Finite Element Analysis of Fully Developed Unsteady MHD Convection Flow in a Vertical Rectangular Duct with Viscous Dissipation and Heat Source/Sink

Transcription

1 Journal of Applied Science and Engineering, Vol. 18, No. 2, pp (2015) DOI: /jase Finite Element Analysis of Fully Developed Unsteady MHD Convection Flow in a Vertical Rectangular Duct with Viscous Dissipation and Heat Source/Sink Naikoti Kishan* and Balla Chandra Shekar Department of Mathematics, Osmania University, Hyderabad, Telangana State, India Abstract The combined effect of viscous and Ohmic dissipations on unsteady, laminar magneto convection fully developed flow in a vertical rectangular duct considering the effects of heat source/ sink is investigated. Finite element method based on Galerkin weighted residual approach is used to solve two dimensional governing momentum and energy equations for unsteady, magneto convection flow in a vertical rectangular duct. The investigations are conducted for the effects of various flow parameters such as buoyancy parameter N, Hartmann number M, aspect ratio A, circuit parameter E and heat source/sink parameter. The results indicate that the flow pattern and the temperature field are significantly dependent on the above mentioned parameters. It is shown that buoyancy parameter, Hartmann number and aspect ratio parameter increase both the velocity and temperature for open circuit (E 0) but decrease for short circuit (E = 0). Key Words: MHD, Fully Developed, Rectangular Duct, Heat Source/Sink, Finite Element Method, Electric Field 1. Introduction *Corresponding author. kishan_n@rediffmail.com Fully developed flow in ducts has been the focus of extensive investigation for many decades due to its wide range of applications in the design of compact heat exchanger and other heat transfer equipments, such as boilers of power-generating equipment, fossil fuel-fired industrial furnacesformaterial processing, etc. Aung and Worku [1,2] discussed the theory of combined free and forced convection, respectively, in a vertical channel with flow reversal conditions for both developing and fully developed flows. The fully developed flow of uniformly conducting and incompressible fluids through ducts under the action of a transverse magnetic field is attracting considerable interest at the present time. Magnetohydrodynamic generators, pumps and accelerators are devices of practical importance in which conducting fluids are passed through transverse magnetic fields with applied electric field and involve large values of buoyancy parameter. A conducting fluid in a pipe or duct is forced to move by the Lorentz force created when mutually perpendicular magnetic field and electric current are applied perpendicular to the pipe or duct. The Lorentz force alters the flow profile and consequently the heat transfer characteristics of the flow. Therefore, a detailed study of the influence of magnetic field, applied electric field and buoyancy on magnetoconvection in a duct or channel with walls maintained at different temperatures is necessary. Giuseppe and Michele [3] studied fully developed mixed magnetohydrodynamic convection in a vertical square duct. Most of the magnetohydrodynamic generators and pumps have a rectangular crosssection. Exact solutions have been obtained for laminar flows of uniformly conducting incompressible fluids through rectangular ducts with thin conducting walls under transverse magnetic fields by Chang & Lundgren

2 144 Naikoti Kishan and Balla Chandra Shekar [4], Hunt [5] and Uflyand [6]. The effect of electric and magnetic fields on the laminar MHD flow is studied by Romig [7]. Analytical solutions for the flow and temperature fields of simplified MHD problems were investigated by Sutton & Sherman [8], Alpher [9] and Nigam & Singh [10]. The viscous dissipation effect can become important in internal Winter [11] and external Shadid and Eckert [12] flow of polymers. Shadid and Eckert [12] reported that there is a significant increase in temperature of molten polymers due to viscous dissipation. Asymptotic solution for MHD flow between two infinite parallel plates is presented by Nigam and Singh [10]. For rectangular vertical duct, Buhler [13] analyzed the fluid flow problem in magnetic field with or without buoyancy effect. Yakubenko [14] analyzed numerically the flow of an electrically conducting liquid in a rectangular duct for the effects of magnetic field and induced current. Prabal Talukdar and Mitesh Shah [15] carried out numerical simulations for laminar mixed convective fluid flow and heat transfer in ducts. The influence of heat source/sink in moving fluids is important in view of several practical problems such as fluids undergoing exothermic or endothermic chemical reaction. Possible heat generation effects may alter the temperature distribution, consequently the particle diposition rating nuclear reactors, electronic chips and semiconductor wafers. Steady and unsteady convective flow of a viscous incompressible heat generation/absorption fluid is important due to substantial difference between the temperatures of surface and ambient fluid in so many fluid flow problems of physical interest. And those concerned fluids with heat generation/absorbtion can be found in the works of Sparrow and Cess [16] and Chamkha [17]. Jha and Ajibade [18] considered free convection flow of heat generating/absorbing fluid in a vertical porous channel due to periodic heating of the walls and temperature dependent heat generation/absorption. Kamel [19] studied unsteady MHD convection flow through a porous medium bounded by an infinite vertical porous plate with temperature-dependent source/sink. Recently, Umavathi et al. [20] analyzed the MHD effects of free convective, fully developed and laminar flow in a vertical rectangular duct under the influence of ohmic heating and viscous dissipation. Umavathi and Liu [21] studied a problem of steady, laminar mixed convective flow and heat transfer of an electrically conducting fluid through a vertical channel with heat source or sink. Limited attentions have been received on the study of unsteady MHD viscous, incompressible fully developed flow in a vertical rectangular duct with heat source/ sink. Therefore, in this paper, we solve unsteady MHD flow of a viscous, incompressible, laminar and electrically conductive fluid in a two dimensional vertical rectangular duct for a fully developed regime in the presence of uniform magnetic field, applied normal to the walls of the duct and uniform electric field, applied perpendicular to the magnetic field with heat source/sink parameter under the combined effect of viscous and Ohmic dissipations. The governing equations are solved by finite element method based on Galerkin weighted residual scheme. 2. Mathematical Formulation The unsteady, laminar free convection flow of a conducting fluid with constant properties in the entrance region of an infinitely vertical duct of rectangular crosssection is considered. The physical configuration of the system examined and of the co-ordinate axes is shown in Figure 1. The fluid flow is along Z-axis, i.e. only the velocity component W in Z-direction is non-vanishing. The length and width of the rectangular cross-section of the duct are a and b respectively. T 1 and T 2 are the constant temperatures of the walls Y = 0 and Y = b respectively, where T 2 > T 1. The other two walls of the duct are insu- Figure 1. Physical configuration.

3 Fully Developed Unsteady MHD Convection Flow in a Vertical Rectangular Duct with Heat Source/Sink 145 lated, i.e. are maintained at T 0 for X = a and X =0. X A uniform magnetic field B 0 is applied across the duct normal to the walls along Y-direction and the uniform electric field E 0 is applied along X-direction. Fluid rises in the duct driven by buoyancy forces. Hence the flow is due to difference in temperature and the convection sets in instantaneously. Moreover the gradient of T 2 T 1 is perpendicular to the direction of gravity which is called as Oberbeck convection and therefore there is no pressure gradient in the basic equation. All the fluid properties except density in the buoyancy term are considered as constant. Moreover, the effects of Ohmic heating and viscous dissipation are considered in addition to internal heat generation or absorption. The flow is fully developed and the following relations apply here: sink. From the continuity equation 1, one can conclude that W does not depend on Z. Hence, W is a function of ( T1 T2) X, Y. We assume that T0. Equations 2 and 3 2 are solved subject to the following boundary conditions: Using the following dimensionless variables, (4) Equations 2 and 3 can be written as, Under these assumptions and using the Boussinesq approximation, the continuity, momentum and energy equations for the flow and heat transfer through the infinite duct, can be written as Hughes and Young [22], (1) (5) (6) and the boundary conditions 4 become, (2) (7) (3) where T is the fluid temperature, g is the acceleration due to gravity, T 0 is the reference temperature, Q(T T 0 ) is the rate of volumetric internal heat and the other physical quantities are mentioned in the list of symbols. The third and the fourth terms in equation 2 are the buoyant force and the Lorentz force, respectively. Likewise, the last three terms in equation 3 are, respectively, the viscous dissipation, Ohmic heating and heat source/ 1 e where M B 0 b( ) 2 is the magnetic field parameter, N = g T b 0 ( T 2 T 1 ) is the buoyancy parameter, = bq 2 k 0 C p E0 is heat source/sink parameter, E 2 Bg b( T T) 0 T 2 1 is the circuit parameter and A a is the aspect ratio. b

4 146 Naikoti Kishan and Balla Chandra Shekar Here B 0 is the magnetic field strength, e is the electrical conductivity, is the dynamic viscosity, T is the coefficient of thermal expansion, 0 is the density of the fluid, v is the kinematic viscosity, k is the thermal conductivity of the fluid, C p is the specific heat at constant pressure and Q is the heat source/sink. 3. Method of Solution The dimensionless partial differential equations 5, 6 subject to the boundary conditions 7 are solved by weighted residual Galerkin finite element method. In weighted residual approach the unknowns are replaced by approximate trial solutions to obtain the residuals. In the context of a discretized domain the trial solutions are given by polynomial relationships to obtain the residuals. These residuals are then multiplied by weight functions and their integrals over an element domain are set to be zero. Let the approximate solutions of w, are where i are the linear interpolation functions for a triangular element. The Galerkin finite element model for a typical element e is given by where And e is boundary of the element e, n is the unit outward normal. Time derivative terms are approximated by using Crank-Nicolson scheme. The whole domain is divided into 800 triangular elements of equal size. Each element is three nodded, therefore whole domain contains 441 nodes. Each element matrix is of order 6 6, since at each node two functions are to be evaluated. Hence after assembly of the elemental equations, we obtain a system of 882 non-linear coupled equations. To linearize the system of equations the functions w, are incorporated, which are assumed to be known. After imposing the boundary conditions, a matrix of system of linear equations of order is remained. This system has been solved by using Gauss siedel iteration method. The convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the convergence criterion such that n+1 n 10-4, where n is the number of iterations and stands for w and. 4. Results and Discussions where The numerical computations were carried out for lines of equal velocity and isotherms for different flow parameters such as magnetic field parameter M, aspect ratio A, heat source/sink parameter, buoyancy parameter N and circuit parameter E. For Figures 2 4 the circuit parameter is chosen to be E = -1, 0, 1. The width b of duct is fixed (b =1)iny-direction where as the length of the duct in x-direction a is changed in such a way that the aspect ratio A takes the values A =0.5,1,2.Undera global view of these figures, it is observed that the lines of equal velocity are compressed near the walls indicating the larger velocity gradient near the walls. It is observed that the flow with single vortex occupying the entire region is formed when the electric field is non zero (E 0). A strong flow cell is formed because the flow is dominated by the combined effects of magnetic field and electric field than the buoyancy force. It is also observed

5 Fully Developed Unsteady MHD Convection Flow in a Vertical Rectangular Duct with Heat Source/Sink 147 Figure 2. Lines of equal velocity for A = 0.5; A = 1; A = 2 and M = 2; N = 0.5; l = 0; E = -1, 0, 1 (left to right). Figure 3. Lines of equal velocity for M = 2; M = 4; M = 5 and A = 0.5; N = 0.5; l = 0; E = -1, 0, 1 (left to right).

6 148 Naikoti Kishan and Balla Chandra Shekar that the fluid flow is upward for positive values of E and downward for negative values of E. In the absence of electric field (E = 0), the flow is weakened and single vortex of the flow is split into two vortices rotating in the opposite directions. This is because of the presence of the buoyancy force and magnetic force. The flow is downward in the region 0 y 0.5 and upward in the region 0.5 y 1 and the flows look anti-symmetric about the midsection y = 0.5. The effect of aspect ratio A on velocity profiles is shown in Figure 2. The aspect ratio intensifies the upward flow of the case E =1,the downward flow of the case E = -1 and both the downward and upward flows of the case E = 0. It is observed that flow rates are increased because the increasing aspect ratio widens the flow regions of both upward and downward flows. Figure 3 depicts the effect of magnetic field parameter on the velocity profiles. It is determined that the increase in the magnetic field parameter M enhances the velocity profile when E 0. This is because the combined effect of magnetic field and electric field. It is observed that the increase in the magnetic field parameter flattens the velocity profile when E = 0. The strengths of the upward flow (0.5 y 1) and downward flow (0 y 0.5) are decreased. Application of a uniform transverse magnetic field normal to the flow of an electrically conducting fluid gives rise to a resistive force that acts in the direction opposite to that of the flow. This force is called Lorentz force. This resistive force tends to slow down the motion of the fluid. Figure 4 illustrates the velocity profile for the effect of buoyancy parameter N. The buoyancy parameter N increases the velocity profile when the electrical potential becomes non zero while it has no significant effect on the velocity profile when the electrical potential becomes zero. From Figures 2 4 it was observed that the obtained numerical solutions of were found to be in good agreement with the solutions of Umavathi [20] in the absence of heat source/sink. In Figures 2 4, is set to zero. If is set to nonzero e.g. -1 or 1, the line pattern of equal velocity is qualitatively similar to Figures 2 4 and hence not shown here. Figure 5 shows the influence of aspect ratio A on the isotherms for heat source/sink parameter = -1, 0, 1. It is observed that the isotherms are compressed near the Figure 4. Lines of equal velocity for N = 0.5; N = 1; N = 2 and A = 0.5; M =2; =0;E = -1, 0, 1 (left to right).

7 Fully Developed Unsteady MHD Convection Flow in a Vertical Rectangular Duct with Heat Source/Sink 149 Figure 5. Isotherms for A = 0.5; A = 1; A = 2 and M =2;N = 0.5; E =1; = -1, 0, 1 (left to right). cold wall y = 0 and compressibility decreases from cold wall y = 0 to heated wall y = 1. This indicates that heat flux from the fluid to cold wall is increased and heat flux from the fluid to heated wall is decreased. The maximum value of the isotherms increases with the increase of aspect ratio when 0. The effect of A is negligible in the case of heat source ( < 0). Figure 6 is plotted for the effect of magnetic field parameter on dimensionless temperature for = -1, 0, 1. From the figure it is evident that the increment in M leads to accelerate the temperature profiles in both the cases of source as well as sink. It is observed that for the large value of M, larger the maximum value of the isotherms for the cases = -1, 0, 1. It is also observed that the maximum value of the isotherms occurs in the upper region due to the increasing effect of Ohmic heating. For higher values of M, heat flows from the top wall to the fluid and from the fluid to the bottom wall. Since the effect of Ohmic heating is strong the heat flows back from fluid to top wall also. But for lower values of M, heat flows from the fluid to bottom wall only, being the effect of Ohmic heating is not very strong. Thus two groups of isotherms are formed for higher values of M. Figure 7 shows the effect of buoyancy parameter N on the isotherms for = -1, 0, 1. It is noticed that the temperature rises as buoyancy parameter N increases. It is observed that the maximum value of isotherms also increases with the heat source/sink parameter. Itis clearly noticed that the maximum value of isotherms increase rapidly with the increase of N for the cases of source and sink. The effect of buoyancy parameter N is more in the case of sink ( = 1). Interestingly it is noticed from the figure that the maximum value of temperature is attained in case of = 1 compared to other cases. The effects of viscous and Ohmic heating increase with the increase of N. Therefore, for larger values of N, heat flow directs from the fluid to both the top and bottom walls as explained for the effect of magnetic field parameter. Two groups of isotherms are appeared for higher values of N. The effect of heat source/sink on heat transfer is very much significant for the flow where heat transfer is given prime importance.

8 150 Naikoti Kishan and Balla Chandra Shekar Figure 6. Isotherms for M =2; M = 4; M = 5 and A = 0.5; N = 0.5; E =1; = -1, 0, 1 (left to right). Figure 7. Isotherms for N = 0.5; N = 1; N = 2 and A = 0.5; M =2;E =1; = -1, 0, 1 (left to right).

9 Fully Developed Unsteady MHD Convection Flow in a Vertical Rectangular Duct with Heat Source/Sink Conclusions An analysis is performed to investigate the flow and magnetoconvection heat transfer of fully developed flow regime in a vertical rectangular duct in the presence of applied electric field. The governing partial differential equations are solved by using finite element method. Comparisons with previously published work on special cases of the problem were performed and found to be in good agreement. As a result the following are determined: Magnetic field parameter M, aspect ratio A, and buoyancy parameter N increase the velocity profiles when the electric field in non zero while they decrease the velocity profiles when the electric field is zero. It is noteworthy that heat source/sink parameter increases the velocity profiles even when the electric field is zero. The effect of heat source/sink parameter is to increase the temperature profiles. It is also found that the temperature profile increases with the heat source/sink parameter, buoyancy parameter N, magnetic field parameter M and aspect ratio A. In the cases of = 0 and 1, temperature profiles increase with the increase of aspect ratio A. However the effect of aspect ratio is negligible in the case of = -1. Nomenclature B 0 Magnetic field strength, Wb/m 2 C p Specific heat capacity at constant pressure, KJ/Kg C E 0 Electric field, N/C g Acceleration due to gravity, m 2 /s k Thermal conductivity of the fluid, W/m C M Hartmann number or magnetic field parameter N Buoyancy parameter Q Heat generation/absorption coefficient, W/m 3 C T Fluid temperature, C T 0 Reference temperature, C T 1, T 2 Temperatures of the walls of the duct, C W Velocity in Z-direction, m/s X, Y, Z Cartesian coordinates, m T Coefficient of thermal expansion, C -1 Heat source/sink parameter Dynamic viscosity, Ns/m 2 0 Fluid density, Kg/m 3 e Electrical conductivity, S/m Dimensionless temperature Kinematic viscosity, m 2 /s References [1] Aung, W. and Worku, G., Theory of Fully Developed Combined Convection Including Flow Reversal, ASME J Heat Transf., Vol. 108, pp (1986). doi: / [2] Aung, W. and Worku, G., Developing Flow and Flow Reversal in a Vertical Channel with Asymmetric Wall Temperature, ASME J Heat Transf., Vol. 108, pp (1986). doi: / [3] Giuseppe, S. and Michele, C., Fully Developed Mixed Magnetohydrodynamic Convection in a Vertical Square Duct, Numer. Heat Transfer A., Vol. 53, pp (2008). doi: / [4] Chang, C. C. and Lundgren, T. S., Duct Flow in Magnetohydrodynamics, Z. Angew. Math. Phys. (ZAMP), Vol. 12, pp (1961). doi: /BF [5] Hunt, J. C. R., Magnetohydrodynamic Flow in a Rectangular Duct, J. Fluid Mech., Vol. 21, No. 4, pp (1965). doi: /S [6] Uflyand, Y. S., Flow of Conducting Fluid in a Rectangular Channel in a Transverse Magnetic Field, Sov. Phys. Tech. Phys., Vol. 5, p (1961). doi: / [7] Romig, M., The Influence of Electric and Magnetic Field on Heat Transfer to Electrically Conducting Fluids, Adv Heat Transf., Vol. 1, pp (1964). doi: /S (08)70100-X [8] Sutton, G. W. and Sherman, A., Engineering Magnetohydrodynamic, McGraw-Hill, New York (1965). [9] Alpher, R. A., Heat Transfer in Magnetohydrodynamic Flow between Parallel Plates, Int J Heat Mass Transf., Vol. 3, pp (1960). doi: / (61) [10] Nigam, S. D. and Singh, S. N., Heat Transfer by Laminar Flow between Parallel Plates under the Action of Transverse Magnetic Field, Q J Mech Appl Math., Vol. 13, pp (1960). doi: /qjmam/ [11] Winter, H. H., Viscous Dissipation in Shear Flows of Polymers, Adv Heat Transf., Vol. 13, pp (1977). doi: /S (08)

10 152 Naikoti Kishan and Balla Chandra Shekar [12] Shadid, J. N. and Eckert, E. R. G., Viscous Heating of a Cylinder with Finite Length by a High Viscosity Fluid in Steady Longitudinal Flow, Int J Heat Mass Transf., Vol. 35, pp (1992). doi: / (92) [13] Buhler, L., Laminar Buoyant Magnetohydrodynamic Flow in a Rectangular Duct, Phys Fluids., Vol. 10, pp (1998). doi: / [14] Yakubenko, A. E., Calculation of the Flow of an Electrically Conducting Liquid in a Duct of Rectangular Cross Section with Electrodes, Fluid Dynamic, Vol. 5, pp (1970). doi: /BF [15] Prabal, T. and Mitesh, S., Analysis of Laminar Mixed Convective Heat Transfer in Horizontal Triangular Ducts, Numerical Heat Transfer, Part A., Vol. 54, pp (2008). doi: / [16] Sparrow, E. M. and Cess, R. D., Temperature Dependent Heat Sources or Sinks in a Stagnation Point Flow, Appl Sci Res A., Vol. 10, pp (1961). doi: /BF [17] Chamkha, A. J., On Laminar Hydromagnetic Mixed Convection Flow in a Vertical Channel with Symmetric and Asymmetric Wall Heating Conditions, Int Commun Heat Mass Transf., Vol. 45, pp (2002). doi: /S (01) [18] Jha, B. K. and Ajibade, A. O., Free Convective Flow of Heat Generating/Absorbing Fluid between Vertical Porous Plates with Periodic Heat Input, Int. Comm. Heat Mass Transfer., Vol. 36, p. 624 (2009). doi: /j.icheatmasstransfer [19] Kamel, M. H., Unsteady MHD Convection through Porous Medium with Combined Heat and Mass Transfer with Heat Source/Sink, Energy Conversion and Management, Vol. 42, p. 393 (2001). doi: / S (00) [20] Umavathi, J. C., Liu, I. C., Prathap Kumar, J. and Pop, I., Fully Developed Magneto Convection Flow in a Vertical Rectangular Duct, Heat Mass Transfer, Vol. 47, pp (2011). doi: /s [21] Umavathi, J. C. and Liu, I. C., Magnetoconvection in a Vertical Channel with Heat Source or Sink, Meccanica, Vol. 48, pp (2013). doi: / s [22] Hughes, W. F. and Young, F. J., The Electro Magneto Hydrodynamics of Fluids, Wiley, New York (1996). Manuscript Received: Nov. 13, 2014 Accepted: Apr. 29, 2015