Page 419 - Bird R.B. Transport phenomena

Problems 401

in which a 0 = k /pC and
p
o
M=[ Ф(фг 1 and N = (1 + (12C.2-4, 5)

Then solve for the function 3(t).
(c) Now let Ф(т/) = 1 - fry + iT7. Why is this a felicitous choice? Then find the time-depen-
3
dent temperature distribution T(y, t) as well as the heat flux at у = 0.
12C.3. Heat conduction with phase change (the Neumann-Stefan problem) (Fig. 12C.3) . A liquid,
5
contained in a long cylinder, is initially at temperature 7^. For time t ^ 0, the bottom of the
container is maintained at a temperature T , which is below the melting point T . We want to
m
o
estimate the movement of the solid-liquid interface, Z{t), during the freezing process.
For the sake of simplicity, we assume here that the physical properties p, k, and C p are
constants and the same in both the solid and liquid phases. Let &H be the heat of fusion per
f
gram, and use the abbreviation Л = AHr/C (T - T ).
/ p 1 o
(a) Write the equation for heat conduction for the liquid (L) and solid (S) regions; state the
boundary and initial conditions.
(b) Assume solutions of the form:
(12C.3-1)

(12C.3-2)

(c) Use the boundary condition at z = 0 to show that Q = 0, and the condition at z = °° to
show that C = 1 - C . Then use the fact that T s = T = T at z = Z(t) to conclude that Z(t) =
4
3
m
L
Л Viof, where Л is some (as yet undetermined) constant. Then get C and C in terms of Л. Use
3
4
the remaining boundary condition to get Л in terms of Л and © = (T - T )/(T } - T ):
m
o
Q
m
VTTAX exp Л 2 = (12C.3-3)
erf Л 1-erfA
Liquid Liquid
(Initially at
temperature
Moving
throughout) . interface
located at
Z(t)

2 = 0
t<0 t>0
Temperature Fig. 12C.3. Heat conduction
T at z = 0 with solidification.
o

5
For literature references and related problems, see H. S. Carslaw and J. C. Jaeger, Conduction of
Heat in Solids, 2nd edition, Oxford University Press (1959), Chapter XI; on pp. 283-286 the problem
considered here is worked out for the situation that the physical properties of the liquid and solid phases
are different. See also S. G. Bankoff, Advances in Chemical Engineering, Vol. 5, Academic Press, New York
(1964), pp. 75-150; J. Crank, Free and Moving Boundary Problems, Oxford University Press (1984); J. M. Hill,
One-Dimensional Stefan Problems, Longmans (1987).