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, wellstirred, 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.
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 Fall '08
 Gordon,M
 Chemical Engineering, Thermodynamics, Heat, Heat Transfer, Partial differential equation, Fourier coefficients

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