Copyright American Petroleu Provided by IHS under licens No reproduction or networkin Orifice Metering of Natural Gas and Other Related Hydrocarbon Fluids— Concentric, Square edged Orifice Meters Part[.]
General
This section of API MPMS Ch 14.3/AGA Report No 3 serves as a comprehensive application guide for calculating natural gas flow through a flange-tapped, concentric orifice meter, utilizing the U.S customary (USC) inch-pound system of units.
For applications involving international system (SI) of units, a conversion factor can be applied to the results (Q m ,
The values of Qv or Qb, derived from the equations in section 4.3, may yield inconsistent results when intermediate unit conversions are applied A more reliable method is the universal approach outlined in API MPMS Chapter 14.3.1 and the AGA Report.
No 3, Part 1 can be used The meter has to be constructed and installed in accordance with API MPMS
Ch 14.3.2/AGA Report No 3, Part 2.
Definition of Natural Gas
In this document, "natural gas" refers to fluids that encompass both pipeline and production quality gas, characterized by single-phase flow and specific mole percentage ranges as outlined in Table 1 of API MPMS Ch 14.2/AGA Report No 8 For other hydrocarbon mixtures, the broader guidelines in API MPMS Ch 14.3.1/AGA Report No 3, Part 1 may be more suitable It's important to note that the presence of diluents or other mixtures not specified in API MPMS Ch 14.2/AGA Report No 8 can lead to increased uncertainty in flow measurement.
Basis for Equations
This document employs computation methods aligned with API MPMS Ch 14.3.1/AGA Report No 3, Part 1, incorporating the Reader-Harris/Gallagher (RG) equation for the discharge coefficient of flange-tapped orifice meters The equation has been adapted to utilize the widely used USC inch-pound unit system.
Expansion Factor Application
The choice of which expansion factor equation to apply for all existing installations is left to the discretion of the involved parties, who must remain aware of key considerations.
If the difference between the previous revision (1990) of the Buckingham and Bean expansion factor equation and the newly revised equation is 0.25% or less, the expansion factor values from both equations will fall within the uncertainty of the new expansion factor database, leaving the presence of any flow bias uncertain.
If the difference between expansion factor equations is greater than 0.25%, a variable flow bias will occur, influenced by the diameter ratio (β), isentropic exponent (κ), and ratio (x₁), unless the new expansion factor equation is applied.
The referenced documents are essential for the application of this document For dated references, only the specified edition is applicable, while for undated references, the most recent edition, including any amendments, is relevant.
API MPMS Ch 14.2/AGA Report No 8, Compressibility Factors of Natural Gas and Other Related Hydrocarbon Gases ΔP P f
API MPMS Ch 14.3.1/AGA Report No 3, Part 1, Concentric, Square-edged Orifice Meters, Part 1—General Equations and Uncertainty Guidelines
GPA 2145-09 1 , Table of Physical Properties for Hydrocarbons and Other Compounds of Interest to the Natural Gas
General
The symbols and units utilized in this context adhere to the API MPMS Ch 14.3.3/AGA Report No 3, Part 3 standards, which are based on the USC inch-pound system While standard conversion factors may be applicable, specific considerations must be taken into account.
SI units are used, the more generic equations in API MPMS Ch 14.3.1/AGA Report No 3, Part 1 should be used for consistent results.
Symbols and Units
C d orifice plate coefficient of discharge
C d (FT) coefficient of discharge at a specified pipe Reynolds number for flange-tapped orifice meter
C i (CT) coefficient of discharge at infinite pipe Reynolds number for corner-tapped orifice meter
C i (FT) coefficient of discharge at infinite pipe Reynolds number for flange-tapped orifice meter c p specific heat at constant pressure [Btu/(lbm-°F)] c v specific heat at constant volume [Btu/(lbm-°F)]
D meter tube internal diameter calculated at flowing temperature, T f (in.)
D r meter tube internal diameter at reference temperature, T r (in.) d orifice plate bore diameter calculated at flowing temperature, T f (in.) d r orifice plate bore diameter at reference temperature, T r (in.)
E v velocity of approach factor e Napierian constant rounded to six significant figures (2.71828)
G gas relative density (specific gravity)
G r real gas relative density h w orifice differential pressure (also see ∆P) (inches of water column at 60 °F)
Mr air molar mass (molecular weight) of air (28.9625 lbm/lb-mol)
Mr gas molar mass (molecular weight) of gas (lbm/lb-mol)
Mr i molar mass (molecular weight) of component (lbm/lb-mol) m mass (lbm)
N 4 unit conversion factor (discharge coefficient) n number of moles
1 Gas Processors Association, 6526 E 60th Street, Tulsa, Oklahoma 74145, www.gasprocessors.com. base pressure of air (psia) base pressure of gas (psia)
The static pressure of the fluid at the pressure tap is measured in psia, representing the absolute static pressure at the orifice upstream of the differential pressure tap Additionally, it is important to note the absolute static pressure at the orifice downstream of the differential pressure tap, also measured in psia.
Q b volume flow rate at base conditions (ft 3 /hr)
Q m mass flow rate per hour (lbm/hr)
Q v volume flow rate per hour at reference base conditions (ft 3 /hr) q m mass flow rate per second (lbm/s)
R universal gas constant [1,545.35 (lbf-ft)/(lb-mol-°R)]
T b base temperature (°R) base temperature of air (°R) base temperature of gas (°R)
T f temperature of fluid at flowing conditions (°R)
T r reference temperature of the orifice plate bore diameter and/or meter tube inside diameter (68 °F)
T s reference base temperature, set to U.S standard temperature (519.67 °R) flowing velocity at upstream tap (ft/s)
The volume at base conditions (Vb) is measured in cubic feet (ft³) and represents the flowing volume at the upstream tap The variable \( w \) denotes the number of the last component, while \( x \) indicates the ratio of differential pressure to absolute static pressure Additionally, \( x_1 \) and \( x_2 \) represent the ratios of differential pressure to absolute static pressure at the upstream and downstream pressure taps, respectively Lastly, \( x/\kappa \) refers to the acoustic ratio.
Y 1 expansion factor based on upstream absolute static pressure
Y 2 expansion factor based on downstream absolute static pressure
Z b compressibility at base conditions compressibility of the gas at base conditions (P b, T b )
Z f compressibility at flowing conditions (P f, T f ) compressibility at upstream flowing conditions compressibility at downstream flowing conditions
Z s compressibility at reference base conditions (P s, T s ) compressibility of air at 14.73 psia and 60 °F (0.999590)
The article discusses various temperature scales and coefficients related to thermal expansion It defines the Fahrenheit temperature (\(°F\)) and Rankine temperature (\(°R\)), which is calculated as \(459.67 + °F\) Additionally, it introduces the linear coefficient of thermal expansion (\(\alpha\)) measured in inches per inch per degree Fahrenheit, along with specific coefficients for the orifice plate material (\(\alpha_1\)) and the meter tube material (\(\alpha_2\)) Finally, it mentions the ratio of the orifice plate bore diameter to the meter tube internal diameter (\(\beta\)), which is calculated at the flowing temperature (\(T_f\)).
The orifice differential pressure, denoted as ∆P (in psi), is influenced by various factors including the isentropic exponent (κ), which can be categorized into ideal gas (κ_i), perfect gas (κ_p), and real gas (κ_r) types The absolute viscosity of the flowing fluid is measured in lbm/(ft-s), while π represents the universal constant rounded to six significant figures (3.14159) The density of a fluid at base conditions (P_b, T_b) is expressed in lbm/ft³, which includes the density of air and gases under these conditions Additionally, the density at reference base conditions (ρ_s) and at flowing conditions (ρ_t,p) is also measured in lbm/ft³, with specific values for upstream and downstream tap positions Lastly, the mole fraction of a component is represented as φ_i (mol%/100).
NOTE Factors, ratios, and coefficients are dimensionless.
Terminology
One pound-force per square inch (lbf/in²) is the pressure resulting from a one pound-mass (lbm) exerting force over an area of one square inch, influenced by the standard acceleration due to gravity, which is 32.1740 ft/s².
The subscript 1 on the expansion factor (Y₁), flowing density, fluid flowing static pressure, and fluid flowing compressibility signifies that these variables are measured or calculated in relation to the fluid conditions at the upstream differential tap Conversely, variables associated with the downstream differential pressure tap are denoted by the subscript 2, including Y₂, and can be utilized in equations with the same accuracy for calculating flow rates, except for Y₂, which requires a distinct equation.
In this section, the subscript 1 is used to highlight the importance of preserving the relationship among the four variables concerning the selected static pressure reference tap: \$\rho_{b, \text{air}}\$ and \$\rho_{b, \text{gas}}\$ in relation to \$\rho_{t, p}\$.
The temperature of the flowing fluid (T f) is typically measured downstream of the orifice plate to minimize flow disturbance, although it can also be measured upstream at specified locations according to API MPMS Ch 14.3.2/AGA Report No 3, Part 2 It is assumed that there is no significant temperature difference between the two differential pressure tap locations and the measurement point, making a numerical subscript unnecessary.
This document establishes the reference base conditions for natural gas flow measurement in the United States, specifying an absolute static pressure of 14.73 psia and an absolute temperature of 519.67 °R (60 °F) These conditions are crucial for interstate commerce.
60 °F or 519.67 °R (T s ) Although technically incorrect, these reference base conditions are often referred to as standard conditions and designated in symbology by the subscript “s.”
Base conditions, i.e base pressure (P b ) andd base temperature (T b ), are defined by contract or government regulation and may be different from reference base conditions.
API MPMS Chapter 14.3.1 and AGA Report No 3, Part 1 provide general definitions, while Chapter 14.3.2 and AGA Report No 3, Part 2 continue this coverage Additionally, specific definitions related to API MPMS Chapter 14.3.3 and AGA Report No 3, Part 3 are included within the text.
General
The equations presented illustrate the relationship between flow, mass, and volume per unit time, yielding equivalent outcomes This section focuses solely on the USC inch-pound system of units, with numeric constants adapted from API MPMS Ch 14.3.1/AGA Report No 3, Part 1 to align with these units.
The numeric constants for basic flow equations, unit conversion values, and the densities of water and air are detailed in API MPMS Ch 14.3.1/AGA Report No 3, Part 1, as well as in Section 6 and Annex F of this document This section includes tables that present solutions to these equations, utilizing the specified constants and values Additionally, key components of the equations are elaborated in Section 5, while other physical properties are provided in Section 6.
Equations for Mass Flow of Natural Gas
The mass flow equations for natural gas, measured in lbm/hr, can be derived from the density of the flowing fluid, as well as the ideal or real gas relative density, utilizing specific formulas.
The mass flow developed from the density of the flowing fluid ( ) is expressed as follows:
(1) Mass flow developed from the ideal gas relative density, G i , is expressed as follows:
The mass flow equation, derived from the real gas relative density (\(G_r\)), is based on reference conditions of 14.73 psia and 519.67 °R (60 °F) This approach enables the integration of the base compressibility of air at these specific conditions into the equation for density (\(ρ_t p\)).
The flow rate equation includes a numeric constant, and if the assumptions regarding the reference base conditions are inaccurate, the results will carry an additional level of uncertainty The mass flow equation, derived from the relative density of real gas, \( G_r \), is formulated as follows:
C d (FT ) is the coefficient of discharge for flange-tapped orifice meter; d is the orifice plate bore diameter, in inches, calculated at flowing temperature (T f );
E v is the velocity of approach factor;
G i is the ideal gas relative density;
G r is the real gas relative density; h w is the orifice differential pressure, in inches of water at 60 °F; is the flowing pressure at upstream tap, in psia;
Q m is the mass flow rate, in lbm/hr;
T f is the flowing temperature, in °R;
Y 1 is the expansion factor (upstream tap);
Z s is the compressibility at reference base conditions (P s , T s ); is the compressibility at upstream flowing conditions ( , T f ); is the density of the fluid at upstream flowing conditions ( , T f ), in lbm/ft 3
Equations for Volume Flow of Natural Gas
The volume flow rate of natural gas, measured in cubic feet per hour at base conditions, can be calculated using the densities of the gas under flowing and base conditions, along with the ideal or real gas relative density, through specific equations.
The volume flow rate at base conditions, Q b , developed from the density of the fluid at flowing conditions and base conditions (ρ b ) is expressed as follows:
(4) The volume flow rate at base conditions, developed from ideal gas relative density, G i , is expressed as follows:
To accurately utilize the real gas relative density in flow calculations, it is essential that the reference base conditions for determining real gas relative density align with the base conditions used for flow calculations Consequently, the volume flow rate at base conditions, derived from the real gas relative density, \( G_r \), can be expressed accordingly.
If reference base conditions are substituted for base conditions in Equation (4), Equation (5), and Equation (6), then
The volume flow rate at reference base conditions, Q v , can then be determined using the following equations.
The volume flow rate at reference base conditions, developed from the density of the fluid at flowing conditions ( ) and reference base conditions (ρ s ), is expressed as follows:
The volume flow rate at reference base conditions, developed from ideal gas relative density, G i , is expressed as follows:
The volume flow rate equation, denoted as \$Q_v\$, is derived from the relative density of real gases and is based on specific reference conditions These reference conditions are set at 14.73 psia and 519.67 °R for the gas relative density, \$G_r\$.
(60 °F) in its numeric constant Therefore, the volume flow rate at reference base conditions, developed from real gas relative density, G r , is expressed as follows:
C d (FT) is the coefficient of discharge for flange-tapped orifice meter; d is the orifice plate bore diameter calculated at flowing temperature (T f ), in inches;
E v is the velocity of approach factor;
G i is the ideal gas relative density;
G r is the real gas relative density; h w is the orifice differential pressure, in inches of water at 60 °F;
P b is the base pressure, in psia;
- is the flowing pressure (upstream tap), in psia;
P s is the reference base pressure = 14.73 psia;
Q b is the volume flow rate per hour at base conditions, in ft 3 /hr;
Q v is the volume flow rate per hour at reference base conditions, in ft 3 /hr;
T b is the base temperature, in °R;
T f is the flowing temperature, in °R;
T s is the reference base temperature = 519.67 °R (60 °F);
Y 1 is the expansion factor (upstream tap);
Z b is the compressibility at base conditions (P b , T b ); is the compressibility of air at base conditions (P b , T b ); is the compressibility at upstream flowing conditions ( , T f );
The compressibility at reference base conditions is denoted as \$Z_s\$, while the compressibility of air under the same conditions is also represented as \$Z_s\$ The density of the flowing fluid at base conditions, indicated as \$\rho_b\$, is measured in lbm/ft³, while the density at reference base conditions is represented as \$\rho_s\$, also in lbm/ft³ Additionally, the density of the fluid under upstream flowing conditions is expressed as \$\rho_f\$, measured in lbm/ft³.
Volume Conversion from Reference Base to Base Conditions
This document assumes that the reference base and base conditions are identical However, if the base conditions differ from the reference base conditions, the volume flow rate calculated at the reference base conditions can be converted to the volume flow rate at the base conditions using a specific relationship.
P b is the base pressure, in psia;
P s is the reference base pressure, in psia;
Q b is the base volume flow rate, in ft 3 /hr;
Q v is the reference base volume flow rate, in ft 3 /hr;
T b is the base temperature, in °R;
T s is the reference base temperature, in °R;
Z b is the compressibility at base conditions (P b , T b );
Z s is the compressibility at reference base conditions (P s , T s ).
5 Flow Equation Components Requiring Additional Computation
General
Some of the terms in Equation (1) through Equation (9) require additional computation and are developed in this section.
Diameter Ratio (β)
The diameter ratio (β), or beta ratio, is essential for calculating the orifice plate coefficient of discharge (C d), the velocity of approach factor (E v), and the expansion factor (Y) It is defined as the ratio of the orifice bore diameter (d) to the internal diameter of the meter tube (D) For optimal accuracy, it is crucial to utilize the actual dimensions as specified in API MPMS Ch 14.3.1/AGA Report No 3, Part 1 and/or API MPMS Ch 14.3.2/AGA Report No 3, Part 2.
D is the meter tube internal diameter calculated at flowing temperature, T f ;
The reference meter tube internal diameter, denoted as \$D_r\$, is calculated at the reference temperature \$T_r\$ The orifice plate bore diameter, represented by \$d\$, is determined at the flowing temperature \$T_f\$ Additionally, the reference orifice plate bore diameter, indicated as \$d_r\$, is also calculated at the reference temperature \$T_r\$.
T f is the temperature of the fluid at flowing conditions;
The reference temperature, denoted as \$T_r\$, is crucial for determining the orifice plate bore diameter and the internal diameter of the meter tube The linear coefficient of thermal expansion for the orifice plate material is represented by \$\alpha_1\$, while \$\alpha_2\$ denotes the linear coefficient of thermal expansion for the meter tube material, both of which can be found in Table 1 Additionally, \$\beta\$ signifies the diameter ratio.
NOTE α, T f , and T r have to be in consistent units For the purpose of this standard, T r is assumed to be 68 °F.
The orifice plate bore diameter, d r , and the meter tube internal diameter, D r , calculated at T r are the diameters determined in accordance with API MPMS Ch 14.3.2/AGA Report No 3, Part 2.
Coefficient of Discharge for Flange-tapped Orifice Meter [Cd (FT)]
The coefficient of discharge (\(C_d\)) for a flange-tapped orifice meter has been established based on test data and is correlated with the diameter ratio (\(\beta\)), tube diameter, and pipe Reynolds number This document presents the equation for the coefficient of discharge for flange-tapped orifice meters as developed in API MPMS Chapter 14.3.1 and the AGA Report.
No 3, Part 1 has been adapted to the USC inch-pound system of units. β = d D⁄
The coefficient of discharge, \( C_d (FT) \), for concentric, square-edged flange-tapped orifice meters, as developed by RG, consists of distinct linkage terms This equation is valid for nominal pipe sizes of 2 inches and larger, with diameter ratios (\( \beta \)) ranging from 0.1 to 0.75, provided the orifice plate bore diameter (\( d_r \)) exceeds 0.45 inches and the pipe Reynolds numbers (\( Re_D \)) are at least 4,000 Deviations from these parameters increase uncertainty, and for further guidance, refer to API MPMS Chapter 14.3.1 and AGA Report No 3, Part 1, Section 12.4.1 (September 2012).
The RG equation is defined as follows:
C i (CT) = 0.5961 + 0.0291β 2 – 0.2290β 8 + 0.003(1 – β)M 1 (16) tap term = upstrm + dnstrm (17) upstrm = (18) dnstrm = – 0.0116(M 2 – 0.52 )β 1.1 (1 – 0.14A) (19) also
Table 1—Linear Coefficient of Thermal Expansion Material
Linear Coefficient of Thermal Expansion ( α )
NOTE For flowing temperature limits or other materials, refer to ASM International’s Metals Handbook,
The engineering properties of steel, particularly stainless steels, are crucial for various applications For optimal performance, stainless steels should be used in flowing conditions ranging from +32 °F to +212 °F, while Monel is suitable for +68 °F to +212 °F For conditions between –7 °F and +154 °F, it is advisable to consult API MPMS Chapter 12.2.1 When dealing with orifice plates made of stainless steel, the linear coefficient of thermal expansion for Type 304/316 stainless steel is recommended, as it represents the average of both types, especially when the specific grade is unknown.
NOTE Over a temperature range from 32 °F to 130 °F the maximum difference in calculated flow between use of the 304/316 average coefficient and either the 304 or 316 coefficient is less than 0.005 % (50 ppm).
C d (FT) is the coefficient of discharge at a specified pipe Reynolds number for a flange-tapped orifice meter;
C i (CT) is the coefficient of discharge at an infinite pipe Reynolds number for a corner-tapped orifice meter;
C i (FT) is the coefficient of discharge at an infinite pipe Reynolds number for a flange-tapped orifice meter;
D is the meter tube internal diameter calculated at T f , in inches; d is the orifice plate bore diameter calculated at T f , in inches; e is the Napierian constant = 2.71828;
L 1 = L 2 = dimensionless correction for tap location = N 4 /D for flange taps;
Re D is the pipe Reynolds number; β is the diameter ratio = d/D.
Velocity of Approach Factor (Ev)
The velocity of approach factor (\(E_v\)) is a crucial mathematical expression that connects the fluid velocity in the upstream meter tube of an orifice meter to the fluid velocity within the orifice plate bore.
The velocity of approach factor, E v , is calculated as follows:
E v is the velocity of approach factor; β is the diameter ratio = d/D.
Reynolds Number (ReD)
The pipe Reynolds number (Re D) serves as a key correlation parameter that illustrates how the orifice plate coefficient of discharge varies with the meter tube diameter, fluid flow rate, fluid density, and fluid viscosity This dimensionless ratio is calculated using consistent units and is fundamental in fluid dynamics.
NOTE The constant, 12, in the denominator of Equation (26) is required by the use of D in inches.
The fluid velocity, in ft/s, can be obtained in terms of the hourly volumetric flow rate at base conditions from the following relationship:
Substituting Equation (28) into Equation (26) results in the following relationship:
The Reynolds number for natural gas can be approximated by substituting the following relationship for ρ b (see 6.5.3 for equation development) into Equation (29):
By using an average value of 0.0000069 lbm/ft-s for à and substituting the reference base conditions of 519.67 °R, 14.73 psi, and 0.999590 for T b , P b , and Equation (31) reduces to the following:
D is the meter tube internal diameter calculated at the flowing temperature (T f ), in inches;
G r is the real gas relative density;
Q b is the volume flow rate at base conditions, in ft 3 /hr;
Q v is the volume flow rate at reference base conditions, in ft 3 /hr; q m is the mass flow rate, in lbm/s;
Re D is the pipe Reynolds number;
The base temperature, denoted as \( T_b \), is measured in °R, while the velocity of the flowing fluid at the upstream tap location is expressed in ft/s The compressibility of air is evaluated at 14.73 psia and 60 °F, and the compressibility of the gas is determined at base conditions (\( P_b, T_b \)) The absolute dynamic viscosity, represented by \( \mu \), is measured in lbm/ft-s, with \( \pi \) approximated as 3.14159 Additionally, the density of the flowing fluid at base conditions (\( P_b, T_b \)) is indicated as \( \rho_b \) in lbm/ft³, while the density of the fluid at upstream flowing conditions (\( P_f, T_f \)) is also measured in lbm/ft³.
Viscosity is influenced by temperature, relative density, and pressure, with an average value of 0.0000069 lbm/ft-s commonly used for natural gas measurement For temperatures between 30 °F and 90 °F and relative densities from 0.55 to 0.75, viscosity values typically range from 0.0000059 to 0.0000079 lbm/ft-s This property significantly affects the calculated discharge coefficient, and using the average viscosity of 0.0000069 lbm/ft-s may not be appropriate if the fluid's viscosity, temperature, or real gas relative density deviates from these specified ranges.
When the flow rate is unknown, the Reynolds number can be iteratively calculated by assuming an initial discharge coefficient value of 0.60 for a flange-tapped orifice meter, denoted as C_d (FT), and utilizing the computed volume to estimate the Reynolds number.
Expansion Factor (Y)
When gas flows through an orifice, variations in fluid velocity and static pressure lead to changes in density, necessitating an adjustment factor for the coefficient This expansion factor (Y) depends on the diameter ratio (β), the ratio of differential pressure to static pressure at the designated tap, and the isentropic exponent (κ).
The real compressible fluid isentropic exponent, \( \kappa_r \), depends on the fluid's properties as well as its pressure and temperature For ideal gases, the isentropic exponent, \( \kappa_i \), is defined as the ratio of specific heats at constant pressure (\( c_p \)) and constant volume (\( c_v \)), remaining constant regardless of pressure A perfect gas, characterized by constant specific heats, has an isentropic exponent, \( \kappa_p \), which is equivalent to \( \kappa_i \) evaluated under base conditions.
In many applications, the values of the isentropic exponents κ r, κ i, and κ p are nearly identical, indicating that the flow equation is not significantly affected by small variations in the isentropic exponent Consequently, the perfect gas isentropic exponent, κ p, is commonly utilized in flow equations, with a standard value of κ p = κ = 1.3 for natural gas applications However, under specific conditions, such as changes in composition or significant fluctuations in pressure or temperature, users may opt to calculate either a fixed or a dynamic value for the isentropic exponent.
The application of the expansion factor is valid as long as the following dimensionless criterion for pressure ratio is followed:
N 3 is the unit conversion factor (refer to API MPMS Ch 14.3.1/AGA Report No 3, Part 1—2012, Table 4);
The absolute static pressure at the pressure tap is denoted as \( P_f \), while the absolute static pressures at the upstream and downstream pressure taps are represented as \( P_{up} \) and \( P_{down} \), respectively The orifice differential pressure is indicated by \( \Delta P \).
The expansion factor equation for flange taps may be used for a range of diameter ratios from 0.10 to 0.75 For diameter ratios (β) outside the stated limits, increased uncertainty will occur.
5.6.2 Expansion Factor Referenced to Upstream Static Pressure ( Y 1 )
If the absolute static pressure is taken at the upstream differential pressure tap, the value of the expansion factor, Y 1 , can be calculated using the following equation:
When the upstream static pressure is measured,
When the downstream static pressure is measured,
N 3 serves as the unit conversion factor, as outlined in API MPMS Chapter 14.3.1 and AGA Report No 3, Part 1—2012, Table 4 It represents the absolute static pressure measured at the upstream tap in psia, as well as the absolute static pressure at the downstream tap, also in psia.
2 x 1 is the ratio of differential pressure to absolute static pressure at the upstream tap;
Y 1 is the expansion factor based on the absolute static pressure measured at the upstream tap; β is the diameter ratio = d/D; ΔP is the orifice differential pressure; κ is the isentropic exponent (see 5.6.1).
5.6.3 Expansion Factor Referenced to Downstream Static Pressure ( Y 2 )
To determine the upstream pressure, it is advisable to calculate it by adding the measured differential pressure (ΔP) to the static pressure recorded at the downstream differential tap, ensuring proper unit conversion is applied.
If the user opts not to specify Y₁, the downstream expansion factor Y₂ must be calculated This calculation involves determining several key parameters: downstream static pressure, upstream static pressure, downstream compressibility factor, upstream compressibility factor, diameter ratio, and isentropic exponent The downstream expansion factor Y₂ can be computed using a specific equation.
The absolute static pressure at the upstream pressure tap is denoted as \$P_{up}\$, while the absolute static pressure at the downstream pressure tap is represented as \$P_{down}\$ The variable \$x_1\$ signifies the ratio of differential pressure to the absolute static pressure at the upstream tap.
Y 1 is the expansion factor based on the absolute static pressure measured at the upstream tap;
The expansion factor, denoted as \( Y_2 \), is determined by the absolute static pressure recorded at the downstream tap It is influenced by the fluid compressibility at both the upstream and downstream pressure taps, while \( \kappa \) represents the isentropic exponent.
General
Measuring gaseous flow rates in volumetric units necessitates adjustments for pressure, temperature, and deviations from ideal gas laws, known as compressibility Additionally, energy measurement involves calculating the gas heating value, which indicates its energy content This document uses reference base conditions of 14.73 psia and 519.67 °R (60 °F, U.S standard temperature).
Natural gas, being a mixture of various compounds, makes it challenging to calculate certain conversion factors Those factors that cannot be easily calculated can instead be obtained through analysis of gas composition and additional measurements.
For accurate measurements in the field, it is essential to refer to Annex F and API MPMS Ch 14.3.1/AGA Report No 3, Part 1—2012 Utilizing instruments calibrated with standard gas samples will yield results that are equivalent when rigorous methods are employed.
Energy determination can be calculated by multiplying the volume by the heating value per unit volume, or by multiplying the mass by the heating value per unit mass For guidance on how to determine heating value with or without water vapor, consult API MPMS Ch 14.5/GPA 2172 or AGA Report No 5.
Physical Properties
The physical properties shall be taken from the latest edition of GPA 2145.
The compressibility of air at reference base conditions ( ) is 0.999590.
Compressibility
Ideal gases are defined by their adherence to the thermodynamic laws of Boyle and Charles, known as the ideal gas laws In contrast, real gases deviate from these ideal behaviors, necessitating different calculation and interpretation methods.
If subscript 1 represents a gas volume measured at one set of temperature–pressure conditions and subscript 2 represents the same volume measured at a second set of temperature–pressure conditions, then
The numerical constant in Equation (42) is required to convert P, in psia, to units that are consistent with the value of
All gases exhibit some degree of deviation from ideal gas laws, referred to as compressibility and represented by the symbol Z A comprehensive analysis of compressibility and the determination of Z for natural gas is provided in API MPMS Ch 14.2/AGA Report No 8, which includes the methodology outlined in this standard.
The application of Z changes the ideal relationship in Equation (42) to the following real relationship:
144PV nZRT As modified by Z, Equation (43) allows the volume at the upstream flowing conditions to be converted to the volume at base conditions by use of the following equation:
(45) where n is the number of pound-moles of a gas;
P is the absolute static pressure of a gas, in psia;
P b is the absolute static pressure of a gas at base conditions, in psia; is the absolute static pressure of a gas at the upstream tap, in psia;
R is the universal gas constant = 1,545.35 (lbf-ft)/(lb-mol-°R);
T is the absolute temperature of a gas, in °R;
T b is the absolute temperature of a gas at base conditions, in °R;
T f is the absolute temperature of a flowing gas, in °R;
V is the volume of a gas, in ft 3 ;
V b is the volume of a gas at base conditions (P b , T b ), in ft 3 ; is the volume of a gas at flowing conditions ( , T f ), in ft 3 ;
Z is the compressibility of a gas at P and T ;
Z b is the compressibility of a gas at base conditions (P b , T b ); is the compressibility of a gas at flowing conditions ( , T f ).
The value of Z at base conditions (Z b) is essential and must be calculated using the procedures outlined in API MPMS Ch 14.2/AGA Report No 8 for volume determination Methods for determining Z b, applicable for calculating heating value per real unit volume and real relative density at base conditions, are provided in API MPMS Ch 14.5/GPA 2172 or AGA Report No 5 The variations in Z b from these methods fall within the experimental uncertainty of the property data, which, as noted in API MPMS Ch 14.5/GPA 2172, is generally accurate to no better than 1 part in 1000.
In orifice measurement, Z b and appear as a ratio to the 0.5 power This relationship is termed the supercompressibility factor and may be calculated from the following equation:
F pv is the supercompressibility factor;
Z b is the compressibility of the gas at base conditions (P b , T b ); is the compressibility of the gas at flowing conditions ( , T f ).
Relative Density
Relative density, denoted as G, is a crucial factor in various flow equations, representing the dimensionless ratio of a fluid's density to that of a reference gas under identical temperature and pressure conditions In the gas industry, relative density is often categorized as either ideal or real, with air serving as the reference gas and standard conditions set at 14.73 psia and 519.67 °R (60 °F) This value can be obtained through direct measurement or calculated based on the gas composition.
The ideal gas relative density, \( G_i \), is the ratio of the ideal density of a gas to that of dry air under identical reference conditions of pressure and temperature This relationship simplifies to a ratio of molar masses, as the ideal densities are defined under the same conditions The equation for calculating the ideal gas relative density is thus established.
G i is the ideal gas relative density;
Mr air is the molar mass (molecular weight) of air = 28.9625 lbm/lb-mol;
Mr gas is the molar mass (molecular weight) of a gas, in lbm/lb-mol.
6.4.3 Real Gas Relative Density (Real Specific Gravity)
Real gas relative density, denoted as \( G_r \), is the ratio of the real density of a gas to the real density of dry air under identical reference conditions of pressure and temperature For accurate flow calculations, it is essential that the reference conditions used to determine the real gas relative density match the base conditions for the flow calculation At base conditions \((P_b, T_b)\), the expression for real gas relative density is defined accordingly.
Since the pressures and temperatures are defined to be at the same designated base conditions,
The real gas relative density reduces to:
Relative density is commonly calculated from composition Refer to API MPMS Ch 14.5/GPA 2172 or AGA Report
The historical use of real gas relative density in flow calculations introduces uncertainty due to the limitations of gravitometer devices When measuring real gas relative densities, adjustments must be made to ensure that both air and gas measurements are consistent in terms of pressure and temperature Variations in temperature and pressure from base conditions can lead to discrepancies in relative density determinations Additionally, the composition of atmospheric air, along with its molecular weight and density, fluctuates over time and across different geographical locations, further contributing to these variations.
When using recording gravitometers, calibration with reference gases allows for the determination of relative density, whether for ideal or real gases The relationship between the relative density of ideal gases and that of real gases is defined through specific equations.
G i is the ideal gas relative density;
G r is the real gas relative density;
Mr air is the molar mass (molecular weight) of air = 28.9625 lbm/lb-mol;
The molar mass of the flowing gas, denoted as Mr gas in lbm/lb-mol, is a crucial parameter in gas calculations Additionally, the base pressure of air is measured in psia, alongside the base pressure of the gas, also expressed in psia.
The universal gas constant, denoted as R, is valued at 1,545.35 (lbf-ft)/(lb-mol-°R) It represents the base temperature of air and a gas, both measured in °R Additionally, the compressibility of air and gas at base conditions (P b, T b) is also considered in this context.
Density of Fluid at Flowing Conditions
The flowing density (\( \rho_{t,p} \)) is crucial for flow equations, representing the mass per unit volume at specific flowing pressure and temperature at a static pressure tap It can be derived from equations of state or relative density measurements at the tap location While commercial density meters can measure fluid density under flowing conditions, their design often limits accuracy at the specific pressure tap Consequently, it is essential to verify the difference between the measured density and the density at the defined pressure tap to assess the impact of pressure or temperature changes on flow measurement uncertainty.
6.5.2 Density Based on Gas Composition
When the composition of a gas mixture is established, the gas densities, denoted as \$\rho_{t,p}\$ and \$\rho_{b}\$, can be calculated using gas law equations Additionally, the molecular weight of the gas can be derived from the composition data by utilizing the mole fractions of the components along with their respective molecular weights.
The gas law equation, Equation (44), is rearranged to obtain density values:
Mr gas is the molar mass (molecular weight) of the flowing gas, in lbm/lb-mol;
Mr i is the molar mass (molecular weight) of a component, in lbm/lb-mol; m is the mass of a fluid, in lbm; n is the number of moles;
P is the absolute static pressure of a gas, in psia;
Mr gas φ 1 Mr 1 +φ 2 Mr 2 +… φ+ w Mr w φ i Mr i i = 1
= P b is the absolute static pressure of a gas at base conditions, in psia; is the absolute static pressure of a gas at the upstream tap, in psia;
R is the universal gas constant = 1,545.35 (lbf-ft)/(lb-mol-°R);
T is the absolute temperature of a gas, in °R;
T b is the absolute temperature of a gas at base conditions, in °R;
T f is the absolute temperature of a flowing gas, in °R;
V is the volume of a gas, in ft 3 ;
V b is the volume of a gas at base conditions (P b , T b ), in ft 3 ; is the volume of a gas at flowing conditions ( , T f ), in ft 3 ; w is the number of the last component;
Z is the compressibility of a gas at P and T;
The compressibility of a gas at base conditions, denoted as \( Z_b \) (at pressure \( P_b \) and temperature \( T_b \)), differs from its compressibility at flowing conditions, represented as \( Z_f \) (at pressure \( P_f \) and temperature \( T_f \)) Additionally, the density of the gas at base conditions, \( \rho_b \) (measured in lbm/ft³), contrasts with its density at upstream flowing conditions, \( \rho_f \) (also in lbm/ft³) The mole fraction of a component in the gas mixture is indicated by \( \phi_i \).
6.5.3 Density Based on Ideal Gas Relative Density
The gas densities and ρ b may be calculated from the ideal gas relative density, as determined in 6.4.2 The following equations are applicable when a gas analysis is available:
(57) NOTE The molecular weight of dry air, from GPA 2145-09, is given as 28.9625 lbm/lb-mol (exactly).
Substituting for Mr gas in Equation (55) and Equation (56), and ρ b are determined as follows:
= G i is the ideal gas relative density;
Mr air is the molar mass (molecular weight) of air = 28.9625 lbm/lb-mol;
Mr gas is the molar mass (molecular weight) of a flowing gas, in lbm/lb-mol;
P b is the absolute static pressure of the gas at base conditions, in psia; is the absolute static pressure of a gas at the upstream tap, in psia;
R is the universal gas constant = 1,545.35 (lbf-ft)/(lb-mol-°R);
T b is the absolute temperature of a gas at base conditions, in °R;
T f is the absolute temperature of a flowing gas, in °R;
The compressibility of a gas at base conditions, denoted as \$Z_b\$, is defined at pressure \$P_b\$ and temperature \$T_b\$ In contrast, the compressibility at flowing conditions is represented as \$Z_f\$ at pressure \$P_f\$ and temperature \$T_f\$ The density of the gas at base conditions, represented as \$\rho_b\$, is measured in lbm/ft³ and is determined using \$P_b\$, \$T_b\$, and \$Z_b\$ Conversely, the density at upstream flowing conditions, denoted as \$\rho_f\$, is also measured in lbm/ft³ and is calculated based on \$P_f\$, \$T_f\$, and \$Z_f\$.
6.5.4 Density Based on Real Gas Relative Density
The relationship of real gas relative density to ideal gas relative density is given by the following equation:
NOTE The real gas relative density of dry air at base conditions is defined as exactly 1.00000.
Substituting for G i in Equation (59) and Equation (60) results in the following:
To accurately utilize the density equations derived from the real gas relative density, it is essential that the base conditions for both the relative density and the flow calculations are consistent When substituting reference base conditions (P_s, T_s) for the base conditions, the integrity of the results must be maintained.
The gas density based on real gas relative density is given by the following equations:
G i is the ideal gas relative density;
G r is the real gas relative density;
P b is the absolute static pressure of a gas at base conditions, in psia; is the absolute static pressure of a gas at the upstream tap, in psia;
P s is the absolute static pressure of a gas at reference base conditions, in psia;
T b is the absolute temperature of a gas at base conditions, in °R;
T f is the absolute temperature of a flowing gas, in °R;
The absolute temperature of a gas at reference base conditions is denoted as \$T_s\$ in °R Compressibility values are crucial, with \$Z_b\$ representing the compressibility of air at base conditions (P_b, T_b) and \$Z_f\$ indicating the compressibility of a gas at flowing conditions (P_f, T_f) Additionally, \$Z_s\$ refers to the compressibility of air at reference base conditions (P_s, T_s) and \$Z_g\$ for a gas at these same reference conditions The density of a gas at base conditions is represented as \$\rho_b\$ in lbm/ft³, while \$\rho_s\$ denotes the density at reference base conditions (P_s, T_s) Finally, the density of a gas under upstream flowing conditions is indicated as \$\rho_f\$ in lbm/ft³.
The density equations for reference base conditions (\( \rho_s \)) rely on the real gas relative density, necessitating the determination of \( G_r \) at these same conditions, specifically at 14.73 psia and 519.67 °R, which are integral to the numeric constants used.
This annex provides equations and procedures for adjusting and correcting field measurement calibrations of secondary instruments.
Field practices for secondary instrument calibrations and calibration standard applications contribute to the overall uncertainty of flow measurement.
Calibration standards for differential and static pressure instruments are often utilized in the field without adjusting for local gravitational force or correcting the indicated values It is generally more efficient and precise to integrate these adjustments into the flow computation rather than applying minor corrections during calibration Consequently, additional factors are incorporated into the flow equation to account for the necessary calibration standard corrections, either through the office's flow calculation procedure or by the meter technician in the field, but not both.
Four factors are provided that may be used individually or in combination, depending on the calibration device and the calibration procedure used:
F am is the correction for air over the water in the water manometer during the differential instrument calibration;
F pwl is the local gravitational correction for the deadweight tester static pressure standard;
F wl is the local gravitational correction for the water column calibration standard;
F wt is the water density correction (temperature or composition) for the water column calibration standard. These factors expand the base volume flow equation to the following:
Equation (A.1) encompasses all relevant flow factors for gas flow as defined in this standard Certain factors may not apply to all measurement systems and can be set to 1 or disregarded at the user's discretion In cases involving mass flow calculations, specific factors should be incorporated into the chosen equation based on the system, instrumentation calibration, and specific operating procedures.
This annex utilizes specific symbols and units derived from the USC inch-pound system While standard conversion factors may be applied, it is recommended to use the more general equations found in API MPMS Ch 14.3.1/AGA Report No 3, Part 1 for consistent results when employing SI units.
F am correction for air over the water in the water manometer
F pwl local gravitational correction for deadweight tester
F wl local gravitational correction for water column
G r real gas relative density g l local acceleration due to gravity (ft/s 2 ) g o acceleration of gravity used to calibrate weights or deadweight calibrator (ft/s 2 )
H elevation above sea level (ft) h wa differential pressure above atmospheric (inches of water column at 60 °F)
L latitude on Earth’s surface (degrees)
Mr molar mass of gas (lbm/lb-mol)
Mr air molar mass of air (28.9625 lbm/lb-mol)
P atm local atmospheric pressure (psia)
P f absolute pressure of flowing gas (psia) volume flow rate at reference base conditions modified for instrument calibration adjustments (ft 3 /hr)
R universal gas constant [1,545.35 (lbf-ft)/(lb-mol-°R)]
T f absolute temperature of a flowing gas (°R) gas ambient temperature (°R)
Z compressibility of a gas at T and P
Z a compressibility of air at P atm + h wa and 519.67 °R compressibility of air at P atm and 519.67 °R
Z b compressibility of a gas at base conditions (G r , P b , and T b ) compressibility of air at reference base conditions of 14.73 psia and 519.67 °R (0.999590)
The Z factor represents the compressibility of gas under flowing conditions, which are defined by the gas gravity (G r), flowing pressure (P f), and flowing temperature (T f) in degrees Fahrenheit The density of air at pressures above atmospheric is denoted as ρ a (in lbm/ft³), while ρ atm refers to the density of atmospheric air Additionally, ρ g indicates the density of gas or vapor within the differential pressure instrument, and ρ w represents the density of water in the manometer at temperatures other than 60 °F.
A.4 Water Manometer Gas Leg Correction Factor ( F am )
The factor F am corrects for the gas leg over water when a water manometer is used to calibrate a differential pressure instrument:
During the calibration of differential pressure instruments and water U-tube manometers using atmospheric air, the density of air at atmospheric pressure and 60 °F must be calculated using a specific equation.
By substituting local atmospheric pressure (P atm) for absolute pressure (P), using an absolute temperature (T) of 519.67 °R (60 °F), a molecular weight of air (Mr air) of 28.9625, an ideal relative density of air (G i) of 1.0, and a universal gas constant (R) of 1,545.35, we can establish a significant relationship in gas behavior.
The local atmospheric pressure can be determined using the following equations based on NOAA’s U.S Standard
(A.5) where elevation is the height above mean sea level, in ft;
P a is the atmospheric pressure at 60 °F, in psia.
(A.6) where elevation is the height above mean sea level, in m;
P a is the atmospheric pressure at 15 °C, in kPa.
The density of air at any given differential pressure (h wa ) above atmospheric pressure can then be represented by the following:
2 National Oceanic and Atmospheric Administration, U.S Standard Atmosphere, U.S Department of Commerce, National Technical Information Service, October 1976.
The density of water can be obtained from Table A.1 or calculated from the following Patterson and Morris water density equation (refer to API MPMS Ch 11.4.1):
G i is the ideal gas relative density; h wa is the differential pressure above atmospheric, in inches of water at 60 °F;
Mr is the molar mass of a gas, in lbm/lb-mol;
P is the absolute gas pressure, in psia;
P atm is the local atmospheric pressure, in psia;
R is the universal gas constant = 1,545.35 (lbf-ft)/(lb-mol-°R);
T is the absolute gas temperature, in °R;
T w is the temperature of water, in °C;
Z is the compressibility of a gas at P and T ;
The compressibility of air at a pressure of P atm plus the height of water (h wa) and a temperature of 519.67 °R is denoted as Z a, while the compressibility of air at P atm and the same temperature is represented as Z The temperature difference is defined as Δt = T w – T 0 The density of a gas is measured in lbm/ft³, with ρ a representing the density of air at pressures above atmospheric, and ρ atm indicating the density of atmospheric air The maximum density of water at temperature T 0 is ρ o, which is 999.97358 kg/m³ The density of water, ρ w, can be expressed as ρ w = ρ o [1–(AΔt B + Δt² + CΔt³ + DΔt⁴ + EΔt⁵)].
A.5 Water Manometer Temperature Correction Factor ( F wt )
The F wt correction factor adjusts for changes in water density in manometers at temperatures different from 60 °F It is essential to incorporate this correction in flow measurement calculations when a differential instrument is calibrated using a water manometer.
(A.9) where ρ w is the density of water in a manometer at a temperature other than 60 °F, in lbm/ft 3
A.6 Local Gravitational Correction Factor for Water Manometers ( F wl )