Finite Volume Simulation of Steady Laminer Natural Convection Heat Transfer Through a Mercury –filled Triangular Enclosure with an Isothemal Cold Side Walls and an Isothemal Hot Bottom Wall.

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Finite Volume Simulation of Steady Laminer Natural Convection Heat Transfer Through a Mercury -filled Triangular Enclosure with an Isothemal Cold Side Walls and an Isothemal Hot Bottom Wall.
By :

Dr. Ahmed Kadhim Hussein Lecturer-College of Engineering-Mechanical Engineering Department
Babylon University-Babylon City -Hilla -Iraq.

Abstract:
Thermal and flow fields due to laminar steady natural convection in a triangular enclosure having thick conducting sidewalls have been investigated numerically. Inclined left and right side walls are maintained at isothermal cold temperatures while the bottom wall is maintained at isothermal hot temperature. Problem has been analyzed and the non-dimensional governing equations are solved using finite volume approach and employing more nodes at the fluid -solid interface. Triangular enclosure is assumed to be filled with mercury as a liquid metal with a Prandtl number of 0.026. Rayleigh number varies from 10 4 to 10 6 where the flow and thermal fields are computed for various Rayleigh numbers. Consequently, it was observed that the stream function and temperature contours strongly change with high Rayleigh number.The streamline and isotherm plots and the variations of the average Nusselt number at the hot bottom wall and the cold inclined side walls are also presented. The results explained a good agreement with another published results.

Geometry Description, Mathematical Modeling and Assumptions.
The geometry under investigation is a triangular enclosure filled with mercury which its bottom is maintained at a uniform hot temperature (T h ) .The right and left side walls of the enclosure are maintained at a uniform cold temperature (T c ) . The geometry under consideration is shown schematically in where U and V being the non-dimensional velocity components along X and Y axes respectively, P is the dimensionless pressure, Pr is the Prandtl number and Ra is the xx + yy = Q( , ) where F and Q are control functions used to cluster the grid near the walls .The reason is to predict the velocity gradient because there is a friction between the wall and the fluid. The equations are transformed to ( , ) coordinates by interchanging the roles of dependent variables. This yield the following system equations:g 11 x -g 21 x +g 22 x =-J 2 (F +Q ) g 11 y -g 21 y +g 22 y =-J 2 (F +Q ) Where: The steady-state governing equations (3 to 6) are solved by using the finite-volume method using Patankar's algorithm Patankar,1980. A two-dimensional nonuniformly collocated grid system is used. These equations can be written in a general transport equation as follows Patankar,1980: -  12) and (13) ) can be transformed to the following form:- where (U co , V co ) are the contravariant velocity components and S NEW =J S +S N where S N is the source term due to non-orthogonal characteristic of grid system. Also, the transformation coefficients (a 1 , a 2 ) are defined as Hoffmann,1989:-

Grid Sensitivity Test.
The numerical scheme used to solve the governing equations for the current work is a finite volume approach. It provides smooth solutions at the interior domain including the corners. The enclosure is meshed with a non-uniform grid with a very fine spacing near the corners. As shown in Fig.2, the 2-D computational grids are clustered towards the corners. The location of the nodes is calculated using a stretching function so that the node density is higher near the walls of the enclosure. Solutions are assumed to converge when the following convergence criteria is satisfied at every point in the solution domain :- Where represents a dependent variable U, V, P and . In order to obtain grid independent solution, a grid refinement study is performed for uniformly heated bottom wall, (X,0) = 1, and cold inclined side walls, (X,Y) = 0, inside the triangular enclosure with Ra = 10 5 and Pr = 0.026. In the current work, eight combinations (40x40, 50x50, 60x60, 70x70, 80x80, 90x90, 100x100 and 120x120) of control volumes are used to test the effect of grid size on the accuracy of the predicted results.

Results and Discussion.
The

Conclusions.
The following conclusions may be drawn from the results of the present work: