•• EG0600070 Proceedings of the Environmental Physics Conference, 24-28 i _„. HEAT AND MASS TRANSFER EFFECT ON HYDROMAGNETIC FLOW OF A MOVING PERMEABLE VERTICAL SURFACE M.M. Abdelkhaiek Atomic Energy Authority, Nuclear Research Centre, Nuclear Physics Department, Cairo, Egypt, P.O. Box 13759 ABSTRACT Numerical results are presented for the effects of heat and mass transfer on hydromagnetic flow of a moving permeable vertical surface. The surface is maintained at linear temperature and concentration variations. The nonlinear- coupled boundary layer equations were transformed and the resulting ordinary differential equations were solved by perturbation technique. Numerical results for the dimensionless velocity profiles, the temperature profiles, the local friction coefficient and the local Nusselt number are presented for various values of Prandtl number, suction/blowing parameter, Schmidt number, buoyancy ratio and Hartmann number. The effects of the different parameters on the velocity and temperature profiles as well as skin friction and wall heat transfer are evaluated. Favorable comparisons with previously published work confirm the correctness of numerical results. Key Words: Heat and Mass Transfer, Hydromagnetic Flow, Perturbation Technique/ INTRODUCTION Hydromagnetic incompressible viscous flow has many important engineering applications such as magnetohydrodynamic power generator and the cooling of reactors. Heat transfer and its related topics are heavily studied fields. These studies range from over simplified problems to highly complex types of interactions and configurations, which require sophisticated numerical schemes and high-speed computers to obtain reasonably accurate results. However, the common ground for most of these .-jtudies is that they are solved and analyzed by assuming a pure fluid with no contaminants. While this assumption approximates the reality in some cases quite well, especially for low particle contamination levels, it. is not, however, valid in a lot of other cases in which the contaminants in the fluid play a major role in altering the resultant flow and heat transfer characteristics. In many world environments, such as in Egypt, for example, dust storms or fine dust Suspension in the air are encountered for many months during the year. These fine particles of dust penetrate the enclosures and the various devices, and have a serious impact on the performance of many equipments. This example represents a situation of participate suspension where a pure fluid assumption does not accurately represent the reality. The advent of technology in the field of magnetohydrodynamic power generators and magnetohydrodynamic devices, nuclear engineering and thermonuclear power has created a great practical need for understanding the dynamics of conducting fluids. The use of liquid metals as heat transfer agents and as working fluids in magnetohydrodynamic power generators has created a growing interest in the study of liquid metal flows and nature of interaction with magnetic field. The interaction between 185 Proceedings of the Environmental Pliysics Conference, 24-28 Feb. 2004, Mi/tya, Egypt the conducting fluid and magnetic field radically modifies the flow and heat transfer characteristics. The outlook for a direct coal-fired magneiohydrodynamic power generator as potentially significant source of energy seems promising in view of its efficiency, its effect on environment and the availability of needed natural resources. These studies are useful in understanding the effect of the presence of a slag layer on heat transfer characteristics of a coal-fired magnetohydrodynamic generator. Many oilier similar situations are faced in such industries as food processing, various grinding operations, settling of various liquid contaminants in large storage tanks, waslc processing and recycling and many others. There has been considerable work done on various aspects of free convection flow of a clean fluid over vertical and inclined surfaces. There has been considerable interest in studying flow and heat transfer characteristics of electrically conducting and heat generating/ absorbing fluids (see for instance, Moalem(l), Chakrabarti and Gupta(2); Vajravelu and Nayfeh(3); Chiam(4); Chamkha(5); Chandran et al(6); Vajravelu; Hadjinicalaou(7) and Abdelkhalek(8"15). This interest stems from the fact that hydromagnetic flows and heal '/ansfer have been applied in many industries. For example, in many metallurgical processes such as drawing of continuous filaments through quiescent fluids, and annealing and 'inning of copper wires, the properties of the end product depend greatly on the rate of cooling involved in these processes. The problem of magnetohydrodynamic natural convection about a vertical impermeable flat plate is presented by Sparrow and Cess(16) and Wilks and Hunt(l7). Watanabe and Pop(18) studied the case of a wedge. The laminar plume above a line heat source in a transverse magnetic field was studied by Gray(l9). Kafoussias^20-1 investigated the MI-ID free convective flow through a porous medium over an isothermal cone surface. Balder et al(2l) presented non similar solutions for free convection from a vertical permeable plate in porous media. Similarity solutions of natural convection boundary layers adjacent ic vertical and horizontal surfaces in porous media with internal heat generation were studied by Pop(22). Grubka and Bobba(23) investigated heat transfer characteristics of a continuous, stretching surface with variable wall temperature. Ali(24) numerically presented the heat transfer characteristics of a power law continuous stretched surface without and with suction injection. Takhar et al(25) studied a MHD asymmetric flow past a semi-infinite moving plate and numerically obtained the solutions. Yih(26) studied a free convection effect on MHD coupled heat and mass transfer of a moving permeable vertical surface. In the present paper we studied analytically and numerically the heat and mass transfer in a hydromagnetic flow of a moving permeable vertical surface. The nonlinear-coupled boundary layer equations were transformed and the recaUir.g ordinary differential equations were solved by perturbation technique. The effect of different values of parameters, Prandtl number, suction/blowing parameter, Schmidt number, buoyancy ratio and Hartmann number, on the velocity and temperature profiles as well as skin friction and wall heat transfer are evaluated. FORMULATION OF THE PROBLEM Let us consider two-dimensional free convection effect on the steady incompressible laminar magnetohydrodynamic heat and mass transfer characteristics of a linearly moving permeable vertical surface when the velocity of the fluid iar away from the plate is equal to zero. The variations of surface temperature and concentration are linear. All the fluid properties are assumed to be constant except for the density variations in the buoyancy force term: Introducing the boundary layer and Boussinesq approximations, the governing equations can be written as follows: 186 Proceedings of the Environmental Physics Conference, 24-28 Feb. 2004, Minya, Egypt + 0 (1) d x d y du dv d2u = v P,\c~C,) » (2) ox o y o y~ p p dT dT d2r ... u + v = a (3) dx dy d y" 8C dC d2C u + v = D ^ (4) d x d y d y~ where, u,v is the velocity components in x,y directions, respectively, x,y are the tangential (or radial) distance along the wall and the normal distance, respectively, T is the temperature, <y is the electrical conductivity, Bo is the externally imposed magnetic field in the y- direction, /3T is the volumetric coefficient of thermal expansion, j3c is the volumetric coefficient of expansion with concentration, a is the thermal diffusivity, g is the acceleration due to gravity, D is the coefficient of diffusion in the mixture, C is the species concentration. we assumed the magnetic Reynolds number is small. Therefore, the induced magnetic field effect is negligible compared with the applied magnetic field. The Hall effect, the viscous dissipation and the Joule heating terms are also neglected. The boundary conditions are defined as follows: y = 0, v = -vw, u^Bx, T = Tx+ax, C = Cm+bx (5) y-+co, u = 0, T = Ta, C = Ca (6) where v(1. is the uniform surface mass flux, B, a,b are prescribed constants. The stream function is defined by u = and v = , d y d x Put the non dimensional quantities defined by and substituting Eq. (7) into Eqs. (l)-(6), we obtain /'" + ff" - (M + /')/' - -%•(* + JVC) (8) K t 0 + Pr{f6 -f'6) = O (9) C' + Se(fC'-fC) = 0 (10) The boundary conditions are written as follows, 7 = 0: / = /„., /' = 1, 0 = 1, C = \ 7->oo: /' = 0, 0 = 0, C = 0 The velocity components are: it •= Bxf, v - -(B V)°'* f 187 Proceedings of tins Environmental Physics Conference, 24-28 Feb. 2004, Minva, Eg)'pi where, the primes denote the differentiation with respec!. to //, M = — is the magnetic Bp G T gfiTa G T parameter, - •*•• = —-— is the buoyancy parameter. When —-—• = 0 the governing equations are reduced to forced convection limit. However, as -A—> co free convection is dominated. K The buoyancy ratio N = —— measures the relative importance of mass and thermal diffusion fiTa in the buoyancy driven flow. It is apparent that N is zero for thermal driven flow, infinite for mass driven flow, positive for thermally assisting flow, negative for thermally opposing flow, V V V Pr = — is the Prandtl number, and Sc -— is the Schmidt number, fw = "^-^j is the suction / blowing parameter. For the case of suction, vw > 0 and hence fw > 0 . For the case of blowing, vH, <0 and hence /\<0. The resulting differential equations contain arbitrary parameters, the Prandtl number/^., the magnetic field strength (M) and the buoyancy parameter GrTlR], the Schmidt number (Sc )and buoyancy ratio (N). Solution of the resulting semi-infinite domain, nonlinear equations is accomplished with a three part seder. method(27l The employed power series contains a term K that satisfies the boundary conditions and differential equations at infinity, a second term that satisfies the boundary corHitions at zero and is the solution to the initial homogeneous differential equation, and additional terms that are utilized to obtain increased numerical accuracy.