5-Problems, fluid mech

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Problems
Problems
Flow Rate (Discharge)
5.1
Consider filling the gasoline tank of an automobile at a gas station. (a) Estimate the discharge in gpm.
(b) Using the same nozzle, estimate the time to put 50 gallons in the tank. (c) Estimate the crosssectional
area of the nozzle and calculate the velocity at the nozzle exit.
5.2
The average flow rate (release) through Grand Coulee Dam is 110,000 ft
3
/s. The width of the river
downstream of the dam is 100 yards. Making a reasonable estimate of the river velocity, estimate the river
depth.
5.3
Taking a jar of known volume, fill it with water from your household tap and measure the time to
fill. Calculate the discharge from the tap. Estimate the crosssectional area of the faucet outlet, and calculate
the water velocity issuing from the tap.
5.4
A liquid flows through a pipe with a constant velocity. If a pipe twice the size is used with the same
velocity, will the flow rate be (a) halved, (b) doubled, (c) quadrupled. Explain.
Answer:
(c)
5.5
The discharge of water in a 25 cm diameter pipe is 0.05 m
3
/s. What is the mean velocity?
5.6
A pipe with a 16 in. diameter carries water having a velocity of 3 ft/s. What is the discharge in cubic feet per
second and in gallons per minute (1 cfs is equivalent to 449 gpm)?
Answer:
Q
= 4.19 cfs, 1880 gpm
5.7
A pipe with a 2 m diameter carries water having a velocity of 4 m/s. What is the discharge in cubic meters
per second and in cubic feet per second?
5.8
A pipe whose diameter is 8 cm transports air with a temperature of 20°C and pressure of 200 kPa absolute at
20 m/s. Determine the mass flow rate.
Answer:
= 0.239 kg/s
5.9
Natural gas (methane) flows at 20 m/s through a pipe with a 1 m diameter. The temperature of the methane
is 15°C, and the pressure is 150 kPa gage. Determine the mass flow rate.
5.10
An aircraft engine test pipe is capable of providing a flow rate of 200 kg/s at altitude conditions
corresponding to an absolute pressure of 50 kPa and a temperature of 18°C. The velocity of air through
the duct attached to the engine is 240 m/s. Calculate the diameter of the duct.
Answer:
D
= 1.25 m
5.11
A heating and airconditioning engineer is designing a system to move 1000 m
3
of air per hour at 100 kPa
abs, and 30°C. The duct is rectangular with crosssectional dimensions of 1 m by 20 cm. What will be the
air velocity in the duct?
5.12
The hypothetical velocity distribution in a circular duct is
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Problems
where
r
is the radial location in the duct,
R
is the duct radius, and
V
0
is the velocity on the axis. Find the
ratio of the mean velocity to the velocity on the axis.
PROBLEM 5.12
Answer:
5.13
Water flows in a twodimensional channel of width
W
and depth
D
as shown in the diagram. The
hypothetical velocity profile for the water is
where V
s
is the velocity at the water surface midway between the channel walls. The coordinate system is
as shown;
x
is measured from the center plane of the channel and
y
downward from the water surface.
Find the discharge in the channel in terms of
V
s
,
D
, and
W
.
PROBLEM 5.13
5.14
Water flows in a pipe that has a 4 ft diameter and the following hypothetical velocity distribution: The
velocity is maximum at the centerline and decreases linearly with
r
to a minimum at the pipe wall. If
V
max
= 15 ft/s and
V
min
= 12 ft/s, what is the discharge in cubic feet per second and in gallons per minute?
Answer:
Q
= 163 cfs, 73,400 gpm
5.15
In Prob. 5.14, if
V
max
= 8 m/s,
V
min
= 6 m/s, and
D
= 2 m, what is the discharge in cubic meters per second
and the mean velocity?
5.16
Air enters this square duct at section 1 with the velocity distribution as shown. Note that the velocity varies
in the
y
direction only (for a given value of
y
, the velocity is the same for all values of
z
).
a. What is the volume rate of flow?
b. What is the mean velocity in the duct?
c.
What is the mass rate of flow if the mass density of the air is 1.2 kg/m
3
?
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Problems
PROBLEM 5.16
Answer:
Q
= 5 m
3
/s,
V
= 5 m/s, = 6.0 kg/s
5.17
The velocity at section
A-A
is 18 ft/s, and the vertical depth
y
at the same section is 4 ft. If the width of the
channel is 30 ft, what is the discharge in cubic feet per second?
PROBLEM 5.17
5.18
The rectangular channel shown is 1.5 m wide. What is the discharge in the channel?
PROBLEM 5.18
Answer:
Q
= 0.93 m
3
/s
5.19
If the velocity in the channel of Prob. 5.18 is given as
u
= 10[exp (
y
) 1] m/s and the channel width is 2 m,
what is the discharge in the channel and what is the mean velocity?
5.20
Water from a pipe is diverted into a weigh tank for exactly 20 min. The increased weight in the tank is 20
kN. What is the discharge in cubic meters per second? Assume T = 20°C.
Answer:
Q
= 1.70 × 10
3
m
3
/s
5.21
Water enters the lock of a ship canal through 180 ports, each port having a 2 ft by 2 ft cross section. The
lock is 900 ft long and 105 ft wide. The lock is designed so that the water surface in it will rise at a
maximum rate of 6 ft/min. For this condition, what will be the mean velocity in each port?
5.22
An empirical equation for the velocity distribution in a horizontal, rectangular, open channel is given by
u = u
max
(
y
/
d
)
n
, where
u
is the velocity at a distance
y
feet above the floor of the channel. If the depth
d
of
flow is 1.2 m, u
max
= 3 m/s, and n = 1/6, what is the discharge in cubic meters per second per meter of
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Problems
width of channel? What is the mean velocity?
Answer:
q
= 3.09 m
2
/s,
V
= 2.57 m/s
5.23
The hypothetical water velocity in a Vshaped channel (see the accompanying figure) varies linearly with
depth from zero at the bottom to maximum at the water surface. Determine the discharge if the maximum
velocity is 6 ft/s.
PROBLEM 5.23
5.24
The velocity of flow in a circular pipe varies according to the equation , where
V
c
is
the centerline velocity,
r
0
is the pipe radius, and
r
is the radial distance from the centerline. The exponent
n
is general and is chosen to fit a given profile (
n
= 1 for laminar flow). Determine the mean velocity as a
function of
V
c
and
n
.
Answer:
V
= [1/(
n
+ 1)]
V
c
5.25
Plot the velocity distribution across the pipe, and determine the discharge of a fluid flowing through a pipe
1 m in diameter that has a velocity distribution given by . Here
r
0
is the radius of
the pipe, and
r
is the radial distance from the centerline. What is the mean velocity?
5.26
Water flows through a 2.0 in.–diameter pipeline at 80 lb/min. Calculate the mean velocity. Assume
T
= 60°F.
Answer:
V
= 0.979 fps
5.27
Water flows through a 20 cm pipeline at 1000 kg/min. Calculate the mean velocity in meters per second if
T
= 20°C.
5.28
Water from a pipeline is diverted into a weigh tank for exactly 15 min. The increased weight in the tank is
4765 lbf. What is the average flow rate in gallons per minute and in cubic feet per second? Assume
T
= 60°F.
Answer:
Q
= 0.0849 cfs, 37.9 gpm
5.29
A shell and tube heat exchanger consists of a one pipe inside another pipe as shown. The liquid flows in
opposite directions in each pipe. If the speed of the liquid is the same in each pipe, what is the ratio of the
outer pipe diameter to the inner pipe diameter if the discharge in each pipe is the same?
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Problems
PROBLEM 5.29
5.30
The cross section of a heat exchanger consists of three circular pipes inside a larger pipe. The internal
diameter of the three smaller pipes is 2.5 cm, and the pipe wall thickness is 3 mm. The inside diameter of
the larger pipe is 8 cm. If the velocity of the fluid in region between the smaller pipes and larger pipe is
10 m/s, what is the discharge in m
3
/s?
PROBLEM 5.30
Answer:
Q
= 0.0276 m
3
/s
5.31
The mean velocity of water in a 4 in. pipe is 10 ft/s. Determine the flow in slugs per second, gallons per
minute, and cubic feet per second if
T
= 60°F.
Control Volume Approach
5.32
What is a control surface and a control volume? Can mass pass through a control surface?
5.33
What is the difference between an intensive and extensive property? Give an example of each.
5.34
Explain the differences between the Eulerian and Lagrangian descriptions of a flow field.
5.35
What are the shortcomings of describing a flow field using the Lagrangian description?
5.36
What is the purpose of the Reynolds transport theorem?
5.37
Gas flows into and out of the chamber as shown. For the conditions shown, which of the following
statement(s) are true of the application of the control volume equation to the continuity principle?
a.
B
sys
= 0
b.
dB
sys
/
dt
= 0
c.
d.
e.
b
= 0
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