Open access peer-reviewed chapter

# ECOP Criterion for Irreversible Three-Heat-Source Absorption Refrigerators

Written By

Paiguy Armand Ngouateu Wouagfack and Réné Tchinda

Submitted: December 5th, 2011 Reviewed: July 12th, 2012 Published: October 3rd, 2012

DOI: 10.5772/51547

From the Edited Volume

## Thermodynamics

Edited by Ricardo Morales-Rodriguez

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## 1. Introduction

In this chapter, the optimization analysis based on the new thermo-ecological criterion (ECOP) first performed by Ust et al. [1] for the heat engines is extended to an irreversible three-heat-source absorption refrigerator. The thermo-ecological objective function ECOP is optimized with respect to the temperatures of the working fluid. The maximum ECOP and the corresponding optimal temperatures of the working fluid, coefficient of performance, specific cooling load, specific entropy generation rate and heat-transfer surface areas in the exchangers are then derived analytically. Comparative analysis with the COP criterion is carried out to prove the utility of the ecological coefficient of performance criterion.

## 2. Thermodynamics analysis

The main components of an absorption refrigeration system are a generator, an absorber, a condenser and an evaporator as shown schematically in Fig. 1 [2]. In the shown model, Q.His the rate of absorbed heat from the heat source at temperature THto generator, Q.Cand Q.Aare, respectively, the heat rejection rates from the condenser and absorber to the heat sinks at temperatures TCand TAand Q.Lis the heat input rate from the cooling space at temperature TLto the evaporator. In absorption refrigeration systems, usually NH3/H2O and LiBr/H2O are used as the working substances, and these substances abide by ozone depletion regulations, since they do not consist of chloroﬂuorocarbons. In Fig. 1, the liquid rich solution at state 1 is pressurized to state 1’ with a pump. In the generator, the working fluid is concentrated to state 3 by evaporating the working medium by means of Q.Hheat rate input. The weak solution at state 2 passes through the expansion valve into the absorber with a pressure reduction (2–2’). In the condenser, the working fluid at state 3 is condensed to state 4 by removing Q.Cheat rate. The condensed working fluid at state 4 is then throttled by a valve and enters the evaporator at state 4’. The liquid working fluid is evaporated due to heat transfer rate Q.Lfrom the cooling space to the working fluid (4’–5). Finally, the vaporized working fluid is absorbed by the weak solution in the absorber, and by means of Q.Aheat rate release in the absorber, state 1 is reached.

Work input required by the solution pump in the system is negligible relative to the energy input to the generator and is often neglected for the purpose of analysis. Under such assumption, the equation for the first law of thermodynamics is written as:

Q.H+Q.LQ.CQ.A=0E1

Absorption refrigeration systems operate between three temperature levels, ifTA=TC, or four temperature levels whenTATC. In this chapter, by takingTA=TC, the cycle of the working fluid consists of three irreversible isothermal and three irreversible adiabatic processes. The temperatures of the working fluid in the three isothermal processes are different from those of the external heat reservoirs so that heat is transferred under a finite temperature difference, as shown in Fig. 2 where

Q.O=Q.C+Q.AE2
T1and T2are, respectively, the temperatures of the working fluid in the generator and evaporator. It is assumed that the working fluid in the condenser and absorber has the same temperature T3[2]. Q.LCis the heat leak from the heat sink to the cooled space.

The heat exchanges between the working fluid and heat reservoirs obey a linear heat transfer law, so that the heat-transfer equations in the generator, evaporator, condenser and absorber are, respectively, expressed as follows:

Q.H=UHAH(THT1)E3
(
Q.L=ULAL(TLT2)E4
Q.O=UO(AA+AC)(T3TO)E5

whereAH, AL, ACand AAare, respectively, the heat-transfer areas of the generator, evaporator, condenser and absorber, UHand ULare, respectively, the overall heat-transfer coefficients of the generator and evaporator, and it is assumed that the condenser and absorber have the same overall heat-transfer coefficient UO[2].

The absorption refrigeration system does not exchange heat with other external reservoirs except for the three heat reservoirs at temperaturesTH, TLandTO, so the total heat-transfer area between the cycle system and the external heat reservoirs is given by the relationships:

A=AH+AL+AOE6

where

AO=AC+AA.E7

The rate of heat leakage Q.LCfrom the heat sink at temperature TOto the cold reservoir at temperature TLwas first provided by Bejan [3] and it is given as:

Q.LC=KLC(TOTL)E8

where KLCis the heat leak coefficient.

Real absorption refrigerators are complex devices and suffer from a series of irreversibilities. Besides the irreversibility of finite rate heat transfer which is considered in the endoreversible cycle models and the heat leak from the heat sink to the cooled space, there also exist other sources of irreversibility. The internal irreversibilities that result from friction, mass transfer and other working fluid dissipations are an another main source of irreversibility, which can decrease the coefficient of performance and the cooling load of absorption refrigerators. The total eﬀect of the internal irreversibilities on the working fluid can be characterized in terms of entropy production. An irreversibility factor is introduced to describe these internal irreversibilities:

I=ΔS3ΔS1+ΔS2E9

On the basis of the second law of thermodynamics, ΔS3>ΔS1+ΔS2for an internally irreversible cycle, so thatI>1. If the internal irreversibility is neglected, the cycle is endoreversible and soI=1. The second law of thermodynamics for an irreversible three-heat-source cycle requires that:

δQ.T=Q.HT1+Q.LT2Q.OT30E10

From Eq. (9), the inequality in Eq. (10) is written as:

Q.HT1+Q.LT2Q.OIT3=0E11

The coefficient of performance of the irreversible three-heat-source absorption refrigerator is:

COP=Q.LQ.LCQ.H=Q.LQ.H(1Q.LCQ.L)E12

From Eq. (6), it is expressed as:

AL=A1+AHAL+AOALE13

Using Eqs. (3)-(5), Eq. (13) is rewritten as:

AL=A1+Q.HQ.LUL(TLT2)UH(THT1)+Q.OQ.LUL(TLT2)UO(T3TO)E14

Combining Eqs. (1) and (11), the following ratios are derived:

Q.LQ.H=T2(T1IT3)T1(IT3T2)E15
Q.OQ.L=IT3(T1T2)(T1IT3)T2E16

The first is the coefficient of performance of the irreversible three-heat-source absorption refrigeration cycle without heat leak losses.

Substituting Eqs. (15) and (16) into Eq. (14), the heat-transfer area of the evaporator is expressed as a function ofT1, T2and T3for a given total heat-transfer areas :

AL=A1+ULT1(IT3T2)(TLT2)UH(THT1)(T1IT3)T2+ULIT3(T1T2)(TLT2)UO(T3TO)(T1IT3)T2E17

By investigating similar reasoning, the heat-transfer areas of the generator and of condenser and absorber are given respectively by:

AH=A1+UHT2(T1IT3)(THT1)ULT1(IT3T2)(TLT2)+UHIT3(T1T2)(THT1)UOT1(IT3T2)(T3TO)E18
and
AO=A1+UOT1(IT3T2)(T3TO)UHIT3(T1T2)(THT1)+UOT2(T1IT3)(T3TO)ULIT3(T1T2)(TLT2)E19

Substituting Eq. (17) into Eq. (4):

Q.L=A1UL(TLT2)+T1(IT3T2)UH(THT1)(T1IT3)T2+IT3(T1T2)UO(T3TO)(T1IT3)T2E20

Combining Eqs. (8), (12), (15) and (20), the coefficient of performance of the irreversible three-heat-source refrigerator as a function of the temperaturesT1, T2and T3of the working fluid in the generator, evaporator, condenser and absorber is obtained:

COP=T2(T1IT3)T1(IT3T2){1ξ(TOTL)[1UL(TLT2)+T1(IT3T2)UH(THT1)(T1IT3)T2+IT3(T1T2)UO(T3TO)(T1IT3)T2]}E21

where the parameter

ξ=KLCAE22

represents the heat leakage coefficient and its dimension is w/(Km2)

The specific cooling load of the irreversible three-heat-source refrigerator is deduced as:

r=Q.LQ.LCA=[1UL(TLT2)+T1(IT3T2)UH(THT1)(T1IT3)T2+IT3(T1T2)UO(T3TO)(T1IT3)T2]1ξ(TOTL)E23

The specific entropy production rate of the irreversible three-heat-source absorption refrigerator is:

s=σ.A=Q.OQ.LCTOQ.HTHQ.LQ.LCTLAE24

Using Eq. (1) sis rewritten as:

s=(1TL1TO)Q.LCA+(1TO1TH)Q.HQ.LQ.LA+(1TO1TL)Q.LAE25

or

s=(1TO1TH)Q.HA+(1TO1TL)Q.LQ.LCAE26

Substituting Eqs.(8), (15) and (20) into Eq. (25), the specific entropy production rate as a function ofT1, T2and T3is given by :

s=(1TL1TO){ξ(TOTL)[1εrT1(IT3T2)T2(T1IT3)][1UL(TLT2)+T1(IT3T2)UH(THT1)(T1IT3)T2+IT3(T1T2)UO(T3TO)(T1IT3)T2]1}E27

where

εr=(1TOTH)(TOTL1)E28

is the coefficient of performance for reversible three-heat-source refrigerator.

According to the definition of the general thermo-ecological criterion function for different heat engine models [4-9], a two-heat-source refrigerator [10, 11] and three-heat-source absorption refrigerator [2], the new thermo-ecological objective function called ecological coefficient of performance (ECOP)of an absorption refrigerator is defined as:

ECOP=Q.LQ.LCTenvσ.=Q.LQ.LCATenvsE29

Putting Eq.(26) into Eq. (29):

ECOP=1Tenv[1TO1TH+(1TO1TL)1COP]E30

When Eq. (21) is put in Eq. (30), the ecological coefficient of performance of the irreversible three-heat-source absorption refrigerator as a function ofT1, T2and T3is derived as :

ECOP=1Tenv(TO1TL1)1εrT1(IT3T2)T2(T1IT3){1ξ(TOTL)[1UL(TLT2)+T1(IT3T2)UH(THT1)(T1IT3)T2+IT3(T1T2)UO(T3TO)(T1IT3)T2]}1E31

where Tenvis the temperature in the environment conditions.

## 3. Performance optimization for a three-heat-source irreversible absorption refrigerator based on ECOP criterion

The ECOP function given in Eq. (31) is plotted with respect to the working fluid temperatures (T1, T2andT3) for different internal irreversibility parameters as shown in Fig. 3(a), (b) and (c). As it can be seen from the figure, there exists a specificT1, T2and T3that maximize the ECOP function for given Iand ξvalues. Therefore, Eq. (31) can be maximized (or optimized) with respect toT1, T2andT3. The optimization is carried out analytically.

For the sake of convenience, let

x=IT3T1E32
y=IT3T2E33
z=IT3E34

Then Eq. (31) is rewritten as:

ECOP=1Tenv(TO1TL1)1εr(y1)1x{1ξ(TOTE)[yUL(TLyz)+x(y1)UH(THxz)(1x)+yxU(zT)(1x)]}1E35

where

T=ITOE36

and

U=UOI.E37
.

Starting from Eq. (35), the extremal conditions:

ECOPx=0E38
ECOPy=0E39
ECOPz=0E40

give respectively:

1ξ(TOTE)yUL(TLyz)1U(zT)z(y1)UH(THxz)2=0E41
1ξ(TOTE)yUL(TLyz)1U(zT)z(y1)UL(TLyz)2=0E42
y(1x)UL(TLyz)2+x(y1)UH(THxz)2yxU(zT)2=0E43

Combining Eqs (41)-(43), the following general relation is found:

UH(THxz)=UL(TLyz)=U(zT)E44

From Eqs (44), it is derived as:

x=(1+b1)zTHb1TTHE45
y=(1+b2)zTLb2TTLE46

where

b1=UUHE47
b2=UULE48

When Eqs. (45) and (46) are substituted into Eq. (43):

z=TD+b21+b2E49

where

D=1+d1[1TL(1d1)T]1d1E50
d1=ξ(1+b2)2U(TOTL1)E51

Therefore Eqs. (45) and (46) are rewritten as:

x=TTHB1(D+B)E52
y=TTLDE53

where

B=b2b11+b1E54
B1=1+b11+b2E55

Using Eqs. (49), (52) and (53) with Eqs.(32)-(34), the corresponding optimal temperatures of the working fluid in the three isothermal processes when the ecological coefficient of performance is a maximum, are, respectively, determined by:

T1*=THD+b2(1+b1)(D+B)E56
T2*=TLD+b2D(1+b2)E57
T3*=TOD+b21+b2E58

Substituting Eqs. (56)-(58) into Eqs. (21), (23), (27) and (31) the maximum ECOPfunction and the corresponding optimal coefficient of performance, optimal specific cooling load and optimal specific entropy generation rate are derived, respectively, as:

ECOPmax=1Tenv(TO1TL1)E59
×11εrTH(TDTL)[THB1(D+B)T]TL{1ξ(TOTL)THDB12(D+B)(TL+BDT)U*B12(D1)[THB1(D+B)T]TL}1E60
COP*=[THB1(D+B)T]TLTH(TDTL){1ξ(TOTL)THDB12(D+B)(TL+BDT)U*B12(D1)[THB1(D+B)T]TL}E61
r*=U*B12(D1)[THB1(D+B)T]THTHDB12(D+B)(TL+BDT)ξ(TOTL)E62
s*=(1TL1TO){ξ(TOTL)[1εrTH(TDTL)[THB1(D+B)T]TL][U*B12(D1)[THB1(D+B)T]TLTHDB12(D+B)(TL+BDT)]}E63

where

U*=U(1+b1)2E64

From Eqs. (17)-(19) and (56)-(58), it is found that, when the three-heat-source absorption refrigerator is operated in the state of maximum ecological coefficient of performance, the relations between the heat-transfer areas of the heat exchangers and the total heat-transfer area are determined by:

AH*=Ab11+b1B12(D+B)(TDTL)THDB12(D+B)(TL+BDT)E65
AL*=Ab21+b1DB1[THB1T(D+B)]THDB12(D+B)(TL+BDT)E66
AO*=A11+b1B1[THDB1TL(D+B)]THDB12(D+B)(TL+BDT)E67

From Equations (64)-(66), a concise optimum relation for the distribution of the heat-transfer areas is obtained as:

UHAH*+ULAL*=UAO*E68

Obviously, this relation is independent of the heat leak and the temperatures of the external heat reservoirs.

## 4. Comparison with COP criterion

In Fig.4, the variation of the normalized ECOP (ECOP¯=ECOPECOPmax), normalized COP (COP¯=COPCOPmax) and the specific cooling load (r) with respect to the specific entropy generation rate (s) are demonstrated. One interesting observation from this figure is that maximum of the ECOP and COP coincides although their functional forms are different: the coefficient of performance gives information about the necessary heat rate input in order to produce certain amount of cooling load and the ecological coefficient of performance gives information about the entropy generation rate or loss rate of availability in order to produce certain amount of cooling load. The maximum ECOP and COP conditions give the same amount of cooling load and entropy generation rate. It is also seen analytically that the performance parametersT1*, T2*, T3*, A1*, A2*, A3*, r*, s*and COP*=COPmaxat the maximum ECOP and maximum COP are same. Getting the same performance at maximum ECOP and COP conditions is an expected and logical result. Since, for a certain cooling load the maximum COPresults from minimum heat consumption so that minimum environmental pollution. The minimum environmental pollution is also achieved by maximizing theECOP. Although the optimal performance conditions ECOPand COPcriteria are same, their impact on the system design performance is different. The coefficient of performance is used to evaluate the performance and the efficiency of systems. This method only takes into account the first law of thermodynamics which is concerned only with the conversion of energy, and therefore, can not show how or where irreversibilities in a system or process occur. Also, when different sources and forms of energy are involved within a system, the COPcriterion of a system doesn’t describe its performance from the view point of the energy quality involved. This factor is taken into account by the second law of thermodynamics characterized by the entropy production which appears in the ecological coefficient of performance criterion (ECOP). This aspect is of major importance today since that 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. For this important reason, the ECOPcriterion can enhance the system performance of the absorption refrigerators by reducing the irreversible losses in the system. A better understanding of the second law of thermodynamics reveals that the ecological coefficient of performance optimization is an important technique in achieving better operating conditions.

## 5. Conclusion

This chapter presented an analytical method developed to achieve the performance optimization of irreversible three-heat-source absorption refrigeration models having finite-rate of heat transfer, heat leakage and internal irreversibility based on an objective function named ecological coefficient of performance (ECOP). The optimization procedure consists in defining the objective function ECOP in term of the temperatures of the working fluid in the generator, evaporator, condenser and absorber and using extremal conditions to determine analytically the maximum ECOP and the corresponding optimal design parameters. It also established comparative analyses with the COP criterion and shown that the performance parameters at the maximum ECOP and maximum COP are same. The three-heat-source absorption refrigerator cycles are the simplified models of the absorption refrigerators, but the four-heat-source absorption refrigerators cycles are closer to the real absorption refrigerators.

## References

1. 1. Ust, Y.: Ecological performance analysis and optimization of power-generation systems, Ph.D.Thesis Progress Report, Yildiz Technical University, Istanbul. (2004
2. 2. NgouateuWouagfack. P. A.TchindaR.Performanceoptimization.ofthree-heat-source.irreversiblerefrigerator.basedon. a.newthermo-ecological.criterionInternational Journal of Refrigeration1008 EOF1015 EOF2011
3. 3. Bejan, A.: Theory of heat transfer-irreversible refrigeration plant.International Journal of Heat Transfer. 32,1631 EOF1639 EOF1989
4. 4. UstY.SahinB.SogutO. S.Performanceanalysis.optimizationof.anirreversible.Dualcycle.basedon.ecologicalcoeﬃcient.ofperformance. . E. C. O. P.criterionApplied Energy. 82 (1), 23-39 (2005
5. 5. Ust, Y., Sahin, Kodal, A.: Ecological coeﬃcient of performance (ECOP) optimization for generalized irreversible Carnot heat engines.Journal of the Energy Institute. 78 (3), 145-151 (2005
6. 6. Ust, Y., Sahin,B., Kodal, A.: Performance analysis of an irreversible Brayton heat engine based on ecological coeﬃcient of performance criterion.International Journal of Thermal Science. 45 (1), 94-101 (2006
7. 7. UstY.SogutO. S.SahinB.DurmayazA.Ecologicalcoeﬃcient.ofperformance. . E. C. O. P.optimizationfor.anirreversible.Braytonheat.enginewith.variable-temperaturethermal.reservoirsJournal of the Energy Institute. 79 (1), 47-52 (2006
8. 8. UstY.SahinB.KodalA.AkcayI. H.Ecologicalcoeﬃcient.ofperformance.analysisoptimizationof.anirreversible.regenerativeBrayton.heatengine.Applied Energy. 83 (6), 558-572 (2006
9. 9. SogutO. S.UstY.SahinB.Theeﬀects.ofintercooling.regenerationon.thethermo-ecological.performanceanalysis.ofan.irreversible-closedBrayton.heatengine.withvariable-temperature.thermalreservoirs.Journal of Physics D: Applied Physics. 39, 4713-4721 (2006
10. 10. Ust, Y., Sahin, B.: Performance optimization of irreversible refrigerators based on a new thermo-ecological criterion.International Journal of Refrigeration. 30,527 EOF2007
11. 11. Ust, Y.: Performance analysis and optimization of irreversible air refrigeration cycles based on ecological coeﬃcient of performance criterion.Applied Thermal Engineering. 29, 47-55 (2009

Written By

Paiguy Armand Ngouateu Wouagfack and Réné Tchinda

Submitted: December 5th, 2011 Reviewed: July 12th, 2012 Published: October 3rd, 2012