Page 235 - Bird R.B. Transport phenomena

§7.7 Use of the Macroscopic Balances for Unsteady-State Problems 219

EXAMPLE 7.7-2 The liquid in a U-tube manometer, initially at rest, is set in motion by suddenly imposing a
pressure difference p - p . Determine the differential equation for the motion of the
b
a
Manometer manometer fluid, assuming incompressible flow and constant temperature. Obtain an expres-
Oscillations 2 sion for the tube radius for which critical damping occurs. Neglect the motion of the gas
above the manometer liquid. The notation is summarized in Fig. 7.7-2.

SOLUTION We designate the manometric liquid as the system to which we apply the macroscopic bal-
ances. In that case, there are no planes 1 and 2 through which liquid enters or exits. The free
liquid surfaces are capable of performing work on the surroundings, W , and hence play the
m
role of the moving mechanical parts in §7.4. We apply the mechanical energy balance of Eq.
7.4-2, with E set equal to zero (since the manometer liquid is regarded as incompressible). Be-
c
cause of the choice of the system, both w x and w 2 are zero, so that the only terms on the right
side are - W and —E .
m
v
To evaluate dK /dt and E it is necessary to make some kind of assumption about the ve-
v
-m
tot
locity profile. Here we take the velocity profile to be parabolic:
v(r, t) = 2(v)\ 1 (7.7-14)
in which (v) = dh/dt is a function of time, defined to be positive when the flow is from left to
right.
The kinetic energy term may then be evaluated as follows:

d Г f 27T ( R { i p v 2 ) r d r d e d i
•tot =
t at Jo Jo Jo
о г/1 ч d f R j
= 2ТГ1ЛГР) — v 2 r dr
dt Jo
1
2
= 27rLR (|p)|[ |
(7.7-15)
dt

Рь

T
Gas -
Я
Liquid level hit)
at t = О Г hit)
H
Cross-sectional
Liquid level j_ ^ area S = TTR 2
at t = oo |
К Lowest level
reached by
manometer
fluid (z = 0)
Manometer
liquid Fig. 7.7-2. Damped oscillations of
a manometer fluid.

2 For a summary of experimental and theoretical work on manometer oscillations, see J. C. Biery,
AIChE Journal, 9, 606-614 (1963); 10, 551-557 (1964); 15, 631-634 (1969). Biery's experimental data show
that the assumption made in Eq. 7.7-14 is not very good.