Andhra Pradesh, India. (Received December 2011 and Accepted July 2012)

Transcription

1 Emirates Journal for Engineering Research, 7 (), 9-4 (0) (Regular Paper) EFFECT OF INTERNAL HEAT GENERATION, SORET AND DUFOUR ON MHD FREE CONVECTION HEAT AND MASS TRANSFER IN A DOUBLY STRATIFIED DARCY POROUS MEDIUM WITH VISCOUS DISSIPATION K.Govardhan, B. Balaswamy, N. Kishan Department of Mathematics, GITAM UNIVERSITY, Hyderabad,A.P.India Department of Mathematics, Osmania University, Hyderabad , Andhra Pradesh, India. govardhan_kmtm@yahoo.co.in. (Received December 0 and Accepted July 0) تم تنفيذ دراسات عددية لدراسة ظاهرة سورت ودوفور ولزج وتبدد اللزوجة وتولد الحرارة الداخلية على الانتقال الثابت للحرارة و المادة بالحمل الحر للمواي ع المغناطيسية من سطح عمودي في وسط طبقات دارسي المسامية المضاعفة. وقد تم تحويل المعادلات التفاضلية الجزي ية غير الخطية والتي تنظم المسا لة قيد النظر الى نظام المعادلات التفاضلية العادية بواسطة تحويل التشابه والتي يتم حلها عدديا باستخدام الفروق المحددة ضمنيا. وقد تم فحص ا ثار العديد من المعاملات المو ثرة في المسا لة على مجال تدفق الماي ع. لقد تم عرض نتاي ج درجة الحرارة Le, M, N, S r, D f, ε, ε والترآيز للجدار والتي تم الحصول عليها لقيم مختلفة من المعاملات, Q The numerical studies are performed to examine the Soret, Dufour and viscous dissipation and internal heat generation effects on steady MHD, free convection heat and mass transfer from a vertical surface in a doubly stratified Darcy porous medium. The non linear partial differential equations, governing the problem under consideration, have been transformed by a similarity transformation into a system of ordinary differential equations, which is solved numerically by using the implicit finite difference scheme. The effects of various parameters entering into the problem have been examined on the flow field. The results for the wall temperature and concentration obtained are presented for various values of the parameters Le M, N, S, D, ε,, Q, ε r f Key words: free convection, porous medium, MHD, Dufour, Soret and viscous dissipation, heat generation, finite difference method.. INTRODUCTION Combined heat and mass transfer by free convection in a porous media has attracted considerable attention in the last several decades, due to its many important engineering and geophysical applications. A comprehensive review on this area have been made by many researchers some of them are Nield and Bejan[]. Several studies have been found to analyze the influence of the combined heat and mass transfer process by natural convection in a thermal and /or mass stratified porous medium, owing to its wide applications, such as development of advanced technologies for nuclear waste management, hot dike complexes in volcanic regions for heating of ground water, separation process in chemical engineering, etc. Here stratified porous medium means that the ambient concentration of dissolved constituent and/or ambient temperature is not uniform and varies as a linear function of vertical distance from the orgin. The importance of the Soret effect at a low Rayleigh number has been analyzed by Bergman and Srinivasan[], while Hurle and Jakerman[3] Studied the thermo solutal convection due to a large temperature gradient. Effect of doubly stratification on free convection in Darcian porous medium have been studied by Murthy et al. [4]. The science of magneto hydrodynamics (MHD) was concerned with geophysical and astrophysical problems for a number of years. In recent years, the possible use of MHD is to affect a flow stream of an electrically conducting fluid for the purpose of thermal protection, braking, propulsion and control. From the point of applications, model studies on the effect of magnetic field on free convection flows have been made by several investigators. The quality of product depends on the rate of heat transfer and therefore cooling procedure has to be controlled effectively. The MHD flow in electrically conducting fluid can control the rate of cooling and the desired quality of product can 9

2 K.Govardhan, B. Balaswamy and N. Kishan be achieved. P.R Sharma and Singh[5] studied the effect of variable thermal conductivity and heat source/sink on MHD flow near a stagnation point on a linearly stretching sheet. When heat and mass transfer occur simultaneously in a moving fluid, the relation 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 gradient 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 gasses with very light molecular weight (H, He) and of medium molecular weight (N, 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, Dursukanya and Worek[7] studied diffusion thermo and thermal diffusion effects in transient and steady natural convection from a vertical surface where as Kafoussias and Williams[8] studied the same effects on mixed free forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Partha et al[9] studied the effect of magnetic field and double dispersion on free convection heat and mass transport considering the Soret and Dufour effects in a non Darcy porous medium. The MHD free- convection and Mass Transfer flow with Hall current, viscous dissipation, Joule heating and thermal diffusion is studied by.singh, A.K[0] Kishan et al.[] studied the MHD free convection flow of an incompressible viscous dissipative fluid in an infinite vertical oscillating plate with constant heat flux. Anghel et al.[] investigated the Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium. Postelnicu[3] 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. Recently, Lakshmi Narayana and Murthy[4] investigated the effects of Soret and Dufour on free convection heat and mass transfer from a vertical surface in a doubly stratified Darcy porous medium. They have neglected effect of MHD, Viscous dissipation and internal heat generation. The objective of the present paper is to study the effect of heat generation, viscous dissipation on free convection heat and mass transfer characteristics in a Darcian, fluid-saturated, doubly stratified porous medium under the influence of transversely applied magnetic field.. CONVERNING EQUATIONS We consider the combined free convection heat and mass transfer from a vertical wall in a doubly stratified, fluid saturated, Darcy porous medium with thermal diffusion (Soret) and diffusion-thermo (Dufour) effects. Viscous resistance due to the solid boundary is neglected under the assumption that the medium has low permeability. The wall heat and mass fluxes are assumed to be constant, and the porous medium is vertically stratified with respect to both temperature and concentration. The non-uniform transverse magnetic field B is imposed along the y - 0 axis. The governing boundary layer equations, along with the equation of continuity, may written as u u υ + = 0 x K σμ e ρ H 0 () = u β (C Kg { β T ( T T )+ C v u T + T υ = T α x y C )} () D k m C C S T P C + u μ + Q ( T T ) 0 u C + C υ = C υ + D k T m T x C C S P Here the boundary conditions T C y = 0 : υ = 0, k = q, D = q w m y : u 0, T = T ( x), C C ( x) = (3) (4) (5) Thermal and solutal stratifications are considered to have the form T + C + ( x) = T Ax,0 3 ( x) = C Bx,,0 3 Where A and B are constants, varied to alter the intensity of stratification in the medium. The subscripts w, (,0) and indicate the conditions at the wall, at some reference point in the medium, and at the outer edge of the boundary layer respectively. Making use of the following similarity transformation derived using order magnitudes. 30 Emirates Journal for Engineering Research, Vol. 7, No., 0

3 Effect of Internal Heat Generation, Soret and Dufour on MHD Free Convection Heat. y Ra x = Ra x (6) x q x w T T ( x) = Ra 3θ ( η) x k q x m C C ( x) = Ra 3φ( η) (7) x k η =, ψ α ( η ) 3 3 f Where the stream function ψ is defined in the usual way u = ψ ψ, υ = x The governing Equations () - (4) become f + M ) = θ + Nφ ε f + f θ fθ ( (8) θ + D f φ = ( ) 3 E c ( f ) Qθ (9) Le φ + S r Leθ = ( ε f + f φ fφ ) (0) 3 and the boundary conditions (5) transform into η = 0; f = 0, θ =, = ; f = 0 θ = 0, φ = 0 η, The Darcy-Rayleigh number ( K ) Ra = g βt qwx x ( αvk) The diffusivity ratio Le = D α. And the buoyancy ratio φ () N = β Cqm k. β C qw D ( N > 0 indicates aiding buoyancy, where both the thermal and solutal buoyancies are in the same directions, and N < 0 indicates opposing buoyancy, where the solutal buoyancy is in the opposite direction to the thermal buoyancy). With specific forms for T (x) and C (x) considered in the present analysis, the thermal and solutal stratification parameters are given by ε = ε = D f S r 3k T Ra x 3 q w x 3k C Ra x 3 q m x = Dk m C S C P α qw = qm Dk C C S Pα q w and both are constants. q is the Dufour parameter, is the Soret parameter. The magnetic parameter K σμ e H 0 M = Ra 3, x ρ q w x The Eckert number μ αk Ec = Ra 3 x q w x The heat generation parameter Q = α Q x 0 ( Ra x 3 ) 3. MATHEMATICALSOLUTION The set of non-linear ordinary differential equations numbers. (8)- (0) with boundary conditions () have been solved numerically, by using Crack Nicolson implicit finite difference method. A step size of Δ η = 0.0 was selected to be satisfactory for 5 a convergence criteria of 0 in all cases. The value of η was found to each iteration loop by the statement η = η + Δ η. In order to see the effect of step size Δ η we ran the code for our model with two different step sizes Δ η = 0.0, Δ η = 0.00 and each case we found very good agreement between them. The convergence is achieved only in 00 iterations. 4. RESULTS AND DISCUSSIONS Numerical calculations carried out for different values of D S, M, N, ε,, E Q. The, ε, f r c computation were carried out by the crack nicolson finite difrence method with the help of C-programing results are good agreement with the previous results[4]. For the purpose of discussing the effect of various parameters on the flow behavior some numerical calculations have been carried out for nondimensional velocity f, temperatureθ, and concentrationφ. In figures and the velocity profiles f are presented. for fixed ε, ε = 0, D f = 0.09, S 0.0, for the = 0 = r aiding buoyancies (N>0)for different parameters M, Le.The non dimensional velocity f decreases with the increasing the value of Le, as indicated in fig.. It is also observed that the non dimensional velocity f decreases with the increases of magnetic parameter M for the aiding buoyancy ( N = ) from the figure. The non dimensional temperature profiles are platted in the figures. (3)- (). For the following range of parameter 0 ε, ε 0.5; 0 Le 5; -0. N and 0 S r, D ƒ <0.5 (to avoid change of sign of Emirates Journal for Engineering Research, Vol. 7, No., 0 3

4 K.Govardhan, B. Balaswamy and N. Kishan temperature concentration value in the boundary layer). From the figure 3 it is observed that the temperature profiles θ increases with the increase of Lewis number Le for aiding buoyancy ( N = ), where as temperature profiles decreases with the increases of magnetic parameter M for aiding buoyancy ( N = ) is shown in fig. 4. In fig.5, temperature profiles for aiding buoyancy ( N = ) when ε < ε and ε > ε for different Soret number S r is shown, from these, increasing in Soret number S r is observed to decreases the temperature distribution when, ε < where as the temperature profiles ε increases with the increase of Soret number when ε > ε. The effect of Lewis number on the temperature profiles are shown in fig6, the temperature profilesθ increases with the increases of Lewis number Le for opposing buoyancy ( N = 0.5).The fig. (7) are plotted for temperature profiles for ( N = 0.5) (opposing buoyancy) for ε > ε. It has been observed that the temperature profiles increases with the increase of S r for ε > ε, fixed D ƒ = With the increasing of Lewis number Le, the temperature profiles ε < ε and ε ε > is noticed decreases for from. It is observed that when the medium is free from the stratification for fixed S, Le the temperature profiles decreases with increases of N, and increasing the Dufour parameter D decreases f the temperature in both aiding and opposing buoyancies as shown in figure 9. From figure 0 we observed that the increasing in viscous dissipation effects is leads to decreases in temperature profiles. From figures., for both aiding and opposing buoyancy, it is observed that the effect of increasing in the internal heat generation parameter is increases in the temperature profiles.the effect of the magnetic parameter increases the concentration profiles for aiding buoyancies is observed from figs.3. From the figure 4. is observed that with the increase of the Le concentration profiles decreases for aiding and buoyancy. Soret effect on the concentration field is shown in figures 5 and 6. With the increasing in the Soret number the concentration profiles increases for ε < ε and ε ε > and ε = ε r is observed from figs. 5 and 6. From figs. 7&8 observed that the effect of internal heat generation parameter on concentration profiles. From both aiding and opposing buoyancies the increasing in the internal heat generation parameter decreases in the concentration profiles. Figure. velocity profiles for N= (aiding buoyancy) 3 Emirates Journal for Engineering Research, Vol. 7, No., 0

5 Effect of Internal Heat Generation, Soret and Dufour on MHD Free Convection Heat. Figure. Velocity profiles for different magnetic parameter for N=(aiding buoyancy) Figure 3.Temperature profiles for N=(aiding buoyancy) Emirates Journal for Engineering Research, Vol. 7, No., 0 33

6 K.Govardhan, B. Balaswamy and N. Kishan Figure 4.Temparature profiles for different magnetic parameter for N=(aiding buoyancy) Figure 5. Temperature profiles for N=(aiding buoyancy) 34 Emirates Journal for Engineering Research, Vol. 7, No., 0

7 Effect of Internal Heat Generation, Soret and Dufour on MHD Free Convection Heat. Figure 6. Temperature profiles for N=-0.5(opposing buoyancy) Figure 7.Temparature profiles for ε ε > for N=-0.5(opposing buoyancy) Emirates Journal for Engineering Research, Vol. 7, No., 0 35

8 K.Govardhan, B. Balaswamy and N. Kishan Figure 8.Temparature profiles for N=-0.5(opposing buoyancy) Figure 9. Variation of temperature both in aiding and opposing buoyancy 36 Emirates Journal for Engineering Research, Vol. 7, No., 0

9 Effect of Internal Heat Generation, Soret and Dufour on MHD Free Convection Heat. Figure 0.Variation of θ for different Ec parameter for N = (aiding buoyancy) Figure.Variation of θ for different Q parameter for N = (aiding buoyancy) Emirates Journal for Engineering Research, Vol. 7, No., 0 37

10 K.Govardhan, B. Balaswamy and N. Kishan Figure. variation of θ for different Q parameter for N = 0. 5 (opposing buoyancy) Figure 3. Concentration profiles for different magnetic parameter N=(aiding buoyancy) 38 Emirates Journal for Engineering Research, Vol. 7, No., 0

11 Effect of Internal Heat Generation, Soret and Dufour on MHD Free Convection Heat. Figure 4. Concentration profile for N=(aiding buoyancy) Figure 5.Concentration profiles for N=-0.5(opposing buoyancy) Emirates Journal for Engineering Research, Vol. 7, No., 0 39

12 K.Govardhan, B. Balaswamy and N. Kishan Figure 6.variation of φ both opposing and aiding buoyancy Figure 7. Variation of φ for different Q parameter for N = (aiding buoyancy) 40 Emirates Journal for Engineering Research, Vol. 7, No., 0

13 Effect of Internal Heat Generation, Soret and Dufour on MHD Free Convection Heat. Figure 8. Variation of φ for different Q parameter for N = 0. 5 (opposing buoyancy) NOMENCLATURE T temparature C - concentration C - specific heat at constant pressure P C - concentration susceptibility S D - Dufour parameter f u,υ -Velocity components in x and y directions respectively v - kinematic viscosity α - thermal diffusivity D solutal diffusivity x, y cartesian co-ordinates ψ stream function g - acceleration qw - constant heat flux qm - constant mass flux Ra - Darcy-Reyleigh number x θ -dimension less temperature φ -dimensionless concentration K - permeability parameter S - Soret parameter r Le - Lewis number M - magnetic parameter Q - internal heat generation parameter E c -Eckert number N - buoyancy ratio βt - coefficient of thermal expansion β C - coefficient of solutal expansion ε - thermal stratification parameter ε - solutal stratification parameter k - thermal diffusion ratio SUBSCRIPTS w - condition at wall - condition at infinity REFERENCES. NieldD.A. and Bejan A. (006).Convection Porous Media. NewYork: Springer.. Bergman T. L. and Srinivasan R. (989). Numerical Solution of Soret Induced Double Diffusion in an Initially Uniform Concentration Binary Liquid. Int. J.Heat Mass Transfer, 3, Hurle D.T. and Jakerman E. (989). Soret Driven Thermo Solutal Convection.J. Fluid Mech,.447, Murthy P.V.S.N. Srinivasacharya D. and Krishna P.V.S.S.S.R.004. Effect of Doubly Stratification on Free Convection in Darcian Porous Medium. Trans. ASME J. Heat Transfer,6, Sharma P.R and Singh, G. (009). Emirates Journal for Engineering Research, Vol. 7, No., 0 4

14 K.Govardhan, B. Balaswamy and N. Kishan 6. Effect of variable thermal conductivity and heat source/sink on MHD flow near a stagnation point on a linearly stretching sheet, Journal of Applied fluid mechanics,,no, Eckert E.R.G. and DrakeR.M. (97). Analysis of Heat and Mass Transfer.- Newyork: McGraw- Hill. 8. Dursukanya Z.and Worek W.M. (99). Diffusion-Thermo and Thermal Diffusion Effects in transient and steady Natural Convection from Vertical Surfaces. Int. J.Heat Mass Transfer, 35 (8), Kafoussias N.G. and Williams E.M. (995). Thermal Diffusion and Diffusion Thermo Effects on Mixed free-forced Convective and Mass Transfer Boundary Layer Flow With Temperature Dependent Viscosity. Int. J. Engng.Sci,33, Partha M. K. Murthy P.V.S.N. and Raja Sekhar G. p. (006). Soret and Dufour Effects in Non- Darcy Porous Media. Trans. ASME J. Heat Transfer, 8, Singh A.K. (003). The MHD Free-Convection and Mass Transfer Flow with Hall Current, Viscous Dissipation, Joule Heating and Thermal Diffusion is Studied. Indian Journal of Pure and Applied Physics,.4, Kishan N.Srihari K.and Anandrao J. (006). MHD Free Convection Flow of an Incompressible Viscous Dissipative Fluid in an Infinite Vertical Oscillating Plate with Constant Heat Flux.- Journal of Energy, Heat and Mass Transfer,8, Anghel, M. Takhar H.S.and Pop I. (000). Dufour and Soret Effects on Free Convection Boundary- Layer Flow over a Vertical Surface Embedded in a Porous Medium. Stud. Univ. Babes. Bolyai Math, 45, Postelnicu A.(004): Influence of a Magnetic Field on Heat and Mass Transfer by Natural Convection from Vertical Surfaces in Porous Media Considering Soret and Dufour Effects.- Int. J.Heat Mass transfer,.47, Lakshmi Narayana P.A. and Murthy P.V.S.N. (007). Soret and Dufour Effects on MHD Free Convection Heat and Mass Transfer in a Doubly Stratified Darcy Porous Medium, Journal of Porous Media 0(6), Emirates Journal for Engineering Research, Vol. 7, No., 0