Page 105 - Bird R.B. Transport phenomena

90 Chapter 3 The Equations of Change for Isothermal Systems

Fixed
Torsion wire with torsion constant k
t
Bob is suspended Mirror
Outer cylinder v is a function of r and free to rotate
0
rotating /
A,

In this У Fixed
region the\ cylindrical
fluid is surfaces
moving
with
Inner cylinder v = v (r)
stationary e e Rotating
cylindrical
cup
Fluid inside
is stationary
(a) (b)
Fig. 3.6-1. (fl)Tangential laminar flow of an incompressible fluid in the space between two cylinders; the outer
one is moving with an angular velocity il . (b) A diagram of a Couette viscometer. One measures the angular
0
velocity Cl of the cup and the deflection 6 of the bob at steady-state operation. Equation 3.6-31 gives the vis-
0
B
cosity /л in terms of Д, and the torque T = k 6 .
B
z
t
shown in the figure. Because of the arrangement used, end effects over the region including
the bob height L are negligible.
To analyze this measurement, we apply the equations of continuity and motion for con-
stant p and fi to the tangential flow in the annular region around the bob. Ultimately we want
an expression for the viscosity in terms of (the z-component of) the torque 7\ on the inner
cylinder, the angular velocity fl 0 of the rotating cup, the bob height L, and the radii KR and R
of the bob and cup, respectively.
SOLUTION In the portion of the annulus under consideration the fluid moves in a circular pattern. Rea-
sonable postulates for the velocity and pressure are: v = v {r), v = 0, V-. = 0, and p = p{r, z).
() n r
We expect p to depend on z because of gravity and on r because of the centrifugal force.
For these postulates all the terms in the equation of continuity are zero, and the compo-
nents of the equation of motion simplify to
Щ dp
=
r-component ~PT ~~a~ (3.6-20)
d
^-component (3.6-21)

z-component (3.6-22)

The second equation gives the velocity distribution. The third equation gives the effect of
gravity on the pressure (the hydrostatic effect), and the first equation tells how the centrifugal
force affects the pressure. For the problem at hand we need only the ^-component of the
equation of motion. 2

2 See R. B. Bird, С F. Curtiss, and W. E. Stewart, Chem. Eng. Sci., 11,114-117 (1959) for a method of
getting p(r, z) for this system. The time-dependent buildup to the steady-state profiles is given by R. B.
Bird and С F. Curtiss, Chem. Eng. Sci., 11,108-113 (1959).