The measurement of the flow using the differential pressure apparatus is commonly used in the industry, so it is important to know how to calculate the flow rate through an orifice

1. FLOW MEASUREMENT THROUGH ORIFICE
One of the applications of Bernoulli’s equation is
orifice flow meter.
Orifice meter is a device used for measuring the rate
of flow of a fluid flowing through a pipe. It consists
of flat circular plate which has a circular hole, in
concentric with the pipe. The diameter of orifice
may vary from 0.4 to 0.8 times the pipe diameter.
To derive the orifice flow equation, we will start
with Bernoulli’s equation.
𝑃1
𝛾
+ 𝑍1 +
𝑉1
2
2𝑔
=
𝑃2
𝛾
+ 𝑍2 +
𝑉2
2
2𝑔
ℎ =
𝑉2
2
− 𝑉1
2
2𝑔
𝑉2 = √(2𝑔ℎ + 𝑉1
2)
Where h is the differential head
By continuity equation,
𝐴1 𝑉1 = 𝐴2 𝑉2
Cc (Coefficient of contraction), 𝐶𝑐 =
𝐴2
𝐴 𝑜
𝑉1 =
𝐴 𝑜 𝐶𝑐
𝐴1
𝑉2
Hence, 𝑉2 =
√2𝑔ℎ
√(1 −
𝐴 𝑜
2
𝐴1
2 𝐶 𝑐
2)
Thus, discharge flow rate
𝑄 = 𝐴2 𝑉2 = 𝑉2 𝐶𝑐 𝐴 𝑜 =
𝐴 𝑜 𝐶 𝑐√2𝑔ℎ
√1 − (
𝐴 𝑜
2
𝐴1
2 𝐶 𝑐
2)
(1)
By substituting Cc by Cd (Coefficient of discharge)
𝑄 = 𝐶 𝑑
𝐴 𝑜 𝐴1√2𝑔ℎ
√𝐴1
2− 𝐴 𝑜
2
(2)
Where Cd (Coefficient of discharge)
𝐶 𝑑 = 𝐶𝑣
√(1 −
𝐴 𝑜
2
𝐴1
2)
√(1 −
𝐴 𝑜
2
𝐴1
2 𝐶 𝑐
2)
You can calculate the flow rate through an orifice
using equation 1 or equation 2 with finding the
value of Cd using the above equation.
Another equation for calculating the flow rate
through an orifice is:
𝑄 = 𝐶 𝐴 𝑜 √2𝑔 (
𝑃1
𝛾
+ 𝑍1 −
𝑃2
𝛾
− 𝑍2) (3)
Where C, the coefficient of the meter (orifice
coefficient), is given by
𝐶 =
𝐶 𝑣 𝐶 𝐶
√1 − 𝐶 𝑐
2 (
𝐴0
𝐴1
⁄ )
2
(4)
In case of incompressible flow through a horizontal
flow meter installation, use equation 5.
𝑄 = 𝐶 𝐴 𝑜√(
2
𝜌
(𝑃1 − 𝑃2 )) (5)
Orifice types and values of C, Cv, Cd:
Cc is used in case of equation 1, 2.
Cc, Cv are used in case of equation 3, 4.
C is ued in case of equation 5.