Thermal performance investigation of porous fins with convection and radiation under the influence of magnetic field using optimal homotopy asymptotic method

M ATERIAL S CIENCE


𝐴
Cross sectional area of the fins, m

Introduction
For the passive augmentation of the rate of heat transfer in thermal and electronic components, fins are widely used.In practice, the extended surfaces are attached to heat transfer devices and components to facilitate the rate of heat transfer from the prime surface.Further augmentation of the heat transfer has been achieved through the use of porous fins.Such important passive method of heat transfer enhancements has provoked several studies over the past decades.The importance of the extended surfaces has provoked a large volume of research in literatures.The theoretical investigations of thermal damage problems and heat transfer enhancement by the extended surfaces have attest to the facts that the controlling thermal models of the passive devices are always nonlinear.Consequently, the nonlinear thermal models have been successfully analyzed in the past studies with the aids of approximate analytical, semi-analytical, semi-numerical, and numerical methods.In such previous studies, Jordan et al. [8] adopted optimal linearization method to solve the nonlinear problems in the fin while Kundu and Das [9] utilized Frobenius expanding series method for the analysis of the nonlinear thermal model of the fin.Khani et al. [10] and Amirkolaei and Ganji [11] applied homotopy analysis method.In a further analysis, Aziz and Bouaziz [12], Sobamowo [13], Ganji et al. [14] and Sobamowo et al. [15] employed methods of weighted residual to explore the nonlinear thermal behaviour of fins.In another studies, methods of double decomposition and variation of parameter were used by Sobamowo [16] and Sobamowo et al. [17], respectively to study the thermal characteristics of fins.Also, differential transformation method has been used by some researchers such as Moradi and Ahmadikia [18], Sadri et al. [19], Ndlovu and Moitsheki [20], Mosayebidarchech et al. [21], Ghasemi et al. [22] and Ganji and Dogonchi [23] to predict the heat transfer behaviour in the passive devices.With the help of homotopy perturbation method, Sobamowo et al. [24], Arslanturk [25], Ganji et al. [26] and Hoshyar et al. [27] scrutinized the heat flow in the extended surfaces.With the aid of hybrid method of Laplace transformation and Legendre wavelet collocation, Jemiseye et al. [28] presented the transient heat transfer analysis of fin made of functionally graded materials for electronics cooling.Patel and Meher [29] utilized Adomian decomposition Sumudu transform method to study the thermal characteristics of the fin under the influence of magnetic field while Moradi et al. [30] explored homotopy analysis method to analyze the same problem.In another work, Shateri and Salahshour [31] used least-square method for the heat transfer analysis in the longitudinal porous fins with various profiles and multiple nonlinearities.Sobamowo et al. [32] Laplace transform -Exp-function method to develop explicit exact solutions for nonlinear transient thermal models of a porous moving fin.To the best of the authors' knowledge, optimal homotopy asymptotic method (OHAM) has not been applied to the fin problem.Consequently, in this work, the optimal asymptotic homotopy method is applied to carry out thermal analysis of convective-radiative porous fin with temperature-variant internal heat generation under the influence of magnetic field.Parametric studies are carried out and the results are discussed.

Problem formulation
In Fig. 1, it is consideration is given to a porous fin with temperature-invariant thermal properties allowing radiative and convective heat transfer.To thermally describe the behaviour of the passive device, assumptions is made that the heat flow porous medium is filled with fluid of single-phase.The solid portion of the extended surface is homogeneous and isotropic.The fin temperature changes only along its length and the condition of a perfect thermal contact between the prime surface and the fin base is assumed.The unidirectional energy model based on the assumptions and with the aid of Darcy's model is: Eq. ( 1) can be expressed as: The boundary conditions are: The term  can be expressed as a linear function of temperature as: Substitution of Eq. ( 6) into Eq.( 5), results in: It should be noted that: Therefore: The magnetic field is taken to be temperature-dependent since the magnetic field varies temperature.
Using the following nondimensional parameters in Eq. ( 8) on Eqs. ( 3) and ( 7): The following adimensional form of the governing Eq. ( 1) is developed: and the adimensional boundary conditions:

Application of optimal asymptotic homotopy method
In this section, application of optimal asymptotic homotopy method to the nonlinear model.For OHAM (developed by Marinca and Herizanu [33] and [34] and applied in other works [35][36][37][38][39][40], we choose the linear operators from Eq. ( 9) in the form: The initial approximation  () can be obtain as: with the boundary conditions: Last equation has solutions: Nonlinear operators corresponding to Eq. ( 9) and linear operator given in Eq. ( 14) is defined by: where: By substituting Eq. ( 14) into Eq.( 15), we can obtain the expression of   () : If we consider the first-order approximate solution for nonlinear differential Eq. ( 9): where  (,  ) are obtained as: with boundary conditions: Note that the convergence of the approximate solution () depends up on the auxiliary function ℏ(,  ), we can choose ℏ(,  ) as: By solving Eq. ( 18) with boundary condition Eq. ( 19), we obtained: Finally, the solution in Eq. ( 22) is obtained through Eqs. ( 14) and ( 21): where  is unknown parameters which can be obtained with Least-square method (LSM).In our study we choose  = 4.For example, when  = 0.3,  = 0.4,  = 0.3,  = 0.5,  = 0.6 and ԑ, the values of constants are:  = -3.034900727, = 6.159911506,  = -6.848991138, = 2.590773069.Substituting these values in Eq. ( 22), we obtain () in a series form as follow:

Efficiency of the fin
The fin efficiency is the ratio of the rate of heat transfer rate by the fin to the rate of heat transfer that would be if the entire fin were at the base temperature.Efficiency of the fin is an indication of thermal performance and is given by: Using the dimensional parameters in Eq. ( 17), we arrived at:

Results and discussion
The OAHM solutions are simulated for the purpose of graphical illustrations, sensitivity and parametric investigations.Table 1 presents the verifications of results of the OAHM, numerical method (NUM) and differential transformation method (DTM).Although, the DTM provides higher accurate results than OAHM as compared to the results of NM.The higher accuracy is due to the large number of terms (18 terms) in the solutions of DTM as compared to the small number of terms (2 terms).This proves that OAHM is a very convenient mathematical method for the analysis of nonlinear fin thermal models.Also, Table 2 shows the comparison of the results of the present study with the results of the other methods in the previous studies as presented by Patel and Meher [29].From the results in the table, the validity and superiority of the optimal homotopy asymptotic method are established as the method presents better results with the results of the numerical method.
The significance of various parameters of the nonlinear model on the thermal management enhancement of thermal systems using the solutions presented are graphical represented for pictorial discussion in Figs.2-11.The results illustrate that the augmentations of the conductive-radiative, conductive-convective, porosity and magnetic field cause the extended surface adimensional temperature to reduce as a result of increased rate of heat flow via the passive device.The graphical illustrations show that the efficiency and effectiveness of the fin is high at low values of the radiative-conductive, convective-conductive, porosity and magnetic field parameters.
Comparative of the results of methods using OHAM, ADSTM and LSM  = 0.3,  = 0.4, = 0.9,  = 0.5 and  = 0.1 X NUM ADSTM [29] LSM [36] OAHM 0.0 0.781820729 0.781820594 0.781820569 0.781820657 0.1 0.783904154 0.783904049 0.783904760 0.783904121 0.2 0.790166187 0.790166063 0.790165668 0.790166132 0.3 0.800641805 0.800641676 0.800641589 0.800641734 0.4 0.815389668 0.815389550 0.815389906 0.815389592 0.5 0.834492509 0.834492402 0.834492969 0.834492478 0.6 0.858057690 0.858057590 0.858057970 0.858057623 0.7 0.886217964 0.886217887 0.886217828 0.886217919 0.8 0.919132513 0.919132442 0.919132060 0.919132480 0.9 0.956988020 0.956987943 0.956987665 0.956987985 1.0 1.000000000 1.000000000 1.000000000 1.000000000 The impacts of convective-conductive, radiative-conductive and porosity parameters on the adimensional temperature distribution in the passive device is graphically illustrated in Fig. 2. The figure shows that as the convection-radiative increases the adimensional temperature in the fin increases.This also means that the local temperature in the extended surface increases as the conduction-convection parameter increases.It is presented in Fig. 3 about the impact of porosity on the extended surface temperature behaviour.The graphical illustrations show that the amplification of parameter of porosity (Rayleigh number) causes the passive device temperature to be lessened because of the increased permeability allowed by the fin.
Figs. 4 and 5 display the effects of convective-conductive and radiative-conductive parameters on the fin temperature behaviour.It is shown that the rise of the conductive-radiative, and conductive-convective cause the extended surface adimensional temperature to fall as a result of increased rate of heat flow via the fin.The graphical illustrations show that the efficiency and effectiveness of the fin is high at low values of the radiative-conductive, convective-conductive, porosity and magnetic field parameters.The effects of convective-conductive, radiative-conductive, magnetic field and porous parameters on the thermal efficiency of the fin are presented in Figs. 8, 9 and 10 while the effect of porosity or void ratio on the fin thermal efficiency is shown in Fig. 11.It is shown in the figures that when the convective-conductive, radiative-conductive, porosity and magnetic field parameters rise, the passive device efficiency falls.

Conclusions
In this work, optimal asymptotic homotopy method has been used to investigate the heat transfer characteristics of a convective-radiative porous fin with temperature-invariant thermal conductivity.The effect of various parameters of the nonlinear model on the thermal management enhancement of thermal systems have been explored using the solutions presented by the approximate analytical method.The graphical representations of the thermal behaviour of the extended surfaces have been presented and the results have been discussed.The study has showed that the augmentations of the conductive-radiative, conductive-convective, porosity and magnetic field cause the extended surface temperature to reduce as a result of increased rate of heat flow via the passive device.The graphical illustrations show that the efficiency and effectiveness of the fin is high at low values of the radiative-conductive, convective-conductive, porosity and magnetic field parameters.This study will assist in proper thermal analysis of fins and will help in the passive device design.