I AlllDi 573137

NBS BUILDING SCIENCE SERIES 155

Computer Modeling of the Vapor

Compression Cycle With Constant Flow

f^^'^'^ Expansion Device

.U58

155

1983 ARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS

C.2
ERRATA SHEET

NBS Building Science Series 153

Computer Modeling of the Vapor Compression Cycle

with Constant Flow Area Expansion Device

Piotr Domanski, David Didion

Page vii Line 16: is

:

Line 17: is: Line 19; is I Line 23: is

:

Page xi Line 5; is: Line 22: is:

Page xiv Line 7: is

:

Page 39 Line 23: is:

Page 43 Line 2: is:

Page A8 Line 29; is

:

Page 67 Line 1; is

:

Line 18: is:

Page 70 Line 15; is

:

is: adequate

x2)

should be: 121 should be: 122 should be: 136 should be: NBS BSS 155

should be: 69

should be: 61

should be: 61

should be : inadequate

.2 A

should be

should be -itf

should be: should be: water (frost)

should be: (1 - x)^

Page 111 Coefficients in equation (E5) should read:

b = 2[(v,, - V, ) ill

G^Cvy - Vl)2

2(iT JL

(Vy - Vl)'

Page 114 Symbols used in figure Fl are: PEXIT = refrigerant pressure at tube exit

PFLASH = refrigerant flash pressure

PO = refrigerant pressure before expansion device inlet

PI = refrigerant pressure in expansion device inlet X = refrigerant quality before expansion device inlet

Page 116 Symbols used in figure Gl are: H = refrigerant enthalpy at tube end obtained from present

calculation loop

= refrigerant enthalpy at tube end obtained from previous

calculation loop

= refrigerant enthalpy at coil exit obtained from present

calculation loop

= refrigerant enthalpy at coil exit obtained from previous

calculation loop
KilTIOfAI. B'TriEAU OF BTANDARDS NBS BUILDING SCIENCE SERIES 155 ^o^cJd^J

Computer Modeling of the Vapor

Compression Cycle With Constant Flow

Area Expansion Device

Piotr Domanski

David Didion

Building Equipment Division

Center for Building Technology

National Bureau of Standards

Washington, DC 20234

Partially Sponsored by:

Department of Energy

1000 Independence Avenue, S.W.

Washington, DC 20585

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued May 1983
Library of Congress Catalog Card Number: 83-600524

National Bureau of Standards Building Science Series 155

Natl. Bur. Stand. (U.S.), Bldg. Sci. Ser. 155, 162 pages (May 1983)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1983

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $5.50

(Add 25 percent for other than U.S. mailing)
ABSTRACT

An analysis of the vapor compression cycle and the main components of an air source heat pump during steady-state operation has been performed with emphasis

on fundamental phenomena taking place between key locations in the refrigerant

system. The basis of the general heat pump model formulation is the logic

which links the analytical models of heat pump components together in a format

reqiiiring an iterative solution of refrigerant pressure, enthalpy and mass balances

.

The modeling effort emphasis was on the local thermodynamic phenomena which

were described by fundamental heat transfer equations and equation of state

relationships among material properties. In the compressor model several

refrigerant locations were identified and the processes taking place between

these locations accounted for all significant heat and pressure losses. Evaporator and condenser models were developed on a tube-by-tube basis where performance of each coil tube is computed separately by considering the cross-flow

heat transfer with the external air stream and the appropriate heat and mass transfer relationships. A capillary tube model was formulated with the aid of

Fanno flow theory.

The developed heat pump model has been validated by checking computer results

against laboratory test data for full and part load operation for the cooling/

dehumidifying mode as well as the heating mode under frosting .conditions

.

Key Words: air conditioner; capillary tube; coil; compressor; condenser;

evaporator; expansion device; heat pump; modeling; vapor

compression cycle.

iii
PREAMBLE

This report is essentially the dissertation of Piotr Domanski which was submitted to the Catholic University of America in partial fulfillment of his

Ph.D. requirements. David Didion was his major professor and dissertation

director

iv
DISCLAIMER

In view of the presently accepted practice of the building industry in the

United States and the structure of the computer software used in this project, common U.S. units of measurement have been used in this report. In recognition

of the United States as a signatory to the General Conference of Weights and

measures, which gave official status to the SI system of units in 1960, appropriate conversion factors have been provided in the table below. The reader

interested in making further use of the coherent system of SI units is referred

to: NBS SP330, 1972 Edition, "The International System of Units," or E380-72,

ASTM Metric Practice Guide (American National Standard 2210.1).

METRIC CONVERSION FACTORS

Length 1 inch (in) = 25.4 millimeters (mm)

1 foot (ft) = 0.3048 meter (m)

Area 1 ft2 = 0.092903 m2

Volume 1 ft3 = 0.028317 m3

Temperature F = 9/5 C + 32

Temperature

Interval

1°F = 5/9°C or K

Mass 1 pound (lb) = 0.453592 kilogram (kg)

Mass Per Unit

Volume

1 lb/ft3 = 16.0185 kg/m^

Energy

Specific Heat

Gallon

1 Btu = 1.05506 kilojoules (kJ)

1 Btu/[(lb)(''F)] = 4.1868 kJ/[(kg)(K)]

1 gallon = 0.0037854 m^
TABLE OF CONTENTS

Page

ABSTRACT ili

PREAMBLE iv

DISCLAi-MER v METRIC CONVERSION FACTORS v LIST OF TABLES viii

LIST OF FIGURES ix

LIST OF SYMBOLS xi

1. INTRODUCTION 1 1, HEAT PUMP THERMODYNAMIC CYCLE 5

3. OPERATION OF AN AIR SOURCE HEAT PUMP WITH A CAPILLARY TUBE 11

4. PREVIOUS WORK IN STEADY- STATE VAPOR COMPRESSION CYCLE MODELING 17

5. MODELING A HEAT PUMP WITH A CONSTANT . FLOW AREA EXPANSION

DEVICE 21

5.1 MODELING OF THE SYSTEM 21

5.1.1 Structure of Available Heat Pump Models 21

5.1.2 New System Logic 23

5.2 MODELING OF A RECIPROCATING, HERMETIC COMPRESSOR 25

5.2.1 Compressor Operation 25

5.2.2. Theory and Governing Relations 28

5.3 MODELING OF A CONSTANT FLOW AREA EXPANSION DEVICE 39

5.3.1 Available Capillary Tube Performance Data 39

5.3.2 Critical Flow in a Capillary Tube 42

5.3.3 Model Formulation 48

5.4 MODELING OF AN EVAPORATOR AND A CONDENSER 50

5.4.1 Modeling Methodology 50

5.4.2 Heat Transfer Rate for a Tube of a Cross-Flow

Arrangement 53

5.4.3 Refrigerant and Air Mass Flow Rates Associated

with a Tube 56

5.4.4 Overall Heat Transfer Coefficient for a Dry

Finned Tube 57

5.4.5 Forced Convection Heat Transfer Inside a Tube 58

5.4.6 Forced Convection Heat Transfer at the Air-Side of

a Flat-Finned Tube 61

5.4.7 Overall Heat Transfer Coefficient for a Wet

Finned Tube 62

5.4.8 Pressure Drop in a Tube 68

5.5 MODELING OF ADDITIONAL HEAT PUMP COMPONENTS 71

5.6 REFRIGERANT MASS INVENTORY IN A HEAT PUMP 72

6. MODEL VERIFICATION 79

6.1 LABORATORY TESTS 80

6.2 COMPUTER SIMULATION RUNS 80

6.3 VERIFICATION OF THE MASS INVENTORY SIMULATION o.......... 83

vi
TABLE OF CONTENTS (Cont.)

Page

7. SUMMARY AND CONCLUSIONS 89

8. REFERENCES 93

APPENDIX A. CALCULATION OF THE THERMODYNAMIC AND TRANSPORT PROPERTIES

OF REFRIGERANTS R-12 and R-22 97

APPENDIX B. CALCULATION OF PROPERTIES OF MOIST AIR 105

APPENDIX C. CALCULATION OF WATER AND FROST PROPERTIES 107

APPENDIX D. COMPRESSOR SIMULATION SUBROUTINE PERCOM 109

APPENDIX E. CALCULATION OF CRITICAL PRESSURE FOR TWO-PHASE FANNO FLOW .. Ill

APPENDIX F. CAPILLARY TUBE SIMULATION SUBROUTINE EXDEV 113

APPENDIX G. EVAPORATOR AND CONDENSER SIMULATION SUBROUTINES

EVPHXM AND CNDNEW 115

APPENDIX H. THE LOGIC OF THE MAIN PROGRAM, MAIN 117

APPENDIX I. PROGRAM USER'S GUIDE 121

1.1 INPUT DATA CODING 123

1.2 OUTPUT DATA CODING 124

1.3 EXAMPLE OF RUN OF HEAT PUMP SIMULATION PROGRAM

HPSIM 138

APPENDIX J: LISTING OF HEAT PUMP SIMULATION PROGRAM HPSIM 149

(Available through U.S. Department of Commerce,

National Technical Information Service, as Appendix J

to BSS 153)

vii
LIST OF TABLES

Page

1. Coefficients to be Used in Correlation for Fin Efficiency

(Equation (69)) 65

2. Laboratory Test Results of Heat Pumps in the Cooling Mode . 81

3. Laboratory Test Results of System 2 in the Heating Mode 82

4. Cooling Test Simulation Results 84

5. Heating Test Simulation Results ..».oo., 85

6. Discrepancy Between Laboratory Test Results and Computer Simulation

Al. Coefficients and Physical Constants for Freon-12 an Freon-22 to be

Used in Equations (Al) through (A8) . . . o 100

A2. Coefficients for Freon-12 and Freon-22 to be Used in Equation (A9) ... 102

A3. Refrigerant Property Functions and Subroutines 104

CI. Water Property Evaluation Constants which are Used in Equation (CI) .. 108

11. Programs used in a Heat Pump Simulation Model HPSIM 123

12. Data File DATA12 Containing Physical Constants for Refrigerant 12

in the Input Format to Program HPSIM

.

125

13. Data File DATA22 Containing Physical Constants for Refrigerant 22

in the Input Format to Program HPSIM. . 126

14. Heat Pump Input Data Code to Program HPSIM... 127

15. Example of Heat Pump Data in the Format to Program HPSIM.. 135

viii
LIST OF FIGURES

Page

1. Carnot vapor compression cycle 2

2. Theoretical vapor compression cycle 4

3. Oversimplified thermodynamic cycle of a heat pump 6

4. Schematic of a heat pump 8

5. Thermodynamic cycle realized by a heat pump 9

6. Simplified thermodynamic cycle realized by a heat pump in the

CQoling mode at different outdoor conditions 14

7. Schematic of an accumulator 16

8. Overall logic of a program simulating a heat pump with a constant

flow area expansion device 26

9. Schematic of a hermetic compressor 27

10. Typical efficiency versus load curve for a permanent split-capacitor

2 pole electric motor [4] .... ..... 30

11. Simplified indicator diagram for a reciprocating compressor 31

12. Typical speed (RPM) versus load curve for a permanent splitcapacitor 2 pole electric motor [4] 34

13. Pressure and temperature distribution along typical capillary

tube [11] ...... o .............. o 41

14. Mass balance for an element of fluid in one-dimensional steady flow

in a constant area duct 44

15. Momentum balance for an element of fluid in one-dimensional

flow in a horizontal, constant area duct 44

16. Energy balance for an element of fluid in one-dimensional, adiabatic,

steady flow in a horizontal, constant are duct 45

17. Fanno line 45

18. Pressure distribution along different length capillary tubes at the same operating conditions 47

19. Schematic of a heat pump heat exchanger 51

ix
LIST OF FIGURES (Cont.)

Page

20. Example of coll circuitry 52

21. Approximation method for treating a rectangular-plate fin of

uniform thickness in terms of a flat circular-plate fin of equal

area > 54

22. Cross section of a flat-finned tube indicating parameters which

affect the air-side heat transfer coefficient 63

23. Efficiency for a circular-plate fin of uniform thickness 64

24. Refrigerant phase in heat pump components 74

Fl. Logic of constant flow area expansion device simulation program

EXDEV 114

Gl. Flow chart for coil simulation program 116

HI. Heat pump simulation program HPSIM flow chart 118

X
LIST OF SYMBOLS

A = flow cross section area

Af = fin surface area

Ai^ = heat transfer surface drea

A^ji = coefficients in equation (68)

Ap^j^ = pipe inside surface area

^p,m ~ pipe mean surface area

Aq = pipe total outside surface area

al,a2,a3 = coefficients in equation (20)

CP = pressure drop parameter

CQ = heat transfer parameter

C(. = correction factor in equation (10)

Cg = clearance volume, in fraction of compressor stroke volume

Cp = specific heat at constant pressure

Cv = specific heat at constant volume

D = tube inside diameter

Dq = tube outside diameter

Df = fin tip diameter

dl,d2 = dimensions representing arrangement of tubes in a coil, as per

figure 21

E = electrical power input

F1,F2 = dimensionless parameters used in equation (60)

f = Fanning friction factor

f^p L

= friction factor for the liquid portion of two-phase flow, flowing

alone in the tube

m

G = —, mass flux

A

xi
air mass flux between two adjacent fins

heat transfer coefficient, h^, q refers to air-side mean convective

heat transfer coefficient, h^ ^ refers to heat transfer coefficient

for inside tube deposit, hj^ refers to Inside tube convection heat

transfer coefficient

air-side mass transfer coefficient

horse power

enthalpy

latent heat of evaporation or condensation

the mechanical equivalent of heat

flow contraction coefficient

thermal conductivity

tube length

h

Sn2. , Levis number

t^D.o^Pa

mass of refrigerant

mass flow rate *^o:->iv'>oo s-.l?) JJkv?, ^^woc

^ * ^ , Nusselt number -

• k

polytroplc Index i

pressure :. Mi .

- -u:!'. - > ^

^

'

, Prandtl number

...

I

,

'

^ ; heat transfer rate

rate of moisture removal per unit area

rate of moisture removal per unit width of a fin

^ * ^

, Reynolds number

xll
^®tp,L " Reynolds number for the liquid portion of two-phase, flowing alone

In the tube

RPM = compressor number of revolutions per minute

S = tube perimeter

s = entropy

T =» temperature

Tf^^ =» fin base temperature

^f,m ~ n^fi^ii fill temperature

Tj-^g » refrigerant saturation temperature

t = fin thickness

U = overall heat transfer coefficient

V » velocity

Vs = compressor swept volume

V = volume

v = specific volume

W(, = mechanical power available for compression process

Wg = mechanical power output

w^ = humidity ratio of air, w^ i refers to tube row inlet, w^ g refers

to tube row outlet

w^ » humidity ratio of saturated air at temperature of wetted water film

=^1- ^ —^ , Lockhard-Martinelli parameter

L

" J L J L Ml J

* , quality

my + mL

Xp = pipe wall thickness

y = fin height

xiii
= fraction of the tube in the two-phase region for the tube with

liquid and two-phase flow

Zy = fraction of the tube in the superheated vapor region for the tube

with two-phase and superheated vapor flow

z = distance between adjacent fins

a = void fraction

3 = exponent in equation (60)

Y

= isentropic index

a = liquid layer (frost) thickness

Tig = electric motor efficiency

= mechanical efficiency of compressor

rip = polytropic efficiency of compressor

r\y = volumetric efficiency of compressor

y = absolute viscosity

p = density

T = skin friction factor

$ = correction factor for two-phase pressure drop

(j> = fin efficiency

Subscripts

a = air, a,d refers to dry air

e = exit

f = frost or fin

i = inlet

L = liquid

m = mean value

P = constant pressure process

r = refrigerant

xiv
s = constant entropy process

t = total value

V = vapor

V = constant volume process

w = water

1 to 13 = refrigerant key locations in a heat pump, as per figure 4, unless

otherwise epxlained in the text

/

XV

1 . INTRODUCTION

Any power producing cycle, whose processes can be performed in the reverse direction and in the reverse order, will serve as a refrigerating cycle.

The Carnot cycle is the model of perfection for a power producing cycle as well as for a refrigeration cycle. However, it is not practical to realize

this perfect thermodynamic cycle with hardware (see figure 1). First, two

compression machines, one realizing isentropic compression, the other realizing isothermal compression would be required. Second, the small amount of

expansion work available does not support the idea of incorporating a work

producing expander into the system.
CONDENSER

i W = WORK TRANSFER jQg, Q = HEAT TRANSFER

EMTROPY

Figure 1

.

Carnot vapor compression cycle
Thus a simple isenthalic throttling expansion device and only one type of

compressor are used in a normal refrigeration system. This changes the

theoretical cycle as shown in figure 2. This cycle will have a coefficient

of performance smaller than that for the Carnot cycle.

Figure 2 identifies the main elements of a system working on the vapor

compression cycle principle. They are: compressor, condenser, evaporator and

throttling valve. There are many concepts and designs of heat pump components,

however, certain types are predominant. A reciprocating hermetic compressor

is mostly used for vapor compression in small systems. Heat exchangers are usually in the form of staggered tubes with closely packed wavy fins (spinefins or bristle-fins are sometimes used). There are basically two types of

expansion devices in application today. Constant flow area restrictors are distinctly prevailing in small capacity units such as household refrigerators,

window-type air conditioners and central residential air conditioners and heat

pumps. Larger, more expensive residential units and commercial units are mostly

equipped with a variable flow area device called a thermostatic expansion valve

(TXV)

.

The focus of this study was the mathematical modeling and prediction of

performance of the vapor compression refrigeration system commercially known

as a residential air source heat pump. The type of heat pump considered is

the one most commonly available with a reciprocating hermetic compressor, fin

tube heat exchangers, and a constant flow area expansion device.

The present report discusses and analytically describes the phenomena taking

place in the heat pump system. Based on this analysis, mathematical models

of the main components of the heat pump were developed and linked together in

a general heat pump simulation model HPSIM. The model HPSIM simulates the

performance of a heat pump with a capillary tube at given outdoor and Indoor

air conditions with no prefixed assumption on refrigerant state at any point

of the system.

3
Figure 2. Theoretical vapor compression cycle

4
2. HEAT PUMP THERMODYNAMIC CYCLE

The purpose of a heat pump Is to pump heat from a low temperature reservoir to

a high temperature reservoir. From the thermodynamic point of view, the basic

heat pump components needed to make this heat transfer feasible are: two heat

exchangers, a compressor, and an expansion device. A heat pump thermodynamic

cycle can be explained considering the processes that the refrigerant undergoes

in these four components. The most convenient diagram for such explanation and

performance analysis is that of a pressure vs. enthalpy coordinate system, as shown in figure 3. The compressor receives low pressure and temperature refrigerant at state 1 and compresses it to a high pressure. This compression

process is associated with an increase of refrigerant temperature. At state 2,

5
|«2,3

CONDENSER

X

THROTTLING VALVE

EVAPORATOR

COMPRESSOi '1,2

W = WORK TRANSFER

Q = HEAT TRANSFER

ENTHALPY

Figure 3o Oversimplified thermodynamic cycle of a heat pump

6
the high pressure and high temperature vapor enters the condenser. The

refrigerant passing through the condenser rejects heat to the high temperature

reservoir and changes, usually, to a subcooled liquid at state 3. Then, the

refrigerant flows through the expansion device undergoing a drop in pressure

and temperature. Finally, the low pressure, low temperature, and low quality

refrigerant at state 4 enters the evaporator, where it picks up heat from the

low temperature reservoir, reaching a superheated (or high quality) vapor

state 1 at the evaporator exit. In the explanation above, the low and high

temperature reservoirs are the indoor and the outdoor environment, when the

heat pump is operating in the cooling mode.

Besides the compressor, two heat exchangers, and the expansion device, there

are, for practical reasons, many other components in an actual heat pump system. Four modeling consideration, the most important are: tubes connecting

basic elements, 4-way valve enabling refrigerant flow reversal for the unit to

operate in the heating or cooling mode, and an accumulator, which acts as

a protective device for the compressor by storing excess refrigerant during

part load operation and preventing liquid refrigerant from entering the compressor. All the above mentioned components make up an actual vapor compression system. Configuration of such a system and the thermodynamic cycle are illustrated in figure 4 and figure 5, respectively. Refrigerant states in

figure 5 correspond to particular locations in the system marked in figure 4,

which are:

1 - evaporator exit, suction pipe inlet

2 - suction pipe outlet, low pressure 4-way valve inlet

3 - low pressure 4-way valve exit, compressor can inlet

4 - inside compressor can

5 - compressor cylinder at suction

6 - compressor cylinder at discharge

7 - discharge manifold

8 - compressor can exit, high pressure 4-way valve inlet

9 - high pressure 4-way valve exit, discharge pipe inlet

10 - discharge pipe outlet, condenser inlet

11 - condenser outlet, liquid line inlet

12 - liquid line outlet, expansion device inlet

13 - evaporator inlet

7
INDOOR COIL

CHECK VALVE

13<i

INDOOR FAN

120

FLOW RESTRICTOR

ACCUMULATOR

HERMETIC COMPRESSOR

2»- ^ 3 —^—•-

4-WAY VALVE

9

OUTDOOR COIL

17

T s"

OUTDOOR FAN X

Note: 1. Refrigerant flow direction is marked as for cooling operation

2. Numbers 5 and 6 situated in the compressor correspond to

refrigerant state before and after compression process

Figure 4. Schematic of a heat pump

8
ENTHALPY

Figure 5. Thermodynamic cycle realized by a heat pump

(For number location refer to figure 4)

9

3. OPERATION OF AN AIR SOURCE HEAT PUMP WITH A CAPILLARY TUBE

Refrigerant state in key locations of a heat pump and heat pump performance

are functions of many factors. From a macroscopic point of view they can be

listed as follows:

- individual performance of heat pump components

- compatibility of components in the system

- properties of refrigerant

- refrigerant charge

- outdoor and indoor air conditions

11
Each of the above factors have an independent Impact on heat pump capacity and

efficiency.

In this report, a properly charged, commercially available heat pump is

discussed. A customer receives a heat pump correctly precharged by a manu- facturer and information about proper charge by weight of refrigerant is supplied on the heat pump name plates. Simple charts are also available to

facilitate checking the state of charge of unit by reading compressor suction

and head pressures at given operating conditions.

Manufacturers establish proper charge for a given heat pump during a laboratory

test. It is usually a cooling test at standard operating conditions, which

are

outdoor air dry bulb temperature - 95°F

indoor air dry bulb temperature - 80°F

indoor air wet bulb temperature - 67 °F

The measure of the refrigerant charge of the system is refrigerant superheat

at the compressor can inlet. Superheat of about 15°F at this location is

commonly accepted as an indication of a proper amount of refrigerant in the

heat pump.

In a heat pump with a capillary tube, vapor superheat at the compressor inlet

will vary with changing operating conditions. As a matter of fact, no single

refrigerant parameter stays constant when indoor or outdoor air conditions are altered and the system seeks equilibrium under new conditions. To understand

the phenomena that cause refrigerant state changes, consider a heat pump working in the cooling mode at constant indoor air conditions and the system

response to a step increase of the outdoor temperature. For the purpose of the

analysis it is assumed that the overall heat transfer coefficient for both heat

exchangers does not vary. It is also assumed, as a first approximation, that

the latent heat of evaporation, ifg, is constant under the conditions discussed.

A step change in the outdoor air temperature first affects the performance of

the condenser. Since the condenser environment temperature increase, the mean

temperature difference between the air and the refrigerant decreases. Less

heat is rejected by the refrigerant to the air and a smaller enthalpy change

is realized in the coil. Consecutively, higher enthalpy, less subcooled liquid refrigerant enters the capillary tube. The capillary tube is sensitive to

the amount of subcooling and with less subcooling the mass flow rate through

the capillary tube decreases. Since the compressor capacity stays unchanged,

pressure builds up in the condenser. Thus point 11 on the p - h diagram (refer

to figure 5) moves in the direction of higher enthalpy and higher pressure.

The environment of the evaporator was not altered. However, refrigerant

parameters in the evaporator change in response to the change of refrigerant

state at location 11 as well as to the change in refrigerant mass flow rate.

For pressure drop considerations, as a first approximation, the liquid line,

the capillary tube, and the evaporator are viewed as one tube, in which a given

flow experiences a certain pressure drop. As pressure of the refrigerant at

12
location 11 increases, it pulls up the refrigerant pressure in the evaporator.

Conflicting phenomena affect the change of refrigerant enthalpy at point 1.

Increased enthalphy at point 11 and reduced refrigerant mass flow rate work

against the impact of the smaller temperature difference between the air and

the refrigerant causing refrigerant enthalpy to slightly decrease. The significance of the change of the refrigerant state at location 1 depends on the move towards smaller vapor specific volume. .Since the compressor, as a first

approximation, is a constant intake volume pump, increase in gas density

results in a higher mass flow rate.

The indicated phenomena will balance themselves further until the refrigerant

in the key locations of the system acquires thermodynamic states that satisfy

simultaneously all heat pump components. Since under steady-state operation,

refrigerant mass flow rate through the compressor and the capillary tube have

to match, the pressure in the condenser will rise to an appropriate level.

Again, higher pressure in the condenser implies some increase of pressure (and

saturation temperature) in the evaporator. Smaller temperature difference

between the indoor air and the refrigerant, and higher refrigerant mass flow

rate result in a smaller refrigerant enthalpy increase in the evaporator.

Refrigerant state at point 1 with respect to point 11 is determined by this

enthalpy change and appropriate pressure drop. Finally, when steady-state

conditions are obtained, the following changes in refrigerant thermodynamic

states can be noticed: higher refrigerant pressure in the condenser, higher

refrigerant pressure in the evaporator, less superheat in the evaporator exit,

and less refrigerant liquid subcooling at the condenser exit. A change of

saturation temperature in the condenser corresponds approximately to a change

in the outdoor air temperature. The change of pressure in the evaporator

equals to a fraction (approximately 15 percent) of the pressure change in the

condenser. Though the refrigerant is circulated in the system at a higher mass flow rate, capacity of the heat pump is decreased due to smaller refrigerant enthalpy change in the evaporator. Efficiency also drops since, in addition to a smaller cooling effect, energy input for the compression process

increases due to higher refrigerant mass flow rate and higher compression rate.

A decrease of the outdoor air temperature would induce the opposite trends.

A change of refrigerant parameters is illustrated in figure 6, where results

of three heat pump tests are plotted on the pressure-enthalpy diagram. (Test

results are those of system 2 discussed later in section 6.1). In these tests,

the indoor air conditions were held constant while the outdoor air dry bulb

temperature was changed inducing modification of the realized thermodynamic

cycle.

For the heating mode operation, refrigerant flow is reversed by reset of the

4-way valve. Flow direction of the refrigerant in the system is the opposite

to that in the cooling mode, with the exception of the compressor. The indoor

coil becomes a condenser while the outdoor coil serves as an evaporator. It

is worthwhile to note the effect this change has on the system. Since pressures in heat pump heat exchangers are functions of environment temperatures,

the pressure in the evaporator is now much lower than during cooling operations.

Consequently, the density of the refrigerant at the evaporator exit is smaller

and less refrigerant is being pumped by the compressor. In order to maintain a

13
Indoor conditiono (all three tests): air dry bulb temperature - 80°F

air wet bulb temperature - 67°F

Outdoor dry bulb temperature: 105 °F

95°F

32°F

Note: 1. Saturation lines have been generated by means of equations

presented in appendix A.

2. Refrigerant states in a heat pump were obtained from

laboratory tests 2.1, 2.2., and 2.3 discussed in section 6.1.

Figure 6. Simplified thermodynamic cycle realized by a heat pump in the

cooling mode at different outdoor conditions

14
proper pressure in the condenser, another, properly sized capillary tube is now

used in the system, while the one used in the cooling mode is bypassed.

An indoor coil is usually much smaller than an outdoor coil. Need for good

dehuraidif ication performance is the primary reason. The indoor coil working

as the condenser cannot hold as much refrigerant as the outdoor coil during

cooling operation. Since the condenser holds the bulk of working refrigerant

in the system, excess refrigerant has to be stored somewhere during heating

operation. For storage purposes in heat pumps equipped with a capillary

tube, an accumulator is provided.

A schematic of an accumulator is given in figure 7. The accumulator is a closed

container with two pipes. The longer, bent pipe has a small diameter hole on

the bottom side, and is connected to the compressor. The other pipe connects

the tank with the evaporator exit.

If superheated vapor leaves the evaporator, vapor only is contained in the

accumulator and no special function is fulfilled by the accumulator in this case. If wet vapor enters the accumulator, liquid droplets accumulate on the

bottom of the tank, while saturated vapor enters the pipe leading towards the

compressor. Some liquid refrigerant enters the vapor stream through the hole

in the pipe, driven by the static pressure difference in the liquid-vapor

stream interface. It should be noted that during steady-state operation with

liquid in the accumulator, qualities of the vapor entering and leaving the

accumulator have to be equal. Since the small diameter hole allows for a small

rate of liquid flow into the saturated vapor stream, vapor taken in by the

compressor is close to its saturation state. Such a situation at the compressor inlet prevails in the heating mode since, with a possible exception to

operation at high outdoor temperature, there is always liquid refrigerant

stored in the accumulator.

This discussion has been for the heat pump employing a constant flow area expansion device. Thermostatic expansion valves, TXV's, are variable opening

orifice devices. The amount of opening is varied according to some control

function, usually constant vapor superheat at the evaporator exit. In heat

pumps employing TXV's, vapor superheat at the compressor inlet does not depend

on operating conditions but rather has a constant preset value.

15
Figure 7 . Schematic of an accumulator

16
4. PREVIOUS WORK IN STEADY-STATE VAPOR COMPRESSION CYCLE MODELING

Modeling of the vapor compression cycle has to be performed by modeling

essential processes taking place in the system. A review of state-of-the-art

steady-state vapor compression cycle modeling will be conducted at two levels.

On one level, presently available heat pump models will be discussed. On

another level, the discussion will deal with the available models of particular

components of a heat pump. The latter discussion will take place in later

sections

.

Although some equipment manufacturers have highly developed proprietary models,

one of the first sophisticated steady-state models of the vapor compression

17
cycle was published at Massachusetts Institute of Technology [1]. The MIT

model simulates a heat pump in the heating mode equipped with a thermostatic

expansion valve, TXV, maintaining a constant refrigerant vapor superheat at the

evaporator exit. Input data required to the program, besides heat pump data,

are indoor air conditions, saturation refrigerant temperature at the evaporator

exit, and refrigerant vapor superheat at the same location. Heat pump capacity,

power input, and outdoor air temperature are obtained as results of a simulation run. If heat pump performance data at given operating conditions are required, several simulation runs have to be performed in order to be able to

interpolate to get desired results.

The Oak Ridge National Laboratory developed a heat pump simulation model [2]

based on the study performed at MIT. The ORNL program includes a new com- pressor model, more suitable for use by an averaging user. In addition to

a thermostatic expansion device available in the MIT version, a capillary tube

model has been incorporated, hence a heat pump equipped with either type of an expansion device can be simulated by the ORNL program, the general structure

of MIT model has been retained. Input data to the program include refrigerant

saturation temperature and refrigerant vapor superheat at the evaporator outlet

while as in the MIT model, outdoor air temperature is one of the outputs of the

programl./ . A limitation appears in the simulation of a system employing a constant flow area expansion device. For a heat pump with a capillary tube, superheat is usually known only at one set of standard operating conditions (function

of refrigerant charge only). Then, when a change in operating conditions occurs,

the superheat will also change. The ORNL model does not provide a means for

calculating the change in superheat; it is required as input data, which can

be approximated at best. An additional limitation also results from the

capillary tube model. The model depends on curve fitting of capillary tube

capacity charts presented in ASHRAE Guide and Data Book [4]. These charts

cover performance of a long capillary tube with subcooled liquid entering.

Thus , the model cannot handle either a capillary tube when there is incomplete

condensation in the condenser (that does happen, particularly during heating

operation), or very short constant bore flow restrictors. A capillary tube

model that could handle all cases described above does not exist in literature

to the author's knowledge. The ORNL model was validated in the heating mode

only.

A mathematical model of a heat pump was also developed at the National Bureau

of Standards [3]. The model simulates the performance of a heat pump employing

a TXV as an expansion device. The general logic of the model is similar to those

of the MIT and ORNL models. Heat pump data, indoor air conditions, refrigerant

saturation temperature, and refrigerant vapor superheat at the evaporator exit

are required as input. Output consists of performance information (capacity.

±! At the stage of completion at this report documentation on a new heat pump

model was published by Oak Ridge National Laboratory (December 1981) [55].

This latest model, representing an upgraded version of the model described

in [2], allows the user to specify heat pump operating conditions.

18
power input, efficiency) and the outdoor temperature. The NBS model was validated in the heating mode only.

The above review outlines the available, general models of a heat pump, their

applicability and limitations. Considering the formulation of a model of a heat pump with a capillary tube, the following conclusions can be made:

1. The only program available in the literature for simulating a heat pump

with a capillary tube is provided by ORNL. The program has limitations

which result from the logic of the program and from the capillary tube

model itself. As input data, refrigerant vapor superheat at the evaporator

exit is required, and the program is unable to simulate the heat pump when

a two-phase flow condition exists at the capillary tube inlet. Also, the

model cannot handle a short capillary tube.

2. The logic in the existing models is adequate for simulation of a heat pump

with TXV, however, it is insufficient for modeling a system employing a capillary tube. A new logic allowing for calculation of the vapor

superheat should be formulated.

3. There is no capillary tube model available in the literature which can simulate its performance regardless of refrigerant state (two-phase or subcooled liquid) at the tube inlet, and which can simulate tubes of a short length. Such a model should be formulated.

4. Models of a compressor, an evaporator, and a condenser are available in

the literature. They may be used in original or modified form in any new simulation heat pump model.

5. None of the available models accepts operating conditions as input data.

A model providing heat pump performance data at given operating conditions

is most desirable and appropriate logic should be developed.

6. All discussed models have been verified in the heating mode only.

Verification of a model in heating and cooling operation would be desirable.

These conclusions led to the development of the new model described in this

report. It was felt that the model should simulate an air-to-air heat pump

equipped with a constant flow area expansion device with no prefixed refrigerant

state in any system location required in a set of input data.

19
(
5. MODELING A HEAT PUMP WITH A CONSTANT FLOW AREA EXPAI^SION DEVICE

5.1 MODELING OF THE SYSTEM

5.1.1 Structure of Available Heat Pump Models

During heat pump operation, refrigerant parameters at any location are established at a level which satisfies the respective components of the system

There is a one-to-one relationship between working medium parameters and opera

ing conditions, i.e., for given outdoor and indoor air conditions, there is no alternative but just one set of refrigerant parameters at any location within

21
the system which constitute steady state operation. This unique set of

refrigerant parameters has to be sought by a heat pump simulation program in an iterative process.

In order to set up a heat pump system logic, balances taking place during

steady state operation have to be recognized. From the fact that a thermodynamic cycle is a closed loop, an analysis of a heat pump cycle on a pressureenthalpy diagram allows statement of two balances:

- Enthalpy Balance

This balance implies that energy input to the system, i.e., to the

evaporator, the suction line and the compressor, has to be equal to

energy rejected by the system, i.e., by the high pressure line and

the condenser.

- Pressure Balance

This balance implies that increase of refrigerant pressure during the

compression process has to be equal to the total pressure drop during

other processes forming the cycle. As pressure drop and mass flow rate

are strictly related, this balance may be restated that pressure drop

through each component has to be adequate so mass flow rate in each

component is the same.

These two balances are sufficient to set up logic for a program simulating the

performance of a heat pump equipped with a thermostatic expansion valve. This

was done for all three models reviewed [1], [2], [3]. The logic of these models

is very similar, so only one, the N.B.S. model logic will be described.

For the heating mode simulation, besides heat pump data, indoor air conditions,

refrigerant saturation temperature, and refrigerant vapor superheat at the

evaporator exit are required. These data (including refrigerant state at the

evaporator exit) are fixed for the run. A set of input data also includes com- pressor head pressure and outdoor air temperature, which are input as initial

estimates.

The compressor is simulated first to estimate refrigerant mass flow rate. Then

an enthalpy balance is sought simulating the condenser and the evaporator. If

refrigerant enthalpy at the condenser outlet and the evaporator inlet do not

match, outdoor air temperature is altered and the evaporator is simulated until

the balance is reached. It is assumed reasonably well, that a change in outdoor

temperature has negligible effect on compressor performance. Next, the thermostatic expansion valve is simulated. If refrigerant mass flow rate through the

TXV is not equal to that through the compressor, the compressor discharge

pressure is changed and the whole procedure is repeated again.

The above logic is relatively simple, since all iterative processes take place

with a fixed refrigerant state at the evaporator exit. The evaporator environment temperature is calculated and is part of the simulation run output. If

22
performance of the unit at particular outdoor conditions is needed, a few

simulation runs have to be performed in order to allow for interpolation. Of

course, this results in increased computing time and imposes a need to reduce

output data. A similar logic can be used to simulate the cooling mode, however,

with similar inefficiencies.

5.1.2 New System Logic

Two balances, an enthalpy balance and a pressure balance, are sufficient to set

up the logic for a program simulating steady state performance of a heat pump

with a thermostatic expansion valve because this expansion device maintains a constant refrigerant vapor superheat at evaporator outlet. A constant flow area expansion device does not provide any fixed control of refrigerant superheat or any other flow parameter. It can be generally stated, that for heat

pumps with capillary tubes during cooling mode operation, the vapor superheat

at the compressor can inlet increases with rising indoor temperature and humidity, and with decrease of outdoor temperature. Logically then, the superheat

decreases with respective opposite trends. During cooling operation, superheat

may vary from 0 to 40°F. During heating mode operation, very low or no superheat is observed at the compressor inlet.

In as much as refrigerant superheat is a new unknown, an additional balance has

to be used to establish superheat values for each operating condition. It is

proposed, that superheat be evaluated by means of conservation of mass, as expressed by the equation:

2- / pdV = 0 (1)

Dt V

where p = density

V = system volume

Equation (1) implies, that the amount of working medium in the system is the same at all times, for any operating conditions.

The logic is based on three balances, enthalpy, pressure, and refrigerant mass

as explained below. It provides means for calculation of vapor superheat and

allows for heat pump simulation at implied operating conditions. Explanation

of the logic, for sake of clarity, is done considering only four main components

of a heat pump, i.e., a compressor, a condenser, a capillary tube, and an evaporator.

Required input consists of outdoor and indoor conditions, and heat pump data.

The simulation process starts with guessed refrigerant pressure and vapor

superheat at the compressor inlet, and the compressor discharge pressure.

Using these data, compressor performance is simulated yielding refrigerant mass

23
flow rate. Next, the condenser and the capillary tube are simulatedl/ and a mass flow balance is sought by comparing refrigerant mass flow rate through the

compressor and capillary tube. If compared mass flows are not equal, simulation of the compressor, the condenser, and the capillary tube is redone with an unchanged refrigerant state at the compressor can inlet and a modified guess of

compressor discharge pressure. Increasing this pressure reduces refrigerant mass flow rate through the compressor. Then the condenser works with this

smaller mass flow rate and at a higher saturation temperature. Consequently,

refrigerant reaching the expansion device inlet is at higher pressure and has

more subcooling. Both factors promote greater mass flow rate through the capillary tube. Thus, increase in discharge pressure has a clear and opposite impact

on mass flow rates through the compressor and through the capillary tube and

the appropriate discharge pressure can be found >r which mass flow balance

exists

.

Once mass flow balance is reached, simulation of the evaporator is performed

with the known refrigerant state at the evaporator exit and the refrigerant

mass flow rate. Since the thermodynamic process in a capillary tube may be

viewed as adiabatic, refrigerant enthalpy at the evaporator inlet should be

equal to the enthalpy at the condenser outlet. If these enthalpies are not

equal (enthalpy balance is not reached), a new calculation starts from the

beginning with a modified guess on the refrigerant pressure at the compressor

can inlet. From the condenser operation point of view, a change in compressor

suction presure induces a change in refrigerant mass flow rate and modification

of the condenser saturation temperature caused by the system flow balance

search. These two changes have opposite effects on refrigerant enthalpy at the

condenser exit, leaving it only slightly altered. On the other hand, the same change of compressor suction pressure has a strong effect on refrigerant

enthalpy at the evaporator inlet by a change in the evaporator saturation temperature and refrigerant mass flow rate, both working to change the enthalpy in

the same direction. Thus an appropriate suction pressure at the compressor

inlet can always be found for which an enthalpy balance exists.

Once enthalpy and pressure balances are established, two out of three

refrigerant parameters guessed at the beginning are determined. However, these

two parameters, compressor suction and discharge pressure, are only as good as the third guessed parameter, the refrigerant superheat at the compressor inlet,

which has not yet been verified. For this parameter verification, the refrigerant mass balance in the system is used. Specifically, a refrigerant mass inventory is made. It is based on refrigerant states in the system which are found to satisfy enthalpy and pressure balances with asumed vapor superheat at

the compressor can inlet. Then the amount of refrigerant obtained from mass inventory calculations is compared to the actual refrigerant mass input (known

tl Simulation of a capillary tube can be performed at this time even without

precise knowledge of the evaporator inlet pressure. For a heat pump pressure operation range, capillary tube capacity is a weak function, if not

independent of evaporator pressure. For the first calculation pressure at

the evaporator inlet may be assumed to be the same as at the evaporator

outlet. Refer to section 5.3.

24
as design parameter) . If the amount of refrigerant calculated is smaller than

refrigerant input into the heat pump, the superheat guess has to be decreased

and all calculations have to be repeated from the beginning.

The logic described above is presented graphically in figure 8. Details about

the above program flow chart can be found in appendix H. Refrigerant mass inventory is discussed in section 5.6.

5.2 MODELING OF A RECIPROCATING, HERMETIC COMPRESSOR

5.2.1 Compressor Operation

The compressor is mechanically the most complex component of a heat pump. A

reciprocating, hermetic compressor is most commonly used in heat pump systems.

A schematic of this type of compressor is shown in figure 9. The compressor

consists of a shell containing an electric motor, a cylinder/piston assembly

with valves and manifolds and pipes. The electric motor is coupled to the

compressor eccentric. Lubricating oil is collected on the bottom of the can

and is in free contact with the refrigerant.

Flow of the refrigerant is from location 3 to location 8, as marked in figure 9.

Low pressure refrigerant enters the compressor can and is directed towards the

electric motor to cool motor windings. Enthalpy of the refrigerant changes due

to this and other heat transfer with other surrounding surfaces; namely, the

compressor can, manifolds, the discharge pipe and the cylinder body. Then the

refrigerant at state 4 undergoes the cylinder process, which in case of positive displacement compressors consists of: expansion through the suction valve

and mixing the residual gas, compression, expansion through the discharge valve,

and re-expansion of residual gas. Compressed, high temperature and high pressure refrigerant vapor leaves the compressor through the discharge manifold and

discharge line after giving up some heat to the low pressure refrigerant at

state 4.

Numbers marked in figure 9 identify key compressor locations and corresponding

refrigerant states. Even during steady state operation it is difficult to talk

about steady parameters in the compressor can, where, pressure changes in a manner of pulses. This pulsation is initiated by the periodic compression and

suction processes and valve behavior. Because a certain pressure difference is

needed for valve opening, pressure in the cylinder at the end of compression is

considerably above discharge manifold pressure. Similarly, pressure during

intake falls below suction manifold pressure. Particularly strong pressure

peaks are observed in case of spring equipped valves. These pressure differences vary from compressor to compressor because of valve and manifold design,

but also vary for the same compressor with compressor speed, compression

pressure ratio, refrigerant, and mass flow rate.

Since processes taking place in a compressor are dynamic in nature, the most

correct way of modeling a compressor would require dynamic simulation of valve

motion, cylinder-manifold pressure interaction, and cylinder heat transfer.

Such models exist and are described in [5], [6], and [7]. However, these models

25
INPUT: HEAT PUMP DATA

OPERATING CONDITIONS COMPRESSOR SUCTION PRESSURE COMPRESSOR HEAD PRESSURE VAPOR SUPERHEAT AT COMPRESSOR INLET , REFRIGERANT CHARGE

SIMULATE COMPRESSOR

UJ

Be

a.a

s

SIMULATE CONDENSER

SIMULATE CAPILLARY TUBE

NO

NO

X NO

PRINT RESULTS

STOP 7

Overall logic of a program simulating a heat pump with

constant flow area expansion device

26
ELECTRICAL

MOTOR

Schematic of a heraetic compressor

27
usually require some very detailed experimental data or design information, not

necessarily readily available for the prospective user of a heat pump simulation

program. That is why a relatively simpler, but certainly still adequate

compressor model from [3] was adopted for this study.

5.2.2 Theory and Governing Relations

Several assumptions are made for model formulation. The basic assumption is

that the highly dynamic process in the compressor results in steady gas parameters through the can. Pressure and temperature in the cylinder and in suction

and discharge manifolds are assumed to have constant values. Refrigerant is

considered to have uniform thermodynamic properties throughout the space

assigned to the particular location. Refrigerant flow through the compressor

is assumed to be one-dimensional, so one-dimensional steady flow equations can

be used. Though the model does not simulate valve dynamic behavior, it allows

for valve representation by a steady difference between manifold and cylinder

pressures.

In order to derive equations governing compressor balances, the thermodynamic

irreversibilities taking place in the hermetically sealed compressor have to be

identified. These losses can be put into four categories:

1. Those associated with incomplete conversion of electric energy into a mechanical energy available tor vapor compression.

2. Those associated with non-isentropic conversion process inside the

cylinder.

3. Those due to heat transfer at different locations.

4. Those due to pressure drop at difference locations.

All of these losses contribute to the overall compressor efficiency on the

order of 60 percent (first law basis) and have to be considered in the modeling

effort

.

Conversion of Electrical to Mechanical Energy

Electric energy is supplied to an electric motor to be converted into mechanical

work. This conversion has losses due to windage, friction, winding resistance,

and hysteresis, which are accounted for by a motor efficiency rie* By definition

(2)

where E = electric power input

Wg = mechanical power output

28
Electric motor efficiency, ne> depends on load and is customarily given as a function of the fraction of actual mechanical power output to the maximum power

output. In heat pumps with usual power requirements below 5 hp, single phase

electric motors are mostly used. They are permanent split-capacitor or capacitor-start capacitor-run types. A typical efficiency versus load curves

for a permanent split-capacitor 2 pole electric motor is shown in figure 10

[14].

The electric motor is coupled to the compressor eccentric. Moving compressor

parts experience friction and a power loss accounted for by the compressor

mechanical efficiency nm* By definition

e where = mechanical power available for compression

A common value of compressor mechanical efficiency falls in range of 0.95 to

0.98.

Compression Process

A simplified Indicator diagram Is shown in figure 11. It shows assumed constant

suction and discharge pressures in a cylinder. In modeling the cylinder process,

both compression and re-expansion processes are assumed to be polytropic with

the same polytropic Index, n, following the equation:

P • v'^ = const (4)

where n = polytropic index

P = pressure

V = refrigerant vapor specific volume

Since constant pressure and temperature are assumed during the discharge process,

this Implies no change in specific volume between points "c" and "d" (figure

11); consequently, the compression and re-expansion polytropic curves will coincide. This further means, that the net work required for compression of the

residual gas is zero.

The refrigerant enthalpy Increase during polytropic compression, I5 - 15 (refer

to figure 5), can be evaluated by the equation derived from the expressions for

Isentroplc and polytropic work of compression at the same compression ratio.

Equating these expressions results in the equation:

^

fe)^'^' -1

(5)

Y - 1 .P

A] - 1

P5

29
^ ' ' 0 0.2 0.4 0.6 0.8 1.0

LOAD (FRACTION]

Figure 10. Typical efficiency versus load curve for a permanent

split-capacitor 2 pole electric motor [4]

30
PRESSURE

DISCHARGE

MANIFOLD

PRESSURE

SUCTION

MANIFOLD

PRESSURE

A - B SUCTION

B - C COMPRESSION

C - D DISCHARGE

D - A RE-EXPANSION

CYLINDER VOLUME

11.

Figure 11. Simplified indicator diagram for a reciprocating compressor

31
where I5 - refrigerant enthalpy before compression

I5 = refrigerant enthalpy after compression

Igg = refrigerant enthalpy after Isentroplc compression, defined by

pressure P5 and entropy S5

n = polytroplc Index

P5 = suction pressure

P5 = discharge pressure

Y = Isentroplc Index

The Isentroplc Index, Y> and the polytroplc Index n, are related by the

polytroplc efficiency of the compressor:

Y -1

P n - 1 n

(6)

The Isentroplc Index, Yj equals the ratio of specific heats, as In the

following formula:

Y - ^

(7)

Cv

where Cp - specific heat at constant pressure

Cv = specific heat at constant volume

The specific heats of ideal gases are constant, thus the Isentroplc index of an ideal gas is also constant. For real gases as refrigerant vapors, specific

heats vary along the compression path. To accommodate this fact, the Isentroplc

index can be evaluated by taking the average of the respective specific heat

ratios at points 5 and 6s.

The refrigerant enthalpy Increase during polytroplc compression could also be

calculated using isentroplc efficiency and enthalpy Increase during isentroplc

compression at the same compression ratio. However, isentroplc efficiency is

sensitive to the compression pressure ratio while polytroplc efficiency is more

consistent from one application to another and provides more consistent representation of average compressor performance [8]. The imperfection of using

isentroplc efficiency may be traced to the general thermodynamic relation:

= T (8)

\ 3s / p

where 1 = enthalpy

P = pressure

s = entropy

T = temperature

requiring that pressure lines diverge on a Molller chart.

32
The refrigerant mass flow rate pumped by a compressor can be calculated by the

following formula:

m^ = 60 » RPM ' Vs

where rnj- = refrigerant mass flow rate

RPM = compressor speed (number of revolutions per minute)

Vs = compressor swept volume per revolution

V5 = refrigerant specific volume at piston before compression

Tiy = volumetric efficiency

The RPM of a compressor is equal to that of an electric motor and is a function

of load on the motor. A typical speed (RPM) versus load curve for a permanent

split-capacitor 2-pole elctric motor is shown in figure 12.

With assumptions stated so far for the compression and re-expansion processes,

the following formula for volumetric efficiency of a compressor can be derived:

1

(10)

where C^. = correction factor for leakage from piston, valves and for

throttling effect, assumed to be 0.96, [9]

Cg = clearance volume, in fraction of stroke volume

Heat Transfer Relations

Electrical energy supplied to an electric motor is in part transferred to the

refrigerant and in part is dissipated to the compressor environment. This

compressor heat balance can be expressed by the following equation:

E + mr (i3 - is) " Qc,A = 0 (H)

where E = electric power input

13,13 = refrigerant enthalpy at respective locations (refer to figure 9)

Qq^P^ = rate of heat rejected to ambient air

Reference is made to figure 9, where key locations of refrigerant in the hermetic

compressor are marked. In order to solve the compressor heat balance represented

by equation (11), the following heat transfer losses are considered in the

compressor model:

1. Heat transfer between the compressor can and ambient air, Qc,a

2. Heat transfer between the refrigerant in the compressor can and the

compressor can, Q4^c

33
Figure 12. Typical speed (RPM) versus load curve for a permanent

split-capacitor 2 pole electrical motor [4].

34

I
3. Heat transfer between the inlet refrigerant and the suction toanifold

and valve, Q4^5

4. Heat transfer between the discharge refrigerant and the discharge

valve and manifold, Q6^7

5. Heat transfer between the refrigerant in the compressor can and the

refrigerant in the discharge line, Qy^s

According to general heat transfer equation of P^clet:

Q = U • Ah • AT (12)

The heat transfer rate, Q, for a given temperature difference, AT, can be

calculated if the overall coefficient of heat transfer, U, and heat transfer

area, A|^, are known. The heat transfer between the compressor can and ambient

air (item 1) is governed by free convection (radiation is negligible). A simple

correlation is available to estimate the free convection heat transfer

coefficient in air [10], i.e.,

h a AtO-333 (13)

where h = convection heat transfer coefficient

AT = temperature difference between air and heat transfer

surface

Hence, the heat transfer rate between ambient air and the compressor shell,

Qc,A> ^® calculated by the equation:

QC,A = CQc,A(Te - T^)^-^^^ d^)

where CQc a

~ heat transfer parameter

T^ = ambient air temperature

Tj, = compressor shell temperature

Heat transfer inside the compressor shell between the shell and refrigerant

vapor (item 2) is governed by forced convection, for which the non-dimensional

heat transfer parametric expression in terms of Nusselt, Reynolds and Prandtl

numbers is in the form:

Nu « ReO*8 • PrO*333 (15)

where jju = ^ * ^ Nusselt number

k

Pr = y

* Prandlt number

k

Re = ^

' Reynolds number

35
m

r G = refrigerant mass flux

A

L = characteristic dimension

k = refrigerant thermal conductivity

y = refrigerant dynamic viscosity

Using equation (15) the forced convection heat transfer coefficient, h, can be

expressed as:

h = C . m^O'8 , ^0.667 . . ^ - 0.467

where C = constant of proportionality, a fuction of wetted surface

geometry

Combining equations (12) and (16) yields the following expression for heat

transfer rate between the compressor can and refrigerant:

n -PA TM 0»8 1, 0.667 ^„ 0.333 , - 0.467 _ t ^ %,C ~ ^ ' \ ' • "^4 • ^P4 ' H ^^4 T^)

or

A — rr\ 0.8 1 0.667 0.333 — 0.467 /rp _ t \ /'^7^

^4,0 ~ '-'^4,0 * TMr * H • ^P4 * 1^4 ^c*

where CQ4^c ~ heat transfer parameter

Obviously, heat transfer rates ^ and Q4 q are equal. Derivations of equations

for the heat transfer rates between the iniet refrigerant and the suction manifold and valve, Q4 5, and between the discharge refrigerant and the discharge

manifold valve, Q5 7, is similar to the derivation just performed for Q4 q, since

in these two cass heat transfer is also by forced convection. Resulting

expressions for the heat transfer rates are:

r. - rn n, 0«8 . 0.666 0.333 - 0,467 _ t ^ (18)

n - rn TM 0'8 1. 0.667 p„ 0.333 - 0.467 /rp _ t ^ ^6,7 - CQ6,7 • °^r • ^,7 * ^P6,7 ' % ,7 ^^7

Heat transfer between refrigerant in the compressor can and refrigerant in the

discharge line (item 5) is modeled as forced convection heat transfer between

the fluids separated by a barrier non-resistant to heat flow. Assuming that

the temperature of refrigerant in the shell does not change (as a result of

other heat transfers in the can and mixing) and applying the logarithmic mean

temperature difference, the following expression for heat transfer rate Q7 3 can be derived:

Q7,8 - Cq7.8 • (T7 - T8)/l„Il^ (20)

36
where al = (Cp^ • Cp^ ^g)^-^^^^^ . ^^^^)0.667

a2 = . Cp^^gO'^Sa . k^^g^*^^^

^3 = 1^7,8°*''^^ • Cp^^-^^^ • ^40-667

Subscripts 3 through 8 refer to refrigerant key locations in the compressor as per figure 9. If separated by a comma, the average value is implied.

The heat transfer to/from the following refrigerant changes the refrigerant

enthalpy as per equation:

q = mj. ' M (21)

where Ai = refrigerant enthalpy change

Combining equations (18), (19), (20), and (21) yields the following expressions

for the refrigerant enthalpy change during flow between respective locations of

the compressor:

1 - 1 - rn . k 0-667 . p_ 0.333 /t _ t \/fm 0-2 . 0.467v

(22)

i - i - rn .V 0-667 . p„ 0.333 fr - r ^/rm 0-2 . ,, 0.466/

^6 ^7 - ^^6,7 H,7 • ^P6,7 ^^7 V^^'^r ' ^6,7 > (23)

- '8 = CQ6,7 (T7 - l8)/(»r'''' • >

where al, a2, a3 are as in equation (20)

Derived heat transfer relations contain heat transfer parameters CQc a> ^Q4 5>

CQ5 7, and CQ7 g. These parameters are functions of heat transfer surface geometry mainly and have to be found empirically. If required laboratory test data

of a given compressor are not available, typical compressor test data can be

used. A large number of experimental compressor measurements have been published.

The summary of these data and a list of references can be found in [1]

•

Pressure Drop Relations

Total pressure drop AP^ot experienced by a flowing fluid results from pressure

drops due to friction, momentum change, and gravity, i.e.,

^Ptot = ^Pfriction + ^^accel + ^Pgravity (25)

Pressure drop due to gravity in the hermetic compressor may be disregarded

based on an order of magnitude analysis. On the same grounds, pressure drop of

flowing refrigerant between certain compressor locations may be attributed to

either viscous effect or dynamic effect.

Pressure drop due to the dynamic effect, i^'Paccel' proportional to velocity

head , i.e.,

37
^^accel - P • V2 (26)

which can be written, using the continuity equation:

^Paccel = CP . m//p (27)

where .CP = pressure drop parameter

nij. = refrigerant mass flow rate

V = refrigerant velocity

p = refrigerant density

The relation for the pressure drop due to viscous effect, Pfriction*

derived from the classical Fanning equation for pressure drop in a tube:

^Pfriction = 2f . p . V2 . L/D (28)

where f = friction factor

D = tube diameter

L = tube length

The friction factor, f, in the equation above is approximately proportional to

the Reynolds number to the -0.2 power. Considering this fact and applying the

equation of continuity, equation (28) becomes:

^Pfriction = CP . yO.2 . ^^0.8/^ (29)

Pressure drop of the refrigerant in the compressor can is modeled by evaluating

individual pressure drops between refrigerant key locations as per figure 9.

Based on equations (27) and (29), and attributing pressure drop between particular compressor locations either to the viscous or friction effect, the

following pressure drop relations are proposed:

P^ - P4 = CP3 ^

• m^. /p^ 4 for compressor can inlet (30)

P4 - P^ = CP4 5

• /P4 5 suction manifold and (31)

' * valve

Pg - Py = CP^

-J • nij. /p^ J for discharge valve and (32)

' ' manifold

Py - Pg = CPy g

• mj.-"-*^ • My 8 dlscharge line (33)

Pressure drop parameters CP3^4, CP4 5, CP^ 7, and CP7 3 have to be found

experimentally for a given compressor or can be calculated using test data for

a similar compressor. For sources of such data, the reader may refer to [1].

Equations derived in this section make it possible to carry out an energy

balance of a hermetic compressor solving it in an iterative process for electrical energy input, for heat lost to the ambient air, and for refrigerant parameters in compressor key locations. The variety of designs of refrigerant flow

passages in the compressor caused the modeling of pressure drop and heat transfer

38
to be done in an approximate manner in this general compessor model. In spite

of several assumptions that were made to simplify the modeling process and

reduce computing time, the model still retains sufficient details of the underlying physical principles to allow designers to determine which specific changes

in the compressor design lead to increased efficiency of the compressor and the

heat pump system.

5.3 MODELING OF A CONSTANT FLOW AREA EXPANSION DEVICE

A constant flow area expansion device, as used in heating/air conditioning

systems, is commonly called a capillary tube or a refrigerant flow restrictor.

Usually it is a small bore tube of length as short as one-half inch up to a few

feet, connecting the outlet of the condenser (or a liquid line) to the inlet of

the evaporator. The main task, of the constant flow area expansion device is to

maintain the minimum pressure at the condenser at which all the flowing refrigerant can condense. Many researchers have investigated flow through a capillary tube and a bibliography on the subject can be found in [4] . The objective

of this research was to explain capillary tube phenomena and develop a mass flow rate prediction method. None of the metods found in the literature were computer oriented or could analyze both single-phase and two-phase flow in the

restrictor. Thus there was a need to develop a more comprehensive capillary

tube routine that could be incorporated into a heat pump simulation program.

5.3.1 Available Capillary Tube Performance Data

Capillary tube is a traditionally accepted name for a constant flow area expansion device used in heat pump systems. This name is adequate and misleading since for tube diameters in the neighborhood of 1/16 of an inch,

capillary forces are negligible for the task the capillary tube performs. The

pressure drop consists practically of:

- loss due to sudden contraction at the entrance

- loss due to flow in the tube

- loss due to sudden enlargement at the exit to the evaporator

The flow of refrigerant through a constant bore tube is more complex than the

geometric simplicity of the device would first indicate. The pressure and temperature distribution along a typical capillary tube is shown in figure 13.

Bolstad's description of the flow is as follows:

At the entrance to the tube, section 0-1, there is a slight pressure drop

which was usually unreadable on the gages . From point 1 to point 2 the

pressure drop is linear. In the portion of the tube 0-1-2 the refrigerant

is entirely in the liquid state and at point 2 the first buble of vapor

forms. From point 2 to the end of the tube the pressure drop is not

linear, the pressure drop per unit length increasing as the end of the

tube is approached. For this portion of the tube, both the saturated

liquid and saturated vapor phases are present, the percent and volume of

vapor increasing in the direction of flow ....

39
With a saturation temperature scale corresponding to the pressure scale

superimposed along the vertical axis, it is possible to plot the observed

temperatures in a more meaningful way than if a uniform temperature scale were used. The temperature is constant for the first portion of the tube

0-1-2. At point 2, the presence has dropped to the saturation pressure

corresponding to this temperature. Further pressure drop beyond point 2

is accompained by a corresponding drop in temperature, the temperature

being saturation temperature corresponding to the pressure. As a consequence, the pressure and temperature lines coincide from point 2 to the

end of the tube [11]

.

The point of the tube where the first bubble forms is called a bubble point or

a flash point. The pressure at the point is called the flash pressure. Bolstad

also presented an analytical method of solution for adiabatic flow through the

capillary based on the Fanno flow theory.

Mikol [12] performed a capillary tube investigation from which his conclusions

can be summarized as follows:

- Fluid flow through small bore tubes conforms to continuum flow as established for large bore tubes and pipes.

- the friction factor correlation of Moody and any others consistent

with Moody's correlation [13] is applicable to single-phase flow in

small bore tubes.

- the phenomena of metastability , persistence of the liquid state to

pressure less than the saturation pressure corresponding to its local

temperature, has been found to occur.

- the phenomenon of choked flow in two-phase flow occurs in the same way and for the same reasons as in the case of gaseous flow. Sonic

velocity occurs at the tube exit.

One of Mikol 's findings, existence of superheated liquid in a small portion of

a tube was not observed by Bolstad [11]. However, it was reported by Cooper

[14] and Rezk [15] . Investigators have found that delayed evaporation is

affected by initial disturbances and flow agitation, but there is not enough

data in the current literature to assess all the factors promoting/eliminating

this phenomena and making it possible to consider metastability in a capillary

tube model at this time.

Refrigerant mass flow rate through the capillary tube always increases with an increase of inlet pressure. It also increases with a decrease in external

outlet pressure down to a critical pressure below which the flow does not

change. Sensitivity of the flow to external outlet pressure decreases

significantly at low outlet pressure before the flow is choked.

Pressure and temperature distributions along a capillary tube as shown in figure

13 occur at design operating conditions of a long (a few feet) capillary. Part

of the capillary is filled with flowing liquid, while two-phase flow exists in

40
CONDENSER CAPILLARY TUBE EVAPORATOR

DISTANCE FROM CAPILLARY INLET (Ft)

Figure 13. Pressure and temperature distribution along typical

capillary tube [11]

41
the other part. However, there are also other possible modes of operation,

i.e., with only two-phase flow in the capillary (the case of Incomplete condensation in the condenser) or with only liquid flow (the case of a short restrictor). All these cases are observed in practice and have to be simulated by a general model of the constant flow area expansion device.

Based on the experimental evidence and the theory of large tube fluid mechanics,

the following assumption were set up for model formulation:

1. The capillary tube is a straight, horizontal, constant inner diameter

tube

2. Flow in the capillary is one-dimensional and homogeneous

3. Flow in the capillary is adiabatic

4. Flow resistance in the capillary tube can be subdivided into

a. resistance due to the entrance effect

b. resistance due to flow in a tube which consists, in the general

case, of single-phase liquid flow resistance from the entrance

to the flash point, and two-phase mixture flow resistance in the

rest of the tube. Existence of delayed evaporation phenomena

is neglected. Resistance due to the exit effect is neglected as meaningless for a choked flow and insignificant for a non-choked

flow [11]

5. Choked flow phenomenon for two-phase is governed by the same laws as for single-phase flow, and can be modeled accordingly.

5.3.2 Critical Flow in a Capillary Tube

Refrigerant mass flow rate through a given capillary tube will depend on the

inlet refrigerant state and on pressure that will be established at the tube

outlet. This pressure may equal the evaporator pressure or may have a higher

value if the flow is choked. Since pressure at the capillary tube exit is one

of the parameters affecting the flow, it has to be known for proper refrigerant

mass flow rate prediction.

The assumption that the flow in the capillary is one-dimensional and homogeneous

enables the two-phase flow in the capillary to be treated as single-phase flow

with uniform properties at any cross-section of the flow and allows use of the

single-phase, one-dimensional form of the governing equations as presented in

figure 14, 15, and 16.

Adiabatic flow through a capillary tube is a classical example of so called

Fanno flow; adiabatic flow with friction in a constant area duct. The energy

equation for such flow has the form:

di + V • dV = 0 or 1^=1+ V^/2 = const (34)

42
which, when combined with the equation of continuity yields:

io = i + — • v^/2 = i + • v^/2 (35)

where A = tube cross-sectional area m

G = —E- , refrigerant mass flux

A

i = refrigerant enthalpy

io = refrigerant stagnation enthalpy

THj. = refrigerant mass flow rate

V = flow velocity

V = refrigerant specific volume

Graphical representation of equation (35) on the enthalpy-entropy diagram, as shown in figure 17, is called a Fanno line. Fanno flow as an irreversible

adiabatic process, can sequentially exist only in the direction of increasing

entropy. The upper branch of the Fanno line corresponds to subsonic, accelerating flow while the lower branch is the supersonic, decelerating flow. Both

flows tend towards point C where the sonic velocity is reached.

The Fanno line implies that fluid cannot reach the velocity of sound inside a constant area duct, because, if this happened, further flow in the duct would

have to be associated with a decrease of entropy, as per figure 17, and that

would be in violation of the Second Law of Thermodynamics. Thus, the only

location of a tube at which sonic velocity can be reached is the tube exit, and

choking will never occur inside the capillary regardless of external outlet

pressure. Evaluation of the critical pressure for two-phase Fanno flow is

discussed in appendix E.

It is intuitively felt, based on compressible gas theory, that if the critical

pressure is greater than the pressure in an evaporator, the critical pressure

will be established at the capillary tube outlet. However, it is not clear

what pressure would actually exist at the end of a capillary which is fed with

a subcooled liquid, if the calculation choking pressure is higher than the saturated liquid pressure resulting from the liquid temperature^^/ . Before this is

explained, some information on sonic velocity phenomena is reviewed.

The velocity of sound is calculated by the formula:

0.5

(36)

s

3/ It is assumed here, for discussion purposes only, that choking pressure can

be found independently from refrigerant mass flow rate and then mass flow

rate can be calculated. In fact, mass flow rate and choking pressure (and

other factors of operation) are interrelated and have to be found

simultaneously in an iterative process.

43
I 1

pAV j PAV + ^ PAVldL

^1

L - - '

|-«— dL *^

CONTINUITY EQUATION: dfpVj = 0

Figure 14. Mass balance for an element of fluid in one-dimensional

steady flow in a constant area duct

PA

pAV

PA + T— (PA)dL o L

pAV^ +-TT-fpAv2)dL

O L

I I

dL

MOMENTUM EQUATION: AdP + pAVdV + rSdL = 0

Figure 15. Momentum balance for an element of fluid in one-dimensional

steady flow in a horizontal, constant area duct

44
Q = 0 W = 0

PVA(u + y + P

P pVAlu +- +y) 6 L :PVA(u +

dL

ENERGY EQUATION: dlu +^ + — ' +y =

'o

Figure 16. Energy balance for an element of fluid in one-dimensional

adiabatic, steady flow in a horizontal, constant area duct

^

ENTROPY

Figure 17 . Fanno line

45
where a velocity of sound

P pressure

density

entropy

P

s

For a liquid, in which the density is hardly affected by pressure, the velocity

of sound assumes a very high value, typically in the range of 5000 ft/s. Liquid

flow in capillary tubes is never accelerated to such high velocities. However,

the velocity of sound for a two-phase state containing a small amount of vapor

is drastically smaller, e.g., sonic velocity of two-phase flow of quality less

than 0.0001 and temperature of 100°F is on the order of 1 ft/s. Sonic velocity

increases with temperature and quality and assumes a value of about 500 ft/s

close to the critical state. Thus, the velocity of sound in a liquid is much

different than in the two-phase state and a sharp discontinuity in sonic

velocity takes place at the boundary of single and two-phase flow [16]

In the case of a capillary tube working at design conditions, there is a singlephase liquid flow in the beginning part of the tube and single component twophase mixture flow in the latter tube portion. Though there is basically one substance flowing, the relation between pressure and density along the path of

flow will be different for each part of the flow in the tube. Theoretically, a separate Fanno line could be drawn for each flow, with an intersection point

corresponding to a flash point. Also, two sonic velocities could be considered,

but a limitation of sonic velocity in the liquid is disregarded as it is too

high to be reached in a capillary tube under any operational conditions.

Consider several capillary tubes of the same bore but different lengths. All

the capillaries receive subcooled liquid at the same condition and are working

at the same pressure in the evaporator. The various possible flows are shown

in figure 18. If a capillary tube is sufficiently long (a), it will be able to

realize a pressure drop down to the evaporator pressure and choking will not occur. With a shorter capillary tube such as (b) and (c), the mass flow rate

is greater, and the critical pressure which is established at the tube exit,

exceeds the pressure in the evaporator. Tube (d) has the length at which the

critical pressure equals the flash pressure. The critical pressure in the case

of tube (e) is greater than the flash pressure and the latter is established at

the tube exit due to the flash that will occur from the sudden expansion.

Thus, any tube shorter than tube (d) will have flash pressure at the exit. To

prove this last case, which is least evident, other flow situations than that

proposed must be considered. First, eliminated the possibility of pressure at

the tube exit being greater than the flash pressure because otherwise it would

imply the liquid is choked at other than its critical pressure, which is

physically impossible. Another alternative is to assume that the liquid reaches

saturation pressure within the tube and changes into a two-phase mixture. With

the two-phase mixture critical pressure being higher than saturation pressure,

it would mean a change from single-phase subsonic flow into two-phase supersonic flow. Such flow is also impossible since it would require the flow to

experience a decrease in quality in its progress and would be converted back to

liquid flow immediately. The liquid flow also could not proceed any further

because vapor bubbles would form. Thus the only possible situation for flow

46
i

CAPILLARY TUBE LENGTH

* Actual flash pressure is not constant. It Increases slightly with

capillary tube length.

Figure 18. Pressure distribution along different length capillary

tubes at the same operating conditions

47
with the critical pressure greater than the liquid saturation pressure is to be

choked at the tube exit where the flow is at the liquid saturation pressure.

5.3.3 Model Formulation

Refrigerant flow from the liquid line into the capillary tube experiences a pressure drop due to sudden contraction. This pressure drop consists of an acceleration loss and entrance friction loss and is usually expressed by a decrease in the Bernoulli head and a contraction coefficient referred to the

kinetic energy of the flow in the section of smaller flow area:

(37)

Subscripts in eq. (37) refer to sections shown in figure 13. Combining with

the equation of continuity, the above can be rearranged to:

^1^

Pq - = AP = (1 + K)Po^i '-j- (38)

The contraction coefficient, K, given in the literature is stricly empirical

and is represented as a function of the contraction area ratio. Several sources

are in disagreement about its value. The value of K = 0.15, used here is from

a derivation based on Kay's general formula [17]. It was calculated for a normal range of contraction area ratios for capillary tubes with slightly

beveled entrances.

The equation of motion for steady flow in a constant cross-section area pipe

has the following form:

p'A'dV+A'dP+T«S»dL=0 (39)

The skin friction coefficient, x, can be expressed in terms of the friction

coefficient, f, and the velocity head:

T = 1 f . p . v2 (40)

2

The flow velocity term, V, can be eliminated by means of the equation of

continuity:

d(V • p) = 0 (41)

Substituting and rearranging; the equation of motion assumes the following form:

2 ^i+1 ^i+1

^ • / p • dP + 1 • / f • dL + In -^-i— = 0 (42) mr D p^+i

Pi Li

48
As discussed before, flow in a capillary tube, in the general case, can be

subdivided into two parts separated by a flash point: the liquid flow part and

the two-phase mixture part. The same equation will be applicable for both

flows through in the case of liquid flow, it can be simplified on grounds of

incompressibility . In fact, it reduces to the Fanning pressure drop formula in

the following form:

AP

=^ g2 . L

(43)

where the friction factor, f, can be evaluated by the empirical formula:

f = 0.046 • Re"0-2 (44)

For the two-phase mixture flow, equation (42) has to be solved in its full

form. This was done by Whitesel, [18] and [19], but with significant over- simplified approximations for refrigerant properties. In solving equation (42),

difficulty arises with evaluating the first term because it depends directly on

the pressure-density relation along the path of flow. However, the relation

can be obtained by considering the adiabatic flow case. The specific volume at a given pressure can be expressed in terms of the property values for saturated

liquid and vapor and in terms of quality:

V = vl + x(Vv - vl) (45)

where v = specific volume

X = quality

Subscripts L and V are for liquid and vapor, respectively

The quality of the flow in the Fanno path can be found as explained in appendix

E. Thus integration of refrigerant density over a given pressure interval can

be done numerically. Still another problem is faced in evaluating the second

term of equation (42), which includes a two-phase friction factor as a function

of tube length. Erth [20] made an effort to correlate two-phase average friction factor in a capillary tube. His regression analysis, based on four sets

of data from four different experiments, yielded the following correlation for

a two-phase flow mean friction factor, f^, as a function of the inlet conditions

only:

=-^^exp[(l - xi )/2.4] (46)

Re^*^

where x^ = quality of refrigerant entering capillary tube

Re = ^ * P

, two-phase Reynolds number (47)

PL ^i (^^V ~ ^l)

Using this mean friction factor, f^j, equation (42) may be written for the twophase portion of the flow in the following form:

49
0.5

mj. = A

I

f„ • (L3 - 4) + m (P2/P3)

(48)

where ' numbers used as subscripts denote location consistent with figure 13.

The above equations have to be solved in a highly iterative process since

choking pressure, friction factor, fraction of capillary tube length with liquid

and two-phase flow, and the velocity head used to correct enthlapy are functions

of refrigerant mass flow rate which has yet to be found.

5.4 MODELING OF AN EVAPORATOR AND A CONDENSER

5.4.1 Modeling Methodology

There are two heat exchangers in a heat pump: an indoor coil and an outdoor

coil. Both coils are made in a similar way and both serve as an evaporator or condenser depending on the heat pump opertional mode. Schematic of a typical

heat pump heat exchanger is shown in figure 19. It consists of a set of finned

tubes connected in a specifically designed circuit configuration.. An example

of the coil circuitry is illustrated in figure 20. The refrigerant is pumped

through the tubing while air is forced to flow over the outside of the coil.

Heat exchanger models in the MIT and ORNL heat pump simulation programs, [1]

and [2], are very similar. Coil performance calculations in these models are based on the ef f ectiveness-NTU method for a cross-flow heat exchanger with both

fluids umixed. These models can rigorously simulate only certain regular tube

arrangements and are Insenltive to different coil circuitry patterns. The heat

exchanger models in [3] are based on the so called tube-by-tube simulation

method which depends on imaginary isolation of one tube with appropriate fin

surfaces from the coil assembly and calculating the performance of this tube

independentlyA/ . The heat transfer to/from a tube is calculated with the aid

of heat exchanger cross-flow theory. This is done for each tube in a proper

sequence and the summation results in total coil capacity. An even more detailed

method of heat exchanger simulation would be to apply the finite-difference

technique. Such a model would subdivide an analyzed tube of the coil into

arbitrary small elements and would solve heat transfer equations for each element marching in space from the tube inlet to the tube outlet. Since for this

study the cost of computing time required for such simulation would be

prohibitive, the tube-by-tube approach was adopted.

In the tube-by-tube simulation method, the performance of one tube is calculated

at one time and the general finned coil problem is reduced to a single tube problem. In order to conduct heat transfer calculations, fin surfaces associated

ZJ At the stage of completion of this report a new tube-by-tube computer model

of a condenser coil was published by ORNL [56]

.

50
Figure 19. Schematic of a heat pump heat exchanger

51
32

1 323334

I

SUBCOOLED LIQUID OR LOW

QUALITY VAPOR SIDE MANIFOLD

1716

1514

1344

r

65

1 66 75

1 76 23

64 1 35 « 1 45 22

96 1 4 I 36 12 46

1 77 21 97 5 67 1 11 • 47 78 ' 20

98 68 1 42 I 48 110 1 19 99 7 37 1 74 > 49 111 1 51 100 1 8 > 69 1 106 1 18 79 1 83

101 I 9 1 38 1 107 1 50 80 1 114 102 1 10 1 39 1 108 1 82 81 1 115 70 1 41 i 40 1109 1 113 112 1 116

71 1 73 1 72

103 1105 1104

57

I 56

I 24

I 55

I 54

53

5284

85

< 86

! 87

I 88

I 120

1 119

I 118 1117

2526

I 58

I 59

I 27 i_28

29

30

3163

95

I 126

I 125

I 124

i 123

1 122

60

616294

93

9290

89

121

^ ^ ^ ^ ^ ^ ^ ^ ^ ^

I SUPERHEATED VAPOR SIDE MAINIFOLD ^ REFRIGERANT FLOW DIRECTION IN CONDENSER

Figure 20 . Example of coil circuitry

52
with the tubes must be defined. Following Carrier and Anderson [22], It was assumed that fin area served by each tube Is equivalent In performance to a circular-plate fin of equal area. Thus a single tube is considered with a circular fin of diameter, D^, as shown in figure 21.

5.4.2 Heat Transfer Rate for a Tube in a Cross-Flow Arrangement

Usually a heat pump coll employs some form of cross-flow arrangement. If a separate tube Is considered, the problem is one of pure cross-flow. Fortunately,

this kind of arrangement has received much attention in theoretical investigations. According to the Peclet heat transfer equation:

Q = U • Ah • AT (49)

where Aj^ = heat transfer surface area

U = overall coefficient of heat transfer

AT = temperature difference

In case of a pure cross-flow arrangement, one of two mean temperature differences

applies for AT, [23]:

t2 - ti

To - ti In^^ 1- Ti - twhen temperature of one of the

fluids does not change (temperature of that fluid is

denoted by T)

(50)

m

In

t2 - ti

Tl - T2

^2 - ^1

T, - T, T2 _ t, ^+ In ^

^ t2 - ti Ti - ti

when temperature of both

fluids changes

(51)

where T = temperature of one fluid

t = temperature of another fluid

ATjQ = mean temperature difference

subscripts 1 and 2 refer to tube inlet (upstream) and outlet

(downstream) conditions

The enthalpy change of the fluid will be according to the equation:

Q = m • (12 - ii)

or

Q = m • Cp • (T2 - Ti) (52)

53
Figure 21. Approximation method for treating a rectangular-plate

fin of uniform thickness in terms of a flat circularplate fin of equal area

54
where Cp = specific heat of fluid

i = enthalpy

m = mass flow rate

Looking at any vapor compression cycle P-h or T-s diagram, it can be realized

that the single-phase and two-phase refrigerant flow usually exists in a given

heat exchanger. Also, both flow patterns can actually exist in one tube. That means the refrigerant flowing in the tube may maintain approximately constant

temperature (two-phase) or its temperature may vary (single-phase). Not only

the mean temperature difference between fluids but also the refrigerant pressure

drop is affected by the flow pattern. Equations presented below, derived from

equations (49), (50), (51), and (52), allow for detailed consideration of these

problems. The heat transfer rate for each mentioned flow condition can be

calculated as follows:

- single-phase flow only, refrigerant is superheated or subcooled at both

inlet and outlet

Q = Cpr • mr(T^^i - Ta,i)(l - exp( -

\ (1 - exp(- ^

' )))) (53)

Pr * r Pa * a

- two-phase flow only, refrigerant is in two-phase at both inlet and

outlet

Q = Cpa . (1 - exp(

' ))(Tr,g " Ta,i) (54)

"a a - superheated vapor at tube inlet, two-phase at the tube outlet

U . Q = m^-dr i - ir V) + Cpa • ma(l - Zy) (1 - exp(- ^))(Tr,g - i) (55)

where = fraction of the tube length in the superehated region which

can be calculated by the equation:

-''^

'

• ''''' - cp'(< [

z = L^ii (56)

V U . Aq

CPa • mad " exp(l - — —))

^Pa • "'a - two-phase at tube inlet, subcooled liquid at the tube outlet

Cp^ . m,(l-Z^„) U . AQ = Cp, . .,(T,,g - T,,i)(l - axp(- \ (1 - exp(-^^_^))))

r *^a a + m^d^^i - i^^L) (57)

55
where Zj-p = fraction of the tube length in the two-phase region which can

be calculated by the equation:

'°r^^r,i " ^r,L^

Ztp = U • A- (58)

Cpa • mad - exp(-

^p . I

^^^'^r.i- Ta,i)

In equations (53) through (58) the following nomenclature was used:

Aq = total exterior surface area associated with the tube wetted by air

Cpa = air specific heat at constant pressure

Cpj. = refrigerant specific heat at constant pressure

^r i ~ refrigerant enthalpy at the tube inlet

ij. L

= enthalpy of refrigerant saturated liquid

ij.^y = enthalpy of refrigerant saturated vapor

= air mass flow rate associated with the tube

rnj. = refrigerant mass flow rate in the tube

£ = air temperature upstream of the tube

Tj-^g = refrigerant saturation temperature

Tj. i = refrigerant temperature at the tube inlet

U = overall tube heat transfer coefficient

A desirable feature in the above equations is that they do not involve any of

the outlet temperatures or any of the outlet fluid parameters. Hence, the heat

transfer rate can be calculated based on total heat transfer area, overall tube

heat transfer coefficient, inlet fluid parameters, and their mass flow rates.

5.4.3 Refrigerant and Air Mass Flow Rates Associated with a Tube

Refrigerant Mass Flow Rate

During flow through a heat pump heat exchanger, the refrigerant undergoes a change of phase in the course of evaporation or condensation. The change of

phase is associated with a dramatic change of density which affects the velocity and pressure drop of the working fluid. In order to prevent a high pressure drop, tubes in some heat pump coils are connected to form branched circuits.

An example of such coil circuitry has been shown in figure 20.

Refrigerant flow direction marked in figure 20 is for the coil working as a condenser. Superheated vapor enters the vapor side manifold and is distributed

into 10 tubes. On its flow path, refrigerant merges several times and finally

merges in the liquid manifold to enter the liquid line. For the coil operating

as an evaporator, the direction of flow is opposite to that marked in the

figure. Low quality refrigerant enters the coll and the flow is subdivided

into three circuits. On its way through the coil the refrigerant evaporates

and splits several times on its way towards the exist where it is finally collected into one bigger diameter vapor line. The mass flow rates through the

particular circuits of the coil are self-adjusting so the pressure at merging

(splitting) tubes is the same.

56
To perform a simulation of a coil by the tube-by-tube method, refrigerant mass flow rate for each tube has to be known. Since total refrigerant mass flow

rate supplied to the coil is known, the problem reduces to the determination of

refrigerant distribution. In [3], refrigerant distribution is found by considering the coil working as an evaporator and assuming that refrigerant splits

equally among tubes at any splitting point of tubing circuitry. Thus the fraction of the total flow is assigned to each tube based on circuit geometry and

is used in both the heating and cooling mode simulation.

The method for determination of refrigerant flow distribution used in [3] was checked by examining calculated refrigerant pressures at the ends of all circuit branches. These pressures should be equal if the assumed distribution is

right; however, the pressures were found to be different by as much as 3 psi.

Of course, refrigerant flow distribution could be corrected in an iteration

process. This would require repeating all coil simulation calculations several

times, hence making coil simulation prohibitively expensive.

Another approach was tried to determine refrigerant flow distribution. Since

most of the coil total pressure drop may be expected to result from superheat

vapor and two-phase flow, it was assumed that the refrigerant flow is uniformly

distributed among tubes connected to the vapor side manifold and that mass flow

rates in other tubes may be found by following the refrigerant path with direction of flow as marked in figure 20. The resulting distribution was checked as before. The maximum pressure discrepancy found was equal to 0.3 psi which

represented less than a 0.2°F variation in the saturation temperature between

merging tubes. This was considered satisfactory and the method was adopted in

this study for determining refrigerant mass flow rate distribution in a coil.

Air Mass Flow Rate

Air mass flow rate was assumed to be distributed uniformly over the whole

coil face regardless of the coil and fan respective locations, so each coil was associated with the same air mass flow. It was also assumed that there were enough turbulence in the air stream passing through the coil to provide

effective mixing so that air of uniform properties enters each tube bank.

5.4.4 Overall Heat Transfer Coefficient for a Dry Finned Tube

Dry fined tube analysis is applicable to a condenser and also to an evaporator

if no dehumidifcation takes place. The overall heat transfer coefficient, U,

for a dry finned tube can be derived by summing the individual resistances

between the refrigerant and the air, [23]:

U = Ap.ihi Ap,ihd,i

o P

Ap , m^p

^(1 - c,o^

Af

(1 - <{)))

(59)

where Af = fin surface area

Aq = total exterior surface area wetted by air

57
Ap^£ = pipe inside surface area

^p,m ~ PiP^ mean surface area

h(, Q = convection heat transfer coefficient at exterior surface

i ~ heat transfer coefficient for inside tube deposit

Aj^ = inside tube convection heat transfer coefficient

kp = thermal conductivity of pipe material

Xp = thickness of pipe wall

T — T

(j) = -J^ _a_

^ fin efficiency

Tf.b - Ta

= air temperature

Tf = fin base temperature

^f,m ~ niean fin temperature

The third term of equation (59) can be evaluated if the heat exchanger material

and geometry are known. The heat transfer coefficient for inside tube deposit,

h^ i> assumed to be 5000 Btu/(h 'ft • F) [23] which allows for evaluation

of 'the second term. Terms 1 and 4, which refer to the inside and outside con- vection resistance respectively, required considerably more analysis to

establish the proper algorithm for determining the heat transfer coefficient.

5.4.5 Forced Convection Heat Transfer Inside a Tube

Analyzing the problem for both an evaporator and a condenser, the following

modes of forced convection are encountered:

- single-phase forced convection

- two-phase forced convection with condensation

- two-phase forced convection with evaporation

The physics of these phenomena are very much different and they have to be

considered separately.

Single-Phase Forced Convection

Single-phase forced convection takes place in a condenser, at the entrance

section where the superheated vapor is being cooled, and a the exit section

where a subcooled liquid is being cooled. It is also applicable in the evaporator, as the superheated vapor passes through the exit tubes. The non- dimensional heat transfer parameter describing this phenomena, Nusselt number,

is related to the non-dimensional Reynolds and Prandtl numbers in the following

form:

Nu = 0.023 • ReO-8 . PrO.333 (60)

where Nu = ^ * ^, Nusselt number

k

Pr =iL_!_C£, Prandtl number

58
Re s G

• Reynolds number

y

Cp = specific heat at constant pressure

D = Inside diameter of a tube

G = refrigerant mass flux

h = convection heat transfer coefficient

k = refrigerant thermal conductivity

y = refrigerant absolute viscosity

Two-Phase Forced Convection with Condensation

The predominant flow pattern during condensation in a heat pump condenser is

annular flow with liquid refrigerant flowing on the pipe wall and vapor refrigerant flowing in the core. Traviss, Baron, and Rohsennow [24] performed experimental and analytical studies of vapor condensation in annular flow with

refrigerants R-12 and R-22. They assumed von Karman universal velocity distribution in the condensate flow (like on a flat plate), calculated pressure drops

using the Lockhart-Martinelli method [25], and applied momentum and heat transfer

analogy for calculation of the condensation heat transfer coefficient. The

correlation proposed by them has the following form:

Re.^*^ • Pr. • Fl^

Nu = — ^ (61)

F2

where jj^ = h « D

h = condensation heat transfer coefficient

D = tube Inside diameter

kx, = thermal conductivity of liquid refrigerant

R = G(l - x)D

G = refrigerant mass flux

x = quality

HL - liquid refrigerant absolute viscosity

L

3 = 1 for Fl < 1, 3 =1.15 for Fl > 1

Fl and F2 in equation (61) are dimensionless parameters expressed as follows:

Fl = 0.15 (X^^-1 + 2.85 X,.^^-^24) (62)

F2 = 0.707 • Pr^ • ^^j^^'^ ^^L < 5°

59
F2 = 5 • Pr^ + 5 • ln(l + PrL(0. 09636 • for 50 < Re^ < 1125

P2 = 5 PrL + 5 ln(l + Pr^) + 2.5 • ln(0. 00313 • Re^0.812

) (63)

for ReL < 1125

Parameter ^x:t> formulated by Lockhart-Martinelli [25] with the assumption of no radial pressure gradient and a smooth pipe, has the following form:

"tt 1 - X

0.9 0.5 0.1

(64)

Parameter X^t Is inversely proportional to flow quality and refers to turbulent

vapor and turbulent liquid flow. Physically it is equal to the square roof of

the ratio of pressure drop of the liquid phase to the frictional pressure drop

of the vapor phase if these phases were flowing independently and alone in the

tube , i.e.:

tt

dP

dL

dP

dL /V

0.5

(65)

Equation (61) is applicable where conditions for annular condensation in a tube exist. Such conditons may be assumed to exist for flow qualities ranging

from 0.1 to 0.9. At qualities larger than 0.9, the whole tube inner surface

is not covered by a liquid film and part of the heat transfer is just that of

single-phase convection. At qualities less than 0.1, flow was observed to be

in the slug regime [24]. It is assumed, that in the quality range 0.0 to 0.1

and 0.9 to 1.0, the heat transfer coefficient changes linearly from a two-phase

flow value to a single-phase flow value and is calculated using linear

interpolation between values obtained from equations (60) and (61).

Two -Phase Forced Convection with Evaporation

Refrigerant enters an evaporator from an expansion device at quality about 20

percent and forms an annular flow instantly. The quality increases with the

proceeding flow and the annular flow pattern is maintained until the quality

reaches about 0.90, at which refrigerant vapor has enough kinetic energy to

gradually destory the liquid layer and patches of dry wall appear.

There are many correlations for calculating the forced convection evaporation

heat transfer coefficient (nucleation effect for mass fluxes taking place in

heat pump evaporators at qualities greater than 20 percent is insignificant,

even if it happens). Anderson, Rich, and Geary [26] performed laboratory measurements to obtain the forced convection evaporation heat transfer coefficient

under conditions closely approximating those existing in refrigerant evaporators.

60
They also examined six generally known correlations checking heat transfer

coefficient predictions with their laboratory results. They found best agreement with Pierre's [27] correlation, which correlate reasonably well with

experimental data at high mass velocities (above 10^ lb/(h • ft^)) and provided

an excellent agremeent at lower mass velocities.

Pierre's correlation, based on experimental work with refrigerants R-12 and

R-22, has the following form:

Nu =V-^ = 0-0009 . G ' D . r '

' r

^ (66)

where D = inner diameter of the tube

G = refrigerant mass flux

h = heat transfer coefficient

ifg = latent heat of evaporation

J = the mechanical equivalent of heat

kL = thermal conductivity of liquid refrigerant

L = length of the tube

Ax = refrigerant vapor quality change in the tube

Ml ~ absolute viscosity of liquid refrigerant

Pierre's correlation is applicable for a Nusselt number value within the

limit:

28.5 < Nu < 753 (67)

which falls in the range of heat pump evaporator operation, and for a quality

range from 0.15 to 0.90. For refrigerant quality greater than 0.09, similar to

the condensation case, the evaporative heat transfer coefficient may be obtained

by averaging between the values yielded by equation (66) and the single-phase

forced convection correlation, equation (60).

5.4.6 Forced Convection Heat Transfer at the Air-Side of a Flat-Finned Tube

In order to evaluate the forced convection heat transfer outside a flat-finned

tube (term 4 of equation (59)), the total exterior surface area, Aq , the fin

area, Af, the air-side heat transfer coefficient, h(,^Q, and fin efficiency, ((>, have to be known.

From the number of air-side heat transfer correlation available in the literature,

the one proposed by Briggs and Young [28] is most applicable here. This corre- lation was developed after extensive tests on 18 tube banks of different fin

geometry. A regression analysis of the test data yielded the following equation:

h, ^ • 0.681 0.333 . , _ . 0.2 /„ ^ 0.1134

Nu=-^^^^- ^^o.l34.Rea -Pra (f) * (f ' ^^^^

where Dq = outside tube diameter

^max ~ mass flux at minimum cross section

hj, Q = air-side mean convective heat transfer coefficient

61
= air thermal conductivity

y • Cp

Pro = — — , Prandtl number

'^a G • D

Rea = — , Reynolds number

^a

t = fin thickness

y = fin height

z = distance between adjacent fins

The geometry parameters affecting the heat tranfer are illustrated in figure

22. The adequacy of equation (68) was further verified by Jones and Russell

[29].

The addition of fins to the tubes greatly increases the outer heat transfer

area but at the expense of decreasing the mean temperature difference between

the surface and the air stream. The parameter, fin efficiency, (() , is used to

rate the thermal effectiveness of a fin. As mentioned in section 5.4.1, it is

assumed in this study for heat transfer analysis, that each tube is served by a circular-plate fin of equivalent surface area, as per figure 21. Gardner [30]

solved the diferential equation for describing the temperature distribution in

a circular fin and presented fin efficiency curves in terms of two parameters.

/2 • h^ ^ \ 0.5

Exact theory results are correlated well by the following equation proposed in

[3] (see also figure 23):

8 D j5 2 2 . h

4.= Z (Ai,i +A2,i -2- + A3,i(-^) )(y(-__^) ) (69)

i=l ^t ^t ^f ^

where q = air-side convective heat transfer coefficient

kf = fin material thermal conductivity

The geometry parameters are indicated in figure 22. The coefficients, A^^i, are given in table 1.

5.4.7 Overall Heat Transfer Coefficient for a Wet Finned Tube

Wet finned tube analysis is applicable to an evaporator when its temperature is

below the dew point temperature of ambient air. In such a case, moisture is

being removed from an air stream and is transferred to an evaporator external

surface. If the evaporator temperature is above 32°F, a water film flows down

the fin under force of gravity. If the exterior evaporator temperature is

below 32°F, frost is accumulated.

62
s s

s s

s

s

s

s

s

s

s

s

s

J u

s

s

s

s

s

s

s

t

yi

Do = TUBE OUTSIDE DIAMETER

Dt = FIN TIP DIAMETER

y = FIN MIGHT

t - FIN THICKNESS

z = DISTANCE BETWEEN ADJACENT FINS

Figure 22. Cross section of a flat-finned tube indicating parameters

affect the air-side heat transfer coefficient

63
Figure 23. Efficiency for a circular-plate fin of uniform thickness.

Comparison of exact theory results with those obtained

by equation (69)

64
Table 1. Coefficients to be Used in Correlation for Fin Efficiency

(Equation (69))

i ^l,i A,2i ^3,1

1 1.0 0.0 0.0

2 -0.22920E-01 -0.13755E+00 0.20130E-01

3 0.16106E+00 0 81890E-•01 -0.11440E-01

4 -0.64975E-I-O0 -0.55500E-01 -0.28753E-•01

5 0.53491E+00 0.18040E-01 0.42477E-•01

6 -0.19286 E+OO 0.36494E-•03 -0.20335E-•01

7 0.32564E-01 -0.10660E-02 0.40947E-02

8 -0.20972E-02 0.12410E--03 -0.29673E-03

65
Looking at the heat transfer phenomena from refrigerant to air, one can realize

that the dehumldif ication process will alter the heat transfer situation on the

external surface of the finned tube, while other processes in the tube and

refrigerant stay unaffected and are governed by relations already proposed in

previous sections. Thus only processes connected with dehumidification need to

be discussed.

The heat transfer rate between the air stream and the water surface is described

by the following equation:

dQ = (hc,o(Ta - Tw) + tiD.oC^a " Ww)ifg,w)dAo (70)

The first term in equation (70) accounts or sensible heat and the second term

accounts for latent heat transfer. Noting that for air at atmospheric pressure

the Lewis number,

K « Le = Sut2- , (71)

^D,oCpa

is close to 1, equation (70) assumes the following form for a flat finned tube:

dQ = he,o(l + ^pg'^r^ "V )(1 (1 - <f))(Ta " Tw)dAo (72) Cp^(T^ - T^)

Symbols used in equations (70), (71), and (72) denote:

Af = fin area

Aq = total external area CPa ~ specific heat of air

h(, Q = air-side forced convection heat transfer coefficient

^D,o ~ air-side mass transfer coefficient

^fg,w ~ latent heat of condensation for water

T^ = temperature of air

= temperature of liquid water (frost)

Q = heat transfer rate

W3 = humidity ratio of air

w^, = humidity ratio of saturated air at T^ temperature

T. - T

^ =

w,ni ^—a ^ efficiency

^w,b ^a

^w,m ~ insai^ temperature of water film (frost)

^w,b ~ temperture of water film (frost) at fin base

The one-dimensional heat conduction across the condensate (frost) film can be

expressed by equation:

dQ = hL . ATl • dAo (73)

66
where _ K , heat transfer coefficient for condensate (frost) film

6 = thermal conductivity of water (frost)

6 = thickness of condensate (frost) film

ATl = temperature difference across condensate (frost) film

Using equations (72) and (73) and referring to equation (59), the following

relation for overall heat transfer coefficient for a wet finned tube can be

derived:

-1

U = hiAp,i ^d,i^p,i A k

"^p ,rorp (,^iffi,w("a-"»))(,.Af(i- CO *))

Cp^a^(T - T ) a

(74)

where symbols used are defined as in equations (59) and (72).

In the above formulation of the overall heat transfer coefficient, it is assumed

that the temperature difference across the liquid film (frost) is uniform. The

calculated value for wet fin efficiency is affected by the change of the airside heat transfer coefficient caused by air passage geometry alternation by

liquid (frost) accumulation, and by released latent heat of condensation (sublimation). The effect of water (frost) conductance on fin efficiency is

neglected. In summary the heat transfer phenomena that occurs during

dehumidif ication on the air-side may be itemized as follows:

(1) layer of wet (frost) offers additional heat flow resistance

(2) air-side heat transfer resistance is decreased due to effect of

condensation

(3) air-side heat transfer coefficient, h^, q, since it is sensitive

to external surface geometry, has an increased value (see

equation (68))

(4) fin efficiency has decreased value as h^, q is increased (see

figure 23)

(5) the cross sectional area of the air flow passage between the fins

has decreased, decreasing the flow rate and the heat exchange.

In order to evaluate water (frost) layer thickness, consider the mass transfer

equation:

^a,d • dwa = - hD^o(Wa - w^)dAo (75)

For the Lewis number equal to 1 and for a finned tube differential surface area,

equation (75) assumes the following form:

^Pa ^o

67
Variation of air humidity ratio can be calculated by integrating equation (76),

which yield&t

- ^c,o ' -^(1 - Wa,e = Wa^i - (w^^i - wj(l - exp

Cv ' m

^ ^^^^

"a a ,d

The rate of moisture removal per unit area, R, can be now calculated:

R = ma^d(wa,i " ^a,e^/^ (78)

where d

~ '^^ss flow rate of dry air

Wg e = humidity ratio of air at tube row exit

Wg^i = humidity ratio of air at tube row inlet

If the evaporator temperature is below the freezing point, moisture removed

from the air stream accumulates on the evaporator external surface in the form

of frost. Its thickness, 6f , can be evaluated by integrating the rate of

moisture removal with respect to time, i.e.:

t 6f = / 1 dt (79)

0 Pf

where t = time

R = rate of moisture removal per unit area

6f = frost layer thickness

= frost density

In case of evaporator temperature above 32° F, condensate flows down on the fin.

Assuming no drag on the liquid layer, the condensate thickness, 6^, can be

found by solving the Navier-Stokes equation. The resulting expression for a finned tube situation is:

y • R'

6f = 1.083 (-^ -)

g

• P (80)

where g = gravitational acceleration

R' = condensation rate per unit width of a fin

yjjj = water dynamic viscosity

= water density

5.4.8 Pressure Drop in a Tube

As expressed by equation (25) j the total pressure drop experienced by a flowing

fluid results from pressure drops due to friction, momentum change, and gravity.

In an actual heat pump heat exchanger, pressure drop due to gravity effect is

very small and may be neglected. Only pressure drop due to friction and due to

68
momentum change will be considered with distinction between different flow

patterns in a tube.

Single-Phase Flow

Frictional pressure drop for a single-phase in a tube can be calculated by the

Fanning equation with the Fanning friction factor as per equations (43) and (44):

AP " • • ^ (81)

D • p

f = 0.046 Re"0-2 (82)

Pressure drop due to momentum change can be calculated by the following equation;

dP = _ g2 dv (83)

dL dL

where G = refrigerant mass flux

L = tube length

V = refrigerant specific volume

Two-Phase Flow with Condensation

Frictional pressure drop for two-phase flow with condensation can be calculated

by the method proposed by Lockhard and Martinelli [25]. They performed a semiempirical study of adiabatic two-phase flow with air and different liquids

including benzene, kerosene, water, and various oils in pipes varying in diameter form 0.586 to 1.017 inch. They related pressure drop of two-phase flow

to the pressure drop of the liquid portion of the flow flowing alone in the

pipe, by the dimensionless parameter X^-|-, i.e.:

= f(Xtt) = * APtp = APl • * (8^)

where APl = frictional pressure drop of the liquid portion of two-phase flow

flowing alone in the tube

AP^p = frictional pressure drop of two-phase flow

9 = correction factor for two-phase pressure drop

Xj.^. = as given by equation (64)

The pressure drop, APl, can be calculated by the single-phase pressure drop

relation with calculated liquid Reynolds number and friction factor as follows:

Re^„ T = <^ - x)G » D (85)

f, , = 0.046 • Re, "0-2 (86) tp,L tp,L

where ftp,L = friction factor for the liquid portion of two-phase flow flowing

alone in the pipe

69
^^tp,L ~ Reynolds number for the two-phase liquid portion flowing alone

in the pipe

X = quality

PL ~ liquid dynamic viscosity

A correction factor for two phase pressure drop, 0, can be correlated by the

following equation:

Z A^Clog X^^)^"^

$ = 10 i=l (87)

where ^1 ~ A2 = -0.87917

A3 = 0.14062

A4 = 0.0010417

A5 = -0.00078125

Combining equations (84), (85), (86), and (87), the two-phase pressure drop

equation assumes the form:

AP^p = 2 f^p^L • - x^)L . $/(D . pl) (88)

The pressure drop due to momentum change for separated two-phase flow can be

estimated by the following equation:

2 2

jr) OA VtT • X Vt (1 - x) dP=-G2d (-V +-Jf- _) (89)

dL dL a Ql - a;

where x = quality

vl = specific volume of liquid

vy = specific volume of vapor

a = void fraction

Void fraction, a, percent of tube filled with vapor, was shown by Lockhard and

Martinelli to be a function of X^^- under any flow conditions for separated flow

with both phases turbulent. Wallis [31] correlated their results in the

following form:

n « -0,378

a = (1 + X^.^0-») (90)

This expression was found to correlate well with data presented in [25] for

values of _< 10. For X^.,. greater than 10, another curve fitted formula is

used:

a = 0.823 - 0.157 . In X^t (91)

70
Two-Phase Flow with Evaporation

The pressure drop for two-phase flow with evaporation can be calculated by the

correlation proposed by Pierre [32], based on his experiments with refrigerants

R-12 and R-22. Pierre's correlation has the following form:

AP = (f L + A2C)g2 . Vjj^ (92)

where f = friction factor (calculated by equation (93))

= mean quality

Ax = quality change

Vpj = vl + Xnj(vY - vl) , mean specific volume

The friction factor to be used in equation (92) was correlated by Pierre from

his experimental data by the following empirical equation:

f = 0.0185 (Re h )0-25 (93)

J • ifg • Ax

where Re = ^ * P , Reynolds number

The correlation proposed by Pierre is in the conventional format for the single

pressure drop formula. The first term of equation (92) is for frictional pressure drop while the second is for pressure drop due to change of momentum. The

formula for the friction factor contains conventionally the Reynolds number

divided by the term J • ifg • Ax/L, referrred to as a boiling number, making

friction factor sensitive to vapor generation rate at the vapor-liquid interface. Pierre's correlation was verified in [26] providing overall better

agreement with experimental data for pressure drop of refrigerants R-12 and

R-22 than the other popular correlation of Martinelli and Nelson [33]

5.5 MODELING OF ADDITIONAL HEAT PUMP COMPONENTS

In the previous sections, modeling of the main heat pump components has been

discussed. The analysis included a compressor, a constant flow area expansion

device, a condenser, and an evaporator. Additional components that have to be

considered for a more accurate heat pump performance prediction by the model

are: vapor suction and discharge lines, liquid line, and a four-way valve.

Their modeling is briefly explained below.

Vapor Lines

A suction line connects an evaporator with a compressor. A compressor and a condenser are connected by a discharge line. In both lines, single-phase flow

exists in the form of either saturated or superheated vapor. In heat pump

systems vapor lines are usually insulated. Heat transfer rates between refrigerant vapor flowing in these pipes and ambient air can be calculated by a general heat transfer equation for a circular duct with insulation [10].

Single-phase forced convection is assumed inside the tube (equation (60)) and

71
free convection is assumed outside the tube. The following equation is used

for calculation of the free convection heat transfer coefficient for a horizontal pipe [38]

A rri 0 • 2 5 h = 0.27 (^) (94)

for Grashof numbers from 10^ to 109

where AT = temperature between tube wall and air

Dq = tube outside diameter

Pressure drop in vapor lines can be calculated by the single-phase pressure

drop equation (43).

Liquid Line

A liquid line connects an evaporator and a condenser. This line is filled with

subcooled liquid or low quality two-phase flow. The pressure drop can be calculated by equations (43) or (88), depending on the flow pattern. Heat loss

from the liquid line to the ambient air is neglected.

Four-Way Valve

The main function of a four-way valve is to direct refrigerant flow from the

compressor to the indoor or outdoor coil depending on the mode of operation

(heating or cooling). The side effects of flow through a four-way valve are changes in the refrigerant thermodynamic state due to pressure drop and heat

exchange. Assuming an adiabatic exterior wall, all heat lost by the discharge

refrigerant is gained by the suction refrigerant. Heat transfer rate and refrigerant pressure drop in the valve can be evaluated by formulas similar to equations (20) and (30), respectively. Heat transfer and pressure drop parameters

in these formulas have to be found empirically.

5.6 REFRIGERANT MASS INVENTORY IN A HEAT PUMP

The mass, M, of a substance occupying a known volume, V, may be determined by:

M = / p • d¥ (95)

¥

where p = local density

In reality, the mass of refrigerant in the system can be fund by estimating

masses of the refrigerant in each system component and adding them up. For

this purpose, equation (95) can be written in the form:

M = E Mi (96)

Mi = Vi . p^^i

72
where Mj^ = mass of refrigerant in particular component 1

Vj^ = internal volume of component i

Pm,i ~ mean fluid density in component i

The heat pump components taken into account in the mass inventory calculations

are shovm in figure 24. The refrigerant phase in each of the components as indicated in the figure is based on the following consideration:

Discharge line - receives and is filled with superheated vapor from the

compressor

Condenser - receives superheated vapor from a discharge line. In the

course of passage through the condenser tubes, vapor temperature is brought

to saturation temperature. Starting at this point, a thin condensed liquid

layer forms on the tube walls. Depending on the mass flux, this liquid

film may be swept and entrained within the vapor as a mist forming a dispersed flow. With more condensed vapor, the velocity decreases, a permanent liquid film is formed on the wall, and flow proceeds further as annular

flow or semiannular flow. The quality decreases and flow slows down in

the direction of flow. Gravity forces result in a stratifying effect and

liquid flowing in the bottom of the tube periodically wets the upper tube

wall (slug flow). If full condensation is reached, there will be subcooled

liquid at the condenser exit.

Thus there are four two-phase flow patterns in the refrigerant condensers,

i.e.:

dispersed flow, annular flow, semi-annular flow, slug flow

as identified by Traviss and Rohsenow [34]. Soliman and Azer [35]

identified nine flow regimes; however, five of them may be considered as transition regimes and can be put into the above basic four categories.

The effect of return bends connecting condenser tubes was investigated

in [36]. It was found that this effect is insignificant and that the

annular flow pattern "recovers" almost immediately in a tube after passing

a return bend.

Liquid line - receives refrigerant from a condenser and delivers it to an expansion device. Entering the small bore restrictor at the end of the

liquid line, the flow experiences a sudden contraction. The entrance presure drop depends upon the mass flow rate and fluid density. This along

with capillary sensitivity to the amount of subcooling, provides a flow

controlling mechanism for the system. It is common practice to design a capillary tube so the refrigerant mass flow rate through it would balance

with the mass flow rate through the compressor when a liquid seal exists

before the capillary tube. Temporary existence of vapor before the capillary tube would decrease the mass flow rate and build up the pressure.

This would provide better conditions for condensation in the condenser and

thus cause a return to a subcooled liquid or very low quality two-phase

flow in the liquid line.

73
LIQUID LINE LIQUID PHASE OR LOW

QUALITY TWO-PHASE

Figure 24. Refrigerant phase in heat pump components

74

Ml
Capillary tube - due to small internal volume may be disregarded for mass inventory purposes.

Evaporator - receives refrigerant from a capillary tube in the form of

homogeneous flow of about 20 percent quality, at sonic or close to sonic

velocity. This flow experiences a sudden deceleration and is assumed to

form an annular or semi-annular flow pattern very quickly having a void

fraction about 0.8 [37] . The liquid phase flows along the tube in the

form of an annulus and the vapor phase flows in the core. This type of

flow prevails in most of the evaporator. With increasing quality, the

velocity of the vapor increases and becomes eventually high enough to

entrain small liquid drops and tear the liquid film apart. This results

in a dry mist flow at qualities close to 1. Finally, the flow may reach

the evaporator exit in a superheated vapor state.

Suction line - receives refrigerant from the evaporator as a mist flow or superheated vapor flow and the same flow pattern exists at the exit.

Accumulator - receives the refrigerant from a suction line after some heat was added in the four-way valve. Saturated vapor or superheated

vapor may be expected. (This does not account for refrigerant liquid

that may be accumulated on the accumulator bottom.)

Compressor can - receives refrigerant from the accumulator as a saturated

or superheated vapor which is further superheated due to heat transfer

from an electric motor, discharge line, and compressor cylinder body.

According to the above analysis, refrigerant in the accumulator, compressor

can, suction line, and discharge line is in (or close to) a homogeneous vapor

state. In the liquid line, homogeneous liquid flow or two-phase flow exists

but in a state so close to saturation that both liquid and vapor would travel

at about the same velocity. For mass inventory purposes, the flow may be

assumed to be homogenous. The mass of refrigerant in these components can be

then calculated in a similar manner since refrigerant densities in the components described above will be known as a result of the simulation program

iteration process.

In the case of both coils, the refrigerant flow in part is homogeneous; however,

in most of the coil some type of annular flow prevails. For a separated flow

regime like an annular flow, density of the flowing fluid results from vapor

and liquid densities and the fractions of tube volume occupied by the liquid

and vapor phase:

Pm = „^ = — ^^r^ = Pl(1 - a) + Pv • « (97)

t ^t

where M = mass

V = occupied volume

p = density

a = Vv/Vt, void fraction

75
subscripts refer to:

L = liquid phase

m = mean value

t = total value

V = vapor phase

It should be pointed out that for a two-phase flow of given quality, mean

density depends not only on the thermodynamic parameters affecting densities of

the liquid and vapor, but also on the ratio of mean velocities of the vapor

and liquid referred often as the slip ratio. This can easily be noticed by

looking at the equation for void fraction, which can be derived considering

mass flow rates of each phase:

a = i (98)

1 - X Vy py ^ Vl PL

where V = velocity

Calculation of the void fraction has received much less attention than the

calculation of heat transfer or pressure drop, although all three of these

phenomena are undoubtedly inter-related. The most often referred method for

void fraction calculation is that of Lockhard and Martinelli [25] based on their experimental void fraction data reported along with pressure drop results.

Lockhard and Martinelli correlated void fraction with the dimensionless parameter, Xj-t • Their experiment dealt with adiabatic flow; however, in the conclusions of their paper they suggested that their pressure drop correlation could

be used for prediction of pressure drop during evaporation and condensation as well. On similar grounds, their void fraction data should be applicable beyond

the adiabatic case.

There is another, earlier, method available for void fraction calculation by

Martinelli and Nelson [33], derived with the same assumptions for water/steam

evaporating flow. The method is similar to the Lockhard-Martinelli method so the Lockhard and Martinelli method was used in this study for calculation of

the void fraction in both evaporator and condenser coils.

Correlations for void fraction based on [25] have already been given in section

5.4.8 (equations (90) and (91)).

a = (1 + Xj.,.^*^) " ^-^^^ for Xj.^. < 10 (99)

a = 0.823 - 0.157 In Xtt for Xtt > 10 (100)

Parameter Xj-t > as defined by equation (64), is a function of flow quality,

specific volume, and viscosities of the liquid and vapor, and is not sensitive

to the mass flux which in turn affects the slip ratio.

76
Experimental data of Staub and Zuber [37] for evaporating refrigerant R-22

indicate that void fraction increases with increased mass flow rate. Compari~

son of their data with void fraction predicted by the Martinelli-Nelson method

showed a discrepancy which may lead to underestimation of the two-phase flow

mean density by as much as 300 percent. (Note that a small difference in void

fraction results in a large error in mean density prediction as density of the

vapor and liquid are of different orders of magnitude.) The results of Staub

and Zuber also indicated that approximate agreement with the Martinelli-Nelson

method could be obtained at high mass fluxes, more than five times higher than

those observed in heat pump evaporators.

In discussing the accuracy of a refrigerant mass prediction method, it should

be noted that it also depends upon an accurate measurement of the internal

volume of the system. Internal volume can be easy calculated for straight

pipes. For valves, bends, etc., it can be only approximated.

Mass inventory of a heat pump system, as explained in section 5.1.2, is intended

to be used for iteration of refrigerant superheat at the compressor can inlet.

The discussions above indicated that mass inventory calculations may have errors from two different sources. These sources are: inaccurate density prediction

in the two-phase region, and inaccuracte internal volume measurement. However,

these inaccuracies are not of great importance and can be ignored, since the

program does not really need prediction of the absolute value of refrigerant mass in the system, but rather requires sensitivity in relative mass predictions

in a heat pump with change of operating conditions . This requirement should be

satisfied by the Lockhard-Martinelli method and very precise knowledge of the

internal volume should not be that important. Since refrigerant vapor superheat

at the compressor can inlet is known for a heat pump operating in the cooling

mode at design conditions (outdoor temperature 90°F, indoor dry bulb temperature

SO'F, and wet bulb 67°F) and does not have to be iterated upon, refrigerant mass in the system can be calculated at this operating condition. This calculated

mass of refrigerant may then be used as input data for refrigerant vapor superheat iteration at other outdoor or indoor air conditions at which heat pump

simulation results are required.

77

6. MODEL VERIFICATION

A verification of the computer model has been done by comparison of computer

predictions with performance data obtained from laboratory tests conducted in

NBS environmental chambers . Four heat pumps were available for computer model

evaluation. They are denoted here-ln as systems 1, 2, 3, and 4. System 1,

system 2, and system 3 consisted of the same compressor, outdoor coil, con- necting tubing, and expansion device. The only variable element was an indoor

section (containing an Indoor fan and coil). In the case of system 2, the

Indoor unit was an original match for the rest of the components. For system

1 and for system 3, the Indoor sections were undersized and oversized, respectively. System 4 was a matched indoor/outdoor system of another manufacturer.

79
6.1 LABORATORY TESTS

An identical series of four tests was conducted in the cooling mode on each of

the four systems. The series consisted of a test at an outdoor temperature of

105°F and tests A, B, and C of the NBS recommended test procedure [39] . Thermal conditions during the above tests covered practically the whole range of

the cooling mode operating conditions for a heat pump working in the field.

Additionally, heat pump system 2 was tested in the heating mode at temperature

conditions recommended by [40]

Cooling and heating tests results are presented in table 2 and table 3,

respectively. Symbols used in these and following tables in this section are consistent with symbols used in the main program MAIN (refer to appendix I) and

have the following meaning:

COP - coefficient of performance

ELUSE - energy consumption rate

P3 - refrigerant pressure at compressor inlet

P8 - refrigerant pressure at compressor outlet

RHOA - relative humidity of outdoor air

RHRA - relative humidity of indoor air

RMASS - refrigerant mass flow rate

TOA - temperature of outdoor air

TRA - temperature of indoor air

TSUP3 - superheat at compressor inlet

QLOAD - heat pump capacity

The capacity reported in table 2 is the one measured by the refrigerant enthalpy

method [41]. In this method, the unit capacity is derived from the refrigerant

enthalpy change and refrigerant volumetric flow rate measurements with adjustment for heat added to the indoor section by the indoor fan. Capacity obtained

by the refrigerant enthalpy method is directly comparable to the capacity

obtained from the vapor compression cycle simulation.

During heating mode tests, no liquid subcooling existed in the liquid line of

the heat pump. Vapor bubbles existing in the flow made it impossible to measure volumetric flow rate of refrigerant. In this case, system capacity was determined by the air-side enthalpy difference method [41]

6.2 COMPUTER SIMULATION RUNS

Each laboratory test performed was simulated using the computer program HPSIM.

Since laboratory tests were performed first, the computer simulations could

be made with the same thermal operating conditions that actually existed in the

laboratory during the tests.

The first run executed for each system was at design operating conditions (test

2) with refrigerant superheat at the compressor inlet equal to that of laboratory

test. The refrigerant charge calculated during this run was used later as input

data for other runs requiring superheat iteration.

80
Table 2. Laboratory Test Results of Heat Pumps in the Cooling Mode

TRA RHRA QLOAD ELUSE COP RMASS PJ TbUP-J noPo

_ _ _ Btu/h Watt _ Ib/h psia °F psia

1

1 1 iiiH . y

*7Q Q /y .o .dZ. Q o o o o Duy J i .yiz 3/

Q ^ C OD .D Z . ZO O ^ A ZBb .U

1

1 o 1 • ^ IQ Q /y .o . loU C£L A oZ . / 11 A ii . UO R A Q Z j4 . J

1

1 o Q O A /9 .D .52 Q O C A O Jo54Z 45d4 Z .4/4 C A c: "7 "7 C

/ / .0 O O A Z J .4 ziD .y

1 1.4 82.1 79.5 .19 33203 4511 2.157 534 73.0 1.0 210.5

2 2.1 104.9 80.0 .52 34841 5212 1.959 589 88.9 2.3 292 .2 2 2.2 94.9 80.0 .52 37548 4938 2.228 578 85.0 13.8 259 .5 2 2.3 82.1 80.0 .52 40105 4629 2.538 555 79.3 30.9 220.2

2 2.4 82.1 79.8 .20 37156 4583 2.375 550 76.3 9.4 216.6

3 3.1 105.2 79.9 .51 37963 5542 2.007 640 95.5 1.2 305.2

3 3.2 95.2 79.8 .52 40946 5276 2.274 622 90.9 12.8 270.3

3 3.3 81.9 79.9 .51 42842 4930 2.546 602 85.1 21.9 230.8

3 3.4 82.2 79.9 .21 38551 4867 2.321 591 80.2 1.1 224.9

4 4.1 104.9 80.0 .51 31595 4613 2.007 499 89.3 20.0 316.7

4 4.2 95.0 80.0 .51 32640 4351 2.198 483 84.8 25.0 280.1

4 4.3 82.2 80.0 .51 33656 3982 2.476 449 76.6 32.5 236.2

4 4.4 82.1 80.0 .18 31437 3971 2.320 445 75.2 25.9 233.6

81
Table 3. Laboratory Test Results of System 2 in the Heating Mode

SYSTEM TEST TOA RHOA TRA QLOAD ELUSE COP P3 TSUP3 P8

°F °F Btu/h Watt psia °F psia

2 2.5 46.6 .73 70.2 42648 4763 2.624 71.4 0.0 239.9

2 2.6 35.0 .82 69.9 35553 4471 2.330 60.4 0.0 218.2

2 2.7 17.1 .70 70.0 24958 3956 1.848 45.6 0.0 187.9

82
Cooling simulation results are presented in table 4. Heating simulation

results are presented in table 5. A comparison of results is given in table 6.

This table shows that the maximum discrepancy found between laboratory test

results and computer simulation results was:

capacity - 3.4 percent

energy consumption rate - 7.3 percent

COP - 5.5 percent

The largest deviation of refrigerant conditions was:

refrigerant mass flow - 5.2 percent

refrigerant suction pressure - 3.8 psi

refrigerant superheat at suction - 8.4°F

refrigerant discharge pressure - 23 psi

Agreement between results was generally very good. The only serious difference

was in the case of refrigerant discharge pressure where the discrepancy was found to be as much as 23 psi, which corresponds to about 7.5°F of difference

in refrigerant saturation temperature in the condenser. This disagreement took

place, however, only in the heating mode of operation when the indoor coil was

the condenser. Discharge pressure results for cooling operation agreed much

better.

In the course of the analyses of this 23 psi pressure difference the following

conclusions were made:

- there was a good agreement between refrigerant suction pressure results

and capacity results for cooling mode operation, which indicates that

the indoor coil is simulated well while operating as an evaporator.

- there was a significant difference in the heat transfer rates in the

performed tests in the indoor and outdoor coil working as a condenser.

Data indicated that the heat flux was twice as large in the indoor coil

than in the outdoor coil, which has considerably more surface area and

volume. Additionally, the refrigerant mass flow rate during the heating

mode was smaller than during the cooling mode making the refrigerant

quality variation even greater in the indoor coil.

If it is assumed that these differences are due to deficiencies in the simulation

program in its present form, then it is concluded that the condensing heat transfer coefficient correlation used in HPSIM provides a lower value for the indoor

coil than actually exists and is responsible for the high refrigerant discharge

pressure obtained in the computer simulation runs.

6.3 VERIFICATION OF THE MASS INVENTORY SIMULATION

Verification of the mass inventory method can be done on different levels. One

of the ways is to compare the refrigerant charge in a system tested in a laboratory to the charge computed during a simulation run. This was done for system

1. The system simulation resulted in a charge of 6.1 lb. while the actual heat

83
Table 4. Cooling Test Simulation Results

SYSTEM TEST TOA TRA RHRA QLOAD ELUSE COP RMASS P3 TSUP3 P8

°F °F Btu/h Watt Ib/h ps ia °F psia

1 1.1 104.9 79.8 .52 33750 5248 1.884 557 86.9 2.0 288.0

1 1.2 95.2 79.8 .52 35580 5000 2.085 545 82.6 11.0 255.4

1 1.3 82.0 79.6 .52 37510 4650 2.364 525 76.6 25.0 214.9

1 1.4 82.1 79.5 .19 33690 4572 2.159 506 73.4 1.0 212.8

2 2.1 104.9 80.0 ,52 36010 5401 1.954 586 90.6 5.0 291.8

2 2.2 94.9 80.0 .52 37890 5129 2.165 574 86.1 13.8 258.2

2 2.3 82.1 80.0 .52 39950 4782 2.448 551 79.9 28.0 218.5

2 2.4 82.1 79.8 .20 36570 4740 2.261 544 77.7 1.0 217.4

3 3.1 105.1 79.9 .51 37810 5690 1.947 627 95.3 1.0 298.8

3 3.2 95.2 79.8 .52 40400 5422 2.184 615 91.1 12.8 265.0

3 3.3 81.9 79.9 .51 42940 5067 2.483 598 85.1 2.2 222.3

3 3.4 82.2 79.9 .21 38680 4990 2.271 577 81.6 1.1 222.0

4 4.1 104.9 80.0 .51 32040 4644 2.022 514 90.5 19.8 326.9

4 4.2 95.0 80.0 .51 33070 4379 2.213 498 86.9 25.0 291.7

4 4.3 82.2 80.0 .51 33970 4009 2.482 466 80.4 33.2 248.0

4 4.4 82.1 80.0 .18 32020 3952 2.374 456 77.1 21.3 244.0

84
Table 5. Heating Teat Simulation Results

SYSTEM TEST TOA RHOA TRA QLOAD ELUSE COP P3 TSUP3 P8

Of °F Btu/h Watt psia °F psia

2 2.5 46.6 .73 70.2 42170 4776 2.587 70.3 0.0 262.9

2 2.6 35.0 .82 69.9 35600 4363 2.391 59.8 0.0 239 .5

2 2.7 17.1 .70 70.0 24400 3668 1.949 43.6 0.0 203.0

85
Table 6. Discrepancy between Laboratory Test Results

and Computer Simulation Results

TEST OLOAD ELUSE COP RMASS P3 TSUP3 P8

% % % % ps i °F ps i

1.1 1.6 3.0 -1.5 -2.6 0.3 0.2 2.0

1 .2 -1.5 3.0 -4.4 -2.6 -0.1 0.0 1.1

1.3 -2.7 1.9 -4.4 -3.8 -1.0 1.6 1.0

1 .4 1.5 1.4 0.1 -5.2 0.4 0.0 2.3

2.1 3.4 3.6 0.3 -0.5 1.7 2.7 -0.4

2.2 0.9 3.9 -2.9 -0.6 1.1 0.0 1.3

2.3 -0.4 3.3 -3.5 -0.7 0.6 2.9 -1.7

2.4 -1.6 3.4 -4.8 -1.1 1.4 8.4 -0.8

2.5 -1.1 0.3 -1.4 -1.1 0.0 23.0

2.6 0.1 -2.4 2.6 -0.6 0.0 21.3

2.7 -2.2 -7.3 5.5 -2.0 0.0 15.1

3.1 -0.4 2.7 -3.0 -0.2 -0.2 0.2 -6.4

3.2 -1.3 2.8 -4.0 -1.1 0.2 0.0 -5.3

3.3 0.2 2.8 -2.5 -0.6 0.0 0.1 -6.7

3.4 0.3 2.5 -2.2 -2.3 1.4 0.0 -2.9

4.1 1.4 0.7 0.7 3.0 1.2 -0.2 10.2

4.2 1.3 0.6 0.7 3.1 2.1 0.0 11.6

4.3 0.9 0.7 0.2 3.8 3.8 0.7 11.8

4.4 1.9 -0.3 2.2 2.5 1.9 4.6 10.4

86
pump was charged with 8.3 lb of refrigerant. For the same system, the outdoor

coil was weighed during cooling mode operation giving 5.4 lb of refrigerant

while the computed charge for this coil was 4.8 lb. Due to the test installation, it was not possible to weight the indoor coil; however, estimates based

on the differences in the above data indicate that almost three times more refrigerant existed in an indoor coil (working as evaporator) than that calculated. A discrepancy of the same order of magnitude for the Martinelle-Nelson

method was reported by Staub and Zuber [37]. Though better agreement between

a laboratory and calculated charge would promote more confidence in the

Lockhard-Martinelli method, it should be recognized that, as explained in

section 5.6, the absolute value of the computed charge is not critical since

the program iterates a new system status with change in operating conditions

based on refrigerant mass inventory performed at design conditions. What is

most important for modeling purpose is that the mass inventory based on the

Lockhard-Martinelli method provides a means for calculation of change of the

refrigerant vapor superheat at the compressor can inlet sufficiently well for

accurate heat pump performance prediction.

87

7. SUMMARY AND CONCLUSIONS

The main goal of this study was to formulate an analytical model of an airsource heat pump equipped with a constant flow area expansion device (capillary

tube) that could provide accurate heat pump performance predictions.

In the first step of this study, operation of a heat pump was analyzed and

previous work in steady-state vapor compression cycle modeling was reviewed.

Three mathematical models of a heat pump were found in the literature and their

applicability and limitations were studied. Two of the models could only simulate systems equipped with a thermostatic expansion valve (TXV) which maintains

constant refrigerant vapor superheat at an evaporator outlet. Operation of a

89
heat pump with a capillary tube is different since refrigerant vapor superheat

varies at that location with changing operating conditions. The third heat pump

simulation program, originally formulated for a system with TXV, could accommodate a capillary tube; however, without the ability to calculate refrigerant

vapor superheat (assuming its value is always known). Also it was found, that

the capillary tube model utilized in this program was adequate only when a subcooled liquid was entering the capillary tube entrance and would not simulate

the case of two-phase entering the tube which happens particularly during the

heating mode. None of the existing models used operating conditions as input.

Thus, two major conclusions were drawn. First that a new capillary tube model

had to be developed which would not carry any modeling restrictions. Second, some method should be formulated for calculation of refrigerant vapor superheat

at the compressor suction side. It was also concluded that a program performing

heat pump simulation at imposed operating conditions would be desirable.

In the- second step of this study, new logic for a heat pump simulation program

was proposed which would satisfy the identified needs. The new logic was based

on analysis of heat pump behavior at changing operating conditions and allowed

for calculation of refrigerant vapor superheat at the compressor can inlet by

the refrigerant mass inventory simulation method. Analysis of the main heat

pump components was performed and phenomena taking place were described analytically by fundamental equations or applicable correlations. A comprehensive

literature search was done and the best correlations^/ available were selected

for the model. These equations were used to formulate individual models of

particular heat pump components, including a constant flow area expansion device

not available in the literature. Finally, individual models were incorporated

into an overall heat pump model, HPSIM.

Verification of the model was conducted against laboratory test data of three

heat pump systems in the cooling mode and one system in both heating and cooling

mode covering the whole range of heat pump operating conditions. The maximum

difference between laboratory results and model predictions was found to be 3.4

percent for capacity and 5.5 percent for coefficient of performance.

The model HPSIM developed in this study is the only model in the open

literature that is capable of simulating a heat pump with a constant flow area expansion device at imposed operating conditions without restrictions on refrigerant state at any system location. (A possible exception is the assumption

that refrigerant saturated vapor is leaving an accumulator when liquid is collected on its bottom. However, this assumption is felt to be quite reasonable

and does not cause modeling difficulties.) The model having a demonstrated accu- racy of performance prediction, allows for a wide variety of model applications.

As shown in the verification process, the model was used to evaluate the performance of systems with mismatched components. Also it may be used to test

how certain modifications of components (e.g., coil tube diameter, fin thickness, fin spacing, coil face area, capillary tube diameter or length, etc.)

5/ Those which appeared to have experimental verification in the conditions

most similar to those found during operation of a heat pump.

90
affect overall performance of a heat pump filled with different refrigerants.

This latter application has led to advanced work which will require modification to the refrigerant thermodynamic and transport property equations to

include mixtures.

The program HPSIM occupies 31000 decimal words of core memory of a UNIVAC 1108

computer. Computing time depends on size of coils and refrigerant state input

guesses. On average, computing time should not exceed 200 seconds. It is felt

that the program can be helpful in testing, design, and development of heat

pump systems by providing relief from some expensive laboratory tests and by

providing consistently reliable results.

91

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McGraw-Hill, Inc., New York, New York, 1966.

53. Biguris, G. and Wenzel, L.A., Industrial Engineering Chemical

Fundamentals, Vol. 9, No. 1, 1970.

54. International Critical Tables, McGraw-Hill Book Co., pg. 216, New York,

1929.

55. Fischer, S.K. and Rice, C.K., A Steady-State Computer Design Model for

Air-to-Air Heat Pumps, ORNL/CON-80, Oak Ridge National Laboratory, Oak

Ridge, Tennessee, 1981.

56. Ellison, R.D., Creswick, F.A., Fisher, S.K., and Jackson, W.L., A

Computer Model for Air-Cooled Refrigerant Condensers with Specified

Refrigerant Circulating, ASHRAE Transactions, Vol. 87, 1981.

96
APPENDIX A. CALCULATION OF THE THERMODYNAMIC AND TRANSPORT PROPERTIES OF

REFRIGERANTS R-12 and R-22

Thermodynamic property equations for refrigerants have been presented in

numerous publications. The equation of state was developed by Martin and Hou

[43]. A review of correlations and equations has been given by Martin [44],

Constants for these relations were published in [45] and [46] for R-12 and

R-22, ^respectively . Downing [47] assembled refrigerant thermodynamic property

equations along with constants for R-12, R-22, and ten other refrigerants.

Equations presented by Martin [2] have been supported by Du Pont and have

become the accepted thermodynamic properties of refrigerants. They have been

used by other authors in references [1], [2], [3], [48] and by several heat

pump manufacturers.

The four basic refrigerant property equations used in the calculations are:

1. Vapor pressure at saturation

2. liquid density at saturation

3. pressure - volume - temperature behavior

4. specific heat of vapor

Refrigerant constants for these equations can be found in table Al. Other

symbols are explained with each equation.

The saturation vapor pressure as a function of temperature is

ln(P) = AG + ^+ CG • ln(T) .+ DG • T + EG(FG - T) in(FG - T) (Al)

T T

where P = vapor saturation pressure (psia)

T = vapor saturation temperature (R)

The saturated liquid density equation is:

T, BPL rr, CPL T, DPL , T .^PL , X Pt = AL + BL(1 --^) +CL(1-^) +DL(1--^) +EL(1--!-) (A2) ^ TC TC TC TC

o where p-^ = saturated liquid dens ity (Ib/ft-^)

T = temperature of saturated liquid (R)

The constant volume specific heat at zero pressure equation is:

Cv = AC + BC • T + CC • t2 + DC • T^ + J|C_ (A3)

t2

97
where Cv = constant volume specific heat at zero pressure (Btu/lb • F)

T = gas temperature (R)

The equation of state is:

p = Al * T + A2 + B2 « T + C2 « exp(-AK » T » TC)

(v -Bl) (v - Bl)2

+ A3 + B3 » T + C3 » exp(-AK « T/TC) + A4 + B4 « T + C4 » exp(-AK » T/TC)

(v - Bl)3 (v - Bl)^

+ A5 + B5 « T + C5 » exp(-AK » T/TC)

(v - Bl)5

+ [C6 + B6 « T + C6 * exp(-Al T/TC) ] exp( -2ALPHA * v) (^4)

[CI + exp(-ALPHA)]

where P = vapor pressure (psia)

T = vapor temperature (R)

V = vapor specific volume (ft-^/lb)

TC = refrigerant critical temperature (R)

Using these four equations and other exact thermodynamic relations, a complete

set of vapor refrigerant thermodynamic properties can be determined under any

conditions

.

The exact thermodynamic relations required are:

2

Cv = Cv + / T • (-^) dv (A5)

aT^ V

(IP)

2

. Cp = Cv - T (A6)

(IP)

3v T

s = / dT + / (||) dv + Y (A7)

*

i = /Cv'dT + P«v + / [T(-2% - P]dv + X (A8) 3T y

where the new symbols mean

Cv = vapor specific heat at constant volume

Cp = vapor specific heat at constant pressure

i = vapor enthalpy

s = vapor entropy

X,Y = constant (see table Al)

98
The derivation of equations (A5), (A6), (A7), and (A8) may be found in

reference [49].

Besides the above equations, algorithms have to be used to calculate the

specific heat at constant pressure of liquid, as well as the viscosity and

conductivity of the liquid and vapor. There are many correlations available

(see reference [1], [2], [3], [50], [51]) to calculate these refrigerant properties. It was decided to use those proposed by Chi [3], developed in the

form of the 5th power polynomial, because they are accurate within +0.5

percent and easy to handle in one subroutine:

6

PROP = E A(I,J) . T^"l (A9)

3=1

where A(I,J) = twelve coefficients for calculated property (six for T < 100°F,

J = 1 to 6 and six for T > 100°F, J = 7 to 12) 1=1 for liquid viscosity (Ib/h.ft)

1=2 for vapor viscosity (lb/(h«ft))

1=3 for liquid conductivity (Btu/(h»f t ^F)

)

1=4 for vapor conductivity (Btu/(h»f t»F))

1=5 for liquid specific heat at constant pressure (Btu/(lb»F))

T = saturated temperature (F)

PROP = calculated property at saturated temperature

The coefficients to be used in equation (A9) are given in table A2. Equation

(A9) allows for the computation of the refrigerant property values at saturation temperature. For those in the superheated state, there is a need to account

for the effect of temperature on viscosity and conductivity. This can be done

by using the power law, since viscosity and conductivity may be said to be proportional to T*^. The exponent, n, varies with temperature and is assumed in

this document to equal 0.5 as called for by the kinetic theory of gases [52],

In conclusion, eight functions and subroutines are used in the HPSIM heat pump

simulation program to determine the refrigerant thermodynamic and transport

properties. The equations described in this appendix are applied with constants

presented in tables Al and A2 . The listing of these subprograms are given along

with other subprograms, in appendix J. Input and output data are explained in

the form of comment statements in each routine. For a quick reference, a list

and application of refrigerant property routines is also given in table A3.

99
Table Al. Coefficients and Physical Constants for Freon-12 and Freon-22

to be used in Equations (Al) through (A8)

Coefficients and

Physical Constants FREON-12 FREON-22

TC 6.9330000E+02 6 .6450000E+02

PC 5.9690000E+02 7 .2191000E+02

VC 2.8700000E+02 3.0525000E-02

AG 9.1835883E+01 6.7598246E+01

BG -7.9131381E+03 -8 .8538843E+03

GG -1.2471522E+01 -7 .8610310E+00

DG 1.0892245E-02 5 .0448235E-03

EG 0. 4.4574600E-01

FG 0. 6.8610000E+02

AL 3.4840000E+01 3 .2760000E+01

BL 1.8601368E+01 5 .4634409E+01

CL 2.1983963E+01 3 .6748920E+01

DL 5.3341175E+01 -2 .2292560E+01

EL -3.1509939E+00 2 .0473289E+01

BPL 1. 3.3333333E-01

CPL 5.0000000E-01 6 .6666667E-01

DPL 3.3333333E-01 1.

EPL 2. 1.3333333E+00

Al 8.8734000E-02 1.2409800E-01

Bl 6.5093890E-03 2 .OOOOOOOE-03

CI 0 . 0

A2 -3.4097270E+00 -4 .3535470E+00

B2 1.5943480E-03 2 .4072520E-02

100
Table Al . (Continued)

Coefficients and

Physical Constants FREON-12 FREON-22

C2 -5.6762767E+01 -4 .4066868E+01

A3 6.0239450E-02 -1 .7464000E-02

B3 -1.8796180E-05 7 .6278900E-05

C3 1.3113990E+00 1 .4837630E+00

A4 -5.4873700E-04 2 . 3101420E-03

B4 0. -3.6057230E-06

04 0 . 0

A5 0. -3.7240440E-05

B5 3.468834OE-09 5 .3554650E-08

05 -2.5439070E-05 -1 .8450510E-04

A6 0. 1.3633870E+08

B6 0. -1.6726130E+05

06 0. 0.

ALPHA 0. 5.4820000E+02

AK 5.4750000E+00 4 .2000000E+00

AC 8.0945000E-03 2 .8128360E-02

BO 3.3266200E-04 2 .2554080E-04

CO -2.4138960E-07 -6 .5096070E-08

DO 6.7236300E-11 0.

EC 0. 0.

FC 0. 2.5634100E+02

X 3.9556551E+01 6.2400900E+01

Y -1.6537940E-02 -4 .5333500E-02

101
Table A2. Coefficients for Freon-12 and Freon-22 to be Used in

Equation (A9)

Coefficients

A(:i ,1)

M,1 .2)

A(;i ,3)

A(,1 ,4)

Al ;i .5)

A(,1 .6)

A(,1 .7)

A(^1 .8)

A(,1 .9)

A( ^1 ,10)

Al ^1 .11)

A(,1 ,12)

A( 2 ,1)

A(.2 ,2)

A( 2 ,3)

A(^2 A)

A( 2 ,5)

A( 2 6)

A( 2 ,7)

A( 2 ,8)

A( 2 ,9)

A( 2 ,10)

A( 2 ,11)

A( 2 ,12)

A( 3 1)

A( 3 .2)

A( 3 .3)

A(.3 A)

A( 3 ,5)

A( 3 ,6)

A( 3 ,7)

A( 3 ,8)

A( 3 ,9)

A(.3 ,10)

A(.3 .11)

A(.3 ,12)

A(.4 .1)

A([4 .2)

A( 4 ,3)

A(.4 ,4)

A( 4 ,5)

A(:4 ,6)

A(> ,7)

Freon-12

0.75800E+00

0.44230E-02

0,22659E-04

-0.80937E-07

0.48640E-09

-0.38993E-11

0.16672E+01

0.37691E-01

0.48998E-03

-0.32598E-05

0.10701E-07

0.14069E-10

0.26200E-01

0.58000E-04

0.0

0.0

0.0

0.0

0.10065E+01

0.37971E-01

0.53504E-03

0.36443E-05

0.12032E-07

0.15478E-10

0.49000E-01

0.11950E-03

0.36320E-07

0.52080E-09

0.31690E-11

0.31689E-13

0.33483E+00

0.10635E-01

0.14902E-03

0.10214E-05

0.34038E-08

0.44300E-11

0.43000E-02

0.17000E-04

0.0

0.0

0.0

0.0

0.12814E+00

Freon-22

0.64600E+00

0.29194E-02

0.12164E-04

-0.74985E-07

0.83951E-09

-0.37512E-11

0.69684E+01

0.24319E+00

0.35924E-02

-0.26187E-04

0.93884E-07

0.13301E-09

0.26600E-01

0.63804E-04

0.10761E-06

0.32061E-08

0.43463E-10

0.13175E-12

0.66330E+00

0.25757E-01

0.37913E-03

0.27734E-03

0.10053E-07

0.14474E-10

0.63000E-01

0.15820E-03

0.12289E-06

0.17453E-08

0.52685E-11

0.10856E-13

0.46705E+00

0.19421E-01

0.28507E-03

0.20429E-05

0.71963E-08

0.99460E-11

0.48000E-02

0.19881E-04

0.24815E-08

0.28518E-09

0.62001E-11

0.31001E-13

0.51539E+00

102
Table A2 . (Continued)

Coefficients Freon-12 Freon-22

A(4,8) 0.48691E-02 0.18936E-05

A(4,9) 0.68452E-04 0.26762E-03

A(4,10) 0.46711E-06 0.18936E-05

A(4,ll) 0.15484E-08 0.65891E-08

A(4,12) 0.20038E-11 0.90041E-11

A(5,l) 0.21700E+00 0.2710UE+00

A(5,2) 0.14160E-03 0.24054E-03

A(5,3) 0.64705E-06 0.38936E-07

A(5,4) 0.55390E-08 0.23481E-07

A(5,5) 0.13779E-10 0.97345E-10

A(5,6) 0.17912E-12 0.44953E-12

A(5,7) 0.15409E+01 0.49002E+00

A(5,8) 0.64802E-01 0.83123E-02

A(5,9) 0.91255E-03 0.13105E-03

A(5,10) 0.62141E-05 0.96844E-06

A(5,ll) 0.20422E-07 0.36462E-08

A(5,12) 0.26255E-10 0.52089E-11

103
Table A3. Refrigerant Property Functions and Subroutines

Property Function

or Subroutine

CPCV

ITRPR

SATP

SATPR

SATT

SATVF

VPSV

VPVHS

Application

Calculate specific heat at constant volume and at

constant pressure, specific heat ratio, and sonic

velocity of refrigerant vapor from temperature

and pressure.

Iterate refrigerant vapor thermodynamic properties

from given pressure and specific volume or enthalpy

or entropy.

Calculate refrigerant saturation pressure from given

temperature.

Calculate dynamic viscosity, thermal conductivity of

liquid and vapor, and refrigerant liquid specific

heat at saturation from given temperature.

Calculate refrigerant saturation temperature from

given pressure.

Calculate specific volume of saturated liquid

refrigerant from given temperature.

Calculate specific volume of vapor from given

temperature and pressure.

Calculate refrigerant vapor thermodynamic properties

from given temperature and pressure.

104
APPENDIX B. CALCULATION OF PROPERTIES OF MOIST AIR

For the selection of moist air property equations, it was assumed that air

dry bulb temperature, T, relative humidity, and pressure would be known and

the properties that would be the output were: humidity ratio, specific heat

at constant pressure, gas constant, dynamic viscosity, and thermal conductivity. Presented here are the psychometric equations in their fundamental

form. They are derived assuming moist air to be a mixture of two independent

perfect gases, so that the perfect gas equation of state and Dalton's rule can be applied. The transport properties, dynamic viscosity, and thermodynamic conductivity of air are assumed to be negligibly affected by the

moisture content. Correlations for these two properties were based on dry

air data.

Relations of this appendix were applied in air properties subroutine AIRPR.

The humidity ratio is determined by:

w = 0.622 PSAT/(PATM - (|) • PSAT) (Bl)

where PATH = atmospheric pressure

PSAT = saturation pressure of water vapor at temperature T w = humidity ratio (lb water/lb dry air)

<j) = relative humidity (-)

The saturated water vapor pressure, PSAT (psi), is calculated by the

polynominal approximation:

' For T > 32°F

PSAT = exp(19.504 - 10.431z - 0.2755z2 + 0.039402^)

For 32 < to < 180°F

PSAT = exp(13.4353 - 5.0988z - 1.6896z2 + 0.17829z3)

For T >^ 180°F

PSAT = exp(16.8255 - 14.213z + 7 .5568z2 - 4.01506z3 + 0.17692z^)

1000

where z = (B2)

460 + T

The specific heat at constant pressure is:

Cp = (Cpdry + 0.444)7(1 + w) (B3)

where Cp = specific heat at constant pressure of moist air (Btu/(lb moist

air • F))

Cpdry = specific heat at constant pressure of dry air (Btu/(lb • F)) w = humidity ratio (lb water/ lb dry air)

105
The specific heat at constant pressure of dry air is approximated by the

following polynominal:

Cp, = 0.2478786 - 0.4204563 • 10"^ • TR + 0.567857 • 10"^ • TR^

^ - 0.14936056 Tr3 (B4)

where TR = T + 460*'F is the dry bulb air temperature on the Rankine scale

The gas constant is:

R = (53.34 + 85.76 • w)/(l + w) (B5)

Ibf • ft

where R = moist air gas constant ( ) lb • R

w = humidity ratio (lb water/lb dry air)

The dynamic viscosity and thermal conductivity values are obtained from the

respective equations:

y = 5.5029 • lO'^ + 8.7157 • lO'^ TR-2.9464 • 10-8'rR2 + 6.25 • 10~'^^TR^ (B6)

k = -2.853 • 10-^ + 3.268 • lO'^ tr -8.253 • 10-9tr2 + 1.239 • IQ-'^^TK^ (B7)

where k = moist air thermal conductivity (Btu/(h»F» f t ))

TR = 460 + T = dry bulb air temperature on the Rankine scale (R)

106
APPENDIX C. CALCULATION OF WATER AND FROST PROPERTIES

The equations presented below are for calculation of the properties of water

and frost deposited on a heat pump evaporator outer surface. These properties,

for specific conditions, are either solely a function of temperature or can be

assumed to be constant. The equations below are used in the water and frost

properties subroutine WATPR.

The following fourth degree polynomial expression is used for calculating water

density, conductivity, dynamic viscosity and latent heat of condensation:

5

PROP = Z A(I) • T^-l (CI)

1=1

where A(I) = five constants per calculated property

PROP = calculated property

T = water temperature (F)

Constants A(I) for each property are given in table CI. The specific heat of

water at constant pressure is assumed to be independent of temperature and to

have a constant valu of Cp = l_Btu— . lb • F

The density of frost, as proposed in [53], is calculated by the following

equation:

Pf = exp(bl + b2 • (TW - TP)) (C2)

where bl = 11.9521 + 0.02422 TPR + 35.5498 WA -9.1742 • 10-7 VA +

3.1138 • 10-9 VA • TPR -0.03838

b2 = 13.1606 - 0.02133 TPR -81.955 WA/32.018 - TP

TP = tube temperature (F)

TPR = TP + 460. tube temperature (R)

TW = water (frost) temperature (F)

WA = air humidity ratio

VA = air velocity (ft/sec)

P£ = density of frost (lb/ft )

The frost conductivity, as proposed in [54], is calculated by the equation:

= 0.012138 + 3.8909 10"^ • pj + 5.1409 10"^ • p^^ (C3)

where kf = frost conductivity (Btu/(h*F 'f t))

pf = frost density (lb/ft"')

Frost heat of sublimation, hsugi,, and frost specific heat, Cf , are assumed to

have the following constant values:

hsuBL = 1219.0 Btu/lb

Cf = 0.46 Btu/(lb»F)

107
Table CI. Water Property Evaluation Constants which

are Used in Equation (CI)

L. ax cuJLa u ea Property

A(I)

Density

Ib/ft^

Dynamic

viscosity

lb/(h'ft)

Thermal

conductivity

Btu/(h«F'ft)

Latent heat

of condensation

Btu/lb

A(l) 0.11647 E+03 0.79422 E+03 -0.27694 0.31514 E+04

A(2) -0.40054 0.47589 E+01 0.45215 E-03 0.13714 E+02

A(3) 0.10815 E-02 0.10622 E-01 0.49008 E-05 0.35945 E-01

A(4) 0.12387 E-05 0.10416 E-04 0.88613 E-08 0.43525 E-04

A(5) 0.49002 E-09 0.37690 E-08 0.41387 E-11 0.19695 E-07

108
APPENDIX D. COMPRESSOR SIMULATION SUBROUTINE PERCOM

Reference Is made to figure 4 where the configuration of the main heat pump

components is shown. It was decided to cover in one subroutine called PERCOM

all refrigerant processes from point 1 to point 10. The refrigerant path

covered by this subroutine consists of:

- flow through a suction tube 1-2 and discharge tube 9-10

- flow through a reversing valve on suction side 2-3 and discharge side

8-9

- flow through a compressor 3-8

Input data to PERCOM is detailed in comment statements inserted in the program.

Basically it consists of:

--connecting tube parameters

- four-way valve parameters

- compressor parameters

- ambient air conditions

- refrigerant state at the compressor can inlet (point 3) - refrigerant pressure at discharge (point 6)

Compressor and four-way valve parameters have to be found from a laboratory

test. For a four-way valve, these parameters are heat transfer and pressure

drop parameters and can be calculated with aid of the program VALPAR. In the case of a compressor, parameters to be found are:

Tie ~ electric motor efficiency versus load

RPM = electric motor speed (RPM) versus load

Tija = compressor mechanical efficiency

Cg = compressor effective clearance

rip = compressor polytropic efficiency

CQ4,5, CQ5J, CQ7^8» ^Q3,C' CQc,A = ^^^^ transfer parameters

^^3,4 > ^^4,5 > ^^6,7 » ^^7,8 ~ pressure drop parameters

One of the parameters, the compressor mechanical efficiency, r\^, is assumed

here to be constant and equal to 0.96. The rest of the parameters can be calculated by means of subroutine COMPAR. Efficiency, ng, and RPM as functions

of mechanical load are both part of typical electric motor characteristics for

its class, (see figures 10 and 12). Coefficients describing these curves are required as an input to COMPAR. In the course of these calculations, new coefficients are computed which retain the shape of the characteristics curves,

although a change in absolute values of these coefficients does occur according

to test data supplied. The rest of the parameters (compressor effective clearance, compressor polytropic efficiency, five heat transfer parameters, and four

pressure drop parameters) are calculated by COMPAR using test data and relations

presented in section 5.2.2. Subroutine COMPAR is called by the main program

MAIN. Logic of PERCOM may be followed by looking at the program listing with

Inserted comment statements. Based on a known refrigerant state at the can inlet (point 3), refrigerant parameters at the suction valve (point 5) are guessed and since compression pressure is given, the compression can be

109
computed. Using the refrigerant mass flow rate from this computation and

applying equations given in section 5.2.2, enthalpy and pressure balances are conducted. If balance is not reached, the guess of the refrigerant state at

the suction valve, point 5, is adjusted. Calculations are repeated until

balances are satisfied.

Once refrigerant states at compressor stations 3 to 8 and the mass flow rate are known, other refrigerant paths are computed. Simulation of a four-way

valve and pipes are done by routines VALVE4 and PIPEPI respectively.

Output from subroutine PERCOM is detailed in the program comment statements

and consists of refrigerant thermodynamic properties for points 1 to 10, refrigerant mass flow rate, compressor RPM, energy consumption rate, electric motor

efficiency, compression efficiency, and volumetric efficiency.

110
APPENDIX E. CALCULATION OF CRITICAL PRESSURE FOR TWO-PHASE FANNO FLOW

In this appendix, the procedure is explained for calculation of critical

pressure of a flow if mass flow rate and stagnation enthalpy are given. The

procedure focuses on the fact that Fanno flow assumes maximum entropy at the

flow critical pressure. Hence, the point of maximum entropy on the Fanno line

is being sought and once determined, the choking pressure is found.

The entropy of two-phase flow is calculated by the equation:

s = SL + x(sv - sl) (El)

where s = entropy

x = quality

subscripts L and V refer to liquid and vapor phase, respectively.

Since the entropy of saturated liquid and of saturated vapor, sl and sy, are known for a given pressure, the entropy of two-phase flow can be found if the

quality is known.

The quality for Fanno flow can be found using the energy equation:

° 2

where G = mass flux

i = enthalpy

Iq = stagnation enthalpy

V = specific volume

Two-phase specific enthalpy and specific volume are:

i = iL + x(iv - 11) (E3)

V = Vl + x(vv - Vl) (E^)

Substituting and rearranging, the following quadratic equation can be obtained:

x2+x»b+c=0 (E5)

2(v -v)«v 'G^+i -i

where b =

2a - i ) - e = L o_ + ^

2

g2 ^ (v^ - Vl)2

Equations (E5) allows calculation on the quality for given Fanno flow (mass flux

and stagnation enthalpy known) at a particular pressure. This quality can then

111
be used to find entropy by equation (El). Thus the maximum entropy and the

critical pressure of the flow can be found numerically^

Search for the maximum entropy along the Fanno flow path was tried by two

different iteration methods. The Newton-Raphson method appeared to be unstable

for single-precision calculations. Stability of this method could be assured

by performing calculations in the double precision mode. Such calculations

consume twice as much computing time as single-precision calculations and would

require keeping some refrigerant property subroutines as double precision rou- tines. It was felt to be more reasonable to use the binary search method. The

binary search method is not as efficient as the Newton-Raphson method; however,

it provides a stable solution in single-precision calculations for maximum

entropy.

112
APPENDIX F. CAPILLARY TUBE SIMULATION SUBROUTINE EXDEV

The purpose of the capillary tube simulation routine is to calculate the

refrigerant mass flow rate when capillary dimensions, refrigerant inlet state,

and pressure in the evaporator are given. Depending on the above input, there are several possible capillary tube operation modes. For example, the refrigerant can be either a single or a two-phase state prior to the tube entrance.

If there is a two-phase mixture at the inlet (a highly undesirable but possible condition under very low load operating conditions), two-phase flow will

exist along the whole tube length.

For a subcooled liquid at the inlet, there are three flow alternatives:

- liquid flow only along the whole tube length

- liquid flow in the front portion of the tube and two-phase flow in

the further portion of the tube

- two-phase flow only along the whole tube length

Each of the above flow situations can exist with or without choking condition

at the tube exit. The capillary tube simulation routine EXDEV developed in

this study can distinguish among all of the above operational modes. During

the course of the calculations, the refrigerant critical pressure is found by

routine PCHOKE using a binary search iteration method. Then equations (38),

(43), and (48) are used to compute the refrigerant mass flow rate. The

density-pressure integral in equation (48) is evaluated by routine TSIMP based

on a Simpson's rule. An iteration solution is required to obtain the final

simulation result since choking pressure, friction factor, fractions of tube

length with liquid and two-phase flow, and the velocity head used to correct

enthalpy are functions of refrigerant mass flow rate which has to be found.

The Newton-Raphson method is used in the iteration for the final EXDEV solution.

The logic of the routine EXDEV is presented in figure Fl. The input and output

data are listed in comment statements in this routine.

If3
f INPUT: EXPANSION DEVICE LENGTH & INNER DIA. REFRIG. ENTHALPY BEFORE TUBE ENTRANCE

REFRIG. PRESSURE BEFORE TUBE ENTRANCE

REFRIG. PRESSURE IN EVAPORATOR

GUESSED REFRIG. MASS FLOW RATE

FIND ALL REFRIG. STATE PARAMETERS

BEFORE EXPANSION DEVICE ENTRANCE

(liquid at inlet)

ASSIGN REFRIG. SAT. PRESSURE

AT TUBE OUTLET

FIND MASS FLOW RATE Ml

1 FIND PFLASH PFLASH = PO

r

FIND PEXIT

1

FIND PI

NO

CALC. LENGTH OF TUBE

WITH SU6C00LED LIQUID

FIND LENGTH OF TUBE

WITH 2-PHASE MIXTURE

1* PFLASH^

YES

|2-phase

flow only

in the tube]

ADJUST Ml

CALC. REFRIG. MASS FLOW RATE M2

FOR 2-PHASE PORTION OF TUBE

Figure Fl. Logic of constant flow area expansion device simulati

program EXDEV

114
APPENDIX G. EVAPORATOR AND CONDENSER SIMULATION SUBROUTINES EVPHXM AND CNDNEW

The tube-by-tube approach to coil simulation has been explained in section 5.4

where applicable heat, mass, and momentum transfer equations have also been presented. Though the evaporator and condenser simulation programs require some- what different heat transfer relations, their logic is the same as shown in

figure Gl.

The iilput to coil simulation program is coil design data, refrigerant flow

distribution sequence, and air and refrigerant inlet conditions. During the

first bank, of tubes' performance calculations, the air parameters upstream of

each tube row are estimated. The program then computes the heat transfer rate

and pressure drop tube-by-tube from the inlet to the exit of the coil. Mean

refrigerant properties in each tube are used to calculate heat transfer rates

and pressure drops applying equations described in section 5.4. An iterative

procedure is used for each tube since only refrigerant conditions at tube inlet are known when the calculations start, and initially the average refrigerant

propetties in a tube have to be assumed to be those at the inlet conditions.

Upstream air temperature and humidity ratio may not be accurately known at the

time tube performance is being calculated, so a second iterative loop is used

to update air-side data after each round. It should be realized that, while

in the condenser case, update of air-side data means update of air temperature

and humidity ratio; for the evaporator it also includes liquid (frost) layer

thickness. When refrigerant enthalpy values at the coil exit obtained from two

consecutive loops are within the imposed tolerance, the calculations of coil

performance are completed and results printed.

Both the evaporator coil simulation model EVPMXM and condenser coil simulation

model CNDNEW comply with the logic and equations previously described^/ . Input

and output data for these programs are detailed in comment statements at the

beginning of each subroutine. Other inserted comment statements along with the

flow chart (figure Gl) facilitate the understanding of both program's

organization.

6/ For reasons of organization of the overall heat pump simulation program,

EVPHXM performs evaporator calculations backwards from the evaporator

exit to the inlet.

115
QarT])

'INPUT: COIL GEOMETRY

REFRIGERANT FLOW DISTRIBUTION i AIR INLET CONDITIONS REFRIGERANT INLET CONDITIONS

I ESTIMATE AIR PARAMETERS UPSTREAM EACH TUBE DEPTH ROW

PRINT RESULTS

Figure Gl. Flow chart for coil simulation program

116
APPENDIX H. THE LOGIC FOR THE MAIN PROGRAM, MAIN

The principle of the logic for the heat pump model has been explained in section

5.1.2. It is based on three balances:

- enthalpy balance

- pressure balance

- mass balance

The enthalpy balance and pressure (mass flow rate) balance are interdependent

and have to be found in a simultaneous iteration process. Their solution provides refrigerant properties in the system which are required for mass inventory calculations. The balances are performed in the manner as shown in figure

HI, which presents the logic developed during this study and contained in the

main program MAIN.

The objective of the logic is to iterate refrigerant states at key locations

of a heat pump for given outdoor and indoor conditions. The addresses of the

key locations used in the program MAIN are consistent with those marked in

figures 4 and HI:

1 - evaporator exit, suction pipe inlet

2 - suction pipe outlet, low pressure 4-way valve inlet

3 - low pressure 4-way valve exit, compressor can inlet

4 - inside compressor can

5 - compressor cylinder at suction

6 - compressor cylinder at discharge

7 - discharge manifold

8 - compressor can exit, high pressure 4-way valve inlet

9 - high pressure 4-way valve exit, discharge pipe inlet

10 - discharge pipe outlet, condenser inlet

11 - condenser outlet, liquid line inlet

12 - liquid outlet, expansion device outlet

13 - evaporator inlet

The thermodynamic processes shown in figure HI and figure 5 are covered by

five subroutines: PERCOM, CNDNEW, LQLINE, EXDEV and EVPHXM. The iteration

process starts with a guessed refrigerant pressure and vapor superheat at the

compressor can inlet and with guessed compressor discharge pressure (points

3 and 6). From these data, the compressor performance simulation subroutine

PERCOM computes the refrigerant mass flow rate through the compressor and

refrigerant parameters from point 1 to 10. The condenser and liquid line

subroutines CNDNEW and LQLINE are called next to perform calculations which

yield refrigerant states at point 11 and 12. Subsequently, the expansion

device simulation program EXDEV computes refrigerant mass flow rate through

the flow restrictor. At this point, the mass flow rate through the compressor,

RMASS, and mass flow rate through the expansion device, XMASS, are compared

and mass flow balance is sought. This is done by resuming calculations by

PERCOM, CNDNEW, LQLINE, and EXDEV, holding constant refrigerant parameters at

the compressor can inlet (point 3) and adjusting refrigerant saturation

117
3 rH

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point

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point

13

in

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te

through

t

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t

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.

calculated

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te

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enthalpy

at

enthalpy

at

refrig.

put

mass

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ra

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sat.

temp,

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sat.

temp,

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iss

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refrig

itory

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00 DO O 00 <i} to Ui

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m i fH iH I W EC I

118
temperature at discharge (point 6) until an appropriate saturation temperature

at point 6 is found at which RMASS and XMASS are equal.

The evaporator performance simulation routine, EVPHXM, was chosen in this logic

to be run after the mass flow balance between the compressor and expansion

device has been found. (EVPHXM requires the most computing time of all the

subroutines.) The subroutine EVPHXM computes evaporator performance in a backward scheme from the actual flow direction with the refrigerant state at point

1 and mass flow rate as input, yielding the refrigerant state at point 13 as output. As the thermodynamic processes in the liquid line and flow restrictor are adiabatic, enthalpy at point 12, H12 , should be equal to that at evaporator

inlet, H13. If they are not equal, the refrigerant state at point 3 (i.e.,

saturation temperature) is adjusted and all calculations have to be repeated

from the beginning. (The other parameter defining the refrigerant state at

point 3, the refrigerant superheat, is left unchanged.)

Once an enthalpy balance and mass flow balance are established, two out of three

refrigerant parameters guessed at the beginning are temporarily obtained. To

verify the third guessed parameter, i.e., the refrigerant vapor superheat at

the compressor can inlet, the refrigerant mass inventory in the system is used.

Each time it is performed, it provides a result based on refrigerant states in

the system found after solving enthalpy and pressure balances with the assumed

vapor superheat at point 3. If the amount of refrigerant obtained from mass inventory calculations is smaller than the refrigerant put into the system, the

superheat guess has to be decreased and all calculations have to be repeated

from the beginning. If the superheat is equal to zero and the mass inventory

result is still smaller than refrigerant mass input, it means that some of the

refrigerant is stored in the form of liquid in the accumulator and saturated

vapor is entering the compressor. This situation or the equality between the mass inventory results at design and actual operating conditions marks an end

of the iteration process and the results are printed.

In the course of the computing process, the main program gathers information

about the heat pump components' performance, updated at each iteration loop,

and applies them to anticipate changes of some parameters caused by a change

in some state property. This allows for the iteration process to converge

faster with each iteration loop. The Newton-Raphson iteration procedure is

used in the solution of the main program.

119

APPENDIX I. PROGRAM USER'S GUIDE

HPSIM is written in Fortran IV and makes use of standard Fortran mathematical

functions. It is built around the main program, MAIN, and 41 subprograms for

heat pump component simulation, and heat transfer, fluid mechanics, and fluid

property calculation. Relations used in the heat pump component simulation

subprograms, and fluid mechanics and heat transfer routines are described in

the main text of this report. Equations used in the refrigerant, moist air,

water, and frost property routines are described in appendices A, B, and C.

All subprograms and routines used in heat pump simulation program are listed in

alphabetic order in table II, with a short statement of their application. The

program HPSIM occupies 31000 decimal words of core memory of Univac 1108

computer

.

I.l INPUT DATA CODING

There are five categories of input values for the HPSIM runs, namely:

- heat pump data;

- refrigerant physical constants;

- run controlling data;

- indoor/outdoor air conditions;

- refrigerant parameters (guesses).

Refrigerant physical constants and heat pump design data are read into the

program from a disk file. File 7 and file 8 are used in reading statements

which input these data for the refrigerant properties and heat pump design data,

respectively. The physical constants for refrigerants 12 and 22 in proper input

format are given in tables 12 and 13. The heat pump design data input code is

presented in table 14. Included are the Fortran sjnnbols with their explanation.

One example of heat pump input data in original format is given in table 15.

Run control data, indoor/outdoor air conditions, design data, and refrigerant

property guesses are input at the terminal. They are clearly solicited by the

program and responses have to be given as follows:

1. Request PERFORMANCE(l) or COMPRESSOR PARAMETERS( 2) , IPTP =

Response: 1 for heat pump performance evaluation

or 2 for compressor parameters evaluation

2. Request

ANSWER 1 FOR YES OR 0 FOR NO

DO YOU WANY ANY INPUT DATA PRINTED? LPF =

Response0 for no input data print

or 1 for input data printout (request for specification of

desired data will follow)

121
3. Request: NSYS = 1 FOR HEATING, NSYS = 2 FOR COOLING

LONG = 1 IF REFRIGERANT CHARGE BALANCE REQUIRED,

OTHERWISE PUT LONG = 0

Response: as explained in the request

4. Request: TG3, TSUP3, TG6

Response: TG3 = saturation temperature of refrigerant vapor at

compressor can inlet (F)

TSUP 3 = refrigerant vapor superheat at compressor can

inlet (F)

TG6 = saturation temperature of refrigerant vapor in

compressor cylinder after compression (all guesses,

subject to iteraction)

5. Request: POA, TOA, RHOA, PRA, TRA, RHRA = ?

Response: POA = outside air pressure (psia)

TOA = outside air temperature (F)

RHOA = outside air relative humidity (decimal fraction)

PRA = indoor air pressure (psia)

TRA = indoor air temperature (F)

RHRA = indoor air relative humidity (decimal fraction)

6. Request: I = 999 for DISCONTINUATION?

Response: 999 to terminate HPSIM runs, otherwise enter any other

number, then system will repeat requesting data from 1

to 5 for the next HPSIM run.

All input data symbols follow general Fortran practice. Variables beginning

with I to N are integers, and all other letters represent real numbers. Free

field input format is used for all input variables.

1.2 OUTPUT DATA CODING

Output data consists of refrigerant thermodynamic states at 13 heat pump key

locations, identified on figure 4, along with results describing the heat pump

performance. Symbols used for output data have the following meaning:

T,P,H,S,X - refrigerant temperature (F), pressure (psia), enthalpy (Btu/lb)

,

entropy, (Btu/(lb • F), quality (-)

122
Table II. Programs Used in a Heat Pump Simulation Model HPSIM

PROGRAM

AIRHT

AIRPR

CALMAS

CNDHTC

CNDNEW

COMPAR

CPCV

DISFLW

DYNADP

DYNDP2

EVPDP

EVPHTl

EVPHXM

EXDEV

FEEDP

FINEFF

FLASH

ITRPR

LQLINE

MAIN

OVLHTC

PURPOSE

Calculate air-side finned tube heat transfer coefficient

Calculate moist air properties

Calculate mass of refrigerant in a coil

Calculate condensation heat transfer coefficient inside a tube

Simulate condenser performance

Calculate compressor performance parameters

Calculate specific heat of vapor refrigerant

Determine refrigerant flow distribution in a coil

Calculate dynamic pressure drop for single-phase flow

Calculate dynamic pressure drop for two-phase flow

Calculate frictional evaporation pressure drop

Calculate evaporation heat transfer coefficent

Simulate evaporator performance

Simulate expansion device performance

Calculate correction factor for 2-phase pressure drop

Calculate fin efficiency

Calculate flashing temperature and pressure for Fanno flow

Iterate refrigerant vapor thermodynamic properties from given

pressure and other than temperature property

Calculate pressure drop in a liquid line

Read input data, iterate heat pump simulation solution, print

results

Calculate overall heat transfer coefficient for dry finned

tube

123
PROGRAM PURPOSE

PCHOKE Calculate critical pressure for two-phase Fanno flow

PERCOM Simulate compressor performance

PIPEPI Simulate pipe performance

PMASS Calculate refrigerant mass in a tube

RHOM Calculate two-phase mixture density on Fanno flow path

SATP Calculate refrigerant saturation pressure

SATPR Calculate refrigerant properties at saturation

SATT Calculate refrigerant saturation temperature

SATVF Calculate specific volume of saturated refrigerant liquid

SFANNO Calculate two-phase mixture entropy for Fanno flow

SPriDP Calculate frictional single-phase pressure drop in a tube

SPHDPl Calculate frictional single-phase pressure drop in a tube

SPHTC Calculate single-phase heat transfer coefficient

TSIMP Integrate using Simpson's rule

UPWAO Calculate overall conductance for wet finned tube

VALPAR Calculate reversing valve performance parameters

VALVE4 Simulate reversing valve performance

VPSV Calculate refrigerant vapor specific volume

VPVHS Calculate refrigerant parameters at and above saturation

WATPR Calculate water and frost properties

XTTCAL Calculate Lockhard-Martinelli parameter

124
Table 12. Date File DATA12 Containing Physical Constants for Refrigerant 12

in the Input Format to Program HPSIM

0$0$Q$*DATA12( 1 )

1 FREON-12

2 6. 9330O00E+02 , 5 . 969000CE+02 , 2 . 8700000E - 02

3 4.5967000E+02, 1 .8505300E-01 . 2 . 7 1 828 1 8E+00

4 9. 1335883E+01 . - 7 . 91 3 1 38 1 E+03 . -1 .2471522E+01

5 1 .0892245E-02.0. ,0.

6 120. .312.

7 3.484000QE+01 , 1 .8691368E+01 .2. 1983963E+01

8 5.3341 175E+01 . -3. 1509939E+00, 1

9 5.0000000E-01 .3.3333333E-01 ,2.

10 8.8734000E-02.6.5093890E-03 ,0.

1 1 -3.4097270E+00, 1 . 5943480E - 03 , -5 . 6762767E+01

12 6.0239450E-02, -1 . 87961 80E- 05. 1.31 13990E+C0

13 -5.4873700E-04,D. .0. 14 0. .3.4688340E-09. - 2 . 5439070E - 05

15 0. .0. .0. 16 3. . 5. 4750000E+00

17 B.0945000E-C3.3.32(p6200E-04, -2.4138960E-07

18 6. 7236300E- 1 1 .0. .0. 19 3.9556551 E+01 .- 1 .6537940E-02

20 0.75800E+00, -0.44230E-02.0.22659E-04

21 -0.80937E-07.0.48640E-09. -0.38993E- 1

22 0. 16672E+01 . -0.37691 E -01 .0.4a998E-03

23 -0. 32598E-05.0. 10701 E -07 , -0. 14059E- 1

24 0.26200E-01 .0.58000E-04.0.0

25 0. .0. .0. 26 -0. 10065E+01 .0. 37971 E-01 , -0.53505E-03

27 0.56446E-05. -0. 12032E-07.0. 15478E-10

28 0.49000E-01 . -0. 1 1 95GE - 03 . 0 . 36320E - 07

29 0.52080E-09. -0. 31690E- 1 1

, -0 .316B9E- 1

30 0.33483E+00, -0. 10635E-01 .0. 1 4902 E- 03

31 -0. 1021 4E -05.0 . 34028E-0B . -0.44300E- 1

32 0.43000E-02.O. 17000E-04.0.0

33 0 . . 0 . , 0

34 -0. 12814E+00, 0.48691 E-02 , -0.68452E-04

35 0.4671 1E-06, -0. 1 5484E - 08 . 0 . 20038E - 1

36 0.21700E+00.0. 1 41 60E - 03 , 0 . 64705E - 05

37 0. 55390E-08. -0. 13779E- 1 0,-0.17912E-12

38 -0. 15410E+01 .0.64802E-01 , -0 . 91 255E-03

39 0.62141E-05, -0.20422E-G7.0.26255E-10

125
Table 13. Data File DATA22 Containing Physical Constants for Refrigerant

22 in the Input Format to Program HPSIM

Q$Q$Q$*DATA22( 1

)

1 FREON-22

2 6.645000CE+02 ,7.2191 OOOE + 02 , 3 . 0525000E - 02

3 4.5967000E+02.1 .8505300E-01 , 2 . 7 1 828 1 8E + 00

4 6. 7598246E+01 , - 8 . 8538843 E+03 , - 7 . 861 031 OE+00

5 5.0448235E-03,4.4574700E-01 , 6 . 86 1 OOOOE+02

6 140. ,240.

7 3. 2760000E+01 . 5 . 4634409E+0 1 . 3 . 6748920E+01

8 -2 . 2292550E+01 , 2 . 0473289E+01 . 3 . 3333333E -01 9 6.6656667E-01 . 1 . , 1 . 3333333E+00

10 1 .2409800E-01 . 2 . OOOOOOOE - 03 . 0

.

11 -4.3535470E+00.2.4072520E-03, - 4 . 4066868E+01

12 -1 .7464000E-02.7.6278900E-05, 1 .4837630E+00

13 2.3101420E-03, - 3 . 6057230E - 06 , 0

.

14 -3 . 7240440E-05 , 5 . 3S54650E - 08 , M . 845051 0E-C4

15 1 .3633870E+08, -1 . 6726 1 20E+05 , 0

.

16 5.4820000E+02,4.2000000E+00

17 2.8128360E-02.2.255403GE-04, -6 . 5096070E-08

18 0. ,0. , 2 . E734100E+02

19 6.2400900E-i-01 , -4.53335G0E-02

20 0 . 64600E+00 , - 0 . 29 1 94 E - 02 , 0 . 1 2 1 64 E - 04

21 -0 . 74985E-07 .0 . 83951 E - 09 , - 0 . 375 1 2

E - 1 1 22 0.69684E01 , - 0 . 243 1 yEOO , 0 . 35924E - 02

23 -0. 261 87E- 04.0. 93884 E- 07 .-0.1 3301 E -09

24 0.26600E-01 , 0 . 6380-JE - 04 , 0 . 1 0761 E- 06

25 -0. 32061 E- 08. 0 . 43463E- 1 0.-0.13175E-12

26 -0.66 3 30E00,0.25757E-01 . -0.37913E-03

27 0.27734E- 05,-0. 1 0053 E- 07,0. 14474E-10

28 0.63000E-01 , -0. 1 5820 E- 03 ,-0.1 2289E-06

29 0. 17453E-08, -0.52635E-1 1,-0. 10857E- 13

30 -0.46705E00,0. 19421E-01 , -0. 28507 E- 03

31 0.204 2 9E-C5. - 0 . 7 1 963 E - 03 , 0 . 99460E - 1

1

32 0. 48000E-02 ,0. 1 9881 E- 04 , 0. 2481 5E-08

33 0. 2851 8E-09. -0 . 62001 E-11 .0.31001E-13

34 0.51539E00. -0. 1 8601 E- 01 , 0. 26762 E- 03

35 -0. 1 8936E- 05,0. 65891 E -03 . -0. 90041 E- 1

1

35 0. 271 OOE+00,0. 24054E-03 . 0. 38936E-07

37 0.23481E-07.-0.97345E-10.0.44953E-12

38 0.49002E00. - 0 . 831 23E - 02 . 0 . 1 3 1 05E - 03

39 •0.95084E-06,0.35462E-08, -0.52089E-1

1

126
Table 14. Heat Pump Input Data Code to Program HPSIM

All input same line

Line 1

line 2;

Line 3;

Line 4;

Line 5;

Line 6

Line 7

data are in FORTRAN free field input format with data values on the

separated by commas.

ATITLE

ATITLE = title, maximum 80 characters

ELEFUL, SWPVOL, ETAPLY, CLREFF

ELEFUL = compressor motor energy input rate at max. rated load

(kW)

SWPVOL = compressor swept volume per revolution (in^)

ETALPY = compressor polytropic efficiency (-)

CLREFF = compressor clearance volume as fraction of stroke volume

(-)

EMETA(

EMETA(

EMETA(I

EMETA(I

EMOPT(I

EMOPT(I

EMOPT(I

EMPOT(I

EMRPM(

EMRPM(

. I = 1,5

= compressor motor efficiency in fraction at fraction of

full load specified by EMPOT(I), I = 1,5 (-)

I = 6,11

= compressor motor efficiency in fraction at fraction of

full load specified by EMOPT(I), I = 6,11 (-)

, I = 1,5

= compressor motor full load fraction (-)

, I = 6,11

= compressor motor full load fraction (-)

, I = 1,6

= coefficient for compressor motor RPM calculations (-)

Line 8 : CPC34, CPC45, CPC67, CPC78

CPC34 = pressure drop parameter at compressor can inlet

((Ibf • h2)/(lb • in2 • ft^))

CPC45 = pressure drop parameter at compressor suction valve

((Ibf • h2)/(lb • in2 . ft^))

127
CPC67 = pressure drop parameter at compressor delivety valve

((Ibf • h2)/(lb • in2 . ft^))

CPC78 = pressure drop parameter at compressure can exit

((Ibf • h2-2)/(ib . In2 . ft2-8))

Line 9 ;

' CQC4C, CQCCOA, CQC45 , CQC67 , CQC78

CQC4C = parameter for compressor can wall - refrigerant vapor heat

transfer (ft0«2)

CQCCOA = parameter for compressor can - ambient air heat transfer

(Btu/h • f1*333)

CQC45 = suction valve heat transfer parameter (ft^*2)

CQC67 = delivery valve heat transfer parameter (ft^*2)

CQC78 = heat transfer parameter at can exit (ft^*2)

Line 10 ; CQ, CPDR

CQ - parameter for 4-way valve heat transfer (ftO.2)

CPDR = pressure drop parameter for 4-way valve

((Ibf • h2)/(lb • in2 . ft^))

Line 11 ; ACCH, ACCD, VCAN, REFIN

ACCH = accumulator height (in)

ACCD = accumulator inner diameter (in)

VCAN = volume of compressor can filled by refrigerant (in^)

REFIN = refrigerant charge (lb)

Line 12 ; NDEP(l), NROW(l)

NDEP(l) = number of indoor coil tube depth rows (-)

NROW(l) = rumber of tubes per indoor coil depth row (-)

Line 13 ; DI(1), D0(1), DT(1), RPCH(l), DPCH(l), WIDTH(l)

DI(1) = inner diameter of indoor coil tubes (in)

D0(1) = outer diameter of indoor coil tubes (in)

DT(1) = indoor coil fin tip diameter (in)

128
RPCH(l) = pitch between tubes of the same depth in indoor coil (in)

DPCH(l) = depth pitch for indoor coil tubes (in)

WIDTH(l) = indoor coil width (equal tube length) (in)

Line 14 : FPCH(l) , FTK(l), FMK(l), TMK(l) , AMAS(l)

FPCH(l) = indoor coil fin pitch (in)

FTK(l) = indoor coil fin thickness (in)

FMK(l) = indoor coil fin material thermal conductivity

(Btu/(ft • h • F))

TMK(l) = indoor coil tube material thermal conductivity

(Btu/(ft • h • F))

AMAS(l) = air mass flow rate through indoor coil (Ib/h)

Line 15 ; CONST(l), CPOW(l), ANGLE(l)

CONST(l) = constant for air side heat transfer correlation for

indoor coil equal to 0.134 (-)

GPOW(l) = constant for air side heat transfer correlation for

indoor coil equal to 0.681 (-)

ANGLE(l) = angle between indoor coil face and air streamlines (rad)

Line 16: EIDFAN

EIDFAN = indoor fan energy input rate (kW)

Line 17: NREPTI

NREPTI = number of repeating sections in indoor

Line 18: NTUBE(1,I) I = 1,5

NTU1JE(1,1) = number of tubes in first row in each section of indoor

coil (-)

NTUBE(1,2) = number of tubes in second row in each section of indoor

coil (-)

NTUBE(1,3) = number of tubes in third row in each section of Indoor

coil (-)

NTUBE(1,4) = number of tubes in fourth row in each section of indoor

coil (-)

129
NTUBE(1,5) = number of tubes in fifth row in each section of indoor

coil (-)

Line 19 ; IFR0M(1,I), I = 1,10

IFR0M(1,1) = number of tube of indoor coil from which tube 1 receives

refrigerant when indoor coil works as evaporator (-)

IFR0M(1,2) = number of tube of indoor coil from which tube 2 receives

refrigerant when indoor coil works as evaporator (-)

IFR0M(1,3) =

IFR0M(1,9) =

IFR0M(1,10) = number of tube of indoor coil from which tube 10 receives

refrigerant when indoor coil works as evaporator (-)

Line 20 ; IFR0M(1,I), I = 11,20

IFR0M(1,I) = number of tube indoor coil from which tube I receives

refrigerant when indoor coil works as evaporator (-)

Line 21; IFR0M(1,I), I 21,30

Line 22; IFR0M(1,I)

,

I 31,40

Line 23: IFR0M(1,I)

,

I 41,50

Line 24; IFR0M(1,I)

,

I 51,60

Line 25; IFR0M(1,I), I 61,70

Line 26; IFR0M(1,I)

,

I 71,80

Line 27; IFR0M(1,I)

,

I 81,90

Line 28: IFR0M(1,I)

,

I 91,100

Line 29; IFR0M(1,I)

,

I 101,110

Line 30; IFR0M(1,I)

,

I 111,120

Line 31; IFR0M(1,I)

,

I 121,130

IFR0M(1,I) = number of indoor coil from which tube I receives

refrigerant when indoor coil works as evaporator (-)

130
Line 32 ; NDEP(2), NR0W(2)

NDEP(2) = number of outdoor coil tube row depths (-)

NR0W(2) =» number of tubes per outdoor coll depth row (-)

Line 33 ; DI(2), D0(2), DT(2), RPCH(2), DPCH(2), WIDTH(2)

DI(2) = Inner diameter of outdoor coll tubes (In)

D0(2) =« outer diameter of outdoor coll tubes (In)

DT(2) = outdoor coll fin tip diameter (In)

RPCH(2) = pitch between tubes of the same depth In outdoor coll (In)

DPCH(2) = depth pitch for outdoor coll tubes (In)

WITDTH(2) = outdoor coll width (equal tube length) (In)

Line 34 ; FPCH(2), FTK(2), FMK(2), TMK(2), AMAS(2)

FPCH(2) = outdoor coll fin pitch (In)

FTK(2) = outdoor coll fin thickness (In)

FMK(2) = outdoor coil fin material thermal conductivity

(Btu/(ft • h • F))

TMK(2) = outdoor coll tube material thermal conductivity

(Btu/(ft • h • F))

AMAS(2) = air mass flow rate through outdoor coil (Ib/h)

Line 35 ; C0NST(2), CP0W(2), ANGLE(2)

C0NST(2) = constant for air side heat transfer correlation for

outdoor coll equal to 0.134 (-)

CP0W(2) = constant for air side heat transfer correlation for outdoor

coll equal to 0.681 (-)

ANGLE(2) = angle between outdoor coll face and air streamlines (rad)

Line 36 ; EIDFAN

EIDFAN = outdoor fan energy input rate (kW)

LINE 37 ; NREPTO

NREPTO = number of repeating sections in outdoor coil (-)

131
Line 38 ; NTUBE(2,I), I = 1,5

NTUBE(2,1) = number of tubes in first row in each section of outdoor

coil (-)

NTUBE(2,2) = number of tubes in second row in each section of outdoor

coil (-)

NTUBE(2j3) = number of tubes in third row in each section of outdoor

coil (-)

NTUBE(2,4) = number of tubes in fourth row in each section of outdoor

coil (-)

NTUBE(2,5) = number of tubes in fifth row in each section of outdoor

coil (-)

Line 39 : IFR0M(2,I), I = 1,10

IFR0M(2,1) = number of tube of outdoor coil from which tube 1

receives refrigerant when outdoor coil works as evaporator (-)

IFR0M(2,2) = number of tube of outdoor coil from which tube 2

receives refrigerant when outdoor coil works as evaporator (-)

IFR0M(2,3) =

IFR0M(2,9) =

IFROM(2,10) = number of tube of outdoor coll from which tube 10

receives refrigerant when outdoor coll works as evaporator (-)

Line 40 ; IFR0M(2,I), I = 11,20

IFR0M(2,I) = number of tube of outdoor coil from which tube I receives refrigerant when outdoor coil works as evaporator (-)

Line 41; IFR0M(2,I)

,

I = 21,30

Line 42; IFR0M(2,I)

,

I = 31,40

Line 43; IFR0M(2,I)

,

I = 41,50

Line 44; IFR0M(2,I)

,

I = 51,60

132
Line 45: IFR0M(2,I), I = 61,70

Line 46: IFR0M(2,I), I = 71»80

Line 47: IFR0M(2,I), I = 81,90

Line 48: IFR0M(2,I), I = 91,100

Line 49: IFR0M(2,I), I = 101,110

Line 50: IFR0M(2,I), I = 111,120

Line 51: IFR0M(2,I), I = 121,130

IFR0M(2,I), = number of tube of outdoor coil from which tube I receives refrigerant when outdoor coil works as evaporator (-)

Line 52 : CAPIDl, CAPLl, NCPLl, CAPID2, CAPLl , NCPL2

CAPIDl = inner diameter of cooling operation expansion device (in)

CAPLl = length of cooling operation expansion device (in)

NCPLl = number of cooling operation expansion devices (-)

CAPID2 = inner diameter of heating operation expansion device (in)

CAPL2 = length of heating operation expansion device (in)

NCPL2 = number of heating operation expansion devices (-)

Line 53 : YL, YD, YKl, YDl, YK2 , YD2

YL = length of compressor-outdoor coil tubing (in)

YD = inner diameter of compressor-outdoor coil tubing (in)

YKl = thermal conductivity of compressor-outdoor coil tubing material

(Btu/(ft • h • F))

YDl = outer diameter of compressor-outdoor coil tubing (in)

YK2 = thermal conductivity of compressor-outdoor coil tubing

insulation (Btu/(ft • h • F))

YD2 = outer diameter of compressor-outdoor coil tubing insulation

(Btu/(ft • h • F))

133
Line 54: RL, RD, RKl, RDl, RK2 , RD2

RL = length of compressor-indoor coil tubing (in)

RD = inner diameter of compressor-indoor coil tubing (in)

RKl = thermal conductivity of compressor-indoor coil tubing

material (Btu/(ft • h • F))

RDl = outdoor diameter of compressor-indoor coil tubing (in)

RK2 = thermal conductivity of compressor-indoor coil tubing insulation

(Btu/(ft • h • F))

RD2 = outer diameter of compressor-outdoor coil tubing insulation

(Btu/(ft • h • F))

Line 55: RYL, RYD

RYL = liquid line length (in)

RYD = liquid line diameter (in)

Note on coil input data:

As a coil depth it is understood direction perpendicular to a coil surface

facing incoming air. Line 13 and 33: for fin tip diameter DT(I) refer to

figure 21. Lines 19-31 and 39-51; tubes are numbered consecutively from the

first tube in the first row (facing incoming air) to the last tube in the last

row. Enter 0 (zero), if considered tube receives refrigerant from coil inlet

port; enter 999, if the tube is nonexistant.

134
Table 15. Example of Heat Pump Data in the Input Format to Program

HP SIM

0$0$0S*SYST42( 1

)

1 SYSTEM 2 ***

2 5. .5.228,0.744,0. 1743

3 0.1 996 .0. 3355 , 0 .5446 , 0 . 6540 , 0 . 6958

4 0. 7233, 0. 74 14, 0.7519, 0.7 595. 0. 762 3,0. 7633

5 0.05,0. 1 ,0.2.0.3,0.4

6 0.5,0.6,0.7.0.8.0.9, 1 .0

7 3578. , 23 . 53 , -831 . 4 .3542 .

. -6140. . 3234.

8' 4.7 73E-7. 1 . 353E - 5 . 2 . 7 55E - 4 . 9 . 436E - 5 ? 37 . 16 , 105.7 .0 . 231 2 . 0 . 221 3 . 3 . 1 70

10 0.905. 1 .637E-05

11 10. ,4. . 100. .0. 12 3.29

13 . 3 1 5, . 375 . 1 .055. 1 . ,0. 875 , 1 8

.

14 .0769 , .0055 , 1 18 ., 118 .. 5800

.

15 . 1 34. . 681 . .642

16 .640

17 1 10 29,30,29.0,0

19 2, 32. 4,0, 34. 5, 6, 7. 0,9

20 10,11.12.13,16,17,18,19,20,21

21 22,23,24,25.55.0.26.27,58.31

22 1,3,3,33,34,35,36.37,38.39

23 40,41 .42. 14.15,47,48,49,50,51

24 52 , 53 . 54 , 25 . 56 . 57 , 27 . 28 . 29 . 30

25 60,61,62.63,64,65,66,67,68.69

26 70. 71 . 43 , 44 ,45 , 46 . 78 . 79 , 80 . 81 27 82 , 83 , 84 , 85 , 86 , 87 , S8 . 59 . 999 , 999

28 999,999,999,999,999,999. 999,999,999,999

29 999 , 999 , 999 , 999 , 999 , 999 . 999 , 999 , B99 , 999

30 999 , 999, 999 .999 . 999 , 999 . 99S , 999, 999 , 999

31 999 , 999 , 999 , 999 , 999 . 999 , 999 , 999 , 999 , 999

32 2,36

33 0.315.0.375, 1 . 128. 1 ., 1 . .64.

34 D. 062 5, 0.0055. 1 18. . 1 18. . 16600.

35 .134, .681 , .671

36 .510

37 6 38 6,6,0.0,0

39 2,3.4,5.6,0,1 .7,0,9

40 10 . 1 1 , 999 , 999 , 999 . 999 , 999 , 999 , 999 . 999

41 999 . 999, 999 .999 . 999 . 999 , 999 . 999 , 9 99 . 999

42 999 , 999, 999 ,999 , 999 , 999 . 999 . 999 , 999 , 999

43 999 ,999 , 999 ,999 ,999 . 999 . 999 . 999 , 999 , 999

44 999 .999, 999. 999 .999 . 999 . 999 . 999 , 999 . 999

45 999 , 999 , 999 , 999 , 999 , 999 . 999 , 999 , 999 , 999

46 999,999,999.999,999.999,999,999,999,999

47 999.999.999.999.999.999.999,999.999.599

48 999 , 999 , 999 . 999 , 999 , 999 . 999 , 999 , 999 . 999

49 999 ,999, 999 , 999 , 999 , 999 . 999 , 999 . 999 . 999

50 999 . 999. 999 .999 . 999 , 999 , 999 , 999 , 999 , 999

51 999 , 999 , 999 , 999 , 999 , 999 , 999 . 999 , 999 , 999

52 0. 078 ,0.5,1 ,0. 073 .0. 5 .

1

53 50. .0.68,223. ,0.750,0.05. 1 .75

54 100. ,0.68.223. ,0.750,0 05, 1 .75

55 180. ,0.311

END PRT

135
COP - coefficient of performance (-)

ELUSE - total heat pump energy consumption rate (kW)

QLOAD - net capacity of the heat pump (Btu/h)

RMASS - refrigerant mass flow rate (Ib/h)

TMASS - refrigerant charge (lb)

TG3 - refrigerant saturation temperature at compressor can inlet

(F)

TG6 - refrigerant saturation temperature at compressor delivery

valve (F)

TSUP3 - refrigerant superheat at compressor can inlet (F)

The last three output data TG3 , TG6 , and TSUP3 , are final values iterated from

TG3, TG6, and TSUP3 supplied as guessed input values.

1.3 EXAMPLE OF RUN OF HEAT PUMP SIMULATION PROGRAM HPSIM

The following is a computer printout for a HPSIM run corresponding to laboratory

test No. 2.2 of this study (refer to table 4).

Input in this run was as follows

- refrigerant data - constants for R-22 as given in table 1.3

- heat pump data - as given in table 1.5.

- run controlling data, operating conditions, guessed refrigerant

parameters as shown on printout (lines 1-150, 606)

The solution was iterated in 4 energy balance loops (lines 151-252, 253-355,

356-458, 459-561). Final simulation results can be found at lines 570 through

601. Computation of this example required 112 seconds on Univac 1108 computer.

In the given example there was no need to iterate refrigerant superheat at com- pressor inlet. For cases, where such iteration is required, iterations

presented here would be repeated 3 or 4 times.

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BIBLIOGRAPHIC DATA

SHEET (See instructions)

1 PIIRI ir'ATIDN DR

REPORT NO. NBS BSS 155

2 P^rforminff Orffan RcDOrt No. 3. Publication Date May 1983

4. TITLE AND SUBTITLE

COMPUTER MODELING OF THE VAPOR COMPRESSION CYCLE

WITH CONSTANT FLOW AREA EXPANSION DEVICE

5. AUTHOR(S)

Piotr Domanskl and David Didion

6. PERFORMING ORGANIZATION (If joint or other thon NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stote, ZIP)

Department of Energy

1000 Independence Avenue, S.W.

Washington, D C 20585

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 83-600524

I I Document describes a computer program; SF-185, FIPS Software Summary, is attaclied. 11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gnifleant bibliography or literature survey, mention it here) An analysis of the vapor compression cycle and the main components of an air source heat pump during steady-state operation has been performed with emphasis on fundamental phenomena taking place between key locations in the refrigerant system.

The basis of the general heat pump model formulation is the logic which links the

analytical models of heat pump components together in a format requiring an iterative solution of refrigerant pressure, enthalpy and mass balances.

The modeling effort emphasis was on the local thermodynamic phenomena which

were described by fundamental heat transfer equations and equation of state relationships among material properties. In the compressor model several refrigerant

locations were identified and the processes taking place between these locations

accounted for all significant heat and pressure losses. Evaporator and condenser

models were developed on a tube-by-tube basis where performance of each coil tube

is computed separately by considering the cross-flow heat transfer with the external air stream and the appropriate heat and mass transfer relationships. A

capillary tube model was formulated with the aid of Fanno flow theory.

The developed heat pump model has been validated by checking computer results

against laboratory test data for full and part load operation for the cooling/

dehumidifying mode as well as the heating mode under frosting conditions.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Air conditioner; capillary tube; coil; condenser; compressor; evaporator; expansion

device; heat pump; modeling; vapor compression cycle

13. AVAILABILITY

Unlimited

I I For Official Distribution. Do Not Release to NTIS

|X I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[_|] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

162

15. Price

*U.S. GOVERNMENT PRINTING OFFICE! 1983-0-380-997/5057

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