1. Aaron Johnson National Institute of Standard and Technology Gaithersburg, MD 20899 CFV Measurement Conference September 20, 2013 Poitiers, France The Critical Flow Function and Beyond (Real Gas Corrections for CFVs)

2. Objectives 1)To suggest the use of REFPROP Thermodynamic Database for calculating C* 2)To introduce real gas corrections for large  = d/D applications 3)To present results from measuring C* experimentally

3. Outline Background on CFV Theory & Definitions Listing of Most Common Methods for computing C*  Approximate Analytical Techniques  Tables and Curve Fits  Thermodynamic databases Overview of REFPROP Evaluation of various methods used for computing C* Real Gas Effects in large  = d/D Applications Measurement of C* Discussion

4. Baseline CFV Flow Model 0 T u R 4 0 P 2 d M   th m  * C i Three Basic Assumptions 1) One-dimensional flow 2) Isentropic flow 3) Perfect Gas (Z=1, c P =const) Steady Navier-Stokes Flow Model * C i                        12 1 2 1 + Error Factorization ideal critical flow function         k = 1, 2, or 3 C d, k = 1 –  C d,k

5. Higher Order CFV Models Three Different Models based on Different Simplifications of the Navier-Stokes Equations = f 2 (Re, ,  ) = f 1 ( ,  )  = r/r c d1d1 C  1 m  th m  1) Correction for Sonic Line Curvature (Kliegel) Inviscid Flow (no B.L.) Perfect Gas (Z=1 & c P =const) Assumptions 2 & 3 enforced d2d2 C 2 m   th m  2) Correction for B.L. (Tang, Geropp) 1D flow (flat sonic line) Perfect Gas (Z=1 & c P =const) Assumptions 1 & 3 enforced d3d3 C 3 m   th m  3) Correction for Real Gas Effects (Johnson) 1D flow (flat sonic line) Inviscid Flow (no B.L.) Assumptions 1 & 2 enforced 0 T u R 4 * C 0 P 2 d M   th m  i  * C R * C i = f 2 (Re, ,  ) d2d2 C 2 m   th m  2) Correction for B.L. (Tang, Geropp) 1D flow (flat sonic line) Perfect Gas (Z=1 & c P =const) Assumptions 1 & 3 enforced Basic Equations s = const h 0 = h + ½ u 2 = const = f 1 ( ,  )  = r/r c d1d1 C  1 m  th m  1) Correction for Sonic Line Curvature (Kliegel) Inviscid Flow (no B.L.) Perfect Gas (Z=1 & c P =const) Assumptions 2 & 3 enforced 3 m  4 2 d    u    0 T u R 4 * C 0 P 2 d M   R

6. * * R d CiCi C C 3  All Real Gas Behavior Accounted for in 3 d C Divide by 3 d C to eliminate Real Gas Effects How to Define C d Independent of Real Gas Effects? = f 1 ( ,  ) d1d1 C (Correction for Sonic Line Curvature) = f 2 (Re, ,  ) d2d2 C (Correction for B.L.) Physically C d < 1 C R eliminates real gas effects (if sufficiently accurate)( *

7. Numerous Numerous C* Models are Used by End Users Accuracy between different models can vary significantly Many C* models are tailored for a specific gas type  End user must acquire different models for each gas type The numerous C* models can be confusing to end users  Functional expressions for C* based on approximate analytical solutions  Tables and Curve fits provided in CFV Standards (ISO 9300 and ASME)  Various published C* values for different gases N 2, Air, CO 2, Ar, He, and others (R.C. Johnson 1965, NASA TND-2565 ) Steam (Owen & Amini, 1994 and 1997) wet air (Aschenbrenner, 1983, Britton et. al. 1998) dry & humid air, natural gas, methane and other gases (Sullivan, 1980’s) N 2, Ar, CH 4, dry & humid air, and natural gas (Stewart et. al. 1999, 2000) Tables of Thermodynamic and Transport Properties (Hilsenrath, 1960)  Thermodynamic Databases GERG (2004) AGA 8 (1992) AGA 10 (based on AGA 8, 2003) REFPROP 9.1

8. Overview of REFPROP REFPROP is an acronym for REFerence fluid PROPerties Based on the most accurate pure fluid and mixture models currently available  Maintained by NIST (Eric W. Lemmon)  Continuously updated (next version is being developed)  More than 50 pure fluids  Flexibility to create your own mixture (e.g., wet air, natural gas) REFPROP 9.1 Includes multiple Thermodynamic Databases  GERG-2004 Model (Prof. Dr. Wolfgang Wagner and Dr. Oliver Kunz) 22,000 experimental natural gas data and natural gas like multicomponent data  Modified GERG-2004 Model (Default Model in REFPROP) NIST modified the original GERG model making it more accurate Mixture parameters are identical to GERG Model Pure fluid equations of state are more complex and more accurate  AGA8 (1992) REFPROP Platform and Interface Capabilities  Stand alone graphical user interface (GUI)  Compatible with Excel, Fortran, Visual Basic, C++, MatLab, LabVIEW, Delphi Computes over 75 Thermodynamic properties (gas, liquid, and two phase)  Density, Specific Heat, Enthalpy, Compressibility Factor, etc.  C* is one of the properties applicable to the gas phase for CFV flows

9. Objectives 1)To suggest the use of REFPROP Thermodynamic Database for calculating C* 2)To introduce real gas corrections for large  = d/D applications 3)To present results from measuring C* experimentally

10. Rapidly Converging C* Computations 1D Steady Navier-Stokes Flow Model  Isentropic flow: s(, ) = s(, )  Isoenergetic flow: h(, ) = h(, ) + a(, ) 2 /2 T0T0 P0P0 T*T* P*P* T*T* P*P* T*T* P*P* T0T0 P0P0 s*s* s*s* Numerical Method Updated Temperature

11. CFVs are Used to Determine Flow PT Flow CFV (Critical Flow Venturi) C d C* T0T0 RuRu 4 P0P0 d 2  m   M C* – Critical flow function used during CFV application C d – based on flow calibration of CFV C* m  P 0 d 2  M T 0 R u 4 C d  Uncertainty in m depends on level of correlation between C d & C* - used during calibration

12. 1D Steady Navier-Stokes Flow Model  Isentropic flow: s(, ) = s(, )  Isoenergetic flow: h(, ) = h(, ) + a(, ) 2 /2 N/ATables N/ACurve Fits Real Gas Polytropic Process Ideal Gas Critical Flow Function Formula Thermo. Expressions Model Definition of and Methods used for Computing C* Fits ofTables of T0T0 P0P0 TtTt PtPt TtTt PtPt TtTt PtPt T0T0 P0P0 REFPROP 8.0  Modified GERG (R8,NIST)  GERG (R8,GERG)  AGA 8 (R8,AGA) AGA 10 (AGA10) REFPROP 7.0 (R7) ISO 9300 CFV Standard (2005)

13. -10 -5 0 5 10 050100150200 Evaluation of the Ideal Gas Model for C* 1)  = const2)  =  (T)3)  =  (T,P) How to implement the method? T 0 = 293.15 K P ref = 101.325 kPa CH 4 dry air He N2N2 H2H2 Ar O2O2 CO 2 + 9.3% + 0.2% + 2.1% + 0.5% + 2.7% - 2.1% - 1.8% - 2.2% Max Error C i *

14. Evaluation of the Polytropic Model for C* How to implement the method? CH 4 dry air He N2N2 H2H2 Ar O2O2 CO 2 + 4.9% + 2.7% + 0.5% + 2.7% + 0.8% + 2.1% + 2.6% - 0.3% -10 -5 0 5 10 050100150200 T 0 = 293.15 K P ref = 101.325 kPa + 0.2% + 2.1% + 0.5% + 2.7% - 2.1% - 1.8% - 2.2% + 9.3% Max Error C i * Max Error C p * 1) 2)

15. 1 st Ed. 1990 Evaluation of ISO 9300 Tabulated C* Values ISO 9300 C* Tables (2005) Range of Gas Types and Conditions  7 Gas Types: (CO 2, Ar, N 2, Ar, CH 4, Air, Steam)  T 0 range: 200 K to 600 K  P 0 range: up to 20 MPa Uncertainty of C* = 0.1 % (k = 2)  Generally good agreement with R8 NIST  Interpolation Errors can be Significant Table B.1: C* values for Methane 0.89018 2.7 % 0.91403230 0.83585 0.99220 C ISO * 2 nd Ed. 2005 240 220 T 0 (K) P 0 = 8 MPa 0.83585 0.99220 0 % C R8,NIST * % Diff. R8 NIST ISO Table Interpolation Error ≈ 2.7 % C* Uncertainty Limited gas types and P 0 and T 0 range Not practical to Tabulate Mixtures  Wet Air  Natural Gas

16. Composition Mole Percent 1– 2.50 – 2.5 Carbon Dioxide 0 – 1.50 – 3 Nitrogen 0 – 0.30 – 0.20 – 0.15Hexane + 0 – 0.50 – 0.40 – 0.2Pentane 0.3 – 1.50.2 – 10 – 0.5Butane 1.5 – 40.8– 30.2 – 2Propane 8 – 11.54.5 – 81 – 4.5Ethane 79 – 8884 – 9389 – 98Methane Range 3Range 2Range 1 Components Evaluation of ISO 9300 Curve Fit C* Values (Natural Gas Mixtures) Fit developed by Stewart (2000) Fit Variables: C* = C*(T 0, P 0, x k )  Fitted T 0 range: 270 K to 320 K  Fitted P 0 range: up to 12 MPa  Composition: 8 components valid for 3 ranges of gas composition Expanded Uncertainty  Depends on gas composition 0.1 % for a natural gas mixture that entirely fits into any one of the 3 ranges 0.15 % for a natural gas mixture that does not fit all eight components, but fits within one of the 3 ethane ranges The curve fit is very complex different fit for each of the 3 composition ranges The fit for each range has 85 parameters A total of 255 parameters!

17. Reference Natural Gas Mixtures (AGA Report 8) 0.159900000 0.03500000 0.59950000.03930.0664 1.5987000.0480.04430.0324 0.003000.05090.03210.0473 0.0030.1520.1040.35060.15630.1007 0.02010.1510.10.34860.10370.0977 0.17740.8950.6052.30150.8280.4596 1.89854.3033.38.49194.52791.8186 0.01547.5850.9851.49540.46760.5956 0.0055.70213.4651.00683.12840.2595 95.484581.21281.44185.906390.672496.5222 CEESI Iowa High N 2 -CO 2 High N 2 EkofiskAmarillo Gulf Coast n-Pentane i-Pentane n-Butane i-Butane Helium Hydrogen n-Hexane Propane Ethane Carbon Dioxide Nitrogen Methane Component Composition Mole Percent Range 1 (0.1%) Range 2E (0.15%) Composition Mole Percent Range 3 (0.1%) Range 1E (0.15%) Range 1E (0.15%) Stewart Comp. Range

18. Gulf Coast (Range 1) Ekofisk (Range 3) Amarillo (Range 2E) High N 2 - CO 2 (Range 1E) Methane High N 2 (Range 1E) CEESI Iowa 0.00 0.10 0.20 -0.20 -0.10 0.30 -0.30 020406080100 Evaluation of ISO 9300 Curve Fit Gulf Coast & Ekofisk Curve Fits < 0.1 % Unc. Other gases generally, within 0.15 % Unc. Limit (Natural Gas Mixtures) T 0 = 293.15 K P ref = 101.325 kPa

19. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST (Pure Methane Gas) T 0 = 293.15 K P ref = 101.325 kPa Max Diff. 0.0 % -0.021 % 0.0 % Max diff. occurs for AGA 10 (-0.021 %)

20. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST (Gulf Coast ≈ 96.5 % Methane Gas) T 0 = 293.15 K P ref = 101.325 kPa Max Diff. -0.018 % +0.029 % +0.018 % +0.006 % 0.0 % Max diff. occurs for AGA 10 (+0.029 %)

21. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST (Amarillo ≈ 90.7 % Methane Gas) T 0 = 293.15 K P ref = 101.325 kPa Max Diff. -0.046 % -0.024 % +0.005 % +0.006 % 0.0 % Max diff. occurs for REFPROP 7 (-0.046 %)

22. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST (Ekofisk ≈ 85.9 % Methane Gas) T 0 = 293.15 K P ref = 101.325 kPa Max Diff. -0.064 % +0.084 % +0.09 % +0.011 % 0.0 % Max diff. occurs for AGA 10 (+0.084 %)

23. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases (High N 2 ≈ 81.4 % Methane & 13.5 % Nitrogen ) REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST T 0 = 293.15 K P ref = 101.325 kPa Max Diff. -0.064 % -0.057 % -0.048 % -0.006 % 0.0 % Max diff. occurs for REFPROP 7 (-0.064 %)

24. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases (High N 2 -CO 2 ≈ 81.2 % Methane, 5.7 % Nitrogen, & 7.6 % Carbon Dioxide ) REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST T 0 = 293.15 K P ref = 101.325 kPa Max Diff. -0.016 % 0.026 % +0.012 % +0.006 % 0.0 % Max diff. occurs for AGA 10 (+0.026 %)

25. -0.100 -0.075 -0.050 -0.025 0.000 0.025 0.050 0.075 0.100 020406080100 Comparison of C* for Selected Thermodynamic Databases REFPROP 7 AGA10 R8, AGA8 R8,GERG R8,NIST (CEESI Iowa ≈ 95.5 % Methane, 0.04 % Hydrogen, & 0.16 % Helium ) T 0 = 293.15 K P ref = 101.325 kPa Max Diff. -0.018 % -0.026 % -0.018 % -0.005 % 0.0 % Max diff. occurs for AGA 10 (-0.026 %)

26. Conclusions Ideal gas & polytropic C* values can introduce significant errors (> 2 % depending on gas type and operating conditions) Interpolating tabulated C* values can cause significant error Curve Fits for Natural gas C* values  Generally not as accurate as any of the thermodynamic databases  Limited operating conditions and gas compositions  Very complicated (more than 255 fit coefficients) Thermodynamic Databases  C* values of the GERG and Modified GERG < 0.01 % for all gases  C* values agreed with R8 NIST to better than 0.1 % for all natural gas mixtures  Largest differences occurred at lower methane concentrations  Discontinuity in AGA 10 C* Values Future Work Repeat the analysis at more temperatures

27. Objectives 1)To suggest the use of REFPROP Thermodynamic Database for calculating C* 2)To introduce real gas corrections for large  = d/D applications 3)To present results from measuring C* experimentally

28. Real Gas Corrections for Large  = d/D > 0.25 CFV applications measure the recovery temp. (T m ) and the static pres. (P) Stagnation conditions T 0 and P 0 are necessary to compute o Critical flow function; C r * = C r * (T 0, P 0 ) o Mass flow; Stagnation conditions are based on 1) Ideal gas or 2) Polytropic gas o Ideal Gas; o Polytropic Gas; Large errors can result for  > 0.25 when real gas effects are significant P TmTm Flow CFV (Critical Flow Venturi) d D m 0u d * 0 4 2 TR CCPd M    &&

29. Required Inputs 1)Diameter ratio:  = d/D 2)Measured Temperature: T m 3)Measured Pressure: P How do you compute P 0 and T 0 for large  ? 1D Steady Navier-Stokes Flow Model 1)Isentropic Flow: s(T 0, P 0 ) = s(T *, P * ) 2)Isoenergetic Flow: h(T 0, P 0 ) = h(T *, P * ) + a(T *, P * ) 2 /2 3)Isentropic Flow: s(T 0, P 0 ) = s(T, P) 4)Isoenergetic Flow: h(T 0, P 0 ) = h(T, P) + u 2 /2 5)Mass Conservation:  (T *, P * )a(T *, P * )  2 =  (T, P)u 6)Recovery Factor (RF): RF = (T m – T )/ (T 0 – T )

30. Evaluation of Ideal and Polytropic Gas Models Comparison Parameters 1)% Difference T 0 2)% Difference P 0 3)% Difference C* 4)% Difference (Theoretical Mass Flux) % Diff x Example: % Diff T 0

31. % Diff T 0 for Methane (Ideal Gas Model)

32. % Diff P 0 for Methane (Ideal Gas Model)

33. % Diff C* for Methane (Ideal Gas Model)

34. % Diff Mass Flux for Methane (Ideal Gas Model)

35. % Diff T 0 for Methane (Polytropic Gas Model)

36. % Diff P 0 for Methane (Polytropic Gas Model)

37. % Diff C* for Methane (Polytropic Gas Model)

38. % Diff Theoretical Mass Flux for Methane (Polytropic Gas Model)

39. Objectives 1)To suggest the use of REFPROP Thermodynamic Database for calculating C* 2)To introduce real gas corrections for large  = d/D applications 3)To present results from measuring C* experimentally

40. A Technique for Measuring C R * Measure CFV mass flow with low uncertainty standard in gas A Measure CFV mass flow with low uncertainty standard in gas B at the same Reynolds number: Determine C B using the following expression: * o Gas A is selected so that it behaves closely to ideal gas o C* can be calculated by REFPROP at low uncertainty o Gas B has significant real gas effects

41. A Technique for Measuring C R (Cont) * Advantages No geometric dependence o it can be applied to small CFVs at high pressures o Analytical CFV theory can be used to estimated d C d Ratio approaches unity at high Reynolds numbers All of the dependents can be measured at low uncertainty Requirement Low uncertainty primary standard capable of measuring multiple gas compositions

42. CFV (d = 0.387 mm) Calibrated using NIST 34 L PVTt std. (2002) Nitrogen used for Gas A (P 0 = 200 to 650 kPa) Argon used for Gas B (P 0 = 200 to 650 kPa) CFV theory used to for C d ratio Both gases calibrated over Reynolds numbers from 7 000 to 30 000 Comparison of Measured C meas to Computed C REFPROP ** -0.10 -0.05 0.00 0.05 0.10 0200400600800 Argon Helium

43. Some Disccussion Topics 1)How are you currently computing C*? 2)Does it make sense to standardize the software used to compute C*? 3)The ISO 9300 includes a multi-parameter curve fit for select gases to correct the CFV mass flux for large . Would it be useful to have REFPROP make these corrects? 4)Other questions and points of dissusion