Page 415 - Bird R.B. Transport phenomena

Problems 397
(d) Then solve the equation for g(r)).
(e) Finally, evaluate the constant C u and thereby complete the derivation of the temperature
distribution.

12B.3. Heating of a wall (constant wall heat flux). A very thick solid wall is initially at the tempera-
ture T . At time t = 0, a constant heat flux q is applied to one surface of the wall (at у = 0),
o
0
and this heat flux is maintained. Find the time-dependent temperature profiles T{y, t) for
small times. Since the wall is very thick it can be safely assumed that the two wall surfaces are
an infinite distance apart in obtaining the temperature profiles.
(a) Follow the procedure used in going from Eq. 12.1-33 to Eq. 12.1-35, and then write the ap-
propriate boundary and initial conditions. Show that the analytical solution of the problem is

T(y, 0 - T = j exp(-u )du) (12B.3-1)
2
o
y/VAat
(b) Verify that the solution is correct by substituting it into the one-dimensional heat conduc-
tion equation for the temperature (see Eq. 12.1-33). Also show that the boundary and initial
conditions are satisfied.
12B.4. Heat transfer from a wall to a falling film (short contact time limit) 2 (Fig. 12B.4). A cold liq-
uid film flowing down a vertical solid wall, as shown in the figure, has a considerable cooling
effect on the solid surface. Estimate the rate of heat transfer from the wall to the fluid for such
short contact times that the fluid temperature changes appreciably only in the immediate
vicinity of the wall.
(a) Show that the velocity distribution in the falling film, given in §2.2, may be written as
2
v z = v zmSiX [2{y/3) — (y/8) ], in which v zmax = pgS /2/x. Then show that in the vicinity of the
2
wall the velocity is a linear function of у given by
1ГУ (12B.4-1)

Downflowing
liquid film
enters at uniform
temperature, T o

Outer edge of
film is at у = 8

Solid surface,
at constant Note that the fluid
temperature T^ temperature is different
from T only in the
o
neighborhood of the
wall, where v is Fig. 12B.4. Heat transfer to a film
almost linear. z falling down a vertical wall.

2
R. L. Pigford, Chemical Engineering Progress Symposium Series, 51, No. 17, 79-92 (1955). Robert
Lamar Pigford (1917-1988), who taught at both the University of Delaware and the University of
California in Berkeley, researched many aspects of diffusion and mass transfer; he was the founding
editor of Industrial and Engineering Chemistry Fundamentals.