Entransy analyses of thermal processes with variable thermophysical properties

International Journal of Heat and Mass Transfer 90 (2015) 1244–1254

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Entransy analyses of thermal processes with variable thermophysical properties Bing Zhou, XueTao Cheng, WenHua Wang, 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 9 April 2015 Received in revised form 2 June 2015 Accepted 8 July 2015

Keywords: Entransy Variable thermophysical property Heat transfer Heat-work conversion Optimization

a b s t r a c t Entransy method is developed for the analyses of thermal processes with variable thermophysical properties. The entransy and enthalpy entransy balance equations for thermal processes with variable thermophysical properties are derived based on the definitions of differential entransy and enthalpy entransy. The definitions of the entransy and enthalpy entransy for an object are presented. The optimization principles for heat transfer and heat-work conversion processes with variable thermophysical properties are discussed with the concept of entransy. Entransy analyses for a two-stream heat exchanger and a closed Brayton cycle with variable thermophysical properties are conducted and compared with the entropy generation analyses. For the heat exchanger, the results show that both the minimum entransy-dissipation-based thermal resistance and the minimum revised entropy generation number correspond to the maximum heat exchanger effectiveness when the inlet condition of the heat exchanger is prescribed, but the minimum entropy generation rate and the minimum entropy generation number do not. For the closed Brayton cycle, the results show that both the maximum entransy loss rate and the minimum entropy generation rate correspond to the maximum output power and both the maximum entransy loss coefficient and the minimum entropy generation number correspond to the maximum heat-work conversion efficiency when the inlet conditions of the streams and the environmental temperature are prescribed. When the inlet temperature of the hot stream increases, larger entransy loss rate and larger entransy loss coefficient still correspond to larger output power and larger heat-work conversion efficiency respectively. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Effective energy utilization is becoming a more and more important topic due to the energy situation nowadays [1–4]. The analyses and optimizations of heat transfer and heat-work conversion are of great importance for thermal systems to improve their energy utilization efficiency and reduce the costs. Therefore, great attention has been attracted to this field and some relevant optimization theories have been developed [5–8]. From the viewpoint of thermodynamics, practical heat transfer processes are irreversible and their performance can be improved by reducing their irreversibility. Entropy generation is an important measure of irreversibility [5]. Therefore, the processes or systems can be optimized by reducing entropy generation. This is the widely used entropy generation minimization method [5,6]. In heat transfer, the entropy generation minimization has been used for the optimizations of various kinds of heat transfer processes ⇑ Corresponding author. Fax: +86 10 62788702. E-mail address: [email protected] (X. Liang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.037 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

[9–12], such as heat sinks, heat exchangers, etc. However, entropy generation paradox was noted when it was applied to the counter flow heat exchangers [5,12–14]. For instance, Shah and Skiepko [14] found that the maximum effectiveness of the heat exchangers cannot always be obtained by minimizing entropy generation. Therefore, the entropy generation minimization does not always lead to the optimal heat transfer because the irreversibility is not always related to the objective of heat transfer optimization. In heat-work conversion, entropy generation was related to exergy loss and minimizing entropy generation means minimizing the ability loss of doing work. Under this consideration, the entropy generation minimization is widely used to optimize thermodynamic systems [5,6,15–17]. However, some researches also showed that the entropy generation minimization cannot always lead to the optimal performance of the discussed systems [18– 20]. For instance, in the analyses of the refrigeration systems, Klein and Reindl [18] found that minimizing entropy generation rate does not always result in the same design as maximizing the system performance unless the refrigeration capacity is fixed. Further analyses by Cheng and Liang [20] showed that the

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minimum entropy generation rate corresponds to the maximum output power of the system only when the net exergy flow rate into the system is fixed. From the further understanding of the mechanism of convective heat transfer, Guo et al. [7] found that the performance of convective heat transfer not only depends on the flow velocity of the fluid and temperature difference between the fluid and wall, but also depends on the degree of the synergy between the velocity field and heat flux field. The better the synergy is, the better the heat transfer will be. This is the field synergy principle, which provides a better explanation of the mechanism of the existing techniques for heat transfer enhancement and has been widely used for the analyses and optimizations of convective heat transfer [21–25]. In recent years, Guo et al. [8] introduced the concept of entransy that represents the potential of thermal energy. During heat transfer, thermal energy is conserved, but entransy is dissipated. Therefore, entransy dissipation becomes another measure of heat transfer irreversibility. Based on this concept, the extremum entransy dissipation principle was proposed, which indicates that the maximum entransy dissipation rate corresponds to the maximum heat transfer rate when the heat transfer temperature difference is prescribed, and the minimum entransy dissipation rate corresponds to the minimum heat transfer temperature difference when the heat transfer rate is prescribed. Furthermore, the extremum entransy dissipation principle was generalized into the minimum entransy-dissipation-based thermal resistance principle, which shows that the minimum thermal resistance leads to the optimal heat transfer. The entransy theory has been used to optimize various kinds of heat transfer processes and systems [26– 41], such as heat conduction [28,29], convective heat transfer [30,31], radiative heat transfer [32,33], heat transfer with vaporization processes [34] and the optimization design of heat exchangers [35–38] and other thermal systems [39–41]. The results showed that the entransy dissipation or entransy-dissipation-based thermal resistance is related to the objective of heat transfer optimizations and no paradox was found when entransy theory was applied to heat exchanger analyses. On the other hand, the application of entransy theory to heat-work conversion systems was also discussed. Chen et al. [39] found that the extremum entransy dissipation do not correspond to maximum output power of the heat-work conversion system. Cheng and Liang [42] considered the entransy change due to heat-work conversion and proposed the concept of entransy loss (net system entransy input), which is the net entransy inputted into the system. Researches demonstrated that larger entransy loss is related to larger output work in heat-work conversion processes [43–48], such as the Carnot cycle [44], Rankine cycle [45], Stirling cycle [46], etc., under appropriate preconditions. Based on the concept of entransy loss, the entransy loss coefficient was proposed for the optimization of the heat-work conversion efficiency of systems [49]. Furthermore, the concept of enthalpy entransy was also developed and applied to the analyses of open thermodynamic systems [50]. However, the existing studies related to the entransy theory are under the assumption that the thermophysical properties, such as the specific heat, etc., are constants or can be treated as constants [8]. For thermal processes with variable thermophysical properties, the present definition of entransy cannot be derived from its differential definition, and the present entransy balance equations for heat transfer [8,26,27] cannot be derived, either. The development of entransy theory towards the thermal processes with variable thermophysical properties is needed. In the present work, we derive the entransy balance equations for thermal processes with variable thermophysical properties (specific heat, density and thermal conductivity) and extend the entransy theory to analyze those processes. The definitions of entransy and enthalpy entransy which are consistent with their

differential definitions are proposed, and the optimization principles for heat transfer and heat-work conversion with variable thermophysical properties are developed and applied to the analyses of two practical examples. 2. Entransy balance equations and entransy definitions for thermal processes with variable thermophysical properties 2.1. Entransy balance equations for thermal processes with variable thermophysical properties The internal energy balance equation for a fluid element in thermal system is [51]

de Q_ 1 U_ 1 ¼  r  q þ  pr  u; dt q q q q

ð1Þ

d dt

¼ @t@ þ ðu  rÞ is the substantial derivative, e is the internal energy of the fluid per unit mass, q is the density of the fluid, Q_ is the internal heat source in the fluid per unit volume, q is the heat flux on the boundary of the fluid element, U_ is the viscous dissipawhere

tion rate in the fluid per unit volume, p is pressure, u is the velocity vector, and t is time. Multiplying Eq. (1) by the temperature of the fluid element, T, results in

T

de U_ 1 Q_ 1 ¼ T  T r  q þ T  pT r  u: dt q q q q

ð2Þ

According to the definition of differential entransy, there is [52]

dg ¼ Tde;

ð3Þ

where g is the entransy per unit mass. Based on the Leibniz rule and the Fourier’s law, the second term on the right-hand side of Eq. (2) can be changed into

1

q

Tr  q ¼

1

q

½r  ðqTÞ  qrT ¼

1

q

½r  ðqTÞ þ kðrTÞ2 ;

ð4Þ

where k is thermal conductivity. Noting that pr  u is the expansion power per unit volume per unit time, we can define

_ ¼ pr  u: dW

ð5Þ

Substituting Eqs. (3)–(5) into Eq. (2) yields

dg 1 U_ 1 Q_ 1 _ ¼ T  kðrTÞ2  r  ðqTÞ þ T  TdW; dt q q q q q

ð6Þ

where the left term is the entransy change rate of the fluid element, the first term on the right-hand side is the entransy input rate to the fluid element due to the internal heat source, the second term is the entransy dissipation rate in the fluid element, the third term is the entransy flow into the fluid element due to the heat exchange at the element boundary, the fourth term is the entransy input rate to the fluid element due to the viscous dissipation, and the last term is the entransy output rate of the fluid element due to the expansion power output. This is the entransy balance equation for the fluid element, which is always tenable whether the thermophysical properties of the fluid change or not. According to the relationship between enthalpy and internal energy, there is [51]

h ¼ e þ p=q;

ð7Þ

where h is the enthalpy of the fluid per unit mass. The energy balance equation for the element based on enthalpy can be obtained [51],

dh Q_ 1 U_ 1 dp ¼  rqþ þ : dt q q q q dt

ð8Þ

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Multiplying Eq. (8) by the temperature of the fluid element, T, yields

dh U_ 1 dp Q_ 1 ¼ T  Tr  q þ T þ T : dt q q q q dt

T

ð9Þ

According to the definition of differential enthalpy entransy, there is [50]

dg H ¼ Tdh;

ð10Þ

where gH is enthalpy entransy per unit mass. Noting that  dp is the dt technical power per unit volume per unit time, we can define

_ t ¼  dp : dW dt

ð11Þ

the right-hand side is the entransy input rate to the region due to the internal heat source, the second term on the right is the entransy dissipation rate of the whole region, the third term on the right is the entransy flow into the system due to the heat exchange at the region boundary, the forth term on the right is the entransy input rate to the region due to the viscous dissipation, and the last term on the right is the entransy output rate of the region due to the expansion power output. For the enthalpy entransy balance equation, integrating Eq. (12) over the whole region and using the same method as that for the entransy balance equation, we can get

Z V

@ðqg H Þ dV þ @t

Z

qug H  ndS ¼

S

By use of Eqs. (4), (10) and (11), Eq. (9) can be changed into

dg H 1 U_ 1 Q_ 1 _ t; ¼ T  kðrTÞ2  r  ðqTÞ þ T  TdW dt q q q q q

ð12Þ

q

V



dg dV ¼ dt

Z

_ ½T Q_  kðrTÞ2  r  ðqTÞ þ T U_  TdWdV;

ð13Þ

V

where V represents the volume of the region. According to the Leibniz rule, Gauss law and continuity equation @@tq þ r  ðquÞ ¼ 0, the left term of Eq. (13) can be changed into

Z 

q

V

  Z  dg @g dV ¼ q þ qu  rg dV dt @t V Z Z Z @ðqgÞ @q ¼ dV  g dV þ r  ðqugÞdV @t @t V V V Z  g r  ðquÞdV V Z Z @ðqgÞ ¼ dV þ qug  ndS; @t V S

ð14Þ

where S represents the boundary surface of the region. According to the Gauss law, the third term on the right-hand side of Eq. (13) can be changed into



Z

r  ðqTÞdV ¼ 

V

Z

qT  ndS:

Z T Q_ dV  kðrTÞ2 dV V V Z Z  qT  ndS þ T U_ dV V ZS _  TdW t dV:

ð17Þ

V

where the left term is the enthalpy entransy change rate of the fluid per unit mass, the last term on the right-hand side is the enthalpy entransy output rate of the fluid element due to the technique power output, and the other terms in the equation have the same meaning as those in Eq. (6). This is the enthalpy entransy balance equation for the fluid element, which is also always tenable whether the thermophysical properties of the fluid change or not. Considering that practical processes always happen in a specific region, we can get the entransy and enthalpy entransy balance equations for the whole region by integrating the equations for the fluid element over the whole region. For the entransy balance equation, integrating Eq. (6) over the whole region yields

Z 

Z

ð15Þ

S

This is the enthalpy entransy balance equation for the whole region, where the first term on the left-hand side is the enthalpy entransy change rate of the region, the second term on the left is the enthalpy flow out of the region through boundary with mass flow, the last terms on the right-hand side is the entransy output rate of the region due to the technical power output, and the other terms have the same meaning as those in Eq. (16). Based on the definitions of differential entransy and enthalpy entransy, there is

dg H dh de dðp=qÞ dg 1 _  1 TdW _ t: ¼T ¼T þT ¼ þ TdW dt dt dt dt q dt q

ð18Þ

Combining with Eqs. (6) and (12), it can be seen that Eqs. (6) and (12) are equivalent to each other. Therefore, Eqs. (16) and (17) are also equivalent to each other. In practical analyses, the choice of the equations for application depends on the conditions. For heat transfer, if it happens under a constant volume condition, Eq. (16) is more convenient for application because the term related to expansion power is zero. If it happens under a constant pressure condition, Eq. (17) is more convenient for application because the term related to the technical power is zero. If it happens neither under a constant volume condition nor a constant pressure condition, both Eqs. (16) and (17) can be used by calculating the expansion power or the technical power. For heat-work conversion problems, the concern is the expansion power for closed systems and Eq. (16) should be used. If the working medium system is an open one, the concern is the technical power and Eq. (17) should be used. For solid systems, the enthalpy balance equation is not needed since there is no flow in the mediums. The entransy balance equations for solid systems can be obtained by neglecting the terms related to the convection and viscous dissipation in related equations.

Then, Eq. (13) becomes

Z V

@ðqgÞ dV þ @t

Z S

qug  ndS ¼

Z

Z T Q_ dV  kðrTÞ2 dV V V Z Z  qT  ndS þ T U_ dV V ZS _  TdWdV:

2.2. The calculation of entransy 2.2.1. The entransy of an object and calculation of entransy change According to the definition of the differential entransy, the entransy per unit mass at a certain state can be calculated by

ð16Þ

V

This is the entransy balance equation for the whole region, where the first term on the left-hand side is the entransy change rate of the region, the second term on the left is the entransy flow out of the region through boundary with mass flow, the first term on

g¼

Z 0

g

dg ¼

Z

e

Tde:

ð19Þ

0

Generally, in practical application, we only need to calculate the entransy change from one state to another, rather than the absolute entransy. Therefore, we can use the concept of relative

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entransy by defining a reference state at Tref, where the corresponding internal energy is eref and entransy is gref. Then, the expression of entransy can be changed into

g ¼ g ref þ

Z

e

Tde:

ð20Þ

eref

For the solid, fluid and ideal gas system (or those can be treated as ideal gas systems), according to the thermodynamic relations, there is [53]

de ¼ cv dT;

ð21Þ

where cv is the specific heat at constant volume. Then, the entransy per unit mass can be changed into

g ¼ g ref þ

Z

e

Tde ¼ g ref þ

eref

Z

T

T ref

cv TdT:

ð22Þ

Especially, if we assume Tref = 0, gref = 0 and cv is a constant, the expression of the entransy can be simplified into g ¼ 12 cv T 2 ; which is the traditional definition of entransy in Ref. [8]. When temperature changes from T1 to T2, the entransy change per unit mass for an object is

Dg ¼ g T 2  g T 1 ¼

Z

T2

cv TdT:

T1

ð23Þ

According to the definition of the differential enthalpy entransy, the enthalpy entransy per unit mass at a certain state can be calculated by

gH ¼

Z

gH

dg H ¼

Z

h

Tdh:

ð24Þ

0

0

Similarly, we can define a reference state at Tref, where the corresponding enthalpy is href and enthalpy entransy is gH-ref. Then, the enthalpy entransy per unit mass can be expressed by

g H ¼ g H-ref þ

Z

Fig. 1. The thermal equilibrium process between two objects.

Guo et al. [8] and Cheng et al. [54] have demonstrated that the total entransy of the two objects is decreasing gradually during the thermal equilibrium process. If the specific heats of the two objects are not constants but change with their states, the entransy change of the system during this process is calculated below. Based on the definition of entransy, the total entransy of the system at the initial state can be calculated by

Ginit ¼ m1 g 1-ref þ

Z

ð25Þ

Ginter ¼ m1 g 1-ref þ

Z

href

For the constant pressure process or the ideal gas system (or those can be treated as ideal gas systems), there is [53]

dh ¼ cp dT;

g H ¼ g H-ref þ

Z

h

href

Tdh ¼ g H-ref þ

Z

cv 2 TdT ; ð29Þ

!

T 2-m

cv 2 TdT :

ð30Þ

The entransy change of the system can be obtained as

Z

DG ¼ Ginter  Ginit ¼ m1

T 1-m

cv 1 TdT þ m2

Z

T 2-m

T2

cv 2 TdT:

ð31Þ

T

cp TdT:

Suppose T1 < T2, there is T1 < T1-m < T2-m < T2. So we have

ð27Þ

T ref

m1

T2

ð28Þ

T 1-m

cv 1 TdT < m1 T 1-m < m1 T 2-m

Z Z

T 1-m T1

m2

Z

T 2-m

T2

cv 2 TdT < m2 T 2-m

Z

cv 1 dT

T 1-m T1

T1

2.2.2. Entransy change during a thermal equilibrium process between two objects As shown in Fig. 1, there are two objects whose mass and temperature are m1, T1 and m2, T2 respectively at the initial state. After they contact with each other, heat is transferred from the high temperature object to the low temperature one, and finally both objects reach their equilibrium state at temperature Tm. The volumes of the two objects remain constant during the process. For this problem, if the specific heats of the two objects are constants,

Z

T1

which sion of enthalpy entransy can be simplified into g H ¼ is just the definition of enthalpy entransy in Ref. [50]. When temperature changes from T1 to T2, the change of enthalpy entransy per unit mass for an object is

cp TdT:

Z

T ref

1 c T 2; 2 p

Dg H ¼ g H-T 2  g H-T 1 ¼

T ref

cv 1 TdT

T ref

T1

If we assume Tref = 0, gH-ref = 0 and cp is a constant, the expres-

Z

!

T2

!

T 1-m

þ m2 g 2-ref þ

ð26Þ

where cp is the specific heat at constant pressure. Hence, the enthalpy entransy per unit mass can be changed into

þ m2 g 2-ref þ

cv 1 TdT

T ref

Z

where subscripts 1 and 2 represent the two objects respectively, cv1 and cv2 are the specific heats at constant volume, whose values are positive and related to their states. Assume that after arbitrary amount of heat is transferred, the temperatures of the two objects change to T1-m and T2-m respectively. The entransy of the system becomes

h

Tdh:

!

T1

cv 1 dT;

ð32Þ

cv 2 dT:

ð33Þ

T 2-m T2

Combination of Eqs. (31)–(33) gives

 Z DG < T 2-m m1

T 1-m

T1

cv 1 dT þ m2

Z

T 2-m T2

 cv 2 dT :

ð34Þ

On the other hand, according to the energy conservation of the two objects, there is

m1

Z

T 1-m

T1

cv 1 dT þ m2

Z

T 2-m

T2

Therefore, Eq. (34) gives

cv 2 dT ¼ 0:

ð35Þ

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DG < 0:

ð36Þ

The above analyses show that the irreversible heat transfer always leads to the decrease in entransy, no matter the properties of the objects change or not. Therefore, the total entransy of the system reaches its minimum value when the system reaches its equilibrium state. 3. Entransy optimization principles for thermal processes with variable thermophysical properties 3.1. The heat transfer processes For a steady heat convection system shown in Fig. 2, the fluid flows into the system through boundary SI and flows out of the system through boundary SO. During this process, what we concerned is the heat transfer rate Q_ from the fluid to the wall through SW. At t

steady state, the entransy balance equation shown in Eq. (16) can be simplified by neglecting the unsteady term,

Z

qug  ndS ¼

S

Z

Z Z T Q_ dV  kðrTÞ2 dV  qT  ndS V V S Z Z _ þ T U_ dV  TdWdV: V

ð37Þ

V

TL ¼

G_ dis ¼ Q_ t ðT H  T L Þ:

ð38Þ

For the fluid region, the total heat released to the wall can be expressed by

SI þSO

Q_ dV þ

V

Z

U_ dV 

V

Z

_ dWdV:

Z

q  ndS:

ð39Þ

ð40Þ

SW

According to the energy conservation of the whole region, there is

Q_ t ¼ Q_ H ¼ Q_ L :

ð41Þ

The equivalent temperatures for the fluid region and wall are defined respectively as

 Z TH ¼  SI þSO

qug  ndS þ

Z

T Q_ dV þ V

Z

T U_ dV 

Z

V

_ TdWdV

ð45Þ

Defining the equivalent temperatures of the system and applying the same method as those in Eqs. (39)–(43), the entransy dissipation rate of the system can also be expressed as G_ dis ¼ Q_ t ðT H  T L Þ; which is also the mathematical expression of the extremum entransy dissipation principle. A thermal resistance can be defined based on entransy dissipation as below [8]

V

For the wall, the total heat absorbed from the fluid region is

Q_ L ¼

Z

kðrTÞ2 dV Z Z Z Z ¼ T Q_ dV  qug H  ndS  qT  ndS þ TU_ dV V S S V Z _  TdW t dV:

G_ dis ¼

V

V

Z

ð44Þ

V

kðrTÞ2 dV Z Z Z Z ¼  qug  ndS þ T Q_ dV  qT  ndS þ TU_ dV V S V ZS _  TdWdV:

que  ndS þ

ð43Þ

The above equation indicates that the maximum entransy dissipation rate corresponds to maximum heat transfer flow rate when the equivalent heat transfer temperature difference of the system is prescribed, and the minimum entransy dissipation rate corresponds to the minimum equivalent heat transfer temperature difference when the heat transfer flow rate is prescribed. This is the extremum entransy dissipation principle for convective heat transfer and it is the same as the case discussed in Ref. [8] when the value of work is negligible compared with heat exchange. The extremum entransy dissipation principle is also applicable for the processes with variable thermophysical properties. If we use the enthalpy entransy balance equation to describe the system, the entransy dissipation of the above system can be calculated by

V

Z

 Q_ t ;

which are the heat flux weighted average temperatures. Then the entransy dissipation of the system can be expressed by

Z

Q_ H ¼ 

qT  ndS SW

Then, the entransy dissipation of the system can be obtained by

G_ dis ¼

Z

 Q_ t ;

R¼

T H  T L G_ dis ¼ : Q_ t Q_ 2t

ð46Þ

It can be seen that the extremum entransy dissipation principle can be simplified into the minimum entransy-dissipation-based thermal resistance principle which indicates that the minimum thermal resistance corresponds to the optimal heat transfer. The extremum entransy dissipation principle and minimum entransy-dissipation-based thermal resistance above are derived for the convective heat transfer with variable thermophysical properties. They also can be used for heat conduction by neglecting the terms related to convection and viscous dissipation in the related equations.

V

ð42Þ

3.2. The heat-work conversion processes The entransy balance analyses for a typical steady heat-work conversion process is conducted below. The system is shown in Fig. 3. It is composed of a working cycle and the heat sources, where the heat sources are fluid streams (hot streams and cold streams) that exchange heat with the working medium through heat exchangers. The working medium absorbs heat flow rate Q_ 1 from the hot streams and releases heat flow rate Q_ 2 to the cold _ The used streams are streams, and the cycle output a power W. finally dumped into the environment with temperature T0 and the corresponding total heat flow rate released to the environment is Q_ 0 .

Fig. 2. A steady heat convection system.

Assume that the working medium is in a closed system or can be equivalent to a closed system and the specific heats of the fluid

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Then, the entransy balance equation of the whole system can be obtained by adding Eqs. (47)–(49) together,

Z

T 0 Q_ 0 þ

qug H  ndS ¼  SF

Z

kðrTÞ2 dV 

Z

VF



kðrTÞ2 dV

VM

Z

_ t dV: TdW

ð51Þ

VM

Based on the definition of the entransy loss (net system entransy input) [42], the entransy loss of the present system is

G_ loss ¼ G_ dis þ G_ w ¼ 

Z SF

Fig. 3. A typical heat-work conversion process.

streams are functions of temperature. By neglecting the internal heat source, viscous dissipation and time-related terms, the entransy balance equation for the working medium region can be simplified into



Z

qT  ndS ¼

Z

SM

2

kðrTÞ dV þ

Z

VM

_ TdWdV;

where VM represents the working medium region and SM represents the boundary surface of the working medium region where heat is exchanged between the heat sources and the working medium. Eq. (47) indicates that part of the entransy flow into the working medium is dissipated during heat transfer, and the rest is converted to expansion work entransy rate. The pressure losses in the heat exchangers are usually small, the heat transfer in the heat exchangers can be treated as the constant pressure processes and can be described by the enthalpy entransy balance equation. By neglecting the internal heat source, viscous dissipation and the time-related terms, the enthalpy entransy balance equation for the heat source regions (hot stream region and cold stream region) can be simplified into



Z SF

qug H  ndS ¼

Z

kðrTÞ2 dV þ

VF

Z

qT  ndS þ

SF1

Z

qT  ndS; ð48Þ

where VF represents the heat source regions, SF represents the fluid inlet and outlet areas of the heat source regions, SF1 represents the boundary surfaces of the heat source regions where heat is exchanged between the heat sources and the working medium and it corresponds to SM for the working medium, and SF2 represents the boundary surfaces of the heat source regions where heat is transferred from the used streams to the environment. Eq. (48) indicates that part of the entransy flow released by the heat sources is dissipated due to heat transfer, another part is transferred to the working medium, and the rest is transferred to the environment. For the environment, the entransy flow rate into the environR ment is  SE qT  ndS; where SE represents the boundary surface of the environment through which heat is transferred to the environment from the used streams and it corresponds to SF2 for the heat source regions. As the temperature of the environment is always T0, there is



Z

qT  ndS ¼ T 0

SE

Z

NG ¼

q  ndS ¼ T 0 Q_ 0 :

R G_ loss  SF qug H  ndS  Q_ 0 T 0 R ¼ :  SF qug H  ndS G_ in

_ ¼ W

Z

quh  ndS  Q_ 0 :

ð54Þ

SF

And the heat-work conversion efficiency of the system can be calculated by

g¼



R

quh  ndS  Q_ 0 :  SF quh  ndS SF

R

ð55Þ

Substituting Eq. (54) into Eq. (52) results in

G_ loss ¼ G_ dis þ G_ w Z Z _ 0: ¼ qug H  ndS  T 0 quh  ndS þ WT

SM

qT  ndS þ

Similarly, substituting Eq. (55) into Eq. (53) results in

R NG ¼ 1 þ ðg  1ÞT 0 R

quh  ndS : qug H  ndS SF SF

SF1

qT  ndS þ

Z SF2

qT  ndS þ

ð49Þ

Z SE

qT  ndS ¼ 0:

ð57Þ

It can be seen that the maximum entransy loss rate always corresponds to the maximum output power and the maximum entransy loss coefficient always corresponds to the maximum heat-work conversion efficiency of the system when the inlet conditions of the heat sources and the environmental temperature are prescribed, which means that the maximum entransy loss principle proposed in Ref. [42] and the maximum entransy loss coefficient principle proposed in Ref. [49] can also be used for the analyses of heat-work conversion processes with variable thermophysical properties.

SE

Z

ð56Þ

SF

The continuity of the heat flux and temperature on the heat transfer boundaries leads to

Z

ð53Þ

According to the energy conservation, the output power of the system can be calculated by

SF

SF2

ð52Þ

where G_ dis and G_ w are the entransy dissipation rate and work entransy rate of the system respectively. The entransy loss coefficient is defined as the ratio of the entransy loss to the total entransy flow into the system [49]. The entransy loss coefficient of the present system is

ð47Þ

VM

qug H  ndS  Q_ 0 T 0 :

ð50Þ Fig. 4. A two-stream heat exchanger.

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4. Application to the analyses of a heat exchanger

cp-a ðTÞ ¼ 103

For a two-stream heat exchanger shown in Fig. 4, the temperature of the hot stream at inlet is TH-in and reduces to TH-out at the outlet, while the temperature of the cold stream at inlet is TL-in and increases to TL-out at the outlet. Assume that the mass flow _ H and m _L rates of the hot and cold streams are prescribed at m respectively. The specific heats at constant pressure of the streams are functions of temperature. Suppose that there are no heat sources and the flow velocities of the streams in the heat exchanger are low, the flow can be treated as a constant pressure process and the viscous dissipation and technical power output are negligible. From the enthalpy entransy balance equation, the entransy dissipation rate of the whole heat exchanger can be calculated by

G_ dis ¼

Z

kðrTÞ2 dV ¼ 

V

_H ¼m

Z S

Z

qug H  ndS

T H-in

_L cp-H TdT þ m

Z

T H-out

T L-in

cp-L TdT;

ð58Þ

T L-out

where cp-H and cp-L are the specific heats at constant pressure for the hot and cold streams respectively. The heat exchange rate of the heat exchanger is

_H Q_ ex ¼ m

Z

T H-in

_L cp-H dT ¼ m

Z

T H-out

T L-out

cp-L dT:

ð59Þ

T L-in

The entransy-dissipation-based thermal resistance of the heat exchanger can be calculated by

R¼

G_ dis : Q_ 2

 i1 n X T iai ; 1000 i¼0

ð67Þ

where ai is the ith coefficient of the polynomial function and can be found in Ref. [56]. The specific heat of the kerosene at constant pressure is also a function of temperature [57],

cp-k ðTÞ ¼ 488:7 þ 5:0855T:

ð68Þ

The variations of the heat exchange rate, the entropy generation rate, the entransy-dissipation-based thermal resistance, the entropy generation number and the revised entropy generation number with heat exchanger effectiveness are calculated based on the above parameters and are shown in Fig. 5. It can be seen that the heat exchange rate increases linearly with increasing heat exchanger effectiveness, while the entransy-dissipation-based thermal resistance decreases monotonously, which means that the minimum entransy-dissipation-based thermal resistance principle can be used for the optimization of the heat exchanger under the present condition. The entropy generation rate and the entropy generation number first increase and then decrease with increasing heat exchanger effectiveness, while the revised entropy generation number decreases monotonously. Therefore, the minimum revised entropy generation number can be used as the objective of the heat exchanger optimization under the present condition, while the minimum entropy generation rate and the minimum entropy generation number cannot. The results above are the same as those obtained in previous work by Cheng and Liang [58], where the heat exchanger with constant thermophysical properties was discussed.

ð60Þ

ex

The entropy generation rate, entropy generation number [5] and revised entropy generation number [55] that are usually used for the evaluation of heat exchangers can also be calculated by

_H S_ gen ¼ m

Z

T H-out

T H-in

cp-H _L dT þ m T

Z

T L-out

T L-in

cp-L dT; T

ð61Þ

NS ¼ S_ gen =C min ;

ð62Þ

NRS ¼ S_ gen T 0 =Q_ ex ;

ð63Þ

where Cmin is the minimum heat capacity flow rate. Considering that the specific heats of streams are not constant, we define the minimum heat capacity flow rate as

C min ¼ min

( R _ H TT H-in cp-H dT m L-in

T H-in  T L-in

;

_L m

R T H-in T L-in

cp-L dT

T H-in  T L-in

5. Application to the analyses of a heat-work conversion process Shown in Fig. 6 is a closed Brayton cycle that is used for the recovery of the waste heat from a hot stream with inlet temperature TH-in. The working medium absorbs heat from the hot stream through the counter flow heat exchanger 1 (HE1), and after expansion in the expander, the working medium releases heat to the cold stream with inlet temperature TL-in through the counter flow heat exchanger 2 (HE2). Both the used hot and cold streams are finally dumped into the environment with temperature T0. The heat transfer processes in the heat exchangers can be treated as constant pressure processes, and the specific heats of the streams are functions of temperature.

) :

ð64Þ

The maximum possible heat exchange rate of the heat exchanger can be shown as

Q_ pmax ¼ C min ðT H-in  T L-in Þ:

ð65Þ

According to the definition of the effectiveness of heat exchanger, e, there is

Q_ ex ¼ eQ_ pmax ¼ eC min ðT H-in  T L-in Þ:

ð66Þ

An example is shown below. For a heat exchanger used in the aero-engine for the cooling of the cooling air by kerosene, the hot stream is the compressed air with inlet temperature TH-in = 900 K, and the cold stream is the kerosene with inlet temperature TL-in = 300 K. The mass flow rates of the compressed air _ H ¼ 10 kg/s and m _ L ¼ 2 kg/s respectively. The and kerosene are m specific heat at constant pressure of the air is a function of temperature [56],

Fig. 5. Variations of the heat exchange rate, the entropy generation rate, the entransy-dissipation-based thermal resistance, the entropy generation number and the revised entropy generation number with the heat exchanger effectiveness.

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According to Eq. (52), the entransy loss rate of the system can be expressed as

G_ loss ¼ 

Z

qug H  ndS  Q_ 0 T 0

SF

Z

_H ¼m

Z

T H-in

_L cp-H TdT þ m

T0

T L-in

cp-L TdT  Q_ 0 T 0 :

ð73Þ

T0

Therefore, combining with Eq. (70), Eq. (73) can be changed into

Z

_H G_ loss ¼ m

Z

T H-in

_L cp-H TdT þ m

T0

 Z _H  T0 m

T L-in

cp-L TdT

T0 T H-in

_L cp-H dT þ m

T0

Z

T L-in

cp-L dT



_ 0: þ WT

ð74Þ

T0

According to Eq. (53), the entransy loss coefficient of the system is

Fig. 6. A closed Brayton cycle system for waste heat recovery.

The T-s diagram of the Brayton cycle described above is shown in Fig. 7. For the present system, the structure parameters, the inlet conditions of the fluid streams and the mass flow rate of the working medium are prescribed, the operation temperatures of the cycle is to be optimized to obtain the maximum output power and the maximum heat-work conversion efficiency. For the four operation temperatures (T1, T2, T3 and T4) in the Brayton cycle, the following relationship should be satisfied [53]

R R _ H TT H-in cp-H TdT þ m _ L TT L-in cp-L TdT  Q_ 0 T 0 G_ loss m 0 0 NG ¼ ¼ : R R _ H TT H-in cp-H TdT þ m _ L TT L-in cp-L TdT G_ in m 0 0 Combing with Eq. (72), Eq. (75) can be changed into

NG ¼

G_ loss G_ in

¼ 1 þ ðg  1ÞT 0

R R _ H TT H-in cp-H dT þ m _ L TT L-in cp-L dT m 0 0 : R T H-in R _ L TT L-in cp-L TdT _ H T cp-H TdT þ m m 0

T 2 =T 1 ¼ T 3 =T 4 :

ð69Þ

On the other hand, the heat transfer relationships of the HE1 and HE2 should also be satisfied. Therefore, only one of the four operation temperatures is to be optimized since the other three can be calculated from the optimized one. The temperature T1 is selected for optimization in the analyses below. According to the working condition above, the output power of the system can be expressed by

_ ¼m _H W

Z

T H-in

_L cp-H dT þ m

Z

T0

T L-in

cp-L dT  Q_ 0 ;

ð70Þ

where cp-H and cp-L are the specific heats at constant pressure for the hot and cold streams respectively and Q_ 0 is the heat released by the

_H S_ gen ¼ m

Z

T H-out

_L cp-H dT þ m

T0

Z

g¼

R T H-in T0

cp-L dT:

_H m

R T H-in T0

R T L-in T0

_L cp-H dT þ m

cp-L dT  Q_ 0

R T L-in T0

cp-L dT

ð71Þ

:

cp-H _L dT þ m T

Z

T0 T L-in

Z

T0

cp-H _L dT þ m T

Z

cp-L Q_ 0 dT þ : T T0

ð77Þ

T0

T0

ð78Þ

T0

The entropy generation number of the system can be calculated

T0

_L cp-H dT þ m

T0

cp-L dT T H-in T L-in T   Z T H-in Z T L-in _ 0: _H _L T 0  W=T þ m cp-H dT þ m cp-L dT

_H S_ gen ¼ m

T L-out

by

The heat-work conversion efficiency of the system is

_H m

Z

Combing with Eq. (70), Eq. (77) can be changed into

streams to the environment, which can be calculated by

_H Q_ 0 ¼ m

ð76Þ

0

It can be seen that the maximum entransy loss rate and the maximum entransy loss coefficient always correspond to the maximum output power and the maximum heat-work conversion efficiency of the system respectively when the inlet conditions of the streams and the environmental temperature are prescribed. The entropy generation rate of the system can be calculated by

T H-in

T0

ð75Þ

ð72Þ

NS ¼ ¼

S_ gen C min _H m

Z

T0

T H-in

cp-H _L dT þ m T

Z

T0

T L-in

cp-L Q_ 0 dT þ T T0

!, C min ;

ð79Þ

_ H cp-H ; m _ L cp-L g is minimum heat capacity flow where C min ¼ minfm rate of the heat streams at their inlets. Combing with Eq. (72), Eq. (79) can be changed into

NS ¼

Fig. 7. T-s diagram of the closed Brayton cycle.

Z T0 _L cp-H cp-L m dT þ dT þ ð1  gÞ T C min T L-in T T H-in R R _ H TT H-in cp-H dT þ m _ L TT L-in cp-L dT m 0 0  : T 0 C min

_H m C min

Z

T0

ð80Þ

It can be seen that the minimum entropy generation rate and the minimum entropy generation number always correspond to the maximum output power and the maximum heat-work conversion efficiency of the system respectively when the inlet conditions of the streams and the environmental temperature are prescribed.

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A numerical example is given below. Assume that the hot stream is the waste gas of the gas turbine engine with inlet tem_ H ¼ 20 kg/s. The cold perature TH-in = 700 K, mass flow rate m stream is air with inlet temperature TL-in = 300 K, mass flow rate _ L ¼ 20 kg/s. The working medium is helium with mass flow rate m _ M ¼ 4 kg/s. The environment temperature is T0 = 300 K. For HE1, m the heat transfer coefficient is kH = 500 W/(m2 K), and the heat transfer area is AH = 100 m2. For HE2, the heat transfer coefficient is kL = 500 W/(m2 K), and the heat transfer area is AL = 100 m2. The specific heat at constant pressure for the waste gas is a function of temperature and the fuel/air ratio of the gas turbine engine. Assume that the fuel of the gas turbine engine is kerosene, the function can be shown as [56]

cp-g ðT; f Þ ¼ cp-a þ

f hc ; 1þf p

ð81Þ

where cp-a is specific heat of air at constant pressure, f is the fuel/Air ratio whose value is 0.5 in the present work, hcp is also a function of temperature,

hcp ¼ 103

n X i¼0

 ibi

T 1000

Fig. 9. Variations of the heat-work conversion efficiency, the entransy loss coefficient and the entropy generation number with the temperature T1.

i1 ;

ð82Þ

where bi is the ith coefficient of the polynomial function and can be found in Ref. [56]. The specific heat of helium is a constant. The heat transfer of the HE1 and HE2 are calculated by the discretization method. The variations of the output power, the entransy loss rate and the entropy generation rate with the temperature T1 are calculated under the above condition. It can be seen from Fig. 8 that both the output power and the entransy loss rate of the system increase first and then decrease with increasing T1, and reach their maximum values simultaneously when T1 = 338 K. At the same time, the entropy generation rate decreases first and then increases with increasing T1 and reaches its minimum value when T1 = 338 K. Therefore, both the maximum entransy loss principle and the entropy generation minimization can hold for the optimization of the operation temperature to get the maximum output power in this case. Under the same condition, the variations of the heat-work conversion efficiency, the entransy loss coefficient and the entropy generation with the temperature T1 are also calculated and shown in Fig. 9. It can be seen that both the heat-work conversion efficiency and the entransy loss coefficient reach their maximum values at T1 = 338 K, while the entropy generation number reaches its minimum values at T1 = 338 K. Therefore, both the maximum entransy loss coefficient and the minimum entropy

Fig. 10. Variations of the output power, the entransy loss rate and the entropy generation rate with the temperature TH-in.

Fig. 11. Variations of the heat-work conversion efficiency, the entransy loss coefficient and the entropy generation number with the temperature TH-in.

Fig. 8. Variations of the output power, the entransy loss rate and the entropy generation rate with the temperature T1.

generation number correspond to the maximum heat-work conversion efficiency of the system under this condition, which is consistent with the theoretical analyses above.

B. Zhou et al. / International Journal of Heat and Mass Transfer 90 (2015) 1244–1254

Furthermore, if the working temperature T1 is prescribed at 338 K and the inlet temperature of the hot stream TH-in increases, the variations of the output power, the entransy loss rate and the entropy generation rate are shown in Fig. 10. It can be seen that the output power, the entransy loss rate and the entropy generation rate all increase monotonously with increasing TH-in, which means that larger entransy loss rate still corresponds to larger output power under this condition. The variations of the heat-work conversion efficiency, the entransy loss coefficient and the entropy generation number are shown in Fig. 11. It can be seen that the all the three parameters also increase monotonously with increasing TH-in, which indicates that larger entransy loss coefficient still corresponds to larger heat-work conversion efficiency under this condition.

6. Conclusions The entransy concept is extended and applied to the analyses of thermal processes with variable thermophysical properties in this paper. Based on the definitions of differential entransy and enthalpy entransy, the entransy and enthalpy entransy balance equations for thermal processes with variable thermophysical properties are established. The analyses show that the entransy and enthalpy entransy balance equations are consistent with each other in essence and both can be used for the analyses of heat transfer and heat-work conversion processes. The entransy change during a thermal equilibrium process between two objects with variable thermophysical properties is calculated, and the result shows that the irreversible heat transfer always leads to the decrease of entransy. The optimization principles for heat transfer and heat-work conversion processes with variable thermophysical properties are investigated. For heat transfer processes with variable thermophysical properties, the extremum entransy dissipation principle and the minimum entransy-dissipation-based thermal resistance principle are proved to be tenable either. A two-stream heat exchanger with variable thermophysical properties is analyzed with the entransy theory, and the entropy generation analyses are also conducted for comparison. The results show that both the minimum entransy-dissipation-based thermal resistance and the minimum revised entropy generation number correspond to the maximum heat exchanger effectiveness when the inlet condition of the heat exchanger is prescribed, while the minimum entropy generation rate and the entropy generation number do not. For heat-work conversion processes using streams with variable thermophysical properties as heat source and sink, the maximum entransy loss principle is discussed. A closed Brayton cycle is analyzed with the entransy theory and the entropy generation minimization method. The results demonstrate that both the maximum entransy loss rate and the entropy generation rate correspond to the maximum output power of the cycle when the inlet conditions of the streams and the environmental temperature are prescribed, and both the maximum entransy loss coefficient and the minimum entropy generation number correspond to the maximum heat-work conversion efficiency of the cycle under the same condition. When the inlet temperature of the hot stream increases, larger entransy loss rate still correspond to larger output power and larger entransy loss coefficient still correspond to larger heat-work conversion efficiency.

Conflict of interest None declared.

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Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant No. 51376101) and Science Fund for Creative Research Groups of China (No. 51321002).

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