Convective Fins Problem with Variable Thermal Conductivity: An Approach Based on Embedding Green’s Functions into Fixed Point Iterative Schemes

1 Introduction

Fins are the extensions of the surfaces from which the heat transfers to the ambient fluid. These extended surfaces have shown a significant improvement of convective heat transfers that are frequently encountered in various applications. A detailed review of the convective heat transfer with fins and their industrial applications is provided by Kern and Kraus [1].

The mathematical model of heat transfers with fins is extensively studied in the literature. Heat transfer with constant thermal conductivity results in a linear equation that could be solved analytically. Yet, it was studied numerically by a number of researchers (see [2], [3] and references therein). On the other hand, large temperature variations would cause changes in thermal conductivity, producing a nonlinear equation whose closed form solution is difficult to obtain [4]. Various numerical methods were employed to tackle the nonlinear equation. Ganji et al. [5] applied the perturbation method, homotopy perturbation method, and variational iteration method to solve the problem. Chiu and Chen [6] implemented the decomposition method. Joneidi et al. [7] used differential transform method to handle the problem. Homotopy analysis method was proposed by Domairry and Fazeli [8] and Khani et al. [9]. Adomian decomposition method (ADM) was implemented by Arslanturk [10]. For other related papers that apply Adomian decomposition approach to heat transfer problems, see [11], [12], [13]. The tanh method has also been utilised to tackle nonlinear heat conduction problems (see [14] and the references therein). In [15] the convective-radiative T-shape fin with variant thermal conductivity is analytically studied by Torabi and Aziz. A regular perturbation solution for a straight convecting fin with temperature-dependent thermal conductivity is presented by Aziz and Huq in [16]. The spectral collocation method is investigated for the heat transfer in the following various demonstrations: a continuously moving convective-radiative rod with variable thermal conductivity [17], a convective-radiative fin with temperature-dependent properties [18] and a radiative-conductive porous fin with temperature-dependent properties [19].

In this study, we present a strategy that is based on the implementation of fixed point iterative scheme, namely, Krasnoselskii–Mann’s, to a designated integral operator that is expressed in terms of Green’s function. Our approach is motivated and inspired by the work of the authors in [20], [21], [22] who introduced this novel iterative scheme for solving a wide spectrum of nonlinear initial and boundary value problems. The article is organised as follows. Section 2 presents the mathematical model describing heat transfer with fins. Section 3 briefly introduces Green’s functions and derives the scheme that embeds them into fixed point iteration formula. Section 4 presents the numerical results and demonstrates their accuracy by considering the residual error. Section 5 includes a conclusion that briefly summarises the results.

2 Description of the Problem

In this section, we present a brief physical interpretation of the problem (see [8] for further details). Consider a straight fin with perimeter P, length b, and cross-sectional area A_c. The thermal conductivity of the fin depends linearly on temperature T as follows:

where k_a denotes the thermal conductivity of the fin at the ambient fluid temperature T_a and λ represents the variation of thermal conductivity. The fin, which is attached to a base surface of temperature T_b, extends into the ambient fluid, and its tip is insulated. The one-dimensional energy balance equation is written as

where h is the heat transfer coefficient. The equation is then reduced by introducing the following parameters

The resulting equation is

subject to the boundary conditions

3 Description of the Iterative Method

In this section, we start with an overview of the Green’s function of a class of second order nonlinear boundary value problems (BVP), followed by the derivation of the algorithm that involves Green’s functions and fixed point iteration schemes.

4 Numerical Results

In this section, we will present our numerical results for the equation of rectangular purely convecting fin with temperature-dependent thermal conductivity. To apply the iterative algorithm (18) for our model problem given in (4) and (5), we rewrite (4) as follows:

where for this case f(t, u, u′, u″)=−ϵ(u′)²−ϵuu″. Thus, the iterative scheme (18) for our problem becomes

As for the initial iterate u₀, and as mentioned earlier, it is obtained by solving u″−N²u=0 subject to the boundary conditions in (5) which yields u₀=sech(N) cosh(Nt).

First, the Green’s function, corresponding to the linear differential operator u″−N²u=0, is to be constructed. Its two linearly independent solutions are u₁(t)=e^Nt and u₂(t)=e^−Nt. Thus, upon using the properties of the Green’s function, given in (11) and (12)–(14), it will take the following form:

Substituting the latter Green’s function into algorithm (22) yields the iterative scheme

for n=0, 1, 2, …, where the initial iterate u₀ is chosen as described above and is given by

To find the best choice of α_n, we minimise the L² norm of the residual error, R_n(t; α_n), of the nth iteration, u_n. The residual error for the first iteration is given by

The corresponding L² norm is given by

which is to be minimised for α₁. Taking, for example, the case ϵ=0.3 and N=0.5, we get α₁=0.901975. In Figure 1 we plot the α₁ curve that confirms this optimal value.

Figure 1:

L² norm of the residual error, R₁(t; α₁), that depicts best choice of α₁.

The subsequent tables depict our numerical results for different selections of the parameters ϵ, variation of the thermal conductivity, and N, the thermo-geometric fin parameter. In addition, the evaluation of fin efficiency, η, is displayed and compared with the results of the ADM. Tables 1–3 display the approximate values of u(t) obtained by our proposed Mann–Green Method (MGM) for a specified value of ϵ and various values of N. Corresponding to each N and ϵ, the nth iteration is carried out until the residual error reaches a value of at most 10⁻²⁰, which is the value of tolerance or the stopping criteria. Tables 4–6 report a comparison of the fin efficiency η, for different values of ϵ, between MGM and ADM. The results imply that the convergence of the method is slower for larger values of N. Yet, the CPU time remains small and negligible. It is also observed that when the thermal conductivity of the fins material, ϵ, increases the temperature u(t) increases as well.

5 Conclusion

In this paper, we introduced a novel technique to solve second order nonlinear heat transfer equation associated with variable thermal conductivity condition. The method incorporates Green’s function and Mann’s fixed point scheme. The solutions for selected values of the parameters in the equation and the corresponding fin efficiency were presented. In the absence of the exact solution, the high accuracy of the results was confirmed using residual error computations. In regard to the benefits and/or advantages of the proposed approach, unlike other existing methods, our proposed scheme provides highly accurate approximations with reasonable CPU time. Moreover, for the majority of other methods, such as differential transform method, spectral method, perturbation method, variation iteration method, and decomposition method, the approximations deteriorate when moving away from the initial endpoint, while when employing our strategy the errors are almost uniformly distributed over the domain. The Green’s function is a piecewise-defined function that satisfies the boundary conditions at both ends of the interval; thus, it is designed for boundary value problems and a domain decomposition will be unnecessary. This is dissimilar to other methods which rely solely on one endpoint, so that the accuracy worsens as we approach the other endpoint.