Page 85 - Bird R.B. Transport phenomena

70 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow

2C.4 Falling-cylinder viscometer (see Fig. 2C.4). 6 A falling-cylinder viscometer consists of a long
vertical cylindrical container (radius R), capped at both ends, with a solid cylindrical slug (ra-
dius KR). The slug is equipped with fins so that its axis is coincident with that of the tube.
One can observe the rate of descent of the slug in the cylindrical container when the lat-
ter is filled with fluid. Find an equation that gives the viscosity of the fluid in terms of the ter-
minal velocity v 0 of the slug and the various geometric quantities shown in the figure.

Cylindrical
slug descends
with speed v 0

Cylindrical container Fig. 2C.4 A falling-cylinder viscom-
filled with fluid eter with a tightly fitting solid cylin-
der moving vertically. The cylinder
is usually equipped with fins to
maintain centering within the tube.
The fluid completely fills the tube,
and the top and bottom are closed.

(a) Show that the velocity distribution in the annular slit is given by
V (2C.4-1)
2
V к )\п(1/к)
in which f = r/R is a dimensionless radial coordinate.
(b) Make a force balance on the cylindrical slug and obtain
- p)g(xR) 2
(р 0
In (2C.4-2)
2v 0
in which p and p are the densities of the fluid and the slug, respectively.
0
(c) Show that, for small slit widths, the result in (b) may be expanded in powers of e = 1 - к
to give

(Po - p)gR s 2 3
13 9 (2C.4-3)
6Vn 20**"
See §C2 for information on expansions in Taylor series.
2C.5 Falling film on a conical surface (see Fig. 2C.5). 7 A fluid flows upward through a circular
tube and then downward on a conical surface. Find the film thickness as a function of the dis-
tance s down the cone.

6
J. Lohrenz, G. W. Swift, and F. Kurata, AIChE Journal, 6, 547-550 (1960) and 7, 6S (1961); E. Ashare,
R. B. Bird, and J. A. Lescarboura, AIChE Journal 11, 910-916 (1965).
R. B. Bird, in Selected Topics in Transport Phenomena, CEP Symposium Series #58, 61,1-15 (1965).
7