Page 62 - Bird R.B. Transport phenomena

§2.2 Flow of a Falling Film 47

Reynolds number should be used to delineate the flow regimes. We shall have more to
say about this in §3.7.
This discussion illustrates a very important point: theoretical analysis of flow sys-
tems is limited by the postulates that are made in setting up the problem. It is absolutely
necessary to do experiments in order to establish the flow regimes so as to know when
instabilities (spontaneous oscillations) occur and when the flow becomes turbulent.
Some information about the onset of instability and the demarcation of the flow regimes
can be obtained by theoretical analysis, but this is an extraordinarily difficult subject.
This is a result of the inherent nonlinear nature of the governing equations of fluid dy-
namics, as will be explained in Chapter 3. Suffice it to say at this point that experiments
play a very important role in the field of fluid dynamics.

3
2
4
3
EXAMPLE 2.2-1 An oil has a kinematic viscosity of 2 X 10 m /s and a density of 0.8 X 10 kg/m . If we want
to have a falling film of thickness of 2.5 mm on a vertical wall, what should the mass rate of
Calculation of Film flow of the liquid be?
Velocity
SOLUTION
According to Eq. 2.2-21, the mass rate of flow in kg/s is
Pg&W (0.8 X 10 )(9.80)(2.5 X 10~ ) W
3
3 3
_ 4 (2.2-24)
ZV — 3(2 X 10" )
To get the mass rate of flow one then needs to insert a value for the width of the wall in
meters. This is the desired result provided that the flow is laminar and nonrippling. To
determine the flow regime we calculate the Reynolds number, making use of Eqs. 2.2-21
and 24
4(0.204)
= 5.1 (2.2-25)
(2 X 10~ )(0.8 x 10 )
3
4
This Reynolds number is sufficiently low that rippling will not be pronounced, and therefore
the expression for the mass rate of flow in Eq. 2.2-24 is reasonable.

EXAMPLE 2.2-2 Rework the falling film problem for a position-dependent viscosity fi = fi e ax/5 , which arises
o
when the film is nonisothermal, as in the condensation of a vapor on a wall. Here /x is the vis-
0
Falling Film with cosity at the surface of the film and a is a constant that describes how rapidly JX decreases as x
Variable Viscosity increases. Such a variation could arise in the flow of a condensate down a wall with a linear
temperature gradient through the film.
SOLUTION The development proceeds as before up to Eq. 2.2-13. Then substituting Newton's law with
variable viscosity into Eq. 2.2-13 gives

(2.2-26)
This equation can be integrated, and using the boundary conditions in Eq. 2.2-17 enables us to
evaluate the integration constant. The velocity profile is then
_ gS cosffr Л Л f
2
P x (2.2-27)
2
Mo I \ a a ) \cS
As a check we evaluate the velocity distribution for the constant-viscosity problem (that is,
when a is zero). However, setting a = 0 gives oo - oo in the two expressions within parentheses.