solutions to HW #6

solutions to HW #6 - ChemE 109 - Numerical and Mathematical...

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ChemE 109 - Numerical and Mathematical Methods in Chemical and Biological Engineering Fall 2007 Solution to Homework Set 6 1. The following conservation equation describes the steady-state temperature distribution in a metal square: 2 T ∂x 2 + 2 T ∂y 2 = 0 subject to the following boundary conditions: ∂T ∂y ( x, 0) = 0 , ∂T ∂y ( x, 1) = 0 T (0 ,y ) = 0 , T (1 ,y ) = f ( y ) (a) The centered finite difference method will be employed to compute the temperature profile in the rectangle. Set up the grid in ( x,y ) dimensions, with ( i,j ) serving as local indices, as shown below: (i-1, j) (i, j) (i+1, j) (i, j-1) (i, j+1) Use three equidistant interior nodes and the same discretization step in both x and y dimensions (i.e. i = 1 ,..., 5, j = 1 ,..., 5 and Δ x = Δ y ). Show the exact form of the equations needed to compute T ij , for i = 2 , 3 , 4 and j = 2 , 3 , 4. (b) Use the method of separation of variables to compute an analytic series solution for the above problem. Solution: (a) In the figure below, the shadowed area is the metal square. j = 0 , 6 are fictitious nodes due to the boundary conditions on x-sides. The unknown variable is T j i , where the upper 1
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x 1 2 3 4 5 6 2 3 4 5 y j=0 i=1 T i j subindex denotes x-coordinate and the lower subindex denotes y-coordinate. The black dots in the figures denote the unknown nodes. Since the lattice is equidistantly divided Δ x = Δ y = h where h = 1 4 = 0 . 25. Discretization of the nodes and boundary conditions using centered finite difference method 2 T ∂x 2 = T j i +1 - 2 T j i + T j i - 1 h 2 2 T ∂y 2 = T j +1 i - 2 T j i + T j - 1 i h 2 ∂T ∂y = T j +1 i - y j - 1 i 2 h B. C. 1: ∂T ∂y ( x, 0) = 0 T 1 i = T 3 i , i = 0 , 1 , 2 , 3 , 4 B. C. 2: ∂T ∂y ( x, 1) = 0 T 3 i = T 5 i , i = 0 , 1 , 2 , 3 , 4 2
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B. C. 3: T (0 ,y ) = 0 T i 1 = 0 , i = 0 , 1 , 2 , 3 , 4 B. C. 4: T (1 ,y ) = f ( y ) T i 5 = f ( y i ) , i = 0 , 1 , 2 , 3 , 4 Node at ( i , j ), i = 2 , 3 , 4 and j = 1 , 2 , 3 , 4 , 5: T j i +1 - 2 T j i + T j i - 1 h 2 + T j +1 i - 2 T j i + T j - 1 i h 2 = 0 The equations are linear regardless of the format of f ( y ), so Jacobi or Gauss-Seidal are applicable to solve the solution. (b) Use the method of separation of variables to compute an analytic series solution for the above problem. This is an elliptic PDE
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solutions to HW #6 - ChemE 109 - Numerical and Mathematical...

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