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A parameter optimization method for actual thermal system
International Journal of Heat and Mass Transfer 108 (2017) 1273–1278
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A parameter optimization method for actual thermal system Zeshao Chen a, Gang Wang b,⇑, Chuan Li a a b
University of Science and Technology of China, Hefei, Anhui 230027, China School of Energy and Power Engineering, Northeast Electric Power University, Jilin, Jilin 132012, China
a r t i c l e
i n f o
Article history: Received 30 July 2016 Received in revised form 8 September 2016 Accepted 23 December 2016
Keywords: Thermodynamics Equivalent temperature Finite-time thermodynamic analysis method Thermal system optimization Efficiency improvement
a b s t r a c t The objective of the parameter optimization of thermal system is to increase the system efficiency and decrease the cost of unit output power under the condition which the total output power can meet the actual requirements in energy applications. In this paper, a new parameter optimization method for actual thermal system was proposed, which was called the Equivalent Transformation Constant State Finite-time Thermodynamic Analysis Method and had two calculation steps. The first step was choosing the optimum efficiency under the principle of taking into account both output power and efficiency weights, and the second step was calculating the optimum total heat transfer coefficient ratio and other concerned parameters with the optimum efficiency fixed. The related calculation principles and formulas for parameter optimizations of this method were given. That provided the guiding theoretical basis for actual thermal system efficiency increase and cost decrease of unit output power. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Thermal system is a device which outputs work and discharges waste heat to the environment by the cyclic heat absorption of working medium from heat source. It mainly includes high and low temperature heat exchangers, expansion engine, liquid circulating pump. The main parameters of thermal system are heat source temperatures, heat transfer temperature difference and rates of heat exchangers, output power and system efficiency. The objective of the parameter optimization of thermal system is to increase the system efficiency and decrease the cost of unit output power under the condition which the total output power can meet the actual requirements in energy applications. The efficiency objective of thermal system given by classical thermodynamics is the Carnot engine efficiency, which is gc = 1 TL/TH. TL and TH are the temperatures of high and low temperature heat sources, respectively. The Carnot engine efficiency is obtained under three specific conditions, which are: the heat exchanger has no heat transfer temperature difference, the heat exchange area of heat exchanger or the heat exchange time between working medium and heat source is infinite, and the total output power is equal to 0. But for the actual thermal system, all of the three specific conditions above are invalid. In 1975, Curzon and Ahlborn gave the relationship between efficiency and output power of the heat engine with finite rate and cycle time based on the endo-reversible Carnot cycle model, and gave the ⇑ Corresponding author. E-mail address: [email protected] (G. Wang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.082 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
endo-reversible Carnot cycle heat engine efficiency under the maximum output power (gca) [1]. Then some other researchers made lots of efforts on the finite-time thermodynamics analysis method and related research work [2–9]. For instance, a method for the optimization of thermal systems, based on finite time thermodynamics, was proposed by Stitou and Feidt [10], and some finite time optimization analyses of refrigerators were carried out [11,12]. As the alternating finite-time analysis method simplifies many practical problems, some of the corresponding analysis results are unacceptable in engineering. For now, the relationship between efficiency and output power of actual thermal system is still obtained by experiments. Thus a parameter optimization theory is imperative, which must be convenient for actual thermal system applications. In this paper, a new parameter optimization method for actual thermal system was proposed, which was named the Equivalent Transformation Constant State Finite-time Thermodynamic Analysis Method. The related principles, operating steps and calculation formulas for this method were given, aiming at providing the guiding theoretical basis for system efficiency increase and cost decrease of unit output power from the viewpoints of thermodynamics and heat transfer. 2. Equivalent thermodynamic transformation of thermal system Equivalent thermodynamics transformation is the basis of equivalent transformation finite-time thermodynamics analysis method [13,14]. A superheated steam cycle thermal system was
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T R;H ¼
/H T H;in T H;out ¼ Q m;H DsH lnðT H;in =T H;out Þ
ð2Þ
T R;L ¼
/L T L;out T L;in ¼ Q m;L DsL lnðT L;out =T L;in Þ
ð3Þ
2.2. High and low temperatures of equivalent Carnot cycle The typical cycle of thermal system is presented in Fig. 1, which included high temperature heat absorption, gas expansion, low temperature heat release and liquid squeeze processes. 1-2-3-4 and 1-20 -3-40 were theoretical and actual cycles of superheated steam, respectively. The theoretical cycle was comprised of constant pressure high temperature heat absorption, constant entropy gas expansion, constant pressure low temperature heat release and constant entropy liquid squeeze processes. There was Ds14 = Ds13 = Ds23 in the theoretical cycle, so the Rankine cycle could be equivalently transformed to the Carnot cycle comprised of two constant temperature processes and two constant pressure processes just by transforming the temperatures of high and low temperature constant pressure processes of theoretical cycle to equivalent high and low temperatures. The transforms of the two equivalent temperatures were:
Fig. 1. T-s curve of superheated Rankine cycle.
T R;1 ¼
DqH Dq14 h1 h4 /H ¼ ¼ ¼ Ds14 s1 s3 s1 s3 Q m ðs1 s3 Þ
ð4Þ
T R;2 ¼
DqL Dq23 h2 h3 /L ¼ ¼ ¼ Ds23 s1 s3 s1 s3 Q m ðs1 s3 Þ
ð5Þ
Fig. 2. T-q relationship corresponding to Fig. 1.
chosen as the instance here. The temperature-entropy (T-s) curve and heat transfer processes (T-q) are presented in Figs. 1 and 2. The purpose of equivalent thermodynamics transformation was to give the relationships of heat quantity, entropy change and temperature of different processes in Figs. 1 and 2 according to the thermodynamics definition equation of entropy (dq = Tds). For a reversible process from the initial State a to final State b, the equivalent thermodynamics temperature TR, ab of a-b process was:
T R;ab ¼ Dqab =Dsab
ð1aÞ
where Dqab was the heat change of unit mass of working medium from State a to State b, including the heat quantities transferred with outside system and generated by endo-irreversible process. Dsab was the entropy change of unit mass of working medium from State a to State b, which was a function of state and had nothing to do with the process. For a constant state, assuming the heat flux rate transferred in a-b process was / and the theoretical cycle working medium mass flow rate was Qm, then the equivalent thermodynamics temperature of a-b process was:
T R;ab ¼
/ Q m Dsab
ð1bÞ
where the units of TR, ab, /, Qm and Dsab were K, W, kg/s and J/(K kg), respectively. 2.1. Equivalent temperature of heat source For Fig. 2, it was assumed that the mass flow rates of fluid in high and low temperature heat sources were Qm,H and Qm,L, the inlet and outlet temperatures of fluid in the two heat sources were TH,in, TH,out, TL,in and TL,out, and the heat transfer flux rates of high and low temperature heat exchangers were /H and /L . By the transformation of Dqab in Eqs. (1), the equivalent temperatures of the two heat sources of thermal system were as follows:
As the cycle of actual heat engine was irreversible, the expansion and squeeze processes were both not constant pressure processes. Normally, we defined constant entropy expansion coefficient (gT = w0 exp/wexp) and constant entropy squeeze coefficient (gc = w0 sq/wsq) to express the expansion and squeeze processes, where w0 was the actual work quantity. When the Rankine cycle was equivalently transformed to the Carnot cycle, 1-20 -3-40 could be divided to two recombination processes. 3-40 -1 process included high temperature heat absorption and liquid squeeze processes, and 1-20 -3 was comprised of low temperature heat exchange and gas expansion processes. The entropy values at the initial and final points of the two recombination processes had nothing to do with the processes and the absolute values of the two entropy differences were the same (Ds13 = Ds31). By the two complex processes, the equivalent high and low temperatures of the equivalent Carnot cycle could be given as:
T 0R;1 ¼
Dq0H h1 h4 h1 h3 ðh4 h3 Þ=gc /H ¼ ¼ ¼ 0 Ds13 s1 s3 s1 s3 Q m ðs1 s3 Þ
ð6Þ
T 0R;2 ¼
Dq0L h1 h3 ðh1 h2 ÞgT /L ¼ ¼ 0 Ds13 s1 s3 Q m ðs1 s3 Þ
ð7Þ
where Q0 m was the mass flow rate of actual cycle working medium (Q0 m/Qm = DqH/Dq0 H). The equivalent transformation of actual thermal system was completed by Eqs. (2)–(7). 3. Efficiency and output power of equivalent endo-reversible Carnot cycle heat engine 3.1. Constant state finite-time analysis method According to the thermodynamics transformation introduced in Section 2, the actual thermal system presented in Figs. 1 and 2 could be equivalent to the endo-reversible Carnot cycle model [15], which is presented in Fig. 3.
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With Eqs. (14) and (15), we could obtain:
P ¼ K 1 ½T H ð1 gÞT 2 jðT 2 T L Þ
ð16Þ
According to the third equation of Eq. (14), solving dP/dT2 = 0, there was:
ð1 þ jÞ2 T 22 2T L jð1 þ jÞT 2 þ ðT L jÞ2 T H T L ¼ 0
ð17Þ
Solving T2 at the maximum output power point by Eq. (17), the result was as following:
T 2;m ¼
pffiffiffiffiffiffiffiffiffiffiffi TLj þ T HTL 1þj
ð18Þ
Putting Eq. (18) back into Eqs. (14) and (15), we obtained the expressions of the maximum output power and corresponding efficiency, which were as follows:
Fig. 3. Constant state equivalent endo-reversible Carnot cycle model.
In Fig. 3, P was the net output power. In order to facilitate writing, we use TH and TL instead of TR,H and TR,L, and use T1 and T2 instead of TR,1 and TR,2 (or T00 R,1 and T00 R,2) in the following content.
Pmax ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T H K 2 ð1 T L =T H Þ 1þj
gm ¼ gca ¼ 1
ð19Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi T L =T H
ð20Þ
3.2. Energy flux rate equations The set of energy flux rate equations of constant state endoreversible cycle were:
P ¼ /H /L
ð8Þ
/H ¼ K 1 ðT H T 1 Þ
ð9Þ
/L ¼ K 2 ðT 2 T L Þ
ð10Þ
where K1 and K2 were the equivalent total heat transfer coefficients of heat absorption and release exchangers. The total heat transfer coefficient was the product of heat exchange coefficient (k) and heat exchange area. For equivalent Carnot heat engine, there was:
/L =/H ¼ T 2 =T 1
ð11Þ
and the cycle efficiency was:
g ¼ 1 T 2 =T 1
ð12Þ
4. Parameter optimization The objective of parameter optimization of heat engine should be to increase the efficiency and decrease the cost of unit output power. In this study, the guiding theory of parameter optimization was given from the viewpoints of thermodynamics and heat transfer theories. A two-step parameter optimization method was proposed here, of which the first step was the preliminary choosing of optimum efficiency and the second was increasing the output power by adjusting j under the principle of seeking cost minimization of unit output power. 4.1. Efficiency optimization The first step of the two-step parameter optimization method was to ascertain the optimum efficiency (gopt) of thermal system. The optimal dimension of gopt was as follows:
When TH, TL, K1 and K2 were known, there were still six unknown parameters, which were /H , /L , P, g, T1 and T2. If one of the unknown parameters was chosen, the function relationship between the chosen and other five unknown parameters could be given. As there were five equations above, if an extremal condition was provided, the fixed solution at the extremal point would be obtained. Using Eqs. (9)–(11), we could obtain:
gca < gopt < gc
/L K 2 ðT 2 T L Þ T 2 ¼ ¼ /H K 1 ðT H T 1 Þ T 1
where f was the weight factor of gc and was from 0.25 to 0.75. Yan proposed to consider the extremal value of product (w) of dimensionless output power (p) and efficiency (g) as the optimized objective parameter of endo-reversible cycle heat engine [16], which could be expressed as:
ð13aÞ
Assuming j = K2/K1 (j was the total heat transfer coefficient ratio), there was:
T1 ¼
THT2 jðT 2 T L Þ þ T 2
ð13bÞ
1
ð21aÞ
pffiffiffiffiffi
sL < gopt < 1 sL
ð21bÞ
where gc was the Carnot engine efficiency and there was sL = TL/TH. The weight equation of efficiency optimization was [15]:
gopt ¼ f gc þ ð1 f Þgca
w ¼ gp ¼ 1
ð22Þ
s2 jðs2 sL Þ ½1 ð1 þ jÞs2 þ sL j ½jðs2 sL Þ þ s2 ð23Þ
Then the output power could be expressed as:
P ¼ /H /L ¼ K 1 ðT H T 1 Þ K 2 ðT 2 T L Þ T2 jðT 2 T L Þ gs2 ¼ K1TH 1 ¼ K1TH g TH ½jðT 2 T L Þ þ T 2 1g
where there was s2 = T2/TH. The extremal value of w could be obtained by analytical method and it could be expressed as the following fitted equation which had the same form as Eq. (22):
ð14Þ
gopt ¼ 0:3gc þ 0:7gca
ð24Þ
ð15Þ
The difference between results calculated by Eq. (24) and graphic method was smaller than 1%. But in this study, our recommendatory principle for the efficiency optimization was equal
where the expression of g was:
g ¼ 1 ½jðT 2 T L Þ þ T 2 =T H
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weight principle, which meant that we gave the equal considerations to both Carnot efficiency and maximum output power. The recommendatory equation was:
gopt ¼ ðgc þ gca Þ=2
ð25Þ
4.2. Output power at optimum efficiency point Putting Eq. (25) into the relational expression of output power and efficiency of heat engine, we obtained the output power (Popt, j) corresponding to the optimum efficiency (gopt):
Popt;j ¼ gopt K 1 T H ½1
1 gopt þ sL j ð1 gopt Þð1 þ jÞ
ð26Þ
When j was equal to 1, there was:
" Popt;j¼1 ¼ gopt K 1 T H
1 gopt þ sL 1 2ð1 gopt Þ
# ð27Þ
Popt;j 2j ¼ Popt;j¼1 1 þ j
4.3.1. Principle of equipment cost minimization of unit output power The equipment cost minimization of unit output power was chosen as one principle to optimize the total heat transfer coefficient ratio (j). The equipment cost in was comprised of costs of high and low temperature heat exchangers, expansion engine, liquid circulation pump and other auxiliary equipment. The cost of high temperature heat exchanger (rH) was considered as the comparison reference when j was equal to 1 and efficiency was optimum. When j was equal to 1, the cost coefficients of high and low temperature heat exchangers, expansion engine, liquid circulation pump and other auxiliary equipment were assumed to be 1, a, b, c and d. And when j was not equal to 1, the cost of high temperature heat exchanger did not change, and the cost changes of liquid circulation pump and other auxiliary equipment related to j were ignored. If the dimensionless equipment cost of thermal system was expressed by rr,opt,j, there was:
rr;opt;j ¼
then the self-contrast output power (Pr,opt) corresponding to the optimum efficiency could be expressed as follows:
Pr;opt ¼
4.3. Total heat transfer coefficient ratio optimization
ð28Þ
The calculation results of the two kinds of optimum efficiency selection methods dealt by normalization are presented in Fig. 4, in which there were Pr = P/Pmax, gr = g/gc and wr = w/wmax. The relationship between normalized self-contrast output power Pr and self-contrast efficiency of heat engine is shown in Fig. 4. Point M was the maximum output power point, where the self-contrast efficiency was 0.6461. If the temperature of high temperature heat source was 1000 K and the environment temperature was 300 K, the efficiency of Carnot heat engine was 0.7 but the one of actual heat engine was just 0.4522. gr,M 1.0 was the dominant area for efficiency and Point N (gr,N = 0.7571, Pr, N = 0.9371) was the extremal point of wr curve. Point O was the efficiency optimized reference point recommended by the authors, where there was gr,O = Pr,O = 0.8230. The efficiency could be increased to 0.5760 after other parameter optimizations of actual heat engine by this point. So far, super critical cycle has a higher equivalent temperature and the efficiency can be from 0.437 to 0.440, for which there is still increasing potential.
Fig. 4. Solving graph of optimized point of endo-reversible Carnot cycle heat engine.
ropt;j ¼ 1 þ aj þ bP r;opt þ c þ d rH
ð29Þ
The equipment cost of unit output power (x) was defined as the ratio of self-contrast output power (Pr,opt) and dimensionless equipment cost (rr,opt,j), which could be expressed as:
x¼
rr;opt;j Pr;opt
¼bþ
ð1 þ aj þ c þ dÞð1 þ jÞ 2j
ð30Þ
According to Eq. (30), solving dx/dj = 0, we obtained the optimum total heat transfer coefficient ratio jopt was:
jopt ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þcþd a
ð31Þ
The relationship between equipment cost of unit output power and total heat transfer coefficient ratio is presented in Fig. 5. The conditions of two curves in Fig. 5 were: (1) a = 1, b = 1, c + d = 1; (2) a = 1, b = 0.8, c + d = 1. The optimum total heat transfer coefficient ratio values at minimum cost points of the two curves were 1.437 and 1.547, respectively. 4.3.2. Principle of output power increase After the optimum efficiency (gopt) was finalized, the output power could be increased by changing K1 or j according to Eq. (14). Theoretically, a bigger output power could be obtained by
Fig. 5. j for the principle of equipment cost minimization of unit output power.
Z. Chen et al. / International Journal of Heat and Mass Transfer 108 (2017) 1273–1278
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4.4. Optimization formulas of other parameters After the optimum efficiency (gopt), optimum output power (Popt) and optimum total heat transfer coefficient ratio (jopt) were obtained, other optimum parameters of the thermal system could be also given. The optimum equivalent heat release temperature was:
T 2;opt ¼ T H
1 gopt þ sL jopt 1 þ jopt
ð33Þ
The optimum equivalent heat absorption temperature was:
T 1;opt ¼ T 2;opt =ð1 gopt Þ
ð34Þ
The optimum heat absorption rate was:
/H;opt ¼ Popt =gopt Fig. 6. j optimization under principle of output power increase.
The optimum total heat transfer coefficient of high temperature heat exchanger was:
increasing j. The principle of output power increase with the optimum efficiency (gopt) fixed can be illustrated by Fig. 6. There are four kinds of functions in Fig. 6, which are dimensionless equivalent temperature (s), efficiency (g), dimensionless output power (p) and dimensionless heat transfer flux rate (/). In Fig. 6, dimensionless equivalent heat absorption and release temperatures (s1,opt and s2,opt) corresponding to the optimum efficiency both decreased slowly with j increasing. The optimum efficiency (gopt) and efficiency corresponding to the maximum output power (gca) were both horizontal lines. Three curves corresponding to the optimum efficiency, which were dimensionless output power (popt), dimensionless heat absorption and release flux rates (/1,opt and /2,opt), all increased with j increasing. As j increased, the dimensionless maximum output power (pm, j=1) also increased. According to Fig. 6, to keep the optimum efficiency fixed, Point B should be the objective for output power optimization. Compared with Point A, the dimensionless output power increase of Point B was about 21.5% and the total heat transfer coefficient ratio increase should be about 54.7% (j2 = 1.547). The total heat transfer coefficient was the product of heat exchange coefficient (k) and heat exchange area. When heat exchange coefficient was fixed, Point B could be achieved by increasing 54.7% of the heat exchange area of low temperature heat exchanger. That would obtain a much better economic efficiency. 4.3.3. Method of making self-contrast output power equal to one If there were no actual values of a, b, c and d, the suggested optimum total heat transfer coefficient ratio (jopt) could be obtained by making Popt,j equal to the maximum output power with j = 1 (Pmax, j=1) and the optimum efficiency (gopt) fixed. The derived expression of jopt was:
jopt ¼
g2ca ð1 gopt Þ 2gopt ð1 gopt Þ 2gopt sL g2ca ð1 gopt Þ
ð35Þ
K 1;opt ¼ /H;opt =ðT H T 1;opt Þ
ð36Þ
The optimum total heat transfer coefficient of low temperature heat exchanger was:
K 2;opt ¼ jopt K 1;opt
ð37Þ
The optimum heat release rate was:
/L;opt ¼ K 2;opt ðT 2;opt T L Þ
ð38Þ
5. Conclusions In this paper, aiming at the optimization of the relationship between efficiency and output power of actual thermal system, the Equivalent Transformation Constant State Finite-time Thermodynamic Analysis Method was proposed, which was a parameter optimization method for actual thermal system efficiency improvement. It had two steps, of which the first was choosing the optimum efficiency with the principle of taking into account both output power and efficiency weights, and the second was calculating the optimum total heat transfer coefficient ratio and other concerned parameters with the optimum efficiency fixed. The related principles and calculation formulas for parameter optimizations were given. That provided the guiding theoretical basis for actual thermal system efficiency increase and cost decrease of unit output power from the viewpoints of thermodynamics and heat transfer. Acknowledgements The authors appreciate the support of the Natural Science Foundation of China (Grant No. 51376167). References
ð32Þ
In Fig. 5, the optimum total heat transfer coefficient ratio at minimum unit output power cost was 1.547, which was equal to the one calculated by Eq. (32) and also in the optimization range (1 < jopt < 2.5) given by Fig. 6. So Eq. (32) could be used as the calculation formula of the optimum total heat transfer coefficient ratio. With the optimum efficiency fixed, the new optimized thermal system could obtain a bigger output power than the previous one, but also had a bigger cost. Thus the adjustment constraint condition of j should be that the benefit obtained by output power increase must be bigger than the cost increment.
[1] F.L. Curzon, B. Ahlborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43 (1975) 22–24. [2] L. Chen, F. Sun, Present state and perspectives of finite time thermodynamics of energy system, Adv. Mech. 4 (1992) 479–488. [3] J. Chen, Z. Yan, The characteristics of finite-time thermodynamics theory and some important marks in development, J. Xiamen Univ. (Nat. Sci.) 40 (2001) 232–241. [4] A. Mousapour, A. Hajipour, M.M. Rashidi, et al., Performance evaluation of an irreversible Miller cycle comparing FTT (finite-time thermodynamics) analysis and ANN (artificial neural network) prediction, Energy 94 (2016) 100–109. [5] J. Lin, Z. Xu, S. Chang, et al., Finite-time thermodynamic modeling and analysis of an irreversible Miller cycle working on a four-stroke engine, Int. Commun. Heat Mass Transf. 54 (2014) 54–59. [6] Y. Ge, L. Chen, F. Sun, Finite-time thermodynamic modelling and analysis of an irreversible Otto-cycle, Appl. Energy 85 (2008) 618–624.
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[7] P.K. Bhardwaj, S.C. Kaushik, S. Jain, General performance characteristics of an irreversible vapour absorption refrigeration system using finite time thermodynamic approach, Int. J. Therm. Sci. 44 (2005) 189–196. [8] P.A.N. Wouagfack, R. Tchinda, Finite-time thermodynamics optimization of absorption refrigeration systems: a review, Renew. Sustain. Energy Rev. 21 (2013) 524–536. [9] B.L. Lostec, J. Millette, N. Galanis, Finite time thermodynamics study and exergetic analysis of ammonia-water absorption systems, Int. J. Therm. Sci. 49 (2010) 1264–1276. [10] D. Stitou, M. Feidt, Nouveaux critères pour l’optimisation et la caractérisation des procédés thermiques de conversion énergétique, Int. J. Therm. Sci. 44 (2005) 1142–1153. [11] P.K. Bhardwaj, S.C. Kaushik, S. Jain, Finite time optimization of an endoreversible and irreversible vapour absorption refrigeration system, Energy Convers. Manage. 44 (2003) 1131–1144.
[12] T. Zheng, L. Chen, F. Sun, et al., Performance optimization of an irreversible four-heat-reservoir absorption refrigerator, Appl. Energy 76 (2003) 391–414. [13] Z. Chen, W. Xie, P. Hu, et al., An effective thermodynamic transformation analysis method for actual irreversible cycle, Sci. China, Ser. E 43 (2013) 1230– 1235. [14] Z. Chen, W. Xie, P. Hu, et al., Heat pump performance analysis software based on the equivalent thermodynamic transformation analysis method, Fluid Mach. 41 (2013) 71–75. [15] Z. Chen, Advanced Engineering Thermodynamics, second ed., Press of University of Science and Technology of China, Hefei, Anhui Province, China, 2014. [16] Z. Yan, The relation between optimal efficiency and power of a Carnot heat engine, J. Eng. Thermophys. 6 (1985) 1–5.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A parameter optimization method for actual thermal system Zeshao Chen a, Gang Wang b,⇑, Chuan Li a a b
University of Science and Technology of China, Hefei, Anhui 230027, China School of Energy and Power Engineering, Northeast Electric Power University, Jilin, Jilin 132012, China
a r t i c l e
i n f o
Article history: Received 30 July 2016 Received in revised form 8 September 2016 Accepted 23 December 2016
Keywords: Thermodynamics Equivalent temperature Finite-time thermodynamic analysis method Thermal system optimization Efficiency improvement
a b s t r a c t The objective of the parameter optimization of thermal system is to increase the system efficiency and decrease the cost of unit output power under the condition which the total output power can meet the actual requirements in energy applications. In this paper, a new parameter optimization method for actual thermal system was proposed, which was called the Equivalent Transformation Constant State Finite-time Thermodynamic Analysis Method and had two calculation steps. The first step was choosing the optimum efficiency under the principle of taking into account both output power and efficiency weights, and the second step was calculating the optimum total heat transfer coefficient ratio and other concerned parameters with the optimum efficiency fixed. The related calculation principles and formulas for parameter optimizations of this method were given. That provided the guiding theoretical basis for actual thermal system efficiency increase and cost decrease of unit output power. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Thermal system is a device which outputs work and discharges waste heat to the environment by the cyclic heat absorption of working medium from heat source. It mainly includes high and low temperature heat exchangers, expansion engine, liquid circulating pump. The main parameters of thermal system are heat source temperatures, heat transfer temperature difference and rates of heat exchangers, output power and system efficiency. The objective of the parameter optimization of thermal system is to increase the system efficiency and decrease the cost of unit output power under the condition which the total output power can meet the actual requirements in energy applications. The efficiency objective of thermal system given by classical thermodynamics is the Carnot engine efficiency, which is gc = 1 TL/TH. TL and TH are the temperatures of high and low temperature heat sources, respectively. The Carnot engine efficiency is obtained under three specific conditions, which are: the heat exchanger has no heat transfer temperature difference, the heat exchange area of heat exchanger or the heat exchange time between working medium and heat source is infinite, and the total output power is equal to 0. But for the actual thermal system, all of the three specific conditions above are invalid. In 1975, Curzon and Ahlborn gave the relationship between efficiency and output power of the heat engine with finite rate and cycle time based on the endo-reversible Carnot cycle model, and gave the ⇑ Corresponding author. E-mail address: [email protected] (G. Wang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.082 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
endo-reversible Carnot cycle heat engine efficiency under the maximum output power (gca) [1]. Then some other researchers made lots of efforts on the finite-time thermodynamics analysis method and related research work [2–9]. For instance, a method for the optimization of thermal systems, based on finite time thermodynamics, was proposed by Stitou and Feidt [10], and some finite time optimization analyses of refrigerators were carried out [11,12]. As the alternating finite-time analysis method simplifies many practical problems, some of the corresponding analysis results are unacceptable in engineering. For now, the relationship between efficiency and output power of actual thermal system is still obtained by experiments. Thus a parameter optimization theory is imperative, which must be convenient for actual thermal system applications. In this paper, a new parameter optimization method for actual thermal system was proposed, which was named the Equivalent Transformation Constant State Finite-time Thermodynamic Analysis Method. The related principles, operating steps and calculation formulas for this method were given, aiming at providing the guiding theoretical basis for system efficiency increase and cost decrease of unit output power from the viewpoints of thermodynamics and heat transfer. 2. Equivalent thermodynamic transformation of thermal system Equivalent thermodynamics transformation is the basis of equivalent transformation finite-time thermodynamics analysis method [13,14]. A superheated steam cycle thermal system was
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T R;H ¼
/H T H;in T H;out ¼ Q m;H DsH lnðT H;in =T H;out Þ
ð2Þ
T R;L ¼
/L T L;out T L;in ¼ Q m;L DsL lnðT L;out =T L;in Þ
ð3Þ
2.2. High and low temperatures of equivalent Carnot cycle The typical cycle of thermal system is presented in Fig. 1, which included high temperature heat absorption, gas expansion, low temperature heat release and liquid squeeze processes. 1-2-3-4 and 1-20 -3-40 were theoretical and actual cycles of superheated steam, respectively. The theoretical cycle was comprised of constant pressure high temperature heat absorption, constant entropy gas expansion, constant pressure low temperature heat release and constant entropy liquid squeeze processes. There was Ds14 = Ds13 = Ds23 in the theoretical cycle, so the Rankine cycle could be equivalently transformed to the Carnot cycle comprised of two constant temperature processes and two constant pressure processes just by transforming the temperatures of high and low temperature constant pressure processes of theoretical cycle to equivalent high and low temperatures. The transforms of the two equivalent temperatures were:
Fig. 1. T-s curve of superheated Rankine cycle.
T R;1 ¼
DqH Dq14 h1 h4 /H ¼ ¼ ¼ Ds14 s1 s3 s1 s3 Q m ðs1 s3 Þ
ð4Þ
T R;2 ¼
DqL Dq23 h2 h3 /L ¼ ¼ ¼ Ds23 s1 s3 s1 s3 Q m ðs1 s3 Þ
ð5Þ
Fig. 2. T-q relationship corresponding to Fig. 1.
chosen as the instance here. The temperature-entropy (T-s) curve and heat transfer processes (T-q) are presented in Figs. 1 and 2. The purpose of equivalent thermodynamics transformation was to give the relationships of heat quantity, entropy change and temperature of different processes in Figs. 1 and 2 according to the thermodynamics definition equation of entropy (dq = Tds). For a reversible process from the initial State a to final State b, the equivalent thermodynamics temperature TR, ab of a-b process was:
T R;ab ¼ Dqab =Dsab
ð1aÞ
where Dqab was the heat change of unit mass of working medium from State a to State b, including the heat quantities transferred with outside system and generated by endo-irreversible process. Dsab was the entropy change of unit mass of working medium from State a to State b, which was a function of state and had nothing to do with the process. For a constant state, assuming the heat flux rate transferred in a-b process was / and the theoretical cycle working medium mass flow rate was Qm, then the equivalent thermodynamics temperature of a-b process was:
T R;ab ¼
/ Q m Dsab
ð1bÞ
where the units of TR, ab, /, Qm and Dsab were K, W, kg/s and J/(K kg), respectively. 2.1. Equivalent temperature of heat source For Fig. 2, it was assumed that the mass flow rates of fluid in high and low temperature heat sources were Qm,H and Qm,L, the inlet and outlet temperatures of fluid in the two heat sources were TH,in, TH,out, TL,in and TL,out, and the heat transfer flux rates of high and low temperature heat exchangers were /H and /L . By the transformation of Dqab in Eqs. (1), the equivalent temperatures of the two heat sources of thermal system were as follows:
As the cycle of actual heat engine was irreversible, the expansion and squeeze processes were both not constant pressure processes. Normally, we defined constant entropy expansion coefficient (gT = w0 exp/wexp) and constant entropy squeeze coefficient (gc = w0 sq/wsq) to express the expansion and squeeze processes, where w0 was the actual work quantity. When the Rankine cycle was equivalently transformed to the Carnot cycle, 1-20 -3-40 could be divided to two recombination processes. 3-40 -1 process included high temperature heat absorption and liquid squeeze processes, and 1-20 -3 was comprised of low temperature heat exchange and gas expansion processes. The entropy values at the initial and final points of the two recombination processes had nothing to do with the processes and the absolute values of the two entropy differences were the same (Ds13 = Ds31). By the two complex processes, the equivalent high and low temperatures of the equivalent Carnot cycle could be given as:
T 0R;1 ¼
Dq0H h1 h4 h1 h3 ðh4 h3 Þ=gc /H ¼ ¼ ¼ 0 Ds13 s1 s3 s1 s3 Q m ðs1 s3 Þ
ð6Þ
T 0R;2 ¼
Dq0L h1 h3 ðh1 h2 ÞgT /L ¼ ¼ 0 Ds13 s1 s3 Q m ðs1 s3 Þ
ð7Þ
where Q0 m was the mass flow rate of actual cycle working medium (Q0 m/Qm = DqH/Dq0 H). The equivalent transformation of actual thermal system was completed by Eqs. (2)–(7). 3. Efficiency and output power of equivalent endo-reversible Carnot cycle heat engine 3.1. Constant state finite-time analysis method According to the thermodynamics transformation introduced in Section 2, the actual thermal system presented in Figs. 1 and 2 could be equivalent to the endo-reversible Carnot cycle model [15], which is presented in Fig. 3.
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With Eqs. (14) and (15), we could obtain:
P ¼ K 1 ½T H ð1 gÞT 2 jðT 2 T L Þ
ð16Þ
According to the third equation of Eq. (14), solving dP/dT2 = 0, there was:
ð1 þ jÞ2 T 22 2T L jð1 þ jÞT 2 þ ðT L jÞ2 T H T L ¼ 0
ð17Þ
Solving T2 at the maximum output power point by Eq. (17), the result was as following:
T 2;m ¼
pffiffiffiffiffiffiffiffiffiffiffi TLj þ T HTL 1þj
ð18Þ
Putting Eq. (18) back into Eqs. (14) and (15), we obtained the expressions of the maximum output power and corresponding efficiency, which were as follows:
Fig. 3. Constant state equivalent endo-reversible Carnot cycle model.
In Fig. 3, P was the net output power. In order to facilitate writing, we use TH and TL instead of TR,H and TR,L, and use T1 and T2 instead of TR,1 and TR,2 (or T00 R,1 and T00 R,2) in the following content.
Pmax ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 T H K 2 ð1 T L =T H Þ 1þj
gm ¼ gca ¼ 1
ð19Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi T L =T H
ð20Þ
3.2. Energy flux rate equations The set of energy flux rate equations of constant state endoreversible cycle were:
P ¼ /H /L
ð8Þ
/H ¼ K 1 ðT H T 1 Þ
ð9Þ
/L ¼ K 2 ðT 2 T L Þ
ð10Þ
where K1 and K2 were the equivalent total heat transfer coefficients of heat absorption and release exchangers. The total heat transfer coefficient was the product of heat exchange coefficient (k) and heat exchange area. For equivalent Carnot heat engine, there was:
/L =/H ¼ T 2 =T 1
ð11Þ
and the cycle efficiency was:
g ¼ 1 T 2 =T 1
ð12Þ
4. Parameter optimization The objective of parameter optimization of heat engine should be to increase the efficiency and decrease the cost of unit output power. In this study, the guiding theory of parameter optimization was given from the viewpoints of thermodynamics and heat transfer theories. A two-step parameter optimization method was proposed here, of which the first step was the preliminary choosing of optimum efficiency and the second was increasing the output power by adjusting j under the principle of seeking cost minimization of unit output power. 4.1. Efficiency optimization The first step of the two-step parameter optimization method was to ascertain the optimum efficiency (gopt) of thermal system. The optimal dimension of gopt was as follows:
When TH, TL, K1 and K2 were known, there were still six unknown parameters, which were /H , /L , P, g, T1 and T2. If one of the unknown parameters was chosen, the function relationship between the chosen and other five unknown parameters could be given. As there were five equations above, if an extremal condition was provided, the fixed solution at the extremal point would be obtained. Using Eqs. (9)–(11), we could obtain:
gca < gopt < gc
/L K 2 ðT 2 T L Þ T 2 ¼ ¼ /H K 1 ðT H T 1 Þ T 1
where f was the weight factor of gc and was from 0.25 to 0.75. Yan proposed to consider the extremal value of product (w) of dimensionless output power (p) and efficiency (g) as the optimized objective parameter of endo-reversible cycle heat engine [16], which could be expressed as:
ð13aÞ
Assuming j = K2/K1 (j was the total heat transfer coefficient ratio), there was:
T1 ¼
THT2 jðT 2 T L Þ þ T 2
ð13bÞ
1
ð21aÞ
pffiffiffiffiffi
sL < gopt < 1 sL
ð21bÞ
where gc was the Carnot engine efficiency and there was sL = TL/TH. The weight equation of efficiency optimization was [15]:
gopt ¼ f gc þ ð1 f Þgca
w ¼ gp ¼ 1
ð22Þ
s2 jðs2 sL Þ ½1 ð1 þ jÞs2 þ sL j ½jðs2 sL Þ þ s2 ð23Þ
Then the output power could be expressed as:
P ¼ /H /L ¼ K 1 ðT H T 1 Þ K 2 ðT 2 T L Þ T2 jðT 2 T L Þ gs2 ¼ K1TH 1 ¼ K1TH g TH ½jðT 2 T L Þ þ T 2 1g
where there was s2 = T2/TH. The extremal value of w could be obtained by analytical method and it could be expressed as the following fitted equation which had the same form as Eq. (22):
ð14Þ
gopt ¼ 0:3gc þ 0:7gca
ð24Þ
ð15Þ
The difference between results calculated by Eq. (24) and graphic method was smaller than 1%. But in this study, our recommendatory principle for the efficiency optimization was equal
where the expression of g was:
g ¼ 1 ½jðT 2 T L Þ þ T 2 =T H
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weight principle, which meant that we gave the equal considerations to both Carnot efficiency and maximum output power. The recommendatory equation was:
gopt ¼ ðgc þ gca Þ=2
ð25Þ
4.2. Output power at optimum efficiency point Putting Eq. (25) into the relational expression of output power and efficiency of heat engine, we obtained the output power (Popt, j) corresponding to the optimum efficiency (gopt):
Popt;j ¼ gopt K 1 T H ½1
1 gopt þ sL j ð1 gopt Þð1 þ jÞ
ð26Þ
When j was equal to 1, there was:
" Popt;j¼1 ¼ gopt K 1 T H
1 gopt þ sL 1 2ð1 gopt Þ
# ð27Þ
Popt;j 2j ¼ Popt;j¼1 1 þ j
4.3.1. Principle of equipment cost minimization of unit output power The equipment cost minimization of unit output power was chosen as one principle to optimize the total heat transfer coefficient ratio (j). The equipment cost in was comprised of costs of high and low temperature heat exchangers, expansion engine, liquid circulation pump and other auxiliary equipment. The cost of high temperature heat exchanger (rH) was considered as the comparison reference when j was equal to 1 and efficiency was optimum. When j was equal to 1, the cost coefficients of high and low temperature heat exchangers, expansion engine, liquid circulation pump and other auxiliary equipment were assumed to be 1, a, b, c and d. And when j was not equal to 1, the cost of high temperature heat exchanger did not change, and the cost changes of liquid circulation pump and other auxiliary equipment related to j were ignored. If the dimensionless equipment cost of thermal system was expressed by rr,opt,j, there was:
rr;opt;j ¼
then the self-contrast output power (Pr,opt) corresponding to the optimum efficiency could be expressed as follows:
Pr;opt ¼
4.3. Total heat transfer coefficient ratio optimization
ð28Þ
The calculation results of the two kinds of optimum efficiency selection methods dealt by normalization are presented in Fig. 4, in which there were Pr = P/Pmax, gr = g/gc and wr = w/wmax. The relationship between normalized self-contrast output power Pr and self-contrast efficiency of heat engine is shown in Fig. 4. Point M was the maximum output power point, where the self-contrast efficiency was 0.6461. If the temperature of high temperature heat source was 1000 K and the environment temperature was 300 K, the efficiency of Carnot heat engine was 0.7 but the one of actual heat engine was just 0.4522. gr,M 1.0 was the dominant area for efficiency and Point N (gr,N = 0.7571, Pr, N = 0.9371) was the extremal point of wr curve. Point O was the efficiency optimized reference point recommended by the authors, where there was gr,O = Pr,O = 0.8230. The efficiency could be increased to 0.5760 after other parameter optimizations of actual heat engine by this point. So far, super critical cycle has a higher equivalent temperature and the efficiency can be from 0.437 to 0.440, for which there is still increasing potential.
Fig. 4. Solving graph of optimized point of endo-reversible Carnot cycle heat engine.
ropt;j ¼ 1 þ aj þ bP r;opt þ c þ d rH
ð29Þ
The equipment cost of unit output power (x) was defined as the ratio of self-contrast output power (Pr,opt) and dimensionless equipment cost (rr,opt,j), which could be expressed as:
x¼
rr;opt;j Pr;opt
¼bþ
ð1 þ aj þ c þ dÞð1 þ jÞ 2j
ð30Þ
According to Eq. (30), solving dx/dj = 0, we obtained the optimum total heat transfer coefficient ratio jopt was:
jopt ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þcþd a
ð31Þ
The relationship between equipment cost of unit output power and total heat transfer coefficient ratio is presented in Fig. 5. The conditions of two curves in Fig. 5 were: (1) a = 1, b = 1, c + d = 1; (2) a = 1, b = 0.8, c + d = 1. The optimum total heat transfer coefficient ratio values at minimum cost points of the two curves were 1.437 and 1.547, respectively. 4.3.2. Principle of output power increase After the optimum efficiency (gopt) was finalized, the output power could be increased by changing K1 or j according to Eq. (14). Theoretically, a bigger output power could be obtained by
Fig. 5. j for the principle of equipment cost minimization of unit output power.
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4.4. Optimization formulas of other parameters After the optimum efficiency (gopt), optimum output power (Popt) and optimum total heat transfer coefficient ratio (jopt) were obtained, other optimum parameters of the thermal system could be also given. The optimum equivalent heat release temperature was:
T 2;opt ¼ T H
1 gopt þ sL jopt 1 þ jopt
ð33Þ
The optimum equivalent heat absorption temperature was:
T 1;opt ¼ T 2;opt =ð1 gopt Þ
ð34Þ
The optimum heat absorption rate was:
/H;opt ¼ Popt =gopt Fig. 6. j optimization under principle of output power increase.
The optimum total heat transfer coefficient of high temperature heat exchanger was:
increasing j. The principle of output power increase with the optimum efficiency (gopt) fixed can be illustrated by Fig. 6. There are four kinds of functions in Fig. 6, which are dimensionless equivalent temperature (s), efficiency (g), dimensionless output power (p) and dimensionless heat transfer flux rate (/). In Fig. 6, dimensionless equivalent heat absorption and release temperatures (s1,opt and s2,opt) corresponding to the optimum efficiency both decreased slowly with j increasing. The optimum efficiency (gopt) and efficiency corresponding to the maximum output power (gca) were both horizontal lines. Three curves corresponding to the optimum efficiency, which were dimensionless output power (popt), dimensionless heat absorption and release flux rates (/1,opt and /2,opt), all increased with j increasing. As j increased, the dimensionless maximum output power (pm, j=1) also increased. According to Fig. 6, to keep the optimum efficiency fixed, Point B should be the objective for output power optimization. Compared with Point A, the dimensionless output power increase of Point B was about 21.5% and the total heat transfer coefficient ratio increase should be about 54.7% (j2 = 1.547). The total heat transfer coefficient was the product of heat exchange coefficient (k) and heat exchange area. When heat exchange coefficient was fixed, Point B could be achieved by increasing 54.7% of the heat exchange area of low temperature heat exchanger. That would obtain a much better economic efficiency. 4.3.3. Method of making self-contrast output power equal to one If there were no actual values of a, b, c and d, the suggested optimum total heat transfer coefficient ratio (jopt) could be obtained by making Popt,j equal to the maximum output power with j = 1 (Pmax, j=1) and the optimum efficiency (gopt) fixed. The derived expression of jopt was:
jopt ¼
g2ca ð1 gopt Þ 2gopt ð1 gopt Þ 2gopt sL g2ca ð1 gopt Þ
ð35Þ
K 1;opt ¼ /H;opt =ðT H T 1;opt Þ
ð36Þ
The optimum total heat transfer coefficient of low temperature heat exchanger was:
K 2;opt ¼ jopt K 1;opt
ð37Þ
The optimum heat release rate was:
/L;opt ¼ K 2;opt ðT 2;opt T L Þ
ð38Þ
5. Conclusions In this paper, aiming at the optimization of the relationship between efficiency and output power of actual thermal system, the Equivalent Transformation Constant State Finite-time Thermodynamic Analysis Method was proposed, which was a parameter optimization method for actual thermal system efficiency improvement. It had two steps, of which the first was choosing the optimum efficiency with the principle of taking into account both output power and efficiency weights, and the second was calculating the optimum total heat transfer coefficient ratio and other concerned parameters with the optimum efficiency fixed. The related principles and calculation formulas for parameter optimizations were given. That provided the guiding theoretical basis for actual thermal system efficiency increase and cost decrease of unit output power from the viewpoints of thermodynamics and heat transfer. Acknowledgements The authors appreciate the support of the Natural Science Foundation of China (Grant No. 51376167). References
ð32Þ
In Fig. 5, the optimum total heat transfer coefficient ratio at minimum unit output power cost was 1.547, which was equal to the one calculated by Eq. (32) and also in the optimization range (1 < jopt < 2.5) given by Fig. 6. So Eq. (32) could be used as the calculation formula of the optimum total heat transfer coefficient ratio. With the optimum efficiency fixed, the new optimized thermal system could obtain a bigger output power than the previous one, but also had a bigger cost. Thus the adjustment constraint condition of j should be that the benefit obtained by output power increase must be bigger than the cost increment.
[1] F.L. Curzon, B. Ahlborn, Efficiency of a Carnot engine at maximum power output, Am. J. Phys. 43 (1975) 22–24. [2] L. Chen, F. Sun, Present state and perspectives of finite time thermodynamics of energy system, Adv. Mech. 4 (1992) 479–488. [3] J. Chen, Z. Yan, The characteristics of finite-time thermodynamics theory and some important marks in development, J. Xiamen Univ. (Nat. Sci.) 40 (2001) 232–241. [4] A. Mousapour, A. Hajipour, M.M. Rashidi, et al., Performance evaluation of an irreversible Miller cycle comparing FTT (finite-time thermodynamics) analysis and ANN (artificial neural network) prediction, Energy 94 (2016) 100–109. [5] J. Lin, Z. Xu, S. Chang, et al., Finite-time thermodynamic modeling and analysis of an irreversible Miller cycle working on a four-stroke engine, Int. Commun. Heat Mass Transf. 54 (2014) 54–59. [6] Y. Ge, L. Chen, F. Sun, Finite-time thermodynamic modelling and analysis of an irreversible Otto-cycle, Appl. Energy 85 (2008) 618–624.
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Z. Chen et al. / International Journal of Heat and Mass Transfer 108 (2017) 1273–1278
[7] P.K. Bhardwaj, S.C. Kaushik, S. Jain, General performance characteristics of an irreversible vapour absorption refrigeration system using finite time thermodynamic approach, Int. J. Therm. Sci. 44 (2005) 189–196. [8] P.A.N. Wouagfack, R. Tchinda, Finite-time thermodynamics optimization of absorption refrigeration systems: a review, Renew. Sustain. Energy Rev. 21 (2013) 524–536. [9] B.L. Lostec, J. Millette, N. Galanis, Finite time thermodynamics study and exergetic analysis of ammonia-water absorption systems, Int. J. Therm. Sci. 49 (2010) 1264–1276. [10] D. Stitou, M. Feidt, Nouveaux critères pour l’optimisation et la caractérisation des procédés thermiques de conversion énergétique, Int. J. Therm. Sci. 44 (2005) 1142–1153. [11] P.K. Bhardwaj, S.C. Kaushik, S. Jain, Finite time optimization of an endoreversible and irreversible vapour absorption refrigeration system, Energy Convers. Manage. 44 (2003) 1131–1144.
[12] T. Zheng, L. Chen, F. Sun, et al., Performance optimization of an irreversible four-heat-reservoir absorption refrigerator, Appl. Energy 76 (2003) 391–414. [13] Z. Chen, W. Xie, P. Hu, et al., An effective thermodynamic transformation analysis method for actual irreversible cycle, Sci. China, Ser. E 43 (2013) 1230– 1235. [14] Z. Chen, W. Xie, P. Hu, et al., Heat pump performance analysis software based on the equivalent thermodynamic transformation analysis method, Fluid Mach. 41 (2013) 71–75. [15] Z. Chen, Advanced Engineering Thermodynamics, second ed., Press of University of Science and Technology of China, Hefei, Anhui Province, China, 2014. [16] Z. Yan, The relation between optimal efficiency and power of a Carnot heat engine, J. Eng. Thermophys. 6 (1985) 1–5.