053007-132a-hw 6

053007-132a-hw 6 - C HEMICAL E NGINEERING 132A Professor...

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Unformatted text preview: C HEMICAL E NGINEERING 132A Professor Todd Squires Spring Quarter, 2007 Final Homework! Due Friday, June 8 (Thurs., June 7 if you want it corrected before your final on Monday) Problem 1. A circular pipe of radius 1 contains solute with a concentration initially given by c ( r ) = cos πr . The solute diffuses according to ∂c ∂t = D ∇ 2 c (1) and can not escape from the pipe (no-flux). Pose a separable solution and solve the diffusion equation for the concentration. Note: There are some things you will not be able to solve in a simple way: for example, when imposing the no-flux boundary condition, you will need to solve an equation like ∂J ( q n ) ∂r = 0 , (2) for q n . You need only say that the frequencies q n solve that equation; you can’t do any better than that. Similarly, when calculating the coefficients in the series, all you can do is write down the integral. You would need to evaluate all of these things numerically....
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