The Orifice Flow Meter Equation ~ Learning Instrumentation And Control Engineering Learning Instrumentation And Control Engineering

### The Orifice Flow Meter Equation

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The orifice flow meter is one of the most popular flow devices for measuring flow. It has proven its mettle in the both liquid and gaseous applications. In the Natural gas industry, the orifice plate continues to play a dominant role in flow measurement applications.

In the orifice flow measurement application, changes in static pressure, temperature and density are critical. In liquid systems static pressure changes have no effect on liquid density but in gaseous systems, a change in static pressure significantly impacts density due to the compressible nature of gaseous systems. Change in temperature affects both liquid and gaseous densities and as such are
compensated for. See Basics of Flow Measurement with the Orifice Flow Meter I for more details.

To therefore measure flow with the orifice meter, a number of correction factors are applied to a general flow equation for the orifice plate for flange taps(the most popular application) according to ISO 5167-1. Here we present the generalized orifice flow equation for measuring gas volume :

Qv = 218.527 * Cd * Ev *Y1*d^2*(Tb/Pb)* SQRT[(Pf1*Zb*Hw)/(Gr*Zf1*Tf)]

Where:
Qv     =    Standard volume flow rate - SCF/hr
Cd     =    Orifice plate coefficient of discharge
d        =   Orifice plate bore diameter at flowing temperature (Tf) degree Rankine
Gr      =   Specific gravity of the gas
Hw    =    Orifice differential pressure in inches of water column at 60 degF
Ev     =    Velocity of approach factor
Pb     =    Base pressure in psia
Pf1   =     Flowing pressure (upstream tap) in psia
Tb    =     Base temperature in degree Rankine
Tf     =     Flowing temperature in degree Rankine
Y2    =     Expansion factor (downstream tap)
Zb    =     Compressibility at base conditions (Pb,Tb)
Zf1   =    Compressibility (upstream flowing conditions - Pf1, Tf)

The above equation computes volume at standard conditions – 60 degree Rankine and 14.73psia –AGA Report 3. This standard or base condition is normally agreed upon by terms stipulated in the custody transfer application or government regulation.

Correction Factors in the Orifice Flow Equation:
Orifice Plate Coefficient of Discharge – Cd

The ratio between true flow rate and theoretical flow rate for any measured amount of differential pressure is known as the discharge coefficient of a flow-sensing element, in this case, the orifice plate. The discharge coefficients for flange-tapped orifice meters have been empirically determined. The coefficient of discharge depends on the Reynolds number, sensing tap location, meter tube diameter and orifice diameter with some other smaller influences. Each coefficient of discharge applies to the Reynolds number at which it is calculated.

Orifice Plate Bore Diameter – d
The orifice bore diameter, d, used in the above flow equation is the one calculated at the flowing temperature conditions (Tf, Zf1, Pf1) of the orifice plate.

Specific Gravity - Gr
This is the normal specific gravity obtained from a specific gravity test or a recording instrument. It represents the ratio of the relative densities of the gas, divided by that of air at the same conditions.

Differential Pressure across the Orifice Plate -Hw
This is a measure of the pressure drop across the orifice and is measured in inches of water column at 60 degree Rankine.

Velocity of Approach Factor – Ev
This factor corrects for the change in velocity between the upstream meter tube and the velocity in the orifice bore.

Base Pressure (psia) – Pb
To define the volume of gas being measured, the base pressure must be defined. This can either be set by the custody transfer contract, government regulation or agreement by the two parties to the measurement. The American Gas Association Report 3 on which the above flow equation is based uses 14.73 psia as its base pressure.

Flowing Pressure - Pf1 or 2
The pressure is measured at either the upstream (1) or downstream (2) tap. Downstream pressure measurement is used in most natural gas measurement applications.

Base Temperature - Tb
As mentioned before, base conditions can either be set by the contract, government regulation or agreement by the two parties to the measurement. The base temperature is usually in degree Rankine. To convert degrees Fahrenheit to degrees Rankine use:

Degree Rankine = Degree Fahrenheit + 459.67. Most natural gas business use 519.67°R (i.e. 60°F + 459.67°) as the base temperature.

Flowing Temperature – Tf
The flowing temperature is normally measured downstream from the orifice and must represent the average temperature of the flowing stream in degrees Rankine.

Expansion Factor Y1 or 2
The expansion factor corrects for the density change between the measured tap density and the density at the plane of the orifice face.

Compressibility, Zb, at Base Conditions (Tb, Pb)
This correction factor is very close to one so in the past it has been ignored. However, it is now required to correct for the gas compressibility from the base pressure to absolute zero pressure at 60° F.

Compressibility, Zf, at Flowing Conditions (Tf, Pf)
Real gases compress more than the ideal gas law predicts and this must be corrected for when gas is measured at high pressure and temperatures other than the base conditions(Tb, Pb).This correction, when applied outside of the square root radical is called super compressibility. In round numbers at ambient temperature the compressibility affects volume by 0.5 percent per 100 psi of pressure