Relaxation Effects in Small Critical Nozzles

Aaron N. Johnson; Charles L. Merkle; Michael R. Moldover; John D. Wright

doi:10.1115/1.2137346

Journal of Fluids Engineering | Volume 128 | Issue 1 | TECHNICAL PAPERS

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TECHNICAL PAPERS

Relaxation Effects in Small Critical Nozzles

Aaron N. Johnson, Charles L. Merkle, Michael R. Moldover and John D. Wright

[+] Author and Article Information

Aaron N. Johnson

National Institute of Standards and Technology, 100 Bureau Drive, Stop 8361, Gaithersburg, Maryland, 20899-8361aaron.johnson@nist.gov

Charles L. Merkle, Michael R. Moldover, John D. Wright

National Institute of Standards and Technology, 100 Bureau Drive, Stop 8361, Gaithersburg, Maryland, 20899-8361

Standard reference conditions are at $293.15 K$ and $101.325 kPa$ .

The measured value of the throat diameter was adjusted by less than one micron by matching the experimental $C_{d}$ data to the computed results for $N_{2}$ .

Even after correcting for real gas behavior the discharge coefficient has a weak dependence on $γ$ that diminishes with increasing Reynolds number.

The time marching approach differs from commonly used iterative approaches which omit the time derivative term when applied to steady state problems.

The contribution of the electronic energy is negligible over the temperature range of interest.

Greater than unity $C_{d}$ values are sometimes caused by inaccurate values of the CFV throat diameter. This is especially true for small-sized CFVs where the throat diameter is more difficult to measure.

J. Fluids Eng 128(1), 170-176 (Aug 10, 2005) (7 pages) doi:10.1115/1.2137346 History: Received July 10, 2003; Revised August 10, 2005

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Abstract

We computed the flow of four gases (He, $N_{2}$ , $C O_{2}$ , and $S F_{6}$ ) through a critical flow venturi (CFV) by augmenting traditional computational fluid dynamics (CFD) with a rate equation that accounts for $τ_{relax}$ , a species-dependent relaxation time that characterizes the equilibration of the vibrational degrees of freedom with the translational and rotational degrees of freedom. Conventional CFD $(τ_{relax} = 0)$ underpredicts the flow through small CFVs (throat diameter $d = 0.593 mm$ ) by up to 2.3% for $C O_{2}$ and by up to 1.2% for $S F_{6}$ . When we used values of $τ_{relax}$ from the acoustics literature, the augmented CFD underpredicted the flow for $S F_{6}$ by only 0.3%, in the worst case. The augmented predictions for $C O_{2}$ were within the scatter of previously published experimental data $(\pm 0.1 %)$ . As expected, both conventional and augmented CFD agree with experiments for He and $N_{2}$ . Thus, augmented CFD enables one to calibrate a small CFV with one gas (e.g., $N_{2}$ ) and to use these results as a flow standard with other gases (e.g., $C O_{2}$ ) for which reliable values of $τ_{relax}$ and the relaxing heat capacity are available.

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References

Wright, J. D., 1998, “The Long Term Calibration Stability of Critical Flow Nozzles and Laminar Flowmeters,” NCSL Conference Proceedings , Albuquerque, NM, pp. 443–462.

Wright, P. H., 1985, “The Application of Sonic Nozzles to the Automated Accuracy Testing of Gas Flow Meters,” Proc. International Conference on Flow Measurement , Australia, pp. 152–156.

Nakao, S., Hirayama, T., and Takamoto, M., 1997, “Effects of Thermalphysical Properties of Gases on the Discharge Coefficients of the Sonic Venturi Nozzle,” Proceedings of the 1997 ASME Fluids Engineering Division Summer Meeting , Vancouver, British Columbia, Canada, June 22–26.

Hall, I. M., 1962, “Transonic Flow in Two-Dimensional and Axially-Symmetric Nozzles,” Q. J. Mech. Appl. Math., XV , Pt. 4, pp. 487–508.

Smith, R. E., and Matz, R. J., 1962, “A Theoretical Method of Determining Discharge Coefficients for Venturis Operating at Critical Flow Conditions,” J. Basic Eng., 19 , pp. 434–446.

Tang, S. P., 1969, “Theoretical Dependence of the Discharge Coefficients of Axisymmetric Nozzles Under Critical Flows,” Technical Report PR-118-PU, Department of Mechanical Engineering, Princeton Univ., Princeton, NJ.

Tang, S., 1969, “Discharge Coefficients for Critical Flow Nozzles and Their Dependence on Reynolds Numbers,” Ph.D. thesis, Princeton Univ., Princeton, NJ.

Geropp, D., 1971, “Laminare Grenzschichten In Ebenen Und Rotationssymmetrischen Lavalduesen,” Deutsche Luft-Und Raumfart, Forschungsbericht, pp. 71–90.

Johnson, A. N., Espina, P. I., Mattingly, G. E., Wright, J. D., and Merkle, C. L., 1998, “Numerical Characterization of the Discharge Coefficient in Critical Nozzles,” "Proceedings of the 1998 NCSL Workshop Symposium", Albuquerque, NM.

Johnson, A. N., Wright, J. D., Nakao, S., Merkle, C. L., and Moldover, M. R., 2000, “The Effect of Vibrational Relaxation on the Discharge Coefficient of Critical Flow Venturis,” Flow Meas. Instrum. [CrossRef], 11 , pp. 315–327.

Johnson, A. N., 2000, “Numerical Characterization of the Discharge Coefficient in Critical Nozzles,” Ph.D. thesis, The Pennsylvania State University, University Park, PA.

ISO 9300: (E)., 1990, “Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles,” Geneva Switzerland.

Bhatia, A. B., 1967, "Ultrasonic Absorption: An Introduction to the Theory of Sound Absorption and Dispersion in Gases, Liquids, and Solids", Oxford University Press, Inc., London.

O’Connor, L. C., 1954, “Thermal Relaxation of Vibrational States in Sulfur Hexafluoride,” J. Acoust. Soc. Am., 2 (3), pp. 361–364.

Estrada-Alexanders, A. F., and Trusler, J. P. M., 1999, “Speed of Sound in Carbon Dioxide at Temperatures between (200and450)K and Pressures up to 14MPa,” J. Chem. Thermodyn., 31 (5), pp. 1589–1601.

Breshears, W. D., and Blair, L. S., 1973, “Vibrational Relaxation in Polyatomic Molecules: SF6,” J. Chem. Phys. [CrossRef], 59 (11), pp. 5824–5827.

John, J. E., 1984, "Gas Dynamics", 2nd ed., Allyn and Bacon, Inc., Boston.

Anderson, J. D., 1982, "Modern Compressible Flow with a Historical Perspective", McGraw-Hill, Inc., New York.

Shapiro, A. H., 1954, "The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. II", The Ronald Press Co., New York.

Johnson, R. C., 1963, “Calculation of Real-Gas Effects in Flow Through Critical-Flow-Nozzles,” ASME paper no. 63-WA-71.

Johnson, R. C., 1964, “Calculations of Real-Gas Effects in Flow Through Critical Nozzles,” J. Basic Eng., 1964 , p. 519.

Johnson, R. C., 1965, “Real-Gas Effects in Critical-Flow-Through Nozzles and Tabulated Thermodynamic Properties,” Nasa Technical Note D-2565, National Aeronautics and Space Administration, Washington, DC

Johnson, R. C., 1968, “Real-Gas Effects in Critical Flow Through Nozzles and Thermodynamic Properties of Nitrogen and Helium at Pressures to 300×105 Newtons Per Square Meter (Approx. 300atm),” NASA Technical Note SP-3046, National Aeronautics and Space Administration, Washington, DC.

Johnson, R. C., 1971, “Real Gas Effects in Flowmetering,” Symposium on Flow , ISA, Pittsburgh, PA.

White, F. M., 1991, "Viscous Fluid Flow", McGraw-Hill Inc., New York.

Hirsch, C., 1988, "Numerical Computation of Internal and External Flows: Volume 1, Fundamentals of Numerical Discretization", John Wiley & Sons, Inc., New York.

Hirsch, C., 1990, "Numerical Computation of Internal and External Flows: Volume 2, Computational Methods for Inviscid and Viscous Flows", John Wiley & Sons, Inc., New York.

Davis, D. F., and Snider, A. D., 1991, "Introduction to Vector Analysis, Sixth Edition", Wm. C. Brown Publishers, Dubuque, IA.

Hilsenrath, J., Beckett, C. W., Benedict, W. S., Fano, L., Hoge, H. J., Masi, J. F., Nuttall, R. L., Touloukian, Y. S., and Wooley, H. W., 1955, “Tables of Thermal Properties of Gases,” U.S. Department of Commerce NBS Circular 564.

McCarty, R. D., and Arp, V. D., 1990, “A New Wide Range Equation of State for Helium,” Adv. Cryog. Eng., 35 , pp. 1465–1475.

Arp, V. D., McCarty, R. D., and Friend, D. G., 1998, “Thermophysical Properties of Helium-4 from 0.8to1500K with Pressures to 2000MPa,” NIST Technical Note 1334, Boulder, CO.

Hands, B. A., and Arp, V. D., 1981, “A Correlation of Thermal Conductivity Data for Helium,” Cryogenics [CrossRef], 21 (12), pp. 697–703.

Lambert, J. D., 1977, "Vibrational and Rotational Relaxation in Gases", Oxford University Press, Oxford.

Kruger, C. H., and Walter, G. V., 1965, "Introduction to Physical Gas Dynamics", John Wiley & Sons, Inc., New York.

Anderson, D. A., Tannehill, J. C., and Pletcher, R. H., 1984, "Computational Fluid Mechanics and Heat Transfer", Hemisphere Publishing Corporation, New York.

Buelow, P., Venkateswaran, S., and Merkle, C. L., 1994, “The Effect of Grid Aspect Ratio on Convergence,” AIAA J., 32 , pp. 2401–2406.

Feng, J., and Merkle, C. L., 1990, “Evaluation of Preconditioning Methods for Time Marching Systems,” AIAA paper no. 90-0016, AIAA 28th Aerospace Sciences Meeting, Reno, NV.

Buelow, P. E. O., 1995, “Convergence Enhancement of Euler and Navier-Stokes Algorithms,” Ph.D. thesis, Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA.

DuChateau, P., and Zachmann, D., 1989, “Applied Partial Differential Equations,” Harper and Row Publishers, Inc., New York.

Edwards, C. H., and Penny, D. E., 1989, “Elementary Differential Equations With Boundary Value Problems,” Prentice-Hall, Inc., Englewood, NJ.

Figures

Comparison of experimental calibration data of four gases with equilibrium (open symbols in (b)) and nonequilibrium CFD data (closed symbols) over a Reynolds number range from 2000 to 40,000

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Figure 1

Comparison of experimental calibration data of four gases with equilibrium (open symbols in (b)) and nonequilibrium CFD data (closed symbols) over a Reynolds number range from 2000 to 40,000

Critical nozzle used in this study. The diameter of the throat was d=0.593mm.

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Figure 2

Critical nozzle used in this study. The diameter of the throat was d=0.593mm.

Effective critical flow function versus reference value of Γ* evaluated at the nozzle throat for CO2 (left) and SF6 (right)

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Figure 3

Effective critical flow function versus reference value of Γ* evaluated at the nozzle throat for CO2 (left) and SF6 (right)

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Johnson AN, Merkle CL, Moldover MR, Wright JD. Relaxation Effects in Small Critical Nozzles. ASME. J. Fluids Eng. 2005;128(1):170-176. doi:10.1115/1.2137346.

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