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MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition
International Journal of Thermal Sciences 49 (2010) 1813e1820
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition O.D. Makinde a, *, A. Aziz b a b
Faculty of Engineering, Cape Peninsula University of Technology, P. O. Box 1906, Bellville 7535, South Africa Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 February 2010 Received in revised form 17 April 2010 Accepted 10 May 2010 Available online 12 June 2010
A numerical approach has been used to study the heat and mass transfer from a vertical plate embedded in a porous medium experiencing a first-order chemical reaction and exposed to a transverse magnetic field. Instead of the commonly used conditions of constant surface temperature or constant heat flux, a convective boundary condition is employed which makes this study unique and the results more realistic and practically useful. The momentum, energy, and concentration equations derived as coupled second-order, ordinary differential equations are solved numerically using a highly accurate and thoroughly tested finite difference algorithm. The effects of Biot number, thermal Grashof number, mass transfer Grashof number, permeability parameter, Hartmann number, Eckert number, Sherwood number and Schmidt number on the velocity, temperature, and concentration profiles are illustrated graphically. A table containing the numerical data for the plate surface temperature, the wall shear stress, and the local Nusselt and Sherwood numbers is also provided. The discussion focuses on the physical interpretation of the results as well their comparison with the results of previous studies. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Vertical plate Porous medium Convective boundary condition Heat and mass transfer Magnetic field
1. Introduction The study of hydromagnetic boundary layer flow with heat and mass transfer over a vertical surface embedded in a porous medium is important in many engineering situations such as the concurrent buoyant upward gas-liquid flow in packed bed electrodes [1], sodium oxide-silicon dioxide glass melt flows [2], reactive polymer flows in heterogeneous porous media [3], electrochemical generation of elemental bromine in porous electrode systems [4] and the manufacture of intumescent paints for fire safety applications [5]. A comprehensive survey of magneto-hydrodynamic studies and their technological applications can be found in the book by Moreau [6]. Several interesting computational studies of reactive MHD boundary layer flows with heat and mass transfer have appeared in the recent years [7e12]. Chamkha and Khaled [13] reported similarity solutions for hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media. Merkin and Chaudhary [14] used an asymptotic analysis to study natural convection boundarylayer flow on a vertical surface with exothermic catalytic chemical reaction and concluded that the flow is controlled by the activation energy, the heat of reaction and Prandtl and Schmidt numbers. * Corresponding author. Tel./fax: þ27 21 5550775. E-mail addresses: [email protected] (O.D. Makinde), [email protected] (A. Aziz). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.05.015
Makinde [15] used a shooting numerical technique to analyze MHD boundary-layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux at the plate surface. Except for a few, boundary layer flows have been studied using either a constant surface temperature or a constant heat flux boundary condition. Aziz [16] has recently studied the Blassius flow over a flat plate with a convective thermal boundary condition and established the condition which the convection heat transfer coefficient must meet for a similarity type solution to exist. In a contemporaneous study, Bataller [17] studied the boundary layer flow over a convectively heated flat plate with a radiation term in the energy equation. Because the convective boundary condition is more general and realistic especially with respect to several engineering and industrial processes like transpiration cooling process, material drying, etc., it seems appropriate to use the convective boundary condition to study other boundary layer flow situations. This paper considers MHD mixed convection from a vertical plate with heat and mass transfer and a convective boundary condition at the plate. It is assumed that the plate is embedded in a uniform porous medium and is exposed to a transverse magnetic field. The problem is solved numerically and results are presented for the velocity, temperature, and concentration profiles together with the local skin friction, the plate surface temperature and the local heat and mass transfer rates. The results are believed to be applicable to realistic engineering situations cited earlier.
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Nomenclature (u; v) g K Cp T TN Tf Gc M Pr Sh Bi k (u, v) D v0 CN
Cw B0 Gr U Ec Sc v0 (x,y)
velocity-components gravitational acceleration permeability parameter specific heat at constant pressure fluid temperature free stream temperature hot fluid temperature mass transfer Grashof number magnetic field parameter Prandtl number Sherwood number Biot number thermal conductivity dimensionless velocity-components mass diffusivity wall suction velocity free stream concentration
Greek symbols dimensionless temperature cimensionless concentration thermal expansion coefficient electrical conductivity concentration expansion coefficient fluid density fluid kinematic viscosity reaction rate parameter reaction rate coefficient
q f b s b r y l g
2. Mathematical model
v0
We consider a steady, laminar, hydromagnetic coupled heat and mass transfer by mixed convection flow of a cold fluid at temperature TN over an infinite vertical plate embedded in a porous medium. It is assumed that the left surface of the plate is heated by convection from a hot fluid at temperature Tf which provides a heat transfer coefficient hf. The cold fluid on the right side of the plate is assumed to be Newtonian and electrically conducting. Except for the density, all other fluid properties are assumed to be independent of temperature and chemical species concentration. A uniform magnetic field of strength B0 is imposed normal to the plate (along the y-axis) as shown in Fig. 1. Since the magnetic Reynolds number is very small for most fluid used in industrial applications, we assume that the induced magnetic field is negligible. Under the Boussinesq and boundary-layer approximations, the momentum, energy balance and concentration equations can be written in as [10,15e17],
v0
vu v2 u ¼ y 2 þ g bðT T N Þ þ g bðC C N Þ vy vy ð1Þ
y vu 2 sB20 vT k v2 T þ þ ðU uÞ2 ; ¼ rCp vy2 Cp vy rCp vy
hf , T f
vC v2 C ¼ D 2 gðC C N Þ; vy vy
(2)
u
B0
(3)
where x and y are, respectively, the directions along and perpendicular to the surface, u is the velocity component along the plate, U is the uniform free-stream velocity, v0 is the constant suction velocity, g is the acceleration due to gravity, r; y; k and s are, the fluid density, kinematic viscosity, the thermal conductivity and the electrical conductivity, respectively. The local temperature is T, C is the species concentration and the subscript N denotes free-stream conditions. The other parameters are the coefficient of volume expansion for heat transfer b, the coefficient of volumetric expansion with respect to species concentration b, the molecular diffusivity D, and the reaction rate coefficient g. The appropriate boundary conditions for this flow are
u ¼ 0; u/U;
sB2 y þ ðU uÞ þ 0 ðU uÞ; r K v0
concentration at plate surface magnetic field strength thermal Grashof number free stream velocity Eckert number Schmidt number constant wall suction coordinates
vT k ¼ hf T f T ; vy T/T N ;
C ¼ Cw;
at
y ¼ 0;
C/C N ; as y/N;
(4)
(5)
where the convective heating process at the plate is characterized by the hot fluid temperature T f and the heat transfer coefficient hf. Introducing the following dimensionless quantities,
y u U T TN C CN ; U¼ ; q¼ ; f¼ ; Sc ¼ ; v0 v0 D T T C C N w N f gb T f T N y ryCp gy2 g bðC w C N Þy ; Pr ¼ ; l ¼ 2; Gr ¼ ; Gc ¼ 3 3 k v0 v0 Dv0 sB20 y v20 K v20 hf y ; Bi ¼ ; K ¼ 2 ; Ec ¼ M¼ : kv0 y rv20 CP T f T N y¼
v0 y
y
; u¼
(6)
Eqs. (1)e(5) may be written in dimensionless form as follows.
x
v0
Porous medium v T ,C ,u T∞ , c ∞ ,U g
y Fig. 1. Flow configuration and coordinate system.
du d2 u Uu þ Gr q þ Gcf þ MðU uÞ þ ¼ ; dy K dy2
(7)
2 dq 1 d2 q du þ Ec þMEcðU uÞ2 ; ¼ 2 Pr dy dy dy
(8)
O.D. Makinde, A. Aziz / International Journal of Thermal Sciences 49 (2010) 1813e1820
df 1 d2 f l f; ¼ Sc dy2 Sc dy u ¼ 0;
dq ¼ Bi½q 1; f ¼ 1; dy
u ¼ U; q ¼ f ¼ 0 as y/N;
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(9)
at y ¼ 0;
(10) (11)
where Bi is the Biot number, Ec is the Eckert number, Gc and Gr are the mass transfer and thermal Grashof numbers, respectively, K is the porosity parameter, M is the Hartmann number, l is the reaction parameter, Pr is the Prandtl number and Sc is the Schmidt number. The quantities u, q, f represent the dimensionless velocity, dimensionless temperature, and dimensionless concentration, respectively. Here, we emphasis that the Biot number (Bi) is the ratio of the internal thermal resistance of the plate to the boundary layer thermal resistance. When Bi ¼ 0, the left side of the plate with hot fluid is totally insulated and no convective heat transfer to the cold fluid on the right side takes place. The set of equations (7)e(9) under the boundary conditions (10) and (11) were solved numerically by applying the Nachtsheim and Swigert [18] shooting iteration technique together with a sixth-order RungeeKutta integration algorithm. Besides the velocity, temperature, and concentration data, the numerical solution also yielded the values of q(0), 0 u0 (0),q ð0Þ and f0 ð0Þ which are, respectively, proportional to the plate surface temperature, the local skin-friction coefficient, the local Nusselt number and the local Sherwood number. 3. Results and discussion The numerical computations were carried out for positive values of the buoyancy parameters Gr > 0 (which corresponds to the cooling problem) and Gc > 0 (which indicates that the free stream concentration is less than the concentration at the plate surface). The cooling problem is often encountered in engineering applications such as the cooling of electronic components and nuclear reactors. The Prandtl number was fixed at Pr ¼ 0.72 which corresponds to air. The values of Schmidt number (Sc) were chosen to be Sc ¼ 0.24, 0.62, 0.78, 2.62 which represent, respectively, the diffusion of H2, H2O, NH3, and propyl benzene in air. The computations have been carried out for a range of values of the other parameters appearing in the problem.
Fig. 2. Velocity profiles for Sc ¼ 0.62, Gr ¼ Gc ¼ K ¼ 1, Ec ¼ M ¼ l ¼ 0.1; _____Bi ¼ 0, ooooBi ¼ 0.1, þþþBi ¼ 0.5, ..Bi ¼ 1, ****Bi ¼ 10.
convective process at the plate. The effect of the permeability parameter K on the velocity profiles is illustrated in Fig. 3. As the permeability of the medium increases, it enhances the upward flow profoundly and shifts the location of the peak velocity further away from the plate. Fig. 4 shows how the velocity profile is affected by the Eckert number. Since the Eckert number by definition is proportional to the square of the suction velocity v0, Fig. 4 also implicitly depicts the effect of the suction velocity on the upward flow in the boundary layer. The peak velocity is seen to increase as the suction velocity increases but its location is hardly affected by the increase in the suction velocity. The effect of the suction on the
3.1. Velocity profiles The vertical or the x-component of the velocity, u, profiles are displayed in Figs. 2e9 and follow the pattern characteristic of natural convection boundary layer flow over a vertical plate. The velocity increases from a zero value at the plate to a peak value within the boundary layer and then decreases gradually to the specified freestream velocity value of U ¼ 0.5. It is worth mentioning that the Biot number (Bi) is the ratio of the internal thermal resistance of a solid to the boundary layer thermal resistance. When Bi ¼ 0 (i.e. without Biot number) the left side of the plate with hot fluid is totally insulated, the internal thermal resistance of the plate is extremely high and no convective heat transfer to the cold fluid on the right side of the plate takes place. Fig. 2 shows the velocity profile with or without Biot number. It is interesting to note that without Biot number (i.e. Bi ¼ 0) the peak velocity is low. As the convection Biot number increases, the plate thermal resistance reduces. Consequently, the peak velocity and the velocities in the neighbourhood of the peak increase significantly. This effect is triggered by the stronger buoyancy forces induced as a result of the increase in the strength of the
Fig. 3. Velocity profiles for Sc ¼ 0.62, Gr ¼ Gc ¼ 1, Bi ¼ Ec ¼ M ¼ l ¼ 0.1; _____K ¼ 0.1, ooooK ¼ 1, þþþK ¼ 5, ..K ¼ 10.
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Bi ¼ M ¼ l ¼ 0.1;
Fig. 6. Velocity profiles for Sc ¼ 0.62, K ¼ Gr ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Gc ¼ 0.1, ooooGc ¼ 1, þþþGc ¼ 2, ..Gc ¼ 3.
velocity profile is not as pronounced as the effect of permeability of the medium as observed in Fig. 3. The effect of thermal Grashof number on the velocity distribution can be gleaned from Fig. 5. Because the thermal Grashof number is the ratio of buoyancy to viscous forces in the boundary layer, the increase in its value suggests an increase in the buoyancy forces relative to the viscous forces and this is clearly reflected in the progressive increase in the vertical (upward) velocity of the flow. The increase in mass transfer Grashof number produces a similar effect on the velocity profiles as
shown in Fig. 6. It is interesting to note that for Gc ¼ 3, that is, for high concentration between the wall and the freestream, the vertical component of the velocity in a portion of the boundary layer is larger than unity which implies that the vertical velocity exceeds the wall suction velocity. Attention is now turned to the effects of the Hartmann number M, the Schmidt number Sc, and the reaction parameter l. Fig. 7 shows that as the strength of the magnetic field increases, i.e. M increases, it causes a decrease in the vertical (upward) velocity in the boundary
Fig. 5. Velocity profiles for Sc ¼ 0.62, K ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Gr ¼ 0.1, ooooGr ¼ 1, þþþGr ¼ 3, ..Gr ¼ 5.
Fig. 7. Velocity profiles for Sc ¼ 0.62, K ¼ Gr ¼ Gc ¼ 1, Bi ¼ Ec ¼ l ¼ 0.1; _____M ¼ 0.1, ooooM ¼ 0.5, þþþM ¼ 1, ..M ¼ 2.
Fig. 4. Velocity profiles for Sc ¼ 0.62, K ¼ 1, Gr ¼ Gc ¼ 0.5, _____Ec ¼ 0.1, ooooEc ¼ 0.5, þþþEc ¼ 1, ..Ec ¼ 1.5.
O.D. Makinde, A. Aziz / International Journal of Thermal Sciences 49 (2010) 1813e1820
Fig. 8. Velocity profiles for K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Sc ¼ 0.24, ooooSc ¼ 0.62, þþþSc ¼ 0.78, ..Sc ¼ 2.62.
layer. The vertical velocity also decreases when the Schmidt number Sc is increased as illustrated in Fig. 8. Since Sc is proportional to the density of the diffusing species, the diffusion of heavier species in air tends to retard the upward flow velocity. The increase in the reaction parameter l also tends to reduce the vertical velocity as can be seen in Fig. 9. Thus the effect of increase in the values of M, Sc and l (Figs. 7e9) is to reduce the upward flow velocity while the increase in the values of Bi, K, Ec, Gr, Gc promotes an increase in the vertical velocity (Figs. 2e6). The results presented in Figs. 2e9 and the
Fig. 9. Velocity profiles for Sc ¼ 0.62, K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ 0.1; ____l ¼ 0.1, ooool ¼ 0.5, þþþl ¼ 1, ..l ¼ 5.
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Fig. 10. Temperature profiles for Sc ¼ 0.62, Gr ¼ Gc ¼ K ¼ 1, _____Bi ¼ 0, ooooBi ¼ 0.1, þþþBi ¼ 0.5, ..Bi ¼ 1, ****Bi ¼ 10.
Ec ¼ M ¼ l ¼ 0.1;
conclusions drawn here for MHD flow with a convective surface boundary condition are in agreement with the ones reported in [8,10,17] for MHD flows with constant surface temperature or constant surface heat flux type boundary conditions. 3.2. Temperature profiles The temperature profiles corresponding to the velocity profiles of Figs. 2e9 are provided in Figs. 10e16. In each figure, the maximum temperature occurs at the plate and decreases to reach
Fig. 11. Temperature profiles for Sc ¼ 0.62, _____K ¼ 0.1, ooooK ¼ 1, þþþK ¼ 5, ..K ¼ 10.
Gr ¼ Gc ¼ 1,
Bi ¼ Ec ¼ M ¼ l ¼ 0.1;
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Fig. 12. Temperature profiles for Sc ¼ 0.62, K ¼ 1, Gr ¼ Gc ¼ 0.5, Bi ¼ M ¼ l ¼ 0.1; _____Ec ¼ 0.1, ooooEc ¼ 0.5, þþþEc ¼ 1, ..Ec ¼ 1.5.
Fig. 14. Temperature profiles for Sc ¼ 0.62, _____Gc ¼ 0.1, ooooGc ¼ 1, þþþGc ¼ 2, ..Gr ¼ 3.
the freestream temperature (q ¼ 0) at the edge of the thermal boundary layer. Both the plate surface temperature and the thickness of the thermal boundary thickness increase as the parameters Bi, K, Ec, Gr, Gm increase (Figs. 9e14). Moreover, it is interesting to note that the fluid temperature on the right side of the plate increases with an increase in the Biot number, since as Bi increases, the thermal resistance of the plate decreases and convective heat transfer to the fluid on the right side of the plate increases. Compared with the other parameters, the effect of permeability parameter K on the temperature profile appears to be small (See
Fig. 11). The effect of increase in Sc (Fig. 15) and l (Fig. 16), however, is to decrease the plate surface temperature as well the thickness of the thermal boundary layer.
M ¼ Bi ¼ Ec ¼ l ¼ 0.1;
Fig. 15. Temperature profiles for K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Sc ¼ 0.24, ooooSc ¼ 0.62, þþþSc ¼ 0.78, ..Sc ¼ 2.62.
Fig. 13. Temperature profiles for Sc ¼ 0.62, _____Gr ¼ 0.1, ooooGr ¼ 1, þþþGr ¼ 3, ..Gr ¼ 5.
K ¼ Gc ¼ 1,
K ¼ Gr ¼ 1,
M ¼ Bi ¼ Ec ¼ l ¼ 0.1;
3.3. Concentration profiles The dependence of the concentration profile on Sc is illustrated in Fig. 17. For each value of Sc, the dimensionless concentration decreases from a value of one at the plate to a value of zero at the edge of the concentration boundary layer. As Sc increases, the
O.D. Makinde, A. Aziz / International Journal of Thermal Sciences 49 (2010) 1813e1820
Fig. 16. Temperature profiles for Sc ¼ 0.62, ____l ¼ 0.1, ooool ¼ 0.5, þþþl ¼ 1, ..l ¼ 5.
K ¼ Gr ¼ Gc ¼ 1,
M ¼ Bi ¼ Ec ¼ 0.1;
species concentration drops more rapidly as the thickness of the concentration boundary layer decreases. Since the increase in Sc implies a decrease in the molecular diffusivity D, the concentration decay over a shorter distance is understandable. Similarly, one would anticipate that the heavier species (higher Sc) would have a shorter penetration depth (thinner boundary layer). Fig. 18 shows the effect of reaction rate parameter l on the concentration profiles. The increase in the reaction rate parameter means an increase in the consumption of the species and hence a reduction in the
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Fig. 18. Concentration profiles for Sc ¼ 0.62, K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ 0.1; ____l ¼ 0.1, ooool ¼ 0.5, þþþl ¼ 1, ..l ¼ 5.
general concentration through the boundary layer and the depletion of concentration over a shorter distance as predicted by Fig. 18. 3.4. Skin friction, heat transfer, mass transfer parameters and surface temperature 0
The quantities u0 ð0Þ; q ð0Þ; f0 ð0Þ are measures of the skin friction, heat transfer, and mass transfer at the plate, respectively, while the quantity q(0) gives the plate surface temperature. Table 1 records the values of these quantities for a range of values of the parameters Bi, Sc, Ec, M, K and l with Pr ¼ 0.72, Gr ¼ 0.1, Gc ¼ 0.1 and U ¼ 0.5. The skin friction increases as the Biot number Bi, the Eckert number Ec, M increase but decreases when Sc, K and l increase. The heat transfer rate from the plate increases with the increase in values of Bi, Sc, K, and l but decreases with the increase in Ec and M. For Bi ¼ 0, no convective heat transfer takes place and the rate of heat transfer at the plate surface is zero as expected. The surface mass transfer is seen to increase with the increase in Sc and l. Finally, the plate temperature increases as Bi increases very significantly as the Bi increases from 0.1 to 10. For Bi ¼ 0.1, the plate temperature is hardly Table 1 Values of f0 ð0Þ, u0 (0), q0 (0)and q(0) with Gr ¼ Gc ¼ 0.1, Pr ¼ 0.72, U ¼ 0.5.
Fig. 17. Concentration profiles for K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Sc ¼ 0.24, ooooSc ¼ 0.62, þþþSc ¼ 0.78, ..Sc ¼ 2.62.
Bi
Sc
Ec
M
K
l
f0 ð0Þ
u0 (0)
q0 (0)
q(0)
0 0.1 1.0 10 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.24 0.24 0.24 0.24 0.62 0.78 2.62 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24
0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 2.0 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 10 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 10 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 2.0
0.459012 0. 459012 0.459012 0.459012 0.752957 0.892139 2.657627 0.459012 0.459012 0.459012 0.459012 0.459012 0.459012 1.127174 1.539295
1.891555 1.894933 1.907655 1.917383 1.892262 1.891148 1.882032 1.907091 1.920915 1.962039 2.526743 0.930812 0.759132 1.889451 1.886928
0.000000 0.083565 0.398302 0.638954 0.083583 0.083590 0.083641 0.044871 0.000865 0.083172 0.080068 0.085574 0.085843 0.083600 0.083615
0.048228 0.164348 0.601697 0.936104 0.164168 0.164096 0.163589 0.551288 0.991347 0.168276 0.199315 0.144253 0.141569 0.163991 0.163843
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affected by the changes in Sc. In other words, the molecular diffusivity or the density of the diffusing species has very little effect on the plate temperature. The increase in viscous dissipation (Ec) by a factor of 2 elevates the plate from a value of 0.551288 to 0.991347. The strength of the magnetic field (M) has slight impact on the plate temperature. For example, when the magnetic field is increased by a factor of 10, the plate temperature increases from 0.168276 to 0.199315. Similarly, the impact of the permeability of the medium (K) on the plate temperature is small. An increase in permeability parameter from 0.1 to 10 lowers the plate temperature from 0.199315 to 0.141569. The reaction rate (l) barely affects the plate temperature as can be seen from the last two entries in Table 1. 4. Conclusion The influence of convective boundary condition on hydromagnetic mixed convection with heat and mass transfer past a vertical plate embedded in a porous medium is numerically studied in this work. The left surface of the plate is heated by convection from a hot fluid while the cold fluid on the right side of the plate is assumed to be Newtonian and electrically conducting. A table containing the numerical data showing the effects of various parameters controlling the system on the plate surface temperature, the wall shear stress, and the local Nusselt and Sherwood numbers is also provided. Our results reveal among others that both the fluid velocity and temperature increase with an increase in the convective heat transfer parameter i.e. the Biot Number (Bi). Physical significance and application of Biot number with respect to boundary layer flow problems can be found in several engineering and industrial processes such as drying of material, transpiration cooling, etc. Acknowledgements The authors gratefully acknowledge the valuable suggestions of the anonymous referees. ODM would like to thank the National Research Foundation of South Africa Thuthuka programme for financial support. References [1] K. Takahashi, R. Alkire, Mass transfer in gas-sparged porous electrodes. Chemical Engineering Communications 38 (1985) 209e227.
[2] Y.A. Guloyan, Chemical reactions between components in the production of glass-forming melt. Glass and Ceramics 60 (2003) 233e235. [3] H. Liu, K.E. Thompson, Numerical modelling of reactive polymer flow in porous media. Computers and Chemical Engineering 26 (2002) 1595e1610. [4] J. Qi, R.F. Savinell, Analysis of flow-through porous electrode cell with homogeneous chemical reactions: application to bromide oxidation in brine solutions. Journal of Applied Electrochemistry 23 (1993) 873e886. [5] P. Ducrocq, S. Duquesne, S. Magnet, S. Bourbigot, R. Delobel, Interactions between chlorinated paraffins and melamine in intumescent paint- investing a way to suppress chlorinated paraffins from the formulations. Progress in Organic Coatings 57 (2006) 430e438. [6] R. Moreau, Magnetohydrodynamics. Kluwer Academic Publishers, Dordrecht, 1990. [7] O.D. Makinde, Thermal stability of a reactive viscous flow through a poroussaturated channel with convective boundary conditions. Applied Thermal Engineering Vol. 29 (2009) 1773e1777. [8] O.D. Makinde, P. Sibanda, Magnetohydrodynamic mixed convective flow and heat and mass transfer past a vertical plate in a porous medium with constant wall suction. ASME e Journal of Heat Transfer 130 (2008) 11260 (1e8). [9] O.D. Makinde, A. Ogulu, The effect of thermal radiation on the heat and mass transfer flow of a variable viscosity fluid past a vertical porous plate permeated by a transverse magnetic field. Chemical Engineering Communications 195 (12) (2008) 1575e1584. [10] R. Rajeswari, B. Jothiram, V.K. Nelson, Chemical reaction, heat and mass transfer on nonlinear MHD boundary layer flow through a vertical porous surface in the presence of suction. Applied Mathematical Sciences 3 (50) (2009) 2469e2480. [11] O.A. Bég, J. Zueco, R. Bhargava, H.S. Takhar, Magnetohydrodynamic convection flow from a sphere to a non-Darcian porous medium with heat generation or absorption effects: network simulation. International Journal of Thermal Sciences 48 (2009) 913e921. [12] S. Ahmad, N.M. Arifin, R. Nazar, I. Pop, Mixed convection boundary layer flow past an isothermal horizontal circular cylinder with temperature-dependent viscosity. International Journal of Thermal Sciences 48 (2009) 1943e1948. [13] A.J. Chamkha, A. Abdul-Rahim Khaled, Hydromagnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid saturated porous medium. International Journal of Numerical Methods of Heat & Fluid Flow 10 (5) (2000) 455e476. [14] J.H. Merkin, M.A. Chaudhary, Free-convection boundary layers on vertical surfaces driven by an exothermic surface reaction. Quarterly Journal of Mechanics and Applied Mathematics 47 (1994) 405e428. [15] R.C. Bataller, Radiation effects for the Blassius and Sakiadis flows with a convective surface boundary condition. Applied Mathematics and Computation 206 (2008) 832e840. [16] A. Aziz, A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1064e1068. [17] O.D. Makinde, On MHD boundary-layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux. International Journal of Numerical Methods for Heat & Fluid Flow 19 (3/4) (2009) 546e554. [18] P.R. Nachtsheim, P. Swigert, Satisfaction of the asymptotic boundary conditions in numerical solution of the system of nonlinear equations of boundary layer type, NASA TND-3004 (1965).
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition O.D. Makinde a, *, A. Aziz b a b
Faculty of Engineering, Cape Peninsula University of Technology, P. O. Box 1906, Bellville 7535, South Africa Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 February 2010 Received in revised form 17 April 2010 Accepted 10 May 2010 Available online 12 June 2010
A numerical approach has been used to study the heat and mass transfer from a vertical plate embedded in a porous medium experiencing a first-order chemical reaction and exposed to a transverse magnetic field. Instead of the commonly used conditions of constant surface temperature or constant heat flux, a convective boundary condition is employed which makes this study unique and the results more realistic and practically useful. The momentum, energy, and concentration equations derived as coupled second-order, ordinary differential equations are solved numerically using a highly accurate and thoroughly tested finite difference algorithm. The effects of Biot number, thermal Grashof number, mass transfer Grashof number, permeability parameter, Hartmann number, Eckert number, Sherwood number and Schmidt number on the velocity, temperature, and concentration profiles are illustrated graphically. A table containing the numerical data for the plate surface temperature, the wall shear stress, and the local Nusselt and Sherwood numbers is also provided. The discussion focuses on the physical interpretation of the results as well their comparison with the results of previous studies. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Vertical plate Porous medium Convective boundary condition Heat and mass transfer Magnetic field
1. Introduction The study of hydromagnetic boundary layer flow with heat and mass transfer over a vertical surface embedded in a porous medium is important in many engineering situations such as the concurrent buoyant upward gas-liquid flow in packed bed electrodes [1], sodium oxide-silicon dioxide glass melt flows [2], reactive polymer flows in heterogeneous porous media [3], electrochemical generation of elemental bromine in porous electrode systems [4] and the manufacture of intumescent paints for fire safety applications [5]. A comprehensive survey of magneto-hydrodynamic studies and their technological applications can be found in the book by Moreau [6]. Several interesting computational studies of reactive MHD boundary layer flows with heat and mass transfer have appeared in the recent years [7e12]. Chamkha and Khaled [13] reported similarity solutions for hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media. Merkin and Chaudhary [14] used an asymptotic analysis to study natural convection boundarylayer flow on a vertical surface with exothermic catalytic chemical reaction and concluded that the flow is controlled by the activation energy, the heat of reaction and Prandtl and Schmidt numbers. * Corresponding author. Tel./fax: þ27 21 5550775. E-mail addresses: [email protected] (O.D. Makinde), [email protected] (A. Aziz). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.05.015
Makinde [15] used a shooting numerical technique to analyze MHD boundary-layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux at the plate surface. Except for a few, boundary layer flows have been studied using either a constant surface temperature or a constant heat flux boundary condition. Aziz [16] has recently studied the Blassius flow over a flat plate with a convective thermal boundary condition and established the condition which the convection heat transfer coefficient must meet for a similarity type solution to exist. In a contemporaneous study, Bataller [17] studied the boundary layer flow over a convectively heated flat plate with a radiation term in the energy equation. Because the convective boundary condition is more general and realistic especially with respect to several engineering and industrial processes like transpiration cooling process, material drying, etc., it seems appropriate to use the convective boundary condition to study other boundary layer flow situations. This paper considers MHD mixed convection from a vertical plate with heat and mass transfer and a convective boundary condition at the plate. It is assumed that the plate is embedded in a uniform porous medium and is exposed to a transverse magnetic field. The problem is solved numerically and results are presented for the velocity, temperature, and concentration profiles together with the local skin friction, the plate surface temperature and the local heat and mass transfer rates. The results are believed to be applicable to realistic engineering situations cited earlier.
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Nomenclature (u; v) g K Cp T TN Tf Gc M Pr Sh Bi k (u, v) D v0 CN
Cw B0 Gr U Ec Sc v0 (x,y)
velocity-components gravitational acceleration permeability parameter specific heat at constant pressure fluid temperature free stream temperature hot fluid temperature mass transfer Grashof number magnetic field parameter Prandtl number Sherwood number Biot number thermal conductivity dimensionless velocity-components mass diffusivity wall suction velocity free stream concentration
Greek symbols dimensionless temperature cimensionless concentration thermal expansion coefficient electrical conductivity concentration expansion coefficient fluid density fluid kinematic viscosity reaction rate parameter reaction rate coefficient
q f b s b r y l g
2. Mathematical model
v0
We consider a steady, laminar, hydromagnetic coupled heat and mass transfer by mixed convection flow of a cold fluid at temperature TN over an infinite vertical plate embedded in a porous medium. It is assumed that the left surface of the plate is heated by convection from a hot fluid at temperature Tf which provides a heat transfer coefficient hf. The cold fluid on the right side of the plate is assumed to be Newtonian and electrically conducting. Except for the density, all other fluid properties are assumed to be independent of temperature and chemical species concentration. A uniform magnetic field of strength B0 is imposed normal to the plate (along the y-axis) as shown in Fig. 1. Since the magnetic Reynolds number is very small for most fluid used in industrial applications, we assume that the induced magnetic field is negligible. Under the Boussinesq and boundary-layer approximations, the momentum, energy balance and concentration equations can be written in as [10,15e17],
v0
vu v2 u ¼ y 2 þ g bðT T N Þ þ g bðC C N Þ vy vy ð1Þ
y vu 2 sB20 vT k v2 T þ þ ðU uÞ2 ; ¼ rCp vy2 Cp vy rCp vy
hf , T f
vC v2 C ¼ D 2 gðC C N Þ; vy vy
(2)
u
B0
(3)
where x and y are, respectively, the directions along and perpendicular to the surface, u is the velocity component along the plate, U is the uniform free-stream velocity, v0 is the constant suction velocity, g is the acceleration due to gravity, r; y; k and s are, the fluid density, kinematic viscosity, the thermal conductivity and the electrical conductivity, respectively. The local temperature is T, C is the species concentration and the subscript N denotes free-stream conditions. The other parameters are the coefficient of volume expansion for heat transfer b, the coefficient of volumetric expansion with respect to species concentration b, the molecular diffusivity D, and the reaction rate coefficient g. The appropriate boundary conditions for this flow are
u ¼ 0; u/U;
sB2 y þ ðU uÞ þ 0 ðU uÞ; r K v0
concentration at plate surface magnetic field strength thermal Grashof number free stream velocity Eckert number Schmidt number constant wall suction coordinates
vT k ¼ hf T f T ; vy T/T N ;
C ¼ Cw;
at
y ¼ 0;
C/C N ; as y/N;
(4)
(5)
where the convective heating process at the plate is characterized by the hot fluid temperature T f and the heat transfer coefficient hf. Introducing the following dimensionless quantities,
y u U T TN C CN ; U¼ ; q¼ ; f¼ ; Sc ¼ ; v0 v0 D T T C C N w N f gb T f T N y ryCp gy2 g bðC w C N Þy ; Pr ¼ ; l ¼ 2; Gr ¼ ; Gc ¼ 3 3 k v0 v0 Dv0 sB20 y v20 K v20 hf y ; Bi ¼ ; K ¼ 2 ; Ec ¼ M¼ : kv0 y rv20 CP T f T N y¼
v0 y
y
; u¼
(6)
Eqs. (1)e(5) may be written in dimensionless form as follows.
x
v0
Porous medium v T ,C ,u T∞ , c ∞ ,U g
y Fig. 1. Flow configuration and coordinate system.
du d2 u Uu þ Gr q þ Gcf þ MðU uÞ þ ¼ ; dy K dy2
(7)
2 dq 1 d2 q du þ Ec þMEcðU uÞ2 ; ¼ 2 Pr dy dy dy
(8)
O.D. Makinde, A. Aziz / International Journal of Thermal Sciences 49 (2010) 1813e1820
df 1 d2 f l f; ¼ Sc dy2 Sc dy u ¼ 0;
dq ¼ Bi½q 1; f ¼ 1; dy
u ¼ U; q ¼ f ¼ 0 as y/N;
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(9)
at y ¼ 0;
(10) (11)
where Bi is the Biot number, Ec is the Eckert number, Gc and Gr are the mass transfer and thermal Grashof numbers, respectively, K is the porosity parameter, M is the Hartmann number, l is the reaction parameter, Pr is the Prandtl number and Sc is the Schmidt number. The quantities u, q, f represent the dimensionless velocity, dimensionless temperature, and dimensionless concentration, respectively. Here, we emphasis that the Biot number (Bi) is the ratio of the internal thermal resistance of the plate to the boundary layer thermal resistance. When Bi ¼ 0, the left side of the plate with hot fluid is totally insulated and no convective heat transfer to the cold fluid on the right side takes place. The set of equations (7)e(9) under the boundary conditions (10) and (11) were solved numerically by applying the Nachtsheim and Swigert [18] shooting iteration technique together with a sixth-order RungeeKutta integration algorithm. Besides the velocity, temperature, and concentration data, the numerical solution also yielded the values of q(0), 0 u0 (0),q ð0Þ and f0 ð0Þ which are, respectively, proportional to the plate surface temperature, the local skin-friction coefficient, the local Nusselt number and the local Sherwood number. 3. Results and discussion The numerical computations were carried out for positive values of the buoyancy parameters Gr > 0 (which corresponds to the cooling problem) and Gc > 0 (which indicates that the free stream concentration is less than the concentration at the plate surface). The cooling problem is often encountered in engineering applications such as the cooling of electronic components and nuclear reactors. The Prandtl number was fixed at Pr ¼ 0.72 which corresponds to air. The values of Schmidt number (Sc) were chosen to be Sc ¼ 0.24, 0.62, 0.78, 2.62 which represent, respectively, the diffusion of H2, H2O, NH3, and propyl benzene in air. The computations have been carried out for a range of values of the other parameters appearing in the problem.
Fig. 2. Velocity profiles for Sc ¼ 0.62, Gr ¼ Gc ¼ K ¼ 1, Ec ¼ M ¼ l ¼ 0.1; _____Bi ¼ 0, ooooBi ¼ 0.1, þþþBi ¼ 0.5, ..Bi ¼ 1, ****Bi ¼ 10.
convective process at the plate. The effect of the permeability parameter K on the velocity profiles is illustrated in Fig. 3. As the permeability of the medium increases, it enhances the upward flow profoundly and shifts the location of the peak velocity further away from the plate. Fig. 4 shows how the velocity profile is affected by the Eckert number. Since the Eckert number by definition is proportional to the square of the suction velocity v0, Fig. 4 also implicitly depicts the effect of the suction velocity on the upward flow in the boundary layer. The peak velocity is seen to increase as the suction velocity increases but its location is hardly affected by the increase in the suction velocity. The effect of the suction on the
3.1. Velocity profiles The vertical or the x-component of the velocity, u, profiles are displayed in Figs. 2e9 and follow the pattern characteristic of natural convection boundary layer flow over a vertical plate. The velocity increases from a zero value at the plate to a peak value within the boundary layer and then decreases gradually to the specified freestream velocity value of U ¼ 0.5. It is worth mentioning that the Biot number (Bi) is the ratio of the internal thermal resistance of a solid to the boundary layer thermal resistance. When Bi ¼ 0 (i.e. without Biot number) the left side of the plate with hot fluid is totally insulated, the internal thermal resistance of the plate is extremely high and no convective heat transfer to the cold fluid on the right side of the plate takes place. Fig. 2 shows the velocity profile with or without Biot number. It is interesting to note that without Biot number (i.e. Bi ¼ 0) the peak velocity is low. As the convection Biot number increases, the plate thermal resistance reduces. Consequently, the peak velocity and the velocities in the neighbourhood of the peak increase significantly. This effect is triggered by the stronger buoyancy forces induced as a result of the increase in the strength of the
Fig. 3. Velocity profiles for Sc ¼ 0.62, Gr ¼ Gc ¼ 1, Bi ¼ Ec ¼ M ¼ l ¼ 0.1; _____K ¼ 0.1, ooooK ¼ 1, þþþK ¼ 5, ..K ¼ 10.
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Bi ¼ M ¼ l ¼ 0.1;
Fig. 6. Velocity profiles for Sc ¼ 0.62, K ¼ Gr ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Gc ¼ 0.1, ooooGc ¼ 1, þþþGc ¼ 2, ..Gc ¼ 3.
velocity profile is not as pronounced as the effect of permeability of the medium as observed in Fig. 3. The effect of thermal Grashof number on the velocity distribution can be gleaned from Fig. 5. Because the thermal Grashof number is the ratio of buoyancy to viscous forces in the boundary layer, the increase in its value suggests an increase in the buoyancy forces relative to the viscous forces and this is clearly reflected in the progressive increase in the vertical (upward) velocity of the flow. The increase in mass transfer Grashof number produces a similar effect on the velocity profiles as
shown in Fig. 6. It is interesting to note that for Gc ¼ 3, that is, for high concentration between the wall and the freestream, the vertical component of the velocity in a portion of the boundary layer is larger than unity which implies that the vertical velocity exceeds the wall suction velocity. Attention is now turned to the effects of the Hartmann number M, the Schmidt number Sc, and the reaction parameter l. Fig. 7 shows that as the strength of the magnetic field increases, i.e. M increases, it causes a decrease in the vertical (upward) velocity in the boundary
Fig. 5. Velocity profiles for Sc ¼ 0.62, K ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Gr ¼ 0.1, ooooGr ¼ 1, þþþGr ¼ 3, ..Gr ¼ 5.
Fig. 7. Velocity profiles for Sc ¼ 0.62, K ¼ Gr ¼ Gc ¼ 1, Bi ¼ Ec ¼ l ¼ 0.1; _____M ¼ 0.1, ooooM ¼ 0.5, þþþM ¼ 1, ..M ¼ 2.
Fig. 4. Velocity profiles for Sc ¼ 0.62, K ¼ 1, Gr ¼ Gc ¼ 0.5, _____Ec ¼ 0.1, ooooEc ¼ 0.5, þþþEc ¼ 1, ..Ec ¼ 1.5.
O.D. Makinde, A. Aziz / International Journal of Thermal Sciences 49 (2010) 1813e1820
Fig. 8. Velocity profiles for K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Sc ¼ 0.24, ooooSc ¼ 0.62, þþþSc ¼ 0.78, ..Sc ¼ 2.62.
layer. The vertical velocity also decreases when the Schmidt number Sc is increased as illustrated in Fig. 8. Since Sc is proportional to the density of the diffusing species, the diffusion of heavier species in air tends to retard the upward flow velocity. The increase in the reaction parameter l also tends to reduce the vertical velocity as can be seen in Fig. 9. Thus the effect of increase in the values of M, Sc and l (Figs. 7e9) is to reduce the upward flow velocity while the increase in the values of Bi, K, Ec, Gr, Gc promotes an increase in the vertical velocity (Figs. 2e6). The results presented in Figs. 2e9 and the
Fig. 9. Velocity profiles for Sc ¼ 0.62, K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ 0.1; ____l ¼ 0.1, ooool ¼ 0.5, þþþl ¼ 1, ..l ¼ 5.
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Fig. 10. Temperature profiles for Sc ¼ 0.62, Gr ¼ Gc ¼ K ¼ 1, _____Bi ¼ 0, ooooBi ¼ 0.1, þþþBi ¼ 0.5, ..Bi ¼ 1, ****Bi ¼ 10.
Ec ¼ M ¼ l ¼ 0.1;
conclusions drawn here for MHD flow with a convective surface boundary condition are in agreement with the ones reported in [8,10,17] for MHD flows with constant surface temperature or constant surface heat flux type boundary conditions. 3.2. Temperature profiles The temperature profiles corresponding to the velocity profiles of Figs. 2e9 are provided in Figs. 10e16. In each figure, the maximum temperature occurs at the plate and decreases to reach
Fig. 11. Temperature profiles for Sc ¼ 0.62, _____K ¼ 0.1, ooooK ¼ 1, þþþK ¼ 5, ..K ¼ 10.
Gr ¼ Gc ¼ 1,
Bi ¼ Ec ¼ M ¼ l ¼ 0.1;
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Fig. 12. Temperature profiles for Sc ¼ 0.62, K ¼ 1, Gr ¼ Gc ¼ 0.5, Bi ¼ M ¼ l ¼ 0.1; _____Ec ¼ 0.1, ooooEc ¼ 0.5, þþþEc ¼ 1, ..Ec ¼ 1.5.
Fig. 14. Temperature profiles for Sc ¼ 0.62, _____Gc ¼ 0.1, ooooGc ¼ 1, þþþGc ¼ 2, ..Gr ¼ 3.
the freestream temperature (q ¼ 0) at the edge of the thermal boundary layer. Both the plate surface temperature and the thickness of the thermal boundary thickness increase as the parameters Bi, K, Ec, Gr, Gm increase (Figs. 9e14). Moreover, it is interesting to note that the fluid temperature on the right side of the plate increases with an increase in the Biot number, since as Bi increases, the thermal resistance of the plate decreases and convective heat transfer to the fluid on the right side of the plate increases. Compared with the other parameters, the effect of permeability parameter K on the temperature profile appears to be small (See
Fig. 11). The effect of increase in Sc (Fig. 15) and l (Fig. 16), however, is to decrease the plate surface temperature as well the thickness of the thermal boundary layer.
M ¼ Bi ¼ Ec ¼ l ¼ 0.1;
Fig. 15. Temperature profiles for K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Sc ¼ 0.24, ooooSc ¼ 0.62, þþþSc ¼ 0.78, ..Sc ¼ 2.62.
Fig. 13. Temperature profiles for Sc ¼ 0.62, _____Gr ¼ 0.1, ooooGr ¼ 1, þþþGr ¼ 3, ..Gr ¼ 5.
K ¼ Gc ¼ 1,
K ¼ Gr ¼ 1,
M ¼ Bi ¼ Ec ¼ l ¼ 0.1;
3.3. Concentration profiles The dependence of the concentration profile on Sc is illustrated in Fig. 17. For each value of Sc, the dimensionless concentration decreases from a value of one at the plate to a value of zero at the edge of the concentration boundary layer. As Sc increases, the
O.D. Makinde, A. Aziz / International Journal of Thermal Sciences 49 (2010) 1813e1820
Fig. 16. Temperature profiles for Sc ¼ 0.62, ____l ¼ 0.1, ooool ¼ 0.5, þþþl ¼ 1, ..l ¼ 5.
K ¼ Gr ¼ Gc ¼ 1,
M ¼ Bi ¼ Ec ¼ 0.1;
species concentration drops more rapidly as the thickness of the concentration boundary layer decreases. Since the increase in Sc implies a decrease in the molecular diffusivity D, the concentration decay over a shorter distance is understandable. Similarly, one would anticipate that the heavier species (higher Sc) would have a shorter penetration depth (thinner boundary layer). Fig. 18 shows the effect of reaction rate parameter l on the concentration profiles. The increase in the reaction rate parameter means an increase in the consumption of the species and hence a reduction in the
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Fig. 18. Concentration profiles for Sc ¼ 0.62, K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ 0.1; ____l ¼ 0.1, ooool ¼ 0.5, þþþl ¼ 1, ..l ¼ 5.
general concentration through the boundary layer and the depletion of concentration over a shorter distance as predicted by Fig. 18. 3.4. Skin friction, heat transfer, mass transfer parameters and surface temperature 0
The quantities u0 ð0Þ; q ð0Þ; f0 ð0Þ are measures of the skin friction, heat transfer, and mass transfer at the plate, respectively, while the quantity q(0) gives the plate surface temperature. Table 1 records the values of these quantities for a range of values of the parameters Bi, Sc, Ec, M, K and l with Pr ¼ 0.72, Gr ¼ 0.1, Gc ¼ 0.1 and U ¼ 0.5. The skin friction increases as the Biot number Bi, the Eckert number Ec, M increase but decreases when Sc, K and l increase. The heat transfer rate from the plate increases with the increase in values of Bi, Sc, K, and l but decreases with the increase in Ec and M. For Bi ¼ 0, no convective heat transfer takes place and the rate of heat transfer at the plate surface is zero as expected. The surface mass transfer is seen to increase with the increase in Sc and l. Finally, the plate temperature increases as Bi increases very significantly as the Bi increases from 0.1 to 10. For Bi ¼ 0.1, the plate temperature is hardly Table 1 Values of f0 ð0Þ, u0 (0), q0 (0)and q(0) with Gr ¼ Gc ¼ 0.1, Pr ¼ 0.72, U ¼ 0.5.
Fig. 17. Concentration profiles for K ¼ Gr ¼ Gc ¼ 1, M ¼ Bi ¼ Ec ¼ l ¼ 0.1; _____Sc ¼ 0.24, ooooSc ¼ 0.62, þþþSc ¼ 0.78, ..Sc ¼ 2.62.
Bi
Sc
Ec
M
K
l
f0 ð0Þ
u0 (0)
q0 (0)
q(0)
0 0.1 1.0 10 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
0.24 0.24 0.24 0.24 0.62 0.78 2.62 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24
0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 2.0 0.1 0.1 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 10 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 10 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1.0 2.0
0.459012 0. 459012 0.459012 0.459012 0.752957 0.892139 2.657627 0.459012 0.459012 0.459012 0.459012 0.459012 0.459012 1.127174 1.539295
1.891555 1.894933 1.907655 1.917383 1.892262 1.891148 1.882032 1.907091 1.920915 1.962039 2.526743 0.930812 0.759132 1.889451 1.886928
0.000000 0.083565 0.398302 0.638954 0.083583 0.083590 0.083641 0.044871 0.000865 0.083172 0.080068 0.085574 0.085843 0.083600 0.083615
0.048228 0.164348 0.601697 0.936104 0.164168 0.164096 0.163589 0.551288 0.991347 0.168276 0.199315 0.144253 0.141569 0.163991 0.163843
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affected by the changes in Sc. In other words, the molecular diffusivity or the density of the diffusing species has very little effect on the plate temperature. The increase in viscous dissipation (Ec) by a factor of 2 elevates the plate from a value of 0.551288 to 0.991347. The strength of the magnetic field (M) has slight impact on the plate temperature. For example, when the magnetic field is increased by a factor of 10, the plate temperature increases from 0.168276 to 0.199315. Similarly, the impact of the permeability of the medium (K) on the plate temperature is small. An increase in permeability parameter from 0.1 to 10 lowers the plate temperature from 0.199315 to 0.141569. The reaction rate (l) barely affects the plate temperature as can be seen from the last two entries in Table 1. 4. Conclusion The influence of convective boundary condition on hydromagnetic mixed convection with heat and mass transfer past a vertical plate embedded in a porous medium is numerically studied in this work. The left surface of the plate is heated by convection from a hot fluid while the cold fluid on the right side of the plate is assumed to be Newtonian and electrically conducting. A table containing the numerical data showing the effects of various parameters controlling the system on the plate surface temperature, the wall shear stress, and the local Nusselt and Sherwood numbers is also provided. Our results reveal among others that both the fluid velocity and temperature increase with an increase in the convective heat transfer parameter i.e. the Biot Number (Bi). Physical significance and application of Biot number with respect to boundary layer flow problems can be found in several engineering and industrial processes such as drying of material, transpiration cooling, etc. Acknowledgements The authors gratefully acknowledge the valuable suggestions of the anonymous referees. ODM would like to thank the National Research Foundation of South Africa Thuthuka programme for financial support. References [1] K. Takahashi, R. Alkire, Mass transfer in gas-sparged porous electrodes. Chemical Engineering Communications 38 (1985) 209e227.
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