Diffusion-thermo and thermal-diffusion effects on free convective heat and mass transfer flow in a porous medium with time dependent temperature and concentration

International Journal of Applied Engineering Research, Jan, 2007 by M.S. Alam, M.M. Rahman, M. Ferdows, Koji Kaino, Eunice Mureithi, A. Postelnicu

Abstract

The diffusion-thermo and thermal-diffusion effects on unsteady free convection and mass transfer flow along an accelerated vertical porous plate embedded in a porous medium have been studied numerically taking the plate temperature and concentration to be functions of time. The governing nonlinear partial differential equations are transformed into a set of coupled ordinary differential equations, which are solved numerically by applying Nachtsheim-Swigert shooting iteration technique along with sixth order Runge-Kutta integration scheme. The effects of various parameters entering into the problem have been examined on the flow field for a hydrogen-air mixture as a non-chemical reacting fluid pair. The numerical results have shown that the above-mentioned effects have to be taken into consideration in the fluid, heat and mass transfer processes.

Keywords: Free convection, Porous medium, Vertical plate, Dufour effect, Soret effect.

Introduction

Convective flow through porous media has many important important applications, such as heat transfer associated with heat recovery from geothermal systems and particularly in the field of large storage systems of agricultural products, heat transfer associated with storage of nuclear waste, exothermic reaction in packedded reactors, heat removal from nuclear fuel debris, flows in soils, petroleum extraction, control of pollutant spread in groundwater, solar power collectors and porous material regenerative heat exchangers.

Coupled heat and mass transfer finds applications in a variety of engineering application, such as the migration of moisture through the air contained in fibrous insulation and grain storage installations, filtration, chemical catalytic reactors and processes, spreading of chemical pollutants in plants and diffusion of medicine in blood veins. A Comprehensive reviews on this area have been made by many researchers such as Nield and Bejan [1], Ingham and Pop [2, 3], Bejan and Khair [4] and Trevisan and Bejan [5].

Most of the above studies, however, considered constant plate temperature and concentration and have neglected the diffusion-thermo and thermal-diffusion terms from the energy and concentration equations respectively. When heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of more intricate nature. It has been found that an energy flux can be generated not only by temperature gradients but by composition gradients as well. The energy flux caused by a composition gradient is called the Dufour or diffusion-thermo effect. On the other hand, mass fluxes can also be created by temperature gradients and this is the Soret or thermal-diffusion effect. In general, the thermal-diffusion and diffusion-thermo effects are of a smaller order of magnitude than the effects described by Fourier's or Fick's law and are often neglected in heat and mass transfer processes. However, exceptions are observed therein. The thermal-diffusion (Soret) effect, for instance, has been utilized for isotope separation, and in mixture between gases with very light molecular weight ([H.sub.2], He) and of medium molecular weight ([N.sub.2], air) the diffusion-thermo (Dufour) effect was found to be of a considerable magnitude such that it cannot be ignored (Eckert and Drake [6]). In view of the importance of these above mentioned effects, Dursunkaya and Worek [7] studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface whereas Kafoussias and Williams [8] studied the same effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Recently, Anghel et al. [9] investigated the Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium. Very recently, Postelnicu [10] studied numerically the influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects.

Therefore, the objective of this work is to investigate the Diffusion-thermo and thermal-diffusion effects on unsteady free convection and mass transfer flow past an accelerated vertical porous flat plate embedded in a porous medium with time dependent temperature and concentration.

Mathematical Formulation

We consider an unsteady free convection and mass transfer flow of a viscous incompressible fluid past an infinite vertical porous plate in a porous medium. The flow is assumed to be in the x-direction, which is taken along the vertical plate in the upward direction, and the y-axis is taken to be normal to the plate. Initially the plate and the fluid are at same temperature [T.sub.[infinity]] in a stationary condition with concentration level [C.sub.[infinity]] at all points. At time t > 0 the plate is assumed to be moving in the upward direction with a velocity U(t) and the plate temperature and concentration are raised to T(t) and C(t) respectively. The physical model and co-ordinate system is shown in the following fig. A.

[FIGURE A OMITTED]

It is assumed that the plate is infinite in extent and hence all physical quantities depend on y and t only. Thus accordance with the above assumptions and Boussinesq's approximation, the basic equations relevant to the problem are:

[partial derivative]v / [partial derivative]y = 0, (1)

[partial derivative]u / [partial derivative]t v [partial derivative]u / [partial derivative]y = [upsilon] [[partial derivative].sup.2]u / [partial derivative][y.sup.2] g[beta](T - [T.sub.[infinity]]) g[[beta].sup.*] (C - [C.sub.[infinity]]) - [upsilon] / K' u, (2)