Page 80 - Bird R.B. Transport phenomena

Problems 65

as shown in Fig. 2B.6. Note that the "momentum in" and "momentum out" arrows are al-
ways taken in the positive coordinate direction, even though in this problem the momentum
is flowing through the cylindrical surfaces in the negative r direction.
(a) Show that the velocity distribution in the falling film (neglecting end effects) is

v = 4/х (2B.6-1)
7
(b) Obtain an expression for the mass rate of flow in the film.
(c) Show that the result in (b) simplifies to Eq. 2.2-21 if the film thickness is very small.

2B.7 Annular flow with inner cylinder moving axially (see Fig. 2B.7). A cylindrical rod of diame-
ter KR moves axially with velocity v along the axis of a cylindrical cavity of radius R as seen
0
in the figure. The pressure at both ends of the cavity is the same, so that the fluid moves
through the annular region solely because of the rod motion.

Cylinder of inside
Fluid at modified radius R Fluid at modified
^
pressure £P 0 pressure 2P 0
Rod of radius KR
moving with velocity v 0

Fig. 2B.7 Annular flow with the inner cylinder moving axially.
(a) Find the velocity distribution in the narrow annular region.
(b) Find the mass rate of flow through the annular region.
(c) Obtain the viscous force acting on the rod over the length L.
(d) Show that the result in (c) can be written as a "plane slit" formula multiplied by a "curva-
ture correction." Problems of this kind arise in studying the performance of wire-coating dies. 1
, , v z In (r/R)
A
Answers: (a) — = — ;
v
o In к

(d)F = 2irLfxv 0 U - г 2 £ - he 2 + * • •) where e = \ - к (see Problem 2B.5)
о
2B.8 Analysis of a capillary flowmeter (see Fig. 2B.8). Determine the rate of flow (in lb/hr)
through the capillary flow meter shown in the figure. The fluid flowing in the inclined tube is

Water

CC1 4
Fig. 2B.8 A capillary flow meter.

J. B. Paton, P. H. Squires, W. H. Darnell, F. M. Cash, and J. F. Carley, Processing of Thermoplastic
1
Materials, E. C. Bernhardt (ed.), Reinhold, New York (1959), Chapter 4.