A Quantitative Study of Non-Linear Convective Heat Transfer Model by Novel Hybrid Heuristic Driven Neural Soft Computing

Heat transfer has a vital role in material selection, machinery efficacy, and energy consumption. The notion of heat transfer is essential in understanding many phenomena related to several engineering fields. Particularly, Mechanical, civil and chemical engineering. The presentation of the heat transfer model in this manuscript is a dedication to the heat transfer characteristics such as conduction, convection, and radiation. The heat energy consumption mainly depends on these characteristics. A better conductive and convective paradigm is required for miniaturization of heat loss or transfer. The phenomenon is mathematically assumed with the required parameters. A new mathematical strategy is also designed and implemented in the manuscript to evaluate the dynamics of heat transfer model. The mathematical approach is the hybrid structure of the Sine-Cosine algorithm and Interior point algorithm. The validation of new technique is evaluated by mean absolute deviation, root mean square errors, and error in Nash–Sutcliffe efficiency. For better illustration, an extensive data set executed by the proposed mathematical strategy is also drawn graphically with convergence plots.


I. INTRODUCTION
With the increase in energy consumption, the miniaturization of heat loss and heat transfer on surfaces and in devices surge the interest of researchers. To improve thermal performance, the researchers work in many engineering applications like thermal energy storage, heat transfer exchangers, Solar collectors, thermal control in electronic devices, etc. These researches contribute to minimizing the consumption of heat energy. In this regard, many best insulators are developed to lower energy consumption, which is considered a better strategy to minimize energy consumption. Various types of insulators are reported by [1]. For heat transfer applications, the understanding of its characteristics and determination of thermal properties in different shaped materials. That's why researchers explore heat transfer by convection, conduction, The associate editor coordinating the review of this manuscript and approving it for publication was Inês Domingues . and radiations. Natural convection, radiation, and conduction are associated with diverse engineering applications [2]. Many researches studied natural convection mechanism in different thermal materials based on various conditions such as under the influence of magnetic field convection is studied by [3], [4], porous media saturatedi fluids studied and reported in [5]. Qi et al reported a study on the cooling process of high heat loss in electronic devices, such as CPUs. In this work, they said that magnetic field intensity could improve the heat exchanger efficiency [6], Nonlinear radiation and cross-diffusion effects on the micropolar nanoliquid flow past a stretching sheet with an exponential heat source [7], nonlinear Boussinesq approximation and non-uniform heat source/sink on nanoliquid flow with convective heat condition:sensitivity analysis [8]. In this article heat transfer is studied in Ti 2 − EG with passive and active controlling of nano-particles [9] while the significance of inclination of magnetic field with nonlinear thermal radiation and VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ exponential space based heat source is reported in [8].
In [10] Nazeer et al. report the impact of on velocity and heat profiles. The numerical results in this study reveal that the velocity profile vanished by non-Newtonian terms and other parameters like magnetic field, viscosity and electro-kinetic while the behavior of velocity versus pressure gradient is opposite. Additionally, the temperature profile increases against the Brinkman number, pressure gradient, Joule heating, wall's temperature, viscosity while reverse behavior is observed via electro-kinetic and magnetic field parameters, heat transportation generation/absorption and radiative heat flux in homogeneous-heterogeneous catalytic reactions of non-Newtonian fluid is studied by [11], radiative flow by convective cylinder is studied in [12], comparative study of nano-particles in viscous fluid flow is reported in [13] and Ijaz Khan et al. reported study of reactive aspect in flow of tangent hyperbolic material [14]. For conduction phenomena, in [15] Unger et al. introduce a novel fin tube exchanger for the improvement of conductive heat transfer in the tube fins. At the same time, convection occurs along its surface. Rathod and Modi present numerical phenomena for the evaluation of thermo-fluid mechanism in a fin heat exchanger consisting of sinusoidal and elliptic winglets [16]. For heat conduction, employment of Eulerian-Lagrangian scheme for hydraulic characteristics of hybrid nanofluid flow inside the micro-pin-fins heat sink is discus by Ambreen et al. in [17]. As conduction and convection, radiation also has a contribution to heat transfer. Inquiring fin parameters and natural convection also deal with heat transfer with radiations. Heat transfer has many application such as PCM pipe bank thermal storage for space heating [18], Thermal performance of self-rewetting gold nanofluids [19], heat pipe cooled device with thermo-electric generator for nuclear power application [20], etc. For such problems researcher introduce many numerical techniques or solved by existent techniques like Entropy generation and heat transfer analysis by Finite difference method [21]. Numerical and experimental analysis of resinflow, heat-transfer [22], Computational modeling and analysis on rotating stretched disk flow with heat transfer [23]. Classical techniques need many information like initial point, gradient, feasible range, etc. But the global search techniques need a little information about the problem; such characteristics grab the researcher's interest. Many research introduces the implementation of meta-heuristics. These meta-heuristics are applicable to heat transfer problems and can be applied to solve a wide range of other engineering problems. A few of them are, supervised learning method for reconstruction in heat transfer problem [24], A physicsinformed machine learning approach for solving heat transfer equation [25], heat transfer in a grooved pipe model by Stochastic Algorithms [26] with this the many hybridizations of unsupervised and supervised techniques are also reported in the literature. Analysis of temperature profiles in longitudinal fin is presented in [27], heat and entropy generation in flow of non-Newtonian fluid is analysed by Artificial neural networking (ANN) in [28], temperature distribution in convective straight fins [29], Multi order Fractional Differential Equations [30], Restoring Moment and Damping Effects Using Neuro-evolutionary Technique [31], non-linear MHD Jeffery-Hamel blood flow model, Optimal power flow solution in power systems [32], [33], Falkner-Skan flow problem is solved by SCA-SQP in [34], unipolar pump flow is evaluated by ANN-SCA-SQP by [35]. In a similar context of heat transfer, this work discusses the conductive, convective, and radiative heat transfer model in a fin. The work discusses different scenarios of conductive, convective, and radiative heat transfer combinations. The presented work is given as: • The mathematical model of conductive, convective, and radiative heat transfer is presented.
• For the solution of the presented mathematical model Artificial Neural Network (ANN) is utilized with the hybridization paradigm of Sine-Cosine Algorithm(SCA) with Interior-point algorithm(IPA) abbreviated as ANN-SCA-IPA.
• The Dynamics of the model are evaluated by the variation of coefficient conductive, radiative and convective in the heat transfer model (HTM).
• The HTM is transformed into an optimization problem with a fitness function that uses the least square errors phenomena.
• Additionally, the ANN-SCA-IPA paradigm is evaluated by various statistical operators. The structure of rest of paper is followed by section II which present formulation of HTM, section III present construction and discuses the ANN base HTM, section IV discusses the designing of fitness function, section V consist of hybridization paradigm, section VI define and explain performance operators, section VII discuss empirical results of HTM, section VIII consist of discussion on performance of ANN-SCA-IPA algorithm and section IX conclude the work.

II. FORMULATION OF NONLINEAR HEAT TRANSFER MODEL
Consider a fin having length b with cross section area A and thermal dependent conductivity as shown in Figure 1. It is considered that fin is relatively long as compare to A and is made up of isotropic solid material. On the surface heat is transfers through convection and radiation while at the bottom uniform temperature is considered while the tip of fin has negligible heat transfer. The transfer of heat on the surface of fins satisfies Stefan-Boltzmann's law. The thermal conductivity k(T ) in the fin is depend on the temperature which is defined as [36] where T a denotes ambient temperature, inner temperature of fin is represented by T , temperature variation is denoted by β. Whenever (T = T a ) the equation (1) reduced to k = k a which denotes thermal conductivity. The heat transfer model is analyzing the convective, conductive and radiative heat transfer mechanism in a longitudinal fin and perimeter P. The efficient sink temperature for radiative heat transfer is T s and fin's base temperature is T b . The convection heat transfer coefficient h and the surface emissivity coefficient E g are assumed to be constants, whereas the conduction coefficient k varies. As the fin is longitudinal, that's why it satisfy one dimensional heat transfer equation and boundary conditions are as follows: [37]- [39]: To make the equation dimensionless following new parameters are defined as here, ε 1 is heat conduction, ε 2 is heat radiation and ν is heat convection. By putting equation (1) into equation (2) and if θ a = θ s = 0, then: dθ dη = 0 at η = 0 and θ = 1 at η = 1 (6)

III. ARTIFICIAL NEURAL NETWORK BASED STRUCTURE OF THE HEAT TRANSFER MODEL
This section discusses the mathematical strategy for the approximation of heat transfer. The procedure is approximated by feed-forward neural network in terms of activation function log-sigmoid given in below equation: The feed-forward is simple and unidirectional network. It's computationally sound and simple to utilize. Additionally, The approximated trail solutionθ(η) and its derivatives first, second, third, and n th orderθ (η),θ (η),θ (η) andθ n (η), respectively, for heat transfer procedure in equation (5) are presented in equation (8) in the model (8), number of neurons are denoted by k which must be multiple of three because the set W = [a, w, ζ ] consist of three components of neurons a, w, ζ . Where a = [a 1 , a 2 , . . . , a k ], w = [w 1 , w 2 , . . . , w k ] and ζ = [ζ 1 , ζ 2 , . . . , ζ k ]. As heat transfer equation (5) is second order ordinary differential equation (ODE), so the putting equations (9)- (11) in equation (5) will give ANN based governing equation of heat transfer. The ANN strategy and parameters such as input, hidden layer, and output are given in Figure 2.

IV. DESIGNING FITNESS FUNCTION
For quantitative evaluation of approximated solution fitness function is designed based on two mean-square errors, it's as follows: : here E 1 denote the cost function, can be expressed as follows: where N denotes points the input points in the grid, the number of inputs depend on step size h in given interval, θ(η),θ (η), andθ are given in equation (8). In similar manner, E 2 is the error in boundary conditions define as: The fitness value depends on the provision of neurons, W = [a, w, ζ ], the set of best neurons will tends the fitness value to zero. Which illustrate the minimum error in the approximated solution.

V. HYBRID TECHNIQUE FOR OPTIMIZATION PROCEDURE
A hybrid technique is adapted for the minimization of the fitness function define in equation (12). For this optimization procedure, two search strategies are used: global and local. For global performance, the Sine-Cosine algorithm (SCA) is used. The SCA was presented by Mirjalili [40]. Many further advancement are reported in literature such as crystal wave guides [41], for training of multi-layer perceptrons [42], unit commitment problem [43], and fuzzy probabilistic c-ordered means [44]. The SCA is inspired by trigonometric ratios Sine and Cosine. The mathematical strategy of SCA is as follows: where Y i represents the current location of the solution in the i t h dimension at the t t h iteration, r 1 /r 2 /r 3 are random quantities, P i refers to the position of the target destination, and || represents absolute value. With parametric values, the equation (15) can be written as: here r 4 is a random choice in the range [0,1]. There are four parameters in equation (16): r 1 , r 2 , r 3 , and r 4 . The direction of movement is indicated by the r 1 , which could be within or outside of the feasible region. The r 2 parameter depicts the length from the region, as well as the direction toward or away from the target. The parameter r 3 represents target value weights, whereas r 4 operates equally the Sine and Cosine components of equation (16). For the enhancement of local optimum performance of SCA, it is hybridized with Interior point algorithm (IPA). The IPA is an efficient technique for solving linear and non-linear optimization problems. The IPA is much better than trust regions and Sequential quadratic programming to solve the iterative problems [45]. The exploitation procedure of IPA refines the quality of solutions. The SCA is one of the population-based techniques which generates multiple solutions. As a starting point for IPA, the best solution is used. Later on in this work, the technique is abbreviated as ANN-SCA-IPA. The hybridization procedure is given in algorithm 1.

VI. PERFORMANCE MATRICES
Performance matrices are used to evaluate the effectiveness of the ANN-SCA-IPA approach. For relatable solutions the reliability and consistency of the proposed technique is necessary. For this purpose the performance measures Mean-absolute deviation (MAD), root-mean square error (RMSE), Nash-Sutcliffe efficiency (NSE) and error in Nash-Sutcliffe efficiency (ENSE). The definition of these performance matrices are as given: The set of input points in a grid is denoted by n, the proposed solution isθ(η), and the reference solution is θ(η). The value of performance metrics MAD, ENSE, and RMSE should be approach zero for a dependable and effective system, whereas the NSE value tends to one. The mathematical definition of the global version of the performance matrices stated above is define as: Algorithm 1 Pseudocode of ANN-SCA-IPA Algorithm where, R denotes the number of multiple independent runs, and E r denotes the fitness value of the ANN-SCA-IPA at r t h number run. All global operators have a standard or optimum value of zero. The global version is based on the mean fitness value. Global operators are abbreviated as: GFIT for fitness, GMAD for MAD, GRMSE for RMSE and, GENSE for ENSE.

VII. EMPIRICAL RESULTS AND DISCUSSION
This section discusses the outcome of dynamics of heat transfer, given in equation (5). The heat transfer problem's dynamics are described on variation of thermal conductivity, thermal convection, and radiation with different temperature distributions. The dynamics are quantitatively executed by ANN-SCA-IPA algorithm. Moreover, the discussion is split into four examples.

A. EXAMPLE 1: CONDUCTIVE HEAT TRANSFER
In this example, the heat transfer is considered only through conduction, 1 . There is no heat transfer by convection (ν = 0) and radiation ( 2 = 0). The example has three cases, in Case I is consider as 1 = 0.1, for case II 1 = 0.6 and for case III 1 = 0.9. The in put point is taken in [0, 1], so for N input points the fitness function given in equations (12)- (14) can updated as: as the minimization is done by ANN-SCA-IPA algorithm for the solution of this example with minimum errors. The best unknown variable called weights of ANN is plugged in equation (9)

B. EXAMPLE 2: HEAT TRANSFER WITH CONSTANT RATE
In this example, the heat transfer is consider through conduction ( 1 ), convection (ν) and radiation ( 2 ) with a same rate. The rate of heat transfer is discussed in three cases. In Case I, the rate is considered 0.3, in case II 0.6, and case III 0.9. The input points are taken in [0, 1], so for N input points VOLUME 10, 2022  the fitness function given in equations (12)- (14) can updated as: as the minimization is done by ANN-SCA-IPA algorithm for the solution of this example with minimum errors. The best unknown variable called weights of ANN found is plugged in equation (9)     are graphically compared with the solution of Runge-Kutta order 4 (RK4) shown in Figure 5a. The solution is plotted with step size 0.1 in the interval [0, 1]. The weights are also drawn with 3D bar graphs in figure 5(b-c). The weights are also reported in table 2. The dots (·) show the results of RK4, and the solid lines represent approximated solutions.
The overlapping of these solutions shows the convergence of ANN-SCA-IPA algorithm.

C. EXAMPLE 3: CONVECTIVE HEAT TRANSFER
In this example, the heat transfer is consider through convection, ν, only. There is no heat transfer by conduction ( 1 = 0) and radiation ( 2 = 0). The example has three cases, in Case I convection consider as ν = 0.3, for case II ν = 0.6 and for case III ν = 0.9. The in put point is taken in [0, 1], so for N input points the fitness function given in equations (12)- (14) can updated as: as the minimization is done by ANN-SCA-IPA algorithm for the solution of this example with minimum errors. The best unknown variable called weights of ANN found is plugged in equation (9)

D. EXAMPLE 4: RADIATIVE HEAT TRANSFER
In this example, the heat transfer is consider through radiation, 2 , only. There is no heat transfer by convection (ν = 0) and conduction ( 1 = 0). The example has three cases, in Case I radiation consider as 2 = 0.3, for case II 2 = 0.6 and for case III 2 = 0.9. The input point is taken in [0, 1], so for N input points the fitness function given in equations (12)- (14) can updated as: 4 m 2 VOLUME 10, 2022 as the minimization is done by ANN-SCA-IPA algorithm for the solution of this example with minimum errors. The best unknown variable called weights of ANN found is plugged in equation (9)     The overlapping of these solutions shows the convergence of ANN-SCA-IPA algorithm.

VIII. EVALUATION OF ANN-SCA-IPA ALGORITHM
To verify the consistency, reliability, and convergence of the proposed ANN-SCA-IPA algorithm. The statistical operators are utilized as defined and discussed in section VI. The values of the statistic operators MAD, RMSE, and ENSE is drawn with a line graph having semi-log on y-axis. Additionally, in a similar manner, the fitness of each case is also drawn. The algorithm is executed 100 times to collect large data set because few runs cannot verify the capability of a technique. The data is arranged in descending order to observe the small variation as well in the data. Figure 8 shows data for MAD of each example. From figure 8a it can observe that the values of MAD in example 1 are varies between 10 −5 −10 −8 , figure 8b shows MAD data for example 2 which lies between 10 −1 − 10 −4 , figure 8c shows data of example 3 which lies in between 10 −2 − 10 −5 and similarly from figure 8d data of example 4 can be observed.
In figure 9, Moreover, in figure 11, if the fitness data is observed it can seem that all runs have much better and considerable fitness, which concludes that according to fitness function, equation (12)

IX. CONCLUSION
In this work, the heat transfer problem is considered. Heat transfer dynamics is executed by variations of different ways of heat transfer such as conduction, convection, and radiation and loss of heat with the same rate of conduction, convection, and radiation. The transfer of heat is assumed in longitudinal Fin. The temperature at the bottom is assumed constant while on the tip of the fin, the heat transfer is negligible. Moreover on the surface of fin heat flows through convection and radiation. For quantitative analysis of heat transfer, a novel hybrid technique is proposed. The technique is a hybridization of population-based approach Sine-Cosine algorithm and local search technique Interior Point algorithm with the utilization of artificial neural network, namely ANN-SCA-IPA algorithm. The SCA generates multiple solutions according to its population size and assigns it to IPA as the initial point.
Furthermore, to validate reliability, consistency, and robustness of the ANN-SCA-IPA algorithm, a large data set is collected and is evaluated by performance operators. The data is drawn with a line graph having semi-log at y-axis. For convergence, the solutions of ANN-SCA-IPA algorithm is compared graphically with the numerical solutions of RK4. And the collected data is plotted by convergence graphs such as boxplot and bar-graphs. The proposed technique can solve more complex physical, chemical and biological problems.