Received 03 February 2013, Accepted 03 April 2013

Transcription

1 International Journal of Mechanical and Materials Engineering (IJMME), Vol. 8 (013), No.1, Pages: NATURA CONVECTION IN A RECTANGUAR CAVITY HAVING INTERNA ENERGY SOURCES AND EECTRICAY CONDUCTING FUID WITH SINUSOIDA TEMPERATURE AT THE BOTTOM WA M. Obayedullah*, M.M.K. Chowdhury and M.M. Rahman Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh *Corresponding author s obayed@math.buet.ac.bd Received 03 February 013, Accepted 03 April 013 ABSTRACT Natural Convection in a rectangular enclosure having internal energy sources and electrically conducting fluid with sinusoidal temperature at the bottom wall has been investigated. The left wall is kept at constant hot temperature but the right wall is at uniform cold temperature. The top wall is adiabatic. The governing partial differential equations are solved by applying weighted residual finite element method. The effects of Hartmann number and internal Rayleigh number on isotherms and streamlines are presented. Also heat transfer from the walls of the cavity is shown graphically. The numerical result demonstrates that the increment of Hartmann number decreases convective heat transmission rate. Keywords: Natural convection, Heat generation, Rectangular cavity, Sinusoidal temperature. Nomenclature A B o c p g H k Nu Nu b Nu s, p Pr Q Ra E Ra I T u, v U, V, y X, Y H/, aspect ratio of the cavity magnetic induction specific heat gravitational acceleration cavity height thermal conductivity cavity width Nusselt number local Nu at the bottom wall local Nu at the side wall fluid pressure Prandtl number heat generation per unit volume eternal Raleigh number internal Raleigh number fluid temperature fluid velocity components dimensionless velocities Cartesian coordinates (m) dimensionless co-ordinates 1. INTRODUCTION Natural convection in rectangular enclosure having internal energy source and electrically conducting fluid with different temperatures at the vertical walls and insulated horizontal walls was etensively investigated eperimentally and numerically (in et al., 000; Mobner and Muller, 1999; Okada and Ozoe, 199; Series and Hurle, 1991). This is because of its application in many physical fields such as solar technology, metal coolants in nuclear reactor and crystal growths in liquids. Gelfgat and Yoseph (001) studied on the linear stability of steady natural convective flow of an electrically conducting fluid within a horizontally elongated rectangular cavity. Sarris et al. (005) conducted an investigation on the natural convective flow in a square enclosure having internal energy sources and electrically conducting fluid. Asaithambi (003), Basak and Ayappa (001) and Nishimura et al. (1995) used finite element method to find the solution of the differential equations for flow and temperature fields. Roy and Basak (005) solved the flow problem using finite element method having penalty parameter with non-uniform and uniform temperatures at one side wall and at the lower wall respectively. Mehmet and Elif (006) investigated natural convective flow in a rectangular enclosure having electrically conducting fluid to find the inclination effect of the enclosure. MHD natural convective flow in an enclosure with partially active side wall has been investigated by Kandaswamy et al. (008). Magnetohydrodynamic natural convective flow in a partitioned enclosure was numerically investigated by Kahveci and Oztuna (009). Billah et al. (011) studied magnetohydrodynmic mied convection fluid flow and heat transfer in an obstructed enclosure having moving lid. Rahman et al. (010) studied the effect of Prandtl and Reynolds numbers on magnetohydrodynmic combined convective heat transfer in a lid-driven enclosure having joule heating effect and a heat conducting circular body in the centre. Hussain (010) investigated mied convective flow and heat transfer from an inclined rectangular enclosure with a sliding wavy hot wall at the top. Combined convection fluid flow in a square cavity having a moving lid and a hot wavy wall was investigated by Hussein and Hussain (010). Combined convective flow within a trapezoidal enclosure having moving lid was numerically studied by Mamun et al. (010). Azwadi and Idris (010) investigated finite different and lattice Boltzmann modeling for simulation of natural convective flow in a square enclosure As per author s knowledge the literature review revealed that non-uniform temperature profile in the wall was not used in the magnetohydrodynamic natural convective flow in rectangular enclosure having heat generating fluid. In this paper the investigation is carried out on the 73

2 MHD natural convective flow in a rectangular cavity having heat generating fluid with sinusoidally heated bottom wall.. GEOMETRY AND EQUATIONS OF MOTION A rectangular enclosure containing incompressible fluid is shown in the Figure 1. The cavity dimensions are defined by for width and H for height. The right and left walls are maintained at cold and hot temperature T c and T T T ) respectively. The bottom wall is h ( h c sinusoidally heated with ( Th Tc ) sin( ) Tc while the remaining wall is considered perfectly insulated. T h y Q Fluid adiabatic ( Th Tc )sin( ) Tc Figure 1 Cavity configuration. The equations of motion are the continuity, momentum and energy equations. The effects of magnetic field, heat generation and thermal buoyancy are taken into considerations. The dimensional form of the steady state of these equations are: u v 0 y B 0 u u 1 p u u u v y y g ( T T g T c v v 1 p v v u v y y y T T T T Q u v y y c p The boundary conditions used are: u(, 0)=u(, H) =u(0,y)=u(, y)=0 v(, 0)=v(, H) =v(0, y)=v(, y)=0 c B0 v ) T(,0)= ( Th Tc ) sin( ) Tc, T(0,y)= T h (1) () (3) (4) T T(,y)= T c, (, H) 0 y where u and v are the and y components of velocity along the horizontal and vertical directions respectively; g denotes gravitational acceleration, p denotes the pressure, T is the fluid temperature, is the fluid density, c p is the fluid specific heat, β denotes the volumetric thermal epansion coefficient, Q denotes the rate of heat generation, B 0 is the magnetic induction, denotes the thermal diffusivity, ν the kinematic viscosity of the fluid and the electrical conductivity.. Eqs.(1) (4) are made non-dimensional by the quantities: X, Y y u v p, U, V, P T Tc, T T where U and V are the dimensionless components of velocity along X and Y directions, and P are the dimensionless temperature and pressure respectively. Eqs.(1) (4) in non-dimensional form are follows. U V X Y 0 U U U V X Y P Pr U U X X Y V V P V V U V Pr X Y Y X Y Ra E Pr Ha PrV P Ra U V X Y X X Y Ra The non-dimensional quantities appeared in the Eqs. (5)-(8) are B0 Pr, Ha, 5 gq Ra I k Ra E h obtained as E I (5) 6 c 7 (8) g ( Th Tc ) and Pr, Ha, Ra E, and Ra I, are Prandtl number, Hartmann number, eternal Rayleigh number, and internal Rayleigh number respectively. The dimensionless boundary conditions are: U(X,0)=U(X,A) =U(0,Y)=U(1,Y)=0 V(X,0)=V(X,A) =V(0,Y)=V(1,Y)=0 (X,0)=sin X, (0,Y)=1, (1,Y)=0 ( X, A) 0 Y 3 74

3 Here A H / is the cavity aspect ratio which is taken as Nusselt number Nu is used as the measurement n of heat transfer, n being normal direction of a plane. Average Nusselt numbers at the vertical and bottom walls are calculated from: A 1 1 Nus NusdY Nub A NubdX NUMERICA METHOD Partial differential equations governing the flow and temperature field are solved by using finite element method. The quadratic triangular element is used to develop the finite element equations. For the velocities and temperature all the si nodes are used and for the pressure only the corner nodes are used. Different types of grid densities have been selected to assess the accuracy of the numerical simulation procedure. The grids chosen are: 1187 nodes, 37 elements; 705 nodes, 47 elements; nodes, 5366 elements; nodes, 6175 elements nodes, 7094 elements. From these values nodes, 5366 elements was found to give sufficient accuracy and hence selected for the simulation study. The nonlinear algebraic equations arising from the finite element formulation are solved by applying the Newton-Raphson iteration technique. For convergency of solutions of all dependent variables the following criteria is used n i, j n1 i, j ERMAX where represents the dependent variables U, V, T and P; the suffies i, j refer to space coordinates and n denotes current iteration.. The value of ERMAX is considered as RESUTS AND DISCUSSIONS The steady natural convective flow in a rectangular cavity is considered. Numerical results are presented considering the effect of internal heat generation and magnetic field. There are four governing parameters in this problem. These are eternal Rayleigh number Ra E, Hartmann number Ha, and internal Rayleigh number Ra I and Prandtl number Pr. The range of Hartmann number Ha is taken from 0 to 50 and the internal Rayleigh number Ra I from 0 to 10 5 while Ra E =10 3 and Pr=0.71 are kept fied. The boundaries of the enclosure are rigid (u=v=0). The top wall is adiabatic ( / Y 0 ) and the bottom wall is sinusoidally heated with ( Th Tc ) sin( ) Tc. The right and left vertical walls of the enclosure are kept at cold and hot temperature T c and Th ( Th Tc ) respectively. (c) (d) The present code was eercised on the work of Sathiyamoorty et al. (007) to check its validity. Some results are presented in Table 1 for comparison. Table 1 Nusselt number comparison for Pr=0.71. Ra E Present Sathiyamoorty et al. (007) % of error The parameters used in this study are Hartmann number, eternal and internal Rayleigh numbers and Prandtl number. The etent of the effect of magnetic field is determined by the magnitude of the Hartmann number. The effect of lateral heat is determined by the magnitude of eternal Rayleigh number. The influence of internal heat generation on the flow is determined by the magnitude of the internal Rayleigh number. Figure Streamline (left) and isotherm (right) showing the effect of internal Rayleigh numbers Ra I =0 Ra I =10 3 (c) Ra I =10 4 (d) Ra I = 10 5 with Ra E =10 3, Ha=0 and Pr= Effects of internal Rayleigh numbers The effect of internal Rayleigh number on the flow has been considered. The flow and temperature distribution has been depicted in Figure where the left column gives the streamlines for increasing values of the internal Rayleigh number Ra I = 0, 10 3,10 4 and 10 5 while the eternal Rayleigh number Ra E = 10 3, the Hartmann number Ha=0 and the Prandtl number Pr=0.71 are kept fied with sinusoidally heated bottom wall and adiabatic 75

4 top wall. The right and left vertical wall are maintained at cold and hot temperature respectively. Figure (a, left) shows the streamlines for Ra I =0, Ra E =10 3 and Ha=0. It is seen that without heat generation there is only one cell. As the internal Rayleigh number increases a secondary cell has been developed in the left side within the cavity as seen in the Figure (c, left). The increasing rate of heat within the cavity due to the increase of internal Rayleigh numbers increases the flow rate in the secondary cell and gradually increases its size until it occupies half of the enclosure which is shown in the Figure (d, left). The effect shown here on the flow of the fluid due to increase of internal Rayleigh number is reasonable since heat generation accelerates the fluid flow and assists the buoyancy force. From the Figure (right column) which represents the isotherms, it is observed that fluid temperature increases significantly due to the increase of internal Rayleigh number. It is noticed from the Figure (a, right)) that there are two parts in the isotherm lines in the cavity. Some isotherms are formed adjacent to the bottom wall of the cavity and gradually they are pushed to the left corner of the bottom wall as the internal Rayleigh number increases. fied. Since Ra I is greater than Ra E, the left wall is uniformly heated and bottom wall is sinusoidally heated, two cells are formed as shown in the Figure 3(a, left). The left cell revolves anticlockwise because of greater internal Rayleigh number and the right cell revolves clockwise as epected. From Figure 3(left column) it is observed that the intensity of the flow decreases due to the increment of Hartmann number. Because of the decrease of the intensity of the flow the left cell i.e the secondary cell is becoming smaller. It is reasonable since magnetic field resists the velocity of the fluid.the influence of the increasing Hartmann number on the isotherms may be viewed from the Figure 3(right column). It is seen that the isotherms become more curved as the Hartmann number increases. This is desired as the magnetic field strength retards the flow of the fluid. (c) (d) Figure 3 Streamline ( left) and isotherm(right) showing the effects of Hartmann number Ha =0 Ha=10 (c) Ha=0 (d) Ha = 50 with Ra E =10 3, Ra I =10 4 and Pr= Effects of Hartmann numbers Figure 3 shows the effect of magnetic field on the flow and temperature of the fluid for different Ha= 0, 10, 0 and 50 where Ra I =10 4, Ra E =10 3 and Pr= 0.71 are kept Figure 4 Nusselt number variation with eternal Rayleigh numbers for sinusoidally heated bottom wall and heated left wall for Hartmann numbers Ha = 0, 10, 0 and Heat transfer rates The influences of eternal Rayleigh number Ra E and Hartmann numbers Ha on the average Nusselt number for sinusoidally heated bottom wall and uniformly heated left wall for Pr=0.71, internal Rayleigh number Ra I =10 4 and Hartmann numbers Ha = 0, 10, 0 and 50 are displayed in Figure 4 via average Nusselt numbers vs eternal Rayleigh numbers plot. Figure 4 shows that heat transfer through the sinusoidally heated wall is negative throughout. It is seen that heat transfer is almost same for the eternal Rayleigh number ranging from 10 3 to and then increases smoothly as the eternal Rayleigh number increases. 76

5 Figure 4 shows that heat transfer through heated left wall is positive and smoothly increases for the eternal Rayleigh numbers from 10 3 to 10 4 then rises rapidly as the eternal Rayleigh numbers increase. As the Hartmann numbers increase the heat transfer from the wall of the enclosure decreases. This happens because the presence of magnetic field dampens the flow. The effects of Ra E and Ra I on average Nusselt numbers for sinusoidally heated bottom wall and heated left wall for Pr=0.71, Hartmann number 0 and internal Rayleigh number Ra I = 10, 10 3, 10 4 and 10 5 are displayed in Figure 5 via average Nusselt numbers vs eternal Rayleigh numbers plot. Figure 5 shows that the heat transfer is negative in all the cases for the sinusoidally heated wall. For the left wall heat treansfer is positive for the internal Rayleigh number Ra I varying from 10 to 10 4 and is negative for 10 5 as shown in the Figure 5. In every case heat transfer is high for Ra I = 10 5 in all the walls. But it is very low in all other cases. Figure 5 Nusselt number variation with eternal Rayleigh numbers for sinusoidally heated bottom wall and heated left wall for internal Rayleigh number Ra I = 10, 10 3, 10 4 and CONCUSIONS Natural convective flow of fluid in a rectangular enclosure having electrically conducting and heat generating fluid has been investigated in this paper by using finite element method. The effect of sinusoidally heated bottom wall with cold right wall and heated left wall has been studied on the fluid flow and heat transfer. The investigation is performed to find the effects of nondimensional parameters namely Hartmann number H and the internal Rayleigh number Ra I on streamlines, isotherms and average Nusselt numbers. Some important findings that are obtained from the present investigations are as follows: Internal Rayleigh number affects the flow and temperature field inside the enclosure significantly. The increasing rate of heat within the cavity due to the increase of internal Rayleigh number leads to form a secondary cell within the cavity and increases in size until it occupies half of the enclosure. Increment of internal Rayleigh number enhances the rate of heat transfer The convective current in the enclosure is reduced as the Hartmann number increases and because of this the size of the secondary cell is reduced. With the increase of the strength of magnetic field the average Nusselt number decreases. REFERENCES Asaithambi, A Numerical solution of the Folkner- Skan equation using piecewise linear functions, Applied Mathematics and Computation 81: Azwadi, C.S.N. and Idris, M.S Finite different and lattice boltzmann modelling for simulation of natural convection in a square cavity, International Journal of Mechanical and Materials Engineering 5 (1): Basak, T. and Ayappa, K.G Influence of internal convection during microwave thawing of cylinders, AIChE Journal 47: Billah, M.M., Rahman, M.M., Saidur, R. and Hasanuzzaman, M Simulation of mhd mied convection heat transfer enhancement in a double liddriven obstructed enclosure, International Journal of Mechanical and Materials Engineering 6 (1): Gelfgat, A.Y. and Yoseph, P.Z Effect of an eternal magnetic field on oscillatory instability of convective flows in rectangular cavity, Physics of Fluids 13: Hussain, S.H Combined convection flow through inclined rectangular enclosure with a sliding wavy hot top surface, International Journal of Mechanical and Materials Engineering 5 (): Hussein, A.K. and Hussain, S.H Mied convection through a lid-driven air-filled square cavity with a hot wavy wall, International Journal of Mechanical and Materials Engineering 5 (): Kahveci, K and Oztuna, S MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure, European Journal of Mechanics 8: Kandaswamy, P, Sundari, S.M. and Nithyadevi, N Magneto convection in an enclosure with partially active vertical walls, International Journal of Heat and Mass Transfer 51: in, S.X., Zhou, M., Azzi, A., Xu, G.J., Wakayama, N.I. and Ataka, M Magnet used for protein crystallization: novel attempts to improve the crystal quality, Biochemical and Biophysical Research Communication 75:

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