Chemical Engineering Hand Written_Notes_Part_33

Chemical Engineering Hand Written_Notes_Part_33 - 68 3....

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68 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES spline approximation is developed on each interval. For example, n cubic splines f tting n +1 data points can be expressed as p 0 ( z )= α 0 , 0 + α 1 , 0 ( z z 0 )+ α 2 , 0 ( z z 0 ) 2 + α 3 , 0 ( z z 0 ) 3 (3.38) ( z 0 z z 1 ) (3.39) p 1 ( z α 0 , 1 + α 1 , 1 ( z z 1 α 2 , 1 ( z z 1 ) 2 + α 3 , 1 ( z z 1 ) 3 (3.40) ( z 1 z z 2 ) (3.41) ....... = .................................................. p n 1 ( z α 0 ,n 1 + α 1 ,n 1 ( z z n 1 ) + α 2 ,n 1 ( z z n 1 ) 2 + α 3 ,n 1 ( z z n 1 ) 3 (3.42) ( z n 1 z z n ) (3.43) There are total 4 n unknown coe cients { α 0 , 0 1 , 0 ....... α 3 ,n 1 } to be determined. In order to ensure continuity and smoothness of the approximation, the following conditions are imposed p i ( z i y i ; i =0 , 1 , 2 ,...,n 1 (3.44) p n 1 ( z n y n (3.45) p i ( z i +1 p i +1 ( z i +1 ); i , 1 , .... n 2 (3.46) d p i ( z i +1 ) dz = d p i +1 ( z i +1 ) dz ; i , 1 , .... n 2 (3.47) d 2 p i ( z i +1 ) dz 2 = d 2 p i +1 ( z i +1 ) dz 2 ; i , 1 , .... n 2 (3.48) which result in 4 n 2 conditions. Two additional conditions are imposed at the boundary points (3.49) d 2 p 0 ( z 0 ) dz 2 = d 2 p n 1 ( z n ) dz 2 which are referred to as free boundary conditions. If the
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Chemical Engineering Hand Written_Notes_Part_33 - 68 3....

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