fluid_mechanics

118 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation
SOLUTION
(2)
If the flow is steady, inviscid, and incompressible we can apply
the Bernoulli equation along the streamline from 112 to 122 to 132 as
follows:
p 1 1 2rV 2 1 gz 1 p 2 1 2rV 2 2 gz 2
p 3 1 2rV 2 3 gz 3
With the tank bottom as the datum, we have z 1 15 ft, z 2 H,
and z 3 5 ft. Also, V 1 0 1large tank2, p 1 0 1open tank2,
p 3 0 1free jet2, and from the continuity equation A 2 V 2 A 3 V 3 , or
because the hose is constant diameter, V 2 V 3 . Thus, the speed of
the fluid in the hose is determined from Eq. 1 to be
V 3 22g1z 1 z 3 2 22132.2 fts 2 2315 1524 ft
(1)
(3)
(1)
35.9 fts V 2
Use of Eq. 1 between points 112 and 122 then gives the pressure p 2
at the top of the hill as
p 2 p 1 1 2rV 2 1 gz 1 1 2rV 2 2 gz 2
From Table B.1, the vapor pressure of water at 60 °F is
0.256 psia. Hence, for incipient cavitation the lowest pressure in
the system will be p 0.256 psia. Careful consideration of Eq. 2
and Fig. E3.10b will show that this lowest pressure will occur at
the top of the hill. Since we have used gage pressure at point 112
1 p 1 02, we must use gage pressure at point 122 also. Thus,
p 2 0.256 14.7 14.4 psi and Eq. 2 gives
114.4 lbin. 2 21144 in. 2 ft 2 2
162.4 lbft 3 2115 H2ft 1 2 11.94 slugsft 3 2135.9 fts2 2
or
g1z 1 z 2 2 1 2rV 2 2
H 28.2 ft
(2)
(Ans)
For larger values of H, vapor bubbles will form at point 122 and the
siphon action may stop.
COMMENTS Note that we could have used absolute pressure
throughout 1p 2 0.256 psia and p 1 14.7 psia2 and obtained
the same result. The lower the elevation of point 132, the
larger the flowrate and, therefore, the smaller the value of H allowed.
We could also have used the Bernoulli equation between 122
and 132, with V 2 V 3 , to obtain the same value of H. In this case
it would not have been necessary to determine V 2 by use of the
Bernoulli equation between 112 and 132.
The above results are independent of the diameter and length
of the hose 1provided viscous effects are not important2. Proper
design of the hose 1or pipe2 is needed to ensure that it will not collapse
due to the large pressure difference 1vacuum2 between the
inside and outside of the hose.
15 ft
F I G U R E E3.10a
(1)
Water
H
(2)
5 ft
F I G U R E E3.10b
By using the fluid properties listed in Table 1.5 and repeating
the calculations for various fluids, the results shown in
Fig. E3.10c are obtained. The value of H is a function of both the
specific weight of the fluid, g, and its vapor pressure, .
H, ft
40
35
30
25
20
15
10
5
0
Alcohol
Water
Fluid
F I G U R E E3.10c
Gasoline
(3)
Carbon tet
p v
3.6.3 Flowrate Measurement
Many types of devices using principles involved in the Bernoulli equation have been developed
to measure fluid velocities and flowrates. The Pitot-static tube discussed in Section 3.5 is an
example. Other examples discussed below include devices to measure flowrates in pipes and