Analyses of entropy generation and heat entransy loss in heat transfer and heat-work conversion

International Journal of Heat and Mass Transfer 64 (2013) 903–909

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Analyses of entropy generation and heat entransy loss in heat transfer and heat-work conversion XueTao Cheng, XinGang Liang ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 14 January 2012 Received in revised form 8 May 2013 Accepted 9 May 2013 Available online 5 June 2013 Keywords: Entropy generation Heat entransy loss Heat transfer Heat-work conversion Optimization

a b s t r a c t Heat entransy loss is defined and both the concepts of entropy generation and heat entransy loss are applied to the analyses of heat-work conversion and heat transfer processes in this paper. In heat-work conversion, it is found that the minimum entropy generation rate relates to the maximum output power under the conditions of the prescribed heat absorption and the equivalent thermodynamic forces corresponding to the heat absorption and release of the system, while the maximum heat entransy loss relates to the maximum output power under the conditions of the prescribed heat absorption and the equivalent temperatures corresponding to the heat absorption and release of the system. In heat transfer, the maximum entropy generation rate is consistent with the maximum heat transfer rate with prescribed equivalent thermodynamic force difference, while the minimum entropy generation rate corresponds to the minimum equivalent thermodynamic force difference with prescribed heat transfer rate. Furthermore, when the concept of heat entransy loss is used, the maximum heat entransy loss rate corresponds to the maximum heat transfer rate with prescribed equivalent temperature difference, while the minimum heat entransy loss rate corresponds to the minimum equivalent temperature difference with prescribed heat transfer rate. The numerical results of some examples verify the theoretical analyses. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The optimization of heat transfer and heat-work conversion has been attracting the attention of researchers due to energy situation. In recent decades, some progresses contribute to the heat transfer optimizations and heat-work conversion optimizations [1–3]. The constructal method was first applied to the Volume-toPoint problem [1]. The basic structure of the high conductivity material that covers a part of the domain is first determined and the aspect ratio of the structure is optimized to reduce its highest temperature. Then the next order of construct that resembles the first order one is expended until the whole domain is covered. Later, the constructal method was used for heat convection optimization [4–6]. However, it was found that a more optimal construct is obtained without the premise that the new-order assembly construct must be assembled by the optimized last-order construct in the constructal theory [7]. The heat flow performance does not essentially improve if the internal complexity of the heat generating area increases, and the construct does not contribute to the heat transfer performance beyond the first order [8,9]. In heat convection, it was noticed that employing a fractal microchannel net⇑ Corresponding author. Tel./fax: +86 10 62788702. E-mail address: [email protected] (X.G. Liang). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.05.025

work in a heat sink does not improve thermal performance relative to that of a heat sink with parallel microchannels [10]. The parallel channel network can achieve a more than fivefold higher performance coefficient at a constant flow rate than the bifurcating tree-like network, and almost four times more heat can be removed for a constant pressure gradient across the networks [11]. The thermodynamic optimization method is widely used in heat transfer and heat-work conversion. From the viewpoint of the second law of thermodynamics, the practical heat transfer processes and heat-work conversion processes are both irreversible. Therefore, entropy generation is a measure of the irreversibility of these processes, and describes the loss of the ability to do work [12]. Therefore, the decrease of entropy generation would decrease the loss of the ability to do work and increase the output work. Researchers have done much work on heat-work conversion optimizations [13–18]. Furthermore, the entropy generation minimization method was also related to the heat transfer optimizations [1,19–21]. However, it was found that the entropy generation minimization is not always related to the largest COP [22] of refrigeration systems or the largest effectiveness of heat exchangers [23,24]. The investigation of the refrigeration systems showed that minimizing the entropy generation rate does not always result in the same design as maximizing the system performance unless the refrigeration capacity is fixed [22]. In heat transfer, entropy generation is not

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Nomenclature A C dU Gdis Gf Gloss k P Q Q_ q Sf

area, m2 heat capacity flow rate, W/K change of internal energy, W entransy dissipation rate, WK heat entransy flow rate, WK heat entransy loss rate, WK heat transfer coefficient, W/m2 thermodynamic force, 1/K heat flow rate, W inner heat source, W/m3 heat flux density, W/m2 entropy flow rate, W/K

monotonically related to the effectiveness of heat exchangers [23,24]. The effectiveness may not increase with the decrease of entropy generation number, but decrease under some conditions. In addition, entropy generation minimization may lead to contradictory results for some of the performance evaluation criteria during the optimization of fully-developed laminar flow through square ducts with rounded corners [25]. The concept of entransy was proposed by the analogy between heat and electrical conductions [3]. Heat flow corresponds to electrical current, thermal resistance to electrical resistance, temperature to electrical voltage, and heat capacity to capacitance [3]. Therefore, entransy is actually the ‘‘potential energy’’ of the heat in a body, corresponding to the electrical energy in a capacitor [3]. Entransy is always dissipated during heat transfer, and entransy dissipation can be used to describe the irreversibility of heat transfer [3,26,27]. With the concept of entransy dissipation, the principles of the extremum entransy dissipation and the minimum thermal resistance were developed [3]. These principles were adopted to optimize heat conduction [3,28–30], heat convection [3,31–33], thermal radiation [34,35], heat exchangers and heat exchanger networks [36–40], and are proved to be effective for heat transfer optimization. When the Volume-to-Point problem was optimized with these principles, the average temperature of the heated domain is lower than those by the constructal method and the entropy generation minimization method [3,29]. When the entransy theory was used to analyze heat exchangers, it was found that a smaller entransy-dissipation-based thermal resistance corresponds to a better heat transfer performance [36,37,40]. The applicability of the entransy theory to heat-work conversion was discussed from different viewpoints. Chen et al. [18] found that the concept of entransy dissipation could not be used to optimize heat-work conversion. Wu [41] defined the concept of conversion entransy and used the definition to optimize thermodynamic cycles. Cheng and Liang [42,45], Cheng et al. [43,44] defined the concept of heat entransy loss, which is the difference between the total input heat entransy and output entransy and is also the sum of the entransy dissipation due to the irreversible heat transfer and the entransy variation due to the work doing processes (i.e. work entransy). Heat entransy loss is the entransy reduction of the system, which is the entransy consumed during the heat-work conversion processes. In the analyses of the irreversible Brayton cycle [42], the endoreversible Carnot cycle [43], the air standard cycle [44], the one-stream heat exchanger networks [45], the Stirling cycle [46] and the Rankine cycle [47], it was found that the concept of heat entransy loss can be used to describe the change in output power for the systems, and the increase in heat entransy loss rate relates to the increase in output power. It could be noted from the above discussions that the thermodynamic optimization method is not always effective in heat-work

Sg T T0 V W

entropy generation rate, W/K temperature, K environment temperature, K volume, m3 output power, W

Subscripts H high temperature stream in into the system L low temperature stream out out of the system

conversion optimization and heat transfer optimization. The entransy theory is appropriate for heat transfer optimization but there is still no definite conclusion for its applicability to heat-work conversion optimization. The present work is to make further investigation on the applicability of the thermodynamic optimization method and the entransy theory on heat transfer optimization and heat-work conversion optimization. 2. Analyses of entropy generation For the common closed thermal system as shown in Fig. 1, Qin is the input heat rate, while Qout is output heat flow rate, and W is the output power. The energy conservation gives

Q in ¼ Q out þ W:

ð1Þ

For any infinitesimal element of the system, the entropy balance equation gives [12]

dS ¼ dSf þ dSg ;

ð2Þ

where dS is the entropy change with time, dSf is the entropy flow rate, and dSg is the entropy generation rate of the system. When the system is steady, dS equals to zero. For a differential volume dV, the entropy flow dSf includes two parts, one is associated with the heat flow rate, and the other is from the inner heat source. Considering both parts, we have

dSf ¼ r 

q Q_ dV þ dV: T T

ð3Þ

where Q_ is the inner heat source in dV, q is heat flux vector, and T is the temperature. The first term on the right-hand side is the net entropy flow associated with the heat transfer at the boundary, and the second term is that associated with the inner heat source.

Fig. 1. Sketch of a common closed thermal system.

X.T. Cheng, X.G. Liang / International Journal of Heat and Mass Transfer 64 (2013) 903–909

The term, r  ðq=TÞ, is the net entropy out of dV through boundary and it is equal to the sum of the entropy generation rate in dV and the entropy flow rate due to the inner heat source in dV if we combine Eqs. (2) and (3) at steady state. From Eqs. (2) and (3), the entropy generation rate of the whole steady system can be expressed as

# Z " Z Z _ q Q_ q Q Sg ¼  r   ndA  dV; þ dV ¼ T T T V A V T

ð4Þ

where A is the surface area of the system and n is the normal vector. Some heat (q > 0) flows into the system through parts of the system surface, while some heat (q < 0) goes out of the system through other part of surface at steady state,

Q in ¼ 

Z

q  ndAin þ

Ain

Q out ¼

Z

Z

Q_ dV;

ð5Þ

V

q  ndAout ;

ð6Þ

905

It could be seen that the minimum entropy generation rate corresponds to the minimum equivalent thermodynamic force difference with prescribed heat transfer rate, while the maximum entropy generation rate corresponds to the maximum heat transfer rate with prescribed equivalent thermodynamic force difference. The required condition for the maximum entropy generation rate to achieve maximum heat transfer rate for given thermodynamic force difference seems to be in contrary to the thermodynamic efficiency increase which requires the minimization of entropy generation. This can be explained as below. When the equivalent thermodynamic force difference is prescribed, the increase of heat transfer rate means the increase of exergy loss and entropy generation. In other words, with improvement of heat transfer, it will provide more heat transfer rate for the same thermodynamic force levels and thus the increase of irreversibility and entropy generation. 3. Analyses of heat entransy loss

Aout

where Ain is the surface area through which the system absorbs heat and Aout is that through which the system releases heat. We define the equivalent thermodynamic force weighted by heat transfer rate as

Sfin Pin ¼ ¼ Q in



Z Ain

q  ndAin þ T

Z V

Q_ dV T

!, Q in ;

ð7Þ

where Sfin is the entropy flow rate into the system. As is known, 1/ T is the thermodynamic force [48–51]. When there is only one heat input temperature, Pin is the thermodynamic force. However, Pin is the equivalent thermodynamic force when there are two or more heat input temperatures. Each 1/T is weighted by the fraction of heat input of this part. When the system releases heat to the outside, the equivalent thermodynamic force weighted by heat transfer rate could be defined as

Pout ¼

Sfout ¼ Q out

Z Aout

 q  ndAout Q out ; T

ð8Þ

where Sfout is the entropy flow rate out of the system. Therefore, we could get that

# Z " q Q_ Sg ¼  r  þ dV T T V " ! Z # Z Z _ q q Q  ndAin þ dV   ndAout ¼  Ain T V T Aout T ¼ ðQ in Pin  Q out Pout Þ:

ð9Þ

According to Eq. (1), there is

Sg ¼ ½Q in Pin  ðQ in  WÞPout  ¼ Q in ðPout  Pin Þ  WPout ¼ Q in DP  WPout ;

ð10Þ

where DP is the equivalent thermodynamic force difference of the system. If the system in Fig. 1 is a heat-work conversion system, Eq. (10) tells us that the minimum entropy generation rate corresponds to the maximum output work with prescribed Qin, Pin and Pout. On the other hand, if the system is a heat transfer system, the output work is zero, and there is

Q ¼ Q in ¼ Q out ;

ð11Þ

where Q is the heat transfer rate of the system. Then, Eq. (10) could be simplified into

Sg ¼ Q DP:

ð12Þ

For any infinitesimal element of the closed system in Fig. 1, the entransy balance equation is set up below. In the infinitesimal element, there may be heat-work conversion or heat transfer process. For the heat-work conversion process, the first law of thermodynamic gives

dQ ¼ dU þ dW;

ð13Þ

where dQ is the absorbed heat flow rate, dU is the change of the internal energy, and dW is the output power. Multiplying Eq. (13) with the temperature leads to

TdQ ¼ TdU þ TdW:

ð14Þ

Considering the definition of the entransy flow and the linear relationship between the internal energy and the temperature, there is

TdQ  TdU ¼ dGf  dG ¼ TdW > 0:

ð15Þ

where dG is the rate of entransy change, dGf is the heat entransy flow rate into the infinitesimal element accompanying heat and fluid streams. The net heat entransy flow rate into the infinitesimal element is bigger than the entransy change in the heat-work conversion process because some entransy is consumed in the work doing process, which is named work entransy [42–47]. For the heat transfer process, the net heat entransy flow into the infinitesimal element is always bigger than the entransy change. It means that entransy dissipation, Gdis, will always exist in heat transfer [3,26,27]. The work entransy and the entransy dissipation are both the entransy reduction of the infinitesimal element, which is treated as the heat entransy loss from the viewpoint of heat entransy input [42–47]. The entransy balance equation of the infinitesimal element could be expressed as

dG ¼ dGf  dGloss ;

ð16Þ

where dGloss is the heat entransy loss. Eq. (16) indicates that the heat entransy loss is equal to the difference between the net input heat entransy and the net increase in entransy of the element. Some net input heat entransy is lost from the viewpoint of net input heat entransy. From the above discussions, it can be found that heat entransy loss is the sum of the entransy dissipation and the work entransy. For a practical thermodynamic process, it can be divided into to two necessary parts: (1) the work doing process in which the heat in converted to work; (2) the heat transfer process whose aim is to provide heat for the work doing process. Inevitably, the work entransy would yield during the work doing process, which leads to the loss of the entransy of the heat reservoirs. Similarly, the

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entransy is dissipated during the heat transfer process, which also leads to the loss of the entransy of the heat reservoirs. Therefore, the heat entransy loss can be considered as the necessary cost for the heat-work conversion process. When the system is steady, dG equals to zero. Then, the integration of Eq. (16) over the whole system gives

Gloss ¼

Z

dGloss ¼

Z

V

dGf :

ð17Þ

V

Based on the definition of the entransy flow rate [3], there is

h i dGf ¼ r  ðqTÞ þ Q_ T dV;

ð18Þ

where the first term on the right-hand side is the net entransy flow associated with the heat transfer, and the second term is that associated with the inner heat source. Then, we can get

Z

Z h

i r  ðqTÞ þ Q_ T dV V V Z Z q  nTdA þ Q_ TdV: ¼

Gloss ¼

dGf ¼

A

ð19Þ

V

When the system absorbs heat from the outside, the equivalent temperature weighted by heat transfer rate could be defined as

T in ¼

Gfin ¼ Q in



Z

q  nTdAin þ Ain

Z

!, Q_ TdV

Q in ;

ð20Þ

V

where Gfin is the entransy flow rate into the system. When the system releases heat to the outside, the equivalent temperature weighted by heat transfer rate could be defined as

T out ¼

Gfout ¼ Q out

Z

q  nTdAout

 Q out ;

ð21Þ

Aout

where Gfout is the entransy flow rate out of the system. Then, based on Eqs. (1), (20), and (21), Eq. (19) can be changed as

Gloss ¼ 

Z

Tq  ndA þ

A

Z

_ QTdV ¼ Q in T in  Q out T out

V

¼ Q in ðT in  T out Þ þ WT out ¼ Q in DT þ WT out ;

ð22Þ

where DT is the equivalent temperature difference. If the system in Fig. 1 is a heat-work conversion system, Eq. (22) shows that the maximum heat entransy loss rate corresponds to the maximum output power with prescribed Qin, Tin and Tout.On the other hand, if the system is a heat transfer system, W is zero and Eq. (12) leads to

Gloss ¼ Q DT:

4. Discussions The output work or the heat-work conversion efficiency is concerned in heat-work conversion, while the heat transfer rate is concerned in heat transfer. From the equations derived in Sections 2 and 3, it can be found that the applicability of the concepts of entropy generation and heat entransy loss is conditional in heat transfer optimization and heat-work conversion optimization. In heat-work conversion optimization, the entropy generation minimization is effective when the heat input into the system and the equivalent thermodynamic forces are prescribed, while the heat entransy loss maximization is effective when the heat input into the system and the equivalent temperatures are prescribed. When the parameters are not prescribed, the concepts of entropy generation and heat entransy loss may not be applicable to heat-work conversion optimization. For instance, when analyzing the refrigeration systems with entropy generation, Klein and Reindl [22] found that minimizing the entropy generation rate does not always result in the same design as maximizing the system performance unless the refrigeration capacity is fixed. Let us discuss an example in Fig. 2. A Carnot engine works between two streams with the inlet temperatures, THin and TLin, the outlet temperatures, THout and TLout, and the heat capacity flow rates, CH and CL, respectively. The working fluid takes heat QH per unit time from the high temperature stream through a heat exchanger, and releases heat QL per unit time to the low temperature stream through another heat exchanger. The cycle output power is W. In the cycle, the top and low temperatures of the working fluid, TH and TL, should be optimized to get the maximum output power. If the two streams are treated as the heating and cooling boundaries of the system, the system can be treated as a closed one. Then, Eqs. (10) and (22) can be applied to analyzing the system. The concept of entropy generation is applied to analyzing this cycle first. If the entropy generation of dumping the used streams into the environment is considered [52], the heat flow into the thermodynamic system can be obtained with the consideration that both the temperatures of the streams would become the temperature of the environment,

Q in ¼ C H ðT Hin  T 0 Þ þ C L ðT Lin  T 0 Þ;

ð24Þ

where T0 is the environment temperature . When CH, CL, THin, TLin and T0 are prescribed, Qin is a definite value. As there is no inner

ð23Þ

The minimum heat entransy loss rate corresponds to the minimum equivalent temperature difference with prescribed heat transfer rate, while the maximum heat entransy loss rate corresponds to the maximum heat transfer rate with prescribed equivalent temperature difference. As there is no output power, there is no work entransy. The only heat entransy loss is from the entransy dissipation. From Eq. (23), the extremum entransy dissipation principles were developed and the principles was concluded to be the principle of the minimum thermal resistance that is defined based on the entransy dissipation and the net heat flow between the source and the sink [3]. If W is negative, the system in Fig. 1 is a heat pump system which drives heat from low temperature heat sources to high temperature heat sources. For the heat in this kind of systems, its heat potential will increase because its temperature increases. Therefore, the entransy of these systems will not decrease but increase. This kind of system needs systematic investigation. Hence, we only focus on the cases in which W is positive in this paper.

Fig. 2. Sketch of an endoreversible Carnot cycle working between the hot and cold streams.

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heat source in the system, the entropy flow rate into the system is equal to the entropy generation rate of the streams, that is,

Sfin ¼

Z

T Hin

T0

C H dT þ T

Z

T Lin

T0

C L dT T Hin T Lin ¼ C H ln þ C L ln : T T0 T0

ð25Þ

Therefore, when the system absorbs heat, the equivalent thermodynamic force is

Pin ¼

Sfin ¼ Q in

 C H ln

T Hin T Lin þ C L ln T0 T0

 Q in :

ð26Þ

When CH, CL, THin, TL-in and T0 are prescribed, Pin is also a definite value. Furthermore, the released heat all gets into environment, so Pout = 1/T0. Then, it can be found that Qin, Pin and Pout are all prescribed. According to Eq. (10), the preconditions of the corresponding relationship between the minimum entropy generation rate and the maximum output power are satisfied, so the minimum entropy generation rate will correspond to the maximum output power. In practical applications, the used stream may be not discharged into the environment immediately after the heat-work conversion, but may be used in other thermodynamic processes, or be stored. In such cases, the entropy generation that induced by dumping the used stream into the environment should not be taken into account. Therefore, the heat absorption only comes from the high temperature stream. It is

Q in ¼ Q H ¼ C H ðT Hin  T Hout Þ:

907

high temperature stream and the working medium eH is 0.8, while effectiveness of the heat exchanger between the low temperature stream and the working medium eL is 0.7. The entropy generation rate, the heat entransy loss rate and the output power under different values of TH are calculated with and without the consideration of dumping the used streams into the environment. The results are shown in Figs. 3 and 4. The maximum output power, maximum heat entransy loss rate, and minimum entropy generation rate are obtained at the same time when the entropy generation rate and the heat entransy loss rate induced by dumping the used stream into the environment are considered. However, neither the minimum entropy generation rate nor the maximum heat entransy loss corresponds to the maximum output power when the entropy generation rate and the heat entransy loss rate induced by dumping the used stream into the environment are not considered. The results verify the theoretical analyses above. In many heat transfer optimizations, the best heat transfer was related to the minimum entropy generation [1,2]. However, when the equivalent thermodynamic force difference of the system is prescribed, the maximum heat transfer rate does not correspond

ð27Þ

Assume that there is a heat exchanger between the high temperature stream and the working medium, and the effectiveness of the heat exchanger is prescribed. Then, it can be found that THout varies with the change of TH. Therefore, Qin is not prescribed. With the same consideration, Qout (QL) is not definite either. The entropy flows into and out of the system and the equivalent thermodynamic forces defined by Eqs. (7) and (8) are not definite either. Therefore, the minimum entropy generation rate may not correspond to the maximum output power according to Eq. (10). The case for heat entransy loss is similar. When the heat entransy loss induced by dumping the used stream into the environment is considered, the entransy flow that gets into the thermodynamic system is

Gfin ¼

1 1 C H ðT 2Hin  T 20 Þ þ C L ðT 2Lin  T 20 Þ: 2 2

ð28Þ

Compared with Eq. (25), it can be found that Gfin is also definite if CH, CL, THin, TLin and T0 are prescribed. Considering Eq. (20) and that there is no inner heat source, we can find that Tin is prescribed. As the released heat is discharged into the environment, Tout is T0. Therefore, Qin, Tin and Tout are all prescribed. The preconditions of the relationship between the maximum heat entransy loss rate and the maximum output power are all satisfied, so the heat entransy loss rate can also be used to optimize the cycle in Fig. 2. If the heat entransy loss induced by dumping the used stream into the environment is not considered, Qin is not prescribed according to Eq. (27) and the equivalent temperatures defined by Eqs. (20) and (21) are not definite either. The preconditions of the relationship between the maximum heat entransy loss rate and the maximum output power are not satisfied, so the maximum heat entransy loss rate does not correspond to the maximum output power. The discussions above explain why the concepts of entropy generation and heat entransy loss could be used to optimize the cycle in Fig. 2 when the entropy generation rate and the heat entransy loss rate induced by dumping the used stream into the environment are considered. Let us look at a numerical example. Assume that CH = 2 W/K, CL = 3 W/K, THin = 350 K, TLin = 310 K, T0 = 300 K, the effectiveness of the heat exchanger between the

Fig. 3. Variations of the normalized entropy generation rate, heat entransy loss rate and output power with TH in the case of dumping the used stream into the environment.

Fig. 4. Variations of the normalized entropy generation rate, heat entransy loss rate and output power with TH in the case of not dumping the used stream into the environment.

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to the minimum entropy generation rate, but the maximum entropy generation rate as shown in Eq. (12). When the heat transfer rate is prescribed, the minimum entropy generation rate corresponds to the minimum equivalent thermodynamic force difference, but not the equivalent temperature difference. Therefore, the entropy generation minimization is not appropriate when the maximum heat transfer rate is the optimization objective. Furthermore, when the objective is not the minimum equivalent thermodynamic force difference or the heat transfer rate is not prescribed, the minimum principle of entropy generation is not appropriate, either. For instance, when the concept of entropy generation was used to analyze heat exchangers, an entropy generation paradox was noted [23] in which the effectiveness of heat exchangers may not increase, but decrease with the decrease of entropy generation number under some conditions. On the other hand, when the concept of heat entransy loss is used in heat transfer optimization, the maximum heat entransy loss rate corresponds to the maximum heat transfer rate with prescribed equivalent temperature difference, while the minimum heat entransy loss rate corresponds to the minimum equivalent temperature difference with prescribed heat transfer rate. Therefore, if the optimization objective is not the maximum heat transfer rate or the minimum equivalent temperature difference, and the equivalent temperature difference and the heat transfer rate are not prescribed, the concept of heat entransy loss may be not appropriate for heat transfer optimization, either. Let us discuss an example of optimization. As shown in Fig. 5, there are two equipments with prescribed heat transfer rates, Q1 and Q2, in the heat transfer process. The heat from the equipments is transferred into one isothermal heat sink, whose temperature is T0. The heat transfer coefficients of the equipments are k1 and k2, and the heat transfer areas are A1 and A2, respectively. Assume that the limiting condition is

A1 þ A2 ¼ A ¼ const:

ð29Þ

The distribution of the given total heat transfer area can be optimized. As the total heat transfer rate of the equipments is prescribed, Eq. (12) tells us that the minimum entropy generation rate corresponds to the minimum equivalent thermodynamic force difference, while Eq. (23) shows that the minimum heat entransy loss rate corresponds to the minimum equivalent temperature difference. If the optimization objective is the minimum equivalent thermodynamic force difference, DP, we can calculate the entropy generation rate with Eq. (4),

  Q1 þ Q2 Q1 Q2  þ T0 T1 T2   Q1 þ Q2 Q1 Q2 : ¼  þ T0 T 0 þ Q 1 =ðk1 A1 Þ T 0 þ Q 2 =ðk2 A2 Þ

Sg ¼

ð30Þ

Then, DP can be calculated by

S S DP ¼ Pout  Pin ¼ fout  fin Q out Q in   1 Q1 Q2 ¼  þ ðQ 1 þ Q 2 Þ: T0 T1 T2

Fig. 5. Sketch of a heat transfer area optimization problem.

ð31Þ

Fig. 6. Variations of the entropy generation rate, the equivalent thermodynamic force difference, the heat entransy loss rate and the equivalent temperature difference with A1.

On the other hand, if the optimization objective is the minimum equivalent temperature difference, DT, the heat entransy loss rate could be calculated by Eq. (19),

Gloss ¼ ðQ 1 T 1 þ Q 2 T 2 Þ  ðQ 1 þ Q 2 ÞT 0 :

ð32Þ

Then, DT can be calculated by

DT ¼ T in  T out ¼

Gfin Gfout  Q in Q out

¼ ðQ 1 T 1 þ Q 2 T 2 Þ=ðQ 1 þ Q 2 Þ  T 0 :

ð33Þ

Assume that Q1 = 100 W, Q2 = 200 W, k1 = 100 W/(m2 K), k2 = 5 W/ (m2 K), A = 1 m2 and T0 = 10 K. The variations of the entropy generation rate, the equivalent thermodynamic force difference, the heat entransy loss rate and the equivalent temperature difference with the heat transfer area are calculated with Eqs. (30)–(33). The results are shown in Fig. 6. It can be found that the entropy generation rate and the equivalent thermodynamic force difference both reach their minimum values when A1 is 0.42 m2, while the heat entransy loss rate and the equivalent temperature difference both reach their minimum value when A1 is 0.10 m2. When the optimization objective is the minimum equivalent thermodynamic force difference, the concept of entropy generation is more appropriate, while the concept of heat entransy loss is more appropriate when the optimization objective is the minimum equivalent temperature difference. The choice of optimization method, entropy or entransy, depends on the optimization objective. 5. Conclusions Both the concepts of entropy generation and heat entransy loss are used to analyze the heat-work conversion and heat transfer processes. It is found that the applicability of entropy generation minimization and heat entransy loss maximization is conditional to heat-work conversion and heat transfer optimizations. When they are are used to optimize heat transfer and heat-work conversion, the preconditions of the corresponding relationships should be satisfied. The minimum entropy generation rate corresponds to the maximum output power for prescribed input heat and equivalent thermodynamic forces corresponding to the heat absorption and release in heat-work conversion, while the maximum heat entransy loss rate corresponds to the maximum output power for prescribed input heat and equivalent temperatures corresponding to the heat absorption and release. On the other hand, when the concept of entropy generation is used in heat transfer

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