Adomian decomposition method for a stepped fin with all temperature-dependent modes of heat transfer

Abstract

The application of Adomian decomposition method (ADM) is demonstrated for a nonlinear problem involving conductive, convective and radiative stepped fin with temperature-dependent parameters. The parameters which have been considered to be temperature-dependent are the thermal conductivity, the surface heat transfer coefficient and the emissivity. The performance of the stepped fin is compared with that of the straight fin. For validating the present method, the results obtained using ADM are compared with relevant results of the differential transformation method available in the literature. Effect of various thermo-physical parameters affecting the temperature and the efficiency are illustrated. Comparisons of performance parameters reveal that for a given set of conditions, the stepped fin performs better than the straight fin.

Introduction

Extended surfaces such as fins are one of the basic methods used to enhance the rate of heat transfer. They possess wide range of applications pertaining to various industrial processes [1] such as air conditioning systems, IC engines, electronic systems, etc. During initial stages of development, the rectangular fins were used to amplify heat transfer rate [2], [3]. But, with the passage of time, study of fins with high efficiency alongwith minimum material (i.e., high surface area to volume ratio) has become important. Therefore, various fin profiles [4], [5], [6], [7], [8] involving trapezoidal, triangular, parabolic, hyperbolic, etc. came into the picture. There is no doubt that a rectangular shaped profile is simplest in design and easy to manufacture, but, its weight is relatively more as compared with other shapes such as parabolic, hyperbolic, triangular, etc. However, despite having low weight feature, due to the curvature in the geometry alongwith safety issues related with sharp edges, parabolic, hyperbolic, triangular shapes are also difficult to manufacture and are not very preferable and thus the trapezoidal shape was considered to be the most optimal alternative shape [7]. But, the efficiency of a trapezoidal fin is found to be considerably lower than a rectangular fin [9]. Therefore, in past many optimization and performance studies of various fin profiles have been carried out. It has been found by Torabi and Zhang [10] that as compared with other profiles, the rectangular fin profile possess the highest surface temperature and efficiency.

It is well-known that when all three modes of heat transfer are present then it becomes somewhat difficult to deal with the problem. Moreover, the dependence of thermo-physical parameters on the local temperature also considerably increases the complexity of the problem. To ease out the problem, many times different thermo-physical parameters are assumed to remain constant with the temperature [11], [12], [13]. In addition to the above-mentioned complexities, the geometry of the system alongwith the boundary conditions also influences the amount of effort required for solving the problem. The exact solution of the governing differential equation with linear boundary condition can be easily obtained by simple mathematics such as variable separable method, Runge–Kutta methods, Bessel functions, etc. The ignorance of complex assumptions violates the actual operating conditions prevailing in the system as many times, the thermo-physical parameters are dependent on the temperature [14], [15], [16].

For solving complex heat transfer problems in fins, recently the analytical/semi-analytical techniques are gaining popularity. Aziz and Huq [17] and Hagen [18] employed the perturbation technique to solve different fins with temperature-dependent parameters. Arslanturk [19] has used the decomposition method for a convective straight fin with temperature-dependent thermal conductivity. The performance analysis of trapezoidal fin has been carried out using the homotopy analysis method (HAM) by Khani and Aziz [7]. For a T-shaped fin with temperature-dependent parameters, the performance and optimization analysis have been carried out using the Adomian decomposition method (ADM) by Kundu and Bhanja [20]. However, the effect of the surface radiation was ignored in their study. HAM was also employed by Domairry and Fazeli to investigate the temperature distribution in a straight fin with a nonlinear governing equation [21]. It is found from the literature that ADM is used for solving wide range of nonlinear problems [22], [23], [24], [25], [26], [27], [28]. Recently, semi-analytical technique has also been proposed by Turkyilmazoglu [29] for nonlinear differential equation of extended surfaces.

ADM is one of the most widely used semi-analytical techniques for highly nonlinear problems. This method was introduced and developed by Adomian [30]. Later on, this technique has been applied to solve various problems involving linear, nonlinear, homogeneous, non-homogeneous terms, etc. The usage of ADM is very effective to reduce the computational effort and to increase the accuracy in solution [23], [31], [32]. It is already mentioned earlier that apart from mathematical difficulties, the geometry of the system also contributes to the amount of intricacy of the problem. The heat transfer analysis of a stepped fin involving either one or more temperature-dependent parameters is one of such cases. Nevertheless, some recent studies on nonlinear problems in stepped fins can be found in the available literature. For example, the study of a conductive and radiative stepped fin was carried out using the homotopy perturbation method (HPM) by Arslanturk [33]. Torabi and Yaghoobi [34] also carried out the analysis for a conductive, convective and radiative stepped fin with variable thermal conductivity using the differential transformation method (DTM).

It is finally observed from the available literature that although ADM has been used for many nonlinear problems, but it is not yet used for studying a stepped fin. Moreover, the available literatures indicate that most of the studies on stepped fins either ignore the temperature dependence of surface radiation or heat transfer coefficient. But, it is important to consider the effect of variable parameters with temperature. In addition to this, past studies involving stepped fins are carried out with other simplified assumptions such as ignorance of sink temperatures, etc., thus further limiting the validity of such cases. Therefore, the present work is aimed at applying ADM for studying a conductive, convective and radiative stepped fin with temperature-dependent thermal conductivity, heat transfer coefficient and surface emissivity. Although, a stepped fin is much easy to manufacture and involves either less or no risk of sharp edges, but, no study seems to be available which demonstrates the rationale of using a stepped fin. Therefore, one of the important objectives of this work is to draw a comparison between the straight and the stepped fin. In the following sections, the problem formulation and the solution methodology are discussed.