Natural convective flows in a horizontal channel provided with heating isothermal blocks: Effect of the inter blocks spacing

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Abstract

A numerical study of laminar steady natural convection induced in a two dimensional horizontal channel provided with rectangular heating blocks, periodically mounted on its lower wall, is carried out. The blocks’ surface temperature, TH, is maintained constant and the former are connected with adiabatic surfaces. The upper wall of the channel is maintained cold at a temperature TC<TH. Fluid flow, temperature fields and heat transfer rates are presented for different combinations of the governing parameters which are the Rayleigh number (102Ra2×106), the blocks’ spacing (1/4C=l/H1), the blocks’ height (1/8B=h/H1/2) and the relative width of the blocks (A=(L-l)/H=1/2). The results obtained in the case of air (Pr = 0.72) show that the flow structure and the heat transfer are significantly influenced by the control parameters. It is found that there are situations where the increase of the blocks’ spacing leads to a reduction of heat transfer.

Introduction

Convective heat transfer in periodic geometries is of a particular interest in different technological and industrial applications related to solar engineering fields and cooling of electronic components. The development of semiconductor technology has led to the realization of components well adapted to the compact architecture of the electronic devices. The miniaturization process presents however a disadvantage insofar as it engenders relatively high volumetric electric dissipation power. The periodic geometries are often constituted by a channel provided with heating blocks periodically distributed on one of its walls or on both walls or even detached from the walls. References by Incropera [1] and Peterson and Ortega [2] show that both numerical and experimental investigations have been largely employed to analyze fluid flow and heat transfer characteristics in such geometries. However, for most previous studies, natural convection seems to be ignored in favor of forced and mixed convection. The problem of natural convection in such geometries, although frequently encountered in engineering applications, has received rather little attention. Natural convection air-cooling is still the preferred choice in the cooling of electronic equipments even if it is reserved to low heat generating devices. In fact, when the power released is relatively weak, the natural convection phenomenon is found to be an efficient means of cooling and presents in addition the advantage of being economical and autonomous. Lin and Hsieh [3] studied experimentally the natural convection effect inside two vertical channels with one common wall containing rectangular heating blocks. Their results indicated that the presence of the blocks inside the system is the origin of the appearance of turbulent zones. Correlations were proposed for the Nusselt number in both channels. Numerical predictions of periodically fully developed natural convection in a vertical channel, with surface mounted heat generating blocks, were presented by Kelkar and Choudhury [4]. The obtained results revealed that the mass flow rate induced by buoyancy forces increases at a rate less than the square root of the channel length. Tanda [5] performed an experimental study of natural convection inside vertical channels formed by a heated ribbed surface and an opposing adiabatic smooth surface. He found that the presence of ribs affects heat transfer performance due to inactive regions just upstream and downstream of each protrusion. The problem of natural convection around horizontal downward facing plate with rectangular grooved fins with various aspect ratios was examined numerically and experimentally by Kwak and Song [6]. They showed that the existence of recirculating flows in the grooves prevents circulation of the main stream flow between the fins which leads to a reduction of the heat transfer rate at the inner surface. A numerical investigation of laminar natural convective flows in a vertical isothermal channel with two rectangular obstacles, symmetrically located on each wall, has been recently carried out by Desrayaud and Fichera [7]. They found that the best position of the blocks for heat extraction depends on the magnitude of the Rayleigh number. They also showed that increasing the length of the blocks has only a limited influence on the heat transfer while increasing its width leads to a drastic decrease of the mass flow rate and heat transfer especially if more than half of the opening is obstructed. More recently, Desrayaud et al. [8] performed a numerical study of natural convection in a system of parallel vertical channels with a single protruding heat module mounted mid-height on a substrate of finite thickness. The effects of Rayleigh number, substrate/fluid thermal conductivity and module dimensions on the heat transfer and fluid flow characteristics have been studied.

Natural convection in horizontal channels provided with solid blocks was also the object of interest. Lee et al. [9] studied a problem of natural convection in a horizontal channel heated from below with six equidistant square adiabatic or isothermal bodies, placed in its interior. They compared the case of a wide aspect ratio cell with periodic side boundaries and six internal bodies against those obtained in the case of a unit cell containing a single body with either no-slip adiabatic side boundaries or periodic side boundaries and also against those corresponding to pure Rayleigh–Bénard convection. The results of the wide aspect ratio case are found to be identical to those of two unit cell cases in the quasi-steady conduction regime. However, the velocity and thermal fields are affected by the differences in the aspect ratio of the enclosure and in the boundary conditions as convective motion sets in. Earlier, Hasnaoui et al. [10] studied natural convection inside a horizontal channel with adiabatic rectangular blocks regularly distributed on its lower wall and separated by isothermal heating surfaces. The numerical study, performed in a representative module, showed that, depending on the Rayleigh number and the relative height of the adiabatic blocks, the steady symmetric solution can be either maintained or destroyed, which indicate the limitation of previous studies based on symmetric conditions. The same geometry was considered by Douamna et al. [11] but the thermally active surfaces where heated with a temperature varying sinusoidally in time. The shape of the curves delineating linear and nonlinear regimes in the a-Ra plane were found to depend strongly on the period of the exciting temperature. In addition, different routes leading to chaos were identified in this study by progressively varying the governing parameters. A numerical study was performed by Amahmid et al. [12] on a similar geometry in which the rectangular heating blocks were maintained at a constant temperature. It was found that the calculation domain choice has an effect on the multiplicity of solutions. Also, the results showed that the flow and temperature fields depend strongly on the Rayleigh number and the relative height of the blocks. Thus, the flow symmetry was not always maintained although the boundary conditions are symmetric. In a similar problem, Bakkas et al. [13] studied the effect of the blocks’ width on natural convection and reported that this parameter has a significant effect on the fluid flow and heat transfer in the channel. More recently, the same authors [14] studied numerically natural convection in the same geometry with the blocks releasing a uniform heat flux. It was observed that the flow structures are different from those obtained in the case of isothermal blocks. In addition, it was found that the temperature may undergo important variations around a fixed block and there are solutions for which the temperature changes considerably between two successive blocks. El Alami et al. [15] examined the chimney effect and the heat transfer induced by natural convection in a two dimensional horizontal channel with isothermally heated blocks and slots. They found that the heat transfer and the flow rate generated by the chimney effect are considerably affected by the blocks height. Correlations evaluating the mean heat transfer and the flow rate aspired by the chimney effect were proposed by the authors. Along this line, a numerical study of natural convection was conducted by the same authors [16] in the same configuration to determine the effect of the blocks spacing on the cooling process of a simulated electronic components. The obtained results prove that the blocks spacing strongly affect the fluid flow and heat transfer characteristics. Arquis and Rady [17] investigated numerically the natural convection heat transfer and fluid flow characteristics from a horizontal fluid layer with finned bottom surface. Useful guidelines to enhance the heat transfer rates from the finned surface have been proposed by the authors. Abourida and Hasnaoui [18] have reported a numerical study of laminar natural convection in an infinite channel discretely heated from below and provided with thin adiabatic partitions periodically distributed on its lower wall. The results obtained showed that up to three different types of flow patterns have been observed in the channel and all the unsteady solutions were of periodic nature for each of the obtained solutions. A numerical study on two dimensional natural convection heat transfer from two isothermal protruding heating blocks located in the lower wall of a horizontal open channel was carried out by Icoz and Jaluria [19]. An increase of heat transfer rate was observed by increasing the distance between the blocks (4–15% of increase was observed when the separation distance was doubled).

In the present work, we present a two dimensional numerical results of natural convection in a horizontal channel provided with heating isothermal blocks periodically mounted on its lower wall. The main objective of this study consists to analyze the effect of the inter block spacing (while keeping unchanged their width) on fluid flow and heat transfer characteristics inside the channel for various blocks height and for a sufficiently wide range of Rayleigh number. The effect of the computation domain on the multiplicity of solutions is also considered.

Section snippets

Mathematical formulation

The studied configuration, sketched in Fig. 1, is an infinite horizontal channel of height H′ provided with indefinite number of uniformly spaced rectangular blocks of height h′ and width 0.5 × H′ placed on its lower wall. The blocks are connected with adiabatic surfaces of length l′ and the temperature of their heating surfaces is maintained constant at TH. The upper wall of the channel is maintained cold at a temperature TC (TC<TH). Due to the periodic nature of the problem, the simulations

Numerical method

Eqs. (1), (2), (3) were discretized by using a finite difference procedure. Forward finite difference in time and central differences in space were used. The integration of Eqs. (1), (2) was ensured by the Alternate Direction Implicit method (ADI). The vorticity at the solid walls was calculated using the Wood’s relation (Roache [20]). Values of the stream function at all grid points were obtained with Eq. (3) via a point successive over relaxation method (PSOR). The velocities at all grid

Results and discussions

In this section, fluid flow, temperature fields and heat transfer rates are examined for wide ranges of Rayleigh number (102Ra2×106), relative height of the blocks (1/8B1/2) and relative spacing between adjacent blocks (1/4C1). Solutions of the considered problem were obtained in the stationary regime by considering air as a working fluid (Pr = 0.72). The relative width of the blocks is maintained constant (A = 1/2).

Conclusion

A numerical study of natural convection induced in a periodic horizontal channel provided with heating isothermal rectangular blocks on its lower wall is performed. A particular interest has been given to the effect of the space between the blocks and their height on the flow structure and heat transfer inside the channel. The effect of the computation domain on the multiplicity of solutions is also studied. The calculations performed in the SD showed that the symmetry of the flow is generally

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