lec15_10162006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #15: Differential Algebraic Equations (DAEs). Introduction: Optimization. Differential-Algebraic Equations (DAEs) ) ( * ) ( 0 )) ( ), ( ( x G x x M t x t x = = ± ± If M is invertible, ) ( ) ( 1 x G x M x = ± . ode15i ode15s ordinary ODE Quasi-Steady-State Assumption (QSSA) * make stiff equation into algebraic OH + CO H + CO ⎯→ 1 k 2 O d[OH] / dt = k 2 [H][O 2 ] – k 1 [OH][CO] + 2k 3 [O][H 2 O] H + O 2 OH + O O ⎯→ 2 k d[H] / dt = k 1 [OH][CO] – k 2 [H][O 2 ] O + H 2 O 2OH ⎯→ 3 k originally had 6 Differential equations now have 2 algebraic and 4 differential equations takes you from ODE system to a DAE system QSSA is not always helpful because ODE is faster and more accurate to solve. Solving a D.A.E. is like solving a stiff equation. QSSA has not removed original stiffness. Another problem of DAE: Consistent Initial Conditions 0 )) ( ), ( ( ) ( ), ( 0 0 0 0 = t x t x t x t x ± ± You need both x and x ± . If x ± is not provided, you have to use Newton’s Method. High index DAEs DAE Eventually you will have a normal ODE if you take enough derivatives. 0 = 0 = dt d High-index refers to system that requires a high degree of ODE 0 2 2 =
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lec15_10162006 - 10.34, Numerical Methods Applied to...

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