Page 216 - Bird R.B. Transport phenomena

200 Chapter 7 Macroscopic Balances for Isothermal Flow Systems
which can be obtained by using integral calculus. Since no fluid crosses plane 1 we know that
w x — 0. The outlet mass flow rate w , as determined from the Hagen-Poiseuille formula, is
2
A
pgl)D p
128/>tL (7.1-5)
The Hagen-Poiseuille formula was derived for steady-state flow, but we use it here since the
volume of liquid in the tank is changing slowly with time; this is an example of a "quasi-
steady-state" approximation. When these expressions for m and w are substituted into Eq.
tot
2
7.1-2, we get, after some rearrangement,
(2R - h)h dh _ ,-p-
h + L dt 128/xL U ;
We now abbreviate the constant on the right side of the equation as A. The equation is easier
to integrate if we make the change of variable H = h + L so that
[H - (2R + L)](H -L)dH_ A ( п л п ,
H df
We now integrate this equation between t = 0 (when h = 2R or H = 2R + L), and £ = f efflux
(when h = 0 or H = L). This gives for the efflux time

(7.1-8)
in which Л is given by the right side of Eq. 7.1-6. Note that we have obtained this result with-
out any detailed analysis of the fluid motion within the sphere.

§7.2 THE MACROSCOPIC MOMENTUM BALANCE

We now apply the law of conservation of momentum to the system in Fig. 7.0-1, using the
same two assumptions mentioned in the previous section, plus two additional assump-
tions: (iii) the forces associated with the stress tensor т are neglected at planes 1 and 2,
since they are generally small compared to the pressure forces at the entry and exit planes,
and (iv) the pressure does not vary over the cross section at the entry and exit planes.
Since momentum is a vector quantity, each term in the balance must be a vector. We
use unit vectors щ and u to represent the direction of flow at planes 1 and 2. The law of
2
conservation of momentum then reads
j t P tot = р№)Ь\Щ ~ P ^2>S u + PiSiUt - p S u + F ^ + m g (7.2-1)
2
2
2
2 2
2
tot
rate of rate of rate of pressure pressure force of force of
increase of momentum momentum force on force on solid gravity
momentum in at plane 1 out at plane 2 fluid at fluid at surface on fluid
plane 1 plane 2 on fluid
Here P tot = JpvdV is the total momentum in the system. The equation states that the total
momentum within the system changes because of the convection of momentum into and
out of the system, and because of the various forces acting on the system: the pressure
forces at the ends of the system, the force of the solid surfaces acting on the fluid in the
system, and the force of gravity acting on the fluid within the walls of the system. The
subscript "s —> f" serves as a reminder of the direction of the force.
By introducing the symbols for the mass rate of flow and the A symbol we finally get
for the unsteady-state macroscopic momentum balance

| Ptot = - д ( | ^ w + pSju + F ^ + m | (7.2-2)
tot