Advances in Mathematical Physics

Volume 2017 (2017), Article ID 1658305, 20 pages

https://doi.org/10.1155/2017/1658305

## Heat Transfer in a Porous Radial Fin: Analysis of Numerically Obtained Solutions

DST/NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, Wits 2050, South Africa

Correspondence should be addressed to C. Harley

Received 23 January 2017; Revised 4 May 2017; Accepted 24 May 2017; Published 27 June 2017

Academic Editor: Igor L. Freire

Copyright © 2017 R. Jooma and C. Harley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A time dependent nonlinear partial differential equation modelling heat transfer in a porous radial fin is considered. The Differential Transformation Method is employed in order to account for the steady state case. These solutions are then used as a means of assessing the validity of the numerical solutions obtained via the Crank-Nicolson finite difference method. In order to engage in the stability of this scheme we conduct a stability and dynamical systems analysis. These provide us with an assessment of the impact of the nonlinear sink terms on the stability of the numerical scheme employed and on the dynamics of the solutions.

#### 1. Introduction

Circular annular fins are extensively used to increase the rate of heat transfer from a heat source for a given temperature difference in heat exchange devices or to reduce the temperature difference between the heat source and the given heat flow rate of heat sink [1]. The use of fins is widely ranging in engineering applications where it is necessary to enhance the heat transfer from a surface to an adjacent coolant so that the fin can perform within the acceptable temperature limits. These engineering applications range from considerably large systems such as industrial heat exchangers to smaller systems such as transistors. Radial fins have been conventionally used as a coolant for internal combustion engines, heat exchanges, compressors, and so forth. Due to these extensive practical applications this is considered a vibrant field of research; this is true even more so since the problems that arise are nonlinear and hence not always solvable via analytical techniques.

Many research articles have investigated the use of porous fins [2]. Even though a porous material fin has low thermal conductivity, a vast area of the material comes into contact with the cooling agent enabling the porous fin to give superior performance [2]. Over the past decades, numerous studies have been conducted on the performance of annular fins [3–7]. Aziz and Rahman [8] examined a fin comprising functionally graded material and analysed the performance on the radial fin with a continuously increasing thermal conductivity in the radial direction. They discovered that the heat transfer as well as the fin efficiency and effectiveness are at their highest maximum values when the thermal conductivity of the fin varies inversely with the square of the radius. Furthermore, they found that the use of a spatially averaged thermal conductivity model is not recommended due to large errors occurring in some cases. Kiwan [9] conducted a thermal analysis of natural convection porous fins by introducing Darcy’s model to construct the energy equation governing the distribution of temperature. He further discovered that, by choosing a precise value for the thermal conductivity ratio and the fin length to thickness ratio, the performance of the porous fin exceeded the performance of the solid fin. A study was conducted by Kiwan and Zeitoun [10] to test the performance of rectangular porous fins mounted around the inner cylinder of a cylindrical annulus by performing a finite volume type numerical study. It was concluded that, in comparison to solid fins, porous fins provided higher transfer rates for similar configurations and that the heat transfer rate from the cylinder equipped with porous fins decreased as the fin inclination increased. Gorla and Bakier [11] investigated natural convection and radiation in porous fins. They found that the radiation transfers more heat in comparison to a similar model without radiation. Abu-Hijleh [12] analysed the effects of using permeable fins on the forced convection heat transfer from a horizontal cylinder. The results obtained were similar to results obtained as per Kiwan and Zeitoun [10] in terms of the permeable fins providing much higher heat transfer rates. To study radial fins, a combination of the Taylor transformation and finite difference approximation was implemented by Yu and Chen [13, 14]. They further performed a study on the optimization of a circular fin with variable thermal parameters. Naidu et al. [15] set forth a numerical study of natural convection from a cylindrical fin placed in a cylindrical porous enclosure. Hence, they conducted a conjugate conduction-convection analysis by solving the heat conduction equation. Moitsheki and Harley [16] studied the transient heat transfer through a longitudinal fin of various profiles by employing classical Lie point symmetry methods. They observed that for long periods of time the temperature profile becomes unusual for the heat transfer in longitudinal triangular and concave parabolic fins. This, however, was corrected by increasing the thermogeometric fin parameter. In recent studies, Darvishi et al. [17] accounted for the effects of radiation and convection heat transfer in a rectangular radial porous fin. This allowed for the heat flow to infiltrate the porous fin enabling a solid-fluid interaction to occur. They concluded that in a model containing radiation more heat is present than in a similar model without radiation. In a similar context, our model takes into account the time rate of change of internal energy and the heat flow due to conduction as well as the heat due to radiation and convection.

Extensive analytical studies have been done via the Differential Transformation Method (DTM) for the solution of problems such as the one under discussion. The DTM, which was first proposed by Zhou [18], is a seminumerical-analytical method applied to linear and nonlinear systems of ordinary differential equations. The method captures the exact solution in terms of a Taylor series expansion. This method has been successfully implemented in engineering applications [19–25]. Ndlovu and Moitsheki [26] derived approximate analytical solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fin with temperature dependent thermal conductivity and heat transfer coefficients. These authors, for the first time, used a two-dimensional DTM for the transient heat conduction problem. Ertürk [27] constructed seminumerical-analytical solutions for a linear sixth-order boundary value problem using the DTM. It was observed that the method served as an effective and reliable tool for such problems. Recently, Torabi et al. [28] analysed the radiative radial fin with temperature dependent thermal conductivity by implementing the DTM as well as the Boubaker polynomials expansion scheme (BPES). Similar to the results obtained by Ertürk [27], suitable results were obtained in predicting the solution for both BPES and DTM. A study of a radial fin in terms of the fin’s thickness with convection heating at the base and the convection-radiative cooling at the tip was conducted by Aziz et al. [29]. Furthermore, they conducted an analysis using DTM and verified the results by comparing it to an exact analytical solution. The preceding literature clearly shows that the Differential Transformation Method has been applied to problems relating to many different fins, but no attempt has been made to apply it when investigating the heat transfer in a porous radial fin.

In terms of numerical investigations, an efficient, accurate, extensively validated, and unconditionally stable method was developed in the mid-20th century by Crank and Nicolson [30] in order to evaluate numerical solutions for nonlinear partial differential equations. Rani et al. [31] obtained a solution for the time dependent nonlinear coupled governing equations with the help of an unconditionally stable Crank-Nicolson scheme for the transient couple stress fluid flowing over a vertical cylinder. They observed that the time taken for the flow to reach steady state increases as the Schmidt and Prandtl values increase and decreases with respect to the buoyancy ratio parameter. Furthermore, Ahmed et al. [32] employed the Crank-Nicolson finite difference scheme to the conservative equations in modelling porous media transport for magnetohydrodynamic unsteady flow. They found that the flow velocity and temperature decrease with an increase in the Darcian drag force. The concept of the Crank-Nicolson scheme combined with the Newton-Raphson method was used by Qin et al. [33] to model the heat flux and to estimate the evaporation in applied hydrology and meteorology. The Crank-Nicolson method was used to expand the differential equations whereas the iterative Newton-Raphson method was used to approximate latent heat flux and surface temperatures. Both these methods proved to be successful. In a similar context, Janssen et al. [34] implemented the Crank-Nicolson scheme to transform a system of differential equations into algebraic equations. The Newton-Raphson method is used to implicitly enhance the model’s efficiency by improving the poor convergence rate. Once again, it can be seen that the Crank-Nicolson scheme with the Newton-Raphson method has not been implemented for heat transfer in a porous radial fin.

As far as we know, there has been no or very little work that has been done on obtaining asymptotic solutions or employing a dynamical systems analysis to the problem presented in this research. The purpose of the asymptotic solution is to reveal the dominant physical mechanisms of the model. It can be seen in Moitsheki and Harley [16, 35] that the impact of the thermogeometric parameter () in terms of its proportionality to the length of the fin () was observed. They found that the heat transfer in the fin seemed to be unstable for small values of () due to the fact that . By investigating the asymptotic solution to the steady state heat transfer in a rectangular longitudinal fin, they were able to validate the above relationship and establish the importance of the fin length. Furthermore, the same authors [16, 35] conducted a small scale dynamical analysis. In order to expand the analysis in [35], Harley [36] employed an in-depth dynamical analysis to monitor the behaviour especially at the fin tip. This dynamical analysis also served as a means of investigating the role and effect of the thermogeometric parameter. In this work we do not derive an asymptotic solution; such a solution was derived for the problem but we did not deem it useful in terms of providing deeper insight into the dynamics observed. We will however employ a dynamical systems analysis, which we find to be of immense use in classifying the dynamics of specific points of the solution and the engagement between parameters.

A vast amount of work has been conducted on the steady state case of problems in this field since analytical methods lend themselves more easily to the solution of these equations. In this research, a time dependent partial differential equation modelling the heat transfer in a porous radial fin will be considered. The equation will be derived and nondimensionalised appropriately in Section 2. As a means of comparison to the computational methods employed we structure a semianalytical solution via the Differential Transformation Method in Section 3. In the sections to follow, Sections 4 and 5, the partial differential equation is solved numerically using the Crank-Nicolson scheme combined with the Newton-Raphson method as a predictor-corrector. In order to assess the effectiveness of this scheme and its limitations we employ a dynamical systems analysis in Section 6; in this manner we are able to engage with the limitations placed on parameter values with regard to obtaining solutions. Section 7 provides insight into the comparative dynamics and stability of the equation obtained via an alternate means of nondimensionalisation. Concluding remarks are made in Section 8.

The importance of this work relates to the detailed analysis of the dynamics of the temperature at the fin tip. Furthermore, we are able to investigate the impact of the nonlinear source terms on the stability of the schemes employed and the solutions that can be obtained. This highlights the care than needs to be taken when choosing to solve equations of this nature, particularly when solving via numerical tools. Parameters of physical importance are shown to have value limitations due to stability requirements. Consequently, while numerical solutions may have been obtained here and by other authors noting that these do not indicate an ability to obtain solutions for all relevant parameter values or that it may be assumed that solutions that seemed to have converged are indeed dynamically stable and physically accurate is needed.

#### 2. Model Derivation

Consider a cylindrical porous radial fin with base radius , tip radius , and thickness as shown in Figure 1. The fin comprises an effective thermal conductivity porous material and permeability . It is assumed that the tip of the fin is adiabatic (i.e., a process that occurs without the transfer of heat/matter between the system and its surroundings) and the base of the fin is maintained at a constant temperature . The internal energy per unit volume with the absolute temperature is denoted by . In accordance with Darcy’s law, the fin makes contact with an ambient fluid which infiltrates the fin. The ambient fluid comprises an effective density of the porous fin , a specific heat of the porous fin , the kinematic viscosity of the ambient fluid , the thermal conductivity of the ambient fluid , and the volumetric thermal expansion coefficient of the ambient fluid . The top and bottom surfaces are presumed to have a constant surface emissivity and emit radiation to the ambient fluid at temperature . This also serves as the radiation heat sink.