Many-objective optimization of cross-flow plate-fin heat exchanger

International Journal of Thermal Sciences 118 (2017) 320e339

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Many-objective optimization of cross-flow plate-fin heat exchanger Bansi D. Raja a, R.L. Jhala b, Vivek Patel c, * a

Department of Mechanical Engineering, Indus University, Gujarat, India Department of Mechanical Engineering, Gujarat Technological University, Gujarat, India c Department of Mechanical Engineering, Pandit Deendayal Petroleum University, Gujarat, India b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 October 2016 Received in revised form 5 May 2017 Accepted 6 May 2017

This paper presents a rigorous investigation of many-objective (four-objective) optimization of cross flow plate-fin heat exchanger and its comparison with multi-objective (two-objective) optimization. Maximization of effectiveness and minimization of total annual cost, total weight and number of entropy generation units are considered simultaneously as objective functions during many-objective optimization. Multi-objective heat transfer search (MOHTS) algorithm is introduced and applied to obtain a set of Pareto-optimal points of many-objective problem. Application example of plate-fin heat exchanger is presented to demonstrate the effectiveness and accuracy of proposed algorithm. Many objective optimization results form a set of solutions in four dimensional hyper objective space and for visualization it is represented on a two dimension objective space. Thus, results of four-objective optimization are represented by six Pareto fronts in two dimension objective space. These six Pareto fronts are compared with their corresponding two-objective Pareto fronts. Different decision making approaches such as LINMAP, TOPSIS and fuzzy are used to select the final optimal solution from Pareto optimal set of the many-objective optimization. Finally, to reveal the level of conflict between these objectives, distribution of each design variables in their allowable range is also shown in two dimensional objective spaces. © 2017 Elsevier Masson SAS. All rights reserved.

Keywords: Plate-fin heat exchanger Many-objective optimization Effectiveness Total cost Total weight Number of entropy generation units

1. Introduction Heat exchangers are one of the important equipments which servers the purpose of energy conservation through energy recovery. Among several types of heat exchangers, most important one is compact heat exchanger. Plate-fin heat exchanger (PFHE) belongs to the category of compact heat exchanger due to its large heat transfer surface area per unit volume [1,2]. Due to their high thermal effectiveness, plate-fin heat exchangers are widely used in air separation plants, liquefaction plants, aerospace, petrochemical industries and cryogenics applications [3]. Design-optimization of PFHE requires an integrated understanding of thermodynamics, fluid dynamics and cost estimation. Generally, objectives involved in the design optimization of PFHE are thermodynamics (i.e. maximum effectiveness, minimum entropy generation rate, minimum pressure drop etc.) and economics (i.e. minimum cost, minimum weight etc.). The conventional design approach for PFHE is time-consuming, and may not lead to an

* Corresponding author. E-mail addresses: [email protected] (B.D. Raja), ramdevsinh.jhala@ marwadieducation.edu.in (R.L. Jhala), [email protected] (V. Patel). http://dx.doi.org/10.1016/j.ijthermalsci.2017.05.005 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.

optimal solution. Hence, application of evolutionary and swarm intelligence based algorithms have gained much attention in design-optimization of PFHE. Earlier, several investigators used various optimization techniques with different methodologies and objective functions to optimize PFHE. However, their investigation was focused on single objective optimization or multi-objective (i.e. two or three objective) optimization. Wen et al. [4] carried out a thermodynamic optimization of PFHE. The authors considered two conflicting objectives namely, Colburn factor and friction factor for optimization and used Genetic algorithm (GA) as an optimization tool. Du et al. [5] focused on a double flow plate-fin heat exchanger for improving its thermal and hydraulic behaviour using GA. Turgut [6] investigated Hybrid Chaotic Quantum behaved Particle Swarm Optimization (HCQPSO) algorithm for minimizing the heat transfer area and total pressure drop of PFHE. Sanaye and Hajabdollahi [7] performed a simultaneous optimization of total cost and effectiveness using a design which featured NSGA-II and PFHE. Rao and patel [8] performed a multi-objective optimization of PFHE with effectiveness and total cost of heat exchanger as objective functions. The authors used modified version of teaching learning based optimization (TLBO) algorithm as an optimization tool.

B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

Nomenclature A a Aff CA Cp Cinv Cope C* dh f G h H ir j kel lf L LINMAP m n Nh Nc NS NTU P Pr

heat transfer area (m2) annual co-efficient factor free flow area (m2) cost per unit area ($/m2) specific heat (J/kg K) initial cost ($) operating cost ($) heat capacity ratio hydraulic diameter (m) Fanning friction factor mass flux velocity (kg/m2 s) convective heat transfer co-efficient (W/m2 K) height of fin (m) rate of interest (%) Colburn factor electricity price ($/MWh) fin offset length (m) heat exchanger length (m) Linear Programming Technique for Multidimensional Analysis of Preference mass flow rate (kg/s) fin frequency (no. of fins/m) number of hot side layer number of cold side layer number of entropy generation unit number of transfer units Pressure (kPa) Prandtl number

Wang and Li [9] introduced and applied an improved multiobjective cuckoo search (IMOCS) algorithm for optimization of PFHE. The authors considered conflicting thermo-economic objectives for optimization. Hajabdollahi [10] investigated the effect of non-similar fins in thermo-economic optimization of plate fin heat exchanger. They considered total annual cost and effectiveness of heat exchanger as objective functions and utilized NSGA-II for optimization. Hadidi [11] employed biogeography-based optimization (BBO) algorithm for optimization of heat transfer area and total pressure drop of the PFHE. Patel and Savsani [12] obtained a Pareto front between conflicting thermodynamic and economic objectives of PFHE by implementing multi-objective improved TLBO (MO-ITLBO) algorithm. Wang et al. [13] presented few layer pattern criterion models to determine optimal stacking pattern of multi-stream plate-fin heat exchanger (MPFHE). Authors have developed these models by employing genetic algorithm and observed that the performance of MPFHE in relation to heat transfer and fluid flow was effectively improved by the optimization design of layer pattern. Zaho and Li [14] developed an effective layer pattern optimization model for multi-stream plate-fin heat exchanger using genetic algorithm. Zhou et al. [15] presented an optimization model for PFHE based on entropy generation minimization method. They considered specific entropy generation rate as an objective function and total heat transfer area of PFHE as a constraint. Yousefi et al. [16] presented a learning automata based particle swarm optimization employed to multi-stage thermo-economical optimization of compact heat exchangers. Several other investigators performed single objective [17e26] or multi-objective (two or three objective) [27e34] optimization of PFHE for thermodynamic [17e20,24e26,28,29],

DP Re fs t td T TOPSIS TAC U V Wt

321

pressure drop (kPa) Reynolds number fin spacing (m) fin thickness (m) depreciation time temperature (K) Technique for Order of Preference by Similarity to Ideal Solution total annual cost overall heat transfer co-efficient (W/m2 K) volume flow rate (m3/s) total weight (kg)

Greek letters ε effectiveness m viscosity (N.s/m2) r density (kg/m3) h compressor efficiency t hour of operation d Dimensionless parameter, a Dimensionless parameter, g Dimensionless parameter,

. t l  1f
t  =n ðH
tÞ t 1
t  =n

Subscripts h hot c cold max maximum min minimum tot total

economic [21e23] or thermo-economic [27,30e34] objectives with different optimization strategies. Thus, it can be observed from literature that researchers have carried out economical optimization, thermodynamic optimization or thermo-economic optimization of PFHE for single or multiobjective (two or three objective) consideration. However, manyobjective optimization of PFHE is not yet observed in literature. Considering this fact, efforts have been put in the present work to perform a many-objective (i.e. four-objective) optimization of PFHE. Many-objective consideration results in more realistic design of PFHE and end user can select any optimal design from it depending on their requirements. Further, as an optimization tool, heat transfer search (HTS) algorithm [35] is implemented in the present work. Heat transfer search is a recently developed meta-heuristic algorithm based on natural law of thermodynamics and heat transfer [35]. In this work, a multi-objective variant of heat transfer search (MOHTS) algorithm is presented to address many-objective optimization problem of PFHE. The proposed algorithm uses a grid-based approach in order to keep diversity in the external archive. Pareto dominance is incorporated into the MOHTS algorithm in order to allow this heuristic to handle problems with several objective functions. Qualities of the solution are computed based on the Pareto dominance notion in the proposed algorithm. Main contributions of the present work are (i) Many-objective optimization of PFHE to maximize effectiveness and minimize total annual cost, total weight and number of entropy generation units simultaneously. (ii) To introduce multi-objective variant of the heat transfer search (MOHTS) algorithm and employed it to solve many-objective optimization problem of PFHE (iii) To

B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

compare results of many-objective (i.e. four-objective) optimization with multi-objective (i.e. two-objective) optimization. (iv) To compare underlying relationship of decision variables between many-objective (i.e. four-objective) optimization and multiobjective (i.e. two-objective) optimization. (v) To select a final optimal solution from the Pareto optimal set of the many-objective optimization with the help of LINMAP, TOPSIS and fuzzy decision making approaches. Remaining sections of this paper are organized as follows. Section 2 presents the thermal-hydraulic modeling and the objective functions formulation of PFHE. Section 3 describes the heat transfer search algorithm. Section 4 explains proposed multi-objective heat transfer search algorithm. Section 5 presents the application example of PFHE. Section 6 describes the results-discussion. Finally, the conclusion of the present work is discussed in section 7.

2 6 6    * ε¼1
6 6exp
NTU 1 þ C 4 2  pffiffiffiffiffiffi  pffiffiffiffiffiffi pffiffiffiffiffiffi  33 * * I þ I 2NTU C * 77 2NTU C C 0 1 6 77 6 77 6 f pffiffiffiffiffiffi   77 6 1
C* X *n 2 * 55 4
2NTU C I C n * C n¼2 =

322

Where C*, NTU are heat capacity ratio and number of transfer units of the PFHE respectively and “I” represents modified Bessel function. Similarly, the number of entropy generation units of PFHE is given by Refs. [17,36],

2. Modeling formulation This section describes thermal hydraulic modeling of PFHE, objective function formulation, design variables and constraints involved in PFHE design optimization.

(1)

Ns ¼

" Cp;h C ln 1
ε min Cmax Cp;h " Cp;c C ln 1 þ ε min Cmax Cp;c

!# Rh DPh ln 1
þ Cp;h Ph;i !! !# Th;i Rc DPc
1
ln 1
Tc;i Cp;c Pc;i 1


Tc;i Th;i

!!




(2) 2.1. Thermal and hydraulic formulation Detailed geometry of PFHE with offset strip fin is shown in Fig. 1. In this work, ε-NTU approach is used to predict the performance of PFHE [7,12]. The PFHE is running under a steady state, and the area distribution and heat transfer coefficients are assumed to be uniform and constant. The number of hot layers is assumed to be one layer less than that of cold layers. Table 1 presents thermal and hydraulic model formulation of PFHE.

Where DPh and DPc are hot side and cold side pressure drops respectively. Similarly, Th,i, Tc,i, Pt,i, and Ps,i are inlet temperature and pressure of hot side and cold side fluid, which are found by utilizing the thermal-hydraulic model of PFHE given in Table 1. Further, the total weight of the PFHE is given by Ref. [30],

Wt ¼ ðLh Lc Ln Þ  rmaterial

(3)

Where Lh, Lc and Ln are hot, cold and no flow lengths respectively. Finally, the total annual cost of heat exchanger is expressed as [8,21],

2.2. Objective functions, design variables and constraints

TAC ¼ aCinv þ Cope : In this work, many-objective optimization is carried out between conflicting objectives. Maximization of heat exchanger effectiveness and minimization of total annual cost, total weight and number of entropy generation units (EGU) of PFHE are considered as objectives. For the cross flow heat exchanger with both fluids unmixed, effectiveness (є) is given by Refs. [7,12],

(4)

Where Cinv and Cope are total capital cost and operating cost of the exchanger respectively. In this work, the MOHTS algorithm is used for many-objective optimization of a PFHE. The many-objective problem of PFHE can be described as follows,

Fig. 1. Plate-fin heat exchanger geometry with offset strip fin.

B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

323

Table 1 Modeling equations for PFHE with offset strip fin. Equations

No.

Remarks

C*

1

Heat capacity ratio

2

Number of transfer units

3

Outlet temperature of fluids

4

Heat transfer rate

5

Hot and cold side heat transfer area

6 7

Total heat transfer area free flow area of hot and cold fluid

j ¼ 0:6522ðReÞ
0:5403 ðaÞ
0:1541 ðdÞ0:1499 ðgÞ
0:0678 ½1 þ 5:269  10
5 ðReÞ1:34 ðaÞ0:504 ðdÞ0:456 ðgÞ
1:055 0:1

8

Colburn factor for offset strip fin

f ¼ 9:6243ðReÞ
0:7422 ðaÞ
0:1856 ðdÞ0:3053 ðgÞ
0:2659 ½1 þ 7:669  10
8 ðReÞ4:429 ðaÞ0:920 ðdÞ3:767 ðgÞ0:236 0:1 8 > Re ¼ Gdh ; a ¼ fs ; d ¼ t ; > > < m H
t lf  > > t 1 > : g ¼ ; fs ¼
t fs n G ¼ Am

9

Friction factor offset strip fin

10

Non-dimension number for PFHE

11

Mass flux of stream

12

Heat transfer co-efficient

13

Hydraulic diameter of offset strip fin

DP ¼ 2fLG rdh

14

Pressure drop

Cinv ¼ CA A2   Cope ¼ kel t DPh V þ kel t DPh V

15

Capital cost

16

Operating cost

¼

Cmin Cmax





1 þ 1 min ¼ CUA ¼ Cmin ðhAÞ ðhAÞc h  8 Cmin > > > < Th;2 ¼ Th;1
ε C ðTh;1
Tc;1 Þ h  > Cmin > > : Tc;2 ¼ Tc;1 þ ε ðTh;1
Tc;1 Þ Cc Q ¼ εCmin ðTh;1
Tc;1 Þ ¼ mh cp;h ðTh;1
Th;2 Þ ¼ mc cp;c ðTc;2
Tc;1 Þ

Ah ¼ Lh Lc Nh ½1 þ 2nh ðHh
th Þ Ac ¼ Lh Lc Nc ½1 þ 2nc ðHc
tc Þ A ¼ Ah þ Ac

Aff ;h ¼ ðHh
th Þð1
nh th ÞLc Nh Aff ;c ¼ ðHc
tc Þð1
nc tc ÞLh Nc

1 NTU

ff

h ¼ j G Cp ðPrÞ
0:667 dh ¼ 2ðfsl

4fslf ðH
tÞ f þðH
tÞlf þðH
tÞtÞþðtfsÞ 2

c

h

f ðXÞ ¼ ðεðXÞ; TACðXÞ; Wt ðXÞ; Ns ðXÞÞ

Maximise=Minimise

i h X ¼ Lc ; Lh ; H; t; n; lf ; Nc 

Lc ; Lh ; H; t; n; lf ; Nc

 min

(5)

0:012 < d < 0:048

(9)

120 < Re < 104

(10)

0:041 < g < 0:121

(11)

(6)

   Lc ; Lh ; H; t; n; lf ; Nc    Lc ; Lh ; H; t; n; lf ; Nc

(7)

max

Where X denotes the vector of design variables to be optimized. In this work, seven design variables which affect the performance of PFHE are considered for optimization. These variables are: (i) cold flow length (ii) hot flow length (iii) fin height (iv) fin thickness (v) fin frequency (number of fins per meter length of the heat exchanger (fins/m)) (vi) fin offset length and (vii) number of cold side layer. Design parameters variation ranges are shown in Table 2. Also, the objective functions which are based on thermal-hydraulic model of Table 1 should satisfy the following constraints.

0:134 < a < 0:997

(8)

Table 2 Lower and upper bound of design variables. Design variables

Lower bound

Upper bound

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer

0.1 0.1 2 0.1 100 1 1

1 1 10 0.2 1000 10 200

Where, a, d and g are dimensionless parameters and given in Table 1. The equations used for calculating Colburn factor (j) and Fanning factor (f) in the present work are valid for above bound. Next section describes the heat transfer search algorithm considered in the present work. 3. Heat transfer search algorithm (HTS) Heat transfer search (HTS) [35] is a recently developed optimization algorithm inspired from the law of thermodynamics and heat transfer. The fundamental law of thermodynamics states that any system always tries to achieve thermal equilibrium with its surroundings. Therefore, any system lagging thermal equilibrium always tries to achieve thermal equilibrium by conducting heat transfer with surrounding as well as within the different parts of the system also. The modes of heat transfer which plays an important role in setting thermal equilibrium are conduction, convection, and radiation. Therefore, the HTS algorithm compose with the ‘conduction phase’, ‘convection phase’, and ‘radiation phase’ to reach at optimum solution. The ‘conduction phase’, ‘convection phase’, and ‘radiation phase’ established thermal equilibrium of the system by conduction, convection, and radiation heat transfer respectively. In, HTS algorithm all three phase take place with equal probability. The equal probability is controlled by the parameter ‘R’ in each generation, which is a uniformly distributed random number, varies between 0 and 1. As the value of R varies between 0 and 1, so for equal

324

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probability, each phase must share the equal portion of R (i.e. as there are three phases, three equal portions are 0e0.3333, 0.3333e0.6666 and 0.6666 to 1) during the course of optimization. Depending upon the value of R, any one of the three phases can be executed to update the solutions in that generation. Experimentations are required to decide which portion of R will execute any of the phases as mentioned above. HTS is a nature-inspired, population-based algorithm [35]. The algorithm starts with a set of solutions to reach at the global optimum value. In HTS, a population is akin to molecules of the system, temperature levels of the molecules represent the value of design variables and energy level of the system represents the fitness value of objective function. The best solution is treated as the surrounding and rest of the solutions are part of the system. Working of each phase of HTS algorithm is explained below for minimization problem. Here, size of the population, number of design variables and generation number are denoted by ‘n’, ‘m’ and ‘g’ respectively. 3.1. Conduction phase

1] is a uniformly distributed random number. COF is the convection factor. 3.3. Radiation phase This phase simulate the radiation heat transfer within the system as well as between system and surrounding. Here, the system (i.e. solution) interacts with the surrounding (i.e. best solution) or within the system (i.e. other solution) to establish a thermal balance. In this phase, solutions are updated as below.

  8 < X new ¼ X old þ R* X old
X old j;i j;i j;i k;i   : X new ¼ X old þ R* X old
X old j;i j;i j;i k;i  gmax =RDF

  If f Xj > f ðXk Þ  ; If f ðXk Þ > f Xj

If g (16)

  8   < X new ¼ X old þ ri * X old
X old If f Xj > f ðXk Þ j;i j;i j;i k;i     ; If g > gmax =RDF : X new ¼ X old þ r * X old
X old If f ðX Þ > f X i j k j;i j;i j;i k;i (17)

This phase simulated the conduction heat transfer between the molecules of the substance. In conduction heat transfer, higher energy level molecules transmit heat to adjacent lower energy level molecules. In the course of optimization with HTS algorithm, higher and lower energy level molecule analogues to a population having higher and lesser objective function value. During conduction phase, solutions are updated according to the following equations.

Where j ¼ 1,2, …,n, j s k, k 2 (1,2, …,n) and k is a randomly selected solution from the population, i 2 (1,2, …,m). R 2 [0.6666, 1] is the probability for selection of radiation phase; ri 2 [0, 1] is a uniformly distributed random number. RDF is the radiation factor. Mathematical deduction of Equations 28e33 are given in the Appendix.

(

4. Multi-objective heat transfer search (MOHTS) algorithm

new old old ¼ Xk;i
R2 Xk;i Xj;i new old old ¼ Xj;i
R2 Xj;i Xk;i

  If f Xj > f ðXk Þ  ; If f ðXk Þ > f Xj

If g  gmax =CDF (12)

(

new old old ¼ Xk;i
ri Xk;i Xj;i new old old ¼ Xj;i
ri Xj;i Xk;i

  If f Xj > f ðXk Þ  ; If f ðXk Þ > f Xj

If g > gmax =CDF (13)

Where j ¼ 1,2, …,n, j s k, k 2 (1,2, …,n) and k is a randomly selected solution from the population. Further, i 2 (1,2, …,m) and i is a randomly selected design variables. R 2 [0, 0.3333] is the probability for selection of conduction phase; ri 2 [0, 1] is a uniformly distributed random number. CDF is the conduction factor. 3.2. Convection phase This phase simulates convection heat transfer between system and surrounding. In convective heat transfer, mean temperature of the system interact with the surrounding temperature to set a thermal equilibrium. In the course of optimization with HTS algorithm, best solution is assumed as a surrounding while rest of the solution compose the system. So, the design variable of the best solution interacts with the corresponding mean design variable of the population. In this phase, solutions are updated according to the following equations. new old Xj;i ¼ Xj;i þ R*ðXs
Xms *TCFÞ



TCF ¼ absðR
ri Þ If g  gmax =COF TCF ¼ roundð1 þ ri Þ If g > gmax =COF

(14)

(15)

Where j ¼ 1,2, …,n, i ¼ 1,2, …,m. Xs is the temperature of the surrounding and Xms is mean temperature of the system. R 2 [0.3333, 0.6666] is the probability for selection of convection phase; ri 2 [0,

Multi-objective heat transfer search (MOHTS) algorithm is a multi-objective version of previously developed heat transfer search algorithm (Single objective version). The MOHTS algorithm generates simultaneous solutions for each objective, identifies the non-dominated solution and stores those solutions in an external archive. While previously used HTS algorithm is a single objective version where all these things are not required as it handle only single objective. Multi-objective heat transfer search (MOHTS) algorithm uses an external archive to store non-dominated solutions for generation of Pareto front. A solution is called non-dominated (non-dominated points) if none of the objective functions can be improved in value without degrading some of the other objective values. For example if Solution A is better than Solution B in one objective, and worse in another, then one can say that the two solutions are non-dominated with respect to each other. The MOHTS algorithm uses ε-dominance based updating method [37] to check the domination of solutions in archive. Pareto front is generated based on the solution kept in external archive. MOHTS algorithm uses grid based approach with fixed size archive for archiving process. The best solutions found during the update are stored in archive. ε-dominance method is used to update archive in every generation. ε-dominance method assumes a space having dimensions equal to the number of objectives of the problem. Now slicing each dimension in a ε to ε size will break the space to boxes like squares, cubes or hyper cubes for two, three and more than three objectives respectively. Further, solutions are hold in these boxes. After that, the boxes (holding solutions) which are dominated by other boxes are removed first. In other words, solutions in those boxes are removed. Then, remaining boxes are examined to contain only one solution. If remaining boxes contain more than one solution then dominated ones are removed from each box. Thus, only non-dominated solutions are retained in the archive. Flow-chart of the MOHTS algorithm is shown in Fig. 2. The ability of MOHTS algorithm is assessed by implementing it on five different multi-objective benchmark problems. The

B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

325

Fig. 2. Flow chart of MOHTS algorithm.

benchmark multi-objective problems considered in this work are SCH, ZDT1, ZDT2, ZDT3, and LZ which possesses different characteristics. SCH is a one-dimensional problem with convex Pareto front. ZDT1 have convex Pareto front, whereas ZDT2 have nonconvex Pareto front. The ZDT3 have discontinuous Pareto front. The Pareto front obtained using the MOHTS algorithms for all five benchmark functions are show in Figure A1 to A5 respectively along with the true Pareto front in the Appendix. It can be noted form results that for all functions, MOHTS algorithm is capable to approximate true Pareto front. 5. Application example The effectiveness of present approach using MOHTS algorithm for many-objective optimization is assessed by analyzing an

application example of PFHE which has been taken from the literature [7,12]. It is intended to design and optimized PFHE (as shown in Fig. 1) used to preheat air with hot gas coming out from the furnace. The hot gas is supplied to the heat exchanger at a temperature of 620 K with mass flow rate of 1.45 kg/s. The air enters in to the heat exchanger with mass flow rate of 1.35 kg/s at a temperature of 315 K. The supply pressure of air and hot gas are 1.2 bar and 1.8 bar respectively. Property values of air and hot gas are considered to be temperature dependent. The PFHE is constructed from aluminium. The economic parameter required for the optimization of PFHE is taken from the reference [7]. So, the objectives are to find out design parameter of PFHE (i.e. cold flow length, hot flow length, fin height, fin thickness, fin frequency, fin offset length and number of cold side layer) for maximum effectiveness and minimum total annual cost, total weight and number of entropy

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Table 3 Control parameters of HTS/MOHTS algorithm. Selection probability of conduction phase: 0e0.3333 Selection probability of convection phase: 0.3333e0.6666 Selection probability of radiation phase: 0.6666e1 Conduction factor: 2 Convection factor: 10 Radiation factor: 2 Maximum number of generation: 100

generation units. 6. Results and discussion Initially, single objective optimization of each objective function is carried out to identify the behaviour of objective function with respect to each other. Control parameters of HTS and MOHTS algorithm used in this investigation are listed in Table 3. Results of single objective optimization are demonstrated in Table 4. It can be

observed form the result that when effectiveness is maximum (i.e. maximum effectiveness consideration) at that time other three objective functions are not at their optimum value. Similar situation is observed when we consider other objectives (i.e. total annual cost, total weight and number of entropy generation units) individually. Thus, results of single objective optimization clearly reveal the conflicting nature of all four objectives with respect to each other. So, many-objective (i.e. four-objective) optimization is carried out in the present work with the help of MOHTS algorithm. For the considered example of PFHE, 150 design points are generated as Pareto optimal points during many-objective optimization. For visualization of these points, the Pareto optimal points of many-objective optimization are represented in two dimension objective space of any two-objective together with the Pareto optimal points of corresponding objectives. In that way, results of four-objective optimization of present application example are represented by six Pareto front in the plan of different combination of two objective functions. Fig. 3 shows the distribution of Pareto optimal points of four-

Table 4 Optimal result with single objective consideration. Objectives

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) No. of entropy generation units

Effectiveness

Total annual cost ($)

Total weight (kg)

No. of entropy generation units

1 1 6.44 0.1 1000 2.1 90 0.9475 5779.29 604.55 0.2224

0.89 1 10 0.2 213.27 8.49 70 0.8206 935.34 377.96 0.3695

0.26 0.27 6.76 0.1 1000 3.97 65 0.8271 6795.36 31.63 0.4032

1 0.9 6.48 0.1 1000 3.19 102 0.9447 4878.87 629.43 0.2208

Fig. 3. Variation of effectiveness with total annual cost in both 4-objective and 2-objective optimization.

B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

327

Table 5 Optimal parameters for sample design point (A-E) in the plan of effectiveness and total annual cost.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

A

B

C

D

E

X1

X2

Y1

Y2

0.98 0.98 6.44 0.10 999.31 2.29 87 0.9452 5772.65 561.26 0.2267

0.70 0.75 6.66 0.10 982.77 2.66 71 0.9216 6059.52 255.12 0.2722

0.53 0.45 6.55 0.10 904.97 4.34 72 0.8805 5099.56 111.13 0.3265

0.43 0.79 6.79 0.12 690.77 5.84 87 0.8782 2631.98 192.48 0.3160

0.81 1.00 6.22 0.20 215.53 10.00 105 0.8214 986.24 442.73 0.3693

0.77 0.96 6.41 0.10 994.74 3.23 63 0.9284 6632.06 313.63 0.2655

0.85 0.76 6.54 0.10 984.48 2.22 70 0.9286 6959.91 302.42 0.2648

0.96 0.95 6.64 0.13 396.01 7.18 104 0.8799 1598.62 506.23 0.3036

0.51 0.50 7.35 0.11 481.27 6.83 73 0.8206 1598.35 99.55 0.3752

objective optimization in a two dimension objective space of effectiveness and total annual cost, together with the Pareto optimal points of the corresponding objectives. Scattered distribution of Pareto optimal points of four-objective optimization are observed as compared to two-objective optimization (where clear Pareto front is observed). As shown in figure, some of the design points of two-objective and four-objective optimization are overlapped on each other. However, for remaining design points, fourobjective optimization results in higher total cost at given effectiveness as compared to two-objective optimization. Optimal design parameters for five sample data points (A to E) corresponding to Fig. 3 are listed in Table 5. It can be observed from Table 5 that variation in fin offset length, number of cold side layer and fin frequency is higher as compared to other design variables for data points A-E. Further, it can be observed from Fig. 3 that there are some points that may looks dominate with respect to other in two dimensional objectives space. For example, X1-X2 with respect to total annual cost and Y1-Y2 with respect to effectiveness. However, these individuals are non-dominated while considering all four objectives simultaneously as observed from results listed in

Table 5. Such facts would be very important to the designer to switch from one optimal solution to another for achieving different trade-off requirements of the objectives. The distribution of Pareto optimal points in the plan of effectiveness and total weight is shown in Fig. 4. In this case also, scattered distribution of Pareto optimal points of four-objective optimization are observed as compared to two-objective optimization. Few sample data points with optimal design parameters corresponding to Fig. 4 are listed in Table 6. Results indicate wide variation in the value of design variables (except fin height and fin thickness) for presented data points. Further, non-dominated nature of solutions with respect to many-objective consideration are presented by sample design points X1-X2 and Y1-Y2 (as shown in Fig. 4) and listed in Table 6. Fig. 5 displays the distribution of Pareto optimal points in the plan of effectiveness and number of entropy generation units. It is clear from figure that Pareto front of two-objective consideration appears as a cluster of points overlapped on each other (top left corner of the figure) while wide spread distribution of Pareto points is observed in four-objective consideration. Thus, the conflicting

Fig. 4. Variation of effectiveness with total weight in 4-objective and 2-objective optimization.

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Table 6 Optimal parameters for sample design point (A-E) in the plan of effectiveness and total weight.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

A

B

C

D

E

X1

X2

Y1

Y2

1.00 0.92 6.45 0.10 998.20 2.19 98 0.9468 5475.92 617.07 0.2212

0.99 0.97 6.72 0.14 580.46 7.44 103 0.9100 2284.54 637.46 0.2639

0.82 0.91 6.49 0.12 361.47 9.49 105 0.8593 1323.03 379.74 0.3287

0.43 0.64 6.68 0.11 939.80 6.17 78 0.8903 4136.42 150.24 0.3096

0.33 0.32 6.95 0.10 789.76 6.03 68 0.8206 3683.96 45.29 0.3887

0.77 0.96 6.41 0.10 994.74 3.23 63 0.9284 6632.06 313.63 0.2655

0.85 0.76 6.54 0.10 984.48 2.22 70 0.9286 6959.91 302.42 0.2648

0.51 0.81 6.62 0.11 619.76 5.70 79 0.8730 2397.24 185.28 0.3206

0.57 0.74 6.79 0.11 553.76 7.44 79 0.8626 1915.88 185.44 0.3300

Fig. 5. Variation of effectiveness with number of entropy generation units in both 4-objective and 2-objective optimization.

nature of these objectives is elevated in four-objective consideration. Optimal design parameters for some selected data points are listed in Table 7. For presented data points, variation in total annual cost and weight of PFHE is observed for simultaneous change of effectiveness and number of entropy generation units. Further,

larger variation in fin frequency, fin offset length and number of cold side layer is observed as compared to other design variables. Moreover, to reveal that solutions are non-dominated, sample design points X1-X2 and Y1-Y2 are shown in Fig. 5 and listed in Table 7.

Table 7 Optimal parameters for sample design point (A-E) in the plan of effectiveness and number of entropy generation units.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

A

B

C

D

E

X1

X2

Y1

Y2

0.99 0.99 6.44 0.10 989.97 2.15 83 0.9448 6015.12 545.62 0.2292

0.69 0.91 6.50 0.12 831.96 4.59 96 0.9200 3471.75 419.89 0.2578

0.53 0.68 6.80 0.10 794.66 5.26 78 0.8881 3226.54 172.79 0.31

0.57 0.72 6.69 0.11 445.26 7.64 98 0.8482 1373.25 208.34 0.3435

0.33 0.33 7.16 0.10 821.44 8.30 72 0.8251 3202.39 49.60 0.3812

1.00 1.00 6.51 0.16 485.70 9.51 106 0.8975 1920.07 662.21 0.2804

0.65 0.85 6.58 0.11 704.59 5.34 82 0.8974 2830.26 273.11 0.2888

0.55 0.68 6.63 0.10 1000 4.89 92 0.9126 4096.61 238.25 0.2745

0.99 0.99 6.35 0.15 509.83 6.78 106 0.9023 2097.13 644.84 0.2747

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329

Fig. 6. Variation of total annual cost with total weight in both 4-objective and 2-objective optimization.

Table 8 Optimal parameters for sample design point (A-E) in the plan of total annual cost and total weight.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

A

B

C

D

E

X1

X2

Y1

Y2

0.27 0.26 6.79 0.10 955.53 3.41 71 0.8268 6149.14 33.46 0.3983

0.76 1.00 6.54 0.10 1000 5.88 58 0.9242 5972.27 301.23 0.2684

0.63 0.71 6.53 0.10 757.15 4.22 78 0.8944 3395.26 209.53 0.2969

0.81 0.87 6.22 0.10 870.40 2.60 106 0.9306 3997.05 467.06 0.2422

0.86 0.89 6.76 0.17 409.17 9.03 102 0.8738 1524.75 478.61 0.3116

0.99 1.00 6.34 0.18 582.52 9.14 109 0.9121 2339.30 789.66 0.2610

1.00 1.00 5.88 0.16 591.10 10.00 111 0.9101 2339.20 715.06 0.2641

0.70 0.75 6.62 0.10 948.03 4.58 69 0.9143 5104.56 246.40 0.2776

0.57 0.67 6.67 0.10 1000 4.91 95 0.9138 3958.52 246.99 0.2713

Fig. 6 shows the distribution of Pareto optimal points in two dimension objective space of total annual cost and total weight. It can be observed from the figure that in one part of the Pareto points of four-objective optimization (top to middle region of left side), wide variation in total annual cost is observed with small change in total weight. While in other part (right to middle region of bottom side), wide variation in total weight is observed with small change in total annual cost. Specification of five sample data points of Pareto solutions are listed in Table 8. It can be observed from results that fin height and fin thickness are more or less constant while significant variation is visible in remaining design variables. Further, results are also reported in Table 8 (corresponding to Fig. 6) to indicate that solutions are non-dominated with respect to manyobjective consideration. Comparison of Pareto optimal points for four-objective and twoobjective optimization in the plan of total annual cost and number of entropy generation units is shown in Fig. 7. Design parameters of few sample data points of this distribution are reported in Table 9. Here also, large variation in the value of design variables (except fin height and fin thickness) is observed. Design points X1-X2 and Y1-

Y2 listed in Table 9 reveal that solutions are non-dominated with respect to many-objective consideration. Distribution of Pareto solutions in the plan of total weight and number of entropy generation units is shown in Fig. 8 for twoobjective and four-objective optimization. It can be observed from figure that in this case majority design points of four-objective optimization are spread in the nearby region of Pareto front of the two-objective optimization. Specification of five sample data points along with design points indicating non-dominated nature of solutions (X1-X2 and Y1-Y2) corresponding to Fig. 8 are listed in Table 10. Results indicate variation in the value of each design variables for considered data points. Based on the comparative results between four-objective and two-objective consideration following points can be noted. The conflicting nature between effectiveness and number of entropy generation units is more in four-objective consideration as compared to other cases. This could be due to higher rise in total annual cost of PFHE with reduction in number of entropy generation units and increment in effectiveness. On the other hand, the conflicting nature between number of entropy generation units and

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Fig. 7. Variation of total annual cost with number of entropy generation units in both 4-objective and 2-objective optimization.

Table 9 Optimal parameters for sample design point (A-E) in the plan of total annual cost and number of entropy generation units.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

A

B

C

D

E

X1

X2

Y1

Y2

0.99 0.99 6.44 0.10 989.97 2.15 83 0.9448 6015.12 545.62 0.2292

0.79 0.85 6.57 0.10 967.99 2.45 64 0.9264 6777.75 288.77 0.2688

0.37 0.41 6.60 0.10 938.37 2.45 74 0.8709 5905.94 74.98 0.3465

0.53 0.68 6.80 0.10 794.66 5.26 78 0.8881 3226.54 172.79 0.3055

0.51 0.50 7.35 0.11 481.27 6.83 73 0.8206 1599.35 99.55 0.3752

0.65 0.85 6.58 0.11 704.59 5.34 82 0.8974 2830.26 273.11 0.2888

1.00 1.00 5.82 0.14 720.26 9.24 116 0.9239 2829.02 766.53 0.2446

0.45 0.52 6.68 0.10 830.16 4.09 68 0.8717 4310.26 100.29 0.3348

0.37 0.59 6.90 0.12 811.58 4.88 76 0.8695 3654.98 112.18 0.3350

total weight is less as compared to other cases. Main reason behind this is almost similar variation in effectiveness and total annual cost is observed with reduction in number of entropy generation unit and total weight of PFHE. Many-objective optimization utilizes different decision making methods in order to select best solution from Pareto optimal points. Different methods utilized for the process of decision making are available in the reference [38,39]. In this work, three decision making methods including LINMAP, TOPSIS and Fuzzy have been used. TOPSIS method works based upon the concept that the chosen solution (i.e. alternative) should have minimum Euclidean distance from the ideal solution (i.e. best solution) and maximum Euclidean distance from the negative-ideal solution (i.e. worst solution). Initially, the normalized decision matrix is prepared which convert the attributes into non-dimensional form. Then, weighted normalized decision matrix is created by assigning weights to each attribute. After that, best and worst solutions are determined. Finally, relative closeness of each solution with respect to best solution is identified by utilizing separation measures which are further arranged in the descending order to obtain rank of each

solution. LINMAP used for assessing weights as well as locating the ideal (i.e. best) solution. In this method, a decision maker has to assume his ideal solution (i.e. best solution) from available solutions. Once the location of ideal solution is decided, one can choose an alternative solution which has the shortest Weighted Euclidean distance from the ideal solution. Fuzzy decision making use the measurement of linguistic variables to demonstrate the criteria performance/evaluation (effect-values) by expressions and each linguistic variable can be indicated by a triangular fuzzy number (TFN) within the scale range 0e100. After that, fuzzy performance matrix of each alternatives can be obtained from the fuzzy performance value of each alternative and based on that fuzzy synthetic decision vector is obtained. Finally, defuzzification of fuzzy synthetic decision vector is carried out to obtain best non-fuzzy performance (BNP) and based on BNP, ranking of alternative is proposed. For the selection of final solution, preferences used by the decision making algorithms in terms of normalized weight is mentioned in Table 11. Final solutions selected by LINMAP, TOPSIS and Fuzzy decision making methods are shown in Figs. 3e8 and

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331

Fig. 8. Variation of total weight with number of entropy generation units in both 4-objective and 2-objective optimization.

Table 10 Optimal parameters for sample design point (A-E) in the plan of total weight and number of entropy generation units.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

A

B

C

D

E

X1

X2

Y1

Y2

0.99 0.96 5.93 0.10 887.18 2.57 107 0.9401 4440.39 624.26 0.2270

0.99 0.98 6.35 0.19 489.62 7.94 108 0.9004 2027.16 729.20 0.2771

0.86 0.89 6.76 0.17 409.17 9.03 102 0.8738 1524.75 478.61 0.3116

0.62 0.72 6.16 0.10 764.56 5.76 106 0.8992 2666.71 277.41 0.2847

0.51 0.50 7.35 0.11 481.27 6.83 73 0.8206 1599.35 99.55 0.3752

0.51 0.81 6.62 0.11 619.76 5.70 79 0.8730 2397.24 185.28 0.3206

0.57 0.74 6.79 0.11 553.76 7.44 79 0.8626 1915.88 185.44 0.3300

1.00 1.00 5.88 0.16 591.10 10.00 111 0.9101 2339.20 715.06 0.2641

0.99 0.97 6.72 0.14 580.46 7.44 103 0.9099 2284.54 637.46 0.2639

listed in Table 12. According to results, solution obtained by LINMAP and TOPSIS are approaching towards each other. To gain a greater insight on underlying relationship of decision variables with four-objective optimization, values of decision variables corresponding to four-objective optimization are plotted for all 150 design points in Figs. 9e15. For comparison purpose, distribution of the design variable in two-objective optimization is also presented in the same figures. Figs. 9 and 10 show the distribution of cold flow length and hot flow length for all 150 design points. Predominant effect of these design variables are observed during simultaneous optimization of

Table 11 Preferences used by decision making algorithms. Objective

LINMAP

TOPSIS

FUZZY

Effectiveness Total annual cost Total weight Number of entropy generation units

0.35 0.35 0.2 0.1

0.35 0.35 0.2 0.1

0.35 0.35 0.2 0.1

(i) effectiveness vs. total weight (ii) total weight vs. total annual cost (iii) total weight vs. number of entropy generation units. Due to this conflicting effect, scatter distribution of these design variables is observed in many-objective optimization. The distribution of fin height for all Pareto points is shown in Fig. 11. It can be observed from the figure that fin height is distributed within a narrow range of the allowable limit for fourTable 12 Decision making of many objective optimization results.

Cold flow length (m) Hot flow length (m) Fin height (mm) Fin thickness (mm) Fin frequency (m
1) Fin offset length (mm) Number of cold side layer Effectiveness Total annual cost ($) Total weight (kg) Number of entropy generation units

LINMAP

TOPSIS

FUZZY

0.77 0.81 6.55 0.12 405.79 4.55 72 0.8564 1793.11 227.85 0.3363

0.59 0.6 6.59 0.12 533.02 6.1 91 0.8574 1823.81 181.63 0.3355

0.62 0.99 6.38 0.12 444.57 5.52 83 0.8649 1786.51 267.1 0.3263

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Fig. 9. Distribution of cold flow length for both 4-objective and 2-objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

Fig. 10. Distribution of hot flow length for both 4-objective and 2-objective objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

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333

Fig. 11. Distribution of fin height for both 4-objective and 2-objective objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

Fig. 12. Distribution of fin thickness for both 4-objective and 2-objective objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

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Fig. 13. Distribution of fin frequency for both 4-objective and 2-objective objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

Fig. 14. Distribution of fin offset length for both 4-objective and 2-objective objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

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335

Fig. 15. Distribution of number of cold side layers for both 4-objective and 2-objective objective ((A) ε vs. TAC (B) ε vs. Wt (C) ε vs. Ns (D) TAC vs. Wt (E) TAC vs. Ns (F) Wt vs. Ns) optimization.

objective optimization. Thus, moderate conflicting effect is produced by fin height during many-objective consideration. The distribution of fin thickness for all Pareto points is shown in Fig. 12. It can be noted from the figure that scattered distribution of fin thickness is observed for (i) effectiveness vs. total annual cost (ii) total annual cost vs. number of entropy generation units. As a result of this, variation in distribution of fin thickness is observed in many-objective optimization. Figs. 13 and 14 show the distribution of fin frequency and fin offset length respectively for all design points. Scatter distribution of these design variables are observed during the simultaneous optimization of (i) effectiveness vs. total annual cost (ii) total annual cost vs. total weight and (ii) total annual cost vs. number of entropy generation units. Due to this conflicting effect, scatter distribution of these design variables is observed in many-objective optimization. Finally, the distribution of number of cold side layer for all Pareto points is shown in Fig. 15. It can be noted from the figure that scattered distribution of number of cold side layer is observed for four-objective optimization.

optimal points of many-objective optimization are represented in two dimension objective space of any two-objective together with the Pareto optimal points of the corresponding objectives. Thus, six Pareto front in the plan of different combination of the twoobjective function are presented in this work. A final optimal solution is selected from the Pareto optimal points using three decision making methods including LINMAP, TOPSIS and Fuzzy. Furthermore, the distribution of each design variables in their allowable range is also shown. The results revealed the level of conflict between these four-objective. Cold flow length, hot flow length, fin frequency, fin offset length and fin thickness are found to be important geometric parameters which caused a strong conflict between the objective functions. Finally, it is observed that the many-objective approach leads to more realistic design of the PFHE as compared to multi-objective approaches.

Appendix 7. Conclusion In the present work, multi-objective transfer search (MOHTS) algorithm has been adapted to solve the many-objective optimization problem. The algorithm's ability is investigated on the fourobjective optimization problem of PFHE involving conflicting objectives. Seven design variables which include geometric parameters are considered for optimization. A set of Pareto-optimal points are obtained for conflicting objectives. For visualization, Pareto

A1: Mathematical deduction of equations used in HTS algorithm 1. Conduction phase:

(

new old old ¼ Xk;i
R2 Xk;i Xj;i new old old ¼ Xj;i
R2 Xj;i Xk;i

  If f Xj > f ðXk Þ  ; If f ðXk Þ > f Xj

If g  gmax =CDF

336

(

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new old old ¼ Xk;i
ri Xk;i Xj;i new old old ¼ Xj;i
ri Xj;i Xk;i

  If f Xj > f ðXk Þ  ; If f ðXk Þ > f Xj

If g > gmax =CDF

2. Convection phase: new old Xj;i ¼ Xj;i þ RðXs
Xms TCFÞ

TCF ¼ absðR
ri Þ If g  gmax =COF TCF ¼ roundð1 þ ri Þ If g > gmax =COF

3. Radiation phase:

  8 < X new ¼ X old þ R* X old
X old j;i j;i j;i k;i   : X new ¼ X old þ R* X old
X old j;i j;i j;i k;i

  If f Xj > f ðXk Þ  ; If f ðXk Þ > f Xj

If g

 gmax =RDF

  8   < X new ¼ X old þ ri * X old
X old If f Xj > f ðXk Þ j;i j;i j;i k;i     ; If g > gmax =RDF : X new ¼ X old þ r * X old
X old If f ðX Þ > f X i j k j;i j;i j;i k;i Heat transfer search (HTS) algorithm exhibits proper trade off between exploration and exploitation during course of optimization. The large change in design variables is corresponding to exploration of search space while small change in design variable is corresponding to exploitation of search space. The search mechanism of the Conduction, Convection and Radiation phase subdivided into two part which is controlled by the by the number of generation (g) which in turn depend on the Conduction factor (CDF), Convection factor (COF) and Radiation factor (RDF) respectively (i.e. g < gmax/CDF,COF,RDF or g > gmax/ CDF,COF,RDF). The search mechanism of the first part of the Conduction, Convection and Radiation phase involve the parameter R. The value of R lies between 0 and 0.3333 for Conduction phase, 0.3333e0.6666 for Radiation phase and 0.6666e1 for Convection phase. The search mechanism of the second part of the Conduction, Convection and Radiation phase involve only random number (ri).

The value of random number lies between 0 and 1. Thus, depend on the value of R and random number (ri), large/small change in value of design variables are take place in the first/second part of the Conduction, Convection and Radiation phase. The large change in design variables is corresponding to exploration of search space while small change in design variable is corresponding to exploitation of search space. Thus the rationale behind using g < gmax/ CDF,COF,RDF or g > gmax/CDF,COF,RDF is to provide proper tradeoff between exploration and exploitation by executing different search mechanism. Experiments are conducted to decide the value of CDF, COF and RDF. After, experimentation following value of CDF, COF and RDF is considered in the present work. CDF ¼ 2, COF ¼ 10, RDF ¼ 2. gmax represent maximum number of generation (i.e. iteration). In each generation (g), algorithm carryout search processes till it reach at gmax. At the end of maximum number of generation, algorithm displays optimize output. Experiments are conducted to decide maximum number of generation in the present work. After experiments, maximum number of generation is set as 100 in the present work. Because, after 100 generation, optimized value of objective functions are remain same while computational effort is increases. TCF is responsible for the proper tradeoff between exploration and exploitation (i.e. large/small change in value of design variables) in the convection phase. Based on this fact, relation of TCF is determined so that it can carry out smaller (TCF ¼ abs (R - ri)) or larger (TCF ¼ round (1 þ ri)) change in the value of design variables. A2: Results of MOHTS algorithm for multi-objective benchmark problems

8 < Mininize ð1Þ SCH Function : SCH ¼ Minimize :

Fig. A1. Pareto front for SCH function.

f1 ðxÞ ¼ x2 f2 ðxÞ ¼ ðx
2Þ2
103  x  103

B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

8 Mininize f1 ðxÞ ¼ x1 > > >  > rffiffiffiffiffiffiffiffiffi > > x1 > > Minimize f ðxÞ ¼ gðxÞ 1
> 2 > > gðxÞ > < ! ð2Þ ZDT1 Function : ZDT1 ¼ Pn > > 9 x > i¼2 i > > > > gðxÞ ¼ 1 þ > > > n
1 > > : n ¼ 30; 0 < x < 1

Fig. A2. Pareto front for ZDT1 function.

8 Mininize f ðxÞ ¼ x 1 1 > > " > #  > > > x1 2 > > Minimize f ðxÞ ¼ gðxÞ 1
> 2 > > gðxÞ > < ! ð3Þ ZDT2 Function : ZDT2 ¼ Pn > > > 9 > i¼2 xi > > > > gðxÞ ¼ 1þ > > > n
1 > : n ¼ 30; 0
Fig. A3. Pareto front for ZDT2 function.

337

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B.D. Raja et al. / International Journal of Thermal Sciences 118 (2017) 320e339

ð4Þ ZDT3 Function : ZDT3 ¼

8 Mininize > > > > > > > > Minimize > > > > < > > > > > > > > > > > > :

f1 ðxÞ ¼ x1

 rffiffiffiffiffiffiffiffiffiffiffiffiffi x1 x1
sinð10px1 Þ f2 ðxÞ ¼ gðxÞ 1
gðxÞ gðxÞ ! Pn 9 i¼2 xi

gðxÞ ¼ 1 þ n ¼ 30;

n
1 0
Fig. A4. Pareto front for ZDT3 function.

ð5Þ LZ Function : LZ ¼

8 > > Mininize > > > > > > <

  2 X jp 2 xj
Sin 6px1 þ n jJ1 j j2J 1   pffiffiffi 2 X jp f2 ðxÞ ¼ 1
x þ 2 xj
Sin 6px1 þ n jJ2 j j2J

f1 ðxÞ ¼ x1 þ

> Minimize > > > > > > > : J1 ¼ fjjj is odd and 2  j  ng;

2

J2 ¼ fjjj is even and 2  j  ng

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339

Fig. A5. Pareto front for LZ function.

References [1] Kays WM, London AL. Compact heat exchangers. New York: McGraw Hill; 1984. [2] Shah RK, Sekulic P. Fundamental of heat exchanger design. New York: John Wiley and Sons; 2003. [3] Li Q, Flamant G, Yuan X, Neveu P, Luo L. Compact heat exchangers: a review and future applications for a new generation of high temperature solar receivers. Renew Sustain Energy Rev 2011;15:4855e75. [4] Wen J, Yang H, Tong X, Li K, Wang S, Li Y. Optimization investigation on configuration parameters of serrated fin in plate-fin heat exchanger using genetic algorithm. Int J Therm Sci 2016;101:116e25. [5] Du J, Ni YM, Fang YS. Correlations and optimization of a heat exchanger with offset fins by genetic algorithm combining orthogonal design. Appl Therm Eng 2016;107:1091e103. [6] Turgut OE. Hybrid Chaotic Quantum behaved Particle Swarm Optimization algorithm for thermal design of plate fin heat exchangers. Appl Math Model 2016;40:50e69. [7] Sanaye S, Hajabdollahi H. Thermal-economic multi-objective optimization of plate fin heat exchanger using genetic algorithm. Appl Energ 2010;87: 1893e902. [8] Rao RV, Patel VK. Multi-objective optimization of heat exchangers using a modified teaching-learning based optimization algorithm. Appl Math Model 2013;37:1147e62. [9] Wang Z, Li Y. Irreversibility analysis for optimization design of plate fin heat exchangers using a multi-objective cuckoo search algorithm. Energ Convers Manage 2015;101:126e35. [10] Hajabdollahi H. Investigating the effect of non-similar fins in thermoeconomic optimization of plate fin heat exchanger. Appl Therm Eng 2015;82:152e61. [11] Hadidi A. A robust approach for optimal design of plate fin heat exchangers using biogeography based optimization (BBO) algorithm. Appl Energ 2015;150:196e210. [12] Patel VK, Savsani VJ. Optimization of a plate-fin heat exchanger design through an improved multi-objective teaching-learning based optimization (MO-ITLBO) algorithm. Chem Eng Res Des 2014;92:2371e82. [13] Wang Z, Li Y, Zhao M. Experimental investigation on the thermal performance of multi-stream plate-fin heat exchanger based on genetic algorithm layer pattern design. Int J Heat Mass Tran 2015;82:510e20. [14] Zhao M, Li Y. An effective layer pattern optimization model for multi-stream plate-fin heat exchanger using genetic algorithm. Int J Heat Mass Tran 2013;60:480e9. [15] Zhou Y, Zhu L, Yu J, Li Y. Optimization of plate-fin heat exchangers by minimizing specific entropy generation rate. Int J Heat Mass Tran 2014;78:942e6. [16] Yousefi M, Darus AN, Hooshyar D. Multi-stage thermal-economical optimization of compact heat exchangers: a new evolutionary-based design approach for real-world problems. Appl Therm Eng 2015;83:71e80. [17] Rao RV, Patel VK. Thermodynamic optimization of cross-flow plate-fin heat exchangers using a particle swarm optimization technique. Int J Therm Sci 2010;49:1712e21. [18] Yousefi M, Enayatifar R, Darus AN. Optimal design of plate-fin heat exchangers by a hybrid evolutionary algorithm. Int Commun Heat Mass 2012;39:258e63. [19] Yousefi M, Enayatifar R, Darus AN, Abdullah AH. Optimization of plate-fin heat exchangers by an improved harmony search algorithm. Appl Therm Eng

2013;50:877e85. [20] Zarea H, Kashkooli FM, Mehryan AM, Saffarian MR, Beherghani EN. Optimal design of plate-fin heat exchangers by a Bees algorithm. Appl Therm Eng 2014;69:267e77. [21] Xie GN, Sunden B, Wang QW. Optimization of compact heat exchangers by a genetic algorithm. Appl Therm Eng 2008;28:895e906. [22] Peng H, Ling X, Wu E. An improved particle swarm algorithm for optimal design of plate-fin heat exchangers. Ind Eng Chem Res 2010;49:6144e9. [23] Yousefi M, Darus AN, Mohammadi H. An imperialist competitive algorithm for optimal design of plate-fin heat exchangers. Int J Heat Mass Tran 2012;55: 3178e85. [24] Ghosh S, Ghosh I, Pratihar DK, Maiti B, Das PK. Optimum stacking pattern for multi-stream plate-fin heat exchanger through a genetic algorithm. Int J Therm Sci 2011;50:214e24. [25] Saechan P, Wongwises S. Optimal configuration of cross flow plate finned tube condenser based on the second law of thermodynamics. Int J Therm Sci 2008;47:1473e81. [26] Zhang L, Yang C, Zhou J. A distributed parameter model and its application in optimizing the plate-fin heat exchanger based on the minimum entropy generation. Int J Therm Sci 2010;49:1427e36. [27] Ayala HVH, Keller P, Morais MF, Mariani VC, Coelho LS, Rao RV. Design of heat exchangers using a novel multi objective free search differential evolution paradigm. Appl Therm Eng 2016;94:170e7. [28] Yin H, Ooka R. Shape optimization of water-to-water plate-fin heat exchanger using computational fluid dynamics and genetic algorithm. Appl Therm Eng 2015;80:310e8. [29] Babaelahi M, Sadri S, Sayyaadi H. Multi-objective optimization of a cross-flow plate heat exchanger using entropy generation minimization. Chem Eng Technol 2014;37:87e94. [30] Yousefi M, Enayatifar R, Darus AN, Abdullah AH. A robust learning based evolutionary approach for thermal-economic optimization of compact heat exchangers. Int Commun Heat Mass 2012;39:1605e15. [31] Ahmadi P, Hajabdollahi H, Dincer I. Cost and entropy generation minimization of a cross-flow plate fin heat exchanger using multi-objective genetic algorithm. J Heat trans-T ASME 2011;133:21801e9. [32] Mishra M, Das PK. Thermo-economic design-optimization of cross flow platefin heat exchanger using genetic algorithm. Int J Exergy 2009;6:237e52. [33] Najafi H, Najafi B, Hoseinpoori P. Energy and cost optimization of a plate and fin heat exchanger using genetic algorithm. Appl Therm Eng 2011;31: 1839e47. [34] Wen J, Ynag H, Tong X, Li K, Wang S, Li Y. Configuration parameters design and optimization for plate fin heat exchangers with serrated fin by multiobjective genetic algorithm. Energ Convers Manage 2016;117:482e9. [35] Patel VK, Savsani VJ. Heat transfer search (HTS): a novel optimization algorithm. Inf Sci 2015;324:217e46. [36] Bejan A. The concept of irreversibility in heat exchanger design: counter flow heat exchangers for gas-to-gas applications. J Heat trans-T ASME 1977;99: 374e80. [37] Deb K, Mohan M, Mishra S. Evaluating the epsilon-domination based multiobjective evolutionary algorithm for a quick computation of Pareto-optimal solutions. Evol Comput 2005;13:501e25. [38] Olson DL. Decision aids for selection problems. Springer Science & Business Median New York; 1996. [39] Hwang CL, Yoon K. Multiple attribute decision making: methods and applications a state-of-the-art survey. New York: Springer Science & Business Media; 2012.