An accurate approach for thermal analysis of porous longitudinal, spine and radial fins with all nonlinearity effects – analytical and unified assessment

Abstract

The present study deals with a thermal analysis of porous fins subjected to internal heat generation, convection, and radiation energy transfer considering an actual system of analysis under the moving condition of the fin. The main aspiration to carryout this analysis is to establish a correct approach to derive the governing equation for energy transfer in porous fins. The physical explanation was given to require the modification of all existing analyses of porous fins to exchange heat with the surroundings. Three types of fins, namely longitudinal, annular, and spine, have been analyzed with the common mathematical expressions. The temperature distribution in fins has been obtained by employing the differential transform method for solving a high class nonlinear governing equation for all temperature dependent design variables adopted to develop an actual case study. The present approximate analytical method has been authenticated by the numerical technique based on the finite difference method. An excellent agreement of results between these two analyses has been investigated. Unlike all the published works allied with the porous fin, the present work establishes a correct thermal analysis with the key parameter porosity. A clear demonstration has been discussed to distinguish between the solid and porous fins which can be only conjunction with the porosity. The present method of analysis is equally suited for the solid fin also considering only the zero value of porosity to convert the porous fin into a solid fin directly. An optimization study has also been carried out for the maximization of heat transfer rate for a given volume of the solid fraction of a porous fin. The optimum results indicate that under a given design condition, there is an optimum value of porosity for which the heat transfer rate attains a maximum and it is always higher than the solid fin. Therefore, the enhancement of heat transfer can be significantly done by using porous fins for a constant volume of solid or weight of a fin. Finally, the results have been presented in a direction adopted to clearly visualize the superiority of the present study over that of the existing analyses, and to exhibit a difference between solid and porous fins.

Introduction

Fins are made of highly conducting material which uses in a large number of applications to enhance the heat transfer from a primary surface by increasing the heat exchange area. The heat transfer occurred between fin surfaces and surrounding is due to convection or convection–radiation depending upon the nature of thermal environmental interactions. In many practical applications like electronic cooling systems, there is a high requirement for dissipating heat quickly to avoid overheating the essential devices for increasing the durability of the component. The conventional solid fins may not have such capability to transfer heat adequately in a practical design constraint. From the design aspect, thermal engineers are always trying to modify the enhancement system so that the ratio of heat transfer rate to its mass of the fin can be increased significantly. To satisfy this design condition, the conventional solid fin can be replaced by a fin with porous medium called porous fin. The main advantage of using porous fins is to increase the effective surface area of the fin through which the fin exchanges heat to the working fluid. However, the effective thermal conductivity of the porous fin diminishes owing to the removal of the solid material and this reduction can be overcome by increasing the effective surface area.

Kiwan and Al-Nimr [1] introduced first an innovative concept for enhancing heat transfer from a fin using porous medium. Their study was clearly demonstrated the incremented heat transfer due to increase the effective area in porous fins through which heat transfers to ambient fluid. After a long gap, Kiwan [2] adopted a simple method of analysis to determine the performance of a porous fin under a natural convection environment. The heat transfer equation of the porous fin was derived by using the energy balance and Darcy's model. Kiwan's simple analysis states that, the heat transfer rate could exceed that of a solid fin. However, Kiwan [2] developed the analysis of heat transfer through a porous fin by considering heat dissipation from the fin due to energy carried out by the fluid passing through the porous medium only. As a porous fin consists of pores and solid, the interaction of heat from the solid surface of a porous fin also occurs due to directly expose to the surroundings. With this physical fact, another investigation was done by the same author for the study of the influence of radiation heat transfer with the natural convection on the heat transfer from a porous fin surface [3]. To establish this study, the Roseland approximation for the radiation heat exchange and the Darcy model to simulate the fluid velocity owing to solid-fluid interactions in the porous medium was taken into account.

After the above development on the porous fin, Kundu and Bhanja [4] extended the work of Kiwan [3] to develop an accurate analytical model using Adomian decomposition method (ADM) for the performance analysis. From the practical implementation point of view, the optimum analysis of a rectangular porous fin was also studied. Different models of prediction were demonstrated to look for a better model to analyze a porous fin from comparative assessments and recommended to be chosen a porous fin when the requirement of heat dissipation rate is adequately high. In continuation, Bhanja and Kundu [5] determined the temperature distribution and thermal performance parameters of a constructal T-shape porous fin. With the absence of convection from the fin surface, effects of porosity, and effective thermal conductivity, Gorla and Bakier [6] studied numerically the heat transfer behavior in porous fins. In order to look for an analytical technique, homotopy perturbation method (HPM), to develop a nonlinear model for heat transfer in straight porous fins, Rahimi Petroudi et al. [7] calculated the energy dissipation from a porous fin. To determine the maximum heat transfer in porous fins of various profiles, namely, rectangular, convex, and exponential types, Kundu et al. [8] presented an ADM analysis. The main result of their studies shows that the heat transfer rate in a negative power factor of an exponential profile is much higher than the rectangular profile but slightly higher than the convex profile. Additionally, they demonstrated the merit to use a porous fin which is always better from the heat transfer rate point of view in comparison to the solid fin at a low porosity and a high flow parameter. For the analysis of a porous pin fin, Bhanja et al. [9] determined the temperature distribution and performance parameters for electronic cooling applications. Through HPM analysis, Saedodin and Shahbabaei [10] solved a nonlinear fin equation for a straight porous fin by omitting the effective thermal conductivity in conduction and the porosity in convection.

Using an inverse approach, Das and Ooi [11] estimated five critical parameters of a natural convection porous fin, from a given temperature distribution. To include the effect of the transient heat transfer through a porous fin, Al-Ghamdi [12] developed an analysis based on the MATLAB software. On account of the internal heat generation in straight porous fins (Si₃N₄ and Al), three analytical methods, namely, differential transform method (DTM), collocation method (CM), and least square method (LSM), were successfully implemented by Hatami et al. [13] to predict the temperature distribution. For circular-convective-radiative porous fins, Hatami and Ganji [14] evaluated the heat transfer and temperature distribution in four different shapes, rectangular, convex, triangular, and exponential, using LSM and fourth order Runge-Kutta method (NUM) for different selected materials such as Al, SiC, Cu, and Si₃N₄. To reduce the computational time, Darvishi et al. [15] conducted a study of the convection heat transfer in porous media of a rectangular profile fin using homotopy analysis method (HAM). By employing Chebyshev spectral collocation method (CSCM), Darvishi et al. [16] studied the effects of transient thermal performance of a rectangular porous fin. Rostamiyan et al. [17] employed some analytical methods, viz. homotopy perturbation method (HPM), variational iteration method (VIM), and perturbation method (PM) to approach the temperature distribution in porous fins. Adopting a moving porous fin of loosing heat simultaneously convection and radiation, Bhanja et al. [18] established an analytical model for temperature distribution, fin efficiency, and optimum design parameters. An inverse problem was solved by Das [19] to estimate five unknown design parameters satisfying a prescribed temperature distribution in a conductive, convective, and radiative cylindrical porous fin.

It is always desirable to reduce heat through the tip surface of a fin of the insulated tip. The effect of Peclet number for slowing down the value of heat transfer rate through the tip surface was noticed by Moradi et al. [20] who conducted a thermal analysis in moving straight porous fins by HAM. Under the fin surface maintaining below the dewpoint temperature of surrounding air, Turkyilmazoglu et al. [21] analyzed the simultaneous heat and mass transfer on a fully wet porous fin of exponential and straight profile. For the dry surface, Ghasemi et al. [22] established a DTM based model for solid and porous rectangular fins with a temperature dependent internal heat generation. For the fully wet fin surface, Hatami et al. [23] employed LSM and NUM for predicting the temperature distribution in a semi-spherical shape of Si₃N₄ porous fin. A triangular profile porous fin was analyzed by Moradi et al. [24] using DTM. Hatami and Ganji [25] developed LSM and NUM methods to determine the refrigeration efficiency for fully wet circular aluminum porous fins of variable sections. With the same methodologies, ceramic materials (SiC and Si₃N₄) in porous fins were investigated by Hatami and Ganji [26]. Based on HAM, Darvishi et al. [27] described the thermal performance of porous radial fin. To determine the optimal condition of porous pin fins with fully wet surfaces of different fin geometries, Vahabzadeh et al. [28] employed LSM.

Sometime, there is a necessity to determine unknown design variables from a given temperature distribution in a porous fin. By applying the differential evolution algorithm based optimization technique, Das and Prasad [29] predicted two thermal parameters such as the porosity and the thermal diffusivity of a fluid from a given temperature field. Using forward analysis, Kundu and Lee [30] determined the minimum shape of porous fins based on the calculus of variation. Their analysis highlights that the volume of optimum profile fins monotonically increases with the porosity for transferring the same heat rate through a fin. Patel and Meher [31] included different fractional orders in the governing equation of longitudinal porous fins analyzed by Adomian decomposition sumudu transform method (ADSTM). HPM was successfully applied by Cuce and Cuce [32] to investigate the thermal performance of a straight porous fin.

Heat transfer through porous fins can be improved by creating magnetohydrodynamics (MHD) field. Hoshyar et al. [33] used the LSM as an analytical tool to predict the temperature distribution in a longitudinal porous fin under an imposed uniform magnetic field. By spectral collocation method (SCM), Darvishi et al. [34] investigated the thermal analysis of fully wet longitudinal porous fins. The step fin always transfers more heat compared to the conventional constant thickness fin. Kundu and Lee [35] developed an appropriate analytical method for annular step porous moving fins. The double differential transform method (DDTM) was employed for the analytical scheme. The optimum analysis was also carried out for the maximization of heat transfer rate through this porous fin under a design constraint of mass. In their study, it was highlighted that the moving condition is a better approach to increase the heat transfer rate through a fin. With convective-radiative longitudinal porous fins with temperature dependent heat transfer coefficients and internal heat generation, Ma et al. [36] studied through SCM for the effects of various geometric and thermophysical parameters on heat transfer characteristics. A fractional order energy balance equation was solved by Patel and Meher [37] with the help of ADSTM, to determine the temperature distribution in longitudinal solid and porous fins. To search an alternative fin shape, Singh et al. [38] presented an approximate analytical method for analyzing porous step fins.

For avoiding various complicated numerical or analytical approximate procedures developed in the above literatures, Turkyilmazoglou [39] targeted a direct solution method based on the series expansion of the temperature in the vicinity of the mounted surface. An unsteady heat transfer analysis was numerically modelled by Jooma and Harley [40] to study the heat transfer in a porous radial fin using the Crank-Nicolson finite difference method. Utilizing the spectral element method (SEM), Ma et al. [41] solved a coupled conductive, convective, and radiative heat transfer in moving rectangular porous fins of trapezoidal, convex parabolic, and concave parabolic profiles. The thermal performance of the irregular porous fins was obtained as a function of the volume adjusted fin efficiency. Patel and Meher [42] considered a Roseland approximation to radiate heat transfer, uniform magnetic field, and Darcy's model to simulate the flow in porous media of rectangular porous fins by deploying ADSTM. By the application of Galerkin's method (GM) and Akbari-Ganji's method (AGM), Asadian et al. [43] investigated the heat transfer in rectangular porous fins. Shateri and Salahshour [44] determined temperature distribution and heat performance of a longitudinal porous fin with different cross-sectional areas by LSM. Das and Kundu [45] predicted the heat generation in porous fins from the surface temperature whereas Baghban et al. [46] estimated the time dependent base temperature of a porous fin from the tip temperature measurement using the inverse analysis.

The effect of the two-dimensional heat transfer in a porous fin having variable thermal conductivity of the solid phase was studied by Shokouhmand et al. [47] using the finite difference method (FDM). The particle swarm optimization (PSO) technique was implemented by Deshamukhya et al. [48] to carryout the optimization of significant variables for the maximum heat transfer through a rectangular porous fin. Three different analytical methods, namely, CM, HPM, and HAM were used by Hoseinzadeh et al. [49] to determine the temperature distribution in a constant cross-sectional porous fin. For an efficient cooling of consumer electronics, Oguntala et al. [50] adopted a functionally graded material in the analysis and optimization of convective-radiative longitudinal porous fin. A heat sink of inclined porous fins exhibits higher thermal performance than a heat sink of vertical porous fins operating under the same thermal conditions with the same geometrical configuration that was observed by Oguntala et al. [51] and pseudo-spectral collocation method (PSCM) was used. The magnetic parameters on the enhancement of the thermal performance of a longitudinal porous fin were analyzed by Oguntala et al. [52] using an efficient iterative approach called Daftardar–Gejiji and Jafari method (DJM). Hoseinzadeh et al. [53] validated the heat transfer through a rectangular porous fin numerically.

The powerful analytical methods, namely, CM, LSM, and HAM were used to solve a nonlinear fin equation by Hoshyar et al. [54] to study the heat transfer behavior in longitudinal porous fins. In case of moving situations, Ndlova and Moitsheki [55] used the differential transform method (DTM) to execute the thermal analysis in porous fins of rectangular and exponential profiles. The symbolic heat transfer model was developed by Abbasbandy and Shivanian [56] for the purpose of the effects of various parameters on the thermal behavior of longitudinal porous fins. For the combined effect of particulate fouling and magnetic field on the efficiency of a convective-radiative porous fin heat sink with a temperature dependent thermal conductivity, Oguntala et al. [57] developed a thermal model in presence of the thermal fouling on the fin surface. To simulate the thermal efficiency, simultaneous effects of surface roughness, porosity, and magnetic field on the performance of a porous microfin were investigated by Oguntala et al. [58] using the CSCM. Deshamukhya et al. [59] carried out an optimization study of a rectangular porous fin with the temperature dependent internal heat generation using ADM and DTM for the insulated and convected tip conditions.

For the effects of periodic thermal conditions and non-Fourier heat transfer, Mehraban et al. [60] investigated numerically the heat transfer in a longitudinal porous fin. The new definition of porous fin efficiency and effectiveness were introduced and estimated by Kiwan [61] for several surface inclination angles to calculate the heat loss from a porous finned surface under a constant heat flux. Using a neuro-computing based stochastic numerical paradigm, Ahmad et al. [62] studied the dynamic temperature distribution in rectangular porous fin by exploiting the strength of artificial neural network (ANN) integrated with global search exploration with genetic algorithm (GA) and efficient local search with the interior-point technique (IPT). Using the second law of thermodynamics to calculate the entropy generation, Khatami and Rahbar [63] established a thermal analysis based on a numerical solution. Three different analytical methods, namely, CM, HPM, and HAM, along with a numerical method were established by Hoseinzadeh et al. [64] to determine the temperature distribution in a rectangular porous fin in which heat is generated as a linear function of temperature.

To improve the thermal behavior of porous fins in the convective and radiative transient heat transfer environments, Oguntala et al. [65] imposed a magnetic field and investigated the thermal analysis of a longitudinal porous fin by a finite volume method. The design analysis based on the particle swarm optimization (PSW) was carried out. Deshamukhya et al. [66] developed a firefly algorithm to study the optimization of significant parameters of aluminum and copper rectangular porous fins. For the betterment of heat transport phenomenon, Gireesha et al. [67] studied nanoparticles embedded water based hybrid nanoliquid flow over porous longitudinal fins moving with a velocity for the heat interaction with the surroundings. The free convection flow around an engineering porous fin with spherical connections was investigated by Mesgarpour et al. [68] experimentally and numerically for effects of different positioning angles for different fin materials on the thermal fin performance.

Complete research works on porous fins [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67] have been briefed in the above paragraphs chronologically. It can be noticed from the literature work that the heat transfer enhancement in porous fins was done by conduction, convection, and radiation mechanisms. For the improvement of heat transfer, functionally graded material, moving condition, and imposition of magnetic filed, were taken. Researchers have also concentrated on analysing steady, unsteady, one-dimensional, and two-dimensional heat transfer in porous fins with dry and wet surfaces. Forward and inverse analyses had been carried out for different profiles. Various solution methodologies, such as, NUM, FDM, ADM, DTM, HPM, CM, LSM, HAM, CSCM, HPM, PM, ADSTM, DDTM, SCM, SEM, GM, AGM, PSO, PSCM, DJM, ANN, GA, IPT, PSW, etc. had been developed for the analysis of heat transfer through a porous fin. From the literatures, it can be found that the focussing to analyze a porous fin was mainly given to establish the different analytical and numerical techniques. However, there was a less effort given on the formation of the governing equation based on the physical circumstances. The following points can be highlighted in connection with the governing equation for the analysis of porous fin or the detailed physical explanations are given below to understand the conception taken by many researchers to formulate their governing equations and understanding the research gaps.

• The main difference of porous fins from the solid fin is due to have a large number of pores associated with the porous fin, which allows the fluid flow through these holes. The porosity is a geometrical factor of a porous medium to represent how much tiny holes present in the solid and it makes solely a difference between solid and porous medium. For example, if a solid has no tiny holes, then porosity, ϕ = 0. For a porous medium, ϕ lies between 0 and 1 and an increase in ϕ, increases the number of tiny holes. It is obvious that there will be no solid fraction for ϕ = 1. How much amount of fluid passes through these tiny holes is also dependent on the value of porosity. In the analysis of porous fin, mass of fluid passing the porous part of the fin, m_f = A_zV_zρ_f was taken by all the researchers [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67] including Kiwan [2] who originally proposed a new concept. In m_f, A_zis the area of porous medium (solid+pores) perpendicular to the flow direction, V_zis the fluid velocity direction, and ρ_f is the fluid density. Here it is noted that A_zis the area containing both the solid and flow-passage system. It is always true that there is no possibility to flow of fluid through the solid phase and thus, m_f will be zero for the completely solid medium where ϕ = 0 to be taken mandatorily and porous medium will be converted into solid. But there was no ϕ term associated with m_f according to all the existing works. On the other hand, all analyses have been done with a constant Darcy's number irrespective the value of porosity. Hence, from the authors’ point of views, the modification may be necessary to be determined m_fwith the inclusion of ϕ.

• The radiative heat flux in the porous medium in the x-direction, q_r − x, is given by the Roseland approximation [69] as

  $$q_{r-x}=-\frac{4\sigma }{3{\beta }_R}\frac{d{T}^4}{dx}$$

where σis the Stefan-Boltzmann constant and β_Ris the mean absorption coefficient. The component of rate of heat conduction was determined in all the published works [3, 4, 6, 18, 24, 30, 33, 35, 42, 44, 50, 52, 53, 65, 66] by the simple multiplication area factor as

$$Q_{r-x}=-A_x\frac{4\sigma }{3{\beta }_R}\frac{d{T}^4}{dx}$$

whereA_xis the area of porous medium perpendicular to the direction of heat flow. It is obvious that the radiative heat transfer occurs in the porous medium due to fluid column existed in the flow passage. Therefore, Q_r − xexists only in the porous medium where the porosity ϕ > 0. Again for the solid phase, Eq. (2) vanishes and it thus exists only in porous medium. Nevertheless, for the porous medium, Eq. (2) was constructed without ϕ. So, it may require to be modified.

• Heat conducted through a porous media in the x-direction(Q_c − x) can be determined by applying the Fourier's law of heat conduction as

  $$Q_{c-x}=-k_{eff}{A}_x\frac{dT}{dx};k_{eff}=\left(1-\varphi \right)k_s+\varphi k_f$$

where k_eff, k_s, and k_f are the effective thermal conductivity in the porous medium, thermal conductivity of solid, and thermal conductivity of fluid, respectively. However, a few research works [5, 16, 21, 22, 23, 31, 55, 57, 64, 65] have been carried out without the effect of k_eff for conducting heat in porous medium. It is convenient that for the thermal analysis of a porous medium having a high porosity, the effect of k_eff instead of taking k_smay be essential to be considered to distinguish between solid and porous mediums correctly.

• To analyze the conduction heat transfer in a solid, heat may be generated inside the solid. Research activities have been performed with taking into consideration of the internal heat generation [13, 22, 26, 36, 37, 41, 43, 45, 49, [51], [52], [53], [54], [55], 56, 59, 62, 64]. In the porous medium, the rate of volumetric internal heat generation can be expressed in the following:

  $$q^{″\prime }=\left(1-\varphi \right)q_s^{″\prime }+\varphi q_f^{″\prime }$$

where, q′′′_s and  q′′′_f are the rate of volumetric heat generation in solid and fluid, respectively. However, in all the published works with the internal heat generation, there were no such conditions taken as in Eq. (4), it was similar to be a complete solid case, and also no assumption was indicated.

• From the porous fin, heat is dissipated from the solid surface to the surrounding due to convection. In many published works [10, 13, 17, 22, 31, 32, 37, 39, 43, 47, 49, 51, 57, 58, 62, 64], the whole surface area was taken for the convection heat transfer from the porous fin to the surrounding with a convective heat transfer coefficient. However, for the porous fin application, the convection through the pore passages due to the flow of fluid occurs. The remaining solid area at surface exposed to the surroundings (i.e. (1 − ϕ)A_s instead of only A_s, where A_sis the surface area) may participate in convection heat transfer. Therefore, it is always convenient to take on only the solid surface, (1 − ϕ)A_s, and the error of analysis may become dominant at a high value of porosity.

• Heat may transfer form the fin surface to surroundings due to radiation. The radiative heat transfer occurs from a body only due to its temperature. Therefore, heat transfer always participates for the radiation from the whole surface area, A_s. But, a few works [14, 26, 35, 38, 41, 45, 54, 60] were done by considering only the solid surface, (1 − ϕ)A_s, which may be inappropriate to establish a correct analysis for the high value of porosity.

• A convective or enthalpy flux is produced in the solid due to motion of the solid. The mathematical expression of the enthalpy flux in the solid for a uniform velocity V is ρ_sc_ps VT. In case of a porous medium, the enthalpy flux can be expressed appropriately as (ρc_p)_eff VT, where (ρc_p)_eff = (1 − ϕ)ρ_sc_ps + ϕρ_fc_pf. There is no work in the literature [18, 20, 35, 41, 55, 67] performed with this condition or any assumption had been mentioned in their analysis for omitting this aspect.

• For the transient temperature response in a porous fin, the storage energy depends upon ρc_pof the porous medium and the mathematical expressions of ρc_p are given above. On the other hand, in the published works [12, 16, 60, 65], it was considered as ρ_sc_ps.

• If the porous fin surface temperature is maintained below the dewpoint temperature of the surround air, the water vapor in a moist air is condensed and as a result, simultaneous heat and mass transfer take place on the fin surface. Under this application, in case of porous fins, the condensate will fill in the pore passages and porous medium may not work properly for enhancing heat transfer. Therefore, the wet porous fin [21, 23, 25, 28] may not be possible to design in actual practices for the enhancement of heat transfer point of view.

• In porous fins, thermal energy is transferred by a fluid which passes through the pore passages to the surroundings and at the same time, heat transfer will be dissipated from the porous surface to the surrounding due to surface heat transfer mechanisms. In some published works [2, 7, 11, 56, 61, 63], fin surface heat transfer was omitted. This may not be a right choice to analyze a porous fin.

• There is no such analysis available with considering three common types of porous fins, name longitudinal, spine, and annular fins, in a single derivation approach.

• Researchers have rarely concentrated on determining the performance of a porous fin by calculating the heat transfer from the fin using the integral approach based on the Newton law of cooling. However, this integral approach is mandatorily necessary to adopt in the case when the heat generation occurs inside a fin.

For the practical design of a porous fin, the above points should be kept in mind and these essential physical aspects are to be considered in any analysis of a porous fin depending upon the requirement and condition. Therefore, the above facts are motivated to conduct the present research work.

In this study, an analysis is concentrated on a porous fin to derive an energy equation based on the above critical points under conduction, convection, and radiation heat transfer environments and the fin is moving in a system with a uniform velocity. Three common types of fins, namely, longitudinal, spine and annular geometries are taken in the analysis with the same mathematical formulation. The internal heat generation occurs in the solid portion of a porous fin which is a linear function of temperature. The differential transform method (DTM) is used to determine the temperature. The analytical results have been validated with those determined by the numerical technique using the finite difference method (FDM). The present mathematical formulation is equally suitable for the solid fin by considering only the porosity ϕ = 0 to satisfy the solid medium. Physically, it is always true that the porous medium is converted into a solid if the porosity is set to be zero. However, in all published works [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], this mandatory design aspect doesn't satisfy. The superiority of the present analysis has been illustrated by presenting the results determined from the present and published approaches. The temperature distribution and heat transfer performances are described as a function of porosity. The difference of results between the present and published approaches is significant based on the design variables taken. The optimization study is also carried out to determine the ability of the maximum heat transfer rate of a porous fin under a constraint volume fraction of solid. Different design aspects in a given environment have been highlighted. The distinction of results of the present and published approaches at the optimum condition has been noticed by plotting some important results.