Peter and Chinedu /
Chemistry International 2(2) (2016) 8088
Model prediction for
constant area, variable pressure drop in orifice plate characteristics in flow
system
Ukpaka Chukwuemeka Peter^{1},* and Ukpaka Chinedu^{2}
^{}
^{1}Department of Chemical/Petrochemical Engineering, Rivers State University
of Science and Technology, Nkpolu, P.M.B. 5080, Port
Harcourt, Nigeria.
^{2}Department of Civil Engineering, University of Port Harcourt, Nigeria
*Corresponding author’s E. mail: chukwuemeka24@yahoo.com
For figures refer to PDF file at: http://bosaljournals.com/chemint/images/pdffiles/29.pdf
ABSTRACT
The effect of density, pressure
drop, viscosity and orifice area on the characteristics of fluid flow was
examined in this paper. Also studied was the effect on the control pressure
change of the constant area variable pressure drop meter as a proportional
derivative control. The mathematical model developed to monitor and predict the
control of the system is given as PPo = 7.8/t – 0.06 + K_{c}
+K_{d}. The change
in control pressure decreases with increase in proportional/derivative gain (K_{c}, K_{d})
as well as increase in time. The Bernoulli’s principle was applied in
describing the design principle, stability analysis and development of
mathematic model of a pressurebased flow meter with a constant area, variable
pressure drop; using an orifice plate with different fluid flowing through it.
The developed formula relates pressure drop with the flow rate of a given fluid
passing through the orifice. The formula obtained is then simulated using different
fluids. In order to control the flow rate, of these fluid flowing through the
model developed was related to a Proportional Derivative control (PD). Thereby
getting knowledge on how the PD controller performs with respect to different
fluids, with change in pressure, density and area of the pipe/orifice was
presented in this paper. Finally information and results on the simulation and
how the PD controller functional parameters of proportional gain and derivative
gain influence the control system was examined in this research.
Keywords: Model, Predict, Monitor, Orifice plate, Pressure, Proportional/derivative
gain
Capsule Summary: Model prediction for constant
area, variable pressure drop in orifice plate characteristics in flow system
was studied and actual results showed good match with theoretical values and
developed method can be used in monitoring and predicting the flow
characteristics of fluid flow in an orifice.
Cite This
Article As: Ukpaka Chukwuemeka Peter and Ukpaka
Chinedu. 2016. Model prediction
for constant area, variable pressure drop in orifice plate characteristics in
flow system. Chemistry
International 2(2) 8088
INTRODUCTION
Flow measurement is an important part of the chemical
industry. It can be done in variety of ways. Both gas and liquid flows can be
measured in volumetric and mass flow rate example litres
per second and kilogram per second. Instruments have been designed to aid in
flow rate calculation. They have common name flow meter and various flow meters
can be used depending on application and industry. These flow meters are
classified according to the principle surrounding their operation. and can thus
be classified as follows: Mechanical
flow meters piston meter, rotating piston meter, variable area meter, turbine
flow meter, woftmann meter, nutating
disk meter, single jet meter, oval gear meter, pelton
wheel and multiple jet meter; Pressure
based meterventri meter, orifice plate, dali tube, pilot
tube, multihole pressure probe meter; optical flow meters, thermal mass flow
meters, electromagnetic flow meters, ultra sonic flow meters, coriols
flow meters and laser droplet flow meter (Alan, 2001; Ang
et al., 2005; Aris, 1994; Ayotamuno et al., 2006; Batchelor,
2002; King, 2010; Ogoni and Ukpaka, 2004; Roberson
and Crowe, 1993; Scott, 2005; Wayne, 2006). In this
paper orifice plate meter in terms of characteristics and effect of control
system was studied specifically, when subjected into disturbance with fluid
flow.
Process control is a means of
controlling the quality of products in a given process. In order to use an
instrument for optimum process control, the principles governing the instrument
must be fully understood. it is therefore of great
importance to study and understand these principles as it applies to
controlling a process. The purpose of the study is to know the principle
governing pressure based meters using a basis of the constant area, variable
pressure drop meter in the measurement of the flow rates of different fluids.
And also to know how this meter will work as a proportional derivative
controller. The objectives of this study are as follows: understanding the
design of the meter, developing and deriving a mathematical model of the meter
based on its design, studying how various fluids are measured and the effects,
and studying the meter as a proportional derivative controller.
The following are the contributions of this study to
knowledge: good understanding of a pressure based meter, good understanding on
the behaviour of an orifice meter as a proportional
derivative controller (PD) and good understanding on the design principles of
an orifice plate and pressure based meters. The scope of this paper covers,
pressure based meters (constant area, variable pressure drop meters) using
orifice plate. Investigation conducted by various research groups revealed that
there are several types of flow meter that rely on Bernoulli’s principle,
either by measuring the differential pressure within a constriction, or by
measuring static and stagnation pressures to derive the dynamic pressure. These
are: venturi meter, orifice plate, dali tube, pilot tube and
multihole pressure probe (Ang
et al., 2006; Bennett, 1993; Cunningham,
1951; Geankoplis, 1993; Liang, 2009; Minorsky, 1992; Yang, 2005).
An orifice plate is a plate with a hole through it, placed in the flow; it
constricts the flow, and measuring the pressure differential across the
constriction gives the flow rate. It is basically a crude form of venturi meter, but with higher energy losses. There are
three type of orifice: concentric, eccentric, and segmental.
Orifice plates are most commonly
used for continuous measurement of fluid flow in pipes. They are also used in
some small river systems to measure flow rates at locations where the river
passes through a culvert or drain. Only a small number of rivers are
appropriate for the use of the technology since the plate must remain
completely immersed i.e the approach pipe must be
full, and the river must be substantially free of debris.
In the natural environment large orifice plates are used to
control onward flow in flood relief dams. in these
structures a low dam is placed across a river and in normal operation the water
flows through the orifice plate unimpeded as the orifice is substantially
larger than the normal flow cross section. However, in floods, the flow rate
rises and floods out the orifice plate which can then only pass a flow
determined by the physical dimensions of the orifice. Flow is then held back
behind the low dam in a temporary reservoir which is slowly discharged through
the orifice when the flood subsides.
THE MODEL ORIFICE PLATE
An
orifice plate is a device used for measuring the rate of fluid flow. It uses
the same principle as a Venturi nozzle, namely
Bernoulli’s principle which states that there is a relationship between the
pressure of the fluid and the velocity of the fluid. When the velocity
increases, the pressure decreases and vice versa.
An orifice plate is a thin plate with a hole in the middle.
It is usually placed in a pipe in which fluid flows. When the fluid reaches the
orifice plate, with the hole in the middle, the fluid is forced to converge to
go through the small hole; the point of maximum convergence actually occurs
shortly downstream of the physical orifice, at the socalled vena contracta point as presented in Figure 1. As it does so,
the velocity and the pressure changes. Beyond the vena contracta,
the fluid expands and the velocity and pressure change once again. By measuring
the difference in fluid pressure between the normal pipe section and at the
vena contracta, the volumetric and mass flow rates
can be obtained from Bernoulli’s equation.
Principles of the Orifice: It is based on Bernoulli’s principle and is derived as
follows;
Bernoulli’s
Equation for Incompressible Fluids for a parcel of fluid moving through a pipe
with crosssectional area “A”, the length of the parcel is d”x”,
and the volume of the parcel A dx. If mass density is
Computational Procedure
The
following operational dimensions were arbitrarily chosen for data related to an orifice meter
characteristics, diameter of the pipe = 240mm, diameter of the orifice 120mm,
differential manometer reading = 400mmHg, discharge coefficient = 0.65, density
of the various fluid
=
0.0113* 1.0327*10.954 = 0.1205m31s
Table 1: Determination of flow rate of different fluid
in an orifice plate

Water 
Petrol 
Palm oil 
Engine oil 

P_{1}P_{2} 
Qtkg/m^{3} 
Q_{actual} kg/m^{3} 
Qt kg/m^{3} 
Q_{actual} kg/m^{3} 
Q_{actual} kg/m^{3} 
Q_{actual} kg/m^{3} 
Q_{actual} kg/m^{3} 
Q_{actual} kg/m^{3} 
400 
0.1205 
0.080 
0.142 
0.092 
0.128 
0.083 
0.128 
0.084 
450 
0.1278 
0.083 
0.151 
0.098 
0.136 
0.088 
0.136 
0.089 
500 
0.1348 
0.090 
0.159 
0.103 
0.143 
0.093 
0.143 
0.093 
550 
0.1413 
0.093 
0.167 
0.108 
0.150 
0.098 
0.151 
0.098 
600 
0.1476 
0.096 
0.174 
0.113 
0.157 
0.102 
0.157 
0.102 
650 
0.1534 
0.100 
0.181 
0.118 
0.163 
0.106 
0. 164 
0.106 
700 
0.1594 
0.104 
0.188 
0.122 
0.169 
0.110 
0.170 
0.110 
750 
0.1650 
0.107 
0.194 
0.126 
0.175 
0.114 
0.176 
0.114 
800 
0.1705 
0.111 
0.201 
0.131 
0.181 
0.118 
0.180 
0.118 
850 
0.1757 
0.114 
0.207 
0.135 
0.186 
0.121 
0.187 
0.112 
But
Q = theoretical
Q_{actual}
= Q_{theoretical} * C_{d}
= 0.1205* 0.65 = 0.078
The
data obtained were fed into the model computer using the developed mathematical
expression in equations (19), (23), (24), (25) and (26).
Application
of constant area, variable pressure drop meter as a proportional derivative
control
The
proportional derivative control works using the form stated below. There by
increasing output, with change in error.
P
= P_{o}+ K_{c}_{ }+ K_{d} dE/dt (27)
where K_{d} gain, dE = change in error, dt
= difference in time and P_{o}
actual measurement.
This
system is able to measure flow rate with pressure drop between the ranges of
400 to 850mm Hg. And it is found that after adjustment, the flow rate changes
by approximately 0.006 for 50mmHg variation in pressure difference. And for
such change, the error difference in flow rate is approximately 7.8 This means, at any given time change, this system will
monitor flow rate using the following model
With density by integrating equation (4 ) with respect to x where C is a constant, sometimes referred to as the Bernoulli
constant. It is not a universal constant, but rather a constant of a
particular fluid system. where the speed is large, pressure is low and vice versa. In the above derivation, no external workenergy
principle is invoked. Rather, Bernoulli’s principle was inherently derived
by a simple manipulation of the momentum equation. A streamtube
of fluid moving to the right.
Indicated are pressure, elevation, flow speed, distance (s), and
crosssectional area. Note that in this Figure 2 elevation is denoted by Z;
thus, (7) Putting equations (7) and (8) together, we have equation
(9) or equation (10) (9) or
Making Q the subject of the formula from equation (17)
we have Equation (18) can be written as; The above expression for Q gives the theoretical volume
flow rate. Introducing the beta factor Since C_{d} ranges
between 0.6 and 0.70 for orifice plate [3] And finally introducing the meter coefficient C which is
defined as Equation (22) can be obtained by multiplying the density
of the fluid by the mass flow rate at any section in the pipe Pressure
Drop Relationship; Making
the pressure drop the subject of formula from equation (20) gives Density
Relationship: Making the density subject of
formula from equation (20) gives Orifice
Area Relationship: Making
the orifice area the subject of the formula from eqn
9 gives Viscosity
Relationship: for flow characteristics we have Q =
Table 2: Theoretical
computation of functional parameters of orifice plate and gain K_{c} K_{d} PP_{o} T_{(see)} PP_{o} 0.1 0.1 1 1.00 7.94 0.2 0.2 2 0.50 4.04 0.3 0.3 3 0.33 3.14 0.4 0.4 4 0.25 2.69 0.5 0.5 4 0.20 2.50 0.6 0.6 6 0.17 2.44 0.7 0.7 7 0.14 2.45 0.8 0.8 8 0.13 1.54 0.9 0.9 9 0.11 2.61 1.0 1.0 10 0.10 2.72
P = P_{o} + K_{c}+ K_{d}7.8/dt This will give P = (P = 0.006)+ K_{c} +K_{d}7.8/dt (28) Equation (28) can be expressed as followed P – P_{o} + 0.06 – K_{c}
–K_{d} = 7.8/dt (29) Rearranging equation (29) and integrating yields Equation (32) can be written as Considering when the proportional gain varies directly
with derivative gain then K_{c} = K_{d}. The values of
0.1 to 1 therefore the following expression can be obtained as illustrated
in table 2.
RESULTS AND DISCUSSION
Figure 4 illustrate the flow rate
characteristics of theoretical and actual against change in pressure for water
medium. It is seen that increase in pressure results to increase in flow rate.
The equation of the best fit is given as Q_{a}
= 13136*
From figure 5, the change in flow
rate of petrol was examined with change in pressure. Results obtained revealed
that increase in flow rate of petrol yielded increase in change of pressure.
The equation of the best fit obtained is Q_{a}
=10550 *
Results obtained in Figure 6
illustrate the flow rate characteristics of palm oil in an orifice plate. From
Figure 6 increase in flow rate was observed with increase in change in
pressure. The equation of the best fit is as follows
Q_{a} = 11789 *
The flow rate against change in
pressure for engine oil is shown in Figure 7. From Figure 7, it is seen that
increase in flow rate was observed with increase in change in pressure. The
variation in flow rate can be attributed to the variation in change in pressure.
The equation of the best fit is given as follows.
Q_{a} = 11943 *
Whereas for theoretical Q_{t}
= 768.65 *
Figure 8 illustrates change in
control pressure of the orifice against proportional gain (K_{c})
and derivative gain (K_{d}).
From Figure 8, it is seen that increase in the proportional/derivative gain (K_{c},K_{d})
resulted to decrease in control pressure of the orifice. The variation in the
control pressure can be attributed to the variable in the
proportional/derivative gain (K_{c}, K_{d}). The equation of
the best fit is given as
 150.8k_{c }+ 17.899 and
R^{2} = 0.9998
Defining the polynomial curve of
Figure 8,
The effect of time on the control
pressure of the orifice was illustrated in Figure 9. From Figure 9, increase in
time resulted to decrease in control pressure of the orifice. The variation in
the control pressure of the orifice can be attributed to the variation in time.
The equation of the best fit for a parabolic curve is shown in Eq. 34.
Defining the polynomial curve of
Figure 9,
Figure 10 illustrate the
relationship between control pressure of t he orifice against inverse of time.
From Figure 10 it is seen that increase in the inverse of time resulted to
increased in the control pressure of the orifice. The variation in the control
pressure of the orifice be attributed to the variation
in the inverse of time (Eq. 35).
CONCLUSION
The following conclusion was drawn
from the findings.
1. The effectiveness of orifice
plate depends on the density, viscosity and other factor.
2. Orifice plate mechanism in
place for measuring flow is faster.
3. The control pressure of the
orifice is very important as once of the governing
factors to reduce error in the proportional and derivative gain (K_{c}, K_{d}).
4. As the proportional gain and
derivative gain increases the control pressure of the orifice decrease.
5. The results of the actual has a
good match with theoretical showing how reliable the developed can be used in
monitoring and predicting the flow characteristics of fluid flow in an orifice.
Nomenclature
Q
= volumetric flow rate (at any crosssection), ( m3/s)
C_{d} = coefficient of discharge,
dimensionless
C =orifice
flow coefficient, dimensionless
A_{1} = crosssectional area of the pipe,
(m^{2})
A_{2} = crosssectional area of the orifice
hole, (m^{2})
d_{1} = diameter of the pipe, (m)
d_{2} = diameter of the orifice hole, (m)
V_{1} = upstream fluid velocity, (m/s)
V_{2} = fluid velocity through the orifice
hole, (rn/s)
P_{1} = fluid upstream pressure, Pa with
dimensions of (kg/m s^{2})
P_{2} = fluid downstream pressure, Pa with
dimensions of (kg/m^{2})
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