METHOD, SYSTEM, AND ELECTRONICS FOR CORRECTING A CORIOLIS FLOW METER MEASUREMENT FOR TEMPERATURE EFFECTS

Abstract

A method (300), system (400), and electronics (20) for correcting a mass flow value in measured using a Coriolis flow meter (100) for temperature effects at a known fluid temperature temp below 0 C are provided. The method comprises receiving a known fluid density ρ.sub.indic, receiving the fluid temperature temp, receiving a time period Tp, determining a Young's modulus temperature correction for density TFy.sub.D based on the known fluid density ρ.sub.indic, the known fluid temperature temp, and the time period Tp, determining a Young's modulus temperature correction for mass flow TFy.sub.M based on a temperature correction constant k and Young's modulus temperature correction for density TFy.sub.D, and correcting the mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M.

Claims

1. A method for correcting a mass flow value {dot over (m)} measured using a Coriolis flow meter (100) for temperature effects at a known fluid temperature temp below 0 C, the method comprising: receiving a known fluid density ρ.sub.ref; receiving the known fluid temperature temp; receiving a time period Tp; determining a Young's modulus temperature correction for density TFy.sub.D based on the known fluid density ρ.sub.ref, the known fluid temperature temp, and the time period Tp; determining a Young's modulus temperature correction for mass flow TFy.sub.M based on a temperature correction constant k and the Young's modulus temperature correction for density TFy.sub.D; and correcting the mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M.

2. A method as claimed in claim 1, wherein the time period Tp is determined based on a measured fluid density ρ.sub.indic.

3. A method as claimed in claim 1, further comprising: receiving a phase difference ΔT, and wherein determining the Young's modulus temperature correction for density TFy.sub.D is further based on the phase difference ΔT.

4. A method as claimed in claim 1, further comprising: receiving a fluid pressure P, and wherein the Young's modulus temperature correction for density TFy.sub.D is further based on the fluid pressure P.

5. A method as claimed in claim 1, wherein the method further comprises: determining an expansion temperature correction for density TFe, and wherein the Young's modulus temperature correction for density TFy.sub.D is further determined based on the expansion temperature correction for density TFe based on a known temperature temp.sub.ref.

6. A method as claimed in claim 1, wherein the temperature correction constant k is between 0.8 and 1.2.

7. A method as claimed in claim 1, wherein the temperature correction constant k is one.

8. A method as claimed in claim 1, wherein correcting a mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M further comprises: determining a mass error value Error.sub.m using the Young's modulus temperature correction for mass TFy.sub.M.

9. A system (400) for correcting a mass flow value {dot over (m)} measured using a Coriolis flow meter (100) for temperature effects at a known fluid temperature temp below 0 C, the system (400) comprising: a fluid density receiving module (402) configured to receive a known fluid density ρ.sub.ref; a fluid temperature receiving module (404) configured to receive the known fluid temperature temp; a period determination module (410) configured to receive a time period Tp; a Young's modulus temperature correction for density determination module (414) configured to determine a Young's modulus temperature correction for density TFy.sub.D based on the known fluid density ρ.sub.ref, the known fluid temperature temp, and the time period Tp; a Young's modulus temperature correction for mass flow determination module (416) configured to determine a Young's modulus temperature correction for mass flow TFy.sub.M based on a temperature correction constant k and the Young's modulus temperature correction for density TFy.sub.D; and a mass flow correction module (418) configured to correct the mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M.

10. A system (400) as claimed in claim 9, wherein the fluid density receiving module (402) is further configured to determine a measured fluid density ρ.sub.indic, and the period determination module (410) is further configured to determine the time period Tp based on the measured fluid density ρ.sub.indic.

11. A system (400) as claimed in claim 9, further comprising: a phase difference determination module (408) configured to determine a phase difference ΔT, and wherein the Young's modulus temperature correction for density determination module (414) is further configured to determine the Young's modulus temperature correction for density TFy.sub.D based on the phase difference ΔT.

12. A system (400) as claimed in claim 9, the system further comprising: a fluid pressure determination module (406) configured to determine a measured fluid pressure ρ.sub.indic, and and wherein the Young's modulus temperature correction for density determination module (414) is further configured to determine the Young's modulus temperature correction for density TFy.sub.D based on the fluid pressure P.

13. A system (400) as claimed in claim 9, wherein the system (400) further comprises: an expansion temperature correction module (412) configured to determine an expansion temperature correction for density TFe based on a known temperature temp.sub.ref, and wherein the Young's modulus temperature correction for density module (414) is further configured to determine the Young's modulus temperature correction for density TFy.sub.D based on the expansion temperature correction for density TFe.

14. A system (400) as claimed in claim 9, wherein the temperature correction constant k is between 0.8 and 1.2.

15. A system (400) as claimed in claim 9, wherein the temperature correction constant k is one.

16. A system (400) as claimed in claim 9, wherein the mass flow correction module (418) is further configured to determine a mass error value Error.sub.m using the Young's modulus temperature correction for mass TFy.sub.M.

17. A meter electronics (20) for correcting a mass flow value {dot over (m)} measured using a meter assembly (10) of a Coriolis flow meter (100) for temperature effects at a known fluid temperature temp below 0 C, the meter electronics comprising a system processor (20b) configured to: receive a known fluid density ρ.sub.ref; receive the known fluid temperature temp; receive a time period Tp; determine a Young's modulus temperature correction for density TFy.sub.D based on the known fluid density ρ.sub.ref, the known fluid temperature temp, and the time period Tp; determine a Young's modulus temperature correction for mass flow TFy.sub.M based on a temperature correction constant k and Young's modulus temperature correction for density TFy.sub.D; and correct the mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M.

18. A meter electronics (20) as claimed in claim 17, wherein the time period Tp is determined based on a measured fluid density ρ.sub.indic.

19. A meter electronics (20) as claimed in claim 17, wherein system processor (20b) is further configured to receive a phase difference ΔT, and wherein determining the Young's modulus temperature correction for density TFy.sub.D is further based on the phase difference ΔT.

20. A meter electronics (20) as claimed in claim 17, wherein the system processor (20b) is further configured: to receive a fluid pressure P, and wherein the Young's modulus temperature correction for density TFy.sub.D is further based on the fluid pressure P.

21. A meter electronics (20) as claimed in claim 17, wherein the system processor 20b is further configured to: determine an expansion temperature correction for density TFe, and wherein the Young's modulus temperature correction for density TFy.sub.D is further determined based on the expansion temperature correction for density TFe based on a known temperature temp.sub.ref.

22. A meter electronics (20) as claimed in claim 17, wherein the temperature correction constant k is between 0.8 and 1.2.

23. A meter electronics (20) as claimed in claim 17, wherein the temperature correction constant k is one.

24. A meter electronics (20) as claimed in claim 17, wherein correcting a mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M further comprises: determining a mass error value Error.sub.m using the Young's modulus temperature correction for mass TFy.sub.M.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The same reference number represents the same element on all drawings. The drawings are not necessarily to scale.

[0039] FIG. 1 depicts Coriolis flow meter 100;

[0040] FIG. 2 depicts system 200, in accordance with an embodiment;

[0041] FIG. 3 depicts method 300, in accordance with an embodiment; and

[0042] FIG. 4 depicts system 400, in accordance with an embodiment.

DETAILED DESCRIPTION

[0043] FIGS. 2-4 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.

[0044] FIG. 2 depicts system 200 in accordance with an embodiment. System 200 may be used for temperature correcting a mass flow value {dot over (m)} measured using a Coriolis flow meter for temperature effects at a fluid temperature below 0 C. For example, system 200 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 {dot over (m)} measured with a Coriolis flow meter.

[0045] System 200 includes Coriolis flow meter 100, meter electronics 20, and process conduit 206. Process conduit 206 carries a flow of fluid to be measured by Coriolis flow meter 100.

[0046] Meter electronics 20 may be used to generate a mass flow value {dot over (m)} for the fluid measured with meter assembly 10 of Coriolis flow meter 100, or to temperature correct a mass flow value {dot over (m)} obtained using meter assembly 10. Meter electronics 20 includes a memory 20a, a system processor 20b, and an interface 20c.

[0047] 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 300 or execute a portion or all of the modules of system 400.

[0048] System processor 20b may be configured to execute computer instructions, which perform a portion or all of the steps described in relation to method 300 or execute a portion or all of the modules described in relation to system 400. 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.

[0049] Interface 20c is configured to communicate with meter assembly 10 of Coriolis flow meter 100. Interface 20c may be configured to communicate with devices external to electronics 20, such as, for example, a pressure sensor, a temperature sensor, or any other sensor known to those of skill.

[0050] In embodiments, system 200 may comprise an additional measurement device 208. In embodiments, additional measurement device 208 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 208 may include a corresponding meter electronics 204, as depicted in FIG. 2. Like meter electronics 20, meter electronics 204 may comprise a memory 204a, system processor 204b, and an interface 204c. In further embodiments, however, additional measurement device 208 may provide signals and information directly to interface 20c of meter electronics 20.

[0051] In further embodiments, system 200 may include a server 202. In embodiments, server 202 may be in communication with interface 20c of meter electronics 20 and/or interface 204c of meter electronics 204. Any portion of the steps described in relation to method 300 or the modules described in relation to system 400 may be stored or executed on server 202.

[0052] FIG. 3 depicts a method 300 in accordance with an embodiment. Method 300 may be used for correcting a mass flow value {dot over (m)} measured using a Coriolis flow meter for temperature effects at a known fluid temperature temp.sub.ref below 0 C. For example, method 300 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 {dot over (m)} measured with Coriolis flow meter 100.

[0053] Method 300 begins with step 302. In step 302, a known fluid density ρ.sub.ref is received. Method 300 continues with step 304. In step 304, the known fluid temperature temp is received. The known fluid density ρ.sub.ref and the known fluid temperature temp may be well understood due to the nature of the fluid being measured.

[0054] Method 300 continues with step 310. In step 310 a time period Tp is received. Time period Tp is the period of time of the vibrating flow tube 130, 130′.

[0055] In embodiments, time period Tp may be measured directly using a vibration sensor coupled to a flow tube 130, 130′, including, for example, one or both of left and right velocity pick-off sensors 170L and 170R of Coriolis flow meter 100.

[0056] In further embodiments of step 310, however, time period Tp may be determined indirectly based on the measured fluid density ρ.sub.indic, the phase difference ΔT, and a fluid pressure P as follows.

[0057] In the method where time period Tp is determined indirectly, step 310 may further comprise steps 306 and 308. In step 306, 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 100 in process conduit 206. In further embodiments, however, fluid pressure P may comprise a pressure measurement that is internal to Coriolis flow meter 100, 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.

[0058] In step 308, a phase difference ΔT may be received. In embodiments, the phase difference ΔT may be determined using velocity pick-off sensors 170L and 170R of Coriolis flow meter 100. In further embodiments, however, phase difference ΔT may be determined indirectly using the measured mass flow value {dot over (m)}, FCF, a combined temperature factor TF, and a fluid temperature temp, as will be understood by those of skill.

[0059] In embodiments, the measured fluid density ρ.sub.indic may be measured with a densitometer. For example, the measured fluid density ρ.sub.indic may be received from additional measurement device 208 in system 200, which may comprise a densitometer.

[0060] In further embodiments, additional measurement device 208 in system 200 may comprise a gas chromatograph that may provide the measured fluid density ρ.sub.indic.

[0061] Coriolis flow meter 100 is typically calibrated at factory conditions at a temperature between 20-30 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 {dot over (m)} and a measured fluid density ρ.sub.indic value for each fluid. Using these measured values for mass flow value {dot over (m)} and a measured fluid density ρ.sub.indic, it is possible to determine calibration constants K.sub.1 and K.sub.2, one constant for each respective fluid.

[0062] Calibration values C.sub.1 and C.sub.2, which are valid for a temperature of 0 C and a pressure of 0 barg, can then be calculated using calibration constants K.sub.1 and K.sub.2 via Equations 2 and 3. Calibration value C.sub.1 is proportional to inertia moment and inversely proportional to flow area of the flow tube 130, 130′:

[00001]$\begin{array}{cc}{C}_1=\frac{D_2-D_1}{\left(K_2^2-K_1^2\right)}.& \left(\mathrm{Equation}2\right)\end{array}$

Calibration value C.sub.2 is proportional to the mass of flow tube 130, 130′ material divided by fluid volume:

[00002]$\begin{array}{cc}{C}_2=\frac{K_1^2}{\left(K_2^2-K_1^2\right)}& \left(\mathrm{Equation}3\right)\end{array}$

In Equations 2 and 3, D.sub.1 is the outer diameter of flow tube 130, 130′, and D.sub.2 is the inner diameter of flow tube 130, 130′.

[0063] A measured fluid density ρ.sub.indic may be determined using Equation 4:

[00003]$\begin{array}{cc}{\rho }_{\mathrm{indic}}=\left(\frac{{\mathrm{TF}}_d*T^2-K_1^2}{K_2^2-K_1^2}\right)*\left(D_2-D_1\right)+D_1-\mathrm{FD}*{\left(\Delta T\right)}^2*10^{-9}+\mathrm{pcd}*P.& \left(\mathrm{Equation}4\right)\end{array}$

In Equation 4, TF.sub.d is a combined temperature correction coefficient for density. FD is a constant to correct the measured fluid density ρ.sub.indic under flowing conditions, as will be understood by those of skill in the art. In Equation 4, pcd is a pressure correction for density.

[0064] Equation 4 can be re-arranged to Equation 5:

[00004]$\begin{array}{cc}{T}_p^2=\frac{\frac{\begin{array}{c}\left({\rho }_{\mathrm{indic}}+\mathrm{FD}\star {\left(\Delta T\right)}^2\star {10}^{-9}+\mathrm{pcd}\star P-D_1\right)\\ \left(K_2^2-K_1^2\right)\end{array}}{\left(D_2-D_1\right)}+K_1^2}{{\mathrm{TF}}_d}& \left(\mathrm{Equation}5\right)\end{array}$

In embodiments, time period squared T.sub.p.sup.2 may be determined based on measured fluid density ρ.sub.indic, fluid pressure P, and phase difference ΔT using Equation 5. In further embodiments, however, the flow effect on the measured fluid density ρ.sub.indic represented by the FD*(ΔT).sup.2*10.sup.−9 term in Equation 5, may be very small, and therefore ignored. The pressure correction for density pcd may also be small, and therefore Equation 5 may be further simplified by making pcd equal to zero. This may provide for the simplified embodiment of Equation 6:

[00005]$\begin{array}{cc}{T}_p^2=\frac{\frac{\left({\rho }_{\mathrm{indic}}\right)\left(K_2^2-K_1^2\right)}{\left(D_2-D_1\right)}+K_1^2}{{\mathrm{TF}}_d}& \left(\mathrm{Equation}6\right)\end{array}$

According to Equation 6, time period squared T.sub.p.sup.2 may be determined based on measured fluid density ρ.sub.indic.

[0065] Method 300 continues with step 314. In step 314, Young's modulus temperature correction for density TFy.sub.D 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.

[0066] In embodiments, Young's modulus temperature correction for density TFy.sub.D 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 TFy.sub.D may be determined based on the known fluid density ρ.sub.ref, the fluid temperature temp, and the time period Tp.

[0067] For example, a known fluid density ρ.sub.ref is related to Young's modulus E(temp,P) via exact theory according to Equation 7:

[00006]$\begin{array}{cc}{\rho }_{\mathrm{ref}}=\frac{12.36}{64*{\pi }^2}*\frac{\left(D_o^4-D_i^4\right)}{L^4*D_i^2}*E\left(\mathrm{temp},P\right)\star T_p^2-\frac{\left(D_o^2-D_i^2\right)}{D_i^2}*{\rho }_{\mathrm{tube}\mathrm{material}}-\mathrm{FD}*{\left(\Delta T\right)}^2*{10}^{-9}.& \left(\mathrm{Equation}7\right)\end{array}$

In Equation 7, FD is the flow effect on density, L is the length of the flow tube 130, 130′, D.sub.o is the outer diameter of flow tube 130, 130′, and D.sub.i is the inner diameter of flow tube 130, 130′. When the temperature temp is 0 C and the fluid pressure P is 0 barg, Equation 7 may be re-written as Equation 8:

[00007]$\begin{array}{cc}{\rho }_{\mathrm{ref}}=\frac{{\mathrm{TF}}_y*{\mathrm{PF}}_{C1}*C_1*T_p^2}{{\left({\mathrm{TF}}_e\right)}^2}-\frac{C_2}{{\left(\mathrm{TFe}\right)}^3*{\mathrm{PF}}_{C2}}-\mathrm{FD}*{\left(\Delta T\right)}^2*10^{-9},& \left(\mathrm{Equation}8\right)\end{array}$

where PF.sub.c1 is a pressure factor which represents a combination of Young's modulus and geometry changes due to fluid pressure PF.sub.c1=1+pc.sub.c1*P, with pc.sub.c1 being the pressure coefficient for constant C.sub.1. In Equation 8, PF.sub.C2 is a pressure factor which relates to the change of fluid volume due to pressure PF.sub.C2=1+pc.sub.C2*P, where pc.sub.C2 is a pressure coefficient for constant C.sub.2. For example, for Micro Motion flow meter model CMF400, pc.sub.c1 is 3.45*10.sup.−5, pc.sub.c2 is 0.99*10.sup.−5, and the pressure effect is −0.145 kg/m.sup.3/bar.

[0068] In Equation 8, TF.sub.y is the temperature factor due to Young's modulus. At cryogenic temperatures, the temperature factor due to Young's modulus TF.sub.y may be non-linear. For example, in the Journal of Applied Physics article, “Stainless steel elastic constants at low temperatures” written by Mr. H. M. Ledbetter, in March 1981, the polynomial of Equation 9 is proposed for stainless steel at cryogenic temperatures:

TF.sub.y=1−tc.sub.y*temp−3.5*10.sup.−7*(temp).sup.2−2*10.sup.−9*(temp).sup.3−1.3*10.sup.−11*(temp).sup.4.  (Equation 9)

In Equation 9, temp represents a temperature, which can be a known or a measured temperature. In embodiments of step 314, known temperature temp.sub.ref may be used to determine the temperature factor due to Young's modulus TF.sub.y.

[0069] In Equation 8, known fluid density ρ.sub.ref 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 314 may further comprise step 312. In step 312, an expansion temperature correction for density TFe may be determined based on empirical data relating to flow tube material expansion.

[0070] 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 1978 provides the following polynomial Equation 10 describing the temperature correction for thermal expansion at cryogenic temperatures:

TF.sub.e=1+16.061*10.sup.−6*temp+5.65*10.sup.−9*temp.sup.2−6.007*10.sup.−11*temp.sup.3.   (Equation 10)

In embodiments of step 312, the known temperature temp.sub.ref may be used to determine the expansion temperature correction for density TF.sub.e.

[0071] Using the known fluid density ρ.sub.ref, the phase difference ΔT, the fluid pressure P, the known fluid temperature temp.sub.ref, and the time period Tp, it is therefore possible to determine the Young's modulus temperature correction for density TF.sub.yd via Equation 11:

[00008]$\begin{array}{cc}{\mathrm{TF}}_{\mathrm{yd}}=\frac{\left({\rho }_{\mathrm{ref}}+\frac{C_2}{{\mathrm{TF}}_e^3*{\mathrm{PF}}_{C2}}+\mathrm{FD}*{\left(\Delta T\right)}^2*10^{-9}\right)\left({\mathrm{TFe}}^2\right)}{T_p^2{\mathrm{PF}}_{C1}{C}_1}.& \left(\mathrm{Equation}11\right)\end{array}$

[0072] Because the Young's modulus of the flow tubes affects the vibration of flow tubes 130, 130′, both the mass flow measurement {dot over (m)} 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 130, 130′ material properties, and the flow tubes 130, 130′ are typically fabricated from steel.

[0073] In further embodiments, however, the flow effect on the fluid density represented by the FD*(ΔT).sup.2*10.sup.−9 term in Equation 11, 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 TF.sub.yd. Setting the flow effect on fluid density FD to zero and pressure factors PFC1 and PFC2 to 1, may provide for the simplified representation of Young's modulus temperature correction for density TF.sub.yd of Equation 12:

[00009]$\begin{array}{cc}{\mathrm{TF}}_{\mathrm{yd}}=\frac{\left({\rho }_{\mathrm{ref}}+\frac{C_2}{{\mathrm{TF}}_{e^{*}}^3}\right)\left({\mathrm{TFe}}^2\right)}{T_p^2{C}_1}.& \left(\mathrm{Equation}12\right)\end{array}$

According to Equation 12, the Young's modulus temperature correction for density TF.sub.yd may be determined based only on known fluid density ρ.sub.ref, the known fluid temperature temp.sub.ref, and the time period Tp.

[0074] Once the Young's modulus temperature correction for density TF.sub.yd is determined, method 300 may continue with step 316. In step 316, a Young's modulus temperature correction for mass flow TFy.sub.M is determined based on a temperature correction constant k multiplied by Young's modulus temperature correction for density TFy.sub.D, as represented by Equation 13:

TF.sub.ym=k*TF.sub.yD  (Equation 13)

The Young's modulus temperature correction for mass flow TFy.sub.M is generally related to torque in the flow tubes and the Young's modulus temperature correction for density TFy.sub.D 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 TFy.sub.M and the Young's modulus temperature correction for density TFy.sub.D 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 non-limiting example, k may be set to a value between 0.8 and 1.2.

[0075] Once the Young's modulus temperature correction for mass flow TFy.sub.M is determined, method 300 may continue with step 320. In step 320, a mass flow value {dot over (m)} determined using Equation 1 with Coriolis flow meter 100 is corrected using the Young's modulus temperature correction for mass flow TFy.sub.M. In embodiments, the mass flow value {dot over (m)} may be corrected using the Young's modulus temperature correction for mass flow TFy.sub.M via any method known to those of skill in the art.

[0076] In embodiments, step 320 may further comprise step 318, In step 318, a mass error value Error.sub.m may be determined using the Young's modulus temperature correction for mass TFy.sub.M and the expansion temperature correction for density TFe determined via steps 312 and 316:

[00010]$\begin{array}{cc}& \left(\mathrm{Equation}14\right)\end{array}$${\mathrm{Error}}_m=\left(\left(\frac{{\mathrm{TF}}_{\mathrm{ym}-\mathrm{cal}}\star {\mathrm{TF}}_{e-\mathrm{cal}}\star {\mathrm{PF}}_{m-\mathrm{cal}}\left(1+\frac{{\mathrm{error}}_{\mathrm{cal}}\%}{100}-\frac{Q_{m-\mathrm{zero}-\mathrm{cal}}}{Q_{m-\mathrm{cal}}}\right)}{{\mathrm{MF}}_{m-\mathrm{cal}}*{\mathrm{TF}}_{m-cal}*{\mathrm{PF}}_{m-cal}}\right)*\left(\frac{{\mathrm{MF}}_{m-\mathrm{oper}}{\mathrm{TF}}_{m-\mathrm{oper}}{\mathrm{PF}}_{m-\mathrm{oper}}}{{\mathrm{TF}}_{\mathrm{ym}}{\mathrm{TF}}_e{\mathrm{PF}}_{m-\mathrm{oper}}}\right)-1\right).$

In Equation 14:

[0077] Q.sub.m-zero-cal is the zero flow mass flow rate measured during factor calibration with a calibration fluid; [0078] Q.sub.m-cal is the mass flow rate measured during factory calibration with the calibration fluid; [0079] PF.sub.m-real-cal is a real pressure factor determined during calibration; [0080] PF.sub.m-cal is an applied pressure factor determined during calibration; [0081] PF.sub.m-oper is an applied pressure factor determined during operation; [0082] PF.sub.m-real-oper is real pressure factor determined during operation; [0083] Error.sub.cal% is the meter error determined during calibration; [0084] MF.sub.m-cal is a meter-specific factor for mass determined during calibration; [0085] MF.sub.m-oper is a meter-specific factor for mass determined during operation; [0086] TF.sub.e-cal is an expansion temperature correction for density determined during calibration; and [0087] TFy.sub.m-cal is a mass temperature correction for density determined during calibration.

[0088] The first part of Equation 14 comes from calibration and reflects the mass error value Error.sub.m at 0° C. and 0 barg, and the second part of Equation 14 comes from operation in the application and reflects the error from 0° C. and 0 barg to operating conditions. In practice, the first part of Equation 14 is small in relation to the second part, however. For that reason, in embodiments Equation 14 may be simplified to Equation 15:

[00011]$\begin{array}{cc}{\mathrm{Error}}_m=\left(\frac{{\mathrm{MF}}_{m-\mathrm{oper}}{\mathrm{TF}}_{m-\mathrm{oper}}{\mathrm{PF}}_{m-\mathrm{oper}}}{{\mathrm{TF}}_{\mathrm{ym}}{\mathrm{TF}}_e{\mathrm{PF}}_{m-\mathrm{oper}}}\right)-1.& \left(\mathrm{Equation}15\right)\end{array}$

[0089] In embodiments, a meter factor MF may be determined to correct the mass flow value {dot over (m)} measured with Coriolis flow meter 100 using Equation 16:

[00012]$\begin{array}{cc}\mathrm{MF}=\frac{1}{1+\frac{{\mathrm{Error}}_m}{100}}.& \left(\mathrm{Equation}16\right)\end{array}$

[0090] A corrected mass flow value {dot over (m)} may then be determined by multiplying the measured mass flow value {dot over (m)} by meter factor MF.

[0091] FIG. 4 depicts system 400. In embodiments, system 400 may be used to correct a mass flow value {dot over (m)} measured using a Coriolis flow meter 100 for temperature effects at a fluid temperature temp below 0 C. System 400 comprises fluid density receiving module 402, fluid temperature receiving module 404, period determination module 410, Young's modulus temperature correction for density determination module 414, Young's modulus temperature correction for mass flow determination module 416, and mass flow correction module 418. In embodiments, system 400 may further comprise fluid pressure determination module 406, phase difference determination module 408, and expansion temperature correction module 412.

[0092] Fluid density receiving module 402 is configured to determine a fluid density p, such as, for example, the known fluid density ρ.sub.ref. For example, fluid density receiving module 402 may execute step 302, described above.

[0093] Fluid temperature receiving module 404 is configured to determine the fluid temperature temp, such as, for example, the known fluid temperature temp. For example, fluid temperature receiving module 404 may execute step 304, as described above.

[0094] Fluid pressure determination module 406 is configured to determine a fluid pressure P. For example, fluid pressure determination module 406 may execute step 306, as described above.

[0095] Phase difference determination module 408 is configured to determine a phase difference ΔT. For example, phase difference determination module 408 may execute step 308, as described above.

[0096] Period determination module 410 is configured to receive a time period Tp. For example, period determination module 410 may execute step 310, as described above.

[0097] Expansion temperature correction module 412 is configured to determine an expansion temperature correction for density TFe. For example, expansion temperature correction module 412 may execute step 312, as described above.

[0098] Young's modulus temperature correction for density determination module 414 is configured to determine a Young's modulus temperature correction for density TFy.sub.D 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 414 may execute step 314, as described above.

[0099] Young's modulus temperature correction for mass flow determination module 416 is configured to determine a Young's modulus temperature correction for mass flow TFy.sub.M based on a temperature correction constant k and Young's modulus temperature correction for density TFy.sub.D. For example, Young's modulus temperature correction for mass flow determination module 416 may execute step 316, as described above.

[0100] Mass flow correction module 418 is configured to correct the mass flow value {dot over (m)} using the Young's modulus temperature correction for mass flow TFy.sub.M. For example, mass flow correction module 418 may execute step 318, as described above.

[0101] 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 {dot over (m)} with errors that are less than 0.10%. Some of the tests conducted by the Applicant provided mass flow errors that were as low as 0.07% and 0.01% for flow meters with flow tubes that are four 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 four inches, to provide very accurate mass flow values m for higher fluid flows.

[0102] 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.

[0103] The detailed descriptions of the above examples are not exhaustive descriptions of all examples contemplated by the inventors to be within the scope of the Application. Indeed, persons skilled in the art will recognize that certain elements of the above-described examples may variously be combined or eliminated to create further examples, and such further examples fall within the scope and teachings of the Application. It will also be apparent to those of ordinary skill in the art that the above-described examples may be combined in whole or in part to create additional examples within the scope and teachings of the Application. Accordingly, the scope of the Application should be determined from the following claims.