Page 244 - Bird R.B. Transport phenomena

228 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

7B.8 Change in liquid height with time (Fig. 7.1-1).
(a) Derive Eq. 7.1-4 by using integral calculus.
(b) In Example 7.1-1, obtain the expression for the liquid height h as a function of time t.
(c) Make a graph of Eq. 7.1-8 using dimensionless quantities. Is this useful?
7B.9 Draining of a cylindrical tank with exit pipe (Fig. 7B.9).
(a) Rework Example 7.1-1, but with a cylindrical tank instead of a spherical tank. Use the
quasi-steady-state approach; that is, use the unsteady-state mass balance along with the
Hagen-Poiseuille equation for the laminar flow in the pipe.
(b) Rework the problem for turbulent flow in the pipe.
2
128fiLR ,
Answer: (a) t ei In 1 + S
PgD 4

Fig. 7B.9. A cylindrical tank with a long pipe attached. The fluid surface
and pipe exit are open to the atmosphere.

7B.10 Efflux time for draining a conical tank (Fig. 7B.10). A conical tank, with dimensions given in
the figure, is initially filled with a liquid. The liquid is allowed to drain out by gravity. Deter-
mine the efflux time. In parts (a)-(c) take the liquid in the cone to be the "system."
(a) First use an unsteady macroscopic mass balance to show that the exit velocity is
V
2=-- 2J- (7B.10-1)
(b) Write the unsteady-state mechanical energy balance for the system. Discard the viscous
loss term and the term containing the time derivative of the kinetic energy, and give reasons
for doing so. Show that this leads to
= V2g(z - z 2) (7B.10-2)
v 2

Liquid surface
at time t

Fig. 7B.10. A conical container from
which a fluid is allowed to drain. The
quantity r is the radius of the liquid sur-
face at height z, and r is the radius of
= z 2
the cone at some arbitrary height z.