Chapter 13
Momentum Analysis of Flow Systems
Review Problems
1359
Water is flowing into and discharging from a pipe Usection with a secondary discharge section
normal to return flow. Net x and y forces at the two flanges that connect the pipes are to be determined.
Assumptions
1
The flow is steady, onedimensional, and incompressible.
2
The weight of the Uturn and
the water in it is negligible.
Properties
We take the density of water to be 1000 kg/m
3
.
Analysis
The flow velocities of the 3 streams are
m/s
51
.
0
]
4
/
m)
05
.
0
(
)[
kg/m
(1000
kg/s
30
)
4
/
(
2
3
2
1
1
1
1
1
D
m
A
m
V
m/s
80
.
2
]
4
/
m)
10
.
0
(
)[
kg/m
(1000
kg/s
22
)
4
/
(
2
3
2
2
2
2
2
2
D
m
A
m
V
m/s
3
.
11
]
4
/
m)
03
.
0
(
)[
kg/m
(1000
kg/s
8
)
4
/
(
2
3
2
3
3
3
3
3
D
m
A
m
V
We take the entire Usection as the control volume. We designate the horizontal coordinate by
x
with the
direction of incoming flow as being the positive direction and the vertical coordinate by
y
. The momentum
equation for steady onedimensional flow is
i
i
e
e
m
m
F
V
V
. We let the
x
and
y
components of the
anchoring force of the cone be
F
Rx
and
F
Ry
, and assume them to be in the positive directions. Then the
momentum equations along the
x
and
y
axes become
0

0
)
(
3
3
3
3
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
V
V
V
V
V
V
m
F
m
F
m
m
A
P
A
P
F
m
m
A
P
A
P
F
Ry
Ry
Rx
Rx
Substituting the given values,
N
666
N
k
F
Rx
666
.
0
m/s
kg
1000
kN
1
m/s)
1
kg/s)(0.5
30
(
m/s
kg
1000
kN
1
m/s)
0
kg/s)(2.8
22
(
4
m)
(0.10
]
kN/m
)
100
150
[(
4
m)
(0.05
]
kN/m
)
100
200
[(
2
2
2
2
2
2
N
90.4
2
m/s
kg
1
N
1
m/s)
3
kg/s)(11.
8
(
Ry
F
The negative value for
F
Rx
indicates the assumed direction is wrong, and should be reversed. Therefore, a
force of 666 N acts on the flanges in the opposite direction. A vertical force of 90.4 N acts on the flange in
the vertical direction.
Discussion
To assess the significance of gravity forces, we estimate the weight of the weight of water in
the Uturn and compare it to the vertical force. Assuming the length of the Uturn to be 0.5 m and the
average diameter to be 7.5 cm, the mass of the water becomes
kg
2
.
2
m)
(0.5
4
m)
(0.075
)
kg/m
1000
(
4
2
3
2
L
D
AL
V
m
whose weight is 2.2
9.81 = 22 N, which is much less than 90.4 N, but still significant. Therefore,
disregarding the gravitational effects is a reasonable assumption if great accuracy is not required.
1360
A fireman was hit by a nozzle held by a tripod with a rated holding force. The accident is to be
investigated by calculating
the water velocity, the flow rate, and the nozzle velocity.
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 Spring '08
 Chung
 Fluid Dynamics, Force, Flow Systems, water jet, Momentum Analysis

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