Exergetic Efficiency Optimization for an Irreversible Heat Pump Working on Reversed Brayton Cycle

PRAMANA c© Indian Academy of Sciences Vol. 74, No. 3— journal of March 2010

physics pp. 351–363

Exergetic efficiency optimization for an irreversibleheat pump working on reversed Brayton cycle

YUEHONG BI1,2, LINGEN CHEN2,∗ and FENGRUI SUN2

1Institute of Civil & Architectural Engineering, Beijing University of Technology,Beijing 100124, People’s Republic of China2Postgraduate School, Naval University of Engineering, Wuhan 430033, People’s Republicof China∗Corresponding author. E-mail: [email protected]; [email protected]

MS received 7 December 2008; revised 18 August 2009; accepted 17 November 2009

Abstract. This paper deals with the performance analysis and optimization for irre-versible heat pumps working on reversed Brayton cycle with constant-temperature heatreservoirs by taking exergetic efficiency as the optimization objective combining exergyconcept with finite-time thermodynamics (FTT). Exergetic efficiency is defined as theratio of rate of exergy output to rate of exergy input of the system. The irreversibili-ties considered in the system include heat resistance losses in the hot- and cold-side heatexchangers and non-isentropic losses in the compression and expansion processes. Theanalytical formulas of the heating load, coefficient of performance (COP) and exergeticefficiency for the heat pumps are derived. The results are compared with those obtainedfor the traditional heating load and coefficient of performance objectives. The influencesof the pressure ratio of the compressor, the allocation of heat exchanger inventory, thetemperature ratio of two reservoirs, the effectiveness of the hot- and cold-side heat ex-changers and regenerator, the efficiencies of the compressor and expander, the ratio ofhot-side heat reservoir temperature to ambient temperature, the total heat exchanger in-ventory, and the heat capacity rate of the working fluid on the exergetic efficiency of theheat pumps are analysed by numerical calculations. The results show that the exergeticefficiency optimization is an important and effective criterion for the evaluation of anirreversible heat pump working on reversed Brayton cycle.

Keywords. Exergetic efficiency; optimization; irreversible; heat pump working on re-versed Brayton cycle; finite-time thermodynamics.

PACS Nos 05.70.Ln; 05.70.-a; 05.60.Cd

1. Introduction

The finite-time thermodynamics (FTT) or entropy-generation minimization (EGM)[1–18] is a powerful tool for analysing and optimizing the performance of thermody-namic processes and cycles. The results obtained for various thermodynamic cyclesusing FTT are closer to real device performance than those obtained using classicalthermodynamics. Due to environmental damage (ozone layer, global warming) by

351

Yuehong Bi, Lingen Chen and Fengrui Sun

CFC refrigerants, the analysis for heat pump working on reversed Brayton cyclewhich uses air as the refrigerant and meets all criteria for a refrigerant being envi-ronmental friendly has been preferred [19–26]. Many important works about heatpumps working on reversed Brayton cycle applying FTT have been published inrecent years [12,27–32]. The performance analysis and optimization for heat pumpcycles were carried out by taking the heating load and the coefficient of performance(COP) as the optimization objectives [26–31]. The heating load density was alsotaken as the optimization objective [32].

In recent years, the research combining classical exergy concept [33–35] withFTT [1–18] is becoming increasingly important. Exergy output optimization andexergetic efficiency optimization for endoreversible cogeneration cycle [36], endore-versible and irreversible Carnot refrigeration cycles [37,38] and irreversible Braytonrefrigeration cycle [39] were carried out. In this paper, exergetic efficiency optimiza-tion for an irreversible heat pump working on reversed Brayton cycle is investigated.The irreversibilities considered in the system include heat resistance losses in thehot- and cold-side heat exchangers and non-isentropic losses in the compression andexpansion processes. The influences of various parameters on the characteristic ofthe heat pump are analysed by numerical examples. The results may help to bet-ter understand the performance of the irreversible heat pump working on reversedBrayton cycle and provide guidelines for the design of practical heat pump plants.

2. Irreversible heat pump working on reversed Brayton cycle model

The diagram of an irreversible heat pump working on reversed Brayton cycle(BCHP) and its surroundings are shown in figure 1. The following assumptionsare made for this model:

Figure 1. Temperature–entropy diagram

of an irreversible heat pump working on

reversed Brayton cycle with constant tem-

perature heat reservoirs.

Figure 2. Effect of heat reservoir temper-

ature ratio on the exergetic efficiency vs.

pressure ratio.

352 Pramana – J. Phys., Vol. 74, No. 3, March 2010

Irreversible heat pump working on reversed Brayton cycle

(i) The working fluid flows through the system in a steady-state fashion. The ir-reversible BCHP consists of two isobaric processes (1-2, 3-4) and two non-isentropicadiabatic processes (2-3, 4-1). The two adiabatic processes (2-3s, 4-1s) are the cor-responding isentropic ones. Adiabatic accessibility of a thermodynamic state fromanother state is an important issue in the study of heat pumps especially the onesbased on irreversible processes.

Therefore, cycle 1-2-3-4-1 is an irreversible one and 1s-2-3s-4-1s is an endore-versible one. The irreversible and endoreversible cycles are distinguished by usingthe efficiencies of the non-isentropic compression and expansion processes. Thecompressor and expander efficiencies are defined as

ηc =(T3s − T2)(T3 − T2)

, ηt =(T4 − T1)(T4 − T1s)

. (1)

For irreversible cycle, the internal irreversibility, i.e., non-isentropic losses in thecompression and expansion processes are considered, ηc < 1 and ηt < 1 are satisfied.When the two adiabatic processes become reversible, the compressor and expanderefficiencies are ηc = ηt = 1, the cycle becomes an endoreversible one (i.e. with thesole irreversibility of heat resistance) (with the loss of only heat resistance).

(ii) The heat reservoirs have infinite thermal capacitance rates. The heat sink isat temperature TH and the heat source at TL.

(iii) The hot- and cold-side heat exchangers are considered to be counter-flowheat exchangers, and their heat conductances (product of heat transfer coefficientand area) are UH and UL, respectively. The working fluid is an ideal gas havingconstant thermal capacitance rate (the product of mass flow rate and specific heat),Cwf .

According to the properties of the heat transfers between the heat reservoir andworking fluid and the theory of the heat exchangers, the rate of heat transfer (QH)released to the heat sink, i.e., the heating load, and the rate of heat transfer (QL)supplied by the heat source, are, respectively, given by

QH =UH(T3 − T4)

ln[(T3 − TH)/(T4 − TH)]= CwfEH(T3 − TH), (2)

QL =UL(T2 − T1)

ln[(TL − T1)/(TL − T2)]= CwfEL(TL − T1), (3)

where U is the heat conductance, E is the effectiveness of the heat exchanger andN is the number of heat transfer units, and are defined as

EH = 1 − exp(−NH), EL = 1 − exp(−NL), (4)

NH =UH

Cwf, NL =

UL

Cwf. (5)

Pramana – J. Phys., Vol. 74, No. 3, March 2010 353


3. Analytical relations

3.1 The heating load and the COP

The second law of thermodynamics requires T2T4 = T1sT3s. Combining eqs (1)–(5)gives

T1 =

EHTHηc(ηtx−1 − ηt + 1) + ELTL(1 − EH)(x + ηc − 1)

×(ηtx−1 − ηt + 1)

ηc − (1 − EH)(1 − EL)(x + ηc − 1)(ηtx−1 − ηt + 1), (6)

T2 =ηc[ELTL + EHTH(1 − EL)(ηtx

−1 − ηt + 1)]ηc − (1 − EH)(1 − EL)(x + ηc − 1)(ηtx−1 − ηt + 1)

, (7)

where x is the isentropic temperature ratio of the working fluid, that is, x =T3s/T2 = (P3/P2)m = πm, where π is the pressure ratio of the compressor,m = (k− 1)/k, and k is the ratio of specific heats. Combining eqs (2)–(7) gives theheating load (QH) and the COP (β) of the cycle

QH =

CwfEH{(x + ηc − 1)ELTL − [ηc − (1 − EL)(x + ηc − 1)×(ηtx

−1 − ηt + 1)]TH}ηc − (1 − EH)(1 − EL)(x + ηc − 1)

×(ηtx−1 − ηt + 1)

, (8)

1 − β−1 =

EL{[ηc − (1 − EH)(πm + ηc − 1)(ηtπ−m − ηt + 1)]

−EH(ηtπ−m − ηt + 1)ηcτ1}

EH{(πm + ηc − 1)EL − [ηc − (1 − EL)(πm + ηc − 1)×(ηtπ

−m − ηt + 1)]τ1}. (9)

When ηc = ηt = 1 is satisfied, the two adiabatic processes become reversible, andthe irreversible cycle becomes endoreversible. The heating load (QH) and the COP(β) of the endoreversible cycle are, respectively

QH = CwfEHEL[TL/(1 − β−1) − TH](EH + EL − EHEL) (10)

β =πm

πm − 1. (11)

Equation (11) indicates that the COP of endoreversible cycle is only dependent onthe pressure ratio. The dimensionless heating load Q̄H is given by

Q̄H =QH

CwfTH

=

EH{(πm + ηc − 1)ELτ1

− [ηc − (1 − EL)(πm + ηc − 1)×(ηtπ

−m − ηt + 1)]}ηc − (1 − EH)(1 − EL)(πm + ηc − 1)(ηtπ−m − ηt + 1)

, (12)



where τ1 = TH/TL is the temperature ratio of the heat reservoirs. When ηc = ηt = 1is satisfied, the dimensionless heating load Q̄H of the endoreversible cycle is

Q̄H =EHEL(πm/τ1 − 1)(EH + EL − EHEL)

. (13)

3.2 The exergetic efficiency

The rate of exergy input Ξin is the negative value of net work transfer rate Wcv

that crosses the system boundary. That is,

Ξin = −Wcv = QH − QL. (14)

The purpose of employing a heat pump system is to release heat for heatingspace. The rate of exergy output utilized is the negative value of exergy transferrate accompanying heat

∑j(T0/Tj − 1)Qj . It gives

Ξout = −∑

j

(1 − T0/Tj)Qj =∫

H

(1 − T0/T )dQ −∫

L

(1 − T0/T )dQ.

(15a)

It is also noted that the values of exergy output rates are different for equivalentheat transfer rates at various boundary temperatures. For the purposes of calcu-lating the heat exergy, the heat reception and heat rejection are assumed to takeplace at TL and TH, respectively.

Ξout = (1 − T0/TH)QH − (1 − T0/TL)QL. (15b)

From the above, the following equation is also obtained:

Ξout = Ξin − Ξd. (16)

The exergetic efficiency is defined as the ratio of rate of exergy output to rate ofexergy input

ηex =Ξout

Ξin. (17)

So, the exergetic efficiency is obtained as

ηex =[(1 − T0/TH)QH − (1 − T0/TL)QL]

QH − QL. (18)

Combining eqs (2), (3), (6), (7) and (18) gives



ηex =

EL(a1 − 1){ηc − (ηtπ−m − ηt + 1)[(1 − EH)(πm + ηc − 1) + ηcEHτ1]}

−EH(a2 − 1){(πm + ηc − 1)EL + [(1 − EL)(πm + ηc − 1)×(ηtπ

−m − ηt + 1) − ηc]τ1}2EH{(πm + ηc − 1)EL + [(1 − EL)(πm + ηc − 1)

×(ηtπ−m − ηt + 1) − ηc]τ1}

−2EL{ηc − (ηtπ−m − ηt + 1)[(1 − EH)(πm + ηc − 1) + ηcEHτ1]}

, (19)

where τ2 = TH/T0 is the ratio of the hot-side heat reservoir temperature to theambient temperature, a1 = 2T0/TL−1 = 2τ1/τ2−1, and a2 = 2T0/TH−1 = 2/τ2−1.When ηc = ηt = 1 is satisfied, the exergetic efficiency of the endoreversible cycle is

ηex =[(1 − a2)πm − (1 − a1)]

[2(πm − 1)]. (20)

Equation (20) indicates that the exergetic efficiency of endoreversible cycle dependson the pressure ratio, the temperature ratio of the heat reservoirs and the ratio ofthe hot-side heat reservoir temperature to the ambient temperature.

4. Effects of design parameters on exergetic efficiency

Equation (16) indicates that when the temperature ratio of the heat reservoir (τ1)and the ratio of hot-side heat reservoir temperature to ambient temperature (τ2)are fixed, the exergetic efficiency (ηex) of the irreversible heat pump working onreversed Brayton cycle is dependent on the external heat transfer irreversibility(EH, EL), the internal irreversibility (ηc, ηt) and the pressure ratio (π).

Figure 2 shows the effect of heat reservoir temperature ratio (τ1) on the exergeticefficiency (ηex) vs. the pressure ratio (π) for k = 1.4, EH = EL = 0.9, ηc = ηt = 0.8and τ2 = 1. It can be seen that the curve of ηex vs. π is parabolic. That is, thereexists an optimum pressure ratio (πopt,ηex), which leads to a maximum exergeticefficiency (ηex max,π). Furthermore, ηex max,π increases at first and then decreaseswith the increase of the temperature ratio of the heat reservoirs (τ1), while theoptimum pressure ratio (πopt,ηex) increases with the increase of the temperatureratio of the heat reservoirs (τ1). The pressure ratio at which exergetic efficiency isequal to zero becomes larger when the temperature ratio of the heat reservoirs (τ1)increases.

For comparison with figure 2, figure 3 indicates the effect of heat reservoir tem-perature ratio (τ1) on the dimensionless heating load (Q̄H) vs. the pressure ratio (π).The figure illustrates that the dimensionless heating load (Q̄H) increases monoto-nously with increase in the pressure ratio (π), and it decreases with the increase ofthe temperature ratio of the heat reservoirs (τ1).

Figure 4 shows the exergetic efficiency (ηex) and the dimensionless heatingload (Q̄H) vs. the COP (β). In the calculations, k = 1.4, τ1 = 1.25, τ2 = 1,EH = EL = 0.9 and ηc = ηt = 0.8 are set. It can be seen that the exergeticefficiency (ηex) is an increasing function of the COP (β), and the curve of thedimensionless heating load (Q̄H) vs. the COP (β) is parabolic. Thus, when theperformance optimization of the heat pump working on reversed Brayton cycle iscarried out by selecting the pressure ratio, increasing the heating load inevitably



Figure 3. Effect of heat reservoir tem-

perature ratio on the dimensionless heat-

ing load vs. pressure ratio.

Figure 4. Exergetic efficiency and the di-

mensionless heating load vs. the COP.

Figure 5. Effect of effectivenesses of the

heat exchangers on the exergetic efficiency

vs. pressure ratio.

Figure 6. Effect of efficiencies of the com-

pressor and expander on the exergetic effi-

ciency vs. pressure ratio.

results in the decrease in the COP if Q̄H is larger than certain value, but optimizingthe exergetic efficiency can simultaneously increase the COP. In this instance, theexergetic efficiency optimization objective is more practical and effective than thetraditional heating load optimization objective.

Figures 5–7 show the effects of the effectivenesses of the hot- and cold-side heatexchangers (EH and EL), the efficiencies of the compressor and expander (ηc andηt), and the ratio of hot-side heat reservoir temperature to ambient temperature(τ2) on the exergetic efficiency (ηex) vs. the pressure ratio (π) characteristics, respec-tively. They indicate that ηex increases with increases of the parameters mentionedabove, respectively. The result shown in figure 6 with ηc = ηt = 1 is that of anendoreversible cycle.

Figures 8 and 9 show the effects of the effectivenesses of the hot- and cold-sideheat exchangers (EH and EL), and the efficiencies of the compressor and expander(ηc and ηt) on the optimum pressure ratio (πopt,ηex) vs. heat reservoir temperature



Figure 7. Effect of the ratio of hot-

side heat reservoir temperature to ambient

temperature on the exergetic efficiency vs.

pressure ratio.

Figure 8. Effect of effectivenesses of the

heat exchangers on the optimum pressure

ratio vs. heat reservoir temperature ratio.

Figure 9. Effect of efficiencies of the

compressor and expander on the optimum

pressure ratio vs. heat reservoir tempera-

ture ratio.

Figure 10. Comprehensive relationships

among exergetic efficiency, distribution of

heat conductance and pressure ratio.

ratio (τ1), respectively. Those figures indicate that the optimum pressure ratio(πopt,ηex) decreases with increase of the parameters mentioned above, respectively.

5. Optimal distribution of heat exchange inventory

For the fixed total heat exchanger inventory UT, that is, when UH + UL = UT,defining the distribution of heat conductance u = UL/UT leads to

UL = uUT, UH = (1 − u)UT. (21)

Figure 10 shows the corresponding three-dimensional diagram among ηex, u andπ. In the calculations, k = 1.4, Cwf = 0.8 kW/K, UT = 5 kW/K, ηc = ηt = 0.8,



Figure 11. Effect of thermal capacity rate

of the working fluid on the optimum dis-

tribution of heat conductance vs. pressure

ratio.


perature ratio on the optimum distribution

of heat conductance vs. pressure ratio.

Figure 13. Effect of total heat exchanger

inventory on the optimum distribution of

heat conductance vs. pressure ratio.


compressor and expander on the optimum

distribution of heat conductance vs. pres-

sure ratio.

τ1 = 1.25 and τ2 = 1 are set. It indicates that the curve of ηex vs. u is alsoparabolic. There exists an optimum allocation (uopt,ηex) of heat conductance cor-responding to maximum exergetic efficiency (ηex max,u) for a fixed pressure ratio.Therefore, there exist an optimum distribution (uopt,ηex) of heat conductance andan optimum pressure ratio (πopt,ηex), which lead to a double maximum exergeticefficiency (ηex max,max).

Figure 11 shows the influence of thermal capacity rate of the working fluid (Cwf)on the optimum distribution (uopt,ηex) of heat conductance vs. pressure ratio (π)for k = 1.4, UT = 5 kW/K, ηc = ηt = 0.8, τ1 = 1.25 and τ2 = 1. Figure 12shows the influence of the heat reservoir temperature ratio (τ1) on the optimumdistribution (uopt,ηex) of heat conductance vs. pressure ratio for k = 1.4, UT =5 kW/K, Cwf = 0.8 kW/K, ηc = ηt = 0.8 and τ2 = 1. Figure 13 shows the



Figure 15. Effect of thermal capacity rate

of the working fluid on the maximum ex-

ergetic efficiency vs. pressure ratio.


perature ratio on the maximum exergetic

efficiency vs. pressure ratio.

Figure 17. Effect of total heat exchanger

inventory on the maximum exergetic effi-

ciency vs. pressure ratio.


compressor and expander on the maximum

exergetic efficiency vs. pressure ratio.

influence of the total heat exchanger inventory (UT) on the optimum distribution(uopt,ηex) of heat conductance vs. pressure ratio for k = 1.4, τ1 = 1.25, Cwf = 0.8kW/K, τ2 = 1 and ηc = ηt = 0.8. Figure 14 shows the influences of efficiencies ofthe compressor and expander (ηc and ηt) on the optimum distribution (uopt,ηex) ofheat conductance vs. pressure ratio for k = 1.4, τ1 = 1.25, τ2 = 1, Cwf = 0.8 kW/Kand UT = 5 kW/K.

The numerical calculations show that uopt,ηex is an increasing function of π, anduopt,ηex increases very quickly when π is smaller, while it almost is unchangeablewhen π gets larger. It decreases with the increase of Cwf or τ1, while it increaseswith the increase of UT or ηc and ηt, and it increases less if UT gets larger. It isalways less than 0.5.

The influence of thermal capacity rate of the working fluid (Cwf) on the maximumexergetic efficiency (ηex max,u) vs. the pressure ratio (π) for k = 1.4, UT = 5 kW/K,ηc = ηt = 0.8, τ1 = 1.25 and τ2 = 1 are shown in figure 15. The influence of the



heat reservoir temperature ratio (τ1) on the maximum exergetic efficiency vs. thepressure ratio (π) for k = 1.4, UT = 5 kW/K, Cwf = 0.8 kW/K, ηc = ηt = 0.8 andτ2 = 1 are shown in figure 16. The influence of the total heat exchanger inventory(UT) on the maximum exergetic efficiency vs. the pressure ratio (π) for k = 1.4,τ1 = 1.25, Cwf = 0.8 kW/K, τ2 = 1 and ηc = ηt = 0.8 are shown in figure 17.The influences of efficiencies of the compressor and expander (ηc and ηt) on themaximum exergetic efficiency vs. the pressure ratio (π) for k = 1.4, τ1 = 1.25,τ2 = 1, Cwf = 0.8 kW/K and UT = 5 kW/K are shown in figure 18.

The numerical calculations show that ηex max,u decreases with the increase of Cwf ,while it increases with the increase of UT or ηc and ηt, and it increases less if UT

gets larger. ηex max,max increases at first and then decreases with the increase of τ1.

6. Conclusion

Optimization of exergetic efficiency for an irreversible heat pump working on re-versed Brayton cycle was performed in this paper. The expression of the exergeticefficiency was deduced based on the theoretical model of the heat pump. Then, theinfluences of the pressure ratio of the compressor, the allocation of heat exchangerinventory, the temperature ratio of the two reservoirs, the effectivenesses of thehot- and cold-side heat exchangers, the efficiencies of the compressor and expander,the ratio of hot-side heat reservoir temperature to ambient temperature, the to-tal heat exchanger inventory and the heat capacity rate of the working fluid on theexergetic efficiency of the heat pump were investigated by detailed numerical exam-ples. Moreover, performance comparisons between exergetic efficiency optimizationobjective and traditional heating load optimization objective were carried out.

In general, there exists an optimum pressure ratio (πopt,ηex) corresponding to anoptimum exergetic efficiency (ηex max,π) and when pressure ratio is chosen as a fixedvalue or just the optimum value there exists an optimum distribution (uopt,ηex) ofheat conductance corresponding to another optimum exergetic efficiency (ηex max,u).Therefore, there exists a double maximum value (ηex max,max) for the exergeticefficiency. The results show that the exergetic efficiency optimization objectiveis more practical and effective than the traditional heating load optimizationobjective.

Acknowledgements

This paper is supported by Scientific Research Common Program of Beijing Mu-nicipal Commission of Education (Project No. KM200710005034), China Post-doctoral Science Fundation (Project No. CPSF20060400837), Beijing Municipal-ity Key Lab of Heating, Gas Supply, Ventilating and Air Conditioning Engi-neering and Program for New Century Excellent Talents in University of P. R.China (Project No. NCET-04-1006). The authors wish to thank the reviewersfor their careful, unbiased and constructive suggestions, which led to this revisedmanuscript.



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Exergetic Efficiency Optimization for an Irreversible Heat Pump Working on Reversed Brayton Cycle

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