fluid_mechanics

8.4 Dimensional Analysis of Pipe Flow 425
TABLE E8.6
250
Location Area ( ft 2 ) Velocity ( fts)
wind tunnel diffuser is interrupted by the four turning corners and
the fan, it may not be possible to obtain a smaller value of K Ldif for
this situation. Thus,
COMMENTS By repeating the calculations with various test
h Ldif K V 2 6
Ldif
2g 0.6 V 2 section velocities, V
6
5 , the results shown in Fig. E8.6c are obtained.
Since the head loss varies as V
2
2g
5 and the power varies as
head loss times V 5 , it follows that the power varies as the cube of
The loss coefficients for the conical nozzle between section 142
and 152 and the flow-straightening screens are assumed to be
K Lnoz 0.2 and K Lscr 4.0 1Ref. 132, respectively. We neglect the
head loss in the relatively short test section.
Thus, the total head loss is
or
h L19 30.21V 2 7 V 2 8 V 2 2 V 2 32
or
Hence, from Eq. 1 we obtain the pressure rise across the fan as
From Eq. 2 we obtain the power added to the fluid as
or
1 22.0 36.4
2 28.0 28.6
3 35.0 22.9
4 35.0 22.9
5 4.0 200.0
6 4.0 200.0
7 10.0 80.0
8 18.0 44.4
9 22.0 36.4
h L19 h Lcorner7 h Lcorner8 h Lcorner2 h Lcorner3
h Ldif h Lnoz h Lscr
0.6V 2 6 0.2V 2 5 4.0V 2 442g
30.2180.0 2 44.4 2 28.6 2 22.9 2 2 0.612002 2
0.212002 2 4.0122.92 2 4 ft 2 s 2 32132.2 fts 2 24
p 1 p 9 gh L19 10.0765 lbft 3 21560 ft2
p a 10.0765 lbft 3 214.0 ft 2 21200 fts21560 ft2
34,300 ft # lbs
p a
h L19 560 ft
42.8 lbft 2 0.298 psi
34,300 ft # lbs
550 1ft #
62.3 hp
lbs2hp
(Ans)
(Ans)
a , hp
200
150
100
50
the velocity. Thus, doubling the wind tunnel speed requires an
eightfold increase in power.
With a closed-return wind tunnel of this type, all of the
power required to maintain the flow is dissipated through viscous
effects, with the energy remaining within the closed tunnel.
If heat transfer across the tunnel walls is negligible, the air
temperature within the tunnel will increase in time. For steadystate
operations of such tunnels, it is often necessary to provide
some means of cooling to maintain the temperature at acceptable
levels.
It should be noted that the actual size of the motor that powers
the fan must be greater than the calculated 62.3 hp because the fan
is not 100% efficient. The power calculated above is that needed
by the fluid to overcome losses in the tunnel, excluding those in
the fan. If the fan were 60% efficient, it would require a shaft
h L
(200 ft/s, 62.3 hp)
0
0 50 100 150
V 5 , ft/s
F I G U R E E8.6c
200 250 300
power of p 62.3 hp10.602 104 hp to run the fan. Determination
of fan 1or pump2 efficiencies can be a complex problem that
depends on the specific geometry of the fan. Introductory material
about fan performance is presented in Chapter 12; additional
material can be found in various references 1Refs. 14, 15, 16, for
example2.
It should also be noted that the above results are only
approximate. Clever, careful design of the various components
1corners, diffuser, etc.2 may lead to improved 1i.e., lower2
values of the various loss coefficients, and hence lower power requirements.
Since is proportional to V 2 , the components with
the larger V tend to have the larger head loss. Thus, even though
K L 0.2 for each of the four corners, the head loss for corner 172
is 1V 7V 3 2 2 18022.92 2 12.2 times greater than it is for corner
132.
8.4.3 Noncircular Conduits
Many of the conduits that are used for conveying fluids are not circular in cross section. Although the
details of the flows in such conduits depend on the exact cross-sectional shape, many round pipe results
can be carried over, with slight modification, to flow in conduits of other shapes.
Theoretical results can be obtained for fully developed laminar flow in noncircular
ducts, although the detailed mathematics often becomes rather cumbersome. For an arbitrary