numerical

numerical - Numerical Methods Due to the increasing...

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Numerical Methods Due to the increasing complexities encountered in the development of modern technology, analytical solutions usually are not available. For these problems, numerical solutions obtained using high-speed computer are very useful, especially when the geometry of the object of interest is irregular, or the boundary conditions are nonlinear. In numerical analysis, two different approaches are commonly used: 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: 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 equations Solve the system of equations using numerical schemes
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The Nodal Networks The basic idea is to subdivide the area of interest into sub-volumes with the distance between adjacent nodes by x and y as shown. If the distance between points is small enough, the differential equation can be approximated locally by a set of finite difference equations. Each node now represents a small region where the nodal temperature is a measure of the average temperature of the region. Example: m,n m,n+1 m,n-1 m+1, n m-1,n y x m-½,n intermediate points m+½,n x=m x, y=n y
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Finite Difference Approximation 2 P 2 1 Heat Diffusion Equation: , k where = is the thermal diffusivity C No generation and steady state: q=0 and 0, 0 t First, approximated the first order differentiation at intermediate q T T k t V T α ρ + = = ⇒ ∇ = & & 1, , ( 1/ 2, ) ( 1/ 2, ) , 1, ( 1/ 2, ) ( 1/ 2, ) points (m+1/2,n) & (m-1/2,n) T x T x m n
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numerical - Numerical Methods Due to the increasing...

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