Target_Nozzle_Characterization_and_Assessment_Report_Final2

Target/Nozzle Characterization Report for Water Jet
Sean W. Tingey
October 21, 2015

Abstract
An experiment was recently performed where a jet of water was directed to impact an object and the
resultant force on the object was measured. Three different target objects in total were used one after
another at five different rates of water flow and two diametrically differing streams, giving eighteen
outputs of force. The independent variables of interest in the experiment were the water volumetric
flow rate through the nozzle, angle of redirected water after impact, and stream diametrical size.
A theoretical model for the water jet/ target system is produced and compared to the experimental
results. The Coanda effect is skirted around by extending control volumes. Percent error is analyzed
and discrepancies are accounted for. After further observation and estimation, a more enhanced
model is developed by combining the energy equation with the momentum equation. Assumptions
are analyzed, and the percent error for the final model is reduced from 45.13% to 30.00%.
1

Introduction
This report will first describe the development of a symbolic model for the water jet/ target system
in terms of the variables of interest. It will further divulge the methods and details of the experiment
with its apparatus. Then it illustrates in greater detail the effect which the variables of interest have
on the force output according to the symbolic model. The percent uncertainty is discussed in regards
to the accuracy of the model as it compares to measured values resulting from the experiment. Key
error causing factors are addressed and accounted for resulting in a more accurate model for the
water jet/ target system. The objective of this report is to outline the arrival of a more accurate
mathematical model of the water jet/ target system through experimentation and numerical analysis.
Theory
To fully understand this experiment, a greater understanding of the momentum equation and uncer-
tainty analysis is required. This understanding will allow an accurate prediction of a force produced,
from the water-jet onto the target, when the independent variables are adjusted. These variables
are: the angle of the redirected water, the diameter of the jet-stream, and the volumetric flow rate,
which are represented by θ, d, and ˙V respectively. The momentum equation is proven by using
the Reynold’s Transport Theorem and Newton’s second law. Reynold’s Transport Theorem uses
the forces acting on a control volume, which are surface forces and body forces. Newton’s second
law states that the sum of the forces acting on the control volume is equivalent to the time rate of
change in linear momentum of the system. After combining the Reynold’s Transport Theorem and
Newton’s second law, the linear momentum equation for any given control volume is
F =
d
dt
ρvdV + ρv(vf · n)dA (1)
The first term in the equation, d
dt ρvdV , is always zero when the control volume is non-accelerating
and non-deforming. Using this principle, Eq. 1 simplifies to
2

F = ρv(vf · n)dA (2)
where ρ(vf · n)dA represents the mass flow rate over an area, and v is the fluid velocity in vector
form. This can be further simplified by making the assumption of steady-state fluid flow:
F =
out
β ˙mv −
in
β ˙mv (3)
This equation illustrates that the sum of the forces acting on a control volume during steady flow
is equal to the difference between outgoing and incoming momentum flow rates. Since ˙m = ˙V ρ
and vy =
˙V
A where A is the circular cross-sectional area of the fluid flow, Eq. 3 can be written in
terms of the independent variables used in the experiment:
Fy = ˙Voutρ
˙Vout
π
4 d2
Cos(θ) − ˙Vinρ
˙Vin
π
4 d2
(4)
where θ accounts for the angle at which the flow is deflected upon impact with the target. d is the
diameter of the fluid flow as shown in Figure1. If we assume the flow to be incompressible, then we
can also assume ˙Vout = ˙Vin, which simplifies Eq. 4 to the form:
Fy =
4ρ ˙V 2
πd2
(1 − Cos(θ)) (5)
where ρ is density, ˙V is volumetric flow rate, and d, remains as the cross-sectional diameter of
the fluid stream.
When any experiment is performed there is always a chance of error. There are three principle
types of error: systematic, illegitimate, and random. The first accounts for human error, the second
for computational errors, and the third for measuring system sensitivity. The first two forms of error
should be avoided and fixed, thus causing an experiment to be rerun. The third is more difficult
3

Figure 1: Schematic for experiment
to avoid and should be calculated through uncertainty analysis. It is important to understand the
difference between error and uncertainty. Error is the difference between the correct value and the
measured value. Whereas, uncertainty describes the range in which a value can still be correct. The
uncertainty in this experiment is contained in the measured independent variables, because they are
not exact. The variables include ˙V , ρ, d, and θ where θ is the angle from the stream to the redirected
fluid flow see Fig 1. The errors from each measurement system will be propagated to get an overall
uncertainty of the calculated force. This is preformed by using Taylor’s theorem to get a basis of
uncertainty propagation, which is
µF = µ ˙V
∂F
∂ ˙V
2
+ µθ
∂F
∂θ
2
+ µρ
∂F
∂ρ
2
+ µd
∂F
∂d
2
(6)
The partial derivatives of Eq. 5 with respect to flow rate, θ, ρ and diameter are taken. Then the
first term must be multiplied by
˙V
˙V
, the second by 1−cos(θ)
1−cos(θ) , the third by ρ
ρ, and the forth by d
d . This
put a F in each of the terms, which can be factored out of the square root as shown
µF =
4ρ ˙V 2(1 − cos(θ))
πd2
2µ ˙V
˙V
2
+
µθsin(θ)
1 − cos(θ)
2
+
µρ
ρ
2
+
2µd
d
2
(7)
Both sides of the equation can be divided by F to get
µF
F
=
2µ ˙V
˙V
2
+
µθsin(θ)
1 − cos(θ)
2
+
µρ
ρ
2
+
2µd
d
2
(8)
This is the uncertainty equation in terms of the measured values for this experiment.
4

Experimental Methods
The momentum apparatus (see figure 2) was used along with two different nozzles to shoot water
at three different targets and measure the force on the target. The first step in the testing procedure
was to select an initial target from the three options (see figure 3). One target had a flat end, one
had a cone shaped end, and the last target had a recessed cone shape. Next a nozzle was selected
and threaded into the momentum apparatus (see figure 4). The pump was then turned on and the
maximum and minimum flow rates were determined for that nozzle. The water was then shut off and
the force was zeroed on the target. The pump was then turned back on and the throttle was used to set
the volumetric flow rate to the first specified value. The target was then centered in the apparatus.
The operator then pressed the ”End Test” button, then waited for five seconds then pressed the
”End Test” button again. Another team member then recorded the average force and the standard
deviation as it was recorded through the force measuring device shown in figure 5. The throttle was
used to set the volumetric flow rate to the second specified value and the experiment was ran again
for this value on the first target. Once all three volumetric flow rates had been experimented on the
first target, the target was then removed and replaced with the second target and then the third. Once
all targets had been experimented on with the first nozzle, it was replaced with the second nozzle
and all three targets were run again at all three different flow rates. Lastly, the theoretical forces
were calculated and compared with the actual forces measured in the experiment. The uncertainties
were also calculated for each measured and calculated force.
5

Figure 2: Apparatus for experiment
Figure 3: Targets for experiment
Figure 4: Nozzles for experiment
6

Figure 5: Force sensor for experiment
Table 1: Forces acting on each target object averaged over all results.
Target θ(°) Forceave,Measured(N) Forceave,Theoretical(N)
Divot 153 0.387 0.591
Flat Plate 90 0.255 0.312
Cone 31 0.102 0.043
Results and Discussion
The calculated results from Equation 5 show that as the angle θ increases from 0° to 180°, the force
also increases see Figure 6. If the graph in Figure 6 continued on from 180° to 360° it would show a
Cosine wave. As can be seen from table 1, the overall trend for the calculated and measured values
suggests that the divot experienced the most force, the cone experienced the least force, and the flat
plate fell between the two. This may suggest that θ had the greatest impression on the force output,
since θ is the only variable not held constant in the evaluation between targets. Since θDivot = 153°
is the highest change in momentum direction of the fluid between the three targets, it could be
assumed that the greater the change in momentum direction of the fluid flow, the greater the force
on the colliding object causing the change.
Table 2: Forces acting on targets at different diametrical flows averaged over all results.
Diameter (mm) Forceave,Measured(N) Forceave,Theoretical(N)
3.88 0.278 0.330
6.95 0.218 0.301
7

Figure 6: Force as a function of theta and volumetric ﬂow rate for two different nozzle diameters
Figure 7: Force as a function of nozzle diameter and volumetric ﬂow rate for all three targets
8

Table 3: Forces acting on targets at different flow rates averaged over all results.
˙Vave (L/min.) Forceave,Measured(N) Forceave,Theoretical(N)
3.6 0.103 0.153
4.9 0.235 0.296
6.3 0.405 0.498
Table 4: Average Change in force with respect to change in the independent variable.
Variable ∆V ariable ∆Forceave,Measured(N) ∆ Forceave,Theoretical(N)
angle θ 122° 0.285 0.548
diameter 3.07 mm 0.06 0.029
Vol. Flow Rt 2.7 L/min. 0.302 0.345
Figure 7 shows how decreasing the diameter of the fluid flow through the nozzle increased the
force experienced by all three targets. Table 2 shows both the experimental and calculated results
illustrating this same trend. Since a decreasing diameter with a constant flow rate would cause
a greater fluid velocity, this data would suggest that the increased velocity of the water caused a
greater force. Although this change in force is not nearly as drastic as the effects from the change
in the shape angle θ.
Figure 7 also shows how increasing the volume flow rate also increases the force. This same
figure illustrates well the trending exponential increase in force with ˙V . Table 3 shows that on
average the measured values again agree with the calculated values in their trends though they are
not exactly identical values. The experiment itself was carried out at five different flow rates while
changing the diameter of flow as well. Therefore, in order to distinguish the effect of the volumetric
flow rate each of the flow rates given in this table are averaged between two different flows having
the same diameter of flow. This information suggests that as the average amount of fluid particles
colliding with the target over a fixed area increases, the average force to counteract their momentum
increases as well.
Finally, from this information we can determine that between the variable upper and lower limits
the independent variable with the most influence over the output force for the measured values is
different than the calculated values. The calculated values would suggest the angle of the object
where collision occurs, θ has the most influence on force. However, for the experimental results, the
9

Table 5: % error for all target/nozzle combinations averaged over all volumetric flow rates.
Target Nozzle Average % Error
Cone Large 23.44%
Small 69.97%
Divot Large 53.56%
Small 57.44%
Flat Large 47.04%
Small 19.31%
variable with the most influence over the output force is the volume flow rate ˙V . This discrepancy
is shown in table 4.
Half of the calculated values did not match the measured values. The results of all measurements
and calculations were plotted graphically with force on the vertical axis verses volumetric flow rate
on the horizontal axis along with their bars of uncertainty. These are given in Figure 8. It can be
seen that for the cone target at a 31° angle with a small diameter of fluid flow, the calculated values
do not lie within the uncertainty of the measured values. The same can be said for the divot target
at a 153° angle for both diameters of flow.
There is a visible trend between the % uncertainty and the Volume Flow Rates while no other
independent variable shows such trends. As Volumetric Flow Rate increases, the average % uncer-
tainty decreases. This trend is illustrated in figure 9. The % uncertainty only changes by 2% with
varying θ and 8% between the two diameters of flow.
On average, the calculated values for force erred from the measured values by 45.13%. Significant
contributing variables to this % error is the cone target with the small diameter of fluid flow at
69.97%, and the divot target for both the large diameter at 53.56% and the small diameter at 57.44%.
These average % errors are compared to the remainder of the target/nozzle combinations in table 5.
It can be assumed that the theoretical model for the water jet/ target system is not accounting for
one or more factor. This unknown factor presents itself at high velocities on the cone shape, and at
all velocities during the divot shape.
After further observation of the cone in the small diameter jet of water, it is clear that the water
10

Figure 8: Graphs illustrating the various margins of uncertainty for all eighteen outputs of force.
11

Figure 9: Graph illustrating the average % uncertainty at the various ﬂow rates.
Figure 10: Image of the ﬂow of water over the cone shaped target.
12

exiting the target control volume is not exiting at the same angle as the shape of the cone see Fig
10. This means that the original assumption that the angle θ of the target is equal to the angle of
deflection is not always true. This wrapping of the fluid around the target is due to the Coanda
effect. The Coanda effect occurs when the surface tension of the fluid must be greater to form a
droplet then it has to be in order to stick to a surface. This is commonly seen when pouring water
from a pitcher. This effect is reduced by either making the departure angle more abrupt, increasing
the exit velocity of the fluid, or using a hydrophobic material.
In order to account for the Coanda effect the true exit angle must be found while accounting for
other factors. However, while observing the fluid flow it is easy to see that measuring a new angle
simply by observation is no easy task. Therefore the projectile motion equation is adopted to help
identify this angle.
y = y0 + vySin(φ)t + gt2
/2 (9)
where vy is the velocity in the y-direction, g is gravity, and t is time. In order to estimate vy,
the jet stream was injected with dye while a slow motion camera recorded the time it took for the
dye to travel a small distance. This test was cumbersome and somewhat inaccurate but assisted in
finding an estimate of velocity in the y direction. The various exit streams should have differing exit
velocities but many are approximated to be between 6.25 and 10m/s.
While testing for velocities it also became apparent that the water did not exit in one single stream
and in fact was border lined chaotic. The flow would shoot upwards as is shown in figure 10 and
then would collapse on itself randomly as can be seen from figure 11. The fluid would impact the
target outside of the original control volume.
Therefore, the original equation needed alterations. After widening the control volume to include
all areas of contact between fluid and target and accounting for what often appeared to be three exit
streams, the original force equation resulted in
Fy = ˙V ρ[Σ(xvCosθ)out − (
4 ˙V
πd2
)in] (10)
where Σ accounts for the summation of any additional exit streams, x is the percent of the mass
13

Figure 11: Image of the chaotic flow of water over the cone shaped target.
flow in that stream, v is the exit velocity, and θ is the exit angle. Equation 10 allows for the addition
of multiple exit streams while not altering the outputs for single streamed results.
The new hypothetical model yielded a percent error of 31.06%. This new model changed the
force output by 55%. The exit angles approximated with the projectile motion equation were 40°,
45°, and 15° at a slow volume flow rate and increased small amounts with increasing flow rates. The
exit velocities also increased with an increased volume flow rate. The cone with the small nozzle
was the only target/nozzle combination which had additional exit streams therefore the force vs.
volume flow rate graph for the cone with the small nozzle is given in figure12. It can be seen from
this graph that two of the three values now lie within the bars of uncertainty, whereas before, none
of them did.
Previously it was assumed that the distance between targets was held constant at 5 cm, however
this was not true for the divot. Though the nozzle measured 5 cm from the closest part of the divot,
the point of first contact of the water was actually 2 cm beyond the closest point. Therefore the
distance between the nozzle and the point of contact is more appropriately around 7 cm. It was
assumed that the velocity from the nozzle to the target was constant. However, due to the force of
14

Figure 12: Theoretical and actual forces vs. volumetric ﬂow rates for the old and new momentum
model.
Table 6: % error for all target/nozzle combinations averaged over all volumetric ﬂow rates before
and after theoretical alterations.
Target Nozzle Average % Error Before Average % Error After
Cone Large 23.44% 23.44%
Small 69.97% 31.06%
Divot Large 53.56% 53.56%
Small 57.44% 57.44%
Flat Large 47.04% 47.04%
Small 19.31% 19.31%
15

gravity on the jet the original assumption of vin = vout may not be correct.
In order to factor in the force of gravity on the jet stream the energy equation will need to be
used. The derivation is somewhat lengthy but solving for velocityout gives the energy equation in
the form
vout = v2
in − 2Zg (11)
where z is the height and g is gravity. Plugging this into equation 10 and moving around variables
gives
F = ˙V ρ[Σ( v2 − 2ZgxCosθ)out − (
4 ˙V
πd2
)in] (12)
where d is still the diameter of the flow from the nozzle, Z is the height from nozzle to point of
contact, g is gravity, and x is the percent mass flow.
However, after further analysis the numbers showed < 1% change in value. Therefore the form
of the equation given in equation 11 will only need to be used when the jet of water is being shot
over a long range. Because the resultant change in force was so small, no figures or tables will be
used to show the new values. Instead, refer to Table 6.
By retracting the assumption that velocityin = velocityout, the force on the divot can be esti-
mated to change 22.3% for the small nozzle and 29.33% for the large nozzle while not affecting the
flat plate. This assumption did not affect the force on the flat plate target since the redirected flow
was completely horizontal. This change in velocity can likely be attributed to the Coanda effect and
was already accounted for in the analysis of the cone.
Calculating this change in velocity required the use of equation 12 and slow motion camera
testing procedures which uncovered the fact that the change in velocity is nearly linearly related
to the volumetric flow rate. In order to measure the velocity leaving the control volume the slow
motion camera was utilized along with dye just as was done previously when using the projectile
motion equation to find the angles. Equation 12 was used to ensure the most accurate results. The
velocities used ranged from 6.25 m/s with the larger nozzle and low volume flow rates to 20 m/s
with the smaller nozzle at higher volume flow rates.
16

Figure 13: Theoretical actual forces vs. volumetric ﬂow rates for the old and new water jet/target
model.
After factoring in the change in velocity between the entrance and exit of the control volume,
the average percent error decreased to 30.00%. The individual percent errors are shown in table 7.
Figure 13 shows how the calculated values now fall within the bars of uncertainty for the measured
values, whereas before they did not. The accuracy varies based on the accuracy of the measurement
for velocityout. The loose estimation for velocityout also makes it difﬁcult to identify other sources
of error.
17

Table 7: % error for all target/nozzle combinations averaged over all volumetric flow rates before
and after theoretical alterations.
Target Nozzle Average % Error Before Average % Error After
Cone Large 23.44% 23.44%
Small 69.97% 31.06%
Divot Large 53.56% 29.33%
Small 57.44% 29.80%
Flat Large 47.04% 47.04%
Small 19.31% 19.31%
Conclusion
Plugging in values to the original force equation (Equation 5 of this report) shows that as θ increases
from 0° to 180°, the force output will also increase since the target will be less streamlined, and
absorb more of the fluid’s momentum. As the diameter of the fluid jet stream decreases the force
output will increase. They are inversely related since decreasing the diameter of the jet stream while
holding the volume flow rate constant increases the velocity of the fluid. The increase in velocity
of the fluid means an increase in momentum of the fluid which is absorbed upon contact. As the
volumetric flow rate increases, the force output increases with it. This is due to the fact that their
will be more particles transferring momentum to the target during a given interval of time. This
relationship is exponential.
There were many discrepancies between the calculated values and the measured values when
analysing the system with a simplified momentum model. Half of the calculated values did not
match the measured values at first. On average, the calculated values for force erred from the
measured values by 45.13%. This discrepancy was largely due to the experiment between the cone
and the small nozzle, and the divot targets with both nozzle sizes.
After further observation of the cone, it became clear that the exiting fluid was not exiting at
the same angle as the target, there were more than one exiting stream, and the flow was somewhat
chaotic. This was largely due to the Coanda effect. In order to account for the Coanda effect, the
control volume must include all points on the target which are contacted by the fluid, the true exit
angles need to be approximated, all exiting streams must be accounted for, and the varying velocities
18

need to be estimated. The original equation was modified to account for these new variables and is
given as Equation 10. This new model changed the output force by 55%.
It was assumed that because the distance between the target and nozzle was so small that the
effect of gravity on the jet stream would be negligible. However, at larger distances the effect of
gravity may slow the velocityin. Using the energy equation these effects can be quantified and are
given as Equation 12. In the case of this experiment where the nozzles were held roughly 5 cm from
the targets, the effects on the force output were < 1%.
By revisiting the fact that due to forces internal to the control volume our velocityout = velocityin,
we can also make changes to the output force on the divot shape though it only produces one exiting
stream. Applying the same equation, equation 12, the force on the divot changes by 22.3% for the
small nozzle and 29.33% for the large nozzle. This assumption does not affect the force on the flat
plate target since the redirected flow is completely in the horizontal direction.
The change in velocity over the control volume was found to be nearly linearly related to the
volume flow rate of the fluid. After factoring in the change in velocities, the average percent error
decreased to 30.00%. The loose estimation for velocityout due to inadequate measuring devices
makes it difficult to identify further sources of error.
The final form of the model for the water jet/ target system is given as equation 12, yet equation
10 will suffice for most cases. If the reader wishes to define the characteristics of the system further,
a good place to start would be to identify methods for more accurately measuring velocities of near
chaotic fluid flows.
19

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