EXPERIMENT NO. 4
CALIBRATION OF AN ORIFICE PLATE FLOWMETER
MECHANICAL ENGINEERING DEPARTMENT
KING SAUD UNIVERSITY
RIYADH
Submitted By:
ABDULLAH IBN ABDULRAHMAN
ID: 123456789
GROUP A
EXPERIMENT PERFORMED ON 29/01/2013
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1
Introduction
In this experiment an orifice plate flow-meter is calibrated and the calculated coefficient
of discharge, Cd , is compared to the given value in the lab report as well as other published
data.
The orifice plate flow-meter provides a simple and low cost method for measuring the flow
rate in a pipe using the pressure drop measurement across the plate. The orifice plate is a
circular plate with a sharp square edge hole in the center inserted in a pipe. When a fluid
is allowed to flow inside the pipe, the orifice plate obstructs the flow which results in fluid
pressure loss. A schematic diagram of an orifice plate installed in a pipe with flow from left
to right is shown in figure 1. The pressure loss is dependent on the orifice diameter, pipe
diameter and the flow rate. Hence the flow rate is less than the theoretical flow rate which
would occur if there were no losses.
Figure 1: Orifice plate schematic
For ideal flow (no viscous effects) the flow rate can be calculated from Bernoulli’s equation,
as follows, [3],
p1 V12
p2 V22
+
+ gz1 =
+
+ gz2
(1)
ρ
2
ρ
2
Where, p, ρ, V , z and g are the fluid pressure, density, velocity, elevation and acceleration
due to gravity, respectively. The subscripts 1 and 2 denote the conditions upstream and
downstream of the plate, respectively, as shown in figure 2
Since the flow is incompressible we can write the flow velocity upstream of the orifice, V1 ,
in terms of the throat velocity, V2 , using the continuity equation given in equation 2, [3],
2
Figure 2: Orifice plate stations
A1 V 1 = A2 V 2
d2
A2
V2 = 22 V2 = β 2 V2
V1 =
A1
d1
(2)
(3)
Where, β = dd21 and d1 , d2 are the pipe and throat diameters, respectively. Substituting
equation 3 into equation 1 assuming the flow is horizontal (z1 = z2 = 0), we get,
"
V22 − V12
= 1−
2
h
4 #
V22 =
2 (p1 − p2 )
ρ
(4)
1 − β 4 V22 =
2 (p1 − p2 )
ρ
(5)
2 (p1 − p2 )
ρ
(6)
d2
d1
i
1
V2 = p
1 − β4
s
Since the flow rate, Q, is given by Q = AV , where A and V are the cross-section area and
average speed at any section, respectively, then we write the flow rate across the orifice plate
as follows,
A2
Q = A2 V 2 = p
1 − β4
s
2 (p1 − p2 )
ρ
(7)
Equation 7, gives an expression for the maximum theoretical volume flow rate assuming ideal
flow. However, since real flows are viscous, hence, the maximum flow rate will be less than
the ideal value due to viscous losses. This is taken into account by multiplying the ideal Q
by a factor, Cd , this factor is the coefficient of discharge. Hence, we can define the coefficient
of discharge as the ratio of actual to ideal volume flow rate.
QActual = Cd QIdeal
C d A2
=p
1 − β4
s
2 (p1 − p2 )
ρ
(8)
The coefficient of discharge depends on the Reynolds’ number based on the pipe’s diameter
as well as the orifice diameter to pipe diameter ratio, β.
The main objective of this experiment is to determine this coefficient. In the next section,
the apparatus and techniques used to determine Cd are discussed. In section 3 the results
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and discussion are presented, then finally conclusions are drawn in section 4
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2.1
Apparatus and Techniques
Apparatus
The orifice plate used is part of SOLTEQ flow-meter measurement apparatus, model FM 101,
shown in Figure 3.
Figure 3: SOLTEQ flow-meter measurement apparatus, FM101.
The SOLTEQ FM 101 flow meter apparatus consists of three different types of volume
flow-meters. It contains a Venturi type flow-meter, an orifice plate flow-meter as well as a
Rotameter. All flow-meters are connected in series. The apparatus also includes a flow control
valve that can be used to adjust the volume flow rate. Water is allowed into the apparatus
through the water inlet and is dumped through the water outlet as shown in Figure 3. The
apparatus is connected to a TQ H-10 Volumetric Bench, which contains a closed circuit water
pump. The water exiting the FM 101 module is dumped in a collection tank and is pumped
back into the water inlet.
The SOLTEQ FM 101 also contains a multi-tube manometer to measure the difference
in pressure across the Venturi and orifice plate flow-meters. The manometer has a range of
450 mm and the graduation is in 1 mm increments.
The orifice plate flow-meter, shown in Figure 4(a), is fitted with two pressure taps located
one tube diameter upstream of the orifice plate and half a tube diameter downstream the plate.
The difference in these pressures is the required ∆p in equation 8. The tube diameter, D, is
equal to 26 mm while the orifice-meter throat diameter, DT , is equal to 16 mm. Therefore
the ratio of diameters, β, is β = DDT = 0.6154.
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(a) Orifice plate flow-meter.
(b) Rotameter.
Figure 4: SOLTEQ FM-101
2.2
Procedures and Techniques
In this experiment, the SOLTEQ FM-101 was connected to the TQ H-10 Volumetric Bench.
The flow-rate was set at three different values 5, 10 and 15 L/min. This was done by adjusting
the flow control valve. The volume flow-rate value was read off the Rotameter. Before any
measurements were taken, the flow was allowed to reach a steady condition such that there are
no air bubbles in the system and all pressure readings were stable. Once the pressure readings
in the manometer were stable and all air bubbles were out, the pressure measurements were
recorded for each volume flow-rate. Each time the valve position changed, the flow was allowed
to stabilize before recording the new pressure readings. The measurements were repeated for
the 10 L/min flow-rate case for uncertainty estimates purposes. It is worth noting that
reading the water height in the manometer was difficult due to the fact that the water used
was clear and the background was white.
3
Results and Discussions
In this section the results will be presented and discussed. The atmospheric pressure and
temperature during the experiment were measured to be 940 ± 1.5 mbar and 22 ± 1 ◦ C,
respectively. Since the working fluid is water, the water temperature was measured to be
20 ± 1 ◦ C. The water density, ρ, and viscosity, µ, at this temperature were taken as
998.2 kg/m3 and 10−3 P a · s, respectively, [2].
The pressure measurements upstream, p1 , and downstream, p2 , of the orifice plate are
shown in Table 1. Also shown in Table 1, are the average velocity U and the Reynolds
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number, ReU defined as follows,
Q
4Q
=
A
πD2
ρU D
ReU =
µ
U=
(9)
(10)
Where, ρ and µ are the water density and viscosity, respectively. Finally, using equation 8
Table 1: Pressure Measurement
Q [L/min]
5
10
15
10
p1 [mmH2 O ]
295
286
283
287
p2 [mmH2 O ]
143
220
263
218
U [m/s]
0.47
0.31
0.16
0.32
12220 8147 4973 8147
ReU
Cd
0.667 0.675 0.613 0.667
we can write the coefficient of discharge, Cd , as follows,
Q
Cd =
A2
s
ρ (1 − β 4 )
2 (p1 − p2 )
(11)
Equation 11 is used to calculate the values of Cd shown in Table 1. The measured discharge
coefficient values are close to the supplied value in the lab manual of 0.63. While recording
the pressure measurements it was noticed that there are some fluctuations in the pressure
values. This might be due to the fact that the pipe length between the Venturi-meter ,and
the orifice flow-meter is too short, less than three pipe diameters. This was amplified when
the flow rate was set to 5L/min, which corresponds to a Re = 4973. This value is in the
transition region, hence the flow is not fully turbulent. To get a better reading there should
be longer pipe connecting the venturi-meter and the orifice plate or the Reynolds number
should be increased. In the next section the uncertainty in measured quantities is presented.
3.1
Uncertainty Analysis
Uncertainty analysis is performed based on two readings of the pressure difference at 10 L/min
flow-rate. The values are listed in Table 2.
Table 2: Pressure Measurement for uncertainty estimates at Q = 10 L/min
Run
1
2
p
Sp
p1 [mmH2 O ] 286 287 286.5 0.707
p2 [mmH2 O ] 220 218 219 1.414
Where, p and Sp are the mean and standard deviation of the measured pressures, respecS
tively. The uncertainty in the measured pressure is given by Up = tα/2,ν √pn , Where t is the
t-distribution and α = 0.05 is the level of significance, and ν = n − 1, where n = 2 is the
number of observations. Hence, the uncertainties in the upstream and downstream pressures
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are given as follows,
Sp
0.707
Up1 = t0.05,1 √ 1 = 12.706 √ = 6.35 mm H2 O
n
2
Sp2
1.414
Up2 = t0.05,1 √ = 12.706 √ = 12.71 mm H2 O
n
2
(12)
(13)
(14)
Using, equation 11 we can estimate the uncertainty in Cd , UCd , as follows, [4],
s
UCd =
∂Cd
UQ
∂Q
2
+
∂Cd
Up
∂p1 1
2
+
∂Cd
Up
∂p2 2
2
(15)
The uncertianty in the measured flow-rate is taken to be 4% F ull Scale, such that δQ =
0.88 L/min. Using equation 11 and the calculated pressure and flow-rate uncertainties we
can evaluate the uncertainty in Cd as UCd = ±0.075.
Since in general the discharge coefficient is a function of the Reynolds number, Re, a plot of
the measured Cd versus the Reynolds number is shown in Figure 5. Also shown in the figure
is a curve fit for experimental data reported in the literature versus Re as given by equation
16, [1]. The length L1 and L2 are the pressure taps locations’ upstream and downstream,
respectively. In this case, L1 = d1 and L2 = d2 .
Figure 5: Discharge coefficient versus Reynolds number.
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Cd = 0.5959 + 0.0312β
2.1
8
− 0.1840β + 0.0029β
L1
+0.0900
D
"
#
2.5
106
Re
!0.75
β4
L2
β3
− 0.0337
4
(1 − β )
D
(16)
The measured coefficient of discharge is Cd = 0.67 ± 0.075 while the value supplied in the lab
manual is 0.63. This is about 6% difference. However, the measured values are within 4.5%
from the empirical relation given by equation 16 for Q = 10, 15 L/min while the difference
is 8% for the case Q = 5 L/min. This is probably due to the low Reynolds number which is
in the transition region. To get higher accuracy we need to make more measurements with
more accurate pressure transducers.
4
Conclusions
In this work the discharge coefficient of an orifice plate flow-meter was estimated at three
values of flow rate, Q = 5, 10 and 15 L/min. The measured values were compared with the
values obtained from the lab manual and from a curve fit of empirical data. The following
conclusions are made,
• The mean coefficient of discharge was estimated to be Cd = 0.67 ± 0.075
• The difference between current measured Cd and values in the literature are within
experimental uncertainty
• Main source of uncertainty is the unsteadiness in pressure measurement. This is due to
the fact that the Reynolds number is in the transition region for low values of flow rates
• The variation in Cd versus the Reynolds number follows the curve fit of empirical data
in the literature
• The results can be improved if a more accurate manometer is used as well as more data
points are taken
• Using a very light dye in the water would improve will eliminate human reading errors
References
[1] Emperical fit for cd : www.brighthubengineering.com/.
[2] Water properties: www.engineeringtoolbox.com/.
[3] Robert W. Fox Alan T. McDonald Philip J. Pritchard. Fluid Mechanics. John Wiley &
Sons, eighth edition, 2012.
[4] Thomas G. Beckwith Roy D. Marangoni John H. Lienhard V. Mechanical Measurements.
Addison Wesley, fifth edition, 1993.
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