Thermal Prediction of Convective-Radiative Porous Fin Heatsink of Functionally Graded Material Using Adomian Decomposition Method

In recent times, the subject of effective cooling have become an interesting research topic for electronic and mechanical engineers due to the increased miniaturization trend in modern electronic systems. However, fins are useful for cooling various low and high power electronic systems. For improved thermal management of electronic systems, porous fins of functionally graded materials (FGM) have been identified as a viable candidate to enhance cooling. The present study presents an analysis of a convective–radiative porous fin of FGM. For theoretical investigations, the thermal property of the functionally graded material is assumed to follow linear and power-law functions. In this study, we investigated the effects of inhomogeneity index of FGM, convective and radiative variables on the thermal performance of the porous heatsink. The results of the present study show that an increase in the inhomogeneity index of FGM, convective and radiative parameter improves fin efficiency. Moreover, the rate of heat transfer in longitudinal FGM fin increases as β increases. The temperature prediction using the Adomian decomposition method is in excellent agreement with other analytical and method.


Introduction
Increased miniaturization in modern consumer electronics has motivated research on thermal management of high-performance microprocessor-based systems by mechanical and electronic designers.High computational performance of electronic systems usually demands increased power and on-chip power density requirements, both of which involve increased heat dissipation.However, to achieve effective cooling of electronic systems, fins are used as passive approach to reduce thermally-induced failures in electronic components.Moreover, porous fins are established to exhibit improved performance over solid fins following the research discovery by Kiwan et al. [1].Consequently, there are diverse works on the subject of fin using different materials, profiles, orientation and operating conditions in the literature [2][3][4][5][6][7][8][9].
Different works, in recent times, have conducted research using the analytical, numerical and experimental methodology to investigate the effects of various thermal parameters on fin performance [10][11][12][13][14][15][16].Seyf et al. carried out computational analysis of nanofluid effects on convective heat transfer of micro-pin-fin heatsinks [17].Fazeli et al. experimentally and numerically examined the effect of using silica nanofluid on heat transfer in a miniature heat sink [18].Other have studied these methodologies to investigate the effects of conventional airflow properties and features of Computation 2019, 7, 19 2 of 17 channel cross-section as an approach to enhance heat transfer via heat sinks.Such studies include Kim and Mudawar which develop analytical heat diffusion models for different microchannel heatsink cross-sectional geometries as a means to investigate heat transfer enhancement [19].Naphon et al. present a numerical investigation of the convective heat flow and pressure drop in mini-fin and microchannel heat sink for central processing unit cooling [20,21].Kim and Kim investigate the fluid flow and the characteristics of heat transfer in cross-cut heatsinks [22].Oguntala et al. present various numerical investigations on the effects of particle deposition, surface roughness, porosity and magnetic field on the thermal performance of porous micro-fin and longitudinal heatsinks [23][24][25].Wan et al. [26] present an experimental analysis of flow and heat transfer in a miniature porous heat sink for high heat flux applications.These studies explore the convective, radiative and various thermal parameters of the heatsink using the properties of conventional airflow, features of channel cross-section, variable geometries and air flow paths.
Nevertheless, the use of materials of changing composition, microstructure, or porosity across the material volume has been identified as a reliable candidate for improving the thermal performance of heat sinks.Such inhomogeneous materials of varying physical properties including electrical, chemical, mechanical, magnetic and thermal properties over the volume of the bulk material are referred to as functionally graded material (FGM).The continuous variations in properties in FGM along a specific axis are based on the porosity and pore size, chemical, and microstructural gradient-structures resulting in its increased popularity for various applications including nuclear, automobile structure, aerospace and optoelectronics.
With the thermal capabilities of FGM, its application as heatsink fin would serve as a viable cooling material.However, an in-depth review of existing work shows that research on the application of FGM for heatsink design is not exhaustive in the literature.Therefore, the present work is motivated by the capabilities of FGM and several established thermal characteristics of porous fins.The present work focuses on the analysis of a porous FGM heatsink operating under a convective-radiative environment for improved cooling low and high power electronic systems.The thermal property of FGM is assumed to follow linear and power-law functions.The developed thermal models are solved using Adomian decomposition method (ADM).The approximate analytical solutions are used to study the effects of inhomogeneity index of FGM, convective and radiative parameters on the thermal performance of the porous heat sink.
The paper is organized as follows: the fin problem is formulated in Section 2. A detailed description of ADM used for the nonlinear heat transfer analysis is presented in Section 3. The fin efficiency is presented in Section 4. The developed results of the present study are presented and discussed in Section 5.The conclusions of the study are summarized in Section 6.

Formulation of the Model
We consider a porous fin made of FGM as shown in Figure 1.The geometrical dimensions of the fin are given as: length L, width W and thickness t, with both faces of the fin exposed to a convective-radiative environment at temperature T ∞ .
Assume the porous medium is homogeneous, isotropic, and saturated with single-phase fluid and the physical properties of solid as well as fluid are considered as constant.The fluid and porous medium are locally in the thermodynamic domain.The surface radiative transfers and non-Darcian effects are negligible.The temperature variation inside the fin is one-dimensional, that is, temperature varies along the length only and remains constant with time.There is no thermal contact resistance at the fin base and the fin tip is of adiabatic type.

Formulation of the Model
We consider a porous fin made of FGM as shown in Figure 1.The geometrical dimensions of the fin are given as: length L, width W and thickness t, with both faces of the fin exposed to a convectiveradiative environment at temperature T  .Using the above assumptions, the energy balance of the fin [23] is expressed as: The mass flow rate of the fluid passing through the porous material is expressed as: .
and from the Darcy model Equation (1) becomes However, as dx→0, Equation (4) reduces to From Fourier's law of heat conduction, the heat flow in the fin with FGM is expressed as: where simplifying Equation (8) gives the governing differential equation of the fin as: The boundary conditions are When the Temperature Difference in the Fin is Small during Heat Flow The scenario considered in this section is one where a small temperature difference exists within the material during the heat flow.Such condition necessitates the use of temperature-invariant physical and thermal properties of the fin.Moreover, under such condition, it is established that the term T 4 can be expressed as a linear function of temperature [27].Therefore, we have; On substituting Equation (11) into Equation (9), we arrived at For the FGM fin, the spatial-dependent thermal conductivities is established [28] as follows: Exponential-law function Power-law function where α < 0 and β > 0.
On substituting the following dimensionless parameters in Equation (15) into Equations ( 12)-( 14) The dimensionless forms of the governing equation in Equation ( 12) is expressed as: Exponential law function Power-law function on expanding Equations ( 16) and (17), we obtain Exponential law function Power-law function and the dimensionless boundary conditions are For a solid fin, i.e., the non-porous fin of FGM, the governing differential equations are Power-law function An approximate analytical solution can easily be developed using Bessel function and mean-value theorem such that Equations ( 21) and ( 22) becomes: Exponential-law function Power-law function It should be noted that the first and second kinds of Bessel and modified Bessel functions are expressed as:

Analysis of Nonlinear Heat Transfer Equation Using the Adomian Decomposition Method
The nonlinearity in the governing Equations ( 18) and ( 19) poses a computational challenge in developing a closed-form solution of the nonlinear heat transfer equations.However, the application of approximate analytical methods such as Adomian decomposition method (ADM) is efficient to overcome such computational challenge.The section discusses the ADM, which is used to solve the developed thermal model.

Principle of ADM
The analytical solutions obtained in Equations ( 23) and ( 24) are closed-form solutions of the linearized form of the nonlinear model of the present study.However, when the nonlinear term is incorporated, the developed analytical scheme (special functions) fails.This necessitates the need for an alternative analytical scheme as the numerical approach is employed for verification.ADM is employed in the present study because it transforms only the nonlinear terms into an Adomian function with all the linear terms preserved which increases the accuracy of the method.However, a strong limitation of ADM is the ability to obtain the right Adomian level, which when obtained, speeds up the convergence of the required solution.
To discuss the principle of ADM, we express its general nonlinear equation in the form: In Equation ( 26), the linear terms are decomposed into L + R, with L taken as the highest-order derivative that is easily invertible and R as remainder of the linear operator of less derivative order than L. In addition, g represents the source term, u is the system output and Nu represents the nonlinear terms, which is assumed to be analytic.L −1 is the inverse operator of L and is defined by a definite integration from 0 to x, i.e., If L is a second-order operator, then L −1 is a two-fold indefinite integral i.e., L −1 could be expressed as: By applying the inverse operator L −1 to both sides of Equation ( 26), and using the given conditions, the resulting equation is expressed as: where µ(x) = λ x + L −1 g and λ x are terms arising from integrating the source term g(x).
The Adomian method decomposes the solution u(x) of Equation ( 29) into a series form as: and the nonlinear term as where A m 's are Adomian's polynomials of u 0 , u 1 , . . ., u m and are obtained for the nonlinearity Nu = f (u) from the recursive formula where ζ is a grouping parameter of convenience.
The ADM defines the solution of the function f (x) to be approximated as: by applying ADM to Equations ( 18) and ( 19), we obtain Exponential law function Power-law function However, the differential forms of Equations ( 34) and ( 35) is expressed as: Exponential law function Power-law function where L X = d 2 dX 2 is the linear second-order differential operator that is invertible.On applying the inverse operator, L −1 X on both sides of Equations ( 36) and (37), we obtain: Exponential law function Power-law function is the dimensionless tip temperature of the fin, which is denoted as θ 0 .
The unknown θ m , m ≥ 1 is decomposed into a sum of components defined by the decomposition Therefore Equations ( 38) and (39) becomes: Exponential law function Power-law function To determine the higher order terms, Equations ( 40) and (41) can be written as with a recursive relationship as: Exponential law function where m ≥ 1.
Power-law function Here, the Adomian polynomials for the nonlinear terms are given as follows Computation 2019, 7, 19 8 of 17 using the Adomian decomposition scheme in Equation ( 45) and considering 15 Adomian terms for the nonlinear part of the model, the approximate analytical solution for the exponential law is expressed as: Moreover, we show that the solution for the power-law for β = −1/2 is: It is worth noting that for homogenous porous fin, β = 0.
θ 0 is the unknown dimensionless tip temperature of the fin, which can be determined by applying Equation (20b).
The values of the thermogeometric parameter, θ 0 are determined from Equations ( 50) and (51) using the Newton-Raphson iterative method.

Fin Efficiency
As established from previous works, the instantaneous total surface heat loss is the sum of the convective and radiative losses together with heat loss due to porosity and is expressed in Equation (41) as: The ideal fin heat transfer is the heat transfer from the fin when the entire fin surface operates at the fin base temperature.Thus, the idea heat transfer from the fin is given as: Therefore, fin efficiency η could be expressed as the ratio of the actual rate of heat transfer of the fin to the rate that would be if the entire fin is at base temperature expressed as: The dimensionless form of Equation ( 30) is given in Equation (31): Computation 2019, 7, 19 10 of 17

Results
The results of ADM for the comparison of the porous fin of homogeneous material (HM) and porous fin of FGM are presented in Figures 2-6.The figures highlight the effects of various thermal parameters of the fin.

v
The dimensionless form of Equation ( 30) is given in Equation (31):

Results
The results of ADM for the comparison of the porous fin of homogeneous material (HM) and porous fin of FGM are presented in Figures 2-6.The figures highlight the effects of various thermal parameters of the fin.Figures 2-4 shows the effect of the inhomogeneity index on the dimensionless temperature profile and heat transfer rate of the fin. Figure 2 highlights the results of the linear-law function, whilst Figures 3 and 4 highlight the results of the power law function.From the parametric result presented in Figures 2-4, it is seen that an increase in the inhomogeneity index improves the rate of heat transfer through the fin.The figures also show that for all values of the convective and radiative parameters, the temperature gradient along the fin with FGM was smaller than with fin of homogeneous material for both linear and power-law function.The inhomogeneity index β increases as the fin temperature gradient decreases, whilst the rate of heat transfer of longitudinal FGM fin increases as β increases.The results show that the temperature profiles of the FGM fin are highly sensitive in the power-law function compared to the linear-law function.Furthermore, it can be seen that the application of FGM is reliable at low thermogeometric, convective and radiative parameters since the difference in temperature profile between FGM fin and HM fin slightly decreases as the dimensionless thermogeometric parameter increases.Therefore, the application of fin of FGM decreases the thermal resistance along the fin such that FGM fin has a higher temperature at the fin tip than HM fin.
Figure 5 shows the effects of porosity on the fin's temperature profile, whereas Figure 6 highlights the effects of porosity on the thermal efficiency of the fin.From Figures 5 and 6, it is can be seen that increase in the porosity parameter causes the fin temperature to decrease rapidly, whilst the heat transfer rate through the fin increases as the fin temperature decreases faster.The rapid decrease in fin temperature is a result of an increase in the porosity parameter.This is due to the fact that as the fin porosity increases, the permeability of the fin increases, which affects the ability of the working fluid to penetrate through fin pores to increase.The increased penetration of the working fluid increases the effect of the buoyancy force, which causes heat flow by convection to increase.Consequently, the increased rate of heat transfer by convection invariably improves the thermal performance of the fin as shown in Figure 7. Practically, the findings of the present study are useful among others for effective determination of coolant flow rate, which is essential in the designing miniaturized heatsinks with smaller volume fan for low energy consumption.


Dimensionless lenght, X  = 0.00 (HM solid)  = 0.00 (HM porous)  = 0.20 (FGM solid)  = 0.20 (FGM porous)  = 0.60 (FGM porous)  = 1.00 (FGM porous)  Figures 2-4 shows the effect of the inhomogeneity index on the dimensionless temperature profile and heat transfer rate of the fin. Figure 2 highlights the results of the linear-law function, whilst Figures 3 and 4 highlight the results of the power law function.From the parametric result presented in Figures 2-4, it is seen that an increase in the inhomogeneity index improves the rate of heat transfer through the fin.The figures also show that for all values of the convective and radiative parameters, the temperature gradient along the fin with FGM was smaller than with fin of homogeneous material for both linear and power-law function.The inhomogeneity index β increases as the fin temperature gradient decreases, whilst the rate of heat transfer of longitudinal FGM fin increases as β increases.The results show that the temperature profiles of the FGM fin are highly sensitive in the power-law function compared to the linear-law function.Furthermore, it can be seen that the application of FGM is reliable at low thermogeometric, convective and radiative parameters since the difference in temperature profile between FGM fin and HM fin slightly decreases as the dimensionless thermogeometric parameter increases.Therefore, the application of fin of FGM decreases the thermal resistance along the fin such that FGM fin has a higher temperature at the fin tip than HM fin.
Figure 5 shows the effects of porosity on the fin's temperature profile, whereas Figure 6 highlights the effects of porosity on the thermal efficiency of the fin.From Figures 5 and 6, it is can be seen that increase in the porosity parameter causes the fin temperature to decrease rapidly, whilst the heat transfer rate through the fin increases as the fin temperature decreases faster.The rapid decrease in fin temperature is a result of an increase in the porosity parameter.This is due to the fact that as the fin porosity increases, the permeability of the fin increases, which affects the ability of the working fluid to penetrate through fin pores to increase.The increased penetration of the working fluid increases the effect of the buoyancy force, which causes heat flow by convection to increase.


Radiative paramter, Nr  = 0.00 (HM solid)  = 0.00 (HM porous)  = 0.20 (FGM solid)  = 0.40 (FGM prous) Results from the present study were compared to other work as presented in Table 1.From Table 1 it is shown that the temperature result of ADM is reliable for temperature prediction of heatsinks as it agrees with numerical and ADM results.

Figure 1 .
Figure 1.Schematic representation of the fin problem under investigation.

Figure 6 .
Figure 6.Fin dimensionless temperature profile for the varying thermogeometric parameter: (a) linear-law function and (b) power-law function.

NomenclatureAφ 3 σ
Fin cross-sectional area, m 2 P Fin perimeter, m h b Heat transfer coefficient at the base of the fin, Wm −2 k −1 c p Specific heat of the fluid passing through porous fin, J/kg-K h Heat transfer coefficient over the fin surface, W/m 2 K H Dimensionless heat transfer coefficient at fin base, Wm −2 k −1 k Thermal conductivity of fin material, Wm −1 k −1 k b Thermal conductivity of fin material at fin base, Wm −1 k −1 k eff Effective thermal conductivity ratio K Porosity or void ratio η Fin efficiency ν kinematic viscosity, m 2 /s ρ Density of the fluid, kg/m Stefan-Boltzmann constant

Table 1 .
Comparison of temperature result.