Page 221 - Bird R.B. Transport phenomena

§7.5 Estimation of the Viscous Loss 205

2
3
quotient (v )/(v) by (v) . For the empirical \ power law velocity profile given in Eq. 5.1-4,
3
2
it can be shown that (v )/(v) = t§f?§(^) , so that the error amounts to about 6%. (iii) We
further omit the brackets and overbars to simplify the notation in turbulent flow.
EXAMPLE 7.4-1 Continue the problem in Example 7.2-1 by accounting for the spreading of the jet as it moves
upward.
Force Exerted by a Jet
(Part b)
SOLUTION
We now permit the jet diameter to increase with increasing z as shown in Fig. 7.2-1 (b). It is
convenient to work with three planes and to make balances between pairs of planes. The sep-
aration between planes 2 and 3 is taken to be quite small.
A mass balance between planes 1 and 2 gives
(7.4-8)
w, = w 2
Next we apply the mechanical energy balance of Eq. 7.4-5 or 7.4-7 between the same two
planes. The pressures at planes 1 and 2 are both atmospheric, and there is no work done by
moving parts W m . We assume that the viscous dissipation term E v can be neglected. If z is
measured upward from the tube exit, then gAh = g(h 2 - hj ~ g{h - 0), since planes 2 and 3
are so close together. Thus the mechanical energy balance gives
2
\(v\ - v ) + gh = 0 (7.4-9)
We now apply the z-momentum balance between planes 2 and 3. Since the region is very
small, we neglect the last term in Eq. 7.2-3. Both planes are at atmospheric pressure, so the
pressure terms do not contribute. The fluid velocity is zero at plane 3, so there are only two
terms left in the momentum balance
mg = v 2w 2 (7.4-10)
From the above three equations we get

from Eq. 7.4-9
v\
2
v 2 ( (m?/w 2) \
= ^~ U - , from Eq. 7.4-10
2
g\ v\ )
_ from Eq. 7.4-8 (7.4-11)
2g
2
2
in which mg and v }w^ = irR pv are known. When the numerical values are substituted into
Eq. 7.4-10, we get h = 0.77 m. This is probably a better result than the value of 0.87 m obtained
in Example 7.2-1, since it accounts for the spreading of the jet. We have not, however, consid-
ered the clinging of the water to the disk, which gives the disk-rod assembly a somewhat
greater effective mass. In addition, the frictional resistance of the rod in the sleeve has been
neglected. It is necessary to run an experiment to assess the validity of Eq. 7.4-10.

§7.5 ESTIMATION OF THE VISCOUS LOSS
This section is devoted to methods for estimating the viscous loss (or friction loss), E v,
which appears in the macroscopic mechanical energy balance. The general expression
for E v is given in Eq. 7.4-4. For incompressible Newtonian fluids, Eq. 3.3-3 may be used
to rewrite £,, as
E v = (7.5-1)