Thermal Analysis of Conductive-Convective-Radiative Heat Exchangers With Temperature Dependent Thermal Conductivity

In this paper, one dimensional mathematical model of convective-conductive-radiative fins is presented with thermal conductivity depending on temperature. The temperature field with insulated tip is determined for a fin in convective, conductive and radiative environments. Moreover, an intelligent soft computing paradigm named as the LeNN-WOA-NM algorithm is designed to analyze the mathematical model for the temperature field of convective-conductive-radiative fins. The proposed algorithm uses function approximating ability of Legendre polynomials based on artificial neural networks (ANN’s), global search optimization ability of Whale optimization algorithm (WOA), and local search convergence of Nelder-Mead algorithm. The proposed algorithm is applied to illustrate the effect of variations in coefficients of convection, radiation heat losses, and dimensionless parameter of thermal conductivity on temperature distribution of conductive-convective and radiative fins in convective and radiative environments. The experimental data establishes the effectiveness of the design scheme when compared with techniques in the latest literature. It can be observed that accuracy of approximate temperature increases with lower values of <inline-formula> <tex-math notation="LaTeX">$N_{c}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N_{r}$ </tex-math></inline-formula> while decreases with increase in <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula>. The quality of solutions obtained by LeNN-WOA-NM algorithm are validated through performance indicators including absolute errors, MAD, TIC, and ENSE.


I. INTRODUCTION
Heat exchangers or fins are also known as extended surfaces which are commonly used as an element of heat dissipation, that improves the performance and efficiency of equipment [1]. Fins have various applications in air conditioning, energy systems equipment, chemical processes, heat exchanger, cooling systems for computer equipment and The associate editor coordinating the review of this manuscript and approving it for publication was Mauro Tucci . refrigeration. Extended surfaces or fins are designed in different shapes for a class of longitudinal fins with a cross section much less than one dimensional (1D) extended surfaces or length directional. In particular, temperature-dependent behavior is revealed by thermal conductivity when dramatic changes in temperature of the fins occurred. This results in a nonlinear fin problem. Another source of nonlinearity arises from radiation. For example, measurable results from an experiment reveal that heat loss due to radiation is around 15-20 percent of the total heat loss along a fin cooled by natural convection and radiation [2]. As a result, radiation heat transfer has a significant impact on heat exchanger performance, particularly at high temperatures [3]. Thus, similar to conduction and convection, radiation has a substantial influence on temperature distribution and is important for increasing the thermal efficiency of fins, especially for devices with a low convection heat transfer coefficient. Heat transfer in fins is related to one dimensional nonlinear problem, where heat transfer coefficient and thermal conductivity are temperature dependent. Effect of variations in thermal conductivity and heat transfer of several nonlinear models has been extensively studied in [4]- [10]. Various techniques have been designed to study the approximate temperature distribution of extended surfaces in convective-conductive nonlinear fin problems. Chiu et al. [11] and Arsalan [12] proposed Adomian decomposition method to model analytical solution in the form of power series. In addition, other numerical methods that are used to find temperature distribution of fins are homotopy perturbation method [13], [14], homotopy analysis method [15], variational iteration method [16], [17], differential transformation method [18], Galerkin's method [19] and the series method [20]. Moitsheki [21] applied classical Lie symmetry techniques to find exact solutions of the fin problem with a power-law temperature-dependent thermal conductivity. Abbasbandy and Shivanian [22] in 2017 calculate closed-form solutions for heat transfer in a straight fin. In recent times, Sun and Li [23] studied the exact solution of the nonlinear fin problem with exponentially temperature-dependent thermal conductivity and heat transfer coefficient. [24]- [30] recently focused on the study of optimization of various nonlinear models representing physical phenomenon. Radiation, in addition to convection, is another source of heat loss. When heat loss through natural convection is comparable to heat loss from an extended (fin) surface, radiation heat loss cannot be neglected. Thus, for the devices having a low convection heat transfer coefficient, convection and radiation heat transfer coefficients play a vital role. Convection and radiation heat transfer must be used to evaluate high performances of convective, conductive, and radiative extend surfaces (fins). Meanwhile, a strong nonlinear impact on temperature is exhibited by radiation heat loss transfer. Most of above listed methods have been designed to study the thermal distribution and performance of conductive, convective and radiative extend surfaces. DTM method is developed by [31] to study convective and radiative fins with thermal conductivity depending on temperature. [32] find the series solution for convective radiative conduction equation of nonlinear fin with temperature-dependent thermal conductivity. [33] studies a radial fin of uniform thickness with convective heating at the base and convectiveradiative cooling at the tip. Generalized variational iteration method is used by Miansari et al. [34] to deal with nonlinear fin problem with radiation heat loss. Atouei uses collocation method [35], Runge-Kutta method [36] and least square method [37] to analyze temperature distribution and performance of radiative-convective semi-spherical extended surfaces. Optimal linearization method (OLM) [38] was developed to find approximate solutions for temperature field in convective and radiative heat transfers. An integral equation method is introduced by Huang and Li [39] to find an analytical and approximate distribution of temperature and fin performance for convective, conductive, and radiative fin. Multiple shape fins along with longitudinal fins has been widely studied such as T-shaped fins [40], [41], 2D orthotropic convection pin fin [42], [43] and stepped fins [44], [45]. By considering, Cattaneo-Christov heat flux model Khan and Alzahrani [28], [46] studied the impact of variable thermal conductivity over a variable thick surfaces. The classical lie point symmetry method is applied by Mhlongo et al. [47] to investigate the behavior of temperature when subjected to heat flow jump and base temperature jump. [48], [49] analyzed the mathematical model of non-Fourier heat conduction on wet extended surfaces.
In recent times, the heat transfer performance of fin gained the attention of researchers due to dramatic changes in the behavior of fin with temperature variations. Thus, it becomes necessary to design a method that can easily calculate the distribution of temperature in a fin. Unlike approaches available in the literature, this paper focuses on strengthening the concept of artificial neural networks (ANN's). ANN based meta heuristic algorithms are used to solve variety of nonlinear problems arising in fluid dynamics [50]- [54], civil engineering [55], [56], wire coating dynamics [57], thermal engineering [58], [59], biomathematics [60]- [62], financial marketing [63]- [65], fuzzy systems [66]- [68] and petroleum engineering [69]. These potential application of stochastic techniques encourage the authors to strengthen computational ability of ANN's based on Legendre neural networks to study the temperature distribution of fins. The innovative contribution of the given study are summarized as follows: • A mathematical model for temperature distribution of fin with thermal conductivity in the conductive, convective and radiative environment is presented.
• A novel computing paradigm is design by using function approximating ability of orthogonal Legendre polynomials with hybridization of the Whale optimization algorithm (WOA) and the Nelder-Mead algorithm (NM). Proposed methodology is named as LeNN-WOA-NM.
• Further, the design scheme is utilized to study the influence of variations in coefficient of radiation and convection.
• The results obtained by design soft computing paradigm are compared with integral method and exact solution which shows the accuracy of design algorithm with minimum absolute errors in the solutions.
• Verification and validation of the performance analysis based on statistics in terms of standard deviations, mean absolute deviations, absolute errors, Theil's inequality coefficient, variance and error in Nash Sutcliffe efficiency for design scheme have been evaluated by executing LeNN-WOA-NM algorithm for 100 independent runs.

II. PROBLEM FORMULATION
Consider a conductive-convective-radiative fin of length L and cross-sectional A with a temperature-dependent thermal conductivity shown in Figure 1. It is assumed that the fin is made of isotropic solid material relatively long compared to its cross-section. Moreover, the temperature at the base of the fin in a convection environment is considered uniform. By convection-radiation, heat is dissipated on the surface of a fin, and heat transfer through the tip of the fin is neglected. Stefan-Boltzmann's law is obeyed by radiation when heat is dissipated from the surface of fins. During the process, the fin is at rest while heat flows in a steady state. Over the entire surface of the fin, convection heat transfer coefficient N c is considered to be uniform while thermal conductivity k(T ) depends on temperature, which is defined as [70] where ambient temperature is presented by T a . When inner temperature T of fins is equal to ambient temperature (T = T a ) then k denotes thermal conductivity. Temperature change is denoted by λ . At any cross section during flow of heat, T is invariant and varies only with longitudinal directions. Therefore, the phenomena presented in Figure 1 satisfies one dimensional non-linear differential equation for heat transfer which is given by Eq (2) [22].
where 0 < X < L, is surface emissivity, h is convection heat transfer coefficient, Stefan-Boltzmann constant is denoted by ρ and sink temperature for radiation is presented by T s . Now, introducing non-dimensional variables as follows temperature at base of fin is denoted by T b and λ is dimensionless parameter so, Eq (2) can be written as further, Eq (5) can be simplified to [71] at fin tip (x = 0), loss of heat is negligible therefore, it is assumed to be insulated. The boundary conditions for conductive, convective and radiative fin with thermal conductivity can be defined as [11], [72] dθ dx (0) = 0, θ(1) = 1.
Despite exact solutions found by [21], [22], [39] for Eq (6) with boundary conditions Eq (7), a novel soft computing technique known as LeNN-WOA-NM algorithm is designed to find analytical solution for conductive, convective and radiative fin with thermal conductivity.

III. APPROXIMATE SOLUTION AND WEIGHTED LEGENDRE NEURAL NETWORK (LeNN) MODAL
The Legendre polynomials are denoted by L n (x), where n denotes order of Legendre polynomials. The set L 1 , L 2 , L 3 , . . . , L n constitutes the set of orthogonal polynomials on [−1, 1]. First eleven Legendre polynomials are given in Table 1. Higher order Legendre polynomials are generated by using a recursive relation given by Eq (8) [72].
We consider trial solution for Eq (6) presenting non-linear fin problem with temperature dependent thermal conductivity as where δ n , ψ n and ξ n are unknown neurons that would be determined in course of solution. Figure 2 shows structure of Legendre neural networks. As Eq (9) is continuous and differentiable therefore θ and θ can be calculated as following Plugging, θ, θ and θ in Eq (6) will model governing differential equation of conductive, convective and radiative fins. Mathematical model in terms of input, hidden and output layers is shown in Figure 4.

A. THE WHALE OPTIMIZATION ALGORITHM
Whale optimization algorithm (WOA) is a nature-inspired stochastic optimization technique developed by Mirjalili and Lewis [73] which mimics the foraging of humpback whales. It is a global search optimizer that utilizes population search space to determine the global optimum solution for optimization problems. Likewise, other population-based meta-heuristic algorithms, WOA, start optimizing the given problem by generating random candidate solutions. It then improves the set with each iteration until a satisfaction criterion for an ending is achieved. WOA is inspired by the bubble net hunting strategy of humpback whales as shown in Figure 3.
From Figure 3(b) it can be observed that hump back whales encircles the prey by moving in spiral path and creating bubbles along the way. The mathematical model for bubble net mechanism is given by where ''p is a random value in [0,1], b is shape of logarithmic spiral and l is a random number in [−1,1]. X * represents best solution obtained so far while D and D are defined by following equations A and C are coefficient vectors and given as follows:   r is a random vector between [0,1] and a decreases linearly from 2 to 0 with the course of iterations. The first component of Eq (12) illustrates the encircling mechanism, whereas the second mimics the bubble-net strategy. The variable p switches between these two components with an equal probability. Output X * depends on the value of p. WOA starts the process with a set of random solutions. At each iteration, update the position of search agents with respect to either a randomly chosen search agent or the best solution obtained so far. Working procedure of the WOA is shown through flow chart in Figure 5. Initial parameter setting for WOA is given in Table 2.

B. NELDER-MEAD ALGORITHM
Nelder-Mead (NM) algorithm is a direct search method known as the downhill simplex method developed by Nelder and Mead in 1965 to solve different problems without any information about the gradient [74]. NM is a single path following a local search optimizer that can find good results VOLUME 9, 2021   if initialized with a better initial solution. A simplex consisting of n+1 vertices is set up to minimize a function f with n dimensions [75]. NM algorithm generates a sequence of simplices by following four basic steps, named reflection, expansion, contraction, and shrink. Initially, the points x 1 , x 2 , . . . , x n+1 are generated and corresponding values of objective function are evaluated.

Sorting:
Objective values for corresponding vertices of simplex are sorted to obtain centroid (x 0 ), worst (x h ), next to worst (x nw ) and best (x b ) values in all points.
Reflection: In this step, reflection point x r is determined by Eq (17).   where α is reflection coefficient.
Expansion: The expansion point x e is computed by using the equation given below If f (x e ) ≤ f (x r ) then x e would be accepted and iteration will be stopped.

Contraction:
If objective value at x r is strictly greater then objective value at x nw then this steps contraction is applied.
a) If f (x r ) < f (x h ) then the outside contraction is applied by using Eq. (19).
then the inside contraction is applied by using Eq. (20).    Shrinkage: It is a final step and the result is calculated by Eq. (21).
where δ is shrink coefficient. The resulting simplex generated by NM algorithm for succeeding iterations can be written as X = x i , i = 1, 2, 3, . . . , n+1. Parameter setting for Nelder-Mead algorithm are given in Table 2.

IV. LeNN-WOA-NM ALGORITHM
The steps for the proposed hybridized algorithm are summarized as: Step 1: Initialization: Randomly generates an initial population using Eq (9), with number of parameters equal to number of neurons in LeNN's structure. Parameters setting to initialize WOA is demonstrated in Table 2.   Step 2: Fitness evaluation: Calculate the fitness value for each individual of candidate space by using Eq (22).
Step 3: Termination criteria: Terminate the process of fitness evaluation, if any of the following termination criteria is achieved.
• When maximum number of predefined iterations in achieved.
Step 4: Ranking: Rank the individuals of the population on the basis of values of the fitness function .
Step 5: Storage: Store the values of weights and fitness function.
Step 6: Initialization of NM : Nelder-Mead algorithm is used for further speedy fine tuning of the results, starting with global best values of δ n , ψ n and ξ n obtained by WOA. Parameters setting for NM algorithm is shown in Table 2.
Step 7: Refinement: NM algorithm uses MATLAB built in function ''FMINSEARCH'' to update the weights and evaluate the fitness function using Eq (22). The execution of the process stops when predefined stopping criteria is attained.
Step 8: Storage: Store the refined best values of ζ n , ψ n and θ n along with fitness. The procedure in executed for 100 independent runs to obtain large set of statistical data. VOLUME 9, 2021    LeNN-WOA-NM algorithm has a simple structure and easy to implement. WOA updates the position of individual using global search ability and bubble net strategy of humpback whales while NM algorithm further complements its local convergence. Since, Legendre polynomials are orthogonal on [−1, 1] so the experimental analysis shows that proposed algorithm converges to best solutions for number of real-world problems by training the weights from the interval [−1, 1]. It has been notices that convergence of design scheme is slightly effected by increasing the domain.

V. CONSTRUCTION OF FITNESS FUNCTION
Fitness function is formulated on the basis of an unsupervised error, which is defined as the sum of mean square errors of Eq (6) and Eq (7) as where N = 1 h and h is a step size.

VI. PERFORMANCE MEASURES
To examine the accuracy and convergence of design scheme (LeNN-WOA-NM), in obtaining solutions for different problems of conductive-convective and radiative fins with thermal conductivity, performance measures are defined in term of fitness evaluation, mean absolute deviation VOLUME 9, 2021    (MAD), Theil's inequality coefficient (TIC) and error in Nash Sutcliffe efficiency (ENSE). Mathematical formulation for these indices are given below [69].
where, n denote a grid points.

VII. NUMERICAL EXPERIMENTATION
In this section, we have defined different problems to study the influence of variations in coefficients of convective heat    loss N c , coefficient of radiative heat lost N r and dimensionless parameter of thermal conductivity λ on temperature distribution of conductive-convective-radiative fins with thermal conductivity. Problems along with different cases studied in this paper are presented in the flow chart through Figure 6.  Problem I: Effect of Variations in N c on temperature distribution with no resemblance of radiation heat loss and λ.
In this problem, the proposed technique is applied to study the influence of N c on temperature distribution of fins with neglected radiation heat loss N r = 0 and dimensionless parameter of thermal conductivity λ = 0. The fitness function for this problem is formulated as where 0 ≤ x ≤ 1. Following four cases are considered. Case I: Eq (28) with N c = 0.5. Case II: Eq (28) with N c = 1.0. Case III: Eq (28) with N c = 2.0. Case IV: Eq (28) with N c = 4.0.
Problem II: Effect of Variations in dimensionless parameter of thermal conductivity λ on temperature distribution with N c = N r = 1.
In this problem, influence of variations in λ on temperature distribution of conductive, convective and radiative fins is considered with N c = N r = 1. Fitness function for this problem is formulated as where 0 ≤ x ≤ 1, different cases for Eq (29) depending on dimensionless parameter of thermal conductivity are considered as follows. In this problem, influence of variations in dimensionless parameter of thermal conductivity on for temperature distribution of conductive, convective and radiative fin is studied with coefficient of convective heat loss (N c = 1 ) and coefficient of radiative heat loss N r = 2. Fitness function    for given problem can be written as Three cases on bases of dimensionless parameter of thermal conductivity λ are considered to study its effect on temperature distribution of fin.     In this problem, effect on variations in thermal conductivity on temperature distribution of conductive, convective and radiative fins is considered with coefficients of convective heat loss N c = 1 and coefficient of radiative heat loss N r = 3. Fitness based error function for this problem is formulated as Following cases on basis of changes in dimensionless parameter of thermal conductivity are considered as follows Case I: Eq (31) with λ = 1.0. Case II: Eq (31) with λ = 2.0. Case III: Eq (31) with λ = 3.0.

VIII. RESULTS AND DISCUSSION
This paper has analyzed the mathematical model for temperature distribution of conductive, convective, and radiative fins with thermal conductivity. The model given by Eq (6) depends on different parameters named as the coefficient of conductive heat loss, coefficient of radiative heat loss, and dimensionless parameter of thermal conductivity. Furthermore, a new soft computing algorithm is designed to study the influence of parameters on temperature distribution of conductive-convective and radiative fins. The behavior of approximate solutions is discussed with trained neurons from [-1,1]. Results obtained by the proposed method are compared with exact solutions, and approximate solution by integration method [71], decomposition method [11], Galerkin method [70] and simplex search method [76].
To study the performance of the proposed technique by obtaining solutions to different cases of each problem, hundred independent simulations have been carried out. Figures 7-14 demonstrates the comparison of best and worst approximate solutions with exact solution along with absolute minimum errors for temperature distribution of each case of problem I, II, III, and IV, respectively. It can be observed from Figure 7 that with neglecting coefficient of radiation and thermal conductivity, temperature distribution of fin decreases and becomes strong convective with increasing coefficient of convection (N c = 0.5, 1, 2, 4). Furthermore, in the presence of radiation heat loss and thermal conductivity from fin surface we consider N r = 1, 2, 3 and λ = 1, 2, 3, since these non dimensionless parameters covers most of practical cases [1]. It can be viewed from Figure 9,11 and 13 that with increasing value of thermal conductivity (λ), temperature at fin tip rises and the accuracy of temperature distribution becomes higher. In addition, it can be witnessed that with increasing value of coefficient of convection, the temperature distribution of fin rises with fixed values of thermal conductivity VOLUME 9, 2021 and coefficient of radiation. Plots of best weights obtained by design scheme for calculating temperature distribution of each case of different problem are shown in Figures 15-18. Box plots for fitness evaluation, MAD, TIC, and ENSE are shown in Figures 19-22. It can be seen that mean values of fitness function and performance indicators lies around 10 −4 to 10 −6 , 10 −3 to 10 −5 , 10 −4 to 10 −5 and 10 −3 to 10 −6 respectively. The bar graphs are shown in Figures 23-26 represent minimum, mean, median, mode, standard deviation, and variance of fitness value and performance indices obtained by proposed algorithm 100 independent runs.
Approximate solutions obtained by the LeNN-WOA-NM algorithm for different cases of problems I, II, III, and IV are compared with the exact solution and integral method as dictated in Tables 3 and 4. Maximum and minimum absolute errors (AE) of the proposed technique for each case of different problems are given in Tables 5-8. Minimum AE's for case I, II, III and IV of problem I lies between 10 −11 to 10 −14 , 10 −12 to 10 −14 , 10 −11 to 10 −14 and 10 −9 to 10 −13 respectively. Minimum AE's for case I, II, and III of problem II lies between 10 −9 to 10 −12 , 10 −10 to 10 −13 and 10 −9 to 10 −11 respectively. Minimum AE's for case I, II and III of problem III lies between 10 −9 to 10 −13 , 10 −10 to 10 −13 and 10 −10 to 10 −12 respectively. Minimum AE's for case I, II and III of problem IV lies between 10 −9 to 10 −10 , 10 −9 to 10 −12 and 10 −10 to 10 −11 respectively. Statistics of fitness value, MAD, TIC, and ENSE in terms of minimum, mean, median, mode, standard deviation and variance are demonstrated through Tables 9-12. It can be seen that the minimum value of fitness function for the problem I, II, III and IV lies around 10 −13 , 10 −10 , 10 −11 and 10 −10 respectively. Unknown neurons in LeNN structure optimized by design algorithm for obtaining best solutions for temperature distribution of conductive, convective and radiative fins with thermal conductivity. Convergence analysis of the proposed algorithm for each case of problem I-IV is given in Table 17. Statistical results and Figures 27 demonstrates the effectiveness and accuracy of the proposed algorithm.

IX. CONCLUSION
This paper has analyzed a mathematical model for temperature distribution of fin with thermal conductivity in the conductive, convective and radiative environment. Furthermore, we have designed an intelligent soft computing paradigm named as LeNN-WOA-NM algorithm. Weighted Legendre polynomials are used to model approximate series solutions for temperature distribution under the influence of variations in thermal conductivity (λ) and coefficients of convective and radiative heat loss N c and N r . We summarize our findings as follows: • In problem I, with the increasing value of the coefficient of convective heat loss, the excess of temperature is getting lower, which decreases the transfer of heat and fin becomes strong convective as shown in Figure 27(a).
• With the increasing value of dimensionless parameter λ in thermal conductivity, the excess of temperature becomes higher, and the transfer of heat increases. Figure 27(b),(c) and (d) represent the behavior of temperature distribution for problem II, III and IV, respectively.
• Approximate solutions obtained by LeNN-WOA-NM algorithm are compared with exact solutions, and integral methods [71]. Tables 21-25 shows the accuracy of proposed technique in obtaining solutions for temperature distributions under influence of N c ,N r and λ.
• Minimum absolute errors in approximate solutions by design algorithm prove that LeNN-WOA-NM is efficient and accurate. Moreover, the values of performance indicators MAD, TIC and ENSE extend the worth of the designed scheme.
• Convergence of proposed algorithm has been proven by boxplots and bar graphs representing the minimum and mean values of performance indicators obtained during 100 independent runs.
• From the above-discussed figures and tables, it should be noted that the lower value of convection and radiation parameter, the higher is the accuracy of approximate solutions, while larger the value of thermal conductivity, the more accurate the approximate temperature distributions for fins.
In the future, the application of Legendre neural networksbased soft computing algorithms can be extended to solve highly nonlinear and stiff models arising in different applications of practical interest.