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021910 sec12-bessel func

# 021910 sec12-bessel func - ChE 120B Bessel Functions and...

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ChE 120B 7 - 1 Bessel Functions and Cylindrical Geometry Steady state temperature distribution in a semi-infinite cylinder. The energy balance in cylindrical coordinates: 2 2 2 2 1 0 T T T r r r z Boundary Conditions : 1, 0 T z 0, finite T z   ,0 T r f r , 0 as T r z z   Assume a separation of variables solution exists: (can be shown using boundary conditions & Sturm-Liouville Thm) T ( r,z ) = R ( r ) Z ( z ) hence 2 2 2 2 0 d R Z dR d T Z R dr r dr dz Divide by RZ to get (primes denote differentials) 2 1 R R Z R r R Z       2 1 R R Z R r R Z       (chosen to give exponentials in Z directions) 2 2 2 1 0 d R dR R dr r dr or 2 2 2 0 d R dR r r R dr dr 2 0 d dR r r R dr dr Remember S-L Equation 2 0 d dy p x s x y r x y dx dx

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