A study of changes in temperature profile of porous fin model using cuckoo search algorithm

Alexandria Engineering Journal (2019) xxx, xxx–xxx

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

A study of changes in temperature profile of porous fin model using cuckoo search algorithm Waseem Waseem a, Muhammad Sulaiman a,*, Saeed Islam a, Poom Kumam b,c,d,*, Rashid Nawaz a, Muhammad Asif Zahoor Raja e, Muhammad Farooq a, Muhammad Shoaib f a

Department of Mathematics, Abdul Wali Khan University Mardan, KP, Pakistan Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand c Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand d Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan e Department of Electrical Engineering, COMSATS University Islamabad, Attock Campus, Attock, Pakistan f Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, Pakistan b

Received 3 July 2019; revised 19 October 2019; accepted 1 December 2019

KEYWORDS Heat distribution; Porous fin; Artificial intelligence; Metaheuristics; Cuckoo search algorithm; Differential equations; Optimization problems

Abstract For analysis of physical properties of different materials, rectangular porous fins are used to examine the heat transformation through a system. In this paper, a metaheuristic is combined with neural computing modelling to study the effects of temperature changes in a porous fin model. Cuckoo search algorithm is used as an efficient optimization technique to find the best weights to reduce the mean squared error in the required temperature profile. The governing partial differential equation is converted into a non-linear ordinary differential equation subject to certain boundary conditions. Two individual cases, of silicon nitride (Si3 N4 ) and Aluminium (Al), are considered. In the proposed procedure, the Cuckoo Search(CS) algorithm is combined with the artificial neural network (ANN), namely CS-ANN, to solve the differential equations and obtain solutions with better accuracy. Ó 2019 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

* Corresponding authors at: Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand (P. Kumam); Departmetn of Mathematics, Abdul Wali Khan University Mardan, KP, Pakistan (M. Sulaiman). E-mail addresses: [email protected] (M. Sulaiman), [email protected] (S. Islam), [email protected] (P. Kumam), [email protected] (R. Nawaz), [email protected] (M.A.Z. Raja), [email protected] (M. Farooq), [email protected] (M. Shoaib). Peer review under responsibility of Faculty of Engineering, Alexandria University. https://doi.org/10.1016/j.aej.2019.12.001 1110-0168 Ó 2019 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: W. Waseem et al., A study of changes in temperature profile of porous fin model using cuckoo search algorithm, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.12.001

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W. Waseem et al.

Nomenclature ANN IPT cp keff js q t kf h Q Vw

artificial neural network interior point technique specific heat of the fluid effective thermal conductivity thermal conductivity of fin actual heat transfer rate thickness of the fin fluid thermal conductivity dimensionless temperature dimensionless heat transfer velocity of fluid passing through a fin at any point

1. Introduction The process of heat transfer means changes in energy states between two bodies while in contact. The variation in temperature occurs until the equilibrium is achieved [1]. For devices operating under conditions of high energy/ temperatures, a reliable cooling system like rectangular porous fins are required to maintain the equilibrium [2]. The heat loss also occurs due to the motion of free electrons in the system [3]. The performance of mechanical systems is severely affected due to increase in friction with loss in energy. Every design of a mechanical system is subject to limits on operating temperatures, and by violating these limits, a system failure occurs [4,5]. To balance the heat transfer in the system, hybridization of methodologies like compound and passive methods are used [6]. Passive techniques are noiseless and produce low budget solutions of the problem [4,5]. Passive techniques are mainly about adding additional components to the system under consideration for maximum heat transfer [6]. These additional components in passive techniques include rectangular porous fins, rough and tempered surfaces. By treatment, we mean coating or changing the texture of surfaces up to a satisfactory level. The rough surfaces are modified so that the flow field turbulence is promoted especially in single-phase flow. Porous fins are used to extend the surfaces in a given system which results in a sufficient amplification of heat transfer [7,8]. Heat convection and conduction occur at the same time. Convection is related to the outside of the surface while heat conduction happens inside the surfaces [9,10]. In the modern era, manufacturing companies are designing compact systems to reduce the volumes of appliances and optimize the space efficiently. For this purpose, engineers are continuously in search of designing the compact devices, like porous fins, which are fitted in plate-fin heat exchangers, conditioners, turbines, cooling parts in different electronic devices [11,12]. Several types of fins are designed by engineers to increase its efficiency and reduce its cost. Fins with rectangular, porous, pin, solid shapes are implemented in different systems to balance the heat transfer. The main aim of scientists is to optimize its cost and design parameters [11,13,14]. Porous fin models are widely used in filtering devices, reactors, lubricating devices and insulations [15], irregular flow stabilization [16] and increasing the production of oil and gas. Porous fin models are famous for their porous medium made of specific

CSA cuckoo Search Algorithm GA genetic algorithm ANOVA analysis of variance h heat transfer coefficient T temperature jR thermal conductivity ratio tb thickness at fin base Ta ambient temperature Tb fin base temperature g fin efficiency PSO particle swarm optimization

material, which are compact and efficient designs as compared to other types of fins. Porous fins are also used in different reactors and solar collectors [11,17]. Many researchers have highlighted the applications of porous fins. By using their porous media, despite their low thermal conductivity, porous fins are considerably helpful in increasing the contact between cooling fluid and the target surface, or in fluid flow mixing. Thus, these devices are useful in heat transfer augmentation [18]. Porous fins permit the best efficiency and a considerable abatement in the weight of a system as compared to nonporous fin. It is necessary to model an energy equation to study the temperature variations in porous fins working under given conditions. It is evident that Kiwan and A.Nimr [19] first studied porous fins. Different characteristics of porous fins were studied by using Darcy’s model. The model investigates the effect of strong fluids while they are injected to the porous media. They introduced a solitary parameter which can further explain the thermal conduction of fluids in porous media. This parameter is the combination of Darcy’s number, ratio of length to thickness of fin (L=t), Rayleigh number, and a function of heat conduction (K). It was observed that by choosing suitable values of (L=t) and k, the resulting design of porous fins yielded better results as compared to fins of other types [20,21]. In [22], Kraus and snider, have studied the design of porous fins as an optimization problem. A study of temperature distribution on porous fins was published in a book written by Meyer [23]. Another study by Hijleh and Bassam [24] considered a regular heat transfer through multiple porous fins fitted in horizontal chambers. A rectangular-shaped porous fin design was observed under normal radiation and heat conduction in [20,25]. On the other hand, triangular-shaped porous fins fitted in a vertical chamber were studied under normal conditions of heat conduction in [26]. Another study by Kundu and Bhanja [27] was carried out by optimization of the problem of rectangular porous fin by maximizing the effectiveness of fins. Also, a T-shaped fin design was considered, and an analytical model was proposed to analyze the temperature distribution and heat conduction in a T-shaped porous fins [28]. Kundu et al. [29,30] studied the various types of porous fins by applying the adomian decomposition technique to obtain the best solution for the design of porous fins. A circular-shaped porous fins design was studied in [31]. In a recent study, Balkier and Gorla [25] investigated the impact of radiation on the performance and design of porous fins

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A study of changes in temperature profile of porous fin model under normal conditions. Mulani in [32] studied the feasibility of Cuckoo Search algorithm for inverse heat conduction problems. Chen Hao-Long investigated the performance of an Improved Cuckoo Search Algorithm for inverse geometry heat conduction systems [33]. 2. Emerging randomized techniques and their multidisciplinary applications The phenomena of energy transportation which involves the heat transfer among different bodies can be modelled with governing differential equations and suitable boundary conditions. Such phenomena can be analyzed for temperature profile, heat flux, and efficiency of the system by solving the modelled governing differential equations. Mostly, by modelling the governing differential equations for real-life problems, the resulting differential equation happens to be very complex and non-linear. In such cases, to obtain an exact solution is not possible by using existing analytical methods for the solution of the differential equation. Differential equation modelling the associated heat transfer involves non-linear terms which make them hard to solve using conventional methods [34,35]. To overcome this difficulty, stochastic iterative techniques are emerging to get a solution to the problems that have very negligible absolute error as compared to the real solution. These techniques are efficient in getting better results in fewer computations [36–38]. Since last decades, numerical computation techniques are becoming popular to solve these hard problems [39]. Initially, Monro and Robbins applied stochastic techniques in [40]. These solvers fall in the category of artificial intelligence (AI), which mostly uses the idea of feed-forward neural network schemes optimization with the help of Evolutionary Algorithms (EAs). Approximate techniques are widely applied to solve linear and non-linear differential equation [41,42]. These techniques are rapidly designed, modified and implemented to solve problems arising in optimization, signal processing units and control systems. Differential numerical algorithms are applied with a combination of metaheuristic techniques. Metaheuristics are universal procedures in terms of applicability to any sort of problems, linear or non-linear. These solution algorithms are based on mathematical models built for approximation of solution through artificial neural networks (ANNs) [43,44]. In [45,46], problems of Bagely-Torvik and Riccati have been tackled by using these methodologies. Recently, Jeffery–Hamid fluid flow problems are solved using ANNs based methodology. Moreover, problems involving painlene transcendental equation [47,48], problems from combustion theory involving non-linear Bratu equation [49] are solved with this methodology. Iftikhar et al. used ANNs based algorithm to solve Blasius equation numerically [50]. All these extensive applications of ANNs based methodology combined with evolutionary algorithms motivated the author to investigate such hard problems. We have chosen the most recently investigated problems of the design of porous fins. We apply the cuckoo search algorithm to solve models designed for such a differential equation using ANNs. These models involve unknown weights that are determined by using the cuckoo search algorithm. In this study, we propose a reliable and efficient methodology to solve porous fin

3 problems in a rectangular shape. The problems considered here are non-linear, which models the heat transfer in porous fins. In our analysis, we have investigated porous fins made of different materials along with heat production and thermal conductivity by finding better solutions using ANNs based cuckoo search optimization algorithm. The solution models are introduced using feed-forward ANNs [51]. Cuckoo search algorithm is mainly applied to design optimization problems with highly non-linear objective functions and constraints [38]. It is hybridized with different stochastic techniques to handle the different problems [38]. In this paper, ANNs based models are combined with CS to find best-trained weights to complete the approximate solution. Many applications of ANNs based models can be seen in [52–54]. 3. Transformation Function Before investigating the problem, we present a detailed formulation for the rectangular-shaped porous fin. L represents its length, t it’s thickness and w its width. These symbols represent the dimensions. It is assumed that the cross-sectional area is a constant with a temperatures dependent heat generation system in the fin. For an effective design of fin, the transverse Biot number should be small [29,35,55,56], and the heat loss in the transverse direction is negligible. Thus, variation in heat is assumed to occur only in a longitudinal direction. We show a diagrammatical presentation of the porous fin problems in Fig. 2. The governing differential equation for the energy transportation is expressed as in Eq. (1), qðxÞ  qðx þ DxÞ þ q ADx ¼ hðp:DxÞ½TðxÞ  T1  _ p ½TðxÞ  T1 ; þ mc

ð1Þ

where flow rate of mass in the fluid is given as m_ ¼ qvw Dxw

ð2Þ

with velocity of fluid passing through a fin at any point [29,56] vw ¼

gKb ½TðxÞ  T1  v

ð3Þ

now substituting m_ in Eq. (1) and rearranging qðxÞ  qðx þ DxÞ gKbqw þ q A ¼ cp ½TðxÞ  T1 2 Dx v þ hp½TðxÞ  T1 ;

ð4Þ

as Dx ! 0 [34,61,62], we have dq gKbqw þ q A ¼ cp ½TðxÞ  T1 2 þ hp½TðxÞ  T1 ; dx v

ð5Þ

, here by using Fourier’s law of heat conduction q ¼ keff A dT dx keff represents the effective thermal conductivity of the porous fin, which considers both solid and fluid parts of the porous fin. It is expressed as keff ¼ /kf þ ð1  /Þks , using this law in Eq. (2), we get d2 T q gKbqw hp cp ½TðxÞ  T1 2  þ  ½TðxÞ  T1  2 k v k k dx eff eff Aeff ¼ 0:

ð6Þ

A denoted the cross sectional area of porous fin and is expressed as A ¼ wt ) Aw ¼ 1t

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d2 T q gKbqw hp cp ½TðxÞ  T1 2  þ  ½TðxÞ  T1  ¼ 0; keff vt kAeff dx2 keff ð7Þ d2 T gKbqw hp q  ½TðxÞ  T1  þ ¼ 0: cp ½TðxÞ  T1 2  2 keff keff vt Akeff dx ð8Þ Now transforming the Eq. (3) with the help of X ¼ ) x L

and h ¼

TðxÞT1 , Tb T1

dX x

¼ L1

we get

d2 h L2 gKbqw L2 hp L2 q cp ½TðbÞ  T1 h2  ¼ 0:  hþ 2 keff vt Akeff keff ðTðbÞ  T1 Þ dX ð9Þ by putting the value of q ¼ q1 ð1 þ ðT  T1 ÞÞ d2 h dX2

2

hpL þ Ak eff

q1 A h hpðTb T1 Þ g

2

hpL þ Ak eff

q1 A hpðTb T1 Þ

¼ 0:

d2 h  Sh h2  ðM2 þ M2 Gg Þh þ M2 G ¼ 0: dX2

ð10Þ

ð11Þ

with boundary conditions [56], hð1Þ ¼ 1; h0 ð0Þ ¼ 0:

ð12Þ

where Sh represents the porosity value and is expressed as 2 gbqcp ðTðbÞT1 Þ Sh ¼ Dka Rr a ðLtÞ and Da ¼ tk2 ; Ra ¼ . Thus above Eq. vkf (11) can be expressed as LðhÞ þ NðhÞ ¼ 0:

ð13Þ

Eq. (13) has two parts; the first part represents the linear portion, while the second part expresses the non-linear terms involved in the porous fin model. To compute the dimensionless value of heat transfer Q through a design of a porous fin, the temperature gradient is used and given in Eq. (14) [57] wkeff 0 Q¼ ðh ðXÞÞX¼1 ¼ Xwðh0 ðXÞÞX¼1 ks where w ¼ Lt , and X ¼

2

d h derivative while dX 2 is the second derivative. Eq. (15) express the approximate solution of Eq. (11) [51],

~ ¼ hðXÞ

2

2

L hp  L kgKbqw cp ½TðbÞ  T1 h2  Ak h eff vt eff

One of the basic objectives of the resulting mathematical model is to choose the best unknown parameters involved in a model solution to represent the system in the best way. For this purpose, optimization algorithms play a key role. In this study, we are modelling the solutions of porous fin problem with the Artificial Intelligence of Neural network (ANN) methodology. ANNs are simulating the biological neurons, which are networked and assembled in the same way. A mathematical model of approximate solution for general non-linear Eq. (11) is formulated, by using feed-forward ANNs, given in Eq. (15). In this mapping with continuous input domain, hðXÞ repredh sent the solution for the problem of porous fin, and dX is its first

ð14Þ

keff . ks

4. Mathematical formulation of approximate solution for the porous fin problem Artificial Neural Network (ANN) models can approximate highly nonlinear functions efficiently, even for large data sets. Because of the excellent approximation properties of ANN [58], it has been used for the solution of various mathematical programming problems [59]; Dillon and O’Malley, [60]. The network comprises an input layer, an output layer, and one or more hidden layer. The input and output layers represent the inputs, X, and the outputs, h, of a system, respectively. ANN models aim to represent the input-output correlation by getting the weights and biases of the nodes in the network [61]. Before describing a physical system, it is necessary to express it in a mathematical form, which is known as mathematical modelling. A mathematical model can explain a system and make it easy to analyze different changes occurring in it. One can predict the behaviour of a system with a proper mathematical model. It is used in many practical applications.

m X ni ½fðdi X þ gi Þ;

ð15Þ

i¼1 m ~ X dhðXÞ d ¼ ½fðdi X þ gi Þ; ni dX dX i¼1 m ~ X d2 hðXÞ d2 ¼ ni 2 ½fðdi X þ gi Þ; 2 dX dX i¼1 .. . m ~ X dn hðXÞ dn ¼ ni n ½fðdi X þ gi Þ dXn dX i¼1

ð16Þ ð17Þ ð18Þ ð19Þ

where W ¼ ðn; d; gÞ ¼ ðn1 ; n2 ;    ; nm ; d1 ; d2 ;    ; dm ; g1 ; g2 ;    ; gm Þ are the optimization decision weights to be chosen by CS, the number of neurons involved in the artificial network are denoted by m and all values of hðXÞ are approximated as 2 ^ hðXÞ. Using a radial based function fRB ðXÞ ¼ eX in equations 15  18, they can be re-written as ~ ¼ hðXÞ

m X 2 ni eðdi Xþgi Þ ;

ð20Þ

i¼1 m ~ X 2 dhðXÞ  2ni di eðdi Xþgi Þ ðdi X þ gi Þ; ¼ dX i¼1

ð21Þ

m ~ X 2 d2 hðXÞ ¼ 2ni d2i eðdi Xþgi Þ ½1 þ 2ðdi X þ gi Þ2 : 2 dX i¼1

ð22Þ

Table 1 Different Parameters and symbols used in equations of Cuckoo Search [38]. Symbols

Name of parameter

pa xtj ; xti  HðuÞ S  a Lðs; kÞ

Switching Parameter Two distinct solutions Random number from uniform distribution Heaveside function Step size Entry-wise product Scalling factor Random steps from levy distribution

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A study of changes in temperature profile of porous fin model Table 2

5

Solutions obtained by different algorithms for X 2 ½01 with step size h ¼ 0:1 for case 1 based on Si3N4.

X

GA15

GA30

GA45

IPT15

IPT30

IPT45

CS15

CS30

CS45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.999552 0.957581 0.921288 0.890175 0.863852 0.84203 0.824497 0.811088 0.801657 0.796062 0.79415

0.999987 0.958049 0.921803 0.890792 0.864628 0.842986 0.825608 0.812294 0.802898 0.797312 0.795457

0.99999 0.95806 0.921826 0.890824 0.86467 0.843047 0.825695 0.812406 0.803025 0.797441 0.795588

1 0.958076 0.921842 0.890842 0.864694 0.843078 0.825733 0.812452 0.803076 0.797494 0.795641

1 0.958076 0.921842 0.890842 0.864693 0.843078 0.825733 0.812452 0.803076 0.797494 0.795641

1 0.958076 0.921842 0.890842 0.864693 0.843078 0.825733 0.812452 0.803076 0.797494 0.795641

1 0.987130 0.975816 0.965990 0.957599 0.950595 0.944934 0.940581 0.937502 0.935669 0.935062

1 0.958239 0.922639 0.892689 0.867915 0.847886 0.832207 0.820523 0.812516 0.807899 0.806413

1.00001 0.94510 0.90152 0.86760 0.84180 0.82275 0.80922 0.80015 0.794597 0.79175 0.79093

Table 3

Solutions obtained by different algorithms for X 2 ½01 with step size h ¼ 0:1 for case 2 based on Aluminium (Al).

X

GA15

GA30

GA45

IPT15

IPT30

IPT45

CS15

CS30

CS45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.99978 0.971822 0.947339 0.926132 0.90804 0.892945 0.880752 0.871385 0.864772 0.860841 0.85951

0.999985 0.972056 0.947623 0.926495 0.908504 0.893504 0.881378 0.872037 0.865415 0.861468 0.860157

1.000001 0.972085 0.947654 0.926529 0.908548 0.893565 0.881459 0.872136 0.865525 0.86158 0.860268

1 0.972084 0.947661 0.926541 0.90856 0.893575 0.881471 0.872154 0.86555 0.861608 0.860297

1 0.972083 0.947661 0.926541 0.908559 0.893574 0.881471 0.872153 0.865549 0.861607 0.860296

1 0.972084 0.947661 0.926541 0.908559 0.893575 0.881471 0.872153 0.865549 0.861607 0.860296

1 0.966996 0.938766 0.914766 0.89457 0.877876 0.864434 0.854089 0.846740 0.8423309 0.840853

1.00000 0.95471 0.91661 0.88487 0.85875 0.83767 0.82112 0.80873 0.80017 0.795191 0.793565

1 0.961112 0.928282 0.900862 0.87831 0.860176 0.84604 0.83555 0.828403 0.82430 0.822987

Table 4

Absolute Errors (MAE) in each solution obtained by different algorithms for case 1 based on Si3N4.

X

GA15

GA30

GA45

IPT15

IPT30

IPT45

CS15

CS30

CS45

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4.670E04 5.110E04 5.670E04 6.790E04 8.520E04 1.060E03 1.240E03 1.370E03 1.420E03 1.440E03 1.510E03

3.870E05 3.920E05 4.740E05 5.730E05 7.150E05 0.0001 0.000142 0.000186 0.000215 0.000222 0.000220

2.640E05 3.210E05 3.530E05 3.760E05 4.230E05 5.230E05 6.90E05 8.80E05 9.920E05 0.000102 0.000101

2.30E08 3.150E07 2.460E07 4.320E07 8.340E07 1.090E06 1.090E06 1.120E06 1.460E06 1.90E06 2.020E06

2.510E08 3.380E07 2.80E07 4.850E07 8.860E07 1.130E06 1.120E06 1.170E06 1.530E06 2.010E06 2.130E06

3.090E08 3.770E07 3.050E07 5.420E07 1.020E06 1.30E06 1.290E06 1.330E06 1.750E06 2.290E06 2.420E06

5.0510E10 5.9392E09 3.0337E10 7.322E09 3.915E09 4.331E10 8.298E09 6.675E09 3.710E10 1.365E08 2.451E09

5.088E11 1.749E09 4.955E11 1.61E08 9.029E10 7.924E09 1.162E08 8.995E10 3.877E09 4.464E09 2.251E09

3.6836E08 1.948E07 9.719E09 1.3973E07 4.3272E08 1.7179E08 1.352E07 1.426E07 3.104E08 2.119E08 2.280E07

Table 5

Absolute Errors (MAE) in each solution obtained by different algorithms for case 2 based on Aluminium (Al).

X

GA15

GA30

GA45

IPT15

IPT30

IPT45

CS15

CS30

CS45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.000227 0.000269 0.000328 0.000417 0.000535 0.000658 0.000754 0.000806 0.000815 0.000802 0.000821

1.910E05 3.030E05 4.20E05 5.030E05 6.0E05 7.910E05 0.00010 0.00013 0.00015 0.00015 0.00015

6.860E06 1.240E05 2.050E05 2.290E05 2.360E05 3.180E05 5.040E05 7.380E05 9.090E05 9.540E05 9.310E05

1.510E08 1.790E07 2.190E07 3.490E07 4.440E07 4.880E07 5.490E07 6.960E07 9.240E07 1.110E06 1.130E06

2.140E08 2.630E07 2.910E07 4.880E07 7.770E07 9.750E07 1.010E06 1.10E06 1.420E06 1.780E06 1.870E06

1.72E08 2.26E07 2.8E07 4.4E07 7.24E07 9.19E07 9.61E07 1.04E06 1.33E06 1.68E06 1.77E06

8.472E08 2.405E08 3.513E08 1.434E08 7.66E09 6.323E09 1.97E10 3.71E08 1.91E07 1.47E07 4.50E07

6.587E09 2.076E07 3.275E07 3.493E08 1.118E07 1.411E07 1.303E09 7.770E08 4.116E08 4.790E08 1.913E09

6.480E09 6.99E08 2.35E07 1.60E08 8.21E08 1.65E07 6.52E08 2.55E09 1.226E07 1.48E07 8.406E08

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6

W. Waseem et al. Select Neural Network Architecture and Cuckoo Search (CS) Algorithm Parameters

Randomly inialize Arficial Neural Network weights and biases with CS

Apply CS operators to generate new set of weights

Arficial Neural Network (ANN)

Training data set output data pair

input

Evaluate the error funcon with each set of weights Compare fitness values of each set and select best set of weights

Maximum Generaon or threshold error reached No

Yes

STAGE I: EVOLVING INITIAL NEURAL NETWORK WEIGHTS AND BIASES USING CUCKOO SEARCH ALGORITHM

Opmized Neural Network Weights and Biases

Neural Network architecture and training parameters

Update weights and baises

No

Fig. 1

Evaluate Training Error (MSE)

Threshold training error or maximum iteraons reached

Training data set input output data pair

STAGE II: TRAINING OF ANN AND FINE TUINING OF WEIGHTS AND BIASES USING CUCKOO SEARCH ALGORITHM

Yes

Trained Arficial Neural Network

Graphical overview of process followed in studying the porous fin design problem.

Choosing suitable weights with the help of CS in equations 19  21, by using adjusted combinations of ANNs, we get approximate solutions to the problem of porous fins with better accuracy [42,51,62,63].

5. Optimization problem The main issue in the design of porous fin problem is to reduce the mean squared error resulting from the approximate solu-

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A study of changes in temperature profile of porous fin model

7 and E2 expresses the error in boundary conditions as in Eq. (25), 2 2 E2 ¼ mean½ðh^0  1Þ þ ðh^1 0 Þ :

ð25Þ

6. Cuckoo search algorithm

Fig. 2

Cuckoo Search is a population-based technique that simulates brood parasitism present in some cuckoo species [38]. This algorithm is equipped with Levy flights and random search equations. Because of its ability to escape from local optima, it has been tested in literature against state-of-the-art algorithms PSO and GA. In terms of solutions to hard optimization problems, CS is shown to be better than its counter-part algorithms [38,64]. To balance the local and global search, it is controlled by different parameters given in Table 1. Random search is carried out by using Eq. (26),

Rectangular porous fin design model [10].

tions ^hðXÞ. This is done by introducing a new objective function, which comprises two errors, one resulting from Eq. (11) and the second error quantity is related to Eq. (12). Mathematically, it is expressed as in Eq. (23) minimize

E ¼ E1 þ E2 ;

ð23Þ

Moreover, E1 is the error resulting from Eq. (11) as, E1 ¼ mean½h^j 00  Sh h^j 2  M2 ð1 þ Gg Þh^j þ M2 G 1 ^ ; hj h

2

ð26Þ

where the symbols used are explained in Table 1. By introducing the levy walk, it enhances the search globally and can be expressed as in Eq. (27) xitþ1 ¼ xti þ aLðs; kÞ;

ð27Þ

where ð24Þ

¼ ^hðXj Þ and Xj ¼ jh the where j ¼ 1; 2; . . . ; N; X P 0; N ¼ interval X 2 ð0; NÞ is divided into N subintervals X 2 ðX0 ¼ 0; X1 ; X2 ; . . . ; Xj ¼ NÞ, here h is used as a step size

Fig. 3

xitþ1 ¼ xti þ as  Hðpa  Þ  ðxtj  xtk Þ;

Lðs; kÞ 

kCðkÞsinðpk=2Þ 1 p s1þk

ðs > 0Þ:

ð28Þ

In Eq. (27) and (28), all the terms and constants are explained in Table 1.

Graphical Illustration of solutions presented in Table 2 for case 1 based. on Si3 N4 .

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W. Waseem et al.

Fig. 4

Graphical Illustration of solutions presented in Table 3 for case 2 based on Al.

Fig. 5

Graphical Illustration of errors presented in Table 4 for case 1.

7. Results and discussion The temperature profile of a rectangular shaped porous fin, made of two types of materials, namely, Si3N4 and Al are studied in this paper. Porous fins are depicted by showing their heat generation and airflow, as in Fig. 2. We have proposed a methodology which is based on Cuckoo search algorithm combined with Artificial Neural networks scheme or in short CSANN. The parameters for cuckoo search are pa ¼ 0:25, popu-

lation size = 25, maximum iteration is 1000 and tolerance is taken 1012 . Numerical solution of Eq. 11,12, which are nonlinear with given boundary condition, are presented in Tables 1–5. We have considered design problems of rectangular porous fins made of two different materials and studied the effects in terms of changes in temperature profiles and heat transfer. A neural network approach represents the general solution of the differential Eq. (11) with boundary conditions in (12) suggested in [42,51,62,63,65]. We give the solutions and their

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A study of changes in temperature profile of porous fin model

Fig. 6

9

Graphical Illustration of errors presented in Table 5 for case 2.

Fig. 7 Convergence plots of MAEs obtained during 100 runs for CS-ANN in case 1 for Si3 N4 , (a) ANN model solutions with m = 15, (b) ANN model solutions with m = 30 and (c) ANN model solution with m = 45.

derivatives in Eq. (15)–(17), (19). The independent variable t varies from 0 to 1 with 10 equally spaced steps with a step size h = 0.1. We construct a fitness function based on the quality of the approximate solutions as in Eqs. (24) and (25). The unknown weights in solutions with different neural numbers, m = 15, 30 and 45 in hidden layers of the ANNs are found by using Cuckoo search algorithm with a better initialization strategy. The flowchart given in Fig. 1 explains the global and local search phases in CS. Three series solutions for m = 15, 30 and 45 are obtained and given in Eq. (29)–(34) for case 1: Si3N4 material,

2 h^jCS15 ðXÞ ¼ 1:45950eð1:1127Xþ3:6898Þ þ    2

þ 1:09371eð0:2873X0:4378Þ

ð29Þ

2 h^jCS30 ðXÞ ¼ 4:9989eð1:1691X4:5150Þ þ   

þ 4:9996eð0:72322X3:62621Þ

2

ð30Þ

2 h^jCS45 ðXÞ ¼ 0:1241eð1:2814Xþ0:9070Þ þ   

 0:25761eð0:2304Xþ2:1662Þ

2

ð31Þ

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W. Waseem et al.

Fig. 8 Convergence plots of MAEs obtained during 100 runs for CS-ANN in case 2 for Al, (a) ANN model solutions with m = 15, (b) ANN model solutions with m = 30 and (c) ANN model solution with m = 45.

Effect of both Si3 N4 and Al materials on temperature distribution.

Fig. 9

As mentioned above, we have presented a complete series solution keeping 14 decimal places in the Appendix, see (A1)–(A3) for case 1 to present reproducible solutions for further analysis. On the other hand, series solutions obtained for case 2 considering Al material, are given in Eqs. (32)–(34) same steps size of h = 0.1 is considered. The temperature profile is varied from t = 0 to t = 1. We give results obtained below for m = 15, 30 and 45. 2 h^jCS15 ðXÞ ¼ 1:4595eð1:1127Xþ3:6898Þ þ   

þ 1:09371eð0:2873X0:4378Þ

2

ð32Þ

2 h^jCS30 ðXÞ ¼ 0:6098eð1:16936Xþ1:7857Þ þ    2

 4:9824eð0:8431X4:06711Þ

ð33Þ

2 h^jCS45 ðXÞ ¼ 0:14148eð4:5987X3:1253Þ þ   

 3:7222eð0:3740X3:3977Þ

2

ð34Þ

For further analysis to reproduce the results, all solutions with 14 decimal places of accuracy are given for case 2 in the Appendix, see Eqs. (A.4)–(A.6). The approximate solutions obtained by CS-ANN are compared with reference solutions reported in [65], for both cases related to Si3N4 and Al, see Tables 2,3. Graphical illustration of solutions and errors are presented in Fig. 3–10. Absolute Errors (AE) are calculated according to the fitness functions given in Eqs. (24) and (25). Table 2 is related to case 1, Si3N4. Our results are compared with the genetic algorithm (GA), and interior-point technique (IPT) [65]. Results obtained by CS-ANN are superior as compared to GA and

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A study of changes in temperature profile of porous fin model

Fig. 10

Table 6

11

Heat transfer comparison for Si3 N4 and Al.

Heat transfer for Aluminium (Al) and Si3N4.

w

Si3 N4

Al

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.0118044578764615 0.0236089157529229 0.0354133736293844 0.0472178315058458 0.0590222893823073 0.0708267472587688 0.0826312051352302 0.0944356630116917 0.106240120888153 0.118044578764615

0 0.0171242923044410 0.0342485846088820 0.0513728769133230 0.0684971692177640 0.0856214615222050 0.102745753826646 0.119870046131087 0.136994338435528 0.154118630739969 0.171242923044410

Fig. 5,6. The convergence analysis is carried out by plotting the best results obtained during 100 runs. It is interesting to note that in both cases when m = 45, the convergence plot is smooth and accurate as compared to plots for m = 15 and m = 30. Thus by choosing a higher value of m one can get a better solution for the type of problem considered in this paper. It is further established in Fig. 10 that by increasing values of w, the heat transfer values are also increased. Thus it is worth to note that nature-inspired metaheuristics combined with ANNs resulted in better results and proved to be efficient in solving porous fin problem. 8. Conclusion

IPT in terms of solutions quality and minimum errors. It is obvious that AE for CS15 ; CS30 , and CS45 as in Table 4 lies between 1.365E8 to 3.03E10, 1.16E08 to 5.08E11, and 1.35E08 to 9.71E09 respectively. For case 2, AE is less than those reported in literature [65]. By varying X from 0 to 1, we have obtained consistently overlapping solutions according to lower AE values for CS15 ; CS30 , and CS45 as 1.91E07 to 6.32E09, 1.11E07 to 1.30E09, 1.48E07 to 6.48E09 respectively. All results listed in Tables 2,3 or keeping in view Tables 4,5, dictates that our solutions are closer to the exact solutions for t = 0–1. This shows that our proposed methodology of CS-ANN is better than those algorithms reported in [65]. Our hybrid procedure is better in terms of getting better solutions for the problem of rectangular-shaped porous fins made up of two types of materials, namely, Si3N4 and Aluminum (Al). Table 6 shows the heat transfer analysis of both cases by using Eq. 14 and varying X from 0 to 1 with step size h = 0.1. It is evident from Table 6 that rectangular porous fins made of Al materials transfers more heat than those fins which are made of Si3N4 materials. To further elaborate the effects of materials on heat transfer, we have plotted values of Table 6 in Fig. 10. Figs. 3 and 4 depicts the values shown in Tables 2 and 3. While values of AE given in Tables 4, 5 are plotted in

In this work, we propose a method based on hybridized scheme of Cuckoo search algorithm with Artificial Neural Networks to study the temperature distribution in rectangular shaped porous fins made up of two types of materials, Si3N4 and Al. Our proposed scheme of CS-ANN led us to solutions with better accuracy as compared to the results reported in the literature by GA, GA-IPT approaches. The convergence analysis of our results points to the better efficiency of CS-ANN. It is interesting to note that by increasing the terms in the series solution and by choosing high values of m; The results are more accurate and the convergence is smooth. We got better results with CS-ANN using less number of function evaluations. We carry temperature profiles and heat transfer analysis by choosing two types of materials for the design of the porous fin. We conclude that porous fins made of Al can transfer more heat than Si3N4. By increasing or decreasing the value of w, it affects the heat transfer capability of porous fins. Funding This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.

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W. Waseem et al.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Case 2 (Al material): Below we present the best solutions obtained by CS-ANN methodology with 14 decimal places of accuracy. Equation (A4-A6) presents solutions for CS15 ; CS30 , and CS45 . 2 h^j CS15 ðXÞ ¼ 1:45950346839321eð1:11270509236516Xþ3:68985686841408Þ

þ 1:62190873089501eð1:17283960163820Xþ2:75836866160135Þ

Acknowledgments

2

þ 1:25137245618232eð0:856187870265878Xþ1:71705627119782Þ

This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Research Cluster (CLASSIC), Faculty of Science, KMUTT.

þ 0:580225846599511eð0:660346245124736Xþ1:71470266645258Þ þ 1:09371834512461eð0:287366882028715X0:437854630951948Þ

Case 1 (Si3N4): Below we present the complete solutions obtained by CS-ANN with 14 decimal places of accuracy to reduce the rounding errors. Eqs. (A1)–(A3) presents solution for CS15 ; CS30 , and CS45 . 2 h^j CS15 ðXÞ ¼ 2:08515807891151eð1:13075379751153Xþ4:15491076096040Þ

þ 4:52144161567194eð0:477905081203140Xþ1:65223859420571Þ

þ 1:83239424807959eð0:892181630039003X2:72752889171589Þ þ 3:84807849951512eð0:310349700827759X2:22134491556254Þ þ 0:788348962390591e

2

þ 4:97379909099054eð1:81120960878403X3:51562430441734Þ

2

 4:07615138060901eð0:811078671706904X4:84231636718166Þ þ 4:64844958923035eð0:314621860279187Xþ1:39844181803842Þ

2

ð0:0328611644217960Xþ0:391504688906355Þ2

2

 3:66092843791736eð1:53647655605159X3:10203664261862Þ

2

ðA1Þ

 2:01729889957249eð1:24759296769073Xþ4:28450946127231Þ  2:88493606043185e

 4:95473235423155eð0:217185861560498X2:83150446917765Þ

2

 2:90631967995135eð1:16507600104340X4:74907582207759Þ

þ 4:99972858552922eð4:99956555168244Xþ3:81510860364607Þ

2

þ 3:82827049404445eð0:501407974983654Xþ1:67219071173188Þ

ð0:238935485196873X1:04097996826441Þ2 2

þ 4:06823995300586eð1:08474937756011X3:02122727204498Þ

2

 4:99994596802706eð2:66510036184703X3:42651381921884Þ

2

þ 4:99969922717422e

ðA2Þ

þ 0:590357848767041eð2:10051129552664X4:29063776003157Þ

2

2

2

2

2

 2:51967671626329eð2:05478716461224X3:03082073145660Þ  2:14968900471378e

2

ð4:78442811366772X4:68647275298980Þ2

 3:72221482671837eð0:374086411360552X3:39776131858651Þ

ð1:24930141266931Xþ1:92273003499438Þ2

þ 0:407164580447141eð0:821031807846069Xþ3:99988889775553Þ

2

 0:0316557913692871eð1:04708372268471Xþ0:252050308913043Þ

2

2

2

 0:267389992133115eð0:757517610577622X0:0764626437931788Þ

2

2

þ 1:70329796605301eð0:223486485384669Xþ0:597665271593962Þ

2

þ 0:749641082253991eð0:245155286942353Xþ3:35844525477262Þ

2

 0:0127582986225872eð0:586249389522472Xþ0:227013799973810Þ  0:255440999145026eð0:136969294820861Xþ2:34093536630688Þ

2

þ 0:318507831307946eð0:549050766056994X1:63013928597910Þ

2

 0:257618350704811eð0:230486188693042Xþ2:16627191301624Þ

2

2

ðA6Þ

References

2

þ 2:93694858921975eð0:414715983332253Xþ2:94394822648103Þ  1:07893423957921eð1:10845912994402Xþ2:30475652666984Þ

2

þ 1:56141003297186eð1:20647390755803Xþ2:53569677571080Þ

2 h^j CS45 ðXÞ ¼ 0:124189106808553eð1:28141381686884Xþ0:907067471955803Þ

þ 1:66410754173173eð0:546376748484809Xþ3:04956755209370Þ

2

 3:41279800947624eð2:55445755716148X4:85552502569572Þ

ð0:723224301415807X3:62621222430224Þ2

 1:80829676269307eð1:03953512816947Xþ2:63138107748218Þ

2

þ 4:95615547541500eð0:0350476211915898Xþ1:38871031139250Þ

þ 4:03240788587708e

þ 1:97441688427573e

2

 0:0852987339736814eð3:16112633048895Xþ4:97007741538652Þ þ 1:67326734759935eð0:329388843324730X1:90059920284836Þ

ð0:173208582433961X1:34829499680880Þ2

ðA5Þ

2

2

 1:37763588261583eð4:99965973352955X4:93734030737973Þ

2

ð2:70197067841339Xþ3:19776397065118Þ2

þ 4:99720914439601eð4:99904768221928Xþ3:70374705662747Þ

þ 1:01916906245486e

2

2 h^j CS45 ðXÞ ¼ 0:141483851421906eð4:59870185736739X3:12532806307375Þ

2 h^j CS30 ðXÞ ¼ 4:99896602018590eð1:16914320919728X4:51509788489403Þ

þ 4:24217516364856eð0:748216633646503Xþ3:98689375151132Þ

2

2

þ 4:99605335413532eð3:21075433876665X3:48523811716128Þ

 4:98243549597753eð0:843178515944178X4:06711132621219Þ

2

ðA4Þ

2

þ 1:91584828434287eð1:83883679243495Xþ4:97157442961746Þ

 2:72501887081263eð1:12692000367242Xþ3:52946860492530Þ

2

2

2

2 h^j CS30 ðXÞ ¼ 0:609854951302354eð1:16936957396582Xþ1:78576085123502Þ

þ 1:60745985787835eð0:234880299623178X1:27362486616377Þ

Appendix A. Detailed Solutions

2

2

ðA3Þ

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Please cite this article in press as: W. Waseem et al., A study of changes in temperature profile of porous fin model using cuckoo search algorithm, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.12.001