Modeling the thermo physical behavior of metallic porous ﬁn on varying convective loads

Background/ Objectives: Porous-permeable structured ﬁns are the principal operational mechanism for enhancing the percentage of heat evolved and dissipated because of their many thermo physical characteristics. Study of thermal gradients on the basis of convective loads in porous ﬁns is important in many engineering ﬁelds. Methods: In the present fractional investigation, well-established optimal homotopy asymptotic method (OHAM) has been applied on thermal system expressed in nonlinear fractional order of ordinary diﬀerential equations for Darcy’s approach for porous-structured ﬁn. Here parameters related to porosity, permeability and convection have been deliberated. In order to study the thermal solicitations, the thermal analysis with insulated tip of copper based alloy is studied. Findings: It is found that porosity of system is inﬂuencing more than other factors. Novelty: This study demonstrates the eﬃciency of OHAM as well. ethylene glycol and kerosene oil). Hall Effect on Couple Stress 3D Nanofluid Flow over an Exponentially Stretched Surface with Cattaneo Christov Heat Flux Mode. (3) Radiative flow of magneto hydrodynamics single-walled carbon nanotube over a convectively heated stretchable rotating disk with velocity slip effect. (5) Darcy Forchheimer nanofluid thin film flow of SWCNTs and heat transfer analysis over an unsteady stretching sheet. (6) Impact of Nonlinear Thermal Radiation on MHD Nanofluid Thin Film Flow over a Horizontally Rotating Disk. (7) Radiative heat and mass transfer analysis of micropolar nanofluid flow of Casson fluid between two rotating parallel plates with effects of Hall current. (8) Uniform magnetic force impact on water based nanofluid thermal behavior in a porous enclosure with ellipse shaped obstacle. (9) Application of Electric Field for Augmentation of Ferrofluid Heat Transfer in an Enclosure with Double Moving Walls. (10) The electrical MHD and hall current impact on micropolar nanofluid flow between rotating parallel plates.


Introduction
The study of thermo-physical properties in convective stream through porous permeable medium is highly commendable in numerous engineering problems. Porous permeable metallic and ceramic materials have a lot of industrials and bio medical situations and no doubt could become modern materials with their vast applications in fields. Several mathematical and investigational studies were carried out to deliver a profounder consideration of the transportation instrument for the temperature transferal confidential the porous permeable medium. Porous permeable media has widespread solicitations like catalytic bed reactors, improving dehydrating effectiveness, sieving, isolation, and dealing petroleum recovery. The thermionic conductive porous permeable substances are used to augment forced convective heat transfer problems in several technological problems like reactor designing, thermal components like exchangers, and parabolic shape solar plates heaters (1)(2)(3)(4)(5) . The rapid and fast expansion of electronic machinery has developed more noticeable modifications https://www.indjst.org/ in several manufacturing, business, and ecological problems. This expansion did not happen without an accompanied progress in heat dissipation (cooling) approaches. Augmentation of natural convection has been and will continue to be fundamental in refining the demonstration of the heat dissipation apparatuses of integrated circuit based technology confined.
Fin or stretched surface is a nice operational component to augment heat transfer from the heated regions and surfaces. It has been extensively functional in atmospheric heat conditioning, automotive, heat transfer equipment, etc. The main objectives of these investigations are dropping the dimensional/geometrical perspectives and material price of fin-based system (3)(4)(5)(6)(7)(8) . The enhancement of heat dissipation capabilities in extended surfaces or fins have been given preferences in research and development that has guided to utilization on the use of porous fins. The enhancement of heat dissipation capabilities in extended surfaces have been a theme of enthusiastic prominence that has directed to widespread investigation on the use of porous finned structures (PFS). Two factors that are most prominent in system are; one is permeability i.e. inter connectivity of pores and scale of pores in PFS and second is convective load. Ghasemi et al. (7,8) had a preliminary investigation on heat transfer in fin. They studied heat transfer of fin as compared to the porous fin escalations heat transfer geometrical benefits for a definite area and shrinkages the material mass by eliminating the solid mass with holes and cavities. It is important to narrate that Kiwan and Al-Nimr (9) initially announced perception of employing PFS to augment heat dissipation. Also Kiwan (10) familiarized the Darcy approach to designate the solid-gas/air interfaces in the PFS. Khaled (11) studied heat dissipation in the quadrangular permeable fin and determined that the PFS improved heat dissipation rate. Kiwan and Zeitoun (12) established that the cylinder-shaped permeable fin augmented thermionic transitions. Ghasemi et al. (13) associated heat transfer presentations of solid permeable fins with 60 thermal controlled heat transfers. Kiwan (14) assessed thermionic radiation possessions in heat dissipation in (PFS). E. Cuce and P. M. Cuce (15) efficaciously used HPM to evaluate fin efficacy and performance analysis of the quadrangular permeable fin. Numerical approach has been used by Ma et al. (16) to investigate thermal capacity of the convective radiative permeable fin. Moradi et al. (17) employed a HPM modelled convective capacity and radiation thermal dissipation in the moveable porous-permeable fin. Bhanja et al. (18) formulated and followed analytical model based approach to attain fin performance and maximum design geometrical values and constraints of a movable porous-structured-fin uncovered in convective-radiative environment. Some other scientists like Das (19) approved the inverse results of convection-radiation base approach for cylinder-shaped porous-permeable fins. The factors of permeability along with porosity and internal heat evolved in the system are linked to many applications. The many physio-thermal characteristics of porous-structures in metals and ceramics have been investigated by the features of the connected solid part and the fluids encompassed in the porous membranes and structures (20)(21)(22)(23)(24)(25)(26)(27)(28)(29) . Mohammad Hemmat Esfe et al. (30) provide a comprehensive review on convective heat transfer of nano fluids in porous media. Liaqat Ali et al. (31) studied thin-film flow of magneto hydrodynamic (MHD) fluid in the presence of thermal conductivity and variable viscosity over a porous steady stretching surface with a magnetic field and radioactive heat fluctuation. Partner L. Ndlovu (32) the variation of temperature distribution and fin efficiency in a porous moving fin of rectangular profile is studied. Zahirhah et al. (33)(34)(35)(36)(37)(38)(39)(40)(41)(42) study the inventive conception of the electrical Magneto hydrodynamics (MHD) rotational flow of Single and Multi-Walled Carbon nanotubes (SWCNTs/MWCNTs) base on the fluids (water, engine oil, ethylene glycol and kerosene oil). Darcy-Forchheimer (2) three-dimensional micropolar rotational nanofluid flow of single wall and multiwall carbon nanotubes base on the fluids (water, engine oil, ethylene glycol and kerosene oil). Hall Effect on Couple Stress 3D Nanofluid Flow over an Exponentially Stretched Surface with Cattaneo Christov Heat Flux Mode. (3) Radiative flow of magneto hydrodynamics single-walled carbon nanotube over a convectively heated stretchable rotating disk with velocity slip effect. (5) Darcy Forchheimer nanofluid thin film flow of SWCNTs and heat transfer analysis over an unsteady stretching sheet. (6) Impact of Nonlinear Thermal Radiation on MHD Nanofluid Thin Film Flow over a Horizontally Rotating Disk. (7) Radiative heat and mass transfer analysis of micropolar nanofluid flow of Casson fluid between two rotating parallel plates with effects of Hall current. (8) Uniform magnetic force impact on water based nanofluid thermal behavior in a porous enclosure with ellipse shaped obstacle. (9) Application of Electric Field for Augmentation of Ferrofluid Heat Transfer in an Enclosure with Double Moving Walls. (10) The electrical MHD and hall current impact on micropolar nanofluid flow between rotating parallel plates.
In the fractional calculus, the concept of non-integral order differentiation and integration has been used. Fractional calculus is the generalization of the classical calculus. From the last many years, many mathematicians, researchers and scientists have observed that the role of non-integer operators is very important in expressing the properties of physical phenomena. Many procedures and apparatus have been proficiently expounded by fractional differentiation and integrals. Additional relative study has been done between classical models and fractional models (20)(21)(22)(23)(24)(25) . OHAM technique is rapidly convergent as compared to other techniques. The reliability and effectiveness of OHAM has been shown in literature with several applications in different fields of sciences and engineering.
The paper is categorized in four sections. In Section 2; the governing equation based upon the heat transfer equation is formulated. In Section 3; Computational remarks for solution based on OHAM are given. Some numerical examples are https://www.indjst.org/ considered in section 4. The findings are tabulated and graphical results are shown in this section.

Governing Equation
A quadrangular fin contour has chosen and deliberated in this work, for investigating the behavior of convective effects, as shown in Figure 1. The convection of system is written in term of differential equation. The geometrical measurements of the fin are given as: its length is described as "L", width is described as "w" and thickness is described as "t". The cross segment area of the fin is not variable rather than fixed. This fin has particular porosity to permit the stream of penetrate through it. The subsequent assumptions are taken to elucidate the objectives.
Energy balance can be written as: The system is translated in Darcy's porous-structure approach with total energy balance of the system is presented as Whereas, the mass-flow rate of the convective-fluid streaming PFS is given as And velocity as flow-steam is given as : In this system, it is presumed that heat energy generated in the fin varies with temperature. Here the concept of energy balance is used for steady state condition, Porous parameter, S k = D a R a k ( L t ) and Convection parameter, m = ( Here, S h is a parameter related to porosity that defines the permeability of the porous structure as well as buoyancy effect. If the value of S h is higher, then it specifies greater interconnected-porousness of the PSF or greater buoyantic forces. Here, m is a convective parameter that designates the consequence of convection from the surface of the fin. The conditions to be considered for the work are for finite length fin with insulated tip θ (0) = 1, dθ dy | y=1 = 0. In order to solve non-linear problems, the idea of homotopy has been combined with perturbation method. In his investigation, Lio (9) did the fundamental work by using homotopy analysis method. A novel technique, which is known as OHAM, was created by Marinca et.al. (29) . The benefit of OHAM is that it establishes its convergence criteria similar to HAM but more pliable. R. Kumar (43) provide the three-dimensional (3D) analysis for Cassonnanofluid and Carreau-nanofluid flows due to a flat body in a magneto hydrodynamic (MHD). Nawaz et al. (44) use the Optimal Homotopy Asymptotic Method for solution of a coupled system of nonlinear partial differential equations. In various research papers, S. Iqbal et al. (45)(46)(47)(48)(49) , have proved usefulness generalization and trust of this method, and achieved trust worthy solutions, and presented important applications in science and engineering. Concept of OHAM has articulated in this paper. It provides logical, trust worthy solution to linear, non-linear, time dependent, time fractional and space fractional differential equations & PDEs.

Basic definitions of fractional calculus
In this section, we give some basic definitions of real valued functions. Definition Definition Riemann-Liouville sense derivative operator of a function f (t) of fractional order α > 0 is defined as Caputo sense derivative operator of a function f (y) of fractional order α > 0 is defined as OHAM Scheme for fractional order heat-transfer differential equation.
According to the OHAM algorithm (16,17) . We shall extend this scheme for fractional heat-transfer differential equation {FPDE} in the following steps. Step-1: Compose the fractional order heat-transfer governing equation as Where Ω is domain. Now equation (10) is decomposed in to J (θ ) = Q (θ ) + T (θ ) https://www.indjst.org/ Where Q is a linear part and T is a non-linear part but it is worth noting that this is not necessary. Here we can take liberty about choice of Q.
Where c i ; i = 1, 2, 3, . . . m is auxiliary control parameter of convergence, and k i (y) ,i = 1, 2, 3, . . . m, may be function y. The selections of k m (y, c i ) may be polynomial, exponential etc. The selection of functions is very important, because the rate of convergence of the solution really depends on the functions. Step-3: Extend Ψ (y, ; p, k i ) in Taylor's series about p, to develop an approximate result as It has been cleared that the rate of convergence of the (13) depends upon auxiliary constants k i . If it is convergent at p=1, one has Step-4: Substituting eq. (14) in eq. (10) yields the following expression for residual: R is a residual of the problem Heat Transfer Equation (ii) Auxiliary constants c i can be calculated as follows If R (y; k i ) = 0, then θ (y; k i ) will be the exact solution. Normally it does not happen, likely in non-linear problems. Step-5: The use of auxiliary constants in equation (14), we can get the rapidly convergent approximate solutions.

Applications of fractional order heat transfer model
In this section, consider the Fractional Order Heat Transfer Problem for {OHAM} algorithm. This represents the accuracy, validity and effectiveness of the extended OHAM algorithm. Let us consider the following heat transfer fractional order model from equation (4) We can make an optimal homotopy in above equation with OHAM algorithm as https://www.indjst.org/ In the following p 0 , p 1 , p 2 and p 3 are zeroth-order, first-order, second-order and third-order problems.
Residual of equation 17, can be written as For the solution of auxiliary constant c 1 , the least squares method involves that setting Provides the values of c 1 constant, where And here after substitution in (27) and simplification, we get R (30) https://www.indjst.org/

Results and Discussions
The OHAM algorithm presented in sec-3 for time fractional order heat-transfer equation and explanation of the formulation in the examples of sec-4, provides extremely valid results for the problems without any spatial discretization. While applying {OHAM}, there is no need to compute higher order solutions. Values of materials properties are taken from (33) as D a = 0.0003; L/t = 10, R a = 10000, K r = 954, ε = 0.5, K s = 411.
In the following Tables 1, 3 Table 1. After using the equation (28) we acquire the following Selecting the real value of c 1 , i.e. c 1 = 0.2762700280567808 ; similarly other real values of c 1 with different values of ∈ are presented in the Table 3.   Table 5. https://www.indjst.org/  After using the equation (28) we acquire the following Selecting the real value of c 1 , i.e. c 1 = 0.2643088748972952 ; similarly other real values of c 1 with different values of s are presented in the Table 9. In the above, Figure 2 represents the thermal response behaviour of case-1 for different values of α, Figure 3 represents the internal heat behaviour of case-2 for different values of ∈, Figure 4 represents the generation numbers of case-3 for different values of G, Figure 5 represents the effect of surface convicting of the fin in case-4 for different values of m, and Figure 6 represents the higher permeability of the porous medium in case-5 for different values of S h. All the above figures represent the accuracy, validity and effectiveness of the extended {OHAM} algorithm. It is cleared that as we proceed along the domain, we obtain consistent validity. All the tables also represent the strength of OHAM based solutions.

Summary and Conclusions
The above fractional thermal porous fin model, is a non-linear second-order ODE. In this work, the model solution is presented numerically by using the OHAM. It is presented that OHAM method offer unpretentious, correct and fitting method for forecasting heat dissipation effects in PFS with the occurrence of a thermionic system. We have two dissimilar types of cases subject to the tip situation of the fin, here the investigation of heat transfer is limited to porous fin of finite-length with insulated tip. Figure 2 shows the thermal gradients that are outcome of consequence of dissimilarity of m on heat energy evolved and temperature propagation by varying values of α. Figures 3 and 4 shows the thermal gradients that are outcome of consequence of dissimilarity of ∈ and G on heat energy evolved and temperature propagation by varying values respectively. It is concluded here that S h value rises, the temperature declines rapidly and the fin rapidly reaches the environmental temperature, as shown in Figure 5 as given in (19)(20)(21)(22)(23)(24) , influenced by m. Figure 6 shows the thermal gradients are mainly influenced by S h and S h is mainly attributed to Darcy factor, and this factor is main cause of heat transfer rate in this thermal system. The results also explain the dissimilarity behavior due to porosity of fin and solidity of fin heat transfer rate with parameter k r (50)(51)(52)(53) . Dissimilarity of proportion of porous fin to solid fin heat transfer rate rises as k r upsurges for both the cases. The outcomes designate that the thermal dissemination of metallic Cu-Al-Ag porous structure is intensely conditional on the values and changes in D a and R a . It is evident from that the scale of thermal gradients and thermionic levels is augmented with declining the Darcy parameter and accordingly clear from Figure 2, it is evident that the amount of dimensionless temperature reduces along the fin span. It is concluded here that S h parameter is mostly affecting the PFS, because the more the permeability, more convection and heat transferability in PFS is effective. Higher the S h , more heat transfer is possible. This study leads to design of porous-finnedstructures (PFS) appropriate for several industrial applications, especially for use of electronics and bio-medical industry (33)(34)(35)(36)(37) .