Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity

doi:10.1016/j.matcom.2007.11.009

Mathematics and Computers in Simulation

Volume 79, Issue 2, November 2008, Pages 189-200

https://doi.org/10.1016/j.matcom.2007.11.009 Get rights and content

Abstract

In this paper, the homotopy analysis method (HAM) has been used to evaluate the efficiency of straight fins with temperature-dependent thermal conductivity and to determine the temperature distribution within the fin. The fin efficieny of the straight fins with temperature-dependent thermal conductivity has been obtained as a function of thermo-geometric fin parameter. It has been observed that the thermal conductivity parameter has a strong influence over the fin efficiency. The series solution is developed and the reccurance relations is given. Comparison of the results with those of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), has led us to significant consequences. The analytic solution of the problem is obtained by using the HAM. The HAM contains the auxiliary parameter $ℏ$ , which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter $ℏ$ , we can obtain reasonable solution for large values of $τ$ and $η$ .

Introduction

Nonlinear phenomena play a crucial role in applied mathematics and physics. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solutions to nonlinear evolution equations have been proposed. The investigation of exact travelling wave solutions to nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. The wave phenomena observed in fluid dynamics, plasma, elastic media, optical fibers, etc. In the past several decades, both mathematicians and physicists have made significant progression in this direction. Unlike classical techniques, the nonlinear problems are solved easily and elegantly without transforming or linearizing the equation by using HAM [1], [2], [3], [10], [11], [18], [19], [20], [21], [22], [23]. It provides an efficient explicit solution with high accuracy, minimal calculations and avodiance of physically unrealistic assumptions.

Fins are used to enhance the heat transfer between a solid surface and its convective by Kern and Kraus [15]. Fins are employed to enhance the heat transfer between the primary surface and its convective, radiatingor convective-radiating environment. Aziz and Hug [6], used the regular perturbation method and a numerical solution method to compute a closed form solution for a straight convecting fin with temperature-dependent thermal conductivity. Razelos and Imre [26] considered the variation of the convective heat-transfer coefficient from the base of a convecting fin to its tip. A method of temperature correlated profiles is used to obtain the solution of optimum convective fin when the thermal conductivity and heat-transfer coefficient are functions of temperature [31]. Laor and Kalman [16] examined straight, spine and annular fins governed by power law-type temperature dependence of the heat-transfer coefficient. Yu and Chen [33] invastigated the optimal fin length of a convective-radiative straight fin with rectangular profile under convective boundary conditions and variable thermal conductivity. Bouaziz et al. [7] presented the efficiency of longitudinal fins with temperature-dependent thermophysical properties. Chiu and Chen [8] used ADM to evaluate the efficiency and the optimal length of a convective rectangular fin with variable thermal conductivity. Arslantürk [4] and Rajabi [25] used the ADM and HPM to evaluate the efficiency of straight fins with temperature-dependent thermal conductivity and to determine the temperature distribution within the fin, respectively. Lesnic and Heggs [17] applied the ADM to determine the temperature distribution witin a single fin with a temperature-dependent heat-transfer coefficient.

The HAM is developed in 1992 by Liao. Liao [18], [19], [20], [21], [22], [23] applied this method to solve many types of nonlinear equations in science and engineering and then, this method has been succesfully applied to solve many other nonlinear evolution equations.The HAM contains an auxiliary parameter $ℏ$ which provides us with a simple way to adjust and control the convergence region of solution series for large values of x and t. Note that the HAM has already been applied for the analytical solution of several other problems [1], [2], [3], [5], [9], [10], [11], [12], [13], [14], [23], [24], [27], [28], [29], [30], [32]. All these problems verify the validity of the HAM. The validity of the HAM is dependent upon whether or not nonlinear problems under consideration contain small parameters. We choose freedom to select related initial approximation by the HAM.

In this paper, the basic idea of HAM is introduced and then it is used to fins with temperature-dependent thermal conducdivity and to determine the temperature distribution within the fin. This problem is solved through the HAM and comparison is made the ADM and the HPM which are obtained by Arslantürk [4] and Rajabi [25], respectively.

Section snippets

The homotopy analysis method

We will apply the HAM to solve nonlinear equation arising in heat transfer. But before this, we will give the basic ideas of HAM. We consider the following equation $\tilde{N} [u (τ)] = 0$ where $\tilde{N}$ is a nonlinear operator, $τ$ denotes independent variables and $u (τ)$ is an unknown function. For simplicity, we ignore all boundary and initial conditions, which can be treated in the similar way. By means of HAM, Liao [18], [19], [20], [21], [22], [23] constructed zero-order deformation equation $(1 - p) £ [ϕ (τ; p) - u_{0} (τ)] = p ℏ$

Problem description

We consider a straight fin with a temperature-dependent thermal conducdivity, arbitrary constant crosssectional area $A_{c}$ , paremeter P and length b (see Fig. 1). The fin is attached to a base surface of temperature $T_{b}$ , extends into a fluid of temperature $T_{a}$ , and its tip is insulated. The one-dimensional energy balance equation is given by $A_{c} \frac{d}{d x} [k (T) \frac{d T}{d x}] - P h (T_{b} - T_{a}) = 0$

The thermal conductivity of the fin material is assumed to be a linear function of temperature according to $k (T) = k_{a} [1 + λ (T - T_{a})],$ where $k_{a}$

The homotopy analysis method

To investigate the exact solution of Eq. (15) and to made comparison with ADM [4] and HPM [25], we choose the linear operator $£ [ϕ (τ; P)] = \frac{\partial^{2} ϕ (τ; P)}{\partial r^{2}}$ with the property $£ [c_{1} + c_{2} τ] = 0,$ where $c_{1}$ and $c_{2}$ are constants. From Eq. (15), we define the nonlinear operator as $N [ϕ (τ; P)] = (1 + ɛ ϕ (τ; p)) \frac{\partial^{2} ϕ (τ; p)}{\partial r^{2}} + ɛ {(\frac{\partial^{2} ϕ (τ; p)}{\partial r^{2}})}^{2} - N^{2} [ϕ (τ; p)]$ Using above definition, we construct the zeroth-order deformation equation: $(1 - p) £ [ϕ (τ; p) - u_{0} (τ)] = p ℏ ϕ (τ; p)$ $ϕ (1; p) = 1, \frac{\partial ϕ (τ; p)}{\partial τ} |_{τ = 0} = 0,$ where $ℏ$ is an auxiliary parameter. Obviously $ϕ (τ; 0) = u_{0} (τ), ϕ$

Fin efficiency

The heat dissipation of a fin can be obtained by integrating the convection heat loss from the fin surface [8]: $Q = \int_{0}^{b} P (T - T_{a}) d x = b (T_{b} - T_{a}) \int_{0}^{1} P u (τ) d τ$

The efficiency of the fins is defined as the ratio of the actual heat transfer rate to the heat-transfer rate of the entire surface that depends on the temperature at the base of the fin, $T_{b}$ . So, $η = \frac{Q}{Q_{ideal}} = \frac{\int_{0}^{b} P (T - T_{a}) d x}{P b (T_{b} - T_{a})} = \int_{0}^{1} u (τ) d τ$

Thus the efficiency of straight fins is obtained as an analytical expression as follows: $η = \frac{1}{2520} (840 - 84 ℏ (N^{2} - 6 ɛ - 20) - 264 C^{3} ℏ$

Concluding remarks

The homotopy analysis method is applied to find the converged approximate solution and the exact solution of Eq. (15). The HAM provides us with a convenient way to control the convergence of approximation series by a proper auxiliary parameter $ℏ$ . This is the most importance properties of the HAM. The results show that we can obtain very accuracy results for nonlinear problems in science and engineering by using HAM. The homotopy analysis method offers many advantages over the ADM, the Homotopy

References (33)

S. Abbasbandy
The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KDV equation
Phys. Lett. A
(2007)
S. Abbasbandy
The application of homotopy analysis method to nonlinear equations arising in heat transfer
Phys. Lett. A
(2007)
S. Abbasbandy
Homotopy analysis method for heat radiation equations
Int. Commun. Heat Mass Transfer
(2007)
C. Arslantürk
A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity
Int. Commun. Heat Mass Transfer
(2005)
M. Ayub et al.
Exact flow of a third grade fluid past a porous plate using homotopy analysis method
Int. J. Eng. Sci.
(2003)
M.N. Bouaziz et al.
Etude des tranferts de chaleur non lineaires dans les ailettes longitudinales, numerical study of nonlinear heat transfer in longitudinal fins
Int. J. Thermal Sci.
(2001)
C.H. Chiu et al.
A decomposition method for solving the convective longitudinal fins with variable thermal conductivity
Int. J. Heat Mass Transfer
(2002)
T. Hayat et al.
The influence of thermal radiation on MHD flow of a second grade fluid
Int. J. Heat Mass Transfer
(2007)
T. Hayat et al.
On the explicit analytic solutions of an Oldroyd 6-constant fluid
Int. J. Eng. Sci.
(2004)
T. Hayat et al.
Poiseuille flows of an Oldroyd 6-constant with magnetic field
J. Math. Anal. Appl.
(2004)

T. Hayat et al.

Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet

Int. J. Heat Mass Transfer

(2007)

T. Hayat et al.

On analytic solution for thin film of a fourth grade fluid down a vertical cylinder

Phys. Lett. A

(2007)

K. Laor et al.

Performance and optimum dimensions of different cooling fins with a temperature-dependent heat transfer coefficient

Int. J. Heat Mass Transfer

(1996)

D. Lesnic et al.

A decomposition method for power-law fin-type problems

Int. Commun. Heat Mass Transfer

(2004)

S.J. Liao

An explicit totally analytic approximation of Blasius’ viscous flow problems

Int. J. Non-linear Mech.

(1999)

S.J. Liao

On the homotopy analysis method for nonlinear problems

Appl. Math. Comput.

(2004)

Cited by (52)

Nonlinear heat transfer analysis of spines using MLPG method
2021, Engineering Analysis with Boundary Elements
Citation Excerpt :
The homotopy perturbation method (HPM) was used by Ganji [10], Rajabi [11] and Languri et al. [12] to solve nonlinear heat transfer problems of different fins. The homotopy analysis method (HAM) was used by Mustapha [13], Domairry and Fazeli [14], Chaudhari et al. [15], and Hosseini et al. [16]. The variational iteration method (VIM) was used by Coskun and Atay [17-19], and Languri et al. [12].
In the current article, an analysis of heat transfer in the spine of different profiles is carried out. A nonlinear differential equation that includes the effect of temperature-dependent thermal conductivity, variable heat transfer coefficient, and surface radiation is solved using a predictor-corrector scheme and the meshless local Petrov-Galerkin (MLPG) method. The moving least square (MLS) method is employed for the approximation of unknown function and the penalty parameter method is used to impose essential boundary conditions. Temperature variation along the length of the spine and rate of heat transfer in steady-state, under convective and convective-radiative environment has been obtained and explained. Performance of the spine under the above conditions is evaluated. Further, the study is extended to calculate volume adjusted performance which presents a comparative thermal performance of different profiles.
On the effect of magnetic field on thermal performance of convective-radiative fin with temperature-dependent thermal conductivity
2018, Karbala International Journal of Modern Science
Citation Excerpt :
Also, the investigation into the effects of the inherent nonlinearities in the developed thermal models (due to its temperature-dependent thermal properties) on the thermal performance of the passive devices has attracted a considerable amount of research works. In fact, in the past few decades, different solutions to the nonlinear equations have been developed using different techniques [1–23] such regular perturbation expansion [1,2], method of successive approximation [3], adomian decomposition method [4,5,8], homotopy perturbation method [6,7,9,12] homotopy analysis method [10,15,16], variational iteration method [11–14], differential transform method [17–23], least square method [24] and Galerkin method of weighted residual [25] to analyse the thermal behaviour and efficiency of fins. Optimal homotopy analysis method (OHAM) is employed by Moitsheki et al. [26] in providing an analytical solution and numerical simulation for one-dimensional steady nonlinear heat conduction in a longitudinal radial fin with various profiles.
The removal of the excess heat by effective cooling technology is very vital for reliable operation, proper functioning and performance of thermal components, equipment and systems. In such phenomena, fins or extended surfaces play an essential and a very important role among various passive and active cooling options. In this paper, the effect of magnetic field on the thermal analysis of convective-radiative straight fin with temperature-dependent thermal conductivity is investigated using a new iterative method proposed by Daftardar-Gejiji and Jafari. The developed analytical solution is used not only to investigate the effects of magnetic parameter but also to establish the effects of convective and radiative parameters on the thermal performance of the fin under the influence of the magnetic field. From the results, it is established that increase in magnetic, convective and radiative parameters increase the rate of heat transfer from the fin and consequently improve the thermal performance of the fin. The results obtained are compared with the results established in the literature and good agreements are found. The analysis serves as a basis for comparison with any other method of analysis of the problem and also as a platform for improvement in the design of fin in heat transfer equipment under the influence of magnetic field.
Analysis of convective longitudinal fin with temperature-dependent thermal conductivity and internal heat generation
2017, Alexandria Engineering Journal
Citation Excerpt :
In the same year, Chowdhury and Hashim [10] applied Adomian decomposition method to evaluate the temperature distribution of straight rectangular fins with temperature-dependent surface flux for all possible types of heat transfer while in the following year, Rajabi [11] applied homotopy perturbation method (HPM) to calculate the efficiency of straight fins with temperature-dependent thermal conductivity. Also, a year later, Mustapha [12] adopted homotopy analysis method (HAM) to find the efficiency of straight fins with temperature-dependent thermal conductivity. Meanwhile, Coskun and Atay [13] utilized the variational iteration method (VIM) for the analysis of convective straight and radial fins with temperature-dependent thermal conductivity.
In this study, analysis of heat transfer in a longitudinal rectangular fin with temperature-dependent thermal conductivity and internal heat generation was carried out using finite difference method. The developed systems of non-linear equations that resulted from the discretization using finite difference scheme were solved with the aid of MATLAB using fsolve. The numerical solution was validated with the exact solution for the linear problem. The developed heat transfer models were used to investigate the effects of thermo-geometric parameters, coefficient of heat transfer and thermal conductivity (non-linear) parameters on the temperature distribution, heat transfer and thermal performance of the longitudinal rectangular fin. From the results, it shows that the fin temperature distribution, the total heat transfer, and the fin efficiency are significantly affected by the thermo-geometric parameters of the fin. Also, for the solution to be thermally stable, the fin thermo-geometric parameter must not exceed a specific value. However, it was established that the increase in temperature-dependent properties and internal heat generation values increases the thermal stability range of the thermo-geometric parameter. The results obtained in this analysis serve as basis for comparison of any other method of analysis of the problem.
Thermal analysis of longitudinal fin with temperature-dependent properties and internal heat generation using Galerkin's method of weighted residual
2016, Applied Thermal Engineering
In this study, heat transfer in a longitudinal rectangular fin with temperature-dependent thermal properties and internal heat generation is analysed using Galerkin's method of weighted residual. The simple, but highly accurate solution was validated by the exact solution for the linear problem, and also by numerical method and differential transformation method in the case of the non-linear problem. The developed heat transfer models were used to investigate the effects of thermo-geometric parameters, coefficient of heat transfer and thermal conductivity (non-linear) parameters on the temperature distribution, heat transfer and thermal performance of the longitudinal rectangular fin. From the results, it shows that the fin temperature distribution, the total heat transfer, the fin effectiveness, and the fin efficiency are significantly affected by the thermo-geometric and thermal parameters of the fin. The analysis revealed that the operational parameters must be carefully chosen to ensure that the fin retains its primary purpose of removing heat from the primary surface. Therefore, the results obtained in this analysis serve as basis for comparison of any other method of analysis of the problem, and they also provide a platform for improvement in fin design of fin in heat transfer equipment.
Analytical thermal study on nonlinear fundamental heat transfer cases using a novel computational technique
2016, Applied Thermal Engineering
In this paper a novel computational technique called Parameterized Perturbation Method (PPM) is used to obtain the solutions of nonlinear fundamental heat conduction equations. Three well known problems in the area of heat transfer are addressed to be solved. An analytical investigation is carried out for: (a) the temperature distribution in a fin with a temperature-dependent thermal conductivity, (b) the cooling of the lumped system with variable specific heat, and (c) the temperature distribution of a convective–radiative fin. The validity of the results of PPM solution was verified via comparison with numerical results obtained using a fourth order Runge–Kutta method. These comparisons revealed that PPM is a powerful approach for solving these problems. Also, the results showed that the main attributions of this method are very straightforward calculations and low computational burden compared to previous analytical and numerical approaches.
General exact solution of the fin problem with variable thermal conductivity
2016, Propulsion and Power Research
In this paper, Lie point symmetry method is used to obtain the general exact solution of the second order nonlinear ordinary differential equation which governing heat transfer in rectangular fin with variable thermal conductivity. Some new forms of thermal conductivity are introduced and the associated exact solution is obtained in each case. The general relation among the fin efficiency, thermal conductivity and thermo-geometric parameter is obtained.

View all citing articles on Scopus

View full text

Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity

Abstract

Introduction

Section snippets

The homotopy analysis method

Problem description

The homotopy analysis method

Fin efficiency

Concluding remarks

Phys. Lett. A

Phys. Lett. A

Int. Commun. Heat Mass Transfer

Int. Commun. Heat Mass Transfer

Int. J. Eng. Sci.

Int. J. Thermal Sci.

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Eng. Sci.

J. Math. Anal. Appl.

Int. J. Heat Mass Transfer

Phys. Lett. A

Int. J. Heat Mass Transfer

Int. Commun. Heat Mass Transfer

Int. J. Non-linear Mech.

Appl. Math. Comput.