Flow Compensation of an Orifice Meter
Background
The calculations that underlie flow measurements typically assume that the fluid being measured is at the same conditions as the fluid used in the meter’s design. Flow meter compensation is the process of adjusting a measured flow for changes in flowing conditions (e.g. fluid temperature, pressure, and composition) to improve accuracy.
Flow meter compensation factors vary with the meter service, flow units, and meter type. As an example, we’ll consider the following case,
- orifice meter (see Figure 1),
- vapor service, and,
- configured to indicate mass flow.
Figure 1: Orifice Meter
Flow Calculation
The mass flowrate of a vapor through an orifice meter can be described by the following,
{\dot{m}}={N}{C_d}{E_v}{Y}{d^2}\sqrt{\rho}\sqrt{\Delta{P}} (1)
where,
\dot{m} is the vapor mass flowrate
N is a units conversion factor
C_d is the discharge coefficient
E_v is the velocity approach factor
Y is the vapor expansion factor
d is the orifice bore size
\rho is the vapor density
\Delta{P} is the pressure drop across the orifice, {P_1}-{P_2}
Orifice meters are designed by specifying typical operating conditions. If we consider that the terms in Equation (1) change little over the range of operating conditions for the meter they can be considered constant and we can calculate a maximum mass flowrate as,
{\dot{m}_{max}={N}{C_d}{E_v}{Y}{d^2}\sqrt{\rho_d}\sqrt{\Delta{P_{max}}}} (2)
and, Equation (1) becomes simply,
\dot{m}_{d} = \dot{m}_{max}\sqrt{\dfrac{\Delta{P}}{\Delta{P}_{max}}} (3)
where,
\dot{m}_{max} is the maximum mass flowrate (i.e. flowrate at full-scale pressure drop)
\rho_d is the vapor density, design conditions
\dot{m}_d is the mass flowrate, design conditions
\Delta{P} is the pressure drop
\Delta{P_{max}} is the full-scale pressure drop
Equation (3) is then implemented online but assumes that the vapor density differs little from the design value. Compensation for deviations in fluid properties from design to flowing conditions is described below.
Flow Compensation
Flow meters don’t operate at the conditions used for meter design. If suitable measurements are available, a compensation factor can be applied to compensate for changes in flowing density (i.e. fluids not at density \rho_d in Equation (2)) and improve meter accuracy.
The density of a vapor can be expressed as,
\rho=M\dfrac{ZRT}{P} (4)
where,
\rho is vapor density
M is molecular weight
Z is ideal gas compressibility factor
R is the ideal gas constant
T is temperature (absolute)
P is pressure (absolute)
The density at flowing conditions can then be calculated as a deviation from design conditions as,
\rho_f=\rho_d\dfrac{M_f}{M_d}\dfrac{P_f}{P_d}\dfrac{Z_d}{Z_f}\dfrac{T_d}{T_f} (5)
where,
\rho_f is the vapor density, flowing conditions
\rho_d is the vapor density, design conditions
M_f is molecular weight, flowing conditions
M_d is molecular weight, design conditions
P_f is operating pressure (absolute), flowing conditions
P_d is operating pressure (absolute), design conditions
T_f is operating temperature (absolute), flowing conditions
T_d is operating temperature (absolute), design conditions
Z_f is ideal gas compressibility factor, flowing conditions
Z_d is ideal gas compressibility factor, design conditions
Therefore, for orifice meters in vapor service configured to indicate mass flow, the compensated vapor mass flowrate is,
\dot{m}_f={K_f}{\dot{m}_d} (6)
K_f=\sqrt{\dfrac{M_f}{M_d}\dfrac{P_f}{P_d}\dfrac{Z_d}{Z_f}\dfrac{T_d}{T_f}} (7)
where,
\dot{m}_f is the mass flowrate, flowing conditions
\dot{m}_d is the mass flowrate, design conditions
K_f is the flow meter compensation factor
Recommendation
Flow meter compensation should be implemented to improve meter accuracy. This is especially true for meters that do not always operate at or near their design conditions. Measurements include fluid temperature, pressure, and composition but intermittent lab analysis and online estimators can also be used.
An extensive list of flow meter compensation factors (K_f) for various flow units and meter types in both liquid and vapor service is given in Table 1.
Table 1: Flow Meter Compensation Factors, K_f
where,
\bar\rho_d is the standard liquid density, design conditions
\bar\rho_f is the standard liquid density, flowing conditions
Note that there are eight (8) situations when flow compensation is never required (K_f=1.0).