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2σ₁=%U₉₅

100 × M₁=±2

100×300 = ±6 kg (95% confidence error)

Therefore, the weight factor, (1/␴12), 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.

FLOW CORRECTIONS

Process industry flowmeters can be classified into three broad categories that include differential-pressure meters, actual volu-metric 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 statisti-cal 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, 2␴.

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 (sys-tematic 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.

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 equip-ment 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 cal-culations. It was noted that the simplified uncertainty equations would provide very good uncertainty estimates.⁶ The mathemati-cal 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 differen-tiation 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 ␦Q/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.

FLOWMETER UNCERTAINTY

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:

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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:

T_D = Design temperature = 300°C PD = Design pressure = 1,500 kPaa

␳D_Std = Design standard density = 950 kg/m³ (normally obtained from process simulation)

␳D = Design actual density = 750 kg/m³ (normally obtained from process simulation).

During actual operation, the measured conditions are:

T_M = Measured temperature = 310°C P_M = Measured pressure = 1,500 kPaa

␳M_Std = Measured standard density = 960 kg/m³ (measured in the laboratory using the sample)

␳M = 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

SG_m= (SG_r)^{2−0.0011×(T}_m^−T_r⁾

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

ρ_m = M

V_C

(

3.964 −1.957T^mT_C

)

where ␳m = Density at measured temperature in kg/m³, M = Molecular weight,

V_C = Critical volume in m³/kg T_m = 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:

Correction Factor = ρ_M ρ_D ^×

Magnetic field (B) L

Examples of various flowmeters used by industry.

FIG. 3

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and the corrected flowrate at standard conditions is given by:

QStdVol_Corr = QStdVol_Meas ⫻ 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 ^600×

740 750×950

960

Uncertainty calculation. A 3-in. orifice meter run with a ␤ ratio of 0.6 is selected for the previous liquid hydrocarbon flow measurement example at a static pressure of 1,500 kPaa and flow-ing 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 ␦x/x at 95% confi-dence 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 mea-surement uncertainty at 95% confidence level is ± 0.76%. For mass flow measurement uncertainty, the standard density variable,

␳Std, in the above equation is excluded, which gives the % U95

value of ± 0.58%.

APPENDIX A

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

Differential pressure flowmeters. The flowmeters that measure differential pressure to calculate the flowrates can be clas-sified 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 diam-eter for venturi and nozzle mdiam-eters,

P1 = Pressure at the upstream pressure tap, P2 = Pressure at the downstream pressure tap

␳1 = 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:

E_u = 1 1−β⁴

where ␤ = 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: 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

ΔP = 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,

ΔP = Pressure drop across the orifice

␳Act = Liquid density at actual temperature and pressure condi-tions, 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.

E_u = 1 1−β⁴

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

Volumetric flowmeters. The flowmeters that directly inter-pret the actual volumetric flow from other measured parameters are called volumetric flowmeters. To interpret the velocity, vortex

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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 volumet-ric 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 veloc-ity, 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:

Q_ActVol= Au = π

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 nonuni-form profile in industrial applications. Combining the above equations the actual flowrate is given as:

Q_ActVol= fπD³

The Strouhal number, S, is about constant across a wide Reyn-olds 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 volu-metric 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 = Kr 

where u = Incoming flow velocity,

 = 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 (nonideali-ties) due to rotor blade design and  is the rotor angular velocity.

r = r_o²+ r_i² 2

where r_o = Blade radius outer edge and r_i = Radius blade root.

Actual volumetric flowrate can be given by:

Q_ActVol= Au = π

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:



t_D Linear distance between transducers (L) Net sound velocity along flow direction

(D / sin ) c u cos t_U Linear distance between transducers (L)

Net sound velocity opposite flow direction

(D / sin ) c u cos where  = 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:

is the instrument factor determined through calibration. There-fore, 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:

Q_ActVol= Au = π

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  BLu where:

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:

u K E

= BL

where K is the instrument coefficient that is usually deter-mined through calibration.

Subsequently, the actual volumetric flow rate is calculated as:

Q_ActVol= Au = π

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

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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 flow-ing out of the meter resists havflow-ing 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:

Q_Mass=

(

K_u− I_uω²

)

τ 2KL² where Ku = Tube stiffness,

K = A shape-dependent factor L = Width,

␶ = Time lag,

␻ = Vibration frequency

Iu = Tube inertia that includes the tube fluid mass. The expres-sion can be simplified as:

Q_Mass=

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 ⫻ ⌬H

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

ACKNOWLEDGMENTS

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

LITERATURE CITED

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).

Appendices B–D can be found at HydrocarbonProcssing.com.

Subodhsen Peramanu has more than 15 years of experi-ence 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 character-ization, 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 pro-cess and equipment design of major projects in refining, bitumen upgrading and oil and gas production facilities. At present, Mr.

Juon Wah’s career in process engineering spans more than 30 years and covers conceptual design, FEED, EPC and detailed pro-cess and equipment design of major projects in refining, bitumen upgrading and oil and gas production facilities. At present, Mr.

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