Mixed convective cooling of a fin in a channel

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Abstract

Mixed convection heat transfer in an inclined parallel-plate channel with a transverse fin located at lower channel wall is investigated numerically. Employing the stream function vorticity transformation, solution of the transformed governing equations for the system is obtained by the control-volume method with non-uniform grid. The results indicate that the optimum aspect ratio of a fin corresponding to the fin with maximum heat transfer rate increases with increasing Re but decreases with K for a fixed fin profile area. Also, the optimum aspect ratio increases with the inclination angle for Gr/Re2 = 10; whereas the effects of orientation on the optimum aspect ratio is not pronounced for Gr/Re2 < 1. In addition, the optimum aspect ratio of a fin with a smaller fin profile area and Re at Gr/Re2 = 0.1 approaches that of analytical solution having the assumption of constant heat-transfer-coefficient.

Introduction

Combined convection of flow in a parallel-plate channel with transverse fins has been the problem of great interest lately because of its practical application in heat exchangers, electronic equipment, nuclear reactors and other thermal devices. The installed fins can provide additional heat-transfer-surface area and improve the mixing of flow. They are frequently used in the heat transfer augmentation techniques. Employing the assumption of constant heat transfer coefficient, the optimization problems of fin or fin array in natural or forced convection have been investigated to obtain the maximum heat transfer rate at a fixed fin volume [1], [2], [3]. However, the cases of non-uniform heat transfer coefficient are frequently encountered in engineering practice. The mixed convection heat transfer in a channel has been studied primary by considering either isothermal or constant heat-flux boundary condition along the surface on which fins are mounted. Therefore, attention has been focused mainly on the flow pattern and convective transport phenomenon within the fluid. A thorough literature survey for laminar internal flow was given by Aung [4]. Considering the smooth vertical channels with asymmetric wall temperatures Aung and Worku [5], [6] presented the numerical solution for developing flow and analytical solution for fully-developed flow as well. The developing flow with flow reversal was further studied by Ingham et al. [7]. They employed a marching procedure to forward the numerical calculation from inlet to fully-developed region where reverse flow may occur. Cheng et al. [8] extended and solved the problem and found the reversed flow under various boundary conditions. Conducting a numerical analysis of fully developed, laminar, mixed or free convection in horizontal or vertical tubes, many investigators [9], [10], [11], [12], [13] studied the heat transfer characteristics and pressure drop in horizontal or vertical tubes.

For the parallel-plate channel with fins, Habchi and Acharya [14] made a numerical investigation of laminar mixed convection in a vertical channel containing a partial rectangular blockage on one channel wall. Applying the finite differences on the conservation equations, Cheng et al. [15] conducted a numerical study of mixed convection in a finned channel at low Reynolds number. Fung and Lazaridis [16] investigated the effects of Reynolds number, Grashof number, and fin height and pitch. A finite element model of mixed convection is described for a periodically fully developed flow in a vertical channel with transverse fins placed at regular intervals. Besides, Al-Sarkhi et al. [17] presented the thermal performance of a vertical shrouded fin under combined free and forced convection. Solving the three-dimensional elliptic governing equations, Yalcin et al. [18] investigated the effects of clearance parameters on the steady-state heat transfer of fin arrays. Recently, Andrew et al. [19] conducted a numerical study to investigate the fluid flow and heat transfer characteristics of a square microchannel with four longitudinal internal fins. Meanwhile, Shaeri and Yaghoubi [20] have made a numerical analysis of three-dimensional laminar fluid flow and convective heat transfer over an array of solid and perforated fins.

Heat transfer from a fin is mainly influenced by the surrounding heat transfer coefficients which are highly dependent of the thermal characteristics of the fluid flow and fins. Therefore, it is of great importance to explore the interaction between buoyancy-induced flow and pressure-driven external flow, orientation of the channel and conductivities of the fin and fluid in the parallel-plate channel. In this study, a systematic and thorough investigation of free, mixed or forced dominantly convective cooling of a single fin mounted upon an isothermal channel wall is presented. Various aspect ratios are investigated to obtain the maximum heat transfer rates of fins for a fixed fin profile area. The effects of the parameters, Gr, Re, K, and inclination angle of the channel upon the flow and heat transfer are numerically examined.

Section snippets

Analysis and formulation

A parallel-plate channel comprising two infinite plates with interspacing H is considered. One plate wall is maintained constant wall temperatures Tw and the other wall is assumed to be adiabatic. The center of the protruding fin, installed on the uniform temperature surface of the channel, is located at ℓi + ℓ/2. The length and height of fin are ℓ and D, respectively. The schematic view of the geometry with inclination considered in the present study is shown in Fig. 1. The gravity force acts

Results and discussion

Numerical results were obtained for Pr = 0.7, 50Re400, and 10K200. The results cover natural to mixed and to forced convection for Gr/Re2 = 10, Gr/Re2 = 1, and Gr/Re2 = 0.1 at various inclination angles. The center of the fin is located at X = 4. In addition, a fixed dimensionless fin profile area is assumed and is set equal to 0.48 initially. It should be pointed out that the computational domain outlet was positioned far beyond the region shown in the figures. Fig. 3a and b plot the stream

Concluding remarks

  • 1.

    For a fixed fin profile area, there exists an optimum aspect ratio, γo, of a fin which dissipates maximum heat transfer in a parallel-plate channel.

  • 2.

    The optimum aspect ratio of a fin increases with Re at a fixed Gr/Re2. For a given Re, γo becomes larger as Gr/Re2 increases.

  • 3.

    For Gr/Re2 = 10, γo increases with the inclination angle, whereas the effects of orientation on γo is not pronounced for Gr/Re2 < 1.

  • 4.

    The optimum aspect ratio of a fin decreases with increasing K for a mixed convection flow in a

Acknowledgement

The financial support of this research by the Engineering Division of National Science Council, Republic of China, through contract NSC 98-2221-E-022-016 is greatly appreciated.

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