Orifice plate: M J Rhoades Understanding the flow dynamics of the system The orifice plate is used in many fluid systems to detect and measure the amount of flow of a fluid moving through piping in various systems. They are relatively inexpensive, and provide adequate accuracy for most applications. Understanding the flow dynamics involved is an important and sometimes confusing part. (From an engineering stand point.) This paper is for those Engineers or students who wish to get a better understanding of the orifice plate and the calculations that go along with it. In order to solve for the data to be used in the construction, calibration of sensing equipment, and passing an engineering exam, an understanding of the system and terms of the flow formulas must be analyzed. See drawing one below for a typical installation and features of an orifice plate flow device. Drawing one: (Cutaway view of an orifice plate installed in a pipe) Pressure taps
P1
P2
P = P1 - P2
Flow direction, @ reduced pressure
Flow Direction A V1, P1
∆
A2 P2, V2 Area of Vena contracta
Orifice plate
Flange holding the orifice plate Orifice plate aperture (opening) the hole may be changed in shape or position to prevent build up of debris or sludge
As can be seen by the drawing, fluid flow inters from the left, and through the hole in the orifice plate. Because A 1 V1 = A2 V2 the velocity must be greater when passing through the orifice with subsequent drop in pressure. The max pressure drop is near the aperture due to the flow restriction created by the orifice, a pressure difference can be measured by a sensor, and then be used to generate an electrical output used for instrumentation, control, or alarms. To figure out a way to take this information above, and calculate the flows through this physical set up, we can use the Bernoulli equation and modify it to accomplish that task. With that done, we will be able to use it to set up the system and calibrate instrumentation accordingly. Before we derive the first equation, let's define some terms and talk about some problems in theory regarding orifice plates. First, the term C d (Discharge coefficient) used in equations is related to the flow characteristics through the actual orifice device. Commonly you will see this term described as the ratio of the orifice aperture diameter to the diameter of the pipe. This ratio is often referred to as the β (beta) ratio, but we are still talking about the C d. β=
= Cd
Having said that however, there are other factors involved in the actual evaluation of what the exact values are. These are the shape of the orifice aperture, the Reynolds number, the vena contracta which is related to the shape, density changes, expansion coefficient, operating temperature, and the actual P of the orifice. We will explore these later on in the discussion. The vena contracta is essentially the area after the orifice where the pressure drop is at maximum and the point where you might consider placing the low pressure tap. There are many different ways to obtain the desired characteristics of the delta P and there are different equations that are also used to describe the flow rate. The one I am going to derive is one of the common ones but by all means is not the only one. The industry is trying to come up with one standard equation to describe the theoretical flow through an orifice, but to my knowledge, that has not happened yet. Until then, it will be up to the engineer to decide his own program for determining flow rates and determining what type of orifice plate configuration to use.
∆
The Bernoulli equation (principle) for an orifice, with incompressible fluid, is as follows:
ℊ ℊ 1
+
2 1
2
+
Z1 =
2
+
2 2
2
+
Z2 Where: P1 = Pressure at inlet to orifice
P2 = Pressure at orifice @ Vena contracta, V 1 = Velocity before the orifice
V2 = Velocity at vena contracta, gravitational constant
= rho, density of what you are pumping,
ℊ
=
Z1,2 = the height difference
ℊ
Since the heights are the same, (orifice plates are installed on straight runs of horizontal pipe) and Z can be ignored.
So then:
− − − 1
1
2 1
+
=
2
2
-
2
2 1
+
2
2( 1
2)
2 2 =
2 1 +
2 2
+
=
2
Rearranging terms gives
2 2
1
2
+
2
2 1
2
=
2 2
2
Multiplying both sides by 2
+ 12 = 22 Rearranging
2( 1
2)
Since A1 V1 = A2 V2 (Continuity equation) I can rewrite V 1 in terms of V2
− − − − − 2
V1 =
1
2
2 1 =
2
Now, squaring both sides
x
2 2 Next, I am going to substitute these terms back in for
2
1
2
2
2 2 =
x 22 +
1
2
1=
2
+
1
1-
2
2
1
2 x 2
2( 1
1
=
2)
2)
2( 1
2 1
2
=
2( 1
2 1
Dividing both sides by 22
Subtracting the area term from both sides
2)
Multiplying both sides by
2 2
1
2 2
1
2( 1
2
2 2
Dividing both sides by the area terms.
− − − − 1
2 2 =
2( 1
2 1
1
1
2 =
2( 2( 1
2 1
1
∆ −
Q=
2 2
2( 2(
2
∆ − 2( 2(
2
∆
)
Adding in the discharge coefficient for the orifice plate.
)
2
2 1
1
Converting to flow instead of velocity.
and, P1 - P2 = P, then:
2
2 1
1
Taking the square of both sides.
2)
2
Since Q = flow volume =
Q=
2
2
This is one of the equations you will see in text books and on the internet for orifice plates. Notice that the equation in this form does not have any time consideration on the right hand side of the equation. This is an expression, and must be modified to calculate flow. The equation can be modified to get answers in flow by understanding what Q is. Q, for an orifice plate is: Q = Cd A2
2
(Note that Q is also used in the
continuity equation with different terms.) Now, rewriting the equation, we get:
Q=
− 2
1
2 1
2
2
because
∆ ≈ 2( 2(
)
2
Here is an example: Say you have a 4 inch pipe and an orifice which has a C d of .6 and a reading of 120 inches head. What is the flow rate in gpm? Q=
− 2
1
inch.
2 1
2
2
A1 =
D1 = 4.026 (Nominal 4"pipe) D2 = .6 x 4.026 = 2.416
2
= 3.1416 x 4.05 = 12.723 inch
A2 = 3.1416 x 1.459 = 4.5835 inch
2
2
Q=
.6
Q=
≈ 2
4.5835
2
.93
.6
32
10
2
2
.03165
3
x 25.3
.93
= .5166
231.87
3
Remember, converting
to Gpm is accomplished by multiplying that number by
∆
448.83. Also to convert to feet head is done by multiplying by 2.311, or to inches head by 27.73 at STP. Now, go on the internet and find Daniel orifice flow calculator and download the free program for calculating flow. Once you have done that select flow, 3 density of water is 62.4 lb/ft , and plug in the pipe size and C d given in the example. You should come up with 233 gpm. You can use this program to approximate any flow condition that you like and determine orifice size or whatever. Here are some other formulas that have been derived or experimentally determined that lead us to the same set of information only different as to how we get there.
Q = CeE
− 2 2
4
2( 1
2)
where Q = flow rate C = discharge coefficient C d e = expansion coefficient
E=
− 1
1
2 1
4
2
D2 = diameter of the orifice P1- P2 = pressure differential across the orifice
= density of what your pumping
Note: This equation must also be converted to account for time on the right hand side, just like we did for the first one. That is if you want to calculate a flow in GPM.
Here is an interesting equation that comes from a manufacturer of orifice plates
− 2
Q=
0
1
4
where: Q = flow rate
1
Constant = to convert to GPM (44.748) by doing this you dont have worry about the time factor in the equation as the other two d = throat diameter of orifice C = discharge coefficient Y = expansion factor Fa = thermal expansion factor hw = differential pressure in inches of water g0 = standard gravity acceleration g = acceleration where the orifice is β = Cd or beta ratio
2 1
= density of what your pumping
Remember in all of this, the Idea is to get a good approximation of the flow through the system, as that is the reason for putting in an orifice, right? So these equations can be used to suit your needs as an engineer. You can also include the Reynolds number if you want in the Cd figure or if you want you can include a flow coefficient to account for the Reynolds number and expansion coefficient. It's really up to you, the engineer. I hope this helped you out understanding the confusion over orifice plate calculations and how complicated it can be. But you as the engineer must determine what best suits your needs. As for students that are just trying to figure things out, use the first equation I gave you to calculate flow through an orifice, it will be close enough for you to understand what's happening.
