FINITE DIFFERENCE

FINITE DIFFERENCE - FINITE DIFFERENCE In numerical analysis...

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Unformatted text preview: FINITE DIFFERENCE In numerical analysis, two different approaches are commonly used: In The finite difference and the finite element methods. In heat transfer problems, the finite difference method is used more often and will be discussed here. The finite difference method involves: discussed Establish nodal networks Derive finite difference approximations for the governing equation at both interior and exterior nodal points Develop a system of simultaneous algebraic nodal Develop equations equations Solve the system of equations using numerical schemes The Nodal Networks Finite Difference Approximation Finite Difference Approximation cont. Finite Difference Approximation cont. A System of Algebraic Equations Matrix Form Numerical Solutions Iteration Example Example (cont.) Example (cont.) Summary of nodal finite-difference relations for various configurations: Case 1 Interior Node Tm,n +1 + Tm,n −1 + Tm +1,n + Tm −1,n − 4Tm,n = 0 Case 2 Node at an internal corner with convection 2(Tm−1,n + Tm,n +1 ) + (Tm+1,n + Tm,n −1 ) + 2 h∆ x h∆ x T∞ − 2(3 + )Tm,n = 0 k k Case 3 Node at a plane surface with convection (2Tm −1,n + Tm ,n +1 + Tm ,n −1 ) + 2 h∆x h∆x T∞ − 2( + 2)Tm ,n = 0 k k Case 4 Node at an external corner with convection (Tm ,n −1 + Tm −1,n ) + 2 h∆x h∆x T∞ − 2( + 1)Tm ,n = 0 k k Case 5 Node at a plane surface with uniform heat flux (2Tm −1,n + Tm ,n +1 + Tm ,n −1 ) + 2q ' ' ∆x − 4Tm ,n = 0 k ...
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