Applied Energy 113 (2014) 982–989

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Applied Energy

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An optimization method for gas refrigeration cycle based on thecombination of both thermodynamics and entransy theory

0306-2619/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.apenergy.2013.08.039

⇑ Corresponding author. Tel.: +86 10 62796332.E-mail address: chenqun@tsinghua.edu.cn (Q. Chen).

Qun Chen ⇑, Yun-Chao Xu, Jun-Hong HaoKey Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

h i g h l i g h t s

� An optimization method for practical thermodynamic cycle is developed.� The entransy-based heat transfer analysis and thermodynamic analysis are combined.� Theoretical relation between system requirements and design parameters is derived.� The optimization problem can be converted into conditional extremum problem.� The proposed method provides several useful optimization criteria.

a r t i c l e i n f o

Article history:Received 8 March 2013Received in revised form 17 August 2013Accepted 17 August 2013

Keywords:Gas refrigeration systemOptimizationThermodynamicsEntransy

a b s t r a c t

A thermodynamic cycle usually consists of heat transfer processes in heat exchangers and heat-work con-version processes in compressors, expanders and/or turbines. This paper presents a new optimizationmethod for effective improvement of thermodynamic cycle performance with the combination ofentransy theory and thermodynamics. The heat transfer processes in a gas refrigeration cycle are ana-lyzed by entransy theory and the heat-work conversion processes are analyzed by thermodynamics.The combination of these two analysis yields a mathematical relation directly connecting system require-ments, e.g. cooling capacity rate and power consumption rate, with design parameters, e.g. heat transferarea of each heat exchanger and heat capacity rate of each working fluid, without introducing any inter-mediate variable. Based on this relation together with the conditional extremum method, we theoreti-cally derive an optimization equation group. Simultaneously solving this equation group offers theoptimal structural and operating parameters for every single gas refrigeration cycle and furthermoreprovides several useful optimization criteria for all the cycles. Finally, a practical gas refrigeration cycleis taken as an example to show the application and validity of the newly proposed optimization method.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Refrigeration cycle, a typical thermodynamic system, has beenwidely utilized in every corner of the world to provide specificenvironmental conditions. Effective improvement of refrigerationcycle performance has been an attractive but tough issue in bothresearch and engineering fields, which on one hand has huge po-tential for energy conservation, but on the other hand is a multi-parameter coupling problem including several different physicalphenomena.

During the past several decades, a large number of approachesfor thermodynamic cycle performance improvement have beenexplored and employed in engineering, which can be classified intotwo categories. One category is to find a better cycle with highertheoretical conversion efficiency between thermal energy and

power including gas–steam combined cycles [1], integrated gasifi-cation combined cycles (IGCC) [2], and combined cooling, heatingand power (CCHP) systems [3]. The other category is to improvethe existing system’s performance, where researchers usually listseveral possible combinations of such structural and operatingparameters as pressures, temperatures and mass flow rates, esti-mate their influences on the system performance, and finally finda better solution by experiments or some simulation-based optimi-zation methods including gradient-based algorithm, genetic algo-rithm and neural network algorithm [4–6]. These methods cangreat improve the system performance, especially very complexsystems, and have successfully reduced not only the energy con-sumption but also the equipment cost. However, compared withthe theoretical optimization methods, these methods share weakerphysical analyses and rely more on the computer to empiricallychoose a better solution among huge amounts of possible parame-ter configurations.

Nomenclature

A area, m2

cp constant pressure specific heat, J kg�1 K�1

G entransy flow rate, W KK heat transfer coefficient, W m�2 K�1

KA thermal conductance, W K�1

m mass flow rate, kg s�1

n polytropic indexp pressure, Paq heat flux, W m�2

Q heat flow rate, WT temperature, KW0 power consumption rate, WP Lagrange function

Ug entransy dissipation rate, W Kk Lagrange multipliern fluid arrangement factor of heat exchanger

Subscriptsa airC compressionE expansionh hot fluid, hot end of thermodynamic cyclein inletl cold fluid, cold end of thermodynamic cycleout outletc adiabatic exponent

Q. Chen et al. / Applied Energy 113 (2014) 982–989 983

In parallel, many researchers optimize thermodynamic cycleperformance from the viewpoint of irreversibility. Curzon and Ahl-born [7] and Salamon et al. [8,9] focused on the power output max-imization of Carnot cycles and mentioned that when the area ofeach heat exchanger is specified, the finite-time irreversibility ofheat transfer should be taken into account. Afterwards, Bejanet al. [10–13] optimized several thermodynamic cycles includingboth power and refrigeration plants by allocating the areas of eachheat exchanger based on the combination of finite-time thermody-namics and entropy theory. Thenceforth, this method attractedattention of many researchers. They advocated meaningful effortson the optimization problems with different objectives throughthe area allocation of heat exchangers in different thermodynamiccycles. For instance, Grazzini and Rinaldi [14], Sahin and Kodal[15], Klein [16], and Ait-Ali [17] focused on the maximization ofthe coefficient of performance and the cooling capacity rates ofrefrigerators. Chen et al. [18] studied the total area variation ofall heat exchangers versus their area allocation at the hot and coldends to minimize the total area. Sarkar and Bhattacharyya [19] de-rived a brief expression of the total heat exchanger area in terms ofworking fluid temperatures for an irreversible refrigeration cycleand obtained the optimal fluid temperatures to minimize the totalarea.

All the aforementioned developments effectively improve thethermodynamic system performance. However, many studiesshare a common hypothesis that the thermodynamic cycle isdivided into three parts, i.e. two heat exchangers at hot and coldends with irreversible heat transfer processes and a thermody-namic cycle between them simplified as ideal Carnot cycle, whichcannot offer any information about such components as workingfluids, turbines and compressors inside the thermodynamic cycle.As a result, they can only optimize heat exchanger performancewithout considering the influence of other design parameters,such as the physical properties and heat capacity rates of workingfluids and the performance of turbines, compressors and expand-ers. Therefore, it is highly desired to develop a new method forthermodynamic cycle optimization with comprehensive consid-erations of both heat-work conversion and heat transferperformance.

Chen et al. [20] applied the finite-time thermodynamics toanalyze an air refrigeration cycle and maximized the COP and thecooling load of the cycle. Thereafter, Chen et al. [21] optimizedthe cycle with the aim of maximizing the exergetic efficiency. Tuet al. [22] maximized the COP of a real air-refrigerator by findingoptimal allocation of heat exchanger inventory and consideredthe influencing factors, e.g. the total heat exchanger inventoryand the efficiency of compressor and expander. Liu et al. [23]discussed the relation between the optimal COP and pressure ratio.

They have made some contributions for the theoretical optimiza-tion of the air refrigeration cycle. However, in order to make thecomplex optimization simpler, they always reduce the degrees offreedom of the system to be single degree via fixing other designparameters, which can lead to good local optimization results butmay miss the global optimal configuration of all the designparameters.

For heat transfer, Guo et al. [24] recently introduced the physi-cal quantities of entransy and entransy dissipation to respectivelyrepresent the heat transfer ability of an object during a time periodand describe the irreversibility of a heat transfer process. Further-more, they proposed an entransy dissipation-based method foroptimization of heat transfer elements [25–29], heat exchangers[30–34], and heat exchanger networks [35–37] in practical engi-neering applications. Besides, Chen et al. extended entransy theoryto analyze and optimize mass transfer [38] and coupled heat andmass transfer processes [39–41] based on the analogy betweenheat and mass transfer.

In order to consider the combined influence of working fluids,heat exchangers, turbines and compressors on thermodynamiccycle optimization, this paper proposes an optimization methodbased on the combination of both thermodynamics and entransytheory. Gas refrigeration cycle is taken as an example and analyzedthrough not only thermodynamic analysis for heat-work conver-sion processes in a compressor and an expander, but also entransyanalysis for heat transfer processes in heat exchangers. Both anal-yses are combined to develop a new method for the structural andoperation parameter optimization. Finally, a practical gas refriger-ation cycle is taken as an example to show the application andsuperiority of the newly proposed method.

2. Analysis of a gas refrigeration cycle

Fig. 1 shows the sketch of a typical gas refrigeration cycle con-sisting of a compressor, an expander and two counter-flow heatexchangers at the hot and cold ends, HXh and HXl. At the coldend, the fluid from low-temperature environment flows into theHXl and heats the gas, i.e. the working fluid in the refrigeration cy-cle, from the temperature T4 to T1. The heated gas leaves the HXl

and enters the compressor, C, where gas is compressed from thepressure p1 to p2 and its temperature correspondingly rises to T2.After compression, the gas enters the HXh and is cooled to T3 bythe fluid from high-temperature environment. Finally, the gasenters the expander, E, and expands from the pressure p3 to p4

and its temperature drops to T4. In Fig. 1, W1 and W2 are the inputand output works of the compressor and the expander, respec-tively, m is the mass flow rate, and the subscripts a, l, h, in and

Fig. 1. The sketch of a gas refrigeration cycle.

984 Q. Chen et al. / Applied Energy 113 (2014) 982–989

out represent the gas, the fluid from high-temperature environ-ment, the fluid from low-temperature environment, inlet and out-let, respectively.

In this cycle, there are such two different categories of thermalprocesses as heat transfer processes in both heat exchangers andheat-work conversion processes in both compressor and expander.Because heat transfer processes depend on the temperature differ-ences of the working fluids in heat exchangers, while heat-workconversion processes always depend on the absolute temperaturesof the working fluids. These two processes need to be analyzed bytwo different theories separately for heat transfer analyses andthermodynamic analyses.

2.1. Entransy analysis

The entransy introduced by Guo et al. [24] is defined asfollowing:

G ¼ 12

UT; ð1Þ

where U and T are respectively the internal energy and the temper-ature of an object. During heat transfer, the entransy will be trans-ported and partly dissipated.

For the counter-flow heat exchanger at the hot end, HXh, theheat transfer rate, Qh, is expressed as:

Q h ¼ macp;aðT2 � T3Þ ¼ mhcp;hðTh;out � Th;inÞ: ð2Þ

where cp,a is the constant pressure specific heat of the gas, and cp,h isthe constant pressure specific heat of the fluid.

Based on the definitions of entransy and entransy dissipation[30,33,34], the variation of entransy flux of specific flow in the heatexchanger equals to the entransy flux at the inlet of the flow minusthat at the outlet:

Gin � Gout ¼12ðmcpT2Þin �

12ðmcpT2Þout; ð3Þ

where G is the entransy flow rate accompanying fluid flow.The entransy dissipation rate during the heat transfer process in

the HXh is calculated by

Ug;h ¼ Ga;in � Ga;out þ Gh;in � Gh;out

¼ 12

macp;aðT22 � T2

3Þ �12

mhcp;h T2h;out � T2

h;in

� �¼ 1

2Q hðT2 � Th;out þ T3 � Th;inÞ ð4Þ

On the other hand, the entransy dissipation rate in the HXh isalso calculated as [30]:

/g;h ¼Z Qh

0ðTa � ThÞdq; ð5Þ

where q is heat flux.Based on Eq. (5), the entransy dissipation ratecan be expressed as the function of the thermal conductance of heatexchanger, i.e. the product of heat transfer coefficient and area KAh,and the heat capacity rates of fluids, macp,a and mhcp,h, which isexpressed as [33]:

Ug;h ¼12

Q 2hnh

eKAhnh þ 1eKAhnh � 1

: ð6Þ

where nh ¼ 1mhcp;h

� 1macp;a

.The combination of Eqs. (4) and (6) gives a fundamental and

mathematical relation described in Eq. (7), which connects the in-let and outlet temperatures of the fluids, i.e. Th,in, Th,out, T2 and T3, tothe thermal conductance of HXh, KAh, the heat capacity rates offluids, macp,a and mhcp,h,

T2 � Th;out þ T3 � Th;in ¼ Q hnheKAhnh þ 1eKAhnh � 1

: ð7Þ

Substituting Eq. (2) into Eq. (7) yields

T2 ¼ Th;in þQ hNh1

2; ð8Þ

and

T3 ¼ Th;in þQ hNh2

2: ð9Þ

where

Nh1 ¼ 1mhcp;h

þ 1macp;a

þ nheKAhnhþ1eKAhnh�1

and Nh2 ¼ 1mhcp;h

� 1macp;a

þ nheKAhnhþ1eKAhnh�1

.

Similarly, the mathematical relation between the inlet andoutlet fluid temperatures and the design parameters in the HXl is

T1 ¼ Tl;in �Q lNl1

2; ð10Þ

and

T4 ¼ Tl;in �Q lNl2

2: ð11Þ

where Nl1¼1

mlcp;l� 1

macp;aþnl

eKAlnlþ1eKAlnl�1

;Nl2¼1

mlcp;lþ 1

macp;aþnl

eKAlnlþ1eKAlnl�1

and

nl¼ 1mlcp;l� 1

macp;a:

The relations described in Eqs. (5)–(11) physically exist for theheat transfer processes in counter-flow heat exchangers no matterwhich theory is used to deduce it. For the counter-flow heatexchanger, both the entransy theory and the heat exchanger effi-ciency can obtain the same relations. However, as studied in theliterature [33], in the LMTD method, it is inevitable to introducea correction factor to adjust the effective temperature differencesfor heat exchangers with more complex structures, e.g. TEMAE-type shell-and-tube heat exchangers. In the effectiveness –NTU method, the fluid with the minimum heat capacity rate hasto be first taken as the benchmark to calculate both effectivenessand NTU. Therefore, the similar relations like Eqs. (5)–(11) canhardly be theoretically deduced. But based on entransy theory,we can still obtain the relations through theoretical deductionwithout introducing any correction factor or benchmark.

2.2. Thermodynamic analysis

For the compressor and the expander in a practical cycle, wecan use the characteristic curves of compressor and expander toestablish the relation between pressures and temperatures. Here,if assuming that the gas satisfies the ideal-gas law and thecompression and expansion processes are both polytropic forsimplicity, the relations between temperatures and pressures are

p2

p1¼ T2

T1

� � nCnC�1

; ð12Þ

and

Q. Chen et al. / Applied Energy 113 (2014) 982–989 985

p3

p4¼ T3

T4

� � nEnE�1

: ð13Þ

where nC and nE are the polytropic indexes of the compression andexpansion processes, respectively.The gas pressure variations inboth the HXh and the HXl are usually much smaller than those inthe compressor and the expander, so they are ignored, i.e. p2 = p3,p1 = p4. Therefore, the combination of Eqs. (12) and (13) gives

T2

T1

� � nCnC�1

¼ T3

T4

� � nEnE�1

: ð14Þ

Meanwhile, during the compression and expansion processes,the heats released from the system to the environment are

Q C ¼ NCmacp;aðT1 � T2Þ; ð15Þ

and

Q E ¼ NEmacp;aðT3 � T4Þ; ð16Þ

where NC ¼ nC�ccðnC�1Þ ; NE ¼ nE�c

cðnE�1Þ, c is the adiabatic exponent. Basedon the energy conservation in the entire system, there is

Q h þ Q C þ Q E ¼ Q l þW0; ð17Þ

where W0 is the net power consumption rate.

2.3. Combination of entransy and thermodynamic analysis

The combination of Eqs. (2), (8), (9), (15)–(17) yields

T2 ¼2Th;in þ NEcNh1½Q l þW0 �macp;aNCT1 þmacp;aNET4�

2�macp;aNEcNh1NCE; ð18Þ

and

T3 ¼2Th;in þ NCcNh2½Q l þW0 � NCmacp;aT1 þ NEmacp;aT4�

2� NCcNh2NCEmacp;a; ð19Þ

where NEc ¼ nEc�cnEc�nE

; NCc ¼ nCc�cnCc�nC

and NCE ¼ NC � NE.

Substituting Eqs. (10), (11), (18), and (19) into Eq. (16) yields

nC

nC � 1ln

2Th;in þ NEcNh1NQ

ð2�macp;aNEcNh1NCEÞ Tl;in � Ql2 Nl1

� �0@

1A

¼ nE

nE � 1ln

2Th;in þ NCcNh2NQ

ð2�macp;aNCcNh2NCEÞðTl;in � Ql2 Nl2Þ

!ð20Þ

where NQ ¼QlþW0�macp;aNCðTl;in� Ql2 Nl1Þþmacp;aNEðTl;in� Q l

2 Nl2Þ.Eq. (20) contains two types of variables: (1) the known conditionsincluding the design requirements, i.e. the cooling capacity rateand the net power consumption rate, and the boundary conditions,i.e. the mass flow rates and inlet temperatures of fluids from thehigh- and low-temperature environments; (2) the design parame-ters including the thermal conductances of each heat exchangerand heat capacity rates of gas. Therefore, Eq. (20) directly connectsall the known conditions with the design parameters without intro-ducing any intermediate variable, which has never appeared in re-lated optimization studies before and makes it possible totheoretically and globally optimize refrigeration cycles.

3. Optimization models for gas refrigeration cycles

For the refrigeration cycle shown in Fig. 1 with a required coolingcapacity and a prescribed power consumption, the working perfor-mance of selected compressor and expander in different operatingconditions are usually given, and the optimization objective is oftento minimize the total area of all heat exchangers. Because the heattransfer area is proportional to the thermal conductance when the

heat transfer coefficient is assumed constant, the optimizationobjective can be converted to minimizing the total thermal conduc-tance of all heat exchangers with the constraint of satisfying bound-ary conditions and design requirements, i.e. Eq. (20).

The conditional extremum method is applied to solve the opti-mization problem above, and a Lagrange function P is constructedas:

P ¼ KAh þ KAl þ knC

nC � 1ln

2Th;in þ NEcNh1NQ

ð2�macp;aNEcNh1NCEÞðTl;in � Ql2 Nl1Þ

!

� nE

nE � 1ln

2Th;in þ NCcNh2NQ

ð2�macp;aNCcNh2NCEÞðTl;in � Ql2 Nl2Þ

!!; ð21Þ

where k is the Lagrange multiplier.The differential of P with respectto KAh yields:

@P@KAh

¼ 1

þ knC

nC � 12ðmacp;aTh;inNEcNCE þ NEcNQ Þ

ð2Th;in þ NEcNh1NQ Þ 2�macp;aNEcNh1NCE� � @Nh1

@KAh

� nE

nE � 12ðmacp;aTh;inNCcNCE þ NCcNQ Þ

ð2Th;in þ NCcNh2NQ Þð2�macp;aNCcNh2NCEÞ@Nh2

@KAh

�¼ 0;

ð22Þ

where@Nh1

@KAh¼ @Nh2

@KAh¼ � 2n2

heKAhnh

ðeKAhnh � 1Þ2.

The differential of P with respect to KAl offers:

@P@KAl

¼ 1þ kQ lnC

nC � 12Th;in þ NEcNh1ðQ l þW0 � NEQ lÞð2Th;in þ NEcNh1NQ Þð2Tl;in � Nl1Q lÞ

@Nl1

@KAl

�

� nE

nE � 12Th;in þ NCcNh2ðQ l þW0 � NCQ lÞð2Th;in þ NCcNh2NQ Þð2Tl;in � Nl2QlÞ

@Nl2

@KAl

�¼ 0; ð23Þ

where @Nl1@KAl¼ @Nl2

@KAl¼ � 2n2

l eKAlnl

ðeKAlnl�1Þ2.

The differential of P with respect to macp,a provides:

@P@ðmacp;aÞ

¼ knC

nC � 1

NEcNh1ND þ NEcNQ@Nh1

@ðmacp;aÞ

2Th;in þ NEcNh1NQ

"

þ Q l

ð2Tl;in � Nl1Q lÞ@Nl1

@ macp;a� �þ NEcNCE

ð2�macp;aNEcNh1NCEÞðNh1

þmacp;a@Nh1

@ðmacp;aÞÞ�� knE

nE � 1

NCcNh2ND þ NCcNQ@Nh2

@ðmacp;aÞ

2Th;in þ NCcNh2NQ

"

þ Q l

ð2Tl;in � Nl2Q lÞ@Nl2

@ðmacp;aÞþ NCcNCE

ð2�macp;aNCcNh2NCEÞ

� Nh2 þmacp;a@Nh2

@ðmacp;aÞ

� ��¼ 0 ð24Þ

where

ND ¼macp;aQ lNC

2@Nl1

@ðmacp;aÞ� NCðTl;in �

Q lNl1

2Þ �macp;aQlNE

2

� @Nl2

@ðmacp;aÞþ NEðTl;in �

QlNl2

2Þ; @Nh1

@ðmacp;aÞ

¼ 2ðeKAhnh � KAhnheKAhnh � 1Þðmacp;aÞ2ðeKAhnh � 1Þ2

;@Nh2

@ðmacp;aÞ

¼ 2ðe2KAhnh � eKAhnh � KAhnheKAhnh Þðmacp;aÞ2ðeKAhnh � 1Þ2

;@Nl1

@ðmacp;aÞ

¼ 2ðe2KAlnl � eKAlnl � KAlnleKAlnl Þðmacp;aÞ2ðeKAlnl � 1Þ2

@Nl2

@ðmacp;aÞ

¼ 2ðeKAlnl � KAlnleKAlnl � 1Þðmacp;aÞ2ðeKAlnl � 1Þ2

Table 1The optimization results for the refrigerator.

Design parameter macp,a (W K�1) KAh (W K�1) KAl (W K�1)P

KA (W K�1)

Optimal value 489.8 86.0 86.0 172.0

Temperature T1 (K) T2 (K) T3 (K) T4 (K)Value 269.7 337.2 332.0 265.6

45 65 85 105 125165

184

203

222

241

260

(62,470) (72,170)

(75,510) (88,110)

(116,65)

(93,320) (112,350)

(152,300)

KA

tota

l [W

K-1

]

univariate analysis

entransy-based

KAh [WK-1]

(197,720)

Fig. 2. The comparison of the design results obtained by the univariate analysis andentransy-based methods.

800 1790 2780 3770 4760 5750

0

630

1260

1890

2520

3150

KA

[W

K-1

]Q

l [W]

KAl

KAh

mac

p,a

ma

cp,

a [

WK

-1]

450

470

490

510

530

Fig. 3. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the cooling capacity rate (adiabatic compression andexpansion).

986 Q. Chen et al. / Applied Energy 113 (2014) 982–989

.Eqs. (20) and (22)–(24) have four unknown quantities, i.e. KAh,KAl, macp,a and k, as many as the governing equations, and thussimultaneously solving the governing equations will easily obtainthe optimal values of design parameters to minimize the totalthermal conductance of the two heat exchangers in a refrigerationcycle.

In this contribution, the minimization of the total heat exchang-ers inventory is expected to act as an example to demonstrate thetheoretical analyzing steps and the fundamental and key optimiza-tion idea. It does not mean that the proposed method can only bevalid for the optimization of total heat exchangers inventory. Usinga similar method as that performed above, we can also optimizethermodynamic cycles with many other objectives, e.g. minimiza-tion of net power consumption, maximization of net power output,minimization of the fixed/operation or total cost and thermody-namic efficiency.

4. Optimization results for the gas refrigeration cycle

4.1. The refrigeration cycle with adiabatic compression and expansionprocesses

For the gas refrigeration cycle with adiabatic compression andexpansion processes, as shown in Fig. 1, where the inlet tempera-tures and heat capacity rates of fluids from the high- and low-tem-perature environments are Th,in = 303 K, Tl,in = 293 K, mhcp,h =500 W K�1 and mlcp,l = 480 W K�1. The cooling capacity rate andthe net power consumption rate are Ql = 2000 W andW0 = 500 W. The working gas is air and its adiabatic exponentequals to 1.4. The problem aims to find the best thermal conduc-tance allocation of each heat exchanger and heat capacity rate ofgas to minimize the total thermal conductance of both heatexchangers in the cycle.

Solving Eqs. (20) and (22)–(24) gives the optimal values for allthe design parameters as listed in Table 1. In this case, the heatexchangers at the hot and cold ends share the total thermal

conductance equally due to the reversible heat-work conversionsduring the adiabatic compression and expansion processes. A sim-ilar result has been obtained by Bejan [13] in his studies about thesystem involving reversible thermodynamic cycles. Besides, basedon Eqs. (8)–(12), we can easily obtain the gas temperatures alongthe cycle, as listed in Table 1, and the optimal gas compression ra-tio, p2/p1 = 2.2.

Fig. 2 shows the total thermal conductances of both heatexchangers with different design parameters. Two different kindsof dots stand for different design parameters, where the dots withcrosses are obtained by univariate analysis and the solid dot by thenewly developed entransy-based method. The numbers in thebrackets near each dot indicate the thermal conductance of thecold-end heat exchanger and the heat capacity rate of gas, respec-tively. It is obvious that the total thermal conductance of both heatexchangers reaches its minimum when the design parameters arethe optimized values listed in Table 1. On the other hand, the uni-variate analysis method can also lead to many different configura-tions, where the total thermal conductances are quite small insome cases, but hardly reach the best solution obtained by the en-transy-based method. Therefore, the entransy-based method issuperior to the univariate analysis in thermodynamic cycleoptimization.

If all the other known parameters remain unchanged except forthe influence factor to be studied (the same hereinafter), Figs. 3and 4 show the optimal thermal conductances in each heat ex-changer and the optimal heat capacity rate of gas versus the cool-ing capacity rate and the net power consumption rate, respectively,where the left and right vertical axes represent the thermal con-ductance of heat exchangers and the heat capacity rate of gas(the same hereinafter). In order to minimize the total thermal con-ductance in the entire refrigeration cycle, the heat capacity rate ofgas keeps constant, while the thermal conductances of both thehot-end and the cold-end heat exchangers should be enlarged withthe same margin with increasing the cooling capacity rate orreducing the net power consumption rate, i.e. increasing the en-

100 400 700 1000 1300 16000

100

200

300

400

500

W0 [W]

KAl

KAh

mac

p,a

450

470

490

510

530

ma

c p,a

[W

K-1

]

KA

[W

K-1

]

Fig. 4. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the net power consumption rate (adiabatic compressionand expansion).

248 258 268 278 288 2980

300

600

900

1200KA

l

KAh

mac

p,a

Tl [K]

450

470

490

510

530m

ac p

,a [

WK

-1]

KA

[W

K-1

]

Fig. 5. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the fluid inlet temperature from the low-temperatureenvironment (adiabatic compression and expansion).

0 170 340 510 680 85080

95

110

125

140

155

170

mlcp,l [WK -1]

50

200

350

500

650

KAl

KAh

mac

p,a

ma

c p,a

[W

K-1

]

KA

[W

K-1

]

Fig. 6. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the heat capacity rate of fluid from the low-temperatureenvironment (adiabatic compression and expansion).

900 1700 2500 3300 4100

0

400

800

1200

1600

2000

KA

[W

K-1

]

ma

c p,a

[W

K-1

]

Ql

[W]

0

80

160

240

320

400

KAl

KAh

mac

p,a

Fig. 7. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the cooling capacity rate (polytropic compression andexpansion).

Q. Chen et al. / Applied Energy 113 (2014) 982–989 987

ergy efficiency ratio. Meanwhile, for a given cooling capacity rate,it is also clear from Fig. 4 that the total heat exchanger thermalconductance and the net power consumption rate both have theirindividual minimum limits, and the minimum limit of net powerconsumption rate is larger than that calculated by an ideal Carnotcycle due to the non-isothermal heat transfer processes in both thehot-end and the cold-end heat exchangers. Furthermore, the meth-od can quantitatively find the minimum limit of the net powerconsumption.

Fig. 5 gives the optimal thermal conductance and the optimalheat capacity rate versus the fluid inlet temperature from thelow-temperature environment. When the low-temperature fluidinlet temperature increases, the thermal conductances of not onlythe cold-end but also the hot-end heat exchangers should be re-duced regardless of the constant high-temperature fluid inlet tem-perature. For prescribed net power consumption rate and coolingcapacity rate, increasing the low-temperature fluid inlet tempera-ture will enlarge the temperature differences in both hot-end andcold-end heat exchangers, and consequently reduce the thermalconductances of both heat exchangers. In this case, the heat capac-ity rate of gas still remains unchanged. Fig. 6 shows the optimalconductances and the optimal heat capacity rate with differentfluid heat capacity rates from the low-temperature environment.Increasing the fluid heat capacity rate will reduce the thermal con-ductances of both hot-end and cold-end heat exchangers, but need

a larger gas heat capacity rate in the cycle. Therefore, for a gasrefrigeration cycle with adiabatic compression and expansion pro-cesses, none but the heat capacity rates of fluids from the high-temperature and low-temperature environments will influencethe optimal heat capacity rate of gas in the cycle.

4.2. The refrigeration cycle with polytropic compression and expansionprocesses

If the compression and expansion processes in the refrigerationcycle are not adiabatic but polytropic, where the polytropic in-dexes of the compression and expansion processes are nC = 1.37,nE = 1.30 and all the other known parameters are the same as thosein the adiabatic case, Fig. 7 displays the variations of the optimalthermal conductances of each heat exchanger and the optimal heatcapacity rate of gas with different cooling capacity rates. Similar tothe optimized results in the adiabatic case, the thermal conduc-tances of both hot-end and cold-end heat exchangers should be en-larged with an increasing cooling capacity rate. However, theoptimal thermal conductances of heat exchangers do not equal toeach other, and the hot-end heat exchanger has a larger thermalconductance than the cold-end one. This is because the polytropicindex of the compression process is larger than that of the expan-sion process, nC > nE, and the heat absorbed during the expansionprocess is more than that released during the compression process,

150 300 450 600 750 9000

350

700

1050

1400

ma

cp,

a [

WK

-1]

KA

[W

K-1

]

KAl

KAh

mac

p,a

W0 [W]

0

60

120

180

240

300

360

Fig. 8. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the net power consumption rate (polytropic compressionand expansion).

1.19 1.26 1.33 1.40120

260

400

540

680

820

KA

[W

K-1

]

nC

105

207

309

411

513

ma

c p,a

[W

K-1

]

KAl

KAh

mac

p,a

Fig. 10. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the polytropic index of compression process.

272 278 284 290 2960

520

1040

1560

2080

2600

KAl

KAh

mac

p,a

Tl [K]

100

130

160

190

220

250

ma

c p,a

[W

K-1

]

KA

[W

K-1

]

Fig. 9. The optimal conductances of each heat exchanger and the optimal heatcapacity rate of gas versus the fluid inlet temperature from the low-temperatureenvironment (polytropic compression and expansion).

988 Q. Chen et al. / Applied Energy 113 (2014) 982–989

which leads to a larger thermal conductance of hot-end heat ex-changer. In addition, it is notable in Fig. 7 that the heat capacityrate of gas increases as the cooling capacity rate promotes, whichis obviously different from the result with adiabatic compressionand expansion, where the heat capacity rate of gas keeps constant.

Fig. 8 gives the variations of the optimal thermal conductancesof each heat exchanger and the heat capacity rate of gas with dif-ferent net power consumption rates. The optimal thermal conduc-tances of each heat exchanger and the optimal heat capacity rate ofgas all decrease with an increasing net power consumption rate. Alarger net power consumption rate in the system enlarges the tem-perature rise during the compression process, which increases thetemperature differences in both hot-end and cold-end heatexchangers, and consequently decreases the thermal conductancesof each heat exchanger as well as the heat capacity rate of gas.Fig. 9 shows the variations of the optimal thermal conductancesof each heat exchanger and the optimal heat capacity rate of gaswith different fluid inlet temperatures from the low-temperatureenvironment. As the fluid inlet temperature promotes, lower ther-mal conductances of both heat exchangers and heat capacity rateof gas are needed.

If the polytropic index of expansion process, nE, is fixed at 1.2,Fig. 10 shows the variations of the optimal thermal conductancesof each heat exchanger and the optimal heat capacity rate of gaswith different polytropic indexes of the compression process.When the polytropic index of the compression process equals to

that of the expansion process, nC = nE = 1.2, the thermal conduc-tances of both exchangers are the same. Increasing the polytropicindex of compression process from 1.2 continuously enlarges thethermal conductances of both heat exchangers in the refrigerationcycle, but does not uniformly increase the heat capacity rate of gas.The heat capacity rate of gas decreases at first to a minimum atabout nC = 1.25, and then begins to increase. Furthermore, in orderto obtain the minimum total thermal conductance, the heat capac-ity rates of gas should be adjusted appropriately to obtain the opti-mal temperature differences in both heat exchangers withdifferent polytropic indexes, and thus the thermal conductancesof both heat exchangers should be enlarged consequently.

In summary, all the aforementioned analyses and discussionsreveal that, for a gas refrigeration cycle with polytropic compres-sion and expansion processes, all the factors including the coolingcapacity rate, the net power consumption rate, the inlet tempera-tures and the heat capacity rates of fluids from the high-tempera-ture and low-temperature environments influence the optimalheat capacity rate of gas in the cycle, which is different from thatin the adiabatic case. Meanwhile, all the influences can be quanti-tatively described via the proposed method. The optimal thermalconductances of heat exchangers do not always equal to eachother, except for the situation when the compression and expan-sion processes are both adiabatic or have the same polytropicindex.

5. Conclusions

A thermodynamic cycle consists of two different types of ther-mal processes: heat transfer processes in heat exchangers andheat-work conversion processes in compressors, expanders, andturbines. By the combination of entransy analysis for heat transferprocesses and thermodynamic analysis for heat-work conversionprocesses, we developed a new method for practical thermody-namic cycle optimization.

For the gas refrigeration cycle studied in this paper, entransyanalysis connects the fluid inlet and outlet temperatures in eachheat exchanger with the cycle design parameters, e.g. the heattransfer area of each heat exchanger and the heat capacity rate ofeach working fluid, thermodynamic analysis connects the fluid in-let and outlet temperatures of each component in the refrigerationcycle with the system requirements, e.g. the cooling capacity rateand the power consumption rate, and then their combination of-fers a mathematical relation directly relating the system require-ments to the design parameters without any intermediatevariable. Based on this relation together with the conditional extre-

Q. Chen et al. / Applied Energy 113 (2014) 982–989 989

mum method, we derive an optimization equation group. Simulta-neously solving this equation group yields the optimal structuraland operation parameters for gas refrigeration cycles to minimizethe total area of heat exchangers or the power consumption.

The newly developed optimization method not only gives theoptimal structural and operation parameters for every single cycle,but also provides several useful optimization criteria for all the cy-cles. For instance, when the compression and expansion processesare adiabatic, the thermal conductances of both heat exchangersshould be the same in the optimal design, and none but the heatcapacity rates of fluids from the high-temperature and low-tem-perature environments will influence the optimal heat capacityrate of gas in the cycle. Conversely, when the compression andexpansion processes are polytropic, the optimal thermal conduc-tances of heat exchangers seldom equal to each other, and all thefactors including the cooling capacity rate, the net power con-sumption rate, and the fluid temperatures influence the optimalheat capacity rate of gas in the cycle.

Therefore, the newly developed method based on the combina-tion of thermodynamics and entransy theory is helpful for globaldesign and optimization of thermodynamic cycles.

Acknowledgements

The present work is supported by the National Natural ScienceFoundation of China (Grant No. 51006060), the National Basic Re-search Program of China (Grant No. 2010CB227305), and the Foun-dation for the Author of National Excellent Doctoral Dissertation ofChina (Grant No. 201150).

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