Page 215 - Bird R.B. Transport phenomena

§7.1 The Macroscopic Mass Balance 199

ity is perpendicular to the relevant cross section, and (ii) at planes 1 and 2 the density
and other physical properties are uniform over the cross section.
The law of conservation of mass for this system is then
d
v
v
T m tot = P\( \)$\ ~ Pi( i)^2 (7.1-1)
at
rate of rate of rate of
increase mass in mass out
of mass at plane 1 at plane 2
Here m tot = JpdV is the total mass of fluid contained in the system between planes 1 and
2. We now introduce the symbol w = p(v)S for the mass rate of flow, and the notation
Aw = w 2 - w x (exit value minus entrance value). Then the unsteady-state macroscopic mass
balance becomes

^-m inl=-bw (7.1-2)

If the total mass of fluid does not change with time, then we get the steady-state macro-
scopic mass balance
Aw = 0 (7.1-3)
which is just the statement that the rate of mass entering equals the rate of mass leaving.
For the macroscopic mass balance we use the term "steady state" to mean that the
time derivative on the left side of Eq. 7.1-2 is zero. Within the system, because of the pos-
sibility for moving parts, flow instabilities, and turbulence, there may well be regions of
unsteady flow.

EXAMPLE 7.1-1 A spherical tank of radius R and its drainpipe of length L and diameter D are completely
filled with a heavy oil. At time t = 0 the valve at the bottom of the drainpipe is opened. How
Draining of a Spherical long will it take to drain the tank? There is an air vent at the very top of the spherical tank. Ig-
Tank nore the amount of oil that clings to the inner surface of the tank, and assume that the flow in
the drainpipe is laminar.

SOLUTION We label three planes as in Fig. 7.1-1, and we let the instantaneous liquid level above plane 2
be h(t). Then, at any time t the total mass of liquid in the sphere is

| | j p (7.1-4)

Airvent

Plane 1

Plane 2

Plane 3 Fig. 7.1-1. Spherical tank with drainpipe.