_b)+D1_v/Q*(b_mx*bp_b+3*b_mx*cp_b-b_mx^2

+3*bp_b*c_mx+c_mx*cp_b-c_mx^2-6*b_mx*c_mx)

f_ibv=yb*P*exp(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_b)/(v_mv-b_mx)

+(ap_b*D1_v)/(R*T)+a_mx/(R*T)*(F_1bv)-ln(Z_mv))

f_ibv1=exp(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_b)/(v_mv-b_mx)


f_ibv2=yb*P

phi_ibv=(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_b)/(v_mv-b_mx)




mwmv=mwa*ya+mwb*yb

156

ya_mass=ya*mwa/mwmv

yb_mass=1-ya_mass

aTcab=((a_Tc_a*Omega_1a)*(a_Tc_b*Omega_1b))^.5

daabdT=-(aTcab*F_a*(1+F_b*(1-Tr_b^.5)))/(2*(Tr_a)^.5*Tc_a)

-(aTcab*F_b*(1+F_a*(1-Tr_a^.5)))/(2*(Tr_b)^.5*Tc_b)


dabdT=(Omega_1b*R^2*Tc_b/Pc_b)*(-F_b*alpha_b^.5*Tr_b^(-.5))


dabsrdT=-(Omega_1b*R^2/Pc_b)^.5*F_b/(2*(Tr_b)^.5)

damxdT2=ya*ya*daadT*(1-k_aa)+ya*yb*(daabdT)*(1-k_ab+(k_ab

-k_ba)*ya)+yb*yb*dabdT*(1-k_bb)+yb*ya*(daabdT)*(1

-k_ba+(k_ba-k_ab)*yb)

damxdT=ya*ya*daadT*(1-k_aa)+(daasrdT*a_b^.5+dabsrdT*a_a^.5)*ya*yb*

(1-k_ab+(k_ab-k_ba)*ya)+yb*yb*dabdT*(1-k_bb)

+(daasrdT*a_b^.5+dabsrdT*a_a^.5)*yb*ya*(1-k_ba+(k_ba-k_ab)*yb)

mixture enthalpy and entropy departures:

h (kJ/kmol) and s (kJ/kmol*K)

h_depv=100000*R*T*(Z_mv-1)-100000*(T*damxdT-a_mx)*(1/(2*N_m))

*ln((Z_mv+M_m)/(Z_mv+Q_m))

s_depv=100000*(-R*ln(P/(Z_mv-B_m))-damxdT*((1/(2*N_m))

*ln((Z_mv+M_m)/(Z_mv+Q_m))))

END

pteosawl.lib

PROCEDURE pteosawl(T,P,xa:f_ial,f_iwl,v_ml,xw,xa_mass,xw_mass,

h_depl,s_depl)

k_aa=0

k_ww=0

k_aw=-0.26328

k_wa=-0.29368

R = 8.314/100000 m^3*bar/mol K

ammonia

omega_a=.25

Tc_a=405.6

Pc_a=112.8

Tr_a=T/Tc_a

F_a=.627090

157

Zeta_a=.282

Omega_2a=.06934


Omega_3a=1-3*Zeta_a



alpha_a=(1+F_a*(1-Tr_a^.5))^2


b_a=Omega_2a*b_Tc_a

c_a=Omega_3a*b_Tc_a

Aa=(a_a*P)/(R^2*T^2)

Ba=b_a*P/(R*T)

Ca=c_a*P/(R*T)

Na=(b_a*c_a+((b_a+c_a)/2)^2)^.5



water

omega_w=.344

Tc_w=647.3

Pc_w=221.2

Tr_w=T/Tc_w

F_w=.689803

Zeta_w=.269

Omega_2w=0.065103

Omega_1w=3*Zeta_w^2+3*(1-2*Zeta_w)*Omega_2w+Omega_2w^2+1-3*Zeta_w

Omega_3w=1-3*Zeta_w

a_Tc_w=(R^2*Tc_w^2/Pc_w)

b_Tc_w=(R*Tc_w/Pc_w)

alpha_w=(1+F_w*(1-Tr_w^.5))^2

a_w=a_Tc_w*alpha_w*Omega_1w

b_w=b_Tc_w*Omega_2w

c_w=b_Tc_w*Omega_3w

Aw=(a_w*P)/(R^2*T^2)

Bw=b_w*P/(R*T)

Cw=c_w*P/(R*T)

Nw=(b_w*c_w+((b_w+c_w)/2)^2)^.5

Mw=((b_w+c_w)/2-Nw)*P/(R*T)

Qw=((b_w+c_w)/2+Nw)*P/(R*T)

xw=1-xa

a_aa=a_a*(1-k_aa)

158

a_aw=(a_a*a_w)^.5*(1-k_aw+(k_aw-k_wa)*xa)

a_ww=a_w*(1-k_ww)

a_wa=(a_w*a_a)^.5*(1-k_wa+(k_wa-k_aw)*xw)

a_mx=xa*xa*a_aa+xw*xw*a_ww+xa*xw*a_aw+xw*xa*a_wa

b_mx=xa*b_a+xw*b_w

c_mx=xa*c_a+xw*c_w

A_m=(a_mx*P)/(R^2*T^2)

B_m=b_mx*P/(R*T)

C_m=c_mx*P/(R*T)

N_m=(b_mx*c_mx+((b_mx+c_mx)/2)^2)^.5



ap_1=-2*(xa*xa*a_a*(1-k_aa)+xa*xw*(a_a*a_w)^.5*(1-k_aw)

+xw*xw*a_w*(1-k_ww)+xw*xa*(a_w*a_a)^.5*(1-k_wa))

ap_2=-3*(xa^2*xw*(a_a*a_w)^.5*(k_aw-k_wa)

+xw^2*xa*(a_w*a_a)^.5*(k_wa-k_aw))

ap_3a=2*xa*xw*(a_a*a_w)^.5*(k_aw-k_wa)

ap_3w=2*xw*xa*(a_w*a_a)^.5*(k_wa-k_aw)

ap_4a=2*xa*a_a*(1-k_aa)+xw*(a_a*a_w)^.5*(1-k_aw+1-k_wa

+(k_wa-k_aw)*xw)

ap_4w=2*xw*a_w*(1-k_ww)+xa*(a_w*a_a)^.5*(1-k_wa+1-k_aw

+(k_aw-k_wa)*xa)


ap_w=ap_1+ap_2+ap_3w+ap_4w

bp_a=b_a-b_mx

bp_w=b_w-b_mx

cp_a=c_a-c_mx

cp_w=c_w-c_mx

c_1=1

c_2=(C_m-1)

c_3=(-2*B_m*C_m-B_m^2-B_m-C_m+A_m)


pp=c_2/c_1

qq=c_3/c_1

rr=c_4/c_1




Q_r=(c_2^2-3*c_3)/9

R_r=(2*c_2^3-9*c_2*c_3+27*c_4)/54

159

theta_r=arccos(R_r*Q_r^(-1.5))

Z_ml=-2*(Q_r^.5)*(cos(theta_r/3))-c_2/3

v_ml=Z_ml*R*T/P


D_l=v_ml^2+v_ml*(b_mx+c_mx)-b_mx*c_mx

D1_l=(1/(-Q)^.5)*ln((2*v_ml+(b_mx+c_mx)-(-Q)^.5)/(2*v_ml

+(b_mx+c_mx)+(-Q)^.5))

F_1al=v_ml/(Q*D_l)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_a


+1/(Q*D_l)*(bp_a*c_mx^2+b_mx^2*cp_a-b_mx*bp_a*c_mx

-b_mx*c_mx*cp_a)+D1_l/Q*(b_mx*bp_a

+3*b_mx*cp_a-b_mx^2+3*bp_a*c_mx+c_mx*cp_a

-c_mx^2-6*b_mx*c_mx)

f_ial=xa*P*exp(ln(v_ml/(v_ml-b_mx))+(b_mx+bp_a)/(v_ml-b_mx)


F_1wl=v_ml/(Q*D_l)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_w

+3*bp_w*c_mx+3*b_mx*cp_w+c_mx*cp_w)

+1/(Q*D_l)*(bp_w*c_mx^2+b_mx^2*cp_w-b_mx*bp_w*c_mx

-b_mx*c_mx*cp_w)+D1_l/Q*(b_mx*bp_w+3*b_mx*cp_w-b_mx^2

+3*bp_w*c_mx+c_mx*cp_w-c_mx^2-6*b_mx*c_mx)

f_iwl=xw*P*exp(ln(v_ml/(v_ml-b_mx))+(b_mx+bp_w)/(v_ml-b_mx)

+(ap_w*D1_l)/(R*T)+a_mx/(R*T)*(F_1wl)-ln(Z_ml))


mww=18.015 water kg/kmol

mwml=mwa*xa+mww*xw

xa_mass=xa*mwa/mwml

xw_mass=1-xa_mass

aTcaw=((a_Tc_a*Omega_1a)*(a_Tc_w*Omega_1w))^.5

daawdT=-(aTcaw*F_a*(1+F_w*(1-Tr_w^.5)))/(2*(Tr_a)^.5*Tc_a)

-(aTcaw*F_w*(1+F_a*(1-Tr_a^.5)))/(2*(Tr_w)^.5*Tc_w)


dawdT=(Omega_1w*R^2*Tc_w/Pc_w)*(-F_w*alpha_w^.5*Tr_w^(-.5))


dawsrdT=-(Omega_1w*R^2/Pc_w)^.5*F_w/(2*(Tr_w)^.5)

damxdT2=xa*xa*daadT*(1-k_aa)+xa*xw*(daawdT)*(1-k_aw+(k_aw

-k_wa)*xa)+xw*xw*dawdT*(1-k_ww)+xw*xa*(daawdT)*(1

-k_wa+(k_wa-k_aw)*xw)

damxdT=xa*xa*daadT*(1-k_aa)+(daasrdT*a_w^.5+dawsrdT*a_a^.5)*xa*

xw*(1-k_aw+(k_aw-k_wa)*xa)+xw*xw*dawdT*(1-k_ww)

160

+(daasrdT*a_w^.5+dawsrdT*a_a^.5)*xw*xa*(1-k_wa+(k_wa

-k_aw)*xw)


h (kJ/kmol) and s (kJ/kmol*K)

h_depl=100000*R*T*(Z_ml-1)-100000*(T*damxdT-a_mx)*(1/(2*N_m))

*ln((Z_ml+M_m)/(Z_ml+Q_m))

s_depl=100000*(-R*ln(P/(Z_ml-B_m))-damxdT*((1/(2*N_m))

*ln((Z_ml+M_m)/(Z_ml+Q_m))))

END

pteosawv.lib

PROCEDURE pteosawv(T,P,ya:f_iav,f_iwv,v_mv,yw,ya_mass,yw_mass,

h_depv,s_depv)

k_aa=0

k_ww=0

k_aw=-0.26328

k_wa=-0.29368

R = 8.314/100000 m^3*bar/mol K

ammonia

omega_a=.25

Tc_a=405.6

Pc_a=112.8

Tr_a=T/Tc_a

F_a=.627090

Zeta_a=.282

Omega_2a=.06934


Omega_3a=1-3*Zeta_a



alpha_a=(1+F_a*(1-Tr_a^.5))^2


b_a=Omega_2a*b_Tc_a

c_a=Omega_3a*b_Tc_a

Aa=(a_a*P)/(R^2*T^2)

Ba=b_a*P/(R*T)

Ca=c_a*P/(R*T)

161

Na=(b_a*c_a+((b_a+c_a)/2)^2)^.5



water

omega_w=.344

Tc_w=647.3

Pc_w=221.2

Tr_w=T/Tc_w

F_w=.689803

Zeta_w=.269

Omega_2w=0.065103

Omega_1w=3*Zeta_w^2+3*(1-2*Zeta_w)*Omega_2w+Omega_2w^2+1-3*Zeta_w

Omega_3w=1-3*Zeta_w

a_Tc_w=(R^2*Tc_w^2/Pc_w)

b_Tc_w=(R*Tc_w/Pc_w)

alpha_w=(1+F_w*(1-Tr_w^.5))^2

a_w=a_Tc_w*alpha_w*Omega_1w

b_w=b_Tc_w*Omega_2w

c_w=b_Tc_w*Omega_3w

Aw=(a_w*P)/(R^2*T^2)

Bw=b_w*P/(R*T)

Cw=c_w*P/(R*T)

Nw=(b_w*c_w+((b_w+c_w)/2)^2)^.5

Mw=((b_w+c_w)/2-Nw)*P/(R*T)

Qw=((b_w+c_w)/2+Nw)*P/(R*T)

yw=1-ya

a_aa=a_a*(1-k_aa)

a_aw=(a_a*a_w)^.5*(1-k_aw+(k_aw-k_wa)*ya)

a_ww=a_w*(1-k_ww)

a_wa=(a_w*a_a)^.5*(1-k_wa+(k_wa-k_aw)*yw)

a_mx=ya*ya*a_aa+yw*yw*a_ww+ya*yw*a_aw+yw*ya*a_wa

b_mx=ya*b_a+yw*b_w

c_mx=ya*c_a+yw*c_w

A_m=(a_mx*P)/(R^2*T^2)

B_m=b_mx*P/(R*T)

C_m=c_mx*P/(R*T)

N_m=(b_mx*c_mx+((b_mx+c_mx)/2)^2)^.5



162

ap_1=-2*(ya*ya*a_a*(1-k_aa)+ya*yw*(a_a*a_w)^.5*(1-k_aw)

+yw*yw*a_w*(1-k_ww)+yw*ya*(a_w*a_a)^.5*(1-k_wa))

ap_2=-3*(ya^2*yw*(a_a*a_w)^.5*(k_aw-k_wa)

+yw^2*ya*(a_w*a_a)^.5*(k_wa-k_aw))

ap_3a=2*ya*yw*(a_a*a_w)^.5*(k_aw-k_wa)

ap_3w=2*yw*ya*(a_w*a_a)^.5*(k_wa-k_aw)

ap_4a=2*ya*a_a*(1-k_aa)+yw*(a_a*a_w)^.5*(1-k_aw+1

-k_wa+(k_wa-k_aw)*yw)

ap_4w=2*yw*a_w*(1-k_ww)+ya*(a_w*a_a)^.5*(1-k_wa+1

-k_aw+(k_aw-k_wa)*ya)


ap_w=ap_1+ap_2+ap_3w+ap_4w

bp_a=b_a-b_mx

bp_w=b_w-b_mx

cp_a=c_a-c_mx

cp_w=c_w-c_mx

c_1=1

c_2=(C_m-1)

c_3=(-2*B_m*C_m-B_m^2-B_m-C_m+A_m)


pp=c_2/c_1

qq=c_3/c_1

rr=c_4/c_1




Q_r=(c_2^2-3*c_3)/9

R_r=(2*c_2^3-9*c_2*c_3+27*c_4)/54

theta_r=arccos(R_r*Q_r^(-1.5))

Z_mv=2*(Q_r)^.5*(.5*cos(theta_r/3)

+(3^.5/2)*sin(theta_r/3))-(c_2/3)

v_mv=Z_mv*R*T/P


D_v=v_mv^2+v_mv*(b_mx+c_mx)-b_mx*c_mx

D1_v=(1/(-Q)^.5)*ln((2*v_mv+(b_mx+c_mx)-(-Q)^.5)/(2*v_mv

+(b_mx+c_mx)+(-Q)^.5))

F_1av=v_mv/(Q*D_v)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_a


+1/(Q*D_v)*(bp_a*c_mx^2+b_mx^2*cp_a-b_mx*bp_a*c_mx

-b_mx*c_mx*cp_a)+D1_v/Q*(b_mx*bp_a+3*b_mx*cp_a

163

-b_mx^2+3*bp_a*c_mx+c_mx*cp_a-c_mx^2-6*b_mx*c_mx)

f_iav=ya*P*exp(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_a)/(v_mv-b_mx)


F_1wv=v_mv/(Q*D_v)*(b_mx^2+c_mx^2+6*b_mx*c_mx+b_mx*bp_w

+3*bp_w*c_mx+3*b_mx*cp_w+c_mx*cp_w)

+1/(Q*D_v)*(bp_w*c_mx^2+b_mx^2*cp_w

-b_mx*bp_w*c_mx-b_mx*c_mx*cp_w)+D1_v/Q*(b_mx*bp_w

+3*b_mx*cp_w-b_mx^2+3*bp_w*c_mx+c_mx*cp_w-c_mx^2

-6*b_mx*c_mx)

f_iwv=yw*P*exp(ln(v_mv/(v_mv-b_mx))+(b_mx+bp_w)/(v_mv-b_mx)

+(ap_w*D1_v)/(R*T)+a_mx/(R*T)*(F_1wv)-ln(Z_mv))



mwmv=mwa*ya+mww*yw

ya_mass=ya*mwa/mwmv

yw_mass=1-ya_mass

aTcaw=((a_Tc_a*Omega_1a)*(a_Tc_w*Omega_1w))^.5

daawdT=-(aTcaw*F_a*(1+F_w*(1-Tr_w^.5)))/(2*(Tr_a)^.5*Tc_a)

-(aTcaw*F_w*(1+F_a*(1-Tr_a^.5)))/(2*(Tr_w)^.5*Tc_w)


dawdT=(Omega_1w*R^2*Tc_w/Pc_w)*(-F_w*alpha_w^.5*Tr_w^(-.5))


dawsrdT=-(Omega_1w*R^2/Pc_w)^.5*F_w/(2*(Tr_w)^.5)

damxdT2=ya*ya*daadT*(1-k_aa)+ya*yw*(daawdT)*(1-k_aw+(k_aw

-k_wa)*ya)+yw*yw*dawdT*(1-k_ww)+yw*ya*(daawdT)*(1

-k_wa+(k_wa-k_aw)*yw)

damxdT=ya*ya*daadT*(1-k_aa)+(daasrdT*a_w^.5+dawsrdT*a_a^.5)*ya

*yw*(1-k_aw+(k_aw-k_wa)*ya)+yw*yw*dawdT*(1-k_ww)

+(daasrdT*a_w^.5+dawsrdT*a_a^.5)*yw*ya*(1-k_wa

+(k_wa-k_aw)*yw)


h (kJ/kg) and s (kJ/kg*K)

h_depv=100000*R*T*(Z_mv-1)-100000*(T*damxdT-a_mx)*(1/(2*N_m))

*ln((Z_mv+M_m)/(Z_mv+Q_m))

s_depv=100000*(-R*ln(P/(Z_mv-B_m))-damxdT*((1/(2*N_m))

*ln((Z_mv+M_m)/(Z_mv+Q_m))))

END

164

ptabvlehsv.lib

PROCEDURE ptabvlehsv(T,P,xa,ya:xa_mass,ya_mass,xb_mass,yb_mass,

hmv,hml,smv,sml,v_ml,v_mv,f_ial,f_iav,f_ibl,f_ibv)

R=8.314 kJ/kmol-K

k_ab=0.2692

k_ba=0.1402

CALL pteosambtl(T,P,xa,k_ab,k_ba:f_ial,f_ibl,v_ml,xb,xa_mass,

xb_mass,h_depl,s_depl)

CALL pteosambtv(T,P,ya,k_ab,k_ba:f_iav,f_ibv,v_mv,yb,ya_mass,

yb_mass,h_depv,s_depv)

Molecular weights



mwml=xa*mwa+xb*mwb

mwmv=ya*mwa+yb*mwb

Ideal Gas Enthalpy and Entropy

c_pa=27.31+.02383*T+1.707e-5*T^2+-1.185e-8*T^3 ideal gas

c_pb=9.487+.3313*T+-1.108e-4*T^2+-2.822e-9*T^3 J/mol*K

Tra=238.8

Trb=272.3

Pref=1

hiar=23184/mwa

hia=c_pa*(T-Tra)/mwa+hiar

siar=96.94/mwa

sia=c_pa*ln(T/Tra)/mwa+siar

hibr=22669/mwb

hib=c_pb*(T-Trb)/mwb+hibr

sibr=82.97/mwb

sib=c_pb*ln(T/Trb)/mwb+sibr

hdv=h_depv/mwmv

hdl=h_depl/mwml

sdv=s_depv/mwmv

sdl=s_depl/mwml

Mixture Enthalpy and Entropy (massbased)

hmv=ya_mass*hia+yb_mass*hib+hdv

hml=xa_mass*hia+xb_mass*hib+hdl

smv=ya_mass*sia+yb_mass*sib+sdv-(ya_mass*R/mwa*ln(ya)

165

+yb_mass*R/mwb*ln(yb))

sml=xa_mass*sia+xb_mass*sib+sdl-(xa_mass*R/mwa*ln(xa)

+xb_mass*R/mwb*ln(xb))

END

ptawvlehsv.lib

PROCEDURE ptawvlehsv(T,P,xa,ya:xa_mass,ya_mass,xw_mass,yw_mass,

hmv,hml,smv,sml,v_ml,v_mv,f_ial,f_iav,f_iwl,f_iwv)

R=8.314 kJ/kmol-K

CALL pteosawl(T,P,xa:f_ial,f_iwl,v_ml,xw,xa_mass,xw_mass,

h_depl,s_depl)

CALL pteosawv(T,P,ya:f_iav,f_iwv,v_mv,yw,ya_mass,yw_mass,

h_depv,s_depv)



mwv=ya*mwa+yw*mww

mwl=xa*mwa+xw*mww



c_pw=32.24+.001924*T+1.055e-5*T^2+-3.596e-9*T^3 J/mol*K

Pref=1

Tra=238.8

Trw=372.5

hiar=23184/mwa


siar=96.94/mwa


hiwr=40761/mww

hiw=c_pw*(T-Trw)/mww+hiwr

siwr=109.4/mww

siw=c_pw*ln(T/Trw)/mww+siwr

hdl=h_depl/mwl

hdv=h_depv/mwv

sdl=s_depl/mwl

sdv=s_depv/mwv

Mixture Molar Enthalpy and Entropy

166

hmv=ya_mass*hia+yw_mass*hiw+hdv

hml=xa_mass*hia+xw_mass*hiw+hdl

smv=ya_mass*sia+yw_mass*siw+sdv-(ya_mass*R/mwa*ln(ya)

+yw_mass*R/mww*ln(yw))

sml=xa_mass*sia+xw_mass*siw+sdl-(xa_mass*R/mwa*ln(xa)

+xw_mass*R/mww*ln(xw))

END

ptamwtsc.lib

PROCEDURE ptamwtsc(T,P,xa:hml,sml)

R=8.314 kJ/kmol-K

CALL pteosawl(T,P,xa:f_ial,f_iwl,v_ml,xw,xa_mass,xw_mass,

h_depl,s_depl)



mwl=xa*mwa+xw*mww



c_pw=32.24+.001924*T+1.055e-5*T^2+-3.596e-9*T^3 J/mol*K

Pref=1

Tra=238.8

Trw=372.5

hiar=23184/mwa


siar=96.94/mwa


hiwr=40761/mww

hiw=c_pw*(T-Trw)/mww+hiwr

siwr=109.4/mww

siw=c_pw*ln(T/Trw)/mww+siwr

hdl=h_depl/mwl

sdl=s_depl/mwl

Mixture Molar Enthalpy and Entropy

hml=xa_mass*hia+xw_mass*hiw+hdl

sml=xa_mass*sia+xw_mass*siw+sdl-(xa_mass*R/mwa*ln(xa)

+xw_mass*R/mww*ln(xw))

167

END

ptamhsv.lib

PROCEDURE ptamhsv(T,P:hl,hv,sl,sv,vl,vv)

mwa=17.03 kg/kmol

R = 8.314/100000 m^3*bar/kmol K

c_pa=27.31+.02383*T+1.707e-5*T^2+-1.185e-8*T^3

Patm=1

Tra=238.8

hir=23184/mwa

hi=c_pa*(T-Tra)/mwa+hir

sir=96.94/mwa

si=c_pa*ln(T/Tra)/mwa+sir

omega_a=.25

Tc_a=405.6

Pc_a=112.8

Tr_a=T/Tc_a

F_a=.627090

Zeta_a=.282

Omega_2a=.06934

Omega_3a=1-3*Zeta_a

Omega_1a=3*Zeta_a^2+3*(1-2*Zeta_a)*Omega_2a

+Omega_2a^2+1-3*Zeta_a



alpha_a=(1+F_a*(1-Tr_a^.5))^2


b_a=Omega_2a*b_Tc_a

c_a=Omega_3a*b_Tc_a

Aa=(a_a*P)/(R^2*T^2)

Ba=b_a*P/(R*T)

Ca=c_a*P/(R*T)

Na=(b_a*c_a+((b_a+c_a)/2)^2)^.5



c_1a=1

c_2a=(Ca-1)

c_3a=(-2*Ba*Ca-Ba^2-Ba-Ca+Aa)

168

c_4a=(Ba^2*Ca+Ba*Ca-Aa*Ba)

ppa=c_2a/c_1a

qqa=c_3a/c_1a

rra=c_4a/c_1a

A_cubica=1/3*(3*qqa-ppa^2)

B_cubica=1/27*(2*ppa^3-9*ppa*qqa+27*rra)

D_cubica=(A_cubica)^3/27+(B_cubica)^2/4

IF D_cubica<0 THEN

phi=arccos(((B_cubica^2/4)/(-A_cubica^3/27))^.5)

ycubic1=2*(-A_cubica/3)^.5*cos(phi/3+2.09439510239*0)



Z_av=ycubic1-ppa/3

Z_al=ycubic2-ppa/3

Z_a3=ycubic3-ppa/3

endif

IF D_cubica>0 THEN

Mcubic=(-B_cubica/2+D_cubica^.5)^(1/3)

Ncubic=(-B_cubica/2-D_cubica^.5)^(1/3)

ycubic1=Mcubic+Ncubic

ycubic2=-.5*(Mcubic+Ncubic)

ycubic3=-.5*(Mcubic+Ncubic)

Z_av=ycubic1-ppa/3

Z_al=ycubic2-ppa/3

Z_a3=ycubic3-ppa/3

endif

daadT=(Omega_1a*R^2*Tc_a/Pc_a)

*(-F_a*alpha_a^.5*Tr_a^(-.5))

hdepl=100000*R*T*(Z_al-1)-100000*(T*daadT-a_a)*(1/(2*Na))

*ln((Z_al+Ma)/(Z_al+Qa))

hdepv=100000*R*T*(Z_av-1)-100000*(T*daadT-a_a)*(1/(2*Na))

*ln((Z_av+Ma)/(Z_av+Qa))

sdepl=100000*(-R*ln(P/(Z_al-Ba))-daadT*((1/(2*Na))

*ln((Z_al+Ma)/(Z_al+Qa))))

sdepv=100000*(-R*ln(P/(Z_av-Ba))-daadT*((1/(2*Na))

*ln((Z_av+Ma)/(Z_av+Qa))))

vl=Z_al*R*T/P

vv=Z_av*R*T/P

Enthalpy and Entropy in kJ/kg

hl=hi+hdepl/mwa

169

hv=hi+hdepv/mwa

sl=si+sdepl/mwa

sv=si+sdepv/mwa

END

170

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VITA

Laura Schaefer was born Laura Michelle Atkinson in Danville, Illinois, on

January 13, 1973. She is the oldest of three children, and was raised in Springfield,

Illinois, The Woodlands, Texas, and Gaithersburg, Maryland. Laura graduated from

Thomas Wootton High School in Rockville, Maryland, in 1991.

Laura attended Rice University in Houston, Texas, where she was a resident

of Wiess College. She held in summer internships at the National Air and Space

Museum in Washington, D.C., and at Lawrence Livermore National Lab in Califor-

nia. During her time at the Air and Space Museum, Laura designed and created a

longitudinal wave device for inclusion in How Things Fly, NASM’s first interactive

gallery. Laura received a B.A. in English and a B.S. in Mechanical Engineering in

May 1995. Upon graduation, she spent three months traveling through Europe by

train.

In September 1995, Laura entered graduate school at the Georgia Institute

of Technology. She received a Master of Science degree in December 1997. Laura’s

M.S. thesis focused on developing an analytical mean temperature difference relation-

ship for refrigerant mixtures that have been designated as replacements for ozone-

depleting CFCs and HCFCs. Laura’s M.S. and Ph.D. research at Georgia Tech was

supported by NSF, DuPont, ASME, and ASHRAE.

Laura married Andrew James Schaefer on June 21, 1997. Andrew is receiving

a Ph.D. in Operations Research from Georgia Tech’s School of Industrial and Systems

Engineering. In her spare time, Laura enjoys reading, gardening, and homebrewing.

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