Chemical Engineering Hand Written_Notes_Part_63

Chemical Engineering Hand Written_Notes_Part_63 - 128 4....

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128 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES De f ning τ = F/V, we can re arrange the above set of equations as dC A 1 dt dC A 2 dt dC A 3 dt = ( k +1 1 )0 0 1 ( k 2 01 ( k 3 ) C A 1 C A 2 C A 3 (1.5) + 1 1 0 0 C A 0 (1.6) (1.7) x =[ C A 1 ,C A 2 A 3 ] T d x dt = A x + BC A 0 where matrices A and B are de f ned in the above equation. Now, suppose initially C A 0 = ¯ C A 0 , till t =0 , and, for t 0 , C A 0 was changed to C A 0 . Then we are required to solve (1.8) d x dt = A x ; x = x (0) at t and generate trajectories x ( t ) ( i.e. C A 1 ( t ) A 2 ( t ) and C A 3 ( t )) over interval [0 ,t f ] . This is a typical problem of dynamic simulation of lumped parameter system. Example 51 . Continuous Fermenter Consider a continuously operated fermenter described by the following set of ODEs (1.9) dX dt = F 1 ( X,S,P, D, S f )= DX + µX (1.10) dS dt = F 2 ( X, S,P, D, S f D ( S f S ) 1 Y X/S µX (1.11) dP dt = F 3 ( f DP +( αµ + β ) X where X represents e f uent cell-mass or biomass concentration, S
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Chemical Engineering Hand Written_Notes_Part_63 - 128 4....

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