INFLUENCE OF VARIABLE PERMEABILITY ON FREE CONVECTION OVER VERTICAL FLAT PLATE EMBEDDED IN A POROUS MEDIUM
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1 INFLUENCE OF VARIABLE PERMEABILITY ON FREE CONVECTION OVER VERTICAL FLAT PLATE EMBEDDED IN A POROUS MEDIUM S. M. M. EL-Kabeir and A. M. Rashad Department of Mathematics, South Valley University, Faculty of science, Aswan, Egypt am_rashad@yahoo.com Received 10 Oct 006; accepted 10 Dec 006 ABSTRACT The influence of variable permeability on free convection flow over a vertical flat plate embedded in a fluid saturated porous medium in the presence of heat sources or sinks and with nonlinear density temperature variation is studied in this paper. Solutions for a class of coupled nonlinear equations are obtained by using the fourth-order Runge-Kutta method with shooting technique for the cases of uniform and variable permeability. Numerical results of rate of heat transfer for different parameters such as; variable suction/blowing, variable wall temperature exponent, heat source/sink, nonlinear density temperature variation (NTD) for both of uniform and variable permeability cases are presented in tabular form. Also many results are obtained and representative set is displayed graphically to illustrate the influence of these various parameters on velocity and temperature fields in the uniform and variable permeability cases. Keywords: Free convection, porous medium, variable permeability, nonlinear density temperature (NDT), heat source/sink. Nomenclature: A Constant f Dimensionless stream function f w The suction/blowing parameter g Acceleration due to gravity H Constant based on Rayleigh number K (y) Variable permeability of the porous medium K (η) Non-dimensional variable permeability of the porous medium Nu x Local Nusselt number P Pressure Q Heat source/sink Ra x Rayleigh number T Temperature u, v Darcian velocity component in the x and y directions V w Transpostion velocity x, y Cartesian coordinates
2 Greek symbols: Influence of Variable Permeability on Free Convection Over Vertical Flat Plate 13 α Effective thermal diffusivity α 0 Heat source/sink parameter β 0, β 1 Thermal expansion coefficients of temperature ε Permeability parameter γ Nonlinear density temperature (NDT) parameter λ The range of exponent ρ Fluid density ψ Dimensional stream function η Similarity parameter ν Viscosity of the fluid µ Fluid kinematic viscosity θ Dimensionless temperature function Subscripts w Evaluated on the wall Evaluated at the outer edge of the boundary layer 1 INTRODUCTION Convective heat transfer and fluid flow through porous media has been topic of several studies in resent years. The interest for such studies is motivated by a wide range of thermal engineering applications such as ground water pollution, solid matrix heat exchangers, nuclear waste repositories, geothermal systems, oil extraction and the cooling of electronic components. The most recent books (Nield and Bejan 1998; Vafai 000; Ingham and Pop 1998) present a comprehensive account of the available information on these flows. From a technological point of view free convection study of fluids with known physical properties is always important, for it can reveal hitherto-unknown properties of fluids of practical interest. In such buoyancy driven flows the exact governing equations are unwieldy so that recourse to approximations such as Boussinesq s is generally called for. Assuming the linear density temperature variation of Boussinesq s approximation (Cheng and Minkowycz 1977; Cheng 1977) studied the free convection flow in a saturated porous medium of the constant permeability, and when the temperature difference between the plate and the ambient fluid is appreciably large. The nonlinear density temperature (NDT) variation in the buoyancy force term was examined by Vajravelu and Sastri (Vajravelu and Sastri 1977) in On other hand, convective boundary-layer flows are often controlled by injecting or withdrawing fluid through a porous bounding heated surface. This can lead to enhanced heating or cooling of the system and can help to delay the transition from laminar to turbulent flow. Previous work on the effects of blowing and suction on free convection boundary layers have been confined to cases with a prescribed wall temperature. Eichhorn (Eichhorn 1960), he obtained those power-law variations in surface temperature and transpiration velocity which give rise to a similarity solution for the flow from a vertical surface. The effect of strong suction and blowing from general body shapes, which admit a similarity solution, has been given by Merkin (Merkin 1975). A transformation of the equations for general blowing and wall temperature variations (Vedhanayagam et al. 1980). Lin and Yu (Lin and Yu 1976) have discussed the case of a heated isothermal horizontal surface with transpiration.
3 14 S. M. M. EL-Kabeir and A. M. Rashad However, all the above investigations were carried out for the fluid having uniform permeability of the porous medium. In studying nonuniform permeability effects a simple exponential function of the distance from the wall was usually employed. For the convective boundary layer problems this function usually leads to nonsimilar boundary layer equations, but enforced self-similarity by allowing the region of increased permeability to grow in size in the same way as the boundary layer itself (Cahndrasekhara et al. 1984; Cahndrasekhara et al. 1985). They have incorporated the variable permeability to study the flow past and through a porous medium and have shown that the variation of porosity and permeability has great influence on velocity fields and on rate of heat transfer. Rees and Pop (Rees and Pop 000) has studied how the variable permeability affects the flow and heat transfer from uniform temperature heated surface. Rees and Bassom (Rees and Bassom 1991) have found some exact solutions for free convective flows over semi-infinite surfaces in a porous media. The aim of the present work is to study how an exponentially decaying permeability affects the free convective boundary layer flow induced by a vertical flat plate embedded in a porous medium, and held with the variable parameters such as suction/blowing. The nonlinear density temperature (NDT) variation in the buoyancy force term and the temperature dependent heat source/sink under the condition variable wall temperature are considered. ANALYSIS We consider the free convection flow over a vertical flat plate embedded in a fluid-saturated porous medium of variable permeability and otherwise uniform temperature, T, where the surface temperature of the plate given by T w = T +Ax λ. The Cartesian coordinates x and y are measured along and perpendicular directions of the stream velocity, respectively. The following assumptions are made to obtain the volume averaged conservation equations; 1. The flow is steady, laminar, incompressible and two-dimensional.. The temperature of the fluid is everywhere below the boiling point. 3. The Boussinesq approximation is valid. This approximation is of two parts: a. It approximates the density difference term between the source (or sink) and at infinity, that is we use NDT variation with the equation of the state given by Vajravelu and Sastri (Vajravelu and Sastri 1977). ρ = ρ [1 β ( T T ) β ( T T ) ] (1) 0 1 where ρ and β are respectively, the density and the volumetric coefficient of thermal expansion. b. The volumetric heat source/sink term in the energy term equation depends on the temperature. 4. The fluid and the porous medium are everywhere in local thermodynamic equilibrium.
4 Influence of Variable Permeability on Free Convection Over Vertical Flat Plate It is assumed here that the permeability, K(y) of the porous medium varies as Rees and Pop (Rees and Pop 000): K( y) y / H = K + ( K w 1) e () where K w is the permeability at the wall, K is the permeability of the ambient, and H is constant based on the modified Rayleigh number Ra x. Under these assumptions the governing equations describing problem are given by Cheng (Cheng 1977). u v + x y = 0 (3) K( y) p u = + ρg µ x (4) T u x T + v y T Q = α + ( T T ) (5) y ( ρ C ) p f where u, v are the Darcy velocities in the x, y directions, respectively, p is the pressure, g being the acceleration due gravity. Cp is the specific heat at constant pressure. α is the equivalent thermal diffusivity. T is the temperature of the fluid in the boundary layer region. The boundary conditions for this problem are: at λ ( x), T = T Ax y = 0, u = 0 v = Vw w + (6) ( x) T = at y =, u = 0 v = Vw, T (7) where V w (x) is the transpiration velocity of the fluid through the surface of the plate, which is positive for injection or blowing of the fluid through the plate and negative for suction or withdrawal. If we define a stream functionψ, in the usual way according to u = ψ / y and v = ψ / x. Then a boundary layer transformation now introduced which is very closed related to that of Cheng and Minkowycz (Cheng and Minkowycz 1977) who were the first to consider constant permeability counterpart of the present problem, we set the dimensionless variables as 1/ y T T ψ = αra x f ( η), η = Ra 1/ x, θ (η) =, and the modified Rayleigh number be x T T w Ra x ρ gβ 0K ( T = µα w T ) x (8)
5 16 S. M. M. EL-Kabeir and A. M. Rashad 1/ From equation (), we choose H = xra x such that the non-dimensional permeability is purely function ofη, and given by η [ 1+ ( ε 1 ] K ( η) = K ) e (9) where ε = K w / K being the permeability parameter. Given how naturally occurring and artificial media are packed near solid boundaries we restrict attention to values of ε which are greater than 1. When ε=1, which corresponds to a uniform permeability, and when are greater than 1 corresponds to nonuniform permeability cases. Under this assumption the governing equations and the conditions can be written as η [( ε 1) e + 1] θ 0 f ( 1+ γθ ) = (10) 1+ λ θ + f θ ( λf α 0 ) θ = 0, (11) f ( 0) =, θ (0) = 1 (1) f w f ( ) = 0, θ ( ) = 0 (13) where ε = K w / K is the permeability parameter, γ = β1 ( T w T ) / β 0 is the nonlinear density temperature (NDT) parameter, α 0 = Qx / αra x ( ρ C p ) f is the heat source/sink parameter 1/ and f w = V w ( x) x /(1 + λ) Rax is the suction/blowing parameter which is positive for suction of the fluid and negative for injection or blowing of the fluid. Finally, the main physical quantity of interest in this problem is the rate of heat transfer at plate (local Nusselt number) can be determined from the expression 1/ Nu = θ (0) (14) x Ra x 3 RESULTS AND DISCUSSION The resulting ordinary differential equations (10) to (11) with the corresponding boundary conditions in equations (1) to (13) are solved numerically by using the fourth-order Rung- Kutta method. Calculations were carried out for several values of the permeability parameter ε, the nonlinear density temperature (NDT) variation parameter γ, the heat source/sink parameter α 0, the suction/blowing parameter f w and the range of exponent λ.
6 Influence of Variable Permeability on Free Convection Over Vertical Flat Plate 17 The numerical results of the variation of rate heat transfer of the local Nusselt number corresponding to the indicated values of ε, γ,α 0, f w and λ. are summarized in Tables 1 to 3, for uniform permeability (ε=1.0) and non uniform permeability (ε=.0, 5.0) cases. We point out to the governing results in tables founded by Cheng and Minkowycz (Cheng and Minkowycz 1977) for the absence of (NDT) with constant surface temperature in the uniform permeability case, i.e. ε=1. It is clearly seen that our results in excellent agreement with those known from the open literature. Tables 1 to 3 contain the variation of the rate of heat transfer for uniform and variable permeability cases. In these tables, it can be seen that an increasing of the values of the parameters ε,γ, f w and λ enhances the rate of the heat transfer. Also, it is clear that, the range local Nusselt number reduced as the (NDT) parameter α 0 increases. Table 1: The values of rate of heat transfer in terms of the local Nusselt number Nu Ra 1/ = θ x x (0) for selected values of f w, α 0, λ and γ in the uniform permeability case (ε=1.0). f w α 0 λ γ=-0.5 γ= γ= * *The value found by Cheng and Minkowycz (Cheng and Minkowycz 1977) θ. is ( 0 ) =
7 18 S. M. M. EL-Kabeir and A. M. Rashad Table : The values of rate of heat transfer in terms of the local Nusselt number 1/ Nu xra x = θ (0) for selected values of f w, α 0, λ and γ in the nonuniform permeability case (ε=.0). f w α 0 λ γ=-0.5 γ= γ= Physically it means that as ε increases the rate of the heat transfer increases in magnitude at any chosen values of α 0, γ, f w and λ ; this is because the increased near wall permeability allows the fluid to advocate heat away more quickly than it would for ε=1, there by thinning the boundary layer and increasing the temperature gradient. Now we discuss the effect of the permeability parameter ε, the nonlinear density temperature variation (NDT) parameter γ, the heat source/sink parameter α 0, the suction/blowing parameter f w and the range of exponent λ on the velocity profiles in Figures 1 to 6.
8 Influence of Variable Permeability on Free Convection Over Vertical Flat Plate 19 Table 3. The values of rate of heat transfer in terms of the local Nusselt number 1/ Nu xra x = θ (0) for selected values of f w, α 0, λ and γ in the nonuniform permeability case (ε=5.0). f w α 0 λ γ=-0.5 γ= γ= Figure 1: The effect of variable permeability on velocity and temperature profiles, respectively.
9 0 S. M. M. EL-Kabeir and A. M. Rashad Figure : Velocity and temperature profiles, respectively, for various values of ε, γ with α 0 =-0., f w = and λ =0.. Figure 3: Velocity and temperature profiles, respectively, for various values of f w with α 0 =0., γ =0.5 and λ =0.5 in both of uniform and nonuniform permeability cases. Figure 4: Velocity and temperature profiles, respectively, for various values of α 0 with f w =-0., γ =-0.5 and λ = in both of uniform and nonuniform permeability cases.
10 Influence of Variable Permeability on Free Convection Over Vertical Flat Plate 1 Figure 5: Velocity and temperature profiles, respectively, for various values of λ with f w =0., γ = and α 0 =0. in both of uniform and nonuniform permeability cases. Figure 6: Local Nusselt number for various values of γ and ε. Figure1 illustrates that the effect of variable permeability on velocity and temperature profiles, it can be seen that the velocity of the fluid increases with an increase in the permeability parameter ε, while the temperature decreases with an increasing of the permeability parameter ε. Also, the thermal boundary-layer thickness decreases with an increasing ε. Figures to 5 illustrate that comparison between the effect of the nonlinear density temperature variation (NDT) parameter γ, the heat source/sink parameter α 0, the suction/blowing parameter f w and the range of exponent λ on the velocity an the temperature profiles in uniform and nonuniform permeability cases (when the permeability parameter ε.=1.0 and 5.0), we can observe that the velocity of the fluid increases with an increase in (NDT) parameter γ, and decreases with an increase in suction/blowing parameter f w in both uniform and nonuniform permeability cases.
11 S. M. M. EL-Kabeir and A. M. Rashad It is interesting to note that, for uniform permeability case (ε.=1.0), the velocity profiles increases with an increase in the heat source/sink parameter α 0, while the opposite with an increase in the range of exponent λ. But for nonuniform permeability case (ε.=5.0), the velocity increases with an increase in the range of exponent λ, while the opposite with an increase in the heat source/sink parameter α 0. Also, it is noteworthy that an increase in the heat source/sink parameter α 0 enhances the temperature profiles, while an increase in the values of the parameters γ, f w and λ reduces the temperature profiles in two cases. 4 CONCLUDSIONS In this paper we study an influence of variable permeability on free convection flow over a vertical flat plate embedded in a fluid saturated porous medium in the presence of heat sources or sinks and with nonlinear density temperature variation. The results of this study were in good agreement with previous studies of uniform permeability case. In the variable permeability case, the results show that the rate of heat transfer is much higher than in uniform permeability case. As the permeability parameter ε increases the rate of the heat transfer increases in magnitude at any chosen values of α 0, γ, f w and λ ; this is because the increased near wall permeability allows the fluid to advocate heat away more quickly than it would for uniform permeability case. They re by thinning the boundary layer and increasing the temperature gradient. The velocity profiles increases with an increase in the heat source/sink parameter α 0 for uniform permeability case, but the opposite in nonuniform permeability case. REFERENCES Nield DA, Bejan A (1998). Convection in Porous Media. nd ed., Springer, New York. Vafai ED (000). Hand book of prous media. Begell House, pp Ingham DB and Pop I (1998). Transport Phenomena in Porous Media. Pergamon Press, Oxford. Cheng P and Minkowycz WJ (1977). Free convection about a vertical flat plate imbedded in a porous medium with application to heat transfer from a dike. J. Geophys. Res., 8, pp Cheng P (1977). The influence of lateral mass flux on free convection boundary layers in saturated porous medium. Int. J. Heat Mass Transfer, 0, pp Vajravelu K and Sastri KS (1977). Fully developed laminar free convection flow between two parallel vertical walls. Int. J. Heat Mass Transfer, 0, pp Eichhorn R (1960). The effect of mass transfer on free convection. J. Heat Transfer, 8, pp
12 Influence of Variable Permeability on Free Convection Over Vertical Flat Plate 3 Merkin HJ (1975). The effects of blowing and suction on free convection boundary layers. Int. J Heat Mass Transfer, 18, pp Vedhanayagam M, Altenkirch RA, and Eichhorn R (1980). A transformation of the boundary layer equations for free convection past a vertical flat plate with arbitrary blowing and wall temperature variations. Int. J. Heat Mass Transfer, 3, pp Lin HT and Yu WS (1976). Free convection on a horizontal plate with blowing and suction. Trans ASME J Heat Transfer, 110, pp Chandrasekhara BC, Hamboodiri PMS, and Hanumanthappa AR (1984). Similarity solutions for buoyancy-induced flows in a saturated porous medium adjacent to Impermeable horizontal surfaces. Wärme und Stoffübertr, 18, pp Chandrasekhara BC and Hamboodiri PMS (1985). Influence of variable permeability on combined free and forced convection about inclined surfaces in porous media. Int. J. Heat Mass Transfer, 8, pp Rees DAS and Pop I (000). Vertical free convection in a porous medium with variable permeability effects. Int. J. Heat Mass Transfer, 43, pp Rees DAS and Bassom AP (1991). Some exact solutions for free convective flows over semiinfinite surfaces in porous media. Int. J. Heat Mass Transfer, 34, pp
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