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Chemical Engineering Hand Written_Notes_Part_115

Chemical Engineering Hand Written_Notes_Part_115 -...

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232 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES (10.59) v ( n ) ( z ) = N n ( z ) = z z n 1 z ; z n 1 z z n 0 Elsewhere Introduction of these trial functions allows us to express the approximate solu- tion as (10.60) b u ( z ) = b u 0 v (0) ( z ) + ...... + b u n v ( n ) ( z ) and now we can work with b u = h b u 0 b u 2 ... b u n i T as unknowns. The optimum parameters b u can be computed by solving equation (10.61) A b u b = 0 where (10.62) ( A ) ij = ¿ dv ( i ) dz , dv ( j ) dz À and dv ( i ) dz = ( 1 / z on interval left of z i 1 / z on interval right of z i ) If intervals do not overlap, then (10.63) ¿ dv ( i ) dz , dv ( j ) dz À = 0 The intervals overlap when (10.64) i = j : ¿ dv ( i ) dz , dv ( i ) dz À = z i Z z i 1 (1 / z ) 2 dz + z i Z z i 1 ( 1 / z ) 2 dz = 2 / z or i = j + 1 : ¿ dv ( i ) dz , dv ( i 1) dz À = z i Z z i 1 (1 / z ) . ( 1 / z ) dz = 1 / z (10.65) i = j 1 : ¿ dv ( i ) dz , dv ( i +1) dz À = z i Z z i 1 (1 / z ) . ( 1 / z ) dz = 1 / z (10.66)
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10. NUMERICAL METHODS BASED ON OPTIMIZATION FORMULATION 233 Thus, the matrix A is a tridiagonal matrix (10.67) A = 1 / z
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