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Chemical Engineering Hand Written_Notes_Part_32

# Chemical Engineering Hand Written_Notes_Part_32 - 66 3...

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66 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES These equations can be transformed into a set of linear (or nonlinear) algebraic equations by using Taylor’s theorem and approximating μ 2 u ∂x 2 ijk = ( u i +1 ,j,k 2 u i,j,k + u i 1 ,j,k ) ( x ) 2 μ 2 u ∂y 2 ijk = ( u i,j +1 ,k 2 u i,j,k + u i,j 1 ,k ) ( y ) 2 μ 2 u ∂z 2 ijk = ( u i,j,k +1 2 u i,j,k + u i,j,k 1 ) ( z ) 2 and so on. Example 40 . Laplace equation represents a prototype for steady state dif- fusion processes. For example 2-dimensional Laplace equation (3.28) 2 T/∂x 2 + 2 T/∂y 2 =0 represents a description of 2-dimensional steady state heat conduction in a solid , where T is temperature and x, y are space coordinates. Equations similar to this arise in many problems of f uid mechanics, heat transfer and mass transfer. In the present case , we consider conduction in a rectangular plate of dimension L x × L y. The boundary conditions are as follows:

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Chemical Engineering Hand Written_Notes_Part_32 - 66 3...

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