Page 87 - Bird R.B. Transport phenomena

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

(a) Begin by analyzing the system without the rotation of the cone. Assume that it is possible
to apply the results of Problem 2B.3 locally. That is, adapt the solution for the mass flow rate
from that problem by making the following replacements:
replace (9> 0 - SP L )/L by -d<3>/dz
replace W by 2rrr = 2TTZ sin /3
thereby obtaining

The mass flow rate w is a constant over the range of z. Hence this equation can be integrated
to give

^ ln ¥ (2C.6-2)

(b) Next, modify the above result to account for the fact that the cone is rotating with angular
velocity П. The mean centrifugal force per unit volume acting on the fluid in the slit will have
a z-component approximately given by
2
2
(Fcentrif.) 2 = Kp^ z sin p (2C.6-3)
What is the value of K? Incorporate this as an additional force tending to drive the fluid
through the channel. Show that this leads to the following expression for the mass rate of flow:
2 L
l\
4тгВ р sin p [<9>i " 9>) + <Щ1 2 sin 2 p)(L\ - L )
3
w = * 2 1 n /T \ (2C.6-4)
Зд |_ ln (L /L )
2 }
Here 9>j = p t + pgL, cos p.
2C.7 A simple rate-of-climb indicator (see Fig. 2C.7). Under the proper circumstances the simple
apparatus shown in the figure can be used to measure the rate of climb of an airplane. The
gauge pressure inside the Bourdon element is taken as proportional to the rate of climb. For
the purposes of this problem the apparatus may be assumed to have the following properties:
(i) the capillary tube (of radius R and length L, with R « L) is of negligible volume but ap-
preciable flow resistance; (ii) the Bourdon element has a constant volume V and offers negli-
gible resistance to flow; and (iii) flow in the capillary is laminar and incompressible, and the
volumetric flow rate depends only on the conditions at the ends of the capillary.

Rate of climb

• Bourdon
Capillary^, ^ / element
tube
Pressure Pressure
outside = p 0 i n s i d e " Pi Fig. 2C.7 A rate-of-climb indicator.

(a) Develop an expression for the change of air pressure with altitude, neglecting tempera-
ture changes, and considering air to be an ideal gas of constant composition. (Hint: Write a
shell balance in which the weight of gas is balanced against the static pressure.)