# Impact of a Jet â€“ Conservation of Linear Momentum

Impact of a Jet â€“ Conservation of Linear Momentum
â€¢ Measure the force on the
plate Fex due to the jet (jetvane system) via balance of
moment (weight beam).
â€¢ Calculate the force on the
plate Fj using the linear
momentum balance.
â€¢Compare measured value Fex
with theoretically derived
value Fj
. Discuss errors causing
the observed difference
between Fex and Fj
S
â‘
â“ª
z
Use conservation of energy to obtain jet velocity
impinging onto the plate:
hL
g
P V
z
g
P V
z ï€« ï€« ï€½ ï€« ï€« ï€«
2 2
2
1 1
1
2
0 0
0
ï§ ï§
pressure; z1
-z0=S
V V gS
g
V
S
g
V
2
2 2

2
1 0
2
1
2
0
ïƒž ï€½ ï€­
ïƒž ï€½ ï€«
Note
V0=Q/A can be obtained from the measurement:
Q: total measured volume in weight tank/duration
1. Calculate the force on the plate by the jet using linear momentum
balance:
F1
minVin
ï€¦
A
ï²
A
ï²
V
ï¶
V ïƒ— AïƒžPositive
ï¶ ï¶
A
ï²
V
ï¶
V ïƒ— Aïƒž Negative
ï¶ ï¶
Conservation of linear momentum in z-direction:
ïƒ¥ ï€½ïƒ¥ ï€­ïƒ¥
z
in in
z
out out
z
F mï€¦ V mï€¦ V
ï€­minV1
ï€½ F1
F1
ï€½ ï€­Fj
ï€¦
,
F1
: force exerted by the plate onto
the jet
Fj
: reactive force; force exert by the
jet onto the plate
ïƒž Positive if flow is out of control surface (C.S.); Negative if flow is into C.S.
1 1
ïœ Fj
ï€½ m
ï€¦
inV ï€½ ï²QV
In a given direction:
net force = net momentum fluxes
Momentum flux = mass flux times velocity
V A
ï¶ ï¶
mass flux =
ï² ï‚·
2. Jet force can be directly measured from the balance of moment of
the weight-beam system:
W
Fsp
a
b
W
Fsp
a
b + x
Fex
x
Fsp
ïƒ—a ï€½ Wïƒ— g ïƒ—b
a F g ï€¨b xï€© Fsp
ïƒ— ï€« 0.15(m)
ïƒ—
ex
ï€½ Wïƒ— ïƒ— ï€«
0.15 m
Note: given in class note W=0.6 Kg
Fex calculated this way can be
compared with that calculated by
linear momentum balance Fj
.
Error Analysis:
Fj=ï²QV1
Error in Fj
can be caused by Q and V1
Fj+ Î”Fj =ï²(Q+Î”Q)(V1+Î”V1
)
Fj+ Î”Fj =ï²QV1+ï²Â·QÂ·Î”V1+ï²Â·Î”QÂ·V1+ ï²Â·Î”QÂ·Î”V1
Normalized by Fj
, we obtain:
1
1
V
V
Q
Q
F
F
j
j ï„
ï€«
ï„
ï€½
ï„
neglect higher
order
0
0.15
W Fj
gx
ï€½
Error in Fj
can be caused by x and W
0.15
0.15
W
0.15
W W
Fj
Î”Fj
ï„ ï„ ï„ ï„
ï„ ï„
ï€« ïƒ— ï€« ïƒ— ï€« ïƒ—
ï€½
ï€½ ï€« ï€«
ï€«
x x x
g
g
x
g
x x
0 neglect
x
x
x x
g
ï„
ï€«
ï„
ïƒž ï€½
ïœ ï„ ï€½ ïƒ— ï€« ï„ ïƒ—
W
W
F
Î”F

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Î”W W
0.15
F
j
j
j
This relation is linear.
i.e., 2% error in Q and 3% error in V1 make
total error in estimating Fj
to be 5%.
For a circular plate:
A V
ï€«
A V
ï€«
A V
ï€­
S
hL
g
P V
z
g
P V
z ï€« ï€« ï€½ ï€« ï€« ï€«
2 2
2
1 1
1
2
0 0
0
ï§ ï§
V V gS
g
V
S
g
V
2
2 2

2
1 0
2
1
2
0
ïƒž ï€½ ï€­
ïƒž ï€½ ï€«
z
2mout V1
F1
z
ïƒ¥ ï€½ ï€­ ï€¦ ï€« ï€¦ ï€­ ï€½
F1
1. Energy balance:
2. Momentum balance:
Mass conservation suggest:
min mout
ï€¦
ï€½ 2 ï€¦
ïƒž F1
ï€½ ï€­2minV1
ï€½ ï€­Fj
ï€¦
Force for circular plate is two times
larger than that of flat plate !

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