Lab 2 - Impact of a Jet
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2050
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Date
Dec 6, 2023
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7
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Impact of a Jet
Sloane Hazzard
10/25/2023-11/2/2023
CIVL 2050
Prof. Dan Lander
2. THEORY
In this experiment – Impact of a Jet – a vertically aligned free jet of velocity jet
𝒱
flows into a
deflector of weight
W
. The deflector changes the direction of the jet by an angle α without altering
its speed. For an incompressible and steady flow, show that the weight of the deflector is balanced
by the change in the jet’s momentum, such that
𝑊
=
𝜌𝑄𝒱
(1 − cos α)
(1)
Assume the weight of the water is negligible and that the effect of friction from the air and the
deflector do not affect the jet speed. [Hint: use a control volume which surrounds the deflector] [1]
Figure 1.
Free Jet Flow and a Deflector [1]
The equation of conservation of momentum can be used to show the weight of the deflector is
balanced by the change in the jet’s momentum.
(2)
The angle of deflection is
α
and the flow that we are analyzing is only in the y-direction therefore the
conservation of momentum equation becomes
(3)
We then substitute -W for F
y
since they are equal magnitudes but in opposite directions, also
substituting v out the conservation of momentum equation becomes
(4)
And since m in the equation is equal to density multiplied by velocity and area we can substitute in
Q and the conservation of momentum equation becomes
𝑊
=
𝜌𝑄𝒱
(1 − cos α)
And considering the equation of
𝑆
theo
=
𝑊
/
𝒱
2
(5)
we get
𝑆
theo
=
𝜌𝐴
(1 − cos α)
(6)
3. APPARATUS
3.1 Diagram of Apparatus
The jet apparatus is a clear acrylic cylinder, a nozzle, and a flow deflector (Figure 2). Water enters
vertically from the top of the cylinder, through a nozzle striking a target, mounted on a stem, and
leaves through the outlet holes in the base of the cylinder. An air vent at the top of the cylinder
maintains the atmospheric pressure inside the cylinder. A weight pan is mounted at the top of the
stem to allow the force of the striking water to be counterbalanced by applied masses. [1]
Figure 2.
F1-16 Impact of Jet Apparatus
Within the lab set up there is a built-in volumetric cylinder (Figure 3) that we used to measure
volume of water at certain times to determine volumetric flow rates.
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Figure 3.
Volumetric Cylinder Measurement Device
3.2 Variables:
D – Orifice Diameter
m – Mass
(kg)
V – Volume (L
→
m
3
)
ρ – Density (kg/ m
3
)
α – Angle of Deflection (θ)
3.3 Measuring Devices:
Phone stopwatch – measures time
4. RESULTS
Table 1.
Estimated Uncertainty
Independent
Variable
X
i
Measurement Device
Least Count
Measure [Units]
Estimated
Error [Units]
Relative
Uncertainty u
xi
[%]
Orifice Diameter
D
Factory Measured
-
±0.00025 m [1]
±31
Mass
m
Factory Measured
-
±0.001 kg [1]
±1.3
Volume
V
Volumetric Cylinder
.5 L
±10mL
±0.056
Time
T
Phone Stopwatch
.02 s
±.01s
±0.017
Density
Ρ
N/A
-
N/A
±0.013
Angle
α
Factory Measured
-
±0.01° [1]
±0.0083
Table 2.
Comparison of experimental slopes and theoretical slopes of 120° and 180° deflector
S(exp)
S(theo)
Difference (%)
Relative Uncertainty u
xi
(%)
Average for 120
0.073517
0.074
0.652507
8.875579
Average for 180
0.085624
0.0932
8.12868
94.93452
Figure 4.
Weight vs. Velocity
2
for 120 and 180 Deflectors
Figure 5.
Force vs. Weight for 120 and 180 Deflectors
5. DISCUSSION
The practical application of this principle is that engineers and designers use the momentum
equation to accurately calculate the force that moving fluid may exert on a solid body. For example,
in hydropower plants, turbines are utilized to generate electricity. Turbines rotate due to force
exerted by one or more water jets that are directed tangentially onto the turbine’s vanes or buckets.
The impact of the water on the vanes generates torque on the wheel, causing it to rotate and to
generate electricity. [1] The experiment does provide a feasible means of verifying the conservation
of momentum equation because it was able to show a direct relationship between jet force and
y = 0.074x - 0.0467
y = 0.0932x - 0.1696
0
1
2
3
4
5
6
-10
0
10
20
30
40
50
60
70
80
Weight, W (N)
Velocity
2
, V
2
(m/s
2
)
Deflector 1 - 120
Deflector 2 - 180
Linear (Deflector 1 - 120)
Linear (Deflector 2 - 180)
y = 6.025x - 5.6332
y = 7.4512x - 7.1774
-10
-5
0
5
10
15
20
25
30
35
40
0
1
2
3
4
5
6
Jet Force, F (N)
Weight, W (N)
Deflector 2 - 180
Deflector 1 - 120
Linear (Deflector 2 - 180)
Linear (Deflector 1 - 120)
weight as seen in Figure 5. As well from the experiment we were able to compare the theoretical
slopes with the experimental slopes which had a .653% difference for the 120
° deflector and an
8.13% difference for the 180° deflector. Since these percentage differences are relatively low it
verifies the conservation of momentum equation. The result of omitting viscosity in this experiment
makes weight and jet force equal meaning there would be no difference in the theoretical and
experimental slopes. If different deflectors were closer to the nozzle,
results would still be the same
since the equation we derived for jet force does not rely on that distance between the deflector and
nozzle. Any differences between the theoretical and experimental results could be the result of
volume and time measurements taken quickly and with error. Since draining the volumetric cylinder
and filling it back up quickly resulted in excess movement in the measuring device it could have
altered the results.
The variable
𝒱
2
is most sensitive to flow rate since it is volume divided time and neither are held
constant. F
y
is most sensitive to
𝒱
2
because force is mass times
𝒱
2
and it is not held constant. And
𝑆
theo
is most sensitive to angle because the only variable that changes in its calculation is the different
deflector angles. The measurement that had the largest impact on
𝒱
2
was relative uncertainty of
volume. For F
y
, volume and time are the largest impacting measurements. And the largest impacting
measurement for S
theo
is angle. Errors could be minimized by taking time and volume measurements
more consistently at exact times or exact volumes. Other sources of error could be turbulent water.
Sources of systematic error might be air resistance, weight of the water, weight of the pan. These
sources of error change the required force of the jet ultimately increasing force slightly and
decreasing slope marginally.
DATA APPENDIX
Table 3:
Raw Data of the 120
° and 180° Deflectors
Deflector 1: 120°
Deflector 2: 180°
Test No. Volume (Liters) Time (s)
Applied Mass
(kg)
Volume (Liters) Time (s)
Applied Mass
(kg)
1
7
59.84
0.05
9
69.02
0.05
2
13
67.09
0.1
10
58.56
0.1
3
15
64.78
0.15
13
61.14
0.15
4
17
64.11
0.2
15
62.43
0.2
5
19
65
0.25
16
59.39
0.25
6
20
62.26
0.3
18
61.17
0.3
7
22
63.87
0.35
20
63.93
0.35
8
23
62.6
0.4
21
62.43
0.4
9
24
61.9
0.45
21
59.53
0.45
10
25
60.79
0.5
23
62.99
0.5
Table 4:
Results Data for 120
° Deflector
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Test No.
Weight
(N)
Flow Rate
Q(m
3
/s)
Velocity
V (m/s)
Velocity
2
V
2
(m/s
2
)
Jet Force
F (N)
S
exp
S
theo
1
0.4905
0.000116979
2.327006
5.414959
0.27075
0.090582
0.074
2
0.981
0.00019377
3.854577
14.85776
1.48578
0.066026
0.074
3
1.4715
0.000231553
4.606186
21.21695
3.18254
0.069355
0.074
4
1.962
0.000265169
5.2749
27.82457
5.56491
0.070513
0.074
5
2.4525
0.000292308
5.814754
33.81137
8.45284
0.072535
0.074
6
2.943
0.000321234
6.390164
40.83419
12.2503
0.072072
0.074
7
3.4335
0.00034445
6.851993
46.9498
16.4324
0.073131
0.074
8
3.924
0.000367412
7.308775
53.4182
21.3673
0.073458
0.074
9
4.4145
0.000387722
7.712794
59.48718
26.7692
0.074209
0.074
10
4.905
0.000411252
8.18086
66.92648
33.4632
0.073289
0.074
Table 5:
Results Data for 180
° Deflector
Test
No.
Weight
(N)
Flow
Rate
Q(m
3
/s)
Velocity
V (m/s)
Velocity
2
V
2
(m/s
2
)
Jet
Force F
(N)
S
exp
S
theo
1
0.4905
0.00013
2.593932
6.728486
0.336424
0.072899
0.0932
2
0.981
0.000171
3.396957
11.53932
1.153932
0.085014
0.0932
3
1.4715
0.000213
4.229695
17.89032
2.683548
0.082251
0.0932
4
1.962
0.00024
4.779572
22.84431
4.568862
0.085886
0.0932
5
2.4525
0.000269
5.359173
28.72073
7.180184
0.085391
0.0932
6
2.943
0.000294
5.853628
34.26496
10.27949
0.085889
0.0932
7
3.4335
0.000313
6.223238
38.72869
13.55504
0.088655
0.0932
8
3.924
0.000336
6.691401
44.77485
17.90994
0.087638
0.0932
9
4.4145
0.000353
7.017372
49.24352
22.15958
0.089646
0.0932
10
4.905
0.000365
7.263523
52.75877
26.37939
0.09297
0.0932
SAMPLE CALCULATIONS
Weight = mass*gravity
→
.05kg*9.81m/s
2
= 0.4905 N
m
3
=.001L
→
.001(7L) = .007m
3
Flow Rate = Volume/Time
→
.007 m
3
/59.84s = 1.17E-4
m
3
/s
Velocity = Flow Rate/Area
→
1.17E-4
m
3
/s/5.027E-5m
2
= 2.33m/s
Velocity
2
= (2.33m/s)
2
= 5.41m/s
2
Jet Force = Mass* Velocity
2
→
.05kg*5.41m/s
2
= 0.271 N
S
exp
= Weight/ Velocity
2
→
.05kg/5.41m/s
2
= 0.0729
BIBLIOGRAPHY
[1] Lander, D., “Impact of a Jet” Instruction Manual. Department of Civil and Environmental Engineering, Rensselaer
Polytechnic Institute, Troy, NY. 2023.