Chemical Engineering Hand Written_Notes_Part_48

Chemical Engineering Hand Written_Notes_Part_48 - 98 3...

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98 3. LINEAR ALGEBRAIC EQUATIONS AND RELATED NUMERICAL SCHEMES Gauss Seidel Iterations (6.7) x ( k +1) i = g i [ x ( k +1) 1 , ........ x ( k +1) i 1 ,x ( k ) i , ............ x ( k ) n ] i =1 , .......... , n Relaxation Method x ( k +1) i = x ( k ) i + ω [ g i ( x k ) x ( k ) i ] A popular method of this type is Wegstein iterations. Given initial guess vector x (0) x (1) i = g i ( x (0) ); ( i , ............. , n ) s ( k ) i = £ g i ( x ( k ) ) g i ( x ( k 1) ) ¤ h x ( k ) i x ( k 1) i i (6.8) ω ( k ) i = s ( k ) i h 1 s ( k ) i i x ( k +1) i = g i ( x ( k ) )+ ω ( k ) i [ x ( k ) i g i ( x ( k ) )] i , ......... , n The iterations can be terminated when (6.9) ° ° x ( k +1) x ( k ) ° ° Example 44 . Consider the ODE-BVP describing steady state conditions in a tubular reactor with axial mixing (TRAM) in which an irreversible 2nd order reaction is carried out. (6.10) 1 Pe d 2 C dz 2 dC dz DaC 2 =0 (0 z 1) dC dz = ( C 1) at z =0; (6.11) dC dz at z ; (6.12) Using method of f nite di f erence, we get following set of n +1 nonlinear algebraic equations 1 C i +1 2 C i + C i 1 ( z ) 2 C i +1 C i 1 2( z ) = DaC 2 i (6.13) or αC i +1 βC i + αC i 1 = DaC 2 i (6.14)
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6. SOLVING NONLINEAR ALGEBRAIC EQUATIONS
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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_48 - 98 3...

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