Page 60 - Bird R.B. Transport phenomena

§2.2 Flow of a Falling Film 45

Velocity
distribution

Momentum
flux

Direction
of gravity

Fig. 2.2-3 Final results for the falling film problem,
showing the momentum-flux distribution and the
velocity distribution. The shell of thickness Ax, over
which the momentum balance was made, is also shown.

The constant of integration is evaluated by using the no-slip boundary condition at the
solid surface:
B.C. 2 at x = 8, v z = 0 (2.2-17)
2
Substitution of this boundary condition into Eq. 2.2-16 shows that C 2 = (pg cos p/2/x)8 .
Consequently, the velocity distribution is

(2.2-18)

This parabolic velocity distribution is shown in Fig. 2.2-3. It is consistent with the postu-
lates made initially and must therefore be a possible solution. Other solutions might be
possible, and experiments are normally required to tell whether other flow patterns can
actually arise. We return to this point after Eq. 2.2-23.
Once the velocity distribution is known, a number of quantities can be calculated:
(i) The maximum velocity v znwx is clearly the velocity at x = 0; that is,
_ pg8 2 cos p
(2.2-19)

(ii) The average velocity (v ) over a cross section of the film is obtained as follows:
z
rW Г8
v dxdy
z
/ . . v _ Jo Jo _ 1 dx
fW Г8 7
dxdy
Jo Jo

Pg8 cos j
2
(2.2-20)
3/x