The choked mass flux density and the choked momentum flux density for the nonideal fluids methane and nitrogen have been calculated using the Soave–Redlich–Kwong equation of state (EoS). For the computation a steady, one-dimensional (1D), isenthalpic and isentropic flow is assumed. The developed algorithm for the calculation of the choked flow properties includes a bounded multidimensional Newton method. A possible second phase emerging in the critical nozzle area is excluded using the saturation properties of the considered fluids. The critical ratios of pressure, density, temperature, and speed of sound are discussed and compared to other publications. Formulations of the choked mass flux density and the choked momentum flux density explicit in Tr, pr, and Zr are given valid for different reduced pressures and temperatures depending on the fluid. Additional computational fluid dynamics (CFD) simulations are carried out in order to validate the findings of the algorithm and the proposed correlations.
Issue Section:
Fundamental Issues and Canonical Flows
References
1.
Tsien
, H.-S.
, 1946
, “One-Dimensional Flows of a Gas Characterized by Van Der Waal's Equation of State
,” Stud. Appl. Math.
, 25
(1–4
), pp. 301
–324
.2.
Van der Waals
, J. D.
, 1873
, Over de Continuiteit Van Den Gas-en Vloeistoftoestand
, Vol. 1
, Sijthoff
, Amsterdam, The Netherlands.3.
Donaldson
, C. D.
, 1948
, “Note on the Importance of Imperfect-Gas Effects and Variation of Heat Capacities on the Isentropic Flow of Gases
,” National Advisory Committee for Aeronautics, Langley Aeronautical Lab, Langley Field, VA, Report No. NACA RM No. L8J14
.https://ntrs.nasa.gov/search.jsp?R=199300854784.
Crown
, J.
, 1949
, “Flow of a Gas Characterized by the Beattie-Bridgeman Equation of State and Variable Specific Heats—Part I: isentropic Relations
,” Naval Ordnance Laboratory, Memorandam
, Vol. 9619
, U.S. Naval Ordnance Laboratory, White Oak, MD.5.
Johnson
, R. C.
, 1971
, “A Set of Fortran 4 Routines Used to Calculate the Mass Flow Rate of Natural Gas Through Nozzles
,” NASA Lewis Research Center, Cleveland, OH, Report No. NASA TM X-2240
.https://ntrs.nasa.gov/search.jsp?R=197100131406.
Johnson
, R. C.
, 1964
, “Calculations of Real-Gas Effects in Flow Through Critical-Flow Nozzles
,” ASME J. Basic Eng.
, 86
(3
), pp. 519
–526
.7.
Johnson
, R. C.
, 1970
, “Calculations of the Flow of Natural Gas Through Critical Flow Nozzles
,” ASME J. Basic Eng.
, 92
(3
), pp. 580
–586
.8.
Johnson
, R. C.
, 1971
, “Real Gas Effects in Flow Metering
,” NASA Lewis Research Center, Cleveland, OH, Report No. NASA-TM-X-52965
.https://ntrs.nasa.gov/search.jsp?R=197100091589.
Johnson
, R. C.
, 1965
, Real-Gas Effects in Critical-Flow-Through Nozzles and Tabulated Thermodynamic Properties
, Vol. 2565
, National Aeronautics and Space Administration
, Washington, DC.10.
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. 300 Atm)
,” NASA Lewis Research Center, Cleveland, OH, Report No. NASA SP-3046
.https://ntrs.nasa.gov/search.jsp?R=1969000676611.
Johnson
, R. C.
, 1972
, “Tables of Critical-Flow Functions and Thermodynamic Properties for Methane and Computational Procedures for Both Methane and Natural Gas
,” NASA Lewis Research Center, Cleveland, OH, Report No. NASA SP-3074
.https://ntrs.nasa.gov/search.jsp?R=1973000658212.
Benedict
, M.
, Webb
, G. B.
, and Rubin
, L. C.
, 1940
, “An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures I: Methane, Ethane, Propane and n-Butane
,” J. Chem. Phys.
, 8
(4
), pp. 334
–345
.13.
Beattie
, J. A.
, and Bridgeman
, O. C.
, 1928
, “A New Equation of State for Fluids
,” J. Am. Chem. Soc., 50
(12), pp. 3133
–3138
.14.
Leung
, J.
, and Epstein
, M.
, 1988
, “A Generalized Critical Flow Model for Nonideal Gases
,” AIChE J.
, 34
(9
), pp. 1568
–1572
.15.
Soave
, G.
, 1972
, “Equilibrium Constants From a Modified Redlich-Kwong Equation of State
,” Chem. Eng. Sci.
, 27
(6
), pp. 1197
–1203
.16.
Maytal
, B.-Z.
, 2006
, “Real Gas Choked Flow Conditions at Low Reduced-Temperatures
,” Cryogenics
, 46
(1
), pp. 21
–29
.17.
Klein
, S. A.
, and Alvarado
, F.
, 2002
, Engineering Equation Solver
, Vol. 1
, F-Chart Software
, Madison, WI
.18.
Ding
, H.
, Wang
, C.
, and Zhao
, Y.
, 2014
, “Flow Characteristics of Hydrogen Gas Through a Critical Nozzle
,” Int. J. Hydrogen Energy
, 39
(8
), pp. 3947
–3955
.19.
Leachman
, J. W.
, Jacobsen
, R. T.
, Penoncello
, S.
, and Lemmon
, E. W.
, 2009
, “Fundamental Equations of State for Parahydrogen, Normal Hydrogen, and Orthohydrogen
,” J. Phys. Chem. Ref. Data
, 38
(3
), pp. 721
–748
.20.
Thompson
, P. A.
, 1971
, “A Fundamental Derivative in Gasdynamics
,” Phys. Fluids
, 14
(9
), pp. 1843
–1849
.21.
Lambrakis
, K. C.
, and Thompson
, P. A.
, 1972
, “Existence of Real Fluids With a Negative Fundamental Derivative γ
,” Phys. Fluids
, 15
(5
), pp. 933
–935
.22.
Cramer
, M.
, and Best
, L.
, 1991
, “Steady, Isentropic Flows of Dense Gases
,” Phys. Fluids A
, 3
(1
), pp. 219
–226
.23.
Kluwick
, A.
, 1993
, “Transonic Nozzle Flow of Dense Gases
,” J. Fluid Mech.
, 247
(1
), pp. 661
–688
.24.
Schnerr
, G.
, and Molokov
, S.
, 1994
, “Exact Solutions for Transonic Flows of Dense Gases in Two-Dimensional and Axisymmetric Nozzles
,” Phys. Fluids
, 6
(10
), pp. 3465
–3472
.25.
Colonna
, P.
, and Guardone
, A.
, 2006
, “Molecular Interpretation of Nonclassical Gas Dynamics of Dense Vapors Under the Van Der Waals Model
,” Phys. Fluids
, 18
(5
), p. 056101
.26.
Harinck
, J.
, Guardone
, A.
, and Colonna
, P.
, 2009
, “The Influence of Molecular Complexity on Expanding Flows of Ideal and Dense Gases
,” Phys. Fluids
, 21
(8
), p. 086101
.27.
Guardone
, A.
, Spinelli
, A.
, and Dossena
, V.
, 2013
, “Influence of Molecular Complexity on Nozzle Design for an Organic Vapor Wind Tunnel
,” ASME J. Eng. Gas Turbines Power
, 135
(4
), p. 042307
.28.
Guardone
, A.
, 2015
, “Effects of Molecular Complexity and Reservoir Conditions on the Discharge Coefficient of Adapted Planar Nozzles
,” J. Phy.: Conf. Ser.
, 633
, p. 012092
.29.
Guggenheim
, E. A.
, 1945
, “The Principle of Corresponding States
,” J. Chem. Phys.
, 13
(7
), pp. 253
–261
.30.
Pitzer
, K. S.
, Lippmann
, D. Z.
, Curl
, R.
, Jr., Huggins
, C. M.
, and Petersen
, D. E.
, 1955
, “The Volumetric and Thermodynamic Properties of Fluids–II: Compressibility Factor, Vapor Pressure and Entropy of Vaporization
,” J. Am. Chem. Soc.
, 77
(13
), pp. 3433
–3440
.31.
Eifler
, W.
, Schlücker
, E.
, Spicher
, U.
, and Will
, G.
, 2009
, Küttner Kolbenmaschinen: Kolbenpumpen, Kolbenverdichter, Brennkraftmaschinen
, Springer-Verlag
, Berlin.32.
Abdi
, M. A.
, Jassim
, E.
, Haghighi
, M.
, and Muzychka
, Y.
, 2010
, “Applications of CFD in Natural Gas Processing and Transportation
,” Computational Fluid Dynamics
, InTech
, London.33.
Akansu
, S. O.
, Dulger
, Z.
, Kahraman
, N.
, and Veziroğlu
, T. N.
, 2004
, “Internal Combustion Engines Fueled by Natural Gas-Hydrogen Mixtures
,” Int. J. Hydrogen Energy
, 29
(14
), pp. 1527
–1539
.34.
Erfan
, I.
, Chitsaz
, I.
, Ziabasharhagh
, M.
, Hajialimohammadi
, A.
, and Fleck
, B.
, 2015
, “Injection Characteristics of Gaseous Jet Injected by a Single-Hole Nozzle Direct Injector
,” Fuel
, 160
, pp. 24
–34
.35.
Saad
, M. A.
, 1985
, Compressible Fluid Flow
, Prentice Hall
, Englewood Cliffs, NJ
, p. 570
.36.
Banholzer
, M.
, Müller
, H.
, and Pfitzner
, M.
, 2017
, “Numerical Investigation of the Flow Structure of Underexpanded Jets in Quiescent Air Using Real-Gas Thermodynamics
,” AIAA
Paper No. 2017–4289
.37.
Hirsch
, C.
, 2007
, Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics
, Butterworth-Heinemann
, Oxford, UK.38.
Gernert
, J.
, Jäger
, A.
, and Span
, R.
, 2014
, “Calculation of Phase Equilibria for Multi-Component Mixtures Using Highly Accurate Helmholtz Energy Equations of State
,” Fluid Phase Equilib.
, 375
, pp. 209
–218
.39.
Ouellette
, P.
, and Hill
, P.
, 2000
, “Turbulent Transient Gas Injections
,” ASME J. Fluids Eng.
, 122
(4
), pp. 743
–752
.40.
Baud
, M.
, 1993
, “Data Analysis, Mathematical Modeling
,” Methods of Immunological Analysis, Vol. 1, N. Staines, ed., VCH Publishers, New York, pp. 656
–671
.41.
Kraposhin
, M. V.
, Banholzer
, M.
, Pfitzner
, M.
, and Marchevsky
, I. K.
, 2018
, “A Hybrid Pressure-Based Solver for Non ideal Single-Phase Fluid Flows at All Speeds
,” Int. J. Numer. Methods Fluids
, 88
, pp. 79–99.42.
Traxinger
, C.
, Banholzer
, M.
, and Pfitzner
, M.
, 2018
, “Real-Gas Effects and Phase Separation in Underexpanded Jets at Engine-Relevant Conditions
,” AIAA
Paper No. 2018–1815
.Copyright © 2019 by ASME
You do not currently have access to this content.
