Page 230 - Bird R.B. Transport phenomena

214 Chapter 7 Macroscopic Balances for Isothermal Flow Systems

EXAMPLE 7.6-4 A rectangular incompressible fluid jet of thickness b emerges from a slot of width c, hits a flat
x
plate and splits into two streams of thicknesses b 2a and b 2b as shown in Fig. 7.6-4. The emerg-
The Impinging Jet ing turbulent jet stream has a velocity v x and a mass flow rate w v Find the velocities and mass
rates of flow in the two streams on the plate. 1

SOLUTION We neglect viscous dissipation and gravity, and assume that the velocity profiles of all three
streams are flat and that their pressures are essentially equal. The macroscopic balances then
give
Mass balance
W
Щ = ™2a + 2b (7.6-31)
Momentum balance (in the direction parallel to the plate)
Viiv l cos в = v w la - v w 2b (7.6-32)
2b
la
Mechanical energy balance
\v\w x = \v\\w 2a + \v\ w 2b (7.6-33)
b
Angular momentum balance (put the origin of coordinates on the centerline of the jet and at
an altitude of \by this is done so that there will be no angular momentum of the incoming jet)

0 = (v w ) • 1(Ъ г - b ) - (v w ) • £(&, - b 2b) (7.6-34)
2b 2b
2a 2a
2a
This last equation can be rewritten to eliminate the b's in favor of the w's. Since w x = pvjo xc
and w 2a = pv 2ab 2ac, we can replace b ] - b 2a by (w^/pv^c) - (w 2a/pv 2ac) and replace h x - b 2b cor-
respondingly. Then the angular momentum balance becomes
(7.6-35)

Slot-shaped
nozzle
Fluid emerges
from jet with

Mass rate of flow ги
г

Velocity v
Velocity v 2b 2a
Mass rate of flow 2a Mass rate of flow
W
2b
Fig. 7.6-4. Jet impinging on a wall and splitting into two streams. The point
O, which is the origin of coordinates for the angular momentum balance, is
taken to be the intersection of the centerline of the incoming jet and a plane
that is at an elevation \b^.

1 For alternative solutions to this problem, see G. K. Batchelor, An Introduction to Fluid Dynamics,
Cambridge University Press (1967), pp. 392-394, and S. Whitaker, Introduction to Fluid Dynamics,
Prentice-Hall, Englewood Cliffs, N.J. (1968), p. 260. An application of the compressible impinging jet
problem has been given by J. V. Foa, U.S. Patent 3,361,336 (Jan. 2,1968). There, use is made of the fact
that if the slot-shaped nozzle moves to the left in Fig. 7.6-4 (i.e., left with respect to the plate), then, for a
compressible fluid, the right stream will be cooler than the jet and the left stream will be warmer.