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 wellestablished 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.
METERING APPLICATIONS
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
secondaryequipment calibration, a complete system mechanical
inspection, an actual throughput test against some agreedupon
standards or any combination of these. The equipment used to
test the meter, such as thermometers, deadweight tests,
pressure gauges, differentialpressure 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 custodytransfer 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.
Custodytransfer measurement is distinct from other
measurement types because of the contractual nature of the
meter. Custodytransfer 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 highintegrity
custodytransfer 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}
Custodytransfer management involves the entire chain from
the custodytransfer 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 custodytransfer procedures. To
solve the flow measurement equation, it is imperative that
every equation parameter be well understood and
represented.
A primary custodytransfer measurement consideration is to
minimize flow variations by maintaining better flow control.
For situations where this may not be possible, a meter with a
wideranging 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 meterswitching 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 custodytransfer
metering and critical control measurement, it is important to
maintain the meter operation within the stated flow capacity
limits.
Errors.
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 custodytransfer 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 custodytransfer 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 massbalance
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 massbalance 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 reconciliation^{3,4} 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 leastsquares 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 s_{i}
is the measurement i standard deviation. The value
1/s^{2} 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:
f_{j}(M_{i} 1
DM_{i}) =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(DM_{i}/ M_{i}) >
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:
M_{1} = stream 1 cumulative measured mass flow
over a period (a day, for example)
M_{2} = stream 2 cumulative measured mass flow
over a period
M_{3} = stream 3 cumulative measured mass flow
over a period
M_{4} = 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:
M_{1} 1 M_{2} ﬁ
M_{3} 1 M_{4}
Therefore, reconciliation is required; adjustments to each
of the measured quantities need to be made to obtain a mass
balance:
(M_{1} 1 DM_{1}) 1
(M_{2} 1 DM_{2}) =
(M_{3} 1 DM_{3}) 1
(M_{4} 1 DM_{4})
where ∆M_{i} is an adjustment
(+ve or –ve) for the measured quantity
M_{i}.
It is evident that there are infinite sets of
∆M_{i}, each of which will give the
desired mass balance, i.e., they will satisfy the equation
above. Of the infinite sets of ∆M_{i} ,
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
∆M_{i} can be formulated as one that
entails minimizing a function (usually referred to as
objective function) subject to some constraints
(massbalance equations). For the simple example here, the
problem of finding that particular set of
∆M_{i} can be formulated as:
Minimize the objective function:
Subject to the constraint:
(M_{1} 1 DM_{1}) 1
(M_{2} 1 DM_{2}) –
(M_{3} 1 DM_{3}) –
(M_{4} 1 DM_{4}) = 0
Where s_{i} is the uncertainty associated with the
instrument that gives the measured quantity,
M_{i}.
The use of uncertainties for reconciliation can be explained
with this example. Assume that a flowmeter
M_{1} with percent uncertainty at 95%
confidence level (%U_{95}) as ±2% is reading 300
kg during a certain time period. Since the 95% confidence level
corresponds to two standard deviations, 2s_{1}, the
standard deviation error, s_{1}, for this measurement
can be calculated as ±3 kg as:
Therefore, the weight factor, (1/s_{1}^{2}),
for measurement M_{1} in the objective
function above is 1/9. Suppose if the reconciled flowrate for
M_{1} 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
M_{1} 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 differentialpressure meters,
actual volumetric flowmeters and mass flowmeters. The
differentialpressure 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,
U_{95}, is a statistical statement of
measurement accuracy that is useful in:
• Defining tolerances for reconciling measurements
with concurrent grosserror detection and elimination
• Estimating accuracies when reporting to government
on measurements that impact royalties and emissions
• Evaluating custodytransfer 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(x_{1}, x_{2},.....x_{N}) 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,
S_{x} is the sensitivity coefficient
associated with the variable and U_{x} 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:
where:
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
P_{D} = Design pressure = 1,500 kPaa
r_{D_Std} = Design standard density = 950
kg/m^{3} (normally obtained from process
simulation)
r_{D} = Design actual density = 750
kg/m^{3} (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
r_{M_Std} = Measured standard density = 960
kg/m^{3} (measured in the laboratory using the
sample)
r_{M} = Measured actual density = 740
kg/m^{3} (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 C_{20} and heavier alkanes, the densities can be
obtained using the method by Fisher:^{8
}
^{
}
where SG_{m} is specific gravity at
measured temperature
SG_{r} is the specific gravity at reference
temperature
T_{m} is measured temperature in Kelvin
T_{r} is reference temperature in °C.
A method for calculating actual density using liquid
critical properties is given by Noor:^{9
}
^{
}
where r_{m} = Density at measured
temperature in kg/m^{3},
M = Molecular weight,
V_{C} = Critical volume in
m^{3}/kg
T_{m} = Measured temperature in Kelvin
T_{C} = 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:
Q_{StdVol_Corr} = Q_{StdVol_Meas} 3
Correction Factor
If the flowmeter indicated (readout) flow is 600 std.
m^{3}/d, then the corrected flowrate at standard
condition is:
= 589.8 Std m^{3}/d ~ 590 std. m^{3}/d
Uncertainty calculation.
A 3in. 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. m^{3}/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, U_{95}, 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 RP31.
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, r_{Std}, in the above
equation is excluded, which gives the % U_{95}
_{}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.

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,
P_{1} = Pressure at the upstream pressure
tap,
P_{2} = Pressure at the downstream pressure
tap
r_{1} = Density at P_{1} pressure
condition.
C_{d} = Discharge coefficient to account for
frictional losses (kinetic energy into heat) due to viscosity
and turbulence effects.
E_{u} 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 innerdiameter ratio.
Y is the expansion factor to account for the
gas compressibility that is given by:
where k is specific heat ratio
C_{P}/C_{V}. For b less than 0.25,
b^{4} 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, P_{t}, and the static pressure,
P_{s}.
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
r_{Act} = Liquid density at actual temperature and
pressure conditions, T, P.
C_{d} is the wedge meter discharge coefficient
to account for frictional losses (kinetic energy into heat) due
to viscosity and turbulence effects.
E_{u} 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
(10^{2}–10^{7}). 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 = Rootmeansquare 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 r_{o} = Blade radius outer edge and
r_{i} = 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
(transittime 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 t_{D} is the sound pulse transittime
(or timeofflight) traveling from the upstream transducer to
the downstream transducer, and t_{U} is the
transittime 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 t_{f} = (t_{U} –
t_{D}) / (t_{U}t_{D}) is the
transittime function and K is the instrument factor
determined through calibration. Therefore, by accurately
measuring the upstream and downstream transittimes,
t_{U} and t_{D}, the flow
velocity, u, can be obtained.
Actual volumetric flowrate is calculated as:
where A is the pipe inner crosssection 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
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:
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 crosssection 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 Utube 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 phaseshift is directly affected by the mass passing
through the tube.
A Ushaped Coriolis flowmeter mass flow is given as:
where K_{u} = Tube stiffness,
K = A shapedependent factor
L = Width,
t = Time lag,
omega = Vibration frequency
I_{u} = Tube inertia that includes the tube
fluid mass. The expression can be simplified as:
where:
is the natural frequency of the Ushaped 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:
Q_{Mass} = 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.
APPENDIX B: FLOWMETER CORRECTION CALCULATIONS
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. m^{3}/h). The equation for
corrected flow is:
Liquid flowmeter (standard volumetric flow
measurement—std. m^{3}/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. m^{3}/h). The equation for
corrected flow is:
Gas flowmeter (standard volumetric flow
measurement—std. m^{3}/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: Q_{ActVol} f_{v}
Turbine meter: Q_{ActVol} omega
Ultrasonic meter: Q_{ActVol} f
_{}(T_{UP}, T_{Down})
Magnetic meter: Q_{ActVol} E
Liquid flowmeter (actual volumetric flow
measurement—act. m^{3}/h). The equation for
corrected flow is:
Liquid flowmeter (standard volumetric flow
measurement—std. m^{3}/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. m^{3}/h). The equation for
corrected flow is:
Gas flowmeter (standard volumetric flow
measurement—std. m^{3}/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: Q_{Mass} t
Thermal meter: Q_{Mass} DH
Liquid flowmeter (actual volumetric
flow—act. m^{3}/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, r_{D} , in the above
equations, then correction factor is equal to 1.
Liquid flowmeter (standard volumetric
flow—std. m^{3}/h). Standard volumetric flow is
calculated using:
The equation for corrected flow is:
Liquid flowmeter (mass
flowmeter—kg/h). Equation for corrected flow is:
Q_{Mass_Corr} =
_{}Q_{Mass_Meas}
The correction factor required for correcting the flowmeter
reading is equal to 1.
Gas flowmeter (actual volumetric flow
measurement—act. m^{3}/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. m^{3}/h). Standard volumetric
flow is:
The equation for corrected flow is:
Gas flowmeter (mass flowmeter—kg/h).
The equation for corrected flow is:
Q_{Mass_Corr} =
_{}Q_{Mass_Meas}
APPENDIX C: PROPAGATION OF UNCERTAINTY BY TAYLOR
SERIES
The sensitivity coefficient associated with each flow
variable is derived analytically by partial differentiation
based on uncertainty propagation by the Taylor series
[ASMEMFC2M, 1983]. The relationship of dependent variable Q
with independent variables x1, x2,..., xN can be expressed in a
general form as:
Q = f (x_{1}, x_{2},...,
x_{N})
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)x_{i}
(delta)x_{j} = 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, S_{x} is the sensitivity
coefficient associated with the variable, and
U_{x} is the uncertainty of the variable.
APPENDIX D: UNCERTAINITY EQUATIONS FOR VORTEX,
TURBINE, ULTRA SONIC AND CORIOLIS
Vortex flowmeter uncertainty is:
Turbine flowmeter uncertainty is:
Ultrasonic flowmeter uncertainty is:
Coriolis flowmeter uncertainty is:
NOMENCLATURE
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
Q—Flowrate
Q_{Mass}—Mass flowrate, kg/h
Q_{ActVol} —Volumetric flowrate at actual
operating conditions, act. m^{3}/h
Q_{StdVol} —Volumetric flowrate at
standard conditions (15°C and 1 atm.), std.
m^{3}/h
Q_{Mass_Meas}—Measured (readout) mass
flowrate from the flowmeter, kg/h
Q_{ActVol_Meas}—Measured actual
volumetric flowrate (readout) from the flowmeter, act.
m^{3}/h
Q_{StdVol_Meas}—Measured standard
volumetric flowrate (readout) from the flowmeter, std.
m^{3}/h
Q_{Mass_Corr} —Corrected mass flowrate
for the flowmeter, kg/h
Q_{ActVol_Corr} —Corrected actual
volumetric flowrate for the flowmeter, act.
m^{3}/h
Q_{StdVol_Corr} —Corrected standard
volumetric flowrate for the flowmeter, std.
m^{3}/h
R—Universal gas constant (R = 8.3145
kPaa m^{3}/kmol K)
S—Sensitivity coefficient
T—Temperature, K
U—Uncertainty
U_{95} —Uncertainty at 95% confidence
level
Z—Compressibility factor for gases (Z = 1 for
ideal gas)
s—Standard deviation
r—Mass density, kg/m^{3
}r_{D} —Design density at actual
conditions from flowmeter specification datasheet
r_{M} —Measured (or calculated) density
at actual conditions during operation
r_{D_Std} —Design density at standard
conditions from flowmeter specification datasheet
r_{M_Std} —Measured density at standard
conditions during operation
Acknowledgments
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.
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 MFC2M, Measurement Uncertainty for Fluid
Flow in Closed Conduits, American National Standard, 1983
(Revised 2006).
^{6} AGA RP31, Orifice Metering of Natural Gas
and Other Related Hydrocarbon Fluids Part 1—General
Equations and Uncertainty Guidelines, American Gas
Association, June 2003. (API MPMS 14.31; ANSI/API 253091 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 nAlkane
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 MFC3M, Measurement of Fluid Flow in
Pipes Using Orifice, Nozzle and Venturi, American National
Standard, 2004.
^{11} ASME MFC6M, Measurement of Fluid Flow in
Pipes using Vortex Flowmeters, American National Standard,
1998 (Revised 2005).
^{12} AGA RP7, Measurement of Natural Gas by
Turbine Meters, American Gas Association, February
2006.
^{13} API MPMS5.3, Measurement of Liquid
Hydrocarbons by Turbine Meters, American Petroleum
Institute, September 2000.
^{14} ASME MFC4M, Measurement of Gas Flow by
Turbine Meters, American National Standard, 1986 (Revised
2008).
^{15} AGA RP9, Measurement of Gas by Multipath
Ultrasonic Meters, American Gas Association, April
2007.
^{16} API MPMS5.8, Measurement of Liquid
Hydrocarbons by Ultrasonic Flow Meters Using Transit Time Technology, American Petroleum
Institute, February 2005.
^{17} ASME MFC5M, Measurement of Liquid Flow in
Closed Conduits Using TransitTime Ultrasonic Flowmeters,
American National Standard, 1985 (Revised 2006).
^{18} ASME MFC16M, Measurement of Liquid Flow in
Closed Conduits with Electromagnetic Flowmeters, American
National Standard, 1995 (Revised 2006).
^{19} AGA RP11, Measurement of Natural Gas by
Coriolis Meter, American Gas Association, January
2003.
^{20} API MPMS5.6, Measurement of Liquid
Hydrocarbons by Coriolis Meters, American Petroleum
Institute, October 2002.
^{21} ASME MFC11M, 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, frontend 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 startup of CNRL Horizon Upgrader
and is working with CNRL Thermal Team as a senior
engineering specialist on insitu 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.
