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Viscous effects on real gases in quasi-one-dimensional supersonic convergent divergent nozzle flows

Published online by Cambridge University Press:  03 November 2022

Julián Restrepo Open the ORCID record for Julián Restrepo [Opens in a new window]  and
José R. Simões-Moreira Open the ORCID record for José R. Simões-Moreira [Opens in a new window]
Julián Restrepo*
Affiliation:
SISEA – Renewable and alternative Energy Systems Laboratory in the Mechl. Eng. Department at Escola Politécnica, Universidade de São Paulo, Av. Prof. Mello Moraes, 2231, 05508-900 São Paulo, SP, Brazil
José R. Simões-Moreira
Affiliation:
SISEA – Renewable and alternative Energy Systems Laboratory in the Mechl. Eng. Department at Escola Politécnica, Universidade de São Paulo, Av. Prof. Mello Moraes, 2231, 05508-900 São Paulo, SP, Brazil
*
†Email addresses for correspondence: restrepo@usp.br, jcrestrepol@hotmail.com
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Abstract

Viscous effects on an ideal gas flow in a supersonic convergent–divergent nozzle are a well-studied subject in classical gas dynamics. However, the ideal assumption fails on fluids that exhibit complex behaviours such as near-critical-region and non-ideal dense vapours. Under those conditions, a realistic equation of state (EOS) plays a vital role for a precise and realistic computation. This work examines the problem for solving the quasi-one-dimensional viscous compressible flow using a realistic EOS. The governing equations are discretised and solved using the fourth-order Runge–Kutta method coupled with a state-of-the-art EOS to calculate the thermodynamic properties. The role of the Grüneisen parameter along with viscous and real gas effects and their influence on the sonic point formation are discussed. The study shows that the flow may not achieve the supersonic regime for any pressure ratio depending on the combination of that parameter and the normalised friction factor. In addition, the analysis yields the discharge coefficient and the isentropic nozzle efficiency, which may achieve maximum values as a function of the stagnation conditions. Finally, the study also evaluates the formation and intensity of normal shock waves by using the Rankine–Hugoniot relations, which now depend on the real gas and viscous effects in opposition to the inviscid solution. Moreover, the methodology used captures the sonic point and shock wave position by a space marching algorithm using the Brent method for scalar minimisation. Experimental data available in the open literature corroborate the approach.

JFM classification

Compressible Flows: Gas dynamics Compressible Flows: Shock waves Compressible Flows: Supersonic flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A14
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Aidoun, Z., Ameur, K., Falsafioon, M. & Badache, M. 2019 Current advances in ejector modeling, experimentation and applications for refrigeration and heat pumps. Part 1: single-phase ejectors. Inventions 4 (1), 15.CrossRefGoogle Scholar
Arina, R. 2004 Numerical simulation of near-critical fluids. Appl. Numer. Maths 51 (4), 409–426.CrossRefGoogle Scholar
Arp, V., Persichetti, J.M. & Chen, G.-B. 1984 The Grüneisen parameter in fluids. Trans. ASME J. Fluids Engng 106 (2), 193–200.CrossRefGoogle Scholar
Beans, E.W. 1970 Computer solution to generalized one-dimensional flow. J. Spacecr. Rockets 7 (12), 1460–1464.CrossRefGoogle Scholar
Bell, I.H., Wronski, J., Quoilin, S. & Lemort, V. 2014 Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library COOLPROP. Ind. Engng Chem. Res. 53 (6), 2498–2508.CrossRefGoogle ScholarPubMed
Bier, K., Ehrler, F. & Niekrawietz, M. 1990 Experimental investigation and computer analysis of spontaneous condensation in stationary nozzle flow of CO$_2$-air mixtures. In Adiabatic Waves in Liquid-Vapor Systems (ed. G.E.A. Meier & P.A. Thompson), pp. 113–127. Springer.CrossRefGoogle Scholar
Bolaños-Acosta, A.F., Restrepo, J.C. & Simões-Moreira, J.R. 2021 Two semi-analytical approaches for solving condensation shocks in supersonic nozzle flows. Intl J. Heat Mass Transfer 173, 121212.CrossRefGoogle Scholar
Bücker, D. & Wagner, W. 2006 Reference equations of state for the thermodynamic properties of fluid phase n-butane and isobutane. J. Phys. Chem. Ref. Data 35 (2), 929–1019.CrossRefGoogle Scholar
Buresti, G. & Casarosa, C. 1989 One-dimensional adiabatic flow of equilibrium gas–particle mixtures in long vertical ducts with friction. J. Fluid Mech. 203, 251–272.CrossRefGoogle Scholar
Cao, X. & Bian, J. 2019 Supersonic separation technology for natural gas processing: a review. Chem. Engng Process. Process. Intensif. 136, 138–151.CrossRefGoogle Scholar
Cinnella, P. & Congedo, P.M. 2007 Inviscid and viscous aerodynamics of dense gases. J. Fluid Mech. 580, 179–217.CrossRefGoogle Scholar
Colonna, P., Nannan, N.R., Guardone, A. & Lemmon, E.W. 2006 Multiparameter equations of state for selected siloxanes. Fluid Phase Equilib. 244 (2), 193–211.CrossRefGoogle Scholar
Colonna, P., Nannan, N.R., Guardone, A. & van der Stelt, T.P. 2009 On the computation of the fundamental derivative of gas dynamics using equations of state. Fluid Phase Equilib. 286 (1), 43–54.CrossRefGoogle Scholar
Cramer, M.S. & Fry, R.N. 1993 Nozzle flows of dense gases. Phys. Fluids A 5 (5), 1246–1259.CrossRefGoogle Scholar
Cramer, M.S., Monaco, J.F. & Fabeny, B.M. 1994 Fanno processes in dense gases. Phys. Fluids 6 (2), 674–683.CrossRefGoogle Scholar
Ferrari, A. 2021 a Analytical solutions for one-dimensional diabatic flows with wall friction. J. Fluid Mech. 918, A32.CrossRefGoogle Scholar
Ferrari, A. 2021 b Exact solutions for quasi-one-dimensional compressible viscous flows in conical nozzles. J. Fluid Mech. 915, A1.CrossRefGoogle Scholar
Gernert, J. & Span, R. 2016 EOS–CG: a Helmholtz energy mixture model for humid gases and CCS mixtures. J. Chem. Thermodyn. 93, 274–293.CrossRefGoogle Scholar
Goodwin, A.R., Sengers, J. & Peters, C.J., ed. 2010 Applied Thermodynamics of Fluids. Royal Society of Chemistry.CrossRefGoogle Scholar
Gori, G., Zocca, M., Cammi, G., Spinelli, A., Congedo, P.M. & Guardone, A. 2020 Accuracy assessment of the non-ideal computational fluid dynamics model for siloxane MDM from the open-source SU2 suite. Eur. J. Mech. B/Fluids 79, 109–120.CrossRefGoogle Scholar
Guardone, A. & Vimercati, D. 2016 Exact solutions to non-classical steady nozzle flows of Bethe–Zel'dovich–Thompson fluids. J. Fluid Mech. 800, 278–306.CrossRefGoogle Scholar
Gyarmathy, G. 2005 Nucleation of steam in high-pressure nozzle experiments. Proc. Inst. Mech. Engrs A 219 (6), 511–521.CrossRefGoogle Scholar
Hodge, B.K. & Koenig, K. 1995 Compressible Fluid Dynamics with Personal Computer Applications. Prentice Hall.Google Scholar
Hoffman, J.D. 1969 Approximate analysis of nonisentropic flow in conical nozzles. J. Spacecr. Rockets 6 (11), 1329–1334.CrossRefGoogle Scholar
Huber, M.L., Perkins, R.A., Laesecke, A., Friend, D.G., Sengers, J.V., Assael, M.J., Metaxa, I.N., Vogel, E., Mareš, R. & Miyagawa, K. 2009 New international formulation for the viscosity of H$_2$O. J. Phys. Chem. Ref. Data 38 (2), 101–125.CrossRefGoogle Scholar
Invernizzi, C.M., Ayub, A., Di Marcoberardino, G. & Iora, P. 2019 Pure and hydrocarbon binary mixtures as possible alternatives working fluids to the usual organic Rankine cycles biomass conversion systems. Energies 12 (21), 4140.CrossRefGoogle Scholar
Johnson, R.C. 1964 Calculations of real-gas effects in flow through critical-flow nozzles. Trans. ASME J. Basic Engng 86 (3), 519–525.CrossRefGoogle Scholar
Jones, O.C. 1976 An improvement in the calculation of turbulent friction in rectangular ducts. Trans. ASME J. Fluids Engng 98 (2), 173–180.CrossRefGoogle Scholar
Klein, S.A., McLinden, M.O. & Laesecke, A. 1997 An improved extended corresponding states method for estimation of viscosity of pure refrigerants and mixtures. Intl J. Refrig. 20 (3), 208–217.CrossRefGoogle Scholar
Kluwick, A. 1993 Transonic nozzle flow of dense gases. J. Fluid Mech. 247, 661–688.CrossRefGoogle Scholar
Kluwick, A. 2004 Internal flows of dense gases. Acta Mech. 169 (1), 123–143.CrossRefGoogle Scholar
Kunz, O. & Wagner, W. 2012 The GERG-2008 wide-range equation of state for natural gases and other mixtures: an expansion of GERG-2004. J. Chem. Engng Data 57 (11), 3032–3091.CrossRefGoogle Scholar
Lemmon, E.W. & Jacobsen, R.T. 2004 Viscosity and thermal conductivity equations for nitrogen, oxygen, argon, and air. Intl J. Thermophys. 25 (1), 21–69.CrossRefGoogle Scholar
Lemmon, E.W., Jacobsen, R.T., Penoncello, S.G. & Friend, D.G. 2000 Thermodynamic properties of air and mixtures of nitrogen, argon, and oxygen from 60 to 2000 K at Pressures to 2000 MPa. J. Phys. Chem. Ref. Data 29 (3), 331–385.CrossRefGoogle Scholar
Louisos, W.F. & Hitt, D.L. 2012 Viscous effects on performance of two-dimensional supersonic linear micronozzles. J. Spacecr. Rockets 45, 706–715.CrossRefGoogle Scholar
Mausbach, P., Köster, A., Rutkai, G., Thol, M. & Vrabec, J. 2016 Comparative study of the Grüneisen parameter for 28 pure fluids. J. Chem. Phys. 144 (24), 244505.CrossRefGoogle ScholarPubMed
Menikoff, R. & Plohr, B.J. 1989 The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61 (1), 75–130.CrossRefGoogle Scholar
More, J.J., Garbow, B.S. & Hillstrom, K.E. 1980 User guide for MINPACK-1 [in Fortran].Google Scholar
Raman, S.K. & Kim, H.D. 2018 Solutions of supercritical CO$_2$ flow through a convergent–divergent nozzle with real gas effects. Intl J. Heat Mass Transfer 116, 127–135.CrossRefGoogle Scholar
Restrepo, J.C., Bolaños-Acosta, A.F. & Simões-Moreira, J.R. 2022 Short nozzles design for real gas supersonic flow using the method of characteristics. Appl. Therm. Engng 207, 118063.CrossRefGoogle Scholar
Schmidt, R. & Wagner, W. 1985 A new form of the equation of state for pure substances and its application to oxygen. Fluid Phase Equilib. 19 (3), 175–200.CrossRefGoogle Scholar
Schnerr, G.H. & Leidner, P. 1994 Internal flows with multiple sonic points. In Fluid- and Gasdynamics, pp. 147–154. Springer.CrossRefGoogle Scholar
Simões-Moreira, J.R. & Shepherd, J.E. 1999 Evaporation waves in superheated dodecane. J. Fluid Mech. 382, 63–86.CrossRefGoogle Scholar
Sirignano, W.A. 2018 Compressible flow at high pressure with linear equation of state. J. Fluid Mech. 843, 244–292.CrossRefGoogle Scholar
Span, R. 2000 Multiparameter Equations of State. Springer.CrossRefGoogle Scholar
Spinelli, A., Cammi, G., Gallarini, S., Zocca, M., Cozzi, F., Gaetani, P., Dossena, V. & Guardone, A. 2018 Experimental evidence of non-ideal compressible effects in expanding flow of a high molecular complexity vapor. Exp. Fluids 59 (8), 126.CrossRefGoogle Scholar
Thol, M., Dubberke, F.H., Baumhögger, E., Vrabec, J. & Span, R. 2017 Speed of sound measurements and fundamental equations of state for octamethyltrisiloxane and decamethyltetrasiloxane. J. Chem. Engng Data 62 (9), 2633–2648.CrossRefGoogle Scholar
Thompson, P.A. 1971 a A fundamental derivative in gasdynamics. Phys. Fluids 14 (9), 1843–1849.CrossRefGoogle Scholar
Thompson, P.A. 1971 b Compressible-Fluid Dynamics. McGraw-Hill.Google Scholar
Tosto, F., Lettieri, C., Pini, M. & Colonna, P. 2021 Dense-vapor effects in compressible internal flows. Phys. Fluids 33 (8), 086110.CrossRefGoogle Scholar
Tsien, H.-S. 1946 One-dimensional flows of a gas characterized by Vander Waal's equation of state. J. Math. Phys. 25 (1–4), 301–324.CrossRefGoogle Scholar
Vimercati, D. & Guardone, A. 2018 On the numerical simulation of non-classical quasi-1D steady nozzle flows: capturing sonic shocks. Appl. Maths Comput. 319, 617–632.CrossRefGoogle Scholar
Virtanen, P., et al. 2020 SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Meth. 17, 261–272.CrossRefGoogle ScholarPubMed
Vogel, E., Küchenmeister, C. & Bich, E. 2000 Viscosity correlation for isobutane over wide ranges of the fluid region. Intl J. Thermophys. 21 (2), 343–356.CrossRefGoogle Scholar
Wagner, W. & Pruß, A. 2002 The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 31 (2), 387–535.CrossRefGoogle Scholar
Wilke, C.R. 1950 A viscosity equation for gas mixtures. J. Chem. Phys. 18 (4), 517–519.CrossRefGoogle Scholar
Yeddula, S.R., Guzmán-Iñigo, J. & Morgans, A.S. 2022 A solution for the quasi-one-dimensional linearised Euler equations with heat transfer. J. Fluid Mech. 936, R3.CrossRefGoogle Scholar
Yeom, G.-S. & Choi, J.-I. 2019 Efficient exact solution procedure for quasi-one-dimensional nozzle flows with stiffened-gas equation of state. Intl J. Heat Mass Transfer 137, 523–533.CrossRefGoogle Scholar
Zucrow, M. 1976 Gas Dynamics. Wiley.Google Scholar