Tamkang Journal of Science and Engineering, Vol. 13, No. 3, pp. 235242 (2010) 235 Steady MHD Natural Convection Flow with Variable Electrical Conductivity and Heat Generation along an Isothermal Vertical Plate P. R. Shrama 1 * and Gurminder Singh 2 1 Department of Mathematics, University of Rajasthan, Jaipur - 302055, India 2 Birla Institute of Technology (Mesra, Ranchi) Ext. Centre Jaipur, 27, Malviya Industrial Area, Jaipur - 302017, India Abstract Aim of the paper is to investigate the effect of temperature dependent electrical conductivity on steady natural convection flow of a viscous incompressible low Prandtl (Pr << 1) electrically conducting fluid along an isothermal vertical non-conducting plate in the presence of transverse magnetic field and exponentially decaying heat generation. The governing equations of continuity, momentum and energy are transformed into ordinary differential equations using similarity transformation. The resulting coupled non-linear ordinary differential equations are solved using Runge-Kutta fourth order method alongwith shooting technique. The velocity and temperature distributions are discussed numerically and presented through graphs. The numerical values of skin-friction coefficient and Nusselt number at the plate are derived, discussed numerically for various values of physical parameters and presented through Tables. Key Words: Steady, MHD, Free Convection, Boundary Layer Flow, Variable Electrical Conductivity, Internal Heat Generation, Skin-Friction Coefficient, Nusselt Number 1. Introduction The free convection flow occurs frequently in nature and present in various physical phenomena such as fire engineering, combustion modeling, nuclear reactor cooling, heat exchangers, petroleum reservoir etc. In such flows, the velocity distribution and temperature distribution are coupled, as the flow arises due to buoyancy force, which is induced by temperature difference between the surface and the fluid. The study of heat transfer is integral part of natural convection flow and belongs to the class of problems in boundary layer theory. The quantity of heat transferred is highly dependent upon the fluid motion within the boundary layer. A large number of physical phenomena involve natural convection (Jaluria [1]), which are enhanced and driven by internal heat *Corresponding author. E-mail: profprsharma@yahoo.com generation. In such flows the buoyancy force is incremented due to heat generation resulting in modification of heat transfer characteristic. The effect of internal heat generation is especially pronounced for low Prandtl number fluid e.g. liquid metals. Magnetohydrodynamics is the study of motion of an electrically conducting incompressible fluid in the presence of magnetic field i.e. an electromagnetic field interacting with the velocity field of an electrically conducting fluid. Hydromagnetic flows have become important due to industrial applications, for instance it is used to deal with the problem of cooling of nuclear reactor by fluid having very low Prandtl number (Michiyoshi et al. [2], Fumizawa [3]). Liquid metals have small Prandtl number of order 0.001~0.1 (e.g. Pr = 0.01 is for Bismuth, Pr = 0.023 for Mercury etc.) and are generally used as coolants because of very high thermal conductivity. They have ability to transport heat even if small temperature difference exists
236 P. R. Shrama and Gurminder Singh between the surface and fluid. Due to this reason, liquid metals are used as coolant in nuclear reactor for disposal of waste heat. It is expected that these intricate engineering applications should work ideally and hence every parameter affecting the fluid flow and heat transfer must be studied carefully. In applications where radioactive material is surrounded by inert alloy or the electromagnetic heating is existent the exponentially decaying heat generation model can be utilized as heat source (Sahin [4], Crepeau and Clarksean [5]). Further, the velocity field and temperature distribution of the liquid metals is modified in the presence of transverse magnetic field because of their high electrical conductivity which is function of temperature and in case of metals it varies inversely with respect to the temperature. Hence temperature dependent electrical conductivity becomes a point of study. Ostrach [6] presented the similarity solution of natural convection along vertical isothermal plate. Gebhart [7] used perturbation technique to analyse the effect of dissipation on natural convection. Soundalgekar [8] studied natural convection flow along vertical porous plate with suction and viscous dissipation. Joshi and Gebhart [9] observed the effect of pressure stress work and viscous dissipation in some natural convection flow: isothermal, uniform flux and plumes. Watanabe and Pop [10] investigated the heat transfer in thermal boundary layer of MHD flow over a flat plate. Crepeau and Clarksean [5] discussed similarity solution of natural convection with internal heat generation, which decays exponentially. Chamkha and Khaled [11] obtained similarity solution of natural convection on an inclined plate with internal heat generation/absorption in presence of transverse magnetic field. Chen [12] studied MHD flow in natural convection on inclined porous surface with variable surface temperature. Molla et al. [13] observed the effect of heat generation/absorption on natural convection along a wavy surface. Molla et al. [14] studied the natural convention flow along a horizontal cylinder in the presence of heat generation. Some of the authors mentioned above have studied MHD natural convection in fluid having low Prandtl fluid neglecting the temperature effect on electrical thermal conductivity. Aim of the present study is to investigate the effects of varying electrical conductivity on free convection flow of a viscous incompressible electrically conducting fluid and heat transfer along an isothermal vertical nonconducting plate in the presence of exponentially decaying heat-generation and transverse magnetic field. 2. Formulation of the Problem Consider steady laminar natural convection flow of a viscous incompressible fluid along a vertical non-conducting plate kept at constant temperature T w, and the fluid has internal volumetric rate of heat generation Q. The x-axis is taken along the plate and y-axis is normal to the plate. Magnetic field of intensity B o is applied in y- direction. It is assumed that the external field is zero, also electrical field due to polarization of charges and Hall effect are neglected. Incorporating the Boussinesq s approximation within the boundary layer, the governing equations of continuity, momentum and energy (Jeffery [15], Bansal [16,17], Schlichting and Gersten [18]), respectively are given by (1) (2) (3) where electrical conductivity * is variable with temperature as given below The boundary conditions are 3. Method of Solution Introducing the stream function (x,y) such that (4) (5) (6)
MHD Natural Convection Flow with Variable Electrical Conductivity 237 1 Gr 4 where (, x y) 4f ( ) and the similarity variable 1 4 y Gr 4. (7) x 4 Following Crepeau and Clarksean [5], the volumetric rate of heat generation is taken as given below (8) where u v is the shear stress and u 0 = y x y 0 1 { gx( T T )} 2 is the local convective velocity. w 5. Nusselt Number The rate of heat transfer in terms of the Nusselt number at the plate is given by (13) Here, magnetic field intensity B o must be propotional to x 1 4 (Chen [12]), in order to eliminate the dependency of M on x. It is observed that the equation (1) is identically satisfied with equation (6). Substituting equations (7) and (8) into the equations (2) and (3), alongwith the equation (4), the resulting coupled non-linear ordinary differential equations are and The boundary conditions are reduced to (9) (10) (11) The governing boundary layer equations (9) and (10) with boundary conditions (11) are solved using Runge- Kutta fourth order technique (Jain et al. [19], Krishnamurthy and Sen [20] and Jain [21]) alongwith double shooting technique (Conte and Boor [22]). 4. Skin-Friction Coefficient Skin-friction coefficient at the plate is given by (12) T where q y 0. y 6. Particular Cases In the absence of magnetic field i.e. M =0,theresults of present paper are reduced to those obtained by Crepeau and Clarksean [5] and Chamkha and Khaled [11]. The former have used commercial software Mathematica while the latter have used implicit finite difference scheme. It is seen from Table 1 that the numerical results of (0) of present paper are in good agreement. 7. Results and Discussion It is observed from Table 2 that with the increase in the value of electrical conductivity parameter (), the skin-friction coefficient increases in the presence or absence of heat generation. However in the case of rate of heat transfer the effect is practically negligible. Hence, it can be concluded that the parameter affects only the skin-friction. The increase in the Prandtl number decreases the skin-friction coefficient and increases the rate of heat transfer. This is attributed to the fact that as the Prandtl number decreases, the thermal boundary layer thickness increases, causing reduction in the temperature gradient (i.e. (0)) at the surface of the plate. The temperature gradient reduces at the surface because low Prandtl fluid has high thermal conductivity, causing the fluid to attain higher temperature thereby reducing heat flux at the surface. Also, for such low Prandtl number, the velocity boundary layer is inside the thermal
238 P. R. Shrama and Gurminder Singh Table 1. Values of (0) for different values of Pr are compared with the results obtained by Chamkha and Kahaled [11] and Crepeau and Clarksean [5] Chamkha and Khaled [11] Crepeau and Clarksean [5] Present work Pr Without Q With Q Without Q With Q Without Q With Q (0) (0) (0) (0) (0) (0) 0.001-0.0264 0.9392-0.0264 0.9391 0.010-0.0800 0.8413-0.0805 0.8236-0.0805 0.8236 0.100-0.2119 0.5656-0.2302 0.5425-0.2301 0.5424 1.000-0.5646 0.0057-0.5671 0.0057-0.5671 0.0058 1000. -1.1720-0.8027- -1.1690-0.7963- -1.1662-0.7962- Table 2. Values of f (0), -(0) for different values of M,, Pr in the presence or absence of heat generation Pr = 0.001 S =0.0 S =1.0 f (0) -(0) f (0) -(0) (M = 0) 1.0414 0.0264 1.3501-0.9391 (M = 1) 0.0 0.7932 0.0232 1.0539-0.9448 0.1 0.8105 0.0234 1.0860-0.9442 0.3 0.8387 0.0237 1.1331-0.9434 0.5 0.8609 0.0239 1.1660-0.9428 (M = 2) 0.0 0.6430 0.0215 0.8565-0.9496 0.1 0.6647 0.0218 0.9000-0.9486 0.3 0.7020 0.0222 0.9684-0.9471 0.5 0.7328 0.0226 01.01927-0.9460 Pr = 0.01 (M = 0) 0.9877 0.0805 1.2629-0.8236 (M = 1) 0.0 0.7647 0.0700 0.9989-0.8394 0.1 0.7798 0.0706 1.0258-0.8380 0.3 0.8045 0.0715 1.0659-0.8358 0.5 0.8239 0.0723 1.0945-0.8343 (M = 2) 0.0 0.6264 0.0621 0.8203-0.8536 0.1 0.6459 0.0630 0.8576-0.8508 0.3 0.6794 0.0646 0.9166-0.8468 0.5 0.7070 0.0658 0.9609-0.8440 Pr = 0.1 (M = 0) 0.8591 0.2301 1.0679-0.5424 (M = 1) 0.0 0.6894 0.2016 0.8690-0.5818 0.1 0.6999 0.2030 0.8864-0.5788 0.3 0.7172 0.2053 0.9133-0.5742 0.5 0.7310 0.2072 0.9332-0.5707 (M = 2) 0.0 0.5799 0.1788 0.7317-0.6162 0.1 0.5944 0.1810 0.7569-0.6107 0.3 0.6191 0.1850 0.7973-0.6022 0.5 0.6395 0.1883 0.8284-0.5958 boundary layer and its thickness reduces as Prandtl number decreases so the fluid motion is confined in more and more thinner layer near surface, which experiences increased drag (skin-friction) by the fluid. In other words there is more straining motion inside velocity boundary layer resulting in the increase of skin-friction coefficient. It is observed from the Table 2 that with the increase in magnetic field intensity, the skin-friction coefficient and the rate of heat transfer decrease. This happens due to the fact in the presence of transverse magnetic field the fluid velocity decreases near the surface; hence the surface experiences reduction in drag. Further, as the fluid velocity near the surface decreases it results in decreased heat removal at the surface. It is observed that in the presence of heat generation, the skin-friction coefficient increases, while the rate of heat transfer reduces considerably for fixed values of M, and Pr. Obviously, in the presence of heat source, buoyancy force increases due to which fluid velocity in boundary layer increase considerably resulting in the increase of drag i.e. skin-friction at the surface. Also in the presence of heat source the fluid is at higher temperature, hence the rate of heat transfer is reduced at the surface. It is seen from Figure 1 that the fluid velocity increase in the presence of volumetric rate of heat generation (S = 1.0) in comparison when heat generation is absent (S = 0.0), which is due to the fact that the volumetric heat generation increases buoyancy force. The effect is more pronounced in the case of lower Prandtl number as seen from Figure 1, since velocity profiles are more separated as Prandtl number decreases. Also, with the increase in Prandtl number the velocity profiles decreases. It is observed from Figure 2 that in the presence of volumetric rate of heat generation, the fluid temperature increases and the affect is more pronounced on fluid with lower Prandtl number. For S = 1.0, the fluid temperature near the plate increases sharply and is even higher than the plate temperature, this is practically important be-
MHD Natural Convection Flow with Variable Electrical Conductivity Figure 1. Velocity distribution versus h when M = 0.0 and e = 0.0. 239 Figure 2. Temperature distribution versus h when M = 0.0 and e = 0.0. cause such an increase in fluid temperature near the plate can effect the material properties of the plate. The fluid temperature decrease with the increase in the Prandtl number and the effect is more pronounced in the case of lower Prandtl number. Figure 3 shows with the increase in magnetic parameter, the fluid velocity decreases. The presence of transverse magnetic field sets in Lorentz force, which results in retarding force on the velocity field and therefore as magnetic parameter increases, the fluid velocity decreases. However, the fluid temperature increases with the increase in the magnetic parameter or heat generation parameter as noted from Figure 4. Figure 5 reflects the effect of electrical conductivity parameter e on the velocity profiles. The fluid with high electrical conductivity experience more retarding force in the presence of transverse magnetic field and therefore the fluid velocity decreases, but as the parameter e in- creases (which mean that electrical conductivity of fluid decreases) the retarding force acting on the fluid declines and hence the fluid velocity increases. Thus the increase in parameter e has a kind of nullifying effect on magnetic parameter. The effect of e is even more for higher value of magnetic parameter as can be seen from apartness in velocity profiles at higher value of magnetic parameter. The comparative study of Figures 5 and 6 shows that in the presence of volumetric rate of heat generation the effect of e is more pronounced because, the increase in fluid temperature means reduction in electrically conductivity (equation 5). Figures 7 and 8 show that with the increase in parameter e the fluid temperature decreases; but effect is marginal, hence for practical purposes it can be neglected. Figures 9 and 10 show the same trend as discussed earlier for Figures 5 and 6, however it can be reasonably pointed through their comparative study that the effect is more pronounced at low Prandtl number be- Figure 3. Velocity distribution versus h when Pr = 0.01 and e = 0.0. Figure 4. Temperature distribution versus h when Pr = 0.01 and e = 0.0.
240 P. R. Shrama and Gurminder Singh Figure 5. Velocity distribution versus h when Pr = 0.001 and S = 0.0. Figure 6. Velocity distribution versus h when Pr = 0.001 and S = 1.0. Figure 7. Temperature distribution versus h when Pr = 0.001, M = 1.0 and S = 0.0. Figure 8. Temperature distribution versus h when Pr = 0.001, M = 1.0 and S = 1.0. Figure 9. Velocity distribution versus h when Pr = 0.01 and S = 0.0. Figure 10. Velocity distribution versus h when Pr = 0.01 and S = 1.0.
MHD Natural Convection Flow with Variable Electrical Conductivity 241 cause low Prandtl number implies higher electrical conductivity. Figures 11 and 12 show the same trend as are seen in Figures 7 and 8 with a note that more affect of electrical conductivity parameter is noted for low Prandtl number. 8. Conclusion 1. Fluid velocity increases in the presence of volumetric rate of heat generation or due to increase in the electrical conductivity parameter, while it decreases due to increase in magnetic field intensity, which is in full agreement with physical phenomenon. 2. Fluid temperature increases in the presence of volumetric rate of heat generation or due to increase in magnetic field intensity, while it decreases due to increase in electrical conductivity parameter but is not Figure 11. Temperature distribution versus when Pr = 0.01, M = 1.0 and S = 0.0. Figure 12. Temperature distribution versus when Pr = 0.01, M = 1.0 and S = 1.0. much effective. 3. The skin-friction coefficient increases with the increase in electrical conductivity parameter or in the presence of volumetric rate of heat generation, while it decreases due to increase in the Prandtl number. 4. The rate of heat transfer increases with the increase in the Prandtl number or electrical conductivity parameter, which is not much effective, while it decreases due to increase in the magnetic field intensity. Nomenclature g: Acceleration due to gravity of the Earth x, y: Cartesian coordinates f: dimensionless stream function g( T w T ) x 3 Gr: Grashof number 2 S: heat generation parameter B o : magnetic field intensity 2 2 Bo x Gr M: magnetic parameter 4 Nu: Nusselt number Pr: Prandtl number C p C f : skin-friction coefficient C p : specific heat at constant pressure T: temperature of the fluid T : temperature of fluid far away from plate T w : temperature of the plate u, v: velocity components along x- andy-directions, respectively Q: volumetric rate of heat generation 1 T T Gr 2 w e 2 x 4 Greek Letters : coefficient of thermal expansion : coefficient of viscosity : coefficient thermal conductivity : density of fluid : T T dimensionless temperature Tw T 1 2
242 P. R. Shrama and Gurminder Singh : : electrical conductivity electrical conductivity parameter : kinematic viscosity : similarity variable : stream function *: variable electrical conductivity Superscript : differentiation with respect to References [1] Jaluria, Y., Natural Convection, Heat and Mass Transfer, Pergamon Press, New York (1980). [2] Michiyoshi, I., Takahashi, I. and Serizawa, A., Natural Convection Heat Transfer from a Horizontal Cylinder to Mercury under Magnetic Field, International Journal of Heat and Mass Transfer, Vol. 19, pp. 10211029 (1976). [3] Fumizawa, M., Natural Convection Experiment with NaK under Transverse Magnetic Field, Journal of Nuclear Science and Technology, Vol. 17, pp. 98105 (1980). [4] Sahin, A. Z., Transient Heat Conduction in Semi-Infinite Solid with Spatially Decaying Exponential Heat Generation, International Communication in Heat and Mass Transfer, Vol. 19, pp. 349358 (1992). [5] Crepeau, J. C. and Clarksean, R., Similarity Solution of Natural Convection with Internal Heat Generation, ASME Journal of Heat Transfer, Vol. 119, pp. 183185 (1997). [6] Ostrach, S., An analysis of laminar free convective flow and heat transfer about a flat plate parallel to direction of the generating body force, NACA Technical Report 1111 (1952). [7] Gebhart, B., Effects of Viscous Dissipation in Natural Convection, Journal of Fluid Mechanics, Vol. 14, pp. 225232 (1962). [8] Soundalgekar, V. M., Effects of Mass Transfer on Free Convective Flow of a Dissipative, Incompressible Fluid Past an Infinite Vertical Porous Plate with Suction, Proc. Indian Academy of Sciences, India, Vol. 84A, pp. 194203 (1976). [9] Joshi, Y. and Gebhart, B., Effect of Pressure Stress Work and Viscous Dissipation in Some Natural Convection Flow, International Journal of Heat and Mass Transfer, Vol. 24, pp. 15771588 (1981). [10] Wantanabe, T. and Pop, I., Thermal Boundary Layers in Magnetohydrodynamic Flow over a Flat Plate in the Presence of a Transverse Magnetic Field, Acta Mechanica, Vol. 105, pp. 233238 (1994). [11] Chamkha, A. J. and Khaled, A. R. A., Similarity Solutions for Hydromagnetic Simultaneous Heat and Mass Transfer by Natural Convection from an Inclined Plate with Internal Heat Generation or Absorption, Heat and Mass Transfer, Vol. 37, pp. 117123 (2001). [12] Chen, C. H., Heat and Mass Transfer in MHD Flow by Natural Convection from a Permeable, Inclined Surface with Variable Wall Temperature and Concentration, Acta Mechanica, Vol. 172, pp. 219235 (2004). [13] Molla, M. M., Hossain, M. A. and Yao, L. S., Natural Convection Flow along a Vertical Wavy Surface with Heat Generation/Absorption, International Journal of Thermal Science, Vol. 43, pp. 157163 (2004). [14] Molla, M. M., Hossain, M. A. and Paul, M. C., Natural Convection Flow from an Isothermal Cylinder in the Presence of Heat Generation, International Journal of Engineering Science, Vol. 44, pp. 949958 (2006). [15] Jeffery, A., Magnetohydrodynamics, Oliver and Boyed, New York, USA (1966). [16] Bansal, J. L., Viscous Fluid Dynamics, Oxford & IBH Pub. Co., New Delhi, India (1977). [17] Bansal, J. L., Magnetofluiddynamics of Viscous Fluids, Jaipur Pub. House, Jaipur, India (1994). [18] Schlichting, H. and Gersten, K., Boundary Layer Theory, McGraw-Hill Book Co., New York, USA (1999). [19] Jain, M. K.; Iyengar, S. R. and Jain, R. K., Numerical Methods for Scientific and Engineering Computation, Wiley Eastern Ltd., New Delhi, India (1985). [20] Krishnamurthy, E. V. and Sen, S. K., Numerical Algorithms, Affiliated East-West Press Pvt. Ltd., New Delhi, India (1986). [21] Jain, M. K., Numerical Solution of Differential Equations, New Age Int. Pub. Co., New Delhi, India (2000). [22] Conte, S. D. and Boor, C., Elementary Numerical Analysis, McGraw-Hill Book Co., New York (1981). Manuscript Received: Dec. 16, 2008 Accepted: Sep. 12, 2009