Fall 2004
1
ChEN 507
Homework Assignment No. 9
(Laplace Transforms)
December 8, 2004
23. An infinitely long solid bar, well insulated on the sides and initially at the uniform
temperature To, suddenly has its front end (i.e., x = 0) placed into a constant temperature
bath at temperature T1. Find the unsteady-state temperature profile T(t, x) in the bar. What
is the rate of heat transfer to the bar as a function of time?
24. At time zero, a slab of thickness 2h at a uniform temperature To is placed into a well-stirred
temperature bath which is insulated an initially at a temperature
θ
o. Starting at the same
time, liquid at a temperature
θ
in is pumped into the bath at a volumetric flow rate,
V, and
liquid at the prevailing bath temperature
θ
, is removed at the same rate. Assume that all of
the resistance to heat transfer between the fluid and the slab is within the slab, and that the
temperature (T) in the slab is a function of time and only one space coordinate (i.e., t and
y).
a. Set up the equations which determine the temperature T(t, y) of the slab.
b. Find an expression for the Laplace transform of the temperature Tsy(; ).
c. Verify that this gives the expected values of the temperature for the limiting cases at
time zero and at a long time.
d. Find expressions for the temperature valid for small but not zero, and for large but not
infinite time.