ChEE 402: Applied Mathematics in Chemical Engineering
Fall 2008
Instructor: R.Z. Guzmán
Assignment No. 10
Date Submitted: 11/10/08
Date Due: 11/17/08
Problem 1.-Consider the transient one-dimensional heat equation
T
t
2
T
y
2
with initial and boundary conditions:
I.C. t = 0 T = To
B.C 1) y = 0 T = To
B.C 1) y = L T = T1
a) Can this problem be solved by separation of variables?, if not (why not?). Will a suitable
transformation of variables at this point would allow a separation of variables solution
(transforming dependent and independent variables as well as boundary conditions)
b) If a suitable system can be obtained that gives homogeneous boundary conditions solve this
resulting problem by separation of variables.
i) Find the eigenvalues (n), the eigen functions F(X) and G() and the function un(, Y). That is
the function un = Fn(X) Gn()
c) Find the temperature distribution U = un as a function and Y solving for all the integration
constants.
d) What is the complete solution for the original problem?
Problem 2. Capillary -Tube Method to Analyze Binary Liquid Diffusivity. In this method a straight,
narrow bore capillary is maintained in a vertical position with the open end pointed upward. A slow
stream of pure solvent is allowed to continuously sweep past the open mouth of the capillary tube.
Designate species A as solute and species B as solvent. After an elapsed time, t, the capillary is removed
and the solution contains is well mixed and then analyzed for composition. To find how much solute A
was extracted. Generally, several such experiments are conducted at different values of t.
If the diffusivity does not change too much with composition (actually, an average diffusivity is
computed corresponding to the average composition between initial and final state), the transient
form of Fick's law for equimolar counter diffusion and constant molar density is:
x
A
tD
AB
2
x
A
z
2
1
