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Renewable and Sustainable Energy Reviews 21 (2013) 524–536

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Renewable and Sustainable Energy Reviews journal homepage: www.elsevier.com/locate/rser

Finite-time thermodynamics optimization of absorption refrigeration systems: A review
´ ´
Paiguy Armand Ngouateu Wouagfack a,n, Rene Tchinda b a b

L2MSP, Department of Physics, University of Dschang, PO Box 67 Dschang, Cameroon
LISIE, University Institute of Technology Fotso Victor, University of Dschang, PO Box 134 Bandjoun, Cameroon

a r t i c l e i n f o

a b s t r a c t

Article history:
Received 18 June 2012
Received in revised form
14 December 2012
Accepted 16 December 2012
Available online 9 February 2013

This paper presents a literature review of the optimization of absorption refrigeration systems based on finite-time thermodynamics. An overview of the various objective functions is presented.
& 2012 Elsevier Ltd. All rights reserved.

Keywords:
Finite-time thermodynamics
Optimization
Endoreversible
Irreversible
Absorption refrigerator

Contents
1.
2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
Optimization based on the coefficient of performance and cooling load criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
2.1.
Three-heat-source absorption refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
2.2.
Four-heat-source absorption refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
2.3.
Solar absorption refrigeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
3. Optimization based on the thermo-economic criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
4. Optimization based on the ecological criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
5. Optimization based on the new thermo-ecological criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

1. Introduction
The absorption refrigeration systems are thermodynamic processes which produce cold thanks to thermal energy. Then, they exchange heat with at least three sources at different temperatures without receiving work. A three-heat-source reversible refrigerator operates between heat hot reservoir, heat cold reservoir and heat sink. When T H , T L and T O denote the temperatures of heat hot reservoir, heat cold reservoir and heat sink respectively, the coefficient of performance for three-heat-source reversible refrigerators is

n

Corresponding author. Tel.: þ237 77 18 58 71.
E-mail address: ngouateupaiguy@yahoo.fr (P.A. Ngouateu Wouagfack).

1364-0321/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rser.2012.12.015 expressed as: er ¼ ½ðT H ÀT O Þ=T H Š½T L =ðT O ÀT L ފ [1]. This expression reveals the product of thermal efficiency of Carnot cycle for heat engines working between T H and T O and coefficient of performance of reversible Carnot refrigerator producing cold at T L and rejecting heat at T O : er ¼ ZC  eC with ZC ¼ ðT H ÀT O Þ=T H and eC ¼ T L =T O ÀT L . In classical thermodynamics, the efficiency of a cycle operating on reversibility principles proposed by Carnot [2] became the upper bound of thermal efficiency for heat engines that work between the same temperature limits. This equally applies to the coefficient of performance of refrigeration cycles that execute a reversed Carnot cycle (Carnot refrigerator). This implies that the coefficient of performance defined above is the maximum coefficient of performance for three-heat-source refrigerators from the point of view of classical thermodynamics. P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

Nomenclature
A
AA
AC
AL
AH
AO
ECOP
I
K LC
KH
KL
KO
ncu
_
QA
_
QC
_
QL
_
QH
_
Q

total heat-transfer area (m2) heat-transfer area of absorber (m2) heat-transfer area of condenser (m2) heat-transfer area of evaporator (m2) heat-transfer area of generator (m2)
AA þAC ecological coefficient of performance internal irreversibility parameter heat leak coefficient (W K) thermal conductance of heat source (W K À 1) thermal conductance of cooled space (W K À 1) thermal conductance of heat sink (W K À 1) national currency unit heat reject load from absorber to heat sink (W) heat reject load from condenser to heat sink (W) heat input load from cooled space to evaporator (W) heat input load from heat source to generator (W)
_
_
Q þQ

R
Rm

cooling load (W) cooling rate at maximum coefficient of performance (W) specific cooling load (W m À 2) specific cooling rate at maximum coefficient of performance (W m À 2) temperature of working fluid in generator (K) temperature of working fluid in evaporator (K) temperature of working fluid in absorber and condenser (K) temperature of working fluid in absorber and condenser (K)

O

r rm T1
T2
T3
T3

C

A

However, since the absorption refrigeration cycles are in direct contact with reservoirs and sink, the heat transfers during the isothermal processes are supposed to be carried out infinitely slowly. Therefore, duration of the processes will be infinitely long and hence it is not possible to obtain a certain amount of cooling
_
load Q L with heat exchangers having finite heat-transfer areas, i.e.
_ ¼ 0 for 0 o A o1. If we require certain amount of cooling load in
Q
an absorption refrigerator executing a reversible cycle, the necessary heat exchanger area would be infinitely large, i.e. A-1 for
_
Q 4 0.
Thus in classical thermodynamics the real absorption refrigerators producing cold with a certain amount of cooling load are compared with the ideal absorption refrigerators developing no cooling load. In other words the performance of an absorption refrigerator of given size (in term of total heat-transfer area) is measured with an ideal absorption refrigerator which would require an infinite total heat-transfer area to produce the same amounts of cooling load. In practice, all absorption refrigeration processes take place in finite-size devices in finite-time; therefore, it is impossible to meet reversibility conditions between the absorption refrigeration system and the surroundings. For this reason, the reversible absorption cycle cannot be considered as a comparison standard for practical absorption refrigeration systems from the view of cooling load on size perspective, although it gives an upper bound for coefficient of performance. The performance bound of classical thermodynamics [3–6] is highly important in theory, but it is usually too rough to predict the coefficient of performance of practical absorption refrigerators.
Therefore, it is necessary to establish the bound of finite-time thermodynamics [7].

TA
TC
T env
TH
TL
TO
UH
UL
UO
_
W

525

temperature of the absorber-side heat sink (K) temperature of the condenser-side heat sink (K) temperature in environmental conditions temperature of the heat source (K) temperature of the cooled space (K)
TA ¼ TC overall heat-transfer coefficient of generator (W K/ m 2) overall heat-transfer coefficient of evaporator (W K/ m 2) overall heat-transfer coefficient of absorber and condenser (W K/m2) power output (W)

Symbol

e eC ZC l s eI er em coefficient of performance for absorption refrigerators coefficient of performance of reversible Carnot refrigerator thermal efficiency of Carnot cycle
Dissipation coefficient of cooling rate
Entropy generation rate (W/K) coefficient of performance for three-heat-source refrigerator affected only by internal irreversibility coefficient of performance for reversible three-heatsource refrigerator coefficient of performance at maximum cooling rate

Subscripts max maximum

The finite-time thermodynamics has been first proposed by
Chambadal [8] and Novikov [9] independently on1957, then popularized in many works including Curzon and Ahlborn [10],
De Vos [11], Sieniutycz et al. [12], Bejan [13–18], Wu [19],Chen
[20], Stitou [21,22], Feidt [23,24], Leff and Teeters [25], Blanchard
[26], Stitou and Feidt [27], Andresen [28], Sieniutycz and Salamon
[29], De Vos [30], Bejan et al. [31], Bejan and Mamut [32], Berry et al. [33], Radcenco [34] and in many review articles including
Sieniutycz and Shiner [35], Chen et al. [36], Hoffmann et al. [37] and Durmayaz et al. [38].
The finite-time thermodynamics tends to model the real systems in a way closer to reality and enable to distinguish the irreversibilities due to internal dissipation of the working fluid from those due to finite-rate heat transfer between the system and the external heat reservoirs and heat sink.
The objective of this paper is to review the present state of optimization of absorption refrigeration processes based on finite-time thermodynamics. The different performance optimization criteria are provided and discussed.

2. Optimization based on the coefficient of performance and cooling load criteria
2.1. Three-heat-source absorption refrigerator
An absorption refrigeration system (equivalent to three-heatreservoir refrigeration system) affected by the irreversibility of finite rate heat transfer may be modeled as a combined cycle which consists of an endoreversible heat engine and an endoreversible

526

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

refrigerator that the model is shown in Fig. 1 where the temperatures of the working fluid in the Carnot engine at two isothermal processes are T 1 and T 3 , the temperatures of the working fluid in the Carnot refrigerator at two isothermal processes are T 2 and T 3 . T 1 , T 2 and T 3 represent the temperatures of the working fluid in the endoreversible three-heat-source refrigerator when the three isothermal processes are carried out, respectively. They are respectively different from the temperatures of the corresponding three reservoirs. A single-stage absorption refrigerator consists primarily of a generator, an absorber, a condenser and an evaporator which is shown in Fig. 2. An equivalent single-stage absorption combined refrigeration system is
_
also shown in Fig. 3. In these figures, Q H is the heat-transfer rate from
_
the heat source at temperature T H to the system, Q L is the heattransfer rate (cooling load) from the cooled space at temperature T L
_
to the system, Q O is the heat-transfer rate from the system to the heat sink at temperature T O ( in the case of three-heat-source model)
_
_ and Q A and Q C are the heat-transfer rates from the absorber and condenser to the heat sink at temperature T A and T C respectively (in the case of four-heat-source model). T 1 ; T 2 ; T 3 and T 4 are the temperatures of the working fluid in the generator, evaporator,
_
absorber and condenser. W is the power output of the heat engine which is the power input for the refrigerator. A single-stage absorption refrigerator normally transfers heat between three temperature levels when T A ¼ T C , but very often among four temperature levels when T A aT C . For the three-heat-source single-stage absorption refrigerator, it is generally assumed that the working fluid in the condenser and absorber has the same temperature T 3 ¼ T 4 . This assumption is reasonable because the working fluid in the condenser and absorber exchanges heat with the heat sink at the same temperature. An absorption refrigerator is endoreversible absorption refrigerator when it is affected only by the external irreversibility of heat conduction between the working fluid and reservoirs.
The first performance optimization studies for absorption refrigeration systems based on finite-time thermodynamics started with the works of Yan and Chen [1,39]. They applied the theory of finite time thermodynamics in heat two-heat-source to optimize the performance of endoreversible three-heat-source refrigerator by taking the coefficient of performance and cooling load as objective

Fig. 2. Absorption refrigeration system [46].

Fig. 3. Equivalent cycle of an absorption refrigeration system [46].

functions. They treated the endoreversible three-heat-source refrigerator as a combined cycle of an endoreversible Carnot engine driving an endoreversible Carnot refrigerator as shown in Fig. 1. For convenience, they assumed that the engine and refrigerator in the combined cycle operate alternatively. Thus, the combined cycle time t supposed to be constant may be expressed as: t ¼ t 1 þt 2

Fig. 1. Sketch of an endoreversible three-heat-source refrigerator, treated as an endoreversible Carnot engine driving an endoreversible Carnot refrigerator [1].

ð1Þ

where t 1 and t 2 are the cycle times of the engine and refrigerator in the combined cycle respectively. They defined the coefficient of performance of the combined cycle as the product of the efficiency of the endoreversible Carnot engine and the coefficient of performance

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

527

and the following optimal distribution of thermal conductance inventory at the Rmax conditions: ðUAÞO ¼

UA
2

ðUAÞH ¼ ðUAÞL ¼

ð8Þ
UA
4

ð9Þ pffiffiffiffiffiffiffiffiffiffi ðUAÞ T ð

Fig. 4. RÀe characteristic of an endoreversible three-heat-source refrigerator [1].

of refrigerator:

e ¼ Zf

ð2Þ

and expressed it with respect to t 1 and t 2 taken as optimization parameters. In order to optimize e, they introduced the Lagrangian function L ¼ e þ lðtÀt 1 Àt 2 Þ and from the Euler–Lagrange equations
@L=@t 1 and @L=@t 2 they derived the fundamental optimum relation between the coefficient of performance and cooling load of endoreversible three-heat-source refrigerator as:


K 1 T H ðT O ÀT L Þðer ÀeÞ

ð3Þ

ðK 2 þ 1Þ2 ð1þ eÞT H ÀK 2 2 T O þð1 þ eÀ1 ÞT L

pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ffi KOð K À KLÞ ffi where K 1 ¼ K H K O =ð K H þ K O Þ2 and K 2 ¼ pffiffiffiffi pffiffiffiffiH pffiffiffiffiffi
KLð

KH þ

KOÞ

From Eq. (3), they obtained the maximum cooling load (Rmax ) and the corresponding coefficient of performance at Rmax conditions (em ) shown in the following equations respectively: pffiffiffiffiffiffi pffiffiffiffiffiffi
K 1 ð T H À T O Þ2 ðT L þ FÞ ð4Þ Rmax ¼
T H ÀT L þ G

em ¼

pffiffiffiffiffiffi pffiffiffiffiffiffi
THÀ TO
TL
pffiffiffiffiffiffi ffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi
 pffiffiffiffiffiffiffiffiffiffiffi
TH
T H T O ÀT L þ K 2 T O ð T H À T O Þ

ð5Þ

where


K 2 T L ðT H ÀT O Þ
T H ÀT O

G ¼ K2ð

" pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi# pffiffiffiffiffiffi pffiffiffiffiffiffi 2 B2 ðT H ÀT O Þ þ B T H ð3 T H þ 2 T O ÀT L = T H Þ 3 T H þ T O
T H À T OÞ þ pffiffiffiffiffiffi pffiffiffiffiffiffi
T H ÀT L
THÀ TO

ð6Þ
They also obtained the behavior of the cooling rate as a function of coefficient of performance as presented in Fig. 4. Like the maximum power output and corresponding efficiency concerning a Carnot engine Rmax and em are two important performance parameters of a practical endoreversible three-heat-source refrigerator. Its optimal operating region is em r e r er [1].
Chen [40] investigated the maximum cooling rate of an endoreversible three-heat-source absorption refrigerator. He also analyzed the influence of irreversibility of finite-rate heat transfer on the performance of the system. The coefficient of performance at the maximum cooling rate was derived and the optimal performance with respect to heat transfer areas of the refrigerator was analyzed.
Bejan et al. [41] optimized the cooling load with respect to the thermal conductance of an equivalent combined endoreversible three-heat-source absorption refrigerator and obtained the analytical expression of Rmax as: ðUAÞR T O
Rmax ¼
8

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#

2
4W max
TL
16T L W max 4W max
TL
À þ1 þ þ À1
À
2 ðUAÞR T O T O ðUAÞR T O T O ðUAÞR T O

ð7Þ

T =T À1Þ

H
O
P O
In Eqs. (7)–(9) W max ¼ is the maximum power
4
output of the endoreversible heat engine in the combined cycle; ðUAÞH , ðUAÞL and ðUAÞO are the overall thermal conductance of the respective heat exchangers; ðUAÞP and ðUAÞR are the total thermal conductance inventory of the heat and refrigerator portions;
UA ¼ ðUAÞH þðUAÞL þðUAÞO ¼ ðUAÞP þðUAÞR is constant and represent the total thermal conductance inventory of the entire installation. Bejan et al. [41] also reported the maximum cooling load per unit of total heat exchanger inventory.
Wijeysundera [42] focused its attention on the three-heatsource refrigerator affected only by external irreversibility of linear heat-transfer law. He investigated its optimal performance by considering the cooling load as optimization objective. He maximized the cooling capacity in term of the temperature of the heat source and the heat sink and obtained:
(
) ðe2 þ g fÀf Àf gÞ þ ½ðe2 þ g fÀf Àf gÞ2 À4ð1þ gÞðe2 Àf gފ1=2
Rmax ¼ bT H
2ð1þ gÞ

ð10Þ pffiffiffi where e ¼ v þ u y, f ¼ v þ yu, g ¼ v þu and the non-dimensional design and operating variables are defined as: f ¼ T L =T H , y ¼ T O =T H , u ¼ g=b, v ¼ a=b.
Wijeysundera [42] derived the following optimum relation pffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 3 =T 1 ¼ T O =T H ð11Þ and the optimal coefficient of performance at the maximum of cooling capacity condition: pffiffiffi pffiffiffi em ¼ qmax ð1 þu y=vÞ=½uð yÀyÞÀqmax Š ð12Þ where qmax ¼ Rmax =ðbT H Þ
The condition given by Eq. (11) is the same as that obtained by
Curzon and Ahlborn for the maximum power output for an endoreversible Carnot cycle.
Wu [43,44] investigated the maximum cooling load of an endoreversible heat-engine-driven refrigerator modeled as the combined cycle of heat engine and refrigerator.
Chen and Yan [45] analyzed the effect of linear phenomenological law on the performance of three-heat-source absorption refrigerator. The heat transfer between the working fluid and the external reservoir obey the following equations:
Q H ¼ K H ðT 1 À1 ÀT H À1 Þt H

ð13Þ

Q L ¼ K L ðT 2 À1 ÀT L À1 Þt L

ð14Þ

Q O ¼ K O ðT O À1 ÀT 3 À1 Þt O

ð15Þ

Chen and Yan [45] defined the coefficient of performance and cooling load as the function of the temperatures T 1 , T 2 and T 3 of the working fluid. By introducing the Lagrangian function
L ¼ e þ lR and using the Euler–Lagrange equations @L=@x, @L=@y and @L=@z where x ¼ T 3 =T 1 , y ¼ T 3 =T 2 and z ¼ T 3 they established the maximum cooling load, the corresponding coefficient of performance and the fundamental optimal relation between the coefficient of performance and cooling load as follows:
Rmax ¼

K 3 er 2 ðT O ÀT L Þ
4ðK 4 er þ 1ÞT O T L

em ¼ er =ð2 þ K 4 er Þ

ð16Þ ð17Þ 528



P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

K 3 eðer ÀeÞðT O ÀT L Þ

ð18Þ

ðK 4 e þ 1Þ2 T O T L

and the optimal coefficient of performance:

pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ffi KH ð KO þ KLÞ where K 3 ¼ K H K L =ð K H þ K L Þ2 ,K 4 ¼ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffi



K 1 ð1ÀxÞðT H xÀT O ÞT L
ÀK LC ðT O ÀT L Þ
½ð1 þK 2 ÞT H xÀK 2 T O Š½1þ K 2 ð1ÀxފÀT L x

Chen et al. [46] carried out similar analysis and studied the effect of the irreversibility of non linear heat transfer condition.
The performance optimization of irreversible three-heatsource refrigerator was carried out by Chen and Wu [47] to investigate the influence of irreversibility of heat leak on the performance of the Newton’s law three-heat-source refrigerator.
Fig. 5 shows the three-heat-source refrigeration cycle model affected by the irreversibility of finite rate heat transfer and heat leakage where Q LC is the heat leak from the environment reservoir to the cooled space occurs during the full cycle time t with a thermal conductance K LC . Chen and Wu [47] modeled their system by the following heat transfer Newton’s law equations:



ð1ÀxÞT L
K LC ðT O ÀT L Þ

R þK LC ðT O ÀT L Þ ð1þ K 2 ÞT H xÀK 2 T O ÀT L

Eliminating x from Eqs. (24) and (25) yields the optimal fundamental relationship between the cooling rate and coefficient of performance. Obviously, the RÀe characteristic of an absorption refrigeration cycle affected by thermal resistances and heat leak losses can be generated by the optimal fundamental relationship between the cooling rate and coefficient of performance as shown in Fig. 6. Fig. 6 presents the coefficient of performance bounds and cooling load bounds of real three-heat-source refrigerator as:

Q H ¼ K H ðT H ÀT 1 Þt H

ð19Þ

Rm rR rRmax and emax Z e Z em

Q L ¼ K L ðT L ÀT 2 Þt L

ð20Þ

Q O ¼ K O ðT 3 ÀT O Þt O

ð21Þ

Q LC ¼ K LC ðT O ÀT L Þt

ð22Þ

KOð

KH þ

KLÞ

where t H , t L and t O are the times of the three isothermal processes during the cycle. In such a refrigeration cycle, which consists of three isothermal processes and three adiabatic processes, the working fluid exchanges only heats with the three external heat reservoirs at temperatures T H , T L and T O during the full cycle time. To give prominence to the time for heat exchange, the time for the adiabatic processes is taken as a negligible quantity because the adiabatic process is not affected by thermal resistance. The cycle time may be approximately given by t ¼ tH þ tL þ tO .
On the basic on this model, Chen and Wu [47] defined the coefficient of performance and the cooling load as:

e ¼ ðQ L ÀQ LC Þ=Q H

ð23aÞ

and
R ¼ ðQ L ÀQ LC Þ=t

ð23bÞ

They maximized the cooling load and the coefficient of performance of Eqs. (23a) and (23b) in term of the temperature of the working fluid in the three isothermal processes and determined analytically the maximum cooling rate and corresponding coefficient of performance, the maximum coefficient of performance and corresponding cooling rate. They derived the optimal cooling rate

Fig. 5. A three-heat-source refrigeration cycle model [47].

ð24Þ

! ð25Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ffi KO ð K À KLÞ ffi where x ¼ T 3 =T 1 , K 1 ¼ K H K O =ð K H þ K O Þ2 , K 2 ¼ pffiffiffiffi pffiffiffiffiH pffiffiffiffiffi
KLð

KH þ

KOÞ

ð26Þ

Jincan Chen [48] used a general irreversible cycle model to investigate the optimal performance of a class of three-heatsource affected by the three main irreversibilities which are finite rate heat transfer between the working fluid and the external heat reservoir, internal dissipation due to the working fluid and heat leakage between heat reservoirs. He established the fundamental optimal relation between the cooling rate and coefficient of performance: Â
À
Á
Ã
R ¼ K 5 e ðT H ÀIT O ÞT L Àe 1 þ C=R T H ðIT O ÀT L Þ h  ð1 þ eÞT L þ K 6 2 ð1þ eÞeð1 þ C=RÞT H



!À1
À
Á e C eC IT O
ÀK 7 2 e 1 þC=R = 1 þ
À
ð27Þ
1þe
1þe R pffiffiffiffiffiffiffiffiffiffiffi 1 þ K L =IK O
K L =I pffiffiffiffiffiffiffiffiffiffiffi 2 , pffiffiffiffiffiffiffiffiffiffiffi, K5 ¼
K6 ¼ where C ¼ Q LC =t,
½1 þ K L =IK H Š
1 þ K L =IK H pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
K L =IK H À K L =IK O pffiffiffiffiffiffiffiffiffiffiffi K7 ¼


K L =IK H

From Eq. (27) the general performance characteristic of irreversible three-heat-source refrigerator are plotted as shown in
Fig. 7. Eq. (27) and the characteristic of Fig. 7 may be used directly to analyze the influence of different irreversibilities on the performance of a three-heat-source refrigerator. For example when I ¼ 1 and K LC ¼ 0, the endoreversible three-heat-source affected by the irreversibility of heat transfer and heat leakage become the endoreversible three-heat-source affected only by the irreversibility of heat transfer [1]. The characteristic of Fig. 7 shows that the coefficient of performance and cooling rate should

Fig. 6. RÀe characteristic of a three-heat-source refrigerator affected by thermal resistances and heat leak losses. Curve a and b correspond to the cases of no heat leak loss and heat leak loss respectively [47].

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

be respectively, constrained by:
Rm r Rr Rmax and emax Z e Z em

ð28Þ

where emax , Rmax , em and Rm determine the upper and lower bounds of the coefficient of performance and cooling rate, respectively of the three-heat-source refrigerator affected by the irreversibility of linear heat transfer law between the working fluid and external reservoir, internal dissipation due to the working fluid and heat leakage between heat reservoirs.
An irreversible three-heat-source refrigerator affected by the irreversibility of finite rate heat transfer and internal dissipation of the working fluid was modeled as a combination of a finite size irreversible Carnot heat engine and an irreversible Carnot refrigerator [49]. The performance optimization work of this model was performed by considering the coefficient of performance as optimization criterion. The optimal overall coefficient of performance was derived and the combined effects of finite-rate heat transfer and internal dissipation on optimal performance were investigated. The performance optimization based on the coefficient of performance and cooling load criteria for an equivalent cycle system of an endoreversible single-stage absorption refrigerator was investigated by Jincan Chen [50]. The absorption cycle and its external heat reservoir considered was modeled as an equivalent combined system which consist of a heat engine operating between heat source at temperature T H and the heat sink at temperature T O and a refrigerator operating between heat sink at temperature T O and the cooled space at temperature T L . Jincan
Chen [50] considered the linear heat transfer law to model the finite-rate heat transfer between the system and its external heat reservoirs: _
Q H ¼ U H AH ðT H ÀT 1 Þ

ð29Þ

_
Q L ¼ U L AL ðT L ÀT 2 Þ

ð30Þ

_
Q C ¼ U O AC ðT 3 ÀT O Þ

ð31Þ

_
Q A ¼ U O AA ðT 3 ÀT O Þ

ð32Þ

where AH , AC , AA and AL , are, respectively, the heat-transfer areas of the generator, condenser, absorber and evaporator; U H and U L are, respectively, the heat-transfer coefficient between the generator and evaporator and the external heat reservoir at temperatures T H and T L . The heat transfer coefficient between two heat exchangers, the absorber and condenser and the heat sink at

529

temperature T O is taken to be U O . This is reasonable assumption because the working fluid in the absorber and condenser exchanges heat with the same heat sink at temperature T O . The total heat-transfer areas of the heat engine and refrigerator in the combined cycle are AH þAA and AC þ AL respectively. The total heat transfer area of the system is A ¼ AH þ AA þ AC þ AL . Chen [50] optimized the coefficient of performance and specific cooling load with respect to the total heat-transfer area of the refrigerator in the combined cycle and derived the fundamental optimum relation: ðT H ÀT O ÞT L ÀeT H ðT O ÀT L Þ ð33Þ ð1 þ eÞT L ÀC 2 eT O þ ð1 þ CÞ2 ð1þ eÞeT H pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C ¼ U H =U r À1. Eq. (33) is the same as the general optimum relation derived from the cycle model of an endoreversible three-heat-source refrigerator [40]. From the external condition @r=@e and Eq. (33), the maximum specific cooling load r max and the corresponding coefficient of performance em were derived. They were given by the following equations respectively: pffiffiffiffiffiffi pffiffiffiffiffiffi
U H ð T H À T O Þ2 T L pffiffiffiffiffiffiffiffiffiffiffi ffi ð34Þ r max ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi
T H ÀT L þ 2CðT H À T H T O Þ þC 2 ð T H À T O Þ2

r ¼ UH e

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1À T O =T H ÞT L pffiffiffiffiffiffiffiffiffiffiffi ffi
T H T O ÀT L þ Cð T H T O ÀT O Þ

em ¼ pffiffiffiffiffiffiffiffiffiffiffiffi

ð35Þ

The optimal distribution of heat-exchanger areas and the optimal temperatures of the working fluid were also determined.
The RÀe characteristic of the system was obtained and it is identical to that presented by Yan and Chen [1]. Jincan Chen
[50] concluded that the coefficient of performance smaller than em is not optimal for endoreversible three-heat-source singlestage refrigerator.
Wijeysundera [51] considered the three-heat-reservoir singlestage absorption cycles affected by internal irreversibility and discusses its specific cooling load optimization and the effect of internal irreversibilities on the performance of the system.
Chen and Schouten [52] established an irreversible singlestage absorption refrigeration cycle model which operates between three temperature levels and includes finite-rate heat transfer between the working fluid and the external heat reservoirs, heat leak from the heat sink to the cooled space, and irreversibilities due to the internal dissipations of the working fluid. This model was used to optimize the coefficient of performance and cooling rate of the system for a given total heattransfer area of the heat exchangers with respect to the temperatures of the working fluid. Chen and Schouten [52] assumed that the heat transfer rate of the system obeys a linear law:
_
Q H ¼ U H AH ðT H ÀT 1 Þ

ð36Þ

_
Q L ¼ U L AL ðT L ÀT 2 Þ

ð37Þ

_
Q O ¼ U O ðAC þAA ÞðT 3 ÀT O Þ

ð38Þ

_
Q LC ¼ K LC ðT O ÀT L Þ

ð39Þ

where K LC is the heat leak coefficient. The irreversible cycle model mentioned above is obviously more general than an endoreversible cycle model, because it includes the major irreversibilities existing usually in real absorption refrigeration systems. Chen and Schouten [52] derived the maximum coefficient of performance (emax ) and corresponding cooling rate (Rm ) as well as, the maximum cooling load (Rmax ) and corresponding coefficient of performance
(em ). They determined the following optimal expressions:
Fig. 7. eÀR characteristic of a three-heat-source refrigerator affected by thermal resistances, heat leak losses and internal irreversibility. Curves a (I ¼ 1,K LC ¼ 0), b
(I 4 1,K LC ¼ 0) and c (I 41,K LC 4 0) are represented [48].



ð1ÀxÞT L
R
_ ð1þ BÞxT H ÀBIT O ÀT L R þ Q LC

ð40Þ

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P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

ð1ÀxÞðxT H ÀIT O ÞUT L
_
ÀQ LC ð41Þ ½ð1 þ BÞxT H ÀBIT O Š½1 þBð1ÀxފÀxT L pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
U O =IU L À U O =IU H pffiffiffiffiffiffiffiffiffiffiffiffi where x ¼ IT 3 =T 1 , B ¼ and U ¼ U O =Ið1þ U O =IU H Þ2




U O =IU H

They obtained the fundamental optimum relation by eliminating x from the above optimal coefficient of performance and cooling load. The optimal temperatures of the working fluid and the optimal region of temperature of the working fluid and the optimum relation for the distribution of the heat transfer areas were also calculated. From Eqs. (40) and (41) the RÀe characteristic curve of the Chen and Schouten model is presented which is shown in Fig. 8. The interpretation of this figure reveals that em and Rm and emax and Rmax represent the lower bounds and the upper bounds of the coefficient of performance and cooling load respectively. Chen  al. [53] studied the effect of heat transfer law of et  qaD T À1 on the optimal performance of an irreversible threeheat-source model [52]. They considered an irreversible threeheat-source single-stage refrigerator with linear phenomenological heat transfer law affected also by the heat leak from the heat sink to the cooled space, and irreversibilities due to the internal dissipations of the working fluid. Therefore the heat transfer rates that govern the considered system are:
_
Q H ¼ U H AH ðT À1 1 ÀT H À1 Þ

ð42Þ

_
Q L ¼ U L AL ðT À1 ÀT L À1 Þ
2

ð43Þ

_
Q O ¼ U O ðAC þAA ÞðT O À1 ÀT 3 À1 Þ

ð44Þ

_
Q LC ¼ K LC ðT L À1 ÀT O À1 Þ

ð45Þ

Applying similar optimization methods, Chen et al. [53] derived the maximum cooling load and corresponding coefficient of performance, as well as the maximum coefficient of performance and corresponding cooling rate, the optimal region of temperature of the working fluid, the optimum relation for the distribution of the heat transfer areas and the fundamental optimum relation between the coefficient of performance and cooling load. The RÀe characteristic curve of an irreversible threeheat-source refrigeration model which include the irreversibility of heat transfer law of qaDðT À1 Þ, heat leak from the heat sink to the cooled space and the internal dissipation of the working fluids was also derived. It is identical to that obtained by Chen and
Schouten [52]. The higher and lower bounds of the coefficient of performance and cooling load with heat transfer law of qaDðT À1 Þwere deduced.

Fig. 8. RÀe characteristic of a three-heat-source single-stage refrigerator affected by thermal resistances, heat leak losses and internal irreversibility [52].

Grosu et al. [54] optimized the coefficient of performance of an irreversible and endoreversible three-heat-reservoir refrigeration machine with finite time constraints on the total conductance of the system.
Baustita and Mendez [55] performed a performance optimization based on the coefficient of performance criterion of an irreversible refrigeration cycle controlled by three heat sources and affected by irreversibility of heat transfer and internal dissipation. They modeled the cycle as a combined cycle of an internally irreversible two heat source engine driving an internally irreversible two heat source refrigerator. A fundamental optimum relation between the coefficient of performance with the cooling effect was derived.
2.2. Four-heat-source absorption refrigerator
Chen [56] performed a finite time optimization analysis of a four-heat-source refrigerator affected by the heat resistance and internal dissipation inside the working fluid to determine the coefficient of performance, the temperature of the working fluid in the generator, absorber, condenser and evaporator and the useful optimal distribution relation of the heat-transfer areas at maximum specific cooling load. The maximum specific cooling
(r max ) and the corresponding coefficient of performance (em ) were derived. r max and em are two important performance parameters of absorption refrigerator operating between four temperature levels. Chen [56] determined then the upper bound for the specific cooling load and the lower bound for coefficient of performance respectively of four temperature level absorption refrigerator. Shi and Chen [57] carried out a similar performance analysis and optimization for an irreversible four-heat-source with linear heat transfer law affected by the internal irreversibility of the working substance.
Bhardwaj et al. [58,59] investigated a performance optimization analysis based on finite-time thermodynamics approach of an equivalent endoreversible and irreversible four-heat-source absorption refrigeration system considering the thermal resistance finite thermal capacitance. In ref. [58] the optimal bounds for coefficient of performance and working fluid temperatures of the system at the maximum cooling capacity was determined. In ref. [59], they derived the maximum cooling load and the corresponding optimal coefficient of performance.
Using finite time thermodynamics, Zheng et al. [60–66] derived the fundamental optimal relation between the coefficient of performance and the cooling load, the maximum coefficient of performance and the corresponding cooling load, as well as the maximum cooling load and the corresponding coefficient of performance, the optimal working fluid temperatures, the optimal distribution relation of heat-transfer surface areas of a fourheat-reservoir affected by the irreversibilities of (i) linear phenomenological heat transfer law [62], (ii) linear phenomenological heat transfer law and heat leak [61], (iii) linear heat transfer law and heat leak [64,66], (iv) linear heat transfer law, heat leak and internal dissipation inside the working fluid [63], (v) linear phenomenological heat transfer law, heat leak and internal dissipation inside the working fluid [60,65]. They studied the effects of the cycle parameters on the coefficient of performance and the cooling load. Fig. 9 shows the four-heat-source refrigeration cycle model affected by the irreversibility of finite rate heat transfer and heat leakage. The RÀe characteristic of an irreversible four-heat-source model with Newton heat transfer law and the
RÀe characteristic of an irreversible four-heat-source model with linear phenomenological heat transfer law obtained in Refs.
[60,63] are identical to those obtained in Refs. [52,53] for an irreversible three-heat-source model with Newton heat transfer

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

531

ÀDT A ÀT L ŠT H À2ðD þ 1ÞT A T H 3 þT H T A 1=2 ðDT A þ T L Þ ¼ 0

ð50Þ

The resolution of Eq. (50) yields the optimum solar collector temperature. When this minimum value is substituted into Eq.
(49) the optimum operating coefficient of performance for the solar endoreversible absorption refrigerator is determined.
The performance optimization research of the irreversible
Carnot absorption refrigerator coupled to the corrugated sheet collector (CSC) have been conducted by Goktun and Ozkaymak
[69]. The considered model is affected by the internal irreversibility in the working fluid. The optimal overall system coefficient of performance as the function of CSC temperature was [69]:

e ¼ ½0:68À7ðT H ÀT A Þ=IŠ½ðT L =T H ÞðRT H ÀT A Þ=ðT A ÀRT L ފ

Goktun and Ozkaymak [69] derived the optimum CSC temperature and the maximum overall coefficient of performance respectively as:

Fig. 9. A four-heat-source refrigeration cycle model [63].

law and irreversible three-heat-source model with linear phenomenological heat transfer law. These optimum characteristics may be used directly to analyze the influence of major irreversibility on the performance of an irreversible four-heat-reservoir absorption refrigeration. It can be seen from these figure that the coefficient of performance and cooling load should be constrained by: Rm r Rr Rmax and emax Z e Z em

ð47Þ

Researches into heat-engine-driven combined vapor compression and absorption refrigerator have been conducted by Goktun
[67] for space cooling to investigate their optimal performances.
2.3. Solar absorption refrigeration
Flat-plates collectors are commonly used to valorize the solar energy. The solar energy absorbed by solar collectors can be utilized to drive absorption refrigeration for cooling purposes. In a solar operated absorption refrigeration system made of a solar collector and a refrigeration cycle, the energy supplied to the generator is directly from the solar collector. The absorption refrigerator is solar absorption refrigeration when solar collector is coupled to the generator so that, the energy required by the generator is the solar energy absorbed by the solar collector. So the solar absorption refrigerator model is obtained from the model of Fig. 2 by replacing the ‘‘Heat source’’ by the ‘‘Solar collector’’ [68]. The overall coefficient of performance of the solar absorption refrigerator is equal to the product of the efficiency of the solar collector (Zs ) and the coefficient of performance of the absorption refrigerator (ea ) [68]

e ¼ Zs ea

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
T H ¼ ðT A = RÞ a þ 1

ð52Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi emax ¼ ½0:68T L R=T A ÀRT L Šf1 þ ð1=aÞ½ð1=RÞ þ 1À2 ða þ 1Þ=RŠg

ð53Þ

where a ¼ 0:68I=7T A is the operating parameter of a CSC.
Fig. 10 shows the variation of the maximum overall coefficient of performance for CSC-driven irreversible Carnot absorption refrigerator with a for different value of I.
The Wijeysundera solar-powered absorption refrigeration model [70,71] is shown schematically in Fig. 11. Wijeysundera derived the maximum cooling capacity and the corresponding coefficient of performance.
Fath et al. [72] derived the maximum specific cooling load and the corresponding coefficient of performance of a four-heatsource solar absorption refrigerator.
Lin and Yan [73] and Vargas et al. [74] investigated the performance analysis and optimization of a solar-driven refrigerators. They obtained the optimal bounds of coefficient of performance and cooling capacity.
Coefficient of performance and cooling load criteria are used to evaluate the performance and the efficiency bounds of the absorption refrigerators. However, they do not give the performance limit from the view point of the thermo-economical design.

ð48Þ

Wu et al. [68] presented an endoreversible solar absorption refrigerator model. They derived the optimal coefficient of performance of an endoreversible solar absorption refrigerator with respect to operating temperature of the solar collector T H : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1À T A =T H ÞT L pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ð49Þ e ¼ pffiffiffiffiffiffiffiffiffiffiffi
T H T A ÀT L þD T A ð T H À T A Þ where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
U A AA =U L AL ð U H AH À U L AL Þ pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ffi D¼
U H AH þ U A AA
Wu et al. [68] maximized the coefficient of performance given in Eq. (49) with respect to the solar collector temperature T H and obtained the following optimum relation: ðD þ 1ÞT H 2 T A 1=2 þ 2ðT A ÀT L ÞT H 3=2 þ T A 1=2 ½ðD þ 1ÞT H

ð51Þ

Fig. 10. The variation of emax with a for different values of I [69].

532

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

Fig. 11. Schematic diagram of solar-powered absorption system [70,71].

3. Optimization based on the thermo-economic criterion
Finite-time thermo-economic optimization is a further step in performance analysis of absorption systems based on finite-time thermodynamics to include their economic analyses. For an economical design, Sahin and Kodal [75] have introduced a new finite time thermo-economic performance criterion, defined as the cooling load for refrigerators and the heating load for heat pumps per unit total cost (total of investment and energy consumption costs). Based on this criterion, they investigated the economic design conditions of single stage and two stage vapor compression refrigerators and heat pumps [75–78]. The finite-time thermo-economic optimization technique, first introduced by Sahin and Kodal [75], has been extended to irreversible absorption refrigerators [79,80]. The objective function for an irreversible three-heat-source absorption refrigerator with not heat leak losses is defined, as [79]:
F ¼ R=ðC i þ C e Þ

internal irreversibility and the economical parameters increase.
Moreover when e ¼ eI ¼ ½T L ðIT O ÀT H ފ=½ T H ðT L ÀIT O ފ, the thermoeconomic objective function of the three-heat-source refrigerator becomes zero. Therefore, the upper bound of the coefficient of performance given above when there is no finite rate heat transfer irreversibility does not have very much instructive significance for practical applications.
Kodal et al. [79] maximized the objective function with respect to the working fluid temperature and derived the optimum working fluid temperature, the optimum coefficient of performance and the optimum specific cooling load. The optimal distribution of the heat exchangers areas are also obtained for a given total heat transfer area ( i.e. A ¼ AH þAL þAO ). The effects of the internal irreversibility, the economic parameter (k ¼ a=b) and the external temperatures on the global and optimal economic performances were discussed.
Qin et al. [80] analyzed and optimized the thermo-economic performance of an endoreversible four-heat-reservoir absorption

ð54Þ

where C i and C e refer to annual investment and energy consumption costs, respectively. The investment cost of the absorption system is assumed to be proportional to the system size, which may be considered as the total heat transfer areas.
C i ¼ aðAH þAL þ AO Þ

ð55Þ

where the proportionality coefficient for the investment cost of the system, a, is equal to the capital recovery factor times the investment cost per unit heat transfer area, and its dimension is ncu/(year m2). The annual energy consumption cost is proportional to the heat rate input, i.e.
_
C e ¼ bQ H

Fig. 12. Variation of the thermoeconomic objective function for three-heat-source refrigerator with respect to the coefficient of performance, for various I values [79].

ð56Þ

where the coefficient b is equal to the annual operation hours times price per unit energy, and its dimension is ncu/(year kW).
Substituting Eqs. (55) and (56) into Eq. (54) yields


R
_
½aðAH þ AL þAO Þ þbQ H Š

ð57Þ

It should be noted that the objective function defined by Eq.
(57) stands as a more general form for some of the objective functions used in the literature. For example, for the special case of a ¼ 1 and b ¼ 0, the objective function becomes the specific cooling load and for a ¼ 0 and b ¼ 1, it becomes the coefficient of performance. Kodal et al. [79] obtained the variation of the objective function for the irreversible three-heat-source absorption refrigerator with respect to the coefficient of performance for various internal irreversibility parameter I values and various economical parameter k ¼ a=b values which are shown in Figs. 12 and 13 respectively. It can be observed that the maximum thermo-economic objective function and the optimal coefficient of performance reduce as the

Fig. 13. Variation of the thermoeconomic objective function for three-heat-source refrigerator with respect to the coefficient of performance, for various k values [79].

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

refrigeration cycle assuming a linear heat transfer law. In this study the total heat transfer surface area of the four-heatexchangers is assumed to be constant. The optimal relation between the thermo-economic criterion and the coefficient of performance, the maximum thermo-economic criterion, and the corresponding coefficient of performance (eF ) and specific cooling-load (r F ) were derived. They derived the optimal relation between the thermo-economic criterion and the coefficient of performance of an endoreversible absorption refrigerator.
Applying this optimal relation, the thermo-economic performance characteristics of an endoreversible four-heat-source absorption refrigerator with a linear heat transfer law can be derived. Figs. 14 and 15 show the FÀe characteristic curve and the
FÀr characteristic curve respectively of the endoreversible absorption refrigerator. According to these figures the optimal region of the endoreversible absorption refrigerator should obey the following relations:
F r rF r F max , eF r e r er ,r F r r rr max

ð58Þ

where F r is the thermo-economic criterion for the maximum specific cooling load ðr max Þ, and er is the coefficient of performance for the maximum specific cooling load ðr max Þ of the endoreversible absorption refrigerator. Qin et al. [80] also provided numerical examples to discuss the effects of the cycle parameters on the characteristic of the cycle.
The thermo-economical objective function F is used to reduce as well as possible the costs in the design and the industrial facility of the absorption refrigerators and then to make the savings in their thermal consumption of energy. However, like the coefficient of performance and cooling load criteria, he only

533

takes into account the first law of thermodynamics and therefore he does not describe the performance of the absorption refrigerators from the view point of the inevitable degradations of energy which occur in the system during the refrigerating cycle of the working fluid. This aspect is taken into account by the second law of thermodynamics and appears in the thermo-ecological criteria. 4. Optimization based on the ecological criterion
Angulu-Brown [81] proposed an ecological optimization function E which is expressed as:
_
E ¼ W ÀT L s

ð59Þ

_ where W is the power output and s is the entropy generation rate. Yan [82] discussed the results of Angulu-Brown [81] and suggested that it may be more reasonable to use:
_
E ¼ W ÀT O s

ð60Þ

if the cold reservoir temperature is not equal to the environment temperature T O . The optimization of ecological function is therefore claimed to achieve the best compromise between the workenergy rate (e.g. power of an engine, cooling rate of a refrigerator, or heating rate of a heat pump) and its dissipation which is produced by entropy generation in the system and its surroundings. Chen et al. [83] and Yan et Lin [84] reported a similar ecological optimization of the three-heat-source absorption refrigerator. They defined the ecological criterion function E of a refrigerator as:
E ¼ RÀlT O s

ð61Þ

where l is the dissipation coefficient of the cooling rate. The physical meaning of the dissipation coefficient of cooling rate l is that, in theory, if the rate of availability T O s were not lost, it would produce a cooling rate lT O s through a reversible Carnot refrigerator operating between T O and T L . This shows that l is equal to the coefficient of performance of the reversible Carnot refrigerator, i.e.

l ¼ T L =ðT O ÀT L Þ

Fig. 14. Variation of the thermoeconomic objective function for an endoreversible four-heat-source refrigerator with respect to the coefficient of performance [80].

ð62Þ

In their performance ecological optimization work for an irreversible three-heat-source absorption refrigerator with linear heat transfer law affected by internal dissipation inside the working fluid, Yan et Lin [84] investigated the maximum ecological optimization criterion E and derived the optimal cooling load, coefficient of performance and entropy production rate i.e.

eE ¼

er eI T H =T L À2þ ½ðeI T H =T L þ 1Þðer T H =T L þ 2ÞðeI þ 1Þðer þ 2ފ1=2 ðer þ 2eI ÞT H =T L þ 2ðT H =T L þ 1Þ ð63Þ RE ¼

sE ¼

Fig. 15. Variation of the thermoeconomic objective function for an endoreversible four-heat-source refrigerator with respect to the specific cooling load [80].



aA
TH
IT O pffiffi À ð1þ IÞ2 T H =T L þ eE À1 1þ eE À1 aAð1ÀT O =T H ÞðeÀ1 ÀeÀ1 Þ r E pffiffi T O ð1þ IÞ2



TH
IT O
À
T H =T L þ eÀ1 1þ eÀ1
E
E

ð64Þ
!
ð65Þ

where eI ¼ ð1ÀIT O =T H ÞT L =ðIT O ÀT L Þ is the upper bound of coefficient of performance for the irreversible three-heat-source absorption refrigerator. eE , RE and sE are three important parameters of an irreversible three-heat-source refrigerator, they determine the ecological optimization performance of the refrigerator. The optimal temperature of the working fluid in the three isothermal processes were also obtained. By numerical example they showed some advantages in using the ecological optimization criterion; it is

534

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

especially beneficial for determining the reasonable use of energy and protecting the ecological environment. The ratios RE =Rmax , sE =sm and ðsE =sm Þ=ðRE =Rmax Þ against I curves are shown in
Fig. 16. It can be observed from this figure that the principal advantage of the ecological optimization criterion over the maximum work-energy rate criterion is a great reduction in the ratio between entropy production and work-energy rate for the refrigerator. Huishan [85] carried out similar ecological performance analysis to determine the influence of finite time heat transfer between the heat sources and the working fluid on the ecological optimal performance of a three-heat-source absorption refrigerator.
Chen et al. [86,87] also reported similar performance characteristic of an irreversible absorption refrigeration system at maximum ecological criterion operating among four temperature levels. They derived the optimal relationship between the ecological criterion and the coefficient of performance.
The thermo-ecological criterion is used to achieve the best compromise between the cooling rate and its dissipations of the absorption refrigerators. However, it may take negative values.
Such an objective function in a performance analysis can be defined mathematically; however, it needs interpretation to comprehend this situation thermodynamically.

5. Optimization based on the new thermo-ecological criterion
Ust [88] has recently introduced a new dimensionless ecological optimization criterion called the ecological coefficient of performance (ECOP) that has always positive values and takes into account the loss rate of availability on the performance. Ust
[88] defined the ECOP as the ratio of power output to the loss rate of availability:
ECOP ¼

_
W
_
T env s

ð66Þ

By employing the ECOP function, many studies have been done for different heat engine models [89–94]. The ECOP function defined for heat engines has been modified for irreversible Carnot refrigerator model by Ust and Sahin [95,96], as the ratio of cooling load to the loss rate of availability:
ECOP ¼

_
R
_
T env s

ð67Þ

The ECOP give information about to the loss rate of availability or entropy generation rate in order to produce a certain cooling.
It should be noted that for a certain cooling load, the entropy generation rate is minimum at maximum ECOP condition. The

Fig. 16. The RE =Rmax , sE =sm and ðsE =sm Þ=ðRE =Rmax Þ against I curves of the irreversible three-heat-source refrigerator [84].

maximum of the ECOP function signifies the importance of getting the cooling load from a refrigerator by causing lesser dissipation in the environment. Therefore the higher the ECOP, we have a better absorption refrigerator in terms of cooling load and the environment considered together.
Very recently Ngouateu Wouagfack and Tchinda [97] extended a thermo-ecological performance analysis based on ECOP criterion given in Refs. [95,96] to an irreversible three-heat-source absorption refrigerator which includes finite-rate heat transfer between the working fluid and the external heat reservoirs, heat leak from the heat sink to the cooled space, and irreversibilities due to the internal dissipations of the working fluid. They determined analytically the maximum of the ecological performance criterion and the corresponding optimal coefficient of performance, cooling load and entropy generation rate for a given total heat-transfer area of the heat exchangers. The corresponding optimal temperatures of the working fluid in the main components of the system and the optimal distribution of the heattransfer areas are also obtained analytically. The influences of the major irreversibilities on the thermo-ecological performances are discussed detailed. Additionally, the variations of the normalized
ECOP and COP with respect to the entropy generation rate have been demonstrated which is shown in Fig. 17. From this figure and analytically, Ngouateu Wouagfack and Tchinda [97] obtained that the maximum of the ECOP and COP coincides.
Ngouateu Wouagfack and Tchinda [98] considered an irreversible three-heat-source absorption refrigeration model with the
_
linear phenomenological heat transfer law of Q aDðT À1 Þ which includes heat leak from the heat sink to the cooled space, and irreversibilities due to the internal dissipations of the working fluid. They optimized the specific cooling load, the COP, the ecological function E and the ECOP and established that the absorption refrigeration cycle working at maximum ECOP conditions has a significant advantage in terms of entropy production rate and coefficient of performance over that working at maximum
E or maximum R conditions.

6. Conclusion
Finite-time thermodynamics optimization is the optimization method of various real thermodynamic processes and devices affected by the irreversibility of heat transfer with their surroundings. It is different from classical thermodynamics optimization which does not take into account the thermal resistances.

Fig. 17. Variation of the normalized ECOP, COP and the specific cooling load with respect to the specific entropy generation rate [97].

P.A. Ngouateu Wouagfack, R. Tchinda / Renewable and Sustainable Energy Reviews 21 (2013) 524–536

In this paper an overview of the performance optimization criteria based on the finite-time thermodynamics for absorption refrigerator systems has been presented. The coefficient of performance, the cooling load, the thermo-economic objective function, the thermo-ecological objective function and the new thermo-ecological objective function have been discussed. It has been seen that the major irreversibilities such as thermal resistance, heat leak and internal irreversibilities due to the dissipation of the working fluid affect the performance of real absorption refrigeration systems. The coefficient of performance, the cooling load, the thermo-economic criteria take into account only the first law of thermodynamics and therefore, they do not describe the performance of the absorption refrigeration cycles from the view point of the thermo-ecological aspect. This factor is taken into account by the second law of thermodynamics characterized by the entropy production which appears in the ecological optimization criterion (E) and the ecological coefficient of performance
(ECOP). With the requirement of a rigorous management of our energy resources, one should have brought to be interested more and more in the second principle of thermodynamics, because degradations of energy, in other words the entropy productions, are equivalent to consumption of energy resources. The ecological function E criterion can take negative values. At this condition, the loss of cooling load is greater than the cooling load produced.
The ECOP criterion is dimensionless and has always positive values. This literature review is a contribution for the development of real absorption refrigeration systems since it may provide a general theoretical tool for their design. Moreover, it is hoped that this contribution will stimulate wider interest in the definition of new performance criteria for the optimization of absorption refrigerators.

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