OPTIMIZATION OF PLATE-FIN-AND-TUBE CONDENSER PERFORMANCE AND DESIGN FOR REFRIGERANT R-410A AIR-CONDITIONER

A Thesis Presented to

The Academic Faculty

By

Kristinn A. Aspelund

In Partial Fulfillment of the Requirements for the Degree

Master of Science in Mechanical Engineering

Georgia Institute of Technology December 2001

ii

OPTIMIZATION OF PLATE-FIN-AND-TUBE CONDENSER PERFORMANCE AND DESIGN FOR REFRIGERANT R-410A AIR-CONDITIONER

Approved:

________________________________ Samuel V. Shelton ________________________________ Sheldon M. Jeter

________________________________ William J. Wepfer

Date Approved____________________

iii

TABLE OF CONTENTS

TABLE OF CONTENTS III

LIST OF FIGURES V

NOMENCLATURE VI

SUMMARY XII

CHAPTER I. INTRODUCTION 1

RESEARCH OBJECTIVES 3

CHAPTER II. THE AIR-CONDITIONER MODEL 4

AIR CONDITIONING SYSTEM AND COMPONENT MODELING 4 Compressor 5 Condenser 9 Condenser Fan 18 Expansion Valve 19 Evaporator 20 Evaporator Fan 22 Refrigerant Mass Inventory 23

CHAPTER III. THE OPTIMIZATION ALGORITHM 27

THE FIGURE OF MERIT 27 SIMPLEX SEARCH METHOD 28 ONE ITERATION OF THE NELDER-MEAD SIMPLEX SEARCH ALGORITHM 29 SOFTWARE TOOLS 31

CHAPTER IV. OPTIMIZATION OF PARAMETERS 34

iv

OPTIMIZATION PARAMETERS 34 OPTIMAL CONDENSER DESIGN 44

CHAPTER V. CONCLUSIONS AND RECOMMENDATIONS 46

CONCLUSIONS 46 GENERAL DESIGN GUIDELINES 49 RECOMMENDATIONS 49

REFERENCES 51

v

LIST OF FIGURES

Figure 2-1: The Actual Vapor-Compression Refrigeration Cycle 6

Figure 2-2: Typical Cross Flow Heat Exchanger with 5 tubes per circuit, 3 circuits and 6 rows. 12

Figure 2-3: Hexagonal Fin Layout and Tube Array 17

Figure 3-1: Nelder-Mead simplex and all possible new points. 32

Figure 4-1: Optimization parameters 35

Figure 4-2: Seasonal COP as a function of maximum condenser cost for varying outer tube diameter and fixed 7.5 ft2 frontal area. 37

Figure 4-3: Seasonal COP, for an air-conditioner with a 5/16 tube condenser, as a function of maximum condenser cost for varying frontal area. 39

Figure 4-4: Number of horizontal rows with 5/16 tubing and max cost at $20 and COP versus frontal area. 40

Figure 4-5: Number of rows with 5/16 tubing versus condenser cost for fixed 7.5 ft2 frontal area. 41

Figure 4-6: Horizontal tube spacing ratio for 5/16 tubing versus condenser cost for fixed 7.5 ft2 frontal area. 42

Figure 4-7: Fin spacing versus number of rows. 43

Figure 4-8: Search for the optimal solution for a 5/16 tube condenser with fixed frontal area of 7.5 ft2 and maximum material cost of $20. 45

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NOMENCLATURE

Ac = Minimum free-flow cross sectional area.

Aci = Cross sectional area of the refrigerant-side of the tube.

Afin = Total fin surface area.

Afr,con = Frontal area of condenser.

Ao = Total air-side heat transfer area including the fin and tube areas.

C = Heat capacity.

Cmax = Maximum heat capacity between that of the air and the refrigerant.

Cmin = Minimum heat capacity between that of the air and the refrigerant.

COP = Coefficient of Performance

COPseas = Seasonal Coefficient of Performance

COP@82F = Coefficient of Performance for an air-conditioner running in 82 F ambient temperature.

cp = Specific heat at constant pressure.

cp,eff = Effective specific heat at constant pressure.

Cr = Ratio of the minimum heat capacity to the maximum heat capacity

H = Condenser height.

h1 = Specific enthalpy of refrigerant entering the compressor.

h2 = Actual specific enthalpy of refrigerant exiting the compressor.

h2a = Specific enthalpy of refrigerant exiting the superheated portion of the compressor.

vii

h2b = Specific enthalpy of refrigerant entering the sub-cooled portion of the compressor.

h2s = Ideal specific enthalpy of refrigerant exiting the compressor.

h3 = Specific enthalpy of refrigerant entering the expansion valve.

h4 = Specific enthalpy of refrigerant exiting the expansion valve.

ah = Air-side heat transfer coefficient.

rh = Refrigerant-side heat transfer coefficient.

k = Thermal conductivity.

k = Number of iterations.

L = Length.

l = Integral variable evaporating tube length.

Lcon,sc = Tube length of the sub-cooled portion of the condenser tubes.

Lcon,sh = Tube length of the superheated portion of the condenser tubes.

Levap,sh = Tube length of the superheated portion of the evaporator tubes.

Lsat = Tube length of the saturated portion of the heat exchanger tubes.

Ltot = Total tube length of the heat exchanger tubes.

m = Refrigerant mass flow rate through the compressor.

satam , = Mass of flow rate of air flowing over the saturated portion of the condenser.

totam , = total mass flow rate of air flowing over the condenser.

mcon,sc = Mass of refrigerant in the sub-cooled portion of the condenser.

mcon,sh = Mass of refrigerant in the superheated portion of the condenser.

viii

mes = Extended surface geometric parameter.

mevap,sh = Mass of refrigerant in the superheated portion of the evaporator.

n = Number of parameters

Nc = Number of parallel flow circuits.

Ntpc = Number of tubes per circuit

NTU = Number of transfer units.

PD = Piston Displacement.

Pe = Perimeter.

Prat = Ratio of condenser saturation pressure to the evapuratior saturation pressure

Q = Rate of total heat transferred between the refrigerant and the air.

qcon,sat = Amount of heat per unit mass transferred between the air and the refrigerant in the saturated portion of the condenser.

qcon,sc = Amount of heat per unit mass transferred between the air and the refrigerant in the sub-cooled portion of the condenser.

qcon,sh = Amount of heat per unit mass transferred between the air and the refrigerant in the superheated portion of the condenser.

maxQ = Maximum possible amount of heat transferred between the refrigerant and the air.

r = Outer radius of tube.

R = Aspect ratio.

Rcv,pd = Ratio of clearance volume to the piston displacement.

Re = Equivalent radius for a hexagonal fin.

Rf,a = Air-side heat exchanger fouling factor.

ix

Rf,r = Refrigerant-side heat exchanger fouling factor.

Rw = Tube wall thermal resistance.

Tc,i = Temperature of cold fluid entering the heat exchanger.

Th,i = Temperature of hot fluid entering the heat exchanger.

Trat = Ratio of condenser saturation temperature to the evapuratior saturation temperature.

Tsc = Sub-cool exiting condenser.

UA = Overall heat transfer coefficient.

W = Condenser width.

wa,com = Actual compressor work per unit mass of refrigerant.

confW , = Condenser fan power.

ws,com = Isentropic compressor work per unit mass of refrigerant.

v = Specific volume.

v1 = Refrigerant specific volume entering the compressor.

v2 = Refrigerant specific volume exiting the compressor.

Va,con = Velocity of the air flowing over the condenser.

vl = Specific volume of the fluid in the liquid phase.

vv = Specific volume of the fluid in the vapor phase.

x = Vapor quality.

xi = Vapor quality at the inlet of the heat exchanger.

Xf = Fin spacing

Xl = Transverse tube spacing.

Xt = Tube spacing normal to air flow.

x

Z = Number of rows.

= Coefficient of the empirical relation for determining the equivalent circular radius for hexagonal fins.

= Expansion.

hlat = Change in the latent enthalpy.

hsens = Change in the sensible enthalpy.

htot = Change in the total enthalpy.

Pa,con = Pressure drop on the air-side of the condenser.

= Fin effectiveness.

= Contraction.

= Fin parameter that is a function of the equivalent circular radius of a hexagonal fin

c = Compressor thermal efficiency.

f = Fin efficiency.

fan,con = Condenser fan efficiency.

s = Surface efficiency.

s,a = Air-side surface efficiency.

s,r = Refrigerant-side surface efficiency.

v = Compressor volumetric efficiency.

= reflection.

v = Density of the fluid in the vapor phase.

= Shrinkage.

xi

= Coefficient of the empirical relation for determining the equivalent circular radius for hexagonal fins.

xii

SUMMARY

Residential air-conditioning equipment currently uses HCFC refrigerant R-22.

Production of the refrigerant will be banned in 2010 except to service existing equipment.

The refrigerant R-410a is a strong candidate to replace R-22. While there is limited

information available on R-410a condenser coil design, a model of an air-conditioning

system with a focus on the finned-tube condenser design details using R-401a as the

working fluid has previously been developed by Wright (2000). The model evaluates the

performance for a specific and detailed condenser design, e.g. frontal area, tube diameter,

air velocity, etc.

An optimization algorithm for the fin-tube condenser design is needed. Due to

computational speed limitations an exhaustive search for the optimal design is not

practical. This research developed design search techniques to find the optimal

condenser design and controllable operational parameters with various constrains for a

given figure of merit. The Simplex Search Method (Nelder et. al., 1965) was

implemented to search and optimize the eight primary condenser design parameters. This

study found an optimum condenser design for various frontal area and cost constrains.

The software developed is appropriate for engineering design use in the air-conditioning

industry.

1

CHAPTER I.

INTRODUCTION

In the last decade public awareness on the destruction of the stratospheric ozone

layer grew and the most harmful materials were banned. Under the terms of the Montreal

Protocol, the United States agreed to meet certain obligations that have brought

challenges to the Heating Ventilation Air Conditioning and Refrigeration (HVAC&R)

industry. Chlorofluorocarbons (CFCs) were used to a large extent as refrigerants but

have high ozone-depletion potential (ODP) and they were completely phased out in the

USA in 1995. Though not harmless environmentally friendlier and inexpensive

hydrochlorofluorocarbons (HCFCs), such as HCFC-22 (R-22), are exclusively used as a

refrigerants in residential heat pumps and air-condition systems. However in the USA

the Environmental Protection Agency (EPA) has published regulations prohibiting the

production of R-22 after 2010 except for servicing equipment produced prior to 2010.

After 2020 the production of R-22 will be completely banned (EPA, 2001).

Due to zero ODP and many favorable performance characteristics, e.g., good

cycle efficiency, non-flammability and high working pressure, R-410a is a strong

candidate as a replacement for R-22. There is however limited information about

condenser coil design for air-conditioners using R-410a as a working fluid.

2

Due to its global warming impact environmental regulations have also focused on

the emission of CO2. Many countries have agreed to reduce their CO2 production. This

must be accomplished by reducing energy usage through higher efficiency energy

systems. In a warm climate, residential air-conditioners are responsible for a major

portion of a households total energy usage and since they are only run when the outside

temperatures are high, a peak electrical demand occurs only on hot days. Utilities must

invest in an electric power generation and distribution infrastructure to meet the air-

conditioner peak demand (Wenzel et. al 1997). This, along with public awareness, has

created pressure for the efficiency of air-conditioning equipment to improve.

Wright (2000) developed a detailed model for an air-conditioning system using R-

410a as a working fluid. The model has detailed simulation of the components of the air-

conditioner system for various designs, including the compressor, the condenser, the

evaporator and the expansion valve. The condenser is the focus of the model

incorporating the best available simulations for the air-side and refrigerant-side pressure

drops and heat transfer coefficients. While the effects of varying some design parameters

were studied an exhaustive design optimization search would have taken months to

execute.

3

Research Objectives

The primary objective of the current work is to study and optimize the geometric

design and operating parameters for a finned-tube condenser of a vapor compression

residential air-conditioning system using R-410a as a working fluid. More specifically

to:

Find and implement a design optimization search technique for the air-

conditioner condenser design.

Apply the optimization technique through software development to optimize

controllable operational and geometric design parameters.

Develop design guidelines for a condenser coil design

Develop a software tool for condenser designers to design coils for optimized

air-conditioner performance.

4

CHAPTER II.

THE AIR-CONDITIONER MODEL

The air-conditioner model used in the current study was developed by Wright

(Wright, 2000). The following development of the model is based on Wrights (2000)

thesis entitled Plate-Fin-and-Tube Condenser Performance and Design for Refrigerant

R-410A Air-Conditioner. This development is detailed here in this study for

completeness.

Air Conditioning System and Component Modeling

Heating, Ventilating, and Air-Conditioning (HVAC) systems that provide a

cooling effect depend on a refrigeration cycle. Both the control and performance of

HVAC systems are significantly affected by the performance of the refrigeration cycle.

Therefore a basic understanding of the refrigeration cycle is needed in the design and

optimization of HVAC systems. Of the three basic refrigeration cycles (vapor

compression, absorption, and thermo-electric), the cycle typically used in the HVAC

industry is the vapor compression cycle. Vapor compression refrigeration has many

complex variations, but only the basic compression cycle will be discussed here.

5

The vapor compression refrigeration cycle modeled for this study is shown in

Figure 2-1. As the figure shows, 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 2b), and then cooled to below the saturation point until sub-cooled

liquid is present (State 3). The high-pressure liquid then flows through the expansion

valve into the evaporator (State 4). The refrigerant then absorbs heat from warm indoor

air that is blown over the evaporator coils. The refrigerant is completely evaporated

(State 4a) and super heated above the saturation temperature before entering the

compressor (State 1). The indoor air is cooled and dehumidified as it flows over the

evaporator and returned to the living space.

Compressor

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

refrigerant. The compressor is the major energy-consuming component of the

refrigeration system, and its performance and reliability are significant to the overall

performance of the HVAC system. In general there are two categories of compressors:

dynamic compressors and displacement compressors. Dynamic compressors convert

angular momentum into a pressure rise and transfer this pressure rise to the vapor

(McQuiston and Parker, 1994).

6

Figure 2-1: The Actual Vapor-Compression Refrigeration Cycle

Saturated Sub-cooled Superheated 2b 2a3

Expansion Valve

Saturated Superheated

4 4a

Compressor

Condenser

Evaporator

1

2

2b3

4

2a

2

4a

1

S

T

7

Positive displacement compressors increase the pressure of the vapor by reducing the

volume in a closed space. For this study, scroll type positive displacement compressors,

which dominate the residential air-conditioning industry, are considered.

The amount of specific work (work per unit mass of refrigerant) done by an ideal

compressor can be expressed with the following:

( )12, hhw scoms = (2-1)

where h1 is the refrigerant enthalpy entering compressor and h2s refrigerant enthalpy for

isentropic compressor. For a non-ideal compressor, the actual amount of work done

depends on the efficiency,

( )12,, hhwwc

comscoma ==

(2-2)

where c is the compressor isentropic efficiency. The subscripts hx refer to the state point

x on Figure 2-1. For a scroll type compressor, Klein and Reindl (1997) have determined

that the thermal efficiency is related to the pressure ratio and a temperature ratio by the

following relationship:

ratratratratratratc TPTTPP 061.331.503.111281.0814.325.6022 ++= (2-3)

8

where Prat is the pressure ratio and Trat is the temperature ratio, which are defined by

the following relationships:

evapsat

condsatrat P

PP

,

,= (2-4)

evapsat

condsatrat T

TT

,

,= (2-5)

The coefficients in this correlation are based on saturated temperatures and not on the

actual temperatures at the inlet and outlet of the compressor.

The volumetric efficiency is another important consideration in selecting and

modeling compressors. The volumetric efficiency is the ratio of the mass of vapor that is

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

were equal to the compressor piston displacement. The volumetric efficiency is

expressed as:

= 1

vv1

2

1,v pdcvR (2-6)

where v is the compressor volumetric efficiency, Rcv,pd is the ratio of clearance volume

to the piston displacement, v1 is the specific volume entering the compressor and v2 the

specific volume at the compressor exit. The volumetric efficiency is used to determine

9

the mass flow rate of the refrigerant through the compressor,

m , for a given compressor

size by the following expression,

2v

PDm v= (2-7)

where PD is the Piston Displacement (Threlkeld, 1970).

Condenser

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

outside air. Although there are many configurations of heat exchangers, finned-tube heat

exchangers are the type most commonly used 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 are of the cross-flow, plate-fin-and-tube

type. A schematic of this heat exchanger is shown in Figure 2-2. The tubing of the

condenser is one of the decisions that designer has to make. On Figure 2-2 tubes having

flow in the same direction are shown to have the same color. The flow can flow through

many parallel through many tubes simultaneously in a parallel circuit. The tubes are then

connected at the ends by bends. It is assumed that since the bends have no fins the heat

transfer is zero.

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

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

10

three sections: superheated, saturated, and sub-cooled. The amount of heat per unit mass

of refrigerant rejected from each section can be expressed as the difference between the

refrigerant enthalpy at the inlet and at the outlet of each section:

,22, ashcon hhq = (2-8)

,22, basatcon hhq = (2-9)

and

.32, hhq bsccon = (2-10)

The total heat rejected from the hot fluid, which in this case is the refrigerant, to the cold

fluid, which is the air, is dependent on the heat exchanger effectiveness and the heat

capacity of each fluid:

( )icih TTCQ ,,min = (2-11)

where is the heat exchanger effectiveness; Cmin is the smaller of the heat capacities of

the hot and cold fluids, Ch and Cc respectively; Th,i is the inlet temperature of the hot

11

fluid; and Tc,i is the inlet temperature of the cold fluid. The heat capacity C, is expressed

as:

pcmC = (2-12)

where

m is the mass flow rate of fluid and cp is the specific heat of the fluid. The heat

capacity, C, is the extensive equivalent to the specific heat, and it determines the amount

of heat a substance absorbs or rejects for a given temperature change.

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

the tube length, L, corresponding to each specific section. For example, the mass of air

flowing over the saturated section of the condenser can be found by the following

relation:

tot

sat

tota

sata

LL

mm

=

,

,

(2-13)

The heat exchanger effectiveness discussed earlier in this chapter is the ratio of the actual

amount of heat transferred to the maximum possible amount of heat transferred,

maxQQ

= (2-14)

12

Figure 2-2: Typical Cross Flow Heat Exchanger with 5 tubes per circuit, 3

circuits and 6 rows.

w

d

h

Xl

Xt

Air flow

13

The heat exchanger effectiveness is dependent on the temperature distribution within

each fluid and on the paths of the fluids as the heat transfer takes place, i.e. parallel-flow,

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

mass flow is separated into a number of discrete tubes and does not mix between fluids.

Furthermore, the plates of the heat exchanger prevent mixing of the air flowing over the

fins. Therefore, air at one end of the heat exchanger will not necessarily be the same

temperature as the 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 expression (Incropera & DeWitt, 1996):

( ) ( )( )[ ] ,1exp1exp1 78.022.0

= NTUCNTU

C rr (2-15)

where Cr is the heat capacity ratio:

.max

min

CCCr = (2-16)

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

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

effectiveness for any heat exchanger configuration is expressed as:

14

( ).exp1 NTU= (2-17)

The NTU is a function of the overall heat transfer coefficient, U, and is defined as

,minC

UANTU = (2-18)

where A is the heat transfer area upon which the overall heat transfer coefficient, U, is

based. The overall heat transfer coefficient accounts for the total thermal resistance

between the two fluids and is expressed as follows.

,111

,,

",

,

",

, rrrsrrs

rfw

aas

af

aaas AhAR

RA

RAhUA

++++= (2-19)

where Rf,(a or r) is the fouling factor, Rw is the wall thermal conduction resistance, s(a or r)

is the surface efficiency defined below in equation 2-27, andh is the convective heat

transfer coefficient. There are no fins on the refrigerant side of the condensing tubes.

Therefore, the refrigerant side surface efficiency is 1. Neglecting the wall thermal

resistance, Rw (this value is usually 3 orders of magnitude lower than the other

resistances), and the fouling factors, Rf,(a or r), the overall heat transfer coefficient reduces

to:

15

.111

,

+=

rraaas AhAhUA

(2-20)

In the sub-cooled region the Dittus-Boelter equation is used to determine the

refrigerant side heat transfer coefficient (Incropera & DeWitt, 1996). In the super-heated

region the heat transfer correlation developed by Kays and London (1984) is used.

Correlation developed by Shah (1979) is then used for the two-phase flow condensing

heat transfer correlation. The work of McQuiston (McQuiston and Parker, 1994) is used

to evaluate the air-side convective heat transfer coefficient.

To determine the overall surface efficiency for a finned tube heat exchanger, it is

first necessary to determine the efficiency of the fins as if they existed alone. For a plate-

fin-and-tube heat exchanger with multiple rows of staggered tubes, the plates can be

evenly divided into hexagonal shaped fins as shown in Figure 2-3. Schmidt (1945)

analyzed hexagonal fins and determined that they can be treated as circular fins by

replacing the outer radius of the fin with an equivalent radius. The empirical relation for

the equivalent radius is given by

( ) ,3.027.1 2/1= rRe (2-21)

where r is the outside tube radius. The coefficients and are defined as

16

r

X t2

= (2-22)

and

,4

12/12

2

+= tl

t

XXX

(2-23)

where Xl is the tube spacing in the direction parallel to the direction of air flow, and Xt is

the tube spacing normal to the direction of air flow.

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

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

thickness. Therefore, the standard extended surface parameter, mes can be expressed as,

,2

2/12/1

=

=

kth

kAPehm a

ces (2-24)

where ha is the air-side heat transfer coefficient, k is the thermal conductivity of the fin

material, Pe is the fin perimeter, Ac is the fin cross sectional area, and t is the thickness of

the fin. For circular tubes, a parameter can be defined as:

17

Figure 2-3: Hexagonal Fin Layout and Tube Array

Tube Spacing Normal toAir Flow

Xt

Transverse Tube Spacing

Xl

Air Flow

18

.ln35.011

+

=

rR

rR ee (2-25)

The fin efficiency, f, for a circular fin is a function of mes, Re, and f, and can be

expressed as

( ) .tanh

ees

eesf Rm

Rm= (2-26)

The total surface efficiency of the fin, s is therefore expressed as:

( ),11 fo

fins A

A = (2-27)

where Afin is the total fin surface area, Ao is the total air-side surface area of the tube and

the fins.

Condenser Fan

Natural convection is not sufficient to attain the heat transfer rate required on the

air-side of the condenser used in a reasonably sized residential air-conditioning system.

Therefore a fan must be employed to maintain the airflow at a sufficient rate. Although

much of the electrical power consumed by the total system is due to the compressor, the

19

condenser fan also requires a significant amount of power. The power required by the

fan is directly related to the air-side pressure drop across the condenser and to the

velocity of air across the condenser:

confan

confrconaconaconf

APVW

,

,,,,

=

(2-28)

43 hh = (2-29)

where Va,con is the air velocity over the face of the condenser, Pa,con is the air-side

pressure drop over the condenser, Afr,con is the frontal area of the condenser, and fan,con is

the condenser fan Isentropic efficiency. The work of Rich (1973) and Zukauskas and

Ulinskas (1998) are used to evaluate the air-side pressure drop over the finned tubes in air

cross-flow.

Expansion Valve

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

Under normal operating conditions, the expansion valve opens and closes in order to

maintain a fixed amount of superheat in the exit of the evaporator. In this study, the

superheat is set at the typical 10 F. Because the expansion valve can only pass a limited

volume of refrigerant, it cannot maintain the specified superheat at the evaporator exit if

the refrigerant is not completely condensed into liquid. If incomplete condensation in the

20

condenser occurs, the vapor refrigerant backs up behind the expansion valve and the

condenser pressure increases until the refrigerant is fully condensed. As a result, in some

cases the expansion valve cannot regulate the refrigerant mass flow rate, and cannot

maintain a fixed superheat at the evaporator exit. Wright found that this can occur when

the air-conditioner is run at low ambient temperature. In that case the evaporator

superheat varies above the desired 10 F.

Evaporator

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

lower its temperature and humidity. Because the refrigerant enters the evaporator as a

liquid-vapor mixture, it is only divided into saturated and superheated sections. No sub-

cooled section is necessary. The analysis of the thermodynamic parameters of the

evaporator is nearly identical to that of the condenser. However, the dehumidification

process involving the evaporator results in some modifications of the analysis. To

maintain the simplicity of the evaporator heat transfer model, the evaporator coil is

assumed to be dry in calculating the air-side heat transfer coefficient. However, because

the air flowing over the evaporator is cooled to a temperature below the wet bulb

temperature, some of the heat rejected by the air causes water to condense out of the air

rather than simply lowering the temperature of the air. Therefore, the specific heat must

be modified to account for this condensation. The total enthalpy change of the air is thus

the sum of the enthalpy change due to the decrease in temperature (sensible heat), and the

enthalpy change due to condensation (latent heat).

21

latsenstot hhh += (2-30)

If the specific heat for dry air is utilized in the model for the evaporator, the

resulting exit temperatures will be too low for complete vaporization. Therefore, an

effective specific heat that takes into account both the latent heat and the sensible heat

must be utilized. Using an effective specific heat will result in a more accurate

determination of the evaporator exit temperature without the complications associated

with using the standard equations for air-water mixtures. Since the evaporator is not the

focus of this study, this approximation should not affect the condenser optimization

methodology.

Dividing (2-30) by the air temperature change gives the following:

T

hT

hT

h latsenstot

+

=

(2-31)

The ratio of the sensible heat enthalpy change to the temperature change is by definition,

the specific heat, cp. Therefore, after substituting cp into (2-31) and rearranging, the

following expression is obtained:

T

hcc latpeffp

+=, (2-32)

22

where cp is the specific heat ratio for dry air and cp,eff is the effective specific heat. To

maintain indoor humidity, the latent heat accounts for approximately 25% of the total

enthalpy change of the air flowing over an evaporator. The effective specific heat can

thus be expressed in terms of the specific heat for dry air only,

.33.175.0

25.0, p

tot

senslatpeffp ch

hT

hcc =

+= (2-33)

Evaporator Fan

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

coils would present unwelcome complications in the overall analysis. In addition to

affecting the heat transfer, wet coils also have an effect on the air-side pressure drop.

Although there are correlations available for determining the pressure drop over wet

coils, they are cumbersome to use and again, the evaporator is not the primary focus of

this investigation.

After the air flows over the evaporator, it enters a series of ducts that then return

the air back inside the living space. The power required by the evaporator fan depends

on the losses in these ducts and can vary from installation to installation. Therefore, the

default power requirement specified by the Air-conditioning and Refrigeration Institute

(ARI, 1989) of 365 Watts per 1000 ft3/minute of air will be used.

23

Refrigerant Mass Inventory

The amount of sub-cooling at the condenser exit are controlled by the system

operating conditions and the quantity of refrigerant mass in the system. The mass of

refrigerant in the tubes connecting the components is neglected. Since the compressor

contains only vapor, the mass of refrigerant in the compressor is also neglected.

Therefore the calculated total mass of refrigerant in the system includes the mass in the

sub-cooled, saturated, and superheated portions of the condenser, and in the saturated and

superheated portions of the evaporator.

The following text outlines the procedure for finding the refrigerant mass in the

saturated portion of the evaporator. The same procedure is also used to determine the

mass of refrigerant in the saturated portion of the condenser, however the boundary

conditions are different.

The mass of refrigerant can be expressed as

.v

=

L

cidlAm (2-34)

where, Aci is the cross sectional area of the refrigerant-side of the tube, and v is the

specific volume, which at saturated conditions is a function of quality expressed as

( ) ( ) .v1vv vl xx += (2-35)

24

The boundary conditions for the saturated portion of the evaporator are

( ) ixlx == 0 (2-36)

and

1)( == Llx (2-37)

where l is integral variable evaporating tube length and L is the total evaporating tube

length. Using the boundary conditions and assuming the quality varies linearly with tube

length, the following expression results

( ) .1 ii xlLxlx += (2-38)

Substituting (2-38) into (2-35) yields an expression for the specific volume as a function

of length,

( ) ( ) ( ).vv1vvvv lvilvil Lxlxl

++= (2-39)

25

For a uniform tube cross sectional area, substituting (2-39) into (2-34) yields

( ) ( )

.vv1vvv

1

0,

=

=

++

=

Ll

llv

ilvil

cievapsat dl

Lxlx

Am (2-40)

Integrating (2-40) yields the following expression

( )( ) ( ) ( ) .vv1vvvln

vv10v

,

Ll

llv

ilvil

li

cievapsat L

xlxx

LAm=

=

++

= (2-41)

Substituting for l, the expression for the final mass in the saturated portion of the

evaporator is expressed as:

( )( ) ( ) .vvvvln

vv1,

,

+=

llvi

v

lvi

evapsatcievapsat xx

LAm (2-42)

The mass of refrigerant in the superheated portions of the condenser and evaporator are

expressed simply as:

shconcivshcon LAm ,, = (2-43)

26

and

.,, shevapcivshevap LAm = (2-44)

Finally, the mass of refrigerant in the sub-cooled section of the condenser is expressed as

.,, scconcivsccon LAm = (2-45)

By using the above relations for the air-conditioning system components in a system

simulation program it is possible to evaluate the performance of an total air-conditioning

system for varying condenser design.

27

CHAPTER III.

THE OPTIMIZATION ALGORITHM

The figure of merit

To quantitatively evaluate the performance of an air-conditioning system there

must be a quantitative figure of merit established. The efficiency of vapor compression

refrigeration cycles is expressed as the Coefficient of Performances (COP). The COP is a

ratio of the rate of the cooling capacity to the electrical and mechanical power used to

drive the system.

=

i

e

WQ

COP (3-1)

The weighted average of the COP over a summer is referred to as the seasonal

COP (COPseas). Wright (2000) showed that for a air-conditioning system the seasonal

COP is very close and nearly identical to the COP for ambient temperature of 82 F.

Using COP@82F instead of the seasonal COP requires fewer calculation therefore

increases calculation speed and stability. In this research it is assumed that COPseas is

equal to COP@82F.

28

Simplex search method

In selection of an optimization algorithm the fact that the model of the air-

conditioner is complicated and solved numerically in EES must be considered. The

Simplex search method presented by Nelder and Mead (1965) has been widely used to

optimize complicated functions. This method was chosen for this study because it is

robust and relatively simple to implement and gives fairly good results even though it is

not yet known whether the Nelder-Mead method can be proved to converge to an

optimum value in all cases (Lagarias, et. al., 1998). Even though the Simplex search

method will find a good solution for the design of the condenser, like for other search

methods it cannot be proven that it is the global optimal design.

Because the Nelder-Mead algorithm uses only function values, to minimize

scalar-value function of n real variables it falls into the class of Direct Search Methods

(Reklaits, et. al., 1983). Each kth iteration (k > 0) of the simplex direct search method

begins with a simplex, specified by its n + 1 vertices and the associated function values.

Since the desired solution is the maximum seasonal COP of the air-conditioner, the COP

is calculated for all the vertices and they are then ordered and labeled x1(k) ,, xn+1(k) ,

such that

)COP( )COP( )COP( (k)1(k)2

(k)1 + nxxx (3-2)

where x1(k) is the best point or vertex in the simplex while the xn+1(k) is the worst.

29

To start the search, one base point is chosen. Preferably the base point is within

the optimization constrains and in the range of the air-conditioner simulation program.

The other vertices of the starting simplex are then set by adding % to one parameter for

each vertex so the initial vertex will span the whole space. Here was set to be 30 %.

In the Nelder-Mead method there are four scalar parameters defined: coefficients

of reflection (), expansion (), contraction () and shrinkage (). The recommended

values by Nelder and Mead (1965) are nearly universally used in the standard algorithm

(Lagarias, et. al., 1998):

= 1, = 2, = 0.5, and = 0.5 (3-3)

One iteration of the Nelder-Mead Simplex search algorithm

1. Order. Order the n +1 vertices to satisfy )COP( )COP( )COP( 121 + nxxx .

2. Reflect. When =

=

n

in

1/ixx is the center of the n best points, compute the

reflection point xr from

11 )1()( ++ +=+= nnr xxxxxx , (3-4)

Calculate COP for xr. If COP1 > COPr > COPn accept xr as new point and terminate

the iteration.

30

3. Expand. If COPr > COP1 calculate the expansion point xe and COP(xe) where:

11 )1()()( ++ +=+=+= nnr xxxxxxxxxe , (3-5)

If COPe > COPr accept xe and terminate the iteration, else accept xr and terminate the

iteration.

4. Contract.

a. Outside. If COPn > COPr > COPn+1 calculate:

11 )1()()( ++ +=+=+= nnrc xxxxxxxxx , (3-6)

If COPc > COPr then accept xc and terminate the iteration else perform

shrinking.

b. Inside. If COPr < COPn+1 calculate:

11 )1()( ++ +== nncc xxxxxx , (3-7)

If COPcc > COPn+1 accept xcc and terminate the iteration else perform

shrinking.

31

5. Shrinking. Calculate COP(vi) where vi = x1 + (xi - x1) and i = 2, , n+1. Then

next simplex has vertices x1, v2, , vn+1

Nelder and Mead did not discuss any tie-breaking rules, however, in this study

points with the same value are going to be ordered so that the newest vertex is the best. If

a parameter of the vertex doesnt fall within constraints the calculated COP is divided by

a relatively big number. The search comes to a halt when the new vertex is close to the

average point. This does not necessarily mean that the volume of the simplex is getting

close to zero, i.e. the vertices are converging to the same point, but rather that the simplex

is not changing between the latest two iterations. When the search comes to a stop it is

restarted with the previous best point as a starting point. This is repeated until the COP

for the best point in the restarted solution is the same as the COP for the base point.

Software Tools

Wrights (2000) air-conditioner model used in the current study was programmed in

EES (Engineering Equation Solver) (Klein and Alvarado 2001), a software tool that has

built-in thermodynamic and transport property relations. It solves numerically, multiple

equations for an equal number of unknown. While EES is useful for simulating the air-

conditioner it is not suitable for performing the optimization Simplex search. However,

EES does have the ability to communicate with other programs using Dynamic Data

Exchange (DDE) supported by many other programs.

32

Figure 3-1: Nelder-Mead simplex in two dimensions and all possible new points.

xr

x

xe

xc

xcc

x2

x3

x1

y1

y2

33

Therefore, in the current study, the Simplex search scheme was programmed in Microsoft

Visual Basic using Microsoft Excel to organize the inputs and the outputs of the search.

When starting the Simplex search program in Visual Basic, it copys the initial

simplex from the Excel sheet, opens the EES program pastes into a parametric table, and

then instructs EES to solve the table. Once solved the COP values are pasted back into

Excel again and the new vertex is calculated and sent again to EES. The COP value is

again sent back to Excel. The Simplex search program will send vertex information from

Excel to EES and the COP back to Excel until the simplex has converged.

34

CHAPTER IV.

OPTIMIZATION OF PARAMETERS

Optimization Parameters

The optimization parameters can be divided into two groups, geometric design

and controllable operational parameters. Geometric design parameters are decided when

the condenser is designed and after it is built they cannot be changed, when on the other

hand, it is possible to change the controllable operational parameters without much effort

after the condenser is built. As seen in Figure 2-1 both the height and the depth of the

condenser are dependent on other parameters. The height is dependent on number of

circuits, number of tubes per circuit and the tube vertical spacing. The depth is then

dependent on number of rows and horizontal tube spacing. Many of the geometric design

parameters are positive integer numbers but the Simplex search method only works for

continuous parameters. Even though it may not have a practical meaning, all parameters

are assumed to be continuous to optimize the design with the Simplex method. The

optimal solution found gives an idea about optimal design of a condenser that is possible

to build. The optimal integer design is then found by fixing the integer parameters and

performing a manual search close to the optimal solution.

35

Wright (2000) showed, by calculating COPseas for various degrees of sub-cool, that

the COP is not sensitive of varying sub-cool. In practice the sub-cool is set by charging

the system with refrigerant until a certain value of sub-cool condition is met with 95 F

air blowing through the coil. Also, Wright showed that 15 F sub-cool at 95 F is

approximately the optimal sub-cool. To reduce optimization parameters and speed up

calculations the sub-cool was fixed to 15 F at 95 F in the current study.

Figure 4-1: Optimization parameters

Geometric Design Parameters Operational Parameters W [ft] Width of condenser. Va,con [ft/s] Air velocity over condenser.

Xf [in] Spacing between fins . Tsc [F] Sub-cool exiting condenser.

Xl [in] Vertical tube spacing.

Xt [in] Horizontal tube spacing.

Nc Number of parallel flow circuits.

Ntpc Number of tubes per parallel flow circuit.

Z Number of rows

D [in] Tube diameter

36

Condenser coils in current air-conditioning systems are dominantly produced with

5/8 and 1/2 diameter tubes, but 3/8 and 5/16 tubes are also used (AAON, 2001). The

design was optimized for each of the tube diameters with constraints put on the cost and

the frontal area of the condenser. Wrights model found that the cost of the condenser

coil is directly linked to the material cost, which is calculated from the geometric design.

A fixed frontal area of 7.5 ft2 has been selected for a typical condenser design of a

30,000 Btu/hr air-conditioning system. A constraint is needed on the aspect ratio, i.e.

frontal width divided by frontal height:

3=HWR (4-1)

The maximum of 3 is a common design for the condenser of a residential air-conditioner

since they are often designed as a cube unit package where the condenser covers 3 sides.

Figure 4-2 shows that, for fixed frontal area and varying maximum material cost

in U.S. Dollars, the smallest tube, diameter 5/16 gives the best results. Smaller tubes

like 1/4 is not likely to give significantly better results. Also from Figure 4-2 it can be

seen that the optimum COPseas reaches a maximum where spending more money on a

bigger condenser will not result in higher efficiency.

37

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

15 20 25 30 35 40 45 50 55 60 65

Max Cost (USD)

CO

P sea

sona

l

5/16(in)3/8(in)1/2(in)5/8(in)

Figure 4-2: Seasonal COP as a function of maximum condenser cost for varying

outer tube diameter and fixed 7.5 ft2 frontal area.

38

As shown in Figure 4-3, the COP increases with increasing frontal area and the

best solution would have just one row, one circuit, and one tube per circuit, hence a

condenser composed of one long fined tube. It can be seen from Figure 4-4 that

increasing the frontal area for a condenser with maximum cost constraint makes the

condenser thinner until the condenser will become just one horizontal row. This will

reduce the air-side pressure drop and also the air that would flow across each tube would

be at the ambient temperature. That would though make the condenser far to big for most

residential applications and the size constrains are set to keep the design inside

practical limits. When the Simplex search was run with no constrains at all then the

solution was as expected one straight tube. This is because there is pressure drop in the

tube bends but there is no heat transfer.

For increasing maximum cost constraints values for a condenser with a fixed

frontal area the only way to increase the efficiency is to add more rows. Figure 4-5

shows that more rows are added until the air-side pressure drop becomes so high that

adding more rows will reduce the seasonal COP. From figure 4-6 it seems that when

adding more rows, the spacing between the rows becomes slightly smaller. Conversely,

it can be seen from Figure 4-7, when adding more rows, the spacing between the fins

becomes larger. This allows the air to pass easier through the condenser which has a

reducing effect on the air-side pressure drop.

39

4.18

4.2

4.22

4.24

4.26

4.28

4.3

4.32

4.34

15 20 25 30 35 40 45 50 55

Max Cost (USD)

CO

P sea

sona

l

Area= 8.5Area = 7.5Area = 6.5

Figure 4-3: Seasonal COP, for an air-conditioner with a 5/16 tube condenser,

as a function of maximum condenser cost for varying frontal area.

40

Figure 4-4: Number of horizontal rows with 5/16 tubing and max cost at $20

and COP versus frontal area.

0

1

2

3

4

5

6

7

6 8 10 12 14

Fixed Frontal Area [ft2]

Num

ber o

f row

s

4.19

4.2

4.21

4.22

4.23

4.24

4.25

4.26

4.27

4.28

4.29

CO

P

RowsCOP

41

0

5

10

15

20

25

15 20 25 30 35 40 45 50 55 60 65

Max Cost (USD)

Num

ber o

f Row

s

Figure 4-5: Number of horizontal rows with 5/16 tubing versus condenser cost

for fixed 7.5 ft2 frontal area.

42

0.7

0.8

0.9

1

1.1

1.2

1.3

15 20 25 30 35 40 45 50 55 60 65

Max Cost (USD)

Hor

izon

tal S

paci

ng/D

iam

eter

Figure 4-6: Horizontal tube spacing ratio for 5/16 tubing versus condenser cost

for fixed 7.5 ft2 frontal area.

43

8

9

10

11

12

13

14

0 5 10 15 20 25

Number of rows

fins/

in

Figure 4-7: Fin spacing versus number of horizontal rows.

44

Optimal Condenser Design

As discussed above, the Simplex method is not able to optimize the design using

integer values for the integer parameters. However the Simplex search method gives a

solution for the optimal design of a hypothetical fin-plate condenser with decimal values

that should be close to the optimal integer design. In order to find the optimal integer

solution, the Simplex search was run with number of rows, number of parallel circuits

and the number of tubes per circuit fixed to integer values on either side of the optimal

solution. The Simplex search was then used to find an optimal COP for each of the

possible design combinations of the three integer parameters. Figure 4-8 shows the

search for the optimal integer design of a condenser with 7.5 ft2 fixed frontal area and

5/16 tube and maximum coil cost set to $20.

In Wrights (2000) study, the vertical and horizontal tube spacing was held fixed

to 1.25 and 1.08 respectively, which are currently the widely used values in air-

conditioning condensers. In the current study it was shown that the optimal vertical tube

spacing is slightly larger but the horizontal optimal spacing is about 1/3 of the

conventional value. The tighter horizontal spacing makes it possible to fit in more tubes

with the same air-side pressure drop and the higher vertical spacing gives the air a higher

flow area through the condenser.

45

Figure 4-8: Search for the optimal solution for a 5/16 tube condenser with fixed

frontal area of 7.5 ft2 and maximum material cost of $20.

Tubes/circuits #rows #circuits Fin/inch Vair Spacingv Spacingh Width Height Cost COP [ft/s] [in] [in] [ft] [in] [USD] 4 5 2 14.23 14.38 2.37 0.46 4.74 18.98 19.98 4.154 5 3 13.29 12.83 1.68 0.41 4.46 20.19 19.97 4.214 6 2 13.34 14.44 2.36 0.37 4.73 18.92 19.99 4.174 6 3 13.17 13.16 1.68 0.29 4.46 20.17 19.99 4.205 5 2 15.14 13.63 1.90 0.39 4.74 18.98 20.00 4.205 5 3 11.38 13.13 1.44 0.41 4.16 21.65 19.98 4.215 6 2 12.11 14.06 1.89 0.36 4.72 18.90 19.99 4.215 6 3 12.86 13.91 1.67 0.30 3.59 25.04 19.97 4.22

46

CHAPTER V.

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

The objective of the current work was to study and optimize the geometric design

and operating parameters for the finned-tube condenser of a 30,000 BTU/hr vapor

compression residential air-conditioning system using R-410a as a working fluid with

coil cost, aspect ratio and frontal area constraints. An optimization search technique was

implemented for the air-conditioner model. Also software was developed to optimize the

condenser design.

Consistent with Wrights study (2000), the condenser with the smallest tubing,

5/16 diameter, gives the best air-conditioner efficiency at any cost and frontal area.

Also consistent with Wrights study is that larger frontal area results in higher efficiency

until the design becomes one straight finned tube. On the other hand, to keep

computational time practical Wright used fixed spacing between the tubes. With the

Simplex search method the spacing was a design variable, and the best horizontal tube

spacing design was shown to be approximately three times smaller than the conventional

value with the vertical tube spacing 50-100% farther apart than conventional designs.

The result is that for a fixed frontal area of 7.5 ft2 the optimal number of horizontal rows

47

with no cost constraint is around 20. Adding a cost constraints of $20 the optimal design

is a 5/16 condenser with 5 tubes per circuits, 3 parallel circuits 12.9 fins per inch, 6

horizontal rows and with the coil 3.6 ft wide and 25 high. The horizontal and vertical

spacing between the tubes should be 0.3 and 1.7 respectively. The COP for this design

was calculated to be 4.2.

In the previous study (Wright, 2000), the best design found for a finned-tube

condenser using R-410a as a working fluid was also a condenser with 5/16 tubing but

having 3 rows of tubes 12 fins per inch and 5 tubes per circuit. Comparing the result

COP from Wrights study with the COP from the current study, using the Simplex Search

method has given a better design than both of those found for a fixed frontal area

problem. With a fixed cost of $26 Wright found a design with the COP equal to 4.23

compared to the 4.22 COP value found in the current study with the cost constraint set to

$20. Therefore the Simplex search method found a design that gives the same COP for

23% less condenser cost. For comparison with an air-conditioner using R-22 as a

working fluid (Saddler, 2000) studies the design in similar fashion as Wright. The best

design found for a condenser using R-22 had a COP of 4.18, 6 tubes per circuit and 12

fins per inch for a condenser with 5/16 tubes. As did Wright, Sadler fixed the tube

spacing as well. Therefore the COP for an air-conditioning system is similar for a system

using R-410a as one using R-22. Since R-410a is environmentally friendlier than R-22 it

is a very good replacement candidate

48

More specifically the conclusions drawn from this study are following:

The best tube diameter for the condenser of an air-conditioning system using R-

410a as a working fluid is 5/16, the smallest studied.

Even though it is not yet possible to prove that the Simplex search will give the

best global design available, it did find a better design than that found in the

previous manual search study and the software can be used by designers to find an

optimal design of an air-conditioner condenser.

For higher cost constraints, a condenser with fixed frontal area will have

increasing number of rows, until the number of rows reaches a point when the

airflow resistance becomes dominant (for 5/16 it is around 20). The higher heat

transfer created by additional rows beyond that point has less effect on the COP

than the increasing air-side pressure drop.

49

General Design Guidelines

1. A single row condenser has the highest efficiency for fixed cost and

unconstrained frontal area.

2. To minimize the pressure drop due to tube bends it is desirable to have the

aspect ratio as high as possible to minimize the number of tube bends.

3. For all the tube diameters and frontal area studied, coil cost higher than $30

will not significantly increase the condenser performance.

4. The smallest tube studied, 5/16, gives the best performance.

5. Airflow velocity should be approximately to 13 ft/s.

6. Horizontal tube spacing for 5/16 tubes should be approximately 0.3 and

vertical spacing approximately 1.7.

Recommendations

As previously described, the goal of this study was to find and implement an

optimization technique to the previously existing finned-tube air-conditioning system

model. The fins used in this study are flat plate fins, but in the industry enhanced surface

fins have become common. Therefore, for future studies, enhanced surface fins are a

recommended addition to the air-conditioning condenser simulation program. Though

the correlation used for calculating the fin efficiency is stated by Schmidt (1945) to be

50

accurate for non-symmetric fins, it should be validated for the highly non-symmetric

optimal design in the current study.

The circuiting of the condenser coil has a major effect on the air-conditioning

system total performance. In future studies there is an opportunity to enhance the air-

conditioner simulation program and allowing more design options for the circuiting such

as varying the number of tubes per parallel circuit through the condenser.

Due to limitations in air-side pressure drop and heat transfer correlations,

condensers with smaller tube diameter than 5/16 have not been studied. Since the

optimum in this study occurs for a condenser with 5/16 tubing it could be that a smaller

diameter would give higher COP. This could depend on the on the available smaller

diameter tubing wall thickness. It is therefore recommended that condensers with smaller

tubes be included in future studies, with valid pressure drop and heat transfer

correlations.

This study uses a system component models developed from experimental

correlations to simulate the air-conditioning system performance. No experiments have

been run to verify the system model. In future studies it is recommended that the air-

conditioner system model be compared to experimental system performance values.

51

REFERENCES

AAON, AAON Heating and Air-Conditioning Products web site, http://www.aaon.com/, 2001

EPA, United States Environmental Protection Agency: Ozone Depletion Home Page, http://www.epa.gov/ozone/, 2001.

Incropera, F. P. and DeWitt, D. P., Fundamentals of Heat and Mass Transfer, 4th Edition, John Wiley & Sons, New York, p. 445, 1996.

Kays, W. M. and London, A. L., Compact Heat Exchangers, 3rd Edition, McGraw-Hill, New York, 1984.

Klein, S. A. and Reindl, D. T., The Relationship of Optimum Heat Exchanger Allocation and Minimum Entropy Generation for Refrigeration Cycles, Proceedings of the ASME Advanced Energy Systems Division, vol. 37, pp. 87-94, 1997.

Lagarias, J. C., Reeds, J. A., Wright, M. H. and Wright, P. E., Convergence properties of the Nelder-Mead Simplex method in low dimensions, SIAM J. Optim., Vol. 9, No1, pp. 112-147, 1998.

McQuiston, F. C. and Parker, J. P., Heating Ventilating and Air-Conditioning-Analysis and Design, John Wiley & Sons, New York, 1994.

Nelder, J. A. and Mead, R., A Simplex Method for Function Minimization, Computer J., 7 308-313 (1965)

Reklaitis, G. V., Ravindran, A., Ragsdell, K. M., Engineering Optimization, Methods and Applications, John Wiley and Sons, 1983.

Rich, D. G., The Effect of Fin Spacing on the Heat Transfer and Friction Performance of Multi-Row, Smooth Plate Fin-and-Tube Heat Exchangers, ASHRAE Transactions, vol. 79, pt. 2, pp. 137-145, 1973.

Schmidt, T. E., La Production Calorifique des Surfaces Munies dailettes, Annexe Du bulletin De LInstitut International Du Froid, Annexe G-5, 1945.

Shah, M. M., A General Correlation For Heat Transfer During Film Condensation Inside Pipes, Heat and Mass Transfer, vol. 22, pp. 547-556, 1979.

Threlkeld, J. R., Thermal Environmental Engineering, 2nd Edition, Prentice Hall International, New York, pp.55, 1970.

Wenzel, T. P., Koomey, J. G., Rosenquist, J. G., Sanchez, M., and Hanford, J. W., Energy data sourcebook for the U.S. residential sector, Lawrence Berkeley National Laboratory, 1997. LBL-40297.

52

Wright, M. F., Plate-Fin-and-Tube Condenser performance and Design for Refrigerant R-410a Air-Conditioner, M.S. Thesis Georgia Institute of Technology, May 2000

Zukauskas, A. and Ulinskas, R., Banks of Plain and Finned Tubes, Heat Exchanger Design Handbook, G. F. Hewitt Edition, Begell House, Inc., New York, pp. 2.24-1 2.24-17, 1998.

Research ObjectivesAir Conditioning System and Component ModelingCompressorCondenserCondenser FanExpansion ValveEvaporatorEvaporator FanRefrigerant Mass Inventory

The figure of meritSimplex search methodOne iteration of the Nelder-Mead Simplex search algorithmSoftware ToolsOptimization ParametersOptimal Condenser DesignConclusionsGeneral Design GuidelinesRecommendations