Non-Similar Solutions for Heat and Mass Transfer from a Surface Embedded in a Porous Medium for Two Prescribed Thermal and Solutal Boundary Conditions

Transcription

1 INTERNATIONAL JOURNAL OF CHEMICAL REACTOR ENGINEERING Volume Article A56 Non-Similar Solutions for Heat and Mass Transfer from a Surface Embedded in a Porous Medium for To Prescribed Thermal and Solutal Boundary Conditions Abdelraheem Mahmoud Aly Ali Chamkha South Valley University, abdelraheem 00@yahoo.com The Public Authority for Applied Education and Training, achamkha@yahoo.com ISSN Copyright c 010 The Berkeley Electronic Press. All rights reserved.

2 Non-Similar Solutions for Heat and Mass Transfer from a Surface Embedded in a Porous Medium for To Prescribed Thermal and Solutal Boundary Conditions Abdelraheem Mahmoud Aly and Ali Chamkha Abstract This ork studies the effects of homogeneous chemical reaction and thermal radiation on coupled heat and mass transfer by free convection from a surface embedded in a fluid-saturated porous medium. To different cases of thermal and solutal boundary conditions, namely the prescribed surface temperature and concentration () case and the prescribed heat and mass flues () case are considered. The governing boundary-layer equations are formulated and transformed into a set of non-similar equations. The resulting equations are solved numerically by an accurate and efficient implicit finite-difference method. The obtained results compared ith previously published ork and found to be in ecellent agreement. A representative set of results is displayed graphically to illustrate the influence of the radiation parameter, chemical reaction parameter and the permeability of the porous medium on the velocity, temperature and concentration fields as ell as the local skin-friction coefficient, local Nusselt number and the local Sherood number. KEYWORDS: chemical reaction, radiation, isothermal surface, heat flu, mass flu, adiabatic surface, plane plume, porous medium A. Aly is from the Department of Mathematics, Faculty of Science, South Valley University, Qena.

3 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 1 1. Introduction Simultaneous heat and mass transfer from different geometries embedded in porous media has many engineering and geophysical applications such as geothermal reservoirs, drying of porous solids, thermal insulation, enhanced oil recovery, packed bed catalytic reactors, cooling of nuclear reactors, and underground energy transport. Convective flos in porous media have been etensively eamined during the last several decades due to many practical applications hich can be modeled or approimated as transport phenomena in porous media. Moreover, coupled heat and mass transfer problems in the presence of chemical reactions are of importance in many processes and have, therefore, received considerable amount of attention in recent years. Chemical reactions can occur in processes such as drying, distribution of temperature and moisture over agricultural fields and groves of fruit trees, damage of crops due to freezing, evaporation at the surface of a ater body, energy transfer in a et cooling toer and flo in a desert cooler. Chemical reactions are classified as either homogeneous or heterogeneous processes. This depends on hether they occur at an interface or as a single-phase volume reaction. A homogeneous reaction is one that occurs uniformly throughout a given phase. On the other hand, a heterogeneous reaction takes place in a restricted area or ithin the boundary of a phase. Soundalgekar (1977) presented an eact solution to the flo of a viscous fluid past an impulsively-started infinite plate ith constant heat flu and chemical reaction. The solution as derived by the Laplace transform technique and the effects of heating or cooling of the plate on the flo field ere discussed through the Grashof number. Das, et al. (1994) studied the effects of mass transfer on the flo past impulsively-started infinite vertical plate ith constant heat flu in the presence of a chemical reaction. Muthucumarasamy and Ganesan (1998) solved the problem of unsteady flo past an impulsively-started isothermal vertical plate ith mass transfer numerically by an implicit finite-difference method. Muthucumarasamy and Ganesan (001) and Muthucumarasamy and Ganesan (1999) also considered the problem of unsteady flo past an impulsively-started vertical plate ith uniform heat and mass flu and variable temperature and mass flu, respectively. Diffusion of a chemically-reactive species from a stretching sheet as studied by Andersson, et al. (1994). Anjalidevi and Kandasamy (1999) and Anjalidevi and Kandasamy (000) analyzed the effects of a chemical reaction and heat and mass transfer on laminar flo along a semi-infinite horizontal plate in the presence or absence of a magnetic field. The study of the effect of radiation on MHD flo and heat transfer problem has become more important industrially. At high operating temperatures, Published by The Berkeley Electronic Press, 010

4 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 radiation effects can be quite significant. Many processes in engineering areas occur at high temperatures. a knoledge of the radiative heat transfer becomes very important for design of the pertinent equipment. Nuclear poer plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are eamples of such engineering areas. Bestman (1990) eamined the natural convection boundary layer ith suction and mass transfer in a porous medium. Makinde (005) eamined the transient free convection interaction ith thermal radiation of an absorbing-emitting fluid along a moving vertical permeable plate. Carey and Mollendorf (1978) presented a boundary- layer similarity analysis for the laminar natural convection from a vertical isothermal surface. Their analysis as applicable to liquids herein the viscosity variations are large compared to other fluid properties. Pantokratoras (006) discussed the Falkner-Skan flo ith constant all temperature and variable viscosity. The therrmal radiation effect on natural convection in laminar boundary layer flo ith constant properties over an isothermal flat plate as investigated by Ali, et al. (1984). Again, the radiative mode of heat transfer becomes important hen the temperature difference beteen the plate surface and ambient is large. Yih (001) studied the radiation effect on free convection over a vertical cylinder embedded in porous media. El-Hakiem and Rashad (007) used the Rosseland diffusion approimation in studying the effect of radiation on free convection from a vertical cylinder embedded in a fluid-saturated porous medium. Chamkha and Ben-Nakhi (008) considered MHD mied convection-radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour effects. El-Hakiem (009) studied radiative effects on non-darcy natural convection from a heated vertical plate in saturated porous media ith mass transfer for a non-netonian fluid. Murthy, et al. (005) investigated mied convection heat and mass transfer ith thermal radiation in a non-darcy porous medium. Salem (006) investigated coupled heat and mass transfer in Darcy- Forchheimer mied convection from a vertical flat plate embedded in a fluidsaturated porous medium under the effects of radiation and viscous dissipation. El-Amin, et al. (008) studied the effects of chemical reaction and double dispersion on non-darcy free convection heat and mass transfer. Mansour, et al. (008) analyzed the effects of chemical reaction and thermal stratification on MHD free convection heat and mass transfer over a vertical stretching surface embedded in a porous media considering Soret and Dufour numbers. Postelnicu (007) reported on the influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. The main objective of the present investigation is to study the effects of radiation and chemical reaction on natural convection flos of a Netonian fluid along a vertical surface embedded in a porous medium for to thermal and solutal

5 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 3 boundary conditions, namely the prescribed surface temperature and concentration () condition and the prescribed heat and mass flues () condition. It is assumed that the temperature differences ithin the flo are sufficiently small so that T 4 may be epressed as a linear function of temperature. The value of this ork lies in the use of different transformations for the and cases and obtaining the same non-similar equations for both cases.. Mathematical Analysis Consider steady to-dimensional heat and mass transfer by natural convective flo of a viscous incompressible fluid along a vertical surface of different thermal and solutal boundary conditions embedded in a fluid-saturated porous medium in the presence of a chemical reaction and thermal radiation effects. The flo is taken to be in the direction of -ais, hich is taken along the plate and the y-ais normal to it. The temperature of the quiescent ambient fluid, T at large values of y, is taken to be constant. The fluid properties are assumed to be constant ecept the density in the buoyancy term of the momentum equation and the chemical reaction is assumed to be homogeneous, i.e. taking place in the flo and of first order. Under these assumptions and taking the Boussinesq approimation, the governing equations for the problem under consideration can be ritten as: u + υ = y 0, (1) u u u + υ y u = ν y + gβ T ν () C k ( T T ) + gβ ( C C ) u, 1 T u C u T k T 1 q r + υ =, y ρ C y ρc y (3) P P C C + υ = D k ( C C ), c (4) y y here and y are the coordinate distances along and normal to the surface, respectively. u is the -component of velocity, υ is the y-component of velocity, T is the fluid temperature, C is the solute concentration, ρ is the fluid density, ν is the fluid kinematic viscosity, k 1 is the permeability of the porous medium, Published by The Berkeley Electronic Press, 010

6 4 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 k, C c P and g are the rate of chemical reaction, fluid specific heat at constant pressure and the acceleration due to gravity, respectively. β T is the thermal epansion coefficient, β c is the concentration epansion coefficient, D is the mass diffusion coefficient, C and T are the free stream dimensional concentration and temperature, respectively. q r is the radiative heat flu. The radiative heat flu term is simplified by using the Rosseland approimation [see Sparro and Cess (196)] as q r 4σ 0 = * 3k T y 4, (5) here σ 0 and * k are Stefan-Boltzmann constant and mean absorption coefficient, respectively. For sufficiently small temperature differences and folloing many previous investigators such as Ali et al. (1984), Mansour (1989), Mansour (1990), Perdikis and Raptis (1996), Raptis and Massalas (1998), Elbashbeshy and Bazid (000), Ganesan and Loganathan (00), El-Arabay (003) and Siddheshar and Mahabalesar (005), Equation (5) is linearized using Taylor series epansion about T. The obtained Taylor series epansion for 4 T neglecting higher order terms is given by T = 4T T 3T, (6) Using Equations (5) and (6) in the energy equation (3) e obtain T u T ν 4 T + υ = 1 +, (7) y Pr 3 R y * ρν C here Pr = P k k is the Prandtl number and R = is the dimensionless 3 k 4σ 0 T radiation parameter. The relevant boundary conditions for each of the cases of the problem under consideration are: (a) Prescribed Surface Temperature and Concentration () case: u = 0, υ = 0, T = T (), C = C () at y = 0 T = C u = 0, = T, C as y

7 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 5 (b) Prescribed Heat and Mass Flues () case: T q () C m () u = 0, υ = 0, =, = at y = 0 y k y ρd u = 0, T = T, C = C as y (8) The suitable transformations for the case can be ritten as y ξ =, (, y) Gr 1/ 4 1/ =, ψ, y = νgr T T C C θ ( η, ) =, φ( η, ) =, T T C C Gr 4 η ( ) f ( η,), 3 gβt (T T ) =, ν β C N = β T ( C C ) ( T T ) T = 0 T T, C C K =,, 1 / k 1 Gr C = 0 ν kc Sc =, γ = (9) 1/ D ν Gr hile the suitable transformations for the case are given by y 5 ξ =, (, ) Gr 1 / 5 η y =, ψ (, y) = ν Gr 1/ f ( η, ), q 5 m = Gr 1/ 5 T T θ, C C = Gc 1/ φ, q = q 0, m = m0 k ρd gβt q 4 gβc m 4 Gc 1 Gr =, Gc =, N =, K k ν ρdν Gr k =, ( ) / 5 1 Gr ν k c Sc =, γ = (10) / 5 D ν Gr Substituting Equations (9) or (10) in Equations (1), (), (4) and (7) leads to the same general equation: f f f + f f f Kf +θ+ Nφ ξ f f = 0 (11) ξ ξ 1 Pr θ f θ + f θ f θ ξ f θ = 0 3R ξ ξ (1) Published by The Berkeley Electronic Press, 010

8 6 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 1 φ f φ + f φ f φ γφ ξ f φ = 0 (13) Sc ξ ξ here K and γ are the permeability parameter and chemical reaction parameter, respectively, N is the buoyancy ratio and Sc is the Schmidt number. The dimensionless boundary conditions for each case become: (a) Prescribed Surface Temperature and Concentration () case: f f = = 0, θ = 1, φ = 1 on η = 0 η f = 0, θ = 0, φ = 0 as η (14a) η b) Prescribed Heat and Mass Flues () case: f θ φ f = = 0, = 1, = 1 on η = 0 η η η f = 0, θ = 0, φ = 0 as η (14b) η The above system of non-similar equations (11)-(13) ith the boundary conditions (14) have been solved numerically by using an accurate implicit iterative finite-difference method for various values of parameters. The local skin-friction coefficient, local Nusselt number and the local Sherood number are important parameters for this problem. They are defined for each case as follos: (a) Prescribed Surface Temperature and Concentration () case: The all shear stress and the local skin-friction coefficient C f can be ritten as τ C f u = μ y y= 0 = ρν Gr 3 / 4 f ''( ξ,0) τ Gr 3 / 4 = = f ( ξ,0) or C C Gr 3 / 4 f (,0) f = f = ξ ρν / (15)

9 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 7 The surface heat flu q () and the local Nusselt number Nu are defined by: q T 4σ 0 T * 3 k y ' k 4 [ (,0)]( ) [ ] 1 θ ξ T T Gr 4 1 +, 4 = k = y y= 0 y= 0 3R Nu Nu = q ( T T ) = 1/ 4 Nu Gr = ' θ ( ξ,0) k 4 = ' 1 + θ 3R 4 ( Gr ) 1/ 1, ( ξ,0) 4 + or 3R (16) The surface mass flu m and the local Sherood number Sh are given by: m Sh C = ρd y = ( C C ) m y= 0 = ρ D ' ρd 4 [ φ ( ξ,0)]( C C ) ( Gr ) 1/, ' 4 (,0)( ) 1/ 4 ' = φ ξ Gr, or Sh = Sh Gr 1/ = φ ( ξ,0) (17) b) Prescribed Heat and Mass Flues () case: In a similar ay, the local skin-friction coefficient, local Nusselt number and the local Sherood number for this case take on the respective forms: C f Gr 3 / 5 5 = f ( ξ,0) or C = C Gr 3 / = f ( ξ,0) f f (18) Nu = 5 1 Gr 1/ (,0), θ ξ or Nu = Nu Gr 1/ 5 = θ 1 ( ξ,0) (19) 1/ 5 1 Sh = φ( ξ,0), Gr or Sh = Sh Gr 1/ 5 = φ 1 ( ξ,0) (0) 3. Numerical Method and Validation The heat and mass transfer problem for the surface embedded in a porous medium represented by Equations (11) through (13) are nonlinear and, therefore, must be solved numerically. The standard implicit finite-difference method discussed by Blottner (1970) has proven to be adequate and gives accurate results for such equations. For this reason, it is employed in the present ork and graphical results based on this method are presented subsequently. Published by The Berkeley Electronic Press, 010

10 8 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 Equations (11)-(13) are discretized using three-point central difference formulae ith f ' replaced by another variable V. The η direction is divided into 196 nodal points and a variable step size is used to account for the sharp changes in the variables in the region close to the cylinder surface here viscous effects dominate. The initial step size used is Δη = and the groth factor K = such that Δη n = KΔη n 1 (here the subscript n is the number of nodes minus one). This gives η ma 35 hich represents the edge of the boundary layer at infinity. The ordinary differential equations are then converted into linear algebraic equations that are solved by the Thomas algorithm discussed by Blottner (1970). Iteration is employed to deal ith the nonlinear nature of the governing equations. The convergence criterion employed in this ork as based on the relative difference beteen the current and the previous iterations. When this difference or error reached 10-5, then the solution as assumed converged and the iteration process as terminated. It is possible to compare the results obtained by this numerical method ith the previously published ork of Erbas and Ece (1998) and Ece and Erbas (1999) for free convection flo of poer-la fluids along a vertical plate ith variable all temperature and heat flu. Comparisons ere made for Netonian fluids. Tables 1 and sho the results of these comparisons. It is evident from these comparisons that these results are in ecellent agreement. This lends confidence in the numerical results to be reported net. Table 1. Comparison of values of C f and Nu ith Erbas and Ece (1998) for Pr = 100, K=0, N = 0, R =, δ = 0 and γ = 0 for prescribed surface temperature and concentration () case. C f Nu ξ Erbas and Ece (1998) Present ork Erbas and Ece (1998) Present ork Table. Comparison of values of C f and Nu ith Ece and Erbas (1999) for Pr = 100, K=0, N = 0, R =, δ = 0 and γ = 0 for prescribed heat and mass flues () case. C f 1/Nu ξ Ece and Erbas (1999) Present ork Ece and Erbas (1999) Present ork Results and Discussion In order to get a clear insight of the physical problem, numerical results are displayed ith the help of graphical illustrations. A representative set of results is

11 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 9 shon in Figs Computations ere carried out for various values of physical parameters such as the chemical reaction parameterγ, permeability parameter K and the radiation parameter R. Figures 1-3 sho the influence of the chemical reaction parameter γ on the velocity, temperature and concentration profiles in the boundary layer for both the Prescribed Surface Temperature and Concentration () and the Prescribed Heat and Mass Flu () cases, respectively. For both cases of surface boundary conditions, increasing the chemical reaction parameter produces a decrease in the species concentration. In turn, this causes the concentration buoyancy effects to decrease as γ increases. Consequently, less flo is induced along the plate resulting in decreases in the fluid velocity in the boundary layer. Also, increasing the chemical reaction parameter leads to increase in the temperature profiles. In general, it is also observed that aay from the free stream conditions, the velocity, temperature and concentration profiles are higher for the case than those corresponding to the case. In addition, as epected the all temperature and concentration change as γ changes for the case hile they are fied for the case. These behaviors are clearly depicted in Figs f' γ = 0, 1,, 3, 4, 5 K=1, N=1, Pr=0.7 R=1, Sc= η Fig. 1. Effects of chemical reaction parameter on the velocity profiles Published by The Berkeley Electronic Press, 010

12 10 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A K=1, N=1, Pr=0.7 R=1, Sc=0. θ γ = 0, 1,, 3, 4, η Fig.. Effects of chemical reaction parameter on the temperature profiles K=1, N=1, Pr=0.7 R=1, Sc=0. φ γ = 0, 1,, 3, 4, η Fig. 3. Effects of chemical reaction parameter on the concentration profiles

13 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 11 Figures 4-6 elucidate the influence of the chemical reaction parameter γ on the aial distributions of the local skin-friction coefficient C f, local Nusselt number Nu and the local Sherood number Sh for both the Prescribed Surface Temperature and Concentration () and the Prescribed Heat and Mass Flu () cases, respectively. As seen from Figs. 1-3, the all slope of the velocity profile for the and cases, hich corresponds to the local skinfriction coefficient, decreases as the chemical reaction parameter γ increases. This causes the local skin-friction coefficient C f to decrease for both the and the cases. Similarly, the local Nusselt number (local Sherood number) hich is directly proportional to the all slope of the temperature (concentration) profile for the case and inversely proportional to the all temperature (concentration) for the case are predicted to decrease (increase) as the chemical reaction parameter γ increases. In addition, it is clear that the dependence of all of C f, Nu and Sh on the dimensional aial distance ξ for both the and cases. It should be noted here that for the parametric conditions employed to obtain the results in Figs. 4-6, the values of C f and Sh are higher hile the values of Nu are loer for the case than for the case...0 γ = 0, 1,, 3, 4, C f K=1, N=1, Pr=0.7 R=1, Sc=0. γ = 0, 1,, 3, 4, Fig. 4. Effects of chemical reaction parameter on the local skin-friction coefficient ξ Published by The Berkeley Electronic Press, 010

14 1 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A γ = 0, 1,, 3, 4, 5 Nu K=1, N=1, Pr=0.7 R=1, Sc=0. γ = 0, 1,, 3, 4, Fig. 5. Effects of chemical reaction parameter on the local Nusselt number ξ γ = 0, 1,, 3, 4, Sh K=1, N=1, Pr=0.7 R=1, Sc= Fig. 6. Effects of chemical reaction parameter on the local Sherood number ξ

15 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 13 Figures 7 and 8 display the effects of the thermal radiation parameter R and the permeability parameter K on the velocity and temperature profiles in the boundary layer for the and cases, respectively. Decreasing the thermal radiation parameter R (i.e. increasing radiation effects) produces a significant increase in the thermal state of the fluid causing its temperature to increase. This increase in the fluid temperature induces, through the effect of thermal buoyancy, more flo in the boundary layer causing the velocity of the fluid there to increase. This is true for both the and cases. In addition, the presence of a porous medium presents an obstacle to flo resulting in reduced flo. Therefore, as the permeability parameter K increases, the fluid velocity decreases and the fluid temperature increases for both the and cases. f' K=0 R=1, 5, 10, 0 N=1, Pr=0.7 γ=1, Sc=0. 0. K= η Fig. 7. Effects of radiation and permeability parameters on the velocity profiles Published by The Berkeley Electronic Press, 010

16 14 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 θ K=0 R=1, 5, 10, 0 K= N=1, Pr=0.7 γ=1, Sc= K=0 K= η Fig. 8. Effects of radiation and permeability parameters on the temperature profiles Figures 9 and 10 sho the effects of the thermal radiation parameter R and the permeability parameter K on the aial distributions of the local skin-friction coefficient C f and the local rate of heat transfer or Nusselt number Nu for both the and cases, respectively. It is observed that as the thermal radiation parameter R increases, the local skin-friction coefficient and the local Nusselt number decrease for both the and cases. In addition, it is predicted that the local skin-friction coefficient decreases hile the local Nusselt number increases as the permeability parameter K increases for both the and cases. Again, the values of C f are higher hile the values of Nu are loer for the case than for the case.

17 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer K=0 R=1, 5, 10, 0 C f K= N=1, Pr=0.7 γ=1, Sc=0. K=0 K= Fig. 9. Effects of radiation and permeability parameters on the local skin-friction coefficient ξ 1.1 Nu N=1, Pr=0.7 γ=1, Sc=0. K = 0, R=1, 5, 10, Fig. 10. Effects of radiation and permeability parameters on the local Nusselt number ξ Published by The Berkeley Electronic Press, 010

18 16 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 Figures illustrate the effects of the buoyancy ratio N on the velocity, temperature and concentration profiles for both the and cases, respectively. For N=0 the flo is induced only by the thermal buoyancy effect due to temperature gradient. Hoever, for N>0 the flo is induced by both the temperature and concentration gradients. In this case, the flo is aided by the concentration buoyancy effect. Therefore, as N increases, the flo along the plate increases at the epense of decreased temperature and concentration values. These decreases in both the temperature and concentration values are accompanied by corresponding decreases in both the thermal and solutal (concentration) boundary layer thicknesses. The increased flo is characterized by the formation of distinctive peaks in the velocity profiles in the immediate vicinity of the plate surface as N increases. Hoever, the velocity appear to decrease far donstream as N increases causing decreases in the hydrodynamic boundary layer thickness. The behaviors are true for both the and cases. f' N= 0, 0.5, 1,, 3, 4 K=1, Pr=0.7 R=1, γ=1, Sc= η Fig. 11. Effects of buoyancy parameter on the velocity profiles

19 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 17 θ N= 0, 0.5, 1,, 3, 4 K=1, Pr=0.7 R=1, γ=1, Sc= η Fig. 1. Effects of buoyancy parameter on the temperature profiles φ N= 0, 0.5, 1,, 3, η Fig. 13. Effects of buoyancy parameter on the concentration profiles Published by The Berkeley Electronic Press, 010

20 18 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 Finally, Figs illustrate the influence of the buoyancy ratio N on the distributions of the local skin-friction coefficient C f and the local Nusselt and Sherood numbers Nu and Sh along the plate for both the and cases, respectively. Obviously, increasing the buoyancy ratio has the tendency to increase the flo hich causes more friction and increased tendency for heat and mass transfer. This is depicted in the increases in all of C f, Nu and Sh as N increases in Figs for both the and cases. Again, the values of C f and Sh are higher hile the values of Nu are loer for the case than for the case N= 0, 0.5, 1,, 3, 4 K=1, Pr=0.7 R=1, γ=1, Sc=0. C f ξ Fig. 14. Effects of buoyancy parameter on the local skin-friction coefficient

21 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer N= 0, 0.5, 1,, 3, Nu K=1, Pr=0.7 R=1, γ=1, Sc= Fig. 15. Effects of buoyancy parameter on the local Nusselt number ξ K=1, Pr=0.7 R=1, γ=1, Sc=0. N= 0, 0.5, 1,, 3, 4 Sh ξ Fig. 16. Effects of buoyancy parameter on the local Sherood number Published by The Berkeley Electronic Press, 010

22 0 International Journal of Chemical Reactor Engineering Vol. 8 [010], Article A56 5. Conclusions The problem of natural convective flo along a vertical surface embedded in a porous medium in the presence of homogeneous chemical reaction and thermal radiation effects as investigated. To different cases of thermal and solutal boundary conditions, namely the prescribed surface temperature and concentration () case and the prescribed surface heat and mass flues () case ere studied. The governing boundary-layer equations ere formulated, non-dimensionalized and then transformed into a set of non-similarity equations hich ere solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. It as found that, in general, the fluid velocity, temperature and concentration profiles for the ere predicted to be higher than those for the case. In addition, the local skin-friction coefficient increased as the buoyancy ratio increased and it decreased as a result of increasing either of the chemical reaction parameter, the permeability parameter or the thermal radiation parameter. Moreover, the local Nusselt number increased due to increases in the either of the buoyancy ratio or the permeability parameter hile it decreased as the chemical reaction parameter or the thermal radiation parameter increased. Furthermore, the local Sherood number as predicted to increase as a result of increasing either of the buoyancy ratio or the chemical reaction parameter. Nomenclature C Concentration T Ambient Temperature C p Specific heat diffusivity Horizontal co-ordinate D Mass Diffusivity y Vertical co-ordinate f ' Free stream velocity Gr Local Grashof number Greek Symbols Gc Local modified Grashof number β T Coefficient of thermal epansion g Acceleration due to gravity β C Coefficient of concentration epansion k 1 Permeability of porous medium γ Chemical reaction parameter k Thermal conductivity ν Kinematic viscosity * k Mean absorption coefficient φ Dimensionless concentration m Mass flu θ Dimensionless temperature N Buoyancy ratio ψ Stream function Nu Nusselt number ρ Density of the fluid Pr Prandtl number σ 0 Stefan-Boltzmann constant

23 Mahmoud Aly and Chamkha: Solutions for Heat and Mass Transfer 1 q Local all heat flu Subscripts q r Radiative heat flu Refers to all condition Sc Schmidt number Refers to ambient condition Sh Sherood number Superscript T Temperature ' Differentiation ith respect to η T Wall temperature Local value References Ali, M. M., Chen, T. S. and Armaly, B. F., Natural convection radiation interaction in boundary layer flo over horizontal surfaces, AIAA Journal, 1984,, Andersson, H. I., Hansen, O. R. and Holmedal, B., Diffusion of a chemically reactive species from a stretching sheet, Int. J. Heat Mass Transfer, 1994, 37, Anjalidevi, S. P. and Kandasamy, R., Effects of chemical reaction heat and mass transfer on laminar flo along a semi infinite horizontal plate, Heat Mass Transfer, 1999, 35, Anjalidevi, S. P. and Kandasamy, R., Effects of chemical reaction heat and mass transfer on MHD flo past a semi infinite plate. Z. Ange. Math. Mech., 000, Bestman, A. R., Natural convection boundary layer ith suction and mass transfer in a porous medium, Int J Energy Res., 1990, 14, Blottner, F. G., Finite-difference methods of solution of the boundary-layer equations, AIAA Journal, 1970, 8, Carey, V. P. and Mollendorf, J. C., Natural convection in liquids ith temperature dependent viscosity, Proceedings of the Sith International Heat Transfer Conference, Toronto, Ontario, Canada, 1978, 5, Chamkha, A. J. and Ben-Nakhi, A., MHD mied convection-radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour effects, Heat Mass Transfer, 008, 44, Published by The Berkeley Electronic Press, 010

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