Problem 4.78

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Problem 4.78 [Difficulty: 2]
Rx 
y 
x 
CS
Given: Water flow through elbow
Find: Force to hold elbow
Solution:
Basic equation: Momentum flux in x direction for the elbow
Assumptions: 1) Steady flow 2) Incompressible flow 3) Atmospheric pressure at exit 4) Uniform flow
Hence Rx p1g A1⋅+ V1 ρ− V1⋅ A1⋅( )⋅ V2 ρ V2⋅ A2⋅( )⋅−= Rx p1g− A1⋅ ρ V12 A1⋅ V22 A2⋅+⎛⎝ ⎞⎠⋅−=
From continuity V2 A2⋅ V1 A1⋅= so V2 V1
A1
A2
⋅= V2 10
ft
s
⋅ 4
1
⋅= V2 40
ft
s
⋅=
Hence Rx 15−
lbf
in2
⋅ 4× in2⋅ 1.94 slug
ft3
⋅ 10 ft
s
⋅⎛⎜⎝
⎞
⎠
2
4⋅ in2⋅ 40 ft
s
⋅⎛⎜⎝
⎞
⎠
2
1⋅ in2⋅+
⎡⎢⎣
⎤⎥⎦×
1 ft⋅
12 in⋅
⎛⎜⎝
⎞
⎠
2
× lbf s
2⋅
slug ft⋅×−= Rx 86.9− lbf⋅=
The force is to the left: It is needed to hold the elbow on against the high pressure, plus it generates the large change in x momentum