Chemical Engineering Hand Written_Notes_Part_66

Chemical Engineering Hand Written_Notes_Part_66 - 134 4....

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134 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES and F ( x ) is a n × 1 vector, the associated sensitivity equations are de f ned as (1.47) d Φ ( z ) dz = ∂F x ¸ Φ ( z ); Φ (0) = I where (1.48) Φ ( z )= x ( z ) x 0 ¸ represents the n × n sensitivity of solution vector x ( z ) with respect to the initial conditions and I denotes identity matrix. In the present case, the sensitivity equations are d Φ dz = " 01 2 D a P e x 1 P e # Φ ( z ) (1.49) Φ ( z )= ∂x 1 ( z ) ∂x 1 (0) ∂x 1 ( z ) ∂x 2 (0) ∂x 2 ( z ) ∂x 1 (0) ∂x 2 ( z ) ∂x 2 (0) (1.50) These equations have to be integrated from z =0 to z =1 to evaluate (1.51) [ df / d s ] s = s ( k ) = Φ 21 (1) = ∂x 2 (1) ∂x 1 (0) Example 57 . Converting a PDE to an ODE-BVP [ 6 ] : Consider 2-D steady state heat transfer problem (1.52) 2 T ∂x 2 + 2 T ∂y 2 =0 x =0 : T = T ; x =1: T = T (1.53) y =0 : T = T ; y =1: k dT dx = h ( T T )
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_66 - 134 4....

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