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Optimization principles for two-stream heat exchangers and two-stream heat exchanger networks
Energy 46 (2012) 386e392
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Optimization principles for two-stream heat exchangers and two-stream heat exchanger networks 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
a b s t r a c t
Article history: Received 21 March 2012 Received in revised form 17 May 2012 Accepted 9 August 2012 Available online 5 September 2012
This paper presents the relationships for the heat transfer performance of two-stream heat exchangers (THEs) and two-stream heat exchanger networks (THENs) with the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the entransy-dissipation-based (EDB) thermal resistance. The results indicate that the effectiveness only increases with decreasing revised entropy generation number, entransy dissipation number and EDB thermal resistance when the heat capacity flow rates and the inlet temperatures are both fixed. The heat transfer rate increases with decreasing thermal resistance, increasing entropy generation and increasing entransy dissipation for prescribed inlet temperatures and prescribed ratios of the heat transfer rate to the heat capacity flow rates. The effectiveness has monotonic relations with all six parameters if the heat transfer rate is prescribed. Only the EDB thermal resistance has monotonic relation with the heat transfer performances for all the discussed cases. A lower resistance gives a better heat transfer performance. Other heat transfer examples also show that the concept of EDB thermal resistance has a wider suitability to the heat transfer optimizations discussed in the paper than the other five concepts. 2012 Elsevier Ltd. All rights reserved.
Keywords: Two-stream heat exchangers Two-stream heat exchanger networks Entropy generation Entransy dissipation Entransy-dissipation-based thermal resistance
1. Introduction As nearly 80% of the world’s total energy consumption is related to heat transfer processes [1], heat transfer analyses, enhancement and optimization are always important since improving heat transfer will significantly reduce energy consumption [2e5]. Twostream heat exchangers (THEs) and two-stream heat exchanger networks (THENs) are widely used in industry [6e8], so their optimization is very important for engineers. Many theories have been developed in recent decades for optimizing THEs and THENs, such as entropy generation minimization [9], the uniformity principle of temperature difference field [10], and entransy theory [11,12]. From the thermodynamic viewpoint, practical heat transfer processes are irreversible and entropy generation would always occur. The minimum entropy generation was found for the steady thermal systems [13]. Afterward, the minimum entropy generation method was used in optimal heat transfer designs and was specifically applied to optimize THE designs [9]. However, an entropy generation paradox has been observed. The heat exchanger effectiveness does not always
* Corresponding author. Fax: þ86 10 62788702. E-mail address: [email protected] (X.G. Liang). 0360-5442/$ e see front matter 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.08.012
increase with decreasing entropy generation number, but decreases for some conditions [9]. Shah and Skiepko [14] showed that the heat exchanger effectiveness can be the maximum, an intermediate value or the minimum at the maximum entropy generation for eighteen different kinds of THEs. Qian et al. [1] also observed that the entropy generation number first increases and then decreases with increasing number of heat transfer units in THEs. Cheng et al. [15,16] analyzed the entropy generation of THEs and THENs and demonstrated that the entropy generation first increases and then decreases with increasing effectiveness and heat transfer rate. To remove this paradox, Hesselgreaves [17] developed a revised entropy generation number for THE analysis based on the work of Witte and Shamsundar [18]. Hesselgreaves [17] showed that the effectiveness continuously increases with decreasing revised entropy generation number and there is no paradox between the revised entropy generation number and the effectiveness. The entropy generation number and the revised entropy generation number have the advantage that they can express the irreversibility induced by the flow resistance though there are arguments on the application of entropy generation to the analyses of THEs. The concept of entropy generation was used to analyze THEs with pressure drop [17]. The uniformity principle of temperature difference field was also developed to optimize THE designs [10]. A more uniform
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temperature difference field corresponds to higher heat exchanger effectiveness for a fixed number of heat transfer units and a fixed ratio of the heat capacity flow rates. The principle has been verified by numerical simulations, theoretical analyses and experiments [10,19]. Many researches have shown that the principle is suitable for THE optimization designs [20,21]. Entransy theory [11] can also be applied to heat exchange design. The concept of entransy can be deduced by an analogy between heat conduction and electrical conduction or by the entransy balance equation derived from the energy conservation equation. The physical meaning of entransy can be seen in the electrical conduction analogy, in which it corresponds to the electrical potential energy in a capacitor [11]. Thus, entransy is the “potential energy” of the heat stored in a body, similar to the electrical energy stored in a capacitor. Cheng et al. [22] and Guo et al. [11] proved that the entransy will always decrease during any practical heat transfer process. Therefore, the entransy decrease, named entransy dissipation, can be used to describe the irreversibility of a heat transfer process [11,22e24]. Guo et al. [11] derived the minimum entransy dissipation principle for prescribed heat flux boundary conditions and the maximum entransy dissipation principle for prescribed temperature boundary conditions. These two principles are referred to as the extremum entransy dissipation principle. Guo et al. [11] further defined the equivalent thermal resistance of the heat transfer system based on the entransy dissipation and heat flow. The extremum entransy dissipation principle was then evolved into the minimum entransydissipation-based (EDB) thermal resistance principle. The minimum EDB thermal resistance principle states that a lower thermal resistance leads to better heat transfer. The extremum entransy dissipation principle and the minimum EDB thermal resistance principle have been used to optimize heat conduction processes [11,25,26], heat convection processes [2,11,24,27] and thermal radiation processes [28,29]. The uniformity principle of temperature difference field for THEs was proved by Song et al. [30] and Cheng et al. [15] using these principles. Song et al. [30] verified that the maximum heat transfer rate always corresponds to a uniform temperature difference field with the prescribed entransy dissipation, while the minimum entransy dissipation corresponds to the uniform temperature difference field with the prescribed heat transfer rate. Cheng et al. [15] established the mathematical relationship between the uniformity factor for the temperature difference field and the effectiveness for THEs and found that a larger temperature difference field uniformity factor always goes with a smaller EDB thermal resistance and a higher effectiveness. The entransy dissipation concept has also been applied to analyses of THEs [15,31e35]. The best performance was found to correspond to the minimum entransy dissipation when the heat transfer rate was prescribed [15,31e35]. However, the best performance does not always match the extremum entransy dissipation if the specified parameter is not the heat transfer rate [15] because the precondition for the extremum entransy dissipation is not satisfied. Guo et al. [12] introduced an effectiveness-thermal resistance method for THE analyses. They defined the thermal resistance of THEs based on the entransy dissipation and found that the effectiveness increases with decreasing EDB thermal resistance. No paradox similar to the entropy generation one has been observed when the EDB thermal resistance was applied to THE analyses [1,12,15]. There are several reports on the application of the EDB thermal resistance concept to THEs and THENs [1,12,15,16,31e37]. Chen et al. [36] analyzed a THEN connected by a hydronic fluid based on the minimum EDB thermal resistance to show that minimizing the thermal resistance results in the maximum heat transfer rate. Qian et al. [37] investigated three THEN designs to show that the maximum heat transfer rate corresponds to the
387
minimum EDB thermal resistance. Cheng and Liang [16] compared the concepts of entropy generation, entransy dissipation and EDB thermal resistance to show that only the EDB thermal resistance is always appropriate for the optimization of THENs, while the applicability of the entropy generation and the entransy dissipation is conditional. The perfect parameter for heat transfer optimization should have a monotonic and single valued functional relationship with the performance. The above discussion shows that there are arguments on the suitableness of these parameters and principles for the optimization of THEs and THENs. The uniformity principle of temperature difference field is not discussed here because it has already been proven by using the entransy theory [15]. The focus of this paper is then on the comparison of the relationships between the various design parameters and the heat transfer performance to demonstrate their suitability for optimizing THEs and THEN designs. 2. Relationships for heat transfer optimization parameters A generalized THE is shown in Fig. 1 [12]. The inlet temperatures are Tin-h and Tin-c, the outlet temperatures are Tout-h and Tout-c and the heat capacity flow rates are Ch and Cc for the hot and the cold streams. Assume that the fluids in the heat exchanger are incompressible, the influence of viscous dissipation on the system entransy and entropy could be ignored, and there is no heat exchange between the heat exchanger and the environment. The entransy dissipation Gdis [12,15], the entransy dissipation number NG [33], the EDB thermal resistance R [12,15] and the entropy generation Sg are expressed as [9]
Gdis ¼
1 1 1 1 2 2 2 2 ; þ Cc Tinc þ Cc Toutc Ch Tinh Ch Touth 2 2 2 2
Gdis Gdismax 1 1 1 1 2 2 2 2 þ Cc Tinc þ Cc Toutc Ch Tinh Ch Touth 2 2 2 2 ¼ ; Q ðTinh Tinc Þ
(1)
NG ¼
(2)
R ¼ Gdis =Q 2 1 1 1 1 2 2 2 2 Q 2; ¼ þ Cc Tinc þ Cc Toutc Ch Tinh Ch Touth 2 2 2 2 (3)
Fig. 1. A generalized THE [12].
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and Gdis-max is the maximum entransy dissipation defined by Guo et al. [33],
satisfactory in the vicinity of ε 0.5 in the balanced counter flow heat exchanger because ε can be sometimes around ε 0.5 in practical applications. The perfect parameter for the optimization of a target quantity should have a monotonous and single valued functional relationship with this quantity. To eliminate the entropy generation paradox, Hesselgreaves [17] suggested a revised entropy generation number for THE analyses. For the generalized THE in Fig. 1, the revised entropy generation number is defined as [17].
Gdismax ¼ Q ðTinh Tinc Þ:
NRS ¼ Tinc Sg =Q ;
Sg ¼ Ch ln
Touth Toutc þ Cc ln ; Tinh Tinc
(4)
where Q is the heat transfer rate in the heat exchanger,
Q ¼ Ch ðTinh Touth Þ ¼ Cc ðToutc Tinc Þ;
(5)
(6)
In THENs such as shown in Fig. 2, heat is delivered from the hot stream to the cold stream through various heat exchangers, distributors, mixers and intermediate fluids. Using the same assumptions as for the generalized THE, Cheng and Liang [16] derived expressions for the entransy dissipation, the EDB thermal resistance and the entropy generation for a generalized THEN, which are the same as Eqs. (1), (3) and (4) because these expressions are determined only by the inlet and outlet parameters with the intermediate heat exchange processes fully represented by the outlet parameters. Similarly, the entransy dissipation number of THEN could also be defined as Eq. (2). The effectiveness of the generalized THEN is then the same as the definition of the heat exchanger effectiveness [16],
ε ¼
Q C ðT Touth Þ Cc ðToutc Tinc Þ ¼ h inh ¼ ; Qmax Cmin ðTinh Tinc Þ Cmin ðTinh Tinc Þ
(7)
where Qmax is the maximum possible heat transfer rate and
Cmin ¼ minðCh ; Cc Þ:
(8)
Bejan [9] defined the entropy generation number for heat exchangers as
NS ¼
Sg ¼ Cmin
Ch ln
Touth Toutc þ Cc ln Tinh Tinc
Cmin ;
(9)
The entropy generation paradox was found [9] when the entropy generation number was used to analyze a balanced counter flow heat exchanger. The heat exchanger effectiveness, ε, does not always increase with decreasing NS, but decreases with decreasing NS for ε ˛ [0, 0.5]. Especially, NS is zero when ε is zero. Bejan [9] explained that the behavior in the ε/0 extreme is neither expected nor intuitively obvious and that the vanishing NS seen in the limit is first and foremost a sign that the heat exchanger disappears as an engineering component. Therefore, the ε/0 limit will never happen in practical applications. However, this explanation is not
Fig. 2. A generalized THEN.
(10)
When the heat capacity flow rates and the stream inlet temperatures are prescribed, ε increases with decreasing NRS so there is no paradox between NRS and ε. This conclusion is also true for THENs since the expression for the entropy generation in THENs is the same as that for THEs. The investigation of the effectiveness-thermal resistance for THEs by Guo et al. [12] showed that the EDB thermal resistance always decreases with increasing heat exchanger effectiveness. Cheng et al. [15,16] observed that for analyses of the generalized THE and the generalized THEN, only the thermal resistance among the concepts of entropy generation, entransy dissipation and thermal resistance can give consistent guidance for optimizing THEs and THENs with either prescribed heat transfer rate or prescribed inlet temperatures and heat capacity flow rates. Guo et al. [33] analyzed the entransy dissipation number in THEs, and showed that it can also evaluate the performance of THEs. Several typical cases are presented here to compare the relationships of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the EDB thermal resistance for THE and THEN designs. In the first case, the heat capacity flow rates and the inlet temperatures for the THE and the THEN are prescribed. Let Ch ¼ 5 W/K, Cc ¼ 8 W/K, Tin-h ¼ 360 K, and Tin-c ¼ 300 K. The variations of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the EDB thermal resistance with the effectiveness are shown in Fig. 3. The revised entropy generation number, the entransy dissipation number and the EDB thermal resistance all decrease with increasing effectiveness. The entropy generation, the entropy generation number and the entransy dissipation first increase and then decrease with increasing effectiveness.
Fig. 3. Variations of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB thermal resistance with the effectiveness.
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Fig. 3 shows that the entransy dissipation, the entropy generation and the entropy generation number have extremum. This extremum can be derived from Eqs. (1), (4) and (9),
Q ¼ ðTinh Tinc Þ=ð1=Ch þ 1=Cc Þ:
(11)
Substituting Q into Eq. (5) yields,
Touth ¼ Toutc ¼ ðCh Tinh þ Cc Tinc Þ=ðCh þ Cc Þ:
(12)
Thus, from Eq. (12) and Fig. 3, the effectiveness of THEs and THENs increases with increasing entropy generation, entropy generation number and entransy dissipation when the hot stream outlet temperature is higher than that of the cold stream. On the other hand, the effectiveness increases with decreasing entropy generation, entropy generation number and entransy dissipation when the hot stream outlet temperature is lower than that of the cold stream. For example, in parallel flow heat exchangers, the hot stream outlet temperature will never be lower than that of the cold stream; hence, a larger entropy generation, a larger entropy generation number and a larger entransy dissipation all indicate better heat transfer in parallel flow heat exchangers. The maximum entransy dissipation principle is then suitable for parallel flow heat exchangers, but the minimum entropy generation method is not. For the second case, the prescribed parameters are the two inlet temperatures, the ratio Q/Ch and the ratio Q/Cc instead of the heat capacity flow rates. There are
rh ¼ Q =Ch ¼ const;
(13)
rc ¼ Q =Cc ¼ const ;
(14)
where rh and rc are the ratios. The effectiveness is constant in this case and the discussion focuses on the heat exchange. Combining of Eqs. (1)e(6), (9), (10), (13), (14) gives
1 Gdis ¼ Q ðrh þ rc Þ þ ðTinh Tinc Þ ; 2
NG ¼
Gdis Gdismax
1 Q ðrh þ rc Þ þ ðTinh Tinc Þ 2 ¼ Q ðTinh Tinc Þ
ðrh þ rc Þ ¼ þ 1; 2ðTinh Tinc Þ R ¼
Sg ¼ Q
1 ðr þ rc Þ þ ðTinh Tinc Þ 2 h
(16) Q;
1 r 1 rc þ ln 1 þ ; ln 1 h rh rc Tinh Tinc
1 r 1 rc þ ln 1þ ln 1 h Tinh Tinc Cmin rh rc 8 > 1 rh 1 rc > > þ ln 1þ ;Cmin ¼ Ch > < rh r ln 1 T rc Tinc h inh ¼ ; > 1 rh 1 rc > > > þ ;C r ln 1 ln 1þ ¼ C c c min : r Tinh Tinc rc h
NS ¼
Q
(15)
389
entransy dissipation, a smaller EDB thermal resistance, and a larger entropy generation, while the entropy generation number, the revised entropy generation number and the entransy dissipation number remain constant. The revised entropy generation number is the entropy generation per unit heat transfer rate with prescribed Tin-c. According to Eq. (18), the entropy generation increases linearly with increasing heat transfer rate, leading to a constant revised entropy generation. For the entransy dissipation number, Guo et al. [33] showed that it has relation with the effectiveness and the ratio between the minimum and the maximum heat capacity flow rates of the THEs. When the prescribed parameters are the stream inlet temperatures, the ratio Q/Ch and the ratio Q/Cc, both the effectiveness and the ratio between the minimum and the maximum heat capacity flow rates are fixed according to Eq. (7) and the entransy dissipation number is also kept constant. Consider a numerical example with rh ¼ 5 K, rc ¼ 8 K, Tin-h ¼ 320 K and Tin-c ¼ 300 K. The variations of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the EDB thermal resistance with the heat transfer rate are shown in Fig. 4. The numerical results show that the entropy generation, the entransy dissipation and the thermal resistance vary with the heat transfer rate in the THE and THEN, while the entransy dissipation number, the entropy generation number and the revised entropy generation number do not. In the third case, the heat transfer rate is prescribed. Chen et al. [32], Guo et al. [34] and Cheng et al. [15] have shown that the entransy dissipation concept can be used to optimize the THEs. For the THENs, Cheng and Liang [16] demonstrated that both the entropy generation and the entransy dissipation decrease with increasing effectiveness. Since the expressions for the effectiveness, the entropy generation, the entransy dissipation and the entransy dissipation number for THEs are the same as those for the THENs, the entropy generation and the entransy dissipation can both be used to optimize the THEs in this case. With the prescribed heat transfer rate, the entransy dissipation, the entransy dissipation number and the thermal resistance behave in the same manner, while the behaviors of the entropy generation, the entropy generation number and the revised entropy generation number are also the same. Therefore, all six concepts, the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB
(17)
(18)
NRS ¼ Tinc
1 r 1 rc þ ln 1 þ : ln 1 h rh rc Tinh Tinc
(19)
(20)
From Eqs. (13) and (14), proportionally increasing in Ch and Cc will increase Q. From Eqs. (15)e(20), a larger Q will lead to a larger
Fig. 4. Variations of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB thermal resistance with the heat transfer rate.
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thermal resistance, are suitable for optimizing THE and THEN designs with a prescribed heat transfer rate. Applications of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number to optimizing THE and THEN designs are all conditional. Only the EDB thermal resistance has monotonic and single valued relationships with the THE and THEN performance for all the cases discussed here. It decreases with increasing effectiveness when the effectiveness is to be optimized, and decreases with increasing heat transfer rate when the effectiveness is fixed. 3. Discussion The suitability of the entropy and entransy principles for optimizing THE and THEN designs is discussed above. This section analyzes the suitability of these concepts for optimizing other heat transfer cases. The fourth case is a one-dimensional heat transfer process as shown in Fig. 5, where the high temperature, TH, and the low temperature, TL, are prescribed. k1 is the product of the equivalent thermal conductivity and the section area of heat transfer for the upper part, while k2 is that of the lower part. The heat transfer mode can be conduction, convection or thermal radiation. Considering that there is a limitation on the amount of materials in some engineering designs, we use the following limiting condition on the optimization
k1 þ k2 ¼ k ¼ const:
(22)
where d1 and d2 are the thicknesses of the upper and lower parts. The entropy generation number cannot be used in this case because there is no heat capacity flow. The entransy dissipation, the entransy dissipation number, the EDB thermal resistance, the entropy generation and the revised entropy generation number are given by
Gdis ¼ Q ðTH TL Þ;
NG ¼
Gdis Q ðTH TL Þ ¼ ¼ 1; Gdismax Q ðTH TL Þ
R ¼
d1 d2 T TL þ ¼ H ; k1 k2 Q
(25)
1 1 ; TL TH
(26)
Sg ¼ Q
(21)
The optimization objective of the heat transfer process is the maximum heat transfer rate. The heat flow in Fig. 5 can be expressed as,
d d Q ¼ ðTH TL Þ= 1 þ 2 ; k1 k2
Fig. 6. Variations of the heat transfer rate, entransy dissipation, EDB thermal resistance, entropy generation, entransy dissipation number and revised entropy generation number with k1.
(23)
(24)
NRS ¼
TL Sg 1 1 T ¼ 1 L : ¼ TL TL TH Q TH
(27)
The heat flow increases with decreasing thermal resistance and increasing entransy dissipation and entropy generation. The revised entropy generation number, however, is constant with prescribed temperatures. For the simple heat transfer process in Fig. 5, the entropy generation rate per unit heat transfer rate does not depends on the heat transfer rate, but the high and low temperatures. The revised entropy generation number is constant once the temperatures are prescribed. For the entransy dissipation number, it is also constant because the maximum entransy dissipation equals to the practical entransy dissipation in this case. Fig. 6 shows the variation of the heat transfer rate, the entransy dissipation, the EDB thermal resistance, the entropy generation and the revised entropy generation number with k1 for k ¼ 100 Wm/K, TH ¼ 350 K, TL ¼ 300 K and d1 ¼ d2 ¼ 1 m. The heat transfer rate, the entropy generation and the entransy dissipation reach maximum at the same k1, while the EDB thermal resistance reaches a minimum. The revised entropy generation number and the entransy dissipation number remain constant. Therefore, they cannot be used to optimize the heat transfer process shown in Fig. 5 because they are not related to the heat transfer rate.
Table 1 Five cases discussed in the present work. Cases
Case description
Case I
THEs and THENs with prescribed stream inlet temperatures and heat capacity flow THEs and THENs with prescribed stream inlet temperatures and prescribed ratios of the heat transfer rate to the heat capacity flow rates THEs and THENs with prescribed heat transfer rate One dimensional heat transfer (Fig. 5) Volume-to-Point problem [38]
Case II
Fig. 5. One-dimensional heat transfer diagram.
Case III Case IV Case V
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Table 2 The relationships of the entropy generation Sg, entropy generation number NS, revised entropy generation number NRS, entransy dissipation Gdis, entransy dissipation number NG and EDB thermal resistance R with the heat transfer performance for the five cases. Concept
Case I
Case II
Case III
Case IV
Case V
Optimization objective
Sg NS NRS Gdis NG R
non-monotonic non-monotonic monotonic non-monotonic monotonic monotonic
monotonic constant constant monotonic constant monotonic
monotonic monotonic monotonic monotonic monotonic monotonic
non-monotonic not applicable constant non-monotonic constant monotonic
monotonic not applicable monotonic monotonic e monotonic
min e min min e min
The last case is a conduction problem with a prescribed heat transfer rate. We take the volume-to-point problem [38] as an example. The volume has a uniform heat source and the boundary is adiabatic except for a small piece (point) which is fixed at a constant temperature. The entropy generation number is again not applicable since there is no heat capacity flow. The entransy dissipation is equivalent to the thermal resistance, while the entropy generation is equivalent to the revised entropy generation number because the heat transfer rate (that is, the total heat production in the volume) is prescribed. Chen et al. [38] found that the minimum EDB thermal resistance principle gives a smaller average temperature but a larger difference in the thermodynamic potential (reciprocal of temperature) in the domain than the minimum entropy generation principle. The entropy generation is reduced by decreasing the thermodynamic potential difference, not the temperature difference. Minimizing the entropy generation does not reduce the heat transfer temperature difference or increase the heat transfer rate, but reduce the difference in the thermodynamics potential (i.e., the reciprocal of the absolute temperature) [16]. The minimum EDB thermal resistance corresponds to the minimum average temperature for the volume-to-point problem [11,38]. The concepts of entransy dissipation and EDB thermal resistance are then more suitable for heat transfer optimization than the concepts of entropy generation and revised entropy generation number when the heat transfer rate is prescribed. For the entransy dissipation number, its applicability to the Volume-to-Point problem needs further discussion because there is no report on this topic. The five cases discussed in this paper are listed in Table 1. The relationships of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number, the EDB thermal resistance and the heat transfer performance are given in Table 2. The EDB thermal resistance is the most suitable for optimizing heat transfer problems while the other concepts are only suitable for certain classes of problems. The physical meaning of the EDB thermal resistance is the resistance to the heat flow for a given temperature difference. Naturally, a smaller resistance will give better heat transfer. 4. Conclusions The suitability of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB thermal resistance for optimizing THE and THEN designs is discussed in this paper. The investigations demonstrate that only the revised entropy generation number, the entransy dissipation number and the EDB thermal resistance decrease with increasing effectiveness for THEs and THENs among the six parameters for prescribed heat capacity flow rates and stream inlet temperatures. Only the EDB thermal resistance decreases with increasing heat transfer rate while the entropy generation and entransy dissipation increase for prescribed inlet temperatures and the ratios of the heat transfer
D(1/T) D(1/T) D(T) or max (Q) D(T) or max (Q)
rate to the heat capacity flow rates. The entropy generation number, the entransy dissipation number and the revised entropy generation number remain constant. All six parameters decrease with increasing effectiveness for THEs and THENs with prescribed heat transfer rate. The results further show that the EDB thermal resistance is the most suitable for optimizing of THE and THEN designs. It decreases with increasing effectiveness if the effectiveness is to be optimized, or decreases with increasing heat transfer rate if the effectiveness is fixed. The application of all six concepts to other heat transfer problems is also discussed with two examples. For one-dimensional heat transfer problem with prescribed temperatures, the maximum entropy generation, the maximum entransy dissipation and the minimum thermal resistance all correspond to the maximum heat transfer rate while the revised entropy generation number and the entransy dissipation number remain constant. The minimum EDB thermal resistance and the maximum entransy dissipation give the same result, but the minimum entropy generation and the revised entropy generation are not applicable. For the volume-to-point problem, the minimum EDB thermal resistance and the minimum entransy dissipation result in a smaller average temperature in the volume than the minimum entropy generation and the minimum revised entropy generation number. The physical meaning of the EDB thermal resistance is clear and it is directly related to the heat transfer while the entropy generation is directly related to the irreversibility. The EDB thermal resistance is the only parameter among the parameters analyzed here suitable for optimizing all five heat transfer problems. Acknowledgement The present work was supported by the Natural Science Foundation of China (Grant No. 51106082) and the Tsinghua University Initiative Scientific Research Program. The authors thank Professor Zeng-Yuan Guo for his helpful discussions and suggestions. References [1] Qian XD, Li ZX. Analysis of entransy dissipation in heat exchangers. Int J Thermal Sci 2011;50:608e14. [2] Chen Q, Wang M, Pan N, Guo ZY. Optimization principles for convective heat transfer. Energy 2009;34:1199e206. [3] Wang RZ, Xia ZZ, Wang LW, Lu ZS, Li SL, Li TX, et al. Heat transfer design in adsorption refrigeration systems for efficient use of low-grade thermal energy. Energy 2011;36:5425e39. [4] Kaluri RS, Basak T. Entropy generation due to natural convection in discretely heated porous square cavities. Energy 2011;36:5065e80. [5] Xu MT. The thermodynamic basis of entransy and entransy dissipation. Energy 2011;36:4272e7. [6] Cheng XT, Xu XH, Liang XG. Application of entransy to optimization design of parallel thermal network of thermal control system in spacecraft. Sci China Tech Sci 2011;54:964e71. [7] Hauer A. Adsorption system for TES-design and demonstration projects. Therm Energy Stor Sust Energy Cons 2007;234:409e27. [8] Kreider JF. Heating and cooling of buildings: design for efficiency. New York: McGraw-Hill; 1994. [9] Bejan A. Advanced engineering thermodynamics. 2nd ed. New York: John Wiley & Sons; 1997.
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Optimization principles for two-stream heat exchangers and two-stream heat exchanger networks 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
a b s t r a c t
Article history: Received 21 March 2012 Received in revised form 17 May 2012 Accepted 9 August 2012 Available online 5 September 2012
This paper presents the relationships for the heat transfer performance of two-stream heat exchangers (THEs) and two-stream heat exchanger networks (THENs) with the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the entransy-dissipation-based (EDB) thermal resistance. The results indicate that the effectiveness only increases with decreasing revised entropy generation number, entransy dissipation number and EDB thermal resistance when the heat capacity flow rates and the inlet temperatures are both fixed. The heat transfer rate increases with decreasing thermal resistance, increasing entropy generation and increasing entransy dissipation for prescribed inlet temperatures and prescribed ratios of the heat transfer rate to the heat capacity flow rates. The effectiveness has monotonic relations with all six parameters if the heat transfer rate is prescribed. Only the EDB thermal resistance has monotonic relation with the heat transfer performances for all the discussed cases. A lower resistance gives a better heat transfer performance. Other heat transfer examples also show that the concept of EDB thermal resistance has a wider suitability to the heat transfer optimizations discussed in the paper than the other five concepts. 2012 Elsevier Ltd. All rights reserved.
Keywords: Two-stream heat exchangers Two-stream heat exchanger networks Entropy generation Entransy dissipation Entransy-dissipation-based thermal resistance
1. Introduction As nearly 80% of the world’s total energy consumption is related to heat transfer processes [1], heat transfer analyses, enhancement and optimization are always important since improving heat transfer will significantly reduce energy consumption [2e5]. Twostream heat exchangers (THEs) and two-stream heat exchanger networks (THENs) are widely used in industry [6e8], so their optimization is very important for engineers. Many theories have been developed in recent decades for optimizing THEs and THENs, such as entropy generation minimization [9], the uniformity principle of temperature difference field [10], and entransy theory [11,12]. From the thermodynamic viewpoint, practical heat transfer processes are irreversible and entropy generation would always occur. The minimum entropy generation was found for the steady thermal systems [13]. Afterward, the minimum entropy generation method was used in optimal heat transfer designs and was specifically applied to optimize THE designs [9]. However, an entropy generation paradox has been observed. The heat exchanger effectiveness does not always
* Corresponding author. Fax: þ86 10 62788702. E-mail address: [email protected] (X.G. Liang). 0360-5442/$ e see front matter 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.08.012
increase with decreasing entropy generation number, but decreases for some conditions [9]. Shah and Skiepko [14] showed that the heat exchanger effectiveness can be the maximum, an intermediate value or the minimum at the maximum entropy generation for eighteen different kinds of THEs. Qian et al. [1] also observed that the entropy generation number first increases and then decreases with increasing number of heat transfer units in THEs. Cheng et al. [15,16] analyzed the entropy generation of THEs and THENs and demonstrated that the entropy generation first increases and then decreases with increasing effectiveness and heat transfer rate. To remove this paradox, Hesselgreaves [17] developed a revised entropy generation number for THE analysis based on the work of Witte and Shamsundar [18]. Hesselgreaves [17] showed that the effectiveness continuously increases with decreasing revised entropy generation number and there is no paradox between the revised entropy generation number and the effectiveness. The entropy generation number and the revised entropy generation number have the advantage that they can express the irreversibility induced by the flow resistance though there are arguments on the application of entropy generation to the analyses of THEs. The concept of entropy generation was used to analyze THEs with pressure drop [17]. The uniformity principle of temperature difference field was also developed to optimize THE designs [10]. A more uniform
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temperature difference field corresponds to higher heat exchanger effectiveness for a fixed number of heat transfer units and a fixed ratio of the heat capacity flow rates. The principle has been verified by numerical simulations, theoretical analyses and experiments [10,19]. Many researches have shown that the principle is suitable for THE optimization designs [20,21]. Entransy theory [11] can also be applied to heat exchange design. The concept of entransy can be deduced by an analogy between heat conduction and electrical conduction or by the entransy balance equation derived from the energy conservation equation. The physical meaning of entransy can be seen in the electrical conduction analogy, in which it corresponds to the electrical potential energy in a capacitor [11]. Thus, entransy is the “potential energy” of the heat stored in a body, similar to the electrical energy stored in a capacitor. Cheng et al. [22] and Guo et al. [11] proved that the entransy will always decrease during any practical heat transfer process. Therefore, the entransy decrease, named entransy dissipation, can be used to describe the irreversibility of a heat transfer process [11,22e24]. Guo et al. [11] derived the minimum entransy dissipation principle for prescribed heat flux boundary conditions and the maximum entransy dissipation principle for prescribed temperature boundary conditions. These two principles are referred to as the extremum entransy dissipation principle. Guo et al. [11] further defined the equivalent thermal resistance of the heat transfer system based on the entransy dissipation and heat flow. The extremum entransy dissipation principle was then evolved into the minimum entransydissipation-based (EDB) thermal resistance principle. The minimum EDB thermal resistance principle states that a lower thermal resistance leads to better heat transfer. The extremum entransy dissipation principle and the minimum EDB thermal resistance principle have been used to optimize heat conduction processes [11,25,26], heat convection processes [2,11,24,27] and thermal radiation processes [28,29]. The uniformity principle of temperature difference field for THEs was proved by Song et al. [30] and Cheng et al. [15] using these principles. Song et al. [30] verified that the maximum heat transfer rate always corresponds to a uniform temperature difference field with the prescribed entransy dissipation, while the minimum entransy dissipation corresponds to the uniform temperature difference field with the prescribed heat transfer rate. Cheng et al. [15] established the mathematical relationship between the uniformity factor for the temperature difference field and the effectiveness for THEs and found that a larger temperature difference field uniformity factor always goes with a smaller EDB thermal resistance and a higher effectiveness. The entransy dissipation concept has also been applied to analyses of THEs [15,31e35]. The best performance was found to correspond to the minimum entransy dissipation when the heat transfer rate was prescribed [15,31e35]. However, the best performance does not always match the extremum entransy dissipation if the specified parameter is not the heat transfer rate [15] because the precondition for the extremum entransy dissipation is not satisfied. Guo et al. [12] introduced an effectiveness-thermal resistance method for THE analyses. They defined the thermal resistance of THEs based on the entransy dissipation and found that the effectiveness increases with decreasing EDB thermal resistance. No paradox similar to the entropy generation one has been observed when the EDB thermal resistance was applied to THE analyses [1,12,15]. There are several reports on the application of the EDB thermal resistance concept to THEs and THENs [1,12,15,16,31e37]. Chen et al. [36] analyzed a THEN connected by a hydronic fluid based on the minimum EDB thermal resistance to show that minimizing the thermal resistance results in the maximum heat transfer rate. Qian et al. [37] investigated three THEN designs to show that the maximum heat transfer rate corresponds to the
387
minimum EDB thermal resistance. Cheng and Liang [16] compared the concepts of entropy generation, entransy dissipation and EDB thermal resistance to show that only the EDB thermal resistance is always appropriate for the optimization of THENs, while the applicability of the entropy generation and the entransy dissipation is conditional. The perfect parameter for heat transfer optimization should have a monotonic and single valued functional relationship with the performance. The above discussion shows that there are arguments on the suitableness of these parameters and principles for the optimization of THEs and THENs. The uniformity principle of temperature difference field is not discussed here because it has already been proven by using the entransy theory [15]. The focus of this paper is then on the comparison of the relationships between the various design parameters and the heat transfer performance to demonstrate their suitability for optimizing THEs and THEN designs. 2. Relationships for heat transfer optimization parameters A generalized THE is shown in Fig. 1 [12]. The inlet temperatures are Tin-h and Tin-c, the outlet temperatures are Tout-h and Tout-c and the heat capacity flow rates are Ch and Cc for the hot and the cold streams. Assume that the fluids in the heat exchanger are incompressible, the influence of viscous dissipation on the system entransy and entropy could be ignored, and there is no heat exchange between the heat exchanger and the environment. The entransy dissipation Gdis [12,15], the entransy dissipation number NG [33], the EDB thermal resistance R [12,15] and the entropy generation Sg are expressed as [9]
Gdis ¼
1 1 1 1 2 2 2 2 ; þ Cc Tinc þ Cc Toutc Ch Tinh Ch Touth 2 2 2 2
Gdis Gdismax 1 1 1 1 2 2 2 2 þ Cc Tinc þ Cc Toutc Ch Tinh Ch Touth 2 2 2 2 ¼ ; Q ðTinh Tinc Þ
(1)
NG ¼
(2)
R ¼ Gdis =Q 2 1 1 1 1 2 2 2 2 Q 2; ¼ þ Cc Tinc þ Cc Toutc Ch Tinh Ch Touth 2 2 2 2 (3)
Fig. 1. A generalized THE [12].
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and Gdis-max is the maximum entransy dissipation defined by Guo et al. [33],
satisfactory in the vicinity of ε 0.5 in the balanced counter flow heat exchanger because ε can be sometimes around ε 0.5 in practical applications. The perfect parameter for the optimization of a target quantity should have a monotonous and single valued functional relationship with this quantity. To eliminate the entropy generation paradox, Hesselgreaves [17] suggested a revised entropy generation number for THE analyses. For the generalized THE in Fig. 1, the revised entropy generation number is defined as [17].
Gdismax ¼ Q ðTinh Tinc Þ:
NRS ¼ Tinc Sg =Q ;
Sg ¼ Ch ln
Touth Toutc þ Cc ln ; Tinh Tinc
(4)
where Q is the heat transfer rate in the heat exchanger,
Q ¼ Ch ðTinh Touth Þ ¼ Cc ðToutc Tinc Þ;
(5)
(6)
In THENs such as shown in Fig. 2, heat is delivered from the hot stream to the cold stream through various heat exchangers, distributors, mixers and intermediate fluids. Using the same assumptions as for the generalized THE, Cheng and Liang [16] derived expressions for the entransy dissipation, the EDB thermal resistance and the entropy generation for a generalized THEN, which are the same as Eqs. (1), (3) and (4) because these expressions are determined only by the inlet and outlet parameters with the intermediate heat exchange processes fully represented by the outlet parameters. Similarly, the entransy dissipation number of THEN could also be defined as Eq. (2). The effectiveness of the generalized THEN is then the same as the definition of the heat exchanger effectiveness [16],
ε ¼
Q C ðT Touth Þ Cc ðToutc Tinc Þ ¼ h inh ¼ ; Qmax Cmin ðTinh Tinc Þ Cmin ðTinh Tinc Þ
(7)
where Qmax is the maximum possible heat transfer rate and
Cmin ¼ minðCh ; Cc Þ:
(8)
Bejan [9] defined the entropy generation number for heat exchangers as
NS ¼
Sg ¼ Cmin
Ch ln
Touth Toutc þ Cc ln Tinh Tinc
Cmin ;
(9)
The entropy generation paradox was found [9] when the entropy generation number was used to analyze a balanced counter flow heat exchanger. The heat exchanger effectiveness, ε, does not always increase with decreasing NS, but decreases with decreasing NS for ε ˛ [0, 0.5]. Especially, NS is zero when ε is zero. Bejan [9] explained that the behavior in the ε/0 extreme is neither expected nor intuitively obvious and that the vanishing NS seen in the limit is first and foremost a sign that the heat exchanger disappears as an engineering component. Therefore, the ε/0 limit will never happen in practical applications. However, this explanation is not
Fig. 2. A generalized THEN.
(10)
When the heat capacity flow rates and the stream inlet temperatures are prescribed, ε increases with decreasing NRS so there is no paradox between NRS and ε. This conclusion is also true for THENs since the expression for the entropy generation in THENs is the same as that for THEs. The investigation of the effectiveness-thermal resistance for THEs by Guo et al. [12] showed that the EDB thermal resistance always decreases with increasing heat exchanger effectiveness. Cheng et al. [15,16] observed that for analyses of the generalized THE and the generalized THEN, only the thermal resistance among the concepts of entropy generation, entransy dissipation and thermal resistance can give consistent guidance for optimizing THEs and THENs with either prescribed heat transfer rate or prescribed inlet temperatures and heat capacity flow rates. Guo et al. [33] analyzed the entransy dissipation number in THEs, and showed that it can also evaluate the performance of THEs. Several typical cases are presented here to compare the relationships of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the EDB thermal resistance for THE and THEN designs. In the first case, the heat capacity flow rates and the inlet temperatures for the THE and the THEN are prescribed. Let Ch ¼ 5 W/K, Cc ¼ 8 W/K, Tin-h ¼ 360 K, and Tin-c ¼ 300 K. The variations of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the EDB thermal resistance with the effectiveness are shown in Fig. 3. The revised entropy generation number, the entransy dissipation number and the EDB thermal resistance all decrease with increasing effectiveness. The entropy generation, the entropy generation number and the entransy dissipation first increase and then decrease with increasing effectiveness.
Fig. 3. Variations of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB thermal resistance with the effectiveness.
X. T. Cheng, X. G. Liang / Energy 46 (2012) 386e392
Fig. 3 shows that the entransy dissipation, the entropy generation and the entropy generation number have extremum. This extremum can be derived from Eqs. (1), (4) and (9),
Q ¼ ðTinh Tinc Þ=ð1=Ch þ 1=Cc Þ:
(11)
Substituting Q into Eq. (5) yields,
Touth ¼ Toutc ¼ ðCh Tinh þ Cc Tinc Þ=ðCh þ Cc Þ:
(12)
Thus, from Eq. (12) and Fig. 3, the effectiveness of THEs and THENs increases with increasing entropy generation, entropy generation number and entransy dissipation when the hot stream outlet temperature is higher than that of the cold stream. On the other hand, the effectiveness increases with decreasing entropy generation, entropy generation number and entransy dissipation when the hot stream outlet temperature is lower than that of the cold stream. For example, in parallel flow heat exchangers, the hot stream outlet temperature will never be lower than that of the cold stream; hence, a larger entropy generation, a larger entropy generation number and a larger entransy dissipation all indicate better heat transfer in parallel flow heat exchangers. The maximum entransy dissipation principle is then suitable for parallel flow heat exchangers, but the minimum entropy generation method is not. For the second case, the prescribed parameters are the two inlet temperatures, the ratio Q/Ch and the ratio Q/Cc instead of the heat capacity flow rates. There are
rh ¼ Q =Ch ¼ const;
(13)
rc ¼ Q =Cc ¼ const ;
(14)
where rh and rc are the ratios. The effectiveness is constant in this case and the discussion focuses on the heat exchange. Combining of Eqs. (1)e(6), (9), (10), (13), (14) gives
1 Gdis ¼ Q ðrh þ rc Þ þ ðTinh Tinc Þ ; 2
NG ¼
Gdis Gdismax
1 Q ðrh þ rc Þ þ ðTinh Tinc Þ 2 ¼ Q ðTinh Tinc Þ
ðrh þ rc Þ ¼ þ 1; 2ðTinh Tinc Þ R ¼
Sg ¼ Q
1 ðr þ rc Þ þ ðTinh Tinc Þ 2 h
(16) Q;
1 r 1 rc þ ln 1 þ ; ln 1 h rh rc Tinh Tinc
1 r 1 rc þ ln 1þ ln 1 h Tinh Tinc Cmin rh rc 8 > 1 rh 1 rc > > þ ln 1þ ;Cmin ¼ Ch > < rh r ln 1 T rc Tinc h inh ¼ ; > 1 rh 1 rc > > > þ ;C r ln 1 ln 1þ ¼ C c c min : r Tinh Tinc rc h
NS ¼
Q
(15)
389
entransy dissipation, a smaller EDB thermal resistance, and a larger entropy generation, while the entropy generation number, the revised entropy generation number and the entransy dissipation number remain constant. The revised entropy generation number is the entropy generation per unit heat transfer rate with prescribed Tin-c. According to Eq. (18), the entropy generation increases linearly with increasing heat transfer rate, leading to a constant revised entropy generation. For the entransy dissipation number, Guo et al. [33] showed that it has relation with the effectiveness and the ratio between the minimum and the maximum heat capacity flow rates of the THEs. When the prescribed parameters are the stream inlet temperatures, the ratio Q/Ch and the ratio Q/Cc, both the effectiveness and the ratio between the minimum and the maximum heat capacity flow rates are fixed according to Eq. (7) and the entransy dissipation number is also kept constant. Consider a numerical example with rh ¼ 5 K, rc ¼ 8 K, Tin-h ¼ 320 K and Tin-c ¼ 300 K. The variations of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number and the EDB thermal resistance with the heat transfer rate are shown in Fig. 4. The numerical results show that the entropy generation, the entransy dissipation and the thermal resistance vary with the heat transfer rate in the THE and THEN, while the entransy dissipation number, the entropy generation number and the revised entropy generation number do not. In the third case, the heat transfer rate is prescribed. Chen et al. [32], Guo et al. [34] and Cheng et al. [15] have shown that the entransy dissipation concept can be used to optimize the THEs. For the THENs, Cheng and Liang [16] demonstrated that both the entropy generation and the entransy dissipation decrease with increasing effectiveness. Since the expressions for the effectiveness, the entropy generation, the entransy dissipation and the entransy dissipation number for THEs are the same as those for the THENs, the entropy generation and the entransy dissipation can both be used to optimize the THEs in this case. With the prescribed heat transfer rate, the entransy dissipation, the entransy dissipation number and the thermal resistance behave in the same manner, while the behaviors of the entropy generation, the entropy generation number and the revised entropy generation number are also the same. Therefore, all six concepts, the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB
(17)
(18)
NRS ¼ Tinc
1 r 1 rc þ ln 1 þ : ln 1 h rh rc Tinh Tinc
(19)
(20)
From Eqs. (13) and (14), proportionally increasing in Ch and Cc will increase Q. From Eqs. (15)e(20), a larger Q will lead to a larger
Fig. 4. Variations of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB thermal resistance with the heat transfer rate.
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thermal resistance, are suitable for optimizing THE and THEN designs with a prescribed heat transfer rate. Applications of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number to optimizing THE and THEN designs are all conditional. Only the EDB thermal resistance has monotonic and single valued relationships with the THE and THEN performance for all the cases discussed here. It decreases with increasing effectiveness when the effectiveness is to be optimized, and decreases with increasing heat transfer rate when the effectiveness is fixed. 3. Discussion The suitability of the entropy and entransy principles for optimizing THE and THEN designs is discussed above. This section analyzes the suitability of these concepts for optimizing other heat transfer cases. The fourth case is a one-dimensional heat transfer process as shown in Fig. 5, where the high temperature, TH, and the low temperature, TL, are prescribed. k1 is the product of the equivalent thermal conductivity and the section area of heat transfer for the upper part, while k2 is that of the lower part. The heat transfer mode can be conduction, convection or thermal radiation. Considering that there is a limitation on the amount of materials in some engineering designs, we use the following limiting condition on the optimization
k1 þ k2 ¼ k ¼ const:
(22)
where d1 and d2 are the thicknesses of the upper and lower parts. The entropy generation number cannot be used in this case because there is no heat capacity flow. The entransy dissipation, the entransy dissipation number, the EDB thermal resistance, the entropy generation and the revised entropy generation number are given by
Gdis ¼ Q ðTH TL Þ;
NG ¼
Gdis Q ðTH TL Þ ¼ ¼ 1; Gdismax Q ðTH TL Þ
R ¼
d1 d2 T TL þ ¼ H ; k1 k2 Q
(25)
1 1 ; TL TH
(26)
Sg ¼ Q
(21)
The optimization objective of the heat transfer process is the maximum heat transfer rate. The heat flow in Fig. 5 can be expressed as,
d d Q ¼ ðTH TL Þ= 1 þ 2 ; k1 k2
Fig. 6. Variations of the heat transfer rate, entransy dissipation, EDB thermal resistance, entropy generation, entransy dissipation number and revised entropy generation number with k1.
(23)
(24)
NRS ¼
TL Sg 1 1 T ¼ 1 L : ¼ TL TL TH Q TH
(27)
The heat flow increases with decreasing thermal resistance and increasing entransy dissipation and entropy generation. The revised entropy generation number, however, is constant with prescribed temperatures. For the simple heat transfer process in Fig. 5, the entropy generation rate per unit heat transfer rate does not depends on the heat transfer rate, but the high and low temperatures. The revised entropy generation number is constant once the temperatures are prescribed. For the entransy dissipation number, it is also constant because the maximum entransy dissipation equals to the practical entransy dissipation in this case. Fig. 6 shows the variation of the heat transfer rate, the entransy dissipation, the EDB thermal resistance, the entropy generation and the revised entropy generation number with k1 for k ¼ 100 Wm/K, TH ¼ 350 K, TL ¼ 300 K and d1 ¼ d2 ¼ 1 m. The heat transfer rate, the entropy generation and the entransy dissipation reach maximum at the same k1, while the EDB thermal resistance reaches a minimum. The revised entropy generation number and the entransy dissipation number remain constant. Therefore, they cannot be used to optimize the heat transfer process shown in Fig. 5 because they are not related to the heat transfer rate.
Table 1 Five cases discussed in the present work. Cases
Case description
Case I
THEs and THENs with prescribed stream inlet temperatures and heat capacity flow THEs and THENs with prescribed stream inlet temperatures and prescribed ratios of the heat transfer rate to the heat capacity flow rates THEs and THENs with prescribed heat transfer rate One dimensional heat transfer (Fig. 5) Volume-to-Point problem [38]
Case II
Fig. 5. One-dimensional heat transfer diagram.
Case III Case IV Case V
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391
Table 2 The relationships of the entropy generation Sg, entropy generation number NS, revised entropy generation number NRS, entransy dissipation Gdis, entransy dissipation number NG and EDB thermal resistance R with the heat transfer performance for the five cases. Concept
Case I
Case II
Case III
Case IV
Case V
Optimization objective
Sg NS NRS Gdis NG R
non-monotonic non-monotonic monotonic non-monotonic monotonic monotonic
monotonic constant constant monotonic constant monotonic
monotonic monotonic monotonic monotonic monotonic monotonic
non-monotonic not applicable constant non-monotonic constant monotonic
monotonic not applicable monotonic monotonic e monotonic
min e min min e min
The last case is a conduction problem with a prescribed heat transfer rate. We take the volume-to-point problem [38] as an example. The volume has a uniform heat source and the boundary is adiabatic except for a small piece (point) which is fixed at a constant temperature. The entropy generation number is again not applicable since there is no heat capacity flow. The entransy dissipation is equivalent to the thermal resistance, while the entropy generation is equivalent to the revised entropy generation number because the heat transfer rate (that is, the total heat production in the volume) is prescribed. Chen et al. [38] found that the minimum EDB thermal resistance principle gives a smaller average temperature but a larger difference in the thermodynamic potential (reciprocal of temperature) in the domain than the minimum entropy generation principle. The entropy generation is reduced by decreasing the thermodynamic potential difference, not the temperature difference. Minimizing the entropy generation does not reduce the heat transfer temperature difference or increase the heat transfer rate, but reduce the difference in the thermodynamics potential (i.e., the reciprocal of the absolute temperature) [16]. The minimum EDB thermal resistance corresponds to the minimum average temperature for the volume-to-point problem [11,38]. The concepts of entransy dissipation and EDB thermal resistance are then more suitable for heat transfer optimization than the concepts of entropy generation and revised entropy generation number when the heat transfer rate is prescribed. For the entransy dissipation number, its applicability to the Volume-to-Point problem needs further discussion because there is no report on this topic. The five cases discussed in this paper are listed in Table 1. The relationships of the entropy generation, the entropy generation number, the revised entropy generation number, the entransy dissipation, the entransy dissipation number, the EDB thermal resistance and the heat transfer performance are given in Table 2. The EDB thermal resistance is the most suitable for optimizing heat transfer problems while the other concepts are only suitable for certain classes of problems. The physical meaning of the EDB thermal resistance is the resistance to the heat flow for a given temperature difference. Naturally, a smaller resistance will give better heat transfer. 4. Conclusions The suitability of the entropy generation, entropy generation number, revised entropy generation number, entransy dissipation, entransy dissipation number and EDB thermal resistance for optimizing THE and THEN designs is discussed in this paper. The investigations demonstrate that only the revised entropy generation number, the entransy dissipation number and the EDB thermal resistance decrease with increasing effectiveness for THEs and THENs among the six parameters for prescribed heat capacity flow rates and stream inlet temperatures. Only the EDB thermal resistance decreases with increasing heat transfer rate while the entropy generation and entransy dissipation increase for prescribed inlet temperatures and the ratios of the heat transfer
D(1/T) D(1/T) D(T) or max (Q) D(T) or max (Q)
rate to the heat capacity flow rates. The entropy generation number, the entransy dissipation number and the revised entropy generation number remain constant. All six parameters decrease with increasing effectiveness for THEs and THENs with prescribed heat transfer rate. The results further show that the EDB thermal resistance is the most suitable for optimizing of THE and THEN designs. It decreases with increasing effectiveness if the effectiveness is to be optimized, or decreases with increasing heat transfer rate if the effectiveness is fixed. The application of all six concepts to other heat transfer problems is also discussed with two examples. For one-dimensional heat transfer problem with prescribed temperatures, the maximum entropy generation, the maximum entransy dissipation and the minimum thermal resistance all correspond to the maximum heat transfer rate while the revised entropy generation number and the entransy dissipation number remain constant. The minimum EDB thermal resistance and the maximum entransy dissipation give the same result, but the minimum entropy generation and the revised entropy generation are not applicable. For the volume-to-point problem, the minimum EDB thermal resistance and the minimum entransy dissipation result in a smaller average temperature in the volume than the minimum entropy generation and the minimum revised entropy generation number. The physical meaning of the EDB thermal resistance is clear and it is directly related to the heat transfer while the entropy generation is directly related to the irreversibility. The EDB thermal resistance is the only parameter among the parameters analyzed here suitable for optimizing all five heat transfer problems. Acknowledgement The present work was supported by the Natural Science Foundation of China (Grant No. 51106082) and the Tsinghua University Initiative Scientific Research Program. The authors thank Professor Zeng-Yuan Guo for his helpful discussions and suggestions. References [1] Qian XD, Li ZX. Analysis of entransy dissipation in heat exchangers. Int J Thermal Sci 2011;50:608e14. [2] Chen Q, Wang M, Pan N, Guo ZY. Optimization principles for convective heat transfer. Energy 2009;34:1199e206. [3] Wang RZ, Xia ZZ, Wang LW, Lu ZS, Li SL, Li TX, et al. Heat transfer design in adsorption refrigeration systems for efficient use of low-grade thermal energy. Energy 2011;36:5425e39. [4] Kaluri RS, Basak T. Entropy generation due to natural convection in discretely heated porous square cavities. Energy 2011;36:5065e80. [5] Xu MT. The thermodynamic basis of entransy and entransy dissipation. Energy 2011;36:4272e7. [6] Cheng XT, Xu XH, Liang XG. Application of entransy to optimization design of parallel thermal network of thermal control system in spacecraft. Sci China Tech Sci 2011;54:964e71. [7] Hauer A. Adsorption system for TES-design and demonstration projects. Therm Energy Stor Sust Energy Cons 2007;234:409e27. [8] Kreider JF. Heating and cooling of buildings: design for efficiency. New York: McGraw-Hill; 1994. [9] Bejan A. Advanced engineering thermodynamics. 2nd ed. New York: John Wiley & Sons; 1997.
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