Optimization strategies of heat transfer systems with

International Journal of Heat and Mass Transfer 112 (2017) 137–146

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

Optimization strategies of heat transfer systems with consideration of heat transfer and flow resistance Qun Chen ⇑, Yi-Fei Wang, Meng-Qi Zhang 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 27 December 2016 Received in revised form 20 April 2017 Accepted 25 April 2017

Keywords: Heat transfer system Heat transfer enhancement Flow resistance reduction Pareto Optimality Equivalent thermal circuit diagram

a b s t r a c t Performance optimization of heat transfer systems with consideration of both heat transfer and flow resistances is critical for energy conservation. This paper compares two different optimization strategies to trade off heat transfer enhancement and flow resistance reduction. One is to convert multiple objectives to a single one by some individual optimization criteria, such as minimization of entropy generation and other dimensionless entropy generation-based numbers, and the other is to select the maximum heat transfer rate or the minimum flow resistance as the objective directly and adapt the others as constraints by Pareto Optimality. After optimizing a practical multi-loop heat exchanger network (HXN), the optimized results by the former strategy with individual optimization criteria are probably unsuitable for practical operations. Nevertheless, it needs the energy conservation and heat transfer equations of all heat exchangers as the constraints, which makes the optimization more complex. Oppositely, the latter strategy provides a series of maximum heat transfer rates with different flow resistances, i.e. Pareto front, which can be applied according to different practical requirements. The equivalent thermal circuit diagram offers the systematical constraint without involving any intermediate fluid temperatures. The systematic constraint shows advantages and convenience to optimize HXN with consideration of heat transfer and flow resistance, which is pivotal and general for the synergy of heat transfer enhancement and flow resistance reduction in heat transfer system optimization. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Heat transfer enhancement is promising in almost all thermal systems for energy conservation, but it does not always work due to the increased fluid flow resistance [1,2]. That is, flow resistance should also be taken into consideration for thermal system optimization to achieve the synergy of heat transfer and flow resistance [3]. Because heat transfer processes occur in heat exchangers and heat exchangers are the basic units in thermal systems, it is necessary to consider flow resistance from three levels: (1) the flow resistance during a heat transfer process, (2) the pressure drop in a heat exchanger, and (3) the pumping power consumption in a thermal system. For heat transfer processes, in order to estimate the heat transfer performance from the aspects of both heat transfer and viscous effects, Bejan [4] introduced the concept of entropy generation into heat transfer analysis, and used the entropy generation minimization (EGM) as a criterion for heat transfer process optimization, where the total entropy generation was the sum of those caused ⇑ Corresponding author. E-mail address: [email protected] (Q. Chen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.04.117 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

by both heat transfer and flow resistance. Many researchers applied the entropy generation rate and some other dimensionless entropy generation numbers as criteria to analyze or optimize heat transfer components. For example, Poulikakos and Bejan [5] optimized the fin geometry by minimize the entropy generation, Sekulic et al. [6] analyzed the irreversibility phenomena associated with heat transfer and fluid friction in laminar flows, Sara et al. [7] optimized several rectangular channels with square pin fins with the minimization of entropy generation, and Siavashi et al. [8] analyzed the heat transfer characteristics of natural convection processes in porous media with entropy generation. Meanwhile, Paoletti et al. [9] proposed the criterion of Bejan number and Mahmud et al. [10] derived the corresponding expression for some typical convective heat transfer processes to evaluate the ratio of entropy generations due to heat transfer and flow resistance. Thereafter, it is natural to apply the entropy generation-based criteria in heat exchanger analysis to take both heat transfer and flow resistance into consideration, and then obtain the target of lower investment and operating costs [11–14]. Recently, Manjunath and Kaushik [15] reviewed more than 100 literatures on the second law of thermodynamic analysis of heat exchangers to highlight the importance of second law investigation for heat

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Nomenclature A a Be b cp d f H Hs Hd K k L m P Q R S Sg T a, b, c

area, m2 characteristic parameter of VSP Bejan number dynamic coefficient of head loss specific heat at constant pressure, J/(kg K) diameter, m Darcy friction-factor head loss, m static head, m dynamic head, m minor loss coefficient heat transfer coefficient, W/(m2 K) length, m mass flow rate, kg/s energy consumption, W heat transfer rate, W thermal resistance, K/W cross-sectional areas of the pipe, m2 entropy generation, W/K temperature, K Lagrange multipliers

exchangers, where entropy generation, exergy destruction, and Bejan number were acknowledged as the basic performance criteria. Laskowski et al. [16] applied the entropy generation rate as a thermodynamic objective to optimize the diameter of condenser tubes. Besides, heat exchangers always serve as fundamental components in thermal systems [17], where flow resistances exist in both heat exchangers and transport pipelines [18]. Therefore, several flow resistance models for different thermal systems were deduced. They were also introduced into the optimization objectives with the aid of entropy generation. For instance, Huang et al. [19] optimized the mass flow rates of working fluids and the geometrical parameters of pipes in a vertical U-tube ground heat exchanger network (HXN) with the criterion of entropy generation minimization. Shojaeefard et al. [20] optimized a fin-and-flat tube condenser with entropy generation number to obtain the optimal heat transfer rate with fixed pressure drop. Simultaneously enhancing heat transfer and reducing flow resistance are actually a multi-objective optimization problem. The aforementioned studies focused on introducing different evaluation criteria, including entropy generation rate [4,5,13,14,21] and entropy generation-based dimensionless numbers [9,15], to convert multiple objectives to a single one. However, due to different practical applications, the optimization objectives of different thermal systems differ. In this case, a single optimization criterion is hard to correspond to all different optimization objectives. It is why several different entropy generation-based numbers are proposed to make them have a better correspondence with practical optimization objectives as is indicated in Manjunath and Kaushik’s review paper [15], but no one could cover all the cases. What’s more, the least total entropy generation rate in the entire system were always obtained by minimize the entropy generation of each component separately. However, optimization of complex heat transfer systems cannot be done by using the optimized results with specific objectives obtained from subsequent heat transfer processes and elemental exchangers when operating standalone [22]. On the other hand, besides converting multiple objectives to a single one by regarding flow resistance as a part of criteria, the

g n

P

q

u

dimensionless number heat exchanger effectiveness Lagrange function density, kg/m3 dimensionless number

Subscripts c cold fluid d dynamic e evaporator h hot fluid i inlet; the ith one m mixing process o outlet p pressure differential s static t total Dp pressure difference DT temperature difference

multi-objective optimization can also be solved through Pareto Optimality [23], a method to provide a series of optimal solutions for further selection with different preferences after optimization. In this optimization method, one objective is selected for optimization, e.g. heat transfer rates, and the other objectives, e.g. pressure drops, are given a set of choices and be considered as constraints. For optimization of a heat transfer process, Mereu et al. [24] maximized the overall thermal conductance of a 2-D space filled with a stack of heat generating boards under different pressure drops, which is cooled by forced convection. Yilmaz et al. [25] optimized the geometrical structures of ducts to maximize the convective heat transfer coefficient with a given pressure loss. In these studies, some simplifications were required to find the explicit relations between the heat transfer rates and the pressure drops. Instead, Chen et al. [26,27] derived a general quantitative relation between the boundary heat transfer coefficient and other local physical parameters over the entire heat transfer domain by the concept of entransy dissipation [28], which was hard to derive by conventional heat transfer analysis. Based on this relation together with the variational principle, they obtained different optimal fluid flow fields for different heat transfer processes to guide the design of different heat transfer facilities [29]. Similarly, the pressure drop was also adopt as a constraint for heat exchanger optimization. Mioralli and Ganzarolli [30] optimized the heat capacity rates of working fluids in a rotary regenerator with fixed pressure drops, and Kara and Guraras [31] optimized the heat transfer area of a shell-and-tube heat exchanger with various specified pressure drops. In addition, in order to consider the flow resistances in fluid transportation pipelines simultaneously, Rhee et al. [32] provided an emulation method to evaluate the performance of a hydronic radiant heating system by combining the hydronic balance with a flow limit valve and a pressure differential control valve. Wemhoff and Frank [33] optimized the energy efficiency of a heating, ventilating, and air conditioning system with consideration of flow resistance balance. In summary, there are two main strategies to trade off heat transfer enhancement and flow resistance reduction. One is to convert multiple objectives to a single one by regarding the flow resistance as a part of different evaluation criteria, where entropy

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generation rate and some other dimensionless entropy generationbased numbers are widely applied. The other is to adapt the heat transfer rate or the pressure drop as constraints to realize multiobjective optimization. The former strategy sets the weights of multiple objectives in advance of optimization, while the latter provides a series of optimal solutions for further selections based on different practical applications. In this paper, the above two strategies to trade off heat transfer enhancement and flow resistance reduction are compared by optimizing a practical multi-loop HXN, where the characteristic parameters of pumps and pipelines are obtained experimentally. Adapting (1) the minimization of entropy generation rate or entropy generation-based numbers as objective, (2) the maximum heat transfer rate as objective directly with different given total pump power consumption rates by Pareto Optimality, and (3) the minimization of entropy generation rate or entropy generationbased numbers as objective with the constraint of prescribed flow resistances, respectively, give different optimal operating parameters for the HXN. Comparison of the optimized results points out a suitable and convenient strategy to trade off heat transfer enhancement and flow resistance reduction in practical HXNs.

2. The structure of a heat exchanger network Fig. 1 is the sketch of a typical HXN, which is referenced to the experimental HXN in Ref. [34], which consists of two counter-flow plate heat exchangers, three variable speed pumps (VSPs), a thermostatic hot water tank and a chiller. In the sketch, T is the temperature, the subscripts h and c represent hot and cold fluids, i and o mean inlets and outlets of heat exchangers, and 1, 2 and e stand for heat exchanger 1, heat exchanger 2 and the evaporator, respectively. In this HXN, the VSPs drive the working fluids to circulate in each loop, and transfer heat from the thermostatic hot

Th1,o Heat Exchanger 1

Tc1,o

3. Optimization by the strategy of converting multiple objectives to a single one The enhancement of heat transfer and the reduction of flow resistance are two objectives for the optimization of the HXN, where the common strategy is converting these two objectives into a single objective. Due to the additivity, entropy generation rate can be applied to evaluate the irreversibilities of heat transfer processes, components, and further complex systems. Meanwhile, for the HXN, the total entropy generation rate is the sum of those caused by both heat transfer processes and flow transportation processes in all the components. Therefore, entropy generation rate [4,5,13,14,21] and entropy generation-based dimensionless numbers [9,15], are widely applied as criteria for heat transfer optimization. 3.1. Expressions of entropy generation and entropy generation-based numbers The entropy generation rates caused by the finite temperature difference in a heat exchanger is expressed as [21,35]

Sg;DT ¼ mh cp;h ln

T h;o T c;o þ mc cp;c ln : T h;i T c;i

ð1Þ

where m represents the mass flow rate, and cp represents the constant pressure specific heat. Ignoring the heat leakage in the pipelines, Tc1,o, Tc1,i, Tc2,o, Tc2,i, and Th1,i equals Th2,i, Th2,o, The,i, The,o, and Th, respectively. In this case, the total entropy generation rate in the HXN caused by the finite temperature differences is

Tank Th1,i

tank to heat exchanger 1, heat exchanger 2, and finally to the evaporator of the chiller. For this practical HXN, the heat transfer areas of each heat exchanger and the characteristic curves of pumps and pipelines are fixed, which can be determined experimentally. If the tank temperature, Th, and the evaporation temperature, Te, are constant, the optimization objective is to provide the optimal mass flow rates of working fluids in each pipeline to enhance heat transfer and reduce flow resistance simultaneously.

VSP1

Sg;DT;t ¼ Sg;DT;hx1 þ Sg;DT;hx2 þ Sg;DT;e þ Sg;DT;tank ¼

Q Q  : T e T h1;i

ð2Þ

On the other hand, the entropy generation rate caused by the flow resistance in a typical counter-flow plate heat exchanger, shown in Fig. 2, is

Tc1,i

Sg;Dp ¼ Sg;Dp;hx þ Sg;Dp;pipe :

ð3Þ

Th2,o

Th2,i

VSP2

Heat Exchanger 2 Tc2,o

Tc2,i

The,i

a

The,o VSP3

Evaporator Te

Te Chiller

Fig. 1. The sketch of the heat exchanger network [34].

Tc,i

mc

Tc,o d

b

c

g

f

h

e Th,o

mh

Th,i

Fig. 2. The sketch of a typical counter-flow plate heat exchanger.

140

Sg;Dp;hx ¼

Q. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 137–146

  fL X þ K i : 2g q2 S2 d

mh Dpf g mc Dpbc T c;o T h;o þ : ln ln qc ðT c;o  T c;i Þ T c;i qh ðT h;o  T h;i Þ T h;i

ð4Þ

b¼

mc Dpab mc Dpcd mh Dpef mh Dpgh þ þ þ : qc T c;i qc T c;o qh T h;i qh T h;o

ð5Þ

Then the total entropy generation rate in the HXN caused by the flow resistance is

Sg;Dp;pipe ¼

1

ð13Þ



where Sg,Dp,hx is the entropy generation rate caused by the flow resistance in the heat exchanger [21,35], Sg,Dp,pipe is the entropy generation rate caused by the flow resistance in the transportation pipe, q stands for the density, and Dp represents the pressure differential. The subscripts a–b, b–c, c–d, e–f, f–g, and g–h stand for the corresponding pipelines in Fig. 2. It can be assumed that the flow resistances of parts a–b, b–c, and c–d, are all equal, and the flow resistances of parts e–f, f–g, and g–h, are also equal. In the whole circulating loop, for example, the circulating loop 2 driven by VSP2, the flow resistance in the pipe part a–d is assumed to be equal to that in the remaining part of the circulating loop d–a. Then the flow resistances of different pipeline parts can be deduced as

Sg;Dp;t ¼

 1 T h1;o 1 1 m1 gðHs1 þ b1 m21 Þ þ þ ln 3ðT h1;o  T h1;i Þ T h1;i 3T h1;i 3T h1;o   1 T h2;o 1 1 m2 gðHs2 þ b2 m22 Þ þ þ þ ln 3ðT h2;o  T h2;i Þ T h2;i 3T h2;i 3T h2;o   1 T he;o 1 1 þ m3 gðHs3 þ b3 m23 Þ: þ þ ln 3ðT he;o  T he;i Þ T he;i 3T he;i 3T he;o ð14Þ

Combination of Eqs. (2) and (14) gives the total entropy generation rate in the HXN Sg;t ¼ Sg;DT;t þ Sg;Dp;t 2



3 m1 gðHs1 þ b1 m21 Þ h1;i h1;i h1;o 6  7  6 7 Q Q T h2;o 1 1 1 2 7 ¼  þ6 ln þ þ gðH þ b m Þ þ m 2 s2 2 2 3ðT 3T 3T T Þ T 7: h2;o h2;i h2;i h2;i h2;o T e T h1;i 6 4  5  T he;o 1 1 1 2 þ 3ðT T Þ ln T þ 3T þ 3T m3 gðHs3 þ b3 m3 Þ 1 3ðT h1;o T h1;i Þ

T

ln Th1;o þ 3T1 þ 3T1

1 1 Dpab ¼ Dpbc ¼ Dpdd ¼ Dpad ¼ Dploop;ada : 3 6

ð6Þ

1 1 Dpef ¼ Dpf g ¼ Dpgh ¼ Dpeh ¼ Dploop;ehe : 3 6

ð7Þ

ð15Þ

For the HXN, the entropy generation rates caused by the flow resistances in the heat exchanger and the pipeline are deduced as

Besides the entropy generation rate, in order to optimize the heat transfer systems with some specific objectives, some other entropy generation-based dimensionless numbers are also applied in thermal system optimization extensively. For instance, dividing the entropy generation rate by the ratio of the heat transfer rate to the environment temperature, Witte and Shamsundar [39] defined a evaluation parameter, gW–S, as

Sg;Dp;hx;t ¼

he;o

m1 DpVSP1 T h1;o m2 DpVSP2 T h2;o þ ln ln 3q1 ðT h1;o  T h1;i Þ T h1;i 3q2 ðT h2;o  T h2;i Þ T h2;i þ

Sg;Dp;pipe;t ¼

m3 DpVSP3 T he;o : ln 3q3 ðT he;o  T he;i Þ T he;i

ð8Þ

gWS ¼

m1 DpVSP1 m1 DpVSP1 m2 DpVSP2 m2 DpVSP2 þ þ þ 3q1 T h1;i 3q1 T h1;o 3q2 T h2;i 3q2 T h2;o þ

m3 DpVSP3 m3 DpVSP3 þ : 3q3 T he;i 3q3 T he;o

ð9Þ

where DpVSP1, DpVSP2, and DpVSP3 are the pressure drops of each pipeline, provided by VSP1, VSP2, and VSP3, respectively. Then the total entropy generation rate in the HXN caused by the flow resistance is deduced as

Sg;Dp;t ¼ Sg;Dp;hx;t þ Sg;Dp;pipe;t 2 3 T T T m1 DpVSP1 ln Th1;o þ 3q mðT2 DpVSP2 ln Th2;o þ 3q mðT3 DpVSP3 ln The;o : 3q1 ðT h1;o T h1;i Þ h1;i h2;o T h2;i Þ h2;i he;o T he;i Þ he;i 2 3 5 ¼ 4 m Dp þ 31q TVSP1 þ m31qDTpVSP1 þ m32qDpTVSP2 þ m32qDTpVSP2 þ m33qDpTVSP3 þ m33qDpTVSP3 1 h1;i

1 h1;o

2 h2;i

2 h2;o

3 he;i

3 he;o

ð10Þ Meanwhile, the pressure drop is always evaluated by the concept of head loss, H, which is a function of mass flow rate [36,37]

! X f Li X Dp m2 i H¼ þ Ki : ¼ Hs þ Hd ¼ Hs þ qg di 2g q2 S2 i i

ð11Þ

where Hs is the static head unrelated to the fluid flow rate, Hd is the dynamic head depending on the fluid flow rate, f is the Darcy friction-factor, L is the length of pipe, d is the pipe diameter, and K is the minor loss coefficient of a detail structure, such as regular 90° elbow, sharp entrance, sharp exit, and valve. The subscript i means different pipeline sections. For a given pipeline, the static head, the density of the working fluid, and the length/diameter of the pipe are all fixed in advance, and the Darcy friction-factor can be considered constant with a small variation of flow velocity. Then the head loss is simplified, with a coefficient b, as [34,38] 2

H ¼ Hs þ bm ;

ð12Þ

he;i

Sg;t T e : Q

he;i

he;i

he;o

ð16Þ

In order to describe the relative importance of the two irreversibility mechanisms caused by heat transfer and flow resistance, Bejan [40] defined the concept of irreversibility distribution ratio, u, which is the ratio of fluid flow irreversibility to heat transfer irreversibility

u¼

Sg;DP;t : Sg;DT;t

ð17Þ

Meanwhile, Paoletti et al. [9] defined Bejan number, Be, as the ratio of the entropy generation rate caused by heat transfer to the total entropy generation

Be ¼

Sg;DT;t 1 ¼ : Sg;DT;t þ Sg;DP;t 1 þ u

ð18Þ

3.2. Optimization with the extremum of entropy generation rate and entropy generation-based numbers For the HXN shown in Fig. 1, the total entropy generation rate are functions of multi-parameters, including all the mass flow rate, i.e. m1, m2, and m3, and many intermediate temperatures, i.e. Th1,o, Th2,i, Th2,o, The,i, and The,o. In order to seek the minimum entropy generation rate, two methods are extensively applied. One is to minimize the entropy generation rate in each component, such as each heat exchanger. In this method, the starting-point is that entropy generation minimization of each component leads to the least total entropy generation rate in the entire system, since the total entropy generation rate in the entire system is the sum of the entropy generation rate in each component. As is mentioned in the introduction, optimization of a complex heat transfer system cannot be done by using the optimized results with specific objec-

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tives obtained from subsequent heat transfer processes and elemental exchangers when operating standalone [22]. The other method is to minimize the entropy generation rate with some system constraints, including energy conservation and heat transfer equations shown in Eqs. (19) and (20).

Q ¼ m1 cp;1 ðT h1;i  T h1;o Þ ¼ m2 cp;2 ðT h1;i  T h2;o Þ ¼ m3 cp;3 ðT he;i  T he;o Þ: Q ¼ ðkAÞ1

ð19Þ

ðT h1;o  T h2;o Þ  ðT h1;i  T h2;i Þ T

T

ln Th1;o T h2;o h1;i

¼ ðkAÞ2

h2;i

ðT h2;o  T he;o Þ  ðT h2;i  T he;i Þ T

T

ln Th2;o T he;o h2;i

¼ ðkAÞe

he;i

T he;i  T he;o T

T

ln T he;i Tee

:

ð20Þ

he;o

where k is the heat transfer coefficient, and A is the heat transfer area. For seeking the extremum of entropy generation rate and entropy generation-based numbers, different Lagrang functions are built as



result illustrates that the strategy by converting multiple objectives of enhancing heat transfer and reducing flow resistance into a single objective with the extremum of entropy generation rate cannot always work for heat transfer systems. Optimization with the extremum of u, Be, and gW-S provides the optimized mass flow rates of each working fluid, listed in Table 2. The total power consumption rate is much larger than the heat transfer rate. For example, when u reaches the extremum, the total power consumption (7539.07 W) is 5.3 times of the heat transfer rate (1432.9 W). In addition, in these optimized cases, the rotation speeds of the three VSPs are all overload. The results imply that the aforementioned entropy generation-based numbers are not always suitable in practice.







PF ¼ F þ a1 Q  m1 cp;1 ðT h1;i  T h1;o Þ þ a2 Q  m2 cp;2 ðT h2;i  T h2;o Þ 

 þ a3 Q  m3 cp;3 ðT he;i  T he;o Þ 2 3 ðT  T Þ  ðT  T Þ h1;o h2;o h1;i h2;i 5 þ a4 4Q  ðkAÞ1 T T ln Th1;o T h2;o h1;i h2;i 2 3 ðT  T Þ  ðT  T Þ h2;o he;o h2;i he;i 5 þ a5 4Q  ðkAÞ2 T T ln Th2;o T he;o h2;i he;i 2 3 T he;i  T he;o 5 þ a6 4Q  ðkAÞe F ¼ Sg ; u; Be ; gWS ; T T ln T he;i Tee

4. Optimization by the strategy of Pareto Optimality For optimization with multiple objectives, Pareto Optimality [23] treats one target as the optimization objective, and regards other objectives as constraints. For the optimization problem mentioned above, i.e. to enhance heat transfer and reduce flow resistance simultaneously, the heat transfer rate of the HXN can be treated as the optimization objective, while the flow resistance is regarded as the constraint. 4.1. The thermal circuit diagram of the HXN

ð21Þ

he;o

where ai (i = 1, 2, 3, . . ., 6) are Lagrange multiplies. Making the partial derivations of these 4 Lagrange functions with respect to mi, Q, Th1,o, Th2,i, Th2,o, The,i, and The,o, equal to zero, offers the partial derivation equations, where the ones of entropy generation rate are Eqs. (a1)–(a9), presented in Appendices A.1. The partial derivations equations and the constraints, i.e. Eqs. (19) and (20), can be combined as an equation set, which has 14 equations together with 14 unknown parameters, i.e. mi, ai, b, iTh1,o, Th2,i, Th2,o, The,i, and The,o. Solving the equation set simultaneously by some traditional numerical methods will get the optimal values of all the unknown parameters. For the given HXN shown in Fig. 1, the thermal conductances of heat exchanger 1, 2 and the evaporator, are 182.6 W/K, 143.6 W/K, and 56.0 W/K, respectively. The temperature in thermostatic hot water tank and the evaporator are 47.0 °C (320.15 K) and 2.0 °C (275.15 K), and the specific heat at constant pressure of the working fluids in the three loops are 4196 J/(kg K), 1304 J/(kg K), and 4196 J/(kg K), respectively. Table 1 offers the resistance characteristic parameters of each pipeline. The equation set for seeking the extremum of entropy generation rate, i.e. Eqs. (a1)–(a9), (19) and (20), has no solution, which means there is no extremum of the entropy generation rate for the HXN. When the total mass flow rate of all working fluids is zero, i.e. the system does not work, the total entropy generation rate is the minimum. Oppositely, when the total mass flow rate is infinite, the total entropy generation rate is the maximum. This Table 1 The resistance characteristic parameters of each pipeline [34]. Loop

Hs (m)

b (m s2/kg2)

Loop 1 Loop 2 Loop 3

0.57 0.10 0.36

2623.6 1205.2 413.0

In addition to the constraint of flow resistance, the constraints of heat transfer processes are also required. In Section 3, the introduced intermediate temperatures, i.e. Th1,o, Th2,i, Th2,o, The,i, and The,o, require the detail equations of each heat transfer process, including energy conservation equations and heat transfer equations, shown in Eqs. (19) and (20). However, if the heat transfer constraints among the design requirements, the operating parameters and the structural parameters of the HXN is built from the system level, it is unnecessary to introduce each intermediate temperature. Based on the electrical circuit analogy for analysis of heat transfer processes in HXNs [41–44], Fig. 3 gives the equivalent thermal circuit diagram [42,43] to clearly present the heat flow directions in the HXN.

T h1;i  T e ¼ Q ðR1 þ R2 þ Re þ Rm Þ:

ð22Þ

where R1, R2, Re, and Rm are the entransy dissipation-based thermal resistances (EDTR) of the heat transfer processes in heat exchanger 1, heat exchanger 2, evaporator and the mixing process in the water tank, respectively.

R1 ¼

n1 exp½ðkAÞ1 n1  þ 1 ; 2 exp½ðkAÞ1 n1   1

n1 ¼

1 1  ; m1 cp;1 m2 cp;2

ð23Þ

R2 ¼

n2 exp½ðkAÞ2 n2  þ 1 ; 2 exp½ðkAÞ2 n2   1

n2 ¼

1 1  ; m2 cp;2 m3 cp;3

ð24Þ

Re ¼

ne exp½ðkAÞe ne  þ 1 ; 2 exp½ðkAÞe ne   1

ne ¼

Rm ¼

1 ; 2m1 cp;1

1 ; m3 cp;3

ð25Þ

ð26Þ

where n1, n2, and ne are the flow arrangement factors of heat exchanger 1, heat exchanger 2 and evaporator, m means the mixing process, and the numbers 1, 2, and 3 represent the working fluids driven by the VSP1, VSP2, and VSP3, respectively. Eq. (22) gives the direct relationship between the design requirements, the operating parameters and the structures parameters of the HXN, which can be seen as the system heat transfer constraint without any intermediate temperatures. It simplifies

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Table 2 The optimized results with the extremum of u, Be, and gW–S. Criteria

m1 (kg/s)

m2 (kg/s)

m3 (kg/s)

Pt (W)

Q (W)

u Be

0.830 0.830 0.476

0.544 0.544 0.676

0.611 0.611 0.020

7539.07 7539.07 6428.20

1432.9 1432.9 1385.7

gW-S

Q

Th1,i

Te R2

R1

Rm

Re

Fig. 3. The equivalent thermal circuit diagram of the HXN.

the optimization model in physics and the corresponding solution process in mathematics [44]. 4.2. Optimization for the maximum heat transfer rate with given total power consumption rates For the optimization of the HXN, if the total power consumption rate of all VSPs is given, shown in Eq. (30)

X mi gðHs;i þ bi m2i Þ  Pt

i ¼ 1; 2; 3;

ð27Þ

i

Optimization can be made to find the maximum heat transfer rate, where correspondingly Lagrange function is constructed

PQ ¼ Q þ a½T h  T e  QðRm þ R1 þ R2 þ Re Þ "

X mi gðHs;i þ bi m2i Þ  Pt þb

#

i ¼ 1; 2; 3;

ð28Þ

i

where a and b are the Lagrange multiplies. The partial derivations of Eq. (28) with respect to mi, and Q, equal to zero, offers equations, i.e. Eqs. (a10)–(a13), presented in Appendix A.2. These partial derivations equations and the constraints, i.e. Eqs. (22) and (27), can be combined as a equation set, which has totally 6 equations together with 6 unknown parameters, i.e. mi, a, b, and Q. The equation set can be solved by some traditional numerical methods to provide the optimal values of all the unknown parameters. With the same characteristic parameters of each components, the temperature of thermostatic hot water tank and the evaporator 1440 1430

Q/W

5. Optimization by the strategy based on Pareto Optimality with entropy generation or entropy generation-based numbers as objective Based on Pareto Optimality, if the total power consumption rate of all VSPs is given, as shown in Eq. (27), optimization can also be made to find the extremum of entropy generation rate or the extremum of entropy generation-based numbers. However, due to the existence of intermediate temperatures, i.e. Th1,o, Th2,i, Th2,o, The,i and The,o, in the expressions of entropy generation rate and entropy generation-based numbers, the heat transfer constraint based on the electrical circuit analogy, i.e. Eq. (22), is not enough, which requires the detailed constraints of each heat exchanger, i.e. energy conservation equations and heat transfer equations, Eqs. (19) and (20). With the objectives of entropy generation rate or entropy generation-based numbers, respectively, the Lagrange functions are built as







P0F ¼ F þ a1 Q  m1 cp;1 ðT h1;i  T h1;o Þ þ a2 Q  m2 cp;2 ðT h2;i  T h2;o Þ

1420



1410

Pareto front of Q-Pt

1400

Extremum of Extremum of Extremum of

1390 1380 20

as 47.0 °C (320.15 K) and 2.0 °C (275.15 K). Fig. 4 shows the maximum heat transfer rates with different total power consumption rates of all VSPs, varying from 30 to 130 W. The curve of the maximum heat transfer rates with different total power consumption rates can be summarized as a Pareto front, where the points on the curve provide the maximum heat transfer rates under the corresponding total power consumption rates. Based on the Pareto front, operation parameters can be chosen for specific requirements. For example, if the heat transfer rate of this HXN system is required as 1430 W, the Pareto front can provide the optimal operating condition with the least total power consumption, i.e. 100 W. In addition, the points below the curve mean the operating parameters are less efficient for energy conservation, which can be optimized for more heat transfer rates or less total power consumption rates, and the points above the curve are impossible to achieve for this HXN. Fig. 4 also gives the optimal cases obtained from extremum of u, Be, and gW-S, which are below the Pareto front obviously. These results mean less heat transfer rates are provided with the same power consumption rates or more power are consumed with the same required heat transfer rates, which illustrates that the parameters obtained from criteria extremum do not always satisfy the optimization requirements in practice.

40

60

80

100

120

W-S

140 6000 7000 8000

P/W Fig. 4. The curve of the optimal heat transfer rate with different total power consumption rates.





þ a3 Q  m3 cp;3 ðT he;i  T he;o Þ 2 3 ðT h1;o  T h2;o Þ  ðT h1;i  T h2;i Þ5 4 þ a4 Q  ðkAÞ1 T T ln Th1;o T h2;o h1;i h2;i 2 3 ðT  T Þ  ðT  T Þ h2;o he;o h2;i he;i 5 þ a5 4Q  ðkAÞ2 T T ln Th2;o T he;o h2;i he;i 2 3 " # X T  T he;i he;o 2 5þb þ a6 4Q  ðkAÞe m gðH þ b m Þ  P i s;i i i t T T ln T he;i Tee i he;o

i ¼ 1; 2; 3

F ¼ Sg ;

u; Be ; gWS ;

where ai (i = 1–6) and b are the Lagrange multiplies.

ð29Þ

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Q. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 137–146

1.2

0.9 0.8

1.1

0.7 0.6

Be

1.0

0.5

W-S

W-S

0.4

Be

Sg/(W/K)

Sg

0.3

0.9

0.2 0.8 20

40

60

80

100

0.1 140

120

Pt/W Fig. 5. The curve of Sg, u, Be and gW-S with different total power consumption rates.

1440 1430 1420

Q/W

1410 Q Sg

1400

W-S

1390

Be 1380 1370 20

40

60

80

100

120

140

P/W Fig. 6. The heat transfer rates obtained from the extremum of heat transfer rate, entropy generation rate, and entropy generation-based numbers with different total power consumption rates.

The partial derivation equations of the Lagrange function with entropy generation rate, with respect to mi, Q, Th1,o, Th2,i, Th2,o, The,i, and The,o, are presented in Appendices A.3. These partial derivations equations and the constraints, including Eqs. (19), (20), and (27), are combined as the optimization equation set, which has totally 15 equations together with 15 unknown parameters, i.e. mi, ai, b, Th1,o, Th2,i, Th2,o, The,i, and The,o. Solving the equation set will get the optimal values of all the unknown parameters with different total power consumption rates of all VSPs, varying from 30 to 130 W. Fig. 5 provides the curves of Sg, u, Be and gW–S, with different total power consumption of all VSPs, varying from 30 W to 130 W.

The curves of these criteria with different total power consumption rates can also be seen as Pareto fronts, however, these Pareto fronts cannot exhibit the variation tendencies of specific requirements, such as the requirement for the largest heat transfer rate under the given power consumption. Fig. 6 provides the comparison of the heat transfer rates obtained from the extremum of heat transfer rate, entropy generation rate, and entropy generationbased numbers, with different total power consumption rates. It’s obvious that, under all the given total power consumption rates, the heat transfer rates obtained from the extremum of heat transfer rate are larger than those obtained from the extremum of entropy generation rate and other entropy generation-based numbers. With the assumption of the same working fluid properties, Table 3 lists the heat transfer rates obtained from the extremum of heat transfer rate, entropy generation rate, and other entropy generation-based numbers, in other two working conditions, where the deviations of the optimized heat transfer rates occur. In specific working conditions, such as low temperatures and high heat transfer rates, the deviations might be larger and more obvious. For example, when the temperature difference between the hot and cold fluids becomes larger, i.e. Th1,i is 355.15 K and Te is 275.15 K, the differences of heat transfer rates with different objectives become larger. when the working temperatures become lower, i.e. Th1,i is 220.15 K and Te is 175.15 K, the differences of heat transfer rates with different objectives also become larger. The results illustrate the optimization objectives of specific requirements cannot be replaced by entropy generation rate or other entropy generation-based numbers. Entropy generation rate and entropy generation-based numbers are not the general objectives for the trade-off of the heat transfer and the flow resistance in heat transfer systems, where the objectives should be chosen based on specific requirements. In addition, the application of entropy generation or entropy generation-based numbers inevitably brings many intermediate temperatures [44], including Th1,o, Th2,i, Th2,o, The,i, and The,o, into the optimization, because entropy generation and entropy generation-based numbers are defined based on absolute temperatures. These intermediate temperatures, which are actually unnecessary to know for the global system optimization. Meanwhile, the constraints from the component levels, i.e. the energy conservation equations and heat transfer equations of each heat exchanger, Eqs. (19) and (20), obstruct the global optimization and burden the mathematical computations. On the contrary, the global system heat transfer constraint through the thermal circuit diagram, i.e. Eq. (22), can discover the direct relations of each independent design parameter in the system, which eliminates unknown intermediate fluid temperatures, reduces the corresponding constraint equations from (15) to (6), and simplifies the optimization processes [44]. Despite the various objectives and requirements, the global system constraint is not varying. That is, the global system constraint, not the optimization objective, is the key to trade off the enhancement of heat transfer and the reduction of flow resistance in heat transfer systems.

Table 3 The optimized heat transfer rates with different objectives under two specific conditions. Condition

Pt 100 W; Th1,i 355.15 K; Te 275.15 K Pt 100 W; Th1,i 220.15 K; Te 175.15 K

Heat transfer rate of different objectives (W) Q

Sg

u

Be

gW–S

2543.9

2515.1 (1.2%) 1402.6 (2.0%)

2495.7 (1.9%) 1390.5 (2.9%)

2495.7 (1.9%) 1390.5 (2.9%)

2490.2 (2.2%) 1383.3 (3.4%)

1430.6

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Q. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 137–146

@ PSg m2 gðHs2 þ b2 m22 Þ T h2;o m2 gðHs2 þ b2 m22 Þ ¼ ln  @T h2;i T h2;i 3T h2;i ðT h2;o  T h2;i Þ 3ðT h2;o  T h2;i Þ2

6. Conclusions This paper compares two different optimization strategies for heat transfer systems to trade off heat transfer enhancement and flow resistance reduction. One is to convert the multiple objectives to a single one by some optimization criteria, such as entropy generation rate and entropy generation-based numbers, and the other is to treat the heat transfer rate as optimization objective directly and regard the flow resistance as a constraint with the aid of Pareto Optimality. The extremum of entropy generation rate or other entropy generation-based numbers cannot always provide the optimal results with various different objectives, which reveals the limitation of the former strategy. The latter strategy based on Pareto Optimality and the systematic constraints with equivalent thermal circuit diagram shows the advantages and convenience to optimize HXN with various different requirements. The global heat transfer constraints of each independent design parameters without involving any intermediate fluid temperature, reducing the corresponding constraint equations and simplifies the optimization processes, which is pivotal and general for the synergy of heat transfer enhancement and flow resistance reduction in heat transfer system optimization.



m2 gðHs2 þ b2 m22 Þ 2

6 4

3T 2h2;i T

T

T h1;i T h2;i

ln Th1;o T h2;o þ ðT h1;i

 a2 m2 cp;2  a4 ðkAÞ1

h2;i

2 h1;o T h2;o Þ



T

3 ðT h1;o  T h2;o  T h1;i þ T h2;i Þ 7 5 2 T

ln Th1;o T h2;o h1;i h2;i 2 3 T T T T  ln Th2;o T he;o þ ðT h2;iT he;iÞ2 ðT h2;o  T he;o  T h2;i þ T he;i Þ h2;i he;i 6 7 h2;o he;o  a5 ðkAÞ2 4 5 ¼ 0:  2 T T ln Th2;o T he;o h2;i

he;i

ða6Þ @ PSg m2 gðHs2 þ b2 m22 Þ T h2;o m2 gðHs2 þ b2 m22 Þ ¼ ln þ 2 @T h2;o T h2;i 3T h2;o ðT h2;o  T h2;i Þ 3ðT h2;o  T h2;i Þ 

m2 gðHs2 þ b2 m22 Þ 2

6 4

3T 2h2;o T

T

þ a2 m2 cp;2  a4 ðkAÞ1 T h1;i T h2;i

 ln Th1;o T h2;o þ ðT h1;i

2 6  a5 ðkAÞ2 4

h2;i

2 h1;o T h2;o Þ



T

3 ðT h1;o  T h2;o  T h1;i þ T h2;i Þ 7 5 2 T

ln Th1;o T h2;o h1;i

T

T

h2;i

h2;i

T h2;i T he;i

ln Th2;o T he;o  ðT he;i

2 h2;o T he;o Þ



T

3 ðT h2;o  T he;o  T h2;i þ T he;i Þ 7 5 ¼ 0: 2 T

ln Th2;o T he;o h2;i

Acknowledgements

he;i

ða7Þ

The present work is supported by the National Natural Science Foundation of China (Grant No. 51422603), the National Key Technology R&D Program of China (Grant No. 2015BAA01B03), the National Natural Science Foundation of China (Grants Nos. 51356001 and 51321002). Appendix A A.1. partial derivations of the Lagrange function of Eq. (21)

@ PSg m3 gðHs3 þ b3 m23 Þ T h2;o m3 gðHs3 þ b3 m23 Þ ¼ ln  @T he;i T h2;i 3T h2;i ðT h2;o  T h2;i Þ 3ðT h2;o  T h2;i Þ2 2 3 T he;i T e T he;o T e m3 gðHs3 þ b3 m23 Þ 6ln T he;o T e  T he;i T e 7   a6 ðkAÞe 4  2 5  a2 m3 cp;3 T T 3T 2he;i ln T he;i Tee he;o 2 3 T T T T ln Th2;o T he;o þ ðT h2;iT he;iÞ2 ðT h2;o  T he;o  T h2;i þ T he;i Þ h2;i he;i 6 7 h2;o he;o  a5 ðkAÞ2 4 5¼0  2 T T ln Th2;o T he;o h2;i

he;i

ða8Þ

The partial derivations of the Lagrange function of Sg in Eq. (21), with respect to mi, Q, Th1,o, Th2,i, Th2,o, The,i, and The,o, provide

  @ PSg 1 T h1;o 1 1 ¼ gðHs1 þ 3b1 m21 Þ þ þ ln 3ðT h1;o  T h1;i Þ T h1;i 3T h1;i 3T h1;o @m1  a1 cp;1 ðT h1;i  T h1;o Þ ¼ 0;

ða1Þ

  @ PSg 1 T c1;o 1 1 ¼ gðHs2 þ 3b2 m22 Þ þ þ ln 3ðT c1;o  T c1;i Þ T c1;i 3T c1;i 3T c1;o @m2  a2 cp;2 ðT h2;i  T h2;o Þ ¼ 0;

ða2Þ

  @ PSg 1 T c2;o 1 1 ¼ gðHs3 þ 3b3 m23 Þ þ þ ln 3ðT c2;o  T c2;i Þ T c2;i 3T c2;i 3T c2;o @m3  a3 cp;3 ðT he;i  T he;o Þ ¼ 0;

ða4Þ

h2;i

he;i

ða9Þ

@ PQ ¼ 1  aðRm þ R1 þ R2 þ Re Þ ¼ 0; @Q

þ a1 m1 cp;1  a4 ðkAÞ1

3T 2h1;o 2 3 T T T T ln Th1;o T h2;o  ðT h1;iT h2;iÞ2 ðT h1;o  T h2;o  T h1;i þ T h2;i Þ h1;i h2;i 6 7 h1;o h2;o 4 5 ¼ 0:   T h1;o T h2;o 2 ln T T h1;i

A.2. partial derivations of the Lagrange function of Eq. (28) The partial derivations of the Lagrange function, Eq. (28), with respect to Q and mi, obtain

@ PSg m1 gðHs1 þ b1 m21 Þ T h1;o m1 gðHs1 þ b1 m21 Þ ¼ ln þ 2 @T h1;o T h1;i 3T h1;o ðT h1;o  T h1;i Þ 3ðT h1;o  T h1;i Þ 

he;o

þ a2 m3 cp;3  a5 ðkAÞ2 2 3 T T T T  ln Th2;o T he;o þ ðT h2;iT he;iÞ2 ðT h2;o  T he;o  T h2;i þ T he;i Þ h2;i he;i 6 7 h2;o he;o 4 5¼0  2 T T ln Th2;o T he;o

ða3Þ

@ PSg 1 1 þ a1 þ a2 þ a3 þ a4 þ a5 þ a6 ¼ 0 ¼  T e T h1;i @Q

m1 gðHs1 þ b1 m21 Þ

@ PSg m3 gðHs3 þ b3 m23 Þ T he;o m3 gðHs3 þ b3 m23 Þ ¼ ln þ @T he;o T he;i 3T he;o ðT he;o  T he;i Þ 3ðT he;o  T he;i Þ2 2 3 T he;i T e T he;i T he;o 2  ln þ m3 gðHs3 þ b3 m3 Þ T he;o T e T he;o T e 7 6   a6 ðkAÞe 4 5   T he;i T e 2 3T 2he;o ln T T e

h2;i

ða5Þ

 @ PQ 1 exp½ðkAÞ1 n1  þ 1 ¼ bðHs;1 g  3b1 m21 Þ  aQ 2 @m1 m1 cp;1 exp½ðkAÞ1 n1   1 # n1 ðkAÞ1 exp½ðkAÞ1 n1  þ 2

¼ 0; m1 cp;1 exp½ðkAÞ n1   1 2 1

ða10Þ

ða11Þ

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Q. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 137–146

@ PQ ¼ bðHs;2 g  3b2 m22 Þ @m2 2 3 1 exp½ðkAÞ1 n1  þ 1 n1 ðkAÞ1 exp½ðkAÞ1 n1   2 2 2 6 2m2 cp;2 exp½ðkAÞ1 n1   1 m2 cp;2 ðexp½ðkAÞ n   1Þ 7 6 7 1 1 þ aQ 6 7¼0 4 5 1 exp½ðkAÞ2 n2  þ 1 n2 ðkAÞ2 exp½ðkAÞ2 n2   þ 2m22 cp;2 exp½ðkAÞ2 n2   1 m22 cp;2 ðexp½ðkAÞ2 n2   1Þ2

ða12Þ @ PQ ¼ bðHs;3 g  3b3 m23 Þ @m3 2 3 1 exp½ðkAÞ2 n2  þ 1 n2 ðkAÞ2 exp½ðkAÞ2 n2   2 2 2 6 2m3 cp;3 exp½ðkAÞ2 n2   1 m3 cp;3 ðexp½ðkAÞ n   1Þ 7 6 7 2 2 þ aQ 6 7¼0 4 5 1 exp½ðkAÞe ne  þ 1 ne ðkAÞe exp½ðkAÞe ne   þ 2 2 2 2m3 cp;3 exp½ðkAÞe ne   1 m3 cp;3 ðexp½ðkAÞe ne   1Þ

ða13Þ A.3. The partial derivations of the Lagrange function of Eq. (29) The partial derivations of the Lagrange function of Sg in Eq. (29), with respect to mi, Q, Th1,o, Th2,i, Th2,o, The,i, and The,o, are



@ P0Sg 1 T h1;o 1 ¼ gðHs1 þ 3b1 m21 Þð þ ln 3ðT h1;o  T h1;i Þ T h1;i 3T h1;i @m1  1 þ Þ þ bg ðHs1 þ 3b1 m21 Þ  a1 cp;1 ðT h1;i  T h1;o Þ ¼ 0; 3T h1;o

@ P0Sg m2 gðHs2 þ b2 m22 Þ T h2;o m2 gðHs2 þ b2 m22 Þ ¼ ln þ 2 @T h2;o T h2;i 3T h2;o ðT h2;o  T h2;i Þ 3ðT h2;o  T h2;i Þ 

m2 gðHs2 þ b2 m22 Þ 2

6 4

3T 2h2;o T

T

þ a2 m2 cp;2  a4 ðkAÞ1 T h1;i T h2;i

 ln Th1;o T h2;o þ ðT h1;i

h2;i

2 h1;o T h2;o Þ

3 ðT h1;o  T h2;o  T h1;i þ T h2;i Þ 7 5 2

 T T ln Th1;o T h2;o h1;i

h2;i

2 3 T T T T ln h2;o he;o  h2;i he;i ðT  T he;o  T h2;i þ T he;i Þ 6 T h2;i T he;i ðT h2;o T he;o Þ2 h2;o 7  a5 ðkAÞ2 4 5 ¼ 0:   T h2;o T he;o 2 ln T T h2;i

he;i

ða20Þ

@ P0Sg m3 gðHs3 þ b3 m23 Þ T h2;o m3 gðHs3 þ b3 m23 Þ ¼ ln  @T he;i T h2;i 3T h2;i ðT h2;o  T h2;i Þ 3ðT h2;o  T h2;i Þ2 2 3 T he;i T e T he;o T e ln  m3 gðHs3 þ b3 m23 Þ T T T T e e7 6 he;i  a6 ðkAÞe 4 he;o  2 5 T T 3T 2he;i ln T he;i Tee he;o

 a2 m3 cp;3  a5 ðkAÞ2 2 3 T T T T ln Th2;o T he;o þ ðT h2;iT he;iÞ2 ðT h2;o  T he;o  T h2;i þ T he;i Þ h2;i he;i 6 7 h2;o he;o 4 5¼0  2 T T ln Th2;o T he;o h2;i

he;i

ða21Þ

ða14Þ  @ P0Sg 1 T c1;o 1 ¼ gðHs2 þ 3b2 m22 Þð þ ln 3ðT c1;o  T c1;i Þ T c1;i 3T c1;i @m2  1 Þ þ bg ðHs2 þ 3b2 m22 Þ  a2 cp;2 ðT h2;i  T h2;o Þ ¼ 0; þ 3T c1;o ða15Þ 

 @ P0Sg 1 T c2;o 1 ln ¼ gðHs3 þ 3b3 m23 Þ þ @m3 3ðT c2;o  T c2;i Þ T c2;i 3T c2;i   1 þ bg ðHs3 þ 3b3 m23 Þ  a3 cp;3 ðT he;i  T he;o Þ ¼ 0; þ 3T c2;o ða16Þ 0 Sg

@P 1 1 þ a1 þ a2 þ a3 þ a4 þ a5 þ a6 ¼ 0 ¼  T e T h1;i @Q

m1 gðHs1 þ b1 m21 Þ 2

6 4

3T 2h1;o T

T

h1;i

h2;i

2 h1;o T h2;o Þ

3 ðT h1;o  T h2;o  T h1;i þ T h2;i Þ 7 5 ¼ 0: 2

 T T ln Th1;o T h2;o h1;i

h2;i

ða18Þ 0 Sg

@P m2 gðHs2 þ b2 m22 Þ T h2;o m2 gðHs2 þ b2 m22 Þ ¼ ln  @T h2;i T h2;i 3T h2;i ðT h2;o  T h2;i Þ 3ðT h2;o  T h2;i Þ2 m2 gðHs2 þ b2 m22 Þ

 a2 m2 cp;2  a4 ðkAÞ1 3T 2h2;i 3 2 T T T T ln Th1;o T h2;o þ ðT h1;iT h2;iÞ2 ðT h1;o  T h2;o  T h1;i þ T h2;i Þ h1;i h2;i 7 6 h1;o h2;o 4 5  2 T T ln Th1;o T h2;o h1;i h2;i 2 3 T T T T  ln Th2;o T he;o þ ðT h2;iT he;iÞ2 ðT h2;o  T he;o  T h2;i þ T he;i Þ h2;i he;i 6 7 h2;o he;o  a5 ðkAÞ2 4 5 ¼ 0:   T h2;o T he;o 2 ln T T 

h2;i

h2;i

he;i

ða22Þ

References

þ a1 m1 cp;1  a4 ðkAÞ1

T h1;i T h2;i

ln Th1;o T h2;o  ðT

he;o

þ a2 m3 cp;3  a5 ðkAÞ2 2 3 T T T T  ln Th2;o T he;o þ ðT h2;iT he;iÞ2 ðT h2;o  T he;o  T h2;i þ T he;i Þ h2;i he;i 6 7 h2;o he;o 4 5¼0  2 T T ln Th2;o T he;o

ða17Þ

@ P0Sg m1 gðHs1 þ b1 m21 Þ T h1;o m1 gðHs1 þ b1 m21 Þ ¼ ln þ 2 3T @T h1;o T h1;i h1;o ðT h1;o  T h1;i Þ 3ðT h1;o  T h1;i Þ 

@ P0Sg m3 gðHs3 þ b3 m23 Þ T he;o m3 gðHs3 þ b3 m23 Þ ¼ ln þ @T he;o T he;i 3T he;o ðT he;o  T he;i Þ 3ðT he;o  T he;i Þ2 2 3 T he;i T e T he;i T he;o  ln þ m3 gðHs3 þ b3 m23 Þ T T T T e e 6 7 he;o he;o   a6 ðkAÞe 4 5  2 T T 3T 2he;o ln T he;i Tee

he;i

ða19Þ

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