Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 9 (4 ) 37 376 th International Conference on Mechanical Engineering, ICME 3 Free Convective Heat Transfer Flow of Temperature Dependent Thermal Conductivity along a Vertical Flat Plate with Heat Generation A.K.M. Safiqul Islam a, M. A. Alim b * and Md. Rezaul Karim c a Department of Mathematics, Gowripur Govt. College, Mymensingh, Bangladesh b Department of Mathematics, Bangladesh University of Engineering and Technology,Dhaka-, Bangladesh, c Department of Mathematics, Jagannath University Dhaka-, Bangladesh Abstract The natural convection heat transfer flow of temperature dependent thermal conductivity on an electrically conducting fluid along a vertical flat plate with heat generation have been investigated in this paper. The governing equations with associated boundary conditions for this phenomenon are converted to dimensionless forms using a suitable transformation. The transformed non-linear equations are then solved using the implicit finite difference method. Numerical results of the velocity and temperature profiles for Prandtl number and heat generation parameter, skin friction and local rate of heat transfer profiles for different values of the thermal conductivity variation parameter, Prandtl number and heat generation parameters are presented graphically. Detailed discussion is given for the effects of the aforementioned parameters. Significant effects are found in velocity and temperature profiles for Prandtl number. Remarkable effects are found in skin friction and heat transfer for heat generation parameter. Also significant effect is found in heat transfer for thermal conductivity variation parameter. 4 The The Authors. Published Published by Elsevier by Elsevier Ltd. This Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3./). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Selection Engineering and peer-review and Technology under responsibility (BUET). of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) Keywords: thermal conductivity variation, skin friction, heat transfer.. Introduction Heat generation is a volumetric phenomenon. Model studies of the free and mied convection flows have earned reputations because of their applications in nuclear engineering problems. Also the problems of various types of shapes over or on a free convection boundary layer flow have been studied by many researchers. Amongst them Pozzi and Lupo [] investigated the coupling of conduction with laminar convection along a flat plate. Gebhart [] investigated the effect of dissipation natural convection. Rahman and Alim [3] analyzed numerical study of MHD * Corresponding author. Tel.: +88-5534568; fa: +88-86346. E-mail address: maalim@math.buet.ac.bd 877-758 4 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3./). Selection and peer-review under responsibility of the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET) doi:.6/j.proeng.4..864
37 A.K.M. Safi qul Islam et al. / Procedia Engineering 9 ( 4 ) 37 376 free convective heat transfer flow along a vertical flat plate with temperature dependent thermal conductivity. Alim et al. [4] analyzed the combined effect of viscous dissipation & joule heating on the coupling of conduction & free convection along a vertical flat plate. Miraj et al. [8] studied effect of radiation on natural convection flow on a sphere in presence of heat generation. The present study is to incorporate the idea of heat transfer effects of temperature dependent thermal conductivity on free convection flow along a vertical flat plate with heat generation. Nomenclature C p f g G r T f Specific heat at constant pressure Dimensionless stream function Acceleration due to gravity The Grashof number Temperature of the fluid Co-efficient of thermal epansion Dimensionless temperature Viscosity of the fluid Kinematic viscosity Density of the fluid inside the boundary layer Electrical conductivity of the fluid. Mathematical Formulation of the Problem Consider a steady laminar free convection flow of an electrically conducting, viscous and incompressible fluid insulated T l interface T b b T, u g v y Fig.. Physical model and coordinate system along a vertical flat plate of length l and thickness b. Assumed that the temperature at the outer surface of the plate is maintained at a constant temperature T b, where T b > T, the ambient temperature of the fluid. The y -ais i.e. normal direction to the surface and -ais is taken along the flat plate. The coordinate system is shown in Figure-. The governing equations of such flow along a vertical flat plate under the Boussinesq approimations [ ( T b T )], where and T are the density and temperature respectively outside the boundary layer. For the present problem the continuity, momentum and energy equations take the following form
A.K.M. Safi qul Islam et al. / Procedia Engineering 9 ( 4 ) 37 376 373 u v y u u u u v g( T ) f T y y Tf Tf Tf Q u v ( f ) ( Tf T ) y Cp y y Cp () () (3) The temperature dependent thermal conductivity, which is used by Rahman [3] as follow f [ ( Tf T)], Where is the thermal conductivity of the ambient fluid and is a constant, defined as f T. The appropriate boundary condition is to be satisfied by the above equations are f Tf s u, v ; Tf T(,), ( Tf Tb) on y, ; u, Tf Tas y, y b f (4) By using some dimensionless quantities from equations ()-(3), we get the following dimensionless equations u u (5) y u u u u v (6) y y u v ( ) Q y Pr y Pr y (7) C p Where Pr is the Prandtl number, ( T b T ) is the non-dimensional thermal conductivity variation Ql parameter and Q is the dimensionless heat generation parameter. The corresponding boundary C pgr conditions (4) then take the following form u, v, ( ) p on y, ; u, as y, y b 4 Here p Gr sl is the conjugate conduction parameter. To solve the equations (6) and (7) subject to the boundary conditions (8), the following transformations are proposed by Merkin & Pop [5] 4 5 5 5 5 ( ) f(, ); y ( ) ; ( ) h(, ) (9) (8)
374 A.K.M. Safi qul Islam et al. / Procedia Engineering 9 ( 4 ) 37 376 Here is the similarity variable and is the non-dimensional stream function which satisfies the continuity equation and is related to the velocity in the usual way as u and v. Moreover, h (,) represents y the dimensionless temperature. The equations (6) and (7) are transformed for the new co-ordinate system. 6 5 6 5 f f f ff f h f f ( ) ( ) () 5 5 65 h hh h fh Q3( ) h Pr Pr Pr ( ) h f fh f h 5( ) () where prime denotes partial differentiation with respect to. The boundary conditions of (8) then take the form 5 5 ( ) h(, ) f(,) f(,) ; h(,) ; f(, ), h(, ) 4 9 5 ( ) ( ) h(, ) () Equations () and () with the boundary conditions () are solved by applying implicit finite difference method with Keller-bo elimination scheme [7]. The skin friction coefficient Molla [6] in non-dimensional form is 3 C 5( ) f f(, ) (3) The values of local rate of heat transfer in the non-dimensional form is 4 Nu ( ) h(, ) 3. Results and Discussion In this simulation the values of the Prandtl number Pr are considered to be.73,.,.5 and. that corresponds to hydrogen, steam, sulfur dioide and ammonia respectively. Detailed numerical results of the velocity, temperature, skin friction and heat transfer profiles obtained for different values of the thermal conductivity variation parameter, Prandtl number and heat generation parameter are presented graphically. The velocity and temperature fields obtained from the solutions of the equations () and () and the skin friction and rate of heat transfer obtained from the solutions of the equations (3) and (4) are depicted in figures to 6. Figures and illustrate the velocity and temperature against η for different values of Pr with =. and Q =.. From figure, it is observed that the velocity decreases as well as its position moves toward the interface with the increasing values of Pr. From figure, it is seen that the temperature profiles shift downward with the increasing Pr. In figures 3 and 3 describe the velocity and temperature against η for different values of heat generation parameter Q with =. and Pr =.73. From figure 3, it can be observed that the velocity increases as well as its position moves upward the interface with the increasing values of Q. From figure 3, it is seen that the temperature also the same as increasing within the boundary layer. It means that the velocity boundary layer and the thermal boundary layer thickness epand for large values of Q. Figures 4 and 4 illustrate the (4) Nu against with Pr =.73 and Q = effect of the on the skin friction coefficient C f and local heat transfer rate.. It is seen that the skin friction increases monotonically along the upward direction of the plate for a particular values of. It is also seen that that the skin friction increases for the increasing values of. The same result is observed for the heat transfer rate from figure 4. This is to be epected because the higher value of accelerates the fluid flow and increases the temperature. Figures 5 and 5 deal with the effect of Pr on skin friction and rate of heat transfer against with =. and Q =.. It is observed from figure 5 that the skin friction increases
A.K.M. Safi qul Islam et al. / Procedia Engineering 9 ( 4 ) 37 376 375 monotonically for a particular value of Pr. It can also be noted that the skin friction decreases for the increasing values of Pr. From figure 5, it can be seen that the rate heat transfer increases when Pr increases along the positive direction for a particular value of Pr. Important effect is found in skin friction for Pr. From figures 6 and 6 we see the effect of Q on the skin friction and rate of heat transfer against with =. and Pr =.73. It is noted that the skin friction increases for the increasing values of Q. On the other hand heat transfer decreases for the increasing values of Q. Velocity.6.4. Pr =.73 Pr =. Pr =.5 Pr =. 4 6 Temperature.5.5 Pr =.73 Pr =. Pr =.5 Pr =. 3 4 5 Fig. Velocity and Temperature profiles against η for different values of Pr with γ =. and Q =.. Velocity.9.6.3 Q =. Q =.5 Q =.36 Q =.5 4 6 Temperature.5.5 Q =. Q =.5 Q =.36 Q =.5 4 6 Fig. 3 Velocity and Temperature profiles against η for different values of Q with γ =. and Pr =.73. Skin friction 4 3 5 5 Heat transfer.5.5 4 6 Fig. 4 Local skin friction C f and Rate of heat transfer Skin friction.5.5 Pr =.73 Pr =. Pr =.5 Pr =. 4 6 Nu against for different values of with Pr =.73 and Q =.. Heat transfer.9.6 Pr =.73 Pr =. Pr =.5 Pr =..3 3 4 Fig. 5 Local skin friction coefficient and Heat transfer rate against for different values of Pr with =. and Q =..
376 A.K.M. Safi qul Islam et al. / Procedia Engineering 9 ( 4 ) 37 376 Skin friction 3 Q =. Q =.5 Q =.36 Q =.5 3 Heat transfer.8.6.4. Q =. Q =.5 Q =.36 Q =.5 3 6 9 Fig. 6 Local skin friction coefficient and Local rate of transfer against for different values of Q with =. and Pr =.73. 4. Validity of the Present Work Table depicts the comparisons of the present numerical results of the skin friction with those obtained by Pozzi and Lopo [] and Merkin and Pop [5] respectively. Here, = and Q = and Pr =.733 with 5 is chosen. There is an ecellent agreement among the present results with Pozzi and Lopo [] and Merkin and Pop [5]. Table: Comparison of the present numerical results of skin friction C f with Prandtl number Pr =.733, = and Q = against. /5 = Pozzi and Lupo [] Merkin and Pop [5] Present work.4.6.7.8.9..7.337.43.53.635.74.7.337.43.53.635.745.7.34.43.58.633.748 5. Conclusion From this investigation the following conclusions may be drawn The velocity within the boundary layer increases for decreasing values of Pr and for increasing values of Q. Significant effect is found in velocity profile for Prandtl number. The temperature within the boundary layer increases for increasing values of Q and for decreasing values of Pr. Important effect is found in temperature profile for Prandtl number. The local skin friction coefficient decreases for the increasing values of Pr and increases for increasing values of and Q. Very important effect is found in skin friction for Pr. An increase in the values of and Pr leads to an increase in heat transfer rate. On the other hand, this decrease for increasing values of Q. Significant effect is found in heat transfer rate of.remarkable effects are found in skin friction and rate of heat transfer for Q. The opposite result is observed for Q in local skin friction and the heat transfer distribution. References [] Pozzi, A. and Lupo, M., The coupling of conduction with laminar convection along a flat plate. Int. J. Heat Mass Transfer, Vol. 3, No. 9, pp.87-84, (988). [] Gebhart, B. Effect of dissipation on natural convection, J. Fluid Mechanics, Vol. 4, No., pp. 5 3, (96). [3] Rahaman, M.M and Alim, M. A. Numerical study of magnetohydrodynamic (MHD) free convective heat transfer flow along a vertical flat plate with temperature dependent thermal conductivity, Journal of Naval Architecture and Marine Engineering, JNAME, Vol.6, No., pp.6-9, (June, 9). [4] Alim, M. A., Alam, Md. M., Mamun A.A. and Hossain, Md. B., Combined effect of viscous dissipation & joule heating on the coupling of conduction & free convection along a vertical flat plate, Int. Commun. In Heat & Mass Transfer, Vol. 35, No. 3, pp. 338-346, (8). [5] Merkin J.H and Pop I., Conjugate free convection on a vertical surface, Int. J. Heat Mass Transfer, Vol. 39, pp.57-534, (996). [6] Molla M. Md, Rahman A.and Rahman T. L., Natural convection flow from an isothermal sphere with temperature dependent thermal conductivity. J. Architecture and Marine Engineering, Vol., pp.53-64, (5). [7] Keller, H. B. Numerical methods in boundary layer theory, Annual Rev. Fluid Mech. Vol., pp. 47-433, (978). [8] Miraj, M., Alim, M. A. and Mamun, MAH. Effect of radiation on natural convection flow on a sphere in presence of heat generation, International Communications in Heat and Mass Transfer, Vol.37, pp. 66-665, ().