On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions

doi:10.1016/S0017-9310(01)00342-8

International Journal of Heat and Mass Transfer

Volume 45, Issue 12, June 2002, Pages 2509-2525

https://doi.org/10.1016/S0017-9310(01)00342-8 Get rights and content

Abstract

The problem of hydromagnetic fully developed laminar mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions in the presence or absence of heat generation or absorption effects is considered. Through proper choice of dimensionless variables, the governing equations are developed and three types of thermal boundary conditions are prescribed. These thermal boundary conditions are isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel. Analytical solutions for the velocity and temperature profiles for various special cases of the problem are reported. In addition, closed-form expressions for the Nusselt numbers and reversal flow conditions at both the left and right channel walls are derived. The general problem which includes the effects of both viscous dissipation and Joule heating is solved numerically by an implicit finite-difference scheme. Favorable comparisons of special cases with previously published work are obtained. A selected set of graphical results illustrating the effects of the various parameters involved in the problem including viscous and magnetic dissipations on the velocity and temperature profiles as well as flow reversal situations and Nusselt numbers is presented and discussed.

Introduction

Mixed convection flow in a vertical channel has been the subject of many previous investigations due its possible application in many industrial and engineering processes. These include cooling of electronic equipment, heating of the Trombe wall system, gas-cooled nuclear reactors and others. Tao [17] analyzed laminar fully developed mixed convection flow in a vertical parallel-plate channel with uniform wall temperatures. Aung and Worku [1], [2] discussed the theory of combined free and forced convection in a vertical channel with flow reversal conditions for both developing and fully developed flows. Aung and Worku [2] assumed that the walls of the channel were having asymmetric temperatures. The case of developing mixed convection flow in ducts with asymmetric wall heat fluxes was analyzed by the same authors [3]. A comprehensive review of the literature dealing with mixed convection in internal flow was reported by Aung [4]. Cheng et al. [9], Hamadah and Wirtz [11] and Ingham et al. [12] also reported on flow reversal situations in mixed convection in a vertical channel for different wall heating conditions. Kou and Lu [13] analyzed mixed convection in a porous medium channel and discussed the conditions for flow reversal situations.

The use of electrically conducting fluids under the influence of magnetic fields in various industries has led to a renewed interest in investigating hydromagnetic flow and heat transfer in different geoemetries. For example, Sparrow and Cess [15] considered the effect of a magnetic field on the free convection heat transfer from a surface. Raptis and Kafoussias [14] analyzed flow and heat transfer through a porous medium bounded by an infinite vertical plate under the action of a magnetic field. Garandet et al. [10] discussed buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Chamkha [7] analyzed free convection effects on three-dimensional flow over a vertical stretching surface in the presence of a magnetic field.

The study of internal heat generation or absorption in moving fluids is important in view of several physical problems such as those dealing with chemical reactions and those concerned with dissociating fluids (see, for instance, [18], [19]). Other investigations dealing with internal heat generation or absorption can be found in the works of Sparrow and Cess [16] and Chamkha [7].

The aim of the present work is to extend studies available in the literature and especially the work of Barletta [5] on laminar fully developed mixed convection in a vertical parallel-plate channel by including internal heat generation or absorption and magnetic field effects. This will be done for three types of left–right walls thermal conditions. These conditions are the isothermal–isothermal, isoflux–isothermal and the isothermal–isoflux thermal wall conditions.

Section snippets

Governing equations

Consider steady laminar mixed convective flow in a parallel-plate vertical channel in the presence of a magnetic field and a heat source or sink. The vertical plates are separated by a distance L and are maintained at either constant temperature or constant heat flux (see Fig. 1). Let x and y represent the vertical and horizontal distances, respectively, with the origin being at the center of the channel. Let the magnetic field be applied in the horizontal direction normal to the flow

Analytical solutions

Various analytical solutions for special cases of the problem described above are possible only in the absence of both viscous and magnetic dissipation terms. These are reported for the following nine special cases:

Case 1. Hydromagnetic mixed convection flow in a vertical channel with isothermal walls.

For this special case the viscous and magnetic dissipations and the heat generation or absorption effect are neglected (Br=0 and φ=0). Using this, Eq. (8) reduces to $U^{IV} −M^{2} U″=0.$ Solution of Eq. (11)

Heat transfer aspects

The Nusselt numbers at each of the channel walls are important physical characteristics. These can be defined for the three different thermal boundary conditions considered in the present work as follows:

Reversed flow conditions

Depending on the value of the mixed convection parameter Gr/Re and the wall thermal boundary conditions, a flow reversal condition may occur. It is beneficial to understand when this situation occurs and determine a reversed flow zone associated with each of the problems discussed earlier. The occurrence of a reversed flow condition is ensured when the slopes of the velocity profile at the walls have the same sign. That is $[U^{′} (−1/4)][U^{′} (1/4)]>0.$ The critical condition for flow reversal at the

Selected graphical results

Fig. 2 displays typical velocity profiles U in a vertical channel with asymmetric (R_t=1.0) isothermal–isothermal wall heating conditions for different values of the mixed convection parameter Gr/Re. For a vanishing value of Gr/Re, the usual symmetric Hagen–Poiseuille velocity profile is obtained. For an upward flow, increases in the value of Gr/Re have the tendency to increase the momentum of the flow close to the hot right wall causing the velocity profile to become asymmetric. As mentioned

Conclusion

This work focused on the laminar fully developed mixed convective flow of an electrically conducting fluid in a vertical channel in the presence of a magnetic field and heat generation or absorption effects. Three different combinations of thermal left–right wall conditions were prescribed. These thermal left–right wall conditions were isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux conditions. Various analytical solutions for the velocity and temperature profiles for

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On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions

Abstract

Introduction

Section snippets

Governing equations

Analytical solutions

Heat transfer aspects

Reversed flow conditions

Selected graphical results

Conclusion

Int. J. Heat Mass Transfer

Int. J. Heat Fluid Flow

Int. Commun. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Eng. Sci.

Int. Commun. Heat Mass Transfer

Theory of fully developed combined convection including flow reversal

ASME J. Heat Transfer

Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures

ASME J. Heat Transfer

Mixed convection in ducts with asymmetric wall heat fluxes

ASME J. Heat Transfer