Patent Description:
Coriolis mass flowmeters utilize Coriolis forces induced by fluid flowing through one or more vibrating tubes to measure mass flow rate. <FIG> depicts example Coriolis flow meter <NUM> comprising a meter assembly <NUM> and meter electronics <NUM>. Meter assembly <NUM> responds to changes in a process fluid flow. Meter electronics <NUM> is connected to meter assembly <NUM> via leads <NUM>, and provides density, volumetric flow rate, and mass flow rate information to operators over meter electronics interface <NUM>, in addition to other information.

Meter assembly <NUM> includes manifolds <NUM> and <NUM>', flanges <NUM> and <NUM>', two parallel flow tubes <NUM> and <NUM>', driver <NUM>, and velocity pick-off sensors <NUM> and 170R. Flow tubes <NUM> and <NUM>' bend at two symmetrical locations along their length and are essentially parallel throughout their length. Brace bars <NUM> and <NUM>' serve to define an axis about which each flow tube oscillates.

When flanges <NUM> and <NUM>' are connected via inlet end <NUM> and exit end <NUM>' to a process line (not shown), process fluid enters inlet end <NUM> of the meter through flange <NUM> and is conducted through manifold <NUM>. Manifold <NUM> divides and routes the process fluid through flow tubes <NUM> and <NUM>'. Upon exiting flow tubes <NUM> and <NUM>', the process fluid is recombined in a single stream by manifold <NUM>' and routed to outlet end <NUM>', connected by flange <NUM>' to the process line (not shown).

Both flow tubes <NUM> and <NUM>' are driven by driver <NUM> in opposite directions in a first out-of-phase bending mode of the flowmeter. Driver <NUM> may comprise any one of many well-known arrangements, such as a magnet mounted to flow tube <NUM>' and an opposing coil mounted to flow tube <NUM> and through which an alternating current is passed for vibrating both flow tubes. A suitable driver voltage is applied by meter electronics <NUM> to driver <NUM>. In further embodiments, Coriolis flow meter <NUM> may comprise more than one driver <NUM>, providing a multiple-input arrangement that can generate other bending modes.

While Coriolis flow meter <NUM> depicts a dual, curved flow tube design, this is not intended to be limiting. Those of skill understand that other examples of Coriolis flow meters <NUM> may include one, or any number of flow tubes. Those of skill will further understand that other Coriolis flow meters may include straight flow tubes, or any other configuration.

Meter electronics <NUM> provides the drive signal to driver <NUM> to vibrate flow tubes <NUM> and <NUM>' over leads <NUM>. Meter electronics <NUM> receives the left and right velocity signals from velocity pick-off sensors <NUM> and 170R over leads <NUM>, which can be used to compute the mass flow rate, volumetric rate, and/or density information for the flow passing through meter assembly <NUM>.

The left and right velocity signals from pick-off sensors <NUM> and 170R are used to determine a phase difference ΔT between the pick-off sensors <NUM> and 170R representing the Coriolis forces on the flow tubes. The phase difference ΔT is used to determine a mass flow value ṁ using Equation <NUM>: <MAT> where FCF, the Flow Calibration Factor, and ΔT<NUM>, the zero offset, are determined during factory calibration. The FCF captures the stiffness of the one or more flow tubes <NUM>, <NUM>', which is directly proportional to the mass flow rate of the fluid flowing through the tube. The FCF is determined by flowing water at ambient conditions through the Coriolis mass flowmeter and comparing the indicated mass to the mass measured by a reference flow meter.

Corrections are typically made to the Coriolis flow meter <NUM> mass flow measurements after installation at a customer site to account for differences between the customer site and factory environmental conditions. For example, changes in temperature and fluid pressure can change the stiffness of flow tubes <NUM>, <NUM>', which can introduce errors in the meter mass flow and density measurements.

Temperature corrections required for mass flow and density measurements higher than <NUM> C are different than the temperature corrections required below <NUM> C temperatures. The temperature correction made to measured mass flow values ṁ for changes in stiffness due to Young's modulus higher than <NUM> C is approximately linear. For temperatures below <NUM> C, corrections to mass flow measurement ṁ are typically better represented by a polynomial equation. Example of corrections to mass flow measurement known in the prior art are, for example, disclosed by <NPL>. , a Coriolis flowmeter is calibrated at a reference condition and the flow calibration factor is corrected considering the non-linearity of Young's modulus and thermal expansion change with temperature, in order to provide accurate mass flow measurement at cryogenic temperatures.

It has been empirically observed that the temperature correction for density measurements is not the same as the temperature correction based on mass flow measurements for temperatures between <NUM> and <NUM> C. However, it has been difficult to characterize the change in flow tube stiffness based on Young's modulus below <NUM> C due to the limitations in flow rate available in cryogenic calibration facilities. Thus far, it has only been possible to acquire empirical data to characterize changes in flow tube stiffness based on temperature for smaller flow meters, or those with flow tubes that are <NUM> centimetres (<NUM> inches) or smaller.

There is a demand for more precise mass flow measurements at sub-zero and cryogenic temperatures. One possible application is high volume flows of liquified natural gas at a temperature of -<NUM> C.

It is highly desirable to provide more accurate fluid measurements with Coriolis flow meters at sub-zero and cryogenic temperatures.

A method for correcting a mass flow value ṁ measured using a Coriolis flow meter for temperature effects at a known fluid temperature temp below <NUM> C is provided. The method comprises receiving a known fluid density ρref; receiving the known fluid temperature temp, receiving a time period Tp, determining a Young's modulus temperature correction for density TFyD based on the known fluid density ρref, the known fluid temperature temp, and the time period Tp, determining a Young's modulus temperature correction for mass flow TFyM based on a temperature correction constant k and the Young's modulus temperature correction for density TFyD, and correcting the mass flow value ṁ using the Young's modulus temperature correction for mass flow TFyM.

A system for correcting a mass flow value ṁ measured using a Coriolis flow meter for temperature effects at a known fluid temperature temp below <NUM> C is provided. The system comprises a fluid density receiving module configured to receive a known fluid density ρref, a fluid temperature receiving module configured to receive the known fluid temperature temp, a period determination module configured to receive a time period Tp, a Young's modulus temperature correction for density determination module configured to determine a Young's modulus temperature correction for density TFyD based on the known fluid density ρref, the known fluid temperature temp, and the time period Tp, a Young's modulus temperature correction for mass flow determination module configured to determine a Young's modulus temperature correction for mass flow TFyM based on a temperature correction constant k and the Young's modulus temperature correction for density TFyD, and a mass flow correction module configured to correct the mass flow value ṁ using the Young's modulus temperature correction for mass flow TFyM.

A meter electronics for correcting a mass flow value ṁ measured using a meter assembly of a Coriolis flow meter for temperature effects at a known fluid temperature temp below <NUM> C is provided. The meter electronics comprising a system processor is configured to receive a known fluid density ρref, receive the known fluid temperature temp, receive a time period Tp, determine a Young's modulus temperature correction for density TFyD based on the known fluid density ρref, the known fluid temperature temp, and the time period Tp, determine a Young's modulus temperature correction for mass flow TFyM based on a temperature correction constant k and Young's modulus temperature correction for density TFyD, and correct the mass flow value ṁ using the Young's modulus temperature correction for mass flow TFyM.

According to a further aspect, the time period Tp may be determined based on a measured fluid density ρindic.

According to a further aspect, the method may further comprise receiving a phase difference ΔT, and determining the Young's modulus temperature correction for density TFyD may be further based on the phase difference ΔT.

According to a further aspect, the method may further comprise receiving a fluid pressure P, and the Young's modulus temperature correction for density TFyD may be further based on the fluid pressure P.

According to a further aspect, the method may further comprise determining an expansion temperature correction for density TFe, and the Young's modulus temperature correction for density TFyD may be further determined based on the expansion temperature correction for density TFe based on a known temperature tempref.

According to a further aspect, the temperature correction constant k may be between <NUM> and <NUM>.

According to a further aspect, the temperature correction constant k may be one.

According to a further aspect, correcting a mass flow value ṁ using the Young's modulus temperature correction for mass flow TFyM may further comprise determining a mass error value Errorm using the Young's modulus temperature correction for mass TFyM.

According to a further aspect, the fluid density receiving module may be further configured to determine a measured fluid density ρindic, and the period determination module is further configured to determine the time period Tp based on the measured fluid density ρindic.

According to a further aspect, the system may further comprise a phase difference determination module configured to determine a phase difference ΔT, and the Young's modulus temperature correction for density determination module may further be configured to determine the Young's modulus temperature correction for density TFyD based on the phase difference ΔT.

According to a further aspect, the system may further comprise a fluid pressure determination module configured to determine a measured fluid pressure Pindic, and the Young's modulus temperature correction for density determination module may be further configured to determine the Young's modulus temperature correction for density TFyD based on the fluid pressure P.

According to a further aspect, the system may further comprise an expansion temperature correction module configured to determine an expansion temperature correction for density TFe based on a known temperature tempref, and the Young's modulus temperature correction for density module may be further configured to determine the Young's modulus temperature correction for density TFyD based on the expansion temperature correction for density TFe.

According to a further aspect, the mass flow correction module may be further configured to determine a mass error value Errorm using the Young's modulus temperature correction for mass TFyM.

According to a further aspect, the system processor may be further configured to receive a phase difference ΔT, and determine the Young's modulus temperature correction for density TFyD may be further based on the phase difference ΔT.

According to a further aspect, the system processor may be further configured to receive a fluid pressure P, and the Young's modulus temperature correction for density TFyD may be further based on the fluid pressure P.

According to a further aspect, the system processor may be further configured to determine an expansion temperature correction for density TFe, and the Young's modulus temperature correction for density TFyD may be further determined based on the expansion temperature correction for density TFe based on a known temperature tempref.

<FIG> and the following description depict specific examples to teach those skilled in the art how to make and use the best mode of the Application. For the purpose of teaching inventive principles, some conventional aspects have been simplified or omitted. Those skilled in the art will appreciate variations from these examples that fall within the scope of the Application. Those skilled in the art will appreciate that the features described below may be combined in various ways to form multiple variations of the Application. As a result, the Application is not limited to the specific examples described below, but only by the claims and their equivalents.

<FIG> depicts system <NUM> in accordance with an embodiment. System <NUM> may be used for temperature correcting a mass flow value ṁ measured using a Coriolis flow meter for temperature effects at a fluid temperature below <NUM> C. For example, system <NUM> may be used to provide temperature corrections, such as those due to Young's modulus, the modulus of elasticity, thermal expansion, or pressure effects, based on temperature on the mass flow value m measured with a Coriolis flow meter.

System <NUM> includes Coriolis flow meter <NUM>, meter electronics <NUM>, and process conduit <NUM>. Process conduit <NUM> carries a flow of fluid to be measured by Coriolis flow meter <NUM>.

Meter electronics <NUM> may be used to generate a mass flow value m for the fluid measured with meter assembly <NUM> of Coriolis flow meter <NUM>, or to temperature correct a mass flow value ṁ obtained using meter assembly <NUM>. Meter electronics <NUM> includes a memory 20a, a system processor 20b, and an interface 20c.

Memory 20a comprises an electronically readable medium or a computer readable medium configured to store computer program instructions. In examples, memory 20a may include a non-transitory medium. Computer program instructions stored on the memory 20a may perform a portion or all of the steps described in relation to method <NUM> or execute a portion or all of the modules of system <NUM>.

System processor 20b may be configured to execute computer instructions, which perform a portion or all of the steps described in relation to method <NUM> or execute a portion or all of the modules described in relation to system <NUM>. In embodiments, system processor 20b may include a single, or any multiple number of processors, as will be understood by those of skill in the art.

Interface 20c is configured to communicate with meter assembly <NUM> of Coriolis flow meter <NUM>. Interface 20c may be configured to communicate with devices external to electronics <NUM>, such as, for example, a pressure sensor, a temperature sensor, or any other sensor known to those of skill.

In embodiments, system <NUM> may comprise an additional measurement device <NUM>. In embodiments, additional measurement device <NUM> may comprise a device capable of providing density measurements, such as a densitometer, a gas chromatograph, an additional Coriolis meter, or any other type of measurement device known to those of skill. In embodiments, additional measurement device <NUM> may include a corresponding meter electronics <NUM>, as depicted in <FIG>. Like meter electronics <NUM>, meter electronics <NUM> may comprise a memory 204a, system processor 204b, and an interface 204c. In further embodiments, however, additional measurement device <NUM> may provide signals and information directly to interface 20c of meter electronics <NUM>.

In further embodiments, system <NUM> may include a server <NUM>. In embodiments, server <NUM> may be in communication with interface 20c of meter electronics <NUM> and/or interface 204c of meter electronics <NUM>. Any portion of the steps described in relation to method <NUM> or the modules described in relation to system <NUM> may be stored or executed on server <NUM>.

<FIG> depicts a method <NUM> in accordance with an embodiment. Method <NUM> may be used for correcting a mass flow value ṁ measured using a Coriolis flow meter for temperature effects at a known fluid temperature tempref below <NUM> C. For example, method <NUM> may be used to provide measurement corrections, such as those that correct for changes associated with changes in Young's modulus, the modulus of elasticity, thermal expansion, or pressure effects, based on temperature on the mass flow value ṁ measured with Coriolis flow meter <NUM>.

Method <NUM> begins with step <NUM>. In step <NUM>, a known fluid density ρref is received. Method <NUM> continues with step <NUM>. In step <NUM>, the known fluid temperature temp is received. The known fluid density ρref and the known fluid temperature temp may be well understood due to the nature of the fluid being measured.

Method <NUM> continues with step <NUM>. In step <NUM> a time period Tp is received. Time period Tp is the period of time of the vibrating flow tube <NUM>, <NUM>'.

In embodiments, time period Tp may be measured directly using a vibration sensor coupled to a flow tube <NUM>, <NUM>', including, for example, one or both of left and right velocity pick-off sensors <NUM> and 170R of Coriolis flow meter <NUM>.

In further embodiments of step <NUM>, however, time period Tp may be determined indirectly based on the measured fluid density ρinaic, the phase difference ΔT, and a fluid pressure P as follows.

In the method where time period Tp is determined indirectly, step <NUM> may further comprise steps <NUM> and <NUM>. In step <NUM>, a fluid pressure P may be received. In embodiments, fluid pressure P may comprise a fluid pressure determined using a pressure transducer positioned just upstream or downstream of Coriolis flow meter <NUM> in process conduit <NUM>. In further embodiments, however, fluid pressure P may comprise a pressure measurement that is internal to Coriolis flow meter <NUM>, or any other fluid pressure measurement known to those of skill in the art. In embodiments, fluid pressure P may comprise a known or estimated fluid pressure.

In step <NUM>, a phase difference ΔT may be received. In embodiments, the phase difference ΔT may be determined using velocity pick-off sensors <NUM> and 170R of Coriolis flow meter <NUM>. In further embodiments, however, phase difference ΔT may be determined indirectly using the measured mass flow value m, FCF, a combined temperature factor TF, and a fluid temperature temp, as will be understood by those of skill.

In embodiments, the measured fluid density ρindic may be measured with a densitometer. For example, the measured fluid density ρindic may be received from additional measurement device <NUM> in system <NUM>, which may comprise a densitometer. In further embodiments, additional measurement device <NUM> in system <NUM> may comprise a gas chromatograph that may provide the measured fluid density ρindic.

Coriolis flow meter <NUM> is typically calibrated at factory conditions at a temperature between <NUM>-<NUM> C. In many cases, a Coriolis flow meter is calibrated using two fluids, such as ambient air and water, by determining a mass flow value ṁ and a measured fluid density ρindic value for each fluid. Using these measured values for mass flow value ṁ and a measured fluid density ρindic, it is possible to determine calibration constants K<NUM> and K<NUM>, one constant for each respective fluid.

Calibration values C<NUM> and C<NUM>, which are valid for a temperature of <NUM> C and a pressure of <NUM> barg, can then be calculated using calibration constants K<NUM> and K<NUM> via Equations <NUM> and <NUM>. Calibration value C<NUM> is proportional to inertia moment and inversely proportional to flow area of the flow tube <NUM>, <NUM>': <MAT> Calibration value C<NUM> is proportional to the mass of flow tube <NUM>, <NUM>' material divided by fluid volume: <MAT> In Equations <NUM> and <NUM>, D<NUM> is the outer diameter of flow tube <NUM>, <NUM>', and D<NUM> is the inner diameter of flow tube <NUM>, <NUM>'.

A measured fluid density ρinaic may be determined using Equation <NUM>: <MAT> In Equation <NUM>, TFd is a combined temperature correction coefficient for density. FD is a constant to correct the measured fluid density ρindic under flowing conditions, as will be understood by those of skill in the art. In Equation <NUM>, pcd is a pressure correction for density.

Equation <NUM> can be re-arranged to Equation <NUM>: <MAT> In embodiments, time period squared Tp<NUM> may be determined based on measured fluid density ρindic, fluid pressure P, and phase difference ΔT using Equation <NUM>. In further embodiments, however, the flow effect on the measured fluid density ρindic represented by the FD * (ΔT)<NUM> * <NUM>-<NUM> term in Equation <NUM>, may be very small, and therefore ignored. The pressure correction for density pcd may also be small, and therefore Equation <NUM> may be further simplified by making pcd equal to zero. This may provide for the simplified embodiment of Equation <NUM>: <MAT> According to Equation <NUM>, time period squared Tp<NUM> may be determined based on measured fluid density ρindic.

Method <NUM> continues with step <NUM>. In step <NUM>, Young's modulus temperature correction for density TFyD is determined. Young's modulus is affected by material expansion and the changing geometry of the flow tube due to temperature and, to a lesser degree, pressure.

In embodiments, Young's modulus temperature correction for density TFyD may be determined using any method known to those of skill in the art. In further embodiments, however, Young's modulus temperature correction for density TFyD may be determined based on the known fluid density ρref, the fluid temperature temp, and the time period Tp.

For example, a known fluid density ρref is related to Young's modulus E(temp,P) via exact theory according to Equation <NUM>: <MAT> In Equation <NUM>, FD is the flow effect on density, L is the length of the flow tube <NUM>, <NUM>', Do is the outer diameter of flow tube <NUM>, <NUM>', and Di is the inner diameter of flow tube <NUM>, <NUM>'. When the temperature temp is <NUM> C and the fluid pressure P is <NUM> barg, Equation <NUM> may be re-written as Equation <NUM>: <MAT> where PFc1 is a pressure factor which represents a combination of Young's modulus and geometry changes due to fluid pressure PFc1 = <NUM> +pcc1*P, with pcc1 being the pressure coefficient for constant C<NUM>. In Equation <NUM>, PFC2 is a pressure factor which relates to the change of fluid volume due to pressure PFC2 = <NUM> + pcc2*P, where pcc2 is a pressure coefficient for constant C<NUM>. For example, for Micro Motion flow meter model CMF400, pcc1 is <NUM> * <NUM>-<NUM>, pcc2 is <NUM> * <NUM>-<NUM>, and the pressure effect is - <NUM>/m<NUM>/bar.

In Equation <NUM>, TFy is the temperature factor due to Young's modulus. At cryogenic temperatures, the temperature factor due to Young's modulus TFy may be non-linear. For example, in the <NPL>, the polynomial of Equation <NUM> is proposed for stainless steel at cryogenic temperatures: <MAT> In Equation <NUM>, temp represents a temperature, which can be a known or a measured temperature. In embodiments of step <NUM>, known temperature tempref may be used to determine the temperature factor due to Young's modulus TFy.

In Equation <NUM>, known fluid density ρref further depends on TFe, an expansion temperature correction for density. The expansion temperature correction for density TFe may be determined using any method known to those of skill in the art. In embodiments, step <NUM> may further comprise step <NUM>. In step <NUM>, an expansion temperature correction for density TFe may be determined based on empirical data relating to flow tube material expansion.

In embodiments, the expansion temperature correction for density TFe may be non-linear. For example, the article "Low temperature thermal expansion of iron-chromium-nickel alloys of different stabilities" published by Academy of Sciences, Ukraine in February <NUM> provides the following polynomial Equation <NUM> describing the temperature correction for thermal expansion at cryogenic temperatures: <MAT> In embodiments of step <NUM>, the known temperature tempref may be used to determine the expansion temperature correction for density TFe.

Using the known fluid density ρref, the phase difference ΔT, the fluid pressure P, the known fluid temperature tempref, and the time period Tp, it is therefore possible to determine the Young's modulus temperature correction for density TFyd via Equation <NUM>: <MAT>.

Because the Young's modulus of the flow tubes affects the vibration of flow tubes <NUM>, <NUM>', both the mass flow measurement ṁ and the fluid density measurement p are affected by changes in Young's modulus. The vibration of the tubes is a function of the flow tube <NUM>, <NUM>' material properties, and the flow tubes <NUM>, <NUM>' are typically fabricated from steel.

In further embodiments, however, the flow effect on the fluid density represented by the FD * (ΔT)<NUM> * <NUM>-<NUM> term in Equation <NUM>, may be very small, and therefore ignored. In addition, pressure factors for C1, PFC1 and PFC2, may also represent small changes in the Young's modulus temperature correction for density TFyd. Setting the flow effect on fluid density FD to zero and pressure factors PFC1 and PFC2 to <NUM>, may provide for the simplified representation of Young's modulus temperature correction for density TFyd of Equation <NUM>: <MAT> According to Equation <NUM>, the Young's modulus temperature correction for density TFyd may be determined based only on known fluid density ρref, the known fluid temperature tempref, and the time period Tp.

Once the Young's modulus temperature correction for density TFyd is determined, method <NUM> may continue with step <NUM>. In step <NUM>, a Young's modulus temperature correction for mass flow TFyM is determined based on a temperature correction constant k multiplied by Young's modulus temperature correction for density TFyD, as represented by Equation <NUM>: <MAT> The Young's modulus temperature correction for mass flow TFyM is generally related to torque in the flow tubes and the Young's modulus temperature correction for density TFyD is generally related to bending in the flow tubes. Initial tests in a calibration lab using a flow meter with stainless steel tubes shaped in a "U" configuration have indicated these temperature corrections to be substantially similar in value. Therefore, in embodiments the temperature correction constant k may be set to one. It is possible, however, that future tests with more sensitive measurements, different tube materials, and/or different tube geometries may reveal that the Young's modulus temperature correction for mass flow TFyM and the Young's modulus temperature correction for density TFyD are different in value. Therefore, in other embodiments, the temperature correction constant k may be determined to be any number other than one. In one nonlimiting example, k may be set to a value between <NUM> and <NUM>.

Once the Young's modulus temperature correction for mass flow TFyM is determined, method <NUM> may continue with step <NUM>. In step <NUM>, a mass flow value ṁ determined using Equation <NUM> with Coriolis flow meter <NUM> is corrected using the Young's modulus temperature correction for mass flow TFyM. In embodiments, the mass flow value ṁ may be corrected using the Young's modulus temperature correction for mass flow TFyM via any method known to those of skill in the art.

In embodiments, step <NUM> may further comprise step <NUM>, In step <NUM>, a mass error value Errorm may be determined using the Young's modulus temperature correction for mass TFyM and the expansion temperature correction for density TFe determined via steps <NUM> and <NUM>: <MAT> In Equation <NUM>:.

The first part of Equation <NUM> comes from calibration and reflects the mass error value Errorm at <NUM> and <NUM> barg, and the second part of Equation <NUM> comes from operation in the application and reflects the error from <NUM> and <NUM> barg to operating conditions. In practice, the first part of Equation <NUM> is small in relation to the second part, however. For that reason, in embodiments Equation <NUM> may be simplified to Equation <NUM>: <MAT>.

In embodiments, a meter factor MF may be determined to correct the mass flow value ṁ measured with Coriolis flow meter <NUM> using Equation <NUM>: <MAT>.

A corrected mass flow value ṁ may then be determined by multiplying the measured mass flow value ṁ by meter factor MF.

<FIG> depicts system <NUM>. In embodiments, system <NUM> may be used to correct a mass flow value ṁ measured using a Coriolis flow meter <NUM> for temperature effects at a fluid temperature temp below <NUM> C. System <NUM> comprises fluid density receiving module <NUM>, fluid temperature receiving module <NUM>, period determination module <NUM>, Young's modulus temperature correction for density determination module <NUM>, Young's modulus temperature correction for mass flow determination module <NUM>, and mass flow correction module <NUM>. In embodiments, system <NUM> may further comprise fluid pressure determination module <NUM>, phase difference determination module <NUM>, and expansion temperature correction module <NUM>.

Fluid density receiving module <NUM> is configured to determine a fluid density ρ, such as, for example, the known fluid density ρref. For example, fluid density receiving module <NUM> may execute step <NUM>, described above.

Fluid temperature receiving module <NUM> is configured to determine the fluid temperature temp, such as, for example, the known fluid temperature temp. For example, fluid temperature receiving module <NUM> may execute step <NUM>, as described above.

Fluid pressure determination module <NUM> is configured to determine a fluid pressure P. For example, fluid pressure determination module <NUM> may execute step <NUM>, as described above.

Phase difference determination module <NUM> is configured to determine a phase difference ΔT. For example, phase difference determination module <NUM> may execute step <NUM>, as described above.

Period determination module <NUM> is configured to receive a time period Tp. For example, period determination module <NUM> may execute step <NUM>, as described above.

Expansion temperature correction module <NUM> is configured to determine an expansion temperature correction for density TFe. For example, expansion temperature correction module <NUM> may execute step <NUM>, as described above.

Young's modulus temperature correction for density determination module <NUM> is configured to determine a Young's modulus temperature correction for density TFyD based on the fluid density ρ, the fluid temperature temp, and the time period Tp. For example, Young's modulus temperature correction for density determination module <NUM> may execute step <NUM>, as described above.

Young's modulus temperature correction for mass flow determination module <NUM> is configured to determine a Young's modulus temperature correction for mass flow TFyM based on a temperature correction constant k and Young's modulus temperature correction for density TFyD. For example, Young's modulus temperature correction for mass flow determination module <NUM> may execute step <NUM>, as described above.

Mass flow correction module <NUM> is configured to correct the mass flow value ṁ using the Young's modulus temperature correction for mass flow TFyM. For example, mass flow correction module <NUM> may execute step <NUM>, as described above.

Tests on liquified nitrogen at a cryogenic calibration facility using a weighing scale have determined that the methods and system of the present Application provide a corrected mass flow value ṁ with errors that are less than <NUM>%. Some of the tests conducted by the Applicant provided mass flow errors that were as low as <NUM>% and <NUM> % for flow meters with flow tubes that are <NUM> centimetres (<NUM> inches) or less in diameter. The methods and system described in the present Application can be extrapolated to larger meter sizes, or those with flow tube diameters that are greater than <NUM> centimetres (<NUM> inches), to provide very accurate mass flow values ṁ for higher fluid flows.

The methods and system described by the present Application provide temperature corrections that improve the accuracy of mass flow measurements generated with Coriolis flow meters at sub-zero and cryogenic temperatures. The temperature corrections are stable over time, and do not require calibration of the Coriolis flow meter at a cryogenic calibration facility.

Claim 1:
A method for correcting a mass flow value ṁ measured using a Coriolis flow meter (<NUM>) for temperature effects at a known fluid temperature temp below <NUM> C, the method comprising:
receiving a known fluid density ρref;
receiving the known fluid temperature temp;
receiving a time period Tp;
determining a Young's modulus temperature correction for density TFyD based on the known fluid density ρref, the known fluid temperature temp, and the time period Tp;
determining a Young's modulus temperature correction for mass flow TFyM based on a temperature correction constant k and the Young's modulus temperature correction for density TFyD; and
correcting the mass flow value ṁ using the Young's modulus temperature correction for mass flow TFyM.