7/30/2019 10.1.1.32.165

1/161

DESIGN ANALYSIS OF A FINNED-TUBE CONDENSER FOR A RESIDENTIAL

AIR-CONDITIONER USING R-22

A Thesis

Presented to

The Academic Faculty

By

Emma May Sadler

In Partial Fulfillment

of the Requirements for the Degree

Master of Science in Mechanical Engineering

Georgia Institute of Technology

April 2000

7/30/2019 10.1.1.32.165

2/161

ii

DESIGN ANALYSIS OF A FINNED-TUBE CONDENSER FOR A RESIDENTIAL

AIR-CONDITIONER USING R-22

Approved:

____________________________________

S. V. Shelton, Chairman

____________________________________

P. V. Kadaba

____________________________________

A. V. Larson

Date Approved________________________

7/30/2019 10.1.1.32.165

3/161

iii

ACKNOWLEDGEMENTS

This work would not have been completed without the help and support of many

others. In particular, I would like to thank my advisor, Dr. Shelton, for his enthusiasm

for my scholastic, professional, and personal success. He has provided motivation and

insights that have been invaluable to this project and my sanity. Id also like to thank

Monifa Wright and Shawn Klawunder for sharing their resources and discoveries.

I would never have made it this far without my parents who have unconditionally

supported any endeavor that would lead to my happiness, be it a Masters degree or a

career as a goat herder. This thesis is dedicated to my grandparents, Sol and Frieda

Gersen, who have been more concerned about my progress than anyone else.

7/30/2019 10.1.1.32.165

4/161

iv

TABLE OF CONTENTS

Chapter IIntroduction........................................................................................................ 1Background ..................................................................................................................... 1

Considerations................................................................................................................. 1Past Work........................................................................................................................ 2Purpose............................................................................................................................ 4

Chapter IIResidential Air Conditioning Systems.............................................................. 6Refrigeration Cycle ......................................................................................................... 6

Air Conditioner Components .......................................................................................... 8Compressor.................................................................................................................. 8Condenser.................................................................................................................. 11

Condenser Fan........................................................................................................... 12Expansion Valve ....................................................................................................... 13

Evaporator................................................................................................................. 13Evaporator Fan.......................................................................................................... 15

Coefficient of Performance........................................................................................... 16

Seasonal COP................................................................................................................ 16

Chapter IIIHeat Exchangers............................................................................................ 20

Geometry....................................................................................................................... 20NTU-Effectiveness Relations........................................................................................ 23

Refrigerant Side Models................................................................................................ 30Single Phase Heat Transfer Coefficient.................................................................... 30Condensation Heat Transfer Coefficient ................................................................... 33

Evaporative Heat Transfer Coefficient ..................................................................... 35Pressure Drop in Straight Pipe.................................................................................. 36Pressure Drop in Bends............................................................................................. 39

Air Side Models ............................................................................................................ 44Heat Transfer Coefficient.......................................................................................... 44

Pressure Drop............................................................................................................ 46

Chapter IVOptimization of Operating Parameters.......................................................... 52Subcool and Seasonal Effects ....................................................................................... 54

Effect of Varying Air Velocity...................................................................................... 58Effect on Cost Factor..................................................................................................... 60

Chapter VEffects of Geometry with Fixed Cost ............................................................. 62Number of Rows ........................................................................................................... 64Fin Pitch........................................................................................................................ 66

Tube Diameter............................................................................................................... 71Tube Circuiting.............................................................................................................. 75

7/30/2019 10.1.1.32.165

5/161

v

Operating Costs............................................................................................................. 79

Chapter VIEffects of Geometry with Fixed Frontal Area............................................... 85

Number of Rows ........................................................................................................... 85Fin Pitch........................................................................................................................ 90Tube Diameter............................................................................................................... 92Comparing Fixed Area to Fixed Cost ......................................................................... 101

Chapter VIIConclusions and Recommendations .......................................................... 103

Appendix I Mass of Refrigerant in a Heat Exchanger Coil Undergoing Phase Change

............................................................................................................................. 106

Appendix II EES Simulation Program........................................................................... 109

Appendix III Condenser Operating Conditions............................................................. 136

References....................................................................................................................... 146

7/30/2019 10.1.1.32.165

6/161

vi

LIST OF TABLES

Table 1. Distribution of Cooling Load Hours ................................................................... 17Table 2. Euler number coefficients for inverse power series............................................ 49

Table 3. Staggered Array Geometry Factor...................................................................... 50Table 4. Individual row correction factors........................................................................ 51Table 5. Air-Conditioner Design Conditions .................................................................... 52

Table 6. Base Case Condenser and Evaporator Characteristics........................................ 53Table 7. Mass of Refrigerant in Air-Conditioner for Different Subcool Specifications... 54

Table 8. Material Costs ..................................................................................................... 63Table 9. COPs and Area Factors Based on Fin Pitch...................................................... 67

Table 10. Data for Copper Tubes...................................................................................... 71Table 11. COPs and Area Factors Based on Tube Diameter........................................... 72Table 12. Condenser Configurations for Circuiting Analysis........................................... 76

Table 13. Maximum Seasonal COPs for Different Tubes per Circuit with Fixed Cost .. 77Table 14. Pressure Drop Distributions at 83 F................................................................ 78Table 15. COP and Flow Area for Different Circuiting Configurations........................... 80Table 16. Data for Varying Number of Rows at 83 F..................................................... 87Table 17. Overall UA and UA/ Length for Varying Rows at 83 F ................................. 89Table 18. Optimum Operating Conditions For Varying Number of Rows with Fixed Area

................................................................................................................................... 90Table 19. Optimum Operating Conditions For Varying Tube Diameter with Fixed Area94

7/30/2019 10.1.1.32.165

7/161

vii

LIST OF FIGURES

Figure 1. Thermodynamic State of Refrigerant in Refrigeration Cycle.............................. 7Figure 2. Refrigeration Cycle Equipment ........................................................................... 8

Figure 3. Typical Plate Finned-Tube Heat Exchanger...................................................... 11Figure 4. Effective Specific Heat ...................................................................................... 14Figure 5. General Heat Exchanger Dimensions................................................................ 22

Figure 6. Layout of Heat Exchanger Geometry Parameters ............................................. 23Figure 7. Layout of Hexagonal Fins.................................................................................. 28

Figure 8. Effects of Operating Conditions on Evaporator Frontal Area........................... 54Figure 9. Condenser Subcool at Varying Ambient for Different Refrigerant Charges..... 56

Figure 10. Effect of Ambient Temperature on Evaporator Capacity and Mass Flow Rate................................................................................................................................... 57Figure 11. Trends in Seasonal COP vs. COPs at Other Temperatures............................ 58

Figure 12. Effect of Air Velocity on Seasonal COP for Different Subcool Conditions ... 59Figure 13. Effect of Air Velocity on Compressor and Condenser Fan Work................... 60Figure 14. Number of Rows vs. Seasonal COP with Fixed Cost...................................... 64

Figure 15. Number of Rows vs. Compressor Power and Refrigerant Pressure Drop....... 65Figure 16. Rows vs. Air Side Pressure Drop and Fan Power for Fixed Cost at 83 F....... 66

Figure 17. Seasonal COPs for Different Fin Pitches at Optimum Operating Conditionswith Fixed Cost ......................................................................................................... 67

Figure 18. Effect of Fin Pitch on Seasonal COP at Different Air Velocities.................... 68

Figure 19. Effect of Fin Pitch on Frontal Area ................................................................. 69Figure 20. Fin Pitch vs. Airside Pressure Drop at 83F...................................................... 70

Figure 21. Effect of Fin Pitch on Power Requirements at 83F......................................... 70Figure 22. Maximum Seasonal COP for Different Tube Diameters................................. 72Figure 23. Optimal Operating Conditions for Different Tube Diameters......................... 73

Figure 24. Condenser Allocation for Different Tube Diameters at 83F ........................... 74Figure 25. Refrigerant Side Pressure Drop vs. Tube Diameter at 83F ............................. 75

Figure 26. Maximum Seasonal COP for Different Circuiting.......................................... 77Figure 27. Pressure Drop vs. Circuiting at 83 F.............................................................. 78Figure 28. Operating Costs vs. Area Factor for Different Geometry Factors................... 82Figure 29. Operating Costs For Different Tube Diameters and Circuiting at 83 F with

Fixed Cost ................................................................................................................. 83Figure 30. Effect of Varying Fin Pitch for Base Case and Optimum Case at 83 F withFixed Cost ................................................................................................................. 84

Figure 31. Seasonal COP for Varying Rows with Fixed Area.......................................... 86Figure 32. Tradeoffs Between Compressor and Fan Power for Varying Number of Rows

with Fixed Area at 83 F............................................................................................. 87Figure 33. Condenser Allocation for Varying Rows at 83 F........................................... 88

7/30/2019 10.1.1.32.165

8/161

viii

Figure 34. Air Velocity vs. Seasonal COP for Different Row Configurations with Fixed

Area........................................................................................................................... 90Figure 35. Variance of Optimal Air Velocity with Fin Pitch for Fixed Frontal Area....... 91

Figure 36. Condenser Allocation for Varying Fin Pitch at 83F........................................ 92Figure 37. Variance of Optimal Air Velocity with Tube Diameter at Optimum Subcool

for Fixed Frontal Area............................................................................................... 93

Figure 38. Air Side Pressure Drop for Varying Tube Diameters at 83 F........................ 95Figure 39. Compressor and Fan Power Trends vs. Tube Diameter at 83 F .................... 96Figure 40. Operating Costs vs. Cost Factor for Different Geometry Factors ................... 98

Figure 41. Operating Costs For Different Tube Diameters and Circuiting at 83 F withFixed Area................................................................................................................. 99

Figure 42. Optimum Condenser Circuiting for Fixed Area at 83 F with Varying Rows100

Figure 43. Comparison of Area Factor to Cost Factor Based on Number of Rows....... 101Figure 44. Comparison of Area Factor to Cost Factor Based on Fin Pitch.................... 102

Figure 45. Comparison of Area Factor to Cost Factor Based on Tube Diameter........... 102

7/30/2019 10.1.1.32.165

9/161

ix

LIST OF SYMBOLS

Symbol Refers to

a Stanton number coefficient

a Ratio of transverse tube spacing to tube diameter

A Total heat transfer area

Ac Minimum free-flow cross sectional area

Af Total fin surface area

Afr Frontal area

Amin Minimum flow area

Ao Total airside heat transfer area, fins and tubes

AF Area Factor

b Stanton number coefficient

b Ratio of longitudinal tube spacing to tube diameter

B Bend coefficient

BL Building Load

C Heat capacityCF Cost Factor

CLF Cooling Load Factor

COP Coefficient of performance

cp Specific heat

cp,eff Effective specific heat

Cr Heat capacity ratio

Cz Average row correction factor

cz Individual row correction factor

D Tube Diameter

E& Electrical Power Demand

Eu Euler number

f Friction factor

7/30/2019 10.1.1.32.165

10/161

x

FP Fin Pitch

Fr Froude number

G Mass flux

gc Gravitational constant

h Enthalpy

h Heat transfer coefficient

H Heat exchanger height

j Colburn factor

JP j-factor parameter

k Conductivity

k Bend pressure coefficient

k1 Staggered array geometry factor

L Length

m Fin parameter

m Mass

m& Mass flow rate

n Blasius coefficient

NTU Number of transfer unitsNu Nusselt number

pr Reduced pressure

Pr Pressure ratio

PD Piston displacement

PLF Part load factor

Pr Prandtl number

eQ& Evaporator capacity

r Outside tube radius

rb Radius of bend

re Equivalent radius

rr Relative radius

R Ratio of clearance volume to displacement

7/30/2019 10.1.1.32.165

11/161

xi

Rw Wall Resistance

R Fouling factor

Re Reynolds number

SF Size Factor

St Stanton number

t Fin thickness

Tr Temperature ratio

tpc Tubes per circuit

U Overall heat transfer coefficient

v Specific volume

Va Air face velocity

W Heat exchanger width

wa Actual specific compressor work

ws Isentropic specific compressor work

comW& Compressor power

fW& Fan power

x Quality

Xl Longitudinal tube spacing (parallel to air flow)

Xt Transverse tube spacing (normal to air flow)

z Number of rows

z/D Equivalent length

Fin parameter

tt Lockhart-Martinelli parameter

h Enthalpy change

hlatEnthalpy change due to condensation

hsens Enthalpy change due to temperature change

Pa Air-side pressure drop

Pf Refrigerant side two-phase friction pressure drop

Pm Refrigerant side two phase momentum pressure drop

7/30/2019 10.1.1.32.165

12/161

xii

Heat exchanger effectiveness

Roughness

Fin parameter

2 Bend physical property coefficient

2 Two phase bend multiplier

Friction factor

Viscosity

Density

c Compressor thermal efficiency

f Fan efficiencys Surface efficiency

v Volumetric efficiency

Fin parameter

#circ Number of parallel circuits

7/30/2019 10.1.1.32.165

13/161

xiii

SUMMARY

The purpose of this study was to develop an optimization methodology and

software for the detailed design of a finned-tube condenser heat exchanger coil in a

residential air-conditioning unit using the Engineering Equation Solver (EES) software.

The superheat, saturated, and subcool portions of the heat exchanger have been modeled

separately and in detail using appropriate pressure drop and heat transfer fundamental

equations for both the air-side and refrigerant-side of the heat exchangers. The study uses

accurate refrigerant property data for R-22, but can easily be modified to accommodate

other refrigerants. The cooling output and electrical input for the compressor and fans

have been calculated for various ambient temperature conditions. The compressor,

condenser fan, and evaporator components of the cycle are also modeled but in a more

global manner using thermal science laws. Ambient temperature weighting factors used

by the U.S. Department of Energy are used to determine the seasonal coefficient of

performance (COP) of the system.

A base condenser model was arbitrarily chosen and design conditions were

established at 95 F. The operating parameters of condenser subcool and air face velocity

were examined over a wide range of ambient conditions to determine their effects on the

seasonal COP. It was determined that there is a range of subcools and face velocities

where the effects on the seasonal COP were negligible. The COP the system at an

7/30/2019 10.1.1.32.165

14/161

xiv

ambient temperature of 83 F was nearly identical to the seasonal COP and could be used

for quick comparisons.

The effects of changing the tube diameter, tube circuiting, number of rows, and

fin pitch have been investigated for both fixed cost and fixed frontal area. When the

parameters were varied from the base case individually, the changing the number of

circuits to 4 or changing the tube diameter to 1/2 gave the highest COPs. It was

determined that tube diameter and tube circuiting should not be considered separately

because they both affect the refrigerant side pressure drop. When the cost or area was

fixed, the best tube diameter- circuiting configuration was 5 circuits of 5/16 tubing. In

both cases, 4 circuits of 3/8 tubing provided similar performance with better packaging.

In general, the COP will be the highest when the frontal area is maximized and

rows should only be added if there is a frontal area constraint. This is because of the

relationship between the air velocity, depth, and air-side pressure drop. When the cost is

fixed, fewer rows provide better performance. If the frontal area is constrained, adding

rows will increase the performance as long as the refrigerant side pressure drop does not

become too great.

Changing the fin pitch had a relatively negligible effect on the seasonal COP.

The fan power increases as the number of fins increases, but the compressor power

decreases by about the same amount. If cost is fixed, fewer fins provide better

performance. When the area is restricted, more fins provide better performance.

7/30/2019 10.1.1.32.165

15/161

1

CHAPTER I

INTRODUCTION

Background

Refrigeration for personal comfort was first used in 1902. By 1997, 72% of all

American households had air-conditioning and 47% of all households were cooled with

central airi. According to the Air-Conditioning and Refrigeration Institute (ARI), 81% of

all new homes constructed were equipped with central air-conditioning in 1996. ii For a

single family, detached home, the amount of energy dedicated to air-conditioning can be

quite significant. In Atlanta, for example, air-conditioning accounts for approximately

19% of energy costs, which includes both gas and electricity, or 310 dollars per year. It

also accounts for 32% of the total peak power demand of electricity in these homes.iii

Obviously, improving the efficiency of residential air-conditioning units would decrease

utility bills and pollution produced by the power generation.

Considerations

Optimizing an air-conditioning system presents a complex problem for many

reasons. To start, there are many parameters that can be varied for each component. The

effect of varying most parameters is not independent on a component or system basis.

7/30/2019 10.1.1.32.165

16/161

2

Even if a component is optimized for specific operating conditions, like inlet and outlet

conditions, it is necessary to optimize each component at its unique operating conditions

with all other system components. To get a fair comparison among different designs,

operating conditions such as air velocity and refrigerant charge need be optimized for

each design. However, optimizing these parameters will affect the cooling capacity of

the evaporator, skewing the comparison.

Past Work

Until recently, limited computing power and the complex relations for refrigerant

properties had restricted the system design process to experimental testing. In 1975,

James Propst performed a similar study on condenser performance. iv Because of the lack

of computing power, Propst used a simplified model that neglected the refrigerant

properties so the analysis is based on the performance of the air side only. Propst

developed his equations to be solved explicitly and did not depend on any refrigerant side

properties including the condensing temperature. He used a constant refrigerant side

convective heat transfer coefficient, neglected pressure drops in the system, ignored the

superheated and subcooled sections and assumed constant compressor performance.

With increased computing power, it is now possible to create detailed computer

models with accurate refrigerant properties. While manufacturers such as HeatCraft have

created proprietary models for their components, they are not available for general study

and the programs are limited to simulating the performance of the products they sell. The

analytical techniques and assumptions used to develop these models are not known.

7/30/2019 10.1.1.32.165

17/161

3

Also, since most air-conditioner manufacturers outsource their components, the

components are not optimized in the context of the entire system. There have not been

very many recent developments in fundamental heat exchanger modeling. Recent studies

have focused on enhanced fin and wet coil modeling which are not pertinent to flat plate

condenser optimization. Most of the recent heat exchanger studies for air conditioning

applications have focused on the effects of enhanced fin and wet coils. These studies

have not integrated the heat exchanger models in a complete system. Several component

models were reviewed for this study and will be discussed in the chapters where the

component models are developed.

In the past ten years, there have been a few studies that have used modern

technology to evaluate cooling equipment on a system basis. Beans v developed a

computer simulation of a refrigeration cycle using R-12. The program requires the inlet

air properties, cooling load, heat exchanger working pressures and some compressor

characteristics to determine the heat exchanger UA and effectiveness, outlet air

properties, and COP, and the free compressor variables. Using the heat exchanger and

compressor characteristics, the program can also find the COP for off-design inlet air

conditions. Haselden and Chen created a simulation program for air-conditioning

systems focusing on the effects of different refrigerant mixtures. vi This program will

predict the system COP, compressor size, required heat exchanger areas, relevant

temperatures, pressures, and flows. Klein and Reindl have investigated the effects of heat

exchanger allocation between the evaporator and condenser on system performance.vii

The condenser and evaporator are modeled as counterflow heat exchangers, neglecting

7/30/2019 10.1.1.32.165

18/161

4

the superheated and subcooled sections. They also assume the air-side heat transfer

coefficients and fan powers are equal for the condenser and evaporator. Chen et. al have

studied the effect of cooling load on COP using finite-time heat transfer analysis for

steady flow Carnot and Brayton refrigeration cycles.viii They later expanded this study to

include nonisentropic compression and expansion. ix

Purpose

This purpose of this study was to develop an optimization methodology and

software for the detailed design of a finned-tube condenser heat exchanger coil in a

residential air-conditioning unit using the Engineering Equation Solver (EES) software.

The superheat, saturated, and subcool portions of the heat exchanger have been modeled

separately and in detail using appropriate pressure drop and fundamental heat transfer

equations for both the air-side and refrigerant-side of the heat exchangers. The study uses

accurate refrigerant property data for R-22, but can easily be modified to accommodate

other refrigerants. The compressor, condenser fan, and evaporator components of the

cycle are also separately modeled. The cooling output and electrical input for the

compressor and fans have been calculated for various ambient temperature conditions.

Ambient temperature weighting factors used by the U.S. Department of Energy

are used to determine the seasonal coefficient of performance (COP) of the system. The

seasonal COP is the figure-of-merit used to optimize the condenser face velocity, tube

7/30/2019 10.1.1.32.165

19/161

5

diameter, fin spacing, tube circuiting, number of rows and refrigerant charge. Because

there are tradeoffs between capital and operating costs that must be considered if the

system is to succeed on the consumer market, non-dimensional cost and area factors have

been used as constraints. Based on simulation results and considering monetary or

frontal area constraints, optimal condenser configuration recommendations have been

made.

7/30/2019 10.1.1.32.165

20/161

6

CHAPTER II

RESIDENTIAL AIR CONDITIONING SYSTEMS

Before a detailed analysis of the operating conditions and geometry of the

condenser can be attempted, it is necessary to understand how the air conditioning system

works. In this chapter, the overall refrigeration cycle, system components and coefficient

of performance will be discussed.

Refrigeration Cycle

Low pressure, superheated refrigerant vapor from the evaporator enters the

compressor (State 1) and leaves as high pressure, superheated vapor (State 2). This vapor

enters the condenser where heat is rejected to outdoor air that is forced over the

condenser coils. The refrigerant vapor is cooled to the saturation temperature (State 2a),

condensed to a liquid (State 2b), and cooled below the saturation point (State 3). The

high pressure liquid is forced through an expansion valve into the evaporator (State 4).

The pressure in the evaporator is much lower than the pressure in the condenser, so the

refrigerant enters the evaporator as a liquid-vapor mix at low temperature and pressure.

The refrigerant absorbs heat from warm indoor air that is blown over the evaporator coils.

The refrigerant is completely evaporated (State 4a) and heated above the saturation

7/30/2019 10.1.1.32.165

21/161

7

temperature before entering the compressor. The indoor air is cooled and dehumidified

as it flows over the evaporator and returned to the living space. The refrigeration cycle is

shown in Figure 1 and the equipment setup is shown in

Figure 2.

s

T

4

3

2

1

4a

2b 2a

Figure 1. Thermodynamic State of Refrigerant in Refrigeration Cycle

7/30/2019 10.1.1.32.165

22/161

8

3Subcooled Saturated

Saturated

Superheated

Superheated

1

22a

4 4a

CompressorExpansion

Valve

2b

Condenser

Evaporator

Figure 2. Refrigeration Cycle Equipment

Air Conditioner Components

Compressor

The purpose of the compressor is to increase the working pressure of the refrigerant.

Compressors fall into two general categories: positive displacement, which increase the

pressure of the vapor by reducing the volume, and dynamic, which convert angular

momentum into a pressure rise and transfer it to the vaporx. Scroll type, positive

displacement compressors which dominate the residential compressor market were

considered for this study. The amount of specific work done by an ideal compressor can

be found by the energy equation:

( )12, hhw scoms =

7/30/2019 10.1.1.32.165

23/161

9

where: h = refrigerant enthalpy

For a non-ideal compressor, the actual amount of work required depends on the

efficiency.

( )12,

, hhw

wc

coms

coma==

where: c = compressor thermal efficiency

For a scroll type compressor, Klein has determined the thermal efficiency is related to the

reduced pressure and reduced temperature with the following equation. xi

rrrrrrc

TPTTPP 061.331.503.1110281.0814.325.60 22

++=

where: Pr = Pressure ratioevap

cond

rP

PP =

Tr = Temperature ratioevapsat

condsat

rT

TT

,

,=

In his paper, Klein only considers the saturated sections of the heat exchangers.

Therefore, the coefficients in the compressor efficiency correlation are based on the

saturated temperatures rather than the actual inlet and outlet temperatures to the

7/30/2019 10.1.1.32.165

24/161

10

compressor. Since pressure drops are included in the condenser model, the compressor

efficiency is based on the inlet saturation temperature and pressure in the condenser and

the outlet saturation temperature and pressure from the evaporator.

It is important to consider the volumetric efficiency in addition to the thermal

efficiency. The volumetric efficiency is the ratio of the mass of vapor that is compressed

to the mass of vapor that could be ideally compressed if the intake volume were equal to

the piston displacement and filled with evaporator exit state vapor. The volumetric

efficiency can be found by xii

= 1

v

v1

out

in

v R

Where:v = compressor volumetric efficiencyR = ratio piston/ cylinder of clearance volume to swept displacementv = specific volume

The volumetric efficiency is used to determine the mass flow rate of the refrigerant

through the compressor for a given compressor size.xiii

out

vmv

PD=&

where: PD = piston displacement

7/30/2019 10.1.1.32.165

25/161

11

Condenser

The condenser is a heat exchanger that rejects heat from the refrigerant to the

outside air. Heat exchangers come in many configurations, but finned-tube heat

exchangers are most common for residential air conditioning applications. Refrigerant

flows through the tubes and a fan forces air between the fins and over the tubes. The heat

exchangers used in this study will be the plate finned-tube type as shown in Figure 3.

Figure 3. Typical Plate Finned-Tube Heat Exchanger

When the refrigerant leaves the compressor, it enters the condenser as a

superheated vapor and leaves as a sub-cooled liquid. The condenser can be separated

into three sections: superheated, saturated, and sub-cooled. The specific heat rejected

from each section can be found by evaluating the refrigerant enthalpies at the inlets and

outlets.

7/30/2019 10.1.1.32.165

26/161

12

32,

22,

22,

hhq

hhq

hhq

bsccon

basatcon

ashcon

=

=

=

The heat transfer details on the air and refrigerant sides of the heat exchangers

will be discussed in Chapter 3.

Condenser Fan

Because natural convection will not produce sufficient airflow and heat transfer

over a reasonably sized condenser, a fan must be employed to keep the air moving.

Although the compressor uses the majority of the power consumed by the system, the fan

power must also be considered. The power required by the fan is directly related to the

air pressure drop across the condenser and the volume flow rate of the air.

conf

confrconacona

conf

APVW

,

,,,

,

=&

where: Va = air face velocity

Pa = air-side pressure drop

Afr= frontal areaf= fan efficiency

The isentropic efficiency of the combined fan and motor is taken to be 65%. The air-side

pressure drop will be discussed in Chapter 3.

7/30/2019 10.1.1.32.165

27/161

13

Expansion Valve

The expansion valve is used to control the refrigerant flow through the system.

Under normal operating conditions, the thermo-static expansion valve opens and closes to

maintain a fixed superheat exiting the evaporator. In this study, that superheat will be

held at 10 F. Because the expansion valve is designed to pass a certain volume of

refrigerant, it cannot function properly if the refrigerant is not completely condensed.

The vapor refrigerant backs up behind the valve and the condenser pressure increases

until the refrigerant vapor is condensed. When this happens, the expansion valve cannot

regulate the refrigerant superheat exiting the evaporator. Under this condition, it converts

from maintaining superheat to maintaining a saturated liquid leaving the condenser. The

energy equation shows that the enthalpy is constant across the expansion valve.

43 hh =

Evaporator

The purpose of the evaporator is to transfer heat from the room air making the air

cooler and less humid. Because the refrigerant enters the evaporator in as a liquid-vapor

mix, it is only divided into saturated and superheated sections. The analysis for the

evaporator is nearly identical to that of the condenser, but some considerations must be

made for the dehumidification process. To keep the evaporator model simple, the coil is

assumed to be dry, so the air-side heat transfer coefficient is not affected, but the specific

heat is corrected to account for condensation. Because the air flowing over the

7/30/2019 10.1.1.32.165

28/161

14

evaporator is cooled below the wet bulb temperature, some of the heat rejected by the air

results in condensing water out of the air rather than lowering the temperature. The total

enthalpy change of the air is the sum of the enthalpy change due to temperature drop, or

sensible heat, and the enthalpy change due to condensation, or latent heat.

latsenstot hhh +=

As shown in Figure 4, using the specific heat for dry air will result in exit temperatures

that are too low. By using an effective specific heat, a more accurate exit temperature

can be obtained without the complications associated with using an air-water mixture.

Temperature

Enthalpy

Effective

Dry Air

Moist Air

hlat

hsens

Figure 4. Effective Specific Heat

7/30/2019 10.1.1.32.165

29/161

15

Dividing the previous equation for enthalpy by the temperature change gives

T

h

T

h

T

h latsenstot

+

=

By definition, the specific heat is the ratio of the sensible enthalpy change to the

temperature change. Substituting and rearranging results in the following:

T

hcc latpeffp

+=,

where: cp = specific heat for dry air

cp,eff = effective specific heat

The latent enthalpy accounts for about 25% of the total enthalpy change for air flowing

over an evaporator in residential applications. The effective specific heat can be

expressed in terms of the specific heat for dry air only.

p

tot

senstot

peffp ch

h

T

hcc 33.1

75.0

25.0, =

+=

Evaporator Fan

Because the evaporator is not the focus of this study, introducing wet coils would

introduce unwelcome complications. In addition to affecting the heat transfer, wet coils

7/30/2019 10.1.1.32.165

30/161

16

also affect the air-side pressure drop. Although there are correlations available to find the

pressure drop over wet coils, this is not the only issue. After the air flows over the

evaporator, it enters a series of ducts that return it to the inside living space. Because the

power required depends on the losses in the ducts which will change from case to case,

the default used by the Air-conditioning and Refrigeration Institute test standard of 365

W per 1000 cfm of airxiv will be used.

Coefficient of Performance

The coefficient of performance (COP) measures the efficiency of the entire

system. It is the ratio of the heat absorbed by the evaporator to the amount of electrical

energy used by the mechanical components, i.e. the compressor and two fans.

evapfconfcom

e

WWW

QCOP

,,&&&

&

++=

Seasonal COP

The American Refrigeration Institute has determined that the frequency

distribution of temperatures over the summer cooling season is roughly the same across

the country. However, in warmer, southern climates, there are more cooling load

hours, which are defined as the hours when the temperature is above 65 F, per year than

7/30/2019 10.1.1.32.165

31/161

17

in cooler climates. In Atlanta, for example, the number of cooling load hours is

approximately 1300 hours per year, while it is only about 700 hours per year in

Cleveland, OH. Of these hours, the outside temperature will be between 80 F and 84 F

approximately 16.1% of the cooling season in either city. Table 1 shows the distribution

of the cooling load hours.

BinNumbe

r

Temperature

Range (F)

Representative

Temperature

(F)

Fraction of TotalTemperature

Hours

1 65-69 67 0.214

2 70-74 72 0.231

3 75-79 77 0.216

4 80-84 82 0.161

5 85-89 87 0.104

6 90-94 92 0.052

7 95-99 97 0.018

8 100-104 102 0.004

Table 1. Distribution of Cooling Load Hoursxv

The COP changes with the outside air temperature and the overall COP, or

seasonal COP, for an air conditioner depends on the temperatures at which the appliance

runs over an entire year. According to the ANSI/ASHRAE standard,xvi the seasonal COP

for a single speed, single compressor unit is found by:

7/30/2019 10.1.1.32.165

32/161

18

( )

( )

=

8

j

j

8

j

je

seas

TE

TQ

COP

&

&

where: Qe(Tj) = adjusted evaporator capacity at ambient temperature Tj

E(Tj) = adjusted electrical power demand at ambient temperature Tj

( )j = values corresponding to temperature bin j (from Table 1)

The adjusted evaporator capacity and adjusted electrical power demand are based on the

cooling load factor and the fraction of total temperature hours.

( ) ( )jjssje nTQCLFTQ = &

( )( )

PLF

nTECLFTE

jjss

j

=

&&

where: CLF = cooling load factor

Qss(Tj) = steady state evaporator capacity at Tj

nj = fraction of total temperature hours (from Table 1)

Ess(Tj) = steady state electrical power demand at Tj

PLF = part load factor

The part load factor takes into account system cycling. For this study, the system

7/30/2019 10.1.1.32.165

33/161

19

cycling is neglected so the part load factor is equal to 1. The cooling load factor is

defined:

( )( )

( ) ssj

ssj

jss

j

QTBL1CLF

QTBLTQ

TBLCLF

&

&

&

>=

=

where: BL = building load

The building load given by:

( )( )

SF

TQ

65T

35jTBL ODss

OD

j

&

=

where: TOD = outdoor design temperature in Fahrenheit (in this case 95 F)SF = size factor

The size factor determines the amount of over or undersizing of the system and is 1 for

this study because the system designed to meet the requirements at 95 F.

7/30/2019 10.1.1.32.165

34/161

20

CHAPTER III

HEAT EXCHANGERS

Because this study focuses on optimizing the condenser geometry and operating

conditions, the characteristics of heat exchangers must be explored more thoroughly than

other components. In this chapter, the heat transfer and pressure drop models specifically

related to plate finned heat exchangers will be discussed. Plate finned heat exchangers

are made up of in-line or staggered tube bundles that are held in place by continuous,

rectangular fins. For this study, staggered, copper tubes are coupled with aluminum fins.

Geometry

It is important to understand the terminology used to describe the geometric

parameters of the heat exchangers used in this study. The term tubes per circuit is the

number of parallel passages the refrigerant mass flow rate is divided among. If the mass

flow rate is 100 lbm/hr and there is one tube per circuit, 100 lbm/ hr of refrigerant will

pass through that tube. If there are 2 tubes per circuit, then 50 lbm/hr of refrigerant will

flow through each tube. The number of parallel circuits is used to determine the number

of tubes in each row. The number of rows refers to the number of tube rows in the

direction normal to air flow. If the number of parallel circuits is set to 12, and the

number of tubes per circuit is 2, then there will be a total of 24 tubes in each row. The fin

7/30/2019 10.1.1.32.165

35/161

21

pitch is the number of fins per unit length along the axial direction of the tubes. These

parameters and the overall dimensions of the heat exchanger are illustrated in Figure 5

and Figure 6. The model used to determine the air-side heat transfer coefficient depends

on the layout of the tubes, but not on the temperature of the refrigerant which would be

affected by circuiting. The only factors pertinent to the refrigerant side models affected

by the circuiting are the mass flow rate in each tube and the length associated with each

circuit. The layout in Figure 6 meets the requirements of the models and is easy to

conceptualize.

7/30/2019 10.1.1.32.165

36/161

22

Figure 5. General Heat Exchanger Dimensions

7/30/2019 10.1.1.32.165

37/161

23

Figure 6. Layout of Heat Exchanger Geometry Parameters

NTU-Effectiveness Relations

For any heat exchanger, the total heat rejected from the hot fluid, in this case,

refrigerant, to the cold fluid, air, is dependent on the heat exchanger effectiveness and the

heat capacity of each fluid.

( )icih TTCQ ,,min =

7/30/2019 10.1.1.32.165

38/161

24

where: = effectivenessCmin = smaller of heat capacities Ch and CcTh,i = inlet temperature of hot fluid

Tc,i = inlet temperature of cold fluid

The heat capacity, C, the extensive equivalent of the specific heat, determines the

amount of heat a substance absorbs or rejects per unit temperature change.

pcmC=

where: m = masscp = specific heat

The amount of air flowing over each section of the condenser is assumed to be

proportional to the tube length associated with that section. For example:

to t

sat

tota

sata

L

L

m

m=

,

,

The effectiveness is the ratio of the actual amount of heat transferred to the

maximum possible amount of heat transferred.

maxQ

Q

&

&

=

7/30/2019 10.1.1.32.165

39/161

25

The equations used to determine the effectiveness depend on the temperature

distribution within each fluid and on the paths of the fluids as heat transfer takes place, ie.

parallel-flow, counter-flow or cross-flow. In typical condensers and evaporators, the

refrigerant mass flow is separated into a number of tubes and does not mix. As the air

flows through the fins, the plates prevent mixing and air at one end of the heat exchanger

will not necessarily be the same temperature as air at the other end. For a cross-flow heat

exchanger with both fluids unmixed, the effectiveness can be related to the number of

transfer units (NTU) with the following equation:

( ) ( )[ ]{ }

= 1exp1exp1 78.022.0 NTUCNTU

Cr

r

where: Cr = heat capacity ratio

max

min

C

CCr =

In the saturated portion of the condenser, the heat capacity on the refrigerant side

approaches infinity and the heat capacity ratio goes to zero. When Cr=0, the

effectiveness for any heat exchanger configuration is:

7/30/2019 10.1.1.32.165

40/161

26

( )NTU= exp1

The NTU is a function of the overall heat transfer coefficient.

minC

UANTU=

The overall heat transfer coefficient, U, takes into consideration total thermal

resistance to heat transfer between two fluids. Even though the convective heat transfer

coefficients may be different on the air and refrigerant sides of the heat exchanger, the

UA product is the same on either side. This is because all of the heat taken from the

refrigerant must be transferred to the air.

aarr AUAUUA

111==

where: A = total heat transfer area( )r= refrigerant side( )a = airside

Taking all of the thermal resistances into account produces the following

expression for the UA product:

7/30/2019 10.1.1.32.165

41/161

27

rrrsrrs

rf

w

aas

af

aaas AhA

RR

A

R

AhUA ,,

,

,

,

,

111

+

++

+=

where: R= fouling factor

Rw= wall resistances = surface efficiencyh = heat transfer coefficient

Since there are no fins on the refrigerant side of the tubes, the refrigerant side surface

efficiency is 1. Neglecting the wall resistance, Rw, and the fouling factors, R, the overall

heat transfer coefficient reduces to

1

,

11

+=

rraaas AhAhUA

The methods for finding the heat transfer coefficients will be discussed later in

this chapter. To find the overall surface efficiency for a finned tube heat exchanger, it is

first necessary to determine the efficiency of the fins alone. The total air side surface

efficiency is given by:

( )f

o

f

sA

A = 11

where: s = surface efficiencyAf = total fin surface areaAo = total air side surface area, tube and fins

f= fin efficiency

7/30/2019 10.1.1.32.165

42/161

28

The fin efficiency, f, for a circular fin is a function of m, re and .

( )

( )

e

e

fmr

mrtanh=

For a plate fin heat exchanger with multiple rows of staggered tubes, the plates

can be evenly divided into hexagonal shaped fins as shown in Figure 7.

Figure 7. Layout of Hexagonal Fins

7/30/2019 10.1.1.32.165

43/161

29

Schmidtxvii analyzed hexagonal fins and determined that they could be treated like

circular fins by replacing the outer radius of the fin with an equivalent radius. The

empirical relation for the equivalent radius is given by

( ) 213.027.1 = r

re

where: re=equivalent radius of fins

r = outside tube radius

The coefficients and are defined as:

r

Xt

2=

212

2

4

1

+= tL

t

XX

X

where: Xl = tube spacing in direction parallel to air flowXt = tube spacing normal to air flow

Once the equivalent radius has been determined, the equations for standard

circular fins can be used. For the fins in this study, the length is much greater than the

thickness, so a parameterm can be expressed as:

7/30/2019 10.1.1.32.165

44/161

30

21

2

=

kt

hm a

where: ha = air side heat transfer coefficient

k = conductivity of fin materialt = thickness of fins

For circular tubes, a parameter is defined as

+

=

r

r

r

r ee ln35.011

Refrigerant Side Models

Single Phase Heat Transfer Coefficient

To find the single phase heat transfer coefficient, the standard heat transfer

equations and the experimental work of Kays and London were considered. For constant

surface heat flux in the laminar regime, the Nusselt number is a constant.

Nu=4.36

In the turbulent region, the Dittus-Boelter equation holds for fully developed flow in

circular tubes with moderate temperature differences. For refrigerant cooling in the

condenser, the Dittus-Boelter equation is:

7/30/2019 10.1.1.32.165

45/161

31

NuD D

= 0023 0 8 0 3. Re Pr . .

where: ReD = Reynolds number based on diameterPr = Prandtl number

This equation has been confirmed by experimental data for the range:

0.7 Pr160

ReD 10,000

L/D 10

where: L = tube length in single phase region

In the subcooled portion of the condenser, the temperature difference at the inlet

and exit is usually less than 20F, but in the superheated portion, the inlet and exit

temperature can vary by as much as 90F. The temperature differential between the air

flowing over the tubes and, as a result, the inner surface of the tubes and the refrigerant is

also much greater. Under these conditions, the Dittus-Boelter equation does not produce

an accurate value for the heat transfer coefficient. Sieder and Tatexviii have developed a

correlation equation for large property variations based on the mean fluid temperature

and the wall surface temperature.

7/30/2019 10.1.1.32.165

46/161

32

NuD Dm

s

=

0027 0 8 1 3

0 14

. Re Pr ..

where: = viscosity( )m = evaluated at mean fluid temperature

( )s = evaluated at surface

The viscosity s is evaluated at the surface and all other properties are evaluated at the

mean fluid temperature.

Kays and London use empirical data taken from a variety of refrigerants in

circular tubes under different conditions. Unlike the other correlations, Kays and London

have established equations in the transition region. The heat transfer coefficient was

related to the Stanton number, St. The Stanton number is defined by the following:

p

cG

hSt=

St a bPr Re2 3 =

where: cp= specific heat

The coefficients a and b are based on the flow regime.

7/30/2019 10.1.1.32.165

47/161

33

Laminar Re < 3,500 a=1.10647

b=-0.78992

Transition 3,500 < Re < 6,000 a=3.5194 x 10-7b=1.03804

Turbulent 6,000 < Re a=0.2243

b=-0.385

In the laminar and early transition regions, the Kays and London heat transfer

coefficient is lower than the others, but is it is higher in the turbulent region. The Dittus-

Boelter and Sieder and Tate equations assume that the pipe is smooth which would

explain this result. Because the Kays and London relation is based on data taken from

heat exchangers similar to those studied here and because transitional flow has been

addressed, this relation will be used in the simulation.

Condensation Heat Transfer Coefficient

Condensation heat transfer correlations by Shah and Traviss, Rohsenow and

Baronxix were considered for this study. Patexx showed that the results of Shah and the

results of Traviss et al. were not significantly different, however, Traviss model only

applies to annular flow regime while Shahs relation is good in all flow regimes.

Traviss model also requires an iterative scheme while Shahs method is very easy to use.

It is a simple dimensionless correlation which has been verified over a large variety of

experimental data. This model has a mean deviation of about 15% and has been verified

7/30/2019 10.1.1.32.165

48/161

34

for many different condensing fluids, tube sizes and tube orientations. For any given

quality, the two-phase heat transfer coefficient is

( )( )

+=

38.0

04.076.08.0 18.3

1r

LTPp

xxxhh

where: hTP= two phase heat transfer coefficient

x= quality

hL= liquid only heat transfer coefficientpr= reduced pressure

By integrating the two-phase heat transfer coefficient over the length, the mean

two-phase heat transfer coefficient can be determined.

( )( )

( )

+

= 2

138.0

04.076.08.0

12

18.31

L

Lr

LTPM dL

p

xxx

LL

hh

If the quality varies linearly with length which is consistent with constant heat

transfer per unit length, hTPM can be approximated by:

( )

( )2

1

76.2

04.0

76.1

8.3

8.1

1 76.276.1

38.0

8.1

12

x

xr

L

TPM

xx

p

x

xx

hh

+

=

7/30/2019 10.1.1.32.165

49/161

35

For complete condensation, that is x varies from 1 to 0, the mean two-phase heat

transfer coefficient reduces to:

+=

38.0

09.255.0

r

LTPMp

hh

Evaporative Heat Transfer Coefficient

The expression for the average evaporative two-phase heat transfer coefficient is

taken from Tongxxi. This relationship assumes a constant temperature differential

between the wall and the fluid along the length of the pipe.

( )325.0325.0

075.0375.04.08.0

2.00186875.0

ie

ie

l

v

v

l

l

ll

l

levap

xx

xx

k

CpG

D

kh

=

where: D = tube diameter

G = mass flux

k = conductivity

( )l = properties evaluated at saturated liquid stage

( )v = properties evaluated at saturated vapor stage

( )e = properties evaluated at exit

( )i = properties evaluated at inlet

7/30/2019 10.1.1.32.165

50/161

36

Pressure Drop in Straight Pipe

The pressure drop in the superheated and subcooled portions of the condenser can be

found easily by applying the standard pipe pressure drop equation.

pf G L

=2

where: f = friction factor

= density

The friction factor, f, for circular pipe depends on the Reynolds number where the

turbulent expression is taken for the transition region.

000,2ReRe

51.2

7.3log2

1

000,2ReRe

64

211021>

+=

7/30/2019 10.1.1.32.165

51/161

37

dP

dz

dP

dz

dP

dz

dP

dzf g m

=

+

+

The frictional component is found using the following equation:

( ) [ ]dP

dz

G

g D G Df

v

v

c

v

v

tt =

+

2

0.2

0.523

2

0 09 1 2 85

. .

The momentum component is:

( ) ( ) ( )dP

dz

G

g

dx

dz x x x xm c v

v

l

v

l

v

l

=

+

+

2 1 3 2 3

2 1 2 1 2 2 1

Unfortunately, it is very difficult to predict the variation of quality with length, dx/dz so a

linear profile is assumed for simplicity, which is consistent with constant heat transfer per

unit length. The gravitational pressure drop for horizontal tubes is zero. Hillerxxii

integrates the pressure differential over the change in quality for the frictional and

momentum losses. The frictional pressure drop in the two-phase region is reduced to

7/30/2019 10.1.1.32.165

52/161

38

( ) ( )[ ] ei

x

xf xxCxxxCxCP

86.12

3

33.22

3

8.2

2 329.0538.00288.0141.0429.02357.0 ++=

where:

Cl

v

v

l

3

0 0523 0 262

285=

.

. .

CG

C g D

v

c v

2

1 8

1

1 2

009=

. .

.

L

xxC ie

=1

The momentum pressure drop in the two-phase region integrates to:

e

i

x

xl

v

l

v

l

v

l

v

l

v

l

v

cv

m xxg

GP

+=

3231

2

32312

21

The total pressure drop is just the sum of the momentum and frictional pressure drops.

fmtot PPP +=

7/30/2019 10.1.1.32.165

53/161

39

Pressure Drop in Bends

The pressure drop in bends is found by assigning an equivalent length to each bend

based on the flow diameter and the bend radius. For two-phase flow, the method for

finding the pressure drop in bends based on Chisolmxxiii is used. The pressure drop is

calculated for liquid-only flow and correction factors are applied to determine the

approximate two-phase pressure drop. The method predicts the pressure drops for two-

phase flow in horizontal bends rather than the inclined bends found in a typical

condenser. However, the two-phase flow pattern in an inclined bend cannot be

accurately predicted and pressure gradients due to elevation changes are negligible

compared to friction pressure losses, so the horizontal bend model should be sufficient.

Since the bends are not finned and do not have air flowing over them, the heat transfer

and phase change in the bends is neglected.

The first step in computing the pressure drop is to determine the equivalent length of

the bend. The equivalent length is a function of the relative radius, rr.

rr

Drb

i

=

where: rb = radius of bendDi = inner diameter of pipe

7/30/2019 10.1.1.32.165

54/161

40

Chisolm uses the correlation by Beij to determine the equivalent length, z/D. Typical

condensers will have a relative radius between 1 and 3 which corresponds to an

equivalent length between 12 and 15 for 90 bends. The equivalent length for a 180

return bend is about twice that of a 90 bend. For this analysis, the equivalent length of

the return bend will be taken as 26. The single-phase pressure drop on a bend can be

evaluated by substituting the equivalent length for the straight pipe length in the standard

pressure drop equation.

e

bD

zGp

=

2

2

where: pb = pressure drop in bend = friction factor

The friction factor can be determined with Haalands approximationxxiv:

211.1

7.3Re

9.6log8.1

+=

D

where: = pipe roughness

For drawn copper pipes, the pipe roughness is taken to be 0.000005 ft. For two-phase

flow, the pressure drop in a bend is the product of the bend pressure drop for liquid only

and the two-phase multiplier.

7/30/2019 10.1.1.32.165

55/161

41

2

,,, loblobTPb

pp =

where: pb,lo = liquid oly bend pressure drop2 = two-phase multiplier

The two-phase multiplier2 in a bend is:

( ) ( ) ( ) ( ) ( )[ ] b lo bn n nB x x x,

2 2 2 2 2 2 21 1 1= + +

where: b2 = physical property coefficientB = bend coefficient

n = Blasius coefficient

The physical property coefficient 2 for a bend is

b

l

v

v

l

n

2 =

The Blasius coefficient n used to determine 2 and 2 is defined as

n

Lo

vo

l

v

=

ln

ln

7/30/2019 10.1.1.32.165

56/161

42

The friction factors, lo and vo, are found using Haalands approximation. In these

cases, one assumes all of the mass is flowing as either a liquid or a vapor so the mass flux

G used to find the Reynolds number will be the same, but the viscosity will depend on the

refrigerant state.

n

Lo

vo

l

v

=

ln

ln

The B coefficient for bends other than 90 is

[ ]B Bk

k

b

b

= +

1 19090,

The coefficient B90 is defined as

( )B

k R Db90

90

12 2

2

= ++.

,

7/30/2019 10.1.1.32.165

57/161

43

The recovery downstream of bends greater than 90 is assumed to be the same as 90

bends. The pressure coefficient for a 90 bend, kb,90, is used for convenience where

kz

Dbe

,90 =

Assuming homogeneous two-phase flow, the friction factor for two-phase flow is found

using the same Haalands approximation, but the Reynolds number is based on the two-

phase viscosity.

Re =GD

TP

The two-phase viscosity is a function of quality.

( ) TP v l x x= + 1

In the case of 180 bends, the kb,180 is approximately twice kb,90 so B180 reduces to

[ ]B B180 900 51 = +.

7/30/2019 10.1.1.32.165

58/161

44

Air Side Models

Heat Transfer Coefficient

The work of Rich and McQuiston were used to evaluate the air-side convective heat

transfer coefficient for a plate fin heat exchanger with multiple rows of staggered tubes.

The condenser coils are assumed to be dry. The heat transfer coefficient is based on the

Colburn j-factor which is defined as:

j St= Pr2 3

Substituting the appropriate values for the Stanton number gives this relationship for the

convective heat transfer coefficient, h.

hj c G

a

p=

max

Pr2 3

where: Gmax = mass flux through minimum flow area

Gm

Aair

max

min

&=

For cases in this study, the minimum flow area is

( ) ( )( )( )DcirctpcHtFPWA #1min =

7/30/2019 10.1.1.32.165

59/161

45

where: W = width of heat exchanger

FP = fin pitch

H = height of heat exchanger

tpc = tubes per circuit

#circ = number of parallel circuits

McQuiston found the j-factor for a 4 row finned-tube heat exchanger to fit a linear model

based on the parameter JP.

j JP460 2675 1325 10= + . .

and

JP AAD

o

t

=

Re ..

0 4

0 15

The Reynolds number is based on the outside diameter of the tubes, Do, and the

maximum mass flux Gmax. The heat transfer coefficient for heat exchangers with four or

less rows can be found using the following correlation:

( )( )

j

j

nn L

L4

1 2

1 2

1 1280

1 1280 4=

Re

Re

.

.

7/30/2019 10.1.1.32.165

60/161

46

ReL is based on the row spacing.

Remax

L

LG X=

Pressure Drop

The work of Richxxv concludes that the air side pressure drop can be separated into

two components: the pressure drop due to the tubes and the pressure drop due to the fins.

fttot ppp +=

where: pt = pressure drop due to tubespf = pressure drop due to fins

The pressure drop due to the fins can be expressed:

c

f

mffA

AGvfp

2

2

max=

where: ff = fin friction factor

vm = mean specific volumeGmax = mass velocity through minimum area

Af = fin surface areaAc = minimum free-flow cross sectional area

7/30/2019 10.1.1.32.165

61/161

47

In experimental tests, Rich found that the friction factor depends on the Reynolds

number, but is independent of fin spacing. For fin spacing between 3 and 14 fins per

inch, the fin friction factor is

5.0Re70.1

= lff

where the Reynolds number is based on the transverse (in the direction of air flow) tube

spacing.

ll

GX=Re

To find the pressure drop over the tubes, the relationships developed by Zukauskas and

Ulinskasxxvi are used. The pressure drop over the banks of plain tubes is:

zG

Eup ct2

2

=

where: Euc = corrected Euler number

z = number of rows

The corrected Euler number is:

7/30/2019 10.1.1.32.165

62/161

48

EuCkEu zc 1=

where: Eu = Euler numberk1 = staggered array geometry factorCz = average row correction factor

The Euler number is related to the tube friction factor and depends on the

Reynolds number and the tube geometry. For staggered, equilateral triangle banks with

many rows, the Euler number is related to the Reynolds number by a fourth order inverse

power series.

432 ReReReRe

utsrqEu ++++=

The coefficients, q, r,s, t, and u are dependent on the parametera, the ratio of the

transverse tube spacing to tube diameter, and the Reynolds number. The coefficients for

distinct values ofa determined by Zukauskas and Ulinskas from experimental data are

summarized in Table 2.

7/30/2019 10.1.1.32.165

63/161

49

a Reynolds number q r s t u

1.25 3< Re < 103 0.795 0.247 x103

0.335 x103

-0.155 x104

0.241 x 104

103< Re < 2 x 106 0.245 0.339 x104

-0.984 x107

0.132 x1011

-0.599 x1013

1.5 3< Re < 103 0.683 0.111 x 103 -0.973 x102

0.426 x 103 -0.574 x 103

103< Re < 106 0.203 0.248 x

104

-0.758 x

107

0.104 x 1011 -0.482 x

1013

2.0 7< Re < 102 0.713 0.448 x

102

-0.126 x

103

-0.582 x

103

0

102< Re < 104 0.343 0.303 x103

-0.717 x105

0.88 x 107 -0.38 x 109

104< Re < 2 x 106 0.162 0.181 x104

0.792 x108

-0.165 x1013

0.872 x1016

2.5 102< Re < 5 x 103 0.33 0.989 x102

-0.148 x105

0.192 x 107 0.862 x 108

5 x 103< Re < 2 x

106

0.119 0.849 x

104

-0.507 x

108

0.251 x

1012

-0.463 x

1015

Table 2. Euler number coefficients for inverse power series

For non-equilateral triangle tube bank arrays, the staggered array geometry factork1,

must be used as a correction. The staggered array geometry factor is dependent on the

Reynolds number, a and b, the ratio of tube spacing in the direction normal to the air flow

and the tube diameter. The equations fork1 are found in Table 3.

7/30/2019 10.1.1.32.165

64/161

7/30/2019 10.1.1.32.165

65/161

51

The equations for the individual row correction factors are given in Table 4.

Re z* cz

10

7/30/2019 10.1.1.32.165

66/161

52

CHAPTER IV

OPTIMIZATION OF OPERATING PARAMETERS

When comparing the performance of air conditioning systems, it is not valid to assert

that one condenser geometry is better than another if the operating conditions are not

optimized for each configuration. The operating parameters considered for this study are

refrigerant charge and air face velocity over the condenser. Because the performance of

an air-conditioner varies with ambient temperature, design conditions were established at

95 F to provide a fair basis for comparison. These conditions are summarized in Table

5.

Ambient Temperature 95 F

Evaporator Capacity 30,000 Btu/hr

Evaporator Saturation Temperature 45 F

Superheat in Evaporator 10 F

Table 5. Air-Conditioner Design Conditions

To see the effects the operating parameters have on the seasonal COP, a base case

condenser and evaporator coil pair typical for this application was selected. All of the

characteristics of the condenser and all but the width of the evaporator were specified.

These dimensions are given in Table 6.

7/30/2019 10.1.1.32.165

67/161

53

Dimension Condenser Evaporator

Tube spacing (in x in) 1.25 x 1.083 1.00 x 0.625

Tube inner diameter (in) 0.349 0.349

Tube outer diameter (in) 0.375 0.375

Frontal area (ft2) 7.5 n/a

Finned width (ft) 3 n/a

Finned height (ft) 2.5 1.5

Depth (in) 3.25 2.5

Fin pitch (fin/ in) 12 12

# rows 3 4

# circuits 12 9

Tubes per circuit 2 2

Table 6. Base Case Condenser and Evaporator Characteristics

The evaporator frontal area depends on the design conditions and is virtually

independent of operating conditions. In Figure 8, the evaporator frontal area remains

constant for different air velocities and refrigerant charges. The refrigerant charges are

specified by the number of degrees subcool, Tsc, in the condenser at the design

conditions.

7/30/2019 10.1.1.32.165

68/161

54

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12 14 16 18

Air Velocity (ft/s)

EvaporatorFrontalArea(ft

Tsc=5

Tsc=10

Tsc=15

Tsc=20

Figure 8. Effects of Operating Conditions on Evaporator Frontal Area

Subcool and Seasonal Effects

The refrigerant charge is the mass of refrigerant in the system necessary to provide a

specified amount of subcool in the condenser at the design conditions. The relationship

between the specified subcool and refrigerant system mass is demonstrated in Table 7 for

air velocity of 8 ft/s.

Degrees

Subcool

@ 95 F(F)

Mass of

Refrigerantin System

(lbm)

5 3.50

10 4.38

15 5.49

20 6.45

Table 7. Mass of Refrigerant in Air-Conditioner for Different Subcool Specifications

7/30/2019 10.1.1.32.165

69/161

55

The effects of a fixed refrigerant charge must be considered with varying ambient

temperature. As the outdoor air temperature drops, the condensing temperature also

drops and the enthalpy of the refrigerant entering the evaporator is lower. This means

that the inlet quality is lower and more of the refrigerant in the evaporator is in the liquid

state. Since the total mass of refrigerant in the system is held constant, the mass of the

refrigerant in the evaporator increases and the mass of refrigerant in the condenser

decreases as the ambient outdoor temperature decreases. When the mass of the

refrigerant in the condenser drops, the volume fraction of the condenser that is filled with

vapor must increase. If the mass of refrigerant in the condenser drops to the point where

the refrigerant is not completely condensed when it enters the valve, the valve goes wide

open and cannot maintain a fixed superheat in the condenser. Since a negligible amount

of vapor can pass through the expansion valve orifice, a saturated state is forced at the

valve entrance and the subcool in the condenser will be fixed at zero. The superheat in

the evaporator then varies from the specified 10 F. This condition occurs at higher and

higher ambient temperatures as the amount of subcool specified at 95 F decreases as

shown in Figure 9.

7/30/2019 10.1.1.32.165

70/161

56

.

0

5

10

15

20

25

40 50 60 70 80 90 100 110

Ambient Temperature (F)

CondenserSubcool(F)

Tsc=5

Tsc=10

Tsc=15

Tsc=20

Figure 9. Condenser Subcool at Varying Ambient for Different Refrigerant Charges

Because the seasonal COP depends on the performance of the system over a range of

temperature, it is important that the refrigerant charge is high enough to ensure there be

subcool in the condenser at the lower temperatures. When the subcool disappears, the

superheat in the evaporator increases leading to lower density vapor at the compressor

inlet. This lower density vapor causes the mass flow rate to drop significantly, lowering

the evaporator capacity and the COP as shown in Figure 10.

7/30/2019 10.1.1.32.165

71/161

57

385

390

395

400

405

410

415

420

425

50 60 70 80 90 100 110

Ambient Temperature (F)

MassFlowRate(lbm/h

r)

28500

29000

29500

30000

30500

31000

31500

32000

32500

33000

EvaporatorCapacity

(Btu/hr)

Mass Flow Rate Evaporator Capacity

Subcool = 0

Figure 10. Effect of Ambient Temperature on Evaporator Capacity and Mass Flow Rate

The optimum refrigerant charge will be different for each ambient temperature,

but the COP will remain relatively constant at every temperature as long as the subcool is

specified between 10 F and 15 F at 95 F ambient. In the range of 5-20 subcool, the

seasonal COP is within 0.5% of the COP at 83 F ambient. The trends in COP over the

season and at different ambient temperatures are plotted over a range of subcools in

Figure 11.

7/30/2019 10.1.1.32.165

72/161

58

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

0 5 10 15 20 25

Subcool @ 95F

COP

Seasonal

Tamb=77

Tamb=82

Tamb=83

Tamb=87

Figure 11. Trends in Seasonal COP vs. COPs at Other Temperatures

Effect of Varying Air Velocity

As expected, for a fixed amount of subcool at 95, there is an air velocity that

produces the highest seasonal COP. The COP varies exhibits a maximum with the air

velocity for any subcool as shown in Figure 12. For subcools ranging from 5 to 20, the

optimum air velocity is somewhere between 7 ft/s and 10 ft/s. In this range, the seasonal

COP is insensitive to the air velocity; for any refrigerant charge, it varies by less than 1%.

7/30/2019 10.1.1.32.165

73/161

59

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

0 2 4 6 8 10 12 14 16 18

Air Velocity (ft/s)

Tsc=5

Tsc=10

Tsc=15

Tsc=20

Figure 12. Effect of Air Velocity on Seasonal COP for Different Subcool Conditions

Because the seasonal COP varies so little with air velocity, it is difficult to

pinpoint the optimum air velocity for each subcool within more than 0.1 ft/sec. In

practice, this is acceptable because the air speed cannot be specified to a such high

tolerance.

Since the fan work increases proportionally with the cube of the velocity, it does

not initially make sense that the COP would not be affected. However, in this range, as

the fan work is increasing, the compressor work is decreasing by roughly the same

amount, as demonstrated in Figure 13. As the air velocity increases, the condensing

temperature decreases, and the inlet enthalpy to the evaporator also decreases. When this

7/30/2019 10.1.1.32.165

74/161

60

happens, the mass flow rate of refrigerant needed to maintain the design evaporator

capacity drops decreasing the compressor work. Since condensing temperature of the

refrigerant cannot be lower than the air inlet temperature, there is a minimum compressor

work. As the air velocity increases beyond the optimum range, the fan work will grow

exponentially and the decrease in compressor work does not compensate for it.

Air Velocity (ft/s)

Work(BTU/hr

Compressor

Condenser Fan

Total

Figure 13. Effect of Air Velocity on Compressor and Condenser Fan Work

Effect on Cost Factor

Changing the air velocity and refrigerant charge will slightly affect the cost of the

system because different compressors or fans should be used for different heat

exchangers. This would involve cost studies of compressors and fans which is outside

7/30/2019 10.1.1.32.165

75/161

61

the scope of this study. They will be excluded from the cost factor calculation, but the

designer should be aware of the possible effects.

7/30/2019 10.1.1.32.165

76/161

62

CHAPTER V

EFFECTS OF GEOMETRY WITH FIXED COST

When designing a heat exchanger for maximum system COP, the two most

important constraints are the cost of the exchanger and the amount of frontal area it takes

up. It is not possible to keep both the frontal area and cost constant while only varying

one geometry factor, but simultaneously changing more than one variable would make it

difficult to determine the effect each variable has on the system. To examine the tradeoffs

between frontal area and cost, cases with fixed cost and fixed frontal area were

considered.

To compare the relative frontal area of each condenser configuration, the area

factor parameter is defined as the ratio of the frontal area of the test configuration to the

frontal area of the base configuration.

baseAreaFrontal

AreaFrontalAF=

A similar factor, the cost factor, is used to compare the cost of condenser-

evaporator configurations:

7/30/2019 10.1.1.32.165

77/161

63

base

Cost

CF Cost=

The cost of the heat exchanger is largely determined by the cost of the materials xxvii so the

cost factor of each configuration is taken as:

( ) ( ), , , ,Cucon Cu evap Cu Cu Al con Alevap Al Al Cost Vol Vol Cost Vol Vol Cost = + + +

The costs of the materials are summarized in Table 8.

Material Cost ($/lbm)

Copper 0.8

Aluminum 0.7

Table 8. Material Costsxxviii

The heat exchanger cost factor of the base configuration is $35.88. Although the piston

displacement will change slightly for each configuration, it varies from the base case by

no more than 3% under most conditions, so the cost variations of the compressor will be

ignored.

7/30/2019 10.1.1.32.165

78/161

64

Number of Rows

Although altering any geometry factor will change the frontal area of the

condenser with the cost factor fixed, it is easiest to conceptualize this by changing the

number of rows. For these tests, the height of the condenser will remain fixed, but the

width is free to change. Intuitively, a heat exchanger with no bends and the largest

frontal area possible would provide the best performance.

Figure 14verifies this notion.

3.70

3.75

3.80

3.85

3.90

3.95

4.00

4.05

4.10

4.15

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Number of Rows

SeasonalCOP

Fixed Parameters

Fin Pitch =12 fpi

Tube Diameter = 3/8

Tubes/ Circuit = 2

3.703.753.803.853.903.954.00

4.054.104.15

Fixed Parameters

Fin Pitch =12 fpi

Tube Diameter = 3/8

Tubes/ Circuit = 2

Figure 14. Number of Rows vs. Seasonal COP with Fixed Cost

As the number of rows increases, the number of bends also increases. Obviously,

fewer bends in the tubing means less frictional losses and less compressor work. By

7/30/2019 10.1.1.32.165

79/161

65

having the air only flow over one row of tubing, the temperature differential between the

air and the refrigerant is kept to a maximum, decreasing the refrigerant mass flow rate

and compressor work. The refrigerant side pressure drop and compressor power as

functions of the number of rows are plotted in Figure 15.

6100

6200

6300

6400

6500

6600

6700

0 1 2 3 4 5

Number of Rows

CompressorPower(Btu/hr)

0

5

10

15

20

25

30

R22PressureDrop(psi)

Compressor

Power

R22PressureDrop

Figure 15. Number of Rows vs. Compressor Power and Refrigerant Pressure Drop

While this is the main cause for the COP increase, the fan power also decreases as

with the number of rows. The pressure drop will go down as the depth of the air passage,

which is controlled by the number of rows, decreases. For larger number of rows, the

optimal air velocity is higher. This coupled with the increased pressure drop causes the

fan power to nearly double from 1-row to 4-rows as shown in Figure 16. However, if the

velocity remains constant, the fan power does not change.

7/30/2019 10.1.1.32.165

80/161

66

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 1 2 3 4 5

Number of Rows

Air-SidePressureDrop(Psi)

0

50

100

150

200

250

300

350

400

FanPower(Btu/hr)

Pressure Drop

Fan Power

Figure 16. Rows vs. Air Side Pressure Drop and Fan Power for Fixed Cost at 83 F

Fin Pitch

Keeping the cost factor and all design parameters except the frontal area the same

as the base case, the model was run for fin pitches between 8 and 14 fins per inch (fpi).

The maximum seasonal COPs and area factors based on fin spacing are summarized in

Table 9 and shown graphically in Figure 17. Based on these results, the fin spacing has

almost no effect on the seasonal COP or the optimal operating conditions. As seen in

Figure 18, the optimum velocity, for every fin pitch is between 8.5 and 9.5 ft/sec. This

figure is based on 10 F subcool at 95 F outdoor air temperature. Figure 18 also shows

that the seasonal COP varies shows an optimum with fin pitch, but the variation is

marginal.

7/30/2019 10.1.1.32.165

81/161

67

Fin Pitch(fpi)

SeasonalCOP

AreaFactor

8 3.839 1.258

10 3.856 1.114

12 3.862 1.000

14 3.860 0.906

Table 9. COPs and Area Factors Based on Fin Pitch

Effect of Fin Pitch on Seasonal COP at

Optimum Operating Conditions

3.835

3.840

3.845

3.850

3.855

3.860

3.865

6 8 10 12 14 16

Fin Pitch (fins/in)

Sea

sonalCOP

Fixed Parameters

# Rows = 3Tube Diameter = 3/8

Tubes/ Circuit = 2

3.8453.8503.8553.8603.865 Fin Pitch (fins/in)

Fixed Parameters

# Rows = 3Tube Diameter = 3/8

Tubes/ Circuit = 2

Figure 17. Seasonal COPs for Different Fin Pitches at Optimum Operating Conditions with Fixed

Cost

7/30/2019 10.1.1.32.165

82/161

68

3.77

3.78

3.79

3.80

3.81

3.82

3.83

3.84

3.85

3.86

3.87

6 7 8 9 10 11 12

Air Velocity (ft/sec)

SeasonalCOP

FPI=8

FPI=10

FPI=12

FPI=14

Figure 18. Effect of Fin Pitch on Seasonal COP at Different Air Velocities

The maximum COP will occur when fin spacing is increased just before the point

where the airside pressure drop causes the fan work to increase faster than the compressor

work is decreasing, in this case, 12 fins per inch. As long as the operating conditions are

kept in the recommended range of 10-15 degrees subcool and 7-10 ft/sec air face

velocity, the maximum and minimum seasonal COPs will only differ by about 1.6%.

The maximum occurs with 12 fins per inch, 8.8 ft/sec air face velocity and 10 F subcool.

The minimum occurs with 8 fins per inch, 10-ft/sec air face velocity, and 15 F subcool.

Although the fin spacing does not dramatically affect the COP, it does affect the

packaging size. If a more compact heat exchanger is desired, increasing the fin pitch will

decrease the frontal area as shown in Figure 19.

7/30/2019 10.1.1.32.165

83/161

69

0

1

2

3

4

5

6

7

8

9

10

6 7 8 9 10 11 12 13 14 15

Fin Pitch (fins/in)

Figure 19. Effect of Fin Pitch on Frontal Area

As the fin pitch increases, the pressure drop across the fins also increases as seen

in Figure 20. This figure is based on operation at 77 F, but