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: 22 1 0 TT T rr r z   Boundary Conditions :  1, 0 Tz 0, finite ,0 Tr fr a s Trz 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 0 dR ZdR dT ZR dr r dr dz  Divide by RZ to get (primes denote differentials) 2 1 RR Z RrR Z  2 1 Z Z (chosen to give exponentials in Z directions) 2 2 2 1 0 dR R dr r dr or 2 2 2 0 dR dR R dr dr 2 0 dd R R dr dr    Remember S-L Equation   2 0 y px sxy rxy dx dx    
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ChE 120B 7 - 2 Clearly our equation is a SL equation:  p xr 0 sx rx r  weighting function Remember that if the B.C.'s are appropriate, the solutions of this equation will be orthogonal eigenfunctions w.r.t the weight function
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This note was uploaded on 03/22/2011 for the course CHE 120B taught by Professor Zasadinski during the Winter '10 term at UCSB.

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

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