Multi-Objective Optimization of a Cross-Flow Plate

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Multi-Objective Optimization of a Cross-Flow Plate Heat Exchanger Using Entropy Generation Minimization Article in Chemical Engineering & Technology · January 2014 DOI: 10.1002/ceat.201300411

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Entropy generation minimization

87

Mojtaba Babaelahi Somayyeh Sadri

Research Article

Hoseyn Sayyaadi

Multi-Objective Optimization of a Cross-Flow Plate Heat Exchanger Using Entropy Generation Minimization

K. N. Toosi University of Technology, Tehran, Iran.

Multi-objective optimization of a cross-flow plate fin heat exchanger (PFHE) by means of an entropy generation minimization technique is described. Entropy generation in the PFHE was separated into thermal and pressure entropy generation as two objective functions to be minimized simultaneously. The Pareto optimal frontier was obtained and a final optimal solution was selected. By implementing a decision-making method, here the LINMAP method, the best tradeoff was achieved between thermal efficiency and pumping cost. This approach led to a configuration of the PFHE with lower magnitude of entropy generation, reduced pressure drop and pumping power, and lower operating and total cost in comparison to single-objective optimization approaches. Keywords: Cross-flow plate fin heat exchanger, Entropy generation minimization, LINMAP decision-making, Multi-objective optimization, Thermohydraulic design Received: July 06, 2013; revised: August 18, 2013; accepted: October 04, 2013 DOI: 10.1002/ceat.201300411

1

Introduction

The plate fin heat exchanger (PFHE) is a type of heat exchanger that uses plates and finned chambers to transfer heat between hot and cold fluids [1, 2]. Two important issues in the design of heat exchangers are maximum efficiency and minimum cost for a particular application. Optimization of heat exchangers can lead to these objectives. In optimizing thermal systems, many classical and non-classical techniques have been used. Bejan [3] considered heat losses and frictional pressure drops in channels and found that heat losses were reduced while the heat transfer area was increased; however, in this way, pressure drops in channels became higher. The balance of entropy generation in a controlled volume of a gas-gas heat exchanger for calculating the entropy generation was studied by Bejan [4]. Den Buick [5] described the optimal design of a cross-flow heat exchanger and qualified optimal repartition of the transfer area for maximum effectiveness of a heat exchanger. Rao [6] examined all classical and non-classical techniques that could be used in optimizing shell and tube heat exchangers. Venkatrathnam [7] and Abramazon and Ostersetzer [8]

– Correspondence: Prof. Hoseyn Sayyaadi ([email protected]), K. N. Toosi University of Technology, P.O. Box 19395-1999, Tehran 1999 143344, Iran.

Chem. Eng. Technol. 2014, 37, No. 1, 87–94

obtained the optimum design of heat exchangers in separate works by Lagrangian multiplier and an iterative method. Hesselgreaves [9] used an analytical method for size optimization of PFHEs. Gonzales et al. [10] minimized the total cost of air-cooled heat exchangers considering ten decision variables including operating and geometrical parameters. Selba et al. [11] optimized shell and tube heat exchangers using the genetic algorithm from an economic point of view. Allen et al. [12] defined the optimal geometry and flow arrangement for cost minimization of shell and tube condensers. Peng et al. [13] utilized back propagation neural networks in cooperation with the genetic algorithm to obtain an optimal design of PFHEs with objectives including minimum total weight and total annual cost. Similar works for cost minimization of heat exchangers have been performed using classical and non-classical methods [14–21]. With respect to analysis methods, objective functions of all the above-mentioned studies can be divided into two groups in terms of the first law of thermodynamics or combination of the first and second laws of thermodynamics. Sanaye et al. obtained an optimum design for a PFHE using NSGA-II and artificial neural network analysis [22]. Thermal modeling was conducted for the optimal design of compact heat exchangers by Ahmadi et al. [23]. Multi-objective optimization using the genetic algorithm was developed on a plate and frame heat exchanger by Najafi et al. [24]. Rao et al. introduced a modified version of the TLBO algorithm and applied it to multiobjective optimization of PFHEs [25].

© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Nowadays, the second category that involves entropy and exergy has been extended among the researchers. One suitable and interesting method for optimizing heat exchangers and other energy systems is entropy generation minimization (EGM). The concept of EGM was first introduced for modeling and optimizing the technique by Bejan [26]. Later, some researchers [27–30] applied this method for optimizing heat transfer devices. Multi-objective optimization of a cross-flow PFHE was performed. The value of entropy generation was divided into two parts, i.e., entropy generation related to heat transfer and to fluid friction. Since the entropy generations related to heat transfer and pressure drop have conflicting objectives, in which increase in one objective leads to decrease of another and vice versa, in order to find the best values of decision variables, these objectives were optimized simultaneously in a multiobjective optimization process which balanced the two conflicting objectives. The multi-objective optimization of a PFHE was performed using the genetic algorithm while some geometric specifications of the PFHE were considered decision variables and proper constraints were imposed. The Pareto optimal frontier was obtained and a final optimal solution was selected using a class of decision-making tools called LINMAP (linear programming technique for multidimensional analysis of preference) method.

2

Thermal and Hydraulic Design of the Cross-Flow PFHE

A PFHE consists of a block of alternating layers of various fins and flat separators, known as partitioning plates [31, 32]. A schematic view of a simple cross-flow PFHE is given in Fig. 1. In this heat exchanger, two gas streams flow and heat is transferred from a hot gas (fluid a) into a cold gas (fluid b) stream. Data of the streams for hot and cold gas flows are indicated in Tab. 1 [33].

Table 1. Stream data for hot (fluid a) and cold (fluid b) gas flows. Parameters

Fluid a

Fluid b

Mass flow rate [kg s ]

0.8962

0.8296

Inlet temperature [K]

513

277

Inlet pressure [kPa]

100

100

1017.7

1011.8

0.8196

0.9385

Dynamic viscosity [N s m ]

241

218.2

Prandtl number

0.6878

0.6954

Heat duty Q [kW]

160

–1

–1 –1

Specific heat [J kg K ] –3

Density [kg m ] –2

2.1 Thermal Calculation of a PFHE The heat balance between streams can be calculated as follows: Q ma ha;in ha;out mb hb;in hb;out (1) Q ma cp;a Ta;in Ta;out mb cp;b Tb;in Tb;out where Q, m, h, cp, and T denote the rate of heat transfer, mass flow rate of fluid, heat transfer coefficient, and specific heat of fluid and temperature, respectively. On the other hand: Q UAF LMTD

(2)

U, A, and F mean the overall heat transfer coefficient, heat transfer area, and correction factor. The logarithmic mean temperature difference (LMTD) is defined as: LMTD

DT1 DT2 DT1 ln

(3)

DT2

where DT1 Ta;in

Tb;out

(4a)

DT2 Ta;out

Tb;in

(4b)

Thermal resistance related to the plates, which are located between hot and cold streams, is ignored because a thin metal is used; therefore, the overall effective heat transfer area can be estimated from the following equation [34]: Figure 1. Schematic view of a cross-flow PFHE.

Thermohydraulic modeling is performed under the following assumptions: (i) Steady-state flow is considered for both hot and cold sides. (ii) Properties of fluids are constant with respect to temperature. (iii) Thermal resistance related to the partitioning plate between two streams is negligible. (iv) Similar offset-strip fins are considered for both hot and cold sides. (v) Uniform distribution of heat transfer coefficients and heat transfer area through the heat exchanger. (vi) The number of fin layers for stream a is one layer greater than the corresponding number of fin layers for stream b.

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1 1 1 UA hAa hAb

(5)

Flow areas related to fluids a and b are calculated as follows: Affa Lb Na Ha

ta 1

na t a

(6)

Affb La Nb Hb

t b 1

nb tb

(7)

where Aff, L, N, H, t, and n represent the free flow area, heat exchanger length, number of fin layers of fluid, outer height of the fin, fin thickness, and fin frequency, respectively. The hydraulic diameter of the finned passage is given as follows:




Dh

2s

t H

s H

t

h

t

H t t

i

(8)

lf

where s 1=n

DPb

t

(9)

4fb Lb G2b 2fb m2b Lb 2 2 2 2qb Dh;b qb Dh;b La Nb Hb tb 2 1

f 8:12Re

0:74

lf =Dh

0:41

fs=H

For Re > 1500, f is defined as [35]:

Aa La Lb Na 1 2na Ha

t a

(10)

The Re number is expressed as follows:

Ab La Lb Nb 1 2nb Hb

tb

(11)

Re

j St Pr2=3

h Pr 2=3 Gcp

2=3

Ha ta 1 na ta 1 2na ha La

ja ma cp;a Pra 1 Hb tb 1 nb tb FLMTD 2=3 1 2nb hb Lb Q jb mb cp;b Prb

(13)

For Re ≤ 1500, it follows [35]: j 0:53Re

0:5

lf =Dh

0:15

fs=H

tg

0:14

(14)

For Re > 1500, j is defined as [35]: j 0:21Re

0:4

lf =Dh

0:24

t=Dh 0:02

(15)

where lf and h are the lance length and inner height of the fin, respectively.

2.2

f 1:12Re

0:36

lf =Dh

0:65

tg

0:02

t=Dh 0:17

(18)

(19)

GDh m Dh l l Aff

(20)

where DP, f, and q denote the pressure drop, fanning friction factor, and density, respectively. A new detail regarding the hydraulic design of a PFHE is reported in [36].

(12)

j, St, Pr, and G denote the Colburn factor, Stanton number, Prandtl number, and mass flux velocity, respectively. The Colburn factor j depends on parameters such as type of fin, geometry and details of makeup as well as Reynolds number of the stream [32]. Substituting h and A in the heat balance equations (Eqs. (1) and (2)) leads to the following expression: 1

(17)

For Re ≤ 1500, it follows [35]:

Dh and s denote the hydraulic diameter and fin spacing. Heat transfer areas are obtained similarly and the total heat transfer area is calculated by combining Aa and Ab which are determined by:

The heat transfer coefficient (h) is written in terms of the Colburn factor (j) as follows:

nb tb

89

3

Entropy Generation Minimization

Entropy generation minimization (EGM) is based on basic principles of thermodynamics, heat and mass transfer, and fluid mechanics. EGM was introduced earlier by Bosnjakovic in the 1930s [37, 38] and the concept was further developed in the 1970s by Bejan [2]. EGM is used in real applications to detect irreversibilities. These irreversibilities in heat exchangers are related to heat transfer between streams due to temperature differences and frictional pressure drop. As noted by Bejan in [39], the entropy generation rate for two streams (e.g., a and b) can be achieved in terms of temperature and pressure as: Ta;2 Pa;2 S ma cp;a ln Ra ln Ta;1 Pa;1 (21) Tb;2 Pb;2 mb cp;b ln Rb ln Tb;1 Pb;1 where S is the rate of entropy generation. The above equation can be rearranged based on thermal and pressure irreversibilities: Ta;2 Tb;2 mb cp;b ln S ma cp;a ln Ta;1 Tb;1 (22) Pa;2 Pb;2 ma Ra ln mb Rb ln Pa;1 Pb;1

Hydraulic Calculation of a PFHE

The acceptable pressure loss for each stream is specified according to the type of application by customers. In the hydraulic design of heat exchangers, fins and passages are selected such that overall pressure drops of streams never violate these limitations. Uniform distribution of a stream over the width of a layer is provided by proper design of the distributors [32]. Pressure drop on both sides of the heat exchanger can be obtained using factor f as follows: DPa

4fa La G2a 2qa Dh;a

2fa m2a q2a

La Dh;a L2b Na2 Ha


ta 2 1

na ta

(16)

S SDT SDP

(23)

Ta;2 Tb;2 mb cp;b ln ; SDT ma cp;a ln Ta;1 Tb;1 2 1 13 0 0 UAF LMTD UAF LMTD Ta;1 Tb;1 6 C C7 B ma cp;a B mb cp;b 4ma cp;a ln@ A mb cp;b ln@ A5 Ta;1 Tb;1 UAF LMTD UAF LMTD ma cp;a ln 1 mb cp;b ln 1 ma cp;a Ta;1 mb cp;b Tb;1


(24a)

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Pa;2 Pb;2 mb Rb ln ma Ra ln Pa;1 Pb;1 Pa;1 DPa Pb;1 DPb ma Ra ln mb Rb ln Pa;1 Pb;1 DPa DPb ma Ra ln 1 mb Rb ln 1 Pa;1 Pb;1

SDP

Ta;2 Tb;2 Ns;DT ma cp;a ln mb cp;b ln =Cmax Ta;1 Tb;1 UAF LMTD ma cp;a ln 1 ma cp;a Ta;1 UAF LMTD mb cp;b ln 1 =Cmax mb cp;b Tb;1

(28)

(24b) where U, A, F, LMTD, DPa, and DPb are calculated according to Sect. 2.1 and 2.2. Entropy generation is non-dimensionalized through dividing by Cmax, where Cmax is the greatest mcp among two streams, thus it follows: Ns

S Cmax

(25)

(29)

where N is the dimensionless entropy generation. In the current study, the dimensionless entropy generation number was converted into two separate parts as follows: Ns

SDT S DP Cmax Cmax

(26)

Ns Ns;DT Ns;DP

(27)

A lower entropy generation related to heat transfer led to a higher thermal efficiency; however, entropy generation related to fluid friction and pumping cost was increased. Thus, these two types of dimensionless entropy generations were considered as two separate conflicting objective functions.

4

4.2

Decision Variables and Constraints

In the current study, decision variables were the design parameters of the PFHE as follows: (i) fin height (Ha and Hb); (ii) fin thickness (ta and tb); (iii) fin frequency (na and nb); (iv) number of fin layers for a stream (Nb = Na –1); (v) dimensions of heat exchanger (La and Lb); (vi) fin dimension (lfa and lfb). These decision variables are indicated in Fig. 2 and are consistent with the decision variables considered in [33] and [45]. Each decision variable was normally required to be within a given range (Tab. 2) [33, 45].

5

Multi-Objective Optimization

Multi-objective optimization problems usually exhibit a probably uncountable set of solutions to assess the status of vectors showing the best possible trade-offs in the objective function space [40]. Pareto optimality is the key concept to express the relationship between multi-objective optimization results in order to determine a solution which is actually one of the best possible trades-offs [41, 42]. In multi-objective optimization, a process of decision-making is required for selection of the final optimal solution from available solutions of the Pareto frontier. In this paper, the LINMAP decision-making method was employed [43, 44]. In this method, the solution with minimum distance from an imaginary solution, called the ideal point, was selected as the final desired optimal solution [44]. Details of MOEA (multiobjective evolutionary algorithm) and LINMAP methods are described in [44].

4.1

Non-dimensioned entropy generation related to fluid friction: Pa;2 Pb;2 mb Rb ln =Cmax ma Ra ln Ns;DP Pa;1 Pb;1 DPa DPb mb Rb ln 1 =Cmax ma Ra ln 1 Pa;1 Pb;1

Cost Analysis

Cost analysis was performed based on the estimation method of Muralikrishna and Shenoy [21]. In this formulation, the cost of the heat exchanger is divided into capital and operational costs as follows: Total cost = capital cost (heat exchanger core + pump a + pump b) + operating cost of (pump a + pump b). n Cost Af Ca Cb AcHT Ce Cf Wa d o C h pow Ce Cf Wb d Wa Wb g

(30)

where W, AHT, Ci (i ∈ {a, b, e, f, pow}, and d are pumping power, heat transfer area, and constant variables, respectively.

Objective Functions

The two objective functions defined in this paper were as follows: Non-dimensioned entropy generation related to heat transfer:

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Figure 2. Schematic view of the considered decision variables for PFHE.




91

Table 2. Range of decision variables. Decision variable

Range

Fin height Ha [m]

0.002–0.01

Fin thickness ta [m]

0.0001–0.0002

Fin frequency na [fins/m]

100–1000

Number of fin layers Na

1–10

Dimensions of heat exchanger La [m]

0.1–1

Fin dimension lfa [m]

0.001–0.01

Fin height Hb [m]

0.002–0.01

Fin thickness tb [m]

0.0001–0.0002

Fin frequency nb [fins/m]

100–1000

Dimensions of heat exchanger Lb [m]

0.1–1

Fin dimension lfb [m]

0.001–0.01

These parameters were calculated by thermal and hydraulic analysis. The annual operating time of the systems was assumed to be 8000 h. Other coefficients and parameters in Eq. (33) are defined in [21] and presented in Tab. 3. Table 3. Cost coefficiency of a heat exchanger [21]. Parameter

Value

2

Af [m ]

0.322

Ca [US $]

30 000

Cb

750

c

0.8

Ce

2000

Cf

5

d

6

Cpow [US $ Wh ]

0.00005

g

0.7

tance from the ideal point might be considered a desirable final solution. Since the axes in Fig. 3 did not have a similar scale, it was impossible to compare the values corresponding to each value (i.e., Ns,DT and Ns,DP). Thus, both Ns,DT and Ns,DP had to be normalized to become comparable. It follows: Ns;DT i n i q P Ns;DT 2 i Ns;DT

(31)

Ns;DP i n i q Ns;DP P 2 Ns;DP i

(32)

where i and n are indices for each individual solution on the Pareto frontier and normalized value, respectively. The normalized Pareto frontier is demonstrated in Fig. 4. In the next step, the closest point of the normalized Pareto frontier to the

0.68 –1

Figure 3. Pareto frontier: the best trade-off values for two objective functions.

Results and Discussion

Objective functions, decision variables, and constraints as introduced in Sect. 4.1 and 4.2 were optimized in a multiobjective optimization process using MOEA. Fig. 3 illustrates the Pareto frontier in the objective space. As mentioned previously in the multi-objective optimization problems, all solutions located on the Pareto frontier are a potential final solution. Therefore, selection of a final optimal solution from the set of the Pareto frontier needs a process of decision-making. In this paper, the LINMAP method [44] of decision-making was employed. Actually, the LINMAP method worked based on the definition of an imaginary point, namely the ideal point. At this point, both objective functions had their optimum value independent from other objectives. In the LINMAP method, a solution with the closest special dis-


Figure 4. Schematic view of the final optimal solution selection from the normalized Pareto frontier using the LINMAP decisionmaking method.


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ideal solution was selected as a desirable final optimal solution. The procedure is schematically illustrated in Fig. 4. The non-dimensioned Pareto frontier was curve-fitted using a polynomial. The following expression denotes Ns,DP as a function of Ns,DT and Ns,DP for all optimal solutions located on the Pareto frontier: 5 n n n 21:377Ns;DT 6 72:015 Ns;DT Ns;DP 4 3 n n 96:133 Ns;DT 65:673 Ns;DT 2 n n 25:397 Ns;DT 6:2065Ns;DT 0:9987

(33)

If Ns,DT was desired as a function of Ns,DP, then: 5 n n n 15:370Ns;DP 6 54:776 Ns;DP Ns;DT 4 3 n n 78:386 Ns;DP 58:145 Ns;DP 2 n n 24:490 Ns;DP 6:3477Ns;DP 1:0225

(34)

Quantities of decision variables, objective functions, and key parameters of the PFHE for the final solution are presented in Tab. 4, in which they are compared with the corresponding results obtained in the literature by other researchers [33, 45] using different approaches. The same constraints and decision variables which were considered in [33] and [45] were taken into account in this paper.

The main difference between the present approach and [33] and [45] was related to the separation of entropy generation due to temperature difference and pressure losses in the current work. Values of Ns,DT and Ns,DP for the final optimal solution were 0.017972 and 0.028655, respectively. Therefore, values of thermal efficiency and pumping power were evaluated. These values for the final optimal solution were 0.9161 and 7.869 kW, respectively. As already explained, Ns,DT and Ns,DP were separated and considered as two distinct objective functions. However, in [33] and [44], Ns (= Ns,DT + Ns,DP) was taken as a single objective function. Comparison of the results obtained in the present approach, in [33] and in [45] showed that the heat transfer area of the optimized heat exchanger was a little greater than the corresponding heat transfer area in [33] but a little smaller than the corresponding heat transfer area in [45]. The total non-dimensioned entropy generation for the optimal heat exchanger of this study was significantly less than the corresponding total entropy generation of a similar PFHE obtained in [33] and [44] (26.4 % less than the corresponding Ns in [33] and 12.1 % less than the corresponding Ns in [45]). Further pressure drop, pumping power, and operating cost of the optimal PFHE of the current study were much lower than the corresponding values in [33] and [45]. Pumping power and operating cost of the PFHE in this study were 31.0 % and 32.9 % less than the corresponding pumping power in [33] and [45], respectively (similar results valid for the operating cost). The capital cost of the heat exchanger surface was almost similar for all cases; however, the total cost ob-

Table 4. Decision variables, objective function, and parameters for the final optimal solution. Current study Parameters

Reference [29]

Reference [41]

Stream a

Stream b

Stream a

Stream b

Stream a

Stream b

H [mm]

9.90

9.90

9.53

9.53

9.80

9.80

t [mm]

0.130

0.130

0.146

0.146

0.10

0.10

n [fins/m]

505.0

539.0

534.9

534.9

442.9

442.9

N

10

9

8

9

10

9

L [mm]

989.00

983.00

994.00

887.00

925.00

996.00

lf [mm]

5.00

3.30

6.30

6.30

9.80

9.80

A [m ]

0.0897

0.0803

0.0614

0.07739

0.0923

0.0772

DP [kPa]

4.234

2.962

7.100

4.120

1.750

11.100

Ns,DT

0.017972

–

–

Ns,DP

0.028655

–

–

Ns

0.046627

0.063332

0.053028

Pumping power [kW]

7.869

11.406

11.726

Effectiveness e

0.9329

0.9329

0.9329

2

Capital cost

4023.8

Operation cost

4496.6

6517.7

6700.6

Capital cost of surface [US $]

9718.5

9709.8

9718.4

Total cost [US $]

18238.9

20259.7

20450.4

Pumping cost [US $]

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4032.2

4031.4




tained in this work was 10.0 % and 12.1 % lower than the corresponding total cost of PFHEs introduced in [33] and [45], respectively. Tab. 4 indicates that the approach of the current paper with separation of the thermal entropy generation from the hydraulic entropy generation and their consideration as two separate objective functions led to improvement in most factors of the PFHE compared to traditional approaches in [33] and [45]. However, the cost was not an objective of this research. Economic features of the optimized PFHE using the presented approach were better than the corresponding economic parameters in [33] and [45].

7

Q R Re s S St t T U W

A multi-objective optimization of a cross-flow PFHE was presented based on EGM. It was found that both thermal entropy generation and pressure entropy generation of the cross-flow PFHE were in optimum state. Therefore, multi-objective optimization of this type of heat exchangers led to the best tradeoff between entropy generation related to heat transfer and entropy generation related to fluid friction. It was shown that this approach, in terms of separating entropy generation, enabled a better design of the PFHE with less total entropy generation, lower pressure drop and pumping power, less operating cost, and less total cost compared to the approach that considered total entropy generation as an objective function of the single-objective optimization.

[1] [2] [3] [4] [5] [6] [7] [8]

[9]

Symbols used [m2] [m2] [m2] [W K–1] [W kg–1K–1] [m] [–]

G h h H j l L LMTD

[kg m–2s–1] [W m–2K–1] [m] [m] [–] [m] [m] [–]

m n Na, Nb Ns P DP Pr

[kg s–1] [fins/m] [–] [–] [N m–2] [N m–2] [–]

heat transfer area free flow area heat transfer area heat capacity rate specific heat of fluid hydraulic diameter fanning friction factor, correction factor mass flux velocity heat transfer coefficient inner height of the fin height or outer height of the fin Colburn factor lance length of the fin heat exchanger length logarithmic mean temperature difference mass flow rate of fluid fin frequency number of fin layers of fluid a or b dimensionless entropy generation pressure pressure drop Prandtl number


[kg m–3] [N m–2s–1]

density viscosity

References

The authors have declared no conflict of interest.

A Aff AHT C cp Dh f

rate of heat transfer specific gas constant Reynolds number fin spacing (1/n–1) rate of entropy generation Stanton number (h/GCp) fin thickness temperature overall heat transfer pumping power

Greek symbols q l

Conclusions

[W] [J kg–1K–1] [–] [m] [W K–1] [–] [m] [K] [W m–2K–1] [W]

93

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

B. Kim, B. Sohn, Int. J. Heat Fluid Flow 2006, 27, 514–521. D. A. Reay, Heat Recovery Syst. CHP 1994, 14, 459–474. A. Bejan, Int. J. Heat Mass Transfer 1978, 21, 655–658. A. Bejan, Entropy Generation through Heat and Fluid Flow, Wiley, New York 1982. V. den Buick, J. Heat Transfer 1991, 113, 341–347. K. R. Rao, Ph. D. Thesis, Indian Institute of Science, Bangalore 1991. G. Venkatrathnam, Ph. D. Thesis, Indian Institute of Technology, Kharagpur 1991. B. Abramzon, S. Ostersetzer, in Proc. of the 1st Int. Conf. on Aerospace Heat Exchanger Technology (Ed: A. Hashemi), Elsevier, New York 1993, 349–368. J. E. Hesselgreaves, in Proc. of the 1st Int. Conf. on Aerospace Heat Exchanger Technology (Ed: A. Hashemi), Elsevier, New York 1993, 391–399. M. T. Gonzalez, N. C. Petracci, M. Urbican, Heat Transfer Eng. 2001, 22, 11–16. R. Selbas, O. Kizilkan, M. Reppich, Chem. Eng. Process. 2006, 4, 268–275. B. Allen, L. Gosselin, Int. J. Energy Res. 2008, 10, 958–969. H. Peng, X. Ling, Appl. Therm. Eng. 2008, 5, 642–650. P. Wildi-Tremblay, L. Gosselin, Int. J. Energy Res. 2007, 9, 867–885. B. V. Babu, S. A. Munawar, Chem. Eng. Sci. 2007, 14, 3720– 3739. A. C. Caputo, P. M. Pelagagge, P. Salini, Appl. Therm. Eng. 2008, 10, 1151–1159. Y. Ozcelik, Appl. Therm. Eng. 2007, 11, 1849–1856. L. Valdevit, A. Pantano, H. A. Stone, A. G. Evans, Int. J. Heat Mass Transfer 2006, 21, 3819–3830. G. N. Xie, B. Sunden, Q. W. Wang, Appl. Therm. Eng. 2008, 8, 895–906. I. Ozkol, G. Komurgoz, Numer. Heat Transfer, Part A 2005, 3, 283–296. K. Muralikrishna, U. V. Shenoy, Chem. Eng. Res. Des. 2000, 58, 161–167. S. Sanaye, H. Hajabdollahi, Appl. Energy 2010, 87, 1893– 1902.


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H. Sayyaadi et al

[23] P. Ahmadi, H. Hajabdollahi, I. Dincer, J. Heat Transfer 2011, 133, 021801–021810. [24] H. Najafi, B. Najafi, Appl. Therm. Eng. 2011, 31, 1839–1847. [25] R. Venkata Rao, V. Patel, Appl. Math. Model. 2013, 37, 1147– 1162. [26] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, FL 1948. [27] R. T. Ogulata, F. Doba, Int. J. Heat Mass Transfer 1998, 41, 373–381. [28] J. Vargas, A. Bejan, Int. J. Heat Fluid Flow 2001, 22, 657–665. [29] R. Culham, Y. Muzychka, IEEE Trans. Compon. Packag. Technol. 2001, 24, 159–165. [30] M. Iyengar, IEEE Trans. Compon. Packag. Technol. 2003, 26, 62–70. [31] D. Jainender, Design of Compact Plate Fin Heat Exchanger, National Institute of Technology, Rourkela, India 2009. [32] M. A. Taylor, Plate-Fin Heat Exchangers – Guide to their Specification and Use, Heat Transfer and Fluid Flow Service, ORA, UK 1987. [33] M. Mishra, P. K. Das, S. Sarangi, Appl. Therm. Eng. 2009, 29, 2983–2989.

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[34] The Standards of the Brazed Aluminum Plate-Fin Heat Exchanger Manufacturers Association, ALPEMA, 2000. www.alpema.org [35] H. M. Joshi, R. L. Webb, Int. J. Heat Mass Transfer 1987, 30, 69–84. [36] W. Wang, S. Zhang, J. Yang, J. Zheng, X. Ding, M. Chou, P. Tang, X. Zhan, Chem. Eng. Technol. 2013, 36 (4), 657–664. [37] F. Bosnjakovic, Arch. Wärmewirtsch. Dampfkesselwes. 1938, 19 (1), 1–2 (in German). [38] F. Bosnjakovic, Technische Mitteilungen, Essen 1939, 32 (15), 439–445 (in German). [39] A. Bejan, J. Heat Transfer 1977, 99, 374–380. [40] Technical Assessment Guide (TAGTM), Electric Power Research Institute, Palo Alto, CA 1993. [41] C. M. Fonseca, P. J. Fleming, in Handbook of Evolutionary Computation (Eds: T. Back, D. B. Fogel, Z. Michalewicz), Oxford University Press, New York 1997. [42] D. A. Van Veldhuizen, G. B. Lamont, Evolutionary Computation 2000, 2, 125–147. [43] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI 1975. [44] H. Sayyaadi, R. Mehrabipour, Energy 2012, 38, 362–375. [45] R. V. Rao, V. K. Patel, Int. J. Therm. Sci. 2010, 49, 1712–1721.



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