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Applied Energy 85 (2008) 634–640

www.elsevier.com/locate/apenergy

APPLIED

ENERGY

Optimal exergy-efficiency of a thermoacoustic coolerwith a complex heat-transfer exponent

Qing Li a,b, Lingen Chen a,*, Feng Wu a,c, Fangzhong Guo d, Deliang Guo c

a Postgraduate School, Naval University of Engineering, Wuhan 430033, PR Chinab Technical Institute of Physics and Chemistry, Chinese Academy of Science, Beijing 100080, PR China

c School of Science, Wuhan Institute of Technology, Wuhan 430073, PR Chinad School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China

Available online 3 December 2007

EEE

TTT Abstract

A heat-transfer model of the behaviour of a thermoacoustic cooler and its surrounding heat-reservoirs is described.Both the real part and the imaginary part of the heat-transfer exponent affect strongly the optimal performance of thecooler. Both the exergy-efficiency and the cooling load decrease with increase of the imaginary part of the complex expo-nent when the real part is fixed. The optimal zone on the performance of the thermoacoustic cooler is obtained by numer-ical analysis. The results obtained herein will be useful for the selection of the operation parameters for a realthermoacoustic cooler.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Thermoacoustic cooler; Complex heat-transfer exponent; Exergy-efficiency; Cooling load

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1. Introduction

With the development of electronic and low-temperature superconductivity systems, the cooling of them isa vital issue to ensure that they operate in the desired temperature range. A micro-miniature thermoacousticcooler without moving parts, which is fabricated on a substrate by micro electro-mechanical (MEMS) tech-nology, is a new approach to solve the cooling issue of chips [1,2].

The principal parts of a thermoacoustic cooler are the stack and two adjacent heat-exchangers. The acous-tic wave carries the working gas back and forth within these components. A longitudinal pressure oscillating inthe sound channel induces a temperature oscillation in time with angular frequency x. The gas temperaturecan be taken as complex [3,4]. It results in a time-averaged heat exchange with complex exponent betweenthe gas and the environment by the hot and cold heat-exchangers.

The exergy-efficiency eE, which is defined by the first and second laws of thermodynamics is an importantperformance index for a cooler cycle. The energy profit is high when eE is large and the cooling effect is good

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0306-2619/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apenergy.2007.10.004

* Corresponding author. Tel.: +86 27 83615046; fax: +86 27 83638709.E-mail addresses: lgchenna@yahoo.com, lingenchen@hotmail.com (L. Chen).

Q. Li et al. / Applied Energy 85 (2008) 634–640 635

when the cooling load r is large. Therefore, the need to develop the performances of the cooling devices is onereason why the maximum exergy-efficiency, the maximum cooling load and the relationship between eE and r

are discussed. In the present paper, we will derive expressions for these important parameters based on a gen-eralized heat-transfer law _Q / DðT nÞ, where n is a complex number. The expressions are used to discuss theoptimal relationship between eE and r. The effect of the complex heat-transfer exponent on the optimal per-formance of the thermoacoustic cooler is analyzed.

2. The model of thermoacoustic cooler

The energy flow in a thermoacoustic cooler is schematically illustrated in Fig. 1. The following assumptionsare made for the cooler behavioural model in this paper:

The cooler operates between a constant-temperature heat source with temperature TL and a constant-tem-perature heat sink with temperature TH. Because of the heat-transfer, the time-averaged temperatures of theworking gas are TH0 and TL0 in the two heat-exchangers, which differ from the reservoir temperatures (TH andTL). The second law of thermodynamics requires TH0 > TH > TL > TL0. As a result of thermoacoustic oscil-lation, the temperatures (THC and TLC) of the working gas can be expressed in complex form as

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T HC ¼ T H0 þ T 1eixt ð1ÞT LC ¼ T L0 þ T 2eixt ð2ÞTTT

where T1 and T2 are the first-order acoustic quantities, i ¼ffiffiffiffiffiffiffi�1p

. Here the reservoir temperatures TH and TL

are assumed to be invariant.The heat flows released to the heat sink and absorbed from the heat source are assumed to be:AAA

_Q0HC ¼ aF 1ðT nHC � T n

HÞsgnðn1Þ ð3Þ_Q0LC ¼ bF 2ðT n

L � T nLCÞsgnðn1Þ ð4Þ

with sign function CCC

sgnðn1Þ ¼

1 n1 > 0

�1 n1 < 0

�ð5Þ

III

where n = n1 + n2i is a complex heat-transfer exponent, a is the overall heat-transfer coefficient and F1 is thetotal heat-transfer surface area of the hot heat-exchanger, b is the overall heat-transfer coefficient and F2 is thetotal heat-transfer surface area of the cold heat-exchanger. Here the imaginary part n2 of n indicates the relax-ation of the heat-transfer process. Defining _QHC 6

_Q0HC and _QLC 6_Q0LC as the time averages of _Q0HC and _Q0LC,

respectively, Eqs. (3) and (4) can be rewritten asPLPLPL

Fig. 1. Energy flows in a thermoacoustic cooler.

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636 Q. Li et al. / Applied Energy 85 (2008) 634–640

_QHC ¼aF

1þ fT n

H0 � T nH

� �sgnðn1Þ ð6Þ

_QLC ¼bFf

1þ fT n

L � T nL0

� �sgnðn1Þ ð7Þ

where f = F1/F2 and F = F1 + F2. Here, the total heat-transfer surface area F of the two heat-exchangers isassumed to be a constant.

The rate of heat leakage between the heat sink and the heat source is assumed to be [5]

_Qe ¼ a1ðT mH � T m

LÞsgnðmÞ ð8Þ

where sgn(m) is a sign function: both the heat conductance a1 and heat-transfer exponent m are positive realconstants.

As a result of the heat leakage _Qe, the heat flow _QH absorbed by the heat sink and the heat flow _QL releasedby the heat source (i.e. cooling load) are, respectively,

_QH ¼ _QHC � _Qe ð9Þ_QL ¼ _QLC � _Qe ð10Þ

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In addition to the heat resistance between the working gas and the heat reservoirs and heat leakage between

the heat-reservoirs, it is assumed that there are internal irreversibilities in the system due to miscellaneous fac-tors, such as friction and non-equilibrium activities inside the cooler. Based on the second law of thermody-namics, the total consequence of these irreversible-effects can be reduced to the increase of entropy generationfor the heat sink in the cycle under the same cooling load. In other words, the rate of heat flow ð _QHCÞ from thegas to the sink for a real thermoacoustic cooler is larger than that for an endoreversible [6,7] thermoacousticcooler ð _QHCEÞ. A factor / for the internal irreversibility degree, to characterize the additional internal miscel-laneous irreversible-effect, is defined as follows:

ATATAT

/ ¼

_QHC

_QHCE

P 1 ð11ÞCCC

III The cooler, as it satisfies the above assumptions, is called the generalized irreversible thermoacoustic cooler

with a complex heat-transfer exponent. The corresponding research for performance analysis and optimiza-tion of generalized irreversible Carnot coolers were performed by Chen et al. [8–12] and related researchappears in Refs. [13–34].

3. Optimal relationship between exergy-efficiency and cooling load

For an endoreversible thermoacoustic cooler [7], the second law of thermodynamics gives

PLPLPL

_QHCE

T H0

¼_QLC

T L0

ð12ÞUUU

Substituting from Eq. (11) and x ¼ T H0

T L0into Eq. (12) yields

_QHC ¼ /x _QLC ð13Þ

The rate of output exergy produced by the cooler isDDD

_E ¼ _QL

T 0

T L

� 1

� �� _QH

T 0

T H

� 1

� �¼ eH

_QHC � eL_QLC þ _QeðeL � eHÞ ð14Þ

where T0 is the ambient temperature: eL ¼ 1� T 0

T Land eH ¼ 1� T 0

T Hare the Carnot coefficients of the absorbing

heat and removing heat, respectively. The exergy-efficiency of the cooler is defined as [7]

eE ¼_E_W¼

_E_QH � _QL

ð15Þ

Q. Li et al. / Applied Energy 85 (2008) 634–640 637

where _W ¼ _QH � _QL ¼ _QHC � _QLC is the power input. Comparing Eqs. (6)–(15) yield the complex cooling loadr 0 ð _QLÞ and the complex exergy-efficiency e0E of the cooler as follows:

r0 ¼ bFf1þ f

T nLxn � T n

H

d/xþ fxn sgnðn1Þ � _q ð16Þ

e0E ¼/xeH � eL

/x� 1þ _qðeL � eHÞðr0 þ _qÞð/x� 1Þ ð17Þ

with _q ¼ _Qe

F .From Eqs. (16) and (17), we obtain the real part of the cooling load and the real part of the exergy-

efficiency:

rðf Þ ¼ Reðr0Þ ¼ bFf1þ f

A1/xdþ fxn1 ½A1 cosðn2 ln xÞ þ A2 sinðn2 ln xÞ�ðd/xÞ2 þ 2d/fxn1þ1 cosðn2 ln xÞ þ f 2x2n1

� _q ð18Þ

eEðf Þ ¼ Reðe0EÞ ¼ ð/x� 1Þ�1 /xeH � eL þ_qðeL � eHÞ½rðf Þ þ _q�½rðf Þ þ _q�2 þ ½Iðf Þ�2

( )ð19ÞEEE

where

A1 ¼ Re T nLxn � T n

H

� �sgnðn1Þ

A2 ¼ Im T nLxn � T n

H

� �sgnðn1Þ

Iðf Þ ¼ Imðr0Þ ¼ bFf1þ f

A2/xdþ fxn1 ½A2 cosðn2 ln xÞ � A1 sinðn2 ln xÞ�ðd/xÞ2 þ 2d/fxn1þ1 cosðn2 ln xÞ þ f 2x2n1

TTT

AAA Here Re() and Im() indicate the real and imaginary parts of the appropriate complex definitions.

Eqs. (18) and (19) indicate that both exergy-efficiency eE and cooling load r of the cooler are functions ofthe heat-transfer surface area ratio (f) for given values of TH, TL, T0, a, b, n1, n2, /, _q and x. Taking the deriv-ative of eE(f) or r(f) with respect to f and setting it equals to zero i.e. (oeE

of ¼ 0 or orof ¼ 0, respectively), we can

find that, when f satisfies the following equationCCC

f0 ¼ �1

4b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8y þ b2 � 4c

q� �þ 1

2

1

4b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8y þ b2 � 4c

q� �2

� 4 y � by � dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8y þ b2 � 4c

p !" #0:5

ð20Þ

III

both the exergy-efficiency eE and cooling load r approach optimal values

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r ¼ rðf0Þ ¼

bFf0

1þ f0

A1/xdþ f0xn1 ½A1 cosðn2 ln xÞ þ A2 sinðn2 ln xÞ�ðd/xÞ2 þ 2d/f0xn1þ1 cosðn2 ln xÞ þ f 2

0 x2n1

� _q ð21Þ

eE ¼ eEðf0Þ ¼ ð/x� 1Þ�1 /xeH � eL þ_qðeL � eHÞ½rðf0Þ þ _q�½rðf0Þ þ _q�2 þ ½Iðf0Þ�2

( )ð22Þ

PPP

where UUU

b ¼ 2A1/dx1�n1

A1 cosðn2 ln xÞ þ A2 sinðn2 ln xÞ

c ¼ 2A1/2d2x2�2n1 cosðn2 ln xÞ þ A1/dx1�n1

A1 cosðn2 ln xÞ þ A2 sinðn2 ln xÞ � /2d2x2�2n1 � 2/dx1�n1 cosðn2 ln xÞ

d ¼ �2d2x2�2n1

e ¼ c3

108þ A1/

3d3x3�3n1ð0:125b2 � 0:5cÞA1 cosðn2 ln xÞ þ A2 sinðn2 ln xÞ �

d2

8

y ¼ � e2þ e

2

� 2

� c2

36

� �3" #0:5

8<:

9=;

1=3

þ � e2� e

2

� 2

� c2

36

� �3" #0:5

8<:

9=;

1=3

þ c6

DDD

638 Q. Li et al. / Applied Energy 85 (2008) 634–640

The parameter equations defined by Eqs. (21) and (22) give the fundamental relationship between the opti-mal exergy-efficiency and the cooling load of the cooler.

4. Discussion

The optimal exergy-efficiency versus cooling-load characteristic is dependent on TH, TL, T0, b, d, n1, n2, /and _q. From Eqs. (21) and (22) it may be seen that both the exergy-efficiency and the cooling load decreasewith increases of / or _q when TH, TL, T0, b, n1, n2 and d are fixed. The effects of the complex exponentn = n1 + in2 on the optimal exergy efficiency versus cooling load characteristic with TH = 300 K,TL = 250 K, T0 = 295 K, d = 1, _q ¼ 3 W, and / = 1.05 are shown in Figs. 2 and 3. They indicate that theeE versus r characteristics of a generalized irreversible thermoacoustic cooler with a complex heat-transferexponent is a loop-shaped curve. For all n1 and n2, eE = eEmax when r = r0 and r = rmax when eE = eE0. Forexample, when n1 = 1, the eE bound (eEmax) corresponding to n2 = 1.2, 1.4, and 1.6 are 0.1218, 0.1043, and0.0815, respectively, and the values of the maximum cooling load rmax, corresponding to n2 = 1.2, 1.4, and1.6, are 10.3167, 7.1945, and 4.1110, respectively.

The optimization criteria of the thermoacoustic cooler can been obtained from parameters eEmax, r0, rmax

and eE0 as follows:

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eE0 6 eE 6 eEmax and r0 6 r 6 rmax ð23Þ TTT

When the real part n1 = 1 is given, both eE and r decrease with an increase in the imaginary part n2, asshown in Fig. 2. It follows that the imaginary part n2 of the complex heat-transfer exponent n indicates anenergy dissipation. Fig. 3 gives the effect of the real part n1 on eE versus r characteristics when the imaginarypart n2 = 1.6 is fixed. It is worth nothing that n1 affects strongly the optimal performance.

If _q ¼ 0, and / = 1, the optimal relationship between eE and r becomes

AAA

Fig. 2. Optimal exergy-efficiency versus cooling-load characteristic with n1 = 1, n2 = 1.6, n2 = 1.4 and n2 = 1.2.

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Fig. 3. Optimal exergy-efficiency versus cooling-load characteristic with n2 = 1.6, n1 = 1, n1 = 4 and n1 = �1.

Q. Li et al. / Applied Energy 85 (2008) 634–640 639

CATECATECATE

r ¼ bFf0

1þf0

A1xdþf0xn1 ½A1 cosðn2 ln xÞþA2 sinðn2 ln xÞ�ðdxÞ2þ2df0xn1þ1 cosðn2 ln xÞþf 2

0x2n1

x ¼ eE�eL

eE�eH

8<: ð24Þ

III

Eq. (24) is the optimal relation of an endoreversible thermoacoustic cooler with complex heat-transfer

exponent.

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5. Conclusion

The irreversible-cycle model of the thermoacoustic cooler with a complex heat-transfer exponent estab-lished in this investigation reveals the effects of heat resistance, heat leakage, internal irreversibility andcomplex heat-transfer exponent on the exergy-efficiency and the cooling load using constantsbðaÞ; _q;/; n1 and n2. The heat-transfer exponent for a thermoacoustic cooler must be a complex numberdue to the thermal relaxation induced by the thermoacoustic oscillation. The effects of the complexheat-transfer exponent on the optimal performance for a thermoacoustic cooler are discussed in this paper:the optimal zone of the thermoacoustic cooler with a complex heat-transfer exponent is also analyzed. Theresults obtained herein are helpful in the selection of the optimal mode of operation of the real thermoa-coustic cooler.

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Acknowledgement

This paper is based upon a study supported by the National Natural Science Fund Of China under contractNo. E060107-50276064.

640 Q. Li et al. / Applied Energy 85 (2008) 634–640

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