Problem 4.139
[Difficulty: 4]
Given:
Jet impacting a splitter vane
Find:
Mass flow rate ratio; new speed U
Solution:
Apply momentum equation to inertial CV
Assumptions: No pressure force; neglect water mass on vane; steady flow wrt vane; uniform flow; no change of speed wrt the vane
Basic equation
Given data
V2
5
m
s
⋅
=
A
7.85 10
5
−
⋅
m
2
⋅
=
U1
0
m
s
⋅
=
θ
30 deg
⋅
=
ρ
999
kg
m
3
⋅
=
For constant speed wrt the vane, the jet velocity at each location is
VU
−
For no vertical force, y momentum becomes
0v
1
m
1
−
()
⋅
v
2
m
2
⋅
+
v
3
m
3
⋅
+
=
where v
i
and m
i
are the vertical components
of velocity and mass flow rates, respectively,
at the inlet and exits, wrt the vane
coordinates
Hence
00 VU
−
m
2
⋅
+
−
(
) sin
θ
⋅
m
3
⋅
−
=
or
m
2
m
3
sin
θ
⋅
=
m
2
m
3
sin
θ
=
1
2
=
Note that
m
1
ρ
A
⋅
−
⋅
=
m
1
1.18
kg
s
=
and
m
1
m
2
m
3
+
=
so
m
1
m
3
sin
θ
⋅
m
3
+
=
m
3
m
1
1
1
sin
θ
+
=
m
3
m
1
2
3
=
m
3
0.784
kg
s
=
m
2
0.392
kg
s
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 Fall '07
 Lear
 Fluid Dynamics, Fluid Mechanics, Force, Mass, Velocity, Mass flow rate, θ

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