Page 226 - Bird R.B. Transport phenomena

210 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

SOLUTION (a) Mass balance. For steady flow the mass balance gives
VL\ = IV 2 ОГ P]V ] S l (7.6-1)
For a fluid of constant density, this gives
V\ \
(7.6-2)
(b) Momentum balance. The downstream component of the momentum balance is

- v 2 w 2 ) - p 2 S 2 ) (7.6-3)
The force F^ is composed of two parts: the viscous force on the cylindrical surfaces parallel
s
to the direction of flow, and the pressure force on the washer-shaped surface just to the right
of plane 1 and perpendicular to the flow axis. The former contribution we neglect (by intu-
ition) and the latter we take to be p\{S — Sj) by assuming that the pressure on the washer-
2
shaped surface is the same as that at plane 1. We then get, by using Eq. 7.6-1,
- v 2 ) - p 2 S 2 ) (7.6-4)
Solving for the pressure difference gives
\ - v 2 ) (7.6-5)
or, in terms of the downstream velocity,

(7.6-6)

Note that the momentum balance predicts (correctly) a rise in pressure.
(c) Angular momentum balance. This balance is not needed. If we take the origin of coor-
dinates on the axis of the system at the center of gravity of the fluid located between
planes 1 and 2, then [t } X u j and [r 2 X u ] are both zero, and there are no torques on the
2
fluid system.
(d) Mechanical energy balance. There is no compressive loss, no work done via moving
parts, and no elevation change, so that

E v = 4v\ ~ z> + p (P\ ~ Pi) (7.6-7)
v
Insertion of Eq. 7.6-6 for the pressure rise then gives, after some rearrangement,

(7.6-8)

which is an entry in Table 7.5-1.
This example has shown how to use the macroscopic balances to estimate the friction loss
factor for a simple resistance in a flow system. Because of the assumptions mentioned after
Eq. 7.6-3, the results in Eqs. 7.6-6 and 8 are approximate. If great accuracy is needed, a correc-
tion factor based on experimental data should be introduced.

EXAMPLE 7.6-2 A diagram of a liquid-liquid ejector is shown in Fig. 7.6-2. It is desired to analyze the mixing
of the two streams, both of the same fluid, by means of the macroscopic balances. At plane 1
Performance of a the two fluid streams merge. Stream la has a velocity v and a cross-sectional area ^S and
u
Liquid-Liquid Ejector stream lb has a velocity \v and a cross-sectional area \S V 0 Plane 2 is chosen far enough down-
Q
stream that the two streams have mixed and the velocity is almost uniform at v . The flow is
2