The accepted one-dimensional equation for mass flow through a concentric, square edged orifice meter is stated in the Equations below. The derivation is based on conservation of mass and energy, one-dimensional fluid dynamics, and empirical functions such as equations of state and thermodynamic process statements. Any derivation is accurate when all the assumptions used to develop it are valid. As a result, an empirical orifice plate coefficient of discharge is applied to the theoretical equation to adjust for multidimensional viscous fluid dynamic effects. In addition, an empirical expansion factor is applied to the theoretical equation to adjust for the reduction in fluid density that a compressible fluid experiences when it passes through an orifice plate.
The orifice meter is a mass meter from which a differential pressure signal is developed as a function of the velocity of the fluid as it passes through the orifice plate bore. Manipulation of the density variable in the equation permits calculation of flow rate in either mass or volume units. The volumetric flow rate at flowing (actual) conditions can be calculated using the following equation:
The volumetric flow rate at base (standard) conditions can be calculated using the following equation:
Β β Linear Coefficient of Thermal Expansion of Orifice Plate Material [in/inββ]
The meter tube internal diameter, D, is defined as the diameter at flowing conditions and can be calculated using the following equation:
Note: For flowing temperature conditions outside those stated above and for
other materials, refer to the American Society for Metals Handbook.
α΅For flowing conditions between -100Β°F and +300Β°F, refer to ASME PTC 19.5.
α΅For flowing conditions between -7Β°F and + 154Β°F, refer to API Manual of
Petroleum Measurement Standards, Chapter 12, Section 2.
The concentric, square-edged, flange-tapped orifice meter coefficient of discharge, ,πΆ-π.(FT), equation, developed by Reader-Harris/Gallagher (RG), is structured into distinct linkage terms and is considered to best represent the current regression data base. The equation is applicable to nominal pipe sizes of 2 inches (50 millimeters) and larger; diameter ratios (π½) of 0.1-0.75, provided the orifice plate bore diameter, d” is greater than 0045 inch (11.4 millimeters); and pipe Reynolds numbers (,π
π-π·.) greater than or equal to 4000.
Where:
π½ β Diameter Ratio = π/π·
πΆ
π(πΉπ) β Coefficientof Discharge at a Specific Reynolds number for FlangeβTapped Orifice Meter
πΆ
πΒ (πΉπ) β Coefficientof Discharge at infinite Reynolds number for FlangeβTapped Orifice Meter
πΆ
π(πΆπ) β Coefficientof Discharge infinite Reynolds number for CornerβTapped Orifice Meter
πΏ
1Β β dimentionless correction for the tap location
π β Napoerian Constant=2.71828
π
4Β = 1.0 when D is in Inches=25.4 when D is in millimeters
π
π
π·Β β pipe Reynolds number
Reynoldβs Number
The RG equation uses pipe Reynolds number as the correlating parameter to represent the change in the orifice plate coefficient of discharge, Cd, with reference to the fluid’s mass flow rate (its velocity through the orifice), the fluid density, and the fluid viscosity. The pipe Reynolds number can be calculated using the following equation:
π· β Meter Tube Internal Diameter [in]
π β Universal Constant=3.14159
π
πΒ β Mass Flow Rate
π
π
π·Β β pipe Reynolds number
π β Absolute Viscosity of Fluid
Velocity of Approach Factor
The velocity of approach factor, ,πΈ-π£. , is calculated as follows:
Where:
π½ β Diameter Ratio
πΈ
π£Β β Velocity Approach Factor
π β Orifice Plate Bore Diameter ,in.
π· β Meter Tube Internal Diameter [in]
This assumption defines the expansion as reversible and adiabatic (no heat gain or loss). Within practical operating ranges of differential pressure, flowing pressure, and temperature, the expansion factor equation is insensitive to the value of the isentropic exponent. As a result, the assumption of a perfect or ideal isentropic exponent is reasonable for field applications.
The application of the expansion factor is valid if the following dimensionless pressure ratio criteria are followed:
Where:
π
3Β β Unit Conversion Factor
π
π1Β β Absolute Static Pressure at Upstream Tap [psi]
π
π2Β β Absolute Static Pressure at Downstream Tap [psi]
βπ β Orifice Differential Pressure [in π»
20] at 60β
Upstream/Downstream Expansion Factor
The upstream expansion factor requires determination of the upstream static pressure, the diameter ratio, and the isentropic exponent. If the absolute static pressure is taken at the upstream differential pressure tap, then the value of the expansion factor, π
1Β , shall be calculated as follows:
When the upstream static pressure is measured,
When the downstream static pressure is measured,
π
1Β β Upstream Expansion Factor
π
3Β β Unit Conversion Factor
π₯
1Β β Ratio of Differential Pressure to Asolute Static Pressure
π
π1Β β Absolute Static Pressure at Upstream Tap [psi]
π
π2Β β Absolute Static Pressure at Downstream Tap [psi]
βπ β Orifice Differential Pressure ,in ,π»-2.0.at 60β
π β Isentropic Exponent
π½ β Diameter Ratio
The downstream expansion factor requires determination of the downstream static pressure, the upstream static pressure, the downstream compressibility factor, the upstream compressibility factor, the diameter ratio, and the isentropic exponent. The value of the downstream expansion factor, π
2, shall be calculated using the following equation:
When the upstream static pressure is measured,
When the downstream static pressure is measured,
π
1Β β Upstream Expansion Factor
π
2Β β Downstream Expansion Factor
π
3Β β Unit Conversion Factor
π₯
1Β β Ratio of Differential Pressure to Asolute Static Pressure
π
π1Β β Absolute Static Pressure at Upstream Tap [psi]
π
π2Β β Absolute Static Pressure at Downstream Tap [psi]
π
π1Β β Fluid Compressibility at Upstream Tap [psi]
π
π2Β β Fluid Compressibility at Downstream Tap [psi]
βπ β Orifice Differential Pressure ,in ,π»-2.0.at 60β
π β Isentropic Exponentπ½ β Diameter Ratio
Supercompressibility
In orifice measurement, ,π-π.and ,π-π1. appear as a ratio to the 0.5 power. This relationship is termed the supercompressibility factor and may be calculated from the following equation:
Base Pressure Factor
If contract base pressure differs from the reference base conditions (14.73 psia and 60β), the initial calculated flow rate at reference base conditions will require conversion to the flow rate at the given base conditions.
πΉππΒ = 14.73psia/ππ
If contract base temperature differs from the reference base conditions (14.73 psia and 60 ), the initial calculated flow rate at reference base conditions will require conversion to the flow rate at the given base conditions.