Page 83 - Bird R.B. Transport phenomena

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

(a) Assume that locally the velocity distribution in the gap can be very closely approximated
by that for flow between parallel plates, the upper one moving with a constant speed. Verify
that this leads to the approximate velocity distribution (in spherical coordinates)

(ir/2) - в
(2B.11-1)
This approximation should be rather good, because ф is so small.
0
(b) From the velocity distribution in Eq. 2B.11-1 and Appendix B.I, show that a reasonable
expression for the shear stress is

т (2B.11-2)
оф
This result shows that the shear stress is uniform throughout the gap. It is this fact that makes
the cone-and-plate viscometer quite attractive. The instrument is widely used, particularly in
the polymer industry.
(c) Show that the torque required to turn the cone is given by
T z = ! (2B.11-3)
This is the standard formula for calculating the viscosity from measurements of the torque
and angular velocity for a cone-plate assembly with known R and ф .
0
(d) For a cone-and-plate instrument with radius 10 cm and angle ф equal to 0.5 degree, what
0
torque (in dyn • cm) is required to turn the cone at an angular velocity of 10 radians per
minute if the fluid viscosity is 100 cp?
Answer: (d) 40,000 dyn • cm
2B.12 Flow of a fluid in a network of tubes (Fig. 2B.12). A fluid is flowing in laminar flow from A
to В through a network of tubes, as depicted in the figure. Obtain an expression for the mass
flow rate w of the fluid entering at A (or leaving at B) as a function of the modified pressure
< >
drop 3 — ЯР . Neglect the disturbances at the various tube junctions.
В
A
4
- <3f> B )R P
Answer: w =
20/iL
Fluid out

All tubes have the same Fig. 2B.12 Flow of a fluid in a network with
radius R and same length L branching.

5
2C.1 Performance of an electric dust collector (see Fig. 2C.1) .
(a) A dust precipitator consists of a pair of oppositely charged plates between which dust-
laden gases flow. It is desired to establish a criterion for the minimum length of the precipita-
tor in terms of the charge on the particle e, the electric field strength %, the pressure difference

5
The answer given in the first edition of this book was incorrect, as pointed out to us in 1970 by
Nau Gab Lee of Seoul National University.