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SUMMARY
The title of this experiment is orifice and free jet flow. The
objectives are to determine the coefficient of velocity, the coefficient of
contraction and the coefficient of discharge for two small orifices.
The method used throughout this experiment is taking the
measurement of jet trajectories under steady flow conditions. The
coefficient of velocity is estimated from the gradient of the graph root
of yH against x. The coefficient of discharge is obtained from plotting
the square root of the head against the volumetric flow rate. The
coefficient of contraction is from calculation using the equation
Cc=Cd/Cv.
From the experiment that has been carried out, it is found that
the coefficient of velocity, Cv can be taken to an average, since the
values are similar for each diameter of orifice and different heads
taken. Whereas values of the coefficient of discharge, Cd cannot be
taken to an average since both values of Cd for 3 mm and 6 mm orifice
sizes are not similar to each other. Other details of the experiment can
be found in the following sections.
1
Theory
In practical situations, flow through a constriction into free space
is not very common though it does occur in liquid distributors, such as
in packed distillation column. However, similar situations occur in more
common situations, e.g. an orifice plate for flow measurement. In such
enclosed locations, the jet characteristics are harder to study, so free
jet is a convenient experimental model.
Orifice is an aperture through which fluid passes and its
thickness (in the direction of flow) is very small in comparison with its
other measurements. An orifice is used for flow-metering purposes has
a sharp edge so there are minimum contact with the fluid.
Based on Bernoulli equation, fluid that moves from the liquid
surface in the tank to the jet leaving from the orifice, it is seen that the
loss of potential is balanced by the gain in kinetic energy. Assumption
has been made that the pressure is the same at liquid surface and in
the jet.
Hence taking initial velocity in the tank as zero, per unit mass of fluid :
gH = vi2 / 2 (1)
Where vi is the ideal fluid velocity. If actual velocity, v in the discharge
plane was measured it would differ from the ideal velocity vi
We define a Coefficient of Velocity (Cv) as
2
Cv = v / vi (2)
The actual velocity can be deduced from the jet trajectory by
resolving the trajectory in the x and y directions. The horizontal
component x can be assumed remain constant, neglecting air
resistance, so that in time t, it travel by a distance of
x = vt
(3)
The vertical component changes under the influence of gravity, so that
at a time t, it is represented by :
y = gt2 / 2 (4)
Combining those equations 1-4 gives an expression Cv in terms of x, y,
and H
x = 2 Cv √yH
(5)
Therefore graph √yH versus x gives slope 2 Cv.
Also when a jet discharged from a sharp edged orifice, the jet is
smaller in diameter than the orifice.
This leads to the definition of the Coefficient of Contraction (Cc)as :
Cc = Jet Area = Ac (6)
Orifice Area Ao
The Coefficient of Discharge (Cd) is defined as the ratio of actual
flow rate Q (m3/s) to the ideal flow rate, which can be calculated from
the orifice area. The ideal mass flow rate is that which would occur if
the ideal velocity of flow (Vi) existed through the full area of the
orifice :
Cd = Q = Q .
3
Ao vi Ao √ 2gH
Alternatively Cd can be expressed in terms of the coefficient of velocity
and contraction :
Cd = CcCv
RESULTS
Table 1.1 : Orifice : 3mm Head : 395mm
Needle x (m) y (m) √yH (m)0 0.00 0 01 0.05 0.022 0.0932 0.10 0.029 0.1073 0.15 0.040 0.1264 0.20 0.051 0.1425 0.25 0.065 0.1606 0.30 0.087 0.1857 0.35 0.092 0.191
Table 1.2 : Orifice : 3mm Head : 260mm
Needle x (m) y (m) √yH (m)
0 0.00 0 0
1 0.05 0.025 0.081
2 0.10 0.035 0.095
3 0.15 0.048 0.112
4 0.20 0.066 0.131
5 0.25 0.091 0.154
6 0.30 0.115 0.173
7 0.35 0.144 0.193
4
Table 2.1 : Orifice : 6mm Head : 395mm
Needle x (m) y (m) √yH (m)0 0.00 0 01 0.05 0.024 0.0972 0.10 0.035 0.1173 0.15 0.042 0.1284 0.20 0.054 0.1465 0.25 0.069 0.1656 0.30 0.088 0.1867 0.35 0.090 0.189
Table 2.2 : Orifice : 6mm Head : 255mm
Needle x (m) y (m) √yH (m)
0 0.00 0 0
1 0.05 0.025 0.080
2 0.10 0.033 0.092
3 0.15 0.045 0.107
4 0.20 0.066 0.130
5 0.25 0.091 0.152
6 0.30 0.119 0.174
7 0.35 0.150 0.196
5
Table 3.1 : Orifice : 3mm
Needle
Head (m)√Head
(m)^0.5
Volume collected
(ml)
Volume collected
(m3)
Time (s)
Flowrate (m3/s)
1 0.395 0.6285 141 0.000141 100.000014
1
2 0.380 0.6164 136 0.000136 100.000013
6
3 0.360 0.6000 134 0.000134 100.000013
4
4 0.340 0.5831 132 0.000132 100.000013
2
5 0.320 0.5657 130 0.000130 100.000013
0
6 0.300 0.5477 128 0.000128 100.000012
8
7 0.280 0.5292 124 0.000124 100.000012
4
8 0.260 0.5099 120 0.000120 100.000012
0
Table 3.2 : Orifice : 6mm
Needle
Head (m)√Head
(m)^0.5
Volume collected
(ml)
Volume collected
(m3)
Time (s)
Flowrate (m3/s)
1 0.395 0.6285 220 0.000220 40.000055
0
2 0.375 0.6124 206 0.000206 40.000051
5
3 0.355 0.5958 204 0.000204 40.000051
0
4 0.335 0.5788 198 0.000198 40.000049
5
5 0.315 0.5612 194 0.000194 40.000048
5
6 0.295 0.5431 186 0.000186 40.000046
5
7 0.275 0.5244 182 0.000182 40.000045
5
8 0.255 0.5050 176 0.000176 40.000044
0
6
Graph 1: Jet trajectories for two orifice sizes at the different flow rates
Jet Trajectories For Both Orifices at Different Head
-0.16
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.00 0.10 0.20 0.30 0.40
Vertical Distance, x (m)
Hor
izon
tal D
istance, y
(m)
Orifice=3mm,Head=395mm
Orifice=3mmHead=260mm
Orifice=6mmHead=395 mm
Orifice=6mmHead=255 mm
The coefficient of velocity, Cv can be estimated by plotting the
root of yH against x, where the graph gives a gradient of Cv.
7
Graphs of √yH against x
Graph 2.1 Orifice : 3mm Head : 260mm
√yH against x
y = 0.5693x + 0.0043
0.0000
0.0500
0.1000
0.1500
0.2000
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
x (m)
Roo
t of yH (m
)
Graph 2.2 Orifice : 3mm Head : 395mm
8
√yH against x
y = 0.5179x + 0.0385
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
x (m)
Roo
t of yH (m
)
Graph 2.3 Orifice : 6mm Head : 255mm
√yH against x
y = 0.504x + 0.0294
0
0.05
0.1
0.15
0.2
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
x (m)
Roo
t of yH (m)
Graph 2.4 Orifice : 6mm Head : 395mm
9
√yH against x
y = 0.5172x + 0.0424
0
0.05
0.1
0.15
0.2
0.25
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
x (m)
Roo
t of yH (m)
Below are the graphs of the square root of the head against the
volumetric flow rate. The gradient of the graph will give the value for
the discharge coefficient.
Graph 3.1 √head against Flow Rate for Orifice = 3mm
Square Root of Head against Volumetric Flowrate
y = 62001x - 0.2373
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.0000115 0.0000120 0.0000125 0.0000130 0.0000135 0.0000140 0.0000145
Volumetric Flowrate (m3/s)
Squar
e Root of Hea
d
Graph 3.2 √head against Flow Rate for Orifice = 6mm
10
Square Root of Head against Volumetric Flowrate
y = 11843x - 0.0109
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.00004 0.00004 0.00004 0.00005 0.00005 0.00005 0.00005 0.00005 0.00006
Volumetric Flowrate (m3/s)
Squar
e Root of Hea
d
DISCUSSION
Generally, the graphs of the jet trajectories for the two orifice
sizes at the different flow rates are decreasing. The slope of the graph
is much higher for the orifice of diameter 3mm compared with the
orifice of diameter 6mm. From the graph in Figure 1 and 2, it can seen
that the higher the flow rates, the less the gradient of the slope and
the less the degree of the decrease of the graph. The graph also shows
that the diameter of the orifice do not greatly affect the trajectory of
the water flowing out from the orifice. The trajectories differ greatly
because of the flow rates.
By using the coefficients of discharge and velocity, the jet
diameter at the vena contracta can be estimated. Based on both
coefficients obtained, the value for the jet diameter at the vena
contracta is 1.2460 x 10-3 m, which also equivalents to 1.2 mm. This
value seems not very reasonable as the real diameter used to find this
11
value is 6 mm. As soon as the water flows out from the orifice, the jet
diameter will get smaller.
Unlike the coefficient of velocity, values obtained for coefficient
of discharge for both orifice sizes are not similar to each other. This
means that the values cannot be taken to an average. This is because
both coefficient values are obtained from two different graphs, where
the coefficient of velocity obtained from the graph of root of yH against
the flow rate, whereas the coefficient of discharge is obtained from the
graph of root of H against the flow rate.
Like the coefficient of velocity, the values obtained for coefficient
of discharge, Cd for both orifice sizes also seem not similar to each
other. This situation may caused by any error that occurred during
taking the reading of the y value. However, if the average of the entire
gradient is calculated, the value of Cd obtained is 0.5106. From the
literature, the common value for Cd is in the range of 0.60 to 0.65.
Therefore, common errors done by human could affect the
readings and as well as the theory which would be inappropriate value
to be obtained from experiment.
CONCLUSION
In conclusion, it can be said that the diameter of the orifice have not much effect
on the trajectory of the water flowing out from the orifice. But, the head of the water is
found to have more effect on the trajectory than the diameter of the orifice. The
coefficient of velocity, Cv can be estimated by plotting the root of yH against x, where the
graph gives a gradient of Cv .The coefficient of velocity of water is taken to an average
since all values are similar. Therefore, the objective of this experiment has been obtained.
12
Appendix
Coefficient of Velocity, Cv:
The coefficient of velocity, Cv can be estimated by plotting the root of yH against x,
where the graph gives a gradient of Cv. From graph 2.3 and 2.4, which is for 6mm orifice
plate, the average value for Cv is 0.5106.
Coefficient of Discharge, Cd:
The value of Cd can be obtained by calculating the gradient of the straight line of √H
against Q graph. From graph 3.2, the value for Cd is 11.84.
Coefficient of Contraction, Cc:
Jet diameter at the vena contracta, Dc:
13
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