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Improve material balance by using proper flowmeter corrections

10.01.2011  |  Peramanu, S.,  Canadian Natural Resources Ltd., Calgary, CanadaWah, J. C. ,  Canadian Natural Resources Ltd., Calgary, Canada

Here are guidelines to increase accuracy for flow measurements


Process plants frequently encounter mass imbalances. These can be attributed to various factors, but often they lead back to inappropriate measuring devices, improper calibration, incorrect installation or incorrect interpretation of the measured flows. There are well-established guidelines available to ensure appropriate flowmeter selection based on the process conditions and control requirements.

Instrument vendors follow the industry standards and approved procedures for flowmeter calibration, calculating the calibration factors based on the data provided in the flowmeter specification datasheets. Engineering and construction service contractors often follow vendor guidelines and standard practices to correctly install the flowmeters. This means that most flowmeter installations, therefore, meet accepted project standards and specifications.

What’s the flowrate?

However, measured flow interpretation, normally a process or production engineer’s responsibility, is often done without proper directions or guidelines. Although it appears straightforward that the flowmeter measures the flowrate and the flowrate value is read from the display, significant error can be introduced if the flow measurement conditions are not understood and appropriate correction factors are not applied. Accurate stream flow interpretation and critical mass balance reconciliation require understanding flowmeter characteristics and their associated measurement uncertainties. This is of particular importance where mass balances may be used for highly sensitive process control operations, production accounting or government reporting on royalties and emissions.

This article provides a background on the importance of accurate measurements, a description of measurement errors and the role of uncertainties in mass balance and reconciliation. Flowmeter correction equations are derived for differential pressure flow, volumetric flow and mass flowmeters, and flow correction factors are provided for various units of measurements (UOM). Flowmeter uncertainty equations are derived for differential pressure flow, volumetric flow and mass flowmeters.


To achieve the most accurate flow measurement (minimum uncertainty), proper flow system operation and maintenance must be practiced so that meter accuracy capabilities are realized. Periodic maintenance, testing and recalibration are essential because the calibration will shift over time due to wear, damage or contamination.

The maintenance may be only a secondary-equipment calibration, a complete system mechanical inspection, an actual throughput test against some agreed-upon standards or any combination of these. The equipment used to test the meter, such as thermometers, dead-weight tests, pressure gauges, differential-pressure gauges, chromatographs and provers (used for throughput tests), must have accuracy certification and should be approved and agreed upon by the interested parties. Having operators who have had experience with similar metering systems also increases the calibration and test procedure confidence levels. Test equipment itself should be recertified periodically by the agency or manufacturer that originally certified the equipment.

Custody transfer operations.

In custody-transfer measurement, the measurement furnishes quantity and quality information that can be used as the basis for a change in ownership and/or a change in responsibility for materials.

Custody-transfer measurement is distinct from other measurement types because of the contractual nature of the meter. Custody-transfer metering may require accuracy of ± 0.1% or better, whereas control measurement may be accepted at a ± 2% accuracy and operational measurement may require a ± 5% accuracy. A high-integrity custody-transfer measurement system is a result of careful design based on the specific application requirements comprising fluid control, conditioning, metering, computation and a means of traceable site data validation.1, 2

Custody-transfer management involves the entire chain from the custody-transfer metering conceptualization to the final production or sale data reporting. For example, in the upstream oil and gas sector, measurement includes all intermediate steps such as measurement and sampling guidelines, operational procedures, data processing, data transmission and reconciliation, allocation or custody-transfer procedures. To solve the flow measurement equation, it is imperative that every equation parameter be well understood and represented.

A primary custody-transfer measurement consideration is to minimize flow variations by maintaining better flow control. For situations where this may not be possible, a meter with a wide-ranging flow capacity is needed. If a single meter with the required flow capacity to cover the intended operating range with minimum uncertainty does not exist, using multiple meters with some type of meter-switching control is required. Most meters operate with a specified uncertainty within the stated flow capacity limits that is typically from 25% to 95% of the flowmeter maximum capacity. For custody-transfer metering and critical control measurement, it is important to maintain the meter operation within the stated flow capacity limits.


Other than some operating problems and poor maintenance that may affect the measurement, the main cause of error is the fluid characteristics and errors in fluid density calculation. For gases, mixtures are more accurately measured if the stream has relatively constant composition. This allows specific PVT tests to be run, or data may be available for common mixtures from previous work. If the mixture is changing rapidly, a densitometer or a mass meter may be required to determine an accurate measurement.

During times when a custody-transfer meter is out of service or registering inaccurately, a procedure must be in place for measuring or estimating deliveries. This procedure may need to be in accordance with regulatory standards if the meter flow is used for regulatory reporting. An example of this is the recent USA EPA Greenhouse Gas Mandatory Reporting Rule issued Sept. 22, 2009. A typical accuracy limit from ± 0.5% to ± 2% may be used, but may be set closer or wider depending on the specific meter costs and measurement ability.

For custody-transfer meters, a prover system or master meter should be used for throughput testing and recalibration. The best throughput test can be run directly in series with a prover. The prover can come in many forms, but essentially it involves a basic volume that has been certified by a government or industrial group. Since most meters are not totally linear, tests may have to be run over the meter’s operating range to calculate the calibration factors dependent on the flow capacity.

Commercial mass balance software.

The characteristics and strengths of commercial mass-balance software may include:
• Graphically aided input that is user friendly and intuitive
• Interactive diagnostics and feedback on input errors
• Flexibility to select the measurement units desired by the user
• Flexibility for the user to select start and end times to perform the reconciliation
• Facility to construct mass-balance units based on plant configuration user input
• Reconciliation processes perform linear, nonlinear and inequality constraints on the measurement data to produce reconciled measurements and unmeasured flow estimates
• Algorithm for efficient iteration and fast convergence to the solution. Some algorithms may include the Monte Carlo method to generate sets of random values for measurement errors within a prescribed range (±%uncertainty) that are solved and iterated in the reconciliation algorithm
• Algorithm to ensure numerical robustness and prevent numerical runaway
• A reconciled mass balance at the processing unit level or group of processing units, progressing all the way up to the plant level
• Ability to reconcile total mass and selected component fractions at the same time
• Reporting tool to verify the present error in balance for each node and generate the customized reports.

Mass balance and reconciliation.

Data reconciliation3,4 improves process data accuracy by adjusting the measured values so that they satisfy the process constraints. The amount of adjustment made to the measurements is minimized since the random errors are expected to be small. Data reconciliation can be formulated by the following constrained weighted least-squares optimization problem:

Minimize the function (known as objective function):


Where n is the number of measurements, ∆M is the difference between the reconciled and measured values of measurements, i and si is the measurement i standard deviation. The value 1/s2 which is an inverse of variance (square of standard deviation) is the weight factor representing the accuracy of the respective measurements. Since a higher value of standard deviation implies that the measurement is less accurate, the above choice gives larger weights to more accurate measurements.

The above objective function minimization is subject to constraints:

fj(Mi 1 DMi) =0 j = 1,....m

Where f is the balance equation for the measurements (i = 1,....n) and m is the number of balance equations.

During the reconciliation process, the measurements containing systematic bias or gross errors are detected by comparing the difference between the reconciled and measured values to the measurement uncertainties.

If Abs(DMi/ Mi) > Uncertainty, measurement i has gross error.

The measurements containing gross errors are either eliminated or appropriately compensated for data reconciliation to be effective, as shown in Fig. 1. In this example, the reconciliation involves:
M1 = stream 1 cumulative measured mass flow over a period (a day, for example)
M2 = stream 2 cumulative measured mass flow over a period
M3 = stream 3 cumulative measured mass flow over a period
M4 = measured mass inventory gain or depletion over the same period


  Fig. 1. Flow reconciliation example for a storage tank. 

It is evident that because of measurement errors these measured quantities do not balance, namely:

M1 1 M2M3 1 M4

Therefore, reconciliation is required; adjustments to each of the measured quantities need to be made to obtain a mass balance:

(M1 1 DM1) 1 (M2 1 DM2) = (M3 1 DM3) 1 (M4 1 DM4)

where ∆Mi is an adjustment (+ve or –ve) for the measured quantity Mi.

It is evident that there are infinite sets of ∆Mi, each of which will give the desired mass balance, i.e., they will satisfy the equation above. Of the infinite sets of ∆Mi , the one particular set that corresponds to the least amount of total adjustment is required. This suggests that the problem to find this particular set of ∆Mi can be formulated as one that entails minimizing a function (usually referred to as objective function) subject to some constraints (mass-balance equations). For the simple example here, the problem of finding that particular set of ∆Mi can be formulated as:

Minimize the objective function:


Subject to the constraint:

(M1 1 DM1) 1 (M2 1 DM2) –

(M3 1 DM3) – (M4 1 DM4) = 0

Where si is the uncertainty associated with the instrument that gives the measured quantity, Mi.

The use of uncertainties for reconciliation can be explained with this example. Assume that a flowmeter M1 with percent uncertainty at 95% confidence level (%U95) as ±2% is reading 300 kg during a certain time period. Since the 95% confidence level corresponds to two standard deviations, 2s1, the standard deviation error, s1, for this measurement can be calculated as ±3 kg as:


Therefore, the weight factor, (1/s12), for measurement M1 in the objective function above is 1/9. Suppose if the reconciled flowrate for M1 is 307 kg, then the reconciled error (difference between the reconciled value and the measured value) is 7 kg. This value is greater than the 6 kg error (95% confidence) calculated. This means that measurement M1 has a gross error and should be eliminated or properly compensated for effective reconciliation.


Process industry flowmeters can be classified into three broad categories that include differential-pressure meters, actual volumetric flowmeters and mass flowmeters. The differential-pressure meters include orifice, venturi, nozzle, wedge, pitot tube and annubar; volumetric flowmeters include vortex, turbine, ultrasonic and magnetic; and mass flowmeters include Coriolis and thermal meters. The meter operating principles and flow equations are provided in Appendix A.

For any of these flowmeters, the vendor should make sure that the flowmeters measured outputs are in the UOM requested in the flowmeter specification datasheet. For this, the vendor calculates the conversion factor by using the design density data (or pressure, temperature and molecular weight data for gases) specified on the datasheet to output measured values in the desired UOM.

During process operation, the measured density (or P, T, MW and z for gases) values may not be the same as the values on the datasheet. Therefore, the measured flowrates should be corrected to account for the measured process conditions. The correction factors for various flowmeters using different UOM are provided in Table 1. The details of the flowmeter correction calculation are available in Appendix B.


Flow uncertainty equations. Uncertainty, U95, is a statistical statement of measurement accuracy that is useful in:
• Defining tolerances for reconciling measurements with concurrent gross-error detection and elimination
• Estimating accuracies when reporting to government on measurements that impact royalties and emissions
• Evaluating custody-transfer metering performance.

Uncertainty is a measurement process characteristic. It provides an estimate of the error band within which the true value for that measurement process must fall with high probability.5 It is based on the probability of 95% that is twice the standard deviation, 2s. The 95% confidence level for the estimated flowmeter uncertainty is in accordance with prudent statistical and engineering practice.

Flowmeter uncertainty is actually a function of both bias (systematic or gross error) and precision (random error), as shown in Fig. 2. Flowmeter part manufacturers follow rigorous testing and calibration to remove or randomize the measurement biases. In Canada, they follow the standards by Measurements Canada, and in the US, the test method follows the National Bureau of Standards (National Institute of Standards and Technology). The values used for the precision may be obtained from manufacturer’s specifications for the respective equipment provided that the values are adjusted to reflect operating conditions.


  Fig. 2. Bias and precision errors. 

To calculate the uncertainty values, the significance of each variable (parameter) in the flow calculation equation is examined and is related to flow measurement. It is assumed that the meter has been properly installed, operated and maintained. It is also assumed that the systematic equipment biases are randomized within the database, which means that variations in the equipment and laboratories will not impose any bias in the equations’ ability to represent reality.

For practical considerations, the pertinent variables are assumed to be independent to enable simpler uncertainty calculations. It was noted that the simplified uncertainty equations would provide very good uncertainty estimates.6 The mathematical relationships among the variables establish the sensitivity of the metered quantities to each of these variables. Each variable that influences the flow measurement uncertainty has a specific sensitivity coefficient. The uncertainty for a general equation Q = f(x1, x2,.....xN) can be derived analytically by partial differentiation based on propagation of uncertainty by the Taylor series.

Refer to Appendix C for derivation using the Taylor series. The uncertainty in Q can be given as:


This can be represented in a simpler form as:


where dQ/Q is the uncertainty in Q, Sx is the sensitivity coefficient associated with the variable and Ux is the variable uncertainty. The uncertainty equations are derived for differential pressure, volumetric and mass flowmeters in Appendix D using the flow equations representing the basic operating principle.


Uncertainty for orifice, venture or nozzle meter measuring in standard flow is given by:


The same equation above can be used for a wedge meter; however, the deviation in the equivalent diameter, d, for a wedge meter is calculated by using:




Uncertainity equations for vortex, turbinem ultrasonic and coriolis flowmeters are in Appendix D.

Measured flowrate correction.

An orifice meter is used in a refinery to measure the flowrate of liquid hydrocarbons and it is calibrated to indicate (readout) flowrate in standard volumetric flows. For example, design stream conditions indicated on the flowmeter specification datasheet are:
TD = Design temperature = 300°C
PD = Design pressure = 1,500 kPaa
rD_Std = Design standard density = 950 kg/m3 (normally obtained from process simulation)
rD = Design actual density = 750 kg/m3 (normally obtained from process simulation).

During actual operation, the measured conditions are:
TM = Measured temperature = 310°C
PM = Measured pressure = 1,500 kPaa
rM_Std = Measured standard density = 960 kg/m3 (measured in the laboratory using the sample)
rM = Measured actual density = 740 kg/m3 (measured, or calculated using an appropriate correlation).

For liquid flows, if the measured densities are not available at the actual operating conditions, the established correlations can be used. It should be noted that these correlations may result in some error in the density predictions.

Actual liquid hydrocarbon stream density can be estimated using the equation by Yawas:7


For C20 and heavier alkanes, the densities can be obtained using the method by Fisher:8


where SGm is specific gravity at measured temperature
SGr is the specific gravity at reference temperature
Tm is measured temperature in Kelvin
Tr is reference temperature in °C.

A method for calculating actual density using liquid critical properties is given by Noor:9


where rm = Density at measured temperature in kg/m3,
M = Molecular weight,
VC = Critical volume in m3/kg
Tm = Measured temperature in Kelvin
TC = Critical temperature in Kelvin.

From Table 1, the correction factor for the orifice meter with indicated (readout) liquid flowrate at standard conditions is given by:


and the corrected flowrate at standard conditions is given by:

QStdVol_Corr = QStdVol_Meas 3 Correction Factor

If the flowmeter indicated (readout) flow is 600 std. m3/d, then the corrected flowrate at standard condition is:


= 589.8 Std m3/d ~ 590 std. m3/d

Uncertainty calculation.

A 3-in. orifice meter run with a b ratio of 0.6 is selected for the previous liquid hydrocarbon flow measurement example at a static pressure of 1,500 kPaa and flowing temperature of 310°C. Differential pressure recorded for the flow is 25 kPa and the flowrate is 590 std. m3/h.

The variable sensitivity coefficients can be calculated using the orifice uncertainty equation:


The uncertainty values for the variables dx/x at 95% confidence level, U95, can be obtained from industry standards and procedures (AGA, API, ASME, ASTM) and/or manufactures’ specifications for the equipment or parts. For each variable, the uncertainty listed in Table 2 represents random errors only, which are obtained from AGA RP-3-1.


Based on the calculations, the standard volumetric flow measurement uncertainty at 95% confidence level is ± 0.76%. For mass flow measurement uncertainty, the standard density variable, rStd, in the above equation is excluded, which gives the % U95 value of ± 0.58%.


The operating principles and flowmeter equations are listed in this appendix for the flowmeters as shown in Fig. 3. 


  Fig. 3. Examples of various flowmeters used by industry. 

Differential pressure flowmeters.

The flowmeters that measure differential pressure to calculate the flowrates can be classified as differential pressure flowmeters.

Orifice, venturi and nozzle flowmeters. For fluid flow in an orifice, venturi or nozzle flowmeter, the actual volumetric flowrate can be given as:10


where d is the orifice diameter for an orifice meter or throat diameter for venturi and nozzle meters,
P1 = Pressure at the upstream pressure tap,
P2 = Pressure at the downstream pressure tap
r1 = Density at P1 pressure condition.
Cd = Discharge coefficient to account for frictional losses (kinetic energy into heat) due to viscosity and turbulence effects.
Eu is the velocity approach factor that relates the flowing fluid velocity in the meter approach section (upstream meter tube) to the orifice/throat fluid velocity:


where b = d / D is the orifice bore (or throat for the venturi and nozzle) to pipe inner-diameter ratio.

Y is the expansion factor to account for the gas compressibility that is given by:


where k is specific heat ratio CP/CV. For b less than 0.25, b4 value approaches zero in the equation.

Pitot tube or annubar flowmeters (for velocity less than 30% of sonic velocity). For fluid flow in a Pitot tube flowmeter, the actual volumetric flowrate can be given as:


Where: K = Instrument coefficient that is usually determined through calibration,
D = Pipe inside diameter
∆P = Pressure drop measured by the Pitot tube, which is the difference between the total (stagnation) pressure, Pt, and the static pressure, Ps.

Wedge flowmeter (used for liquid flows only). For liquid flow in a wedge flowmeter, actual volumetric flowrate can be calculated using the orifice equation:


where d is equivalent orifice diameter that is calculated using equivalent beta ratio:


where H = wedge segment opening height,
D = Pipe inside diameter,
∆P = Pressure drop across the orifice
rAct = Liquid density at actual temperature and pressure conditions, T, P.
Cd is the wedge meter discharge coefficient to account for frictional losses (kinetic energy into heat) due to viscosity and turbulence effects.
Eu is the velocity approach factor that relates the flowing fluid velocity in the wedge meter approach section (upstream meter tube) to the fluid velocity in the wedge section.


where b is d/D which is equivalent orifice to pipe inner diameter ratio.

Volumetric flowmeters.

The flowmeters that directly interpret the actual volumetric flow from other measured parameters are called volumetric flowmeters. To interpret the velocity, vortex meters use vortex shedding frequency; ultrasonic meters use sound transit time; and magnetic meters use voltage induced in the fluid (conductive) flowing through an imposed magnetic field.

Vortex flowmeter. A vortex flowmeter measures the volumetric flowrate by using the vortex shedding frequency caused by a flow barrier.11

Strouhal number, S, is related to vortex shedding frequency by S = fw / u

where f = Vortex shedding frequency that depends on flow velocity, fluid viscosity and flow barrier dimensions (bluff, which is either a cylinder or a square column) used to create vortex
w = Flow barrier width (bluff)
u = Fluid velocity in the bluff section.

Actual volumetric flowrate can be given by:


where D is the pipe inner diameter

and B is the blockage factor that is defined as the pipe bore area less the bluff body blockage area, divided by the pipe full bore area:


where K factor is used to compensate for the pipe flow nonuniform profile in industrial applications. Combining the above equations the actual flowrate is given as:


The Strouhal number, S, is about constant across a wide Reynolds number range of (102–107). The S value depends on the bluff width to the pipe inner diameter ratio. S = 0.18 for w/D = 0.1; S = 0.26 for w/D = 0.3; and S = 0.44 for w/D = 0.5.

Turbine flowmeter. A turbine flowmeter measures the volumetric flow by counting the rotor revolutions (rotor angular velocity) that turns in proportion to the flow velocity.12–14 The equation for a turbine meter can be given as:

utan(theta) = Kr(omega)

where u = Incoming flow velocity,
Theta = Angle between the pipe axis (incoming flow direction) and the turbine blades,
r = Root-mean-square value of the blade inner and outer radii to represent the average radius,
K = Instrument factor to compensate velocity loss (nonidealities) due to rotor blade design and omega is the rotor angular velocity.


where ro = Blade radius outer edge and ri = Radius blade root.

Actual volumetric flowrate can be given by:


Ultrasonic flowmeter. An ultrasonic flowmeter measures the volumetric flow by using sound pulse transit time in the flow medium caused by doplar effect.15–17 A typical ultrasonic flowmeter (transit-time flow measurement) system utilizes two ultrasonic transducers that function as both transmitter and receiver. The flowmeter operates by alternately transmitting and receiving a sound energy burst between the two transducers and measuring the transit time that it takes for sound to travel between the two transducers. The difference in the transit time measured is directly and related to the liquid velocity in the pipe.

If tD is the sound pulse transit-time (or time-of-flight) traveling from the upstream transducer to the downstream transducer, and tU is the transit-time from the opposite direction, the equations can be given as:


where theta = Angle between the transducer axis to the flow direction,
c = Sound speed in the liquid,
D = Pipe inside diameter
u = Flow velocity averaged over the sound path. Solving the above equations leads to:


where tf = (tU – tD) / (tUtD) is the transit-time function and K is the instrument factor determined through calibration. Therefore, by accurately measuring the upstream and downstream transit-times, tU and tD, the flow velocity, u, can be obtained.

Actual volumetric flowrate is calculated as:


where A is the pipe inner cross-section area.

Magnetic flowmeter. Magnetic flowmeter operation is based on Faraday’s Law that states that the voltage induced across any conductor as it moves at right angles through a magnetic field is proportional to the conductor velocity.18 To apply this principle the fluid being measured must be electrically conductive.

The voltage, E, generated in a conductor is given by:

E (alpha) BLu

E = Voltage generated in a conductor
B = Magnetic field strength perpendicular to the flow direction
L = Distance between the electrodes (usually equal to pipe inside diameter in most construction)
u = Conductor velocity.

The fluid velocity can be given by:


where K is the instrument coefficient that is usually determined through calibration.

Subsequently, the actual volumetric flow rate is calculated as:


where A is the pipe inner cross-section area and D is the pipe inside diameter.

Mass flowmeters.

A coriolis flowmeter directly measures the mass flow based on the inertial forces exerted on the tube vibrations.19–21 When an oscillating excitation force is applied to the tube, causing it to vibrate, the fluid flowing through the tube will induce a twist (or rotation) to the tube because of the Coriolis acceleration acting in opposite directions on either side of the applied force.

In a U-tube coriolis meter, the flow is guided into the U shaped tube that is vibrated using an actuator. The vibration is commonly introduced by electric coils and measured by magnetic sensors. When the tube is moving upward during the first half of a cycle, the fluid flowing into the meter resists being forced up by pushing down on the tube. On the opposite side, the liquid flowing out of the meter resists having its vertical motion decreased by pushing up on the tube. This action causes the tube to twist. When the tube is moving downward during the second half of the vibration cycle, it twists in the opposite direction. The two vibrations are shifted in phase (time lag) with respect to each other, and the degree of phase-shift is directly affected by the mass passing through the tube.

A U-shaped Coriolis flowmeter mass flow is given as:


where Ku = Tube stiffness,
K = A shape-dependent factor
L = Width,
t = Time lag, 
omega = Vibration frequency
Iu = Tube inertia that includes the tube fluid mass. The expression can be simplified as:




is the natural frequency of the U-shaped tube system.

Thermal flowmeter.

A thermal flowmeter measures the mass flow based on heat absorption. As molecules of a moving fluid come into contact with a heat source, they absorb heat and cool the source. At increased flowrates, more molecules come into contact with the heat source absorbing even more heat. The heat dissipated from the source in this manner is proportional to the number of molecules of a particular gas (its mass), the gas thermal characteristics, and its flow characteristics. The mass flow of a thermal mass flowmeter can be given as:

QMass = K × ΔH

where K is the instrument coefficient which is usually determined through calibration, and ∆H is the amount of heat dissipated from the heat source.


Flow correction for differential pressure flowmeters.

Actual flow for differential pressure meters is calculated using:


Assuming incompressible fluid in the range of interest (constant fluid density between the flowmeter pressure taps) the actual flowrate measured by the flowmeter can be given as:


Liquid flowmeter (actual volumetric flow measurement—act. m3/h). The equation for corrected flow is:


Liquid flowmeter (standard volumetric flow measurement—std. m3/h). Standard volumetric flow is calculated using:


The equation for corrected flow is:


Liquid flowmeter (mass flow measurement—kg/h). Mass flow is calculated using:


The equation for corrected flow is given by:


Gas flowmeter (actual volumetric flow measurement—act. m3/h). The equation for corrected flow is:


Gas flowmeter (standard volumetric flow measurement—std. m3/h). Standard flowrate is:


The equation for corrected flow is given by:


Gas/steam flowmeter (mass flow measurement—kg/h). The equation for mass flow is:


The equation for corrected flow is:


Flow correction for volumetric flowmeters.

Actual flow measured by the volumetric flow is:
Vortex meter: QActVol fv
Turbine meter: QActVol omega
Ultrasonic meter: QActVol f (TUP, TDown)
Magnetic meter: QActVol E

Liquid flowmeter (actual volumetric flow measurement—act. m3/h). The equation for corrected flow is:


Liquid flowmeter (standard volumetric flow measurement—std. m3/h). Standard volumetric flow is calculated using:


The equation for corrected flow is:


Liquid flowmeter (mass flow measurement—kg/h). Mass flow is calculated using:


The equation for corrected flow is:


Gas flowmeter (actual volumetric flow measurement—act. m3/h). The equation for corrected flow is:


Gas flowmeter (standard volumetric flow measurement—std. m3/h). Standard volumetric flow is calculated using:


The equation for corrected flow is:


Gas/steam flowmeter (mass flow measurement—kg/h). The equation for mass flow is:


The equation for corrected flow is:


Flow correction for mass flowmeters.

Mass flowrates for the mass flowmeters is:
Coriolis meter: QMass t
Thermal meter: QMass DH

Liquid flowmeter (actual volumetric flow—act. m3/h). Actual volumetric flow is calculated using:


The equation for corrected flow is:


If the coriolis meter has an integral density measurement and uses the operating density (or flowing density) instead of design density, rD , in the above equations, then correction factor is equal to 1.

Liquid flowmeter (standard volumetric flow—std. m3/h). Standard volumetric flow is calculated using:


The equation for corrected flow is:


Liquid flowmeter (mass flowmeter—kg/h). Equation for corrected flow is:

QMass_Corr = QMass_Meas

The correction factor required for correcting the flowmeter reading is equal to 1.

Gas flowmeter (actual volumetric flow measurement—act. m3/h). Actual volumetric flow is calculated using:


The equation for corrected flow is:


If the coriolis meter has an integral density measurement and uses the operating density (or flowing density) in the above equations instead of design density, then correction factor is equal to 1.

Gas flowmeter (standard volumetric flow measurement—std. m3/h). Standard volumetric flow is:


The equation for corrected flow is:


Gas flowmeter (mass flowmeter—kg/h). The equation for corrected flow is:

QMass_Corr = QMass_Meas


The sensitivity coefficient associated with each flow variable is derived analytically by partial differentiation based on uncertainty propagation by the Taylor series [ASME-MFC-2M, 1983]. The relationship of dependent variable Q with independent variables x1, x2,..., xN can be expressed in a general form as:

Q = f (x1, x2,..., xN)

Using the Taylor series the uncertainty in Q can be expressed as:


Squaring on both sides, the equation can be written as:


Given that the variables are uncorrelated (independent), i.e., (delta)xi (delta)xj = 0 (i j), all the values of the summation in the above equation will be zero. Therefore, the uncertainty in Q can be given as:


which can be represented in a simpler form as:


where (delta)Q/Q is the uncertainty in Q, Sx is the sensitivity coefficient associated with the variable, and Ux is the uncertainty of the variable.


Vortex flowmeter uncertainty is:


Turbine flowmeter uncertainty is:


Ultrasonic flowmeter uncertainty is:


Coriolis flowmeter uncertainty is:



Subscript D­—Design values from flowmeter specification datasheet
Subscript M—Measured values during plant operation
M—Molecular weight
P—Pressure, kPaa
P—Pressure drop, kPa
QMass—Mass flowrate, kg/h
QActVol —Volumetric flowrate at actual operating conditions, act. m3/h
QStdVol —Volumetric flowrate at standard conditions (15°C and 1 atm.), std. m3/h
QMass_Meas—Measured (readout) mass flowrate from the flowmeter, kg/h
QActVol_Meas—Measured actual volumetric flowrate (readout) from the flowmeter, act. m3/h
QStdVol_Meas—Measured standard volumetric flowrate (readout) from the flowmeter, std. m3/h
QMass_Corr —Corrected mass flowrate for the flowmeter, kg/h
QActVol_Corr —Corrected actual volumetric flowrate for the flowmeter, act. m3/h
QStdVol_Corr —Corrected standard volumetric flowrate for the flowmeter, std. m3/h
R—Universal gas constant (R = 8.3145 kPaa m3/kmol K)
S—Sensitivity coefficient
T—Temperature, K
U95 —Uncertainty at 95% confidence level
Z—Compressibility factor for gases (Z = 1 for ideal gas)
s—Standard deviation
r—Mass density, kg/m3
rD —Design density at actual conditions from flowmeter specification datasheet
rM —Measured (or calculated) density at actual conditions during operation
rD_Std —Design density at standard conditions from flowmeter specification datasheet
rM_Std —Measured density at standard conditions during operation


The authors thank their colleague Ken Fernie, P.Eng., for review and valuable comments on custody transfer metering, and Andrew Nelson, Production management manager from Matrikon Inc., his for review and valuable input on flow meter uncertainties.


1 Spitzer, D. W., Flow Measurement: Practical Guides for Measurement and Control, 2nd Edition, Research Triangle Park, NC: ISA, 2001.
2 Upp, E. L. and P. J. LaNasa, Fluid Flow Measurement: A Practical Guide to Accurate Flow Measurement, Gulf Professional Publishing, 2nd Edition, 2002.
3 Romagnoli, J. A. and M. C. Sanchez, “Data Processing and Reconciliation for Chemical Process Operations,” Process Systems Engineering, Vol. 2, Academic Press, 1st Edition, 1999.
4 Ozyurt, D. B. and R. W. Pike, “Theory and Practices of Simultaneous Data Reconciliation and Gross Error Detection for Chemical Processes,” Computers and Chemical Engineering, 28, pp. 381–402, 2004.
5 ASME MFC-2M, Measurement Uncertainty for Fluid Flow in Closed Conduits, American National Standard, 1983 (Revised 2006).
6 AGA RP-3-1, Orifice Metering of Natural Gas and Other Related Hydrocarbon Fluids Part 1—General Equations and Uncertainty Guidelines, American Gas Association, June 2003. (API MPMS 14.3-1; ANSI/API 2530-91 Part 1; Gas Processors Association GPA 8185 Part 1).
7 Yawas, C. L., et al, “Equation for Liquid Density,” Hydrocarbon Processing, Vol. 70, No 1, January 1991, pp. 103–106.
8 Fisher, C. H., “How to Predict n-Alkane Densities,” Chemical Engineering, Vol. 96, No 10, pp. 195, October 1989.
9 Noor, A., “Quick Estimate of Liquid Densities,” Chemical Engineering, Vol. 88, No. 7, pp. 111, 6th April 1981.
10 ASME MFC-3M, Measurement of Fluid Flow in Pipes Using Orifice, Nozzle and Venturi, American National Standard, 2004.
11 ASME MFC-6M, Measurement of Fluid Flow in Pipes using Vortex Flowmeters, American National Standard, 1998 (Revised 2005).
12 AGA RP-7, Measurement of Natural Gas by Turbine Meters, American Gas Association, February 2006.
13 API MPMS-5.3, Measurement of Liquid Hydrocarbons by Turbine Meters, American Petroleum Institute, September 2000.
14 ASME MFC-4M, Measurement of Gas Flow by Turbine Meters, American National Standard, 1986 (Revised 2008).
15 AGA RP-9, Measurement of Gas by Multipath Ultrasonic Meters, American Gas Association, April 2007.
16 API MPMS-5.8, Measurement of Liquid Hydrocarbons by Ultrasonic Flow Meters Using Transit Time Technology, American Petroleum Institute, February 2005.
17 ASME MFC-5M, Measurement of Liquid Flow in Closed Conduits Using Transit-Time Ultrasonic Flowmeters, American National Standard, 1985 (Revised 2006).
18 ASME MFC-16M, Measurement of Liquid Flow in Closed Conduits with Electromagnetic Flowmeters, American National Standard, 1995 (Revised 2006).
19 AGA RP-11, Measurement of Natural Gas by Coriolis Meter, American Gas Association, January 2003.
20 API MPMS-5.6, Measurement of Liquid Hydrocarbons by Coriolis Meters, American Petroleum Institute, October 2002.
21 ASME MFC-11M, Measurement of Fluid Flow by Means of Coriolis Mass Flowmeters, American National Standard, 1989 (Revised 2003).

The authors 


Subodhsen Peramanu has more than 15 years of experience in conceptual, front-end design and detailed engineering of upgrading and refining processes. He has authored papers on topics including hydrogen separation and economics, bitumen characterization, and asphaltene solubility and reversibility. Dr. Peramanu was involved in commissioning and start-up of CNRL Horizon Upgrader and is working with CNRL Thermal Team as a senior engineering specialist on in-situ oil recovery. He holds a BChemEng degree in chemical engineering from Institute of Chemical Technology (formerly UDCT), Mumbai, MTech degree from Indian Institute Technology, Kanpur and PhD from University of Calgary. 


Juon Wah’s career in process engineering spans more than 30 years and covers conceptual design, FEED, EPC and detailed process and equipment design of major projects in refining, bitumen upgrading and oil and gas production facilities. At present, Mr. Wah is a consultant on process design and plant operations. At the time of writing, he was working on an expansion project for the Horizon Upgrading complex of CNRL. Mr. Wah holds a BSc degree in chemical engineering from the University of Birmingham, UK, and a Diplôme d’Ingénieur in chemical engineering and petroleum refining from the IFP, France.  

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Anatoliy Pavlov

I would like to suggest use of a new heat-type (no known Thermal Mass) fast-acting gas-liquid MASS FLOW METER for immersion in unsteady flow. Details in Email: pavlov__ag@mail.ru Anatoliy Pavlov Independent inventor.

C. Voigt

Dear Sir,

could you please sent me through the names of the standards which have been used to calculate the correction factor for gas flow at std m3/hr in table 1? Thank you very much in advance,

C. Voigt


Dear Sir,
Please let me know the principle of correction factor applied in the measurement of correction factor and the calculation of correction factor for volumentric flow through orifice meter.
With Regards

Richard von Brecht

This is an amazing article. to me. I'm unfamiliar with some of the nomenclature. I'd be interested seeing an expanded version of your paper.

Muhammad FarhatUllah

It is really handy and valuable article.Thanks

William Blanco

To both of you, excellent and valuable work

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