che132a-hw-5 - Chemical Engineering 132A Analytical Methods...

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Chemical Engineering 132A Analytical Methods of Chemical Engineering Professor Mike Gordon, Fall 2011 Homework #5 : Due Wed. 30 November 2011 The last one! (1) Set up the solution to the problems below. “Setting up” means you should apply the BCs/ICs and all, but leave your Fourier coefficients as integrals of something. Think very carefully about problem (b)…something interesting should happen when you try to find the Fourier coefficients. Plot the solution for problem (b) with 1 = α for t=0 and t=0.01. Does your solution make intuitive sense? Why? (a) (b) Steady state only T(x,y,t) = 100 °C (2) Time to think a bit…. .A cylindrical metal block (radius: r=1 and length: L=1) is taken out of a 100 °C oven and suddenly plunged into a very large, well-stirred, ice water bath (0 °C) at time t=0. Your job is to figure out how the block is going to cool off as a function of time by developing a solution for the temperature inside the block at all points. Solve the heat equation in cylindrical coordinates to determine T(r,z,t). Assume that the thermal diffusivity is 0.01. (a) What is the PDE you must solve and what are the boundary/initial conditions? (b) Although we did not work a problem exactly like this in class, you have all the necessary tools to attack it. Solve the PDE. Find the general solution and apply the boundary/initial conditions. Make sure you explicitly state (i) the eigenvalues of the problem, (ii) the overall solution, and (iii) how to determine the Fourier coefficients. (c) Examine your solution using Mathematica. Create functions to generate the eigenvalues and Fourier
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che132a-hw-5 - Chemical Engineering 132A Analytical Methods...

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