Energy Conversion and Management 58 (2012) 163–170
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Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman
Computation of effectiveness of two-stream heat exchanger networks
based on concepts of entropy generation, entransy dissipation
and entransy-dissipation-based thermal resistance
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 6 December 2011
Accepted 28 January 2012
Available online 22 February 2012
Keywords:
Two-stream heat exchanger networks
Effectiveness
Entransy dissipation
Entropy generation
Thermal resistance
a b s t r a c t
The two-stream heat exchanger networks (THENs) are widely used in industry. The effectiveness of the
THENs is analyzed in this paper. The general expressions for the entransy dissipation, the entransy-dissipation-based thermal resistance and the entropy generation for a generalized THEN are developed. It is
found that the expressions are independent of the specific constitution of the THENs. Only the entransydissipation-based thermal resistance always decreases monotonously with the increase in effectiveness,
while the entransy dissipation and the entropy generation do not. Therefore, the entransy-dissipationbased thermal resistance is most applicable for the optimization of the THENs.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
The worldwide appeals for energy conservation draw great
attention on the effective energy utilization and exchange.
Improvement on heat transfer performance is of great significance
to the reduction of the energy consumption as nearly 80% of the
total energy consumption is related to heat transfer [1]. The
two-stream heat exchanger network (THEN) is one of typical
equipments that has wide applications in thermal engineering,
such as room heating and air conditioning [2], ground-coupled
heat pump [3], and loop units in power plants [4]. Therefore, optimization of heat transport in the THENs is an important topic.
For the optimization of heat transport, many theories have been
developed during the recent years, such as the thermodynamic
optimization [5] and the entransy theory [6]. In the optimization
of the distribution of the limited high conducting materials in
the Volume-to-Point conduction problem, Bejan [7] developed
the constructal theory, in which the basic, imagined structure of
the high conductivity material is given manually, and then the aspect ratio of the structure is optimized through theoretical derivation and numerical simulation to decrease the highest temperature
of the heated domain. The constructal theory has been extended to
optimize heat convection [8–11]. Furthermore, Bejan [7,12] related
the optimized heat transfer to the minimum production of entropy
generation, and extended this idea into other heat transfer sys⇑ Corresponding author. Fax: +86 10 62788702.
E-mail address: [email protected] (X.G. Liang).
0196-8904/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.enconman.2012.01.016
tems, such as the optimization of the parameters in heat exchangers. A lot of research work has been reported on heat transfer
optimization based on the minimum entropy generation principle
[13–16]. As the objective of this kind of optimization is to minimize the entropy generation, it is called thermodynamic optimization. However, it is observed in some cases that the construct
beyond the first order does not contribute to the performance
improvement [17–20]. In the second law analysis of different complexity levels of the tree network of conducting paths, Ghodoossi
[17,18] found that the heat flow performance does not essentially
improve if the internal complexity of the heat generating area increases. It is reported that the hierarchical constructal designs from
simple towards more complex tree-shaped internal structures does
not improve the heat flow performance in general, it improves the
performance only up to the first order construct. In heat convection
with prescribed flow rate, Escher et al. [20] found that the parallel
channel network provides more than fivefold increase in performance coefficient as compared to the bifurcating tree-like
network, with fourfold increase in heat removal rate at a specific
pressure gradient across the network.
An entropy generation paradox has been observed when the
minimum entropy generation principle was used to analyze heat
exchangers [12]. The heat exchanger effectiveness does not always
increase with decrease in entropy generation number. Shah and
Skiepko [21] have analyzed the relationship between the effectiveness and entropy generation for heat exchangers and found that
the heat exchanger effectiveness can be maximum, intermediate
or minimum at the maximum entropy generation. Therefore, the
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X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (2012) 163–170
Nomenclature
A
C
C
c
Gdis
Ns
n
Q
q
R
Sg
T
DT
area (m2)
heat capacity flow rate (W/K)
ratio between the smaller and bigger heat capacity flow
rate of the streams
specific capacity (J/(kg K))
entransy dissipation (W K)
revised entropy generation number
normal vector
heat transfer rate (W)
heat flux (W/m2)
thermal resistance (K/W)
entropy generation (W/K)
temperature (K)
temperature difference (K)
minimum entropy generation principle is not always applicable to
heat exchanger optimization.
During the last decade, the concept of entransy has been developed based on the analogy between heat conduction and electrical
conduction, in which heat flux corresponds to electrical current,
thermal resistance to electrical resistance, temperature to electric
voltage, and heat capacity to capacitance [6]. Therefore, entransy
is actually the potential energy of the heat in a body, corresponding
to the electrical energy in a capacitor [6]. Based on this concept,
Guo et al. [6], Han et al. [22] and Cheng et al. [23,24] proved that
the entransy would always decrease during the thermal equilibrium processes. Furthermore, Cheng et al. [24] found that the
entransy of an isolated system that does not involve any work
always decreases during heat transfer. The decrease of entransy
is called entransy dissipation which could be used to describe
the irreversibility of heat transfer [6,22–25]. It is found that the
maximum entransy dissipation corresponds to the maximum heat
transfer rate when the heat transfer temperature difference is prescribed, while the minimum entransy dissipation corresponds to
the minimum heat transfer temperature difference when the heat
transfer rate is prescribed. Furthermore, Guo et al. [6] defined an
equivalent thermal resistance based on the entransy dissipation
and heat flux. Then, the extreme entransy dissipation principle is
equivalent to the minimum thermal resistance principle [6].
These principles were used in heat conduction [6,26,27], heat convection [6,25,28], thermal radiation [29,30] and the design of heat
exchangers [1,31–33].
The entransy theory has been found to be more applicable to
the optimization of heat transfer processes [1,6,29,32] than the
thermodynamic optimization. For instance, the Volume-to-Point
problem in heat conduction has been analyzed based on the
entransy theory and the minimum entropy generation principle
[6]. It has been found that the distribution of the high conductivity
material optimized by the entransy theory would lead to lower
average temperature than that optimized by the minimum entropy
generation. For heat exchangers optimization, it has been shown
that the entransy-dissipation-based thermal resistance decreases
monotonously with the increase in effectiveness [1,32,33].
For some typical THENs, the entransy theory has also been applied to analyze the heat transfer processes. Chen et al. [34] analyzed a THEN connected by a hydronic fluid based on the minimum
entransy-dissipation-based thermal resistance, and found that
minimizing the thermal resistance results in the maximum heat
transfer rate in the THEN. Qian et al. [35] analyzed three constitutions
of THENs and found that the maximum heat transfer rate between
two streams corresponds to the minimum entransy-dissipation-
u
V
velocity (m/s)
volume (m3)
Greek symbols
effectiveness
k
thermal conductivity (W/m/K)
q
density (kg/m3)
e
Subscripts
c
cold stream
h
hot stream
i
inlet of streams
max
the maximum value
min
the minimum value
o
outlet of streams
based thermal resistance. In this paper, we apply the entransy theory
to investigate generalized THENs and makes comparisons between
entransy dissipation, the entransy-dissipation-based thermal resistance and the entropy generation.
2. Analysis of THENs
A THEN may consist of heat exchangers, distributors, mixers,
and medial fluids. The three THENs analyzed by Qian et al. [35] is
shown in Figs. 1–3, where the heat capacity flow rates of the hot
stream and the cold stream are Ch and Cc, the inlet and outlet temperatures of the hot stream are Thi and Tho, while those of the cold
stream are Tci and Tco, respectively. In Fig. 1, a medial fluid works
between the hot and cold streams. It absorbs heat from the hot
stream with inlet temperature Tm1 through heat exchanger 1, and
releases heat to the cold stream with inlet temperature Tm2 through
heat exchanger 2. In Fig. 2, there is no medial fluid. The hot and cold
streams are distributed into two heat exchangers. In Fig. 3, the
THEN has a medial fluid, two distributors and two mixers. The medial fluid absorbs heat from the hot stream through heat exchanger 3.
Then, the medial fluid is distributed into two heat exchangers to
release heat to the cold stream.
Based on the three THENs, a generalized THEN is proposed as
shown in Fig. 4, where there may be many heat exchangers, distributors, mixers, and medial fluids. A scheme like this is analyzed
in this paper. During a heat transfer process, the temperature of the
hot stream decreases from Thi to Tho, while that of the cold
stream increases from Tci to Tco.
It is assumed that there is no heat exchange between the THEN
and the environment, the fluids are incompressible, and the influence of viscous dissipation on heat exchange, entropy generation
and entransy could be ignored. With these assumptions, the effectiveness of the generalized THEN could be defined, and the
Fig. 1. THEN with a medial fluid [35].
X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (2012) 163–170
165
where
C min ¼ minðC h ; C c Þ:
ð4Þ
Similar to the definition of effectiveness of a heat exchanger
[36], the effectiveness of the THEN could be defined as
e¼
Q
C h ðT hi T ho Þ
C c ðT co T ci Þ
¼
¼
:
Q max C min ðT hi T ci Þ C min ðT hi T ci Þ
ð5Þ
It is evident that higher effectiveness means higher heat transfer rate at prescribed inlet temperatures of the two streams or
smaller difference between the inlet temperatures of the two
streams at a prescribed heat transfer rate. Therefore, the effectiveness defined in Eq. (5) is a measure of the performance of the THEN
of Fig. 4.
For the generalized THEN, the expressions of the entransy dissipation, the entransy-dissipation-based thermal resistance, and the
entropy generation are
Fig. 2. THEN with mixers [35].
1 1 2
C h T hi T 2ho þ C c T 2ci T 2co :
2
2
1 1 2
C h T hi T 2ho þ C c T 2ci T 2co =Q 2 :
R ¼ Gdis =Q 2 ¼
2
2
Gdis ¼
Sg ¼ C h ln
T ho
T co
þ C c ln
;
T hi
T ci
ð6Þ
ð7Þ
ð8Þ
respectively. It could be found that the expressions are the same as
those reported in Ref. [34]. The detailed derivation of the expressions is in the Appendix A at the end of this paper.
Combinations of Eqs. (1), (6)–(8) yield
i
1 h 2
C h T hi ðT hi Q =C h Þ2
2
i
1 h
þ C c T 2ci ðT ci þ Q =C c Þ2
2
1
¼ ðT hi T ci ÞQ Q 2 ð1=C h þ 1=C c Þ:
2
Gdis ¼
Fig. 3. THEN with a medial fluid and mixers [35].
1
R ¼ Gdis =Q 2 ¼ ðT hi T ci Þ=Q ð1=C h þ 1=C c Þ:
2
Q
Q
þ C c ln 1 þ
:
Sg ¼ C h ln 1 C h T hi
C c T ci
ð10Þ
ð11Þ
For the THEN in Fig. 4, there are two kinds of optimization problems: the inlet temperatures of the two streams are prescribed or
the heat transfer rate is prescribed. For both kinds of problems,
the objective is to maximize the effectiveness of the exchanger.
When the inlet temperatures of the two streams are prescribed,
from Eq. (5),
Fig. 4. Sketch of a generalized THEN.
expressions of the entransy dissipation, the entransy-dissipationbased thermal resistance, and the entropy generation can be derived. In the case that there is no heat source in the THEN and no
heat exchange between the THEN and the environment, the energy
conservation gives
Q ¼ C h ðT hi T ho Þ ¼ C c ðT co T ci Þ;
ð9Þ
ð1Þ
Q ¼ eQ max ¼ eC min ðT hi T ci Þ:
ð12Þ
Substituting Eq. (12) into Eq. (9) gives
1
Gdis ¼ eC min ðT hi T ci Þ2 ½eC min ðT hi T ci Þ2 ð1=C h þ 1=C c Þ
2
1
ð13Þ
¼ C min ðT hi T ci Þ2 e e2 ð1 þ C Þ ;
2
where Q is the heat transfer rate between the hot and cold streams
in the THEN. The temperature of the hot stream would never be
lower than Tci, while that of the cold stream would never be higher
than Thi. The maximum temperature change of the two streams is
where
jDT max j ¼ T hi T ci :
It can be seen that the entransy dissipation is a quadratic function of the effectiveness and reaches its maximum value when
ð2Þ
Therefore, the maximum possible heat transfer rate is
Q max ¼ C min jDT max j ¼ C min ðT hi T ci Þ;
ð3Þ
C ¼
e¼
minðC h ; C c Þ
:
maxðC h ; C c Þ
1
< 1:
1 þ C
ð14Þ
ð15Þ
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X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (2012) 163–170
As the value range of the effectiveness is [0, 1], the maximum
entransy dissipation does not correspond to the maximum effectiveness of the THEN.
The entropy generation is given by
C min ðT hi T ci Þ
C ðT T Þ
Sg ¼ C h ln 1 e þ C c ln 1 þ min hi ci e :
ð16Þ
C h T hi
C c T ci
Letting the derivative of Eq. (16) equal zero, we get
@Sg
C min ðT hi T ci Þ
C min ðT hi T ci Þ
¼ C h
þ Cc
¼ 0:
C h T hi C min ðT Hi T ci Þe
C c T ci þ C min ðT hi T ci Þe
@e
ð17Þ
Therefore,
e¼
Cc Ch
minðC h ; C c Þ maxðC h ; C c Þ
1
< 1:
¼
¼
C min ðC c þ C h Þ C min ½minðC h ; C c Þ þ maxðC h ; C c Þ 1 þ C ð18Þ
It could be proved that
2
@ Sg
< 0:
@ e2
ð19Þ
The entropy generation reaches its maximum value when Eq.
(18) is satisfied. However, keeping in mind that the value range
of the effectiveness is [0, 1], the entropy generation would not always decrease with the increase of the effectiveness. The entropy
generation paradox thus exists in the THEN as well.
For the entransy-dissipation-based thermal resistance, we have
R¼
1
1 1
1
1 1 1
¼
þ
ð1 þ C Þ :
C min e 2
eC min 2 C h C c
ð20Þ
This relationship is the same as that of the two-stream heat
exchangers reported in Ref. [32]. The thermal resistance would decrease monotonously with the increase in effectiveness and hence
it can describe the performance of the THEN.
Let us consider a numerical example. Let Ch = 2 W/K, Cc = 3 W/K,
Thi = 350 K, and Tci = 300 K. The variations of the entransy dissipation, the thermal resistance, and the entropy generation with
the effectiveness are calculated from Eqs. (13), (16), and (20),
respectively and the results are shown in Fig. 5. It is seen that
the entransy dissipation and the entropy generation both get to
their maximum value when the effectiveness is 0.6. For the entransy dissipation, the extreme entransy dissipation principle tells
us that the maximum entransy dissipation corresponds to the
maximum heat transfer rate with the prescribed equivalent heat
transfer temperature difference, while the minimum entransy dissipation corresponds to the minimum heat transfer temperature
difference when the heat transfer rate is prescribed [6]. In the
present case when the inlet temperatures of the streams are given,
neither the equivalent heat transfer temperature difference nor the
heat transfer rate is prescribed and consequently, the extreme entransy dissipation does not correspond to the best performance of
the THEN.
For the entropy generation, when the effectiveness is smaller
than 0.6, the entropy generation does not decrease, but increase
with the increase in effectiveness. This is a paradox of entropy generation in the THEN. Entropy generation is not directly related to
heat transfer rate but is related to the loss of the ability of doing
work during an irreversible process. Less entropy generation
means less loss of the ability of doing work. However, in THENs,
what we are concerned of is not the loss of the ability of doing
work, but the effectiveness or the heat transfer rate. Therefore,
the minimum entropy generation principle is not always applicable to the design optimization of heat exchangers.
For the thermal resistance, it decreases monotonously with the
increase in effectiveness. Therefore, it is the appropriate parameter
that could be used to optimize the performance of the THEN.
Fig. 6 demonstrates the variation of the thermal resistance with
the effectiveness at different values of Cc. The minimum thermal
resistance always corresponds to the maximum effectiveness,
and there is no paradox similar to the entropy generation. Thus,
the minimum principle of entransy-dissipation-based thermal
resistance is a more general rule in heat transfer optimization than
the extreme entransy dissipation principle and the minimum entropy generation principle.
When the heat transfer rate is prescribed, Eq. (12) tells us that
higher effectiveness means smaller difference between the inlet
temperatures of the two streams. Substituting Eq. (12) into Eq.
(9), we get
Gdis ¼
Q2 1 1
ð1 þ C Þ :
C min e 2
ð21Þ
The entransy dissipation decreases monotonously with the increase in effectiveness. The entransy dissipation extreme principle
holds good in this case. The entropy generation is
Sg ¼ C h ln 1 Q
C h T hi
þ C c ln 1 þ
Q
:
½T hi Q=ðeC min ÞC c
ð22Þ
for prescribed Thi, or is
Sg ¼ C h ln 1 Q
C h ½Q =ðeC min Þ þ T ci þ C c ln 1 þ
Q
T ci C c
:
ð23Þ
for prescribed Tci. The entropy generation also decreases monotonously with the increase in effectiveness. The minimum entropy
Fig. 5. Variation of entransy dissipation, thermal resistance, and entropy generation
with effectiveness of THEN when the inlet temperatures of the streams are
prescribed.
Fig. 6. Variation of thermal resistance with effectiveness of THEN at different heat
capacity flow rates of cold stream.
X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (2012) 163–170
167
Fig. 7. Variation of entransy dissipation, thermal resistance, and entropy generation
with effectiveness of THEN when the heat transfer rate is prescribed.
Fig. 8. Variation of effectiveness, entransy dissipation, thermal resistance and
entropy generation with thermal conductance of heat exchanger 2 of Fig. 1.
generation principle also holds good in this case. The thermal resistance, Eq. (20), is tenable for all the cases.
The following numerical example accords with above discussion. Let Ch = 2 W/K, Cc = 3 W/K, Q = 10 W, and Tci = 300 K. The
results, shown in Fig. 7, indicate that the entransy dissipation,
the thermal resistance, and the entropy generation are all monotonically related to the performance of the THEN in such a case.
When the heat transfer rate is prescribed, the extreme entransy
dissipation principle is applicable, and the minimum entransy dissipation corresponds to the best performance of the heat transfer
process [6]. This is the reason why the entransy dissipation
decreases monotonously with the increase of the effectiveness.
On the other hand, as the optimization objective of the extreme
entransy dissipation principle is to decrease the heat transfer
temperature difference, the entropy generation is also decreased
due to the decrease of the heat transfer temperature difference.
Cheng et al. [37] drew a similar conclusion when they investigated
the Volume-to-Point problem. When the outlet boundaries are
isothermal (there could be more than one outlet of heat but their
temperatures must be the same), the optimization results of the
extreme entransy dissipation principle and the minimum
entropy generation principle are the same. This is the reason
why the entropy generation could also represent the performance
of the THEN when the heat transfer rate of the streams is
prescribed.
From the above discussions of both kinds of problems, it could
be concluded that only the thermal resistance could represent the
change of the effectiveness of the THEN without limitations. Neither the entransy dissipation nor the entropy generation would always change monotonously with the increase in effectiveness, so
their application is conditional.
Let us consider a numerical example for the THEN sketched in
Fig. 1. Let Thi = 350 K, Tci = 300 K, Ch = 2 W/K, Cc = 3 W/K, the heat
capacity flow rate of the medial fluid is 2.5 W/K, and the thermal
conductance of heat exchanger 1, F1, is 10 W/K. The variations of
the effectiveness, the entransy dissipation, the thermal resistance,
and the entropy generation with the thermal conductance of heat
exchanger 2, F2, are shown in Fig. 8. It can be seen that the
effectiveness increases monotonously with the increase of F2,
while the thermal resistance decreases monotonously. However,
the entransy dissipation and the entropy generation both increase
first and then decrease. This example also indicates that the
application of entransy dissipation and the entropy generation is
conditional to the performance of the THEN, while the thermal
resistance is not.
To improve the application of entropy generation optimization
to heat exchangers, Hesselgreaves [38] developed a revised
entropy generation number for two-stream heat exchanger analysis based on the work of Witte and Shamsundar [39]. For any
two-stream heat exchanger, the expression of the revised entropy
generation number is defined as [38]
Ns ¼
T Li Sg
;
Q
ð24Þ
It was found that the effectiveness varies monotonically with
Ns. There is no paradox between Ns and the effectiveness. As the
expressions of the effectiveness and entropy generation for twostreams heat exchanger networks are the same as those of twostream heat exchangers, the revised entropy generation number
can also be used to describe the performance of two-stream heat
exchanger networks for given heat capacity flow rates.
Compared with the revised entropy generation number, the entransy-dissipation-based thermal resistance has a direct relation
with the heat transfer rate in the heat exchanger and its physical
meaning is clear for heat transfer. Let’s consider the case in which
both Q/Ch and Q/Cc keep constant. The resistance of Eq. (10) can be
rewritten as
1
R ¼ Gdis =Q 2 ¼ T hi T ci ðQ =C h þ Q=C c Þ =Q :
2
ð25Þ
where ðQ =C h þ Q =C c Þ on the right is constant for constant Q/Ch and
Q/Cc. Hence, R decrease with the increase of Q, indicating that lower
thermal resistance corresponds to larger heat transfer rate. On the
other hand, we can also rewrite the revised entropy generation
number from Eq. (11),
Ns ¼
Sg T ci
Ch
Q
Cc
Q
þ
:
¼ T ci
ln 1 ln 1 þ
C h T hi
T ci C c
Q
Q
Q
ð26Þ
The revised entropy generation number is constant for given inlet
temperature in such case. Raising Ch and Cc proportionally will
make Q become larger but the revised entropy generation number
remains constant. The revised entropy generation number has no
direct relation with the heat transfer rate. It is not related to the difference of the stream temperatures but the difference of the reciprocal of the log-mean temperatures as indicated by Eq. (17) in
Ref. [38]. The target of minimizing Ns is not to reduce the heat transfer difference or increase heat transfer rate, but to reduce the difference of thermodynamics potential that is the reciprocal of absolute
temperature. On the other hand, thermal resistance is closely connected with the stream temperature difference and heat flow which
are the most important parameters in heat transfer optimizations.
The different objectives of the thermal resistance and the entropy
optimizations have been shown clearly in the example of the Volume-to-Point problem [40] in which the uniform heat source in
168
X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (2012) 163–170
the square domain is specified. The entransy dissipation is equivalent to thermal resistance and the entropy generation is equivalent
to Ns because the total heat flow rate is prescribed in this example.
It was found that the minimum thermal resistance principle results
in smaller average and maximum temperatures in the domain but a
larger difference of thermodynamic potential (reciprocal of temperature) than the minimum entropy generation does. The entropy
generation is reduced by decreasing the difference of thermodynamic potential, not by temperature difference.
Gdis1 ¼ Z
ðqTÞ n1f dAf s Z
Af s
ðqTÞ n1m dAms ;
where Afs is the area of the interface with the normal vector n1f
between the solid and the two streams, and Ams is the area of
the interface with the normal vector n1m between the solid and
the medial fluids.
For the heat convection region of the medial fluids, the governing equation is
qcðu rTÞ ¼ r q þ U;
3. Conclusions
The general expressions for entransy dissipation, entransydissipation-based thermal resistance and entropy generation for
a generalized THEN are developed. The relationships between
effectiveness and entransy dissipation, entransy-dissipationbased thermal resistance and entropy generation in THENs are
derived and analyzed. When the inlet temperatures of the two
streams of the THEN are prescribed, it is found that the entransy-dissipation-based thermal resistance decreases monotonously with the increase in effectiveness, while the entransy
dissipation and entropy generation do not. When the heat transfer rate of the THEN is prescribed, the entransy dissipation, the
entransy-dissipation-based thermal resistance and the entropy
generation all decrease monotonously with the increase in effectiveness. The overall conclusion is that the thermal resistance is
most suitable to optimize the effectiveness and describe the
performance of the THEN.
Acknowledgements
The present work is supported by the Natural Science Foundation of China (Grant No. 51106082) and the Tsinghua University
Initiative Scientific Research Program.
ðA:5Þ
Ams
ðA:6Þ
where q is the density of the fluid, c is the specific heat capacity, u is
the velocity of the fluid, and U is the viscous dissipation function.
Multiplying Eq. (A.6) by the temperature T gives [6]
1
qc½u rðT 2 Þ ¼ r ðqTÞ þ q rT þ UT;
2
ðA:7Þ
Apply the Fourier’s law to Eq. (A.7), and integrate it over the region [6]
Gdis2 ¼
Z
kðrTÞ2 dV 2
V2
¼
Z
V
1
qc½u rðT 2 ÞdV 2
Z
r ðqTÞdV þ
V
Z
UTdV;
ðA:8Þ
V
For the incompressible medial fluid,
r u ¼ 0:
ðA:9Þ
Based on the Gauss divergence theorem and Eq. (A.9), the
entransy dissipation of the heat convection region of the medial
fluids could be expressed as
Z Z
1
qcT 2 u n2 dA2 ðqTÞ n2 dA2
2
A
Ams
Z2
þ
ðUTÞdV 2 ;
Gdis2 ¼ ðA:10Þ
V2
Appendix A
The heat transfer process in the generalized THEN in Fig. 4 is
composed of three parts. The first part is the heat conduction
through the solid structure, the second part is the heat convection
between the medial fluids and the solid channel surface, and the
third part is the heat convection between the solid channel surface
and the hot and cold streams.
For the heat conduction, the governing equation is
r q ¼ 0;
ðA:1Þ
where q is the heat flux vector. Multiplying Eq. (A.1) by the temperature T leads to [6]
r ðqTÞ q rT ¼ 0:
½r ðqTÞ q rTdV 1 ¼
V1
Z
ðqTÞ n1 dA1 Z
A1
q rTdV 1 ¼ 0;
ðA:3Þ
V1
where V1 is the volume of the heat conduction region, A1 is the surface area of the region, and n1 is the normal vector of the surface of
the region. According to the Fourier’s law, the entransy dissipation
in the heat conduction region is [6]
Gdis1 ¼
Z
V1
u n2 ¼ 0:
kðrTÞ2 dV 1 ¼ Z
ðqTÞ n1 dA1 ;
Gdis2 ¼ where k is the thermal conductivity. As there is no heat exchange
between the THEN and the environment, the heat flux q only exists
at the interface between the solid and the two streams or between
the solid and the medial fluids. Therefore, we get
Z
ðqTÞ n2 dAms :
ðA:12Þ
Ams
For the heat convection between the solid surface and the two
streams, Eq. (A.7) is applicable. Accordingly,
Gdis3 ¼ Z Z
Z
1
qcT 2 u n3 dA3 ðqTÞ n3 dA3 þ ðUTÞdV 3 ;
2
A3
A3
V2
ðA:13Þ
where V3 is the volume of the heat convection region of the two
streams, n3 is the normal vector of the surface of the region, and
A3 is the area of the surface of the region which is the sum of the
area Afs and the area of the inlet and the outlet of the two streams.
Ignoring the influence of the viscous dissipation on the entransy of
the region and the heat conduction at the inlet and the outlet of the
two streams, Eq. (A.13) reduces to
ðA:4Þ
A1
ðA:11Þ
Substituting Eq. (A.11) in Eq. (A.10) and ignoring the influence
of viscous dissipation on the entransy of the region, we get
ðA:2Þ
Applying the Gauss divergence theorem to the integration of Eq.
(A.2) over the whole heat conduction region yields [6]
Z
where V2 is the volume of the heat convection region of the medial
fluids, A2 is the area of the surface of the region, and n2 is the normal
vector of the surface. As there are no outlets or inlets for the medial
fluids, A2 is Ams, hence,
Gdis3 ¼ Z Z
1
qcT 2 u n3 dA3 ðqT Þ n3 dAf s :
2
A3
Af s
ðA:14Þ
Assuming that the velocity and the temperature of the fluids at
the inlet and the outlet are uniform and the velocity of the fluids in
the normal direction at the interface is zero, we get
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X.T. Cheng, X.G. Liang / Energy Conversion and Management 58 (2012) 163–170
Gdis3
1
1
¼
qh ch uhi Ahi T 2hi qh ch uho Aho T 2ho
2
2
1
1
þ
qc cc uci Aci T 2ci qc cc uco Aco T 2co
2
2
Z
ðqT Þ n3 dAf s
Sg3 ¼ Sout Sin þ
1 1 2
C h T hi T 2ho þ C c T 2ci T 2co
2
2
Z
ðqTÞ n3 dAf s ;
DSfluid ¼
Af s
Z
Qf
0
ðA:15Þ
where the subscripts h, c, i and o stand for hot stream, cold stream,
inlet and the outlet of the streams, respectively.
Combination of Eqs. (A.5), (A.12), and (A.15) yields the entransy
dissipation of the THEN,
ðqTÞ ðn1m þ n2 ÞdAms
Ams
1
1
þ C h ðT 2hi T 2ho Þ þ C c ðT 2ci T 2co Þ:
2
2
ðA:24Þ
where Sin is the entropy that the fluids bring into the region, while
Sout is that the fluids take away from the region. For any stream with
inlet temperature Ti and outlet temperature To, the entropy change
is
Af s
Gdis ¼ Gdis1 þ Gdis2 þ Gdis3
Z
Z
¼
ðqTÞ ðn1f þ n3 ÞdAf s q
n3 dAf s :
T
Af s
Af s
¼
Z
ðA:16Þ
dQ f
¼
T
Z
To
Ti
C f dT
To
¼ C f ln :
Ti
T
ðA:25Þ
where Qf is the change of the internal energy, Cf is the heat capacity
flow rate of the stream. Hence, we could get
Sg3 ¼ DSfluid-H þ DSfluid-L þ
Z
Af s
¼ C h ln
T ho
T co
þ C c ln
þ
T hi
T ci
q
n3 dAf s
T
Z
Af s
q
n3 dAf s :
T
ðA:26Þ
where DSfluid-H and DSfluid-L are the entropy changes of the hot and
the cold streams, respectively. Combining Eqs. (A.17), (A.18), (A.22),
(A.23), and (A.26), we get
Sg ¼ Sg1 þ Sg2 þ Sg3 ¼ C h ln
T ho
T co
þ C c ln
:
T hi
T ci
ðA:27Þ
At the interface between the solid surface and the fluids,
n1m þ n2 ¼ 0;
ðA:17Þ
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1
1
C h ðT 2hi T 2ho Þ þ C c ðT 2ci T 2co Þ:
2
2
Gdis ¼
ðA:19Þ
Qian et al. [35] defined the entransy-dissipation-based thermal
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2
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ðA:20Þ
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ðA:21Þ
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entropy generation rate of the heat transfer processes. At steady
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two streams or between the solid surface and the medial fluids,
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Sg1 ¼
Z
dSg1 ¼ V1
Z
Z
dSf 1 ¼
V1
q
n1f dAf s þ
¼
Af s T
Z
Z
V1
r
Z
q
q
dV 1 ¼
n1 dA1
T
A1 T
q
n1m dAms :
Ams T
ðA:22Þ
For the heat convection region of the medial fluids, the entropy
generation is
Sg2 ¼
Z
Ams
q
n2 dAms :
T
ðA:23Þ
The heat convection region of the two streams is composed of
the two parts, the entropy flux at the interface and the entropy flux
at the fluid inlet and fluid outlet. Therefore, the entropy generation
is
170
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