Performance prediction of a vapour-compression heat-pump

APPLIED

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www.elsevier.com/locate/apenergy

Applied Energy xxx (2004) xxx–xxxENERGY

Performance prediction of avapour-compression heat-pump

Adnan S€ozen a,*, Erol Arcaklio�glu b, Ali Erisen b,M. Ali Akc�ayol c

a Mechanical Education Department, Technical Education Faculty, Gazi University,

06503 Teknikokullar, Ankara, Turkeyb Mechanical Engineering Department, Engineering Faculty, Kırıkkale University,

71450 Kırıkkale, Turkeyc Computer Engineering Department, Engineering and Architecture Faculty,

Gazi University, 06570 Maltepe, Ankara, Turkey

Accepted 18 December 2003

Abstract

The performance of the heat pump was predicted using a fuzzy-logic controller under

various working-conditions and mixing ratios of R12/R22 refrigerant mixtures, instead of

requiring an expensive and time consuming experimental study [Int. J. Ref. 13 (1990) 163].

Fuzzy-logic’s linguistic terms provide a feasible method for defining the performance of the

heat pump. Input data for the fuzzy logic are mixing ratio, evaporator-inlet temperature and

condenser pressure. In the comparison of performance, results obtained via analytic equations

and by means of the fuzzy-logic controller, the coefficient of performance (COP), and rational

efficiency (RE) for all working situations differ by less than 1.5% and 1%, respectively. The

statistical coefficient of multiple determinations (R2-value) equals to 0.9988 for both the COP

and the RE. With these results, we believe that fuzzy logic can be used for the accurate pre-

diction of the COP and the RE of the heat pump.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Heat pump; Performance; Fuzzy logic; Prediction

*Corresponding author. Tel.: +90-312-212-6820; fax: +90-312-212-0059.

E-mail address: [email protected] (A. S€ozen).

0306-2619/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apenergy.2003.12.013

mail to: [email protected]

2 A. S€ozen et al. / Applied Energy xxx (2004) xxx–xxx

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1. Introduction

Exact theoretical performance analysis of the heat pump is too complex due to the

equations of the thermodynamic properties of the working fluid and the simulation

programs [1]. Therefore, as the thermodynamic analysis requires many variables, a

fuzzy-logic controller is used. Classical control methods can be implemented in well-defined systems to achieve good performance-predictions for the systems. To control

a system, an accurate mathematical model of its behaviour has to be obtained.

Systems with non-linear behavioural structures cannot be easily modeled. Fuzzy-

logic control has adaptive characteristics, which can achieve robust responses to

uncertainties, parameter variations, and load disturbances. Fuzzy logic, or fuzzy-set

theory, was the first presented by Zadeh [2]. Since the introduction of fuzzy logic,

many researchers have studied modeling of complex systems, and fuzzy-logic con-

trollers have been used to control ill-defined, non-linear, or imprecise systems [2–5].During the past decade, fuzzy logic has found a variety of applications in various

fields ranging from industrial-process control [6] to medical diagnosis [7], and to

securities in trading [8]. Most notably, a fuzzy-logic system has been applied to

control non-linear, time-varying, ill-defined systems [6], to control systems whose

dynamics are exactly known, such as servomotor position control [9,10] and robot-

arm control [11], and to manage complex decision-making or diagnosis systems [8,9].

Quail and Adnan [12] have described fuzzy-logic applications in dozens of household

appliances and Takagi [13] has presented a large suite of electronic components inthe general area of image-processing equipment. Camera manufacturers have ap-

plied fuzzy logic to decide upon which object, in the field-of-view, the camera should

be focused, and then to control the auto-focus mechanism to focus on that object

[14]. Fuzzy-logic controllers have been applied to electrical machines to control

speed and position successfully. A fuzzy-logic controller has been developed for a

brushless direct-current motor [15]. Cheok and Ertugrul [16] have described a fuzzy-

logic controller for a switched-reluctance motor.

In this study, we conducted experiments on the vapour-compression heat-pumpusing different ratios of R12/R22 refrigerant mixtures. The conducted experiments

and corresponding calculated coefficient of performance (COP) and rational effi-

ciency (RE) results are used in the application of a fuzzy-logic controller. After-

wards, we briefly describe the fuzzy logic. Subsequently, we calculated the COP and

the RE of the pump using numerical methods, as explained in Section 3, while

Section 4 describes the modeling of the system with fuzzy logic and finally Section 5

contains the results and conclusions.

2. Fuzzy-logic controller

Fuzzy-logic explains the operational laws by means of linguistic (rather than

complex mathematical) equations. Since it is difficult to model many systems accu-

rately with these complex equations, usage of traditional control methods are gen-

erally not viable. But, fuzzy-logic’s linguistic terms provide a feasible method for

Rule Base

Decisionmaking logic

Database

DefuzzificationFuzzification

Fuzzy inference

Inputs Outputs

Fig. 1. Fuzzy-logic controller block diagram.

A. S€ozen et al. / Applied Energy xxx (2004) xxx–xxx 3

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defining the operational characteristics of such systems [6]. The main advantage of

the fuzzy logic is that it does not require a complex mathematical model and thus, it

has the advantage of using relatively simple mathematical calculations for the rule

processing [17]. In addition, it has been shown that the fast universal computationscheme corresponds exactly to the operations in fuzzy-logic methods using max–min

composition [18]. In a fuzzy-logic controller, the dynamic behaviour of a fuzzy

system is characterized by a set of linguistic-description rules based on expert

knowledge [19]. The expert knowledge is usually in the form of:

IF (input1 is big) and/or (input2 is small)� � �(inputN is medium) THEN

(output1 is negative big) and (output2 is positive small)� � �(outputM is

zero).

Basically, fuzzy-control rules provide a convenient way for expressing control

policy and domain knowledge. Furthermore, several linguistic variables might be

involved in the antecedents (before then) and the conclusions (after then) of these

rules. Fuzzy-logic controllers can be considered as a special class of symbolic con-

trollers. The configuration of the fuzzy-logic controller block diagram is shown in

Fig. 1. The three features of symbolic controllers are fuzzification, fuzzy inference

and defuzzification. In the fuzzification step, the real variables are translated intolinguistic variables by using fuzzy-set theory. In the fuzzy-inference step, ‘‘if-then’’

rules, that define the system behaviour, are evaluated. The defuzzification step

translates the linguistic result obtained from the fuzzy inference into again a real

value by using the rule base provided.

2.1. Fuzzification

Multiple measured crisp inputs first have to be mapped into fuzzy-membershipfunctions. This process is called fuzzification [20]. The fuzzification process requires a

good understanding of all the variables. Fuzzy-logic’s linguistic terms are most often

expressed in the form of logical implications, such as If-Then rules [6]. These rules

m

u

1

Val1 Val2 Val3

1

uVal1 Val2 Val3 Val4

1

u

m

Xp

w

m

(a) (b) (c)

Fig. 2. Triangle (a), trapezoid (b), and bell (c) membership functions.


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define a range of values known as fuzzy-membership functions. These may be in theform of a triangle, a trapezoid, a bell (as seen in Fig. 2) or another appropriate form.

lðuiÞ is a membership function, which is used to calculate the membership degree of

the crisp input value. The range of lðuiÞ is between zero and unity [20].

Triangle membership function is defined by Eq. (1). Val1, Val2, and Val3 define

triangle membership functions’ limits

lðuiÞ ¼

ui � Val1

Val2� Val1; Val16 ui 6Val2;

Val3� uiVal3� Val2

; Val26 ui 6Val3;

0; otherwise:

8>>>>><>>>>>:

ð1Þ

Trapezoid membership function is defined by Eq. (2). Val1, Val2, Val3, and Val4

define the trapezoid membership functions’ limits

lðuiÞ ¼

ui � Val1

Val2� Val1; Val16 ui 6Val2;

1; Val2 < ui < Val3;

Val4� uiVal4� Val3

; Val36 ui 6Val4;

0; otherwise:

8>>>>>>>>><>>>>>>>>>:

ð2Þ

Bell membership functions are defined by parameters Xp, w and m as follows:

lðuiÞ ¼ 1 1

,þ jui � Xpj

w

� �2m!; ð3Þ

where Xp is the mid-point, and w is the width of the bell function. mP 1, anddescribes the convexity of the bell function.

The inputs of the fuzzy controller are expressed in several linguistic levels. An

example can be seen in Fig. 3. These levels can be described as positive big (PB),

µNB NM NS Z PS PM PB

input0

1

-30 -20 -10 0 10 20 307.5

0.25

0.75

Fig. 3. Seven levels of fuzzy membership function.


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positive medium (PM), positive small (PS), zero (Z), negative small (NS), negativemedium (NM), negative big (NB), or another. Each level is described by a fuzzy set.

The linguistic-variable input allows for the translation of a measured input into its

linguistic description. For example, a measured input of 7.5 is translated into the

linguistic variable of (0.25*Z, 0.75*PS) which can be interpret as ‘‘positive small or

may be zero’’. An input of 7.5 is a member in the fuzzy sets of the terms:-

• positive small to the degree of 0.75,

• zero to the degree of 0.25.

Membership degree is calculated by using a triangle membership function as definedby Eq. (1).

In general, experience and expertise are required for the implementation of

fuzzification in complex systems.

2.2. Fuzzy inference

The second phase of the fuzzy-logic controller is its fuzzy inference where the

knowledge base and the decision-making logic reside [21]. The rule base and thedatabase form the knowledge base. The database contains descriptions of the input

and output variables. In this study, parameters of the membership functions for the

input and output variables are stored in the database. The count of data in the

database depends on how many inputs and outputs are used for the fuzzy-logic

controller and how many fuzzy sets are used in each input and output. The decision-

making logic evaluates the control rules. The control rule base can be developed to

relate the output actions of the controller to the obtained inputs. Fuzzy inference

uses linguistic terms (fuzzy sets), selected by fuzzification, for producing outputlinguistic terms by using the rule base. The rule base contains if-then rules between

inputs and output.

2.3. Defuzzification

The outputs of the inference mechanism are fuzzy-output variables. The fuzzy-

logic controller must convert its internal fuzzy-output variables into crisp values, so

that the actual system can use these variables. This conversion is called defuzzifi-cation [22]. One may perform this operation in several ways. One of the most

common ways is the use of the height method.


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In the height method [23], the centroid of each membership function for each rule

is firstly evaluated [19]. The final output, U0, is then calculated as the average of the

individual centroids, weighted by their heights as follows:

U0 ¼Pn

i¼1 uilðuiÞPni¼1 lðuiÞ

; ð4Þ

where lðuiÞ is the minimum/maximum (depend on and/or operator) value of the

membership degree of input values.

3. Experimental set-up description and related calculations

The experiments have been performed for a vapour-compression heat pump. The

main components of this system are the hermetic compressor, condenser with water-

cooled evaporator, with air cooling and a thermostatic expansion-valve.

The parameters measured are the temperatures of the refrigerant entering and

leaving the compressor, condenser, and evaporator; temperatures of the cooling

water entering and leaving the condenser; temperatures of the air (dry and wet-

thermometer values) entering and leaving the evaporator; inlet and outlet pressuresof the evaporator and condenser; the electric current, voltage and cosu values for

the compressor; and the flow rates of the refrigerant, cooling water and air.

The evaporator’s inlet-temperatures selected are )4.7, )3, 0.5, 3.5, 5, 6, 8.6 and

10 �C; and condenser pressures ranged from 800 to 1400 kPa. Initially, 500 g of R12

was loaded as a reference in the experiments, and values of R12 were measured.

Afterwards, experiments were done using the R12/R22 mixture in composition

ranges of 0.1–0.5 mol with R22 fractional increases of 0.1.

The required parameters to be used in the fuzzy logic are as follows: COP is theratio of heating capacity to the compressor work. The heating capacity is obtained

by measuring the inlet and outlet temperatures and the mass-flow rate of the cooling

water. The compressor work is calculated using the voltage, electric current and

cosu values, measured from the experimental plant. The RE is obtained by an ex-

ergy balance. For heat pumps, there are no streams of matter crossing the system

boundary. So, for the heat pump, the exergy terms associated with the flow of matter

are zero. Only the exergy input is the compressor work [24].

For all measurements, numerical calculations of performance values such as COP,total irreversibility, and RE have been performed. When the system reaches the

steady-state condition, for the mixing ratio, the evaporator temperature and con-

denser pressure values – in the heat pump – the other measurement are accom-

plished. Therefore, by using these three variables we may obtain the performance of

the system from a fuzzy-logic controller.

4. Application of fuzzy-logic controller

To estimate the COP and the RE in the system by using fuzzy-set theory, the

condenser pressure (kPa), evaporator temperature (�C), R12% (mass ratio) are

Rule Base

Database

Generation ofinferencerelations

Fuzzification Decisionmaking

DefuzzificationCrisp inputvariables

Fuzzy inputvariables

Fuzzy outputvariables

Crisp outputvariables

R12%,kPa,˚C dRE, dCOP

Fig. 4. The configuration of the fuzzy-logic estimator.

µ

700 1400800 900 1000 1100 1200 1300

Membership functions of kPa

-4.7 10.0-3.0 0.5 3.5 5.0 6.0 8.6

Membership functions of °C

0.5 0.6 0.7 0.8 0.9 1.0

Membership functions of R12%

µ

4.104 7.919

Membership functions of COP

µ

0.5202 0.9398

Membership functions of RE

Fig. 5. The membership functions of kPa, �C, R12%, COP and RE variables.


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employed. The fuzzy estimator is designed to process fuzzy-quantities only. There-

fore, the crisp input values of kPa, �C and R12% must be converted to fuzzy sets

before being used. This process is called the fuzzification operation. Fuzzy outputs

were calculated using the max–min composition method. Then crisp outputs of the

COP and the RE were determined by the centre-of-gravity method.

The inputs and outputs of the fuzzy estimator are expressed in several linguisticlevels. Each level is described by a fuzzy set. Fig. 4 shows the configuration of the

fuzzy block.

Fuzzy-input variables are kPa, �C, R12% and fuzzy-output variables COP and

RE. Eight linguistic levels for kPa, �C, 6 linguistic levels for R12%, as well as 26

linguistic levels for COP and RE have been chosen. In this study, a total of 254

experimentally-obtained values are stored in the database: among these there are

24 for kPa, 24 for �C, 18 for R12%, 78 for COP and 78 for RE. The mem-

bership functions of the fuzzy variables (kPa,�C, R12%, COP, RE) are shown inFig. 5.

To explore the effectiveness of the proposed modeling, the results obtained from

fuzzy-based modeling and experimental data have been compared.

5. Results and conclusions

Figs. 6 and 7 show a parity plot between the compressibility factors generated byexperimental data and those computed by fuzzy logic for the COP, and the RE for

all working parameters, respectively. R2-values obtained are equal to 0.9988 for both

the COP and the RE.

R2 = 0.9988

3.000

4.000

5.000

6.000

7.000

8.000

9.000

10.000

11.000

3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 11.000

Predicted Value (COP)

Act

ual

Dat

a (C

OP

)

Fig. 6. Comparison of the actual data and fuzzy logic results for the COP.

R2 = 0.9988

0.5000

0.5500

0.6000

0.6500

0.7000

0.7500

0.8000

0.8500

0.9000

0.9500

1.0000

0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000

Predicted Value (RE)

Act

ual

Dat

a (R

E)

Fig. 7. Comparison of the actual data and fuzzy logic results for the RE.


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It is clear that the fuzzy-logic model gives a very accurate representation of the

statistical coefficient of multiple determinations (R2-value) over the full range of

operating conditions. The standard deviations of COP and RE in Figs. 8–23 indicate

the high accuracy of the fuzzy logic. The maximum percentage difference between the

simulated and predicted results is 1%. Consequently, we believe that fuzzy logic can

be used successfully for this type of prediction.

Since performance values obtained by analytical functions are very close to thecalculated values by using the fuzzy-logic controller, they cannot easily be

Fig. 8. Variation of the dCOP with R12% at different condenser pressures for Te ¼ �4:7 �C.

Fig. 9. Variation of the dCOP with R12% at different condenser pressures for Te ¼ �3 �C.

Fig. 10. Variation of the dCOP with R12% at different condenser pressures for Te ¼ 0:5 �C.


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graphically shown together. For this reason, the following parameters (Eqs. (5) and

(6)) have been calculated as deviations in values, and these have been shown

graphically:

Fig. 12. Variation of the dCOP with R12% at different condenser pressures for Te ¼ 5 �C.



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dCOP ¼ COPfuzzy � COPcalculated

COPcalculated

; ð5Þ

dRE ¼ REfuzzy �REcalculated

REcalculated

: ð6Þ

Fig. 16. Variation of the dRE with R12% at different condenser pressures for Te ¼ �4:7 �C.



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Values of the deviation in the coefficient of performance (dCOP) are given in Figs. 8–

15 for different working-temperatures. According to the results obtained, deviations

in coefficient of performance of the system are in the range 0.5–1.5% at different

Fig. 18. Variation of the dRE with R12% at different condenser pressures for Te ¼ 0:5 �C.

Fig. 17. Variation of the dRE with R12% at different condenser pressures for Te ¼ �3 �C.


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working temperatures. Maximum and minimum deviations of COP are given in

Table 1.

Deviations in rational efficiency (dRE) are given in Figs. 16–23 for different

working-temperatures. According to the results obtained, deviations in rational

Fig. 20. Variation of the dRE with R12% at different condenser pressures for Te ¼ 5 �C.



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efficiency of the system are in the range 0.5–1% at different working temperatures.

Maximum and minimum deviations of RE are given in Table 2.

Use of the fuzzy-logic controller in a refrigeration system shows that simulation

values can be obtained with a simple approach instead of experimental studies.




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The advantages of fuzzy logic are speed of calculation, simplicity and capability of

learning from examples. All of these advantages have been observed in our appli-

cation. These features enable us to use fuzzy logic for thermal systems and will help

Table 1

Maximum and minimum deviations of COP

R12% Pc (kPa) Te (�C) COP COPFLC dCOP

Minimum 0.7 1000 6.0 6.893 7.004 )0.111Maximum 1.0 1200 0.5 4.963 4.868 0.095

Table 2

Maximum and minimum deviation values of RE

R12% Pc (kPa) Te (�C) RE REFLC dRE

Minimum 0.8 1000 3.5 0.7959 0.8055 )0.0096Maximum 0.7 1300 5.0 0.6295 0.6210 0.0085



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those studying in this field. So, experimental studies can be reduced to a minimum

where the use of fuzzy logic is appropriate. Therefore, in other applications,involving heat pumps, the use of the fuzzy logic in similar circumstances can be

recommended as a test-bed.

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Documents

Performance prediction of a vapour-compression heat-pump