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Chemical Engineering Hand Written_Notes_Part_64

Chemical Engineering Hand Written_Notes_Part_64 - 130 4...

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130 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES where x R m and F ( x ) is a m × 1 function vector. Introducing a parameter λ such that (0 λ 1) , we de fi ne (1.18) F ( x ( λ )) = (1 λ ) F ( x (0) ) where x (0) represents some arbitrary initial guess. Obviously, at λ = 1 we have F ( x ) = 0 . Di ff erentiating w.r.t. λ and rearranging, we get d x = [ ∂F x ] 1 F ( x (0) ) (1.19) x (0) = x (0) (1.20) Integrating above ODE-IVPs from λ = 0 to λ = 1 produces the solution of nonlinear the equations at λ = 1 . 1.3. Solutions of Parabolic / Hyperbolic PDE’s. 1.3.1. Finite Di ff erence Method. Use of fi nite di ff erence method to solve par- abolic or hyperbolic equations with fi nite spatial boundaries results in set of coupled linear / nonlinear ODE-IVPs. Example 52 . Consider the ODE-BVP describing steady state con- ditions in a tubular reactor with axial mixing (TRAM) in which an irreversible 2nd order reaction is carried out. (1.21) ∂C ∂t = 1 Pe 2 C ∂z 2 ∂C ∂z DaC 2 (0 z 1) t = 0 : c ( z, 0) = 0 (1.22) ∂C (0 , t ) ∂z = Pe ( C (0 , t ) 1) at z = 0; (1.23) ∂C
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