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 Th e
met method hod
used used thr through oughou outt
thi this
exp experim erimen entt
is
taki aking the 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, C v 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, C d cannot be taken to an average since both values of Cd for 3 mm and 6 mm orifice sizes are not similar to each e ach 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 o ccur 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 (C v) as
2
Cv = v / v i
(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 = gt 2 / 2
(4)
Combining those equations 1-4 gives an expression Cv in terms of x, y, and H x = 2 C v √yH
(5)
Therefore graph √yH versus x gives slope 2 C v.
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
=
Orifice Area
A c
(6)
A o
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
A o vi
A o √ 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 0 1 2 3 4 5 6 7
x (m) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
y (m) 0 0.022 0.029 0.040 0.051 0.065 0.087 0.092
√yH (m) 0 0.093 0.107 0.126 0.142 0.160 0.185 0.191
y (m) 0 0.025 0.035 0.048 0.066 0.091 0.115 0.144
√yH (m) 0 0.081 0.095 0.112 0.131 0.154 0.173 0.193
Table 1.2 : Orifice : 3mm Head : 260mm Needle 0 1 2 3 4 5 6 7
x (m) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
4
Table 2.1 : Orifice : 6mm Head : 395mm Needle 0 1 2 3 4 5 6 7
x (m) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
y (m) 0 0.024 0.035 0.042 0.054 0.069 0.088 0.090
√yH (m) 0 0.097 0.117 0.128 0.146 0.165 0.186 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)
1 2 3 4 5 6 7 8
0.395 0.380 0.360 0.340 0.320 0.300 0.280 0.260
√Head (m)^0.5
Volume collected (ml)
0.6285 0.6164 0.6000 0.5831 0.5657 0.5477 0.5292 0.5099
141 136 134 132 130 128 124 120
Volume collected (m3) 0.000141 0.000136 0.000134 0.000132 0.000130 0.000128 0.000124 0.000120
Tme (!)
"lo#$ate (m3%!)
10 10 10 10 10 10 10 10
0.0000141 0.0000136 0.0000134 0.0000132 0.0000130 0.0000128 0.0000124 0.0000120
Table 3.2 : Orifice : 6mm Needle
Head (m)
1 2 3 4 5 6 7 8
0.395 0.375 0.355 0.335 0.315 0.295 0.275 0.255
√Head (m)^0.5 0.6285 0.6124 0.5958 0.5788 0.5612 0.5431 0.5244 0.5050
Volume collected (ml) 220 206 204 198 194 186 182 176
Volume collected (m3) 0.000220 0.000206 0.000204 0.000198 0.000194 0.000186 0.000182 0.000176
Tme (!)
"lo#$ate (m3%!)
4 4 4 4 4 4 4 4
0.0000550 0.0000515 0.0000510 0.0000495 0.0000485 0.0000465 0.0000455 0.0000440
Graph 1: Jet trajectories for two orifice sizes at the different flow rates
6
Jet Trajectories or !oth "ri#ices at $i##ere%t &ea' 0.00 -0.020.00
) e c % a t s i $ l a t % o . i r o &
0.10
0.20
0.30
0.0
-0.0 -0.06
Orifice=3mm, Head=395mm Orifice=3mm Head=260mm
-0.0!
Orifice=6mm Head=395 mm
-0.10 -0.12
Orifice=6mm Head=255 mm
-0.1 -0.16
(ertical $ista%ce) * +,-
The coefficient of velocity, C v can be estimated by plotting the root of yH against x, where the graph gives a gradient of C v.
Graphs o# /y& a0ai%st *
7
Graph 2.1 Orifice : 3mm Head : 260mm /y& a0ai%st * 0.2000
0.1500 & y # o 0.1000 t o o R
" # 0.5693$ % 0.003
0.0500
0.0000 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
* +,-
Graph 2.2 Orifice : 3mm Head : 395mm /y& a0ai%st * 0.2500 0.2000 & 0.1500 y # o t o 0.1000 o R
" # 0.51&9$ % 0.03!5
0.0500 0.0000 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
* +,-
Graph 2.3 Orifice : 6mm
8
Head : 255mm / y& a0ai%st * 0.2
0.15 & y # o t o o R
0.1
" # 0.50$ % 0.029
0.05
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
* +,-
Graph 2. Orifice : 6mm Head : 395mm / y& a0ai%st * 0.25 0.2 & y 0.15 # o t o 0.1 o R
" # 0.51&2$ % 0.02
0.05 0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
* +,-
9
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 a(ainst )low *ate for Orifice # 3mm Square Root of Head against Volumetric Fl owrate 0"0 y = 62001x - 02!"!
060
H f o 0$0 t o o 0#0 R e 0!0 r a u 020 q S
010 000 0000011$
00000120
0000012$
000001!0
000001!$
000001#0
000001#$
Volumetric Flowra te (m3/s)
Graph 3.2 'head a(ainst )low *ate for Orifice # 6mm Square Root of Head against V olumetric Flowrate 0"0 y = 11%#!x - 0010&
060
H f o 0$0 t o o 0#0 R e 0!0 r a u 020 q S
010 000 00000#
00000#
00000#
00000$
00000$
00000$
00000$
00000$
000006
Volumetric Flowra te (m3/s)
10
$SUSS"4 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 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
11
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 C d 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.
"4LUS"4 In conclusion, it can be said that the diameter of the orifice have not much effect on the trajector of the !ater flo!in" out from the orifice# $ut, the head of the !ater is found to have more effect on the trajector than the diameter of the orifice# %he coefficient of velocit, &v can be estimated b 'lottin" the root of ( a"ainst ), !here the "ra'h "ives a "radient of &+ #%he coefficient of velocit of !ater is ta*en to an avera"e since all values are similar# %herefore, the objective of this e)'eriment has been obtained#
,--endi$ Coefficient of Velocity, C v:
%he coefficient of velocit, &v can be estimated b 'lottin" the root of ( a"ainst ), !here the "ra'h "ives a "radient of &v# +rom "ra'h 2#3 and 2#4, !hich is for 6mm orifice 'late, the avera"e value for &v is 0#5106# Coefficient of Discharge, C d:
%he value of &d can be obtained b calculatin" the "radient of the strai"ht line of ( a"ainst - "ra'h# +rom "ra'h 3#2, the value for &d is 11#84#
12
Coefficient of Contraction, C c: C c
=
=
=
C d C v
0#5106 11#84
0#0431
Jet diameter at the vena contracta, D c: C c
Ac =
Ac =
Ao
= C c × Ao
0#0431 ×
π
×
4
A c = 1#219 ×10
6#0 ×10
.6
m
2
/ /
2
Ac
Jetdiamete r = 2
3
−
/
π
=
2(
1#219 ×10
6
−
)
π =
1#2460 ×10
3
−
m
13
14
