Autors: Sahami M., Ghassemi H., Terziev, A. K., Pitchurov, G. T.
Title: Homogeneous condensation in high-speed flows: a review on droplet nucleation and growth models
Keywords: Droplet growth, Modifications, Non-equilibrium condensation, Nucleation models, Two-phase flow

Abstract: Droplet formation and growth occur in various industrial equipment such as steam turbines, supersonic separators, and thrusters. By the decrease in temperature through the nozzle of these equipment, non-equilibrium condensation may occur. Due to the complexity and practical application of this phenomenon, simplifying assumptions are applied in numerous studies during the derivation of nucleation and growth models. These assumptions do not correspond well with the thermodynamics of condensation and evaporation phenomena. Consequently, it is imperative to comprehensively present the non-equilibrium condensation physics and concepts to modify the models of droplet nucleation and growth in high-speed mixtures. In this review, the fundamentals of non-equilibrium condensation and its applications are mentioned first. Then, the most important corrections present in the literature on nucleation and droplet growth modeling are compiled and evaluated in terms of accuracy. The results show that the surface tension corrections, and Kantrowitz models provide the best accuracy in predicting flow and condensation parameters among the nucleation models. In addition, it is demonstrated that the general Bakhtar model, which simultaneously satisfies the mass and energy conservation for the droplets, provides a more reliable theory and accuracy than the common droplet temperature approximation models and can be used in multi-component flows.

References

  1. N. Yasugahira K. Namura R. Kaneko T. Satoh Erosion resistance of titanium alloys for steam turbine blades as measured by water droplet impingement Titanium Steam Turbine Blading 10.1016/B978-0-08-037301-0.50025-7
  2. P.H. Niknam H. Mortaheb B. Mokhtarani Dehydration of low-pressure gas using supersonic separation: experimental investigation and CFD analysis J Nat Gas Sci Eng 52 202 214 10.1016/j.jngse.2017.12.007
  3. M. Sahami H. Ghassemi Effects of non-equilibrium condensation on the nozzle performance of a cold gas thruster Acta Astronaut 197 200 216 10.1016/j.actaastro.2022.05.032
  4. F. Bakhtar K. Zidi Nucleation phenomena in flowing high-pressure steam: experimental results Proc Inst Mech Eng, Part A: J Power Eng 203 3 195 200 10.1243/PIME_PROC_1989_203_027_02
  5. C. Luijten K. Bosschaart M. Van Dongen High pressure nucleation in water/nitrogen systems J Chem Phys 106 19 8116 8123 10.1063/1.473818
  6. S. Jinno Y. Fukuda H. Sakaki A. Yogo M. Kanasaki Mie scattering from submicron-sized CO2clusters formed in a supersonic expansion of a gas mixture Opt Express 21 18 20656 20674 10.1364/OE.21.020656
  7. G. Lamanna, (2002) " On nucleation and droplet growth in condensing nozzle flows, " Ph.D. thesis, Applied physics and science education, Technische Universiteit Eindhoven, Eindhoven, https://doi.org/10.6100/IR539104.
  8. J.C. Restrepo A.F. Bolaños-Acosta J.R. Simões-Moreira Condensation shock topologies in carbon dioxide and a non-condensable gas mixture in supersonic nozzles Phys Fluids 10.1063/5.0202444
  9. J. C. Restrepo Lozano and J. R. Simões-Moreira, (2022) Theoretical and experimental study of carbon dioxide content removal from dry air by supersonic gas separation technique, PhD, Universidade de São Paulo, São Paulo. Recuperado de, https://www.teses.usp.br/teses/disponiveis/3/3150/tde-15082023-094336/.
  10. J. B. Young, (1980) Spontaneous Condensation of Steam in Supersonic Nozzles. Part 1: Nucleation and Droplet Growth Theory. Part 2: Numerical Methods and Comparison with Experimental Results, Cambridge Univ. (England). National Aeronautics and Space Administration, Washington, DC, Technical Report, URL: https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/N8113306.xhtml.
  11. J. Diemand R. Angélil K.K. Tanaka H. Tanaka Large scale molecular dynamics simulations of homogeneous nucleation J Chem Phys 139 7 074309 10.1063/1.4818639
  12. D. Guo X. Cao H. Cao J. Bian Heterogeneous condensation mechanism of methane-hexane binary mixture Energy 256 10.1016/j.energy.2022.124627 124627
  13. J. Bian D. Guo Y. Li W. Cai Y. Hua X. Cao Homogeneous nucleation and condensation mechanism of methane gas: a molecular simulation perspective Energy 249 10.1016/j.energy.2022.123610 123610
  14. J. Bian G. Ding D. Guo H. Cao Y. Liu X. Cao Surface crystallization mechanism of n-hexane droplets Energy 263 10.1016/j.energy.2022.125921 125921
  15. A. Bolaños-Acosta J. Restrepo J. Simões-Moreira Two semi-analytical approaches for solving condensation shocks in supersonic nozzle flows Int J Heat Mass Transf 173 10.1016/j.ijheatmasstransfer.2021.121212 121212
  16. D. Martínez-Ruiz (2023) On the structure of steady one-dimensional liquid-fueled detonations, Phys Fluids, vol. 35, no. 8, https://doi.org/10.1063/5.0162358.
  17. C. Wen H. Ding Y. Yang Numerical simulation of nanodroplet generation of water vapour in high-pressure supersonic flows for the potential of clean natural gas dehydration Energy Convers Manage 231 10.1016/j.enconman.2021.113853 113853
  18. J. Bian X. Cao W. Yang M.A. Edem P. Yin W. Jiang Supersonic liquefaction properties of natural gas in the Laval nozzle Energy 159 706 715 10.1016/j.energy.2018.06.196
  19. M. Sahami H. Ghassemi Numerical modeling of the non-equilibrium condensation in multi-component flows through the high expansion ratio nozzles Adv Space Res 10.1016/j.asr.2024.06.004
  20. A. F. Bolaños Acosta, (2022) Two approximations for solving non-equilibrium condensation shocks in supersonic nozzle flows, Master thesis, Escola Politécnica, Engenharia Mecânica de Energia e Fluídos, Universidade de São Paulo, Universidade de São Paulo, São Paulo, Brazil, URL: https://www.teses.usp.br/teses/disponiveis/3/3150/tde-04052023-083246/pt-br.php.
  21. A. Pandey, (2014) Numerical modelling of non-equilibrium condensing steam flows, Master thesis, Faculty of Mechanical, Maritime and Materials Engineering, Department of Process and Energy, Delft University of Technology (TU Delft), http://resolver.tudelft.nl/uuid:34c2fd1c-9fcd-4b32-8253-f91a944da8e5.
  22. F. Bakhtar D. Ryley K. Tubman J. Young Nucleation studies in flowing high-pressure steam Proc Inst Mech Eng 189 1 427 436 10.1243/PIME_PROC_1975_189_053_02
  23. E. Lakzian A. Masjedi Slip effects on the exergy loss due to irreversible heat transfer in a condensing flow Int J Exergy 14 no. 1 22 37 10.1504/IJEX.2014.059511
  24. S.R. Shooshtari A. Shahsavand Maximization of energy recovery inside supersonic separator in the presence of condensation and normal shock wave Energy 120 153 163 10.1016/j.energy.2016.12.060
  25. P.H. Niknam D. Fiaschi H. Mortaheb B. Mokhtarani An improved formulation for speed of sound in two-phase systems and development of 1D model for supersonic nozzle Fluid Phase Equilib 446 18 27 10.1016/j.fluid.2017.05.013
  26. S. Hamidi M. Kermani Numerical study of non-equilibrium condensation and shock waves in transonic moist-air and steam flows Aerosp Sci Technol 46 188 196 10.1016/j.ast.2015.07.011
  27. M.A.F. Aliabadi E. Lakzian A. Jahangiri I. Khazaei Numerical investigation of effects polydispersed droplets on the erosion rate and condensation loss in the wet steam flow in the turbine blade cascade Appl Therm Eng 164 10.1016/j.applthermaleng.2019.114478 114478
  28. X. Luo, (2004) Unsteady flows with phase transition, Ph.D. thesis, Applied Physics and Science Education, Technische Universiteit Eindhoven, Eindhoven, https://doi.org/10.6100/IR576382.
  29. R. McGraw Description of aerosol dynamics by the quadrature method of moments Aerosol Sci Technol 27 2 255 265 10.1080/02786829708965471
  30. R. Fan D.L. Marchisio R.O. Fox Application of the direct quadrature method of moments to polydisperse gas–solid fluidized beds Powder Technol 139 1 7 20 10.1016/j.powtec.2003.10.005
  31. S. Hamidi M. Kermani Numerical study of water production from compressible moist-air flow J App Fluid Mech 9 1 333 341
  32. R. Jansen, S. Gimelshein, N. Gimelshein, and I. Wysong, (2010) Modeling of homogeneous condensation in high density thruster plumes, In 40th Fluid Dynamics Conference and Exhibit, Chicago, Illinois, United States of America (USA), p. 5011: American Institute of Aeronautics and Astronautics (AIAA), https://doi.org/10.2514/6.2010-5011.
  33. S. Adler A. Warshavsky A. Peretz Low-cost cold-gas reaction control system for the Sloshsat FLEVO small satellite J Spacecr Rockets 42 2 345 351 10.2514/1.5128
  34. S. Bonifacio A.R. Sorge D. Krejci A. Woschnak C. Scharlemann Novel manufacturing method for hydrogen peroxide catalysts: a performance verification J Propul Power 30 2 299 308 10.2514/1.B34959
  35. P. Peeters G. Pieterse M. Van Dongen Multi-component droplet growth. Part II. A theoretical model Phys Fluids 16 no. 7 2575 2586 10.1063/1.1751192
  36. J. Bian Y. Liu X. Zhang Y. Li L. Gong X. Cao Co-condensation and interaction mechanism of acidic gases in supersonic separator: a method for simultaneous removal of carbon dioxide and hydrogen sulfide from natural gas Sep Purif Technol 322 10.1016/j.seppur.2023.124296 124296
  37. H. Vehkamäki, (2006) Classical nucleation theory in multicomponent systems. Springer Berlin Heidelberg. https://doi.org/10.1007/3-540-31218-8_5.
  38. K.F. Kelton A.L. Greer Nucleation in condensed matter: applications in materials and biology Elsevier 10.1016/c2009-0-04500-0
  39. P.G. Hill H. Witting E.P. Demetri Condensation of metal vapors during rapid expansion J Heat Transfer 85 4 303 314 10.1115/1.3686115
  40. A. A. Obeidat, (2003) Nucleation theory using equations of state, Ph.D. thesis Dissertation - Citation, Physics, University of Missouri-Rolla, Rolla ProQuest Dissertations Publishing, URL: https://physics.mst.edu/media/academic/physics/documents/faculty/wilemski/AOfinal_thesis.pdf.
  41. C.T.R. Wilson XI. Condensation of water vapour in the presence of dust-free air and other gases Philosophical Trans Royal Soc London Series A, Contain Papers Math Phys Charact 189 265 307 10.1098/rsta.1897.0011
  42. C. Wen N. Karvounis J.H. Walther Y. Yan Y. Feng Y. Yang An efficient approach to separate CO2 using supersonic flows for carbon capture and storage Appl Energy 238 311 319 10.1016/j.apenergy.2019.01.062
  43. P. Sherman Condensation augmented velocity of a supersonic stream AIAA J 9 no. 8 1628 1630 10.2514/3.6397
  44. A. Shapiro The dynamics and the thermodynamics of compressibility fluid flow. Vol. 1 New York: Ronald Press 1 953
  45. B. Yao X. Han H. Shi X. Wu Q. Li Z. Han Shock waves characteristics and losses estimation of non-equilibrium condensation flow in nozzle and steam turbine cascade Appl Therm Eng 258 10.1016/j.applthermaleng.2024.124579 124579
  46. K. Matsuo S. Kawagoe K. Sonoda K. Sakao Studies of condensation shock waves: part 1, mechanism of their formation Bulletin of JSME 28 241 1416 1422
  47. R. Hermann Der Kondensationsstoß in Überschall-Windkanaldüsen Luftfahrtforschung 19 6 201 209
  48. L. Prandtl, "General considerations on the flow of compressible fluids," Washington, United States of America (USA), Technical Memorandum 1936, https://ntrs.nasa.gov/api/citations/19930094611/downloads/19930094611.pdf.
  49. A. White J. Young Time-marching method for the prediction of two-dimensional, unsteadyflows of condensing steam J Propul Power 9 4 579 587 10.2514/3.23661
  50. Y. Kawamura and M. Nakagawa (2012) Investigation on oblique shock waves occurred in the supersonic carbon dioxide two-phase flow, Presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, HEFAT 2012, Malta, URL: http://hdl.handle.net/2263/44307.
  51. E. Lakzian S. Shaabani Analytical investigation of coalescence effects on the exergy loss in a spontaneously condensing wet-steam flow Int J Exergy 16 no. 4 383 403 10.1504/IJEX.2015.069115
  52. M. Sahami H. Ghassemi Intensifying CO2 condensation in the flue gas through the supersonic separators by hydrogen enriching: a computational study Chem Eng Process-Process Intensif 203 10.1016/j.cep.2024.109872 109872
  53. C.F. Colebrook T. Blench H. Chatley E. Essex J. Finniecome Correspondence. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. (includes plates) J Institut Civ Eng 12 8 393 422 10.1680/ijoti.1939.14509
  54. Z. Han X. Han Z. Wang Numeric simulation of wet-steam two-phase condensing flow in a steam turbine cascade J Braz Soc Mech Sci Eng 39 1189 1199 10.1007/s40430-016-0655-y
  55. P. Wiśniewski S. Dykas H. Miyazawa T. Furusawa S. Yamamoto Modified heat transfer correction function for modeling multiphase condensing flows in transonic regime Int J Heat Mass Transfer 201 10.1016/j.ijheatmasstransfer.2022.123597 123597
  56. F. Bakhtar K. Zidi Nucleation phenomena in flowing high-pressure steam Part 2: theoretical analysis Proc Inst Mech Eng, Part A: J Power Energy 204 4 233 242 10.1243/PIME_PROC_1990_204_032_02
  57. S. Dykas W. Wróblewski Single-and two-fluid models for steam condensing flow modeling Int J Multiphase Flow 37 9 1245 1253 10.1016/j.ijmultiphaseflow.2011.05.008
  58. H. Liu Y. Li G. Liu Numerical investigation of oil spray lubrication for transonic bearings J Braz Soc Mech Sci Eng 40 1 14 10.1007/s40430-018-1317-z
  59. Y. Liu X. Cao D. Guo H. Cao J. Bian Influence of shock wave/boundary layer interaction on condensation flow and energy recovery in supersonic nozzle Energy 263 10.1016/j.energy.2022.125662 125662
  60. J. Bian X. Cao W. Yang X. Song C. Xiang S. Gao Condensation characteristics of natural gas in the supersonic liquefaction process Energy 168 99 110 10.1016/j.energy.2018.11.102
  61. K. Cui Y. Song H. Chen F. Chen Numerical method for non-equilibrium phase transition in low pressure stage of steam turbine J Braz Soc Mech Sci Eng 38 2149 2159 10.1007/s40430-015-0452-z
  62. E. Michaelides Particles, bubbles & drops: their motion, heat and mass transfer World Scientific 10.1142/6018
  63. B. Avancini BdA B.S. Carmo Numerical modeling of high-speed flows of fluid mixtures with phase change through nozzles J Therm Anal Calorim 149 1 219 231 10.1007/s10973-023-12716-x
  64. J. Bian X. Cao W. Yang D. Guo C. Xiang Prediction of supersonic condensation process of methane gas considering real gas effects Appl Therm Eng 164 10.1016/j.applthermaleng.2019.114508 114508
  65. D.-Y. Peng D.B. Robinson A new two-constant equation of state Ind Eng Chem Fundam 15 no. 1 59 64 10.1021/i160057a011
  66. J. Bian Z. Zhao Y. Liu R. Cheng X. Zang X. Cao Condensation characteristics of ammonia vapor during supersonic separation: a novel approach to ammonia-hydrogen separation Fuel 359 10.1016/j.fuel.2023.130401 130401
  67. R. Stryjek J. Vera PRSV—An improved peng-robinson equation of state with new mixing rules for strongly nonideal mixtures Can J Chem Eng 64 2 334 340 10.1002/cjce.5450640225
  68. B. Ervens M. Cubison E. Andrews G. Feingold Prediction of cloud condensation nucleus number concentration using measurements of aerosol size distributions and composition and light scattering enhancement due to humidity J Geophys Res Atmos 10.1029/2006JD007426
  69. F. Bakhtar J. Young A. White D. Simpson Classical nucleation theory and its application to condensing steam flow calculations Proc Inst Mech Eng C J Mech Eng Sci 219 12 1315 1333 10.1243/095440605X8379
  70. H. Kirols D. Kevorkov A. Uihlein M. Medraj Water droplet erosion of stainless steel steam turbine blades Mater Res Express 4 8 10.1088/2053-1591/aa7c70 086510
  71. C. Lettieri D. Paxson Z. Spakovszky P. Bryanston-Cross Characterization of nonequilibrium condensation of supercritical carbon dioxide in a de Laval nozzle J Eng Gas Turbines Power 140 4 10.1115/1.4038082 041701
  72. T. Wittmann S. Lück C. Bode J. Friedrichs Modelling the condensation phenomena within the radial turbine of a fuel cell turbocharger Int J Turbomachinery Propuls Power 6 3 23 10.3390/ijtpp6030023
  73. W. Sun L. Niu L. Chen S. Chen X. Zhang Y. Hou Numerical study on the spontaneous condensation flow in an air cryogenic turbo-expander using equilibrium and non-equilibrium models Cryogenics 73 42 52 10.1016/j.cryogenics.2015.11.006
  74. S. Vahaji A. Date S.C. Cheung J. Tu A. Akbarzadeh M. Oreijah Experimental analysis of two-phase flow nozzle for desalination and power generation system Procedia Eng 49 324 329 10.1016/j.proeng.2012.10.144
  75. D. Majidi F. Farhadi Supersonic separator’s dehumidification performance with specific structure: experimental and numerical investigation Appl Therm Eng 179 10.1016/j.applthermaleng.2020.115551 115551
  76. K. Ariafar D. Buttsworth N. Sharifi R. Malpress Ejector primary nozzle steam condensation: Area ratio effects and mixing layer development Appl Therm Eng 71 1 519 527 10.1016/j.applthermaleng.2014.06.038
  77. M.A.F. Aliabadi M. Bahiraei Effect of water nano-droplet injection on steam ejector performance based on non-equilibrium spontaneous condensation: a droplet number study Appl Therm Eng 184 10.1016/j.applthermaleng.2020.116236 116236
  78. R. Kumar D.A. Levin Simulation of homogeneous condensation of small polyatomic systems in high pressure supersonic nozzle flows using Bhatnagar–Gross–Krook model J Chem Phys 134 12 10.1063/1.3569762 124519
  79. J. Zhong M.I. Zeifman D.A. Levin Sensitivity of water condensation in a supersonic plume to the nucleation rate J Thermophys Heat Transf 20 3 517 523 10.2514/1.18477
  80. T. Bai Y. Lu Z. Wen J. Yu Numerical study on the structural optimization of R290 two-phase ejector with a non-equilibrium CFD model Int J Refrig 170 287 301 10.1016/j.ijrefrig.2024.11.027
  81. C. Zhang C. Li W. Jia H. Xie J. He Numerical study on the differences in condensation characteristics in throttle valves with different structures Cryogenics 133 10.1016/j.cryogenics.2023.103696 103696
  82. M.Z. Faizullin A.V. Vinogradov V.P. Koverda Properties of gas hydrates formed by nonequilibrium condensation of molecular beams High Temp 52 6 830 839 10.1134/s0018151x14050046
  83. J. Kindracki K. Tur P. Paszkiewicz Ł. Mężyk Ł. Boruc P. Wolański Experimental research on low-cost cold gas propulsion for a space robot platform Aerosp Sci Technol 62 148 157 10.1016/j.ast.2016.12.001
  84. J. Sangiovanni T. Barber S. Syed Role of hydrogen/air chemistry in nozzle performance for a hypersonic propulsion system J Propul Power 9 1 134 138 10.2514/3.11495
  85. M. Grabe, R.-D. Boettcher, S. Fasoulas, and K. Hannemann (2010) Numerical simulation of nozzle flow into high vacuum using kinetic and continuum approaches, In New Results in Numerical and Experimental Fluid Mechanics VII: Contributions to the 16th STAB/DGLR Symposium Aachen, Germany 2008, pp. 423–430: Springer. https://doi.org/10.1007/978-3-642-14243-7_52.
  86. C. Azevedo A. Sinátora Erosion-fatigue of steam turbine blades Eng Fail Anal 16 7 2290 2303 10.1016/j.engfailanal.2009.03.007
  87. A. Nasikas, (1994) Method and mechanism for the supersonic separation of droplets from a gas stream, ed: Google Patents, https://patents.google.com/patent/US5306330A/en.
  88. V. Alfyorov L. Bagirov L. Dmitriev V. Feygin S. Imayev J.R. Lacey Supersonic nozzle efficiently separates natural gas components Oil Gas J 103 20 53 58
  89. V. Balepin (2017) Supersonic post-combustion inertial CO2 extraction system final report, Alliant techsystems operations LLC, Ronkonkoma, NY 77 Raynor Ave, Ronkonkoma, NY 11779, United States of America (USA), URL: https://www.netl.doe.gov/node/7096.
  90. R. Secchi G. Innocenti D. Fiaschi Supersonic swirling separator for natural gas heavy fractions extraction: 1d model with real gas EOS for preliminary design J Nat Gas Sci Eng 34 197 215 10.1016/j.jngse.2016.06.061
  91. X. Cao X. Song Q. Chu L. Mu Y. Li J. Bian An efficient method for removing hydrogen sulfide from natural gas using supersonic Laval nozzle Process Saf Environ Prot 129 220 229 10.1016/j.psep.2019.07.008
  92. M. Betting Supersonic separator gains market acceptance World Oil 4 197 200
  93. J. Bian X. Cao W. Yang S. Gao C. Xiang A new liquefaction method for natural gas by utilizing cold energy and separating power of swirl nozzle AIChE J 66 no. 2 10.1002/aic.16811 e16811
  94. S. Yousefi M. Changizian S.S. Bahrainian Modeling condensation of Methane-Water vapor mixtures in the Hedbäck nozzle: a comparative study of 2D and 3D geometries Appl Therm Eng 10.1016/j.applthermaleng.2024.125121
  95. X. Duan Z. Zhang Z. Zhao X. Cao J. Bian Supersonic expansion and condensation characteristics of hydrogen gas under different temperature conditions Chin J Chem Eng 69 220 226 10.1016/j.cjche.2023.12.024
  96. B. Chen Y. Zeng E. Luo N. Peng A. Zou An innovative method for helium refrigeration liquefaction utilizing transonic nozzle Energy 314 10.1016/j.energy.2024.134283 134283
  97. B. Chen A. Zou Y. Zeng E. Luo Study on the helium liquefaction characteristics in the laval nozzle Cryogenics 10.1016/j.cryogenics.2024.104017
  98. V. Vlasenko, V. Slesarenko, and K. Bashirov (2018) Experimental study of a combined supersonic separator, presented at the 2018 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon). https://doi.org/10.1109/FarEastCon.2018.8602656.
  99. J. Restrepo and J. R. Simões-Moreira (2019) Operational behaviour of supersonic separators for real gas mixtures of methane and carbon dioxide, from the homogeneous nucleation point of view, presented at the International Conference on Offshore Mechanics and Arctic Engineering: ASME 2019 38th International Conference on Ocean, Offshore and Arctic Engineering, Glasgow, Scotland, United Kingdom (UK). https://doi.org/10.1115/OMAE2019-96315.
  100. R. Maniarasu S.K. Rathore S. Murugan A review on materials and processes for carbon dioxide separation and capture Energy Environ 34 1 3 57 10.1177/0958305X211050984
  101. M. Haghighi K.A. Hawboldt M.A. Abdi Supersonic gas separators: review of latest developments J Nat Gas Sci Eng 27 109 121 10.1016/j.jngse.2015.08.049
  102. L. de Oliveira Arinelli T.A.F. Trotta A.M. Teixeira J.L. de Medeiros F.A. Ofélia de Queiroz Offshore processing of CO2 rich natural gas with supersonic separator versus conventional routes J Nat Gas Sci Eng 46 199 221 10.1016/j.jngse.2017.07.010
  103. M. Haghighi (2010) Supersonic separators: a gas dehydration device, Master thesis, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, URI: http://research.library.mun.ca/id/eprint/8935.
  104. M. A. Abdi, E. Jassim, M. Haghighi, and Y. Muzychka (2010) Applications of CFD in natural gas processing and transportation. Computational Fluid Dynamics. https://doi.org/10.5772/7096.
  105. J. Chen Z. Huang Numerical study on carbon dioxide capture in flue gas by converging-diverging nozzle Fuel 320 10.1016/j.fuel.2022.123889 123889
  106. X. Han T. Wang Z. Wang J. Chen Z. Huang Effect of the thermal insulation layer on non-equilibrium condensation in the nozzle for carbon capture Chem Eng Process-Process Intensif 208 10.1016/j.cep.2024.110124 110124
  107. K. Bier, F. Ehrler, and G. Theis (1990) Spontaneous condensation in stationary nozzle flow of carbon dioxide in a wide range of density, In: Meier, G.E.A., Thompson, P.A. (eds) Adiabatic Waves in Liquid-Vapor Systems. International Union of Theoretical and Applied Mechanics: IUTAM Symposium Göttingen, Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83587-2_12.
  108. A.M. Teixeira L. de Oliveira Arinelli J.L. de Medeiros F.A. Ofélia de Queiroz Economic leverage affords post-combustion capture of 43% of carbon emissions: supersonic separators for methanol hydrate inhibitor recovery from raw natural gas and CO2drying J Environ Manag 236 534 550 10.1016/j.jenvman.2019.02.008
  109. H. Ding Y. Zhang Y. Dong C. Wen Y. Yang High-pressure supersonic carbon dioxide (CO2) separation benefiting carbon capture, utilisation and storage (CCUS) technology Appl Energy 339 10.1016/j.apenergy.2023.120975 120975
  110. X. Cao D. Guo W. Sun P. Zhang G. Ding J. Bian Supersonic separation technology for carbon dioxide and hydrogen sulfide removal from natural gas J Clean Prod 288 10.1016/j.jclepro.2020.125689 125689
  111. W. Sun X. Cao W. Yang X. Jin Numerical simulation of CO2 condensation process from CH4-CO2 binary gas mixture in supersonic nozzles Sep Purif Technol 188 238 249 10.1016/j.seppur.2017.07.023
  112. H. Vijayakumaran T.A. Lemma CFD modelling of non-equilibrium condensation of CO2 within a supersonic nozzle using metastability approach J Nat Gas Sci Eng 85 10.1016/j.jngse.2020.103715 103715
  113. R. Kung L. Cianciolo J. Myer Solar scattering from condensation in Apollo translunar injection plume AIAA J 13 no. 4 432 437 10.2514/3.49725
  114. P. Surmacz Green propulsion research and development at the institute of aviation: problems and perspectives J KONES 23 337 344 10.5604/12314005.1213534
  115. J. Cardin and J. Acosta (2000) Design and test of an economical cold gas propulsion system, presented at the Proceedings of the Small Satellite Conference. https://digitalcommons.usu.edu/smallsat/.
  116. M. Brownell (2014) Design and analysis of a cold gas propulsion system for stabilization and maneuverability of a high altitude research balloon, Mechanical and Aerospace Engineering, Western Michigan University, URL: https://scholarworks.wmich.edu/honors_theses/2396/.
  117. K. Kim, L.-H. Chen, B. Cera, M. Daly, and E. Zhu (2016) Hopping and rolling locomotion with spherical tensegrity robots, In 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea (South), pp. 4369–4376: IEEE, https://doi.org/10.1109/IROS.2016.7759643.
  118. J.M. Martínez T. Lafleur On the selection of propellants for cold/warm gas propulsion systems Acta Astronaut 10.1016/j.actaastro.2023.07.031
  119. A. Gallagher-Rogers, J. Zhong, and D. Levin (2007) Simulation of homogeneous ethanol condensation in supersonic nozzle flows using DSMC, In 39th AIAA Thermophysics Conference, Miami, Florida, United States of America (USA), 4159, Miami, Florida: American Institute of Aeronautics and Astronautics (AIAA), https://doi.org/10.2514/6.2007-4159.
  120. K. Karthikeyan S. Chou L. Khoong Y. Tan C. Lu W. Yang Low temperature co-fired ceramic vaporizing liquid microthruster for microspacecraft applications Appl Energy 97 577 583 10.1016/j.apenergy.2011.11.078
  121. W. Olson (1960) Recombination and condensation in nozzles, presented at the Advances in Aeronautical Sciences, Proceedings of Second International Congress in the Aeronautical Sciences, Zurich, Switzerland, Sept. 12–16, URL: https://www.icas.org/ICAS_ARCHIVE/ICAS1960/Page%20849%20Olson.pdf.
  122. W. Louisos and D. Hitt (2010) Assessing the potential for condensation in supersonic micronozzle flows, in 10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, Chicago, Illinois, United States of America (USA), p. 5059: American Institute of Aeronautics and Astronautics (AIAA), https://doi.org/10.2514/6.2010-5059.
  123. B. Greenfield, W. Louisos, and D. Hitt (2011) Numerical simulations of multiphase flow in supersonic micro-nozzles, In 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, United States of America (USA), p. 189: American Institute of Aeronautics and Astronautics (AIAA), https://doi.org/10.2514/6.2011-189.
  124. D. G. Fahrenheit (1724) VIII. Experimenta & observationes de congelatione aquæ in vacuo factæ a DG Fahrenheit, RS S, Philosophical Transactions of the Royal Society of London, vol. 33, no. 382, pp. 78–84, https://doi.org/10.1098/rstl.1724.0016.
  125. J. W. Gibbs (1879) On the equilibrium of heterogeneous substances, URL: http://www.ub.uni-heidelberg.de/archiv/13220.
  126. M. Tammaro F. Di Natale A. Salluzzo A. Lancia Heterogeneous condensation of submicron particles in a growth tube Chem Eng Sci 74 124 134 10.1016/j.ces.2012.02.023
  127. I. Ford Statistical mechanics of nucleation: a review Proc Inst Mech Eng C J Mech Eng Sci 218 8 883 899 10.1243/0954406041474183
  128. V. I. Kalikmanov (2013) Classical nucleation theory, in Nucleation theory: Springer, pp. 17–41, https://doi.org/10.1007/978-90-481-3643-8_3.
  129. F. F. Abraham (1974) Homogeneous nucleation theory. Elsevier. https://doi.org/10.1016/b978-0-12-038361-0.x5001-7
  130. J. Schmelzer G. Röpke V.B. Priezzhev Nucleation theory and applications Wiley Online Library 10.1002/3527604790
  131. R.E. Sonntag G.J. Van Wylen Fundamentals of statistical thermodynamics J Chem Education 10.1021/ed043pA912.3
  132. R. Becker W. Döring Kinetische behandlung der keimbildung in übersättigten dämpfen Ann Phys (Berlin) 416 8 719 752 10.1002/andp.19354160806
  133. J. Wölk R. Strey C.H. Heath B.E. Wyslouzil Empirical function for homogeneous water nucleation rates J Chem Phys 117 10 4954 4960 10.1063/1.1498465
  134. A. Afzalifar T. Turunen-Saaresti A. Grönman Origin of droplet size underprediction in modeling of low pressure nucleating flows of steam Int J Multiphase Flow 86 86 98 3543867 10.1016/j.ijmultiphaseflow.2016.07.012
  135. A. Dillmann G. Meier A refined droplet approach to the problem of homogeneous nucleation from the vapor phase J Chem Phys 94 no. 5 3872 3884 10.1063/1.460663
  136. J. Haukohl and L. Spradley (1970) Multi-specie condensation in expanding flows, HUNTSVILLE RESEARCH & ENGINEERING CENTER, https://ntrs.nasa.gov/api/citations/19700026352/downloads/19700026352.pdf.
  137. M. Volmer "Kinetik der Phasenbildung, Steinkopff, Dresden u," Stuttgart Germany 10.1002/bbpc.19400460512
  138. Y.B. Zeldovich On the theory of new phase formation: cavitation, in selected works of Yakov Borisovich Zeldovich, Physicochem Princeton University Press 10.1515/9781400862979.120
  139. J. Feder K. Russell J. Lothe G. Pound Homogeneous nucleation and growth of droplets in vapours Adv Phys 15 57 111 178 10.1080/00018736600101264
  140. H. Wakeshima Time Lag in the Self-Nucleation J Chem Phys 22 9 1614 1615 10.1143/JPSJ.10.374
  141. C. Lettieri D. Yang Z. Spakovszky An investigation of condensation effects in supercritical carbon dioxide compressors J Eng Gas Turbines Power 10.1115/1.4029577
  142. D. Kashchiev Solution of the non-steady state problem in nucleation kinetics Surf Sci 14 1 209 220 10.1016/0039-6028(69)90055-7
  143. S. Kotake I. Glass Flows with nucleation and condensation Prog Aerosp Sci 19 129 196 10.1016/0376-0421(79)90003-4
  144. A. Kantrowitz Nucleation in very rapid vapor expansions J Chem Phys 19 9 1097 1100 10.1063/1.1748482
  145. W. Sun L. Niu S. Chen X. Sun Y. Hou Numerical investigation of nitrogen spontaneous condensation flow in cryogenic nozzles using varying nucleation theories Cryogenics 68 19 29 10.1016/j.cryogenics.2015.01.010
  146. S. Senguttuvan J.-C. Lee Numerical study of wet-steam flow in Moore nozzles J Mech Sci Technol 33 10 4823 4830 10.1007/s12206-019-0923-8
  147. T. Hertwig T. Wittmann P. Wiśniewski J. Friedrichs Modeling condensing flows of humid air in transonic nozzles Experimental Comput Multiphase Flow 5 4 344 356 10.1007/s42757-022-0152-8
  148. J. L. Katz and M. D. Donohue, S. A. R. I. Prigogine, Ed. (1979) A kinetic approach to homogeneous nucleation theory (Advances in Chemical Physics). pp. 137–155, https://doi.org/10.1002/9780470142592.ch3.
  149. J.L. Katz H. Wiedersich Nucleation theory without Maxwell demons J Colloid Interface Sci 61 2 351 355 10.1016/0021-9797(77)90397-6
  150. S.L. Girshick C.P. Chiu Kinetic nucleation theory: a new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor J Chem Phys 93 2 1273 1277 10.1063/1.459191
  151. S. Vosel A. Onischuk P. Purtov T. Tolstikova Classical nucleation theory: account of dependence of the surface tension on curvature and translation-rotation correction factor in aerosols handbook Boca Raton Taylor & Francis Group 10.1201/b12668
  152. E.A. Rad S. Naeemi B. Davoodi Examining the curvature dependency of surface tension in a nucleating steam flow Heat Mass Transfer 56 1 207 217 10.1007/s00231-019-02709-8
  153. J. Kalová R. Mareš Size dependences of surface tension Int J Thermophys 36 2862 2868 10.1007/s10765-015-1851-1
  154. E.M. Blokhuis J. Kuipers Thermodynamic expressions for the Tolman length J Chem Phys 10.1063/1.2167642
  155. L.S. Bartell Tolman's δ, surface curvature, compressibility effects, and the free energy of drops J Phys Chem B 105 47 11615 11618 10.1021/jp011028f
  156. D. Zhukhovitskii Size-corrected theory of homogeneous nucleation J Chem Phys 101 6 5076 5080 10.1063/1.467364
  157. J. Rezazadeh E. Lakzian M.R. Mahpeykar Effect of the droplet surface tension correction on nucleation condensing water vapor flow Modares Mech Eng 16 2 264 274
  158. R.C. Tolman The effect of droplet size on surface tension J Chem Phys 17 3 333 337 10.1063/1.1747247
  159. D.H. Rasmussen Energetics of homogeneous nucleation—approach to a physical spinodal J Cryst Growth 56 1 45 55 10.1016/0022-0248(82)90011-2
  160. G. Benson R. Shuttleworth The surface energy of small nuclei J Chem Phys 19 1 130 131 10.1063/1.1747963
  161. S. Kaur B. Akhouri Predictions of fugacity coefficients of pure substances from equations of state IOP Conf Series: Mater Sci Eng 1091 1 10.1088/1757-899X/1091/1/012019 012019
  162. B.N. Hale Application of a scaled homogeneous nucleation-rate formalism to experimental data at T≪ T c Phys Rev A 33 6 4156 10.1103/PhysRevA.33.4156
  163. B.N. Hale The scaling of nucleation rates Metall Trans A Phys Metall Mater Sci 23 7 1863 1868 10.1007/bf02647536
  164. B.N. Hale Temperature dependence of homogeneous nucleation rates for water: near equivalence of the empirical fit of Wölk and Strey, and the scaled nucleation model J Chem Phys 122 20 10.1063/1.1906213 204509
  165. E.A. Rad M.R. Mahpeykar A.R. Teymourtash Evaluation of simultaneous effects of inlet stagnation pressure and heat transfer on condensing water-vapor flow in a supersonic Laval nozzle Sci Iran 20 1 141 151 10.1016/j.scient.2012.12.009
  166. S. Sinha B.E. Wyslouzil G. Wilemski Modeling of H2O/D2O condensation in supersonic nozzles Aerosol Sci Technol 43 no. 1 9 24 10.1080/02786820802441771
  167. E.F. Allard J.L. Kassner Jr New cloud-chamber method for the determination of homogeneous nucleation rates J Chem Phys 42 4 1401 1405 10.1063/1.1696129
  168. J. J. Thomson, (1888) Applications of dynamics to physics and chemistry. Harvard University: Macmillan, https://archive.org/details/in.ernet.dli.2015.153729.
  169. M. Volmer Über keimbildung und keimwirkung als spezialfälle der heterogenen katalyse Zeitschrift für Elektrochemie und angewandte physikalische Chemie 35 9 555 561 10.1002/bbpc.192900026
  170. N.H. Fletcher Size effect in heterogeneous nucleation J Chem Phys 29 3 572 576 10.1063/1.1744540
  171. X. Liu A new kinetic model for three-dimensional heterogeneous nucleation J Chem Phys 111 4 1628 1635 10.1063/1.479391
  172. G. Gyarmathy The spherical droplet in gaseous carrier streams: review and synthesis Multiphase Sci Technol 10.1615/MultScienTechn.v1.i1-4.20
  173. H. Köhler The nucleus in and the growth of hygroscopic droplets Trans Faraday Soc 32 1152 1161 10.1039/TF9363201152
  174. V.Y. Smorodin P. Hopke Condensation activation and nucleation on heterogeneous aerosol nanoparticles J Phys Chem B 108 26 9147 9157 10.1021/jp037742
  175. Y. Fan F. Qin X. Luo L. Lin H. Gui J. Liu Heterogeneous condensation on insoluble spherical particles: modeling and parametric study Chem Eng Sci 102 387 396 10.1016/j.ces.2013.08.040
  176. M. Stastny and M. Sejna (2004) Numerical simulation of the steam flow with condensation in a nozzle, In International Conference on the Properties of Water and Steam: 14th International Conference on the Properties of Water and Steam in Kyoto, Kyoto, Japan, Citeseer, URL: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=34df4f667393fdcf6203943d9962250c57577a91.
  177. R. Xia D. Li M.A.F. Aliabadi Homogeneous–heterogeneous condensation in supersonic nozzle using Eulerian-Eulerian approach: Numerical investigation of liquid phase modeling Phys Fluids 10.1063/5.0256963
  178. G. Zhang X. Wang J. Chen M. Majkut S. Dykas Supersonic nozzle performance prediction considering the homogeneous-heterogeneous coupling spontaneous non-equilibrium condensation Energy 284 10.1016/j.energy.2023.129274 129274
  179. M.A.F. Aliabadi G. Zhang S. Dykas H. Li Control of two-phase heat transfer and condensation loss in turbine blade cascade by injection water droplets Appl Therm Eng 186 10.1016/j.applthermaleng.2020.116541 116541
  180. B. Peng W. Xu Z. Yu A numerical study on the effect of particle surface wettability on homogeneous and heterogeneous condensation processes in supersonic flows Chem Eng Sci 298 10.1016/j.ces.2024.120401 120401
  181. A. Manka H. Pathak S. Tanimura J. Wölk R. Strey B.E. Wyslouzil Freezing water in no-man's land Phys Chem Chem Phys 14 13 4505 4516 10.1039/C2CP23116F
  182. J. Huang L.S. Bartell Kinetics of homogeneous nucleation in the freezing of large water clusters J Phys Chem 99 12 3924 3931 10.1021/j100012a010
  183. T. Němec Estimation of ice–water interfacial energy based on pressure-dependent formulation of classical nucleation theory Chem Phys Lett 583 64 68 10.1016/j.cplett.2013.07.085
  184. M. Żenkiewicz Methods for the calculation of surface free energy of solids J Achiev Mater Manuf Eng 24 1 137 145
  185. B.J. Murray E.J. Jensen Homogeneous nucleation of amorphous solid water particles in the upper mesosphere J Atmos Solar Terr Phys 72 1 51 61 10.1016/j.jastp.2009.10.007
  186. I. Steinke C. Hoose O. Möhler P. Connolly T. Leisner A new temperature-and humidity-dependent surface site density approach for deposition ice nucleation Atmos Chem Phys 15 7 3703 3717 10.5194/acp-15-3703-2015
  187. M.-S. Liou A sequel to ausm: Ausm+ J Comput Phys 129 2 364 382 1426276 10.1006/jcph.1996.0256
  188. M. Moore (1973) Predicting the fog-drop size in wet-steam turbines, Wet steam, URL: https://cir.nii.ac.jp/crid/1574231874929479936.
  189. B.E. Wyslouzil C.H. Heath J.L. Cheung G. Wilemski Binary condensation in a supersonic nozzle J Chem Phys 113 17 7317 7329 10.1063/1.1312274
  190. K. Oswatitsch Kondensationserscheinungen in Überschalldüsen ZAMM-J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 22 1 1 14 10.1002/zamm.19420220102
  191. D.B. Spalding Combustion of liquid fuels Nature 165 4187 160 160 10.1038/165160a0
  192. F. Peters B. Paikert Measurement and interpretation of growth and evaporation of monodispersed droplets in a shock tube Int J Heat Mass Transf 37 2 293 302 10.1016/0017-9310(94)90100-7
  193. Z. Huang H. Zhang On the interactions between a propagating shock wave and evaporating water droplets Phys Fluids. 10.1063/5.0035968
  194. V. Duke-Walker W.C. Maxon S.R. Almuhna J.A. McFarland Evaporation and breakup effects in the shock-driven multiphase instability J Fluid Mech 908 A13 4190717 10.1017/jfm.2020.871
  195. R. Hernández-Sánchez C. Huete D. Martínez-Ruiz Pathological detonations in mono-disperse spray media Proc Combust Inst 40 1–4 10.1016/j.proci.2024.105505 105505
  196. T. Lu C.K. Law Heterogeneous effects in the propagation and quenching of spray detonations J Propul Power 20 5 820 827 10.2514/1.5561
  197. S. Cheatham K. Kailasanath Numerical modelling of liquid-fuelled detonations in tubes Combust Theor Model 9 1 23 48 10.1080/13647830500051786
  198. C.K. Law Recent advances in droplet vaporization and combustion Prog Energy Combust Sci 8 3 171 201 10.1016/0360-1285(82)90011-9
  199. D. Martínez-Ruiz J. Urzay A. Sánchez A. Liñán F. Williams Dynamics of thermal ignition of spray flames in mixing layers J Fluid Mech 734 387 423 10.1017/jfm.2013.500
  200. S.J. Neek H. Ghassemi M.J.Z. Ganji M. Kamalinejad Oleaster (Elaeagnus Angustifolia L.) low-fibrous extract to powder: Drying kinetics analysis Powder Technol 433 10.1016/j.powtec.2023.119249 119249
  201. M. Asrardel Á. Muelas J. Ballester A pseudocomponent-based approach for the formulation of evaporation surrogates of practical liquid fuels Combust Sci Technol 196 16 3937 3968 10.1080/00102202.2023.2202318
  202. G.R. Chumpitaz C.J. Coronado J.A. Carvalho A.Z. Mendiburu T.A. de Souza Design and study of a pure tire pyrolysis oil (TPO) and blended with Brazilian diesel using Y-Jet atomizer J Braz Soc Mech Sci Eng 41 1 20 10.1007/s40430-019-1632-z
  203. T. Zhang D. Jing S. Ge J. Wang X. Chen Supersonic antigravity aerodynamic atomization dusting nozzle based on the Laval nozzle and probe jet J Braz Soc Mech Sci Eng 42 1 15 10.1007/s40430-020-02411-5
  204. J. Young The condensation and evaporation of liquid droplets at arbitrary Knudsen number in the presence of an inert gas Int J Heat Mass Transf 36 11 2941 2956 10.1016/0017-9310(93)90112-J
  205. G. Gyarmathy (1962) Grundlagen einer Theorie der Nabdampfturbine, Ph.D. thesis, ETH Zürich, https://doi.org/10.3929/ethz-a-000087803.
  206. R. Miller K. Harstad J. Bellan Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations Int J Multiph Flow 24 6 1025 1055 10.1016/S0301-9322(98)00028-7
  207. A. Afzalifar (2012) Numerical simulation of steam in a laval nozzle and a low pressure turbine, Master thesis, Faculty of Technology, Department of Environmental Technology, Lappeenranta University of Technology
  208. P.G. Hill Condensation of water vapour during supersonic expansion in nozzles J Fluid Mech 25 3 593 620 10.1017/S0022112066000284
  209. S. Sazhin Droplets and sprays London Springer
  210. B. Abramzon W.A. Sirignano Droplet vaporization model for spray combustion calculations Int J Heat Mass Transf 32 9 1605 1618 10.1016/0017-9310(89)90043-4
  211. R. Clift J.R. Grace M.E. Weber Bubbles, drops, and particles Cambridge Academic Press
  212. B. E. Poling, J. M. Prausnitz, and J. P. O’connell (2001) Properties of gases and liquids. New York: McGraw-Hill Education, URL: https://www.accessengineeringlibrary.com/content/book/9780070116825.
  213. L. Johns Jr R. Beckmann Mechanism of dispersed-phase mass transfer in viscous, single-drop extraction systems AIChE J 12 1 10 16 10.1002/aic.690120105
  214. S.S. Sazhin Advanced models of fuel droplet heating and evaporation Prog Energy Combust Sci 32 2 162 214 10.1016/j.pecs.2005.11.001
  215. J. Bellan M. Summerfield Theoretical examination of assumptions commonly used for the gas phase surrounding a burning droplet Combust Flame 33 107 122 10.1016/0010-2180(78)90054-8
  216. F. Williams On the assumptions underlying droplet vaporization and combustion theories J Chem Phys 33 1 133 144 10.1063/1.1731068
  217. S. Sazhin T. Kristyadi W. Abdelghaffar M. Heikal Models for fuel droplet heating and evaporation: comparative analysis Fuel 85 12–13 1613 1630 10.1016/j.fuel.2006.02.012
  218. S.K. Aggarwal A.Y. Tong W.A. Sirignano A comparison of vaporization models in spray calculations AIAA J 22 10 1448 1457 10.2514/3.8802
  219. W.A. Sirignano Fluid dynamics and transport of droplets and sprays Cambridge Univ Press 10.1115/1.483244
  220. G.-S. Zhu R.D. Reitz S.K. Aggarwal Gas-phase unsteadiness and its influence on droplet vaporization in sub-and super-critical environments Int J Heat Mass Transf 44 16 3081 3093 10.1016/S0017-9310(00)00349-5
  221. D. Noh S. Gallot-Lavallée W.P. Jones S. Navarro-Martinez Comparison of droplet evaporation models for a turbulent, non-swirling jet flame with a polydisperse droplet distribution Combust Flame 194 135 151 10.1016/j.combustflame.2018.04.018
  222. F. Bakhtar K. Zidi On the self diffusion of water vapour Proc Inst Mech Eng C J Mech Eng Sci 199 2 159 164 10.1243/PIME_PROC_1985_199_107_02
  223. F.P. Incropera D.P. DeWitt T.L. Bergman A.S. Lavine Fundamentals of heat and mass transfer New York Wiley
  224. J. Young The condensation and evaporation of liquid droplets in a pure vapour at arbitrary Knudsen number Int J Heat Mass Transf 34 7 1649 1661 10.1016/0017-9310(91)90143-3
  225. G. Petruccelli A.M. Dolatabadi A. Grönman T. Turunen-Saaresti A. Guardone Numerical investigation of two-phase shock waves in CO2 flows using a modified Hertz-Knudsen model J Eng Gas Turbines Power 10.1115/1.4066602
  226. C.R. Kharangate I. Mudawar Review of computational studies on boiling and condensation Int J Heat Mass Transf 108 1164 1196 10.1016/j.ijheatmasstransfer.2016.12.065
  227. R. Puzyrewski and T. Król (1976) Numerical analysis of Hertz-Knudsen model of condensation upon small droplets in water vapor. Trans. Inst. Fluid Flow Machinery, pp. 70–72, https://imp.gda.pl/files/transactions/070-072/70-72.18._R._Puzyrewski__T._Krol.pdf.
  228. P. Wiśniewski M. Majkut S. Dykas K. Smołka G. Zhang B. Pritz Selection of a steam condensation model for atmospheric air transonic flow prediction Appl Therm Eng 203 10.1016/j.applthermaleng.2021.117922 117922
  229. S. Senoo A.J. White "Numerical simulations of unsteady wet steam flows with non-equilibrium condensation in the nozzle and the steam turbine," presented at the fluids engineering division summer meeting, Miami, Florida United States of america (USA) 10.1115/FEDSM2006-98202
  230. J. Young A. Guha Normal shock-wave structure in two-phase vapour-droplet flows J Fluid Mech 228 243 274 10.1017/S0022112091002690
  231. J.C. Maxwell Capillary action (Encyclopaedia Britannica) UK Cambridge University Press
  232. P. Yi, S. Yang, T. Li, Y. Li, and R. He (2019) An improved non-equilibrium multi-component evaporation model for blended diesel/alcohol droplets and sprays, presented at the ILASS-Americas 30th Annual Conference on Liquid Atomization and Spray Systems, Tempe, Arizona, United States of America (USA), https://www.academia.edu/download/82047368/20_2019.pdf.
  233. C. C. M. Luijten, (1998) Nucleation and droplet growth at high pressure, Ph.D. thesis, Applied Physics and Science Education, Technische Universiteit Eindhoven, Eindhoven, https://doi.org/10.6100/IR516103.
  234. X. Cao Y. Liu X. Zang D. Guo J. Bian Supersonic refrigeration performances of nozzles and phase transition characteristics of wet natural gas considering shock wave effects Case Stud Therm Eng 24 10.1016/j.csite.2020.100833 100833
  235. H. Ding Y. Zhang Y. Yang C. Wen A modified Euler-Lagrange-Euler approach for modelling homogeneous and heterogeneous condensing droplets and films in supersonic flows Int J Heat Mass Transf 200 10.1016/j.ijheatmasstransfer.2022.123537 123537
  236. W. Sun S. Chen Y. Hou S. Bu Z. Ma L. Zhang Numerical studies of nitrogen spontaneous condensation flow in laval nozzles using varying droplet growth models Int J Multiph Flow 121 10.1016/j.ijmultiphaseflow.2019.103118 103118
  237. J.B. Young Two-dimensional, nonequilibrium, wet-steam calculations for nozzles and turbine cascades J Turbomach 114 3 569 579 10.1115/1.2929181
  238. P. Peeters C. Luijten M. Van Dongen Transitional droplet growth and diffusion coefficients Int J Heat Mass Transf 44 1 181 193 10.1016/S0017-9310(00)00098-3
  239. N. Fuchs (1971) Topics in current aerosol research, Int Rev Aerosol Phys Chem, vol. 2, https://cir.nii.ac.jp/crid/1574231875510710656.
  240. S.H. Rajaee Shooshtari A. Shahsavand A theoretical mass transfer approach for prediction of droplets growth inside supersonic Laval nozzle J Chem Petrol Eng 48 1 57 68
  241. H. J. Smolders (1992) Non-linear wave phenomena in a gas-vapour mixture with phase transition, Ph.D. thesis, Applied Physics and Science Education, Technische Universiteit Eindhoven, Eindhoven, https://doi.org/10.6100/IR375903.

Issue

Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 47, 2025, Germany, https://doi.org/10.1007/s40430-025-05984-1

Цитирания (Citation/s):
1. Zhang G., Zhang Q., Yang Y., Jin Z., Dykas S., Development of an OpenFOAM non-equilibrium condensation model for flow loss and performance prediction in steam transonic flows, 2025, Energy, issue 0, vol. 340, DOI 10.1016/j.energy.2025.139183, issn 03605442, eissn 18736785 - 2025 - в издания, индексирани в Scopus и/или Web of Science
2. Zhang G., Zuo Q., Yang Y., Jin Z., Dykas S., Numerical study on the effect of heterogeneous condensation in the primary nozzle on condensation flow and performance of steam ejector, 2025, Energy, issue 0, vol. 341, DOI 10.1016/j.energy.2025.139486, issn 03605442, eissn 18736785 - 2025 - в издания, индексирани в Scopus

Вид: статия в списание, публикация в издание с импакт фактор, публикация в реферирано издание, индексирана в Scopus и Web of Science