Page 217 - Bird R.B. Transport phenomena

§7.2 The Macroscopic Momentum Balance 201

If the total amount of momentum in the system does not change with time, then we get
the steady-state macroscopic momentum balance

m tot g (7.2-3)
(v)
Once again we emphasize that this is a vector equation. It is useful for computing the force
of the fluid on the solid surfaces, F^ , such as the force on a pipe bend or a turbine blade.
s
Actually we have already used a simplified version of the above equation in Eq. 6.1-3.
Notes regarding turbulent flow: (i) For turbulent flow it is_customary to replace (v) by
2
2
2
(v) and (v ) by (v ); in the latter we are neglecting the term (v' ), which is generally small
2
2
with respect to (v ). (ii) Then we further replace (v )/(v) by (v). The error in doing this is
quite small; for the empirical \ power law velocity profile given in Eq. 5.1-4, (v )/{v) =
2
%{v), so that the error is about 2%. (iii) When we make this assumption we will normally
drop the angular brackets and overbars to simplify the notation. That is, we will let
(щ) = v and (v ) = v , with similar simplifications for quantities at plane 2.
2
2
A
EXAMPLE 7.2-1 A turbulent jet of water emerges from a tube of radius R } = 2.5 cm with a speed v x = 6 m/s,
as shown in Fig. 7.2-1. The jet impinges on a disk-and-rod assembly of mass m = 5.5 kg,
Force Exerted by a Jet which is free to move vertically. The friction between the rod and the sleeve will be neglected.
(Part a) Find the height h at which the disk will "float' 7 as a result of the jet. Assume that the water is
1
incompressible.
SOLUTION To solve this problem one has to imagine how the jet behaves. In Fig. 7.2-1 (a) we make the as-
sumption that the jet has a constant radius, R v between the tube exit and the disk, whereas in
Fig. 7.2-1 (b) we assume that the jet spreads slightly. In this example, we make the first as-
sumption, and in Example 7.4-1 we account for the jet spreading.
We apply the z-component of the steady-state momentum balance between planes 1 and
2. The pressure terms can be omitted, since the pressure is atmospheric at both planes. The z
component of the fluid velocity at plane 2 is zero. The momentum balance then becomes
mg = ' - (irR h)pg (7.2-4)
2
When this is solved for h, we get (in SI units)
m (6) 2 5.5 0.87 m (7.2-5)
2
8 pirR , (9.807) 77(0.025) : =

Disk-rod assembly
I t / w i t h m a s s '« \ ]||[
Plane3
^ r ^ " P l a n e 2 Plane2

- Rising water jet -

Fig. 7.2-1. Sketches corre-
sponding to the two solutions
Plane 1 Plane 1 to the jet-and-disk problem.
- Tube with radius R^ In (a) the water jet is assumed
Tube to have a uniform radius R].
In (b) allowance is made for the
(a) (b) spreading of the liquid jet.

1
K. Federhofer, Aufgaben aus der Hydromechanik, Springer-Verlag, Vienna (1954), pp. 36 and 172.