# chapt10 - Supplement Finite Difference 1 Finite difference...

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1 Supplement: Finite Difference 1. Finite difference method The complete governing equations in boundary layer: x u + y v = 0 (1) u x u + v y u = g β (T - T ) + ν 2 2 y u (2) u x T + v y T = α 2 2 y T (3) For an analytical solution, it allows for temperature determination at any point of interest in a medium. But for most the practical heat and mass transfer problems considered, the analytical solution is not available, we have to use the numerical method to get the temperature distribution at only discrete points . The nodal network For numerical simulation, the first step is to select the discrete points. Referring to Figure, this is done by subdividing the medium into a number of small regions and assigning to each a reference point that is at its center. The reference point is called a nodal point or node. The aggregate of points is called a nodal network, grid, or mesh. For two dimensional system, the x and y locations are designated by the m and n indices, respectively. Each node represents a certain region, and its temperature and velocity are a measure of the average temperature of the region. The selection of nodes is rarely arbitrary, usually it depends on the geometric convenience and the desired accuracy.

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2 The numerical accuracy of the calculations depends strongly on the number of the nodes. For a fine mesh, the extremely accurate solutions can be obtained. 100 o C 100 o C 20 o C 20 o C
3 Finite-Difference Form of the Conduction Energy Equation The two-dimensional conduction equation: 2 2 x T + 2 2 y T = 0 (4) Consider the second derivative, 2 2 x T , for the interior point, from the figure, n , m x T 2 2 x x T x T n , m n , m + 2 1 2 1 (5) The temperature gradients may in turn be expressed as n , m x T 2 1 + x T T n , m n , m + 1 (6)

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4 n , m x T 2 1 x T T n , m n , m 1 (7) So, n , m x T 2 2 () 2 1 1 2 x T T T n , m n , m n , m + + (8) Similarly, n , m y T 2 2 2 1 1 2 y T T T n , m n , m n , m + + (9) Using a mesh for which x = y, Then the conduction equation, 2 2 x T + 2 2 y T = 0, can be written as T m,n+1 + T m,n-1 + T m+1,n + T m-1,n - 4T
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chapt10 - Supplement Finite Difference 1 Finite difference...

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