Multi-Objective Optimization of Plate Fin Heat ...

International Journal of Advanced Biotechnology and Research (IJBR) ISSN 0976-2612, Online ISSN 2278–599X, Vol-7, Issue-1, 2016 http://www.bipublication.com Research Article

Multi-Objective Optimization of Plate Fin Heat Exchanger Using Evolutionary Algorithm Hossein zarea1,* - Mohsen Keshavarzi1 1,2

Research and technology department, Gachsaran oil & gas production copmany, Gachsaran, Iran

ABSTRACT In this paper, thermal modeling and optimal design of a plate fin heat exchanger are presented. Based on the applications, design parameters under given restrictions namely, cold stream flow length, hot stream flow length, noflow length, fin pitch, fin offset length and fin height were considered as optimization variables. The multi objective bees algorithm (MOBA) has been applied to obtain the minimum number of entropy generation units (EGU) and total annual cost (sum of investment and operation costs). Since the functions are in conflict with each other, so that, with improvement of one objective, the other objective goes to undesirable end, a single solution can’t satisfy both of the objective functions simultaneously. So, instead of a specific solution, optimal solutions (Pareto curve) are required the designer to choose one of these points as the optimal solution according to his need. In order to show the effectiveness of the proposed algorithm the obtained results of this study compared to a case study of references. The numerical results reveal that results of this study better than references in some design points. Generally, the results of this study have lower total annual costs (988) as well as the number of entropy generation units (0.0921). Keywords: Plate fin heat exchanger, Optimization, Bees algorithm

1. INTRODUCTION Energy recovery is one of the significant means of energy preservation to save the rare energy in today’s world. Heat exchangers are important equipment which servers the aim of energy recovery. A heat exchanger is a device which provided for thermal energy flow between two or more fluids in different temperatures. Various types of heat exchangers are used for different industrial applications and one of the important types is the compact heat exchanger. The compact heat exchanger can be either plate-fin type or tube-fin type [1]. Plate fin heat exchanger (PFHE) is a typical compact heat exchanger that is widely used in many industrial power generation plants, chemical, petrochemical, and petroleum industries [2]. Fins or extended

surfaces are components that decrease size and increase heat transfer are widely used in compact heat exchangers [3, 4]. Some of commonly used fins in these exchangers are plain, wavy, louver, perforated, offset strip and pin fins [5]. According to the literature, the customary objectives in heat exchanger design are associated with minimizing investment and operational costs. Generally, a higher flow velocity means a higher heat transfer coefficient and thus a lower heat transfer area and consequently lower investment cost. Also, it should be noticed that, higher velocity leads to higher pressure drop and thus higher power consumption and hence higher pumping cost. So, before doing any optimal design, the suitable

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objective function is needed. In most cases, an agreement between the investment cost and the operational cost should be attained by the design variables. Thus, minimizing the total annual cost is discussed as the objective function [6]. The design of plate fin heat exchangers is a very significant subject in industrial processes. So, different methods have been utilized for optimal design of plate fin heat exchangers. Traditional methods of optimization do not fare well over a broad spectrum of problem domains. Numerous constraints and objectives makes the heat exchanger optimization problems complicated and hence the traditional optimization techniques are not ideal for solving such problems as they tend to obtain a local optimal solution. Considering the drawbacks of traditional optimization techniques, attempts are being made to optimize the heat exchanger design problem using nature inspired optimization algorithms like genetic algorithm (GA), particle swarm optimization (PSO), artificial bee colony (ABC), bees Algorithm (BA), etc. In the early attempts, Genetic Algorithm based on the random search is widely used in designing and optimization of heat exchangers. Ahmadi et al minimized the number of entropy generation units (EGU) and total annual cost as two objective functions in a PFHE with offset strip fins [7]. Najafi et al, successfully utilized multi-objective optimization using GA to obtain a set of design geometrical to achieve two conflicting objectives, namely total heat transfer rate and total annual cost for PFHEs [8]. Sanaye and Hajabdollahi applied fast and elitist non dominated sorting genetic-algorithm (NSGA-II) to achieve maximum effectiveness and the minimum total annual cost two objective functions in a cross flow PFHE with offset strip fins [2]. Some of researchers have currently proposed application of Particle Swarm Optimization (PSO) algorithm for heat exchangers optimization. Ghanei et al, employed Multi objective PSO (MOPSO) to obtain the maximum effectiveness (heat recovery) and the

.

minimum total cost as two objective functions [9]. In another work, Rao et al, used a teachinglearning based optimization algorithm (TLBO) to obtain maximum effectiveness and minimum total annual cost as two objective functions in two types of heat exchangers, a presented algorithm has better results than a GA [10]. Lately, new and/or hybrid algorithms such as an improved multi-objective cuckoo search (IMOCS) algorithm [11], multi-objective improved-teaching-learning-based optimization (MO-ITLBO) algorithm [12], bio-geographybased (BBO) algorithm [13], and multi-objective free search approach combined with differential evolution (MOFSDE) algorithm [14], applied to multi-objective design optimization of PFHEs. Among the most common population-based optimization algorithms are the Genetic Algorithm, Particle Swarm Optimization and Ant Colony Optimization. In addition to these algorithms, bees-inspired algorithms (such as Artificial Bee Colony (ABC), Bee colony optimization (BCO), Bees Algorithm (BA) and etc.) are being also developed and they are emulating various behaviors of the bees. Each algorithm requires its own algorithm-specific control parameters. For example, the effective working of GA requires determination of optimum algorithm-specific parameters such as crossover rate and mutation rate. Similarly, PSO requires determination of inertia weight, social and cognitive parameters. The bees algorithm requires determination of number of employed bees for elite site, neighborhood search size and number of scout bees. Just recently, in two study, Zarea et al. employed a BA to optimize a cross flow plate-fin heat exchanger with the aim of maximizing the effectiveness and minimizing the total annual cost and number of entropy generation [15, 16]. Also, Pham et al, examined the first application of the Bees Algorithm for optimal design of mechanical problems [17] to [19]. Many studies are performed by researchers about plate fin heat exchanger (PFHE) but the

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multi objective bees algorithm (MOBA) technique has never been used in such studies. The objective of the present study is to conduct a thermal modeling of a plate fin type heat exchanger, as well as multi-objective optimization of the heat exchanger. Therefore, two objective functions are considered here. The number of entropy generation units is optimized while total annual cost is minimized. In addition, a BA optimization technique is applied to provide a set of Pareto multiple optimum solutions.

The following assumptions have been used in this analysis:

2. Thermal Modeling

4. Offset-strip fins of the same specifications are used for both the fluids.

2.1 Heat transfer equations

Figures. 1 and 2, show a view of a cross flow plate fin heat exchanger and offset strip fin with a rectangular cross section, respectively. In this study, while output temperature of fluids is unspecified, the ε–NTU method has been used for rating performance of the heat exchanger in the modeling process.

1. To minimize heat losses to the environment, the number of fin layers for the cold side (Nc) assumed to be one more than those of hot side (Nh). 2. Heat exchanger works in steady state conditions. 3. For a gas to gas heat exchanger, influence of fouling is neglected

The number of entropy generation units is defined as follows: )1( According to the Bejan’s methodology [21], rate of entropy generation can be explained in terms of temperature and pressure:

(2)

Heat transfer areas of the present heat exchanger for the two sides [20] are calculated by: (3) (4) Fig. 1. A schematic of a plate and fin heat exchanger (PFHE)

Aff, is free flow area and in the present work [20] is formulated as: (5) (6) Therefore, total heat transfer area of the heat exchanger is formulated as: (7)

Fig. 2. Typical rectangular offset strip fin core [20]

.

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For the cross flow heat exchanger with both fluids unmixed, effectiveness [22]. is given as:

(16) 2.2 Pressure drop calculations

(8) The fanning friction factor (f) is given by [23] as, And (9) (if Re≤1500) By considering thermal resistance of the wall and fouling factor negligible, the number of transfer units is determined as below: (10) Generally, heat transfer coefficient is explained in terms of Colburn factor [20] which is given by:

(17)

(18) (if Re>1500) so, frictional pressure drops for both sides of the fluid [20] are defined as below: (19)

(11)

(20)

G is the mass flux which is obtained as follows: (12)

Outlet temperature on both side of the fluid flow is obtained as follow: (21)

The Colburn factor (j) required for calculating the heat transfer coefficient is given by Joshi and Webb [23] as,

(13)

(22) Outlet pressure on both side of the fluid flow is similarly obtained as follow:

(if Re≤1500) (23) (24) (14) Finally, the number of entropy generation units is determinated as follows [16, 20, 24]

(if Re>1500) Hydraulic diameter is described for calculating Reynolds number as follows:

(25)

(15) 2.3 Cost estimation And Reynolds number is obtained by: In the present work, effectiveness and total annual cost are considered as two objective functions. For the cost calculation, total annual .

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cost is considered as the sum of investment cost and operating cost. Investment cost is the annualized cost of the heat transfer area while operating cost associates with the electricity cost for the compressors. The same approach for cost estimation considered by [2, 7 , 16]. Ctot  aCinv  Cope

(26)

n Cinv  CA  Atot

(27)

 PVt   PVt  Cope   kel    kel   c   h 

(28)

Here, CA and n are cost per unit surface area and the exponent of nonlinear increase with area increase, respectively. kel, τ and η are the electricity price, hours of operation and compressor efficiency, respectively. Also, a is annual coefficient factor that can be defined as follows:

a

r 1  (1  r ) y

(29)

Where, r and y show interest rate and depreciation time, respectively. 3. Multi-objective optimization A multi-objective problem consists of optimizing (i.e., minimizing or maximizing) several objectives simultaneously with a number of inequality or equality constraint. A maximum of a function f is a minimum of –f. Thus, the general optimization problem may be stated mathematically as [16]. Minimize With

fi(X), i=1,2,…,l

(31)

X=(x1,x2,…,xq)T gj(X) ≤0

Subject to j=1,2,…,p xiL≤xi≤xiu i=1,2,….,q

.

(30)

(32) (33)

In the above formulation, fi(X) are the l objective functions that should be minimized, X is the design vector with q components, gj(X) are p equality and inequality constraints that restrict the choice of design vector X. xiL and xiU define the lower and upper bound for each design variable xi and constitute the design variable space. The solutions that satisfy all of the inequality, equality and bound constraints are known as the feasible solutions. For more details, the reader is referred to [25] to [28]. 3.1 Bees Algorithm optimization

for

multi-objective

The Bees Algorithm (BA) was developed by a group of researchers at the Manufacturing Engineering Centre, Cardiff University [19]. BA emulated the behaviour of honey bees in foraging for pollen and nectar to find the optimal solution for both continuous and combinatorial problem. The algorithm required to five parameters, namely the number of scout bees (n), number of sites selected out of n visited sites (m), number of bees recruited for the selected sites (nep), neighbourhood size (the initial size of each patch, a patch is a region in the search space that includes the visited site and its neighbourhood) (ngh) and number of algorithm steps repetitions or stopping criterion ( ). In this algorithm, solutions are categorized based on the Pareto concept and sorting non-dominated solutions into non-dominated layers as schematically shown in Figure. 3. Steps of basic BA for multi-objective optimization are described in detail in Figure. 4 and its flowchart is illustrated in Figure. 5. For more details, the reader is referred to [17, 19]. 3.2 Concepts of domination and Paretooptimal For finding the Pareto set, most multi-objective optimization algorithms use the concept of domination. In these algorithms, two solutions are compared on the basis of whether one

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1. The solution X is no worse than Y in all objectives. 2. The solution X is strictly better than Y in at least one objective. It is intuitive that if a solution X dominates another solution Y, the solution X is betters that Y in the parlance of multi-objective optimization. However, if none of X or Y can dominate each other, it is customary to say that the two solutions are non-dominated with respect to each other. For a given finite set of solutions, we can perform all possible pair-wise comparisons and find which solutions are non-dominated with respect to each other. At the end, we expect to have a set of solutions, any of two which do not dominate each other and for any solution outside this set, we can always find a solution in this set which will dominate the former. In simple term, this means that the solutions of this set are better compared to the rest of solutions which are called the non-dominated set. Thus, the nondominated set of the entire feasible search space for a multi-objective optimization problem is the Pareto-optimal set for that problem (Figure. 3). Using the concept of domination, one can classify into various non-dominated levels. In most Multi-objective EAs the population needs to be sorted according to an ascending level of non-domination. The best non-dominated solutions are called the non-dominated set of Level 1. Once the best non-dominated set is identified, they are temporarily disregarded from the population. The non-dominated solutions of the reminaing population are then found and are called non-dominated set of level 2. In order to find the non-dominated set of level 3, all nondominated set of level 1 and 2 are disregarded and new non-dominated solutions are found. This procedure is continued until all population members are classified into a non-dominated level.

.

Minimize f2

dominates the other solution or not. A solution X is said to dominate the other solution Y, if both the following conditions are true:

Pareto optimal front Minimize f1 Fig. 3. Schematic of solution layering in MOBA

1. Initialize population with random solutions. 2. Evaluate fitness of the population. 3. While (stopping criterion not met) //Forming new population. 4. Select sites for neighborhood search. 5. Determine the patch size. 6. Recruit bees for selected sites and evaluate fitnesses. 7. Select the representative bee from each patch. 8. Amend the Pareto optimal set. 9. Abandon sites without new information 10. Assign remaining bees to search randomly and evaluate their fitnesses. 11. End While. Fig. 4. Pseudo code of the Bees Algorithm [19]

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cost function constant values are listed in Table 1 and Design parameters (decision variables) and the range of their variations are listed in Table 2. Table 1. The operating conditions of the PFHE (input data for the model) Mass flow rate of hot flow (kg/s)

1.8

Mass flow rate of cold flow (kg/s)

2

Inlet hot temperature (K)

658.15

Inlet cold temperature (K)

306.15

Inlet pressure (hot side) (kPa)

180

Inlet pressure (cold side) (kPa)

120

Price per unit area ($/m2)

100

Exponent of nonlinear increase with

0.6

area increase Hours of operation per year (h/yr)

6000

Price of electrical energy ($/MWh)

25

Compressor efficiency

0.6

Table 2. The design parameters and their range of variation Variables

4. A Case Study To minimize the number of entropy generation unit and the total annual cost (sum of investment and operational costs) six design parameters including fin pitch, fin height, fin offset length, cold stream flow length, no-flow length, and hot stream flow length are selected. An illustrative example from the literatures is considered to demonstrate the application of the proposed algorithm and compare with NSGA-II algorithm [7]. The property values of air and hot gases were considered to be temperature dependent. The PFHE metal was from stainless steel with thermal conductivity kw = 18 W/m K. The specifications of operation conditions and the

.

From

To

Fin pitch (mm)

1

2.5

Fin height (mm)

2.5

8

Fin length (mm)

2

3.5

Hot stream flow length (m)

0.2

0.4

Cold stream flow length (m)

0.7

1.2

No-flow length (m)

0.2

0.4

5. Results and discussion Figure. 6 represents the Pareto optimal curve obtained by using the MOBA for multi-objective optimization and its comparison with the Pareto 16


.

generation units at point A (0.0921) obtained from this study is less than NSGA-II in the value of 4.2 % and 1.9 %, respectively. This means that if any of these two functions (total annual costs and number of entropy generation units) examined separately as an objective function, results of present study will be better than NSGA-II.

Total annual cost ($/year)

optimal curved obtained by [7]. Figure. 6, clearly reveal the conflict between two objective functions, the number of entropy generation units and the total annual cost. Any geometrical changes, which decrease the number of entropy generation units, lead to an increase in the total annual cost and vice versa. Since proposed functions are in conflict with each other only a unique solution cannot be satisfied two-objective functions simultaneously. Therefore, multiobjective optimization has been used for obtaining a series of optimum solutions or Pareto optimum front which are a conflict between lower number of entropy generation units and total annual cost. Optimum values of two objectives for comparison this study to [7], for five typical points from A to E (Pareto-optimal fronts) for input values are given in Table 2 and are listed in Table 3. So, designers can choose the best design parameters according to application and considering their needs. For example, for having a number of more favourable numbers of entropy generation units, the points located on upper-left corner of graphs (point A) should be selected. In the same way, for having a more favourable cost should be used from points located in the bottom-right corner of graphs (point E). So, for having an interaction between the numbers of entropy generation units and total annual costs of heat exchanger should be used from the points located in the concavity of the curve (point C) as the optimal solution for having a favourable numbers of entropy generation units and annual costs, simultaneously. With considering the similar values from literature [7] and corresponding values obtained in this study, it was observed that in design points of B and E, despite having almost the same number of entropy generation units, total annual costs of this study were reduced 9.3 % and 4.2 % in comparison with the results of literature [7]. Generally, results of this study have a lower total annual costs at point E (988) as well as the number of entropy

Number of EGU Fig. 6. The distribution of Pareto-optimal points solution for Entropy generation units and annual cost using MOBA Table 3. The optimum values of number of entropy generation units and the total annual Cost for the design points A–E in Pareto-optimal fronts for input values given in Table 1 Parameter

Number of EGU

Total annual cost ($/yr)

Algorithm

NSGA-II

MOBA

NSGA-II

MOBA

Point A

-

0.0921

-

4492

Point B

0.0939

0.0933

4031.3

3654

Point C

0.1048

0. 1048

1924.6

1954

Point D

0.1148

0.1148

1327.4

1370

Point E

0.13

0.1298

1031

988

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6. Conclusion In the present work, a plate fin heat exchanger (PFHE) is optimized considering two objective functions including the number of entropy generation units and the total annual cost of the system and applying multi-objective Bees Algorithm technique (the number of entropy generation units and total annual cost were minimized). Based on applications, different design parameters including the fin pitch, fin height, fin offset length, cold stream flow length, no-flow length and hot stream flow length are proposed as optimization variables. If number of entropy generation units is decreased, total annual cost is increased, so the considered objectives are conflicting and no single solution can satisfy both objectives simultaneously. Thus multi-objective optimization by using BA is utilized for optimization of the system and obtaining a series of optimal solutions or Pareto optimum front each of which is a trade-off between the lowest number of entropy generation units and the least total annual cost. The major privilege of this research is that an extensive range of optimum solutions has been obtained and designers are able to select the best design parameters according to application and total annual cost of a system. The process of offered design for plate fin heat exchanger in this study using BA can be used for other types of heat exchangers such as fin and tube heat exchangers, shell and tube heat exchangers and recuperates REFERENCES [1]

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