Visualization of Natural Convection in Enclosure. Filled with Porous Medium by Sinusoidally. Temperature on the One Side

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1 Applied Mathematical Sciences, Vol., 2012, no. 97, Visualization of Natural Convection in Enclosure Filled with Porous Medium by Sinusoidally Temperature on the One Side Paweena Khansila Department of Mathematics, Faculty of Sciences Khon Kaen University, Khon Kaen 0002, Thailand Centre of Excellence in Mathematics CHE, Si Ayutthaya Rd., Bangkok 1000, Thailand Supot Witayangkurn 1 Department of Mathematics, Faculty of Sciences Khon Kaen University, Khon Kaen 0002, Thailand kku.ac.th Abstract Visualization of natural convection heat transfer in rectangular enclosure filled with porous media and heated sinusoidal temperature on the left vertical wall is computed by using FlexPDE Student version.17. Here, the lower part of the enclosure is heated while the upper part is cooled. The problem is studied for different values of the Rayleigh number10 Ra 10, aspect ratio0.5 AR 2 1 and Darcy number 10 Da 10.The Prandtl number ( Pr ) is kept constant, The results are presented in the form of streamlines, isotherms and heatlines. For the streamlines, there are two circulations inside the enclosure. The values of streamlines and heatlines increase with increasing Rayleigh and Darcy numbers. Keywords: Natural convection, Porous medium, Sinusoidal temperature. 1 Introduction Natural convection in a cavity filled with fluid saturated porous media can be seen in many application of engineering as reviewed by Nield and Bejan [1]. 1 Corresponding Author.

2 802 P. Khansila and S. Witayangkurn Dividers are used to control the convective flows and temperature in the enclosures, which have many engineering applications. The types of divided enclosures with partitions can be classified in four groups: (a) square enclosures [2-], (b) rectangular enclosures [-7], (c) triangular enclosures [8-10] and (d) the complex enclosure [11]. But studies on control of natural convection heat transfer via partitions were very limited for enclosures filled with porous media. AR cavity aspect ratio ( AR H / L) Nomenclature UV, dimensionless fluid velocities c p heat capacity (J kg -1 K -1 ) x, y Cartesian coordinates Da Darcy number (m -2 ) X, Y dimensionless Cartesian coordinates g acceleration due to gravity (ms -2 ) h dimensional heat function Greek symbols H cavity height (m),dimensionless thermal diffusivity (m 2 s -1 ) heat function volumetric coefficient of thermal k thermal conductivity (W m -1 K -1 ) expansion (K -1 ) K permeability of the porous medium penalty parameter L cavity width (m) kinematic viscosity (ms -1 ) p pressure (Pa) dimensionless temperature P dimensionless pressure fluid density(kg m - ) Pr Prandtl number dimensional stream function (m 2 s -1 ) Ra Rayleigh number dimensionless stream function T temperature (K) Subscripts T temperature difference (K) C cold uv, x and y component of fluid velocities H hot Bassak et al. [2] studied numerically on natural convection in square cavity filled porous medium and effects of various thermal boundary conditions for uniformly and non-uniformly temperature in the bottom wall. Roy and Bassak [] analyzed finite element analysis of natural convection flows in square cavity with non-uniformly wall(s). They showed the flows and temperature distribution of uniformly comparing with non-uniformly in cavity. Varol et al. [] performed numerical analysis of natural convection for a porous rectangular enclosure with sinusoidal varying temperature profile on the bottom wall. They considered different aspect ratios and amplitudes of the sinusoidal temperature function. Bilgen et al. [5] performed a similar previous work but the enclosures were heated from the vertical side. Aydin et al. [] considered discrete heating from the bottom while the side walls are cooled. Ganzarolli and Milanez [7] studied the enclosure heated from below and cooling from the sides. Bassak et al. [8] presented the simulation in differentially heated isosceles triangular enclosure filled with porous media. Varol et al. [9] considered visualization of natural convection in porous

3 Visualization of natural convection 80 non-isothermally heated right-angle triangular enclosure. Khansila and Witayankurn [10] studied numerical modeling of natural convection for steady flows in porous media heat the bottom wall of triangular cavity, which was computed using FlexPDE. Cinnakotla et al. [11] studied L-shape cavity. Heatline technique is an important method to visualize heat transport in enclosures filled clear fluid-saturated porous media. The isotherms are used to show the temperature distribution in a domain. Streamline are used to show the flow field in the enclosure. However, it is easy to realize the direction and intensity of heat transfer particularly in convection problems in which path of heat flux is perpendicular to isotherm due to convection effect. The main objective of the present investigation is to study a natural convection flow in a rectangular enclosure filled with porous medium when the walls are well insulated except the left vertical wall is non-uniformly heated. The visualization results show the effect of Rayleigh number, Darcy number and aspect ratio. According to the provided literature above, streamlines, isotherms and heatlines are shown to visualize the flow, temperature and heat transfer. 2 Definition of the Physical Model The Physical model of the two-dimensional rectangular enclosure filled with porous media is shown in Fig. 1. The temperature of the left wall is considered to be sinusoidal temperature distribution. The other three walls are adiabatic. The left wall is given by 2 y T( y) TC Tsin H. (1) T u 0, v0, 0 y u 0, v0 2 y T( y) TC Tsin( ) H H Porous media g T u 0, v0, 0 x L T u 0, v0, 0 y Fig. 1. Physical model, coordinates and boundary conditions.

4 80 P. Khansila and S. Witayangkurn Governing Equation The physical model of fluid flow is assumed to be constant except the density variations causing a body force term in the momentum equation. The Boussinesq approximation is invoked for the fluid properties involving the variation of density with temperature to the flow field. Further, it is assumed that the temperature of the fluid phase is equal to the porous region in the present investigation. Also, a velocity square term could be incorporated in the momentum equations to model the inertia effect which is more important for non-darcy effect on the convective boundary layer flow over the surface of a body embedded in a high porosity media. However, we have neglected this term in the present study because we are dealing with the natural convection flow in rectangular enclosure filled with a porous medium. The governing equation for steady two-dimensional natural convection in the porous enclosure using conservation of mass, momentum and energy equations can be written as: u v 0, (2) x y u u 1 p u u u v u, () x y x x y K v v 1 p v v u v v g( T T ), () C x y y x y K u T v T T T. (5) x y x y where the following non-dimension variables have been used x y T TC X, ul vl Y,,, U, V, Pr, L L T T H C h K H, ( T H T C) g TL Da 2 pl Ra k 2, P 2 L () equation (2) can be written in terms of the stream function defined as: u, v. y x (7) Thus, eqs. (2)-(5) can be written in non-dimensional forms as: U V, X X (8) U U P U U Pr U V Pr U, X X X Da (9) V V P V V Pr U V Pr V RaPr X X Da, (10)

5 Visualization of natural convection 805 U V. (11) X X In order to solve (9)-(10) we use the penalty finite element method where the pressure ( P ) is eliminated by a penalty parameter and incompressibility criteria by (8) which results in form as: U V P. (12) X 7 The values of that yield consistent solution are 10. By using Eqs.(12), the momentum balance Eqs.(9) and (10) reduce to U U U V U U Pr U V Pr U, (1) X X X X Da V V U V V V Pr U V Pr V RaPr X X X Da. Heat function for a dimensional convection problem can be defined as h T cpv( T TC ) k, (15) x y h T cpu( T TC ) k. (1) y x By employing the dimensional parameters defined by the relations in (), Eqs.(15) and (1) then can be written in non-dimensional form as H V, (17) X H U. (18) X Eliminating the temperature gradients of the above equations by cross differentiation, a Poisson-type equation is constructed for the heat function. H H ( U) ( V). (19) X X Boundary Conditions Boundary conditions for the considered model are depicted on the physical model in Fig. 1. In the model, velocities of uand v are equal to zero for all solid surface. On the adiabatic boundaries, temperature gradient is zero. On the left vertical boundary, a sinusoidal temperature is applied. The lower half is heated while the upper half is cooled. The boundary conditions of heat function can be defined as the walls are zero except the left vertical wall which is sin( X ). (1) Results and Discussion In this study, the results are obtained from different values of the interesting 1 parameters, the Darcy number is varied from 10, the Rayleigh number is 10 to

6 80 P. Khansila and S. Witayangkurn varied from 10 to 10, aspect ratio is varied from 0.5 to 2 and the Prandtl number is kept constant at 0.7. The discussion of the following results concerns the streamlines, isotherms and heatlines which occur inside the square, shallow (a) (b) (c) (d) Fig. 2 : Streamlines (left), isotherms (center) and heatlines (right) for AR 1and 1 5 Da 10 ;(a) Ra 10, (b) Ra 10, (c) Ra 10 and (d) Ra 10. and tall enclosure. The left vertical wall of enclosure is heated by sinusoidal temperature, the lower half is heated while the upper half is cooled.

7 Visualization of natural convection Square enclosure Fig. 2 illustrates the streamlines (left), isotherms (center) and heatlines (right) 1 for the value of Darcy number at 10 and different Rayleigh numbers from10 to 10 as shown in Fig 2(a)-2(d) respectively. In this case, aspect ratio is fixed at 1.0. (a) (b) (c) (d) Fig. : Streamlines (left), isotherms (center) and heatlines (right) for AR 1and 5 Da 10 ; (a) Ra 10, (b) Ra 10, (c) Ra 10 and (d) Ra 10. In case of streamlines, it can be seen that double circulation cells were formed in different rotating directions, the cell of upper half rotates in counter clockwise circulation in but the cell of lower half rotates clockwise circulation. Values of streamlines are increased with increasing Rayleigh number and the magnitudes of

8 808 P. Khansila and S. Witayangkurn cells expand close to the right wall. Moreover, isotherms are compressed to the left vertical wall. In this case, double cells of heatlines occur inside the enclosure and they rotate in clockwise direction. The figures show the maximum and minimum values of streamlines, isotherms and heatlines. We can see that in the (a) (b) (c) (d) Fig. : Streamlines (left), isotherms (center) and heatlines (right) for AR 0.5 and 1 5 Da 10 ; (a) Ra 10, (b) Ra 10, (c) Ra 10 and (d) Ra 10. (a) (b) (c) (d) Fig. 5: Streamlines (left), isotherms (center) and heatlines (right) for AR 0.5 and 5 Da 10 ; (a) Ra 10, (b) Ra 10, (c) Ra 10 and (d) Ra 10.

9 Visualization of natural convection 809 case of Rayleigh numbers from 10 5 to 10, the streamlines, isotherms and heatlines fields are symmetric respect to H 0 lines. (a) (b) (c) (d) Fig. : Streamlines (upper), isotherms (center) and heatlines (lower) for AR and Da 10 ; (a) Ra 10, (b) Ra 10, (c) Ra 10 and (d) Ra 10. Similarly, the individual influence of Rayleigh number is shown in Fig., which the same domain and are heater length with Fig. 2 but different Darcy number. In this case, Fig. 2 and are comparable to see the effects of Darcy number on streamlines, isotherms and heatlines fields. When Darcy number is decreased, the values of flows decreases while the values of heat transfer increase for all cases. 1 Moreover, the forms of streamlines and heatlines are similar to Da 10. For this cases, the isotherm of Ra 10 is similar to Ra 10. As Rayleigh number is increased, the values of streamlines increase while the values of heatlines decrease. It can be seen that the isotherms of Da 10 and Ra 10 is similar to 1 Da 10 an Ra 10.

10 810 P. Khansila and S. Witayangkurn.2 Shallow enclosure Fig. illustrates the streamlines (left), isotherms (center) and heatlines (right) 1 for Da 10 and different Rayleigh numbers from 10 to 10 as shown in Fig (a) (b) (c) (d) Fig. 7 : Streamlines (upper), isotherms (center) and heatlines (lower) for AR 2 5 and Da 10 ; (a) Ra 10, (b) Ra 10, (c) Ra 10 and (d) Ra 10. (a)-(d) respectively. In this case, AR 0.5 is considered. The situation of isotherms is similar to Fig. 2. When aspect ratio becomes smaller, the enclosure is also smaller. The flow fields and heat transfer are smaller but the values of streamlines and heatlines are similar to the case of square enclosure. The visualizations of isotherms are similar to square enclosure but it is smaller than the square enclosure. The streamlines, isotherms and heatlines fields are symmetric respect to H 0 for all Rayleigh numbers. For Da 10 as shown in Fig. 5, the streamlines and heatlines pattern in the case of Ra 10 and Ra 10 are similar while the values are different. As Rayleigh number is increased, the values of streamlines increase while the values of heatlines decrease.

11 Visualization of natural convection 811. Tall enclosure Fig. illustrates the streamlines (upper), isotherms (center) and heatlines 1 (lower) for the value of Darcy number at 10 and different Rayleigh numbers from 10 to 10 as shown in Fig. (a)-(d) respectively. In this case, AR 2.0 is considered, which means that the vertical wall is longer than the horizontal wall. The cells circulation of streamlines and heatlines are in the middle position and they are cycles. The isotherms are similar to the previous figure and the contours distribute entire the model. The upper half of isotherms is negative but the lower half is positive. They are also symmetric respect to H 0. When Darcy number is decreased to 10, the values of streamlines decrease while the values of heatlines increase as shown in Fig. 7. However, the values and visualization of isotherms in Fig 7(a) and 7(b) are similar. 5 Conclusion The visualization of natural convection in rectangular enclosure has been carried out. The left vertical wall is heated and other walls are adiabatic. The equally divided active sidewall is heated and cooled with sinusoidal temperature. Such as, the lower half is heated while the upper half is cooled. The influence of Rayleigh number, of the aspect ratio and Darcy number on the heat transfer characteristic is examined. The results and discussion are presented. The following main conclusions have been drawn. The flow fields, temperature and heat transfer are symmetric with respect to the horizontal plane equally dividing lower heated and the upper cooled sections. For heatlines, it can be seen that there are double circulation cells which they rotate in clockwise direction. All cases, the values of streamlines increase with increasing Rayleigh number while the values of heatlines decrease. For streamlines, it is observed that there are double circulation cells. The direction of the upper half rotates in counterclockwise direction while the lower half rotates in clockwise direction ACKNOWLEDGEMENTS This research is supported by Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The authors would like to thank Department of Mathematics, Faculty of Science, Khon Kean University for computational resources in this work. References [1] D.A. Nield and A. Bejan, Convection in Porous Media, Springer-Verleg, New York, 1999.

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