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Analytical and Experimental Investigation of Rotary-Vane Two-Phase Expanders in Vapor Compression Refrigeration Systems

Permanent Link: http://ufdc.ufl.edu/UFE0022592/00001

Material Information

Title: Analytical and Experimental Investigation of Rotary-Vane Two-Phase Expanders in Vapor Compression Refrigeration Systems
Physical Description: 1 online resource (267 p.)
Language: english
Creator: Mahmoud, Ahmad
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: expander, expansion, rotary, two, vapor
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The refrigeration and air-conditioning community has been searching for environmentally friendly refrigerants and ways to improve energy efficiency and to increase the potential for reductions in system size and weight for some time. These system improvements may manifest themselves in numerous ways but significantly impact terrestrial logistics differently. This study investigates the analytical and experimental utilization of rotary-vane expanders as throttle valve replacements in vapor compression refrigeration systems with a specific application of these expanders in smaller deployable environmental control units. The findings support the conclusion that utilizing two-phase expanders in refrigeration systems is very promising. Mathematical models for a system-level parametric cycle analysis were developed to assess potential gains in the cycle coefficient of performance for R-22, R-134 and transcritical CO2 vapor compression technology. Optimization studies were conducted to determine the optimum performance of a refrigeration system subject to constraints of size and weight of both the condenser heat transfer area and the size and weight of the two-phase expander. An exhaustive literature survey aids and validates the suitability of the type of positive-displacement expander used, after which comprehensive component level thermodynamic and fluid dynamic models of a rotary-vane expander were developed to establish the performance of this expansion device as a function of design and fluid parameters. This included rigorous modeling of irreversible loss mechanisms such as throttling in the intake and exhaust ports, two-phase internal leakage, friction, re-compression, and under or over-expansion due to incorrect sizing of the expander or off-design operation. Results from these models were used to establish the operating principles of a two-phase rotary-vane expander. Experimental data gathered after modifying a conventional chiller with an alternative flow path, where a rotary-vane expander and dynamometer have been installed, were used to improve the fidelity of the analytical model developed. Issues such as real-time control to satisfy system-level constraints, expander sizing and operational speed and inadequate operation were addressed. The analytical model developed along with single-phase experimentation were used to understand and overcome component-level complications and inadequacies due to irreversible effects.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ahmad Mahmoud.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Sherif, Sherif A.
Local: Co-adviser: Lear, William E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022592:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022592/00001

Material Information

Title: Analytical and Experimental Investigation of Rotary-Vane Two-Phase Expanders in Vapor Compression Refrigeration Systems
Physical Description: 1 online resource (267 p.)
Language: english
Creator: Mahmoud, Ahmad
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: expander, expansion, rotary, two, vapor
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The refrigeration and air-conditioning community has been searching for environmentally friendly refrigerants and ways to improve energy efficiency and to increase the potential for reductions in system size and weight for some time. These system improvements may manifest themselves in numerous ways but significantly impact terrestrial logistics differently. This study investigates the analytical and experimental utilization of rotary-vane expanders as throttle valve replacements in vapor compression refrigeration systems with a specific application of these expanders in smaller deployable environmental control units. The findings support the conclusion that utilizing two-phase expanders in refrigeration systems is very promising. Mathematical models for a system-level parametric cycle analysis were developed to assess potential gains in the cycle coefficient of performance for R-22, R-134 and transcritical CO2 vapor compression technology. Optimization studies were conducted to determine the optimum performance of a refrigeration system subject to constraints of size and weight of both the condenser heat transfer area and the size and weight of the two-phase expander. An exhaustive literature survey aids and validates the suitability of the type of positive-displacement expander used, after which comprehensive component level thermodynamic and fluid dynamic models of a rotary-vane expander were developed to establish the performance of this expansion device as a function of design and fluid parameters. This included rigorous modeling of irreversible loss mechanisms such as throttling in the intake and exhaust ports, two-phase internal leakage, friction, re-compression, and under or over-expansion due to incorrect sizing of the expander or off-design operation. Results from these models were used to establish the operating principles of a two-phase rotary-vane expander. Experimental data gathered after modifying a conventional chiller with an alternative flow path, where a rotary-vane expander and dynamometer have been installed, were used to improve the fidelity of the analytical model developed. Issues such as real-time control to satisfy system-level constraints, expander sizing and operational speed and inadequate operation were addressed. The analytical model developed along with single-phase experimentation were used to understand and overcome component-level complications and inadequacies due to irreversible effects.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ahmad Mahmoud.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Sherif, Sherif A.
Local: Co-adviser: Lear, William E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022592:00001


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ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF ROTARY-VANE TWO-
PHASE EXPANDERS INT VAPOR COMPRESSION REFRIGERATION SYSTEMS




















By

AHMAD MOHAMED MAHMOUD


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008




































O 2008 Ahmad Mohamed Mahmoud



































I dedicate this work to God Almighty in partial fulfillment of His mandate to seek Knowledge:
To my loving wife Iman, daughter Salma, family and friends, without whose enduring patience
and support this work would not have been possible.









ACKNOWLEDGMENTS


I would like to begin by thanking God for giving me this great opportunity to learn and for

putting in my path wonderful people who have helped me and shown me the way. I want to

thank all those people, and will try my best to remember them all here.

I want to start by thanking my wife, Dr. Iman M. Al-Naggar. To put it simply, this would

not have been possible without her love, support and daily dose of nagging!

I would like to thank my parents, Dr. Mohamed Abdel-moneim Mahmoud and Mrs. Faten

Ismail Mahmoud for preparing me for and supporting me during graduate school. I also thank

them for their unconditional love and endless prayers that have kept me on track.

I would like to thank my mentors Dr. S. A. Sherif and Dr. William E. Lear for their time,

support, guidance and patience. Their technical and non-technical mentorship drove me to excel

at whatever I did, by following their example, and helped me become the researcher I am today. I

am truly grateful.

I would like to thank my committee members for their guidance and input: Dr. Gary Ihas,

Dr. D. Yogi Goswami, and Dr. Skip Ingley. I have been fortunate to have a diverse,

knowledgeable committee that had answers to my questions, solutions to my problems, and great

advice and encouragement.

I would like to thank all the people in the Sherif and Lear Labs, especially Mr. Ayyoub

Mehdizadeh, that have helped me throughout my years in graduate school, as well as make my

long hours in the lab enj oyable. I would especially like to thank Mr. John Crittenden,

engineering director at Triad Research Corporation Inc., for the guidance and knowledge he

passed on to me regarding experimentation and solving countless technical challenges. I am

honored to have worked with him.










I would also like to thank all the staff that has assisted me greatly at UF (Department of

Aerospace and Mechanical Engineering).

I would like to thank my friends who have made my years in Gainesville more enjoyable:

Mr. Ibrahim Taman, Mr. Ahmed El-Mahdawy, Mr. Ahmed El-Hady El-Mahdawy, Mr. Farouk

Dey, Mr. Safwat Mohammad, Mr. Mahmoud Enani, Mr. Ramadan Ajredini, Dr. Mujahid Abdul-

rahim and Dr. Fares Al-Bitar, to name a few.

I would like to acknowledge the support from the Air Force Research Labs for partially

funding my research and graduate education: Dr. Aly Shaaban, Dr. Ragab Moheisen and Mr.

Reza Salavani.

Finally, I would like to thank the University of Florida and the College of Engineering for

providing such a rich and competitive graduate program in which I am very lucky to have been a

part.











TABLE OF CONTENTS


IM Le

ACKNOWLEDGMENT S .............. ...............4.....


LI ST OF T ABLE S ................. ...............9................


LIST OF FIGURES .............. ...............10....

NOMENCLATURE .............. ...............17....


AB S TRAC T ......_ ................. ............_........2


CHAPTER


1 INTRODUCTION AND OBJECTIVES ................. ...............24................

Introducti on ................. ...............24.................
Obj ectives ................. ...............24.......... .....

2 LITERATURE REVIEW ................. ...............28................


Expanders in Refrigeration and Heat Pump Systems ................ ..............................28
Conventional Heat Pump and Refrigeration Systems .............. ...............29....
Transcritical Carbon Dioxide Systems ................. ...............37................
Gas Refrigeration Cycles and Cryogenics............... ...............4
Expander Selection ................ ...............45.................

3 PARAMETRIC ANALYSIS OF CYCLES .............. ...............46....


Ideal Cycles .............. ...............46....
Base Cycle ................. ...............47.................
Economizer Cycle ................... ........ ..............4
Internal Heat Exchanger (IHX) Cycle ................. ...............50........... ...
Actual Cycles............... ... ....... .. ...........5
Effect of Subcooling on System Performance............... ..............5
Effect of Superheating on System Performance ................. ........... ... ............. .....53
Evaluation of CFC, HFC and Transcritical CO2 Refrigeration Cycles ................ ...............54
Performance and Size Optimization of Compression Refrigeration Systems ................... .....70
Pure Hydrocarbons as Refrigerants .............. ...............83....

4 MATHEMATICAL MODEL OF AN IDEAL ROTARY-VANE EXPANDER. ........._......124


Literature Review of Modeling of Rotary-vane Expanders .........._.._.. ......._.._...........124
Model Development ........._.._.. ...._... ...............125....
Thermophy sical Model ................. ...............126........ ......
Geometric and Kinematic Model............... ...............126.












Thermodynamic Model .............. ...............128....
Charging Process ............. ...............129..._ _........
Expansion Process ....._ .................. ...............131......
Exhaust process .............. ...............132..._ _........
Ideal Expander Evaluation .............. ...............133....

5 PRIMARY AND SECONDARY LOSS MECHANISMS INT ROTARY-VANE TWO-
PHASE REFRIGERATING EXPANDERS .............. ...............154....


Primary Loss Mechanisms ................. ...............154...............
Internal Leakage Paths and Clearances .............. ...............155....
Types of Two-Phase Leakage Losses ................... .......... ...............156.....
Axial Clearance Between Rotor and End Plates .............. ...............161....
Leakage around tips of vanes ................. ...............166.......... ...
Leakage past the sides of the vanes ................... ........ ...... .. ..... ........ ............16
Leakage between the faces of the vanes and the side walls of the rotor' s slots ............168
Leakage in radial gap between the rotor and stator cylinder............. ............__ ...168
Friction M odel .............. ......... ........ .. .... ...........16
Two-Phase Throttling Losses in the Inlet and Exhaust Ports ................ ............ .........180
Conventional Inlet to Expander Cavity .............. ...............184....
Modified Inlet to Expander Cavity ............... ... ...............187.......... ....
Thermodynamic Model of the Actual Expansion Process ................. ........................188
Charging Process ................. ...............188................
Expansion Process ................. ...............189................
Exhaust process .................. ...............190......... ......
Expander Performance Evaluation ............... ....... ..............19
Performance Variation Due to Stator-Cylinder Geometry .............. .....................9
Heat Transfer ................. ...............198......... ......


6 EXPERIMENTAL PROGRAM ................. ...............214................


Experimental Facility............... .... ............21
Chilled Water/Heated Water Loop ................. ...............214...............
Data Acquisition System .............. .. ...............215...
Unmodified Chiller Experimental Procedures............... .............. ...........1
Dynamometer Selection, Data Acquisition and Real-Time Control Scheme.......................218
Rotary-vane Ex pander Selection .............. .... ...............220.
Modified Automotive Air-conditioning Compressor ....._____ .........__ ..............220
Performance of Modified Compressor .......................__ ......__ ...........2
Design Issues ............... ....._ ...............223....
Designed and Machined Expander ....__ ......_____ .......__ .............2
Performance of Machined Expander ....__ ......_____ .......___ ...........25
Modified Air-Motor............... ...............22

Single-Phase Experiments .............. ...............229....
Sum mary ............ ..... .._ ...............2 1...

7 CONCLUSIONS AND RECOMMENDATIONS .............. ...............250....













Internal Leakage Losses .............. ...............250....
Friction ................. ...............251................
Other Issues ............... .... .. ..... .. ........ 5

Leakage ratio, Expander sizing and Integration Concepts .............. .....................5
Recommendations............... ............25

Summary of Contributions .............. ...............256....


LIST OF REFERENCES ................. ...............258................


BIOGRAPHICAL SKETCH .............. ...............267....










LIST OF TABLES


Table page

3-1 Summary of the percent changes in the system COP, refrigerating capacity and
required work input by the addition of an expansion device for the ideal and actual
R-13 4a single-stage, economizer and IHX cycles ................ ...............92........... .

3-2 Percentage of cycle irreversibility due to the various components in basic and
modified R-22, R-134a and transcritical CO2 Systems ................ .........................113

3-3 Hydrocarbons viewed as potential refrigerants ................. ................. ......... ..11

4-1 Summary of rotary-vane literature ................. ...._.._ ...............143 ....

4-2 Geometrical input parameters ................. ...............145...............

4-3 Typical operating conditions for a basic and modified 2 ton system .............. ................148

5-1 Primary leakage paths in a rotary-vane expander ................. ............... ......... ...200

5-2 Governing equations and their discretized form describing one-dimensional adiabatic
Fanno-flow from a reservoir ................. ...............201...............

5-3 Variation of operational and dimensionless parameters to model flow in axial gap as
laminar and with a zeroth-order approximation............... .............20

5-4 Comparisons of contraction coefficients for turbulent single-phase flow.............._.... ....209

5-5 Variation of volumetric efficiency and final pressure as a function on inlet angle in
an expander with throttling losses accounted for. ................. ....._._ ............... ..211

6-1 Experimental Thermodynamic State Points............... ...............241

6-2 Parameters used to calculate saturation properties. ............. ...............244....

6-3 Machining modifications made to GAST NL-32-NCC-1 to optimize geometric
volum e ratio .............. ...............247....










LIST OF FIGURES


FiMr page

3-1 A schematic of the ideal base cycle and the modified base cycle with an expansion
device ................. ...............86....... ......

3-2 T-s and P-h diagrams of the ideal base cycle and the modified base cycle with an
expansion device............... ...............87.

3-3 T-s and P-h diagrams of the economizer cycle and an economizer cycle with an
expansion device as a throttle valve replacement ................. ...............88........... ..

3-4 A schematic of the economizer cycle and an economizer cycle with an expansion
device as a throttle valve replacement ................. ...............89...............

3-5 A schematic of the IHX cycle and the modified IHX cycle with an expansion device
as a throttle valve replacement .............. ...............90....

3-6 T-s and P-h diagrams of the IHX cycle and the modified IHX cycle with an
expansion device as a throttle valve replacement ................. ...............91........... ..

3-7 Variation of ACOP, between the modified cycle with an expansion device and the
ideal cycle (R-134a), with evaporating temperature for A) To 25oC, B) To-45oC for
the base, economizer and IHX cycles ................ ...............93...............

3-8 Variation of AQe and AWnet, between the ideal modified cycle with an expansion
device and an ideal base cycle, with evaporating temperature for various condenser
temperatures in a single-stage vapor-compression refrigeration unit. ............... ...............94

3-9 Variation of ACOP, between the modified cycle with an expansion device and the
ideal cycle, as a function of the evaporating temperature for both R-22 and R-134a
for various expander efficiencies To-45oC .............. ...............95....

3-10 Variation of ACOP, between the modified cycle with an expansion device and the
ideal cycle, as a function of the evaporating temperature for various condenser
temperatures for the actual and ideal cycle configurations of the economizer cycle. .......96

3-11 Variation of ACOP, between the modified cycle w/ an expansion device and the ideal
cycle, as a function of the evaporating temperature for various condenser
temperatures for the actual and ideal cycle configurations of the IHX cycle ................... .97

3-12 Variation of refrigerating efficiency as a function of the evaporating temperature for
both the base (no expander) and ideal and actual modified cycle with and an
expander at a condenser temperature of To-45oC ........._.._ .......___ ......._.._ ......98










3-13 Variation of refrigerating efficiency as a function of the evaporating temperature for
the actual base, IHX and economizer cycles with an expander at a condenser
temperature of Te=25oC .............. ...............98....

3-14 Variation of an ideal system's COP as a function of the degree of subcooling for
various condensing temperatures ................. ...............99......__ ....

3-15 Variation of an ideal system's COP as a function of the degree of subcooling for
various condenser temperatures and expansion processes............... ...............10

3-16 Variation of the refrigerating effect as a function of the degree of subcooling for
various condenser temperatures and expansion processes............... ...............10

3-17 Variation in an ideal system's COP with respect to the evaporator temperature for
different degrees of sub cooling and expansion processes .............. ....................10

3-18 Variation of the difference in system COP,%, between the ideal cycle and modified
cycle with an expansion device as a function of the evaporator temperature for
various condenser temperatures and degrees of sub cooling ................. .....................103

3-19 Variation of the difference of the system COP, %, base cycle and modified cycle
with an expansion device as a function of the degree of superheat in the evaporator
for various condenser temperatures. ............. ...............104....

3-20 Variation of the refrigerating effect for an ideal cycle as a function of the degree of
superheat for various condenser temperatures ................. ...............105........... ...

3-21 T-s diagram of the basic system and the modified system with an expansion device
for a A) conventional refrigeration system and B) transcritical CO2 refrigeration
system .............. ...............106....

3 -22 Variation of system coefficient of performance as a function of heat rej section
pressure for various evaporating temperatures for basic and modified transcritical
CO2 System s ................. ...............107........ ......

3-23 Percentage of cycle irreversibility associated with the heat rej section and expansion
processes as a function of heat rej section pressure for basic and modified transcritical
CO2 Systems ................. ...............107................

3 -24 Coefficient of performance of basic and modified R-22, R-134a and transcritical CO2
systems as functions of the evaporating temperature ................ ......... .................108

3-25 Difference in coefficient of performance between basic and modified R-22, R-134a
and transcritical CO2 Systems as functions of the evaporating temperature ................... .108

3-26 Difference in refrigerating effect between basic and modified R-22, R-134a and
transcritical CO2 Systems as functions of the evaporating temperature ................... ........109










3-27 Difference in system input work between basic and modified R-22, R-134a and
transcritical CO2 Systems as functions of the evaporating temperature ................... ........109

3-28 Volume ratio of the expander employed in modified R-22, R-134a and transcritical
CO2 Systems as functions of the evaporating temperature. .....__. ............ ....... ........1 10

3-29 Difference in coefficient of performance between basic and modified R-22, R-134a
and transcritical CO2 Systems as functions of expander isentropic efficiency ................1 10

3-30 Percentage of cycle irreversibility due to various components in a modified R-22
system as functions of the evaporating temperature ................. .......... ................11 1

3-31 Percentage of cycle irreversibility due to condensing process in basic and modified
R-22 and R-134a systems as functions of the evaporating temperature ................... .......11 1

3-32 Percentage of cycle irreversibility due to the expansion process in basic and modified
R-22, R-134a and transcritical CO2 Systems as functions of the evaporating
tem perature ................. ...............112......... ......

3-33 Dimensionless system irreversibility in basic and modified R-22, R-134a and
transcritical CO2 Systems as functions of the evaporating temperature ................... ........1 12

3-34 Variation of dimensionless COP, expander work output, expander volume ratio and
condenser capacitance as a function of dimensionless refrigerating effect ........._._........113

3-35 Variation of dimensionless expander work output and expander volume ratio as a
function of evaporating temperature for various degrees of subcooling in the
condenser ........._.__...... ._ __ ...............114....

3-36 Variation of dimensionless condenser capacitance as a function of evaporating
temperature for various degrees of subcooling in the condenser. ........._._... ................1 14

3-37 Variation of dimensionless condenser capacitance as a function of degree of
subcooling in the condenser for various values of condenser effectiveness. ................... 115

3-38 Variation of dimensionless expander work output as a function of degree of
sub cooling in the condenser for various expander efficiencies ........._._... ........._......115

3-39 Variation of dimensionless expander volume ratio as a function of degree of
sub cooling in the condenser for various expander efficiencies ........._._... ........._......116

3-40 Variation of dimensionless COP as a function of degree of subcooling in the
condenser for various expander efficiencies ....._.__._ .... ... .___ .....__... .........16

3-41 Variation of dimensionless COP as a function of degree of subcooling in the
condenser for various evaporator degree of superhea. ....._.__._ ........___ ...............117










3 -42 Variation of the refrigerating effect as a function of evaporating temperature for
various refrigerants at a condensing temperature of A) 25oC and B) 45oC for the
ideal base cycle ................. ...............118...............

3-43 Variation of the ideal base cycle coefficient of performance as a function of
evaporating temperature for various refrigerants at a condensing temperature of A)
25oC and B) 45oC ................. ...............119...............

3 -44 Variation of compressor work as a function of the evaporating temperature at a
condensing temperature of A) 25oC and B) 45oC for various refrigerants in an ideal
base cycle ................. ...............120................

3-45 Variation of the increase of COP (when comparing the base cycle and the expander
cycle) as a function of the evaporating temperature for various refrigerants at a
condensing temperature of A) 25oC and B) 45oC ................ ................. ......... .12

3-46 Variation of the decrease in net-work (when comparing the base cycle and the
expander cycle) as a function of the evaporating temperature for various refrigerants
at a condensing temperature of a) 25oC and b) 45oC ................. ......... ................1 22

3-47 Variation of the increase in refrigerating effect (when comparing the base cycle and
the expander cycle) as a function of the evaporating temperature for various
refrigerants at a condensing temperature of a) 25oC and b) 45oC............... .................. 123

4-1 Flow diagram of the main computer program developed ................. .......................142

4-2 Schematic of a generally oriented circular rotary-vane expander with 8 vanes
described by two circular arcs ................. ...............144........... ...

4-3 Variation of the inlet throat area with respect to angular displacement ................... .......146

4-4 Variation of the exit throat area as a function of angular displacement for an 8-vane
circular M VE .............. ...............146....

4-5 Pressure variation as a function of volume in an ideal expander for the cases of ideal,
over and under expansion. ............. ...............147....

4-6 Variation of built-in volume ratio as a function of intake angle for different numbers
of vanes ................. ...............149......... ......

4-7 Variation of ideal volume as a function of angular displacement for different
numbers of vanes ................. ...............149...............

4-8 Variation of ideal and actual volumes as a function of angular displacement. ................150

4-9 Variation of pressure as a function of volume for various intake angles. ................... .....150

4-10 Variation of pressure as a function of volume for different numbers of vanes ...............15 1










4-11 Variation of mass within the expander' s cell volume as a function of angular
displacement for various intake angles ................. ...............151........... ...

4-12 Variation of power as a function of angular displacement for different numbers of
vanes .............. ...............152....

4-13 Variation of power as a function of angular displacement for various intake angles......152

4-14 Variation of power as a function of volume ................. ...............153........... .

5-1 Schematic of nomenclature used and leakage paths of a circular rotary-vane
expander with general orientation and a (a) conventional or (b) modified intake ports..199

5-2 Schematic of a typical leakage path ................. ...............200........... ..

5-3 Shear and pressure driven Couette flow .............. ...............201....

5-4 Non-axisymmetric pressure boundary condition on outer surface of the rotor ...............202

5-5 Free-body diagram of a vane protruding outward from a rotor slot at a local angle 6 ....203

5-6 Schematic of generalized Couette flow in the axial gap between rotor and end-plate....204

5-7 Variation of reaction forces on a vane with no vane-tip curvature as a function of
angular displacement .............. ...............204....

5-8 Variation of reaction forces on a vane with a circular vane-tip profile as a function of
angular displacement .............. ...............205....

5-9 Variation of leakage from/to the expander cavity as a function of angular
displacement due to non-axisymmetric flow between the rotor and stationary end-
plates for different intake angle spreads .............. ...............205....

5-10 Variation of non-axisymmetric leakage from/to the expander cavity for the ideal and
throttling cases .............. ...............206....

5-11 Variation of leakage from/to the expander cavity through the gap between the sides
of the vanes and end-plates for different intake angle spreads ................. ................. .206

5-12 Variation of leakage to the expander cavity from the rotor slot (modified intake)
through the gap between the face of the vanes and rotor slot ................. .........._ ....207

5-13 Variation of the ideal and actual mass flow-rates through the expander as a function
of rotational speed ........... _... ......... ...............207....

5-14 Variation of the ideal and actual mass flow-rates through the expander as a function
of the number of vanes ................. ...............208......_.....










5-15 Comparison of way by which fluid is introduced into the expander' s cavity; A)
conventional intake via intake port and B) modified intake via rotor slots through
end-plates to ensure vane-tip and stator-cylinder contact............... ...............208

5-16 Flow through a sudden contraction............... ..............20

5-17 Schematic of the three phases of vane orientation that occur during the intake process.210

5-18 Variation of pressure in expander cavity as a function of angular displacement for the
ideal and actual (throttling only) cases ................. ...............211..............

5-19 Schematic of the intake port of the modified rotary-vane intake through the end-
plates and into the rotor cavity via rotor slots ................. ...............212........... .

5-20 Schematic of (a) symmetric (b) non-symmetric non-circular rotary-vane expander
comprised of four circular arcs (1-4) and a sealing arc (5) ................. ......................212

5-21 Variation of vane displacement as function of the leading vanes' angular
displacement for both a circular and non-circular MVE .............. ....................21

5-22 Variation of vane velocity as function of the leading vanes' angular displacement for
both a circular and non-circular rotary-vane expander ......____ ........._ ........._....213

6-1 The water heater used to add a constant heat load to the constant temperature water
loop .............. ...............236....

6-2 Unmodified two ton air-cooled barrel type water chiller ................. .......................236

6-3 Instrumentation map and schematic of the experimental set-up for a vapor
compression cycle with a by-pass loop for rotary-vane expander integration.................237

6-4 Variation of the measured pressure for the standard mode of operation in the
unmodified chiller as a function of time ......___ .........___....._ ...........3

6-5 Variation of the measured temperature for the standard mode of operation in the
unmodified chiller as a function of time ....__ ......_____ .......___ ...........3

6-6 Variation of the measured pressure for the cut-off mode of operation in the
unmodified chiller as a function of time ....__ ......_____ .......___ ...........3

6-7 Variation of the measured temperature for the cut-off mode of operation in the
unmodified chiller as a function of time ....__ ......_____ .......___ ...........3

6-8 Variation of the compressor work for both the cut-off and standard modes of
operation in the unmodified chiller as a function of time ....._____ ...... ....__...........240

6-9 Schematic of the cut-away view of the bypass loop and evaporator and the logic
behind the continuous control scheme used to control the degree of superheat .............243











6-10 Flow path of refrigerant in both the compression and expansion processes.. ................. .245

6-11 Double acting, five vane modified automotive compressor .............. .....................4

6-12 Unmodified, modified symmetric and non-symmetric non-circular Expander (All
dimensions in inches)............... ...............248

6-13 Schematic of the experimental set-up for compressed air and R-22 tests with a
rotary-vane expander .............. ...............249....

7-1 Variation of the critical-pressure ratio as a function of condensing temperature for a
two-phase expander and vapor compressor ................. .........._ ....... 257.........

7-2 Expander sizing diagram............... ...............257










NOMENCLATURE

0 reference state, zeroth-order

1 inlet state, location of minimum volume (i.e. cut-off)

2 location of maximum volume (i.e. at exhaust)

A cross-sectional area, m2

a speed of sound, m/s

Acc(6) sliding acceleration of vane, m/s2

act actual

Ak, Bk Fourier constants

amb ambient

b base cycle, boundary work, back-pressure

c Carnot

calc calculated

Cd discharge coefficient

comp compressor

cond condenser

COP coefficient of performance, dimensionless

CP critical point

C, specific heat at constant pressure, kJ/(kg K)

crit critical

cy control volume (i.e. expander cavity)

dt differential in time, s

e eccentricity m, evaporator, electric, exit state

E total energy, kJ

econ economizer cycle










ED expansion device

Eu Euler number

ex start of exhaust port

exh exhaust

exp expansion device

f Darcy friction coefficient

F force, N

f saturated liquid, friction

g saturated vapor, acceleration of gravity m/s2

gc gas cooler

h specific enthalpy kJ/kg, evaporator coolant, isenthalpic

Hv height of vane, m

i inlet, intermediate

I specific irreversibility (kJ/kg), current (Amp)

in inlet state, spread of intake port

int intake

L axial length of expander, m

1 leakage channel length, m

leak corresponding to internal leakage

Lv axial length of vane, m

M Mach number

m mass kg, mean

mi mass flow-rate, kg/s

max maximum

MW molecular weight, kg/kmol










Nv number of vanes

O center, location

o original

out outlet state

P pressure, kPa

p dimensionless pressure, pr= (p p x)l(p, P )

q specific heat transfer, kJ/kg

Q heat transfer rate, kW

F dimensionless radial coordinate, F = r /rR

r reservoir

R rotor

ref refrigerant, refrigerating

Reel Reynolds number based on axial gap width

RR rotor radius, mm

RS(6) distance from stator-cylinder' s center to periphery, m

r, volume ratio (process or built-in)

S gap aspect ratio, stator-cylinder

s specific entropy kJ/kgK, isentropic

sc subcooling

sf sliding friction

sh superheat

sys system

T temperature K, torque Nm

t time, s or minutes (unless otherwise noted)

tv thickness of vane, m (=ts, slot thickness)









u specific internal energy kJ/kg, velocity m/s

u dimensionless radial velocity, u = u airR

v dimensionless azimuthal velocity, v = vlirR

v throttling valve, vane, specific volume m3/kg

V voltage V, volume m3

vd viscous drag

Vel(6) sliding velocity of vane, m/s

w dimensionlesss axial velocity, w= w/04~

w specific work, kJ/kg

W power, kW

x quality, dimensionless

X(6) vane height protruding from slot, m

z dimensionless axial coordinate, z = z /6

A differential

0Z speed of angular rotation, RPM

oc void fraction

P inner to outer radius ratio

8 angle between successive vanes, degrees, leakage channel height, m

Si axial gap between rotor and end-plate, m

82 radial gap between rotor and stator during sealing arc, m

E effectiveness, dimensionless

r efficiency

h leakage path flow coefficient

C1 dynamic viscosity, CIPa s










6 angular displacement/local angle, degrees

8cut angle at which expansion process begins, degrees

6ex spread of the expander' s exhaust port, degrees

6in spread of the expander' s intake port, degrees

p density, kg/m3

co angular velocity, rad/s

9 stream availability or exergy, angle between base line and line through
stator center from rotor' s center, degrees









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF ROTARY-VANE TWO-
PHASE EXPANDERS INT VAPOR COMPRESSION REFRIGERATION SYSTEMS

By

Ahmad Mohamed Mahmoud

August 2008

Chair: S. A. Sherif
Cochair: William E. Lear
Major: Mechanical Engineering

The refrigeration and air-conditioning community has been searching for environmentally

friendly refrigerants and ways to improve energy efficiency and to increase the potential for

reductions in system size and weight for some time. These system improvements may manifest

themselves in numerous ways but significantly impact terrestrial logistics differently. This study

investigates the analytical and experimental utilization of rotary-vane expanders as throttle valve

replacements in vapor compression refrigeration systems with a specific application of these

expanders in smaller deployable environmental control units. The findings support the

conclusion that utilizing two-phase expanders in refrigeration systems is very promising.

Mathematical models for a system-level parametric cycle analysis were developed to

assess potential gains in the cycle coefficient of performance for R-22, R-134 and transcritical

CO2 vapor compression technology. Optimization studies were conducted to determine the

optimum performance of a refrigeration system subj ect to constraints of size and weight of both

the condenser heat transfer area and the size and weight of the two-phase expander.

An exhaustive literature survey aids and validates the suitability of the type of positive-

displacement expander used, after which comprehensive component level thermodynamic and










fluid dynamic models of a rotary-vane expander were developed to establish the performance of

this expansion device as a function of design and fluid parameters. This included rigorous

modeling of irreversible loss mechanisms such as throttling in the intake and exhaust ports, two-

phase internal leakage, friction, re-compression, and under or over-expansion due to incorrect

sizing of the expander or off-design operation.

Results from these models were used to establish the operating principles of a two-phase

rotary-vane expander. Experimental data gathered after modifying a conventional chiller with an

alternative flow path, where a rotary-vane expander and dynamometer have been installed, were

used to improve the fidelity of the analytical model developed. Issues such as real-time control to

satisfy system-level constraints, expander sizing and operational speed and inadequate operation

were addressed. The analytical model developed along with single-phase experimentation were

used to understand and overcome component-level complications and inadequacies due to

irreversible effects.









CHAPTER 1
INTRODUCTION AND OBJECTIVES

Introduction

The refrigeration and air-conditioning community has been searching for environmentally

friendly refrigerants and ways to improve energy efficiency and to increase the potential for

reductions in system size and weight for some time. Cycle improvements may manifest

themselves in many ways such as using multistage compression with inter-cooling and/or flash

gas removal as a means to reduce the work input to the compressor, employing economizers

and/or employing expansion devices to recover work.

There are two advantages to using expansion devices with output work. The first is that

lower enthalpy refrigerant is obtained at the inlet of the evaporator and hence increases the

refrigerating effect of the evaporator. The second is the extra work that can be extracted from the

expansion process, which can then be used to lower the input requirement of the compressor or

to operate auxiliary machinery such as fans or pumps. Both effects serve to increase the

coefficient of performance (COP) and hence raise the energy efficiency of the system as well as

increase the potential for reductions in system size and weight.

Objectives

The proposed overall approach is to combine experimentation with thermodynamic and

fluid dynamic modeling in order to demonstrate the operating principles, to discover any

unanticipated difficulties, and to establish a qualified design code which may subsequently be

used for optimizing air-conditioning and refrigeration systems for a wide range of applications.

A detailed list of these obj ectives is as follows:

*Conduct a literature survey of work on the use of various expansion devices in relevant
engineering cycles. This exhaustive survey would aid and validate the suitability of the
selected type of positive-di splacement expander used.










* Perform system-level parametric analysis of vapor-compression refrigeration systems to
assess potential gains in the cycle COP. This includes examining the factors that affect
reduced power requirements and increased refrigerating capacity.

* Develop a thermodynamic and fluid mechanic component-level model of two-phase
rotary-vane expanders to establish their efficiency as a function of design and fluid
parameters.

* Develop a comprehensive model of primary irreversible effects such as internal leakage,
friction, and throttling within the two-phase expander and present relevant discussion or
models of secondary losses.

* Design, build and modify an experimental refrigeration system to incorporate a rotary-vane
expander based on model inputs including rotary-vane expander sizing and rotational
speed.

* Develop an appropriate real-time control scheme to mimic the operational function of
conventional thermo-static expansion valves, i.e. to regulate mass flow-rate through the
rotary-vane expander, to satisfy system-level constraints.

* Utilize the developed computer program and comprehensive single-phase experimentation
to understand, recommend and possibly overcome component-level complications and
inadequacies due to irreversible effects.

* Utilize the experimental results where necessary to improve the fidelity of the qualified
design code.

Based on the motivation behind and obj ectives of this investigation, a detailed review of

recent literature pertaining to the use of expanders as throttle valve replacements in refrigeration

units was conducted. The review entails the use of two-phase expanders in conventional vapor

compression refrigeration systems with conventional CFC, HCFC and HFC refrigerants as the

working fluids. The review also encompasses the use of two-phase expanders in geothermal

systems and transcritical carbon-dioxide refrigeration cycles as well as single-phase expanders

utilized in organic Rankine power cycles and gas refrigeration cycles. This section will enable

the reader to understand the fundamentals of this technology as well as highlight practical

issues/problems that may have been presented by various researchers in this regard. The type of

expander used was chosen and suitability was based on this exhaustive review.










Analytic parametric modeling of the ideal and actual vapor compression refrigeration cycle

and modified cycles such as the vapor compression cycle with an expander, a liquid-to-suction

heat exchanger and a multistage vapor compression cycle with an economizer was then

conducted. These cycles have been analyzed in both ideal and actual cases and have also been

analyzed with the additional modification of utilizing a two-phase expander as a throttle valve

replacement. The working fluids investigated in this study include: R-12, R-22, R-134a, pure

hydrocarbons and trans-critical carbon-dioxide. Optimization studies were also conducted to

determine the optimum performance of a refrigerating unit subj ect to constraints of size and

weight of both the heat transfer area and the size and weight of the two-phase expander.

The development of a detailed thermodynamic and fluid dynamic model of the rotary-vane

expander followed. The geometry, kinematics and thermodynamics of a circular rotary-vane

expander were described mathematically to determine its ideal performance as a function of

design and fluid parameters. A robust thermodynamic model that takes into account primary loss

mechanisms such as friction and internal leakage and presents a discussion, models when

necessary, of secondary irreversible effects such as throttling in the intake and exhaust ports,

two-phase internal leakage, friction, re-compression and under or over-expansion due to

incorrect sizing of the expander is then described.

An experimental program that has been developed at the University of Florida for testing

the use of a two-phase rotary-vane expander in a refrigeration unit is then detailed. The

equipment and instrumentation used in the lab to gather experimental data to validate and

improve the fidelity of the analytical models are described. The sizing and selection of the

rotary-vane expander and the dynamometer is presented. Finally the means of real-time control

of the expander via a high-speed programmable dynamometer controller is presented. The









developed computer program and comprehensive single-phase experimentation was used to

understand, recommend and possibly overcome component-level complications and

inadequacies due to irreversible effects such as friction and internal leakage. A summary of key

technical challenges, recommendations and conclusions from both the comprehensive modeling

and experimental effort is then presented.









CHAPTER 2
LITERATURE REVIEW

The concept of utilizing expansion devices in various thermodynamic cycles is one that has

been described in scattered technical literature. Example applications in which expanders have

been used to recover and utilize lost work or waste heat include geothermal applications, Organic

Rankine Cycles (ORC), and refrigeration cycles. A thorough understanding of these applications

along with the types of working fluids used, the types of expansion devices used and inherent

loss mechanisms within those expansion devices will allow further advancement of such

concepts.

In literature dealing with expansion devices, the working fluid has been predominantly air,

steam, refrigerant vapor or transcritical carbon-dioxide. In the present study, expansion takes

place in a refrigerant' s two-phase region where additional complexities (e.g. erosion due to

cavitation and increased internal leakage due to significant density variation) arise. Although

complexities exist with the expansion of gases, vapors, and transcritical fluids, the loss

mechanisms of internal leakage, friction, and heat transfer and how they affect overall

performance of different types of expanders are qualitatively comparable to one another.

Understanding and incorporating these effects will aid in developing a mathematical model of a

two-phase expander employed in a vapor compression refrigeration/heat pump system.

Expanders in Refrigeration and Heat Pump Systems

There has been much effort to quantify the amount of savings and the methods by which

the work recovered and additional refrigerating effect could be utilized from the use of expanders

in refrigeration and heat pump systems. Among the researchers who investigated this are Zhang

and Yu (1992), Markoski (2003) and Brasz (1995).










Zhang and Yu (1992) compared ideal and modified refrigeration cycles for R-12, R-22, R-

502 and R-707. They found savings from utilizing two-phase expanders and recommended the

use of expanders. They failed to elaborate on the complexities associated with utilizing

expanders in the two-phase region. They mention only that components of the expander are

small and hence friction losses are large and that problems may arise from lubrication related

issues.

Brasz (1995) introduced the use of a turbo-expander in large (>85 tons) vapor-compression

refrigeration systems utilizing R-134a as the working fluid. According to Brasz, a turbine

efficiency of 39% could overcome the disadvantages of using R-134a instead of low-pressure R-

11 or R-123 refrigerants. He concluded that the application of this technology is applicable

where the expander rotational speed matches that of an existing drive train.

Markoski (2003) presented three alternative methods by which the expansion work from

an expansion engine can be utilized in a vapor compression refrigeration cycle. The author

provides brief guidelines and analyses for the use of the power recovered by the expander in

conventional and non-conventional vapor compression cycles as well as powering a circulation

pump for liquid circulation through a flooded evaporator.

Conventional Heat Pump and Refrigeration Systems

The use of expansion devices in conventional vapor-compression refrigeration cycles has

been investigated by Hays and Brasz (1996, 1998), Smith and Stosic (1995), Zoughaib and

Clodic (2003), Smith et al. (1994, 1996, 1999, 2001a, 2001b), Brasz et al. (2000), Kornhauser

(1990), Fischer (1978), Kim et al. (2004), Tamura et al. (1997), Disawas and Wongwises (2004),

Henderson et al. (2000) and Taniguchi et al. (1983).

In most of the aforementioned studies, unless mentioned otherwise, a throttling valve was

used as the default control mechanism responsible for ensuring the degree of superheat of the









refrigerant leaving the evaporator. The use of a throttling valve would however cause a loss in

the available pressure difference that may be utilized in an expansion device for work recovery.

Tamura et al. (1997) investigated the use of a screw expander in a high temperature (up to

180oC) binary heat pump with steam-water and ammonia as the working fluids. They found a

COP improvement of about 40% when comparing a high-temperature heat pump with one

employing an isentropic screw expander and one without in the operating range of 40-180oC.

Fisher (1978) described the concept of a pivoting-tip rotary-vane compressor and expander

applied to a solar-powered vapor-compression heat pump. The solar heat pump had a capacity of

3 tons and R-12 was selected since the author found that the required compressor and expander

displacements are of reasonable size to provide good efficiencies and low manufacturing costs.

Compression and expansion characteristics were taken into consideration before selecting the

working fluid as to avoid entirely the complexities that arise when expanding into the two-phase

regime. The author also describes the performance of pivoting-tip gas bearings, which were

utilized to reduce vane-tip friction in rotary-vane turbomachinery. Estimated efficiencies of 85%

were assumed for both the compressor and expander to reflect improvements that the pivoting-

tip gas bearings will introduce. A cooling coefficient of performance of 0.56 at design conditions

was achieved.

Taniguchi et al. (1983) presented a detailed analytical and experimental investigation of

helical screw two-phase expanders in large refrigeration systems whose working fluid is R-12.

They stated the added advantage of utilizing positive displacement turbomachinery as two-phase

expanders because they operate without erosion and slip losses between liquid and vapor phases.

The authors investigated the degree of subcooling on the performance characteristics of this

modified vapor compression cycle. Effects of vapor formation during the expansion process as









well as internal leakage losses were accounted for in the theoretical model. The helical screw

expander used in this study had a built-in volume ratio of 5:1. The expansion process volume

ratio of R-12, at the operating conditions, however depended on the degree of subcooling in the

condenser and varied from 13.2:1 and 10.6:1 corresponding to 0 and 12 K of subcooling

respectively. The authors concluded that the isentropic efficiency of the expander increased from

30 to 60% at rotational speeds of 500 and 3000 rpm respectively. The authors predicted

analytically an isentropic efficiency approaching 80% for larger screw expanders.

Kornhauser (1990) analytically investigated the use of a two-phase ej ector as a refrigerant

expander to replace the throttling process in a vapor-compression refrigeration system. The

author cites other expansion devices as expensive and susceptible to damage because of two-

phase flow. The power that is recovered by the expander is not extracted or coupled in any way

but is used to partially compress the refrigerant leaving the evaporator. The effects of both

isentropic expansion and partial compression significantly increase the coefficient of

performance of the system. The system is similar to a two-stage refrigeration unit where the

work extracted from the high pressure stage is used to drive the compression process of the low

pressure stage. A throttling valve expands the refrigerant in the low pressure cycle to the

evaporator pressure. According to the author this expansion is across a small pressure difference

and is negligible when accounting for the maximum work extraction from the cycle. Theoretical

calculations were made to compare the conventional and ej ector refrigeration cycles. A constant

pressure mixing model of the ej ector was used. The author found that for an ideal ejector

expansion cycle, an increase in the cycle coefficient of performance of 13%, 21%, 20%, 17%,

and 30% would be realized for R-11, R-12, R-22, R-113 and R-502 respectively. Results









detailing the decrease in compressor displacement and the performance gains of R-1 14, R-500

and R-717 are also presented.

An interesting obstacle that Kornhauser (1990) eludes to is the fact the conventional

expansion valve would defeat any performance improvements made by the use of an expansion

device. According to Kornhauser (1990), Newton (1972a, 1972b) has patented two methods by

which the liquid flow through the expander may be controlled. Controlling the mass flow-rate

through the ej ector may be accomplished by controlling the specific volume of the entering fluid

by means of inj ecting small amounts of hot gas into the refrigerant exiting the condenser.

Varying the nozzle area could also control the mass flow-rate through the ejector.

Disawas and Wongwises (2004) experimentally investigated the use of a two-phase

ej ector-expansion device in a conventional vapor compression refrigeration system whose

working fluid is R-134a. They used an ejector as the sole method by which expansion occurs

unlike other investigators who used a throttling device. They reported a maximum increase in

COP of 2% and 10% at an evaporating temperature of 16oC and a condensing temperature of 37

and 32oC, respectively. This is not an evaporating temperature typical in air-conditioning

applications. The ej ector system performance, when compared to the conventional system,

decreases greatly as the condensing temperature is increased and the evaporator temperature is

decreased.

Hays and Brasz (1996) presented a theoretical investigation of the power recovered by the

use of a two-phase turbine in a refrigeration system with R-134a, R-22, R-123 and propane as the

working fluids. The calculations were run assuming the turbine had isentropic efficiencies of 55

and 70%. The authors also experimentally investigated the performance of two-phase turbine in a









500 ton R-134a chiller. The authors determined a turbine efficiency of 60-65% and discussed the

turbine rotor loss mechanisms that accounted for this deficiency.

Hays and Brasz (1998) report on the implementation of a stand-alone turbo-expander as

throttle valve replacement for a large 2000-5000 ton centrifugal chiller installation at a

commercial building in Manhattan, New York. They reported that over 70 refrigeration units

employing two-phase turbines are in operation in a wide range of industrial and commercial

applications. They concluded that at an evaporator temperature of 44oF (6.7oC) and a condensing

temperature of 86oF (30oC), 15 and 180 kW of power can be produced by a 500 and 6000 ton

chiller unit respectively. These values are ideal and do not take into consideration the generator

efficiency. The installation cost for one of these large units is approximately $1000/kWe initially

and should decrease to a minimum of $400/kWe for additional units.

Zoughaib and Clodic (2003) investigated the use of a micro-turbine as an expansion device

in a domestic refrigeration unit. The power recovery would then be used to drive an auxiliary

fan(s) that would, through forced convection, ensure that no frost would form on the heat

exchanger of no-frost domestic appliances. The authors parametrically investigated the effect of

subcooling on the modified refrigeration cycle. The authors assumed a micro-turbine isentropic

efficiency of 80% and hence concluded that a 1.1% increase in the coefficient of performance

and 1.12 W of generated power were realized.

Smith et al. (1994) investigated the thermodynamic modeling of a Lysholm machine as a

two-phase expander in large-scale refrigeration systems utilizing R-113 as the working fluid.

They discussed the differences that existed between their model and the one proposed by

Taniguchi et al. (1983). Among the differences that the authors pointed out was the fact that

Taniguchi et al. (1983) assumed that the filling process would take place at a constant pressure.









According to the authors, the filling process does not take place at a constant pressure because

the acceleration of the entering refrigerant causes a pressure drop, which in turn induces flashing

and thus causes higher fluid velocities. The authors estimated that isentropic efficiencies of about

70-80% could be expected.

Smith and Stosic (1995) describe the principle behind a novel machine, which they dubbed

the "expressor" that is comprised of a coupled twin-screw compressor and twin-screw expander

in a single casing. They investigated the use of this expressor unit as a throttle valve replacement

in a conventional large chiller where R-134a is the working fluid. The authors estimated the

expander adiabatic efficiency as 70% and explained that large leakage losses do not affect the

performance of the expander because they are in the direction of the bulk flow. A coefficient of

performance gain of approximately 10% and 7.5% with 0 and SoC subcooling respectively was

realized. A throttle valve is directly upstream of the expressor device. Brasz et al. (2000) discuss

the disadvantages of this mechanism that make this technology rather expensive to implement.

The primary disadvantages mentioned were the need for a timing gear, the high cost of seals and

the need for two sets of rotors to carry out the compression and expansion processes.

Smith et al. (1996) presented the high efficiency design of two-phase screw expanders in

various cycles. Smith (1993) first presented the use of these two-phase expanders in a trilateral

flash cycle (TFC). In this cycle a fluid is pressurized adiabatically, heated at constant pressure to

its boiling point, expanded adiabatically as a two-phase mixture and then condensed at constant

pressure. Smith et al. (1994) then investigated the use of different working fluid mixtures to

increase the power output of a two-phase expander employed in the TFC. Smith et al. (1996)

concluded that if "a small amount of under-expansion is permitted, high speed, low built-in

volume ratio designs, with roughly half the volumetric capacity required for lower speed full










expansion alternatives, attain the highest overall adiabatic efficiencies." They also reported that

from a large air-conditioning unit data they were able to predict a 7-10% increase in the

coefficient of performance of the system. This could be achieved by utilizing the two-phase

screw expander to drive a compressor in a hermetic unit called the "expressor," (see Smith and

Stosic 1995) with an adiabatic efficiency of approximately 70%.

Smith et al. (1999) investigated the feasibility of utilizing a twin screw two-phase expander

in a large (500 ton) chiller unit that operated with R-134a as the working fluid. They attained an

expander adiabatic efficiency of approximately 70%. They listed the most significant factors that

hinder the adoption of two-phase expander in refrigeration systems as poor adiabatic efficiencies

and high cost of construction and installation. They discussed the results found by Smith et al.

(1996) as to the improper selection of the built-in volume of the expander when compared to the

actual expansion process volume ratio.

Brasz et al. (2000) presented the development of a twin screw "expressor" with only one

pair of rotors as a throttle valve replacement for a large water-cooled chiller. The need for a

timing gear and high cost seals was avoided by the use of high profie rotors developed earlier by

the authors. The compression process in the expressor unit recompresses the vapor that is

generated in the expansion process by means of power that is recovered. The recompressed

vapor would then be piped directly to the condenser inlet. The authors discussed the pros and

cons of different methods by which the power recovered may be utilized. They indicated that the

efficiency of the expander-compressor mechanism, expander and compressor is approximately

55%, 70% and 80% respectively. At part load, the speed reduction caused by the reduced flow

via the throttling valve negates the use of a control system. Initially the built-in volume ratio of

the screw expander was designed and built at 2.85:1. The authors found however evidence of









over-expansion when this 2.85:1 expander was used with R-134a and R-113 whose expansion

process volume ratios were 11.4:1 and 12.9:1 respectively. The authors then progressively

reduced the built-in volume ratio of the expander to 1.85:1 to avoid over-expansion.

Smith et al. (2001a) investigated the use of a helical twin screw compressor-expander as a

replacement to the throttle valve in a refrigeration system. Poor adiabatic efficiencies as well as

high manufacturing and installation costs are amongst the most significant factors hindering the

extensive use of two-phase expanders. According to Smith et al. (2001a) they have shown that

the adiabatic efficiency of the expander, a reported maximum of 70%, is significantly higher

because the built-in volume ratio of the expander is less than the volume ratio of the actual

expansion process. They designed, built and tested an expander unit with a built-in expansion

ratio of 2.85:1. The expander provided a 3.6-10.3% increase in the system coefficient of

performance for an actual expansion process volume ratio of 11.4:. i

Smith et al. (2001Ib) presented an economical analysis of two-phase screw expanders that

may be utilized in organic Rankine cycles, Trilateral Flash Cycles and refrigeration units. They

reported a peak adiabatic efficiency of 76%. Although this technology could be utilized in

refrigeration systems, the authors conclude that it would be more significant for Trilateral Flash

Cycles (Smith, 1993 and Smith et al., 1994, 1996)

Kim et al. (2004) presented preliminary results of an investigation of the use of a scroll

expander with a heating structure and a proposal of their use in a refrigeration cycles as two-

phase expanders. When used in high-temperature and high-pressure applications, clearances in

both the radial and axial directions are caused by differential thermal expansion of the scroll

elements. This may result in a decrease in the expander efficiency as well as the expander's

specific power output. The authors investigated the use of a heat pipe that would provide a









uniform temperature throughout the scroll elements by means of heating. According to the

authors, if the scroll expander with a heating structure is used in two-phase expansion, instances

of over expansion will cause the liquid in the expander to evaporate by means of heat from the

refrigerated space. They concluded that this in turn would both increase the power output of the

expander and the cooling capacity of the system due to a higher volumetric expansion ratio

needed in the expander. In this case the expander could also be thought of as a partial

evaporator.

Henderson et al. (2000) theoretically investigated the economics of employing compressor-

expander devices in heat pumps that use R-410 as their working fluid. They investigated using a

screw compressor-expander since it was found that the rotary-vane compressor-expander would

likely fail because of excessive friction at the vane-tip/stator surface and could also be very

noisy. They found that if the device had an 80% isentropic efficiency, the system COP would

increase by 30%. They have not specified the operating conditions or any component efficiencies

in their study.

In a maj ority of the studies, no subcooling in the condenser was allowed as to determine

the maximum possible work extraction from the expansion process. This however would result

in immediate formation of vapor upon the slightest decrease in pressure and increase of velocity.

This in turn could cause cavitational erosion, especially in dynamic expansion devices. This

phenomenon however has not been addressed directly.

Transcritical Carbon Dioxide Systems

The inherent coefficient of performance of a transcritical CO2 CyClC is less than the

coefficient of performance of a conventional vapor compression cycle. Among the methods by

which this deficiency may be overcome is to utilize expanders as throttle valve replacements.

One primary difference between the use of expanders in a transcritical CO2 cycle and a









conventional cycle is the fact that complexities such as cavitation only take course as the velocity

of the incoming refrigerant has decreased (i.e. at later stages in the expansion process). In

conventional vapor-compression cycles these complexities may occur immediately and is a

strong function of the degree of subcooling and the severity of loss mechanisms during the intake

phase of the expansion process. The use of expanders in transcritical carbon dioxide systems

have been investigated by Robinson and Groll (1998a, 1998b), Baek et al. (2005a, 2005b), Li

and Groll (2005), Huff et al. (2002, 2003), Fukuta et al. (2003), Zha et al. (2003), Nickl et al.

(2002, 2003), Stosic et al. (2002), Hays and Brasz (2004), Heyl and Quack (1999), Li et. al

(2000), Driver (1996), Driver and Davidson (1999), Heyl et al. (1998), Ertesvag (2002) and

Preissner (2001).

Robinson and Groll (1998a, 1998b) analytically investigated and compared the

performance of a transcritical CO2 cycle and a conventional R-22 refrigeration cycle with and

without an expansion turbine. They found that use of an internal heat exchanger along with an

expansion device decreased the COP of the transcritical CO2 cycle by 6-8%. They concluded that

the stream availability following the heat rej section process in the condenser is better utilized by

an expansion device rather than an internal heat exchanger. They also found a 23% COP increase

when comparing an R-22 refrigeration cycle and a transcritical CO2 cycle both with expansion

devices with a 60% isentropic efficiency operated at an evaporating temperature of SoC and a

condensing temperature of 35oC.

Baek et al. (2005a, 2005b) studied both analytically and experimentally the effect of the

addition of a newly designed piston-cylinder work-producing device to a transcritical CO2 cyc 0.

Experimentally they found a 10% increase of the cycle COP taking into account both the

increase in evaporator capacity and expansion work. The limiting factor in this study was the










isentropic efficiency of the expander, which was reported to be 10% (experimentally) and 34%

(theoretically). The discrepancy between the two can be attributed to the 30% uncertainty in the

instruments used for measurements as well as the inadequate theoretical internal leakage model

that was used.

Li and Groll (2005) analytically investigated the use of an ej ector-expansion device in a

transcritical CO2 cycle. They found that the cycle COP can be increased by 16% by assuming

that the ej ector has an isentropic efficiency of 90%. They have however not investigated the

potential loss in efficiency due to various loss mechanisms.

Heyl and Quack (1999) presented the process calculations, design and results of a novel

free piston expander-compressor for a CO2 cycle. The expander-compressor device had two

expansion and two compression cylinders and was assumed to have an isentropic efficiency of

85%. They noted an increase of cycle efficiency of 38.7% and 10% with and without work

recovery from the expansion device. They reported an experimental coefficient of performance

gain of 30%. They concluded that a significant disadvantage to this type of compressor-expander

was that the expander and compressor pistons moved with identical strokes, which did not utilize

all of the available expansion work (under-expansion). The operating conditions were not

detailed.

Nickl et al. (2002) presented a "second generation" novel piston expander-compressor that

would attempt to overcome the disadvantages of the expander-compressor presented by Heyl and

Quack (1999). They developed a double rocker arm that would control the different speeds of the

expander and compressor pistons. They concluded that this mechanism would prove to be

complex and expensive. A slight increase of the coefficient of performance relative to the "first

generation" expander-compressor (Heyl and Quack, 1999) was realized.









Nickl et al. (2003) presented a "third generation" novel piston expander-compressor with

three expansion stages for a transcritical CO2 Cycle. According to them, this would eliminate the

identical strokes of the expander and compressor pistons (Heyl and Quack, 1999) in their "first

generation" expander-compressor. It would also eliminate the complex double rocker arm

needed to control the different speeds of the expander and compressor pistons (Nickl et al., 2002)

of the "second generation" piston expander-compressor. The author' s first estimate of the

isentropic efficiency of the expander-compressor was 85% and did not take into account losses

due to heat transfer, pressure drops in valves, internal leakage and friction losses. They

concluded that the isentropic efficiency of the expander decreased only 3% when these losses

were taken into consideration.

Zha et al. (2003) developed a rotary-type rolling piston expander to use as a throttle valve

replacement in a transcritical CO2 CyClC in Small heat pumps and refrigeration systems. They

experimentally determined the design parameters, method of control and determined the loss

mechanisms associated with this expander. They also mentioned that cavitation and liquid

slugging are among the obstacles that hinder the development of CO2 expander technology. They

have estimated the isentropic efficiency of the rolling piston expander to be about 50%. They

have attributed the losses in efficiency to friction (~24%) and internal leakage losses (~25%).

Fukuta et al. (2003) investigated and predicted the performance of a rotary-vane expander

utilized in a transcritical CO2 CyClC aS a replacement to the throttle valve. They cited issues of

slugging and cavitation as the main obstacles to the development of two-phase expander

technology. The prototype expander that was used had a built-in expansion ratio of 2:1. They

initially assumed an expander efficiency of 60%. After taking into account the losses from heat

transfer and internal leakage they calculated an isentropic efficiency of 43%.









Huff et al. (2002) developed an algorithm that estimates the performance of the expansion

and compression process in positive displacement turbomachinery. They have taken into account

internal leakage losses, heat transfer and valve losses in their modeling effort. They have applied

this analysis for scroll, piston, rotary-vane, rotary piston and screw type mechanisms. They have

applied this analysis with particular interest to a transcritical CO2 cycle and have reported a

coefficient of performance gain of 40-70% and a 5-15% increase in capacity. The authors

discussed the importance of the built-in volume ratio of the expansion device in use and the

effect it may have on the high side pressure of the cycle. They concluded that if the rotational

speed and built-in volume ratio of the device is not adjustable then this will ultimately reduce the

benefit of the expander in a transcritical cycle. They proposed utilizing an expander and a

throttling valve either in series or parallel to match the high side pressure with the optimum

pressure. They concluded however that the performance gain will be relatively less than if

variable rotational speed and volume ratio were employed.

Huff et al. (2003) experimentally investigated the use of two R-134a scroll expanders in a

transcritical carbon dioxide cycle. The first prototype was modified by reducing the wall height

of the original compressor scrolls. This modification severely influenced this prototype' s 28%

isentropic efficiency. The highest isentropic and volumetric efficiencies that were reported were

42% (at 1800 rpm) and 68% (at 2200 rpm), respectively, in the case of the second unmodified

expander.

Stosic et al. (2002) investigated the use of a twin screw combined compressor and

expander for a high temperature transcritical CO2 refrigeration systems. They stressed the need

to reduce bearing loads as this would make the expander-compressor device more applicable to a

larger range of pressures. They achieved a 20% reduction in the bearing load by balancing the









loads of the compressor-expander rotors. They also concluded that throttling losses much larger

than those associated with conventional refrigerants exist in this helical type turbomachine. They

reported an idealized 72% increase in the coefficient of performance when applying this

compressor-expander device to a high temperature transcritical CO2 refrigeration cycle.

Hays and Brasz (2004) experimentally investigated the use of a turbine-compressor

mechanism in a 6 ton transcritical CO2 refrigeration or liquefaction cycle. The axial flow turbine

previously presented in Hays and Brasz (1996) as a two-phase expander is used to drive a

compressor in a hermetic turbine-compressor unit. At a rotational speed of 10,000 rpm the

turbine and compressor had efficiencies of 69 and 80% respectively. The turbine had a measured

efficiency of 56% at its maximum speed of 12,800 rpm. The vapor exiting the evaporator is then

compressed in the turbine-compressor mechanism. The dual effects of expansion and partial

compression in the turbine-compressor mechanism increased the system coefficient of

performance by 39%.

Driver (1996) theoretically described the potential of utilizing a compressor-expander

device in a vapor compression system. Driver and Davidson (1999) have developed and tested a

hinged-vane compressor-expander unit for transcritical CO2 CyClCS. They found that if the

compressor-expander device operated at a 100% isentropic efficiency, which is highly unlikely, a

COP improvement of 40% could be realized when compared with a conventional heat pump.

The investigators pointed out that their compressor-expander device would not work with

refrigerants such as R-134a because of sizing issues.

Preissner (2001) investigated the use of scroll expanders in a transcritical CO2 cycle. The

author also performed an experimental comparison of a conventional R-134a and transcritical

CO2 refrigeration cycles employing suction line heat exchangers. He found that CO2 CyClCS falls









short in performance by 8 and 23% for a range of typical operating conditions with particular

emphasis on medium to high outdoor temperatures. With the implementation of a scroll expander

the cycles have an identical coefficient of performance at a condensing temperature of 25oC. At

higher condensing temperatures though, he found that a significant performance gap exists in

favor of the R-134a cycle. The author found internal leakage losses to be the most significant

loss mechanism due to limitations in machining accuracy and geometric design. The expander

was thus not implemented in the cycles.

Li et al. (2000) investigated the use of a vortex tube expansion device in a transcritical CO2

refrigeration cycle. The vortex tube's energy separation effect, known as the Ranque-Hilsch

effect, separates the high pressure gas entering the vortex tube into two low pressure streams one

at a low temperature and the other at a high temperature. The authors compared this expansion

device to a reciprocating piston expansion device already developed by the authors. They

concluded that an increase in the system coefficient of performance of 20% could be achieved if

the piston expander's isentropic efficiency was 50%. The vortex tube expander' s isentropic

efficiency would have to be on the order of 38% to achieve the same performance gain.

Heyl et al. (1998) developed a free piston compressor-expander unit to be employed in a

transcritical CO2 cycle. They found that because of the "full-pressure principle" a work

extraction of 78% in relation to a complete expansion is obtained. Ertesvag (2002) described the

geometry and thermodynamics of a patented rotary-piston machine that can be used as a

compressor-expander device invented by K. Vading. He mentioned the possible use of this

concept in vapor compression systems.









Gas Refrigeration Cycles and Cryogenics

The investigation of the use of expanders in gas refrigeration and cryogenic liquefaction

cycles has been investigated by Gnutek (1979), Gnutek and Kalinowski (1986), and Baron and

Trembley (2003).

Gnutek (1979) analytically investigated the use of a rotary-vane expander in cryogenic

liquefaction and vapor-compression gas refrigeration systems. The van der Waal's equation of

state was used to determine the thermodynamic properties of the gas and therefore determine the

performance of this expander device. The effects of friction, internal leakage losses and heat

transfer from the ambient have been taken into consideration.

Gnutek and Kalinowski (1986) described the use of a two-stage rotary-vane compressor

and a single-stage rotary-vane expander in a gas refrigeration unit using air as the working fluid.

The authors obtained a -98oC evaporating temperature. Analytical equations are presented to

describe the performance of the expander. The authors conclude that a gas refrigeration system

employing a rotary-vane expander and compressor would be significantly smaller in size than

that utilizing a reciprocating piston compressor and expander. The work extracted from the

expander was coupled to a generator.

Baron and Trembley (2003) investigated the use of indirect heat exchange and a turbo-

expander in a cryogenic refrigerator using liquid nitrogen as the refrigerant. According to the

authors, heat transfer to the boiling liquid nitrogen is critical in cryogenic applications in order to

ensure that the latent heat of vaporization is maintained as high as possible. The authors

theoretically found that an expander whose isentropic efficiency is 75% can increase the amount

of heat transferred to the liquid nitrogen by 28%. The authors discussed the pros and cons of

using scroll and rotary-vane type expanders. The authors selected a centrifugal turbo-expander










because of compactness and cost. The author' s estimated that a turbo-expander should not

exceed $10,000 in order to be economically viable.

Expander Selection

Although the concept of utilizing a compressor-expander device is appealing, it cannot be

applied to vapor compression and heat pump systems for the following reasons:

* The necessary built-in volume ratios of the compressor and expander are very different.
The expander may have a volume-ratio 7-8 times that of the compressor.

* There is an issue pertaining to the unequal inlet volume flow-rates, which are a direct result
of unequal inlet densities.

* Cavitation erosion and other complexities may arise from two-phase expansion which is
likely to affect the expansion efficiency and may also have a negative impact on the
compression process and therefore greatly reduce the system COP.

Baek et al. (2002) suggested that rotary-vane expanders have an exceptional potential

being utilized as expansion devices in refrigeration systems. This may be the case because of the

fact that they may be mounted on the compressor shaft for ease of work recovery and have

relatively less complicated valve timing. Rotary-vane expanders are also capable of

accommodating higher rotational speeds. This translates into higher fluid handling capacity and

hence a more compact device. Together with the conclusions mentioned above by the different

investigators, the potential of employing rotary-vane type expanders in conventional vapor

compression systems/heat pumps is very promising.









CHAPTER 3
PARAMETRIC ANALYSIS OF CYCLES

Models for the following refrigeration cycles have been developed in FORTRAN90

utilizing REFPROP 7.0's R-134a, R-22, R-12 and pure hydrocarbon property subroutines:

* Ideal single-stage cycle (henceforward referred to as the base cycle)

*A modified ideal single-stage cycle with an expansion device (the expansion device will
henceforward referred to as ED)

* Actual single-stage cycle with and without an ED

* Ideal cycle with two-stage compression and an economizer (henceforward referred to as
the economizer cycle)

* A modified ideal economizer cycle with an ED

* Actual economizer with and without an ED

* Ideal cycle with an internal heat exchanger (henceforward referred to as the IHX cycle)

* A modified ideal IHX cycle with an ED

Both the evaporating and condensing temperatures were varied for all 8 cases. The system

COP, refrigerating efficiency, refrigerating capacity, and required input work were found to be

greatly dependent on the operating conditions. It is of significant interest to calculate the percent

differences (system COP, refrigerating capacity and required input work) for the cycles with and

without expansion devices.

Ideal Cycles

This section will describe both the thermodynamic processes involved with the base,

economizer and IHX refrigeration cycles and the results when these cycles are modified by the

addition of a two-phase expander as a throttle valve replacement.










Base Cycle

The single stage ideal vapor compression cycle is the simplest cycle and is the basis of

most refrigeration and air-conditioning technology. These cycles have been utilized in cascade

refrigeration systems to achieve higher condensing or lower evaporating temperatures.

A schematic of the base cycle is shown in Figure 3-1. Both the T-s and P-h diagrams are

shown in Figures 3-2. The base cycle consists of four processes that take place in the four

components shown in Figure 3-1. Both the condensing and evaporating temperatures, To and Te,

are initially known since they are usually design variables. This allows the computation of the

corresponding high and low saturation pressures, P, and Pe, of the cycle. The refrigerant at State

1 is ideally in a saturated vapor state (xl=1). The properties at State 1 can be determined from

P1=Pe and xl=1.

In the base cycle, the refrigerant vapor undergoes isentropic compression in the

compressor and exits at State 2. The properties at this state are found utilizing the following

assumptions: P2=P, and s2=S1. The superheated refrigerant then enters the condenser where it

undergoes isobaric heat rejection. The exit, State 3, of the condenser is saturated liquid (x3=0).

The properties at State 3 can be found by P3=P, and x3=0. The saturated liquid refrigerant is then

throttled in an expansion valve. This is an isenthalpic process. The properties at State 4 are thus

determined from h4=h3 and P4=Pe. In the case where an expansion device is utilized as a throttle

valve replacement, the saturated liquid undergoes an isentropic expansion process. The

properties at State 4' are thus determined from s4'=S3 and P4'=Pe. The two phase refrigerant then

undergoes isobaric heat addition in the evaporator. The refrigerant is ideally saturated vapor at

the exit of the evaporator.









The coefficient of performance for the base cycle and the base cycle with an expansion

device (assuming the work output from the expander is ideally coupled to the compressor) are

defined by Equations (3-1) and (3-2), respectively


coP, = n4 (3-1)


q, h -h4
C'OP, = (3 -2)
b,exp comp ex (h h)-(4 -h

Economizer Cycle

The economizer cycle is modified so that the compression and expansion processes are

carried out in two stages. A liquid/vapor separator increases the cooling capacity of the cycle. A

schematic of the T-s and P-h diagrams and cycle is shown in Figures 3-3 and 3-4, respectively.

Both the condensing and evaporating temperatures, To and Te, are initially known since

they are usually design variables. This allows the computation of the corresponding high and

low saturation pressures, P, and Pe, of the cycle. The refrigerant at the evaporator exit, State 1, is

saturated vapor. The properties at State 1 can be determined by xl=1 and P1=Pe. The saturated

vapor then undergoes isentropic compression in the compressor. The properties at the 1st stage

compressor exit, State 2, can be found by S2=S1 and P2=Pi, where Pi is the intermediate pressure

of the cycle. Stoecker (1998) has stated that the optimum intermediate pressure in a two-stage

compression process can be determined by the following relationship




The refrigerant at State 5, the condenser outlet, is saturated liquid. The properties at State

5 can thus be found from PS=P, and xg=0. The refrigerant is then either expanded by an

isenthalpic process (h6=h5, P6=Pi) in the throttle valve or by an isentropic process in the

expansion device (S6'=S5, P6'=Pi). The liquid (noted by f) and vapor (noted by g) are then










separated. The vapor in the amount of x is fed into the inlet of the second compressor. The liquid

in the amount of (1-x) however is expanded again. The liquid refrigerant, at State 6f, is either

expanded by an isenthalpic process (h7=h6f, P7=Pe) in the throttle valve or by an isentropic

process in the expansion device (s7'=S6'f, P7=Pe). The two-phase refrigerant then undergoes

isobaric heat addition in the evaporator and is saturated vapor at State 1. State 3 remains to be

determined. An energy balance of the two streams feeding into State 3 will determine the

enthalpy at State 3 for the cases of isenthalpic and isentropic expansion processes as described

by equations (3-4) and (3-5) respectively.

h3 = 6h6g + 6 )h2 (3 -4)


h3 X6'h6'g + 1 6')h2 (3-5)

The properties at State 3 can then be determined from h3 and P3=Pi. The refrigerant at

State 3 then undergoes isentropic compression in the second compressor. The properties at State

4 can then be determined from s4=S3 and P4=P,.

The coefficient of performance for the economizer cycle and the economizer cycle with an

expansion device (assuming the work output from the expander is ideally coupled to the

compressor) are defined by equations (3-6) and (3-7), respectively


CO%, = (3-6)



COP
econ, exp
W +W -W -W
comp,1 comp,2 exp,1 exp,2
(3-7)
COP (-x h-h)
con"'"exp 6,)>(h2 h)>+(h4 h3)- (h, h6~ 1 6, >h6'f h,,)










Internal Heat Exchanger (IHX) Cycle

The internal heat exchanger (IHX) cycle is also known as the liquid-to-suction heat

exchanger (LSHX) cycle. This cycle incorporates a heat exchanger that subcools the saturated

liquid leaving the condenser and superheats the saturated vapor leaving the evaporator. A

schematic of the IHX is shown in Figure 3-5. Both the T-s and P-h diagrams are also shown in

Figures 3-6 respectively.

Both the condensing and evaporating temperatures, To and Te, are initially known since

they are usually design variables. This allows the computation of the corresponding high and

low saturation pressures, P, and Pe, of the cycle. The analysis of the IHX cycle begins with the

determination of States 3 and 6. The refrigerant is saturated vapor at the exit of the evaporator.

The state properties may be determined from X6=1 and P6=Pe. Upon the refrigerant exiting the

condenser, State 3, it is saturated liquid. The properties at this state point may be determined by

x3=0 and P3=P,.

The effectiveness of the internal heat exchanger is defined as follows

q,, T -T7
&,,= ct 1 (3-8)
max T, T,

For the case of the ideal cycle, the effectiveness is assumed to be equal to unity. This

allows for the calculation of the temperature at State 1 (T1=T3). Using T1 and P1=P, the other

state properties may be determined. State 4 may be determined by performing an energy balance

on the IHX, assuming the mass flow of refrigerant and the specific heat of the refrigerant are

both constant, as follows

qn,4 = q6->1 7 6 = 3 4 (3 -9)

In the case of effectiveness equal to unity, T4= T6, the properties of the refrigerant at State 4

can be determined from T4 and P4=P,. The superheated vapor at State 1 undergoes isentropic









compression in the compressor and exits at State 2. The properties at State 2 can be determined

by S2=S1 and P2=P,. The subcooled liquid at State 4 can either be put through isenthalpic

expansion in the expansion valve (h5=h4, P5=Pe) or isentropic expansion in the expansion device

(sg'=s4, P5'=P4). The two-phase refrigerant then undergoes isobaric heat addition in the

evaporator and exits as saturated vapor at State 6.

The coefficient of performance for the IHX cycle and the IHX cycle with an expansion

device (assuming the work output from the expander to be ideally coupled to the compressor) are

defined by equations (3-10) and (3-11) respectively

q, h6 h,
C'OPH 0 (3-10)
w,, h, h

q, h6 hs,
COP "e (3-11)
IHXex com ex h h4-h

For the base cycle configurations (with and without an ED) at a constant condenser

temperature (25oC), the percent increase realized in the system COP was 8% to 19%, while an

increase of 0.5% to 3.4% in refrigerating capacity was realized. The evaporator temperature was

varied from -12oC to 10oC. The required work input to the compressor was found to decrease by

7.5% to 16%. For higher condenser temperatures, the percent differences in the system COP

were found to increase by 10-21%, 13-24%, 15-27% and 18-30% for condensing temperatures of

30oC, 35oC, 40oC and 45oC, respectively.

Actual Cycles

The primary models: (1) simple vapor compression cycle; (2) vapor compression cycle

with an economizer; and (3) vapor compression cycle with an internal heat exchanger, have been

further modified to include the isentropic efficiencies of both the compressor and expansion

devices. The effects of superheating, sub-cooling and pressure drops (both in the line and in the









heat exchangers) as well as the effectiveness of the heat exchangers have also been considered.

These effects would help achieve a more practical system model.

For the actual base cycle system configurations (with and without an ED) and the case of

constant condensing temperature (25oC) and varying evaporating temperature (-12oC to 10oC),

the percent increase realized of the system COP was 3.4% to 8.9%, while an increase of 0.2% to

1.7% was realized in terms of the refrigerating capacity. The required work input to the

compressor was found to decrease by 3.3% to 7.2%. For a compressor efficiency of 85%, an

expander efficiency of 50% and internal heat exchanger effectiveness of 70%, the values

aforementioned are approximately half the values attained in the ideal case. The results for both

the ideal and actual cycle improvements are presented in Table 3-1 and in Figures 3-7 to 3-13.

It can be concluded from the preliminary results obtained that the maximum potential gain

in the cycle coefficient of performance is obtained for the base cycle. This conclusion supports

the work of Robinson and Groll (1998a) who found that the stream availability following the

heat rej section process in the condenser is better utilized by an expansion device rather than an

internal heat exchanger. Although improvements may be realized in the economizer cycle, the

complexity and capitol cost of a two-stage vapor-compression system hinder the utilization of

this cycle with an expansion device.

The following additional parametric inputs have been varied in order to determine the most

important parameters when selecting the expansion device as well as the optimum operating

conditions.

Effect of Subcooling on System Performance

The degree of subcooling that the refrigerant undergoes in or after the condenser is a key

factor that affects both the power requirement and the refrigerating capacity of a vapor

compression refrigeration cycle. Analysis of the effect of the degree of subcooling on the key










system performance parameters including the refrigerant effect, expander work-output and

overall system COP has been done. An increase in the degree of subcooling was found to

decrease the useful work output from the expander but increases the overall system COP by

increasing the refrigerating effect. Figures 3-14 to 3-18 show the aforementioned trends.

Among the most promising preliminary conclusions that may be drawn from this analysis

is that cavitation erosion may be greatly reduced in the inlet of the expander since, depending on

the degree of subcooling, the refrigerant entering the expander may only be liquid. Hence, vapor

will only start to form when the refrigerant has experienced a sizeable pressure drop in the

expander' s cavity (i.e. at large angular displacements). According to current literature, cavitation

erosion in turbomachinery is prevalent at the inlet and the initial stages of compression and

expansion due to the high velocity and composition associated with the incoming two-phase

flow.

Effect of Superheating on System Performance

The degree of superheat that the working fluid experiences in the evaporator is critical in a

vapor compression refrigeration system to ensure that the working fluid leaving the evaporator is

completely vapor. Besides the well documented benefits of better compressor performance and

longer service life, the following figures (Figures 3-19 and 3-20) analyze whether the degree of

superheat has any significant effect on the system COP, refrigerating effect and the net work

input to the system.

A similar parametric analysis, as in the case of condenser subcooling, reveals that the

increase in the degree of superheat in the evaporator does not significantly increase/decrease the

work input into the system nor increase the refrigerating effect produced by the system. For a

large degree of superheat such as 90C, the gain in the refrigerating effect is roughly 5%. These









results are presented in Figures 3-19 and 3-20. There exists however a trade-off between the

degree of superheat and the heat exchanger area required in order to achieve that superheat.

Evaluation of CFC, HFC and Transcritical CO2 Refrigeration Cycles

Vapor compression refrigeration research has been significantly influenced by the

elimination of ozone-depleting refrigerants such as chlorofluorocarbons (CFCs), the planned

phase out of partially ozone-depleting refrigerants such as hydrochlorofluorocarbons (HCFCs),

and the interest in replacing what was once thought to be harmless refrigerants such as

hydrofluorocarbons (HFCs). Among the frontrunners in replacing these refrigerants is carbon

dioxide. A first and second-law based performance evaluation of conventional refrigeration and

transcritical carbon dioxide systems is presented here. The working fluids investigated include

R-134a, R-22 and R-744 (CO2). The systems investigated include the basic and modified vapor

compression systems. The modified system consists of a two-phase expansion device that is

employed as a throttle valve replacement. The irreversibilities of the different system

components in the basic and modified systems are presented. Comparison of the potential

increases in the refrigerating effect, reductions in the work required to operate the system and

resulting increase in the system coefficient of performance are discussed. Comparison of the

reductions/increases in cycle irreversibility and component contributions due to the employment

of an expansion device are also discussed. Key expander operational parameters, such as the

process volume ratio, are investigated.

A recent revival of carbon dioxide's use as a refrigerant has been given much attention in

recently published literature. This arises from the need to replace ozone depleting CFCs and

HCFCs and the global warming contributors HFCs. In comparison, carbon dioxide is naturally

occurring and is thus an environmentally friendly refrigerant with an ozone depletion potential of

zero. Published literature that discussed the revival of carbon dioxide's promise as a potential









candidate to replace CFCs and HFCs includes those of Lorentzen (1994), Lorentzen and

Pettersen (1993). Due to the low critical temperature of CO2 however, 30.98oC, the heat rejection

process typically takes place at a supercritical temperature in a gas cooler cooled by ambient air.

The critical pressure of CO2 is 7.3 8 MPa. This high heat rej section pressure requires a substantial

increase in the compressor work required to achieve it. This dramatically decreases the system

coefficient of performance. Another inherent theoretical limitation of the basic transcritical CO2

system is the fact that the refrigerant leaving the gas cooler may only be cooled to the ambient

temperature. To overcome this, an internal heat exchanger may be utilized to provide a means of

subcooling. In the case of a conventional vapor-compression system, subcooling is typically

obtained by increasing the area of heat transfer of the condenser. It has been reported that the

performance of transcritical CO2 Systems suggests that the technology is comparable to that of

existing systems (Robinson and Groll, 1998). It has also been suggested that transcritical CO2

component technology must be optimized in order to meet the unique operational characteristics

of carbon dioxide. This technology however differs significantly from conventional refrigeration

systems (Kim et al. (2004)). Although the state-of-the-art of transcritical CO2 Systems is still far

from optimum, the inherent thermodynamic performance of the basic transcritical CO2 System

must be comparable to existing refrigeration technology. The system coefficient of performance,

regardless of the technology used, may be increased by the utilization of expansion devices,

liquid-to-suction (internal) heat exchangers or performing compression in multiple stages, to

name a few. In particular, many researchers have concluded that the isenthalpic throttling

process may be the largest source of irreversibility in a transcritical CO2 System.

This section analyzes and compares the use of an expansion device as a throttle valve

replacement in both conventional and transcritical CO2 Systems. There are two advantages to









using expansion devices with output work. The first is the fact that lower enthalpy refrigerant is

obtained at the inlet of the evaporator, which increases the refrigerating effect. The second is the

extra work that can be extracted from the expansion process, which can then be used to lower the

input power requirement of the compressor. The degree of potential increase of the system's

coefficient of performance however is greatly dependent on the isentropic efficiency of the

expansion device employed. Different expansion devices have been suggested in recently

published literature. Reduction of irreversibilities associated with the expansion process also

leads to an increase in the second-law efficiency of the system. The extent of these

improvements in both conventional systems utilizing R-22 and R-134a as refrigerants and

transcritical CO2 Systems are presented and discussed.

An abundant amount of literature dealing with modified refrigeration systems that employ

expansion devices exists. For the purpose of this section, only the most pertinent of those studies

are examined. Robinson and Groll (1998) analytically investigated and compared the

performance of a transcritical CO2 System and a conventional R-22 refrigeration system with and

without an expansion turbine. They found that the use of an internal heat exchanger along with

an expansion device decreased the COP of the transcritical CO2 System by 6-8%. They

concluded that the stream availability following the heat rejection process in the condenser is

better utilized by an expansion device rather than an internal heat exchanger. They also found a

23% COP increase when comparing an R-22 system and a transcritical CO2 System both with

expansion devices with an isentropic efficiency of 60% and when the system is operated at

evaporating and condensing temperatures of 5 and 50oC, respectively. They concluded that a

conventional R-22 system employing an expansion turbine outperforms all other systems

including the transcritical CO2 System with an internal heat exchanger and an expansion turbine.









Preissner (2001) investigated the use of a scroll expander in a transcritical CO2 System. He

also performed an experimental comparison of conventional R-134a and transcritical CO2

refrigeration systems employing suction-line heat exchangers. He found that the CO2 System's

COP is 8 and 23% less than the R-134a system for a range of typical operating conditions with

particular emphasis on medium-to-high outdoor temperatures. With the implementation of a

scroll expander the systems were found to have an identical coefficient of performance at a

condensing temperature of25oC. However at higher condensing temperatures a significant

performance gap existed in favor of the R-134a system. Preissner (2001) found internal leakage

losses to be the most significant loss mechanism due to limitations in machining accuracy and

geometric design. The expander was thus not implemented in either system.

Component-level models are needed for the analysis of conventional and transcritical CO2

systems with and without an expansion device in place of a throttling valve. A general solution

method has already been presented. Detailed component-level models that may include

geometry, flow arrangement and loss mechanisms for the compressor, condenser, evaporator and

expander may be developed separately and incorporated into this model when required. These

component-level models will provide better estimates of the isentropic efficiencies of the

compressor and expander as well as the heat exchanger size and effectiveness of the condenser

and evaporator, respectively.

In the analysis described here, R-22, R-134a and CO2 are aSSumed to be the working

fluids. The thermodynamic properties of these fluids are provided by means of coupling

REFPROP 7.0's (Lemmon, 2002) property subroutines with the models developed for this

analysis. Pressure losses in all system components and throughout the system's piping have been










neglected. Figure 3-1 shows a schematic of a conventional and transcritical CO2 refrigeration

system with and without an expansion device.

The two systems utilize a compressor, an expansion device and an evaporator. The only

component that differs is the device by which heat is rej ected at the high pressure of the system.

This component is a condenser in a conventional system and a gas cooler in the transcritical CO2

system. Figure 3-21 shows the T-s diagram of each system, respectively. Regardless of the

refrigeration technology used, the required input work to compress saturated vapor at the

evaporating pressure to superheated vapor at the pressure of heat rej section is


w = (3-12)


The isentropic efficiency re of the compressor takes into account irreversibilities such as

internal leakage and friction. Kim et al. (2004) reported that although leakage and frictional

losses associated with carbon dioxide compressors are lower, the current state-of-the-art carbon

dioxide compressor doesn't perform as well as a conventional refrigeration compressor. For the

sake of comparison, isentropic efficiencies of current state-of-the-art R-22 and R-134a

compressors will be used to over-estimate the performance of current CO2 COmpressor

technology. This provides a technology-independent evaluation of the performance of the two

systems.

Selection of the supercritical heat rej section pressure is another vital operational parameter

that significantly influences the operation and performance of a transcritical CO2 System.

Pressure is independent of temperature in the supercritical regime. According to numerous

studies there exists a supercritical pressure at which the system' s coefficient of performance is

maximum. Both graphical and numerical simulations have been proposed to select this heat

rej section pressure (Kauf, 1999). For the purpose of this study, the heat rej section pressure is









incrementally varied from the critical pressure (7.38 MPa) to 15 MPa until the optimum pressure

at which heat rej section takes place is determined for various operating conditions. This procedure

is performed for both basic and modified transcritical CO2 Systems.

In the case of a conventional refrigeration system, the refrigerant undergoes de-

superheating, condensation and subcooling in the condenser. The effects of superheating and

subcooling are considered when determining the heat exchanger effectiveness and the

logarithmic mean-temperature difference (LMTD). In the case of a transcritical CO2 System a gas

cooler is employed to rej ect heat to the ambient and operates at a transcritical temperature and

pressure. In an ideal gas cooler, the refrigerant temperature leaving the cooler, Te,out, would

match the ambient temperature. The refrigerant enters the cooler as a vapor-like substance and

leaves as a liquid-like substance. Following the conclusion of Robinson and Groll (1998)

regarding the stream availability of the refrigerant leaving the cooler, no subcooling of the

refrigerant is assumed. This allows for the maximum extraction of work during the expansion

process.

The heat rej ected in the condenser or gas cooler can be calculated as follows


qcod/g = h7 -h? (3-13)

where h2,a is the actual enthalpy of the superheated/supercritical vapor calculated by means of the

compressor isentropic efficiency. The enthalpy h3 is the enthalpy of the refrigerant leaving the

heat-rej ecting device. For the purpose of this study the refrigerant in the transcritical cycle is

assumed to leave the gas cooler at a trans critical pressure and temperature of T3= Tg,ou,=Tamb+5.

In the analysis of the conventional system the refrigerant is assumed not to experience any

subcooling. The state properties thus correspond to the enthalpy of the saturated liquid at the









condenser saturation pressure. In this study the difference between the ambient temperature and

the condensing temperature in the conventional system is assumed to be 15oC.

In conventional and transcritical CO2 refrigeration systems a throttling/expansion valve is

utilized to expand the refrigerant from the pressure of heat rej section to the evaporating pressure.

This irreversible isenthalpic process may be replaced by the use of an expansion device that

would ideally expand the refrigerant in an isentropic fashion. The potential increase in the

system COP due to both the increase in the refrigerating effect and decrease in the cycle net

work input have been studied extensively in recently published literature for systems that utilize

conventional refrigerants and transcritical CO2 aS working fluids. Different positive displacement

type expansion devices such as helical-screw, scroll, rotary-vane and reciprocating piston have

been investigated.

In this analysis, the type of expansion device used is not specified. The type of device

along with the geometrical and operational characteristics may be programmed in separate

subroutines that may provide actual performance predictions and an accurate estimate of the

expander' s isentropic efficiency. Depending on the inlet conditions to the expander the actual

work output from the expander can be expressed as

w, = r (h, h ,) (3-14)

The use of expander work output can either be coupled to the compressor shaft or utilized

to drive other auxiliary machinery or systems that may contribute other system improvements.

This study considers only the alternative of increased cooling capacity brought on from near

isentropic expansion and ideal coupling of the expander work to the compressor.

The size of an expansion device is primarily a function of the built-in volume ratio of the

expander. The expander should not only match (or be fairly similar to) the process expansion










ratio for typical system operating conditions but should also minimize the amount of wasted

power recovery due to pressure losses incurred during under-expansion. The process volume

ratio is defined as the ratio of the specific volume of the refrigerant at the inlet of the evaporator

to the specific volume at the outlet of the condenser/gas cooler. The expansion process in

positive displacement expansion devices is comprised of the intake, expansion and the exhaust

processes. The built-in volume ratio of an expander is defined as the ratio of the specific volume

at the start of the exhaust process to the specific volume at the end of the intake process. In this

study it is assumed that both the built-in and process volume ratios are identical and defined as

follows


r, = "*"'I (3-15)


where 'va,,,in is the specific volume of the refrigerant after the expansion process (or at the inlet

to the evaporator). Depending on the cycle analyzed, this state may correspond to an isenthalpic

or non-isentropic expansion process.

We will further assume that the degree of superheat the refrigerant experiences in the

evaporator is assumed to be zero unless otherwise indicated. For a given set of operating

conditions the degree of superheat has much more of an influence on the overall performance of

the system. The refrigerant is thus assumed to be saturated vapor upon exiting the evaporator

unless otherwise indicated. An energy balance of the evaporator yields

q, =(4 -h )(3-16)

where h'4,a is the actual enthalpy of the refrigerant entering the evaporator determined by means

of the isentropic efficiency of the expander. In the case of an unmodified cycle h'4,a WOuld be

replaced by h4h=h3









The coefficient of performance is defined as a ratio of the desired output to the required

input. In the case of a modified vapor compression system with an expansion device, the net

work input (compressor work less expander work) is the required input to the system. For

simplicity, the work extracted from the expander is assumed to be ideally coupled to the

compressor. The COP is thus defined as


COP = e"(3-17)
we w

A second-law based comparison of conventional and transcritical CO2 refrigeration

systems is a far better comparison criterion. Exergy or availability analyses reveal how closely

actual refrigeration cycles are to ideal cycles. An exergy analysis also reveals the largest source

of irreversibilities in the system by singling out those components that are most irreversible. The

availability of a flowing fluid stream neglecting both kinetic and potential energy is defined as

IV = (h ho) -To (s -so) (3-18)

The subscript 0 denotes a reference state at which the system's temperature and pressure are in

equilibrium with its surroundings (To=303.15 K ). This state is said to have zero availability if all

other energy forms relative to the surroundings are zero. The specific irreversibility of a steady-

flow device with heat transfer may be expressed as the difference between the reversible work

and the actual work as follows


I = wve- wact ITn- Wex,-q 1 --w (3-19)


where q, and Tr are the specific heat transfer and the reservoir temperature at which heat is

transferred to/from, respectively. The sign of q, depends on the direction of heat transfer.

Utilizing Equations (3 -18) and (3 -19), the specific irreversibilities of the various components can

be derived. The specific irreversibility of an adiabatic compressor may be written as follows










Ic = 1 -V2,a h -haa)= T (s2a -s,)(3 -20)

In the case of a conventional system, the specific irreversibility of the condenser is


Icona V)-odt1 = T, (s, sz,a)- (h3 hz,) (3-21)


whereas, for a gas cooler, Tr in Equation (3-21) is replaced by the ambient temperature, Tamb. I

the ambient temperature is identical to the reference temperature this expression would reduce to

the difference in availability of the entering and leaving streams.

If the expansion process is performed using a throttling/expansion valve the specific

irreversibility of that isenthalpic process is

I = To s ,, -s, (3-22)

whereas in the case of utilizing an expansion device, the specific irreversibility is

lexp 3 a- a)=Trsa ss)(3-23)

The specific irreversibility of the evaporator for both systems is


levap 4. 1) 9ea oT S -~ s)+ (h, h) (3-24)


A system' s total irreversibility is the sum of the irreversibilities associated with its four

components. Irreversibilities resulting from pressure drops in the system's components and

transfer lines have been neglected. Irreversibilities associated with heat transfer in the system's

piping have also been neglected.

Both first and second-law based evaluations of R-22, R-134a and transcritical CO2

refrigeration technology are critical for selecting appropriate technology suitable for a specific

application. For this purpose, simulation models for basic and modified vapor-compression

refrigeration cycles have been developed.









The selection of the optimum heat rej section pressure is necessary to adequately compare

transcritical CO2 Systems to conventional systems. Maximum system COP occurs at the gas

cooler' s optimum heat rej section pressure. Figure 3 -22 illustrates the variation of the system COP

as a function of the heat rej section pressure for both basic and modified systems at evaporating

temperatures of 268 and 273 K. The COP of the basic system can be seen as a near constant for

heat rej section pressures greater than the optimum heat rej section pressure. This is not the case for

the modified system since higher heat rej section pressures adversely affect the performance of the

expansion device. For an evaporating temperature of 273K, it can be seen that the COP is

maximum when the heat rej section pressure is 10.2 and 9.75 MPa for the basic and modified

systems, respectively. For an evaporating temperature of 268K the optimum heat rej section

pressures were found to be 10.35 and 9.75 MPa for the basic and modified systems, respectively.

The optimum heat rej section pressure can be determined in a similar fashion for different

evaporating temperatures and operating conditions. For all operating conditions investigated, the

optimum heat rej section pressure has been assumed to be 10.2 MPa. Although the actual optimum

heat rej section pressure may vary by as much as 5% over the entire range of evaporating

temperatures investigated, the system COP deviates by as much as 1 and 1.7% for the basic and

modified systems, respectively. For the sake of computational time this deviation may be

assumed negligible.

Irreversibilities associated with the heat rej section and expansion processes are the largest

sources of irreversibility in basic transcritical CO2 Systems. Figure 3-23 illustrates the variation

of the percent of cycle irreversibility associated with the heat rej section and expansion processes

as a function of the heat rejection pressure for basic and modified CO2 Systems. For these

operating conditions, the compressor and evaporator contribute approximately 16-17 and 5-6%









of the basic cycle' s total irreversibility, respectively. The compressor and evaporator contribute

approximately 22-27 and 9-10% of the modified system's total irreversibility, respectively. It can

also be seen in Figure 3-23 that the throttle valve contributes 40 to 10% more to the system's

total irreversibility when compared to the gas cooler at heat rej section pressures of 9 and 1 1.25

MPa, respectively. In contrast, the contribution of irreversibility from the gas cooler is

approximately 2% less and 25% more at heat rej section pressures of 9 and 1 1.25 MPa,

respectively. The average irreversibilities of the expansion and heat rej section processes are 24%

lower and 12.5% higher when comparing the modified and basic transcritical CO2 Systems,

respectively.

Component irreversibilities corresponding to an optimum heat rej section pressure are

unlikely to be minimum. From Figures 3-22 and 3-23 it can be seen that the optimum heat

rej section pressure corresponds to a maximum system coefficient of performance and a median

value of the system irreversibility.

A first-law based analysis of R-22, R-134a and transcritical CO2 Systems is required to

evaluate the thermodynamic performance of basic and modified refrigeration systems.

Employing a two-phase expansion device as a throttle valve replacement is the only modification

made to the basic system. This allows for an unbiased comparison of the different refrigeration

technologies.

Figure 3-24 shows the variation of the coefficient of performance of basic and modified R-

22, R-134a and transcritical CO2 Systems as a function of the evaporating temperature. It can be

seen that the COP of the cycles investigated increases with increased evaporating temperature.

The values of system COP for the basic and modified R-22 systems are higher than those of R-

134 systems by as much as 6 and 0.6%, respectively. This maximum difference corresponds to









an evaporating temperature of -30oC. For the range of evaporating temperatures investigated, it

can be seen that the COP of the basic transcritical CO2 System is lower than that of R-22 by 33-

34% and that of R-134a by 29-34%. On the other hand, the COP of a modified transcritical CO2

system is lower than that of an R-22 system by 17.5-22% and that of an R-134a by 18-21%. The

COP of a modified transcritical CO2 System is still 4.2-9% less than that of a basic R-22 system.

The COP of a modified transcritical CO2 System is 2% higher and identical to that of a basic R-

134a system operating at evaporating temperatures of -30 and -25oC, respectively. At higher

evaporating temperatures, however, the COP of a transcritical CO2 System is as much as 9%

lower than that of a basic R-134a system.

When compared to similar R-22 and R-134a technology, the performance improvements of

a modified transcritical CO2 System, in comparison to the basic system, are greater. Figures 3-25,

3-26 and 3-27, may further support this result. Figure 3-25 shows the difference in COP between

the basic and modified R-22, R-134 and transcritical CO2 Systems as a function of the

evaporating temperature, respectively. It can be seen that the difference in COP decreases with

increased evaporating temperature for all three systems. The decrease is not as rapid however for

the transcritical CO2 System. As the evaporating temperature decreases, the modified CO2, R-

134a and R-22 systems outperform the basic systems by 27-30%, 10-23% and 9-18%,

respectively. This increase in the COP can be attributed to an increase in the refrigerating effect

and a decrease in the input work required to operate the system.

The refrigerating effect is defined as the specific evaporator capacity. From Figure 3-26, it

can be seen that as the evaporating temperature decreases, the differences in the refrigerating

effect of the modified and basic systems increase from 5-14%, 1.2-7% and 1.3-9% for the CO2,

R134a and R-22 systems, respectively. From Figure 3-27, the decrease of the required input to









the system can be seen to be dramatically different. The work extracted from the expansion

process is ideally utilized in the reduction of the input work to the compressor. It can be seen that

as the evaporating temperature increases, the difference in required input work to operate a

transcritical CO2 System increases. For increasing evaporating temperatures, the reduction in

input work can be seen to be 22-31%, 14.5-9% and 18.5-10% for transcritical CO2, R-22 and R-

134a systems, respectively. When comparing the basic and modified systems, the sizable

differences in the increase in refrigerating effect and reduction of input work of CO2 pOint to

favorable thermodynamic characteristics of CO2 when compared to those of R-22 and R-134a.

When dealing with modified systems, the most favorable operational characteristic of CO2

is the volume ratio of the expander. This operational variable discussed earlier determines the

size and practical feasibility of employing a two-phase expansion device as a throttle valve

replacement. Figure 3-28 shows the variation of the expansion device' s volume ratio as a

function of the evaporating temperature for CO2, R-22 and R-134a. As can be seen, there is a

sizable difference between the volume ratios for the different working fluids. It can also be seen

that the volume ratio decreases as the evaporating temperature increases since the specific

volume at the evaporator inlet decreases as temperature increases. For the evaporating

temperatures reported (5 to 20oC only), the volume ratio can be seen to decrease from 2.7-1.6,

12.3-6.2, and 20.8-10 for transcritical CO2, R-22, and R-134a cycles, respectively.

Efficient expansion devices are necessary to justify the cost and performance improvement

of modified refrigeration systems when compared to conventional technology. This result is

evident in Figure 3-29, which shows the difference in COP of basic and modified CO2, R-22 and

R-134a systems as a function of the expander' s isentropic efficiency. In the case of CO2, it can

be seen that the improvement in the system COP increases in a much more dramatic fashion









when compared to R-22 and R-134a systems. The increase in the system COP can be seen to be

10.5-33%, 6-19% and 5-17% for CO2, R-134a and R-22 systems, respectively.

A second-law based analysis is needed to determine the irreversibilities of the various

components and to determine the increases/reductions in irreversibilities associated with any

system modification. Figure 3-30 illustrates the component irreversibilities of a modified R-22

system as a function of the evaporating temperature. As can be seen, for the modified R-22

system, the largest sources of irreversibility are the condenser and compressor. Employing an

expansion device has greatly reduced the irreversibilities associated with the basic system's

throttling process. Component irreversibilities for the basic and modified CO2, R-134a and R-22

systems at a specific operating condition are given in Table 3-2. The table shows that the

irreversibilities associated with the throttling process are reduced by 50, 18 and 16.5% for

modified CO2, R-134a and R-22 systems, respectively. Similar figures and tables may be

generated for various operating conditions and system configurations.

Figure 3-31 shows the percentage of system irreversibility due to the condensing process

for basic and modified R-22 and R-134a systems as functions of the evaporating temperature.

The percentage of system irreversibility associated with the gas cooler for both the basic and

modified transcritical CO2 Systems are not included for clarity. It was found that the gas cooler' s

contribution to the system irreversibility is fairly constant and averages 29.5 and 42% for the

basic and modified transcritical CO2 Systems, respectively. The figure also shows that the

percentage of the total system irreversibility associated with the condensing process increases

with evaporating temperature and with the use of an expansion device for both R-22 and R-134a

systems. This increase in irreversibility decreases with increased evaporating temperatures from

10-6% and 11-6% for R-134a and R-22 systems, respectively.










Figure 3-32 shows the percentage of system irreversibility associated with the expansion

process for basic and modified CO2, R-134a and R-22 systems. As can be seen, the percentage

irreversibility due to expansion is on average 47.5 and 24% for the basic and modified

transcritical CO2 Systems, respectively. An average value of 50% reduction in irreversible losses

due to the addition of an expansion device is expected with increasing evaporating temperatures.

Irreversibilities associated with the expansion process in R-134a and R-22 systems decrease with

increased evaporating temperature and may be reduced from 24-11% and 22-10.5%,

respectively.

The ratio of the total system irreversibility to the heat rejection rate yields a dimensionless

system irreversibility that may be used to evaluate and compare the total system irreversibilities

of transcritical CO2, R-22 and R-134a systems. The heat rej section rate has been used to non-

dimensionalize the total system irreversibility since the only difference between conventional

and transcritical refrigeration technology is the heat rej section device. Figure 3-33 shows the

variation of this dimensionless quantity as a function of evaporating temperature for basic and

modified CO2, R-22 and R-134a systems. As can be seen, the basic and modified transcritical

CO2 Systems have the largest dimensionless cycle irreversibilities associated with them.

Modified R-22 and R-134a systems, on the other hand, have the least. The basic CO2 System and

the basic R-22 and R-134a systems have fairly close dimensionless cycle irreversibilities.

Theoretical simulation models have been developed to analyze and evaluate the first and

second-law based performance of basic and modified R-22, R-134a and transcritical CO2 vapor

compression refrigeration systems. The use of a two-phase expansion device as a throttle valve

replacement is the only modification made to the basic system. This allows for an unbiased

comparison of the different refrigeration technologies. The following conclusions can be made:










* The optimum heat rej section pressure corresponds to a maximum system coefficient of
performance and a median value of system irreversibility. For the operating conditions
investigated, the optimum heat rej section pressure of a modified system is on average 5%
lower than that corresponding to a basic transcritical CO2 System.

* The COP of basic and modified transcritical CO2 technology is inferior when compared to
basic and modified R-22 and R-134a systems. The COP of basic and modified transcritical
CO2 Systems are 29 to 34% and 17.5 to 22% lower than R-22 and R-134a systems,
respectively.

* Modified transcritical CO2 Systems can compete with basic R-22 and R-134a technology at
lower evaporating temperatures but perform about 9% less at higher temperatures.

* Favorable thermodynamic characteristics of CO2 are apparent when comparing the relative
increases in COP and refrigerating effect and the reduction in input work to R-22 and R-
134a. For the range of evaporating temperatures investigated, the improvement in the CO2
system COP is 12 to 18% when compared to R-22 and R-134a systems. An increase of 3.8
to 7% in refrigerating effect and a 7.5 to 22% decrease in input work were also realized
when compared to R-22 and R-134a systems.

* The most favorable operational characteristic of CO2 is the volume ratio of the expander
needed to expand the refrigerant to the evaporating pressure. At an evaporating
temperature of SoC the volume ratio of an R-22 and R-134a expander are 4.5 and 7.7 times
that of a CO2 expander, respectively.

* Efficient expansion devices are necessary to justify the cost and performance improvement
of modified refrigeration systems when compared to conventional technology. When
compared to inefficient expansion devices (nexp=25%), efficient expansion devices
(rexp 85%) may increase the system COP by 33%, 19% and 17% for CO2, R-134a and R-
22 systems, respectively.

* The throttling process is the largest source of irreversibility in basic transcritical CO2
systems and the second largest in basic R-22 and R-134a systems. Irreversibilities may be
reduced by as much as 50, 24 and 22% for modified CO2, R-134a and R-22 systems,
respectively, employing a 65% efficient expansion device.

* The basic and modified transcritical CO2 Systems have the largest total irreversibilities
associated with them when compared to R-22 and R-134a systems.

Performance and Size Optimization of Compression Refrigeration Systems

A parametric analysis of key system operational variables in a modified vapor compression

refrigeration system is presented in this section. A highly non-linear relationship is found to exist










between the many factors that influence the optimization of the system' s coefficient of

performance and the associated size and weight of a refrigeration system.

Of these factors, the degree of subcooling that the refrigerant experiences in the condenser

is the most influential. It is the degree of subcooling that determines the necessary area of the

heat exchanger required, ultimately affecting the size and the weight of a refrigeration system.

The degree of subcooling also influences the size of the expander required to expand the

refrigerant from the condenser pressure to the evaporator pressure. As the degree of subcooling

increases, the resultant ratio between the outlet and inlet specific volumes decreases, and hence

the built-in volume ratio of the expander would decrease. A decrease in the latter causes a

reduction in the expander size and weight.

As the degree of subcooling increases, the useful power output of the expander (which may

be coupled to the compressor) decreases. On the other hand, this causes the enthalpy of the

refrigerant entering the evaporator to decrease. This causes an increase in the refrigerating effect

of the system. The extent of the influence of these competing effects on the system's coefficient

of performance needed further analysis. This was done using a simulation code developed for

that purpose.

An optimum degree of subcooling exists where the coefficient of performance of the

system is a maximum and the size and weight of the system are a minimum. It is recognized that

the performance and compactness criteria may not be optimized simultaneously by a single

subcooling choice. Furthermore, the penalties associated with the increase of the heat exchanger

area and the decrease of power output of the expander should be minimized, while the benefits of

the increasing refrigerating effect and the decreasing volumetric ratio of the expander should be

maximized.










The quantifieation of the dependencies of the coeffieient of performance, size, and weight

of a vapor-compression system on design choices such as the degree of subcooling are now

presented.

Component level models are needed for vapor-compression systems that utilize an

expander in place of the conventional throttling valve. A general solution method may be applied

to model and validate experimental data and/or data provided by the manufacturers. The method

presented is also applicable to both simple and modified vapor compression systems. Detailed

component-level models that may include geometry, flow arrangement and loss mechanisms for

the compressor, condenser, evaporator and expander may be developed separately and

incorporated into this model when required. These component-level models will provide better

estimates of the isentropic efficiencies of both the compressor and expander as well as the heat

exchanger size and effectiveness of the condenser and evaporator. In the analysis described in

this section, R-22 is assumed to be the working fluid. The thermodynamic properties of R-22 are

provided by means of coupling REFPROP 7.0's property subroutines with the model developed

for this analysis. Pressure losses in all the system components as well as throughout the system's

piping have been neglected. Figure 3-1 shows a schematic of a simple vapor-compression cycle

with and without an expansion device as a throttle valve replacement.

In general, the required input work to compress saturated vapor at the evaporating pressure

to superheated vapor at the condensing pressure is given by Equation (3-12). The refrigerant

undergoes de-superheating, condensation and subcooling in the condenser. The effects of

superheating and subcooling are considered when determining the heat exchanger effectiveness

and the logarithmic mean-temperature difference (LMTD). As a base case for comparing the

effect of subcooling on system size and performance, the refrigerant leaving the condenser is









assumed to be saturated liquid (i.e. no subcooling in the condenser). This represents the

minimum heat exchanger area required to condense the refrigerant to a saturated liquid state and

allow for the completion of the refrigeration cycle. When subcooling is accounted for, the

present analysis only investigates the subcooling that the refrigerant experiences in the condenser

resulting from an increase in the heat exchanger area. Other subcooling methods such as utilizing

a liquid-to-suction heat exchanger or using integrated or mechanically dedicated subcooling

loops are not analyzed in this study.

The heat rej ected in the condenser can be calculated as follows

qcond (h2,a hj) (3-25)

where h2,a is the actual enthalpy of the superheated vapor calculated by means of the compressor

isentropic efficiency. The enthalpy h3 is the enthalpy of the refrigerant leaving the condenser. In

the base case this corresponds to the enthalpy of the saturated liquid at the condenser saturation

pressure. In cases of subcooling this corresponds to the enthalpy of the subcooled refrigerant

determined at the saturation pressure of the constant-temperature condensing process and the

temperature of the subcooled refrigerant.

The effectiveness of the condenser is generally a function of the heat exchanger geometry,

construction material and surface area. The effectiveness of condensers and evaporators however

is independent of flow arrangement since the refrigerant undergoes an isothermal phase change

process. The effectiveness of the condenser is thus defined as

Co (T T )
cond,out c""n (3 -26)
mm~ C, conrdonr can,

In the case of a condenser, the refrigerant typically has a very high heat capacity because it

is undergoing phase change. The coolant, on the other hand, experiences the maximum










temperature difference and therefore its heat capacity is equal to the minimum heat

capacity, Cmm, = Co In this case, the temperature of the coolant leaving the condenser, Te,out, can

be calculated as follows

Tc,ou, = Tc,an + Econd Tcond,, n Tc,zn) (3 -27)

The relative size of the heat exchanger is calculated from the following Equation

~icond \ I Acond Im (A} A (3-28)

where ATim is the logarithmic mean temperature difference, which is defined as follows

(To -T )-(Tcond -T)
ATm = ond,out can onl n c,out (3 -29)

In( 7ond,out c a,,-nd~n-T~,


Equation (3-29) is applicable to any heat exchanger flow arrangement since the refrigerant

undergoes a phase change process (i.e. correction factor F=1) (Incorpera 2002).

In typical vapor compression refrigeration systems a throttling/expansion valve is utilized

to expand the refrigerant from the condensing pressure to the evaporating pressure. This

irreversible isenthalpic process may be replaced by the use of an expansion device that would

ideally expand the refrigerant in an isentropic fashion. The potential increase in the system COP

due to both the increase in the refrigerating effect and decrease in the cycle net work input have

been studied extensively in recent literature for systems that utilize conventional refrigerants and

transcritical CO2 aS working fluids. Many different positive displacement type expansion devices

such as helical-screw, scroll, rotary-vane and reciprocating piston have been investigated.

Here the type of expansion device used is not specified. The type of device along with the

geometrical and operational characteristics may be programmed in separate subroutines that may










provide actual performance predictions and an accurate estimate of the expander' s isentropic

efficiency.

Depending on the degree of subcooling the refrigerant experiences in the condenser, the

inlet conditions to the expander may be determined assuming that subcooling in the condenser

occurs at a constant pressure. The actual work output from the expander can be expressed as

w, = Fl(h, -h ) (3-30)

The size of an expansion device is primarily a function of the built-in volume ratio of the

expander. The expander should not only match (or be fairly similar to) the process expansion

ratio for typical system operating conditions, but should also minimize the amount of wasted

power recovery due to pressure losses incurred during under-expansion. The process volume

ratio is defined as the ratio of the specific volume of the refrigerant at the inlet of the evaporator

to the specific volume at the outlet of the condenser. This volume ratio is expected to be smaller

when the refrigerant leaving the condenser is subcooled. The complete expansion process in

positive displacement expansion devices is comprised of the intake, expansion and the exhaust

processes. The built-in volume ratio of an expander is defined as the ratio of the specific volume

at the start of the exhaust process to the specific volume at the end of the intake process. In this

study it is assumed that both the built-in and process volume ratios are identical and defined as


z; = "*"'11 (3-31)
cond out

We will further assume that the degree of superheat the refrigerant experiences in the

evaporator is zero unless otherwise indicated. For a given set of operating conditions, the degree

of superheat has much more of an influence on the overall performance of the system. In this

study, the impact of the degree of subcooling on the expander cycle performance and size is of










primary interest. The refrigerant is thus assumed to be saturated vapor upon exiting the

evaporator unless otherwise indicated. An energy balance of the evaporator yields

qeve, = (h, h ,) (3-32)

where h'4,a is the actual enthalpy of the refrigerant entering the evaporator determined by means

of the isentropic efficiency of the expander. The COP is thus defined as


COP = e"(3-33)
we w

The results presented here are in terms of dimensionless quantities. This allows the reader

to capture general trends of the analysis without the distraction of system specific calculations.

The dimensionless groups that will be extensively used in the presentation of the results are

defined as follows

SCOP
Dimensionless COP: COP* (3-34)



Dimensionless expander work output: w, xp exp (3 -3 5)
exp,reference


Dimensionless expander volume ratio: r; = r' (3-36)



fUA\*UA
Dimensionless condenser capacitance: = m UAz~ iz1vfeec (3-37)



For a vapor-compression system of known refrigerating capacity both the overall heat

transfer coefficient, U, of a condenser with a constant effectiveness and the mass flow-rate of

refrigerant may be assumed constant. The condenser capacitance can thus be considered a direct










measure of the heat exchanger area required to reject a given heat load in the system's condenser

if these two conditions are met.

A constant cooling capacity comparison criterion may also be used to constrain the

analysis presented here. In the constant cooling capacity design scenario, the mass flow-rate of

refrigerant through the system would vary depending on the degree of subcooling which will

ultimately reduce the system component and pipe sizing. The reduction in system size and

weight would be significantly less, however, than that achieved by reducing the heat exchanger

area.

Figure 3-34 shows the variation of dimensionless condenser capacitance, expander volume

ratio, expander work output and system COP as functions of the dimensionless refrigerating

effect. In this case, the refrigerant does not experience any subcooling in the condenser. The

refrigerating effect is defined as the difference in the specific enthalpy of the refrigerant at the

exit and the inlet of the evaporator. For a given condensing temperature, the refrigerating effect

increases as the evaporating temperature increases. The data used to plot Figure 3-34 has been

normalized by the refrigerating effect corresponding to the lowest evaporating temperature

analyzed. It can be seen from Figure 3-34 that as the refrigerating effect increases (for instance

due to an increase in the evaporating temperature) the system COP increases dramatically. This

is due to the fact that as the difference in operating temperatures (or pressures) decrease both the

compressor input work required and expander work output decrease. For the operating

temperatures reported, the compressor work input decreases by 74%, while the expander work

output decreases by 84%. Furthermore, for the same temperature range, an increase in the

refrigerating effect of 7.5% is observed. This results in an increase in system COP of about 75%

when compared to a system operating at the lowest evaporating temperature analyzed (223 K).









As the refrigerating effect increases, both the size and work output of the expander decrease. As

mentioned earlier, the expander work output decreases by about 84% whereas the expander

volume ratio decreases by 92%. As the evaporating temperature increases, the benefits of

utilizing an expander as a throttle valve replacement become less significant. The condenser

capacitance increases with the increase in the refrigerating effect. Over the full range of the

refrigerating effect considered, the condenser capacitance can be seen to increase by 27%.

Figure 3-35 shows the variation in the dimensionless expander volume ratio and work

output as functions of the evaporating temperature for different degrees of subcooling in the

condenser. The expander volume ratio and work output decrease with an increase in the

evaporating temperature and the degree of subcooling. At an evaporating temperature of 278 K,

typical in air-conditioning applications, the expander volume ratio is found to be 8% and 16.4%

less than the volume ratio corresponding to the case of no subcooling in the condenser for a

degree of subcooling of 4oC and 8oC, respectively. The expander work output was found to be

15.3% and 29% less than the work output corresponding to the case of no subcooling in the

condenser for a degree of subcooling of 4oC and 8oC, respectively.

Figure 3-36 illustrates the variation in the dimensionless condenser capacitance as a

function of the evaporating temperature for different degrees of subcooling in the condenser. It

can be observed that as the degree of subcooling increases, the variation in the condenser

capacitance increases dramatically. For a typical air-conditioning evaporating temperature of 273

K the condenser capacitance is found to be 24.6% and 54.3% greater than that corresponding to

the case of no subcooling in the condenser for a degree of subcooling of 4oC and 8oC,

respectively. As previously mentioned, there exists a direct correlation between the heat

exchanger area and the condenser capacitance. The increase in the heat exchanger area appears










to be much larger than the decrease in the expander volume ratio and work output. The effects of

system variables such as the isentropic efficiencies of the compressor and the expander, and the

condenser' s effectiveness on the system COP, condenser capacitance, expander volume ratio and

work output are discussed below.

Figure 3-37 shows the variation of the dimensionless condenser capacitance as a function

of the degree of subcooling for different values of the condenser' s effectiveness. The operating

conditions and system variables are listed below Figure 3-37. For the case of no subcooling, the

capacitance was found to be 25% and 56% larger for an effectiveness of 0.75 and 0.85,

respectively, when compared to a system whose condenser has an effectiveness of 0.6. This

results from the fact that as the effectiveness increases, the exit coolant temperature, Te,out, also

increases, resulting in a decrease in the condenser's LMTD. For a constant condenser heat load,

this leads to an increase in condenser capacitance. This increase is attributed to the increase of

the heat exchanger effectiveness and does not take into account the increase in the heat

exchanger area required to produce subcooling. This increase in condenser capacitance may

result from either an increase in the overall heat transfer coefficient or an increase in the

condenser' s area. For a condenser effectiveness of 0.6 it is found that the capacitance must

increase by 13% and 29.5% in order to produce a degree of subcooling of 4oC and 90C,

respectively. This represents an increase in the heat exchanger area of the same magnitude

assuming that the overall heat transfer coefficient of the condenser and the mass flow-rate of the

refrigerant are constant. For an effectiveness of 0.85 it is observed that in order to produce a

degree of subcooling of 4oC and 90C in the condenser, the capacitance must increase by 14.5%

and 33%, respectively. The same trends are expected for different values of the ambient air

temperature Tamb.










Figure 3-3 8 shows the variation of the dimensionless expander work output as a function

of the degree of subcooling in the condenser for different isentropic efficiencies of the expander.

Detailed component level models of an expander would incorporate effects of friction and

internal leakage losses and provide an accurate prediction of the expander' s isentropic efficiency.

As it can be seen, the work output from the expander decreases with both the decrease in

isentropic efficiency and increase of the degree of subcooling. For an ideal expander utilized as a

throttle valve replacement it can be shown that a degree of subcooling of SoC and 10oC results in

a decrease of the work output of the expander by 19% and 3 5%, respectively, when compared to

the case of no subcooling in the condenser. Expanders whose isentropic efficiencies are 60% and

80% will theoretically produce less work than an ideal expander by that exact amount. An

increase in the degree of subcooling in either case results in a decrease in the expander work

output by 19% and 35% for a degree of subcooling of SoC and 10oC, respectively. From this

figure it can be concluded that in order to make the use of an expander as a throttle valve

replacement beneficial, losses that affect the expander' s isentropic efficiency as well as the

degree of subcooling the refrigerant experiences in the condenser must be minimized.

Figure 3-39 shows the variation of the expander volume ratio as a function of the degree of

subcooling for different isentropic efficiencies of the expander. The expander volume ratio is

found to increase as the isentropic efficiency of the expander decreases. Not only does the work

output of the expander decrease, but also a larger expander adds both a weight and size penalty

to the modified vapor-compression system. For the case of an ideal expander, a degree of

subcooling of SoC and 10oC results in an increase in the volume ratio by 10% and 20%,

respectively. When compared to an ideal expander, there exists a 1.67% and 3.33% increase in

the volume ratio for the case of no subcooling in the condenser for expander isentropic










efficiencies of 0.8 and 0.6, respectively. For a degree of subcooling of 10oC, this difference can

be shown to decrease to about a 1.4% and 2.8% increase in the expander volume ratio for

isentropic efficiencies of 0.8 and 0.6, respectively, when compared to an ideal expander. It is

observed that the decrease in the expander work output is much more significant when compared

to the increase in the expander volume ratio for both a decrease in the isentropic efficiency of the

expander and the increase in the degree of subcooling the refrigerant experiences in the

condenser.

Figure 3-40 illustrates the variation of a system' s dimensionless COP as a function of the

degree of subcooling for different isentropic efficiencies of the expander. It is observed that as

the degree of subcooling increases, the COP also increases by 2%, 3.4% and 4.7%. This is the

case for 10oC of subcooling in the condenser for expander isentropic efficiencies of unity, 0.8

and 0.6, respectively when compared to the case of no subcooling. For the latter case, the COP is

found to be 1.7% and 3.4% lower than that corresponding to an ideal expander for expander

isentropic efficiencies of 0.8 and 0.6, respectively. It may thus be concluded that the variation in

COP is significant for larger values of subcooling in the condenser and then becomes more

significant as the isentropic efficiency decreases.

Figure 3-41 shows the variation of the dimensionless COP as a function of the degree of

subcooling for different degrees of superheat in the evaporator. The system parameters are

assumed constant and are listed below the figure. Superheating is beneficial to prevent

compressor slugging. An increase in the degree of superheating is generally associated with an

increase in the refrigerating effect and an increase in the input compressor work required to

compress the superheated vapor to the condensing pressure. It is observed that as the degree of

subcooling increases, the COP increases for all values of superheating analyzed. It is also noted









that as the degree of superheat increases the COP is seen to decrease. This may be associated

with a significant increase in the compressor input work in contrast to gains in the refrigerating

effect. For the case of no subcooling in the condenser it may be shown that as the degree of

superheat increases, the compressor input work increases by 2.8% and 5.3% whereas the

refrigerating effect increases by 2.4% and 15.8% for a degree of superheat of SoC and 10oC,

respectively. The degree of superheat has no direct impact on the performance or size of either

the condenser or expander. The degree of superheat should be selected and minimized in order to

maximize a given COP.

The analysis presented in this section provides a foundation for application-specific

optimization. The degree of subcooling the refrigerant experiences in a system's condenser has a

significant effect on the size and performance of a vapor-compression system utilizing two-phase

expanders as throttle valve replacements. The Eindings of this analysis are as follows

* As the evaporating temperature increases the benefits of utilizing an expander as a throttle
valve replacement become less significant. This can be attributed to significant decreases
in the expander volume ratio and work output.

* The expander volume ratio and work output also decrease with increased condenser
subcooling.

* The work output from the expander decreases with both the decrease in isentropic
efficiency and the increase of the degree of subcooling. Losses that affect an expander' s
isentropic efficiency as well as the degree of subcooling must be minimized.

* The decrease in the expander work output is much more significant when compared to the
increase in the expander volume ratio for both a decrease in the isentropic efficiency of the
expander and the increase in the degree of subcooling.

* The variation in system COP is significant for larger values of subcooling in the condenser
and become more significant as the expander's isentropic efficiency decreases.

* The degree of superheat in the evaporator has no direct impact on the performance or size
of either the condenser or expander. The degree of superheat should be selected and
minimized in order to maximize a given system's COP.










Pure Hydrocarbons as Refrigerants

Hydrocarbons are very lucrative as refrigerant replacements for CFCs, HCFCs and HFCs

since they are abundant, inexpensive and environmentally friendly because they have no ozone

depletion potential. Among the maj or drawbacks of using hydrocarbons as refrigerants is their

high flammability potential. This limits their use to hermetic systems and the slightest leak of a

hydrocarbon refrigerant may prove to be catastrophic. The following is a review of recent

literature pertaining to the use of hydrocarbons as pure refrigerants, in mixtures, and issues

pertaining to flammability. The most promising potential hydrocarbons used as refrigerant

substitutes are listed in the Table 3-3 along with pertinent fluid information.

Many researchers have investigated the performance of pure hydrocarbons and

hydrocarbon mixtures as refrigerants. According to Granryd (2001) several standards allow the

use of hydrocarbons as refrigerants in internally safe, hermetically sealed systems with a charge

limit of 0. 15 kg. This limits the use of hydrocarbons to domestic refrigerators and freezers and

low capacity heat pumps. Larger refrigeration systems may utilize hydrocarbons as refrigerants

as long as there is adequate exhaust and extreme safety precautions are in place. According to

Granryd (2001) hydrocarbons are compatible with conventional materials used to construct

vapor-compression refrigerating systems.

Granryd (2001) theoretically investigated various cycle and heat transfer characteristics for

various hydrocarbon refrigerants including propane, butane, isobutene, propene (propylene),

cyclopropane, Dimethyl ether and a 50% propane-50% isobutane mixture. The results were

compared with cycle and heat transfer characteristics of R-12, R-22 and R-134a. The author

found that the pressure ratios associated with butane and isobutene are higher while propane and

cyclopropane are lower than that of R-22, respectively. The author also notes that in the case of

using butane and isobutane as refrigerants, a necessary degree of superheat is required at the










evaporator exit to ensure that isentropic compression doesn't move the vapor into the two-phase

region. When comparing the cycle coefficient of performance to the Carnot efficiency, butane

was found to have the highest cycle Carnot efficiency. Granryd (2001) also experimentally

investigated the use of propane in a refrigeration unit and compared its cycle performance with

that of R-12, R-22, R-134a and R-152a. He concluded that a 3-5% increase in system COP and a

3-15% decrease in refrigerating capacity exists when propane is compared to conventional

refrigerants. He attributed the decrease in compressor work to the lower molecular weight of

propane.

The use of pure hydrocarbons and mixtures as refrigerants in domestic refrigerators has

been investigated by Hammad and Alsaad (1999), Jung et al. (2000), Tashtoush et al. (2002),

Sekhar (2004) and Wongwises and Chimres (2005).

Purkayastha and Bansal (1998) presented an experimental study of utilizing propane and a

LPG mix (by mass fraction: 98.95% propane, 1.007% ethane and 0.0397% isobutane) as a

suitable replacement for R-22 in a heat pump. They found an increase in system coefficient of

performance of 18 and 12%, and 9 and 4% for propane and LPG, respectively. These values

correspond to condensing temperatures of 35oC and 55oC, respectively, at an evaporating

temperature of 3oC.

Colbourne and Suen (2004a, 2004b) appraised the flammability hazards of hydrocarbon

refrigerants in refrigeration and heat pump systems. Dlugogorski et al. (2002) experimentally

investigated the use of inerting agents with natural gas and a propane-butane mixture. Their

recommendations however have not been tested experimentally in a refrigeration unit.

In the case of pure hydrocarbon refrigerants the parameters under investigation include the

system coefficient of performance, refrigerating effect, work consumption by the compressor and










the effect of both subcooling and superheat on system performance. The case where an expander

is utilized as a throttle valve replacement was also investigated. Comparisons with R-12, R-22

and R-134a were also made to compare the performance of pure hydrocarbons as refrigerants in

both the conventional and modified cycles. The results from these simulations are presented in

Figures 3-42 to 3-47.


























Wexp L Wcomp

Qin
4' 4 11
Te,Pe









Figure 3-1. A schematic of the ideal base cycle (1-2-3-4-1) and the modified base cycle with an
expansion device (1-2-3-4'-1)



























s h


Figure 3-2.


T-s and P-h diagrams of the ideal base cycle (1-2-3-4-1) and the modified base cycle with an expansion device (1-2-3-4'-












Pc Pi Pe Te Ti To


Figure 3-3. T-s and P-h diagrams of the economizer cycle (1-2-3-4-5-6-7-1) and an economizer cycle with an expansion device as a
throttle valve replacement (1-2-3 -4-5-6'-7'-1)




























Wcomp,2










Wcomp,1


Figure 3-4. A schematic of the economizer cycle (1-2-3 -4-5-6-7-1) and an economizer cycle
with an expansion device as a throttle valve replacement (1-2-3-4-5-6'-7'-1)





















comp


IHX


exp


Figure 3-5. A schematic of the IHX cycle (1-2-3-4-5-6-1) and the modified IHX cycle with an
expansion device as a throttle valve replacement (1-2-3 -4-5'-6-1)




















3 For:4 32


31~6 T/Fr




s h


Figure 3-6. T-s and P-h diagrams of the IHX cycle (1-2-3 -4-5 -6-1) and the modified IHX cycle with an expansion device as a throttle
valve replacement (1-2-3 -4-5'-6-1)












Table 3-1. Summary of the percent changes in the system COP, refrigerating capacity and required work input by the addition of an
expansion device for the ideal and actual R-134a single-stage, economizer and IHX cycles

Ideal*** Actual *,***
To, oC Evaporating Range ACOP, % AQe, % AWet, % ** ACOP, % AQe, % AWet, % **
25 -20"C to 10"C 8-18.7 0.4-3.4 7.6-15.9 3.4-8.3 0.2-1.7 3.3-7.2
30 -20"C to 10"C 10.3-21.4 0.76-4.3 9.6-17.8 4.5-9.6 0.4-2.2 4.3-8.2
Base Cycle 35 -20"C to 10"C 12.9-24.2 1.2-5.4 11.9-20 5.6-11 0.6-2.8 5.3-9.2
40 -20"C to 10"C 15.6-27.1 1.8-6.7 14-22 6.8-12.4 0.9-3.5 6.4-10.3
45 -20"C to 10"C 18.5-30.2 2.4-8.2 16.5-24 8.2-14 1.2-4.3 7.5-11.4
25 -20"C to 10"C 5.2-11 0.3-1.7 4.9-9 2.2-4.8 0.2-1.4 2.1-3.7
30 -20"C to 10"C 6.5-12 0.5-2 6.1-10.2 2.9-5.6 0.4-1.7 2.6-4.1
Economizer
35 -20"C to 10"C 8.1-14 0.7-2.6 7.4-11.4 3.5-6.4 0.5-2.2 3.2-4.6
Cycle
40 -20"C to 10"C 9.6-15.5 1-3.2 8.8-12.7 4.3-7.3 0.8-2.7 3.7-5.1
45 -20"C to 10"C 11.4-17.4 1.3-3.9 10.2-14.1 5.1-8.3 1.1-3.3 4.3-5.6
25 -20"C to 10"C 2.6-3.6 0.14-0.6 2.5-3 1.6-2.9 0.1-0.6 1.5-2.4
30 -20"C to 10"C 3-3.9 0.2-0.7 2.8-3.3 1.9-3.3 0.2-0.7 1.8-2.7
IHX Cycle 35 -20"C to 10"C 3.4-4.5 0.3-1 3.23-3.7 2.3-3.7 0.2-0.9 2.1-2.9
40 -20"C to 10"C 3.9-5 0.4-1.2 3.6-4 2.7-4.2 0.6-1.1 2.4-3.2
45 -20"C to 10"C 4.4-5.5 0.6-1.4 4-4.4 3.1-4.7 0.5-1.4 1.7-3.5
*r,=85%, rexp=0.5, SIHX=0.7
** a decrease in the amount of work needed theoretically
*** maximum value corresponds to Te--20"C and minimum value corresponds to Te=100C



































-2 0 2 4 6 8 10

Evaporating Temperature, oC
B

30%

-n Base Tc=45C
-o Econom izer Tc=45C
25%



O~ 20%





15%






0%
-2 0 2 4 6 8 10

Evaporating Temperature, oC


Figure 3-7. Variation of ICOP, between the modified cycle with an expansion device
and the ideal cycle (R-134a), with evaporating temperature for A) Tc=250C,
B) Tc=450C for the base, economizer and IHX cycles


-


i


1 4%
-E1Base Tc=25C
12% rF -+-Economizer Tc=25C
~- -HX Tc=25C
10%-
O






4%-


2%

0%t














10%


BW
pet 'L~
--cc~-
..
: -


-E Tc=25C
-*-Tc=30C


-2 0 2 4 6 8 10

Evaporating Temperature, oC

Figure 3-8. Variation of AQe and AWnet, between the ideal modified cycle with an
expansion device and an ideal base cycle, with evaporating temperature for
various condenser temperatures in a single-stage vapor-compression
refrigeration unit (negative denotes a reduction).


5%
o



-5%







S-25% !
-














15 nTexo=0.7

10


5 I R-134a, Ida nex,=0.75
0 ` + -22 Idea
-202 81




Evaporating Tem perature, OC


Figure 3-9. Variation of ACOP, between the modified cycle with an expansion device
and the ideal cycle, as a function of the evaporating temperature for both R-22
and R-134a for various expander efficiencies Te=45oC (rcomp=85% when not
ideal)














20%
-A Actual, Tc=25C
18% -m-Actual, Tc=45C
-&Ideal, Tc=25C
16%~~ ---Ideal, Tc=45C

14%

8j 12%

.9 10%



6%





0%
-20 -15 -10 -5 0 5 10

Evaporating Temperature, oC




Figure 3-10. Variation ofABCOP, between the modified cycle with an expansion device
and the ideal cycle, as a function of the evaporating temperature for various
condenser temperatures for the actual and ideal cycle configurations of the
economizer cycle. rcomp=85%, nexp=0.5

















6%



5%-



4%-



3% -%



2%-

~--Actual, Tc=25C
-m-ctual, Tc=45C
1%-
~-&-deal, Tc=25C
-E1-deal, Tc=45C

0%
-20 -15 -10 -5 0 5 10

Evaporating Temperature, "C



Figure 3-11i. Variation ofA~COP, between the modified cycle w/ an expansion device and
the ideal cycle, as a function of the evaporating temperature for various
condenser temperatures for the actual and ideal cycle configurations of the
IHX cycle reomp 85%, rlexp 0.5, SIHX=0.7



















o 0.95 11ep 1



0.9-

R-22
0.85-



0.8



0.75 11ex=0.75
-2 O 2 4 6 8 10

Evaporating Temperature, oC

Figure 3-12. Variation of refrigerating efficiency as a function of the evaporating
temperature for both the base (no expander) and ideal and actual modified
cycle with and an expander at a condenser temperature of To-45oC

0.9
ncomp=85%, rexp=0.5,
SIHX=0.7
0.8-



S0.7-



S0.6 -


ct Actual, IHX Cycle
0.5-
-Actual, Base Cycle

-Actual, Economizer Cycle
0.4
-20 -15 -10 -5 0 5 10

Evaporating TemperaturePo


Figure 3-13. Variation of refrigerating efficiency as a function of the evaporating
temperature for the actual base, IHX and economizer cycles with an expander
at a condenser temperature of Te=25oC























_,--i-- --i--s'-'-- --- -


x- --- ---- ....-----x----- -x----.--x------x-----


-=- Tc= 25C
S--t-- Tc=30C
....x---- Tc= 35C
< -aTc=40C
- Tc=45C


Figure 3-14. Variation of an ideal system's COP as a function of the degree of
subcooling for various condensing temperatures (Te=5oC, ATsh=0 oC)























Te=45oC


----- *---


To=250C


- -t- Isenthalpic Expansion
SIsentropic Expansion


ATsubcooling, OC


Figure 3-15. Variation of an ideal system's COP as a function of the degree of
subcooling for various condenser temperatures and expansion processes
(Te=5oC, ATsh=0oC)















To= 450 C


160


c --t-~-k- -Iethlic Epanion


a 120 Te=25oCi'

-- Isentropic Exp~ansion

100
0 2 4 6 8

ATsubcooling, oC




Figure 3-16. Variation of the refrigerating effect as a function of the degree of
subcooling for various condenser temperatures and expansion processes
(ATsh=0oC)
















- -C sc, Isenthalpic
26 -- 5C sc, isenthalpic
-0C sc, Isentropic
-5C sc, Isentropic .*
22








10
5 7 9 11 13 15

Evaporator temperature, oC


Figure 3-17. Variation in an ideal system's COP with respect to the evaporator
temperature for different degrees of subcooling and expansion processes
(ATsh=0oC)















20 c c4"


15 OC

10 5C


10



5 7 9 11 13 15

Evaporator temnperature, oC




Figure 3-18. Variation of the difference in system COP,%, between the ideal cycle and
modified cycle with an expansion device as a function of the evaporator
temperature for various condenser temperatures and degrees of subcooling
(ATsh=0oC)















20



159.....~.....,...~ ,~~~~~,~~~~~,~~~~,~ ~~~,~~~~~g~~~~~


10 ~ ~_ ~ _,_ _,_ _, ,_ _,_ _,_ _


- -- Tc=25C
--o--Tc=35C
-9- Tc= 45C


Asuperheats oC


Figure 3-19. Variation of the difference of the system COP, %, base cycle and modified
cycle with an expansion device as a function of the degree of superheat in the
evaporator for various condenser temperatures.













200


ha180






M 120 ---4--T=5



--A- Tc=45C

100

0 2 T 4 oC 6 8
superheat,




Figure 3-20. Variation of the refrigerating effect for an ideal cycle as a function of the
degree of superheat for various condenser temperatures




























A B


Figure 3-21. T-s diagram of the basic system (1-2-3-4-1) and the modified system with an expansion device (1-2-3-4'-1) for a A)
conventional refrigeration system and B) transcritical CO2 refrigeration system














a Te=273 K


Modified System


-A -Te=268 K

-=- Te=273 K

Te=268 K






11.5


2.5


-c-t .



Basic System


2?-


10. 5


60 ,

50

40

30 ~

20 1

10

0


..s..s-'
Modified System *
_.r.-a Basic system
L ..- -

-r -- -


-a-Gas cooler

--Valve
- Gas cooler

--. ....Expander


ycomp=0.85

rexp=0.65
T3= 313 K

Tevap=273 K
11.5


9 9.5


10 10. 5


Heat Rejection Pressure, MPa


Figure 3-22. Variation of system coefficient of performance as a function of heat
rej section pressure for various evaporating temperatures for basic and modified
transcritical CO2 Systems


Heat Rejection Pressure, MIPa


Figure 3 -23. Percentage of cycle irreversibility associated with the heat rej section and
expansion processes as a function of heat rej section pressure for basic and
modified transcritical CO2 Systems













-. - Basic R-744 System

'.. ....M Modified R-744 System

B Basic R-134a System

SModified R-134a
Sy stem
-a-- Basic R-22 System


4-3 5

EvapoatingTempeatur, O


~- Modified R-22 System

1~,=0.85
S=0.65
0 5T =350C
amb
Tcon=500C
T =400C
gc_out


Figure 3-24. Coeffieient of performance of basic and modified R-22, R-134a and
transcritical CO2 Systems as functions of the evaporating temperature

35%
-R-744


30%

25%

0- 20%
O
<1 15%

10%


'I
- .. ....
- .. -


- --R-134a

- - -R-22




ne,=0.65
Tamb=350C

Tcond= 500C

Tooou=400C


-30 -25 -20 -15 -10 -5


0 5


Evaporating Temperature, OC



Figure 3-25. Difference in coeffieient of performance between basic and modified R-22,
R-134a and transcritical CO2 Systems as functions of the evaporating
temperature











0.16


0.12


-R-744
- --R-134a

....... R-22


ycomp=0.85
rex=0.65
Tamb=350C
Tcond= 500C
Tgeou=400C


'r
.. ~


S0.08


a,0.04


0


Evaporating temperature, oC

Figure 3-26. Difference in refrigerating effect between basic and modified R-22, R-134a
and transcritical CO2 Systems as functions of the evaporating temperature


0%




S-10%


... --- comp=0.85
~. -20% -- ne,=0.65
----...Tamb= 350C

Tcond= 500C

-30% T cot= 40 oC
-30 -20 -10 0

Evaporating Temperature, OC

Figure 3-27. Difference in system input work between basic and modified R-22, R-134a
and transcritical CO2 Systems as functions of the evaporating temperature
(negative denotes a decrease)












-R-1 34a

....... R-744

- --R-22

ycomp= 0.85


Tae=350C
Tcond= 500C

Tgeou=400C


- -


35%

30%

25%

20%

15%

10%


-R-744
--R-134a
....... R-22

ncomp=0.85
Tamb= 350C
Tcond= 500C
T,,,o=400C
Tap=C
85%


5% -F -
25%


45% 65%


Evaporating Temperature, oC


Figure 3-28. Volume ratio of the expander employed in modified R-22, R-134a and
transcritical CO2 Systems as functions of the evaporating temperature


Expander Isentropic Efficiency



Figure 3-29. Difference in coefficient of performance between basic and modified R-22,
R-134a and transcritical CO2 Systems as functions of expander isentropic
efficiency














- - Compressor
-- -- -- Condenser

SExpander
- Evaporator


-,,0.85


T,,=350C
Tcond= 500C


40


r~30


~20

10


Evaporating Temperature, oC

Figure 3-30. Percentage of cycle irreversibility due to various components in a modified
R-22 system as functions of the evaporating temperature


50 A- Basic R-134a System

45 Modified R-134a System

"ji 40 .. x- Basic R-22 System

E 35 p -* ..* Modified R-22 System

8 30 ...x..x ..=0.85
..** De,=0.65
*0r 25 -
a ....-'Tamb=350C
o 20 a Tcond= 500C
-30 -20 -10 0


Evaporating Temperature, oC


Figure 3-31. Percentage of cycle irreversibility due to condensing process in basic and
modified R-22 and R-134a systems as functions of the evaporating
temperature










m Basic R-744 System

. .. Moedified R-744 System

, Basic R-134a System

- A- Modified R-134a System

X Basic R-22 System

- --x- -Modified R-22 System

noo,=0.85
rex=0.65
Ta4=350C
Tcond=500C
Tgeou=400C


50 a

40

30

20 E

10

0


o
E
a-


>


i:-- .. .
-- --x .. -
---X-.~-X- ~~


-20 -10 0

Evaporating Temperature, O


Figure 3-32. Percentage of cycle irreversibility due to the expansion process in basic and
modified R-22, R-134a and transcritical CO2 Systems as functions of the
evaporatmng temperature


I Basic R-744 System

- -4- Modif ied R-744 System

A Basic R-134a System

- -M- Modif ied R-134a System

H Basic R-22 System

- -A- Modif ied R-22 System

cop0.85
e=0.65
Tamb=350C
Tcond=500C

Too ot=400C


0.35


0.3


0.25


0.2


0.15


-30 -20 -10 0

Eva porati ng Te mpe ratu re, og


Figure 3-33. Dimensionless system irreversibility in basic and modified R-22, R-134a
and transcritical CO2 Systems as functions of the evaporating temperature













Table 3-2. Percentage of cycle irreversibility due to the various components in basic and
modified R-22, R-134a and transcritical CO2 Systems


Basic System Modified System
R-744 R-1 34a R-22 R-744 R-1 34a R-22
Compressor, % 17.3 22.2 21.4 24.6 28.2 26.3
Condenser, % 33.6 38.1 42.6 46.7
Gas cooler, % 29.7 -- 42.2
Expansion valve, % 46.5 33 29
Expander, % --- 23.2 14.6 12.5
Evaporator, % 6.5 11.2 11.5 10 14.6 14.5


-condenser capacitance
- -expander volume ratio
- expander w ork output
-COP


1.08


Figure 3-34. Variation of dimensionless COP, expander work output, expander volume
ratio and condenser capacitance as a function of dimensionless refrigerating
effect. Tcond= 450C, Tamb= 25"C, Econd=O 075, rcomp= 085, rexp=0 075


1.02 1.04 1.06
Dimensionless refrige rating effect












Degree of
Subcooling, oC



4

-8


4
-8


-30 -20 -10 0 10


Evaporating tempe rature, oC


Figure 3-35. Variation of dimensionless expander work output and expander volume ratio
as a function of evaporating temperature for various degrees of subcooling in
the condenser. Tcond=450C, Tamb=25"C, Econd=0.75, rcomp=0.85, rx=0.75


Degree of
Subcooling, OC


- 4

-8


0 10


Evaporating temperature, oC




Figure 3-36. Variation of dimensionless condenser capacitance as a function of
evaporating temperature for various degrees of subcooling in the condenser.
Tcond=450C, Tamb=2500, Econd=0.75, rcomp=0.85, rp=0. 75













2.6



a 2.2






S1.4


Condenser
effectiveness
-0.6
-0.75
- 0.85


Figure 3-37. Variation of dimensionless condenser capacitance as a function of degree of
subcooling in the condenser for various values of condenser effectiveness.
Tcond=450C, Tevap=50C, Tamb=25"C, comp=0. 75, r,=0. 75


~~---- ~~


Expander
efficiency

-1
----0. 8
------.0. 6


0.7



0.5


Figure 3-38. Variation of dimensionless expander work output as a function of degree of
subcooling in the condenser for various expander efficiencies. Tcond-450C
Teva=50C, Tamb=3000, Econd=0.75, rcomp=0.85


Degree of subcooling, oC


Degree of subcooling, OC












1.05

1

0.95

0.9

0.85


Expander
efficiency

-1
----0. 8
.......0.6





10


0 2 4 6 8


Degree of subcooling, oC


Figure 3-39. Variation of dimensionless expander volume ratio as a function of degree of
subcooling in the condenser for various expander efficiencies. Tcond-450C
Teva=50C, Tamb=30"C, Econd=0.75, rcomp=0.85


1.04








0.96


Excpander
efficiency
--0.6
-0.8


0.92


Degree of subcooling, oC


Figure 3-40. Variation of dimensionless COP as a function of degree of subcooling in the
condenser for various expander efficiencies. Tcond-450C, Tevap 50C
Tamb=30"C, Econd=O 075, rcomp= 085














Degree of
superheat, OC








0 2 4 6 8 10


MW Toi oC Perit, perit3
"'' kPa kg/m'
Butane (R600) 58.12 151.98 3796 227.84
Ethane (R170) 30.07 32.18 4872 206.58
Isobutane (R600a) 58.12 134.67 3640 224.35
Propane (R-290) 44.1 96.675 4247 218.5
Propene (Propylene) (R1270)42.08 92.42 4664 223.39


1.04


1.02


0.98


Degree of subcooling, OC


Figure 3-41. Variation of dimensionless COP as a function of degree of subcooling in the
condenser for various evaporator degree of superheat. Tcond-450C, Tevap 50C
Tamb=25"C, Econd=0.75, rcomp=0.85, Texp=0.85



Table 3-3. Hydrocarbons viewed as potential refrigerants













A B

400 350

Y"350 300-
R-134a
S300-
t~250 -R2
S2501 -R-1
200 Butane
150 150 Isobutane
a,100 Propane
an100
E -Proplyene
S50 50-
0 0
-5 0 5 10 -5 0 5 10

Evaporating Temperature, OC Evaporating Temperature, oc



Figure 3-42. Variation of the refrigerating effect as a function of evaporating temperature for various refrigerants at a condensing
temperature of A) 25oC and B) 45oC for the ideal base cycle
















A B



17-
m---R-134a
E 15 6 -

a, R-22
0. 1 3-
~5 -Butane
-Isobutane
-E -Propane
.2 4
$ 9 -Proplyene

7 3
-5 0 5 10 -5 0 5 10

Evaporating Temperature, OC Evaporating Temperature, OC



Figure 3-43. Variation of the ideal base cycle coefficient of performance as a function of evaporating temperature for various
refrigerants at a condensing temperature of A) 25oC and B) 45oC
























-65


55 -


45 -


35 -


25 -


-15
1( -5 0 5 1

Evaporating Temperature, oc


-R-134a
- R-22
- R-12
Butane
- Isobutane

- Poay e


-5 0 5

Evaporating Temperature, OC


Figure 3-44. Variation of compressor work as a function of the evaporating temperature at a condensing temperature of A) 25oC and
B) 45oC for various refrigerants in an ideal base cycle



















28% 15%
14%-
26%-
13% --R14
i0 lif 24% --R2
eg 1 2%
R-12
p 22% -11%-
Butane
O 20-10- Isobutane
S18% Propane
Proplyene
g! 1 6% -7%-
S1 4% 6%
-5 0 5 10 -5 0 5 10

Evaporating Temperature, oc Evaporating Temperature, OC




Figure 3-45. Variation of the increase of COP (when comparing the base cycle and the expander cycle) as a function of the
evaporating temperature for various refrigerants at a condensing temperature of A) 25oC and B) 45oC

















c





go
"G
.5 >



a *
oe
cn
Qu


14%
1 3%-
12% -
11% _
10%
9% -
8%-
7%-
6% -
5%


24%

2270

20%

18%

16%

14%

1 2%

1 0%


- R-1 34a
- R-22
R-1 2
Butane
- Isobutane
-Propane
-Proplyene


0 5

Evaporating Temperature, OC


10 -5


0 5

Evaporating Temperature, OC


Figure 3-46. Variation of the decrease in net-work (when comparing the base cycle and the expander cycle) as a function of the
evaporating temperature for various refrigerants at a condensing temperature of a) 25oC and b) 45oC






















t 1.8%

1.5%

S1.3%

*C 0 1.0%
C L
e *C 0.8%

c E 0.5%
-- o
S 0.3%

02 .0%


- R-134a
- R-22
R-12
Butane
- Isobutane
-Propane
-Propylene


-5 0 5 10 -


Evaporating Temperature, OC


Evaporating Temperature, OC


Figure 3-47. Variation of the increase in refrigerating effect (when comparing the base cycle and the expander cycle) as a function of
the evaporating temperature for various refrigerants at a condensing temperature of a) 25oC and b) 45oC









CHAPTER 4
MATHEMATICAL MODEL OF AN IDEAL ROTARY-VANE EXPANDER


Much literature has been reviewed in order to formulate an accurate and rigorous

analytic thermodynamic and fluid model of a rotary-vane expander. Most of the literature

found deals with rotary-vane expanders used predominantly in high-temperature

applications where the working fluid is vapor (namely steam and R-113) and is modeled

using ideal-gas relations. This negates the many additional complexities that arise due to

the necessary formation of vapor from liquid exiting the condenser in vapor compression

cycles. These modeling complexities may include but are not limited to cavitational

erosion, choking, issues dealing with lubrication, and a possible increase of friction and

internal leakage losses due to significant variations in the density and viscosity of the

two-phase working fluid. In order to accurately model a two-phase rotary-vane expander,

the studies conducted in the existing literature must be revisited and modified accordingly

to predict such parameters as the working fluid's pressure, temperature and mass

variations as well as an overall estimate of power losses, volumetric efficiency, power

output and torque of the rotary-vane expander.

Literature Review of Modeling of Rotary-vane Expanders

While literature dealing with modeling and design improvements of various types

of compressors and expanders, such as helical screw, reciprocating piston, scroll,

Wankel-engines Badr et al. (1991a, 1991b, 1991c) and rotary-piston are abundant, this

review is confined to rotary-vane (or sliding-vane) type expanders and compressors,

hence referred to as MVE and MVC, respectively. Smith et al. (1990, 1992) introduced

the principles, operation and testing of a novel type compressor, the Groll rotary-vane

compressor. They found a 19% and 15% increase in volumetric and adiabatic









efficiencies, respectively when compared to a conventional rotary-vane compressor.

There have not been any studies utilizing this as an expansion device.

The positive-di splacement rotary-vane expander has been widely used for waste

heat recovery in Rankine cycles. Badr et al. (1984) cited economics, ease of

manufacturing and low maintenance as the most important parameters in the selection of

a MVE as an expansion device. Among the investigators who developed mathematical

models describing the geometry of rotary-vane type turbomachinery (compressors,

expanders, pumps and air-motors) are Wolgemuth and Olson (1971), Marsters and

Ogbuefl (1972), Badr et al. (1985a, 1985b), Somayajulu (1971), and Ben-Bassat and

Wolgemuth (1972). Mechanical friction in a MVE has been investigated by Beck et al.

(1966), Ben-Bassat and Wolgemuth (1972), Peterson and McGahan (1972), Badr et al.

(1986a), Robertson and Wolgemuth (1978), and Edwards and McDonald (1972). Internal

leakage losses in a MVE have been investigated by Peterson and McGahan (1972),

Jacazio et al. (1979), Badr et al. (1985c), and Robertson and Wolgemuth (1978).

All of the above studies either dealt with the expansion or compression of a

working fluid (air, steam, and vapor) that could be modeled by the ideal-gas equation of

state. The studies in question also dealt with rotary-vane type turbomachinery utilized as

expanders in Rankine cycles as air motors or pumps. Table 4-1 summarizes the type of

application, working fluid, analyses performed, and the type of losses accounted for. The

extent and accuracy of their analyses is beyond the scope of this work.

Model Development

In order to accomplish this, five separate subroutines had to be constructed to

accurately predict the aforementioned parameters. A simplified flow diagram of the main










computer program that has been developed and a brief description, in subsequent

sections, of the subroutines to be utilized is presented in Figure 4-1.

Thermophysical Model

NIST's REFPROP 7.0's refrigerant property subroutines are linked with the

various models and are called whenever thermo-physical data is required. The capability

to both use pure refrigerants and common refrigerant mixtures allows the use of

REFPROP's property subroutines no matter the refrigerant. There is also the capability

to use of a user defined and customized mixture.

Geometric and Kinematic Model

In order to model the performance of a rotary-vane expander, the geometry of the

expander and kinematics of the vanes must be accurately described mathematically. The

basic geometrical and kinematic characteristics, as a function of the angular displacement

of the expander' s rotor, are required to evaluate the performance of the expander. The

parameters needed include:

* Variation of the cell volumes
* Volume expansion ratio, rv= V2/V1 (Figure 4-2)
* Variations of the protrusions of the vanes outside their rotor' s slots
* Vane's sliding velocity
* Vane's accelerations

A review of the literature reveals that many different models have been considered

(Table 4-1). A primary difference among the models is accounting for the vane

thickness. Many researchers have assumed an idealized vane thickness of zero when

calculating the cell volume.

For the purpose of this study, the expander is assumed to have a circular stator-

cylinder and the thickness of the vanes is taken into account initially. It can be shown









from Figure 4-2 that a circular stator cylinder may be modeled as having two arcs (2n-

6seat and 6seal) one of which is a sealing arc. The geometrical input parameters to the

model and nomenclature used throughout this section are listed in Table 4-2 and are

shown on a schematic of a circular expander in Figure 4-2. It should be noted that if the

sealing arc is symmetric about the base line, that y=180o and RS(6)=180o.

Neglecting the thickness of the vanes and assuming that the vanes are in continuous

contact with the stator cylinder, the ideal volume of the expander' s cell as a function of

angular displacement may be written as

R ,(0) = Am. (0) -AB (6- )] L (4-1)

where Acy(6) and Acy(86-) are the areas enclosed between the stator and rotor cylinders

and the datum and the leading and trailing vanes respectively. The area functions have

been calculated using the equations and numerical procedure presented in Badr et al.

(1985). The actual volume of the expander' s cells differs by accounting for the volume of

the extended portions of the leading and trailing vanes outside of the rotor' s slots.

In order to calculate the kinematics of the vanes and the actual volume of the

expander' s cell, the protrusions of the vanes from the rotor' s slots must be computed. The

vanes are assumed to be rigid. R(6) is defined as the radius of the stator-cylinder to the

center of the rotor as a function of angular displacement. For a circular stator-cylinder

this quantity may be simplified and expressed as


R (0)= ecos(y-8)+ r +~e -2esrR ~yesi 7 (4-2)

where Y'is defined as the angle between the arc' s center and that of the rotor. The vane' s

protrusion from the slot may then be defined as










X (0)= R (0 -rR (4-3)

The sliding velocity and the acceleration of the vanes may be determined by

differentiating Equation (4-3) with respect to time once and twice, respectively. A

FORTRAN subroutine has been developed to calculate the geometrical characteristics of

a circular rotary-vane expander for a given set of input parameters.

Thermodynamic Model

To model the performance characteristics of an expander, the principles of

conservation of mass and energy in each of the expander' s cells must be used. This will

allow the prediction of the pressure, temperature and mass variations of the working fluid

in the cells as a function of the rotor' s angular displacement. This will require accurate

calculation of both the variation of the cell volume and the thermo-physical properties

from the geometric and thermo-physical models, respectively. The basic geometrical

relationships modeled then helped describe:

* Variation of the cell volumes
* Volume expansion ratio
* Inlet and exhaust port area variations

As mentioned earlier, in order to model the thermodynamic and throttling

characteristics of an expander, the principles of conservation of mass and energy in each

of the expander' s cells must be used. These would allow the prediction of the pressure,

temperature and mass variations of the working fluid in the cells as a function of the

rotor' s angular displacement.

A computer program has been coded to calculate the mass, pressure and

temperature variations in the expander' s charging, expansion, and exhaust stages. Figure

4-2 shows the various stages of flow of a working fluid through the rotary-vane expander.









The following characteristics are calculated once the inlet state to the expander is

defined and the mass, pressure and temperature variations in the expander are found:

* Volumetric efficiency: an indicator of how well the expander' s cell was filled
during the charging process

* Throttling power factor: a ratio of the indicated power of the expander with only
throttling losses in ports accounted for and the indicated power of an ideal expander

* Throttling Efficiency: A ratio of the specific work of the expander with only
throttling losses accounted for and the specific work of an ideal expander

The throttling losses in the intake and exhaust ports are only presented here briefly

but will be discussed and modeled in the subsequent chapter.

Charging Process

The charging process takes place at angular displacements of 0 <; B < 8,, + 3. The

temperature of the fluid entering the expander can be determined from the degree of

subcooling the fluid undergoes in the condenser. The saturation pressure corresponding to

the saturation temperature of the condenser, the state of the fluid entering the expander

can also be determined. This state is denoted by the sub script (int).

The mass flow-rate of the working fluid that fills the cavity, assuming steady flow

and ignoring changes in potential energy, can be written as

th,,, (0) = Cd Pmt Amt (B)YFmt (4-4)

where the coefficient of discharge is assumed to be unity initially and will be determined

by experimental data. The inlet throat area varies as a function of angular displacement

and was found to obey the linear approximation that Wolgemuth and Olson (1971) have

suggested. A typical linear approximation of the inlet throat variation is shown in Figure

4-3.









The expression for the amount of mass contained in the expander cell at any

angular displacement is the following


m,(0) = m, (0)+i 0 (4-5)


The first law of thermodynamics can be written for the process as follows (taking

heat into the system and work done by the system as positive)


= Q Wb + v inthin (4-6)
dt m

Assuming the changes in kinetic energy and potential energy are neglected,

Ecy=dUy. The pressure within the control volume is also assumed to be uniform.

Multiplying by dt and neglecting friction, internal leakage losses and heat transfer,

Equation (4-6) becomes

AU = -r: (V2 V)+ mine hint (4-7)

Solving for the internal energy of the refrigerant at the new time step, State 2, we find


u2_ 11- 2- ) mnhnt (4-8)
m2

The density of the refrigerant at the new time step, State 2, is calculated as


p2_2 (4-9)


The charging process is assumed to be a constant pressure process (P1=P2=Pint).

This conclusion is supported by much of the literature including Taniguchi et al. (1983)

and Baek (2002). The mass in the control volume at State 2 is thus an unknown. In this

case, the first law of thermodynamics can be written as

AU = m2u 2 1~u 1 1p V2 V)+(m2 1, )hin (4-10)









The mass at State 2 can be expressed however as

na, = p,V, (4-11)

Substituting Equation (4-11) into Equation (4-10) we get

p,V,u, -nzzu, = -p, (y -I )+(pzV, -nzz )hmt (4-12)

where p2 and u2 are the unknowns. Neglecting temperature gradients within the control

volume, these two unknowns may be determined by iterating on the temperature of the

control volume that corresponds to the known pressure, P2, and simultaneously satisfies

the conservation of energy statement, Equation (4-12). The thermodynamic state of the

control volume at the new time step or angular displacement has hence been fully defined

and this procedure is carried out until the cut-off angle (8=8in+6).

Expansion Process

During the expansion process, 0,,, + 3<0 < ex, the mass in the control volume,

mov, stays constant after the cut-off angle 8,a, = 8,m + 3 where 6in is the spread of the intake

port and 8 is the angle between successive vanes (e.g. 6=45o for an eight-vane expander).

Assuming internal leakage losses are negligible, the mass in the control volume,

mov, is assumed to be constant throughout the expansion process. If pressure gradients

within the control volume, changes in kinetic and potential energy, friction and heat

transfer are also neglected, the process between the current and new time step may be

assumed to be isentropic and hence

s, = s, (4-13)

The density at the new time step may be determined from


pt = (4-14)









The thermodynamic properties at the new time step, namely temperature, pressure,

and quality, can be calculated from Equations (4-13) and (4-14) with the use of

REFPROP 7.0. This process is repeated until the end of the expansion process at a

predetermined exhaust angle of 6ex.

Exhaust process

The exhaust process occurs at an exhaust angle 0ex < B < 8end 8out + 6, Where 8out

is the spread of the exhaust port and 8 is the angle between consecutive vanes.

During the exhaust process, the mass flow-rate of the working fluid discharged

from the control volume as a function of angular displacement can be determined from

the following equation

thes (8) = Cd,ex, (0) Acx (0)Ve (4- 15)

where pv is the density of the working fluid in the control volume at the current time

step. The discharge coefficient, Cd,ex, iS an empirically determined constant that takes into

account exit port losses. Here, the discharge coefficient is assumed to be unity. The

variation of the exit throat area, Aex (6), can be seen to follow the linear approximation

proposed by Wolgemuth and Olson (1971) in Figure 4-4.

The amount of mass contained in the control volume at any angular displacement

can then be expressed as





where 0ex is the angle at the end of the expansion process. The first law of

thermodynamics can be written for the process as follows










dEc
= Q -Wbi yj2exhex (4-17)


Neglecting changes in kinetic and potential energy leads to Ecy=Ucy. The exhaust

process is assumed to occur at constant pressure P,, (Sx) = Ex and in an quasi-

equilibrium manner. Neglecting friction, internal leakage losses and heat transfer from

the ambient, Equation (4-17) reduces to

AU = -Pex (V2 V) mexhex (4-18)

Applying the same solution methodology as in the charging process, Equation (4-

18) simplifies to

p2-u 2 1~u 1 -Pex (- -R) (P2L m 1)4 (4-19)

where p2 and u2 are the unknowns. Neglecting temperature gradients within the control

volume, these two unknowns may be determined by iterating on the temperature of the

control volume that corresponds to the known pressure, P2, and simultaneously satisfies

the conservation of energy statement, Equation (4-19). The thermodynamic state of the

control volume at the new time step or angular displacement has hence been fully

defined.

Ideal Expander Evaluation

For a given set of operating temperatures in a refrigeration system, the state of the

refrigerant entering and leaving the expander may be determined. The resultant process

volume ratio is defined as


rv, =evapI1 (4-20)


where v3 is the specific volume of the fluid at the condenser exit. The refrigerant may be

a saturated or subcooled liquid at this state. This process volume ratio is expected to be









smaller when the refrigerant leaving the condenser is subcooled. The built-in (or

geometric) volume ratio of the expander may be expressed as


rv,bln =(4-21)


It should be noted that the terms built-in and geometric volume ratios are used

interchangeably. If internal leakage losses are neglected, this parameter reduces to the

ratio of the expander' s cell volumes since at both 6ex and 6in+a the mass in the cell volume

is constant. If the built-in volume ratio is less than, identical to or greater than the process

volume ratio then under-expansion, ideal expansion and over expansion will occur,

respectively. The expander should not only match (or be fairly similar to) the process

expansion ratio for typical system operating conditions but should also minimize the

amount of wasted power recovery due to pressure losses incurred during under-

expansion. In the case of over-expansion, the higher pressure of the downstream reservoir

may result in refrigerant flowing back into the expander via the exhaust port further

complicating the actual exhaust process. Over-expansion is not accounted for in this

study. Figure 4-5 illustrates the three possible scenarios depending on the built-in and

process volume ratios.

The appropriate selection of the dimensions of the expander is critical to ensuring

adequate operation. In general, turbo-machines are designed such that the volumetric

flow-rate of the working fluid is identical to the volumetric displacement of the device.

For the case of an expansion device employed in a refrigeration system this equality may

be expressed as


4 ~ = viz,mlv,, (4-22)









where 0Z is the rotational speed of the expander in RPM and -fi is the volumetric

displacement per revolution of the expansion device. The right-hand side of Equation (4-

22) represents the volumetric flow-rate of refrigerant in m3/S that enters the expander

after leaving the condenser in either a saturated or subcooled liquid state. The rotational

speed is the primary design variable for given operating conditions and geometry of the

expander. As the rotational speed of the expander increases, frictional losses and internal

leakage typically increase and decrease, respectively. An optimum operating condition

exists where the product of the displacement volume of the expander in m3/rev and its

rotational speed minimize friction, leakage losses and the manufacturing cost of the

expander.



In order to calculate the torque and power developed by the expander, the pressure

forces acting on the vanes must be computed. If the pressure forces are assumed to act at

the midway point of the vane's calculated protrusion from the rotor slots and no internal

leakage losses are assumed past the vane tips, the following expression for the torque

developed by one vane at any angular displacement may be written


S(0) =X, (0) L P,(0) -Pt,(6+ ) rR + (4-23)


If frictional losses are also assumed negligible, the average of the sum of the torque

developed by the expander at any angular displacement multiplied by the rotational speed

is the power that may be extracted from the expander. The resultant power as a function

of angular displacement may be expressed as









The appropriate selection of the geometry and rotational speed of the expander

ultimately depends on the application. Parameters of interest for typical operating

conditions for a basic and modified refrigeration system are tabulated in Table 4-3.

The apparent advantages of employing an expansion device include reduction in

mass flow-rate and process volume ratio, increase in the system COP, refrigerating

efficiency and refrigerating effect. The net-work required by the system also decreases if

the expander work is assumed to be ideally coupled to the compressor shaft. As the

efficiency of the expansion device increases, it can be seen that further decreases in the

mass flow-rate and process volume ratio are obtained. Similar trends for various

operating conditions, component efficiencies, degrees of superheating and subcooling are

expected.

In order to determine the work extraction potential of an expander with a known

geometry, an alternative numerical solution procedure to that detailed in the charging,

expansion and exhaust sections must be utilized. The charging and exhaust processes

were assumed to be isobaric and hence the mass flow-rate of refrigerant is unknown.

There exists, however, a unique value of the incoming and leaving refrigerant mass flow-

rate at which the power extraction is optimum. The mass flow-rate is hence iterated upon

in order to satisfy Equations (4-12) and (4-19).

Cerpnalkovski (1991) suggested typical values of aspect ratio (rR/L) of a rotary-

vane compressor of 0.5-0.8. For the purpose of this study the aspect ratio was assumed to

be 0.46. The eccentricity of the rotor and stator cylinders was also assumed to be 2.6 mm.

The dimensions of the expander are assumed to be 28 mm, 30.6 mm and 12.7 mm for the









rotor radius, stator-cylinder radius and the axial length of the expander respectively. The

sealing arc is assumed to have an arc-length of 6seai=40o.

The most important geometrical design parameter, which ensures adequate size and

performance of any expansion device, is the built-in volume ratio. Figure 4-6 shows the

variation of the built-in volume ratio as a function of the intake angle for various numbers

of vanes. The intake angle is found to be the most critical parameter that influences the

performance of a rotary-vane type expander. For all intake angles analyzed, the

thermodynamic performance and built-in volume ratio are maximum when the exhaust

angle is 6ex=1890. The volume ratio is found to decrease as the intake angle increases

independent of the expander' s number of vanes. As the number of vanes in the expander

increases this decrease is found to increase dramatically. The increase in the number

vanes may also be limited by the dynamics and mechanical strength of the rotor and the

added frictional and internal leakage losses that may result. The expander's intake angle

should hence be designed to optimize both the thermodynamic and geometrical

performance. A single value of the intake angle may not optimize both simultaneously,

however. Application dependent optimization is thus necessary. Although smaller intake

angles result in larger built-in volume ratios, higher throttling losses are typically

associated with them. The expander' s built-in volume ratio can be seen to decrease by

40%, 58.3% and 67% for the entire range of intake angles investigated for 4,8 and 12

vane expanders, respectively. Median values of the intake angle (6in=15o) and number of

vanes (N=8) have been selected in order to avoid the aforementioned complications at

which the volume ratio is ry,b=5.4305.









The displacement volume of a circular rotary-vane expander may be computed for

any given set of input parameters detailed in Table 4-3. The control volume analyzed is

defined as the volume bounded by the stator-cylinder, rotor, leading and trailing vanes

and the two axial end-plates. Figure 4-7 illustrates the ideal cell volume variation as a

function of angular displacement for different numbers of vanes. The displacement

volume per revolution is the cut-off volume multiplied by the number of vanes. Figure 4-

8 shows the difference between the ideal and actual volume. By accounting for the

thickness of the vanes, tv=4.2 mm, a slight shift in the volume curve is observed. This

results in a deviation by as much as 89% and by as little as 1% at initial and at larger

angular displacements. It can also be seen from Figure 4-8 that the maximum ideal and

actual volume occurs at 6=184o and 6=180o respectively.

The condensing and evaporating pressures corresponding to 50oC and OoC

condensing and evaporating temperatures are 1.942 and 0.498 MPa, respectively. This

corresponds to a process volume ratio of 14.74 if the refrigerant is assumed to be

expanded isentropically. Figures 4-9 and 4-10 illustrate the variation of the pressure

within the expander as a function of the volume within the expander for various intake

angles and numbers of vanes, respectively. The pressure is found to be greater than the

desired exhaust pressure primarily because the built-in volume ratio is not equivalent to

the process volume ratio. From Figure 4-9, it can be shown that built-in volume ratios of

8.16, 6.6 and 5.43 corresponding to intake angles of So, 10o and 15o, respectively,

correspond, to exhaust pressures 15%, 32.2% and 40% higher than the desired exhaust

pressure. Figure 4-10, illustrates how the number of vanes affects the pressure variation

in an expander for a given intake angle. It can be shown that built-in volume ratios of









4.083, 5.43 and 6.6 corresponding to 6, 8 and 10 vane expanders result in 53%, 40% and

26% higher exhaust pressures, respectively. These differences are primarily due to

inadequate built-in volume ratios of the expander and result in less desirable

thermodynamic performance. Unless a throttle valve or another means of expanding the

discharging fluid to the evaporating pressure is used, flashing and/or other losses are

imminent. For the purpose of this ideal analysis the losses due to under-expansion are not

included in the models developed and it is assumed that a throttle valve further expands

any under-expanded refrigerant to the evaporator pressure.

Independent of the built-in volume ratio, the optimum performance of an expander

is expected to occur when the mass flow of the working fluid is maximum. The

maximum mass flow-rate will likely have to be greater if the built-in volume ratio of the

expander is inadequate. Accurate models for the mass flow-rate of an actual expander

may be developed in which inlet area, exhaust area, velocity and density variations as a

function of angular displacement are modeled. Figure 4-11 shows the variation of the

mass trapped in the expander' s control volume as a function of angular displacement for

various intake angles. It can be seen that as the intake angle increases, the greater the

amount of mass in the control volume during the expansion process. This results from the

fact that the expander' s cut-off is delayed. These delays in cut-off are represented as lines

(a), (b) and (c) respectively. It is found that the mass in the control volume during

expansion is 50% and 28% higher for intake angles of 6=25o and 6=10o, respectively,

when compared with that of 6= So

Figures 4-12 and 4-13 show the variation of power extracted during the process

expansion as a function of angular displacement for various intake angles and different









numbers of vanes. Figure 4-12 shows that the maximum power extracted prior to the

exhaust process is 1440 W, 742 W and 534 W for 6, 8 and 10 vane expanders,

respectively. At the conclusion of the exhaust process, the net power output from the

expander is 355 W, 305 W and 257 W corresponding to 6, 8 and 10 vane expanders

respectively. It can also be seen from Figure 4-13 that as the intake angle increases the

maximum power extracted prior to the exhaust process is 611 W, 690 W, 742 W and 794

W corresponding to intake angles of So, 100, 15o and 200, respectively. At the conclusion

of the exhaust process, the net power output from the expander can be seen to be 309 W,

307 W, 305 W and 297 W corresponding to intake angles of So, 100, 15o and 200

respectively. Figure 4-14 shows the variation of the power extracted as a function of

volume for a particular set of input parameters. The same general trends of Figures 4-12

and 4-13 is observed.

Theoretical simulation models have been developed to analyze and evaluate the

geometric and thermodynamic performance of a circular rotary-vane expander. The

following conclusions can be made:

* The built-in volume ratio can be seen to decrease by 40%, 58.3% and 67% for the
entire range of intake angles investigated for 4,8 and 12 vane expanders
respectively. The increase of the number vanes may be limited by the dynamics and
mechanical strength of the rotor and the added friction and internal leakage that
may result. Lower intake angles result in larger built-in volume ratios but higher
throttling losses are typically associated with them.

* As the spread of the intake angle decreases and the number of vanes increases,
hence an increase in the built-in volume ratio, the resultant exhaust pressure is
found to reach the evaporating pressure. These differences are primarily due to the
inadequate built-in volume ratio of the expander and result in less desirable
thermodynamic performance.

* Unless a throttle valve or another means of expanding the exhaust fluid down to the
evaporating pressure is used, flashing and/or other losses are imminent when the
refrigerant is under-expanded.










* As the number of vanes increases, the net power output of the expander decreases.
For the cases investigated, as much as 75% of the power extracted prior to the
exhaust process is consumed by the exhaust process. Compared to a 10-vane
expander the net work output of a 6-vane expander is found to be greater by as
much as 27%

* As the intake angle increases, the maximum power extracted prior to the exhaust
process increases. At the conclusion of the exhaust process, the net power output
from the expander is found to deviate only 2% from the mean.

The thermodynamic model developed in this section was ideal in its assumptions

and analysis. It did not take into account losses that may arise due to throttling in the

intake and exhaust ports, two-phase internal leakage, friction, heat transfer, re-

compression or irreversible losses due to over- or under-expansion caused by inadequate

expander sizing. Chapter 5 details the theory and presents models of those

aforementioned loss mechanisms. Once described, the models for these loss mechanisms

will modify the thermodynamic model developed in this chapter to provide an accurate

estimate of the expander' s detailed and overall performance.































Figure 4-1. Flow diagram of the main computer program developed












Table 4-1. Summary of rotary-vane literature


Analysis


No. Investigator Application
1 Peterson and McGahan (1972) Compressor
2 Badr et al. (1985a) Expander in ORC
3 Badr et al. (1985b3) Expander in ORC
4 Badr et al. (1985c) Expander in ORC
5 Badr et al. (1986a) Expander in ORC
6 Badr et al. (1986b3) Expander in ORC
7 Edwards and McDonald (1972) Expander in ROVAC
S8 Robertson and Wolgemuth (1978) Expander in RC
9 Beck et al. (1966) Compressor in VC
10 Marsters and Ogbuefi (1972) Air Motor
11 Somayajulu (1971) Pump
12 Ben-Bassat and Wolgemuth (1972) 2-stage Exp in RC
13 Wolgemuth and Olson (1971) Expander in RC
14 Jacazio et al. (1979) Air Motor
15 Bransford and Stein (1960) Compressor in VC
16 Robertson and Wolgemuth (1975) Expander in RC
17 Barszcz, Z. (1980) Air Motor
VC: Vapor Compression refrigeration system
RC: Rankine Cycle
ORC: Organic Rankine Cycle
ROVAC: ROtary Vane Air refrigeration Cycle
*Keenan and Keyes equation of state for steam


Working Fluid
Air
General
R-113
R-113
R-113
R-113
Air
Steam
R-12 vapor
Air
Air
Steam
Steam
Air
Air and R-12
Steam
Air


Eq of State
Ideal-gas

Ideal-gas
Ideal-gas
Ideal-gas
Ideal-gas
Ideal-gas
Ideal-gas
na
Ideal-gas
Ideal-gas
K & K*
K & K*
Ideal-gas
Ideal-gas
Ideal-gas
General


xx x


x x


xX









Base Line
6=0


to


Charging Process
0<6<86+


Exhaust Process
8ex<6<2n:


Expansion Process
86+8<6<6e


Trailing vane


Figure 4-2. Schematic of a generally oriented circular rotary-vane expander with 8 vanes described by two circular arcs












Table 4-2. Geometrical input parameters

Input Parameter Symbol
Rotor radius, m rR
Eccentricity, m e
Angle between rotor and stator-cylinder centers yr
Stator cylinder radius, m rs= rR 8
Axial length of expander, m L=AR rR
Number of vane s N
Angle between successive vanes, deg 8=3600/N
Thickness of vanes, m tv
Inlet port arc-length, deg 8in
Exhaust port arc-length, deg 6ex
Arc-length of sealing arc, deg 6seal













0.003

0.0025


ai 0.002

i 0.0015

0.001

0.0005

0


0 20 40 60 80

Angular Displacement, deg


Figure 4-3. Variation of the inlet throat area with respect to angular displacement


0.003

0.0025-

0.002-

0.0015-

0.001-

0.0005-



150 200


250 300 350

Angular Displacement, deg


Figure 4-4. Variation of the exit throat area as a function of angular displacement for an
8-vane circular MVE












-Ideal Expansion v, process = v,buit-in


Pint Under Expansion rv, process > v,buit-in

-Over Expansion Tv, process < v,buit-in




Pex




6in 6in +6 6ex V


Figure 4-5. Pressure variation as a function of volume in an ideal expander for the cases
of ideal, over and under expansion.












Table 4-3. Typical operating conditions for a basic and modified 2 ton system

Expander
mi Isenthalpic Isentropic
Expansion nex,=0.45 nex,=0.85 Expansion

,g/s 49.6 48.61 47.77 47.5

va, m3Ikg 0.000924 0.000924 0.000924 0.000924
ry,p 16.3 15.6 14.97 14.74
COP 3.52 3.87 4.23 4.38

rlher, % 64.4 70.8 77.4 80.1
wexp, kJ/kg -2.89 5.46 6.42
wnet, kJ/kg 40.3 37.4 34.8 33.8
Qevap, kJ/kg 141.1 144.7 147.3 148.2

*R-22, 2 ton cooling capacity, Tcond=50oC, Tevap OoC, 11,=0.85, no superheat, no subcooling,
pressure losses in lines and components neglected

















8 L

6 -1--~
4-


-a- N=6
--.--N=8
--N= 10
N= 12


Intake Angle 8in, deg



Figure 4-6. Variation of built-in volume ratio as a function of intake angle for different
numbers of vanes



3.5E-06

3.0E-06-
N=4
E 2.5E-06 AR=0.46
e" FR=27.9 mm
E2.0E-06 -N6e=2.7 mm
1.5-0 -N=8 Si=150

> 1.0E-06-

'E N=12
5.0E-07-

0.0E+00

0 60 120 180 240 300 360

Angular Displacement 6, deg


Figure 4-7. Variation of ideal volume as a function of angular displacement for different
numbers of vanes










149













2.0E-06


1.6E-06


1.2E-06


8.0E-07


4.0E-07


0.0E+00


AR=0.46
r,=27.9 mm
e= 2.7 mml
N=8


El.:., ,,= 1 8 9


0 100 200 300 400

Angular Displace me nt 6, deg


Figure 4-8. Variation of ideal and actual volumes as a function of angular displacement


2500
AR=0.46
2000 -Pinlet r,=27.9 mm
ca ~ e=2.7 mm
150 Cn10 ,=1 50 N=8
60ut= 1890
t~1000 in= 5'

500 ~1)Losses
Pexhaust

0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06

Volum7e, m73


Figure 4-9. Variation of pressure as a function of volume for various intake angles











2500


e lniP


20

1E

S1


00u AR=0.46
\ ~N= 6
4 rR=27.9 mm
500 N-0 e=2.7 mm
6in= 150
)000 ''1w Bout= 1 8 9


j0


0.0E+00 5. 0E-07 1.0E-06 1.5E-06 2. 0E-06 2.5E-06


Volume, m3


Figure 4-10. Variation of pressure as a function of volume for different numbers of vanes




4.0E-04


Expansion


AR=0.46
rR=27.9 mm
e= 2.7 mm
N= 8
R=58 rpm
60ut=1890


3.0E-04


2.0E-04


1.0E-04


0.0E+00


(a() 100 200 300

Angular Displacement 6, deg


Figure 4-11i. Variation of mass within the expander' s cell volume as a function of angular
displacement for various intake angles














AR=0.46
TR=27.9 mm
e= 2.7 mm
R=58 rpm
em=-150
60ut=1890


0.8

0.6

0.4

0.2

0


0 120 240 360


Angular Displacement 0,deg


Figure 4-12. Variation of power as a function of angular displacement for different
numbers of vanes








0. 8 8n20 ,=150
8,n=100O AR=0.46
~ 06 rR=27.9 mm
e=2.7 mm
a N=8
S0.4
o a2=58 rpm
60ut= 1890


0 120 240 360


Angular Displacement 8, deg


Figure 4-13. Variation of power as a function of angular displacement for various intake
angles














0.6
~haustAR=0.46
ExhaustR=27.9 mm
0. e=2.7 mm
g~~ I. take and N= 8
a., Expansion s2=58 rpm
0.2 _Net Work o
Output ~im 150
60ut=1890


0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06

Volume, m3


Figure 4-14. Variation of power as a function of volume









CHAPTER 5
PRIMARY AND SECONDARY LOSS MECHANISMS IN ROTARY-VANE TWO-
PHASE REFRIGERATING EXPANDERS


All turbomachinery operate with energy and mass losses. These losses may be

attributed to two-phase internal leakage, friction, throttling losses in the intake and

exhaust ports, re-compression in the expander cavity and over- or under-expansion due to

inadequate sizing of the geometric volume ratio.

Primary Loss Mechanisms

The aim of this chapter is to develop mathematical models that would account for

the primary loss mechanisms of internal leakage and friction. The size of a rotary-vane

expander is of primary interest due to the influence it has on the magnitude of loss

mechanisms present. While rotary-vane compressors are typically very efficient, at

smaller sizes, the effect of losses such as internal leakage becomes much larger. In rotary-

vane expanders of smaller sizes both the volumetric and adiabatic efficiencies are

expected to be smaller. Depending on the operational and geometrical parameters of the

expander, laminar and viscous two-phase leakage flow within the expander may be

present. Single-phase leakage models available in the literature must be revisited and

modified accordingly. A dynamic frictional model for the expander must also be

developed for ideal operation (i.e. no internal leakage) and modified to account for

internal leakage accordingly. A comprehensive component-level model of inherent

friction and internal leakage losses in a two-phase circular rotary-vane expander used in a

vapor compression refrigeration system is presented. The model establishes the efficiency

and performance of the expander as a function of geometric and fluid parameters.

Accurate modeling and prediction of frictional and internal leakage losses is vital to










being able to accurately estimate the efficiency, rotational speed, and the torque and

power produced by the expander.

Internal Leakage Paths and Clearances

The clearances between the rotor and stator cylinders, end-plates and vanes provide

the necessary seals of the expander cavity. Caution must be exercised when designing an

expander' s assembly clearances. Depending on the operating conditions and geometric

volume ratio, large temperature gradients may also lead to thermal deformations of these

materials. The clearances must be minimized for each application to ensure control of

internal leakage losses yet re-compression of liquid may lead to inadequate operation or

"liquid locking". The various leakage paths are tabulated in Table 5-1.

The volumetric efficiency of an expander may be larger than unity due to

significant increases in the consumption of the working fluid. This significant increase is

primarily a result of internal leakage losses. The magnitude and sources of internal

leakage must be identified and minimized. The magnitude and sources of internal leakage

is significantly dependent upon the way by which the working fluid is introduced into the

expander' s cavity. Figure 5-1 illustrates the difference between the conventional (a) and

modified (b) methods through which the fluid is introduced into the expander's cavity.

The "conventional" intake port is machined directly into the stator-cylinder. The

"modified" intake is comprised of slots machined in both end-plates after which the fluid

is introduced into the expander's cavity via a slot machined into the rotor at a specific

angle from the slot.

Upon the completion of the geometric, kinematic and thermodynamic models, the

predicted pressure and quality of the refrigerant in the expander' s control volume are

used to calculate the leakage of refrigerant through the various leakage flow paths. In









order to model internal leakage between control volumes, the flow areas of the leakage

paths, Aleak, must be estimated. The mass flow-rate of internal leakage can be written in

general form as

rhleak (B) CPId (lea Aleak lea (8 UBI () (5 -1)

where Cd is an empirical discharge coefficient. The density and velocity of the leakage

are a function of the angular displacement at which the leakage occurs and the

thermodynamic state of the fluid. Of primary interest is the computation of the velocity.

The net mass flow leaked into the cavity formed by the adj acent vanes, stator and rotor

cylinders (see Figure 5-1) can be computed as

libleak (8 jleak (B-6 jleak (0) (5-2)

Leakage from Path (6) is neglected. It should be noted that if the sealing arc is

symmetric about the base line then gi=180o and RS(6)=rs.

Types of Two-Phase Leakage Losses

The maj ority of internal leakage models in turbo-machinery presented in recent

literature are developed for single-phase compressible flow. The friction associated with

incompressible leakage losses through a duct or leakage path of constant cross-sectional

area affects only pressure in the direction of the flow. The velocity of the working fluid

remains constant. According to Oothusizen (1997), in compressible flow, friction affects

all of the flow variables, i.e., the changes in pressure cause changes in density which lead

to changes in velocity For two-phase flow this also necessitates vaporization of liquid,

which alters the density of the working fluid significantly. Depending on the operational

and geometrical parameters of the expander, turbulent two-phase leakage flow within the

expander may be present.









In the case of two-phase viscous leakage losses, the two-phase mixture is assumed

to be homogenous in nature. The idealizations adopted under this fairly common model

include assuming that the vapor and liquid velocities are equal. The over-all two-phase

flow is also assumed to behave like a single phase, uniformly mixed, having fluid

properties whose values are, in some sense, mean values for the flow according to Carey

(1992). According to Levy (1999), assuming homogenous flow also allows the ability to

apply all available single-phase flow analyses and empirical correlations The density of

the homogenous two-phase mixture is computed as

1 x 1-x
=-+- (5-3)
P Pg fc

The calculation of other necessary properties for the two-phase mixture, such as the

specific heats and viscosity, can be very troublesome. Many different formulations have

been proposed throughout the literature. Among them,


S= (1-x) p, + x p, (5-4)

1 x 1- x
= -+ (5-5)




P P, Pf

For example, a two-phase mixture of water at P=500 kPa and x=0.2 yields a viscosity of

146.09, 53.52 and 15.968 CIPa s from Equations (5-4), (5-5) and (5-6), respectively.

According to Carey (1992), Equation (5-5) is the most common. Properties can also be

computed by utilizing the concept of the homogenous void fraction ot,










Pfx
a = (5-7)
p, x + (1- x-) p,

and hence the dynamic viscosity can be computed from pu= pf (1- a) + pUgC. This method

will be used since the computed value is identical to that obtained from Equation (5-6). In

the same manner the specific heats and thermal conductivity can be computed.

Viscous effects must be accounted for when computing the pressure drop and mass

flow-rate through radial clearances, e.g. flow through minimum clearance. These

channels are typically long and wide when compared to the height of the channel.

Viscous effects may be neglected when dealing with nozzles or short ducts. The axial

leakage flow between the vanes and the end-plates is an example where viscous effects

may be neglected. The flow of the two-phase compressible leakage, accounting for

friction, can be modeled as adiabatic Fanno-flow.

Figure 5-2 shows a schematic of a typical leakage path. The dimensions of the

leakage path, cross-sectional area and length, are known and depend on the leakage path

under consideration. The exit, (e), could be choked depending on the difference between

the pressure of the reservoir and back pressure at local angles 6 and 6+6, respectively.

Darby (2001) concludes that the ratio of the sonic velocity in a homogenous mixture to

that in a gas alone is much smaller than unity. In this case choking can occur in a two-

phase mixture at a significantly higher downstream pressure. This corresponds to both a

lower pressure drop and mass flux.

The flow is choked when the velocity at the exit plane is equal to the speed of

sound. For a non-flashing liquid and an ideal gas mixture, the maximum mass flow-rate









in the channel can be computed. The isentropic speed of sound for a homogenous two-

phase mixture can be computed as


a =a (5-8)


The governing equations and the discretized form of those equations are presented

in Table 5-2 and are derived from first principles and describe adiabatic Fanno flow from

a reservoir.

It should be noted that the volume of the leakage channel is V=Hwl and the mass

flow-rate through the channel is in = p,u,Hw The Darcy friction factor,J; is assumed

constant over the length of the leakage channel. This is a fairly reasonable assumption

since the flow is typically found to be fully developed over a majority of the channel

length. For Reynolds numbers in the lower range of the turbulent region, the modified

Blasius correlation is


f~ = 0.33 Re-0.25 (5-9)

where the Reynolds number is defined as ReH eH/ve This correlation has been

shown to be in good agreement with various experiments throughout the literature. The

momentum equation can therefore be expressed by

I1.75
(i: -4) 0.165 e 1.25 e"u (5-10)


The methodology used to solve for adiabatic Fanno-flow through a leakage path is

an iterative procedure. The state of inlet to the channel is known. Depending on the

leakage path and the state of the refrigerant, the flow may be of an incompressible liquid

or a compressible two-phase refrigerant. The unknowns, in Equation (5-10), that remain









are ue, thz, Pe and pe. The iterative procedure calls for an initial guess at the exit plane of

the channel where the flow is assumed to be choked. An initial guess of the pressure at

the exit plane may correspond to the average pressure that exists in the adj acent expander

cell. The flow is assumed to be isenthalpic or isochoric in the case of two-phase or liquid

flow, respectively. In the case of the two-phase flow the quality is known. Based on the

initial guesses, the speed of sound can be calculated. If the flow is choked at the exit

plane, i.e ue=a, then P is iterated upon until Equations (5-5), (5-8) and (5-10) are satisfied

simultaneously.

If the flow is viscous and the exit plane is unchoked, the average velocity of the

leakage can be determined by solving generalized Couette flow, see Figure 5-3, the flow

is driven by both the shear of a moving wall and a given pressure differential. Solving the

momentum equation and applying the no-slip and moving wall boundary conditions the

velocity distribution is

1 dP c r
u(y> (y2 SP) (5-11)
pu dxe S

The mean velocity between the plates is defined as


u,, u(y)dy (5-12)
r=0

and after integration is found to be
cor 82 dP
u = (5-13)
"' 2 12, d
If the flow in the leakage path can be modeled as isentropic flow through a nozzle

whose throat is choked then the initial guess used to evaluate the sonic velocity is

P,' = E: and se = s,. The density is then calculated. The pressure and density at isentropic









conditions, at the exit plane, are iterated upon accordingly until the computed speed of

sound is identical to the relationship obtained from the energy equation, ue~ = J2(h -he). In

the case that the isentropic flow is unchoked, the exit plane pressure is equal to the back-

pressure, Pe=Pb and se= sl. The thermodynamic state at the exit plane can then be

established and the velocity at the exit can be determined by Equation (5-8).

Axial Clearance Between Rotor and End Plates

The flow of the fluid between the rotor and end-plates can be modeled as that of a

fluid flowing in a narrow gap between a disc which rotates within a chamber of finite

dimensions. The boundary conditions imposed on the periphery of the rotating disk are

non-axisymmetric in nature due to the pressure distribution of the fluid in the expander' s

cavity. The effects of the vanes and their slots are neglected. The characteristics of the

flow as well as the complexity of the solution can vary significantly depending on the

order of magnitude of several key dimensionless variables. The dimensionless governing

equations yields key dimensionless groups that govern the flow and are found to be the

local rotational (disk) Reynolds number in terms of Si, the Euler number and the gap

aspect ratio, S. If the rotational speed of the rotor is to in rad/s then these parameters are

defined as


S =R (5-14)



Eu=2 (5-15)



Re, = m (5-16)









where AP in the Euler number corresponds to the maximum pressure differential on the

rotor' s outer boundary, i.e. the intake and exhaust conditions. It should also be noted that

in rotating disk problems the rotational Reynolds number is utilized instead of the radial

Reynolds number which is defined as Re = corR61/,l, For the same flow conditions, the

use of the latter is typically three orders of magnitude larger.

Daily and Nece (1960) have identified four different modes of flow for given

geometrical and flow characteristics. Of interest in this study are the laminar and

turbulent cases corresponding to a close gap where boundary layers on the rotor and end-

plate are merged. In this case a continuous variation of the velocity across the axial gap

exists. The rotational Reynolds number must be small in the gap, Re~ <1, for this type of

flow to exist. Separate boundary layers exist when the axial gap exceeds the total

thickness of the two boundary layers. The condition for laminar flow can therefore be

estimated from Daily and Nece (1960) as Re4 -S2 < 105 The Euler number which is

defined as the ratio of the maximum pressure differential to the centrifugal pressure

produced during rotation is assumed large, i.e. Eu -Re~ >1. The gap aspect ratio must also

be substantially larger than unity. If these four conditions can be satisfied simultaneously

then a zeroth-order solution approximates the solution to the Navier-Stokes equations.

This solution consists of a power series of Real, e.g. ur = ur, + Re~ uz + Re~ u, +...,

expanded around the asymptotic solution of Real approaching zero. Bein et al. (1976)

concluded that this is a reasonable assumption for Eu 2 0 .

It must be determined whether the following analysis is valid for the flow of two-

phase fluid in the axial gap. Due to the vaporization of liquid in the expander' s cavity, the

thermodynamic properties vary significantly. Coupled with the significant variation of









intake and exhaust parameters, geometry of the leakage gap and rotational speed of the

expander, the aforementioned constraints may not be satisfied. In order to determine the

operational conditions under which the analysis is valid the following operational

parameters considered are presented in Table 5-3. It can be shown that the flow is laminar

for all parameters investigated at lower rotational speeds, i.e. O2<500 RPM. Depending on

the geometric size of the expander, the expander' s actual rotational speed realistically

doesn't exceed this value. Based on the analysis of these operational conditions, an order

of magnitude analysis of the flow dimensionless parameters is presented in Table 5-3. It

maybe concluded that the zeroth-order solution is appropriate.

The location of the phase interface in rotating, eccentric complex geometries is

very difficult to predict. If sufficient centrifugal forces are produced due to high

rotational speeds it may be assumed that the maj ority of the leakage-flow in the axial gap

is vapor. This arises due to the tendency of the dense liquid to flow in the vicinity of the

stator-cylinder wall due to significant centrifugal forces. Hence the incompressible form

of the general equations with constant properties, evaluated at some mean value, is used.

Neglecting body forces, terms of order 1/S2 and equating terms with Real, the zeroth-

order dimensionless 3-D Navier-Stokes equations in cylindrical coordinates are solved in

order to determine the velocity and pressure field within the narrow gap. If a gap width 81

exists between the rotor and the end plates, then the dimensionless governing equations

can be reduced and written as follows


18(ruo+ 180+ = 0 (5-17)


"2 = Eu Re~ (5-18)










82 0
0= Eu Re~ (5-19)


Co = 0 (5-20)


where the dimensionless variables are defined in the nomenclature. According to

Equation (5-20), Po = P0 (r,0) The boundary conditions can be written as follows,

Fo (T, 8, z = 0) = 0

70 (T, 8, z = 0) = 0

wo(F,0,z=0)= 0 (5-21)

uo (F, 8, = 1)= 0




wo(r, 6,7=1)= 0 (5-22)

Po (r = 1, 8) = PN (0)





where Pcy(6) is a linear piece-wise function that closely approximates the actual pressure

distribution around the rotors outer boundary and is presented in Figure 5-4.

A linear approximation first presented by Bein et al. (1976) and furthered by Badr

(1985) for a typical pressure profile is compared to numerically integrated Fourier

constants in Equation (5-25). It can be shown that 100 terms in the summation of the

Fourier constants yield the actual pressure distribution. Depending on the geometric

volume ratio of the expander, the difference between the linear approximation and can

vary significantly. An expander with a volume ratio of rv=3.916 is analyzed in Figure 5-4









and a maximum difference of 40% is found to exist between. This greatly influences the

accuracy of the computed radial velocity.

The zeroth-order solution of the following simplified set of equations and satisfying

all necessary boundary conditions yields Bein et al. (1976) as such

Eu Re4 dp
IT = -_ Z) (5-24)
S2 dr

Eu Re4 1 87,
2 r 80

Substituting Equations (5-24) and (5-25) into the continuity equation, Equation (5-17),

and applying the boundary conditions in the axial direction yields

V P, = 0 (5-26)

Solving Equation (5-26) by means of separation of variables, the general solution for the

pressure distribution in the gap is

Polf,8) = ao + blnr +

a [(".' + b,, ) cos n9i r + c i"+d,,r") sin n0s (5-27)

The Fourier constants can be determined from the specification of the pressure

distribution on the outer boundary of the rotor (see discussion of Figure 5-4). The radial

velocity distribution can be found by differentiating Equation (5-27) with respect to the

radial direction and substituting that result into Equation (5-23). The total dimensional

leakage mass flow-rate in the radial direction with respect to the rotor is

tleak,1 (0) = 2PN.(0) 4,(r = 1, 8) wr 3,(528









The value of the mass flow-rate is negative if the flow leaves the expander cell.

This model doesn't take into account choking, which is likely, since the velocity

distribution at the rotor' s outer boundary is of primary interest.

Leakage around tips of vanes

Depending upon on the method by which the high pressure refrigerant is introduced

into the expander' s cavity; see Figure 5-1, leakage due to insufficient under-vane fluid

pressure or inadequate force/absence of mechanical springs may lead to significant

leakage losses several orders of magnitude larger than other leakage paths. Continuous

and sufficient contact between the vanes and the stator-cylinder is necessary to subdue or

minimize this type of leakage. The maj ority of leakage of this type may occur during the

intake process where low rotational speeds cause inadequate centrifugal forces to keep

the vanes in contact with the stator-cylinder. The existence of both low rotational speeds

and inadequate under-vane pressure can be predicted by the frictional model developed.

The frictional model is presented later in this chapter. It should be noted that the reaction

force, Fn(6) (see Figure 5-5), is predicted by the friction model and accounts for, among

other things, the sliding velocity and acceleration of the protruding vanes as well as the

under-vane pressure. If the vane has lost contact with the stator-cylinder, the value of this

normal force is negative, and the leakage flow area would be computed as

4ak, (0) = 8, (0)L = X(0)L (5-29)

where X(6) is the instantaneous protrusion of the vane from the rotor slot at the angular

rotation of the rotor where the loss of contact has occurred.

When the vanes establish contact with the cylinder it is assumed that the leakage

area is negligible and that this occurs until the start of the sealing arc. There may exist









leakage around the tips of the vanes during the spread of the sealing arc due to a

clearance gap 83 between the stator-cylinder and rotor that may be due to an assembly

clearance or due to machining limitations. This radial leakage is treated in the next

section.

Due to the short nature of this leakage path, t,, the leakage of the two-phase fluid in

this path is assumed to be that of fluid in a convergent-divergent nozzle, i.e. isentropic

flow. It should be noted that if the incoming fluid is introduced into the rotor' s slots than

the internal leakage associated with weak under-vane pressure can be neglected. For this

design, internal leakage around the tips of the vanes can be neglected. The high-pressure

incoming refrigerant is first used to provide sufficient under-vane pressure so that

continuous contact between the vane and stator-cylinder may be achieved.

Leakage past the sides of the vanes

In general, the axial width of this leakage path may be estimated by accounting for

the thermal expansion of the vanes in addition to the gap that exists due to the thickness

of gaskets, 6, and other assembly clearances. The operational length of the vane can be

computed from the known material's coefficient of linear thermal expansion,

Ly L,,o = aYLY,,oT, and the temperature difference that the vane encounters relative to

ambient conditions. The mean temperature of the vane is computed as the average

temperature of two adj acent cells. It can be shown that even at the moderate temperature

differences realized within an expander's cavity this effect accounts for a 0.3% change in

the length of the vane and is therefore neglected. The axial width of the leakage path is

assumed constant and equivalent to the gasket thickness

s33) 6g (5-30)









The height of this leakage channel is estimated by subtracting the height of the gap

realized due to leakage around the vane tips, if any, upon loss of contact with the stator-

cylinder, 62(6). The total leakage flow area is then calculated as

Aleak (0) = 2 X (0)- 6, (0)] S (0) (5-3 1)

Due to the short nature of this leakage path, ty, the leakage of the two-phase fluid in this

path is assumed to be that of fluid in a convergent-divergent nozzle, i.e. isentropic flow.

Leakage between the faces of the vanes and the side walls of the rotor's slots

The leakage from this path is typically neglected due to the existence of two contact

lines across the axial length of the rotor slots. These contact lines result from the reaction

forces FL(6) and FR(8) (See Figure 5-5). Negative reaction forces predicted by the friction

model represent a loss of contact between the vane and the rotor slot. The control volume

to which the mass and energy are added or subtracted depends on which vane face the

loss of contact occurs. If the fluid intake is through the rotor slot the leakage through this

path may be significant and comparable to other paths. Viscous effects need to be

considered since the leakage flow path is wide and long, Hy. The leakage flow area is

computed as

Aleak -8 (t, t,)L (5-32)

The Leakage from the expander to the cavity beneath the vane may also be comparable if

ts-ty is large.

Leakage in radial gap between the rotor and stator cylinder

As the pressure differential between the intake and exhaust ports increases and the

number of vanes decreases, this leakage is typically the largest contributor to internal

leakage.









This leakage flow path arises if the angle between successive vanes is larger than

the spread of the sealing arc (spread of minimal clearance) between the rotor and stator-

cylinder i.e. 8, > 6 If this is the case, the expander cell is in simultaneous contact with

the intake and exhaust ports. This overlap causes leakage flow from the inlet to the

exhaust ports against the direction of rotation. The severity of this leakage path is a

function of the number of vanes and the mean radial gap height 84 that exists between the

rotor and stator-cylinder due to assembly operations or machining limitations. This

leakage flow path would occur for the leading vane's angular displacement of

0 < 0 < 8- 8,,, The flow in this leakage path is modeled as adiabatic, one-dimensional

Fanno-flow. It should be noted that leakage through this path can only be neglected if

sufficient centrifugal forces or under-vane pressure exists to maintain continuous contact

of the vane with the stator-cylinder. Typically an insufficient under-vane force exists due

to the absence of high-pressure fluid (or presence of low-pressure residual fluid) due to

the theoretical completion of the exhaust process. The magnitude of the normal force

Fn(6) determined by the frictional model aids in this prediction. The flow in this leakage

path would necessarily be choked for large geometric volume ratios or uncontrolled

expansion in the exhaust port if no throttle valve is downstream of the expander.

Friction Model

Prediction of the power output and developed torque of the rotary-vane expander

requires accurate modeling of frictional losses incurred in the expander. The primary

frictional losses in the rotary-vane expander result from:


* Rubbing of the vanes against the stator-cylinder

* Rubbing of the vanes against the walls of their rotor' s slots










*The resultant viscous drag acting on both sides of the rotor from the leakage flows
through the axial gaps between the rotor and the end-plates.

A steady-state mathematical model that predicts frictional losses within a rotary-

vane expander must be coupled with geometric, thermodynamic and to leakage models in

order to provide for a better estimate of the mechanical efficiency and actual power

produced by the expander. A general dynamic model that incorporates the main sources

of friction within the rotary-vane expander is presented in this section. Effects of pressure

forces, reaction forces, sliding friction, viscous drag, gravity and inertial forces must all

be accounted to understand the relevance and magnitude of these effects. Figure 5-5 is a

schematic of a free body diagram of a vane protruding out a rotor slot at a local angle 6.

In order to model frictional losses in a rotary-vane expander, vane forces resulting

from pressure loading, vane inertia and body forces due to radial and Coriolis

accelerations must be determined. Sliding friction must also be considered between the

vanes and the rotor slots and stator-cylinder. These forces and frictional losses

continuously change, in both magnitude and direction throughout the expander due to

continuous changes in complex geometry.

The dynamic model developed assumes that the vanes are rigid and that the

coeffieient of friction is constant. The conditions for dynamic equilibrium are solved

simultaneously to determine the unknown reaction forces based on a free-body diagram

of a single vane. Upon determining the reaction forces, the power required to overcome

the resisting frictional forces can be computed. Below is a detailed description of the

various forces as well as simplifying assumptions that have been made.

A pressure force on the base of the vanes is assumed to act at mid-thickness of the

vane as a result of incoming fluid pressure or a spring force utilized to maintain vane-tip









and stator-cylinder contact. This pressure force at the vane base can be expressed as

follows

Fb = hPinistt,L (5-33)

where h is a proportionality constant that can be evaluated from calculating throttling

losses to or from the cavity beneath the vane. If the fluid is introduced directly into the

rotor slots for the spread of the intake port, 6in, and assuming no throttling losses then

h=1. If the fluid is introduced into the expander' s cavity directly then throttled leakage

flow will provide the pressure force on the vane base (e.g. h=0.75). If correctly sized and

assembled springs or a means for adequate pressurization of the slots are used then h=1.

The pressure forces exerted on the vanes from either side FPR and FPL can be

expressed as FPL8) Piv (6) X (6)L and FPR8 P ov (6 + 6) X (6)L respectively. If

leakage losses through the vane rotor-slot gaps are neglected then these pressure forces

are assumed to act mid-way between the rotor' s outer boundary and stator-cylinder' s

wall. If a gap exists between the vanes and the rotor slots, the orientation of the pressure-

loaded vane, see exaggerated effect in Figure 5-5, provides two sealing surfaces. The

portion of the vane that protrudes from the slot is denoted by X(6).

The reaction forces FL(6) and FR(8) are aSSumed to act at the points of contact of

the vane and the rotor slots on the left and right faces of the slot, respectively. These

forces are actually distributed along a finite distance along the side of the vane and not at

a point depending on the vane-to-slot clearance. Since the vanes are assumed to be rigid,

the friction forces are assumed to act along the surfaces as CL,FL (8) and CpFR e)

respectively. The sub script r denotes the rotor material. The reaction forces Fn(6) and









Ft(6) result from the vane-tip and stator-cylinder contact where F (8) = C1sFt (e) The vanes

are assumed to be in continuous contact with the stator-cylinder. The sub script s denotes

the stator-cylinder material. The assumption that the friction force (e.g. CL,FL ) is

proportional to the normal contact force (e.g. FL) may not accurate since large local

velocity gradients at the interface may severally alter the frictional losses. This may also

be the case at the vane-tip and stator-cylinder interface.

It should be noted that Figure 5-5 is valid for a vane that has no vane-tip curvature.

If the vane-tip is round or altered in any way there will exist symmetric radial forces on

either side of the normal force equal to the pressure on that side of the vane multiplied by

the proj ected area of that surface.

The sliding and centrifugal accelerations of the moving vane from the rotor slot

result in a radial body force Fa(6) which can be expressed as


F, (6) =m m/Acc(6) +r,,,(6) to (5-34)

where rm(6) denotes the distance from the center of the rotor to the center of mass of the

vane. The mean tangential body force Fe(6) that arises due to Coriolis acceleration of the

vane can be expressed as

Fo (6) = 214 Vel(6) co (5-35)

where Vel(6) and Acc(6) represent the sliding velocity and acceleration of the vane

respectively. The mass of the vane is assumed to act at the geometric center of the vane

(i.e. Hy/2 and t,/2).

From the following free-body diagram, Figure 5-5, the three conditions of dynamic

equilibrium can be written as follows










Fb 1rFL (8) rLFR (6)- m,,gcos6+ Fa (6)- F (6) =0 (5-36)


FL (6) + mvg sin 6 Fe (6) FR (6) + FPL(8 FPR(8 sLFn (6) = 0 (5-37)


mvg sin 6 Fe (6) FR()H,-X(6 PL -FR 6

22 2



(5-3 8)

where Equations (5-36) and (5-37) represent the net summation of forces in the radial and

tangential directions equal to zero respectively. Equation (5-3 8) represents the net

summation of moments about point O equal to zero wherein the direction of rotation of

the expander is assumed positive. It should be noted that the direction of some of the

forces change depending on whether the vanes are protruding out during the intake and

expansion processes or forced back into the vane slots during the exhaust process.

The three equations can be simplified, rearranged and written in matrix form as

aX = b (5-39)

where matrix a represents the coefficients of the unknown reaction forces of matrix


X= FR (eF 8 L() and matrix b contains known parameters such as pressure and body forces.



Once this matrix is solved at each angular displacement 6 of the leading vane the

reaction forces can be determined. The reaction forces in matrix X reveal the movement

of the vane in the rotor' s slot and can predict the loss of contact with the stator-cylinder

or faces of the rotor slots. It should be noted that the coefficients in matrix a should be

modified according to the vane's inward or outward movement relative to the slot. The









mechanical efficiency and power losses of the expander due to friction can then be

predicted. The frictional torque per angular displacement 6 due to vane-tip (first term)

and net vane-slot friction (second term) can be computed from


Tse (8) = MsF, (6)[rR + X(6)]+ CtlFL (6)c + FR ()]Vel(6) (5-40)


The total frictional loss, Tst~, in the expander is computed by summing Equation (5-

40) through a complete revolution of the rotor. The contribution of each vane is

accounted for and its contribution to the total torque is then averaged to yield Tstm,.

No rubbing frictional forces exist between the sides of the vanes and the end plates

due to the axial gaps between the rotor and the end-plates. The leakage through those

gaps, as determined previously, will lead to viscous drag.

In order to model the viscous drag due to the axial leakage between the rotor and

end-plates the viscosity of the two-phase fluid must be predicted accurately. The pressure

distribution in the axial gap P(r,6) was derived previously. The pressure is assumed to be

invariant in the axial direction. The state of the two-phase liquid can be determined by

assuming isenthalpic expansion in the radial and azimuthal directions. The flow in the

axial gap is driven by both shear caused by the rotation of the disk and a pressure

differential between the intake and exhaust ports. This results in a combined Couette-

Pouisselle (generalized Couette) flow, see Figure 5-6, in which the radial velocity can be

expressed as

1 dP c
u(Z) = (z2 3,z)+-~z (5-41)
p,,, dr 6,

The total torque produced by viscous drag on both sides of the rotor is expressed by














which can be integrated numerically. The total power loss is then calculated by

PRoss = (Ts,m, + Tyd)bZ (5-43)

where 0Z is the angular speed of the expander in RPM.

Uncertainties regarding the gap size are a significant obstacle to quantifying the

magnitude of leakage losses through certain leakage paths. These uncertainties are

presented later. The base case simulations detailed below are valid for an eight vane

circular expander of general orientation (i.e. yi=161.5 and RS(0) + rs) with a geometric

volume ratio of rv=3.26 corresponding to 6in=3 5o and 6ex=189o whose physical

dimensions mimic a modified GAST NL32-NCC-1 air-motor. The rotational speed of the

expander is O2=300 RPM. The inlet conditions to the expander correspond to a saturated

liquid at a condensing temperature of T=45oC.

As previously mentioned, the estimated frictional forces within the expander are

used to predict the movement of the vanes within their rotor slots. The profile of the

vane-tip is of significant importance in determining the forces acting on the vane-tip.

Figure 5-7 shows the instantaneous variation of reaction forces on a single vane with no

vane-tip curvature during one revolution of the rotor. It is assumed that a circular

pressurizing grove ensures that h=1 through out one revolution. It can be seen that the

normal force tends to the same variation as the pressure inside the expander' s cavity due

to the significant contribution of the under-vane pressure and absence of resisting forces

in the opposite direction. It should be noted that no loss of contact is predicted for any of

the reaction forces. It can be shown that for vanes with no vane-tip curvature an










insignificant under-vane pressure (h=0.1) is necessary to maintain contact with the stator-

cylinder.

Figure 5-8 is similar in nature to Figure 5-7 but for a vane that has curvature at the

vane-tip (i.e. addition of symmetric radial forces about normal force). It can be seen that

the normal reaction force is negative, denoting a loss of contact, during 6in+6<6
corresponding to a maj ority of the expansion process. The increased pressure gradient

across the vane in this case may lead to excessive leakage losses. This information is

passed to the appropriate internal leakage code. The use of materials such as graphite,

erode and match the profile of the stator-cylinder depending on the intensity of the under-

vane pressure. The density of the material should be large enough to provide sufficient

centrifugal forces if and when inadequate under-vane pressures exist.

Simulations were run to find the minimum value of h in order to obtain a positive

reaction force at the vane-tip stator-cylinder interface denoting contact with the stator-

cylinder. Values of h=0.75, 0.85 and 0.95 were simulated. It was found that due to the

curvature of the vane-tip, only a value of h>0.95 would suffice. Figures 5-9 through 5-12

show the variation of leakage to and from the expander cavity by different leakage paths

through one revolution of the rotor.

Figure 5-9 shows the variation of non-axisymmetric leakage from the expander

cavity as a function of angular displacement for different intake angles. It is ensured that

the governing dimensionless variables satisfy the constraints and assumptions that govern

the flow. The use of choked flow relations for this leakage path does not grasp the

underlying physics; magnitude and direction, associated with the flow between a rotating

and stationary disk with non-axisymmetric boundary conditions. It can be shown that if









choked flow conditions are assumed at the rotor boundary then the leakage from, model

cannot predict otherwise, the expander cavity is over predicted throughout the maj ority of

the intake and exhaust processes. The current analysis over-predicts the cell leakage

when compared to the choked flow model during the expansion process solely due to the

accuracy of the pressure profile used to approximate the non-axisymmetric boundary

condition. Figure 5-10 illustrates the variation of the non-axisymmetric leakage as a

function of angular displacement for both the ideal and throttling cases. As the inlet angle

decreases the potential for leakage increases significantly during the charging and

expansion processes.

For all leakage paths considered, the velocity of the leakage flow is calculated

initially using the general Couette flow velocity profile determined above. If the flow is

two-phase, the speed of sound is calculated to provide a reference. If the computed

velocity using the general Couette flow model exceeds the speed of sound, then the

model switches over to a choked flow regime corresponding to a maximum leakage flow-

rate. These general trends are apparent from Figure 5-11.

Figure 5-11 illustrates the total variation of flow to/from the expander cavity by

means of leakage past the sides of the vanes. Contact between the vane-tip and stator-

cylinder is assumed. A constant gap size is assumed. In actual operation, upon loss of

contact, the vanes do not necessarily retract all the way back into the rotor slot leaving

the leakage channel height X(6) as assumed in the above derivation. The leakage mass

flow-rate during the constant pressure intake and exhaust processes can be seen to

increase and decrease, respectively with the oscillating variation of vane protrusion X(6).

This effect is more evident in expanders with a larger geometric volume ratio.










The leakage mass flow-rate through the minimal gap that exists between the rotor

and the stator-cylinder is significant but constant for the duration 0<6<6-6seal, where 6seat

is 400. The contribution of this leakage path to the total leakage flow-rate increases with

the decrease of the number of vanes. The start of the intake port may be delayed, if

necessary, to minimize leakage from this path. The magnitude and direction of the

leakage between the sides of the vanes and the rotor slots depends on the method by

which the fluid is introduced into the expander's cavity. Figure 5-12 shows the variation

of leakage into the expander cavity as a function of angular displacement. This leakage

path only exists for the duration of the intake port. There may be leakage from the

expander cavity via this leakage path during the remainder of one revolution. If the flow

is introduced through the rotor slots, the velocity of this leakage path, assuming no

throttling, would be that of an incompressible liquid in shear driven Couette flow.

Here the plate moves with a velocity corresponding to the instantaneous sliding

velocity of the vane. The leakage mass flow-rate can be seen to increase proportional to

the outward sliding velocity of the protruding vane.

Of primary significance for an adequately designed expansion device is the ratio of

total mass flow-rate accounting for leakage and the corresponding ideal case, i.e. a

leakage ratio me, at The mass flow-rate through the expander accounting for leakage
madeal





maZctual = N, mov (B + 6) (5-44)l









where the first term in the bracket represents the ideal mass flow-rate and index i

represents the leakage path. For the base case simulations the leakage ratio is r=10.9.

This is significantly large and due to the low rotational speed of 300 RPM.

Figure 5-13 shows the variation of the ideal and actual mass flow-rates as a

function of rotational speed. It can be seen that the leakage mass flow-rate decreases

significantly as the rotational speed of the expander is increased. The ideal mass flow-rate

increases in a steady manner. The operational rotational speed of the expander is a design

choice that may be optimized by the proper selection of volumetric displacement for the

inlet flow conditions. The leakage ratio is r=30.6 and 4 at rotational speeds of 100 and

1000, respectively. The actual mass flow-rate curve increases linearly because all of the

internal leakage paths, with the exception of axial flow between the rotor and end-plates,

is assumed to be independent of speed.

Figure 5-14 illustrates the variation of the ideal and actual mass flow-rates as a

function of the number of vanes for an expander operational speed of 500 RPM. The

ideal mass flow-rate decreases since the volume of the cavity and the amount of mass in

the expander cavity at the cut-off angle decreases significantly. The amount of leakage is

seen to increase due to a larger contribution from the leakage mass flow-rate through the

radial gap, minimum clearance, between the rotor and stator-cylinder. Caution must be

exercised when increasing the number of vanes due to possible increases in friction and

decrease of mechanical strength of the rotor.

A thermodynamic and fluid dynamic model of the primary loss mechanisms in a

circular rotary-vane expander has been developed as a function of design and fluid

parameters. The following conclusions can be made:










* The curvature of the vane-tip can severally impact the amount of leakage past the
vane-tips due to loss of contact with the stator-cylinder. Even if no throttling is
assumed in the intake port, there is a loss of contact with the stator-cylinder through
out the duration of the expansion process.

* Vanes should be made out of high-density materials that can erode and mold to the
stator-cylinder geometric curvature. This would eliminate the need for adequate
under-vane forces through intake ports or cumbersome, unreliable springs.

* Leakage losses from a maj ority of leakage paths cannot be necessarily neglected in
two-phase rotary-vane expanders. Unlike single-phase rotary-vane turbomachinery,
significant variation in density and speed of sound with the vaporization of liquid
significantly alters the leakage flow-rates.

* The leakage losses from the non-axisymmetric flow in the axial gap between the
rotor and end-plates is significant. The model developed is relatively ideal in its
assumptions and doesn't take into account separated boundary layers, choking or
turbulent fluid flow. A computational model must solve the compressible Navier-
Stokes equations in order to do so.

* Increase in rotational speed decreases internal leakage losses significantly. Further
work must be made to determine the magnitude of detrimental effects such as
friction and viscous drag.

* A decrease in the number of vanes increases the contribution of leakage through the
minimal radial gap between the rotor and stator cylinder.

* Sizing of the rotary-vane expander for two-phase applications is critical with regard
to determining the optimum operational speed to reduce the loss mechanisms
presented in this study.

* Uncertainties regarding the gap size are a significant obstacle to quantifying the
magnitude of leakage losses through all the leakage paths during operation.

Two-Phase Throttling Losses in the Inlet and Exhaust Ports

The state of refrigerant that enters the expansion device is of great importance.

Unlike the conventional throttling valve, the state of the refrigerant may vary

significantly from condenser exit conditions to the expander cavity. This may be due to

significant throttling losses within the inlet port and internal contractions, enlargements

and bends within the expansion device. This may also be partly due to the fact that the

distance downstream over which the influence of the feature is felt is greatly increased in










two-phase flow. A model for the prediction of the pressure losses of two-phase flow due

to these flow restrictions is presented. An algorithm is presented and developed by which

the actual state and amount of refrigerant entering the expansion device can be computed.

Figure 5-15 illustrates the difference between the conventional and modified methods of

intake.

The refrigerant leaving the condenser may be either saturated or subcooled liquid.

Depending on the degree of subcooling, pressure drops resulting from throttling in the

expander' s intake port and friction due to flow restrictions may cause the refrigerant to

drop into the two-phase region. These pressure losses should be minimized in order to

maximize the pressure differential utilized to produce work in the expander. In the case

of under-expansion, where the geometric volume ratio is not sufficient when compared to

the process volume ratio, pressure losses realized in the intake and exhaust ports are a

necessary loss mechanism after which no further throttling may be needed. The analysis

of throttling in the ports can be divided into two distinct analyses. The difference in

analysis depends solely on the method the refrigerant enters the expander cavity.

The equations for the static pressure drops, frictional pressure losses and

contraction coefficients for various geometries are presented below. Calculating throttling

losses and the actual mass flow-rate is presented later. This will aid in the evaluation of

how well modeling the intake and exhaust processes as isobaric is.

During the intake process the refrigerant goes through numerous pressure drops as

a result of contractions and bends. The following assumptions are made to simplify the

analysis for all geometries unless otherwise specified. The void fraction of the

homogenous two-phase mixture is assumed constant over the feature (i.e. no phase









change). Separation of the two phases occurs and the use of general single-phase

frictional loss methods are no longer valid.

The following correlations, from Collier (1972), are presented to calculate the static

pressure drop, Equation (5-45), and the frictional pressure loss across a contraction,

Equation (5-46). A simplified momentum balance for the combined flow yields


PI- 2 2 1- 1+ (5-45)


Ap, = -1 2 1 X (5-46)
21 1


where Eqluations(5-45) and (5-46) differ by a factor (1 which represents the

conversion of pressure energy to theoretical kinetic energy associated with the

acceleration of the fluid through the contraction. The symbol a is the ratio of the

upstream Al to downstream area A2. G2 is the mass velocity of the refrigerant and is

defined as the velocity at the downstream location to the specific volume at that same

location. Co is the coefficient of contraction and is defined as the ratio of minimum area

A, to downstream area A2. This minimum area Ac, see Figure 5-16, corresponds to the

narrowest cross section where the smallest local pressure value is found.

In order to calculate the coefficient of contraction, the two-phase empirical relation

presented by Ruffell (1978) according to Schmidt and Friedel (1997) is used. The

equation for the contraction coefficient can be written as









11
C, =1- (5-47)
2.080 \1 +0.5371


Equation (5-47) has been compared against the contraction coefficients for turbulent

single-phase flow presented by Perry (1963) in Table 5-4. It should be noted that both

models have been determined experimentally. It has been found that modeled pressure

drop across sudden contractions is satisfactory and that the constant void fraction

assumption is not necessarily accurate (Collier (1972)).

Upon exit from either the 90o bends or the expander cavity, sudden enlargement

may exist in which further pressure losses are realized. For a two-phase mixture, the

change in static pressure and total frictional pressure drop across the enlargement can be

determined from a simplified momentum balance for the combined flow written as


p2 p1 = G2G1-0~ aV, 1+ x" (5-48)



Ap, =, (1- )2 Vp 1+ x (5-49)


where G1 is the mass flux upstream of the sudden enlargement. The homogenous and

constant void fraction pressure drop models presented above may yield satisfactory

predictions since low mass velocities and pressures are expected.

In the case of two-phase flow through a sharp edged orifice, the discharge

coefficient is equivalent to the contraction coefficient if small amounts of frictional

dissipation and velocity profile effects are neglected (Collier (1972)). The pressure drop

equation that results from a simplified momentum balance, assuming a homogenous









mixture and constant void fraction through the orifice, must be modified using a two-

phase multiplier

The pressure drop in a uniform flow area, 90o bend can be calculated from the

following equation presented by Collier (1972)


2i 8t=1 pBp P (5-50)


where APf and AP, are the pressure drops for the liquid and vapor phases and are defined




mil (1- X)2 V
Ap = (5-51)
2CD2A2


Ap, = C2 (5-52)


where the discharge coefficient CD is approximated as the coefficient of contraction Cc

from Equation (5-47). This assumption neglects the small amount of frictional dissipation

and velocity effects that may be incurred during the flow. The constant C is evaluated

from the following general expression


C="i ii V+(C 05 051[ 05f, v ,1 (5-53)


D D
where h=1 and the constant C2 is C, = 1+ 35 for 90o bends or C2 = 1+ 20 for 900
L L

bends with an upstream disturbance.

Conventional Inlet to Expander Cavity

As mentioned previously, this inlet port is typically machined into the stator-

cylinder. The inlet port spread is a design parameter with significant impact on the









volumetric efficiency and thermodynamic performance of rotary-vane technology. This

may be of greater concern in two-phase expanders since pressure drops in two-phase

flows are inherently larger and necessitate the formation of vapor. Depending on several

factors including the spread of the intake port, the angle between successive vanes and

the angular location of the leading vane relative to the intake port, the intake manifold

will be in continuous contact with one or more of the expander cells. It should be noted

that due to this continuous contact pulsation effects are negligible. It is hence of

importance to calculate the actual mass flow-rate of incoming refrigerant into each cell

and the impact it may have on the performance of the expansion device. An optimal

design of the intake port would minimize flow restrictions that would alter flow

characteristics by maximizing the available flow area at each angle of rotation. This

would ensure that an equal velocity is obtained at every opening.

To capture the effect of irreversible losses that are inherent in the intake port due to

flow restrictions, the concept of the discharge coefficient must be introduced. In general

the discharge coefficient is defined as the ratio of the actual to ideal mass flow-rates. The

discharge coefficient is a strong function of the area ratio and the Reynolds number of the

flow. The flow through the conventional intake port can be modeled as that of a two-

phase flow through a sharp-edged orifiee. In the case of two-phase flow through a sharp

edged orifice, the discharge coefficient is equivalent to the contraction coefficient,

Equation 5-47, if small amounts of frictional dissipation and velocity profile effects are

neglected (Collier, 1972). A schematic of the three phases of vane orientation that occur

during the intake process are shown in Figure 5-17.









It can be shown that the time rate of change of volumes on either side of the leading

vane in phase I, for example, differ significantly. This alters the amount of mass that

enters either control volume. The throat that is formed in phases I and III constitutes a

maj or flow restriction. The definition of the discharge coefficient may be altered to

incorporate the effect of the formation of the throat as well as the discrepancy in the time-

rate of change in expander cavity volume. An area averaged, change in volume weighted

component is incorporated into the definition of the discharge coefficient as such

AV 4 A(8)
cd(e = Cc (8) (5-54)
A V2 A,

where Aref is a reference area that denotes a frictionless orifice area that produces an ideal

mass-flow-rate. Figure 5-18 shows the variation pressure within the expander cavity as a

function of the angular displacement of the leading vane for the ideal and actual

(throttling only) cases.

It can be seen from Figure 5-18 that a significant drop in pressure is realized in the

intake and expansion phases. This dramatically reduces the expander's performance by

reducing the volumetric efficiency and generated torque and power. For all else equal, the

volumetric efficiency is defined as the ratio of the actual density at the cut-off volume to

ideal density (i.e. inlet density). The variation in the volumetric efficiency, final pressure

in the expander cavity at the maximum volume, are tabulated in Table 5-5 for various

intake angles.

Stand-alone experiments are needed in order to accurately determine the variation

of the discharge coefficient. One discrepancy that may exist is that at smaller local

rotation angles, the discharge coefficient may in fact approach unity and decrease in a









near linear fashion. This trend is represented as the dotted line in Figure 5-18. This

situation arises from the fact that both the time and volume required to be filled initially

is very small and that the extremely small throat area provides for a entrainment free jet

of working fluid to fill the expander cavity in question. At larger angles of rotation, there

may exist entrained fluid that disrupts and alters the flow.

It should be noted that the throttling losses in the exhaust port are typically found to

be negligible due to the smaller velocities associated with the exhaust fluid and the larger

spread area of the exhaust port.

Modified Inlet to Expander Cavity

The flow of the two-phase refrigerant in the intake is assumed to be homogenous

and gravitational effects on the pressure drops in vertical portions of the intake have been

neglected. The pressure drops associated with the work necessary to protrude the vanes

has also been neglected. This may be the case since the centrifugal forces exerted on the

vane due to rotation of the expander are typically an order of magnitude larger.

In order to model the mass flow of refrigerant into the expander, the velocities of

the fluid in the contractions, bends and enlargements, the unknowns, must be calculated.

Figure 5-19 shows a schematic of the intake of a modified rotary-vane intake. The

solution procedure entails an iterative procedure in which the pressure at location 2 must

be initially estimated as the average of the known pressures at locations 1 and 5.

Using Equation (5-45), the velocity at Location 2 may be calculated. Location 3 is

denoted by an because in fact it represents 4 bends of equal flow area. For the sake of

brevity they are illustrated as one. The velocities within these bends are assumed to be

equal. Using Equation (5-50) the pressure at each of the bends can be calculated. Since

the flow is compressible the quality at each one of these locations can be computed by










assuming an isenthalpic process. Using Equation (5-48) the pressure at Location 4 can be

computed since the mass velocity at 3* is known. The final parameter to be determined is

the velocity at Location 4, which can be computed using


Apexlt = -P4U24 (5-55)


This iterative procedure is repeated until the initial guess of the pressure at

Location 2 converges to within a of the following

p1 ps = Ap,,,t + Ap2oz, + Apenlarge + Pexlt (5-56)

It should be noted that the pressure at the exit can be determined using simulation

models or from experiments. The mass flow-rate can be computed as thz = pu,,A,, where

n=2,3*,4

Thermodynamic Model of the Actual Expansion Process

In this section the effects of internal leakage losses, friction and throttling are taken

into account deriving the appropriate conservation of mass and energy equations. The

solution procedure is detailed.

The ideal predicted pressure, temperature and mass variations in the expander's

control volume are used as initial estimates when calculating any non-ideal effects. The

velocity and hence mass flow-rate of the leakage flows via different leakage paths have

already been presented.

Charging Process

The first law of thermodynamics, taking into account heat transfer generated by

friction, frictional power losses and effects of internal leakage, can be written for the

process as follows









dE
= O, -Wb +Wi~ + minthmt + Cnflleak,n~lhleak,m, yl~eak~outhleak~.out (5-57)


Assuming the changes in kinetic energy and potential energy are neglected

dEcy=dUy. The pressure within the control volume is also assumed to be uniform.

Multiplying by dt Equation (5-57) becomes

AUJ = qf p,(Vi ) +1 W + mnthin +f C nPflneak~ln hleak~leak,outfhleak,ourt (5-58)

Solving for the internal energy of the refrigerant at the new time step, State 2, we find

m2121 + f 1, (V2 -6)+ +w +n mthmtr + Cnleak~,nhleak~lm Cleak,outhleak~-ouit
n2? + nfleak,mn C leak,ourt

(5-59)

The density of the refrigerant at the new time step, State 2, is calculated as

n+ +fnleak,mn lmeak,out
pt (5-60)


The use of REFPROP 7.0 allows the determination of the thermodynamic

properties at the new state point, State 2, namely the temperature, pressure and quality.

The process is repeated until the end of the charging process.

Expansion Process

Unlike the ideal expansion process the mass in the control volume, mov, varies

depending on the magnitude of the internal leakage losses after the cut-off angle

Ocs, = 8mt + 3. The density at State 2 can be determined from

nt, +Cn7leak,mn Cleak,outr
pt (5-61)


An energy balance of the control volume, i.e. leakage, during the expansion process

yields the internal energy at State 2 as











u2_11 f 1V V w 1 lak1hekz ekotla~u (5-62)
2 171~~2 + le~ak,ln -C nlea,out

The thermodynamic properties at the new time step, namely temperature, pressure,

and quality, can be calculated from Equations (5-61) and (5-62) by the use of REFPROP

7.0. This process is repeated until the end of the expansion process at a pre-determined

design angle of 6ex.

Exhaust process

The exhaust process occurs at an exhaust angle 6ex until Bend out,, + 6, Where 8out

is the spread of the exhaust port and 8 is the angle between consecutive vanes. The first

law of thermodynamics can be written for the process as follows

dE
if jb f ex r~hex +f mleak,,znhea,,zn, leak,outhleakout (5-63)


Assuming the changes in kinetic energy and potential energy are neglected,

dEcy=dUy. The pressure within the control volume is also assumed to be uniform. The

refrigerant flow from the control volume is also assumed to be quasi-steady and

homogenous. Multiplying Equation (5-63) by dt and solving for the internal energy of

the refrigerant at the new time step, State 2, we find

m,u, +q,- px (V2 V)+ +w mex he + C mleark~lnhla~n la~otlau
u2 = xlek (5-64)
m2+ mleak,ln lmeak~out

The density of the refrigerant at the new time step, State 2, is calculated as

m2+ mleak,ln lmeak,out
p2 = (5-65)


The use of REFPROP 7.0 allows the determination of the thermodynamic

properties at the new state point, State 2, namely the temperature, pressure and quality.









Expander Performance Evaluation

Upon development of the geometric, thermodynamic, internal leakage, frictional,

throttling and any other loss mechanism, the overall performance of a rotary-vane

expander can be predicted. The following parameters describe the performance of the

expander with respect to different criteria.

Of primary importance to the performance of any positive displacement expansion

device is the geometric (built-in) volume ratio. This parameter is defined as the ratio of

volumes at the exhaust and cut-off angles respectively. The reader is referred to an earlier

chapter dealing with this parameter in further detail. Along with the geometric volume

ratio the geometric code developed allows for design optimization with regards to the

number of vanes, location of intake and exhaust ports, stator-cylinder geometry and size,

rotor size and eccentricity and aspect ratio to name a few.

The pressure, quality and mass variations of the working fluid within the

expander' s cells are predicted by utilizing the thermodynamic model. Models for the

intake and exhaust processes are used in conjunction with the thermodynamic model to

determine throttling losses (termed breathing in the open literature) within the ports.

These throttling losses are a primary function of port design (geometry) and spread (size).

The use of these models also allows us to calculate the volumetric efficiency, resultant

power ratio and efficiency of the multi-vane expander due to throttling losses in the

intake port. The volumetric efficiency may be defined as





where vint is the specific volume of the working fluid within the expander' s cell assuming

no throttling losses (i.e. upstream conditions) and vo, (8,,, + 3) is the ensuing specific









volume of the refrigerant if there exists throttling within the intake port. Both quantities

are calculated at the cut-off angle. As the volumetric efficiency decreases, the effects of

throttling significantly alter the torque generated by the pressure forces exerted on the

vanes by the working fluid and hence lead to a decrease in the useful power output and

efficiency of the expander. The power ratio, P,-, may be defined as the ratio of the power

produced when throttling is accounted for to the ideal power produced by the expander

with no throttling losses.

Since the mass of refrigerant contained within the cell volume of the expander at

the cut-off volume may be significantly less when throttling effects are considered, it may

be advantageous to define a thermodynamic efficiency based on the ratio of the

expander' s actual to ideal specific work. This efficiency may be defined as


TI; w tthoeP (5-67)


where the rotational speed in both cases may be assumed equal. The volumetric

efficiency of the expander limits the accuracy of this assumption.

The internal leakage model developed will allow further characterization and

detailed performance analysis of the expander by predicting the magnitude and direction

of leakage from various cells within the expander. The direction of these losses is a

function of the pressure distribution predicted by the thermodynamic model. The leakage

coefficient is defined as the total leakage mass flow-rate to the ideal mass flow-rate

through the expander


I= nflk(5-68)









Single-phase leakage within the expander increases the pressure of neighboring

cells and inherently increases the work output of the expander. In the case of two-phase

leakage losses, the gaps through which the flow enters, produce areas from which further

throttling takes place. The adjacent cell pressure may increase but so does the quality of

the refrigerant within the cell. This reduces the amount of useful work that can be

extracted from the expansion process. The leakage power factor, Klk, of the expander can

thus be defined as the ratio of power produced by the expander accounting for leakage

and throttling losses and the power produced by the expander if only throttling losses are

accounted for. A thermodynamic leakage efficiency may also be defined as


1_W~k+b _W~k+b mb _I 1 )(-9
wb mlkb Wb Kkl& 5b)

The frictional model developed allows for even further characterization and

performance analysis by predicting various frictional losses throughout the expander.

These frictional losses allow for the calculation of an overall frictional torque that

contributes to a frictional power loss within the expander. Viscous shear stresses due to

leakage also contribute to the total frictional power loss, W, The mechanical efficiency

of the expander can then be defined as

W, W
rm k= (5-70)
W~k+b

The aforementioned performance parameters will aid in application specific design

optimization of rotary-vane two-phase refrigeration expanders.










Performance Variation Due to Stator-Cylinder Geometry

One geometric design parameter that has not been discussed in a thorough manner

is the geometric profile of the expander' s stator-cylinder which may have been alluded to

in previous sections. The variation in this parameter serves a single purpose, to increase

the geometric volume ratio of the expander but must simultaneously satisfy key

geometrical constraints. This section presents a more detailed discussion of the apparent

advantages and not so obvious disadvantages of varying the geometric profile of the

stator-cylinder and key geometrical constraints that must be considered in its design.

The profile of a circular stator-cylinder can be described in general by at most two

circular arcs. If the sealing arc is symmetric about the base line, y=180o and RS(6)=180o

then the profile of the stator-cylinder may be easily described by a circle. The profile of a

non-circular stator-cylinder can generally be described by n-1 circular arcs and one

sealing arc depending on the complexity of the stator-cylinder profile. The centers and

spreads of the circular arcs are known input parameters to the geometrical code. Figure 5-

20 shows a schematic and details the nomenclature associated with a symmetric and non-

symmetric non-circular rotary-vane expander about the base line, 6=0.

As previously shown, the performance of multi-vane expanders can be predicted

upon development of the geometric, thermodynamic, leakage and frictional models

described earlier. The geometry of the stator-cylinder is a prominent design variable that

can greatly affect the thermodynamic and fluid flow characteristics of a multi-vane

expander. The performance of circular and non-circular two-phase multi-vane expanders

is herein compared. Final design choices such as intake and exhaust port location and

spread will ultimately determine whether the stator-cylinder is symmetric or non-










symmetric about the vertical plane that represents the datum. Parametric optimization is

detailed here. A concise comparison between non-circular and circular expanders is

presented below.

From the geometrical code developed, the cell volume variation throughout the

expander is predicted. For all else equal, the eccentricity of the non-circular expander

during the intake process is smaller than that of the circular expander. This corresponds

to a lower rate of increase in cell-volume. This lower rate of increase in cell-volume may

correspond to a larger cell pressure or lower cell quality at the cut-off angle if the cell

pressure during the intake process is assumed variable or uniform respectively. This can

be seen when determining the vane protrusion from the rotor slot as depicted in Figure 5-

21.

For the same geometrical parameters, the non-circular expander has a higher

geometric volume ratio when compared to the circular expander. This is a result of the

significantly larger and smaller volumes corresponding to the exhaust and cut-off angles

respectively. The rate of increase of the cell-volume and spread of the expansion zone are

a function of the non-circular expander' s symmetry and location of the exhaust port. For

a non-symmetric non-circular expander, the angular spread of the expansion zone is

greater, due to location of exhaust angle, and hence the cell-volume increase is slower.

This results in lower internal leakage losses due to lower pressure differentials between

adj acent cells. The symmetric non-circular and circular expanders are typically designed

to have the same exhaust angle. In this case the cell-volume increase within the

symmetric non-circular expander is much faster leading to a larger pressure differential

between adjacent cells.










The pressure distribution around the rotor' s periphery significantly influences

internal leakage losses within the expander. Although the expansion ratio of the

symmetric non-circular expander is significantly larger, the higher cell-volume increase

within both leads to higher internal leakage resulting from greater pressure differentials

between adjacent cells. As a result, a smaller low-pressure region is realized throughout

the spread of the exhaust port within a non-symmetric non-circular expander. This

significantly decreases the internal leakage from high-pressure regions across existing

gaps between the rotor and two end-plates. The spread of the sealing arc is governed by

the curvature of the stator-cylinder. The spread of the sealing arc of a circular expander is

almost always larger than that of a non-circular expander. This corresponds to a lower

pressure gradient across the sealing arc resulting in lower internal leakage across that

radial clearance. In the same light, a symmetric non-circular expander has a larger sealing

arc spread when compared to a non-symmetric non-circular expander.

From the developed thermodynamic model, it can be seen that the non-circular

expander has a higher volumetric efficiency when compared to the circular expander. For

all else equal, a lower increase in the cell volume results in less throttling during the

intake process which in tumn results in a more effective filling process. This in turn results

in a higher power factor.

The developed internal leakage model validates the trends of leakage losses

aforementioned in further detail. The higher cell volume increase of the circular and

symmetric non-circular expander result in higher pressure gradients within the gap

between the rotor and end-plates resulting in higher radial velocities of the fluid at the

rotor' s periphery. The leakage across the high and low pressure intake and exhaust










regions via the sealing arc also increases as the spread of the sealing arc decreases.

Leakage losses past the vane tips and the sides of the vanes, adj acent to end-plates, of a

multi-vane expander tend to be significant. The magnitude of these losses is governed by

how well the vanes remain in contact with the stator-cylinder by means of under-vane

pressure forces or springs. If all else is similar, the leakage past the vane tips in the

circular expander is greater due to a higher increase in cell-volume and thus a greater

pressure differential between adj acent vanes in the intake process (i.e. smaller angular

displacements). This is a direct result of the larger eccentricity during the intake process

when compared to non-circular expanders. This results in a greater required vane

protrusion, X(6), from the vane slot. Inadequate under-vane forces will hence result in

larger leakage flow channels. Internal leakage past the vane sides is also a function of the

pressure differential and vane protrusion and the analysis follows in the same light as

mentioned previously.

Although the symmetric non-circular and circular expanders exhibit approximately

similar working fluid capacities, the internal leakage losses in the former are much more

significant. This is a result of the larger vane protrusions necessary to maintain vane and

stator-cylinder contact. It is also the result of higher-pressure differences between

adj acent cells. Although the circular expander exhibits larger leakage losses when

compared to the non-symmetric, non-circular expander (see above) it's higher capacity

(amount of mass in control volume at cut-off angle) leads to a smaller leakage coefficient

(ratio of mass flow of leakage to total mass flow through expander) and hence higher

leakage efficiency.









The kinematic and friction models developed aid in further performance prediction

and comparison between circular and non-circular multi-vane expanders. The number of

arcs that describe the geometry of the stator-cylinder greatly affect the sliding velocity

and acceleration of the vanes. Since the non-circular stator is described by at least 5 arcs,

there are an increased number of discontinuities and an increased rate of change in vane

sliding acceleration and velocity respectively (Figure 5-22). At larger eccentricities, the

presence of these characteristics results in larger frictional losses. For example the

circular expander would exhibit larger frictional losses during the intake and latter stages

of the exhaust processes due to a larger eccentricity. Due to the higher fluid handling

capacity of the circular expander, and hence power output, the mechanical efficiency of

this device is typically larger than that of a non-circular expanders although higher

frictional losses may be realized.

The choice of stator-cylinder geometry is an open-ended design optimization

problem. Fluid handling and geometric volume ratio requirements as well as physical

constraints when modifying existing machinery also influence this choice. Trade-offs due

to competing effects arising from frictional losses and internal leakage losses are also

expected.

Heat Transfer

The following conclusions were made regarding heat transfer modeling:

* External insulation is deemed necessary
* Order of magnitude analysis reveal insignificance relative to other loss mechanisms
* Moderate temperature differences
* Time available for heat transfer to/from working fluid in control volume is insufficient





a ,ROTOR 1(3)




\\RS(6)


V~2 '2)


Leading vane
TATOR
Trailing vane



Figure 5-1. Schematic of nomenclature used and leakage paths of a circular rotary-vane
expander with general orientation and a (a) conventional or (b) modified
intake ports













Table 5-1. Primary leakage paths in a rotary-vane expander
Path Path Description
Index
(1) Axial gap between the rotor and the two
end-plates
(2) Radial gap between the tips of the vanes
and stator-cylinder
(3) Axial gaps between the sides of the vanes
and the end-plates
(4) Radial gap between the rotor and stator
cylinder, minimum clearance
(5) Gap between the faces of the vanes and the
side walls of the rotor' s slots
(6) Axial clearance beneath the vanes and
between the rotor and the two end-plates


P1=Pav(8) xi, T1

ui=0


Pb=Pav(8+6)


Reservoir


xe, Pe, Me


Figure 5-2. Schematic of a typical leakage path


200















General Form Discretized Form

d p)= 0p1u1 = Peue (5-6)
dx

d(pu2> dP2 P-PA u, = m~ue (5-7)
dx dx 23 ve

dy; d u 12 U2
pu-+ pu =O 0 4= h + (5-8)
dx dx2


Table 5-2. Governing equations and their discretized form describing one-dimensional
adiabatic Fanno-flow from a reservoir


U=co r


Pav(B)


Pav(8+6)


Pav(6)>Pav(8+6)


Figure 5-3. Shear and pressure driven Couette flow










Table 5-3. Variation of operational and dimensionless parameters to model flow in axial
gap as laminar and with a zeroth-order approximation
Operational Parameters Dimensionless Condition
Parameters
351, Eu>20
0 100 25<6 i<65 pm 394>1


2000


1800


1600


1400


1200


1000


0 100 200 300

Angular Displacement, 6


Figure 5-4. Non-axisymmetric pressure boundary condition on outer surface of the rotor






































Figure 5-5. Free-body diagram of a vane protruding outward from a rotor slot at a local
angle 6










Pin



Rotor

7 End-Plate
rR


P r,u

m dr



Figure 5-6. Schematic of generalized Couette flow in the axial gap between rotor and
end-plate



350


o 200
a 150 ...F
0 100 /--F
J 150 / FN



0 45 90 135 180 225 270 315 360

Ang ular Displace menst, a


Figure 5-7. Variation of reaction forces on a vane with no vane-tip curvature as a function
of angular displacement












150
z

( 100
o

50


0 45 90 135 180 225 270 315 360

An g ular Disp lace me nt, a


Figure 5-8. Variation of reaction forces on a vane with a circular vane-tip profile as a
function of angular displacement


0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8


- th,in=60 deg
- th,in=15 deg
- th,in=5 deg


0 100 200


300


400


Angular Displacement, 8


Figure 5-9. Variation of leakage from/to the expander cavity as a function of angular
displacement due to non-axisymmetric flow between the rotor and stationary
end-plates for different intake angle spreads










u,0.2
S0.1
o O
LL.
u, -0.1
u~a-0.2 -T~ k/~hrottling th,in=60

Qe -0.3-Ida
-0. Th rottl ing th,i n=30
-1 -0.5
0 100 200 300

Angular Displacement, a



Figure 5-10. Variation of non-axisymmetric leakage from/to the expander cavity for the
ideal and throttling cases


S0.015

S0.01

-o 0.005


:E -th, in=6 0
S-0.005
~ I ~-th,in=1 5
S-0.01
ca -th,in=5
-0.015
0 100 200 300

Angular Displacement, 8


Figure 5-11. Variation of leakage from/to the expander cavity through the gap between
the sides of the vanes and end-plates for different intake angle spreads


206









































Ideal
--Actual


-.E0
1.2E-03


8.0E-04

6.0E-04

4.0E-04

2.0E-04

0.0E+00


0 5


Angular Displacement, 8

Figure 5-12. Variation of leakage to the expander cavity from the rotor slot (modified
intake) through the gap between the face of the vanes and rotor slot


3.5




2.0
1.5

1.0


0 100 200 300 400 500 600 700 800 900 1000


Expander Rotational Speed, RPMI


Figure 5-13. Variation of the ideal and actual mass flow-rates through the expander as a
function of rotational speed


207

















deall
--Actual


3.0
2.5
2.0
1.5
1.0
0.5
0.0


Numbe r of Va nes
Figure 5-14. Variation of the ideal and actual mass flow-rates through the expander as a
function of the number of vanes











A B

Fiue -5.Cmarsn fwy ywhc fudisitrdcd no h epnerscaiy
A)cneninl nae i ntk or n B oife ntk iartr lt
through ~ ~ ~ ~ ~ ~e en-ltst nue aetpadsaorclne otc


208





































Table 5-4. Comparisons of contraction coefficients for turbulent single-phase flow
1/o 0 0.2 0.4 0.6 0.8 1
C, Perry [1] 0.586 0.598 0.625 0.686 0.79 1
C, Ruffell [2] 0.618 0.637 0.664 0.708 0.790 1.0
% 5.2 6.1 5.9 3.1 0.0 0.0


fully developed
linlet flow s r



Figure 5-16. Flow through a sudden contraction


209





8,,


AV1 AV2
I II



Figure 5-17. Schematic of the three phases of vane orientation that occur during the intake process
































Figure 5-18. Variation of pressure in expander cavity as a function of angular
displacement for the ideal and actual (throttling only) cases



Table 5-5. Variation of volumetric efficiency and final pressure as a function on inlet
angle in an expander with throttling losses accounted for
No Throttling With Throttling
6,, r, Pex, kPa py% Pex, kPa py% P
5 8.704 681.48 100 432.39 52.92% -35.09%
15 5.762 878.52 100 618.86 57.31% 5.62%
25 4.134 1049.29 100 795.70 59.98% 26.59%
35 3.138 1194.38 100 955.93 63.44% 38.90%
45 2.487 1315.36 100 1096.45 65.79% 46.73%
75 1.496 1562.02 100 1402.39 71.42% 58.35%
Teva=5oC, Tcond=45oC, reprocess=10.925, N,=8, Pex, kPa=584.1


2000
1800
1600
1400
1200
1000
800
600
400
200
0


0 6in+8 100 200 300

Ang ular Dis place m ent, deg












Contraction


~' 1Enlareement

Exit



Figure 5-19. Schematic of the intake port of the modified rotary-vane intake through the
end-plates and into the rotor cavity via rotor slots


Figure 5-20. Schematic of (a) symmetric (b) non-symmetric non-circular rotary-vane
expander comprised of four circular arcs (1-4) and a sealing arc (5)














-Circular MVE
-Non-circular MVE


0 '/
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360

Angular Displacement, deg




Figure 5-21. Variation of vane displacement as function of the leading vanes' angular
displacement for both a circular and non-circular MVE


2


-Circular MVE
- Non-circular MVE


-


1


E O





S-3


S 20 40 60 80 100 120 140 160 )0 200 220 80 260 280 300 3 0340 3(


Angular Displacement, deg




Figure 5-22. Variation of vane velocity as function of the leading vanes' angular
displacement for both a circular and non-circular rotary-vane expander









CHAPTER 6
EXPERIMENTAL PROGRAM

This chapter deals with the experimental apparatus that has been designed,

constructed and used initially by Alphonso (2003). The apparatus has been modified and

extended to investigate the performance of a modified refrigeration system that includes

the utilization of a two-phase expander as a throttle valve replacement. This chapter

describes the experimental facility, data acquisition system, the two-phase expander

selection and testing and the real-time control scheme by which the expander is

controlled.

Experimental Facility

Chilled Water/Heated Water Loop

The experimental facility employed in this investigation consists of a constant-

temperature water loop. This is accomplished by using a water chiller for cooling the

water and a water heater for heating it. The system is designed in such a way that the

drop in water temperature due to cooling is equal to the rise in its temperature due to

heating. Figures 6-1 and 6-2 show the system used in the experiments.

The water heater (Vanguard VG 0301CO9767, model no. 6E725, wattage: 4500 /

3380, capacity: 119.9 US. gallons) is used to add a constant heating load to the constant-

temperature water loop. The water heater employs two electrically-heated coils to heat

the water. The coil temperature can be adjusted by employing electronic thermostatic

controls located on the water heater control panel. The experimental procedure requires

the water heater to be used in two modes of operation. The first mode, the Standard

mode, was structured in such a way that the water heater provides a larger heating load to

the water in the constant temperature loop. This corresponds to preset upper and lower










coil temperatures of 550F and 540F, respectively, with a loF temperature differential

applied to both coils. The second mode, the Cut-Off mode, was structured in such a way

that the heater provides a comparatively lower heating load to the water in the constant-

temperature loop. This corresponds to preset upper and lower coil temperatures of 500F

and 490F, respectively, with a loF temperature differential applied to both coils. The

water heater was fitted with a blow-off safety valve to guard against undesirable pressure

build-up. Figure 6-1 shows the water heater as well as controllers and hydronic expansion

tank.

The water chiller (model RTP201, 24000 BTU/hr 208-230 VAC air cooled

condenser) is used to chill the water as it flows through the constant-temperature water

loop.

The constant-temperature water loop also includes a hydronic expansion tank,

besides the water chiller and the water heater. Rubber hoses and copper pipes were used

to connect the water chiller and the water heater. The entire piping of the chilled/heated-

closed loop is well insulated to prevent heat loss/gain from and to the loop.

Data Acquisition System

An instrumentation map and schematic of the modified chiller used for experiments

is presented in Figure 5-3. Temperatures were measured in the experimental facility using

copper constantan (type-T) thermocouples with an uncertainty of + 0.20C. This type is

suitable for low temperature applications as well as temperatures up to 3700C. The

thermocouples were insulated for accuracy. Temperatures of the refrigerant as it flows

through the refrigeration system were measured at the inlet and exit of the condenser and

evaporator. The temperature of the air flowing through the air-cooled condenser of the










water chiller was also measured. In the constant-temperature water loop, the temperature

of the water was measured at the inlet and exit of the water chiller tank and at the outlet

of the pump.

Pressure transducers (Mamac Systems, model no. PR-264, accuracy fl% of the full

scale) were used to measure the pressure of the refrigerant at five locations in the

refrigeration system. These transducers were calibrated to give an output of 0-5 VDC to

the data acquisition board corresponding linearly to a pressure range of 0-2000 kPa. The

transducers were calibrated using a Calpal hand held pressure calibrator. These

transducers were used to measure the refrigerant pressure at the inlet and outlet of the

condenser and evaporator.

Air humidity at the inlet and exit of the air-cooled condenser was measured using a

humidity transducer (Mamac Systems, model no. HU-225, accuracy f3% of the full

scale). The transducers were calibrated to give an output of 0-5 VDC to the data

acquisition board corresponding linearly to a relative humidity range of 0-100%.

The refrigerant mass flow-rate as it flows through the chiller was measured using a

flow-rate transducer (Sponsler Co., Inc., model no. SP711-3). This transducer was

calibrated to give an output of 0-5 VDC to the data acquisition system corresponding

linearly to a flow-rate range of 0- 47.4293 lbm/min. The water loop mass flow-rate was

measured using a flow-rate indicator (Sponsler Co., Inc., model no. SP3/4-CB-PHL-A-

4X). The water flow-rate was read off the digital display of the transducer.

Compressor power was measured employing the voltage and current transducers

across the compressor motor. The voltage was measured using a voltage transducer

(Flex-Core, model no. AVT-300CX5, accuracy f0.25%), which was calibrated to give an










output voltage of 0-5 VDC to the data acquisition board corresponding linearly to a range

of 0-300 AC line voltage. The current flowing through the compressor motor coils was

measured using a current transducer (Flex-Core, model no. CTD-025A, accuracy +0.5%),

which was calibrated to give an output voltage of 0-5 VDC to the data acquisition board

corresponding linearly to a current range of 0-25 Amps.

The outputs of the various transducers were inputted to a data acquisition board

(CIO-EXP 32), which is connected to another data acquisition computer card

(Measurement Computing, model PCI-DASO8) located in the computer. A computer

equipped with state-of-the-art data acquisition software LABVIEW 7.1 was used to

record the data. The location and type of instrumentation used are shown in Figure 6-3.

Unmodified Chiller Experimental Procedures

Testing was performed for the unmodified chiller unit for the standard and cut-off

modes of operation. The standard mode operating conditions were structured in such a

way that the heater provides a larger heating load to the water in the constant-temperature

water loop. This corresponds to a temperature setting in which the upper and lower coil

temperatures are 550F and 540F respectively, with a loF temperature differential applied

to both coils. The cut-off mode conditions were structured in such a way that the heater

provides a lower heating load to the water in the loop. This corresponds to a temperature

setting in which the upper and lower coil temperatures are set at 500F and 490F

respectively, with a loF temperature differential applied to both coils. The chilled water

exit set point temperature was kept at 430F throughout testing.

The water flow-rate in the loop was kept constant by regulating the flow through a

gate valve to a preset level according to the required cooling capacity considerations.









All test runs were conducted under similar ambient conditions for a period of 6

hours. A data point was a result of averaging readings at the rate of 1 Hz for a 60 second

time interval. Figures 6-4 to 6-8 represent sample runs of the chiller in both modes of

operation for a period of 6 hours. States 1, 2, 3 and 4 represent the inlet to the

compressor, condenser, throttle valve and evaporator, respectively.

Dynamometer Selection, Data Acquisition and Real-Time Control Scheme

According to data collected from a parametric analysis of the basic vapor-

compression refrigeration cycle, see Table 6-1, the size of the dynamometer was

determined. The size of the dynamometer is a function of the anticipated power output of

the expander. According to a parametric analysis that was conducted, a dynamometer

with a rating of 375 watts was deemed adequate. This rating represents an over-estimate

of the maximum power output for ideal and actual vapor compression cycles utilizing

ideal two-phase expanders operating at 4oC and 45oC evaporator and condenser

temperatures, respectively.

A hysteresis type dynamometer cooled by means of compressed air was selected.

This versatile dynamometer was deemed to be the ideal type for testing turbomachinery

with relatively low power outputs after numerous consultations with leading

dynamometer manufacturers.

A LabVIEW code was written to acquire and graphically present data collected by

the instrumentation. The LabVIEW code also extrapolates and determines the degree of

superheat of the fluid leaving the evaporator in order to adjust the mass flow-rate through

the expander. An integrated closed-loop high-speed programmable dynamometer

controller adjusts the torque applied to the shaft of the expander. This controls the flow-

rate of refrigerant through the expander and ensures an appropriate degree of superheat at









the compressor suction. Figure 6-9 shows a schematic of the proposed control system and

algorithm.

In order to determine the corresponding saturation temperature at a given

evaporator outlet pressure, the following equation proposed by Bachr and Tillner-Roth

(1995) was solved iteratively using the Newton-Raphson method. Equation (6-1) can be

used for both R-22 and R-134a with the appropriate constants in Table 6-2

P(0) 1
1 Pc j;1 -8 [nB a8 46+,~ 6

where

Pe = Critical Pressure of the Refrigerant


6 = Dimensionless Temperature, O = 1


To = Critical Temperature of the Refrigerant

and the coefficients can be found from Table 6-2.

The solution procedure to determine the degree of superheat can be described as

follows:

* Express Equation (6-1) in the form f(B, P) = 0 .

* Input the Pressure, P(6), into Equation (6-1) at which the corresponding saturation
temperature is desired.

* Input an initial guess as to what that saturation temperature is and calculate the
corresponding dimensionless temperature, 8o.

* Compute the Derivative of Equation (6-1) with respect to the dimensionless
df (B, P)
temperature, and use the Newton-Raphson method to find a new
dB
f (B, P)
estimated value for 6, O = 8,
f '(B, P)










*Continue process until convergence.


Rotary-vane Expander Selection

This section details the different choices of rotary-vane type turbomachinery that

have been modified and used as two-phase expanders.

Modified Automotive Air-conditioning Compressor

A double-acting rotary-vane automotive A/C compressor of unknown capacity at

the time of purchase due to proprietary restrictions was obtained. The automotive

compressor was deemed appropriate after several unsuccessful attempts of contacting

compressor manufacturers for residential and industrial refrigeration systems.

Automotive compressors are typically designed for volatile working conditions and may

be difficult to modify. Modifications to the piping to/from the compressor housing could

also be difficult to modify. Rotary-vane compressors were used extensively in the late

80s and early 90s by automobile manufacturers such as BMW and Mazda.

The rotary-vane compressor selected and purchased is a re-built A/C compressor

that runs on R-12 and can be retrofitted to work with R-22 and R-134a. After conducting

research dealing with the modifications that must be done in order to operate it as an

expander, the modifications deemed necessary included changing fittings, O-rings, and

the reed valve.

Internal leakage losses associated with this modified A/C compressor were initially

thought to be less than leakage losses associated with other type of modified

turbomachinery due to the typical existence of tight tolerances and need for significantly

efficient compressor designs.









Performance of Modified Compressor

Upon operation of the modified system with the expander installed, the expander

was found to perform unexpectedly. Further examination of the internals of the expander

revealed that sealing problems associated with the deformation of two O-rings may have

been the cause. These O-rings seal the high and low pressure pockets within the

compressor/expander (Figure 6-10). Observation of the O-rings, upon completion of

several tests in the refrigeration system, showed signs of elastic deformation. Tests with

compressed air were conducted after which it was determined that the inadequate sealing

provided by the O-rings may not be the only cause of the unfavorable performance of the

expander. Problems associated with the start-up operation (as defined below) of this

expander were then investigated.

It was found after further examination and testing of the internal cavity of the

expander that significant centrifugal force due to rotation is necessary to maintain

continuous contact between the vane-tips and the stator-cylinder. Although pressurizing

slots and a means for pressurized fluid to enter the slots exist for compressor operation,

there is no guarantee that the high pressure fluid enters the pressurizing slots when run as

an expander. This lack of contact is believed to be the primary cause of the expander' s

poor performance since insufficient vane protrusion would hinder the development of

adequate torque from the incoming working fluid acting on the vanes. To overcome this,

a mechanically driven coupling was used to drive the expander shaft to generate

sufficient centrifugal force and help eliminate any start-up issues. Secondly, prototype

mechanical springs, in the form of leaf springs, were designed and machined out of

tempered spring steel and tested in the vane slots. Leaf springs were determined to be the

most suitable type of spring for this application because of both the dynamics of the










application and the dimensions associated with the vane slots. Three different

configurations of the vanes were machined in order to determine the optimum vane

geometry to ensure vane-stator contact with the help of leaf springs placed in the

pressurizing slots. Upon installation in the expander initial testing revealed that the

modified vanes did very little to enhance the performance of the expander although

continuous contact between the vane-tips and stator-cylinder was maintained.

One possibility the expander did not function as expected is the internal flow path

arrangement (Figure 6-11). Since the current expander was initially designed as an

automotive air-conditioning compressor, the dual flow path configuration of the

compressor is not ideal for expansion. Upon testing the expander with refrigerant, the

shaft of the expander would operate as if the shaft were locked. This resistance to motion

was deemed to be the primary result of the compression of liquid due to poor geometrical

design of the intake or exhaust ports. Further analysis of the symmetric, elliptical double-

acting rotary-vane compressor revealed that the minimum tolerances that exist between

the rotor and elliptical stator-cylinder were approximately 25 Clm (0.001"). The

temperature differential between the intake (high side temperature) and the exhaust

(evaporator temperature) may have also caused the rotor to lock due to material thermal

expansion of either the stator-cylinder or rotor. Further experimental measurements are

needed to prove this. This also supports the conclusion that an alternate flow path was

needed to eliminate any unwanted liquid compression and material thermal expansion

that may take place. An alternate flow path configuration would also eliminate the

extremely "tight" tolerances that exist between the rotor and the stator-cylinder at the

minimum clearance.










Design Issues


Several design issues were deemed responsible for the inadequate operation of this

modified double acting compressor. They include the following:

* Unsuitable volume ratio as determined from a modified geometrical code.

* Intake port spread of 8in=10o. This results in significant throttling losses in the
intake port. This compressor exhaust port is a sound design choice.

* Overlap between intake and exhaust ports. Since the compressor has 5 vanes
(6=72o) there exists an approximate 650 overlap (accounting for vane thickness)
that is responsible for excessive leakage past the minimum clearance on either side
of the rotor. A significant driving potential drives this.

* Minimal clearance is 25 Clm. Significant pressure and temperature differentials on
either side of the minimal clearance cause thermal expansion of the materials that
results in the "locking" of the expander shaft.

* Poor design of exhaust port as an expander. Retrofitting the ports is difficult due to
the hermetic nature of the compressor.

Designed and Machined Expander

The design, fabrication and testing of a machined rotary-vane expander was

deemed to be the next most reliable and suitable alternative. It would also aid in

understanding the complex factors involved in selecting geometric design choices for this

specific application. Many factors were taken into consideration when making this

decision and they included:


* The ability to design the rotor and stator-cylinder. During this process the
geometric volume ratio of the expander would be optimized to allow for complete
expansion of the refrigerant and to avoid under- and over-expansion. The
dimensions of the cylinders are constrained by the dimensions of the dynamometer.

* The ability to design the flow path of the refrigerant. This would ensure proper
expansion of the refrigerant and avoid compression and additional pressure losses
which may result from poor flow path design. These inefficiencies would reduce
the amount of power that may be recovered from the system expansion process by
means of an expander.










* The ability to optimize the location and the spread angles of the intake and exhaust
ports. This provides an additional degree of flexibility when optimizing the
geometric volume ratio of the expander.

* Ease of installing instrumentation. Locations for different measuring devices,
primarily pressure taps and thermocouples, would be taken into consideration. This
would allow for a more extensive comparison of theoretical and experimental
results.

* Both the size and the weight of the expander may be optimized.

* The ability to design and fabricate the vanes. The design would account for the
addition of mechanical springs if needed. These springs would keep the vane-tips in
continuous contact with the stator-cylinder. Either compression or leaf type springs
may be utilized with appropriate modifications to the vanes and rotor-slots made.

In order to ensure the adequate design of a rotary-vane expander, the geometric

volume ratio of the expander must be as close as possible, ideally (see Chapter 5 in

regards to over- and under-expansion), to the process volume ratio. Adequate selection of

the expander' s dimensions ensures an appropriate value of the expander' s geometric

volume ratio suitable for the use of this rotary-vane expander as a throttle valve

replacement in vapor compression refrigeration systems. Other primary variables that

will play a critical role in the performance of this expander include the intake port arc

length and width, the exhaust port location, depth, and contour, the number of vanes and

the aspect ratio of the rotor. Other variables that effect the design of the expander and can

lead to poor performance of the expander include bearing design and location, rotor

dynamics and the method by which the vanes will maintain continuous contact with the

stator-cylinder. All of the aforementioned design variables have been addressed

thoroughly and a preliminary designs of the expander was developed with the aid of

recent literature and commercially available rotary-vane type turbomachinery.

Adequate tolerances between the various moving parts were allotted to ensure

proper lubrication and a reduction in internal leakage and frictional losses. The tolerances










were calculated to ensure that internal leakage losses are minimized. Springs were

preliminary deemed to be unnecessary due to adequate centrifugal forces that are

expected in this new design. Generated centrifugal forces are needed to maintain

continuous contact between the vane-tips and the stator-cylinder.

Sizing, selection and the mounting locations of two bearings were also carried out.

Conventional shielded ball bearings were selected since they are inexpensive and are

rated for at least 1,000 lbs in both the radial and axial directions. According to

preliminary estimates of the forces that are expected the rating aforementioned should

suffice.

Initial performance tests performed with compressed air and refrigerant revealed

that leaks from the shaft penetration were still significant. The original dimension of the

end-plate (shaft side) was modified and a sealing disc was designed to be bolted to the

end-plate. The sealing mechanism that was conceived for this application consisted of a

tapered sealing disc that would be bolted to the modified end-plate. The gaps that exist

between the tapered disk and the shaft are sealed with general purpose Teflon

compression packing. Further performance testing showed a significant decrease in the

amount of refrigerant/air that leaks.

Performance of Machined Expander

Further testing with compressed air revealed that there exists a critical pressure

differential between the inlet and exhaust states required to overcome the inertial mass of

the expander and vanes and any starting friction between any of the sliding or moving

parts.

During testing of the newly designed expander in the refrigeration system, it was

apparent that the shaft of the expander didn't lock as was the case with the first expander










tested. However, the newly designed expander didn't function as desired due to

significant external leakage. This would deem experiments unrepeatable and cause

excessive uncontrolled refrigerant leakage to the ambient. The design of the expander

was modified to eliminate leakage losses that are primarily due to imperfections in the

current rotary-shaft seal design. Additional research and possible design/selection of

commercially available rotary shaft seals was deemed necessary but beyond the scope of

this investigation. Other factors that may cause this undesirable performance is the fact

that the expander's internal size and weight (sliding and moving parts) play critical

factors in effecting the value of the critical pressure difference necessary to start the

expander.

The performance characteristics of interest for the two-ton air-cooled chiller used

as the basis for this experimental program are limited. These parameters include, but are

not limited to, the mass flow-rate of the refrigerant, the pressure differential between the

high-side and low-side and the thermodynamic state properties at the exit state of the

condenser. The parametrical relationship that must be satisfied in order for the expander

to operate is that the expander displacement must match the experimental volumetric

flow-rate of the refrigerant in the refrigeration system.

Modified Air-Motor

A properly sized, 4-vane, circular GAST air-motor was then tested as an expander.

The inlet ports are engineered in such a way that adequate under-vane pressure from the

incoming refrigerant helps provide for continuous contact between the vane-tips and

stator-cylinder. This also eliminates severe throttling losses that may arise due to a short

inlet port spread. This is typically the case in conventional expanders, where the fluid is

introduced directly into the expander's cavity. As mentioned previously, the inlet port










spread is among the most influential geometrical design choices which affect the

geometric volume ratio of a rotary-vane expander.

The geometrical characteristics of this unmodified expander were found to be

unsuitable for the purpose of its use as an expansion device. Table 6-3, shows 12

proposed modifications, a maj ority of which have already been machined, in order to

optimize the geometric volume ratio of the expander. Several constraints, namely the

vane height, maximum protrusion from the vane slot, machining complexity and

mechanical strength have been addressed to yield parametrically optimized design

parameters using the models developed to yield a maximum volume ratio.

It should be noted that the inlet start parameter denotes the angular location relative

to the datum at which the intake process begins. In conventional designs, and previous

work, this was Oo. In this design, the intake process begins at 55o. It should also be noted

that the angular locations of the cut-off (minimum) volume V1 and the exhaust

(maximum) volume V2 are alSo presented. The angular location of the cut-off volume is

the sum of the inlet start location, angle between successive vanes 6 and spread of the

inlet port 6in. The angular location of the exhaust volume is a design parameter that

typically corresponds to the location of maximum volume. The ratio of the maximum to

minimum volume is defined as the geometric volume ratio.

It can be seen that the unmodified circular expander's geometric volume ratio is

unity due to a very large inlet port spread (6in=90o) and 4 vanes (6=90o). It has been

previously shown that the geometric volume ratio increases as the number of vanes

increase. Four additional vane slots were machined into the rotor as well as the angular

holes that supplied the incoming fluid into the expander's cavity. This modification









alone had a detrimental effect on the volume ratio since the location of the exhaust port in

a circular expander is 1800. Hence the unmodified expander' s exhaust port begins in the

region of compression that follows the angular location of maximum volume. It can be

seen that the geometric volume ratio of the circular expander increases as the inlet port

spread is decreased (case 2) and the location of the exhaust port is changed (case 3).

Further reduction in the intake port spread (case 4) leads to an increase in volume ratio,

but ultimately the extremely small eccentricity that exists between the rotor and stator-

cylinders limits the amount of expansion that can be achieved. It should also be noted that

only 15% of the vanes' total height protruded from the vanes' slot. It was found that the

complexity of machining modified end-plates and reducing the size of the rotor made

machining a non-circular stator cylinder very attractive. These alternate designs can be

seen in Figure 6-12.

Case 5 represents the first use of a machined symmetric non-circular stator-

cylinder. Optimized geometrical parameters from earlier test cases were used. For the

symmetric non-circular case, the maximum expansion volume occurs at the angular

location of 2200 (1800). It should be noted that the values listed in Table 6-3 correspond

to a generally oriented circular expander, the basis of the design code developed. The

extent that the stator-cylinder could be machined was constrained by an adequate sealing

thickness as well as the minimum vane height in the rotor slot necessary to maintain

adequate operation of the expander. This percentage was found to be approximately 20%.

By comparing cases 5, 6, 7, 8 and 9, it was observed that the angular location of the

intake port start has a tremendous impact on the geometric volume ratio of the expander.

This arises due to the fact that the angular location of the cut-off volume decreases as the










location of the start of the intake port decreases. The current design introduces the fluid

into the expander cavity via a 250 angular cut in the rotor. This requires the intake port

start to be no less that 250. This ensures that the fluid is not introduced into the minimum

volume encountered during the sealing arc spread or directly into the exhaust port.

Further increase in the geometric volume ratio can only be attained if a non-

symmetric non-circular stator-cylinder is used. The maximum geometric volume ratio

obtained is 7.3 (case 12) whereas it is 3.86 (case 7) for the symmetric non-circular

expander. The rate of increase of the volume in a non-symmetric non-circular expander is

substantially less than that corresponding to the symmetric non-circular expander and

hence a larger geometric volume ratio can be achieved.

Single-Phase Experiments

In this section the objectives, experimental set-up and results of the single-phase

tests that have been conducted are described. The obj ectives of these experiments

included the verification of the feasibility of using rotary-vane turbomachinery, air-

motors and compressors, as expanders. In order to do so, the thermodynamic and fluid

mechanic characteristics of both the rotary-vane air-motors, compressors and expanders

were analyzed. The optimum geometrical and operational characteristics of each machine

are presented. Experiments were conducted to understand the working principles and

operational constraints of each machine. This study also helps formulate design concepts

that can be utilized to modify air-motors or compressors into optimized expanders for

single phase flow applications. The experiments will be used to predict and evaluate

modifications made to geometrical parameters such as the optimum intake and exhaust

port locations, their spreads and the geometric volume ratio as well as resulting

performance parameters such as the work produced, isentropic and volumetric









efficiencies. Experimental expertise, including proper assembly and sealing, of rotary-

vane turbomachinery is also expected to be gained and may improve performance and

operation significantly.

Single-phase experiments with compressed air and liquid/vapor R-22 were carried

out on a bench-top experimental set-up with various modifications to a commercially

available rotary-vane air-motor. Dry compressed air supplied to the building is used to

test the device. Saturated vapor or liquid R-22 may also be fed into the expander from a

canister that is in equilibrium. The elevation and orientation of the canister relative to the

expander is important as they determine the state of refrigerant that exits the canister. The

refrigerant used enters the expander from the high pressure supply tank and exits the

expander into an evacuated reclamation tank. Depending on the duration of the test

necessary, the reclamation tank can be maintained at a vacuum pressure maintained by a

vacuum pump. Depending on the geometrical parameters chosen, the device can operate

as an expander. The thickness of the gaskets and rotary-shaft seal design were altered,

when necessary, to accommodate higher working pressures than those recommended by

the manufacturer. The commercially available air-motor is of the non-lubricated type.

The addition of lubrication oil to the working fluid may ensure proper lubrication, sealing

and operation. The inlet and outlet pressures can be adjusted by a pressure regulator and a

valve, respectively. Pressure gauges or transducers, thermocouples, mass flow-rate meter

(rotameter), are installed at the inlet and outlet locations where necessary. The shaft of

the expander is coupled with a hysteresis type dynamometer. The dynamometer is

interfaced with a high speed programmable dynamometer controller which displays the

actual amount of mechanical power extracted, torque produced and rotational speed of










the expander. The controller may be interfaced with a computer to provide a continuous

data stream. The controller is operated in the open-loop mode. In this mode the user has

complete control over the braking load, produced by the dynamometer, by which the

torque and speed of the expander are varied. Figure 6-13 shows a schematic of the single-

phase experimental set-up.

In the following sections, the results of numerous tests are summarized. In certain

instances, inadequate operation or complexities are explained.

Summary

A rotary-vane air-motor with an adequate volume displacement has been retrofitted

to an expander by changing the internal geometry of the device, namely the inlet port arc

spread and start, start of exhaust port and the number of vanes. The geometry of the

stator-cylinder is also very critical to the thermodynamic performance of the expander. A

non-circular stator-cylinder has hence been machined. This stator-cylinder is symmetric

about the vertical base line of the expander. The maximum volume ratio that could be

achieved with such a configuration is 5. Further analysis showed that with a non-

symmetric non-circular stator-cylinder, the volume ratio can be increased to

approximately 7.5. In order to avoid time delays that may result from the manufacturer in

case a new air-motor is needed, experimental testing and confirmation is needed before

the non-symmetric, non-circular stator-cylinder is machined.

A maj or source of problems, external leakage, has been solved. The use of

modified gaskets (both material and thickness) and a modified rotary shaft seal have

ensured adequate sealing of the expander well beyond the rated pressure of 689 kPa.

Additional time and effort has been spent to avoid the manufacture of a costly and heavy

external enclosure. The condenser is expected to operate at a pressure as high as 1.75









MPa. Initial tests in isolation with both air and refrigerant have yielded positive results in

regard to the operational speed, sealing and change of state across the expander. The test

set-up could be varied to increase/decrease the operational pressure ratio across the

expander. Start-up problems were also not obvious, in any of the test cases, since

adequate under-vane pressure from incoming refrigerant helps provide for continuous

contact between the vanes and stator-cylinder.

Difficulties arising from the operation of the modified chiller with the expander

installed have hindered the progress of gathering experimental data. These difficulties,

namely, arise from controlling the back pressure (i.e. evaporator pressure) when the

expander loop is opened. The back pressure has a significant effect on the expander

intake and hence the operation of the expander. A low range (0-827 kPa) pressure

regulator was used downstream of the expander to provide a means for both the control

of the evaporator pressure and adequate operation of the refrigeration system. This also

negates the need for the conventional expansion valve to run in series with the expander.

The regulator doesn't and cannot, however, directly provide any means of control of the

back pressure. These operational difficulties must be addressed. It is interesting to note

that no previous work can be found in the open literature that discusses any operational

difficulties that may arise.

The chiller' s operational parameters were then varied in order to simulate operating

conditions of interest to the funding agency. These operating parameters are namely

higher condensing temperatures and system mass flow-rates. These parameters were

varied in order to provide measurable experimental data that may be used to validate the

analytical models developed.









In order for the rotary-vane expander to exhibit adequate operational

characteristics, a necessary pressure drop is required across the expander. One way of

controlling the pressure drop across the expander is by varying the mass flow-rate of

refrigerant in the system. Assuming that the two-phase flow of refrigerant leaving the

condenser is fully developed and turbulent (and ReD > 2 x 104 ) and that the pipe surface

is approximately smooth, the friction factor, pressure drop and mass flow-rate may be

expressed as

f =0.184Re~/ (6-2)


Ap = f pu x (6-3)
2D

xD2
mi = pus (6-4)


Combining Equations (6-2), (6-3) and (6-4) we find that

Ap ~ Cril9/5 (6-5)

where C is a constant.

It can be seen that the pressure drop across any device scales approximately to the

second power of the mass flow-rate through that device. Retrofitting the current

experimental set-up with a larger compressor, which would in turn increase the mass

flow-rate, is both cost prohibitive as well as potentially problematic or even catastrophic.

This is a result of increased system pressures and working fluid velocities that the

undersized components and piping may realize.

A more promising alternative is to change certain system parameters that would in

essence alter the back-pressure that the expander exhaust senses. In one particular study

in the literature (Badr et al. 1985), the mass flow-rate of cooling water through the










condenser of an Organic Rankine Cycle was varied. The authors claim the ability to vary

the operational pressure ratio from approximately 2.2 < Pin/Pex < 4.3. The magnitude of

change of the cooling water mass flow-rate through the condenser has not been reported.

In a similar manner, a reduction of the back-pressure in a vapor-compression

refrigeration cycle could be achieved by varying the mass flow-rate of hot water through

the evaporator. Initial tests with the experimental rig used have validated the general

trends one should observe if the load is decreased. At the same time initial tests revealed

the capacity of the hot water heater and pump, i.e. loading equipment, may not be

suitable enough to achieve the operational pressure ratios necessary. Preliminary

thermodynamic analyses also predict several system level changes. Further analysis is

required to determine the validity of the use of this alternative as a means of regulating

the back pressure that the expander exhaust senses. This methodology was deemed

attractive for larger refrigeration systems.

It was concluded that the variation of the back-pressure is unpractical and is not

responsible for the inadequate operation of the modified air-motor. Excessive internal

leakage losses were deemed the cause of this failure. Comprehensive single-phase tests

were carried out with the intent of determining an approximation of the performance and

leakage flow-rate through the modified air-motor. Static leakage tests were conducted on

the four vane air-motor in a series of tests to eliminate the leakage paths discussed in

Chapter 5. An order of magnitude analysis was made with the aid of the computer

program developed. It was found that the leakage due to the radial clearance between the

rotor and stator-cylinder, axial gap between the rotor and end-plates and that between the

sides of the vanes and the end-plates accounted for almost 95% of the leakage losses. In










order to simulate the dynamic conditions typically realized within the expander, leaf

springs were used to maintain continuous contact between the vanes and the stator-

cylinder.

Further testing was done in order to experimentally investigate the sources of

internal leakage. The leakage paths were blocked or altered consequentially to determine

the magnitude of each path. The results of this single-phase experimentation revealed the

leakage past the axial gap between the rotor and end-plates was most significant followed

by the leakage past the minimal clearance. The leakage flows from the other paths were

found to be insignificant in comparison. This is the case when continuous contact is

maintained between the vane-tips and the stator-cylinder. These results confirmed the

conclusions made based on the developed internal leakage model. The magnitude of

internal leakage flow-rate were significant when compared to the ideal mass flow-rate

through the expander. The relative magnitude of internal leakage can be decreased if a

larger refrigeration system is used. These trends are discussed in detail in Chapter 7.





(1) 110 gallon water
heater
(2) Hydronic
Expansion Tank
(3) Temperature
Controller
(4) Control Panel
(5) Pressure Relief
Valve
(6) Flow Measuring
Device







Figure 6-1. The water heater used to add a constant heat load to the constant temperature
water loop


Figure 6-2. Unmodified two ton air-cooled barrel type water chiller


























Comp essed DLExpansion








P: Pressure transducer Eaoao
T: Thermocouple
M: Mass-flow rate meter
Q: Volumetric flow rate meter
A: Current transducer
V: Voltage transducer T
H: Humidity transducer
D&L: Dynamometer & Load-cell o iae
from Heater

Figure 6-3. Instrumentation map and schematic of the experimental set-up for a vapor compression cycle with a by-pass loop for
rotary-vane expander integration


















~------------


.~hc~~*nMIV~LhrF~lhOl


__ ~~ ____


I _


_ _____ ~_ I__ _____________ __.__


1600

1400
1200
1000
800

600
400


-P2
-P3
-P4
-P1


20 0
0
0 50 100 150 200 250


300 350


Time, minutes



Figure 6-4. Variation of the measured pressure for the standard mode of operation in the
unmodified chiller as a function of time


90-
80-


60
50 -
40 -
30 -
20


-T2
-T3
-T4
T1


O 50 100 150 200 250 300 350 400

Time, Minutes


Figure 6-5. Variation of the measured temperature for the standard mode of operation in
the unmodified chiller as a function of time














1400

1200

1000

800

600

400

200


-P2
-P3
-P4
P1


- 0 10 10 20 5 0 5 0

-ie iue


Figure 6-6. Variation of the measured pressure for the cut-off mode of operation in the
unmodified chiller as a function of time


- T2
- T3
T4
- T1


0 50 100 150 200 250

Time, minutes


300 350 400


Figure 6-7. Variation of the measured temperature for the cut-off mode of operation in
the unmodified chiller as a function of time

















- -h.l ~h.n--e L ~ .- ~-~ h


2500


16; 2000


S1500
-

S1000
-

500
-

0


-Standard Mode
- Cut-Off Mode


Time, minutes


Figure 6-8. Variation of the compressor work for both the cut-off and standard modes of
operation in the unmodified chiller as a function of time












Table 6-1. Experimental Thermodynamic State Points


Exp. Data Points with 6.60C of subcooling
State P, kPA P, psia T, OC
1 579.3 84.0 13.9
2 1593.7 231.1 78.1
3 1574.2 228.3 34.5
4h 755.9 109.6 16.1
4s 755.9 109.6 13.5
4a nexp=25% 755.9 109.6 13.5


Process
r,


T, OF
57.0
172.6
94.1
61.0
56.3
56.3


Quality
SH
SH
SC
0.136
0.127
0.134


h, kJlkg p, kg/m3 Wex, W


242.3684
242.3684
240.7814
241.9717


1154.006
202.26
202.31
204.85


5.71
64.24 5.70
16.06 5.63
32.12 5.56
48.18 5.49


4a exp=50% o
4a exp=75% o


755.9
755.9


109.6 13.5 56.3 0.132 241.5749 207.5
109.6 13.5 56.3 0.129 241.1782 210.23


*avg conditions
T1: Tsat=4.740C and d~superheat=9.1560C
T3: Tsat=41.070C and dTsubcool=6.60C
Comp. Effic. 0.73
4h: isenthalpic expansion (TXV)
4s: isentropic expansion (Expander 100% eff)


in computing


4a:
mdot, g/s
mdot, lblmin


Expansion with irreversible expander, 25%, 50% and 75% eff)
40.48
5.34


V(m3 /S)= m2(/cg/s)-v(m3/g

Vdot(m3/s) 3.51E-05
Process Volume Ratio r,: Ratio of densities at condenser outlet and evap inlet












Table 6-1 Continued.


no subooling
State P, kPA P, psia
SAME
SAME


Process


T, OC T, OF Quality h, kdlkg p, kg/m3 Wexp, W


3 1574.2 228.3 41.07 105.926 sat L 251.08 1123.8
4h 755.9 109.6 13.52 56.336 0.181 251.08 158.33
4s 755.9 109.6 13.52 56.336 0.169 248.81 158.35 91.89
4a -25% 755.9 109.6 13.52 56.336 0.178 250.51 160.6 22.97
4a -50% 755.9 109.6 13.52 56.336 0.175 249.95 162.94 45.94
4a -75% 755.9 109.6 13.52 56.336 0.172 249.38 165.35 68.92


7.10
7.10
7.00
6.90
6.80















Cool
Compressed-
Air


Multi-vane
Expander By-
pass Loop


Evaporator


Extrapolate degree of
superheat (SH) from
or erties subroutine coded


I ino LaVIEWP: Pressure transducer
T: Thermocouple
D&L: Dynamometer & Load-cell
Is S>10C? P w/DAQSC: Signal Conditioner
DAQ: Data Acquisition Board
FalseTrueCent: High Speed Controller

Increment torque Do not
application on change
dynamometer speed



Figure 6-9. Schematic of the cut-away view of the bypass loop and evaporator and the logic behind the continuous control scheme
used to control the degree of superheat


-I


-
,----










Table 6-2. Parameters used to calculate saturation properties.

Parameter R-22 R-134a
Critical Pressure, kPa 4988.44 4056.32
Critical Temperature, K 369.28 374.18
al -7.1394518 -7.7057291
a2 2.1352753 2.4186313
a3 -1.7610879 -2.1848312
a4 -3.016996 -3.4530733
n1
n2 1.5 1.5
n3 2 2
n4 4.25 4












(1) Outlet Ports from
Compression
Chamber/Inlet Ports to
Expansion Chamber
(2)Discharge Port for
Compressor/Intake Port
for Expander
(3) O-ring that separates
low pressure/high
pressure pockets in
Compressor in the
compression process
(4) O-ring that separates
high pressure pockets
before and after the
exhaust port in the
compression process
(5) Suction Port for
C ompre ssor/Di sch arge
Port for Expander
(6) Exhaust Port in
compression process/
Inlet port in expansion
process


Figure 6-10. Flow path of refrigerant in both the compression and expansion processes





































Figure 6-11. Double acting, five vane modified automotive compressor












Table 6-3. Machining modifications made to GAST NL-32-NCC-1 to optimize geometric volume ratio
Unmodified Modified S metric Non-S mmtric
Circular Non-Circular
Case 0 1 2 3 4 5 6 7 8 9 10 11 12
N, 4 a 8 8 8 8 8 8 8 8 8 8 8
8, deg 90 45 45 45 45 45 45 45 45 45 45 45 45
Inlet Start 55 55 55 55 55 55 .;11 I25 15 55 .;11 2
esn, deg 90 90 35 3 11 20 20 20 20 20 20 20 20
8ex, deg 235 235 2F 3 g i I 180 180 180 180 180 180 8 235 235
Cut-Off, V, 235 190 135 135 120 120 95 90 80 65 120 95 80
Exhaust, Vt 235 235 235 180 180 180 180 180 180 180 235 235 235

Relative to Rotor Center
Displacements relative to Datum




























OOOO



Fg re -2 n oiiemdfe ym ercadnn-y mti o-crua xade Aldmniosi nhs


















Pressure
Regulator


Multi-vane Expander


Cooling
Compressed Air


P: Pressure transducer
T: Thermocouple
M: Mass-flow rate meter
DLC: Dynamometer, Load-cell & Controller


Valve


Figure 6-13. Schematic of the experimental set-up for compressed air and R-22 tests with a rotary-vane expander









CHAPTER 7
CONCLUSIONS AND RECO1V1VENDATIONS

This chapter is a summary of the challenges, recommendations and conclusions of

both the comprehensive modeling and experimental effort.

The successful assessment of the performance of a two-phase rotary-vane expander

utilized in vapor compression refrigeration systems (VCRS) based on the results from

extensive experimentation and a comprehensive component-level thermodynamic and

fluid mechanic model has been made. Key technical challenges are realized from both

primary (internal leakage, friction and throttling) loss mechanisms as well as those of

secondary consequence (re-compression, under or over-expansion, lubrication etc).

Internal Leakage Losses

Leakage losses from a maj ority of leakage paths cannot be necessarily neglected in

two-phase rotary-vane expanders. Unlike single-phase rotary-vane turbomachinery,

significant variations in density and speed of sound, inherent with the vaporization of

liquid, significantly alters the leakage flow-rates. This due to a greater mass flux of two-

phase leakage in expanders when compared to conventional technology (i.e. compressors

and air-motors) that results from the following:

*Choking in leakage paths, i.e. maximum leakage flow-rate, occurs at higher back
pressures. Figure 7-1 illustrates the difference in the critical-pressure ratio required
to realize choked flow for typical operating conditions in expanders and
compressors, respectively. Both processes are isentropic. In the case of the
expander, the fluid is initially a saturated liquid at the condensing temperature. The
fluid then accelerates, necessitating the formation of vapor, and the velocity
approaches that of the speed of sound if the back pressure is low enough. In the
case of the compressor, the saturated vapor is assumed to undergo isentropic
compression to the condensing temperature. The specific heat ratio is larger at
higher condensing temperatures leading to a smaller critical-pressure ratio. It can be
shown that for all practical purposes, the critical-pressure ratio for R-22 vapor in
the temperature range of OoC









* Significant variation of the refrigerant' s density in the two-phase region. Internal
leakage has the greatest impact on a two-phase rotary-vane expander' s performance
during the charging and expansion processes. These are regions where the largest
pressure differentials between cells exists (i.e. the driving potential) and also where
the density of the refrigerant is significantly large. Although the speed of sound (or
velocity if not choked) in a two-phase medium is smaller than that of a vapor
refrigerant, the discrepancy between the density has a much more dominant effect
on the mass flow-rate of leakage.

* Gap sizes in conventional rotary-vane turbomachinery are extremely large. Larger
gaps exist due to the significantly smaller density of the working fluid in the gaps
between the moving parts in conventional rotary-vane machinery. These clearances
must be reduced to within acceptable limits for the proper operation of a two-phase
device. Hence a special set of design criteria must be addressed.

* Uncertainty in gap sizes. Uncertainties in regard to quantifying gap sizes are a
significant obstacle to predicting the magnitude of leakage losses through all the
leakage paths during operation. This can be attributed to bearing end-play which
alters the clearances in two dimensions. Proper design of the bearing housing may
influence this. The deformation of the end-plates due to high pressure stresses will
also alter clearances and may cause separation of boundary-layers and the transition
to turbulence between the rotor and stationary end-plates.

* The leakage losses from the non-axisymmetric flow in the axial gap between the
rotor and end-plates is most significant. The magnitude of leakage losses through
this path during the intake and expansion processes has a significant impact on the
efficiency of the expansion device.

* Radial leakage through the minimum clearance between the stator and rotor
cylinders greatly restricts the design and optimization of the intake angle spread
and location. Depending on degree of subcooling, the density of the incoming fluid
is substantially large. The magnitude of the leakage from this path depends on the
spread and location of the intake port. The detrimental affect of re-expansion losses
in compressors is significantly smaller than that of radial leakage in a two-phase
expander. A decrease in the number of vanes increases the contribution of leakage
through the minimal radial gap between the rotor and stator cylinder. The increase
of the number of vanes however increases the potential of internal leakage via other
paths, increases frictional losses and decreases the mechanical strength of the rotor.

Friction

The curvature of the vane-tip can severally impact the amount of leakage past the

vane-tips due to loss of contact with the stator-cylinder. Even if no throttling is assumed

in the intake port, there is a loss of contact with the stator-cylinder through out the









duration of the expansion process. The method used to provide continuous contact

between the vane-tips and stator-cylinder is essential to optimum performance. A trade-

off exists between the throttling losses and increased frictional losses attributed to the

method by which the incoming working fluid is introduced into the expander cavity.

Vanes should be made out of high-density materials that can erode and mold to the stator-

cylinder geometric curvature. This would eliminate the need for adequate under-vane

forces through intake ports or cumbersome, unreliable springs.

Other Issues

Depending on the flow conditions, throttling losses in two-phase flow expanders

are significantly larger than in single-phase flow expanders. It has been shown that the

volumetric efficiency of a two-phase rotary-vane expander is extremely sensitive to the

design of the intake port. For a poorly designed intake port (low Cd), the volumetric

efficiency is approximately 50-60%. For a single-phase expander the volumetric

efficiency is expected to be 75-90%, due to the insignificant density variation that is

inherent with the vaporization of liquid.

A detailed intake model must be developed if two or more expander cells are in

communication with the intake port. The throat that the expander cell makes with the

intake port is expected to be the maj or flow restriction. Throttling losses in the exhaust

port may be neglected when compared to the charging process since velocities associated

with the discharge process are typically less and the flow area in the exhaust port is

significantly larger.

In vapor compression refrigeration technology the lubricant used is soluble and

circulates freely within the system. A dedicated lubrication loop within a hermetic rotary-

vane expander may significantly reduce friction between sliding parts. The thin film of










lubricant will also reduce gap sizes and alter internal leakage flows. The lubrication of

the shaft bearings is accounted for by using sealed, pre-lubricated ball bearings.

Leakage ratio, Expander sizing and Integration Concepts

In order to harness the maximum amount of work out-put from an expander,

different integration concepts must be evaluated according to operational speed

requirements, ease of coupling and requirements related to capacity control and system

performance. This task includes in-depth off-design calculations that result in

significantly different inlet conditions and mass flow-rates of the working fluid. An

expander is sized so that the volumetric flow-rate through the expander is equal to the

volumetric flow-rate of refrigerant at the condenser exit.

* The amount of leakage is seen to increase due to a larger contribution from the
leakage mass flow-rate through the radial gap, minimum clearance, between the
rotor and stator-cylinder. Caution must be exercised when increasing the number of
vanes due to possible increases in friction and decrease of mechanical strength of
the rotor.

* Sizing of the rotary-vane expander for two-phase applications is critical with regard
to determining the optimum operational speed to reduce the loss mechanisms
presented above.

* Inadequate sizing may lead to under-expansion. Depending on the order of
magnitude of under-expansion, an economizer cycle may be a promising
modification that would allow the use of two expansion devices in series with a
separator between them. In this case there would be greater flexibility in the
geometric design and operational speed of the expander.

* Increase in rotational speed decreases internal leakage losses through the expander
significantly. Extensive detailed modeling and experimentation is needed to
determine the magnitude of detrimental effects such as friction and viscous drag.

* The application of an expansion device in a refrigeration system should be
primarily based on the system mass flow-rate. For a given set of operational
conditions, the cooling capacity will ultimately determine the geometric size of the
expander, the degree of impact of the primary and secondary loss mechanisms and
the rotational speed of the expander.










*Once the rotational speed and the actual work out-put from the expansion device
has been determined, the expander's shaft may be integrated in various
configurations. The generated power from the expander can be:

Coupled to the compressor on the same shaft. The matching of rotational
speed and the complexity of a gearing system otherwise increase the
complexity of this integration concept.

Used to drive auxiliary equipment in the VCRS. This integration concept
allows for the most flexible use of the power generated by the expander.
The vapor compression refrigeration system has many components that
require power such as fans, pumps, control devices etc.

Used to drive external equipment. This is another flexible integration
concept that allows equipment such as air compressors or stand-alone
cycles to be driven by the expander. One promising alternative is to use
the out-put power to drive a dedicated or integrated subcooling cycle.

Figure 7-2 illustrates a typical expander sizing diagram. The diagram accounts for

the nominal cooling capacity of the refrigeration unit, the displacement volume of an

expansion device and the operational speed of that device for a given set of operational

conditions. The two operating points detailed on the figure represent the current design

point of a smaller deployable refrigeration unit and that of the minimum design condition.

The current design point is that of a small expansion device operating at a low rotational

speed due to the smaller cooling capacity of the unit. The minimum design point

represents a design condition where the system mass flow-rate is significantly larger than

the leakage flow-rate, the rotational speed of the expander is sufficiently large to reduce

leakage losses and the geometric size of the expander is sufficiently large for reduced

machining cost associated with less "strict" tolerances between the moving parts.

Recommendations

Great potential for use of a rotary-vane expander as a throttle-valve replacement

has been presented analytically. The absence of complicated geometric profiles (e.g.

screw, scroll) and complex valve timing and controls (e.g. reciprocating piston) make the









rotary-vane expander a viable candidate. Valve timing is inherent through geometry and

the principles of operation. This eliminates the need for complex models for valve

dynamics and pulsation effects.

Due to the aforementioned summary of technical challenges and limitations, the

following can be recommended

* Existing rotary-vane technology is limited by the aforementioned technical
challenges and limitations which prevent a drop-in vane expander from being
developed as a retrofit.

* Specialized design recommendations must be considered when re-engineering
current rotary-vane technology.

* System-oriented optimization that accounts for and overcomes the limitations
alluded to in the set of comprehensive findings presented must be made.

* The leakage losses from the non-axisymmetric flow in the axial gap between the
rotor and end-plates is the most significant leakage path. The model developed is
relatively ideal in its assumptions and doesn't take into account separated boundary
layers, choking or turbulent fluid flow. A computational model must solve the
compressible Navier-Stokes equations in order to do so.

* Although there exists a trade-off between reducing internal leakage losses and
increasing frictional losses, internal leakage losses must be reduced in order to
achieve suitable expander operation. This task should be given priority. As
mentioned, internal leakage losses may be reduced by proper selection of design
parameters and detailed engineering of tolerances and clearances. The use of a
dedicated lubrication system may also significantly alter the characteristics of
internal leakage.

* Detailed analyses of the flow distribution within the complex geometry of two-
phase rotary-vane expanders must be carried using computational fluid dynamics.
The solution of the compressible governing equations in this complex flow field
will yield, among many others, the following

Gradients of pressure and density with the expander cavity

Prediction of viscous dissipation due to the formation of vortices during
the charging and discharging processes

A comprehensive account of all leakage potentials and phenomena










The change of angular momentum caused by 'water-hammer like'
phenomena during the charging process

Accurate prediction of the volumetric efficiency by accounting for all
throttling and cell-interaction effects in the intake port

A detailed account of irreversibilities caused primarily by expansion
waves generated due to under-expansion

The existence of re-compression of liquid which may be caused by various
geometrical features

The inception of cavitation due to the impinging of vapor bubbles on
various surfaces


Summary of Contributions

* Comprehensive literature review of the application of expander technology in
relevant thermodynamic cycles (First of its kind)

* Comprehensive parametric system-level analysis and assessment of potentials and
identification of key system design parameters

* Developed comprehensive (scope and application) simulation model of two-phase
flow in a rotary-vane refrigerating expander

* Identified and made quantitative analytical and experimental assessment of
technical challenges and limitations that prevent a drop-in vane expander from
being developed as a retrofit.

* Acceptable tolerances, geometry and operational parameters are identified.

* Recommended specialized set of design criteria and identified system-oriented
optimization guidelines

Adequate sizing of both system/expander including integration concepts

Reduction of internal leakage losses





0.86

0.8

0.74

0.68

0.62

0.56


-- Tw o-Phase R-22

-a R-22 Vapor


Temperature, oC


Figure 7-1. Variation of the critical-pressure ratio as a function of condensing
temperature for a two-phase expander and vapor compressor


45 Minimum Design
40 /1 Condition
35 -1 Current Design
30 / /
25 /1
20 -1------,'------,c-*-- ------------ ----500 RPM
15 / 250 RPM
10 /
-100 RPM

0 s- ----1000 RPM
0.E+00 5.E-06 1.E-05 2.E-05 2.E-05 3.E-05 3.E-05
yexp= 0.65
Displacement Volume, m3/rev Tcond=45C
Te\a p =5 C


Figure 7-2. Expander sizing diagram










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BIOGRAPHICAL SKETCH

Ahmad M. Mahmoud was born February 14, 1979 in Kitchener, Ontario, Canada,

where his father was a Ph.D. student in the Department of Mechanical Engineering at the

University of Waterloo: the apple has clearly not fallen far from the tree! Shortly after

his father' s graduation, Ahmad's family immigrated to the USA and Ahmad lived in

Upstate New York and Connecticut, until the age of 13, where his father held faculty

positions. In 1992, Ahmad's family moved to Kuwait where his father was hired as an

associate professor of mechanical engineering. Ahmad attended the Universal American

School until his high school graduation and then joined Kuwait University's Department

of Mechanical Engineering as an undergraduate student in Fall of 1997. Ahmad ranked

2/150 in his graduating class. At Kuwait University, Ahmad also met his future wife, Ms.

Iman Al-Naggar, who was then a Molecular Biology undergraduate student at the same

university. The two got engaged in their last year of undergraduate education and

married upon graduation in June of 2002. A couple of months later, Ahmad and Iman

both started graduate school at the University of Florida, where Iman also obtained her

Ph.D. from the College of Medicine' s Interdisciplinary Program in Biomedical Sciences

in Spring 2008.

While at UF, Ahmad worked as a team leader at the Industrial Assessment Center

during his Master' s graduate education and served as president of the Gainesville

ASHRAE Chapter. Ahmad has also been active in a plethora of professional activities

while in graduate school.

Finally, Ahmad became a father when his daughter Salma was born on September

12, 2007.


267





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1 ANALYTICAL AND EXPERIMENTAL INVE STIGATION OF ROTARY-VANE TWOPHASE EXPANDERS IN VAPOR COMPRE SSION REFRIGERATION SYSTEMS By AHMAD MOHAMED MAHMOUD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008

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2 2008 Ahmad Mohamed Mahmoud

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3 I dedicate this work to God Almighty in partial fulfillment of His mandate to seek Knowledge: To my loving wife Iman, daughter Salma, fam ily and friends, without whose enduring patience and support this work would not have been possible.

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4 ACKNOWLEDGMENTS I would like to begin by thanking God for giving me this great opportunity to learn and for putting in my path wonderful people who have he lped me and shown me the way. I want to thank all those people, and will try my best to remember them all here. I want to start by thanking my wife, Dr. Iman M. Al-Naggar. To put it simply, this would not have been possible without her love support and daily dose of nagging! I would like to thank my parents, Dr. Moha med Abdel-moneim Mahmoud and Mrs. Faten Ismail Mahmoud for preparing me for and supporti ng me during graduate school. I also thank them for their unconditional love and endle ss prayers that have kept me on track. I would like to thank my mentors Dr. S. A. Sh erif and Dr. William E. Lear for their time, support, guidance and patience. Their technical a nd non-technical mentorship drove me to excel at whatever I did, by following their example, and helped me become the researcher I am today. I am truly grateful. I would like to thank my committee members fo r their guidance and input: Dr. Gary Ihas, Dr. D. Yogi Goswami, and Dr. Skip Ingley. I have been fortunate to have a diverse, knowledgeable committee that had answers to my que stions, solutions to my problems, and great advice and encouragement. I would like to thank all the people in the Sherif and Lear Labs, especially Mr. Ayyoub Mehdizadeh, that have helped me throughout my y ears in graduate school, as well as make my long hours in the lab enjoyable. I would espe cially like to thank Mr. John Crittenden, engineering director at Triad Research Corporation Inc., fo r the guidance and knowledge he passed on to me regarding experimentation and solving countless techni cal challenges. I am honored to have worked with him.

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5 I would also like to thank all the staff that ha s assisted me greatly at UF (Department of Aerospace and Mechanical Engineering). I would like to thank my friends who have made my years in Gainesville more enjoyable: Mr. Ibrahim Taman, Mr. Ahmed El-Mahdawy, Mr. Ahmed El-Hady El-Mahdawy, Mr. Farouk Dey, Mr. Safwat Mohammad, Mr. Mahmoud Enani, Mr. Ramadan Ajredini Dr. Mujahid Abdulrahim and Dr. Fares Al-Bitar, to name a few. I would like to acknowledge the support from the Air Force Research Labs for partially funding my research and gradua te education: Dr. Aly Shaaba n, Dr. Ragab Moheisen and Mr. Reza Salavani. Finally, I would like to thank the University of Florida and the College of Engineering for providing such a rich and competitive graduate program in which I am very lucky to have been a part.

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6 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES................................................................................................................ .......10 NOMENCLATURE................................................................................................................... ...17 ABSTRACT....................................................................................................................... ............22 CHAPTER 1 INTRODUCTION AND OBJECTIVES................................................................................24 Introduction................................................................................................................... ..........24 Objectives..................................................................................................................... ..........24 2 LITERATURE REVIEW.......................................................................................................28 Expanders in Refrigeration and Heat Pump Systems.............................................................28 Conventional Heat Pump and Refrigeration Systems.....................................................29 Transcritical Carbon Dioxide Systems............................................................................37 Gas Refrigeration Cycles and Cryogenics.......................................................................44 Expander Selection............................................................................................................. ....45 3 PARAMETRIC ANALYSIS OF CYCLES...........................................................................46 Ideal Cycles................................................................................................................... .........46 Base Cycle..................................................................................................................... ..47 Economizer Cycle...........................................................................................................48 Internal Heat Exchanger (IHX) Cycle.............................................................................50 Actual Cycles.................................................................................................................. ........51 Effect of Subcooling on System Performance.................................................................52 Effect of Superheating on System Performance..............................................................53 Evaluation of CFC, HFC and Transcritical CO2 Refrigeration Cycles..................................54 Performance and Size Optimization of Compression Refrigeration Systems........................70 Pure Hydrocarbons as Refrigerants........................................................................................83 4 MATHEMATICAL MODEL OF AN IDEAL ROTARY-VANE EXPANDER.................124 Literature Review of Modeling of Rotary-vane Expanders.................................................124 Model Development.............................................................................................................125 Thermophysical Model.........................................................................................................126 Geometric and Kinematic Model..........................................................................................126

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7Thermodynamic Model........................................................................................................128 Charging Process...........................................................................................................129 Expansion Process.........................................................................................................131 Exhaust process.............................................................................................................132 Ideal Expander Evaluation............................................................................................133 5 PRIMARY AND SECONDARY LOSS MECH ANISMS IN ROTARY-VANE TWOPHASE REFRIGERATING EXPANDERS........................................................................154 Primary Loss Mechanisms....................................................................................................154 Internal Leakage Paths and Clearances................................................................................155 Types of Two-Phase Leakage Losses...................................................................................156 Axial Clearance Between Rotor and End Plates...........................................................161 Leakage around tips of vanes........................................................................................166 Leakage past the sides of the vanes...............................................................................167 Leakage between the faces of the vanes and the side walls of the rotors slots............168 Leakage in radial gap between the rotor and stator cylinder.........................................168 Friction Model................................................................................................................. .....169 Two-Phase Throttling Losses in the Inlet and Exhaust Ports...............................................180 Conventional Inlet to Expander Cavity.........................................................................184 Modified Inlet to Expander Cavity................................................................................187 Thermodynamic Model of the Actual Expansion Process...................................................188 Charging Process...........................................................................................................188 Expansion Process.........................................................................................................189 Exhaust process.............................................................................................................190 Expander Performance Evaluation.......................................................................................191 Performance Variation Due to Stator-Cylinder Geometry...................................................194 Heat Transfer.................................................................................................................. ......198 6 EXPERIMENTAL PROGRAM...........................................................................................214 Experimental Facility.......................................................................................................... ..214 Chilled Water/Heated Water Loop................................................................................214 Data Acquisition System...............................................................................................215 Unmodified Chiller Experimental Procedures......................................................................217 Dynamometer Selection, Data Acquis ition and Real-Time Control Scheme.......................218 Rotary-vane Expander Selection..........................................................................................220 Modified Automotive Air-conditioning Compressor....................................................220 Performance of Modified Compressor...................................................................221 Design Issues..........................................................................................................223 Designed and Machined Expander................................................................................223 Performance of Machined Expander......................................................................225 Modified Air-Motor.......................................................................................................226 Single-Phase Experiments....................................................................................................229 Summary........................................................................................................................ .......231 7 CONCLUSIONS AND RECOMMENDATIONS...............................................................250

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8Internal Leakage Losses.......................................................................................................250 Friction....................................................................................................................... ...........251 Other Issues................................................................................................................... .......252 Leakage ratio, Expander sizing and Integration Concepts...................................................253 Recommendations................................................................................................................ .254 Summary of Contributions...................................................................................................256 LIST OF REFERENCES.............................................................................................................258 BIOGRAPHICAL SKETCH.......................................................................................................267

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9 LIST OF TABLES Table page 3-1 Summary of the percent changes in the system COP, refrigerating capacity and required work input by the addition of an expansion device for the ideal and actual R-134a single-stage, economizer and IHX cycles.............................................................92 3-2 Percentage of cycle ir reversibility due to the various components in basic and modified R-22, R-134a and transcritical CO2 systems....................................................113 3-3 Hydrocarbons viewed as potential refrigerants...............................................................117 4-1 Summary of rota ry-vane literature..................................................................................143 4-2 Geometrical input parameters..........................................................................................145 4-3 Typical operating conditions for a basic and modified 2 ton system..............................148 5-1 Primary leakage paths in a rotary-vane expander............................................................200 5-2 Governing equations and their discretized form describing one-dimensional adiabatic Fanno-flow from a reservoir............................................................................................201 5-3 Variation of operational and dimensionless parameters to model flow in axial gap as laminar and with a zeroth-order approximation...............................................................202 5-4 Comparisons of contraction coeffici ents for turbulent single-phase flow.......................209 5-5 Variation of volumetric efficiency and fi nal pressure as a function on inlet angle in an expander with throttling losses accounted for.............................................................211 6-1 Experimental Therm odynamic State Points.....................................................................241 6-2 Parameters used to calc ulate saturation properties..........................................................244 6-3 Machining modifications made to GA ST NL-32-NCC-1 to optimize geometric volume ratio................................................................................................................... ..247

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10 LIST OF FIGURES Figure page 3-1 A schematic of the ideal base cycle and the modified base cycle with an expansion device......................................................................................................................... ........86 3-2 T-s and P-h diagrams of the ideal ba se cycle and the modified base cycle with an expansion device............................................................................................................... .87 3-3 T-s and P-h diagrams of the econom izer cycle and an economizer cycle with an expansion device as a throttle valve replacement..............................................................88 3-4 A schematic of the economizer cycle and an economizer cycle with an expansion device as a throttle valve replacement...............................................................................89 3-5 A schematic of the IHX cycle and the modified IHX cycle with an expansion device as a throttle valve replacement ..........................................................................................90 3-6 T-s and P-h diagrams of the IHX cy cle and the modified IHX cycle with an expansion device as a throttle valve replacement..............................................................91 3-7 Variation of COP, between the modified cycle w ith an expansion device and the ideal cycle (R-134a), with eva porating temperature for A) Tc=25oC, B) Tc=45oC for the base, economizer and IHX cycles................................................................................93 3-8 Variation of Qe and Wnet, between the ideal modified cycle with an expansion device and an ideal base cycle, with eva porating temperature for various condenser temperatures in a single-stage va por-compression refrigeration unit................................94 3-9 Variation of COP, between the modified cycle w ith an expansion device and the ideal cycle, as a function of the evapor ating temperature for both R-22 and R-134a for various expander efficiencies Tc=45oC........................................................................95 3-10 Variation of COP, between the modified cycle w ith an expansion device and the ideal cycle, as a function of the evapor ating temperature for various condenser temperatures for the actual and ideal cycl e configurations of the economizer cycle........96 3-11 Variation of COP, between the modified cycle w/ an expansion device and the ideal cycle, as a function of the evapora ting temperature for various condenser temperatures for the actual and ideal cy cle configurations of the IHX cycle....................97 3-12 Variation of refrigerating efficiency as a function of th e evaporating temperature for both the base (no expander) and ideal a nd actual modified cycle with and an expander at a condenser temperature of Tc=45oC..............................................................98

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113-13 Variation of refrigeratin g efficiency as a function of the evaporating temperature for the actual base, IHX and economizer cycl es with an expander at a condenser temperature of Tc=25oC.....................................................................................................98 3-14 Variation of an ideal systems COP as a function of the degree of subcooling for various condensing temperatures.......................................................................................99 3-15 Variation of an ideal systems COP as a function of the degree of subcooling for various condenser temperatures and expansion processes...............................................100 3-16 Variation of the refrigerating effect as a function of the degree of subcooling for various condenser temperatures and expansion processes...............................................101 3-17 Variation in an ideal systems COP with respect to the evaporator temperature for different degrees of subcooli ng and expansion processes...............................................102 3-18 Variation of the difference in system COP,%, between the ideal cycle and modified cycle with an expansion device as a func tion of the evaporator temperature for various condenser temperatures and degrees of subcooling............................................103 3-19 Variation of the difference of the syst em COP, %, base cycle and modified cycle with an expansion device as a function of the degree of superheat in the evaporator for various condenser temperatures.................................................................................104 3-20 Variation of the refrigerating effect for an ideal cycle as a function of the degree of superheat for various condenser temperatures.................................................................105 3-21 T-s diagram of the basic system and th e modified system with an expansion device for a A) conventional refrigerati on system and B) transcritical CO2 refrigeration system......................................................................................................................... .....106 3-22 Variation of system coefficient of performance as a function of heat rejection pressure for various evaporating temperatur es for basic and modified transcritical CO2 systems.....................................................................................................................107 3-23 Percentage of cycle irreve rsibility associated with the heat rejection and expansion processes as a function of heat rejection pr essure for basic and modified transcritical CO2 systems.....................................................................................................................107 3-24 Coefficient of performance of basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature......................................................108 3-25 Difference in coefficient of performan ce between basic and modified R-22, R-134a and transcritical CO2 systems as functions of th e evaporating temperature....................108 3-26 Difference in refrigerating effect between basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature...........................109

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123-27 Difference in system input work be tween basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature...........................109 3-28 Volume ratio of the expander employed in modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature...............................................110 3-29 Difference in coefficient of performan ce between basic and modified R-22, R-134a and transcritical CO2 systems as functions of expander isentropic efficiency................110 3-30 Percentage of cycle irre versibility due to various components in a modified R-22 system as functions of th e evaporating temperature........................................................111 3-31 Percentage of cycle irreve rsibility due to condensing pr ocess in basic and modified R-22 and R-134a systems as functions of the evaporating temperature..........................111 3-32 Percentage of cycle irreve rsibility due to the expansion process in basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature.................................................................................................................... ..112 3-33 Dimensionless system irreversibility in basic and modi fied R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature...........................112 3-34 Variation of dimensionless COP, expa nder work output, expander volume ratio and condenser capacitance as a function of dimensionless refrigerating effect.....................113 3-35 Variation of dimensionless expander wo rk output and expander volume ratio as a function of evaporating temperature for various degrees of subcooling in the condenser...................................................................................................................... ...114 3-36 Variation of dimensionless condenser cap acitance as a function of evaporating temperature for various degrees of subcooling in the condenser.....................................114 3-37 Variation of dimensionless condenser capacitance as a function of degree of subcooling in the condenser for various values of condenser effectiveness....................115 3-38 Variation of dimensionless expander work output as a function of degree of subcooling in the condenser for various expander efficiencies.......................................115 3-39 Variation of dimensionless expander volume ratio as a f unction of degree of subcooling in the condenser for various expander efficiencies.......................................116 3-40 Variation of dimensionless COP as a function of degree of subcooling in the condenser for various expander efficiencies....................................................................116 3-41 Variation of dimensionless COP as a function of degree of subcooling in the condenser for various evaporator degree of superhea......................................................117

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133-42 Variation of the refrigerating effect as a function of evaporating temperature for various refrigerants at a c ondensing temperature of A) 25oC and B) 45oC for the ideal base cycle............................................................................................................... .118 3-43 Variation of the ideal base cycle co efficient of performance as a function of evaporating temperature for various refriger ants at a condensing temperature of A) 25oC and B) 45oC.............................................................................................................119 3-44 Variation of compressor work as a func tion of the evaporating temperature at a condensing temperature of A) 25oC and B) 45oC for various refrigerants in an ideal base cycle..................................................................................................................... ....120 3-45 Variation of the increase of COP (when comparing the base cycle and the expander cycle) as a function of the evaporating te mperature for various refrigerants at a condensing temperature of A) 25oC and B) 45oC............................................................121 3-46 Variation of the decrease in net-work (when comparing the base cycle and the expander cycle) as a function of the evaporating temperatur e for various refrigerants at a condensing temperature of a) 25oC and b) 45oC.......................................................122 3-47 Variation of the increase in refrigerating effect (when comparing the base cycle and the expander cycle) as a function of the evaporating temperature for various refrigerants at a condens ing temperature of a) 25oC and b) 45oC....................................123 4-1 Flow diagram of the main computer program developed................................................142 4-2 Schematic of a generally oriented circ ular rotary-vane expander with 8 vanes described by two circular arcs..........................................................................................144 4-3 Variation of the inlet throat area with respect to angular displacement..........................146 4-4 Variation of the exit thro at area as a function of angular displacement for an 8-vane circular MVE...................................................................................................................146 4-5 Pressure variation as a function of volume in an ideal expander for the cases of ideal, over and under expansion................................................................................................147 4-6 Variation of built-in volume ratio as a f unction of intake angle for different numbers of vanes....................................................................................................................... .....149 4-7 Variation of ideal volume as a functi on of angular displacement for different numbers of vanes.............................................................................................................149 4-8 Variation of ideal and actual volumes as a function of angular displacement.................150 4-9 Variation of pressure as a functi on of volume for various intake angles.........................150 4-10 Variation of pressure as a function of volume for different numbers of vanes...............151

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144-11 Variation of mass within the expanders cell volume as a function of angular displacement for various intake angles............................................................................151 4-12 Variation of power as a function of a ngular displacement for different numbers of vanes.......................................................................................................................... ......152 4-13 Variation of power as a function of angul ar displacement for various intake angles......152 4-14 Variation of power as a function of volume....................................................................153 5-1 Schematic of nomenclature used and leakage paths of a circular rotary-vane expander with general orienta tion and a (a) conventional or (b) modified intake ports..199 5-2 Schematic of a typical leakage path.................................................................................200 5-3 Shear and pressure driven Couette flow..........................................................................201 5-4 Non-axisymmetric pressure boundary condition on outer surface of the rotor...............202 5-5 Free-body diagram of a vane protruding outward from a ro tor slot at a local angle ....203 5-6 Schematic of generalized Couette flow in the axial gap between rotor and end-plate....204 5-7 Variation of reaction forces on a vane w ith no vane-tip curvature as a function of angular displacement.......................................................................................................204 5-8 Variation of reaction forces on a vane with a circular vane -tip profile as a function of angular displacement.......................................................................................................205 5-9 Variation of leakage from/to the e xpander cavity as a f unction of angular displacement due to non-axisymmetric flow between the rotor and stationary endplates for different intake angle spreads..........................................................................205 5-10 Variation of non-axisymmetric leakage fr om/to the expander cavity for the ideal and throttling cases............................................................................................................... ..206 5-11 Variation of leakage from/to the expa nder cavity through the ga p between the sides of the vanes and end-plates for different intake angle spreads........................................206 5-12 Variation of leakage to the expander cav ity from the rotor slot (modified intake) through the gap between the face of the vanes and rotor slot..........................................207 5-13 Variation of the ideal and actual mass flow-rates through the expander as a function of rotational speed............................................................................................................207 5-14 Variation of the ideal and actual mass flow-rates through the expander as a function of the number of vanes.....................................................................................................208

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155-15 Comparison of way by which fluid is in troduced into the expanders cavity; A) conventional intake via intake port and B) modified intake vi a rotor slots through end-plates to ensure vane-tip and stator-cylinder contact................................................208 5-16 Flow through a sudden contraction..................................................................................209 5-17 Schematic of the three phases of vane or ientation that occur during the intake process.210 5-18 Variation of pressure in expander cavity as a function of angular displacement for the ideal and actual (thr ottling only) cases............................................................................211 5-19 Schematic of the intake port of the m odified rotary-vane intake through the endplates and into the rotor cavity via rotor slots..................................................................212 5-20 Schematic of (a) symmetric (b) non-symm etric non-circular rotary-vane expander comprised of four circular arcs (1-4) and a sealing arc (5)..............................................212 5-21 Variation of vane displacement as function of the leading vanes angular displacement for both a circul ar and non-circular MVE.................................................213 5-22 Variation of vane velocity as function of the leading vanes a ngular displacement for both a circular and non-circul ar rotary-vane expander....................................................213 6-1 The water heater used to add a constant heat load to the constant temperature water loop........................................................................................................................... .......236 6-2 Unmodified two ton air-cooled barrel type water chiller.................................................236 6-3 Instrumentation map and schematic of the experimental set-up for a vapor compression cycle with a by-pass loop fo r rotary-vane expand er integration.................237 6-4 Variation of the measured pressure fo r the standard mode of operation in the unmodified chiller as a function of time..........................................................................238 6-5 Variation of the measured temperature for the standard mode of operation in the unmodified chiller as a function of time..........................................................................238 6-6 Variation of the measured pressure fo r the cut-off mode of operation in the unmodified chiller as a function of time..........................................................................239 6-7 Variation of the measured temperature for the cut-off mode of operation in the unmodified chiller as a function of time..........................................................................239 6-8 Variation of the compressor work for both the cut-off and standard modes of operation in the unmodified chi ller as a function of time................................................240 6-9 Schematic of the cut-away view of the bypass loop and evapor ator and the logic behind the continuous control scheme used to control the degree of superheat..............243

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166-10 Flow path of refrigerant in both the compression and expansion processes....................245 6-11 Double acting, five vane modified automotive compressor............................................246 6-12 Unmodified, modified symmetric and non-symmetric non-circular Expander (All dimensions in inches).......................................................................................................248 6-13 Schematic of the experimental set-up for compressed air and R-22 tests with a rotary-vane expander.......................................................................................................249 7-1 Variation of the critical-p ressure ratio as a function of condensing temperature for a two-phase expander and vapor compressor.....................................................................257 7-2 Expander sizing diagram..................................................................................................257

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17 NOMENCLATURE 0 reference state, zeroth-order 1 inlet state, location of minimum volume (i.e. cut-off) 2 location of maximum volume (i.e. at exhaust) A cross-sectional area, m2 a speed of sound, m/s Acc( ) sliding acceleration of vane, m/s2 act actual Ak, Bk Fourier constants amb ambient b base cycle, boundary work, back-pressure c Carnot calc calculated Cd discharge coefficient comp compressor cond condenser COP coefficient of performance, dimensionless CP critical point Cp specific heat at cons tant pressure, kJ/(kg.K) crit critical cv control volume (i.e. expander cavity) dt differential in time, s e eccentricity m, evaporat or, electric, exit state E total energy, kJ econ economizer cycle

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18 ED expansion device Eu Euler number ex start of exhaust port exh exhaust exp expansion device f Darcy friction coefficient F force, N f saturated liquid, friction g saturated vapor, acceleration of gravity m/s2 gc gas cooler h specific enthalpy kJ/kg, evapor ator coolant, isenthalpic Hv height of vane, m i inlet, intermediate I specific irreversibility (kJ/kg), current (Amp) in inlet state, spread of intake port int intake L axial length of expander, m l leakage channel length, m leak corresponding to internal leakage Lv axial length of vane, m M Mach number m mass kg, mean m mass flow-rate, kg/s max maximum MW molecular weight, kg/kmol

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19 Nv number of vanes O center, location o original out outlet state P pressure, kPa p dimensionless pressure, exinexppppp q specific heat transfer, kJ/kg Q heat transfer rate, kW r dimensionless radial coordinate, /Rrrr r reservoir R rotor ref refrigerant, refrigerating Re 1 Reynolds number based on axial gap width RR rotor radius, mm RS( ) distance from stator-cylinde rs center to periphery, m rv volume ratio (process or built-in) S gap aspect ratio, stator-cylinder s specific entropy kJ/kgK, isentropic sc subcooling sf sliding friction sh superheat sys system T temperature K, torque Nm t time, s or minutes (unless otherwise noted) tv thickness of vane, m (=ts, slot thickness)

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20 u specific internal ener gy kJ/kg, velocity m/s u dimensionless radial velocity, Ruur v dimensionless azimuthal velocity, Rvvr v throttling valve, vane, specific volume /kg m3 V voltage V, volume m3 vd viscous drag Vel( ) sliding velocity of vane, m/s w dimensionless axial velocity, 1/ww w specific work, kJ/kg W power, kW x quality, dimensionless X( ) vane height prot ruding from slot, m z dimensionless axial coordinate, 1/zz differential speed of angular rotation, RPM void fraction inner to outer radius ratio angle between successive vanes, degrees, leakage channel height, m 1 axial gap between rotor and end-plate, m 2 radial gap between rotor and stator during sealing arc, m effectiveness, dimensionless efficiency leakage path flow coefficient dynamic viscosity, Pa.s

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21 angular displacement/local angle, degrees cut angle at which expansion process begins, degrees ex spread of the expanders exhaust port, degrees in spread of the expanders intake port, degrees density, kg/m3 angular velocity, rad/s stream availability or exergy, angl e between base line and line through stator center from rotors center, degrees

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22 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ANALYTICAL AND EXPERIMENTAL INVE STIGATION OF ROTARY-VANE TWOPHASE EXPANDERS IN VAPOR COMP RESSION REFRIGERATION SYSTEMS By Ahmad Mohamed Mahmoud August 2008 Chair: S. A. Sherif Cochair: William E. Lear Major: Mechanical Engineering The refrigeration and air-condi tioning community has been searching for environmentally friendly refrigerants and ways to improve ener gy efficiency and to increase the potential for reductions in system size and weight for some time. These system improvements may manifest themselves in numerous ways but significantly impact terrestrial logistics differently. This study investigates the analytical and experimental util ization of rotary-vane ex panders as throttle valve replacements in vapor compressi on refrigeration systems with a specific application of these expanders in smaller deployable environmen tal control units. Th e findings support the conclusion that utilizing two-phase expanders in refrigeration systems is very promising. Mathematical models for a system-level para metric cycle analysis were developed to assess potential gains in the cycle coefficient of performa nce for R-22, R-134 and transcritical CO2 vapor compression technology. Optimization st udies were conducted to determine the optimum performance of a refrigeration system s ubject to constraints of size and weight of both the condenser heat transfer area and the size and weight of the two-phase expander. An exhaustive literature survey aids and va lidates the suitability of the type of positivedisplacement expander used, after which comprehensive component level thermodynamic and

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23 fluid dynamic models of a rotary-vane expander we re developed to establish the performance of this expansion device as a function of design and fluid parameters. This included rigorous modeling of irreversible loss mechanisms such as throttling in the intake and exhaust ports, twophase internal leakage, fric tion, re-compression, and under or over-expansion due to incorrect sizing of the expander or off-design operation. Results from these models were used to esta blish the operating principles of a two-phase rotary-vane expander. Experimental data gathered after modifying a conventional chiller with an alternative flow path, where a rotary-vane expander and dynamome ter have been installed, were used to improve the fidelity of the analytical mode l developed. Issues such as real-time control to satisfy system-level constraint s, expander sizing and operational speed and inadequate operation were addressed. The analytical model develope d along with single-phase experimentation were used to understand and overcome component-lev el complications and inadequacies due to irreversible effects.

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24 CHAPTER 1 INTRODUCTION AND OBJECTIVES Introduction The refrigeration and air-condi tioning community has been searching for environmentally friendly refrigerants and ways to improve ener gy efficiency and to increase the potential for reductions in system size and weight for so me time. Cycle improvements may manifest themselves in many ways such as using multista ge compression with inter-cooling and/or flash gas removal as a means to reduce the work input to the compressor, employing economizers and/or employing expansion devices to recover work. There are two advantages to using expansion de vices with output wor k. The first is that lower enthalpy refrigerant is obtained at the in let of the evaporator and hence increases the refrigerating effect of the evapor ator. The second is the extra work that can be extracted from the expansion process, which can then be used to lower the input requirement of the compressor or to operate auxiliary machinery such as fans or pumps. Both effects serve to increase the coefficient of performance (COP) and hence raise the energy efficiency of the system as well as increase the potential for reducti ons in system size and weight. Objectives The proposed overall approach is to comb ine experimentation with thermodynamic and fluid dynamic modeling in order to demonstrat e the operating principles, to discover any unanticipated difficulties, and to establish a qualified design code which may subsequently be used for optimizing air-conditioning and refrigeration systems for a wide range of applications. A detailed list of these objectives is as follows: Conduct a literature surv ey of work on the use of various expansion devices in relevant engineering cycles. This exhaus tive survey would aid and validate the suitability of the selected type of positive-displacement expander used.

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25 Perform system-level parametric analysis of vapor-compression refrigeration systems to assess potential gains in the cycle COP. This includes examining the factors that affect reduced power requirements and in creased refrigerating capacity. Develop a thermodynamic and fluid mechanic component-level model of two-phase rotary-vane expanders to establish their e fficiency as a function of design and fluid parameters. Develop a comprehensive model of primary irreve rsible effects such as internal leakage, friction, and throttling within the two-phase expander and present relevant discussion or models of secondary losses. Design, build and modify an experimental refrig eration system to incorporate a rotary-vane expander based on model inputs including rota ry-vane expander sizing and rotational speed. Develop an appropriate real-t ime control scheme to mimic the operational function of conventional thermo-static expansion valves, i.e. to regulate mass flow-rate through the rotary-vane expander, to satisfy system-level constraints. Utilize the developed computer program and comprehensive single-phase experimentation to understand, recommend and possibly overc ome component-level complications and inadequacies due to irreversible effects. Utilize the experimental results where necessary to improve the fidelity of the qualified design code. Based on the motivation behind and objectives of this investigation, a detailed review of recent literature pertaining to the use of expanders as throttle valve replac ements in refrigeration units was conducted. The review en tails the use of two-phase e xpanders in conventional vapor compression refrigeration systems with conventi onal CFC, HCFC and HFC refrigerants as the working fluids. The review also encompasses th e use of two-phase expanders in geothermal systems and transcritical carbon-di oxide refrigeration cycles as well as single-phase expanders utilized in organic Rankine power cycles and gas refrigeration cycles. This section will enable the reader to understand the f undamentals of this technology as well as highlight practical issues/problems that may have been presented by various researchers in th is regard. The type of expander used was chosen and suitability was based on this e xhaustive review.

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26 Analytic parametric modeling of the ideal a nd actual vapor compressi on refrigeration cycle and modified cycles such as the vapor compre ssion cycle with an expander, a liquid-to-suction heat exchanger and a multistage vapor compression cycle with an economizer was then conducted. These cycles have been analyzed in bot h ideal and actual cases and have also been analyzed with the additional modification of uti lizing a two-phase expande r as a throttle valve replacement. The working fluids investigat ed in this study incl ude: R-12, R-22, R-134a, pure hydrocarbons and trans-critical carbon-dioxide. Optimization studi es were also conducted to determine the optimum performance of a refriger ating unit subject to c onstraints of size and weight of both the heat transfer ar ea and the size and weight of the tw o-phase expander. The development of a detailed thermodynamic and fluid dynamic model of the rotary-vane expander followed. The geometry, kinematics a nd thermodynamics of a circular rotary-vane expander were described mathematically to dete rmine its ideal performa nce as a function of design and fluid parameters. A robust thermodynamic model that takes into account primary loss mechanisms such as friction and internal leak age and presents a disc ussion, models when necessary, of secondary irreversib le effects such as throttling in the intake and exhaust ports, two-phase internal leakage, friction, re-c ompression and under or over-expansion due to incorrect sizing of the expande r is then described. An experimental program that has been develo ped at the University of Florida for testing the use of a two-phase rotary-v ane expander in a refrigeration unit is then detailed. The equipment and instrumentation used in the lab to gather experimental data to validate and improve the fidelity of the analytical models are described. The sizing and selection of the rotary-vane expander and the dynamo meter is presented. Finally the means of real-time control of the expander via a high-speed programmab le dynamometer controller is presented. The

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27 developed computer program and comprehensiv e single-phase experimentation was used to understand, recommend and possibly overcom e component-level complications and inadequacies due to irreversible effects such as friction and internal leakage. A summary of key technical challenges, recomme ndations and conclusions from both the comprehensive modeling and experimental effort is then presented.

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28 CHAPTER 2 LITERATURE REVIEW The concept of utilizing expansion devices in various thermodynamic cycles is one that has been described in scattered tech nical literature. Example applicat ions in which expanders have been used to recover and utilize lost work or waste heat include geothermal applications, Organic Rankine Cycles (ORC), and refrig eration cycles. A thorough understa nding of these applications along with the types of working fluids used, the types of expansion devi ces used and inherent loss mechanisms within those expansion devi ces will allow further advancement of such concepts. In literature dealing with expansion devices, the working fluid has been predominantly air, steam, refrigerant vapor or tran scritical carbon-dioxide. In the present study, expansion takes place in a refrigerants two-pha se region where additional comp lexities (e.g. erosion due to cavitation and increased internal leakage due to significant density variation) arise. Although complexities exist with the expansion of gases, vapors, and transcritical fluids, the loss mechanisms of internal leakage, friction, and heat transfer and how they affect overall performance of different types of expanders ar e qualitatively comparable to one another. Understanding and incorporating these effects will aid in developing a mathematical model of a two-phase expander employed in a vapor comp ression refrigeration/ heat pump system. Expanders in Refrigeration and Heat Pump Systems There has been much effort to quantify th e amount of savings and the methods by which the work recovered and additional refrigerating effect could be utilized from the use of expanders in refrigeration and heat pump systems. Among the res earchers who investigat ed this are Zhang and Yu (1992), Markoski (2003) and Brasz (1995).

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29 Zhang and Yu (1992) compared ideal and m odified refrigeration cycles for R-12, R-22, R502 and R-707. They found savings from utilizing two-phase expanders and recommended the use of expanders. They failed to elaborate on the complexities associated with utilizing expanders in the two-phase region. They men tion only that components of the expander are small and hence friction losses are large and that problems may arise from lubrication related issues. Brasz (1995) introduced the use of a turbo-e xpander in large (>85 tons) vapor-compression refrigeration systems utilizing R-134a as the working fluid. According to Brasz, a turbine efficiency of 39% could overcom e the disadvantages of using R-134a instead of low-pressure R11 or R-123 refrigerants. He c oncluded that the application of this technology is applicable where the expander rotational speed matche s that of an existing drive train. Markoski (2003) presented thre e alternative methods by which the expansion work from an expansion engine can be utilized in a va por compression refrigera tion cycle. The author provides brief guidelines and analyses for the use of the power recovered by the expander in conventional and non-conventional vapor compressi on cycles as well as powering a circulation pump for liquid circulation through a flooded evaporator. Conventional Heat Pump and Refrigeration Systems The use of expansion devices in conventiona l vapor-compression refr igeration cycles has been investigated by Hays and Brasz (1996, 1998), Smith and Stosic (1995), Zoughaib and Clodic (2003), Smith et al. (1994, 1996, 1999, 2001a 2001b), Brasz et al. (2000), Kornhauser (1990), Fischer (1978), Kim et al. (2004), Tamura et al. (1997), Disawas and Wongwises (2004), Henderson et al. (2000) and Ta niguchi et al. (1983). In most of the aforementioned studies, unless mentioned otherwise, a throttling valve was used as the default control mechanism responsible for ensuring the degree of superheat of the

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30 refrigerant leaving the evaporator. The use of a throttling valve would however cause a loss in the available pressure difference that may be utilized in an expansion device for work recovery. Tamura et al. (1997) investigated the use of a screw expander in a high temperature (up to 180oC) binary heat pump with steam-water and ammonia as the working fluids. They found a COP improvement of about 40% when compar ing a high-temperature heat pump with one employing an isentropic screw expander and one without in the ope rating range of 40-180oC. Fisher (1978) described the concept of a pi voting-tip rotary-vane compressor and expander applied to a solar-powered vapor -compression heat pump. The solar heat pump had a capacity of 3 tons and R-12 was selected since the author found that the required compressor and expander displacements are of reasonable size to provide good efficiencies and lo w manufacturing costs. Compression and expansion characteristics were ta ken into consideration before selecting the working fluid as to avoid entirely the complex ities that arise when expanding into the two-phase regime. The author also describes the performa nce of pivoting-tip gas bearings, which were utilized to reduce vane-tip friction in rotary-v ane turbomachinery. Estimated efficiencies of 85% were assumed for both the compressor and expande r to reflect improvements that the pivotingtip gas bearings will introduce. A cooling coeffici ent of performance of 0.56 at design conditions was achieved. Taniguchi et al. (1983) presented a detailed an alytical and experiment al investigation of helical screw two-phase expanders in large refrigeration system s whose working fluid is R-12. They stated the added advantage of utilizing positive displacement turbomachinery as two-phase expanders because they operate without erosion a nd slip losses between liquid and vapor phases. The authors investigated the de gree of subcooling on the performance characteristics of this modified vapor compression cycle. Effects of va por formation during the expansion process as

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31 well as internal leakage losses were accounted fo r in the theoretical model. The helical screw expander used in this study ha d a built-in volume ratio of 5:1. The expansion process volume ratio of R-12, at the operating conditions, however depended on the degree of subcooling in the condenser and varied from 13.2:1 and 10.6:1 corresponding to 0 and 12 K of subcooling respectively. The authors concluded that the isen tropic efficiency of the expander increased from 30 to 60% at rotational speeds of 500 and 3000 rpm respectively. Th e authors predicted analytically an isentropic efficiency a pproaching 80% for larger screw expanders. Kornhauser (1990) analytically investigated the use of a twophase ejector as a refrigerant expander to replace the throttli ng process in a vapor-compressi on refrigeration system. The author cites other expansion devices as expens ive and susceptible to damage because of twophase flow. The power that is r ecovered by the expander is not ex tracted or coupled in any way but is used to partially comp ress the refrigerant leaving the evaporator. The effects of both isentropic expansion and part ial compression significantly increase the coefficient of performance of the system. The system is sim ilar to a two-stage refrigeration unit where the work extracted from the high pressure stage is used to drive the compre ssion process of the low pressure stage. A throttling valve expands the refrigerant in the low pressure cycle to the evaporator pressure. According to the author this expansion is across a small pressure difference and is negligible when accounting for the maximu m work extraction from the cycle. Theoretical calculations were made to compare the conventiona l and ejector refrigeration cycles. A constant pressure mixing model of the ejector was use d. The author found that for an ideal ejector expansion cycle, an increase in the cycle coe fficient of performance of 13%, 21%, 20%, 17%, and 30% would be realized for R-11, R-12, R-22, R-113 and R-502 respectively. Results

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32 detailing the decrease in compressor displa cement and the performance gains of R-114, R-500 and R-717 are also presented. An interesting obstacle that Kornhauser ( 1990) eludes to is the fact the conventional expansion valve would defeat any performance improvements made by the use of an expansion device. According to Kornhaus er (1990), Newton (1972a, 1972b) has patented two methods by which the liquid flow through the expander may be controlled. Controlling the mass flow-rate through the ejector may be accomplished by controll ing the specific volume of the entering fluid by means of injecting small amounts of hot gas into the refrigerant exiting the condenser. Varying the nozzle area could also control the mass flow-rate through the ejector. Disawas and Wongwises (2004) experimentally investigated the use of a two-phase ejector-expansion device in a conventional vapor compression refrigeration system whose working fluid is R-134a. They used an ejector as the sole method by which expansion occurs unlike other investigators who used a throttli ng device. They reported a maximum increase in COP of 2% and 10% at an ev aporating temperature of 16oC and a condensing temperature of 37 and 32oC, respectively. This is not an evapora ting temperature typica l in air-conditioning applications. The ejector system performance, when compared to the conventional system, decreases greatly as the condensi ng temperature is increased and the evaporator temperature is decreased. Hays and Brasz (1996) presented a theoretical investigation of the power recovered by the use of a two-phase turbine in a refrigeration system with R-134a, R-22, R-123 and propane as the working fluids. The calculations were run assuming the turbine had isentropic efficiencies of 55 and 70%. The authors also experimentally investig ated the performance of two-phase turbine in a

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33 500 ton R-134a chiller. The authors determined a turbine efficiency of 60-65% and discussed the turbine rotor loss mechanisms that accounted for this deficiency. Hays and Brasz (1998) report on the implemen tation of a stand-alone turbo-expander as throttle valve replacement for a large 2000-5000 ton centrifugal chiller installation at a commercial building in Manhattan, New York. They reported that over 70 refrigeration units employing two-phase turbines are in operation in a wide range of industrial and commercial applications. They concluded that at an evaporator temperature of 44oF (6.7oC) and a condensing temperature of 86oF (30oC), 15 and 180 kW of power can be produced by a 500 and 6000 ton chiller unit respectively. These values are ideal a nd do not take into consideration the generator efficiency. The installation cost for one of these large units is approximately $1000/kWe initially and should decrease to a minimum of $400/kWe for additional units. Zoughaib and Clodic (2003) inves tigated the use of a micro-tu rbine as an expansion device in a domestic refrigeration unit. The power recove ry would then be used to drive an auxiliary fan(s) that would, through forced convection, ensure that no frost would form on the heat exchanger of no-frost domestic appliances. The aut hors parametrically investigated the effect of subcooling on the modified refrigeration cycle. Th e authors assumed a micr o-turbine isentropic efficiency of 80% and hence concluded that a 1.1% increase in the coefficient of performance and 1.12 W of generated pow er were realized. Smith et al. (1994) investigated the therm odynamic modeling of a Lysholm machine as a two-phase expander in large-s cale refrigeration systems utilizing R-113 as the working fluid. They discussed the differences that existe d between their model and the one proposed by Taniguchi et al. (1983). Among the differences that the authors pointed out was the fact that Taniguchi et al. (1983) assumed that the filling process would take place at a constant pressure.

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34 According to the authors, the filling process does not take place at a constant pressure because the acceleration of the entering re frigerant causes a pressure drop, which in turn induces flashing and thus causes higher fluid velocities. The authors estimated that isentropic efficiencies of about 70-80% could be expected. Smith and Stosic (1995) descri be the principle behind a novel machine, which they dubbed the expressor that is comprised of a coupled twin-screw compressor and twin-screw expander in a single casing. They investigated the use of this expressor unit as a throttle valve replacement in a conventional large chiller where R-134a is the working fluid. The authors estimated the expander adiabatic efficiency as 70% and explaine d that large leakage losses do not affect the performance of the expander because they are in the direction of the bulk flow. A coefficient of performance gain of approximate ly 10% and 7.5% with 0 and 5oC subcooling respectively was realized. A throttle valve is di rectly upstream of the expressor device. Brasz et al. (2000) discuss the disadvantages of this mechanism that make th is technology rather e xpensive to implement. The primary disadvantages mentioned were the need for a timing gear, the high cost of seals and the need for two sets of rotors to carry out the compression and expansion processes. Smith et al. (1996) presented the high effici ency design of two-phase screw expanders in various cycles. Smith (1993) first presented the use of these twophase expanders in a trilateral flash cycle (TFC). In this cycle a fluid is pressuri zed adiabatically, heated at constant pressure to its boiling point, expanded adiabatically as a tw o-phase mixture and then condensed at constant pressure. Smith et al. (1994) then investigated the us e of different working fluid mixtures to increase the power outpu t of a two-phase expander employed in the TFC. Smith et al. (1996) concluded that if a small amount of under-e xpansion is permitted, high speed, low built-in volume ratio designs, with roughly half the vol umetric capacity required for lower speed full

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35 expansion alternatives, attain the highest overall adiabatic efficienci es. They also reported that from a large air-conditioning unit data they we re able to predict a 7-10% increase in the coefficient of performance of the system. This could be achieved by ut ilizing the two-phase screw expander to drive a compressor in a herm etic unit called the expressor, (see Smith and Stosic 1995) with an adiabatic efficiency of approximately 70%. Smith et al. (1999) investigated the feasibility of utilizing a twin screw two-phase expander in a large (500 ton) chill er unit that operated with R-134a as the working fluid. They attained an expander adiabatic efficiency of approximately 70% They listed the most significant factors that hinder the adoption of two-phase ex pander in refrigeration systems as poor adiabatic efficiencies and high cost of construction and installation. They discussed the results found by Smith et al. (1996) as to the improper selecti on of the built-in volume of the expander when compared to the actual expansion process volume ratio. Brasz et al. (2000) presented the development of a twin screw expr essor with only one pair of rotors as a throttle valve replacement for a large wate r-cooled chiller. The need for a timing gear and high cost seals was avoided by the use of high profile rotors developed earlier by the authors. The compression process in the e xpressor unit recompresse s the vapor that is generated in the expansion pro cess by means of power that is recovered. The recompressed vapor would then be piped directly to the conde nser inlet. The authors discussed the pros and cons of different methods by which the power rec overed may be utilized. They indicated that the efficiency of the expander-compressor mechanis m, expander and compressor is approximately 55%, 70% and 80% respectively. At part load, the speed reducti on caused by the reduced flow via the throttling valve negates th e use of a control system. Initia lly the built-in volume ratio of the screw expander was designed and built at 2.85:1. The authors found however evidence of

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36 over-expansion when this 2.85:1 expander was us ed with R-134a and R-113 whose expansion process volume ratios were 11.4:1 and 12.9:1 re spectively. The authors then progressively reduced the built-in volume ratio of the expander to 1.85:1 to a void over-expansion. Smith et al. (2001a) investigated the use of a helical twin screw compressor-expander as a replacement to the throttle valve in a refrigeratio n system. Poor adiabatic efficiencies as well as high manufacturing and installation costs are amo ngst the most significant factors hindering the extensive use of two-phase expanders. According to Smith et al. (2001a) they have shown that the adiabatic efficiency of the expander, a reported maximum of 70%, is significantly higher because the built-in volume ratio of the expander is less than the volume ratio of the actual expansion process. They designed, built and tested an expander unit with a built-in expansion ratio of 2.85:1. The expander provided a 3.6-10.3% increase in the syst em coefficient of performance for an actual expansi on process volume ratio of 11.4:1. Smith et al. (2001b) presented an economical analysis of two-phase screw expanders that may be utilized in organic Rankine cycles, Trilateral Flash Cycles and refrigeration units. They reported a peak adiabatic efficiency of 76% Although this technology could be utilized in refrigeration systems, the authors conclude that it would be more signifi cant for Trilateral Flash Cycles (Smith, 1993 and Smith et al., 1994, 1996) Kim et al. (2004) presented preliminary results of an investigation of the use of a scroll expander with a heating structur e and a proposal of their use in a refrigeration cycles as twophase expanders. When used in high-temperature and high-pressure applications, clearances in both the radial and axial direc tions are caused by differential th ermal expansion of the scroll elements. This may result in a decrease in th e expander efficiency as well as the expanders specific power output. The authors investigated the use of a heat pipe that would provide a

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37 uniform temperature throughout the scroll elemen ts by means of heating. According to the authors, if the scroll expander w ith a heating structure is used in two-phase expansion, instances of over expansion will cause the liquid in the expander to evapor ate by means of heat from the refrigerated space. They concluded that this in turn would both increase the power output of the expander and the cooling capacity of the system due to a higher volumetric expansion ratio needed in the expander. In this case the e xpander could also be t hought of as a partial evaporator. Henderson et al. (2000) theore tically investigated the ec onomics of employing compressorexpander devices in heat pumps th at use R-410 as their working fl uid. They investigated using a screw compressor-expander since it was found th at the rotary-vane compressor-expander would likely fail because of excessive friction at the vane-tip/stator surface a nd could also be very noisy. They found that if the device had an 80% isentropic efficiency, the system COP would increase by 30%. They have not specified the oper ating conditions or any component efficiencies in their study. In a majority of the studies, no subcooling in the condenser was allo wed as to determine the maximum possible work extraction from the expansion process. This however would result in immediate formation of vapor upon the slightest decrease in pressure and increase of velocity. This in turn could cause cavita tional erosion, especially in dy namic expansion devices. This phenomenon however has not been addressed directly. Transcritical Carbon Dioxide Systems The inherent coefficient of pe rformance of a transcritical CO2 cycle is less than the coefficient of performance of a conventiona l vapor compression cycle. Among the methods by which this deficiency may be overcome is to utilize expanders as throttle valve replacements. One primary difference between the use of expanders in a transcritical CO2 cycle and a

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38 conventional cycle is the fact that complexities such as cavitation on ly take course as the velocity of the incoming refrigerant has decreased (i.e. at later stages in the expansion process). In conventional vapor-compression cycles these co mplexities may occur immediately and is a strong function of the degree of s ubcooling and the severity of lo ss mechanisms during the intake phase of the expansion process. The use of e xpanders in transcritical carbon dioxide systems have been investigated by Robinson and Gr oll (1998a, 1998b), Baek et al. (2005a, 2005b), Li and Groll (2005), Huff et al. (2002, 2003), Fukuta et al. (2003), Zha et al. (2003), Nickl et al. (2002, 2003), Stosic et al. (2002), Hays and Brasz (2004), Heyl and Quack (1999), Li et. al (2000), Driver (1996), Driver and Davidson (1999), Heyl et al. (1998), Ertesvag (2002) and Preissner (2001). Robinson and Groll (1998a, 1998b) analyti cally investigated and compared the performance of a transcritical CO2 cycle and a conven tional R-22 refrigeration cycle with and without an expansion turbine. They found that use of an internal heat exchanger along with an expansion device decreased the COP of the transcritical CO2 cycle by 6-8%. They concluded that the stream availability following the heat rejecti on process in the condenser is better utilized by an expansion device rather than an internal he at exchanger. They also found a 23% COP increase when comparing an R-22 refriger ation cycle and a transcritical CO2 cycle both with expansion devices with a 60% isentropic efficiency ope rated at an evaporating temperature of 5oC and a condensing temperature of 35oC. Baek et al. (2005a, 2005b) studied both analytic ally and experimentally the effect of the addition of a newly designed piston-cylinde r work-producing device to a transcritical CO2 cycle. Experimentally they found a 10% increase of the cycle COP taking into account both the increase in evaporator capacity and expansion work. The limiti ng factor in this study was the

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39 isentropic efficiency of the expander, which wa s reported to be 10% (experimentally) and 34% (theoretically). The discrepancy between the two can be attributed to the 30% uncertainty in the instruments used for measurements as well as th e inadequate theoretical internal leakage model that was used. Li and Groll (2005) analytically investigated the use of an ejector-expansion device in a transcritical CO2 cycle. They found that the cycle CO P can be increased by 16% by assuming that the ejector has an isentropic efficiency of 90%. They have however not investigated the potential loss in efficiency due to various loss mechanisms. Heyl and Quack (1999) presented the process calculations, design and results of a novel free piston expander-compressor for a CO2 cycle. The expander-compressor device had two expansion and two compression cylinders and was a ssumed to have an isentropic efficiency of 85%. They noted an increase of cycle efficien cy of 38.7% and 10% w ith and without work recovery from the expansion device. They reporte d an experimental coefficient of performance gain of 30%. They concluded that a significant di sadvantage to this type of compressor-expander was that the expander and compresso r pistons moved with identical strokes, which did not utilize all of the available expansion work (under-e xpansion). The operating conditions were not detailed. Nickl et al. (2002) presented a second gene ration novel piston expander-compressor that would attempt to overcome the disadvantages of the expander-compressor presented by Heyl and Quack (1999). They developed a double rocker arm that would control the different speeds of the expander and compressor pistons. They conclude d that this mechanism would prove to be complex and expensive. A slight increase of the co efficient of performance relative to the first generation expander-compressor (Hey l and Quack, 1999) was realized.

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40 Nickl et al. (2003) presented a third gene ration novel piston expander-compressor with three expansion stages for a transcritical CO2 Cycle. According to them, this would eliminate the identical strokes of the expander and compressor pistons (Heyl and Quack, 1999) in their first generation expander-compressor. It would also eliminate th e complex double rocker arm needed to control the different speeds of the e xpander and compressor pist ons (Nickl et al., 2002) of the second generation pist on expander-compressor. The authors first estimate of the isentropic efficiency of the expander-compressor was 85% and did not take into account losses due to heat transfer, pressure drops in valves internal leakage and friction losses. They concluded that the isentropic effi ciency of the expander decreased only 3% when these losses were taken into consideration. Zha et al. (2003) developed a rotary-type rolli ng piston expander to use as a throttle valve replacement in a transcritical CO2 cycle in small heat pumps and refrigeration systems. They experimentally determined the design paramete rs, method of control and determined the loss mechanisms associated with this expander. Th ey also mentioned that cavitation and liquid slugging are among the obstacles that hinder the development of CO2 expander technology. They have estimated the isentropic efficiency of the rolling piston expander to be about 50%. They have attributed the losses in efficiency to fr iction (~24%) and internal leakage losses (~25%). Fukuta et al. (2003) investigat ed and predicted the performa nce of a rotary-vane expander utilized in a transcritical CO2 cycle as a replacement to the thro ttle valve. They cited issues of slugging and cavitation as the main obstacles to the development of two-phase expander technology. The prototype expander that was used had a built-in expansion ratio of 2:1. They initially assumed an expander efficiency of 60%. After taking into account the losses from heat transfer and internal leakage they calcul ated an isentropic efficiency of 43%.

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41 Huff et al. (2002) developed an algorithm that estimates the performance of the expansion and compression process in positive displacement turbomachinery. They have taken into account internal leakage losses, heat tran sfer and valve losses in their m odeling effort. They have applied this analysis for scroll, piston, rotary-vane, rota ry piston and screw type mechanisms. They have applied this analysis with particul ar interest to a transcritical CO2 cycle and have reported a coefficient of performance gain of 40-70% and a 5-15% increase in capacity. The authors discussed the importance of the built-in volume ratio of the expansion device in use and the effect it may have on the high side pressure of th e cycle. They concluded that if the rotational speed and built-in volume ratio of the device is not adjustable then this will ultimately reduce the benefit of the expander in a tr anscritical cycle. They proposed utilizing an expander and a throttling valve either in series or parallel to match the high side pre ssure with the optimum pressure. They concluded however that the perf ormance gain will be relatively less than if variable rotational sp eed and volume ratio were employed. Huff et al. (2003) experimentally investigated the use of two R-134a scroll expanders in a transcritical carbon dioxid e cycle. The first prototype was mo dified by reducing the wall height of the original compressor scrolls. This modifica tion severely influenced this prototypes 28% isentropic efficiency. The highest isentropic and volumetric efficienci es that were reported were 42% (at 1800 rpm) and 68% (at 2200 rpm), respec tively, in the case of the second unmodified expander. Stosic et al. (2002) investigated the use of a twin screw combined compressor and expander for a high temperature transcritical CO2 refrigeration systems. They stressed the need to reduce bearing loads as this would make the expander-compresso r device more applicable to a larger range of pressures. Th ey achieved a 20% reduction in the bearing load by balancing the

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42 loads of the compressor-expander rotors. They also concluded that throttling losses much larger than those associated with convent ional refrigerants exist in this helical type turbomachine. They reported an idealized 72% increase in the coe fficient of performance when applying this compressor-expander device to a high temperature transcritical CO2 refrigeration cycle. Hays and Brasz (2004) experi mentally investigated the use of a turbine-compressor mechanism in a 6 ton transcritical CO2 refrigeration or liquefaction cycle. The axial flow turbine previously presented in Hays and Brasz (1996) as a two-phase expander is used to drive a compressor in a hermetic turbine-compressor unit. At a rotational speed of 10,000 rpm the turbine and compressor had efficiencies of 69 an d 80% respectively. The turbine had a measured efficiency of 56% at its maximum speed of 12,800 rp m. The vapor exiting the evaporator is then compressed in the turbine-compressor mechanism. The dual effects of expansion and partial compression in the turbine-compressor mechan ism increased the system coefficient of performance by 39%. Driver (1996) theoretically described the potential of utilizin g a compressor-expander device in a vapor compression system. Driver a nd Davidson (1999) have developed and tested a hinged-vane compressor-expander unit for transcritical CO2 cycles. They found that if the compressor-expander device operated at a 100% isen tropic efficiency, whic h is highly unlikely, a COP improvement of 40% could be realized when compared w ith a conventional heat pump. The investigators pointed out that their compressor-expander device would not work with refrigerants such as R-134a because of sizing issues. Preissner (2001) investigated the use of scroll expanders in a transcritical CO2 cycle. The author also performed an experimental compar ison of a conventional R134a and transcritical CO2 refrigeration cycles employing suction line heat exchangers. He found that CO2 cycles falls

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43 short in performance by 8 and 23% for a range of typical operating conditions with particular emphasis on medium to high outdoor temperatures. With the implementation of a scroll expander the cycles have an identical coefficient of performance at a condensing temperature of 25oC. At higher condensing temperatures though, he found th at a significant perfor mance gap exists in favor of the R-134a cycle. The author found inte rnal leakage losses to be the most significant loss mechanism due to limitations in machini ng accuracy and geometric design. The expander was thus not implemented in the cycles. Li et al. (2000) investigated the use of a vortex tube expans ion device in a transcritical CO2 refrigeration cycle. The vortex tubes ener gy separation effect, known as the Ranque-Hilsch effect, separates the high pressure gas entering th e vortex tube into two low pressure streams one at a low temperature and the other at a high temp erature. The authors compared this expansion device to a reciprocating piston expansion devi ce already developed by the authors. They concluded that an increase in the system coeffi cient of performance of 20% could be achieved if the piston expanders isentropic efficiency wa s 50%. The vortex tube expanders isentropic efficiency would have to be on the order of 38% to achieve the same performance gain. Heyl et al. (1998) developed a free piston compressor-expander unit to be employed in a transcritical CO2 cycle. They found that because of th e full-pressure principle a work extraction of 78% in relation to a complete expa nsion is obtained. Ertesv ag (2002) described the geometry and thermodynamics of a patented ro tary-piston machine that can be used as a compressor-expander device invented by K. Vadi ng. He mentioned the possible use of this concept in vapor compression systems.

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44 Gas Refrigeration Cycles and Cryogenics The investigation of the use of expanders in gas refriger ation and cryogenic liquefaction cycles has been investigated by Gnutek (1979) Gnutek and Kalinowsk i (1986), and Baron and Trembley (2003). Gnutek (1979) analytically investigated the use of a ro tary-vane expander in cryogenic liquefaction and vapor-compression gas refrigeration systems. The van der Waals equation of state was used to determine th e thermodynamic properties of the gas and therefore determine the performance of this expander de vice. The effects of friction, in ternal leakage losses and heat transfer from the ambient have been taken into consideration. Gnutek and Kalinowski (1986) described the use of a two-stage rotary-vane compressor and a single-stage rotary-vane e xpander in a gas refrigeration un it using air as the working fluid. The authors obtained a oC evaporating temperature. Analy tical equations are presented to describe the performance of the expander. The au thors conclude that a ga s refrigeration system employing a rotary-vane expander and compressor w ould be significantly smaller in size than that utilizing a reciprocating piston compressor and expander. The work extracted from the expander was coupled to a generator. Baron and Trembley (2003) investigated the us e of indirect heat exchange and a turboexpander in a cryogenic refrigerator using liquid nitrogen as the re frigerant. According to the authors, heat transfer to the boi ling liquid nitrogen is critical in cryogenic applications in order to ensure that the latent heat of vaporization is maintained as high as possible. The authors theoretically found that an expa nder whose isentropic efficiency is 75% can increase the amount of heat transferred to the liqui d nitrogen by 28%. The authors di scussed the pros and cons of using scroll and rotary-v ane type expanders. The authors se lected a centrifugal turbo-expander

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45 because of compactness and cost. The authors estimated that a turbo-expander should not exceed $10,000 in order to be economically viable. Expander Selection Although the concept of utilizing a compresso r-expander device is appealing, it cannot be applied to vapor compression and heat pump systems for the following reasons: The necessary built-in volume ratios of the compressor and expander are very different. The expander may have a volume-ratio 7-8 times that of the compressor. There is an issue pertaining to the unequal inle t volume flow-rates, which are a direct result of unequal inlet densities. Cavitation erosion and other complexities ma y arise from two-phase expansion which is likely to affect the expansion efficiency and may also have a negative impact on the compression process and therefore greatly reduce the system COP. Baek et al. (2002) suggested that rotary-vane expanders ha ve an exceptional potential being utilized as expansion devices in refrigeration systems. This may be the case because of the fact that they may be mounted on the compressor shaft for ease of work recovery and have relatively less complicated va lve timing. Rotary-vane expand ers are also capable of accommodating higher rotational speeds. This transl ates into higher fluid handling capacity and hence a more compact device. Together with th e conclusions mentioned above by the different investigators, the potential of employing rota ry-vane type expanders in conventional vapor compression systems/heat pumps is very promising.

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46 CHAPTER 3 PARAMETRIC ANALYSIS OF CYCLES Models for the following refrigeration cy cles have been developed in FORTRAN90 utilizing REFPROP 7.0s R-134a, R-22, R-12 a nd pure hydrocarbon property subroutines: Ideal single-stage cycle (henceforward referred to as the base cycle) A modified ideal single-stage cycle with an expansion device (the expansion device will henceforward referred to as ED) Actual single-stage cycle with and without an ED Ideal cycle with two-stage compression and an economizer (henceforward referred to as the economizer cycle) A modified ideal economizer cycle with an ED Actual economizer with and without an ED Ideal cycle with an internal heat exchanger (henceforward referred to as the IHX cycle) A modified ideal IHX cycle with an ED Both the evaporating and condensing temperatures were varied for all 8 cases. The system COP, refrigerating efficiency, refrigerating capac ity, and required input work were found to be greatly dependent on the operating c onditions. It is of significant in terest to calculate the percent differences (system COP, refrigerating capacity a nd required input work) for the cycles with and without expansion devices. Ideal Cycles This section will describe both the therm odynamic processes involved with the base, economizer and IHX refrigeration cycles and the results when these cycles are modified by the addition of a two-phase expander as a throttle valve replacement.

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47 Base Cycle The single stage ideal vapor compression cycle is the simplest cycle and is the basis of most refrigeration and air-conditioning technology. Th ese cycles have been utilized in cascade refrigeration systems to achieve higher c ondensing or lower evaporating temperatures. A schematic of the base cycle is shown in Figure 3-1. Both the T-s and P-h diagrams are shown in Figures 3-2. The base cycle consists of four processes that take place in the four components shown in Figure 3-1. Both the c ondensing and evaporat ing temperatures, Tc and Te, are initially known since they are usually design variables. This allows the computation of the corresponding high and low saturation pressures, Pc and Pe, of the cycle. The refrigerant at State 1 is ideally in a saturated vapor state (x1=1). The properties at Stat e 1 can be determined from P1=Pe and x1=1. In the base cycle, the refrigerant va por undergoes isentropic compression in the compressor and exits at State 2. The propertie s at this state are found utilizing the following assumptions: P2=Pc and s2=s1. The superheated refrigerant th en enters the condenser where it undergoes isobaric heat rejection. The exit, State 3, of the condenser is saturated liquid (x3=0). The properties at State 3 can be found by P3=Pc and x3=0. The saturated liquid refrigerant is then throttled in an expansion valve. This is an isen thalpic process. The propert ies at State 4 are thus determined from h4=h3 and P4=Pe. In the case where an expansion device is utilized as a throttle valve replacement, the saturated liquid under goes an isentropic e xpansion process. The properties at State 4 are thus determined from s4=s3 and P4=Pe. The two phase refrigerant then undergoes isobaric heat addition in the evaporator. The refrigerant is ideally saturated vapor at the exit of the evaporator.

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48 The coefficient of performance for the base cycle and the base cycle with an expansion device (assuming the work output fr om the expander is ideally coupled to the compressor) are defined by Equations (3-1) and (3-2), respectively 14 21 in b compq hh COP whh (3-1) 14' ,exp exp2134'()()in b compq hh COP wwhhhh (3-2) Economizer Cycle The economizer cycle is modifi ed so that the compression and expansion processes are carried out in two stages. A liqui d/vapor separator increases the c ooling capacity of the cycle. A schematic of the T-s and P-h diagrams and cycle is shown in Figures 3-3 and 3-4, respectively. Both the condensing and evaporating temperatures, Tc and Te, are initially known since they are usually design variable s. This allows the computation of the corresponding high and low saturation pressures, Pc and Pe, of the cycle. The refrigerant at the evaporator exit, State 1, is saturated vapor. The properties at State 1 can be determined by x1=1 and P1=Pe. The saturated vapor then undergoes isentropic compression in the compressor. The properties at the 1st stage compressor exit, State 2, can be found by s2=s1 and P2=Pi, where Pi is the intermediate pressure of the cycle. Stoecker (1998) ha s stated that the optimum interm ediate pressure in a two-stage compression process can be determin ed by the following relationship icePPP (3-3) The refrigerant at State 5, the condenser outlet, is saturated liquid. The properties at State 5 can thus be found from P5=Pc and x5=0. The refrigerant is th en either expanded by an isenthalpic process (h6=h5, P6=Pi) in the throttle valve or by an isentropic process in the expansion device (s6=s5, P6=Pi). The liquid (noted by f) a nd vapor (noted by g) are then

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49 separated. The vapor in the amount of x is fed in to the inlet of the second compressor. The liquid in the amount of (1-x) however is expanded again. The liquid refrigerant, at State 6f, is either expanded by an isenthalpic process (h7=h6f, P7=Pe) in the throttle valve or by an isentropic process in the expansion device (s7=s6f, P7=Pe). The two-phase refr igerant then undergoes isobaric heat addition in the evaporator and is sa turated vapor at State 1. State 3 remains to be determined. An energy balance of the two str eams feeding into State 3 will determine the enthalpy at State 3 for the cases of isenthalpic and isentropic expansion processes as described by equations (3-4) and (3-5) respectively. 36662(1)ghxhxh (3-4) 36'6'6'2(1)ghxhxh (3-5) The properties at State 3 can then be determined from h3 and P3=Pi. The refrigerant at State 3 then undergoes isentropic compression in the second compre ssor. The properties at State 4 can then be determined from s4=s3 and P4=Pc. The coefficient of performance for the economi zer cycle and the economizer cycle with an expansion device (assuming the wo rk output from the expander is ideally coupled to the compressor) are defined by equations (3-6) and (3-7), respectively 617 ,1,262143(1)() (1)()()in econ compcompqxhh COP wwxhhhh (3-6) ,exp ,1,2exp,1exp,2 6'17' ,exp 6'214356'6'6'7'(1)() (1)()()()(1)()in econ compcomp econ fq COP wwww xhh COP x hhhhhhxhh (3-7)

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50Internal Heat Exchanger (IHX) Cycle The internal heat exchanger (IHX) cycle is also known as the liq uid-to-suction heat exchanger (LSHX) cycle. This cycle incorporat es a heat exchanger that subcools the saturated liquid leaving the condenser a nd superheats the saturated vapo r leaving the evaporator. A schematic of the IHX is shown in Figure 3-5. Bo th the T-s and P-h diagrams are also shown in Figures 3-6 respectively. Both the condensing and evaporating temperatures, Tc and Te, are initially known since they are usually design variable s. This allows the computation of the corresponding high and low saturation pressures, Pc and Pe, of the cycle. The analysis of the IHX cycle begins with the determination of States 3 and 6. The refrigerant is saturated vapor at the exit of the evaporator. The state properties may be determined from x6=1 and P6=Pe. Upon the refrigerant exiting the condenser, State 3, it is saturated liquid. The properties at this state point may be determined by x3=0 and P3=Pc. The effectiveness of the internal heat exchanger is defined as follows 16 max36 act IHXqTT qTT (3-8) For the case of the ideal cycle, the effectiveness is assumed to be equal to unity. This allows for the calculation of the temperature at State 1 (T1=T3). Using T1 and P1=Pe the other state properties may be determined. State 4 ma y be determined by performing an energy balance on the IHX, assuming the mass flow of refrigerant and the specific heat of the refrigerant are both constant, as follows 34611634qqTTTT (3-9) In the case of effectiveness equal to unity, T4=T6, the properties of the refrigerant at State 4 can be determined from T4 and P4=Pc. The superheated vapor at State 1 undergoes isentropic

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51 compression in the compressor and exits at State 2. The properties at State 2 can be determined by s2=s1 and P2=Pc. The subcooled liquid at State 4 can either be put through isenthalpic expansion in the expansion valve (h5=h4, P5=Pe) or isentropic expansion in the expansion device (s5=s4, P5=P4). The two-phase refrigerant then undergoes isobaric heat addition in the evaporator and exits as sa turated vapor at State 6. The coefficient of performance for the IHX cycle and the IHX cycle with an expansion device (assuming the work output from the expander to be ideally coupled to the compressor) are defined by equations (3-10) and (3-11) respectively 65 21 in IHX compqhh COP whh (3-10) 65' ,exp exp2145'()()in IHX compqhh COP wwhhhh (3-11) For the base cycle configurations (with a nd without an ED) at a constant condenser temperature (25oC), the percent increase realized in th e system COP was 8% to 19%, while an increase of 0.5% to 3.4% in refrigerating capac ity was realized. The evaporator temperature was varied from -12oC to 10oC. The required work input to th e compressor was found to decrease by 7.5% to 16%. For higher condenser temperatur es, the percent differences in the system COP were found to increase by 10-21%, 13-24%, 15-27 % and 18-30% for condensing temperatures of 30oC, 35oC, 40oC and 45oC, respectively. Actual Cycles The primary models: (1) simple vapor comp ression cycle; (2) vapor compression cycle with an economizer; and (3) vapor compression cycle with an internal heat exchanger, have been further modified to include the isentropic effi ciencies of both the compressor and expansion devices. The effects of superheating, sub-cooling and pressure drops (both in the line and in the

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52 heat exchangers) as well as the effectiveness of the heat exchangers have also been considered. These effects would help achieve a more practical system model. For the actual base cycle system configurati ons (with and without an ED) and the case of constant condensing temperature (25oC) and varying evaporating temperature (-12oC to 10oC), the percent increase realized of th e system COP was 3.4% to 8.9%, while an increase of 0.2% to 1.7% was realized in terms of the refrigerat ing capacity. The required work input to the compressor was found to decrease by 3.3% to 7. 2%. For a compressor efficiency of 85%, an expander efficiency of 50% and internal heat exchanger effectiveness of 70%, the values aforementioned are approximately half the values attained in the ideal case. The results for both the ideal and actual cycle improvements are presen ted in Table 3-1 and in Figures 3-7 to 3-13. It can be concluded from the preliminary re sults obtained that the maximum potential gain in the cycle coefficient of perf ormance is obtained for the base cycle. This conclusion supports the work of Robinson and Groll (1998a) who f ound that the stream availability following the heat rejection process in the conde nser is better utili zed by an expansion device rather than an internal heat exchanger. Alt hough improvements may be realized in the economizer cycle, the complexity and capitol cost of a two-stage va por-compression system hi nder the utilization of this cycle with an expansion device. The following additional parametric inputs have been varied in order to determine the most important parameters when selecting the expa nsion device as well as the optimum operating conditions. Effect of Subcooling on System Performance The degree of subcooling that th e refrigerant undergoes in or after the condenser is a key factor that affects both the power requirem ent and the refrigerating capacity of a vapor compression refrigeration cycl e. Analysis of the effect of the degree of subcooling on the key

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53 system performance parameters including the re frigerant effect, expander work-output and overall system COP has been done. An increas e in the degree of subcooling was found to decrease the useful work output from the e xpander but increases the overall system COP by increasing the refrigerating e ffect. Figures 3-14 to 3-18 show the aforementioned trends. Among the most promising preliminary conclusions that may be drawn from this analysis is that cavitation erosion may be greatly reduced in the inlet of the expander since, depending on the degree of subcooling, the refrigerant enteri ng the expander may only be liquid. Hence, vapor will only start to form when the refrigerant ha s experienced a sizeable pressure drop in the expanders cavity (i.e. at large angular displacements). According to current literature, cavitation erosion in turbomachinery is prevalent at the inlet and the initial stages of compression and expansion due to the high velocity and com position associated with the incoming two-phase flow. Effect of Superheating on System Performance The degree of superheat that the working fluid ex periences in the evaporator is critical in a vapor compression refrigeration syst em to ensure that the working fluid leaving the evaporator is completely vapor. Besides the well documented benefits of better compressor performance and longer service life, the following figures (Figures 3-19 and 3-20) analyze whether the degree of superheat has any significant effect on the system COP, refrigerating effect and the net work input to the system. A similar parametric analysis, as in the cas e of condenser subcooli ng, reveals that the increase in the degree of superheat in the evap orator does not significantly increase/decrease the work input into the system nor increase the re frigerating effect produced by the system. For a large degree of superheat such as 9oC, the gain in the refrigerating effect is roughly 5%. These

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54 results are presented in Figures 3-19 and 3-20. There exists however a trade-off between the degree of superheat and the heat exchanger area required in order to ach ieve that superheat. Evaluation of CFC, HFC and Transcritical CO2 Refrigeration Cycles Vapor compression refrigeration research has been significantly influenced by the elimination of ozone-depleting refrigerants su ch as chlorofluorocarbons (CFCs), the planned phase out of partially ozone-depleting refrigeran ts such as hydrochlor ofluorocarbons (HCFCs), and the interest in replacing what was once t hought to be harmless refrigerants such as hydrofluorocarbons (HFCs). Among the frontrunner s in replacing these refrigerants is carbon dioxide. A first and second-law based performan ce evaluation of conventional refrigeration and transcritical carbon dioxid e systems is presented here. The wo rking fluids investigated include R-134a, R-22 and R-744 (CO2). The systems investigated incl ude the basic and modified vapor compression systems. The modified system consis ts of a two-phase expansion device that is employed as a throttle valve replacement. The irreversibilities of the different system components in the basic and modified system s are presented. Comp arison of the potential increases in the refrigerating effect, reductions in the work required to operate the system and resulting increase in the system coefficient of performance are discussed. Comparison of the reductions/increases in cycle irreversibility and component contributions due to the employment of an expansion device are also discussed. Key expander operational parameters, such as the process volume ratio, are investigated. A recent revival of carbon dioxides use as a re frigerant has been given much attention in recently published literature. This arises from the need to replace ozone depleting CFCs and HCFCs and the global warming contributors HFCs. In comparison, carbon dioxide is naturally occurring and is thus an environm entally friendly refrigerant with an ozone depletion potential of zero. Published literature that discussed the re vival of carbon dioxides promise as a potential

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55 candidate to replace CFCs and HFCs include s those of Lorentzen (1994), Lorentzen and Pettersen (1993). Due to the lo w critical temperature of CO2 however, 30.98oC, the heat rejection process typically takes place at a supercritical temperature in a ga s cooler cooled by ambient air. The critical pressure of CO2 is 7.38 MPa. This high heat rejec tion pressure requires a substantial increase in the compressor work required to achie ve it. This dramatically decreases the system coefficient of performance. Another inherent th eoretical limitation of the basic transcritical CO2 system is the fact that the refrigerant leaving the gas cooler may only be cooled to the ambient temperature. To overcome this, an internal heat exchanger may be utilized to provide a means of subcooling. In the case of a conventional va por-compression system, subcooling is typically obtained by increasing the area of h eat transfer of the condenser. It has been repo rted that the performance of transcritical CO2 systems suggests that the technol ogy is comparable to that of existing systems (Robinson and Groll, 1998). It ha s also been suggested that transcritical CO2 component technology must be optimized in orde r to meet the unique ope rational characteristics of carbon dioxide. This technology however differs significantly from conv entional refrigeration systems (Kim et al. (2004)). A lthough the state-of-the-a rt of transcritical CO2 systems is still far from optimum, the inherent thermodynamic performance of the basic transcritical CO2 system must be comparable to existing refrigeration t echnology. The system coefficient of performance, regardless of the technology used, may be incr eased by the utilization of expansion devices, liquid-to-suction (internal) heat exchangers or performing compression in multiple stages, to name a few. In particular, many researchers have concluded that th e isenthalpic throttling process may be the largest source of irreversibility in a transcritical CO2 system. This section analyzes and compares the use of an expansion device as a throttle valve replacement in both conven tional and transcritical CO2 systems. There are two advantages to

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56 using expansion devices with output work. The first is the fact th at lower enthalpy refrigerant is obtained at the inlet of the eva porator, which increases the refrig erating effect. The second is the extra work that can be extracted from the expansi on process, which can then be used to lower the input power requirement of the compressor. The degree of potential increase of the systems coefficient of performance however is greatly dependent on the isentropi c efficiency of the expansion device employed. Different expansio n devices have been suggested in recently published literature. Reduction of irreversibilitie s associated with the expansion process also leads to an increase in the second-law efficiency of the system. The extent of these improvements in both conventional systems u tilizing R-22 and R-134a as refrigerants and transcritical CO2 systems are presented and discussed. An abundant amount of literature dealing with modified refrigeration systems that employ expansion devices exists. For the purpose of this section, only the most pertinent of those studies are examined. Robinson and Groll (1998) analytically investigated and compared the performance of a transcritical CO2 system and a conventional R-22 refrigeration system with and without an expansion turbine. Th ey found that the use of an inte rnal heat exchanger along with an expansion device decreased the COP of the transcritical CO2 system by 6-8%. They concluded that the stream availa bility following the heat rejection process in the condenser is better utilized by an expansion devi ce rather than an internal h eat exchanger. They also found a 23% COP increase when comparing an R-22 system and a transcritical CO2 system both with expansion devices with an isentropic efficiency of 60% and when the system is operated at evaporating and condensing temperatures of 5 and 50oC, respectively. They concluded that a conventional R-22 system employing an expans ion turbine outperforms all other systems including the transcritical CO2 system with an internal heat ex changer and an expansion turbine.

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57 Preissner (2001) investigated the use of a sc roll expander in a transcritical CO2 system. He also performed an experimental comparison of conventional R-134a and transcritical CO2 refrigeration systems employing suction-line heat exchangers. He found that the CO2 systems COP is 8 and 23% less than the R-134a system fo r a range of typical ope rating conditions with particular emphasis on medium-to-high outdoor te mperatures. With the implementation of a scroll expander the systems were found to have an identical co efficient of performance at a condensing temperature of 25oC. However at higher condens ing temperatures a significant performance gap existed in favor of the R-134a system. Preissner (2001) found internal leakage losses to be the most significant loss mechanism due to limitations in machining accuracy and geometric design. The expander was thus not implemented in either system. Component-level models are needed for the an alysis of conventiona l and transcritical CO2 systems with and without an expansion device in place of a throttling valve. A general solution method has already been presented. Detailed component-level models that may include geometry, flow arrangement and loss mechanisms for the compressor, condenser, evaporator and expander may be developed separately and incor porated into this model when required. These component-level models will provide better estimat es of the isentropic efficiencies of the compressor and expander as well as the heat exch anger size and effectiveness of the condenser and evaporator, respectively. In the analysis describe d here, R-22, R-134a and CO2 are assumed to be the working fluids. The thermodynamic properties of these fluids are provided by means of coupling REFPROP 7.0s (Lemmon, 2002) property subroutin es with the models developed for this analysis. Pressure losses in all system components and throughout the systems piping have been

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58 neglected. Figure 3-1 shows a schematic of a conventional and transcritical CO2 refrigeration system with and without an expansion device. The two systems utilize a compressor, an expansion device and an evaporator. The only component that differs is the device by which heat is rejected at the high pressure of the system. This component is a condenser in a conventional system and a gas cooler in the transcritical CO2 system. Figure 3-21 shows the T-s diagram of each system, respectively. Regardless of the refrigeration technology used, th e required input work to co mpress saturated vapor at the evaporating pressure to superheated vapor at the pressure of heat rejection is 2,1()s c chh w (3-12) The isentropic efficiency c of the compressor takes into account irreversibilities such as internal leakage and friction. Kim et al. ( 2004) reported that although leakage and frictional losses associated with carbon dioxi de compressors are lower, the current state-of-the-art carbon dioxide compressor doesnt perform as well as a conventional refrigeratio n compressor. For the sake of comparison, isentropic efficiencies of current state-of-the-art R-22 and R-134a compressors will be used to over-estimate the performance of current CO2 compressor technology. This provides a tec hnology-independent evaluation of the performance of the two systems. Selection of the supercritical heat rejection pressure is another vital operational parameter that significantly influences the opera tion and performance of a transcritical CO2 system. Pressure is independent of te mperature in the supercritical regime. According to numerous studies there exists a supercritical pressure at which the systems coefficient of performance is maximum. Both graphical and numerical simulati ons have been proposed to select this heat rejection pressure (Kauf, 1999). For the purpose of this st udy, the heat rejection pressure is

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59 incrementally varied from the critical pressure (7.38 MPa) to 15 MPa until the optimum pressure at which heat rejection takes pl ace is determined for various op erating conditions. This procedure is performed for both basic and modified transcritical CO2 systems. In the case of a conventional refrigera tion system, the refrigerant undergoes desuperheating, condensation and subcooling in th e condenser. The effects of superheating and subcooling are considered when determining the heat exchanger effectiveness and the logarithmic mean-temperature difference (LMTD). In the case of a transcritical CO2 system a gas cooler is employed to reject he at to the ambient and operates at a transcritical temperature and pressure. In an ideal gas cooler, the refr igerant temperature leaving the cooler, Tgc,out, would match the ambient temperature. The refrigerant en ters the cooler as a vapor-like substance and leaves as a liquid-like substance. Followi ng the conclusion of Robinson and Groll (1998) regarding the stream availability of the refr igerant leaving the cool er, no subcooling of the refrigerant is assumed. This allows for the maximum extraction of work during the expansion process. The heat rejected in the condenser or ga s cooler can be calculated as follows /2,3 condgcaqhh (3-13) where h2,a is the actual enthalpy of the superheated/s upercritical vapor calculated by means of the compressor isentropic efficiency. The enthalpy h3 is the enthalpy of the refrigerant leaving the heat-rejecting device. For the purpo se of this study the refrigerant in the transcritical cycle is assumed to leave the gas cooler at a tran scritical pressure and temperature of T3=Tgc,out=Tamb+5. In the analysis of the conventional system the refrigerant is assumed not to experience any subcooling. The state properties thus correspond to the enthalpy of the saturated liquid at the

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60 condenser saturation pressure. In this study the difference between the ambient temperature and the condensing temperature in the conve ntional system is assumed to be 15oC. In conventional and transcritical CO2 refrigeration systems a throttling/expansion valve is utilized to expand the refrigerant from the pressure of heat rejection to the evaporating pressure. This irreversible isenthalpic process may be replaced by the us e of an expansion device that would ideally expand the refrigerant in an isen tropic fashion. The potential increase in the system COP due to both the increase in the refrig erating effect and decrease in the cycle net work input have been studied ex tensively in recently published li terature for systems that utilize conventional refrigerants and transcritical CO2 as working fluids. Different positive displacement type expansion devices such as helical-screw, scroll, rotary-vane and re ciprocating piston have been investigated. In this analysis, the type of expansion devi ce used is not specified. The type of device along with the geometrical and operational char acteristics may be programmed in separate subroutines that may provide actual performan ce predictions and an accurate estimate of the expanders isentropic efficiency. Depending on th e inlet conditions to the expander the actual work output from the expander can be expressed as 4,' 3()aeewhh (3-14) The use of expander work output can either be coupled to the compre ssor shaft or utilized to drive other auxiliary machinery or systems that may contribute other system improvements. This study considers only the al ternative of increased coolin g capacity brought on from near isentropic expansion and ideal coupling of the expander work to the compressor. The size of an expansion device is primarily a function of the built-in volume ratio of the expander. The expander should not only match (or be fairly similar to) the process expansion

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61 ratio for typical system operating conditions but should also minimize the amount of wasted power recovery due to pressure losses incu rred during under-expansi on. The process volume ratio is defined as the ratio of the specific volume of the refrigerant at the inlet of the evaporator to the specific volume at the outlet of the conde nser/gas cooler. The expansion process in positive displacement expansion devices is comprised of the intake, expansion and the exhaust processes. The built-in volume ratio of an expande r is defined as the ratio of the specific volume at the start of the exhaust process to the specific vo lume at the end of the intake process. In this study it is assumed that both the built-in and proc ess volume ratios are identical and defined as follows 3 evapin vv r v (3-15) where vevap,in is the specific volume of the refrigerant after the expansion process (or at the inlet to the evaporator). Depending on the cycle analy zed, this state may correspond to an isenthalpic or non-isentropic expansion process. We will further assume that the degree of s uperheat the refrigerant experiences in the evaporator is assumed to be zero unless othe rwise indicated. For a given set of operating conditions the degree of superheat has much more of an influence on the overall performance of the system. The refrigerant is thus assumed to be saturated vapor upon exiting the evaporator unless otherwise indicated. An energy balance of the evaporator yields 4,' 1()aevapqhh (3-16) where h4,a is the actual enthalpy of the refrigerant entering the evaporator determined by means of the isentropic efficiency of the expande r. In the case of an unmodified cycle h4,a would be replaced by h4h=h3.

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62 The coefficient of performance is defined as a ratio of the desired output to the required input. In the case of a modified vapor compressi on system with an expansion device, the net work input (compressor work less expander work ) is the required inpu t to the system. For simplicity, the work extracted from the expande r is assumed to be ideally coupled to the compressor. The COP is thus defined as e c evapw w q COP (3-17) A second-law based comparison of conventional and transcritical CO2 refrigeration systems is a far better comparison criterion. Exer gy or availability analys es reveal how closely actual refrigeration cycles are to ideal cycles. An exergy analysis also reveals the largest source of irreversibilities in the system by singling out th ose components that are most irreversible. The availability of a flowing fluid stream neglecting both kinetic and potential energy is defined as 000hhTss (3-18) The subscript 0 denotes a reference state at whic h the systems temperature and pressure are in equilibrium with its surroundings (T0=303.15 K ). This state is said to have zero availability if all other energy forms relative to the surroundings ar e zero. The specific irreversibility of a steadyflow device with heat transfer may be expressed as the difference between the reversible work and the actual work as follows 01revactinexitract rT I wwqw T (3-19) where qr and Tr are the specific heat transfer and the reservoir temperature at which heat is transferred to/from, respectively. The sign of qr depends on the direction of heat transfer. Utilizing Equations (3-18) and (3-19), the specific irreversibilities of the various components can be derived. The specific irrevers ibility of an adiabatic compressor may be written as follows

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63 12,12,2,1 caaoa I hhTss (3-20) In the case of a conventional system, the sp ecific irreversibility of the condenser is 2,3032,32,1()()oo condacondaa rrTT IqTsshh TT (3-21) whereas, for a gas cooler, Tr in Equation (3-21) is replaced by the ambient temperature, Tamb. If the ambient temperature is identical to the refe rence temperature this e xpression would reduce to the difference in availability of the entering and leaving streams. If the expansion process is performed usi ng a throttling/expansion valve the specific irreversibility of that isenthalpic process is 4,3 voh I Tss (3-22) whereas in the case of utilizing an expansi on device, the specific irreversibility is ''' exp34,34,04,3 aaa I hhTss (3-23) The specific irreversibility of the evaporator for both systems is 4114411()oo evapevapo rrTT IqTsshh TT (3-24) A systems total irreversibility is the sum of the irreversibilities associated with its four components. Irreversibilities resulting from pre ssure drops in the systems components and transfer lines have been neglecte d. Irreversibilities asso ciated with heat tran sfer in the systems piping have also been neglected. Both first and second-law based evaluati ons of R-22, R-134a and transcritical CO2 refrigeration technology are critic al for selecting appropriate technology suitable for a specific application. For this purpose, simulation m odels for basic and modified vapor-compression refrigeration cycles have been developed.

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64 The selection of the optimum heat rejection pressure is necessary to adequately compare transcritical CO2 systems to conventional systems. Ma ximum system COP occurs at the gas coolers optimum heat rejection pressure. Figure 3-22 illustrates the variat ion of the system COP as a function of the heat rejection pressure fo r both basic and modified systems at evaporating temperatures of 268 and 273 K. The COP of the basi c system can be seen as a near constant for heat rejection pressures greater th an the optimum heat rejection pressure. This is not the case for the modified system since higher heat rejection pr essures adversely affect the performance of the expansion device. For an evaporating temperat ure of 273K, it can be seen that the COP is maximum when the heat rejection pressure is 10.2 and 9.75 MPa for the basic and modified systems, respectively. For an evaporating te mperature of 268K the optimum heat rejection pressures were found to be 10.35 and 9.75 MPa for the basic and modified systems, respectively. The optimum heat rejection pressure can be de termined in a similar fashion for different evaporating temperatures and ope rating conditions. For all operat ing conditions investigated, the optimum heat rejection pressure has been assu med to be 10.2 MPa. A lthough the actual optimum heat rejection pressure may vary by as much as 5% over the entire range of evaporating temperatures investigated, the system COP deviat es by as much as 1 and 1.7% for the basic and modified systems, respectively. For the sake of computational time this deviation may be assumed negligible. Irreversibilities associated with the heat rejection and expansion processes are the largest sources of irreversibility in basic transcritical CO2 systems. Figure 3-23 il lustrates the variation of the percent of cycle irreversibility associated with the heat rejecti on and expansion processes as a function of the heat rejection pressure for basic and modified CO2 systems. For these operating conditions, the compressor and evaporator contribute approximately 16-17 and 5-6%

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65 of the basic cycles total irreversibility, respec tively. The compressor and evaporator contribute approximately 22-27 and 9-10% of the modified syst ems total irreversibility, respectively. It can also be seen in Figure 3-23 that the throttle valve contributes 40 to 10% more to the systems total irreversibility when compared to the gas co oler at heat rejection pressures of 9 and 11.25 MPa, respectively. In contrast the contribution of irreversibility from the gas cooler is approximately 2% less and 25% more at heat rejection pressures of 9 and 11.25 MPa, respectively. The average irreversibilities of th e expansion and heat reje ction processes are 24% lower and 12.5% higher when comparing th e modified and basic transcritical CO2 systems, respectively. Component irreversibilities corresponding to an optimum heat rejection pressure are unlikely to be minimum. From Figures 3-22 a nd 3-23 it can be seen that the optimum heat rejection pressure corresponds to a maximum system coefficient of performance and a median value of the system irreversibility. A first-law based analysis of R-22, R-134a and transcritical CO2 systems is required to evaluate the thermodynamic performance of ba sic and modified refrigeration systems. Employing a two-phase expansion device as a thro ttle valve replacement is the only modification made to the basic system. This allows for an unbiased comparison of the different refrigeration technologies. Figure 3-24 shows the variation of the coeffici ent of performance of basic and modified R22, R-134a and transcritical CO2 systems as a function of the evaporating temperature. It can be seen that the COP of the cycles investigated in creases with increased evaporating temperature. The values of system COP for the basic and modified R-22 systems are higher than those of R134 systems by as much as 6 and 0.6%, respecti vely. This maximum difference corresponds to

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66 an evaporating temperature of oC. For the range of evaporati ng temperatures investigated, it can be seen that the COP of the basic transcritical CO2 system is lower than that of R-22 by 3334% and that of R-134a by 29-34%. On the other hand, the COP of a modified transcritical CO2 system is lower than that of an R-22 system by 17.5-22% and that of an R-134a by 18-21%. The COP of a modified transcritical CO2 system is still 4.2-9% less than that of a basic R-22 system. The COP of a modified transcritical CO2 system is 2% higher and identical to that of a basic R134a system operating at evaporating temperatures of and oC, respectively. At higher evaporating temperatures, however the COP of a transcritical CO2 system is as much as 9% lower than that of a basic R-134a system. When compared to similar R-22 and R-134a technology, the performance improvements of a modified transcritical CO2 system, in comparison to the basic system, are greater. Figures 3-25, 3-26 and 3-27, may further support this result. Fi gure 3-25 shows the difference in COP between the basic and modified R-22, R-134 and transcritical CO2 systems as a function of the evaporating temperature, respectively. It can be seen that the difference in COP decreases with increased evaporating temperature for all three syst ems. The decrease is not as rapid however for the transcritical CO2 system. As the evaporating temper ature decreases, the modified CO2, R134a and R-22 systems outperform the basic systems by 27-30%, 10-23% and 9-18%, respectively. This increase in the COP can be attri buted to an increase in the refrigerating effect and a decrease in the input work required to operate the system. The refrigerating effect is defined as the spec ific evaporator capacity. From Figure 3-26, it can be seen that as the evaporating temperature decreases, the differences in the refrigerating effect of the modified and basic systems incr ease from 5-14%, 1.2-7% and 1.3-9% for the CO2, R134a and R-22 systems, respectively. From Figure 3-27, the decrease of the required input to

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67 the system can be seen to be dramatically di fferent. The work extracted from the expansion process is ideally utilized in the reduction of the input wo rk to the compressor. It can be seen that as the evaporating temperature increases, the di fference in required input work to operate a transcritical CO2 system increases. For increasing eva porating temperatures, the reduction in input work can be seen to be 22-31%, 14.5-9% and 18.5-10% for transcritical CO2, R-22 and R134a systems, respectively. When comparing th e basic and modified systems, the sizable differences in the increase in refrigerating effect and reduction of input work of CO2 point to favorable thermodynamic characteristics of CO2 when compared to t hose of R-22 and R-134a. When dealing with modified systems, the mo st favorable operational characteristic of CO2 is the volume ratio of the expander. This opera tional variable discussed earlier determines the size and practical feasibility of employing a tw o-phase expansion device as a throttle valve replacement. Figure 3-28 shows the variation of the expansion devices volume ratio as a function of the evaporating temperature for CO2, R-22 and R-134a. As can be seen, there is a sizable difference between the volume ratios for the different working fluids. It can also be seen that the volume ratio decreases as the evapora ting temperature increases since the specific volume at the evaporator inlet decreases as temperature increases. For the evaporating temperatures reported (5 to 20oC only), the volume ratio can be s een to decrease from 2.7-1.6, 12.3-6.2, and 20.8-10 for transcritical CO2, R-22, and R-134a cycles, respectively. Efficient expansion devices are necessary to justify the cost and performance improvement of modified refrigeration syst ems when compared to conventi onal technology. This result is evident in Figure 3-29, which shows the di fference in COP of basic and modified CO2, R-22 and R-134a systems as a function of the expander s isentropic efficiency. In the case of CO2, it can be seen that the improvement in the system CO P increases in a much more dramatic fashion

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68 when compared to R-22 and R-134a systems. The in crease in the system CO P can be seen to be 10.5-33%, 6-19% and 5-17% for CO2, R-134a and R-22 systems, respectively. A second-law based analysis is needed to de termine the irreversibil ities of the various components and to determine the increases/reducti ons in irreversibilities associated with any system modification. Figure 3-30 illustrates the component irreversibili ties of a modified R-22 system as a function of the evaporating temper ature. As can be seen, for the modified R-22 system, the largest sources of irreversibility are the condenser and compressor. Employing an expansion device has greatly reduced the irrevers ibilities associated with the basic systems throttling process. Component irreversib ilities for the basic and modified CO2, R-134a and R-22 systems at a specific operating condition are give n in Table 3-2. The table shows that the irreversibilities associated with the thrott ling process are reduced by 50, 18 and 16.5% for modified CO2, R-134a and R-22 systems, respectively. Similar figures and tables may be generated for various operating cond itions and system configurations. Figure 3-31 shows the percentage of system i rreversibility due to the condensing process for basic and modified R-22 and R-134a systems as functions of the evaporating temperature. The percentage of system irreve rsibility associated with the gas cooler for both the basic and modified transcritical CO2 systems are not included for clarity. It was found that the gas coolers contribution to the system irreversibility is fairly constant and averages 29.5 and 42% for the basic and modified transcritical CO2 systems, respectively. The figure also shows that the percentage of the total system irreversibility associated with the condensing process increases with evaporating temperature a nd with the use of an expansio n device for both R-22 and R-134a systems. This increase in irreversibility decreas es with increased evaporating temperatures from 10-6% and 11-6% for R-134a and R-22 systems, respectively.

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69 Figure 3-32 shows the percentage of system i rreversibility associated with the expansion process for basic and modified CO2, R-134a and R-22 systems. As can be seen, the percentage irreversibility due to expansion is on aver age 47.5 and 24% for the basic and modified transcritical CO2 systems, respectively. An average valu e of 50% reduction in irreversible losses due to the addition of an expans ion device is expected with incr easing evaporating temperatures. Irreversibilities associated with the expansion pr ocess in R-134a and R-22 systems decrease with increased evaporating temperature and ma y be reduced from 24-11% and 22-10.5%, respectively. The ratio of the total system irreversibility to the heat rejection rate yields a dimensionless system irreversibility that may be used to evalua te and compare the total system irreversibilities of transcritical CO2, R-22 and R-134a systems. The heat re jection rate has been used to nondimensionalize the total system irreversibility since the only difference between conventional and transcritical refrigeration technology is the heat rejection device. Figure 3-33 shows the variation of this dimensionless quantity as a function of evaporating temperature for basic and modified CO2, R-22 and R-134a systems. As can be seen the basic and modi fied transcritical CO2 systems have the largest dimensionless cycl e irreversibilities associated with them. Modified R-22 and R-134a systems, on the ot her hand, have the least. The basic CO2 system and the basic R-22 and R-134a systems have fairly close dimensionless cycle irreversibilities. Theoretical simulation models have been deve loped to analyze and evaluate the first and second-law based performance of basic and modified R-22, R-134a and transcritical CO2 vapor compression refrigeration systems. The use of a two-phase expansion devi ce as a throttle valve replacement is the only modification made to the basic system. This allows for an unbiased comparison of the different refrigeration technologies. Th e following conclusions can be made:

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70 The optimum heat rejection pressure corres ponds to a maximum system coefficient of performance and a median value of system irreversibility. For the operating conditions investigated, the optimum heat rejection pressure of a modifi ed system is on average 5% lower than that corresponding to a basic transcritical CO2 system. The COP of basic and modified transcritical CO2 technology is inferior when compared to basic and modified R-22 and R-134a systems. The COP of basic and m odified transcritical CO2 systems are 29 to 34% and 17.5 to 22% lower than R-22 and R-134a systems, respectively. Modified transcritical CO2 systems can compete with basic R-22 and R-134a technology at lower evaporating temperatures but perform about 9% less at highe r temperatures. Favorable thermodynamic characteristics of CO2 are apparent when comparing the relative increases in COP and refrigera ting effect and the reduction in input work to R-22 and R134a. For the range of evaporating temperatur es investigated, the improvement in the CO2 system COP is 12 to 18% when compared to R-22 and R-134a systems. An increase of 3.8 to 7% in refrigerating effect and a 7.5 to 22% decrease in input work were also realized when compared to R-22 and R-134a systems. The most favorable opera tional characteristic of CO2 is the volume ratio of the expander needed to expand the refrigerant to the evaporating pressure. At an evaporating temperature of 5oC the volume ratio of an R-22 and R134a expander are 4.5 and 7.7 times that of a CO2 expander, respectively. Efficient expansion devices are necessary to justify the cost and performance improvement of modified refrigeration sy stems when compared to conventional technology. When compared to inefficient expansion devices (exp=25%), efficient expansion devices (exp=85%) may increase the system COP by 33%, 19% and 17% for CO2, R-134a and R22 systems, respectively. The throttling process is the largest source of irreversibility in basic transcritical CO2 systems and the second largest in basic R-22 and R-134a systems. Irreversibilities may be reduced by as much as 50, 24 and 22% for modified CO2, R-134a and R-22 systems, respectively, employing a 65% efficient expansion device. The basic and modified transcritical CO2 systems have the largest total irreversibilities associated with them when compared to R-22 and R-134a systems. Performance and Size Optimization of Compression Refrigeration Systems A parametric analysis of key system operati onal variables in a modified vapor compression refrigeration system is presented in this section. A highly non-linear relation ship is found to exist

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71 between the many factors that influence the optimization of the systems coefficient of performance and the associated size a nd weight of a refrigeration system. Of these factors, the degree of subcooling that the refrigerant experi ences in the condenser is the most influential. It is the degree of subcooling that determines the necessary area of the heat exchanger required, ultimately affecting the size and the weight of a refrigeration system. The degree of subcooling also influences the size of the expander re quired to expand the refrigerant from the condenser pressure to the evaporator pressure. As the degree of subcooling increases, the resultant ratio between the outlet and inlet specific volumes decreases, and hence the built-in volume ratio of the expander would decrease. A decr ease in the latter causes a reduction in the expander size and weight. As the degree of subcooling increases, the us eful power output of th e expander (which may be coupled to the compressor) decreases. On th e other hand, this cause s the enthalpy of the refrigerant entering the evaporator to decrease. This causes an increase in the refrigerating effect of the system. The extent of the influence of th ese competing effects on the systems coefficient of performance needed further analysis. This was done using a simulation code developed for that purpose. An optimum degree of subcooling exists wher e the coefficient of performance of the system is a maximum and the size and weight of th e system are a minimum. It is recognized that the performance and compactness criteria may not be optimized simultaneously by a single subcooling choice. Furthermore, the penalties asso ciated with the increase of the heat exchanger area and the decrease of power out put of the expander should be mi nimized, while the benefits of the increasing refrigerating effect and the decr easing volumetric ratio of the expander should be maximized.

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72 The quantification of the dependencies of the co efficient of performance, size, and weight of a vapor-compression system on design choice s such as the degree of subcooling are now presented. Component level models are needed for va por-compression systems that utilize an expander in place of the conventio nal throttling valve. A general solution method may be applied to model and validate experimental data and/or data provided by the manufacturers. The method presented is also applicable to both simple and modified vapor compression systems. Detailed component-level models that may include geometry, flow arrangement and loss mechanisms for the compressor, condenser, evaporator and expander may be developed separately and incorporated into this model when required. Th ese component-level models will provide better estimates of the isentropic efficiencies of bot h the compressor and expander as well as the heat exchanger size and effectiveness of the condenser and evaporator. In the analysis described in this section, R-22 is assumed to be the worki ng fluid. The thermodynamic properties of R-22 are provided by means of coupling REFPROP 7.0s pr operty subroutines with the model developed for this analysis. Pressure losses in all the sy stem components as well as throughout the systems piping have been neglected. Fi gure 3-1 shows a schematic of a simple vapor-compression cycle with and without an expansion device as a throttle valve replacement. In general, the required input work to compre ss saturated vapor at th e evaporating pressure to superheated vapor at the condensing pressure is given by Equation (3-12). The refrigerant undergoes de-superheating, condensation and su bcooling in the condenser. The effects of superheating and subcooling are considered when determining the heat exchanger effectiveness and the logarithmic mean-temperature difference (LMTD). As a base case for comparing the effect of subcooling on system size and performa nce, the refrigerant l eaving the condenser is

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73 assumed to be saturated liquid (i.e. no subcoo ling in the condenser). This represents the minimum heat exchanger area required to condense the refrigerant to a saturated liquid state and allow for the completion of the refrigeration cy cle. When subcooling is accounted for, the present analysis only investigates the subcooling th at the refrigerant experi ences in the condenser resulting from an increase in th e heat exchanger area. Other subc ooling methods such as utilizing a liquid-to-suction heat exchange r or using integrated or mech anically dedicated subcooling loops are not analyzed in this study. The heat rejected in the condens er can be calculated as follows 3 2h h qa cond (3-25) where h2,a is the actual enthalpy of the superheated vapor calculated by means of the compressor isentropic efficiency. The enthalpy h3 is the enthalpy of the refrigerant leaving the condenser. In the base case this corresponds to the enthalpy of the saturated li quid at the condenser saturation pressure. In cases of subcooling this correspond s to the enthalpy of th e subcooled refrigerant determined at the saturation pr essure of the constant-tempera ture condensing process and the temperature of the subcooled refrigerant. The effectiveness of the condens er is generally a f unction of the heat exchanger geometry, construction material and surface area. The effec tiveness of condensers and evaporators however is independent of flow arrangement since the re frigerant undergoes an isothermal phase change process. The effectiveness of the condenser is thus defined as ) ( ) (, min in c in cond in c out c c condT T C T T C (3-26) In the case of a condenser, the refrigerant typi cally has a very high h eat capacity because it is undergoing phase change. The coolant, on the other hand, experiences the maximum

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74 temperature difference and ther efore its heat capacity is equal to the minimum heat capacity,min cCC In this case, the temperature of the coolant leaving the condenser, Tc,out, can be calculated as follows ) (, in c in cond cond in c out cT T T T (3-27) The relative size of the heat exchanger is calculated from the following Equation condlm condQUAT (3-28) where Tlm is the logarithmic mean temperature di fference, which is defined as follows out c in cond in c out cond out c in cond in c out cond lmT T T T T T T T T, , ,ln ) ( ) ( (3-29) Equation (3-29) is app licable to any heat exchanger flow arrangement since the refrigerant undergoes a phase change proce ss (i.e. correction factor F=1) (Incorpera 2002). In typical vapor compression refrigeration syst ems a throttling/expansion valve is utilized to expand the refrigerant from the condensing pressure to the evaporating pressure. This irreversible isenthalpic process may be replaced by the use of an expansion device that would ideally expand the refrigerant in an isentropic fa shion. The potential increase in the system COP due to both the increase in the re frigerating effect and decrease in the cycle net work input have been studied extensively in recent literature for systems that ut ilize conventional re frigerants and transcritical CO2 as working fluids. Many di fferent positive displacemen t type expansion devices such as helical-screw, scroll, rotary-vane and reciprocating piston have been investigated. Here the type of expansion device used is not specified. The type of device along with the geometrical and operational charact eristics may be programmed in separate subroutines that may

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75 provide actual performance predictions and an a ccurate estimate of the expanders isentropic efficiency. Depending on the degree of subcooling the refr igerant experiences in the condenser, the inlet conditions to the expander may be determined assuming that subcooling in the condenser occurs at a constant pressure. The actual work output from the expande r can be expressed as 4,' 3()seewhh (3-30) The size of an expansion device is primarily a function of the built-in volume ratio of the expander. The expander should not only match (or be fairly similar to) the process expansion ratio for typical system operating conditions, but should also minimize the amount of wasted power recovery due to pressure losses incu rred during under-expansi on. The process volume ratio is defined as the ratio of the specific volume of the refrigerant at the inlet of the evaporator to the specific volume at the outlet of the condenser. This volume ratio is expected to be smaller when the refrigerant leaving the condenser is subcooled. The complete expansion process in positive displacement expansion devices is comprised of the intake, expansion and the exhaust processes. The built-in volume ratio of an expande r is defined as the ratio of the specific volume at the start of the exhaust process to the specific volume at the end of the intake process. In this study it is assumed that both the built-in and proc ess volume ratios are identical and defined as .evapin v condoutv r v (3-31) We will further assume that the degree of s uperheat the refrigerant experiences in the evaporator is zero unless otherw ise indicated. For a given set of operating conditions, the degree of superheat has much more of an influence on the overall performance of the system. In this study, the impact of the degree of subcooling on the expander cy cle performance and size is of

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76 primary interest. The refrigerant is thus a ssumed to be saturated vapor upon exiting the evaporator unless otherwise indicated. An energy balance of the evaporator yields 4,' 1()aevapqhh (3-32) where h4,a is the actual enthalpy of the refrigerant entering the evaporator determined by means of the isentropic efficiency of the ex pander. The COP is thus defined as e c evapw w q COP (3-33) The results presented here are in terms of dime nsionless quantities. This allows the reader to capture general trends of the analysis withou t the distraction of syst em specific calculations. The dimensionless groups that will be extensively used in the presentation of the results are defined as follows Dimensionless COP: refrenceCOP COP COP (3-34) Dimensionless expander work output: referencew w wexp, exp exp (3-35) Dimensionless expander volume ratio: reference v v vr r r, (3-36) Dimensionless condenser capacitance: referencem UA m UA m UA (3-37) For a vapor-compression system of known refr igerating capacity both the overall heat transfer coefficient, U, of a condenser with a constant effectiveness and the mass flow-rate of refrigerant may be assumed constant. The condenser capacitance can thus be considered a direct

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77 measure of the heat exchanger area required to reject a given heat load in the systems condenser if these two conditions are met. A constant cooling capacity comparison criter ion may also be used to constrain the analysis presented here. In the constant cooling capacity design scenario, the mass flow-rate of refrigerant through the system would vary de pending on the degree of subcooling which will ultimately reduce the system component and pi pe sizing. The reduction in system size and weight would be significantly le ss, however, than that achieved by reducing the heat exchanger area. Figure 3-34 shows the variation of dimensionless condenser capacitance, expander volume ratio, expander work output and system COP as functions of the dime nsionless refrigerating effect. In this case, the refr igerant does not experience any subcooling in the condenser. The refrigerating effect is defined as the difference in the specific enthalpy of the refrigerant at the exit and the inlet of the evaporat or. For a given condensing temperat ure, the refrigerating effect increases as the evaporating temperature increases The data used to plot Figure 3-34 has been normalized by the refrigerating effect correspond ing to the lowest ev aporating temperature analyzed. It can be seen from Figure 3-34 that as the refrigerati ng effect increases (for instance due to an increase in the evaporating temperature) the system COP increases dramatically. This is due to the fact that as the difference in operating temperatures (or pressures) decrease both the compressor input work required and expande r work output decrea se. For the operating temperatures reported, the compressor work in put decreases by 74%, while the expander work output decreases by 84%. Furthermore, for the sa me temperature range, an increase in the refrigerating effect of 7.5% is observed. This re sults in an increase in system COP of about 75% when compared to a system operating at the lo west evaporating temper ature analyzed (223 K).

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78 As the refrigerating effect increases, both the si ze and work output of th e expander decrease. As mentioned earlier, the expander work output decreases by about 84% whereas the expander volume ratio decreases by 92%. As the evaporat ing temperature increases, the benefits of utilizing an expander as a throttle valve repl acement become less significant. The condenser capacitance increases with the in crease in the refrigerating effect Over the full range of the refrigerating effect considered, the condenser capacitance can be seen to increase by 27%. Figure 3-35 shows the variati on in the dimensionless expa nder volume ratio and work output as functions of the eva porating temperature for different degrees of subcooling in the condenser. The expander volume ratio and work output decrease with an increase in the evaporating temperature and the degree of subcooling. At an ev aporating temperature of 278 K, typical in air-conditioning applications, the expa nder volume ratio is found to be 8% and 16.4% less than the volume ratio corres ponding to the case of no subc ooling in the condenser for a degree of subcooling of 4oC and 8oC, respectively. The expander work output was found to be 15.3% and 29% less than the work output co rresponding to the case of no subcooling in the condenser for a degree of subcooling of 4oC and 8oC, respectively. Figure 3-36 illustrates the variation in the dimensionless condenser capacitance as a function of the evaporating temperature for differe nt degrees of subcooling in the condenser. It can be observed that as the degree of subcooli ng increases, the varia tion in the condenser capacitance increases dramatically For a typical airconditioning evaporating temperature of 273 K the condenser capacitance is found to be 24.6% and 54.3% greater than that corresponding to the case of no subcooling in the conden ser for a degree of subcooling of 4oC and 8oC, respectively. As previously mentioned, there exists a direct correlation between the heat exchanger area and the condenser capacitance. The increase in the heat ex changer area appears

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79 to be much larger than the decrease in the e xpander volume ratio and wo rk output. The effects of system variables such as the isentropic efficien cies of the compressor and the expander, and the condensers effectiveness on the system COP, condenser capacita nce, expander volume ratio and work output are discussed below. Figure 3-37 shows the variation of the dimens ionless condenser capacitance as a function of the degree of subcooling for different values of the condensers eff ectiveness. The operating conditions and system variables are listed belo w Figure 3-37. For the cas e of no subcooling, the capacitance was found to be 25% and 56% larger for an eff ectiveness of 0.75 and 0.85, respectively, when compared to a system whose condenser has an effectiveness of 0.6. This results from the fact that as the effectivene ss increases, the exit coolant temperature, Tc,out, also increases, resulting in a decrease in the condensers LMTD. For a constant condenser heat load, this leads to an increase in conde nser capacitance. This increase is attributed to the increase of the heat exchanger effectiveness and does not take into account the increase in the heat exchanger area required to produce subcooling. This increase in condenser capacitance may result from either an increase in the overall he at transfer coefficient or an increase in the condensers area. For a condenser effectivenes s of 0.6 it is found that the capacitance must increase by 13% and 29.5% in order to produce a degree of subcooling of 4oC and 9oC, respectively. This represents an increase in the heat exchanger area of the same magnitude assuming that the overall heat transfer coefficien t of the condenser and the mass flow-rate of the refrigerant are constant. For an effectiveness of 0.85 it is observed that in order to produce a degree of subcooling of 4oC and 9oC in the condenser, the capacitance must increase by 14.5% and 33%, respectively. The same trends are exp ected for different values of the ambient air temperature Tamb.

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80 Figure 3-38 shows the variation of the dimens ionless expander work output as a function of the degree of subcooling in the condenser for different isentropic efficiencies of the expander. Detailed component level models of an expa nder would incorporate e ffects of friction and internal leakage losses a nd provide an accurate prediction of the expanders isentropic efficiency. As it can be seen, the work output from the e xpander decreases with both the decrease in isentropic efficiency and increase of the degree of subcooling. For an idea l expander utilized as a throttle valve replacement it can be s hown that a degree of subcooling of 5oC and 10oC results in a decrease of the work output of the expander by 19% and 35%, respectively, when compared to the case of no subcooling in the condenser. Expa nders whose isentropic efficiencies are 60% and 80% will theoretically produce le ss work than an ideal expander by that exact amount. An increase in the degree of subcooling in either ca se results in a decrease in the expander work output by 19% and 35% for a degree of subcooling of 5oC and 10oC, respectively. From this figure it can be concluded that in order to make the use of an expander as a throttle valve replacement beneficial, losses that affect the expanders isentropic efficiency as well as the degree of subcooling the refrigerant experien ces in the condenser must be minimized. Figure 3-39 shows the variation of the expander volume ratio as a func tion of the degree of subcooling for different isentropic efficiencies of the expander. The expander volume ratio is found to increase as the isentropi c efficiency of the expander d ecreases. Not only does the work output of the expander decrease, but also a larger expander adds both a we ight and size penalty to the modified vapor-compression system. For the case of an ideal expander, a degree of subcooling of 5oC and 10oC results in an increase in th e volume ratio by 10% and 20%, respectively. When compared to an ideal expand er, there exists a 1.67% and 3.33% increase in the volume ratio for the case of no subcooling in the condenser for expander isentropic

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81 efficiencies of 0.8 and 0.6, respectiv ely. For a degree of subcooling of 10oC, this difference can be shown to decrease to about a 1.4% and 2. 8% increase in the expander volume ratio for isentropic efficiencies of 0.8 and 0.6, respectivel y, when compared to an ideal expander. It is observed that the decrease in th e expander work output is much more significant when compared to the increase in the expander volume ratio for both a decrease in the isentropic efficiency of the expander and the increase in the degree of s ubcooling the refrigerant experiences in the condenser. Figure 3-40 illustrates the variation of a system s dimensionless COP as a function of the degree of subcooling for different is entropic efficiencies of the expander. It is observed that as the degree of subcooling increases, the COP also increases by 2%, 3.4% and 4.7%. This is the case for 10oC of subcooling in the condenser for expa nder isentropic efficiencies of unity, 0.8 and 0.6, respectively when compared to the case of no subcooling. For the latter case, the COP is found to be 1.7% and 3.4% lower than that corresponding to an ideal expander for expander isentropic efficiencies of 0.8 and 0.6, respectively. It may thus be concluded that the variation in COP is significant for larger values of subcoo ling in the condenser and then becomes more significant as the isentropic efficiency decr eases. Figure 3-41 shows the variation of the dimensionless COP as a function of the degree of subcooling for different degrees of superheat in the evaporator. The system parameters are assumed constant and are liste d below the figure. Superheatin g is beneficial to prevent compressor slugging. An increase in the degree of superheating is generally associated with an increase in the refrigerating effect and an in crease in the input compressor work required to compress the superheated vapor to the condensing pressure. It is observed that as the degree of subcooling increases, the COP increases for all valu es of superheating analyzed. It is also noted

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82 that as the degree of superheat increases the COP is seen to decrease. This may be associated with a significant increase in th e compressor input work in contra st to gains in the refrigerating effect. For the case of no subcooling in the cond enser it may be shown that as the degree of superheat increases, th e compressor input work increases by 2.8% and 5.3% whereas the refrigerating effect increases by 2.4% a nd 15.8% for a degree of superheat of 5oC and 10oC, respectively. The degree of superheat has no direct impact on the performance or size of either the condenser or expander. The de gree of superheat should be sele cted and minimized in order to maximize a given COP. The analysis presented in this section provides a founda tion for application-specific optimization. The degree of subcooling the refrige rant experiences in a sy stems condenser has a significant effect on the size and performance of a vapor-compression system utilizing two-phase expanders as throttle valve replacements. The findings of this analysis are as follows As the evaporating temperature increases the bene fits of utilizing an expander as a throttle valve replacement become less significant. This can be attributed to significant decreases in the expander volume ratio and work output. The expander volume ratio and work output also decrease with increased condenser subcooling. The work output from the expander decrease s with both the decrease in isentropic efficiency and the increase of the degree of subcooling. Losses that affect an expanders isentropic efficiency as well as the de gree of subcooling must be minimized. The decrease in the expander work output is much more significant when compared to the increase in the expander volume ratio for both a decrease in the isentr opic efficiency of the expander and the increase in the degree of subcooling. The variation in system COP is significant for larger values of subcooling in the condenser and become more significant as the expande rs isentropic efficiency decreases. The degree of superheat in the evaporator has no direct impact on the performance or size of either the condenser or expander. The degree of superheat should be selected and minimized in order to maximize a given systems COP.

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83Pure Hydrocarbons as Refrigerants Hydrocarbons are very lucrative as refrig erant replacements for CFCs, HCFCs and HFCs since they are abundant, inexpensive and enviro nmentally friendly because they have no ozone depletion potential. Among the major drawbacks of using hydrocarbons as refrigerants is their high flammability potential. This limits their use to hermetic systems and the slightest leak of a hydrocarbon refrigerant may prove to be catastr ophic. The following is a review of recent literature pertaining to the use of hydrocarbons as pure refrigeran ts, in mixtures, and issues pertaining to flammability. The most promis ing potential hydrocarbons used as refrigerant substitutes are listed in the Table 3-3 along with pertinent fluid information. Many researchers have investigated th e performance of pure hydrocarbons and hydrocarbon mixtures as refriger ants. According to Granryd (2001) several standards allow the use of hydrocarbons as refrigerants in internally safe, hermetically sealed systems with a charge limit of 0.15 kg. This limits the use of hydro carbons to domestic refrig erators and freezers and low capacity heat pumps. Larger refrigeration sy stems may utilize hydroca rbons as refrigerants as long as there is adequate exhaust and extrem e safety precautions are in place. According to Granryd (2001) hydrocarbons are compatible with conventional materials used to construct vapor-compression refrigerating systems. Granryd (2001) theoretically inve stigated various cycle and heat transfer characteristics for various hydrocarbon refrigerants including propane, butane, isobut ene, propene (propylene), cyclopropane, Dimethyl ether a nd a 50% propane-50% isobutane mixture. The results were compared with cycle and heat transfer char acteristics of R-12, R-22 and R-134a. The author found that the pressure ra tios associated with butane and isobu tene are higher while propane and cyclopropane are lower than that of R-22, respectively. The author also notes that in the case of using butane and isobutane as refrigerants, a nece ssary degree of superh eat is required at the

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84 evaporator exit to ensure that isentropic comp ression doesnt move the va por into the two-phase region. When comparing the cycle coefficient of performance to the Carnot efficiency, butane was found to have the highest cy cle Carnot efficiency. Granr yd (2001) also experimentally investigated the use of propane in a refrigeration unit and compar ed its cycle performance with that of R-12, R-22, R-134a and R-152a. He concl uded that a 3-5% increase in system COP and a 3-15% decrease in refrigerating capacity exists when propane is compared to conventional refrigerants. He attributed the decrease in co mpressor work to the lower molecular weight of propane. The use of pure hydrocarbons and mixtures as re frigerants in domestic refrigerators has been investigated by Hammad a nd Alsaad (1999), Jung et al. (2000), Tashtoush et al. (2002), Sekhar (2004) and Wongwises and Chimres (2005). Purkayastha and Bansal (1998) presented an e xperimental study of utilizing propane and a LPG mix (by mass fraction: 98. 95% propane, 1.007% ethane and 0.0397% isobutane) as a suitable replacement for R-22 in a heat pump. They found an increase in system coefficient of performance of 18 and 12%, and 9 and 4% for pr opane and LPG, respectively. These values correspond to condensing temperatures of 35oC and 55oC, respectively, at an evaporating temperature of 3oC. Colbourne and Suen (2004a, 2004b) appraised the flammabili ty hazards of hydrocarbon refrigerants in refrigeration and heat pump systems. Dlugogorsk i et al. (2002) experimentally investigated the use of inerti ng agents with natural gas and a propane-butane mixture. Their recommendations however have not been tested experimentally in a refrigeration unit. In the case of pure hydrocarbon refrigerants th e parameters under investigation include the system coefficient of performance, refrigerati ng effect, work consumption by the compressor and

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85 the effect of both subcooling and superheat on sy stem performance. The case where an expander is utilized as a throttle valve replacement was also investigated. Comparisons with R-12, R-22 and R-134a were also made to compare the perfor mance of pure hydrocarbons as refrigerants in both the conventional and modified cycles. The re sults from these simulations are presented in Figures 3-42 to 3-47.

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86 Figure 3-1. A schematic of the id eal base cycle (1-2-3-4-1) and th e modified base cycle with an expansion device (1-2-3-4-1) wcomp qout 1 2 3 4 4 Te,Pe Tc,Pc qin wexp

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87 Figure 3-2. T-s and P-h diagrams of th e ideal base cycle (1-2-3-4-1) and the modifi ed base cycle with an expansion device (1 -2-3-41) s T 1 2 3 44 P h 1 2 3 4 4

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88 Figure 3-3. T-s and P-h diagrams of the economizer cycle (1 -2-3-4-5-6-7-1) and an economizer cycle with an expansion device as a throttle valve replacem ent (1-2-3-4-5-6-7-1) Pe s T Pc Pi 1 2 3 4 5 4 6 6 6f 5 7 7 h P Te Ti 34Tc 1 5 6 6f 7 7 2

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89 Figure 3-4. A schematic of the economizer cycle (1-2-3-4-5-6-7-1) and an economizer cycle with an expansion device as a throttle valve replacement (1-2-3-4-5-6-7-1) qout 2 5 4 Te,Pe qin Tc,Pc wcomp,1 wcomp,2 1 6 3 6g or 6g 7 7 wexp,2 6 wexp,1 6f or 6f Separator x (1-x)

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90 Figure 3-5. A schematic of th e IHX cycle (1-2-3-4-5-6-1) and the modified IHX cycle with an expansion device as a throttle valve replacement (1-2-3-4-5-6-1) wexp 5 5 6 Tc,Pc wcomp q ou t 2 3 Te,Pe q in 1 4 IHX

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91 Figure 3-6. T-s and P-h diagra ms of the IHX cycle (1-2-3-4-5-6-1) and the modi fied IHX cycle with an expansion device as a th rottle valve replacement (1-2-3-4-5-6-1) T 6 23 1 55 1For: =1 T1=T3 T4=T6 4 s P h 6 2 35 5For: =1 T1=T3 T4=T6 14

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92Table 3-1. Summary of the percent changes in the system COP, refrigerating capacity and required work input by the addition of an expansion device for the ideal and actual R-134a single-stage, economizer and IHX cycles c=85%, exp=0.5, IHX=0.7 ** a decrease in the amount of work needed theoretically *** maximum value corresponds to Te=-20oC and minimum value corresponds to Te=10oC Ideal*** Actual *,*** Tc, oC Evaporating Range COP, % Qe Wnet, COP, % Qe Wnet, 25 -20oC to 10oC 8-18.7 0.4-3.4 7.6-15.9 3.4-8.3 0.2-1.7 3.3-7.2 30 -20oC to 10oC 10.3-21.4 0.76-4.3 9.6-17.8 4.5-9.6 0.4-2.2 4.3-8.2 35 -20oC to 10oC 12.9-24.2 1.2-5.4 11.9-20 5.6-11 0.6-2.8 5.3-9.2 40 -20oC to 10oC 15.6-27.1 1.8-6.7 14-22 6.8-12.4 0.9-3.5 6.4-10.3 Base Cycle 45 -20oC to 10oC 18.5-30.2 2.4-8.2 16.5-24 8.2-14 1.2-4.3 7.5-11.4 25 -20oC to 10oC 5.2-11 0.3-1.7 4.9-9 2.2-4.8 0.2-1.4 2.1-3.7 30 -20oC to 10oC 6.5-12 0.5-2 6.1-10.2 2.9-5.6 0.4-1.7 2.6-4.1 35 -20oC to 10oC 8.1-14 0.7-2.6 7.4-11.4 3.5-6.4 0.5-2.2 3.2-4.6 40 -20oC to 10oC 9.6-15.5 1-3.2 8.8-12.7 4.3-7.3 0.8-2.7 3.7-5.1 Economizer Cycle 45 -20oC to 10oC 11.4-17.4 1.3-3.9 10.2-14.1 5.1-8.3 1.1-3.3 4.3-5.6 25 -20oC to 10oC 2.6-3.6 0.14-0.6 2.5-3 1.6-2.9 0.1-0.6 1.5-2.4 30 -20oC to 10oC 3-3.9 0.2-0.7 2.8-3.3 1.9-3.3 0.2-0.7 1.8-2.7 35 -20oC to 10oC 3.4-4.5 0.3-1 3.23-3.7 2.3-3.7 0.2-0.9 2.1-2.9 40 -20oC to 10oC 3.9-5 0.4-1.2 3.6-4 2.7-4.2 0.6-1.1 2.4-3.2 IHX Cycle 45 -20oC to 10oC 4.4-5.5 0.6-1.4 4-4.4 3.1-4.7 0.5-1.4 1.7-3.5

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93 Figure 3-7. Variation of COP, between the modified cycle with an expansion device and the ideal cycle (R-134 a), with evaporating temperature for A) Tc=25oC, B) Tc=45oC for the base, economizer and IHX cycles 0% 2% 4% 6% 8% 10% 12% 14% -20246810Evaporating Temperature, oCIncrease in COP Base Tc=25C Economizer Tc=25C IHX Tc=25C 0% 5% 10% 15% 20% 25% 30% -20246810Evaporating Temperature, oCIncrease in CO P Base Tc=45C Economizer Tc=45C IHX Tc=45CA B

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94 Figure 3-8. Variation of Qe and Wnet, between the ideal modi fied cycle with an expansion device and an ideal base cycl e, with evaporating temperature for various condenser temperatures in a single-stage vapor-compression refrigeration unit (nega tive denotes a reduction). -25% -20% -15% -10% -5% 0% 5% 10% -20246810Evaporating Temperature, oCChange in Refrigerating Effect and Net Wor k Tc=25C Tc=30C Tc=35C Tc=40C Tc=45C Qe Wnet

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95 Figure 3-9. Variation of COP, between the modified cycl e with an expansion device and the ideal cycle, as a function of the evaporating temperature for both R-22 and R-134a for various expander efficiencies Tc=45oC (comp=85% when not ideal) 0 5 10 15 20 25 -20246810 Evaporating Temperature, oC COP R-134a, Ideal R-22, Ideal ex p =1 ex p =0.75 ex p =0.5

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96 Figure 3-10. Variation of COP, between the modified cycl e with an expansion device and the ideal cycle, as a function of the evaporating temp erature for various condenser temperatures for the actual and ideal cycle configurations of the economizer cycle. 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% -20-15-10-50510Evaporating Temperature, oCChange in COP Actual, Tc=25C Actual, Tc=45C Ideal, Tc=25C Ideal, Tc=45C comp=85%, exp=0.5

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97 Figure 3-11. Variation of COP, between the modified cycl e w/ an expansion device and the ideal cycle, as a function of the evaporating temperature for various condenser temperatures for the actual and ideal cycle configurations of the IHX cycle 0% 1% 2% 3% 4% 5% 6% -20-15-10-50510Evaporating Temperature, oCChange in COP Actual, Tc=25C Actual, Tc=45C Ideal, Tc=25C Ideal, Tc=45C comp=85%, exp=0.5, IHX=0.7

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98 Figure 3-12. Variation of refr igerating efficiency as a function of the evaporating temperature for both the base (no expa nder) and ideal and actual modified cycle with and an expander at a condenser temperature of Tc=45oC Figure 3-13. Variation of refrigerating e fficiency as a function of the evaporating temperature for the actual base, IHX a nd economizer cycles with an expander at a condenser temperature of Tc=25oC 0.75 0.8 0.85 0.9 0.95 1 -20246810Evaporating Temperature, oCRefrigerating Efficienc y R-134a R-22 exp=1 exp=0.75 No Expander comp=85%, exp=0.5, IHX=0.7 0.4 0.5 0.6 0.7 0.8 0.9 -20-15-10-50510 Evaporating Temperature, oCRefrigerating Efficiency Actual, IHX Cycle Actual, Base Cycle Actual, Economizer Cycle oC

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99 4 6 8 10 12 14 16 02468 Tsubcooling, oCCOP Tc=25C Tc=30C Tc=35C Tc=40C Tc=45C Figure 3-14. Variation of an ideal system s COP as a function of the degree of subcooling for various c ondensing temperatures (Te=5oC, Tsh=0 oC)

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100 Figure 3-15. Variation of an ideal system s COP as a function of the degree of subcooling for various condenser temp eratures and expa nsion processes (Te=5oC, Tsh=0oC) 0 2 4 6 8 10 12 14 16 0246810 Tsubcooling, oCCOP Isenthalpic Expansion Isentropic Expansion Tc=25oCTc=45oC

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101 Figure 3-16. Variation of the refrigerati ng effect as a function of the degree of subcooling for various condenser temp eratures and expa nsion processes (Tsh=0oC) 100 120 140 160 180 200 02468 Tsubcooling, oCRefrigerating Effect, kJ/ k Isenthalpic Expansion Isentropic ExpansionTc=45oCTc=25oC

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102 10 14 18 22 26 30 579111315Evaporator temperature, oCCOP 0 C sc, Isenthalpic 5 C sc, isenthalpic 0 C sc, Isentropic 5 C sc, Isentropic Figure 3-17. Variation in an ideal system s COP with respect to the evaporator temperature for different degrees of subcooling and expansion processes (Tsh=0oC)

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103 Figure 3-18. Variation of the difference in system COP,%, between the ideal cycle and modified cycle with an expansion de vice as a function of the evaporator temperature for various condenser temp eratures and degrees of subcooling (Tsh=0oC) 0 5 10 15 20 25 579111315Evaporator temperature, oC COP 0C 0C 5C 5CTsc Tc=25oC Tc=45oC

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104 0 5 10 15 20 25 02468 Tsuperheat, oC COP Tc=25C Tc=35C Tc=45C Figure 3-19. Variation of the difference of th e system COP, %, base cycle and modified cycle with an expansion device as a func tion of the degree of superheat in the evaporator for various condenser temperatures.

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105 100 120 140 160 180 200 02468 Tsuperheat, oCRefrigerating Effect, kJ/kg Tc=25C Tc=35C Tc=45C Figure 3-20. Variation of the re frigerating effect for an idea l cycle as a function of the degree of superheat for vari ous condenser temperatures

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106 Figure 3-21. T-s diagram of the ba sic system (1-2-3-4-1) and the modified system with an expansion device (1-2-3-4-1) for a A) conventional refriger ation system and B) transcritical CO2 refrigeration system s T 1 2s 3 4 s 4 a 4 h 1 2 a Pe vapPcond s T 1 2s 3 4s 4a 4h 2a Pevap Pgc CP A B

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107 Figure 3-22. Variation of system coefficien t of performance as a function of heat rejection pressure for various evaporati ng temperatures for basic and modified transcritical CO2 systems Figure 3-23. Percentage of cycl e irreversibility associated with the heat rejection and expansion processes as a function of h eat rejection pressure for basic and modified transcritical CO2 systems 1.5 2 2.5 3 3.5 99.51010.51111.5 Heat Rejection Pressure, MPa COP Te=273 K Te=268 K Te=273 K Te=268 K Modified System Basic System 0 10 20 30 40 50 60 99.51010.51111.5 Heat Rejection Pressure, MPaIrreversibility % Gas cooler Valve Gas cooler Expander comp=0.85 exp=0.65 T3=313 K Tevap=273 KBasic System Modified System

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108 Figure 3-24. Coefficient of performance of basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature Figure 3-25. Difference in coefficient of pe rformance between basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature 0% 5% 10% 15% 20% 25% 30% 35% -30-25-20-15-10-505 Evaporating Temperature, oC COP R-744 R-134a R-22 comp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC 1 2 3 4 5 -30-25-20-15-10-505 Evaporating Temperature, oCCOP Basic R-744 System Modified R-744 System Basic R-134a System Modified R-134a System Basic R-22 System Modified R-22 Systemcomp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC

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109 Figure 3-26. Difference in refr igerating effect between basi c and modified R-22, R-134a and transcritical CO2 systems as functions of th e evaporating temperature Figure 3-27. Difference in system input work between basic and m odified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature (negative denotes a decrease) 0 0.04 0.08 0.12 0.16 -30-20-100 Evaporating temperature, oC Refrigerating Effect R-744 R-134a R-22comp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC -30% -20% -10% 0% -30-20-100 Evaporating Temperature, oC Input Work R-744 R-134a R-22comp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC

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110 Figure 3-28. Volume ratio of the expande r employed in modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature Figure 3-29. Difference in coefficient of pe rformance between basic and modified R-22, R-134a and transcritical CO2 systems as functions of expander isentropic efficiency 0 5 10 15 20 25 5101520 Evaporating Temperature, oCVolume Ratio R-134a R-744 R-22comp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC 5% 10% 15% 20% 25% 30% 35% 25%45%65%85% Expander Isentropic Efficiency COP R-744 R-134a R-22comp=0.85 Tamb=35oC Tcond=50oC Tgc_out=40oC Tevap=0oC

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111 Figure 3-30. Percentage of cycl e irreversibility due to variou s components in a modified R-22 system as functions of the evaporating temperature Figure 3-31. Percentage of cy cle irreversibility due to condensing process in basic and modified R-22 and R-134a systems as functions of the evaporating temperature 0 10 20 30 40 50 -30-20-100 Evaporating Temperature, oCIrreversibilty, % Compressor Condenser Expander Evaporatorcomp=0.85 exp=0.65 Tamb=35oC Tcond=50oC 20 25 30 35 40 45 50 -30-20-100 Evaporating Temperature, oCCondenser Irreversibility, % Basic R-134a System Modified R-134a System Basic R-22 System Modified R-22 Systemcomp=0.85 exp=0.65 Tamb=35oC Tcond=50oC

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112 Figure 3-32. Percentage of cycl e irreversibility due to the expansion process in basic and modified R-22, R-134a and transcritical CO2 systems as functions of the evaporating temperature Figure 3-33. Dimensionless system irreversib ility in basic and m odified R-22, R-134a and transcritical CO2 systems as functions of th e evaporating temperature 0 10 20 30 40 50 60 -30-20-100 Evaporating Temperature, oCExpansion Process Irreversibility, % Basic R-744 System Modified R-744 System Basic R-134a System Modified R-134a System Basic R-22 System Modified R-22 Systemcomp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC 0.1 0.15 0.2 0.25 0.3 0.35 -30-20-100 Evaporating Temperature, oCTotal Irreversibility/Heat Rejection Rate Basic R-744 System Modified R-744 System Basic R-134a System Modified R-134a Syste m Basic R-22 System Modified R-22 Systemcomp=0.85 exp=0.65 Tamb=35oC Tcond=50oC Tgc_out=40oC

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113 0 1 2 3 4 11.021.041.061.08Dimensionless refrigerating effectDimensionless quantit y condenser capacitance expander volume ratio expander work output COP Table 3-2. Percentage of cycl e irreversibility due to the va rious components in basic and modified R-22, R-134a and transcritical CO2 systems Figure 3-34. Variation of di mensionless COP, expander work output, expander volume ratio and condenser capacitance as a f unction of dimensionless refrigerating effect. Tcond=45oC, Tamb=25oC, cond=0.75,comp=0.85,exp=0.75 R-744R-134aR-22R-744R-134aR-22 Compressor, %17.322.221.424.628.226.3 Condenser, %-33.638.1-42.646.7 Gas cooler, %29.7--42.2-Expansion valve, %46.53329--Expander, %---23.214.612.5 Evaporator, %6.511.211.51014.614.5 Basic SystemModified System

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114 Figure 3-35. Variation of dime nsionless expander work output and expander volume ratio as a function of evaporating temperatur e for various degrees of subcooling in the condenser. Tcond=45oC, Tamb=25oC, cond=0.75,comp=0.85,exp=0.75 Figure 3-36. Variation of dimensionless condenser capacitance as a function of evaporating temperature for various de grees of subcooling in the condenser. Tcond=45oC, Tamb=25oC, cond=0.75,comp=0.85,exp=0.75 0 0.2 0.4 0.6 0.8 1 -40-30-20-10010 Evaporating temperature, oCDimensionless wexp and rv 0 4 8 0 4 8Degree of Subcooling, oCdimensionless rvdimensionless wexp 0 1 2 3 -40-30-20-10010 Evaporating temperature, oCDimensionless condense r capacitance 0 4 8Degree of Subcooling, oC

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115 Figure 3-37. Variation of dime nsionless condenser capacitance as a function of degree of subcooling in the condenser for various values of condenser effectiveness. Tcond=45oC, Tevap=5oC, Tamb=25oC,comp=0.75,exp=0.75 Figure 3-38. Variation of dime nsionless expander work output as a function of degree of subcooling in the condenser for various expander efficiencies. Tcond=45oC, Tevap=5oC, Tamb=30oC, cond=0.75,comp=0.85 1 1.4 1.8 2.2 2.6 02468 Degree of subcooling, oCDimensionless condense r capacitance 0.6 0.75 0.85 Condenser effectiveness 0.3 0.5 0.7 0.9 0246810Degree of subcooling, oCDimensionless expander work output 1 0.8 0.6 Expander efficiency

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116 Figure 3-39. Variation of dime nsionless expander volume ratio as a function of degree of subcooling in the condenser for various expander efficiencies. Tcond=45oC, Tevap=5oC, Tamb=30oC, cond=0.75,comp=0.85 Figure 3-40. Variation of dime nsionless COP as a function of degree of subcooling in the condenser for various expander efficiencies. Tcond=45oC, Tevap=5oC, Tamb=30oC, cond=0.75,comp=0.85 0.92 0.96 1 1.04 0246810Degree of subcooling, oCDimensionless COP 0.6 0.8 1Expander efficiency 0.8 0.85 0.9 0.95 1 1.050246810 Degree of subcooling, oCDimensionless expander volume ratio 1 0.8 0.6 Expander efficiency

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117 Figure 3-41. Variation of dime nsionless COP as a function of degree of subcooling in the condenser for various evaporator degree of superheat. Tcond=45oC, Tevap=5oC, Tamb=25oC, cond=0.75,comp=0.85, exp=0.85 Table 3-3. Hydrocarbons viewed as potential refrigerants MWTcrit, OC Pcrit, kPa crit kg/m3 Butane (R600) 58.12151.98 3796 227.84 Ethane (R170) 30.0732.18 4872 206.58 Isobutane (R600a) 58.12134.67 3640 224.35 Propane (R-290) 44.196.675 4247 218.5 Propene (Propylene) (R1270) 42.0892.42 4664 223.39 0.98 1 1.02 1.04 0246810 Degree of subcooling, oCDimensionless COP 10 5 0Degree of superheat, oC

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118 Figure 3-42. Variation of the refrigerati ng effect as a function of evaporating temp erature for various refrigerants at a cond ensing temperature of A) 25oC and B) 45oC for the ideal base cycle 0 50 100 150 200 250 300 350 400 -5051 0 Evaporating Temperature, oCRefrigerating Effect, kJ/kg 0 50 100 150 200 250 300 350 -50510 Evaporating Temperature, oC R-134a R-22 R-12 Butane Isobutane Propane Proplyene A B

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119 Figure 3-43. Variation of the ideal base cycle coefficient of performance as a function of ev aporating temperature for various refrigerants at a condens ing temperature of A) 25oC and B) 45oC 7 9 11 13 15 17 -5051 0 Evaporating Temperature, oCCoefficient of Performance 3 4 5 6 7 -50510 Evaporating Temperature, oC R-134a R-12 R-22 Butane Isobutane Propane Proplyene A B

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120 Figure 3-44. Variation of compressor work as a function of the evapor ating temperature at a condensing temperature of A) 25oC and B) 45oC for various refrigerants in an ideal base cycle 0 5 10 15 20 25 30 35 40 45 -50510 Evaporating Temperature, oCCompressor Work, kJ/kg 15 25 35 45 55 65 -50510 Evaporating Temperature, oC R-134a R-22 R-12 Butane Isobutane Propane Proplyene A B

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121 Figure 3-45. Variation of the increase of COP (when comparing the base cycle and th e expander cycle) as a function of the evaporating temperature for various refriger ants at a condensing temperature of A) 25oC and B) 45oC 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% -50510 Evaporating Temperature, oC R-134a R-22 R-12 Butane Isobutane Propane Proplyene 14% 16% 18% 20% 22% 24% 26% 28% -50510 Evaporating Temperature, oCIncrease in COP (base cycle and Expander cycle) A B

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122 Figure 3-46. Variation of the decrease in ne t-work (when comparing the base cycle and the expander cycle) as a function of the evaporating temperature for various refriger ants at a condensing temperature of a) 25oC and b) 45oC 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% -50510 Evaporating Temperature, oCDecrease in the net work (between the base cycle and expander cycle) 10% 12% 14% 16% 18% 20% 22% 24% -50510 Evaporating Temperature, oC R-134a R-22 R-12 Butane Isobutane Propane Proplyene A B

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123 Figure 3-47. Variation of the increase in re frigerating effect (when comparing the base cycle and the expander cycle) as a func tion of the evaporating temperature for various refrig erants at a condensing temperature of a) 25oC and b) 45oC 0% 1% 2% 3% 4% 5% 6% -50510 Evaporating Temperature, oCIncrease in the refrigerating effect (when comparing base cycle and expander cycle) R-134a R-22 R-12 Butane Isobutane Propane Propylene 0.0% 0.3% 0.5% 0.8% 1.0% 1.3% 1.5% 1.8% -50510 Evaporating Temperature, oCIncrease in the refrigerating effect (when comparing base cycle and expander cycle) A B

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124 CHAPTER 4 MATHEMATICAL MODEL OF AN IDEAL ROTARY-VANE EXPANDER Much literature has been reviewed in or der to formulate an accurate and rigorous analytic thermodynamic and fluid model of a ro tary-vane expander. Mo st of the literature found deals with rotary-vane expanders us ed predominantly in high-temperature applications where the working fluid is va por (namely steam and R-113) and is modeled using ideal-gas relations. This negates the ma ny additional complexities that arise due to the necessary formation of vapor from liqui d exiting the condenser in vapor compression cycles. These modeling complexities may include but are not limited to cavitational erosion, choking, issues dea ling with lubrication, and a po ssible increase of friction and internal leakage losses due to significant variations in the density and viscosity of the two-phase working fluid. In order to accurately model a tw o-phase rotary-vane expander, the studies conducted in the existing literature must be revisited and modified accordingly to predict such parameters as the work ing fluids pressure, temperature and mass variations as well as an overall estimate of power losses, volumetric efficiency, power output and torque of the rotary-vane expander. Literature Review of Modeling of Rotary-vane Expanders While literature dealing with modeling a nd design improvements of various types of compressors and expanders, such as heli cal screw, reciprocating piston, scroll, Wankel-engines Badr et al. (1991a, 1991b, 1991c) and rotary -piston are abundant, this review is confined to rotary-vane (or s liding-vane) type expanders and compressors, hence referred to as MVE and MVC, respec tively. Smith et al. (1990, 1992) introduced the principles, operation and testing of a nove l type compressor, the Groll rotary-vane compressor. They found a 19% and 15% increase in volumetric and adiabatic

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125 efficiencies, respectively when compared to a conventional rotary-vane compressor. There have not been any st udies utilizing this as an expansion device. The positive-displacement rotary-vane expa nder has been widely used for waste heat recovery in Rankine cycles. Badr et al. (1984) cited economics, ease of manufacturing and low maintenance as the most important parameters in the selection of a MVE as an expansion device. Among the investigators who developed mathematical models describing the geomet ry of rotary-vane type tu rbomachinery (compressors, expanders, pumps and air-motors) are Wo lgemuth and Olson (1971), Marsters and Ogbuefi (1972), Badr et al (1985a, 1985b), Somayajulu (1971), and Ben-Bassat and Wolgemuth (1972). Mechanical friction in a MVE has been investigated by Beck et al. (1966), Ben-Bassat and Wolgemuth (1972), Pe terson and McGahan ( 1972), Badr et al. (1986a), Robertson and Wolgemuth (1978), a nd Edwards and McDona ld (1972). Internal leakage losses in a MVE have been i nvestigated by Peters on and McGahan (1972), Jacazio et al. (1979), Badr et al. (198 5c), and Robertson and Wolgemuth (1978). All of the above studies either dealt with the expansion or compression of a working fluid (air, steam, a nd vapor) that could be modele d by the ideal-gas equation of state. The studies in question also dealt wi th rotary-vane type tu rbomachinery utilized as expanders in Rankine cycles as air motors or pumps. Tabl e 4-1 summarizes the type of application, working fluid, an alyses performed, and the type of losses accounted for. The extent and accuracy of their analys es is beyond the scope of this work. Model Development In order to accomplish this, five separate subroutines had to be constructed to accurately predict the aforementioned parameters. A simplified flow diagram of the main

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126 computer program that has been developed and a brief descripti on, in subsequent sections, of the subroutines to be u tilized is presented in Figure 4-1. Thermophysical Model NISTs REFPROP 7.0s refrigerant prope rty subroutines are linked with the various models and are called whenever thermo -physical data is requ ired. The capability to both use pure refrigerants and common re frigerant mixtures allows the use of REFPROPs property subroutines no matter the refrigerant. There is also the capability to use of a user defined and customized mixture. Geometric and Kinematic Model In order to model the perfor mance of a rotary-vane expander, the geometry of the expander and kinematics of the vanes must be accurately described mathematically. The basic geometrical and kinematic characteristics as a function of the angular displacement of the expanders rotor, are required to ev aluate the performance of the expander. The parameters needed include: Variation of the cell volumes Volume expansion ratio, rv=V2/V1 (Figure 4-2) Variations of the protrusions of th e vanes outside their rotors slots Vanes sliding velocity Vanes accelerations A review of the literature reveals that ma ny different models have been considered (Table 4-1). A primary difference among th e models is accounting for the vane thickness. Many researchers have assumed an idealized vane thickness of zero when calculating the cell volume. For the purpose of this study, the expander is assumed to have a circular statorcylinder and the thickness of the vanes is ta ken into account initia lly. It can be shown

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127 from Figure 4-2 that a circul ar stator cylinder may be m odeled as having two arcs (2-seal and seal) one of which is a sealing arc. The geometrical input parameters to the model and nomenclature used throughout this section are listed in Table 4-2 and are shown on a schematic of a circular expander in Figure 4-2. It should be noted that if the sealing arc is symmetric a bout the base line, that =180o and RS()=180o. Neglecting the thickness of the vanes and assuming that the vanes are in continuous contact with the stator cylinde r, the ideal volume of the expa nders cell as a function of angular displacement may be written as cvcvcvVAAL (4-1) where Acv() and Acv(-) are the areas enclosed between the stator and rotor cylinders and the datum and the leading and trailing va nes respectively. The area functions have been calculated using the equations and nume rical procedure presented in Badr et al. (1985). The actual volume of the expanders cel ls differs by accounting for the volume of the extended portions of the leading and tra iling vanes outside of the rotors slots. In order to calculate the kinematics of the vanes and the actual volume of the expanders cell, the protrusions of the vanes from the rotors slots must be computed. The vanes are assumed to be rigid. R() is defined as the radius of the stator-cylinder to the center of the rotor as a function of angular displacement. For a circular stator-cylinder this quantity may be simp lified and expressed as 2222cos2cossinRRRereere (4-2) where is defined as the angle between the arcs center and that of the rotor. The vanes protrusion from the slot ma y then be defined as

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128 R X Rr (4-3) The sliding velocity and the acceleration of the vanes may be determined by differentiating Equation (4-3) with respect to time once a nd twice, respectively. A FORTRAN subroutine has been de veloped to calculate the geom etrical characteristics of a circular rotary-vane expander for a given set of input parameters. Thermodynamic Model To model the performance characteristics of an expander, the principles of conservation of mass and energy in each of th e expanders cells must be used. This will allow the prediction of the pressure, temperat ure and mass variations of the working fluid in the cells as a function of the rotors angul ar displacement. This will require accurate calculation of both the variati on of the cell volume and the thermo-physical properties from the geometric and thermo-physical m odels, respectively. The basic geometrical relationships modeled th en helped describe: Variation of the cell volumes Volume expansion ratio Inlet and exhaust por t area variations As mentioned earlier, in order to model the thermodyna mic and throttling characteristics of an expander, the principles of conservation of ma ss and energy in each of the expanders cells must be used. Thes e would allow the prediction of the pressure, temperature and mass variations of the worki ng fluid in the cells as a function of the rotors angular displacement. A computer program has been coded to calculate the mass, pressure and temperature variations in the expanders charging, expansion, and exhaust stages. Figure 4-2 shows the various stages of flow of a working fluid thr ough the rotary-v ane expander.

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129 The following characteristics are calculated once the inlet state to the expander is defined and the mass, pressure and temperat ure variations in th e expander are found: Volumetric efficiency: an indicator of how well the expanders cell was filled during the charging process Throttling power factor: a ratio of the i ndicated power of the expander with only throttling losses in ports accounted for and the indicated power of an ideal expander Throttling Efficiency: A ratio of the sp ecific work of the expander with only throttling losses accounted for and the specific work of an ideal expander The throttling losses in the intake and e xhaust ports are only pr esented here briefly but will be discussed and modeled in the subsequent chapter. Charging Process The charging process takes place at angular displacements of 0in The temperature of the fluid entering the expande r can be determined from the degree of subcooling the fluid undergoes in the condenser. The saturati on pressure corresponding to the saturation temperature of the condenser, the state of the fluid entering the expander can also be determined. This state is denoted by the subscript (int). The mass flow-rate of the working fluid th at fills the cavity, assuming steady flow and ignoring changes in potential energy, can be written as int int int) ( ) (V A C md in (4-4) where the coefficient of discharge is assumed to be unity initially and will be determined by experimental data. The inle t throat area varies as a f unction of angular displacement and was found to obey the linear approximati on that Wolgemuth and Olson (1971) have suggested. A typical linear appr oximation of the inle t throat variation is shown in Figure 4-3.

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130 The expression for the amount of mass c ontained in the expander cell at any angular displacement is the following d m m min cv cv 0) ( ) 0 ( ) ( (4-5) The first law of thermodynamics can be written for the process as follows (taking heat into the system and work done by the system as positive) intint cv bdE QWmh dt (4-6) Assuming the changes in kinetic ener gy and potential energy are neglected, Ecv=dUcv. The pressure within the control volume is also assumed to be uniform. Multiplying by dt and neglecting friction, inte rnal leakage losses and heat transfer, Equation (4-6) becomes 121inint()mtUPVVh (4-7) Solving for the internal energy of the refrig erant at the new time step, State 2, we find 11121inint 2 2()mtmuPVVh u m (4-8) The density of the refrigerant at the ne w time step, State 2, is calculated as 2 2 2V m (4-9) The charging process is assumed to be a constant pressure process (P1=P2=Pint). This conclusion is supported by much of the literature including Ta niguchi et al. (1983) and Baek (2002). The mass in the control volume at State 2 is thus an unknown. In this case, the first law of thermodynamics can be written as 221112121int()() UmumupVVmmh (4-10)

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131 The mass at State 2 can be expressed however as 2 2 2V m (4-11) Substituting Equation (4-11) into Equation (4-10) we get 22211121221int()() VumupVVVmh (4-12) where 2 and u2 are the unknowns. Neglecting temperat ure gradients within the control volume, these two unknowns may be determin ed by iterating on the temperature of the control volume that corresponds to the known pressure, P2, and simultaneously satisfies the conservation of energy statement, Equa tion (4-12). The thermodynamic state of the control volume at the new time step or angular displacement has hence been fully defined and this procedure is carried out until the cut-off angle ( = in+ ). Expansion Process During the expansion process, inex the mass in the control volume, mcv, stays constant after the cut-off anglein cut where in is the spread of the intake port and is the angle between successive vanes (e.g. =45o for an eight-vane expander). Assuming internal leakage losses are neg ligible, the mass in the control volume, mcv, is assumed to be constant throughout the expansion process. If pressure gradients within the control volume, changes in kine tic and potential ener gy, friction and heat transfer are also neglected, the process be tween the current and new time step may be assumed to be isentropic and hence 21ss (4-13) The density at the new time step may be determined from 2 2V mcv (4-14)

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132 The thermodynamic properties at the new tim e step, namely temperature, pressure, and quality, can be calculated from Equati ons (4-13) and (4-14) with the use of REFPROP 7.0. This process is repeated until the end of the e xpansion process at a predetermined exhaust angle of ex. Exhaust process The exhaust process occurs at an exhaust angle exendout where out is the spread of the exhaust port and is the angle between consecutive vanes. During the exhaust process, the mass flow -rate of the working fluid discharged from the control volume as a function of a ngular displacement can be determined from the following equation ,()()exdexcvexexmCAV (4-15) where cv is the density of the working fluid in the control volume at the current time step. The discharge coefficient, Cd,ex, is an empirically determined constant that takes into account exit port losses. Here, the discharge coefficient is assumed to be unity. The variation of the exit throat area, exA can be seen to follow the linear approximation proposed by Wolgemuth and Ol son (1971) in Figure 4-4. The amount of mass contained in the cont rol volume at any angular displacement can then be expressed as d m m mexex ex cv cv ) ( ) ( ) ( (4-16) where ex is the angle at the end of the expansion process. The first law of thermodynamics can be written for the process as follows

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133 cv bexexdE QWmh dt (4-17) Neglecting changes in kinetic a nd potential energy leads to Ecv=Ucv. The exhaust process is assumed to occu r at constant pressure cvexexPP and in an quasiequilibrium manner. Neglecting friction, intern al leakage losses and heat transfer from the ambient, Equation (4-17) reduces to 21()exexexUPVVmh (4-18) Applying the same solution methodology as in the charging process, Equation (418) simplifies to 22211212211()()exVumuPVVVmh (4-19) where 2 and u2 are the unknowns. Neglecting temperat ure gradients within the control volume, these two unknowns may be determin ed by iterating on the temperature of the control volume that corresponds to the known pressure, P2, and simultaneously satisfies the conservation of energy statement, Equa tion (4-19). The thermodynamic state of the control volume at the new time step or angular displacement has hence been fully defined. Ideal Expander Evaluation For a given set of operating temperatures in a refrigeration system, the state of the refrigerant entering and leaving the expander may be determined. The resultant process volume ratio is defined as 3 evapin vpv r v (4-20) where v3 is the specific volume of the fluid at the condenser exit. The refrigerant may be a saturated or subcooled liquid at this state. This process volume ratio is expected to be

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134 smaller when the refrigerant leaving the condenser is subcooled. The built-in (or geometric) volume ratio of the expander may be expressed as ex vbin inv r v (4-21) It should be noted that the terms built -in and geometric volume ratios are used interchangeably. If internal leakage losses ar e neglected, this parameter reduces to the ratio of the expanders ce ll volumes since at both ex and in+ the mass in the cell volume is constant. If the built-in volume ratio is less than, identical to or greater than the process volume ratio then under-expansion, ideal e xpansion and over expansion will occur, respectively. The expander s hould not only match (or be fairly similar to) the process expansion ratio for typical system operat ing conditions but should also minimize the amount of wasted power rec overy due to pressure lo sses incurred during underexpansion. In the case of over-expansion, the higher pressure of the downstream reservoir may result in refrigerant flowing back into the expander via the exhaust port further complicating the actual exhaust process. Over-expansion is not accounted for in this study. Figure 4-5 illustrates the three possible scenarios depending on the built-in and process volume ratios. The appropriate select ion of the dimensions of the e xpander is critical to ensuring adequate operation. In general, turbo-mach ines are designed such that the volumetric flow-rate of the working fluid is identical to the volumetric displacement of the device. For the case of an expansion device employed in a refrigeration system this equality may be expressed as ,60drefininVmv (4-22)

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135 where is the rotational speed of the expander in RPM and dVis the volumetric displacement per revolution of the expansion device. The right-hand side of Equation (422) represents the volumetric flow-rate of refrigerant in m3/s that enters the expander after leaving the condenser in either a satura ted or subcooled liquid state. The rotational speed is the primary design variable for gi ven operating conditions and geometry of the expander. As the rotational speed of the expa nder increases, frictional losses and internal leakage typically increase a nd decrease, respectively. An optimum operating condition exists where the product of the disp lacement volume of the expander in m3/rev and its rotational speed minimize friction, leakage losses and the manufact uring cost of the expander. In order to calculate the torque and powe r developed by the expander, the pressure forces acting on the vanes must be computed. If the pressure forces are assumed to act at the midway point of the vanes calculated pr otrusion from the rotor slots and no internal leakage losses are assumed past the vane tips, the following expression for the torque developed by one vane at any angular displacement may be written 2v vvcvcvRX TXLPPr (4-23) If frictional losses are also assumed negligib le, the average of the sum of the torque developed by the expander at any angular displacement multiplied by the rotational speed is the power that may be extracted from the expander. The resultant power as a function of angular displacement may be expressed as 2 21 1221 exp 0()()inex inexin cvexPVV muuPVV W ttt (4-24)

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136 The appropriate selection of the geometry and rotational speed of the expander ultimately depends on the application. Parameters of interest for typical operating conditions for a basic and modified refrigera tion system are tabulated in Table 4-3. The apparent advantages of employing an expansion device include reduction in mass flow-rate and process volume ratio, incr ease in the system COP, refrigerating efficiency and refrigerating effect. The net-wo rk required by the system also decreases if the expander work is assumed to be ideally coupled to the comp ressor shaft. As the efficiency of the expansion device increases, it can be seen that further decreases in the mass flow-rate and process volume ratio ar e obtained. Similar trends for various operating conditions, component efficiencies, degrees of su perheating and subcooling are expected. In order to determine the work extracti on potential of an expander with a known geometry, an alternative numerical solution pr ocedure to that detailed in the charging, expansion and exhaust sections must be ut ilized. The charging and exhaust processes were assumed to be isobaric and hence th e mass flow-rate of refrigerant is unknown. There exists, however, a unique value of the incoming and leaving refrigerant mass flowrate at which the power extr action is optimum. The mass flow -rate is hence iterated upon in order to satisfy Equations (4-12) and (4-19). Cerpnalkovski (1991) suggested typical va lues of aspect ratio (rR/L) of a rotaryvane compressor of 0.5-0.8. For the purpose of this study the aspect ratio was assumed to be 0.46. The eccentricity of the rotor and stator cylinders was also assumed to be 2.6 mm. The dimensions of the expander are assumed to be 28 mm, 30.6 mm and 12.7 mm for the

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137 rotor radius, stator-cylinder radius and the axial length of the expander respectively. The sealing arc is assumed to have an arc-length of seal=40o. The most important geometrical design parameter, which ensures adequate size and performance of any expansion device, is th e built-in volume ratio. Figure 4-6 shows the variation of the built-in volume ratio as a function of the inta ke angle for various numbers of vanes. The intake angle is found to be the most critical parameter that influences the performance of a rotary-vane type expande r. For all intake angles analyzed, the thermodynamic performance and built-in volume ratio are maximum when the exhaust angle is ex=189o. The volume ratio is found to decrea se as the intake angle increases independent of the expanders number of vanes. As the number of vanes in the expander increases this decrease is found to increas e dramatically. The increase in the number vanes may also be limited by the dynamics a nd mechanical strength of the rotor and the added frictional and internal leakage losses that may result. The e xpanders intake angle should hence be designed to optimize both the thermodynamic and geometrical performance. A single value of the intake angle may not optimize both simultaneously, however. Application dependent optimization is thus necessary. Although smaller intake angles result in larger built-in volume ratios, higher throttling losses are typically associated with them. The expanders built-in volume ratio can be seen to decrease by 40%, 58.3% and 67% for the entire range of in take angles investigated for 4,8 and 12 vane expanders, respectively. Medi an values of the intake angle ( in=15o) and number of vanes (N=8) have been selected in order to avoid the aforementioned complications at which the volume ratio is rv,b=5.4305.

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138 The displacement volume of a circular ro tary-vane expander may be computed for any given set of input parameters detailed in Table 4-3. The control volume analyzed is defined as the volume bounded by the stator-cyl inder, rotor, leadi ng and trailing vanes and the two axial end-plates. Fi gure 4-7 illustrates the ideal cell volume variation as a function of angular displacement for different numbers of vanes. The displacement volume per revolution is the cut-off volume multiplied by the number of vanes. Figure 48 shows the difference between the ideal and actual volume. By accounting for the thickness of the vanes, tv=4.2 mm, a slight shift in the volume curve is observed. This results in a deviation by as much as 89% and by as little as 1% at initial and at larger angular displacements. It can also be seen from Figure 4-8 that the maximum ideal and actual volume occurs at =184o and =180o respectively. The condensing and evaporati ng pressures corresponding to 50oC and 0oC condensing and evaporating temperatures ar e 1.942 and 0.498 MPa, respectively. This corresponds to a process volum e ratio of 14.74 if the refrigerant is assumed to be expanded isentropically. Figures 4-9 and 4-10 illust rate the variation of the pressure within the expander as a func tion of the volume within the expander for various intake angles and numbers of vanes, respectively. Th e pressure is found to be greater than the desired exhaust pressure primar ily because the built-in volume ratio is not equivalent to the process volume ratio. From Figure 4-9, it can be shown that built-in volume ratios of 8.16, 6.6 and 5.43 corresponding to intake angles of 5o, 10o and 15o, respectively, correspond, to exhaust pressures 15%, 32.2% and 40% higher than the desired exhaust pressure. Figure 4-10, illustrates how the numbe r of vanes affects th e pressure variation in an expander for a given intake angle. It can be shown that bu ilt-in volume ratios of

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139 4.083, 5.43 and 6.6 corresponding to 6, 8 and 10 vane expanders result in 53%, 40% and 26% higher exhaust pressures, respectively. These differences are primarily due to inadequate built-in volume ratios of th e expander and result in less desirable thermodynamic performance. Unless a throttl e valve or another means of expanding the discharging fluid to the eva porating pressure is used, fl ashing and/or other losses are imminent. For the purpose of this ideal anal ysis the losses due to under-expansion are not included in the models developed and it is a ssumed that a throttle valve further expands any under-expanded refrigerant to the evaporator pressure. Independent of the built-in volume ratio, the optimum performance of an expander is expected to occur when the mass flow of the working fluid is maximum. The maximum mass flow-rate will likely have to be greater if the built-in volume ratio of the expander is inadequate. Accurate models fo r the mass flow-rate of an actual expander may be developed in which inlet area, exhaust area, velocity and dens ity variations as a function of angular displacement are modele d. Figure 4-11 shows the variation of the mass trapped in the expanders control volume as a function of angular displacement for various intake angles. It can be seen that as the intake angle increases, the greater the amount of mass in the control vo lume during the expansion pro cess. This results from the fact that the expanders cut-off is delayed. These delays in cut-off are represented as lines (a), (b) and (c) respectively. It is found that the mass in the control volume during expansion is 50% and 28% hi gher for intake angles of =25o and =10o, respectively, when compared with that of =5o. Figures 4-12 and 4-13 show the variati on of power extracted during the process expansion as a function of angular displacemen t for various intake angles and different

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140 numbers of vanes. Figure 4-12 shows that the maximum power extracted prior to the exhaust process is 1440 W, 742 W and 534 W for 6, 8 and 10 vane expanders, respectively. At the conclusion of the exha ust process, the net power output from the expander is 355 W, 305 W and 257 W corresponding to 6, 8 and 10 vane expanders respectively. It can also be seen from Figur e 4-13 that as the intake angle increases the maximum power extracted prior to the exha ust process is 611 W, 690 W, 742 W and 794 W corresponding to in take angles of 5o, 10o, 15o and 20o, respectively. At the conclusion of the exhaust process, the net power output fr om the expander can be seen to be 309 W, 307 W, 305 W and 297 W correspondi ng to intake angles of 5o, 10o, 15o and 20o respectively. Figure 4-14 show s the variation of the power extracted as a function of volume for a particular set of input paramete rs. The same general trends of Figures 4-12 and 4-13 is observed. Theoretical simulation models have been developed to analyze and evaluate the geometric and thermodynamic performance of a circular rotary-vane expander. The following conclusions can be made: The built-in volume ratio can be seen to decrease by 40%, 58.3% and 67% for the entire range of intake angles investigated for 4,8 and 12 vane expanders respectively. The increase of the numbe r vanes may be limited by the dynamics and mechanical strength of the rotor and the added friction and internal leakage that may result. Lower intake angles result in larger built-in volu me ratios but higher throttling losses are typically associated with them. As the spread of the intake angle decreases and the number of vanes increases, hence an increase in the built-in volume ratio, the resultant exhaust pressure is found to reach the evaporating pressure. Th ese differences are primarily due to the inadequate built-in volume ratio of th e expander and result in less desirable thermodynamic performance. Unless a throttle valve or another means of expanding the exhaust fluid down to the evaporating pressure is used, flashing a nd/or other losses are imminent when the refrigerant is under-expanded.

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141 As the number of vanes increases, the net power output of the expander decreases. For the cases investigated, as much as 75% of the power extracted prior to the exhaust process is consumed by the exha ust process. Compared to a 10-vane expander the net work output of a 6-vane expander is found to be greater by as much as 27% As the intake angle increases, the maxi mum power extracted prior to the exhaust process increases. At the conclusion of th e exhaust process, the net power output from the expander is found to deviate only 2% from the mean. The thermodynamic model developed in this section was ideal in its assumptions and analysis. It did not take into account losses that may ar ise due to thro ttling in the intake and exhaust ports, two-phase internal leakage, friction, heat transfer, recompression or irreversible losses due to overor under-expansion caused by inadequate expander sizing. Chapter 5 details the th eory and presents models of those aforementioned loss mechanisms. Once describe d, the models for these loss mechanisms will modify the thermodynamic model develope d in this chapter to provide an accurate estimate of the expanders detailed and overall performance.

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142 Figure 4-1. Flow diagram of the main computer program developed Geometric and Kinematic Model Thermo-physical Model Friction Model Internal Leakage Model Throttling Model Expander Model Inputs: (from Vapor Compression System Model) -Pressure Temperature Outputs: (to Vapor Compression System Model) -Pressure -Temperature -Power Output -Volumetric Efficiency -Tor q ue

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143Table 4-1. Summary of rotary-vane literature Analysis No. Investigator Application Working Fluid Eq of StateGeometrical Thermodynamic Friction Internal Leakage HeatTransfer Lubrication 1 Peterson and McGahan (1972) Compressor Air Ideal-gas x x x x 2 Badr et al. (1985a) Expander in ORC General x 3 Badr et al. (1985b) Expander in ORC R-113 Ideal-gas x 4 Badr et al. (1985c) Expander in ORC R-113 Ideal-gas x 5 Badr et al. (1986a) Expander in ORC R-113 Ideal-gas x 6 Badr et al. (1986b) Expander in ORC R-113 Ideal-gas x 7 Edwards and McDonald (1972) Expander in ROVACAir Ideal-gas x 8 Robertson and Wolgemuth (1978) Expander in RC Steam Ideal-gas x x x 9 Beck et al. (1966) Compressor in VC R-12 vapor na x 10 Marsters and Ogbuefi (1972) Air Motor Air Ideal-gas x x 11 Somayajulu (1971) Pump Air Ideal-gas x x x 12 Ben-Bassat and Wolgemuth (1972)2-stage Exp in RC Steam K & K* x x x 13 Wolgemuth and Olson (1971) Expander in RC Steam K & K* x x x 14 Jacazio et al. (1979) Air Motor Air Ideal-gas x x 15 Bransford and Stein (1960) Compressor in VC Air and R-12 Ideal-gas x x 16 Robertson and Wolgemuth (1975) Expander in RC Steam Ideal-gas x x x x x 17 Barszcz, Z. (1980) Air Motor Air General x x VC: Vapor Compression refrigeration system RC: Rankine Cycle ORC: Organic Rankine Cycle ROVAC: ROtary Vane Air refrigeration Cycle *Keenan and Keyes equation of state for steam

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144 Figure 4-2. Schematic of a generally orie nted circular rotary-vane expander with 8 vanes described by two circular arcs Sealing arc Trailing vane STATOR Leading vane ROTOR in ex OS OR e rRV2 V1 rS seal RS( ) R( ) Charging Process 0< < + Expansion Process + ex Exhaust Process ex< <2 Base Line =0

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145Table 4-2. Geometrical input parameters Input Parameter Symbol Rotor radius, m rR Eccentricity, m e Angle between rotor and stator-cylinder centers Stator cylinder radius, m rS=rR+e Axial length of expander, m L=AR.rR Number of vanes N Angle between successive vanes, deg =360o/N Thickness of vanes, m tv Inlet port arc-length, deg in Exhaust port arc-length, deg ex Arc-length of sealing arc, deg seal

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146 Figure 4-3. Variation of the inlet throat area with respect to angular displacement Figure 4-4. Variation of the exit throat area as a function of angular displacement for an 8-vane circular MVE 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 020406080100 Angular Displacement, degInlet Throat Area, m2 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 150200250300350400 Angular Displacement, degExit Throat Area, m2

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147 Figure 4-5. Pressure variation as a function of volume in an ideal expander for the cases of ideal, over and under expansion. + exP Pin buit v process vr r, P Vint ex in in Ideal Expansion UnderExpansion Over Expansion in buit v process vr r, in buit v process vr r, + exP Pin buit v process vr r, P Vint ex in in Ideal Expansion UnderExpansion Over Expansion in buit v process vr r, in buit v process vr r, + exP Pin buit v process vr r, P Vint ex in in Ideal Expansion UnderExpansion Over Expansion in buit v process vr r, in buit v process vr r, exP Pin buit v process vr r, P Vint ex in in Ideal Expansion UnderExpansion Over Expansion in buit v process vr r, in buit v process vr r, P Pin buit v process vr r, P Vint ex in in Ideal Expansion UnderExpansion Over Expansion in buit v process vr r, in buit v process vr r, in buit v process vr r, P Vint ex in in Ideal Expansion UnderExpansion Over Expansion in buit v process vr r, in buit v process vr r,

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148 Table 4-3. Typical operating conditions for a basic and modified 2 ton system *R-22, 2 ton cooling capacity, Tcond=50oC, Tevap=0oC, c=0.85, no superheat, no subcooling, pressure losses in lines and components neglected Expander Isenthalpic Expansion exp=0.45 exp=0.85 Isentropic Expansion ,g/s 49.6 48.61 47.77 47.5 v3, m3/kg 0.000924 0.000924 0.000924 0.000924 rv,p 16.3 15.6 14.97 14.74 COP 3.52 3.87 4.23 4.38 ref, % 64.4 70.8 77.4 80.1 wexp, kJ/kg 2.89 5.46 6.42 wnet, kJ/kg 40.3 37.4 34.8 33.8 qeva p kJ/kg 141.1 144.7 147.3 148.2 refm

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149 Figure 4-6. Variation of built-i n volume ratio as a function of intake angle for different numbers of vanes Figure 4-7. Variation of ideal volume as a function of angular displacement for different numbers of vanes 0.0E+00 5.0E-07 1.0E-06 1.5E-06 2.0E-06 2.5E-06 3.0E-06 3.5E-06060120180240300360 Angular Displacement degIdeal Volume, m3 N=4 N=6 N=12 N=8 N=10 A R=0.46 rR=27.9 mm e=2.7 mm in=15oout=189o 0 2 4 6 8 10 1015202530 Intake Angle in, degBuil-in Volume Ratio rv,b N=4 N=6 N=8 N=10 N=12

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150 Figure 4-8. Variation of ideal and actual volumes as a function of angular displacement Figure 4-9. Variation of pre ssure as a function of volume for various intake angles 0 500 1000 1500 2000 25000.0E+005.0E-071.0E-061.5E-062.0E-06Volume, m3Pressure, kPa A R=0.46 rR=27.9 mm e=2.7 mm N=8 out=189oPinletPexhaust} Losses}} in=5oin=10o in=15o 0.0E+00 4.0E-07 8.0E-07 1.2E-06 1.6E-06 2.0E-060100200300400 Angular Displacement degVolume, m3 Ideal ActualAR=0.46 rR=27.9 mm e=2.7 mm N=8 in=15oout=189o

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151 Figure 4-10. Variation of pressu re as a function of volume fo r different numbers of vanes Figure 4-11. Variation of mass within the expanders cell volume as a function of angular displacement for various intake angles 0 500 1000 1500 2000 2500 0.0E+005.0E-071.0E-061.5E-062.0E-062.5E-06 Volume, m3Pressure, kPaPinletAR=0.46 rR=27.9 mm e=2.7 mm in=15oout=189oN=10 N=6 N=8 0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-040100200300 Angular Displacement degmcv( ), kgin=25oin=5oin=10o Expansion Exhaust Intake (a) (b) (c) AR=0.46 rR=27.9 mm e=2.7 mm N =8 =58 rpm out=189o

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152 Figure 4-12. Variation of power as a func tion of angular displacement for different numbers of vanes Figure 4-13. Variation of power as a function of angular displacemen t for various intake angles 0 0.2 0.4 0.6 0.8 1 1.2 0120240360 Angular Displacement ,degPower, kWN=10 N=8 N=6 Net Power Output AR=0.46 rR=27.9 mm e=2.7 mm =58 rpm in=15o out=189o 0 0.2 0.4 0.6 0.8 1 0120240360 Angular Displacement degPower, kWin=20o A R=0.46 rR=27.9 mm e=2.7 mm N=8 =58 rpm out=189oin=5oin=10o in=15o

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153 Figure 4-14. Variation of power as a function of volume 0 0.2 0.4 0.6 0.8 0.0E+005.0E-071.0E-061.5E-062.0E-06Volume, m3Power, kW Intake and Expansion Exhaust Net Work Output AR=0.46 rR=27.9 mm e=2.7 mm N=8 =58 rpm in=15oout=189o

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154 CHAPTER 5 PRIMARY AND SECONDARY LOSS MECH ANISMS IN ROTARY-VANE TWOPHASE REFRIGERATING EXPANDERS All turbomachinery operate with energy and mass losses. These losses may be attributed to two-phase internal leakage, friction, throttling losses in the intake and exhaust ports, re-compression in the expander cavity and overor under-expansion due to inadequate sizing of the geometric volume ratio. Primary Loss Mechanisms The aim of this chapter is to develop ma thematical models that would account for the primary loss mechanisms of internal leak age and friction. The size of a rotary-vane expander is of primary interest due to th e influence it has on the magnitude of loss mechanisms present. While rotary-vane comp ressors are typically very efficient, at smaller sizes, the effect of losses such as in ternal leakage becomes much larger. In rotaryvane expanders of smaller sizes both the volumetric and adiabatic efficiencies are expected to be smaller. Depending on the ope rational and geometrical parameters of the expander, laminar and viscous two-phase le akage flow within the expander may be present. Single-phase leakage models availabl e in the literature must be revisited and modified accordingly. A dynamic frictional model for the expander must also be developed for ideal operation (i.e. no intern al leakage) and modified to account for internal leakage accordingly. A comprehens ive component-level model of inherent friction and internal leakage losses in a two-pha se circular rotary-van e expander used in a vapor compression refrigeration system is presented. The model establishes the efficiency and performance of the expander as a func tion of geometric and fluid parameters. Accurate modeling and prediction of frictiona l and internal leakag e losses is vital to

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155 being able to accurately estimate the effici ency, rotational speed, and the torque and power produced by the expander. Internal Leakage Paths and Clearances The clearances between the rotor and stator cylinders, end-plates and vanes provide the necessary seals of the expander cavity. Ca ution must be exercised when designing an expanders assembly clearances. Depending on the operating conditions and geometric volume ratio, large temperature gradients may al so lead to thermal deformations of these materials. The clearances must be minimized for each application to ensure control of internal leakage losses yet re -compression of liquid may lead to inadequate operation or liquid locking. The various leakage path s are tabulated in Table 5-1. The volumetric efficiency of an expande r may be larger than unity due to significant increases in the consumption of the working fluid. This significant increase is primarily a result of internal leakage losses. The magnitude and sources of internal leakage must be identified and minimized. The magnitude and sources of internal leakage is significantly dependent upon the way by which the working fl uid is introduced into the expanders cavity. Figure 5-1 illustrates th e difference between th e conventional (a) and modified (b) methods through which the fluid is introduced into the expanders cavity. The conventional intake port is machined directly into the stator-cylinder. The modified intake is comprised of slots machin ed in both end-plates after which the fluid is introduced into the expanders cavity via a slot machined in to the rotor at a specific angle from the slot. Upon the completion of the geometric, kinematic and thermodynamic models, the predicted pressure and quality of the refrig erant in the expanders control volume are used to calculate the leakage of refrigerant through the various leakage flow paths. In

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156 order to model internal leakage between cont rol volumes, the flow areas of the leakage paths, Aleak, must be estimated. The mass flow-rate of internal leakage can be written in general form as leakdleakleakleakmCAu (5-1) where Cd is an empirical discharge coefficient. The density and velocity of the leakage are a function of the angular displaceme nt at which the leakage occurs and the thermodynamic state of the fluid. Of primary in terest is the computa tion of the velocity. The net mass flow leaked into the cavity form ed by the adjacent vanes, stator and rotor cylinders (see Figure 5-1) can be computed as leakleakleakmmm (5-2) Leakage from Path (6) is neglected. It s hould be noted that if the sealing arc is symmetric about the base line then =180o and RS( )=rS. Types of Two-Phase Leakage Losses The majority of internal leakage models in turbo-machinery presented in recent literature are developed for single-phase comp ressible flow. The friction associated with incompressible leakage losses through a duct or leakage path of constant cross-sectional area affects only pressure in the direction of the flow. The velocity of the working fluid remains constant. According to Oothusizen ( 1997), in compressible flow, friction affects all of the flow variables, i.e., the changes in pressure cause changes in density which lead to changes in velocity For two-phase flow th is also necessitates vaporization of liquid, which alters the density of the working fluid significantly. Depending on the operational and geometrical parameters of the expander, turbulent two-phase leak age flow within the expander may be present.

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157 In the case of two-phase viscous leakage losses, the two-phase mixture is assumed to be homogenous in nature. The idealizatio ns adopted under this fairly common model include assuming that the vapor and liquid velocities are equal. The over-all two-phase flow is also assumed to behave like a single phase, uniformly mixed, having fluid properties whose values are, in some sense, mean values for the flow according to Carey (1992). According to Levy (1999), assuming hom ogenous flow also allows the ability to apply all available single-phase flow analys es and empirical correlations The density of the homogenous two-phase mixture is computed as 11 g f x x (5-3) The calculation of other necessa ry properties for the two-phase mixture, such as the specific heats and viscosity, can be very tr oublesome. Many different formulations have been proposed throughout th e literature. Among them, 1 f g x x (5-4) 11 g f x x (5-5) 1 g f gfxx (5-6) For example, a two-phase mixture of water at P=500 kPa and x=0.2 yields a viscosity of 146.09, 53.52 and 15.968 Pa.s from Equations (5-4), (5-5) and (5-6), respectively. According to Carey (1992), E quation (5-5) is the most comm on. Properties can also be computed by utilizing the concept of the homogenous void fraction

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158 1f f gx xx (5-7) and hence the dynamic viscosity can be computed from 1fg This method will be used since the computed value is identi cal to that obtained from Equation (5-6). In the same manner the specific heats and thermal conductivity can be computed. Viscous effects must be accounted for when computing the pressure drop and mass flow-rate through radial clearances, e.g. flow through minimum clearance. These channels are typically long a nd wide when compared to the height of the channel. Viscous effects may be neglected when deali ng with nozzles or short ducts. The axial leakage flow between the vanes and the end-plates is an example where viscous effects may be neglected. The flow of the twophase compressible leakage, accounting for friction, can be modeled as adiabatic Fanno-flow. Figure 5-2 shows a schematic of a typical leakage path. The dimensions of the leakage path, cross-sectiona l area and length, are known and depend on the leakage path under consideration. The exit, (e), could be choked depending on the difference between the pressure of the re servoir and back pressure at local angles and + respectively. Darby (2001) concludes that the ratio of the sonic velocity in a homogenous mixture to that in a gas alone is much smaller than un ity. In this case choking can occur in a twophase mixture at a significan tly higher downstream pressure. This corresponds to both a lower pressure drop and mass flux. The flow is choked when the velocity at the exit plane is equal to the speed of sound. For a non-flashing liquid and an idea l gas mixture, the maximum mass flow-rate

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159 in the channel can be computed. The isen tropic speed of sound for a homogenous twophase mixture can be computed as s P a (5-8) The governing equations and the discretized form of those equations are presented in Table 5-2 and are derived from first princi ples and describe adia batic Fanno flow from a reservoir. It should be noted that the vol ume of the leakage channel is V=Hwl and the mass flow-rate through the channel is eemuHw The Darcy friction factor, f is assumed constant over the length of the leakage channe l. This is a fairly reasonable assumption since the flow is typically found to be fully developed over a majority of the channel length. For Reynolds numbers in the lower ra nge of the turbulent region, the modified Blasius correlation is 0.250.33ReHf (5-9) where the Reynolds number is defined as Re H eeuHv This correlation has been shown to be in good agreement with various experiments throughout the literature. The momentum equation can ther efore be expressed by 1.75 2 1 0.751.250.165e eee elu PPu vH (5-10) The methodology used to solve for adiabati c Fanno-flow through a leakage path is an iterative procedure. The state of inle t to the channel is known. Depending on the leakage path and the state of the refrigerant, the flow may be of an incompressible liquid or a compressible two-phase refrigerant. The unknowns, in Equation (5-10), that remain

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160 are ue, m Pe and e. The iterative procedure calls for an initial guess at the exit plane of the channel where the flow is assumed to be choked. An initial guess of the pressure at the exit plane may correspond to the average pres sure that exists in the adjacent expander cell. The flow is assumed to be isenthalpic or isochoric in the case of two-phase or liquid flow, respectively. In the case of the two-pha se flow the quality is known. Based on the initial guesses, the speed of sound can be cal culated. If the flow is choked at the exit plane, i.e ue=a, then P is iterated upon until Equations (5-5), (5-8) and (5-10) are satisfied simultaneously. If the flow is viscous and the exit plane is unchoked, the average velocity of the leakage can be determined by solving generalized Couette fl ow, see Figure 5-3, the flow is driven by both the shear of a moving wall a nd a given pressure differential. Solving the momentum equation and applying the no-sl ip and moving wall boundary conditions the velocity distribution is 21 ()()mdPr uyyyy dx (5-11) The mean velocity between the plates is defined as 01m yuuydy (5-12) and after integration is found to be 2212m mrdP u dx (5-13) If the flow in the leakage path can be modeled as isentropic flow through a nozzle whose throat is choked then the initial guess used to evaluate the sonic velocity is 1 i e P P and 1 e s s The density is then calculated. The pressure and density at isentropic

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161 conditions, at the exit plane, are iterated upon accordingly until the computed speed of sound is identical to the relationship obtained from the energy equation, 12()eeuhh In the case that the isentropic flow is unchoked, the exit plane pr essure is equal to the backpressure, Pe=Pb and se=s1. The thermodynamic state at the exit plane can then be established and the velocity at the exit can be determined by Equation (5-8). Axial Clearance Between Rotor and End Plates The flow of the fluid between the rotor and end-plates can be modeled as that of a fluid flowing in a narrow gap between a disc which rotates within a chamber of finite dimensions. The boundary conditions imposed on the periphery of the rotating disk are non-axisymmetric in nature due to the pressure distribution of the fluid in the expanders cavity. The effects of the vanes and their slots are neglected. The characteristics of the flow as well as the complexity of the solu tion can vary signific antly depending on the order of magnitude of several key dimensi onless variables. The dimensionless governing equations yields key dimensionless groups that govern the flow and are found to be the local rotational (disk) Reynol ds number in terms of 1, the Euler number and the gap aspect ratio, S If the rotational speed of the rotor is in rad/s then these parameters are defined as 1 Rr S (5-14) 22 mR P Eu r (5-15) 2 1Rem (5-16)

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162 where P in the Euler number corresponds to th e maximum pressure differential on the rotors outer boundary, i.e. the intake and exha ust conditions. It should also be noted that in rotating disk problems the rotational Reynolds number is utilized in stead of the radial Reynolds number which is defined as 1Re R mrv For the same flow conditions, the use of the latter is typically three orders of magnitude larger. Daily and Nece (1960) have identified four different modes of flow for given geometrical and flow characte ristics. Of interest in th is study are the laminar and turbulent cases corresponding to a close ga p where boundary layers on the rotor and endplate are merged. In this case a continuous va riation of the velocity across the axial gap exists. The rotational Reynolds number must be small in the gap, 1Re1 for this type of flow to exist. Separate boundary layers exist when the axial gap exceeds the total thickness of the two boundary layers. The c ondition for laminar flow can therefore be estimated from Daily and Nece (1960) as 125Re10 S. The Euler number which is defined as the ratio of the maximum pressu re differential to the centrifugal pressure produced during rotation is assumed large, i.e.1Re1 Eu The gap aspect ratio must also be substantially larger than unity. If these four conditions can be satisfied simultaneously then a zeroth-order solution approximates th e solution to the Navier-Stokes equations. This solution consists of a power series of Re, e.g. 112 012ReRe... uuuu expanded around the asymptotic solution of Re approaching zero. Bein et al. (1976) concluded that this is a reasonable assumption for20 Eu It must be determined whether the followi ng analysis is valid for the flow of twophase fluid in the axial gap. Due to the vapor ization of liquid in the expanders cavity, the thermodynamic properties vary significantly. Co upled with the signi ficant variation of

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163 intake and exhaust parameters, geometry of the leakage gap and rotational speed of the expander, the aforementioned constraints may not be satisfied. In or der to determine the operational conditions under which the anal ysis is valid the following operational parameters considered are presented in Table 53. It can be shown that the flow is laminar for all parameters investigated at lower rotational speeds, i.e. <500 RPM. Depending on the geometric size of the expander, the expa nders actual rotational speed realistically doesnt exceed this value. Based on the anal ysis of these operational conditions, an order of magnitude analysis of the fl ow dimensionless parameters is presented in Table 5-3. It maybe concluded that the zerothorder solution is appropriate. The location of the phase in terface in rotating, eccentric complex geometries is very difficult to predict. If sufficient centrifugal forces are produced due to high rotational speeds it may be assumed that the ma jority of the leakage-flow in the axial gap is vapor. This arises due to th e tendency of the dense liquid to flow in the vicinity of the stator-cylinder wall due to significant centrifugal forces. Hence the incompressible form of the general equations with c onstant properties, evaluated at some mean value, is used. Neglecting body forces, terms of order 1/ S2 and equating terms with Re1, the zerothorder dimensionless 3-D Navier-Stokes equation s in cylindrical coordinates are solved in order to determine the velocity and pressure field within the narrow gap. If a gap width 1 exists between the rotor and the end plates then the dimensionless governing equations can be reduced and written as follows 00 011 0 vw ru rrrz (5-17) 12 00 2Re u p Eu zr (5-18)

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164 12 00 21 Re v p Eu zr (5-19) 00 p z (5-20) where the dimensionless variables are defi ned in the nomenclature. According to Equation (5-20), 00, PPr The boundary conditions can be written as follows, (,,0)0ourz 0(,,0)0 vrz (,,0)0owrz (5-21) (,,1)0ourz 0(,,1) vrzr (,,1)0owrz (5-22) (1,)()ocvPrP 0 0(,)(,)0 P urr r (5-23) where Pcv( ) is a linear piece-wise function that cl osely approximates the actual pressure distribution around the rotors outer boundary and is presented in Figure 5-4. A linear approximation first presented by Be in et al. (1976) an d furthered by Badr (1985) for a typical pressure profile is compared to nume rically integrated Fourier constants in Equation (5-25). It can be s hown that 100 terms in the summation of the Fourier constants yield the actual pressure distribution. Depending on the geometric volume ratio of the expander, the differen ce between the linear approximation and can vary significantly. An expande r with a volume ratio of rv=3.916 is analyzed in Figure 5-4

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165 and a maximum difference of 40% is found to exist between. This gr eatly influences the accuracy of the computed radial velocity. The zeroth-order solution of the following simplified set of equations and satisfying all necessary boundary conditions yields Bein et al. (1976) as such 12 0 0Re () 2 Eu P uzz r (5-24) 12 0 0Re 1 () 2 Eu P vzzrz r (5-25) Substituting Equations (5-24) and (5-25) into the continuity equation, Equation (5-17), and applying the boundary conditions in the axial direction yields 2 00 P (5-26) Solving Equation (5-26) by mean s of separation of variables, the general solution for the pressure distribution in the gap is 000 1(,)ln cos()sinnnnn nnnn nPrabr arbrncrdrn (5-27) The Fourier constants can be determined from the specification of the pressure distribution on the outer boundary of the rotor (see discussion of Fi gure 5-4). The radial velocity distribution can be found by differen tiating Equation (5-27) with respect to the radial direction and substituting that result into Equation (5-23). The total dimensional leakage mass flow-rate in the radial di rection with respect to the rotor is 2 ,10121,leakcvRmurr (5-28)

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166 The value of the mass flow-rate is negativ e if the flow leaves the expander cell. This model doesnt take into account choki ng, which is likely, since the velocity distribution at the rotors outer boundary is of primary interest. Leakage around tips of vanes Depending upon on the method by which the high pressure refrigerant is introduced into the expanders cavity; see Figure 5-1, leakage due to insufficient under-vane fluid pressure or inadequate force/absence of mechanical springs may lead to significant leakage losses several orders of magnitude larger than other leak age paths. Continuous and sufficient contact between the vanes and th e stator-cylinder is ne cessary to subdue or minimize this type of leakage. The majority of leakage of this type may occur during the intake process where low rotational speeds cause inadequate centrifugal forces to keep the vanes in contact with the stator-cylinder. The existence of both low rotational speeds and inadequate under-vane pressure can be pr edicted by the frictiona l model developed. The frictional model is presented later in this chapter. It should be noted that the reaction force, Fn( ) (see Figure 5-5), is pred icted by the friction model and accounts for, among other things, the sliding veloci ty and acceleration of the prot ruding vanes as well as the under-vane pressure. If the vane ha s lost contact with the stato r-cylinder, the value of this normal force is negative, and the leakage flow area would be computed as 2()()leak A LXL (5-29) where X( ) is the instantaneous protru sion of the vane from the rotor slot at the angular rotation of the rotor where the loss of contact has occurred. When the vanes establish contact with the cylinder it is assumed that the leakage area is negligible and that this occurs until the start of the sealing arc. There may exist

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167 leakage around the tips of the vanes during the spread of the sealing arc due to a clearance gap between the stator-cylinder and roto r that may be due to an assembly clearance or due to machining limitations. Th is radial leakage is treated in the next section. Due to the short nature of this leakage path, tv, the leakage of the two-phase fluid in this path is assumed to be th at of fluid in a convergent-di vergent nozzle, i.e. isentropic flow. It should be noted that if the incoming fluid is introduced into the roto rs slots than the internal leakage as sociated with weak under-vane pre ssure can be neglected. For this design, internal leakage around th e tips of the vanes can be neglected. The high-pressure incoming refrigerant is first used to provi de sufficient under-vane pressure so that continuous contact between the vane a nd stator-cylinder may be achieved. Leakage past the sides of the vanes In general, the axial width of this leakage path may be estimated by accounting for the thermal expansion of the vanes in addition to the gap that exists due to the thickness of gaskets, g and other assembly clearances. The operational length of the vane can be computed from the known materials coe fficient of linear thermal expansion, ,, vvovvo L LLT, and the temperature difference that the vane encounters relative to ambient conditions. The mean temperature of the vane is computed as the average temperature of two adjacent cells. It can be shown that even at the moderate temperature differences realized within an expanders cavi ty this effect accounts for a 0.3% change in the length of the vane and is therefore neglected. The axial width of the leakage path is assumed constant and equivale nt to the gasket thickness 3 g (5-30)

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168 The height of this leakage channel is estim ated by subtracting the height of the gap realized due to leakage around th e vane tips, if any, upon loss of contact with the statorcylinder, 2( ). The total leakage flow area is then calculated as 23()2()leakAX (5-31) Due to the short nature of this leakage path, tv, the leakage of the two-phase fluid in this path is assumed to be that of fluid in a c onvergent-divergent nozzle, i.e. isentropic flow. Leakage between the faces of the vanes a nd the side walls of the rotors slots The leakage from this path is typically ne glected due to the exis tence of two contact lines across the axial length of the rotor slots. These contact lines result from the reaction forces FL ) and FR( ) (see Figure 5-5). Negative reaction forces predicted by the friction model represent a loss of contact between the vane and the rotor sl ot. The control volume to which the mass and energy are added or subtracted depends on which vane face the loss of contact occurs. If the fl uid intake is throug h the rotor slot the leakage through this path may be significant and comparable to other paths. Viscous effects need to be considered since the leakage fl ow path is wide and long, Hv. The leakage flow area is computed as ()()leaksvAttL (5-32) The Leakage from the expander to the cavity be neath the vane may also be comparable if ts-tv is large. Leakage in radial gap between the rotor and stator cylinder As the pressure differential between the in take and exhaust ports increases and the number of vanes decreases, this leakage is t ypically the largest contributor to internal leakage.

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169 This leakage flow path arises if the angl e between successive vanes is larger than the spread of the sealing arc (spread of minimal clearance) between the rotor and statorcylinder i.e. seal If this is the case, the expander ce ll is in simultaneous contact with the intake and exhaust ports. This overlap causes leakage flow from the inlet to the exhaust ports against the direction of rotation. The severity of this leakage path is a function of the number of vanes and the mean radial gap height 4 that exists between the rotor and stator-cylinder due to assembly operations or machining limitations. This leakage flow path would occur for the leading vanes angular displacement of 0 s eal The flow in this leakage path is modeled as adiabatic, one-dimensional Fanno-flow. It should be noted that leakage through this path can only be neglected if sufficient centrifugal forces or under-vane pre ssure exists to mainta in continuous contact of the vane with the stator-c ylinder. Typically an insuffici ent under-vane force exists due to the absence of high-pressure fluid (or presen ce of low-pressure re sidual fluid) due to the theoretical completion of the exhaust pr ocess. The magnitude of the normal force Fn( ) determined by the frictional model aids in this prediction. The flow in this leakage path would necessarily be choked for la rge geometric volume ratios or uncontrolled expansion in the exhaust por t if no throttle valve is dow nstream of the expander. Friction Model Prediction of the power output and devel oped torque of the rotary-vane expander requires accurate modeling of frictional loss es incurred in the expander. The primary frictional losses in the rotary-vane expander result from: Rubbing of the vanes agai nst the stat or-cylinder Rubbing of the vanes against the walls of their rotors slots

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170 The resultant viscous drag acting on both side s of the rotor from the leakage flows through the axial gaps between the rotor and the end-plates. A steady-state mathematical model that pred icts frictional losses within a rotaryvane expander must be coupled with geomet ric, thermodynamic and to leakage models in order to provide for a better estimate of th e mechanical efficiency and actual power produced by the expander. A general dynamic mo del that incorporates the main sources of friction within the rotary-vane expander is pr esented in this section. Effects of pressure forces, reaction forces, sliding friction, viscous drag, gravity and inertial forces must all be accounted to understand the relevance and ma gnitude of these effects. Figure 5-5 is a schematic of a free body diagram of a vane pr otruding out a rotor slot at a local angle In order to model frictional losses in a ro tary-vane expander, vane forces resulting from pressure loading, vane inertia a nd body forces due to radial and Coriolis accelerations must be determined. Sliding fric tion must also be considered between the vanes and the rotor slots and stator-cyli nder. These forces and frictional losses continuously change, in both magnitude and direction throughout the expander due to continuous changes in complex geometry. The dynamic model developed assumes that the vanes are rigid and that the coefficient of friction is constant. The c onditions for dynamic equilibrium are solved simultaneously to determine the unknown reaction forces based on a free-body diagram of a single vane. Upon determining the reac tion forces, the power required to overcome the resisting frictional forces can be computed. Below is a detailed description of the various forces as well as simplifying assumptions that have been made. A pressure force on the base of the vanes is assumed to act at mid-thickness of the vane as a result of incoming fluid pressure or a spring force utilized to maintain vane-tip

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171 and stator-cylinder contact. This pressure fo rce at the vane base can be expressed as follows b inletvFPtL (5-33) where is a proportionality consta nt that can be evaluated from calculating throttling losses to or from the cavity beneath the vane. If the fluid is introduced directly into the rotor slots for the spread of the intake port, in, and assuming no throttling losses then =1. If the fluid is introduced into the expa nders cavity directly then throttled leakage flow will provide the pressure force on the vane base (e.g. =0.75). If correctly sized and assembled springs or a means for adequate pressurization of the slots are used then =1. The pressure forces exerted on the vanes from either side FPR and FPL can be expressed as PLcvFPXL and PRcvFPXL respectively. If leakage losses through the vane rotor-slot gaps are neglected then these pressure forces are assumed to act mid-way between the ro tors outer boundary a nd stator-cylinders wall. If a gap exists between the vanes and the rotor slots, the orientation of the pressureloaded vane, see exaggerated effect in Fi gure 5-5, provides two sealing surfaces. The portion of the vane that protrude s from the slot is denoted by X( ). The reaction forces FL( ) and FR( ) are assumed to act at the points of contact of the vane and the rotor slots on the left and right faces of the slot, respectively. These forces are actually distribute d along a finite distance along the side of the vane and not at a point depending on the vane-to-slot clearance. Since the vanes are assumed to be rigid, the friction forces are assume d to act along the surfaces as rLF and rRF respectively. The subscript r denotes th e rotor material. The reaction forces Fn( ) and

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172 Ft( ) result from the vane-tip and stator-cylinder contact where nstFF The vanes are assumed to be in continuous contact with the stator-cylinder. Th e subscript s denotes the stator-cylinder material. The assu mption that the friction force (e.g. rLF) is proportional to the normal contact force (e.g. FL) may not accurate since large local velocity gradients at the inte rface may severally alter the fric tional losses. This may also be the case at the vane-tip and stator-cylinder interface. It should be noted that Figure 5-5 is valid for a vane that has no vane-tip curvature. If the vane-tip is round or altered in any wa y there will exist symmetric radial forces on either side of the normal force equal to the pressure on that side of the vane multiplied by the projected area of that surface. The sliding and centrifugal acc elerations of the moving vane from the rotor slot result in a radial body force Fa( ) which can be expressed as 2 avmFmAcc()r (5-34) where rm( ) denotes the distance from the center of the rotor to the center of mass of the vane. The mean tangential body force Fc( ) that arises due to Cori olis acceleration of the vane can be expressed as cvF2mVel (5-35) where Vel( ) and Acc( ) represent the sliding velocity and acceleration of the vane respectively. The mass of the vane is assumed to act at the geometric center of the vane (i.e. Hv/2 and tv/2). From the following free-body diagram, Figur e 5-5, the three conditions of dynamic equilibrium can be written as follows

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173 brLrRvanFFFmgcosFF0 (5-36) LvcRPLPRsnFmgsinFFFFF0 (5-37) vv vcRvPLPRv vv snvrLrRX HH mgsinFFHXFFH 222 tt FHFF0 22 (5-38) where Equations (5-36) and (5-37) represent th e net summation of forces in the radial and tangential directions equal to zero respect ively. Equation (5-38) represents the net summation of moments about point O equal to zero wherein th e direction of rotation of the expander is assumed positive. It should be noted that the direc tion of some of the forces change depending on whether the vane s are protruding out dur ing the intake and expansion processes or forced back into th e vane slots during th e exhaust process. The three equations can be simplified, r earranged and written in matrix form as aXb (5-39) where matrix a represents the coefficients of the unknown reaction forces of matrix X= L R nF F F and matrix b contains known parameters such as pressure and body forces. Once this matrix is solved at each angular displacement of the leading vane the reaction forces can be determined. The reacti on forces in matrix X reveal the movement of the vane in the rotors slot and can predict the loss of contact with the stator-cylinder or faces of the rotor slots. It should be noted that the coefficients in matrix a should be modified according to the vanes inward or outward movement relative to the slot. The

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174 mechanical efficiency and power losses of the expander due to friction can then be predicted. The frictional tor que per angular displacement due to vane-tip (first term) and net vane-slot friction (second term) can be computed from r sfsnRLRTF()rX()F()F()Vel() (5-40) The total frictional loss, Tsf,, in the expander is com puted by summing Equation (540) through a complete revolution of the ro tor. The contribution of each vane is accounted for and its contributi on to the total tor que is then averaged to yield Tsf,m. No rubbing frictional forces exist between th e sides of the vanes and the end plates due to the axial gaps between the rotor a nd the end-plates. The leakage through those gaps, as determined previously, will lead to viscous drag. In order to model the viscous drag due to the axial leakage between the rotor and end-plates the viscosity of the two-phase flui d must be predicted accurately. The pressure distribution in the axial gap P(r, ) was derived previously. The pressure is assumed to be invariant in the axial direction. The state of the two-phase liquid can be determined by assuming isenthalpic expansion in the radial and azimuthal directions. The flow in the axial gap is driven by both shear caused by the rotation of the disk and a pressure differential between the intake and exhaust ports. This results in a combined CouettePouisselle (generalized Couette) flow, see Figure 5-6, in which the radial velocity can be expressed as 2 1 11 ()()mdPr uzzzz dr (5-41) The total torque produced by viscous drag on both sides of the ro tor is expressed by

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175 10 ,22RRrr vdm r rzdu Trrdr dz (5-42) which can be integrated numerically. The total power loss is then calculated by losssf,mvdPTT (5-43) where is the angular speed of the expander in RPM. Uncertainties regarding the gap size ar e a significant obstacle to quantifying the magnitude of leakage losses through certain leakage paths. These uncertainties are presented later. The base case simulations de tailed below are valid for an eight vane circular expander of general orientation (i.e. =161.5 and ()SRSr ) with a geometric volume ratio of rv=3.26 corresponding to in=35o and ex=189o whose physical dimensions mimic a modified GAST NL32-NCC-1 air-motor. Th e rotational speed of the expander is =300 RPM. The inlet conditions to the expander correspond to a saturated liquid at a condensing temperature of T=45oC. As previously mentioned, the estimated fr ictional forces within the expander are used to predict the movement of the vanes w ithin their rotor slots. The profile of the vane-tip is of significant importance in de termining the forces acting on the vane-tip. Figure 5-7 shows the instantaneous variation of reaction forces on a single vane with no vane-tip curvature during one revolution of the rotor. It is assumed that a circular pressurizing grove ensures that =1 through out one revolution. It can be seen that the normal force tends to the same variation as th e pressure inside the expanders cavity due to the significant contribution of the under-vane pressure and absence of resisting forces in the opposite direction. It should be noted that no loss of contact is predicted for any of the reaction forces. It can be shown that for vanes with no vane-tip curvature an

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176 insignificant under-vane pressure ( =0.1) is necessary to maintain contact with the statorcylinder. Figure 5-8 is similar in nature to Figure 57 but for a vane that has curvature at the vane-tip (i.e. addition of symmetr ic radial forces about normal force). It can be seen that the normal reaction force is negative, denoting a loss of contact, during in+ < < corresponding to a majority of the expansion process. The increased pressure gradient across the vane in this case may lead to excessive leakage losses. This information is passed to the appropriate internal leakage c ode. The use of materials such as graphite, erode and match the profile of the stator-cy linder depending on the intensity of the undervane pressure. The density of the material should be large enough to provide sufficient centrifugal forces if and when inad equate under-vane pr essures exist. Simulations were run to find the minimum value of in order to obtain a positive reaction force at the vane-tip stator-cylinde r interface denoting contact with the statorcylinder. Values of =0.75, 0.85 and 0.95 were simulate d. It was found that due to the curvature of the vane-tip, only a value of >0.95 would suffice. Figures 5-9 through 5-12 show the variation of leakage to and from the expander cavity by different leakage paths through one revolution of the rotor. Figure 5-9 shows the variation of non-ax isymmetric leakage from the expander cavity as a function of angular displacement for different intake angles. It is ensured that the governing dimensionless vari ables satisfy the c onstraints and assumptions that govern the flow. The use of choked flow relations for this leakage path does not grasp the underlying physics; magnitude and direction, as sociated with the flow between a rotating and stationary disk with non-axisymmetric boundary conditions. It can be shown that if

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177 choked flow conditions are assumed at the rotor boundary then the leakage from, model cannot predict otherwise, the expander cavity is over predic ted throughout the majority of the intake and exhaust processes. The curre nt analysis over-predicts the cell leakage when compared to the choked flow model duri ng the expansion process solely due to the accuracy of the pressure pr ofile used to approximate the non-axisymmetric boundary condition. Figure 5-10 illustra tes the variation of the nonaxisymmetric leakage as a function of angular displacement for both the id eal and throttling cases. As the inlet angle decreases the potential for leakage increases significantly during the charging and expansion processes. For all leakage paths considered, the velo city of the leakage flow is calculated initially using the general Couette flow velocity profile determined above. If the flow is two-phase, the speed of sound is calculated to provide a reference. If the computed velocity using the general Couette flow model exceeds the speed of sound, then the model switches over to a choked flow regime corresponding to a maximum leakage flowrate. These general trends are apparent from Figure 5-11. Figure 5-11 illustrates the total variation of flow to/from the expander cavity by means of leakage past the sides of the vane s. Contact between the vane-tip and statorcylinder is assumed. A constant gap size is assumed. In actual operation, upon loss of contact, the vanes do not necessa rily retract all the way back into the rotor slot leaving the leakage channel height X( ) as assumed in the above derivation. The leakage mass flow-rate during the constant pressure inta ke and exhaust processes can be seen to increase and decrease, respectively with the oscillating variation of vane protrusion X( ). This effect is more evident in expander s with a larger geometric volume ratio.

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178 The leakage mass flow-rate through the mini mal gap that exists between the rotor and the stator-cylinder is significan t but constant for the duration 0< < seal, where seal is 40o. The contribution of this l eakage path to the total leak age flow-rate increases with the decrease of the number of vanes. The star t of the intake port may be delayed, if necessary, to minimize leakage from this path. The magnitude and direction of the leakage between the sides of the vanes a nd the rotor slots depends on the method by which the fluid is introduced into the expa nders cavity. Figure 5-12 shows the variation of leakage into the expander cavity as a function of angular displacement. This leakage path only exists for the duration of the in take port. There may be leakage from the expander cavity via this leakage path during th e remainder of one revolution. If the flow is introduced through the rotor slots, the ve locity of this leakage path, assuming no throttling, would be that of an incompre ssible liquid in shear driven Couette flow. Here the plate moves with a velocity co rresponding to the instantaneous sliding velocity of the vane. The leakage mass flow-ra te can be seen to increase proportional to the outward sliding velocity of the protruding vane. Of primary significance for an adequately designed expansion devi ce is the ratio of total mass flow-rate accounting for leakage and the corresponding ideal case, i.e. a leakage ratio actual idealm m The mass flow-rate through the expander accounting for leakage is 5 0160lki actualvcv im mNm (5-44)

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179 where the first term in the bracket repr esents the ideal mass flow-rate and index i represents the leakage path. For the base case simulations the leakage ratio is =10.9. This is significantly large and due to the low rotational speed of 300 RPM. Figure 5-13 shows the variation of the ideal and actual mass flow-rates as a function of rotational speed. It can be seen that the leakage mass flow-rate decreases significantly as the rotational speed of the e xpander is increased. The ideal mass flow-rate increases in a steady manner. The operational rotational speed of th e expander is a design choice that may be optimized by the proper se lection of volumetric displacement for the inlet flow conditions. The leakage ratio is =30.6 and 4 at rotati onal speeds of 100 and 1000, respectively. The actual mass flow-rate cu rve increases linearly because all of the internal leakage paths, with the exception of axial flow between the rotor and end-plates, is assumed to be independent of speed. Figure 5-14 illustrates the variation of th e ideal and actual mass flow-rates as a function of the number of vanes for an expander operational speed of 500 RPM. The ideal mass flow-rate decreases since the volum e of the cavity and the amount of mass in the expander cavity at the cut-off angle decr eases significantly. The amount of leakage is seen to increase due to a larger contribu tion from the leakage ma ss flow-rate through the radial gap, minimum clearance, between the ro tor and stator-cylinder. Caution must be exercised when increasing the number of vane s due to possible increases in friction and decrease of mechanical strength of the rotor. A thermodynamic and fluid dynamic model of the primary loss mechanisms in a circular rotary-vane expander has been de veloped as a function of design and fluid parameters. The following conclusions can be made:

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180 The curvature of the vane-tip can severall y impact the amount of leakage past the vane-tips due to loss of contact with the stator-cylinder. Even if no throttling is assumed in the intake port, there is a loss of contact w ith the stator-cylinder through out the duration of the expansion process. Vanes should be made out of high-density ma terials that can erode and mold to the stator-cylinder geometric curvature. This would eliminate the need for adequate under-vane forces through intake ports or cumbersome, unreliable springs. Leakage losses from a majority of leakag e paths cannot be nece ssarily neglected in two-phase rotary-vane expanders. Unlike si ngle-phase rotary-vane turbomachinery, significant variation in dens ity and speed of sound with the vaporization of liquid significantly alters the leakage flow-rates. The leakage losses from the non-axisymmetr ic flow in the axial gap between the rotor and end-plates is significant. The m odel developed is relatively ideal in its assumptions and doesnt take into acc ount separated boundary layers, choking or turbulent fluid flow. A computational m odel must solve the compressible NavierStokes equations in order to do so. Increase in rotational speed decreases internal leakage losses significantly. Further work must be made to determine the magnitude of detrimental effects such as friction and viscous drag. A decrease in the number of vanes increases the contribution of leakage through the minimal radial gap between the rotor and stator cylinder. Sizing of the rotary-vane expander for two-phase applications is critical with regard to determining the optimum operational speed to reduce the loss mechanisms presented in this study. Uncertainties regarding the gap size ar e a significant obstacle to quantifying the magnitude of leakage losses through a ll the leakage path s during operation. Two-Phase Throttling Losses in the Inlet and Exhaust Ports The state of refrigerant that enters the expansion device is of great importance. Unlike the conventional thrott ling valve, the state of the refrigerant may vary significantly from condenser exit conditions to the expander cavity. This may be due to significant throttling losses within the inlet port and internal contra ctions, enlargements and bends within the expansion device. This ma y also be partly due to the fact that the distance downstream over which the influence of the feature is felt is greatly increased in

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181 two-phase flow. A model for the prediction of the pressure losses of two-phase flow due to these flow restrictions is presented. An algorithm is presented and developed by which the actual state and amount of refrigerant entering the expans ion device can be computed. Figure 5-15 illustrates the difference between the conventional and modified methods of intake. The refrigerant leaving the condenser may be either saturated or subcooled liquid. Depending on the degree of subcooling, pressu re drops resulting from throttling in the expanders intake port and friction due to flow restrictions may cause the refrigerant to drop into the two-phase region. These pressure losses should be minimized in order to maximize the pressure differential utilized to produce work in the expander. In the case of under-expansion, where the geometric volume ratio is not sufficient when compared to the process volume ratio, pressu re losses realized in the intake and exhaust ports are a necessary loss mechanism after which no further throttling may be needed. The analysis of throttling in the ports can be divided into two distinct analyses. The difference in analysis depends solely on the method the refrigerant enters the expander cavity. The equations for the static pressure drops, frictional pressure losses and contraction coefficients for va rious geometries are presented below. Calculating throttling losses and the actual mass flow-rate is presente d later. This will aid in the evaluation of how well modeling the intake and exha ust processes as isobaric is. During the intake process the refrigerant goes through numerous pressure drops as a result of contractions a nd bends. The following assumptions are made to simplify the analysis for all geometries unless otherw ise specified. The void fraction of the homogenous two-phase mixture is assumed constant over the feature (i.e. no phase

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182 change). Separation of the two phases occu rs and the use of ge neral single-phase frictional loss methods are no longer valid. The following correlations, from Collier (1972) are presented to cal culate the static pressure drop, Equation (5-45), and the fric tional pressure loss across a contraction, Equation (5-46). A simplified momentum balance for the combined flow yields 2 2 fg 2f 12 2 cfv Gv 11 pp111x 2Cv (5-45) 2 2 fg 2f f cfv Gv1 p11x 2Cv (5-46) where Equations (5-45) and (5-46) differ by a factor 21 1 which represents the conversion of pressure energy to theoreti cal kinetic energy associated with the acceleration of the fluid through the contraction. The symbol is the ratio of the upstream A1 to downstream area A2. G2 is the mass velocity of the refrigerant and is defined as the velocity at the downstream location to the specific volume at that same location. Cc is the coefficient of contraction and is defined as the ratio of minimum area Ac to downstream area A2. This minimum area Ac, see Figure 5-16, corresponds to the narrowest cross section where the smalle st local pressure value is found. In order to calculate the coefficient of c ontraction, the two-phase empirical relation presented by Ruffell (1978) according to Schmidt and Friedel (1997) is used. The equation for the contraction coefficient can be written as

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183 c1 1 C1 1 2.08010.5371 (5-47) Equation (5-47) has been compared against the contraction coefficients for turbulent single-phase flow presented by Perry (1963) in Table 5-4. It should be noted that both models have been determined experimentally It has been found that modeled pressure drop across sudden contractions is satisfactory and that the constant void fraction assumption is not necessarily accurate (Collier (1972)). Upon exit from either the 90o bends or the expander cavity, sudden enlargement may exist in which further pressure losses are realized. For a two-phase mixture, the change in static pressure and total frictional pressure drop across the enlargement can be determined from a simplified momentum ba lance for the combined flow written as fg 2 211f fv ppG1v1x v (5-48) 2 2 fg 1 ff fv G p1v1x 2v (5-49) where G1 is the mass flux upstream of the sudden enlargement. The homogenous and constant void fraction pressure drop models presented above may yield satisfactory predictions since low mass velociti es and pressures are expected. In the case of two-phase flow through a sharp edged orifice, the discharge coefficient is equivalent to the contracti on coefficient if sma ll amounts of frictional dissipation and velocity profile effects are neglected (Collie r (1972)). The pressure drop equation that results from a simplified momentum balance, assuming a homogenous

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184 mixture and constant void fraction through the orifice, must be modified using a twophase multiplier The pressure drop in a uniform flow area, 90o bend can be calculated from the following equation presented by Collier (1972) gg 12 fffpp pp 1C ppp (5-50) where Pf and Pg are the pressure drops for the li quid and vapor phases and are defined as 2 2 f f 22 Dm1xv p 2CA (5-51) 22 g g 22 Dmxv p 2CA (5-52) where the discharge coefficient CD is approximated as the coe fficient of contraction Cc from Equation (5-47). This assumption neglects the small amount of frictional dissipation and velocity effects that may be incurred dur ing the flow. The constant C is evaluated from the following general expression 0.50.50.5 2fggf gfgvvv CC vvv (5-53) where =1 and the constant C2 is 2135D C L for 90o bends or 2120D C L for 90o bends with an upstream disturbance. Conventional Inlet to Expander Cavity As mentioned previously, this inlet port is typically machined into the statorcylinder. The inlet port spread is a design parameter with significant impact on the

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185 volumetric efficiency and thermodynamic perf ormance of rotary-vane technology. This may be of greater concern in two-phase e xpanders since pressure drops in two-phase flows are inherently la rger and necessitate the formati on of vapor. Depending on several factors including the spread of the intake port, the angle between successive vanes and the angular location of the lead ing vane relative to the intake port, the intake manifold will be in continuous contact with one or more of the expander cells. It should be noted that due to this continuous contact pulsation effects are ne gligible. It is hence of importance to calculate the actual mass flow-ra te of incoming refrigerant into each cell and the impact it may have on the performance of the expansion device. An optimal design of the intake port would minimize fl ow restrictions that would alter flow characteristics by maximizing the available fl ow area at each angle of rotation. This would ensure that an equal veloc ity is obtained at every opening. To capture the effect of irreversible losses that are inherent in the intake port due to flow restrictions, the concept of the discharge coefficient mu st be introduced. In general the discharge coefficient is defined as the ratio of the actual to ideal mass flow-rates. The discharge coefficient is a st rong function of the area ratio and the Reynolds number of the flow. The flow through the conventional inta ke port can be modeled as that of a twophase flow through a sharp-edged orifice. In the case of two-phase flow through a sharp edged orifice, the discharge coefficient is equivalent to the contraction coefficient, Equation 5-47, if small amounts of frictional dissipation and velocity profile effects are neglected (Collier, 1972). A schematic of the th ree phases of vane orientation that occur during the intake process are shown in Figure 5-17.

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186 It can be shown that the time rate of cha nge of volumes on either side of the leading vane in phase I, for example, differ signifi cantly. This alters the amount of mass that enters either control volume. The throat that is formed in phases I and III constitutes a major flow restriction. The definition of the discharge coefficient may be altered to incorporate the effect of the fo rmation of the throat as well as the discrepancy in the timerate of change in expander cavity volume. An area averaged, change in volume weighted component is incorporated into the defin ition of the discharge coefficient as such 1 2() ()()in dc refA V CC VA (5-54) where Aref is a reference area that denotes a frictio nless orifice area that produces an ideal mass-flow-rate. Figure 5-18 shows the variation pressure within the expander cavity as a function of the angular displacement of th e leading vane for the ideal and actual (throttling only) cases. It can be seen from Figure 5-18 that a signi ficant drop in pressure is realized in the intake and expansion phases. This dramati cally reduces the expanders performance by reducing the volumetric efficiency and generated torque and pow er. For all else equal, the volumetric efficiency is defined as the ratio of the actual density at the cut-off volume to ideal density (i.e. inlet density). The variati on in the volumetric efficiency, final pressure in the expander cavity at the maximum volume, are tabulated in Table 5-5 for various intake angles. Stand-alone experiments are n eeded in order to accurate ly determine the variation of the discharge coefficient. One discrepanc y that may exist is that at smaller local rotation angles, the discharge coefficient may in fact approach unity and decrease in a

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187 near linear fashion. This trend is repres ented as the dotted line in Figure 5-18. This situation arises from the fact that both the time and volume required to be filled initially is very small and that the extremely small throat area provides for a entrainment free jet of working fluid to fill the expa nder cavity in question. At larg er angles of rotation, there may exist entrained fluid that di srupts and alters the flow. It should be noted that the throttling losse s in the exhaust port are typically found to be negligible due to the smaller velocities as sociated with the exhaus t fluid and the larger spread area of the exhaust port. Modified Inlet to Expander Cavity The flow of the two-phase re frigerant in the intake is assumed to be homogenous and gravitational effects on the pr essure drops in vertical porti ons of the intake have been neglected. The pressure drops associated with the work nece ssary to protrude the vanes has also been neglected. This may be the cas e since the centrifugal forces exerted on the vane due to rotation of the expander are typically an order of magnitude larger. In order to model the mass flow of refrigerant into the expander, the velocities of the fluid in the contractions, bends and enla rgements, the unk nowns, must be calculated. Figure 5-19 shows a schematic of the intake of a modified rotary-vane intake. The solution procedure entails an iterative procedur e in which the pressure at location 2 must be initially estimated as the average of the known pressures at locations 1 and 5. Using Equation (5-45), the velocity at Lo cation 2 may be calculated. Location 3 is denoted by an because in fact it represents 4 bends of equal flow area. For the sake of brevity they are illustrated as one. The veloc ities within these bends are assumed to be equal. Using Equation (5-50) the pressure at each of the bends can be calculated. Since the flow is compressible the quality at each one of these locations can be computed by

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188 assuming an isenthalpic process. Using Equati on (5-48) the pressure at Location 4 can be computed since the mass velocity at 3* is known. The final parameter to be determined is the velocity at Location 4, wh ich can be computed using 2 441 2exit p u (5-55) This iterative procedure is repeated un til the initial guess of the pressure at Location 2 converges to within of the following 1523'argcontenleexit p ppppp (5-56) It should be noted that the pressure at th e exit can be determined using simulation models or from experiments. The mass flow-rate can be computed as nnmuA where n=2,3*,4 Thermodynamic Model of the Actual Expansion Process In this section the effects of internal l eakage losses, friction and throttling are taken into account deriving the appropriate cons ervation of mass and energy equations. The solution procedure is detailed. The ideal predicted pressure, temperatur e and mass variations in the expanders control volume are used as in itial estimates when calculating any non-ideal effects. The velocity and hence mass flow-ra te of the leakage flows via different leakage paths have already been presented. Charging Process The first law of thermodynamics, taking into account heat transfer generated by friction, frictional power losses and effects of internal leak age, can be written for the process as follows

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189 intint,,,,cv f bfleakinleakinleakoutleakoutdE QWWmhmhmh dt (5-57) Assuming the changes in kinetic ener gy and potential energy are neglected dEcv=dUcv. The pressure within the control vol ume is also assumed to be uniform. Multiplying by dt Equation (5-57) becomes 121intint,,,,() f fleakinleakinleakoutleakoutUqpVVwmhmhmh (5-58) Solving for the internal energy of the refrig erant at the new time step, State 2, we find 11121intint,,,, 2 2,,() f fleakinleakinleakoutleakout leakinleakoutmuqpVVwmhmhmh u mmm (5-59) The density of the refrigerant at the ne w time step, State 2, is calculated as 2,, 2 2leakinleakoutmmm V (5-60) The use of REFPROP 7.0 allows the determination of the thermodynamic properties at the new state point, State 2, name ly the temperature, pressure and quality. The process is repeated until the end of the charging process. Expansion Process Unlike the ideal expansion process the mass in the control volume, mcv, varies depending on the magnitude of the internal leakage losses after the cut-off angle intcut. The density at State 2 can be determined from 1,, 2 2 leakinleakoutmmm V (5-61) An energy balance of the control volume, i. e. leakage, during the expansion process yields the internal en ergy at State 2 as

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190 11121,,,, 2 2,,() f fleakinleakinleakoutleakout leakinleakoutmuqpVVwmhmh u mmm (5-62) The thermodynamic properties at the new tim e step, namely temperature, pressure, and quality, can be calculated from Equations (5-61) and (5-62) by the use of REFPROP 7.0. This process is repeat ed until the end of the expansi on process at a pre-determined design angle of ex. Exhaust process The exhaust process occurs at an exhaust angle ex until out end, where out is the spread of the exhaust port and is the angle between consecutive vanes. The first law of thermodynamics can be writ ten for the process as follows ,,,, cv f bfexexleakinleakinleakoutleakoutdE QWWmhmhmh dt (5-63) Assuming the changes in kinetic ener gy and potential energy are neglected, dEcv=dUcv. The pressure within the control volum e is also assumed to be uniform. The refrigerant flow from the control volume is also assumed to be quasi-steady and homogenous. Multiplying Equation (5-63) by dt and solving for the internal energy of the refrigerant at the new time step, State 2, we find 1121,,,, 2 2,,() f exfexexleakinleakinleakoutleakout leakinleakoutmuqpVVwmhmhmh u mmm (5-64) The density of the refrigerant at the ne w time step, State 2, is calculated as 2,, 2 2 leakinleakoutmmm V (5-65) The use of REFPROP 7.0 allows the determination of the thermodynamic properties at the new state point, State 2, na mely the temperature, pressure and quality.

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191Expander Performance Evaluation Upon development of the geometric, therm odynamic, internal leakage, frictional, throttling and any other loss mechanism, the overall performan ce of a rotary-vane expander can be predicted. The following pa rameters describe the performance of the expander with respect to different criteria. Of primary importance to the performance of any positive displacement expansion device is the geometric (built-in) volume ratio. This parameter is defined as the ratio of volumes at the exhaust and cut-off angles respec tively. The reader is referred to an earlier chapter dealing with this parameter in further detail. Along with the geometric volume ratio the geometric code deve loped allows for design optimization with regards to the number of vanes, location of intake and exha ust ports, stator-cylinder geometry and size, rotor size and eccentricity and aspect ratio to name a few. The pressure, quality and mass variati ons of the working fluid within the expanders cells are predicte d by utilizing the thermodynamic model. Models for the intake and exhaust processes are used in conjunction with the thermodynamic model to determine throttling losses (termed breathing in the open literature) within the ports. These throttling losses are a primary function of port design (geometry) and spread (size). The use of these models also allows us to calculate the volumetric efficiency, resultant power ratio and efficiency of the multi-van e expander due to throttling losses in the intake port. The volumetric e fficiency may be defined as int vcvinv v (5-66) where vint is the specific volume of the working fluid within the expanders cell assuming no throttling losses (i.e upstream conditions) and cvinv is the ensuing specific

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192 volume of the refrigerant if there exists thro ttling within the intake port. Both quantities are calculated at the cut-off angle. As the vol umetric efficiency decreases, the effects of throttling significantly alter th e torque generated by the pre ssure forces exerted on the vanes by the working fluid and hence lead to a decrease in the us eful power output and efficiency of the expander. The power ratio, Pr, may be defined as the ratio of the power produced when throttling is accounted for to the ideal pow er produced by the expander with no throttling losses. Since the mass of refrigerant contained w ithin the cell volume of the expander at the cut-off volume may be signi ficantly less when throttling e ffects are considered, it may be advantageous to define a thermodynami c efficiency based on the ratio