© The Author(s) 2013. This article is published with open access at Springerlink.com csb.scichina.com www.springer.com/scp
*Corresponding author (email: liangxg@tsinghua.edu.cn)
Engineering Thermophysics December 2013 Vol.58 No.36: 46964702 doi: 10.1007/s11434-013-6096-4
Entransy and entropy analyses of heat pump systems
CHENG XueTao & LIANG XinGang
*Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Received March 27, 2013; accepted May 6, 2013
In this paper, heat pump systems are analyzed with entransy increase and entropy generation. The extremum entransy increase principle is developed. When the equivalent temperatures of the high and low temperature heat sources are fixed, the theoretical analyses and numerical results both show that the maximum COP leads to the maximum entransy increase rate for fixed input power, while it leads to the minimum entransy increase rate for fixed heat flow absorbed from the low temperature heat source. The minimum entropy generation principle shows that the minimum entropy generation rate always leads to the maximum COP for fixed input power or fixed heat flow absorbed from the low temperature heat source when the equivalent thermodynamic forc-es of the high and low temperature heat sourcforc-es are given. Further discussions show that only the entransy increase rate always increases with increasing heat flow rate into the high temperature heat source for the discussed cases.
entransy loss, entransy increase,entropy generation, heat pump, optimization
Citation: Cheng X T, Liang X G. Entransy and entropy analyses of heat pump systems. Chin Sci Bull, 2013, 58: 46964702, doi: 10.1007/s11434-013-6096-4
Heat pump systems are common industrial equipments and have many applications [1–15]. For instance, in the heating and air conditioning system, the heat pump is applied to driving the heat from the environment into the room [15]. The optimization design of heat pump systems has received more and more attention because it can improve the system performance and increase the energy utilization efficiency [6–12].
There are different optimization objectives for heat pump systems, such as the thermo-economic performance [6] and the thermodynamic performance optimization [4,5,7–12]. For instance, Quoilin et al. [6] analyzed the thermo-eco- nomic performance of heat pump systems. Chen et al. [4] optimized the piston speed ratios to get the maximum COP for the irreversible Carnot refrigerator and heat pump using the finite time thermodynamics. In this paper, we focus on the thermodynamic performance optimization. As the ther-modynamic processes in heat pump systems are mainly composed of heat transfer processes and thermodynamic cycles, the analyses of the heat transfer processes and
ther-modynamic cycles are very important for the optimization designs. In the past decades, some optimization theories have already been developed and applied to heat transfer and thermodynamic cycles [15–18].
Practical heat transfer processes are irreversible from the thermodynamic viewpoint, and entropy generation will be produced. Many researchers applied the entropy generation minimization method to analyzing and optimizing heat transfer processes [18–20]. However, the entropy generation paradox tells us that the effectiveness of heat exchangers ε
does not always decrease when the entropy generation number increases [18]. When Bejan [18] analyzed a bal-anced counter flow heat exchanger, he explained the entro-py generation paradox as following: when ε→0, the heat exchanger would disappear as an engineering component, and such case does not exist. The entropy generation para-dox could not be removed with this explanation because the effectiveness can be a value in the range of [0, 0.5] in which the entropy generation number and the effectiveness still increase at the same time. In addition, Shah and Skiepko [21] also noticed that ε may be the maximum, an intermediate value or the minimum when the entropy generation reaches
the maximum value. Cheng et al. [22,23] analyzed the en-tropy generation of the heat exchangers and heat exchanger networks with two streams and found that the entropy gen-eration does not decrease monotonically with the increase of the heat transfer rate and effectiveness.
Guo et al. [16] developed the concept of entransy, which describes the heat transfer ability. Entransy dissipation al-ways exists during practical heat transfer processes [16,24]. Guo et al. [16] derived the extremum entransy dissipation principle and the minimum thermal resistance principle, which have been applied to the optimizations of conductive heat transfer [16,25–32], convective heat transfer [33], radi-ative heat transfer [34,35], heat exchangers and heat ex-changer networks [17,22,23,36,37]. In the analyses of heat exchangers with the entransy theory, there is no paradox like the entropy generation paradox [17,22,36].
For thermodynamic cycles, more entropy generation means that more ability to do work is lost [38,39]. Hence, the entropy generation minimization method has been widely applied to the analyses and optimizations of ther-modynamic processes because it can decrease the loss of the ability to do work [38–42]. For instance, Myat et al. [42] showed that the entropy generation minimization leads to the largest COP when they analyzed an absorption chiller. However, there are also some different viewpoints for the applicability of the entropy generation minimization to the optimization of thermodynamic cycles [43,44]. For instance, Klein and Reindl [43] analyzed the refrigeration system and found that the entropy generation minimization does not always lead to the best system performance unless the re-frigeration capacity is given.
The entransy theory is also used to analyze thermody-namic cycles [15,44–50]. Wu [45] defined the conversion entransy by which the thermodynamic processes with work were analyzed. In the recent investigations of thermody-namic cycles, Cheng et al. [15,44,47] defined a new concept, entransy loss rate, that is the difference between the entran-sy flow rate into the entran-system and that out of the entran-system. It is shown that the maximum entransy loss rate leads to the maximum output work for the discussed systems [15,44, 47–50].
The above introduction shows that the applicability of the entropy generation minimization to the analyses and optimizations of heat transfer processes and thermodynamic cycles is limited. For the concept of entransy, it has been applied to the analyses and optimizations of heat transfer and thermodynamic cycles, but there are not many reports.
For the heat pump system, Chen et al. [4] applied the concept of entropy to its optimization. However, Klein and Reindl [43] found that the minimum entropy generation rate does not always lead to the best system performance. So, the applicability of the entropy generation minimization to the heat pump systems needs further discussion. On the other hand, there are few reports on the applicability of the entransy theory to the analyses and optimization designs of
heat pump systems. Therefore, it is also necessary for us to discuss the applicability of the entransy theory to heat pump systems.
1 Extremum entransy increase principle and
minimum entropy generation principle for heat
pump system
As shown in Figure 1, the heat pump system is mainly composed of four parts, which are the cooler, the compres-sor, the cold storage (the heat source with low temperature) and the expander, respectively. The working fluid absorbs heat flow Qin from the cold storage during process 4–1. Then, it is compressed during process 1–2 by the compres-sor. In the next process, the working fluid releases heat flow
Qout during process 2–3 in the cooler (the heat source with high temperature). Finally, the working fluid gets beck to the initial state when the expansion in process 3–4 fin-ishes in the expander. The heat in the cold storage is pumped into the cooler when a cycle finishes, and the input power is Pin.
For the heat pump system, the energy conversation gives
out in in
Q Q P . (1)
The system COP is
out in in in
COPQ P 1 Q P . (2)
Eq. (2) shows that larger Qin leads to larger COP with fixed
Pin, while smaller Pin leads to larger COP with fixed Qin. For the heat pump system, assume that there are n low temperature heat sources and m high temperature heat sources. The temperature of the ith low temperature heat source is Tin-i, while that of the jth high temperature heat
source is Tout-j. The thermodynamic processes of the
work-ing fluid are shown in Figure 2, which can be divided into two parts. One is the heat transfer processes between the working fluid and the heat sources, while the other is the thermodynamic cycle.
During heat transfer, the entransy theory gives [16]
Figure 2 Thermodynamic process of the heat pump system.
in
dis1 1 in- in- in fd in
n i i
i A
G
Q T
q T A , (3)out
dis2 out fd out 1 out-
out-m
j j j
A
G
q T A
Q T , (4)where Gdis1 is the entransy dissipation rate during the heat transfer process between the low temperature heat sources and the working fluid, Gdis2 is that during the heat transfer process between the high temperature heat sources and the working fluid, Qin-i is the heat transfer rate between the ith
low temperature heat source and the working fluid, Qout-j is
that between the jth high temperature heat source and the working fluid, Tf is the temperature of the working fluid, qin and qout are the heat fluxes absorbed and released by the working fluid, and Ain and Aout are the corresponding heat transfer areas.
For the thermodynamic cycle, we have [15,44]
fδ fδ
Q W
G
T Q
T P G , (5)whereGQis the heat entransy flow rate, δQ is the heat flow absorbed by the working fluid, δP is the power output, andGWis the work entransy flow rate. There is
in out in fd in out fd out Q A A G
q T A
q T A . (6) So, we have in out in fd in out fd out W A A G
q T A
q T A . (7)Considering the definition of entransy loss rate and eqs. (3), (4) and (7), we obtain
loss dis1 dis2 1 in- in- 1 out-
out-n m
W i i i j j j
G G G G
Q T
Q T . (8)For Figure 1, we have
in 1 in-n i i Q
Q , (9) out 1 out-m j j Q
Q . (10)The equivalent temperature of the low temperature heat sources can be defined as
in 1 in- in- in
n i i i
T
Q T Q , (11)while that of the high temperature heat sources can be de-fined as
out 1 out- out- out
m
j j j
T
Q T Q . (12)Then, eq. (8) can be changed into loss in in out out
G Q T Q T . (13)
The entransy loss rate is negative for the heat pump sys-tems because Qin is smaller than Qout and Tin is lower than
Tout. Therefore, the system entransy does not decrease, but increases. The power input into the system, δP in eq. (5), is negative, and the work entransy flow is also negative. This means that the work entransy flow gets into the system, which makes the system entransy increase. Therefore, the entransy increase rate can be defined as
inc loss out out in in
G G Q T Q T . (14)
According to eq. (3), we have
inc in in out in in in out in in out
G Q P T Q T Q T T P T . (15)
When Tin and Tout are given, the maximum Ginc leads to the maximum Qin and the maximum COP (see eq.(2)) for fixed Pin, and the minimum Ginc leads to the minimum Pin and the maximum COP for fixed Qin. This is the extremum entransy increase principle of the heat pump system.
On the other hand, the entropy balance equation gives [51]
f g
dSdS δS , (16)
where dSf is the entropy flow, dS is the entropy change, and
δSg is the entropy generation. As the thermodynamic pro-cesses are steady, dS is zero. Therefore, the entropy genera-tion is
g f
δS dS . (17)
Therefore, the entropy generation rate of the system is
out- in-g f-out f-in 1 1 out- in-m j n i j i j i Q Q S S S T T
, (18)where Sf-out is the entropy flow rate that gets out of the system, and Sf-in is that gets into the system. We can
de-fine the equivalent thermodynamic forces of the low and the high temperature heat sources as
in- in-in 1 1 in- 1 in in- in-n i n n i i i i i i i Q Q H Q Q T T
, (19) out-out-out 1 1 out- 1 out
out- out-m j m m j j j j j j j Q Q H Q Q T T
. (20)Then, eq. (18) can be changed into
g out out in in
S H Q H Q . (21)
Considering eq. (1), we have
g out in in in in in out in in out
S H Q P H Q Q H H P H .(22)
As Hout is smaller than Hin, the term in the last bracket is negative. Therefore, when Hin and Hout are given, the minimum Sg leads to the maximum Qin with fixed Pin, while it leads to the minimum Pin with fixed Qin. Consider-ing eq. (2), we can see that the maximum COP leads to the minimum Sg with either fixed Pin or fixed Qin. This is the minimum entropy generation principle of the heat pump system.
As above, we get different optimization principles for heat pump systems. Eq. (15) shows that the maximum COP sometimes leads to the maximum entransy increase rate, and sometimes leads to the minimum entransy increase rate. On the other hand, the minimum entropy generation rate always leads to the maximum COP for fixed equivalent thermodynamic forces of heat sources. Therefore, when COP is the optimization objective of heat pump systems, the minimum entropy generation principle is convenient, though extremum entransy increase principle is also appli-cable.
2 Optimization examples and discussions
2.1 Numerical examples of the heat pump system with reversed Brayton cycle
Let us discuss a heat pump system composed of the re-versed Brayton cycle. For the working fluid, its thermody-namic processes are shown in Figure 3. The reversed Bray-ton cycle works between the low and high temperature heat sources with constant temperatures Tin and Tout, respectively. The temperatures of the working fluid at the state points are
T1, T2, T3 and T4, respectively. The working fluid absorbs heat flow Qin from the low temperature stream under con-stant pressure, then its temperature increases to T1. The next process is an isentropic process and the temperature of the working fluid increases to T2. Then, the working fluid re-leases heat flow Qout to the high temperature heat source under constant pressure, and its temperature decreases to T3.
Finally, the working fluid is expanded and gets back to the initial state. During the whole thermodynamic processes, the input mechanical power is Pin.
For the system in Figure 3, it is assumed that in out const
U U U , (23)
where Uin is the thermal conductance of the heat exchanger between the cold storage and the working fluid, and Uout is that of the heat exchanger between the cooler and the working fluid. Then, the distribution of U is to be optimized to increase the COP.
The heat transfer rates in the heat exchangers are [15]
in f in 4 1 exp in f
Q C T T U C , (24)
out f 2 out 1 exp out f
Q C T T U C , (25)
where Cf is the heat capacity flow rate of the working fluid. The energy conservation gives
in f 1 4 Q C T T , (26)
out f 2 3 Q C T T . (27)In the Brayton cycle, there is [52]
2 1 3 4
T T T T . (28)
When Tin, Tout, Qin and Cf are fixed, the values of Qout, T1,
T2, T3 and T4 can be calculated with eqs. (23)–(28) for every distribution of U. Then, the input power can be obtained from eq. (1), and the corresponding COP can be calculated from eq. (2). On the other hand, when Tin, Tout, Pin and Cf are fixed, Qin, Qout, T1, T2, T3 and T4 can also be calculated with eqs. (1), (23)–(28) for every distribution of U. Then, the corresponding COP can also be obtained with eq. (2). According to eqs. (14) and (21), the entransy increase rate and entropy generation rate could also be calculated.
Let us discuss some numerical examples below. Let
U=10 W/K, Tin=270 K, Tout=300 K, Cf=2 W/K, and Qin=100 W. The variations of the COP, the entransy increase rate and the entropy generation rate with Uin can be seen in Fig-ure 4. The minimum entransy increase rate and the mini-mum entropy generation rate both lead to the maximini-mum
Figure 4 Variations of the COP, the entransy increase rate and the en-tropy generation rate with Uin when Qin is fixed.
COP of the system when Tin, Tout, Cf and Qin are fixed. On the other hand, if the fixed parameter is not Qin, but the in-put power Pin, we assume that Pin=100 W, and the values of
U, Tin, Tout and Cf are the same as those of the first case. The variations of the COP, the entransy increase rate and the entropy generation rate with Uin are shown in Figure 5. It can be seen that the maximum entransy increase rate and the minimum entropy generation rate both lead to the maximum COP of the system. Therefore, both the extremum entransy increase principle and the minimum entropy generation principle can be applied to optimizing the heat pump system. The thermodynamic forces of the high and low temperature heat sources are given when their temperatures are fixed. According to eqs. (15) and (22), the preconditions of the principles are both satisfied. This is the reason why the principles are effective in optimizing the system.
2.2 Discussions
For the heat pump system with the reversed Brayton cycle discussed above, we can make a discussion in which the heat flow rate released to the high temperature heat source is the optimization objective.
When the heat flow rate pumped from the low temperature
Figure 5 Variations of the COP, the entransy increase rate and the en-tropy generation rate with Uin when Pin is fixed.
heat source Qin, the temperatures of the heat sources, Tin and
Tout, are fixed, it can be seen that the entransy increase rate, the entropy generation rate and the heat flow rate into the high temperature heat source all decrease with decreasing input power from eqs. (1), (14) and (21). It means that both the maximum (not the minimum) entropy generation rate and the maximum entransy increase rate lead to the maxi-mum heat flow rate into the high temperature heat source. The variations of Qout, Pin, Ginc and Sg with Uin are shown in Figure 6. It can be seen that the variation tenden-cies of Qout, Ginc and Sg are all same as that of Pin. The results verify the analyses above.
Furthermore, when the input power is fixed, eqs. (1), (15) and (22) show that both the entransy increase rate and the heat flow rate into the high temperature heat source increase with increasing Qin, and the entropy generation rate de-creases. It means that both larger entransy increase rate and smaller entropy generation rate lead to larger heat flow rate into the heat source high temperature.
For the cases discussed above, larger entransy increase rate always leads to larger heat flow rate into the high tem-perature heat source, while smaller entropy generation rate does not always. Therefore, if the optimization objective is the heat flow rate into the heat source with high temperature, the concept of entransy increase rate is more convenient. This is the difference between the concepts of entransy in-crease and entropy generation.
Last, let us analyze the simple heat pump system with reversed Carnot cycle as shown in Figure 7. The working fluid absorbs heat flow Qin at temperature TL from the low temperature heat source whose temperature is Tin, and re-leases heat flow Qout at temperature TH to the high tempera-ture heat source whose temperatempera-ture is Tout. The input power is Pin. According to the Carnot theorem, we have
out in H L
Q Q T T . (29)
Therefore, combining eqs. (1), (14) and (21) leads to
Figure 6 Variations of the entransy increase rate, the entropy generation rate, the heat flow rate into the high temperature heat source and the input power with Uin when Qin is fixed.
Figure 7 Sketch of the heat pump system with reversed Carnot cycle.
out H L in in 1 L H
Q T T Q P T T , (30) H
inc out out in in in out in
L H H in out in L L 1 , T G Q T Q T Q T T T T T P T T T T (31) out in H g in out in L out in H H in L out in L 1 1 1 . Q Q T S Q T T T T T T T P T T T T (32) When the temperatures Tin, Tout, TH and TL are fixed, it can be seen that Qout, Ginc and Sg all increase with in-creasing Qin or Pin. Hence, larger entransy increase rate is always compatible with larger heat flow rate into the high temperature heat source, while smaller entropy generation rate is not always. When heat is removed from a low tem-perature to a high temtem-perature, the total entransy increases. The power of the heat pump becomes heat into the high temperature heat source and contributes to the entransy in-crease. The entropy generation only comes from the heat transfer between heat sources and working fluid because the Carnot cycle is reversible. So, driving more heat from the low temperature heat source in the high temperature one will increase entropy generation.
Let us just only look at the reversed Carnot cycle. As-sume that the absorbed heat flow rate is Qin, the released heat flow rate is Qout, the input power is Pin, and the high and low working temperatures of the working fluid are TH and TL, respectively. In this case, eq. (29) is still tenable. The entransy increase rate is
2 2
inc in L H L 1 in H L
G Q T T T P T T . (33)
For given TH and TL, it can also be seen that both Qout and
inc
G increase with increasing Qin or Pin. However, the
en-tropy generation rate is always zero. For the reversed Carnot cycle, the entropy generation rate does not relate to the re-leased heat flow rate because entropy generation is the measure of the irreversible degree of thermodynamic pro-cess. It does not directly relate to heat or work for the re-versible processes. The entransy increase rate always is al-ways related to heat or power, which can be seen from their expressions.
3 Conclusions
This paper discusses the optimization of heat pump systems by the concepts of entransy increase and entropy generation and proposes the extremum entransy increase principle. The maximum COP leads to the maximum entransy increase rate for fixed input power and to the minimum entransy increase rate for fixed heat flow rate absorbed from the low temperature heat source when the equivalent temperatures of the high and low temperature heat sources are given. On the other hand, the minimum entropy generation rate always leads to the maximum COP for fixed input power or fixed heat flow rate absorbed from the low temperature heat source with given equivalent thermodynamic forces of the high and low temperature heat sources. These different principles are applied to the analyses of the heat pump sys-tem with reversed Brayton cycle.
When the optimization objective is the heat flow rate into the high temperature heat source, it is shown that larger entransy increase rate always leads to larger heat flow rate released into the high temperature heat source, while small-er entropy gensmall-eration rate does not always. The diffsmall-erence between the concepts of entransy increase and entropy gen-eration mechanisms and the mechanisms of the principles are discussed.
This work was supported by the National Natural Science Foundation of China (51376101) and the Tsinghua University Initiative Scientific Re-search Program.
1 David B, Ramousse J, Luo L. Optimization of thermoelectric heat pumps by operating condition management and heat exchanger de-sign. Energy Convers Manage, 2012, 60: 125–133
2 Zhang L, Hihara E, Saikawa M. Combination of air-source heat pumps with liquid desiccant dehumidification of air. Energy Convers Manage, 2012, 57: 107–116
3 Fernández-Seara J, Piñeiro C, Dopazo J A, et al. Experimental analy-sis of a direct expansion solar asanaly-sisted heatpump with integral storage tank for domestic water heating under zero solar radiation conditions. Energy Convers Manage, 2012, 59: 1–8
4 Chen L G, Feng H, Sun F R. Optimal piston speed ratios for irre-versible Carnot refrigerator and heat pump using finite time thermo-dynamics, finite speed thermodynamics and the direct method. J En-ergy Inst, 2011, 84: 105–112
5 Chen L G, Ding Z, Sun F R. Model of a total momentum filtered en-ergy selective electron heat pump affected by heat leakage and its performance characteristics. Energy, 2011, 26: 4011–4018
opti-mization of waste heat recovery Organic Rankine Cycles. Appl Therm Eng, 2011, 31: 2885–2893
7 Sun F, Chen W, Chen L, et al. Optimal performance of an en-doreversible Carnot heat pump. Energy Convers Manage, 1997, 38: 1439–1443
8 Wu C, Chen L G, Sun F R. Optimization of steady flow heat pumps. Energy Convers Manage, 1998, 39: 445–453
9 Bi Y, Chen L G, Sun F R. Heating load, heating load density and COP optimizations for an endoreversible variable-temperature heat reservoir air heat pump. J Energy Inst, 2009, 82: 43–47
10 Bi Y, Chen L G, Sun F R. Heating load, heating load density and COP optimizations for an endoreversible air heat pump. Appl Energy, 2008, 85: 607–617
11 Zhao X L, Fu L, Zhang S G. General thermodynamic performance of irreversible absorption heat pump. Energy Convers Manage, 2012, 52: 494–499
12 Bi Y, Chen L G, Sun F R. Comparative performance analysis for en-doreversible simple air heat pump cycles considering ecological, ex-ergetic efficiency and heating load objectives. Int J Exergy, 2009, 6: 550–566
13 Wang E, Fung A S, Qi C, et al. Performance prediction of a hybrid solar ground-source heat pump system. Energy Buildings, 2012, 47: 600–611
14 Yu X, Wang R Z, Zhai X Q. Year round experimental study on a constant temperature and humidity air-conditioning system driven by ground source heat pump. Energy, 2011, 36: 1309–1318
15 Cheng X T, Liang X G. Entransy loss in thermodynamic processes and its application. Energy, 2012, 44: 964–972
16 Guo Z Y, Zhu H Y, Liang X G. Entransy-A physical quantity de-scribing heat transfer ability. Int J Heat Mass Transfer, 2007, 50: 2545–2556
17 Guo Z Y, Liu X B, Tao W Q, et al. Effectiveness-thermal resistance method for heat exchanger design and analysis. Int J Heat Mass Transfer, 2010, 53: 2877–2884
18 Bejan A. Advanced Engineering Thermodynamics. 2nd ed. New York: John Wiley & Sons, 1997
19 Poulikakos D, Bejan A. Fin geometry for minimum entropy genera-tion in forced convecgenera-tion. J Heat Transfer, 1982, 104: 616–623 20 Erek A, Dincer I. An approach to entropy analysis of a latent heat
storage module. Int J Therm Sci, 2008, 47: 1077–1085
21 Shah R K, Skiepko T. Entropy generation extrema and their relation-ship with heat exchanger effectiveness-number of transfer unit be-havior for complex flow arrangements. J Heat Transfer, 2004, 126: 994–1002
22 Cheng X T, Zhang Q Z, Liang X G. Analyses of entransy dissipation, entropy generation and entransy-dissipation-based thermal resistance on heat exchanger optimization. Appl Therm Eng, 2012, 38: 31–39 23 Cheng X T, Liang X G. Computation of effectiveness of two-stream
heat exchanger networks based on concepts of entropy generation, entransy dissipation and entransy-dissipation-based thermal re-sistance. Energy Convers Manage, 2012, 58: 163–170
24 Cheng X T, Liang X G, Guo Z Y. Entransy decrease principle of heat transfer in an isolated system. Chin Sci Bull, 2011, 56: 847–854 25 Cheng X T, Liang X G. From thermomass to entransy. Int J Heat
Mass Transfer, 2013, 62: 174–177
26 Cheng X T, Liang X G, Xu X H. Microscopic expression of entransy. Acta Phys Sin, 2011, 60: 060512
27 Cheng X T, Xu X H, Liang X G. Homogenization of temperature field and temperature gradient field. Sci China Ser E-Tech Sci, 2009, 52: 2937–2942
28 Zhu H Y, Chen Z J, Guo Z Y. Thermal electric analogy experimental research for the extremum principle of entransy dissipation. Prog Nat Sci, 2007, 17: 1692–1698
29 Xiao Q H, Chen L G, Sun F R. Constructal entransy dissipation rate
and flow-resistance minimizations for cooling channels. Sci China Tech Sci, 2010, 53: 2458–2468
30 Wang W H, Cheng X T, Liang X G. Entransy dissipation and irre-versibility of some thermodynamic processes. Chin Sci Bull, 2012, 57: 4091–4099
31 Guo Z Y, Cheng X G, Xia Z Z. Least dissipation principle of heat transport potential capacity and its application in heat conduction op-timization. Chin Sci Bull, 2003, 48: 406–410
32 Feng H, Chen L G, Sun F R. Constructal entransy dissipation rate minimization for leaf-like fin. Sci China Tech Sci, 2012, 55: 515–526 33 Cheng X T, Zhang Q Z, Xu X H, et al. Optimization of fin geometry in heat convection with entransy theory. Chin Phys B, 2013, 22: 020503
34 Cheng X T, Liang X G. Entransy flux of thermal radiation and its ap-plication to enclosures with opaque surfaces. Int J Heat Mass Trans-fer, 2011, 54: 269–278
35 Cheng X T, Xu X H, Liang X G. Radiative entransy flux in enclo-sures with non-isothermal or non-grey, opaque, diffuse surfaces and its application. Sci China Tech Sci, 2011, 54: 2446–2456
36 Cheng X T, Liang X G. Optimization principles for two-stream heat exchangers and two-stream heat exchanger networks. Energy, 2012, 46: 386–392
37 Wang W H, Cheng X T, Liang X G. Entransy dissipation, entran-sy-dissipation-based thermal resistance and optimization of one-stream hybrid thermal network. Sci China Tech Sci, 2013, 56: 529–536
38 Zhao K H, Luo W Y. Thermotics (in Chinese). Beijing: Higher Edu-cation Press, 2002
39 Ust Y, Sahin B, Safa A. The effects of cycle temperature and cycle pressure ratios on the performance of an irreversible otto cycle. Acta Phys Polonica A, 2011, 120: 412–416
40 Cheng X T, Liang X G. Applicability of minimum entropy generation method to optimizing thermodynamic cycles. Chin Phys B, 2013, 22: 010508
41 Adavbiele A S. Optimization of thermofluid systems with second law. Int J Eng Research Africa, 2010, 1: 67–80
42 Myat A, Thu K, Kim Y D. A second law analysis and entropy gener-ation minimizgener-ation of an absorption chiller. Appl Therm Eng, 2011, 31: 2405–2413
43 Klein S A, Reindl D T. The relationship of optimum heat exchanger allocation and minimum entropy generation rate for refrigeration cy-cles. J Energ Res, 1998, 120: 172–178
44 Cheng X T, Wang W H, Liang X G. Entransy analysis of open ther-modynamic systems. Chin Sci Bull, 2012, 57: 2934–2940
45 Wu J. Potential energy (Entransy) in thermal science and its applica-tion (in Chinese). Ph.D. thesis, Tsinghua University, Beijing, 2009, 72–106
46 Cheng X T, Chen Q, Hu G J, et al. Entransy balance for the closed system undergoing thermodynamic processes. Int J Heat Mass Transfer, 2013, 60: 180–187
47 Cheng X T, Wang W H, Liang X G. Optimization of heat transfer and heat-work conversion based on generalized heat transfer law. Sci China Tech Sci, 2012, 55: 2847–2855
48 Cheng X T, Liang X G. Heat-work conversion optimization of one-stream heat exchanger networks. Energy, 2012, 47: 421–429 49 Zhou B, Cheng X T, Liang X G. Power output analyses and
optimi-zations of the Stirling cycle. Sci China Tech Sci, 2013, 56: 228–236 50 Wang W H, Cheng X T, Liang X G. Entropy and entransy analyses
and optimizations of the Rankine cycle. Energy Convers Manage, 2013, 68: 82–88
51 Moran M J. Availability Analysis: A Guide to Efficient Energy Use. New Jersey: Prentice-Hall Inc, 1982
52 Chen Z S. Advance Engineering Thermodynamics (in Chinese). Bei-jing: Higher Education Press, 2008
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