Page 86 - Bird R.B. Transport phenomena

Problems 71

s = downstream distance
along cone surface,
measured from the
cone apex

Film thickness is
S(s)

s = L

Fluid in with mass Fig. 2C.5 A falling film on a conical
flow rate w surface.

(a) Assume that the results of §2.2 apply approximately over any small region of the cone sur-
face. Show that a mass balance on a ring of liquid contained between s and s + As gives:

or -f (2C.5-1)

(b) Integrate this equation and evaluate the constant of integration by equating the mass rate
of flow zv up the central tube to that flowing down the conical surface at s = L. Obtain the fol-
lowing expression for the film thickness:

3/JLW
8 = (2C.5-2)
2
rrp gL sin 2)3
2C.6 Rotating cone pump (see Fig. 2C.6). Find the mass rate of flow through this pump as a func-
tion of the gravitational acceleration, the impressed pressure difference, the angular velocity
of the cone, the fluid viscosity and density, the cone angle, and other geometrical quantities
labeled in the figure.

Pressure p 2

Pressure p x
2
(poundals ft" )
Fig. 2C.6 A rotating-cone pump. The variable r
Direction of flow
with mass rate of is the distance from the axis of rotation out to
flowzi>(lb,,,/s) the center of the slit.