fluid_mechanics

114 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation
SOLUTION
COMMENTS Note that this problem was solved using
For steady, inviscid, incompressible flow, the Bernoulli equation
p 1 1 2rV 2 1 gz 1 p 2 1 2rV 2 2 gz 2 (1)
pipe, respectively. Although this was convenient (because most
applied between points 112 and 122 is
points (1) and (2) located at the free surface and the exit of the
With the assumptions that p 1 p 2 0, z 1 h, and z 2 0, Eq. 1
becomes
1
2V 2 1 gh 1 2V 2 2
(2)
Although the liquid level remains constant 1h constant2, there is an
average velocity, V 1 , across section 112 because of the flow from the
tank. From Eq. 3.19 for steady incompressible flow, conservation of
mass requires Q 1 Q 2 , where Q AV. Thus, A 1 V 1 A 2 V 2 , or
p
4 D2 V 1 p 4 d2 V 2
Hence,
Equations 1 and 3 can be combined to give
2gh
V 2 B 1 1dD2 219.81 ms 2 210.20 m2
4 B 1 10.01 m0.20 m2 1.98 m s
4
Thus,
V 1 a d D b 2V 2
Q A 1 V 1 A 2 V 2 p 4 10.01 m22 11.98 ms2
1.56 10 4 m 3 s
(3)
(Ans)
of the variables are known at those points), other points could be
selected and the same result would be obtained. For example,
consider points (1) and (3) as indicated in Fig. E3.7b. At (3), located
sufficiently far from the tank exit, V 3 0 and z 3 z 2 0.
Also, p 3 h since the pressure is hydrostatic sufficiently far
from the exit. Use of this information in the Bernoulli equation
applied between (1) and (3) gives the exact same result as obtained
using it between (1) and (2). The only difference is that
the elevation head, z 1 h, has been interchanged with the pressure
head at (3), p 3 / h.
In this example we have not neglected the kinetic energy of
the water in the tank 1V 1 02. If the tank diameter is large compared
to the jet diameter 1D d2, Eq. 3 indicates that V 1 V 2
and the assumption that V 1 0 would be reasonable. The error
associated with this assumption can be seen by calculating the
ratio of the flowrate assuming V 1 0, denoted Q, to that assuming
V 1 0, denoted Q 0 . This ratio, written as
Q
V 2
22gh 31 1dD2 4 4
Q 0 V 2 0 D 22gh
is plotted in Fig. E3.7c. With 0 6 dD 6 0.4 it follows that
1 6 QQ 0 1.01, and the error in assuming V 1 0 is less than
1%. For this example with d/D 0.01 m/0.20 m 0.05, it follows
that Q/Q 0 1.000003. Thus, it is often reasonable to assume
V 1 0.
1
21 1dD2 4
The fact that a kinetic energy change is often accompanied by a change in pressure is shown
by Example 3.8.
E XAMPLE 3.8
Flow from a Tank—Pressure
GIVEN Air flows steadily from a tank, through a hose of diameter
D 0.03 m, and exits to the atmosphere from a nozzle of
diameter d 0.01 m as shown in Fig. E3.8. The pressure in the
tank remains constant at 3.0 kPa 1gage2 and the atmospheric conditions
are standard temperature and pressure.
FIND
Determine the flowrate and the pressure in the hose.
(1)
Air
(2) (3)
p 1 = 3.0 kPa
D = 0.03 m
d = 0.01 m
Q
F I G U R E E3.8
SOLUTION
If the flow is assumed steady, inviscid, and incompressible, we
can apply the Bernoulli equation along the streamline from (1) to
(2) to (3) as
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 assumption that z 1 z 2 z 3 1horizontal hose2, V 1 0
1large tank2,and p 3 0 1free jet2, this becomes
V 3 B
2p 1
r