Theoretical and computational approaches to predicting the viscosity of liquids

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Abstract

Predictive models for the shear viscosity of liquids and gases along with exact equations of state are of great practical importance for hydrodynamic modeling of processes occurring in nature, industrial plants, and machinery. We consider currently proposed theoretical, including atomistic modeling, and semi-empirical approaches to predicting the viscosity of liquids, gases, and their mixtures in a wide range of thermodynamic conditions. Viscosity models of homogeneous liquids in a thermodynamically stable state are described. The dynamics of supercooled and vitrescent liquids and dispersed systems (colloids, emulsions) remain beyond the scope of this review. We discuss the area of applicability of correlation methods for predicting viscosity and the accuracy of various methods in the pressure range up to 1 GPa. Application examples of various approaches for hydrocarbons—model oil and gas, fuel, and lubrication systems—are given.

About the authors

Nikolay Dmitrivich Kondratyuk

Joint Institute for High Temperatures, Russian Academy of Sciences; National Research University Higher School of Economics; Moscow Institute of Physics and Technology (National Research University)

without scientific degree, Researcher

Vasilii Vyacheslavovich Pisarev

Joint Institute for High Temperatures, Russian Academy of Sciences; National Research University Higher School of Economics; Moscow Institute of Physics and Technology (National Research University)

Email: pisarevvv@gmail.com

References

  1. Batschinski A. J., Z. Phys. Chem., 84 (1913), 643
  2. van der Gulik P. S., Mostert R., van den Berg H. R., Physica A, 151 (1988), 153
  3. van der Gulik P. S., Mostert R., van den Berg H. R., Fluid Phase Equilib., 79 (1992), 301
  4. Hildebrand J. H., Science, 174 (1971), 490
  5. Hanley H. J. M., Cryogenics, 16 (1976), 643
  6. Rowlinson J. S., Watson I. D., Chem. Eng. Sci., 24 (1969), 1565
  7. Ely J. F., Hanley H. J. M., Ind. Eng. Chem. Fundamen., 20 (1981), 323
  8. Ely J. F., Hanley H. J. M., A computer program for the prediction of viscosity and thermal conductivity in hydrocarbon mixtures, NBS Technical Note 1039, US Department of Commerce, National Bureau of Standards, Boulder, CO, 1981
  9. Leach J. W., Chappelear P. S., Leland T. W., AIChE J., 14 (1968), 568
  10. Huber M. L., Hanley H. J. M., Transport Properties of Fluids, J. Millat, J. H. Dymond, C. A. Nieto de Castro, IUPAC, Cambridge Univ. Press, Cambridge, New York, 1996, 283
  11. Hanley H. J. M., McCarty R. D., Haynes W. M., Cryogenics, 15 (1975), 413
  12. Ely J. F., Marrucho I. M. F., Equations of State for Fluids and Fluid Mixtures, Experimental Thermodynamics, 5, J. V. Sengers et al., Elsevier, Amsterdam, 2000, 289
  13. Huber M. L., Ely J. F., Fluid Phase Equilib., 80 (1992), 239
  14. Klein S. A., McLinden M. O., Laesecke A., Int. J. Refrigerat., 20 (1997), 208
  15. McCarty R. D., Cryogenics, 14 (1974), 276
  16. Younglove B. A., Ely J. F., J. Phys. Chem. Ref. Data, 16 (1987), 577
  17. Pedersen K. S. et al., Chem. Eng. Sci., 39 (1984), 1011
  18. Pedersen K. S., Fredenslund A., Chem. Eng. Sci., 42 (1987), 182
  19. Assael M. J., Dymond J. H., Tselekidou V., Int. J. Thermophys., 11 (1990), 863
  20. Mulero A. (Ed.), Theory and Simulation of Hard-Sphere Fluids and Related Systems, Lecture Notes in Physics, 753, Springer, Berlin, 2008
  21. Assael M. J. et al., Int. J. Thermophys., 13 (1992), 269
  22. Ciotta F., Trusler J. P. M., Vesovic V., Fluid Phase Equilib., 363 (2014), 239
  23. Nguyen T.-B., Vesovic V., Fluid Phase Equilib., 487 (2019), 58
  24. Wang X., Teja A. S., Fluid Phase Equilib., 425 (2016), 47
  25. Baylaucq A. et al., Petroleum Sci. Technol., 23 (2005), 143
  26. Nguyen T.-B., Riesco N., Vesovic V., Fuel, 208 (2017), 363
  27. Malta J. A. M. S C et al., Fluid Phase Equilib., 505 (2020), 112343
  28. Galliero G., Boned C., Baylaucq A., Ind. Eng. Chem. Res., 44 (2005), 6963
  29. Baylaucq A. et al., Int. J. Thermophys., 24 (2003), 621
  30. Galliero G. et al., Phys. Rev. E, 73 (2006), 061201
  31. Rosenfeld Y., Phys. Rev. A, 15 (1977), 2545
  32. Dzugutov M., Nature, 381 (1996), 137
  33. Lötgering-Lin O., Gross J., Ind. Eng. Chem. Res., 54 (2015), 7942
  34. Pasturel A., Jakse N., J. Phys. Condens. Matter, 28 (2016), 485101
  35. Dyre J. C., J. Chem. Phys., 149 (2018), 210901
  36. Carnahan N. F., Starling K. E., J. Chem. Phys., 51 (1969), 635
  37. Bell I. H., Dyre J. C., Ingebrigtsen T. S., Nat. Commun., 11 (2020), 4300
  38. Fomin Yu. D., Ryzhov V. N., Gribova N. V., Phys. Rev. E, 81 (2010), 061201
  39. Chopra R., Truskett T. M., Errington J. R., J. Phys. Chem. B, 114 (2010), 10558
  40. Soave G., Chem. Eng. Sci., 27 (1972), 1197
  41. Peng D.-Y., Robinson D. B., Ind. Eng. Chem. Fundamen., 15 (1976), 59
  42. Брусиловский A. И., Фазовые превращения при разработке месторождений нефти и газа, Грааль, М., 2002
  43. Chapman W. G. et al., Fluid Phase Equilib., 52 (1989), 31
  44. Gross J., Sadowski G., Ind. Eng. Chem. Res., 40 (2001), 1244
  45. Polishuk I., Ind. Eng. Chem. Res., 53 (2014), 14127
  46. Gerasimov A., Alexandrov I., Grigoriev B., Fluid Phase Equilib., 418 (2016), 204
  47. Григорьев Б. А., Герасимов А. А., Александров И. С., “Актуальные вопросы исследований пластовых систем месторождений углеводородов”, Вести газовой науки, 1(12), Под ред. Б. А. Григорьева, Газпром ВНИИГАЗ, М., 2012, 1
  48. Levashov P. et al., AIP Conf. Proc., 505 (2000), 89
  49. Baled H. O. et al., Fuel, 218 (2018), 89
  50. Taib M. B. M., Trusler J. P. M., J. Chem. Phys., 152 (2020), 164104
  51. Rokni H. B. et al., Fuel, 241 (2019), 1203
  52. Abramson E. H., West-Foyle H., Phys. Rev. E, 77 (2008), 041202
  53. Abramson E. H., J. Phys. Chem. B, 118 (2014), 11792
  54. Abramson E. H., Phys. Rev. E, 80 (2009), 021201
  55. Abramson E. H., High Pressure Res., 31 (2011), 544
  56. Bell I. H., Proc. Natl. Acad. Sci. USA, 116 (2019), 4070
  57. Quiñones-Cisneros S. E., Zeberg-Mikkelsen C. K., Stenby E. H., Fluid Phase Equilib., 169 (2000), 249
  58. Quiñones-Cisneros S. E., Zeberg-Mikkelsen C. K., Stenby E. H., Fluid Phase Equilib., 178 (2001), 1
  59. Quiñones-Cisneros S. E., Zeberg-Mikkelsen C. K., Stenby E. H., Int. J. Thermophys., 23 (2002), 41
  60. Quiñones-Cisneros S. E. et al., AIChE J., 52 (2006), 1600
  61. Bair S., High Temp. High Press., 44 (2015), 415
  62. Kondratyuk N. D., Pisarev V. V., Ewen J. P., J. Chem. Phys., 153 (2020), 154502
  63. Allal A., Boned C., Baylaucq A., Phys. Rev. E, 64 (2001), 011203
  64. Llovell F., Marcos R. M., Vega L. F., J. Phys. Chem. B, 117 (2013), 8159
  65. Llovell F., Marcos R. M., Vega L. F., J. Phys. Chem. B, 117 (2013), 5195
  66. Polishuk I., Yitzhak A., Ind. Eng. Chem. Res., 53 (2014), 959
  67. Yarranton H. W., Satyro M. A., Ind. Eng. Chem. Res., 48 (2009), 3640
  68. Hildebrand J. H., Lamoreaux R. H., Proc. Natl. Acad. Sci. USA, 69 (1972), 3428
  69. Motahhari H., Satyro M. A., Yarranton H. W., Ind. Eng. Chem. Res., 50 (2011), 12831
  70. Polishuk I., Ind. Eng. Chem. Res., 51 (2012), 13527
  71. Polishuk I., Ind. Eng. Chem. Res., 54 (2015), 6999
  72. Abolala M., Peyvandi K., Varaminian F., Fluid Phase Equilib., 394 (2015), 61
  73. Ryshkova O. S., Postnikov E. B., Polishuk I., Ind. Eng. Chem. Res., 58 (2019), 20116
  74. Oliveira C. M. B. P, Wakeham W. A., Int. J. Thermophys., 13 (1992), 773
  75. Audonnet F., Padua A. A. H., Fluid Phase Equilib., 181 (2001), 147
  76. Dymond J. H., Robertson J., Isdale J. D., Int. J. Thermophys., 2 (1981), 133
  77. Canet X., Baylaucq A., Boned C., Int. J. Thermophys., 23 (2002), 1469
  78. Battezzati L., Greer A., Acta Metallurg., 37 (1989), 1791
  79. Kaptay G., Z. Metallkd, 96 (2005), 24
  80. Budai I., Benkõ M. Z., Kaptay G., Mater. Sci. Forum, 537–538 (2007), 489
  81. Gasior W., Calphad, 44 (2014), 119
  82. Chen W. et al., Philos. Mag., 94 (2014), 1552
  83. Beltyukov A., Olyanina N., Ladyanov V., J. Mol. Liq., 281 (2019), 204
  84. Козлов Л. Я., Романов Л. М., Петров Н. Н., Изв. вузов. Черная металлургия, 1983, № 3, 7
  85. Голубев И. Ф., Вязкость газов и газовых смесей, Физматгиз, М., 1959
  86. Hirschfelder J. O., Curtiss C. F., Bird R. B., Molecular Theory of Gases and Liquids, Wiley, New York, 1954
  87. Голубев И. Ф., Гнездилов Н. Е., Вязкость газовых смесей, Изд-во стандартов, М., 1971
  88. Neufeld P. D., Janzen A. R., Aziz R. A., J. Chem. Phys., 57 (1972), 1100
  89. Chung T. H., Lee L. L., Starling K. E., Ind. Eng. Chem. Fundamen., 23 (1984), 8
  90. Wilke C. R., J. Chem. Phys., 18 (1950), 517
  91. Palmer G. E., Wright M. J., J. Thermophys. Heat Transfer, 17 (2003), 232
  92. Herning F., Zipperer L., Gas Wasserfach, 79 (1936), 69
  93. Davidson T. A., A simple and accurate method for calculating viscosity of gaseous mixtures, Report of Investigations 9456, US Department of the Interior, Bureau of Mines, Spokane, WA, 1993
  94. Норман Г. Э., Стегайлов В. В., Матем. моделирование, 24:6 (2012), 3
  95. Tuckerman M. E., Statistical Mechanics: Theory and Molecular Simulation, Oxford Univ. Press, Oxford, 2010
  96. Frenkel D., Smit B., Understanding Molecular Simulation, Computational Science Series, 1, 2nd ed., Academic Press, San Diego, CA, 2002
  97. Ewen J. P. et al., Materials, 9 (2016), 651
  98. Kondratyuk N. D., Norman G. E., Stegailov V. V., J. Chem. Phys., 145 (2016), 204504
  99. Glova A. D. et al., RSC Adv., 9 (2019), 38834
  100. Orekhov N., Ostroumova G., Stegailov V., Carbon, 170 (2020), 606
  101. Белащенко Д. К., УФН, 190 (2020), 1233
  102. Nazarychev V. M. et al., Int. J. Heat Mass Transf., 165 (2021), 120639
  103. Zabaloy M. S., Machado J. M. V., Macedo E. A., Int. J. Thermophys., 22 (2001), 829
  104. Бражкин В. В., УФН, 189 (2019), 665
  105. Lv X. et al., J. Fluorine Chem., 241 (2021), 109675
  106. Galamba N., Nieto de Castro C. A., Ely J. F., J. Phys. Chem. B, 108 (2004), 3658
  107. Galamba N., Nieto de Castro C. A., Ely J. F., J. Chem. Phys., 122 (2005), 224501
  108. Janz G. J., J. Phys. Chem. Ref. Data, 17:Suppl. 2 (1988), 1
  109. Белащенко Д. К., УФН, 183 (2013), 1281
  110. Daw M. S., Baskes M. I., Phys. Rev. B, 29 (1984), 6443
  111. Finnis M. W., Sinclair J. E., Philos. Mag. A, 50 (1984), 45
  112. Canales M., Gonzalez L. E., Padro J. À, Phys. Rev. E, 50 (1994), 3656
  113. Meyer N., Xu H., Wax J.-F., Phys. Rev. B, 93 (2016), 214203
  114. Demmel F., Tani A., Phys. Rev. E, 97 (2018), 062124
  115. Metya A. K., Hens A., Singh J. K., Fluid Phase Equilib., 313 (2012), 16
  116. Кирова Е. М., Норман Г. Э., Писарев В. В., Письма в ЖЭТФ, 110 (2019), 343
  117. Cherne F. J. (III), Deymier P. A., Scr. Mater., 39 (1998), 1613
  118. Cherne F. J., Baskes M. I., Deymier P. A., Phys. Rev. B, 65 (2001), 024209
  119. Mendelev M. et al., Philos. Mag., 88 (2008), 1723
  120. Smirnova D. et al., Modelling Simul. Mater. Sci. Eng., 21 (2013), 035011
  121. O'Connor T. C., Andzelm J., Robbins M. O., J. Chem. Phys., 142 (2015), 024903
  122. Jorgensen W. L. et al., J. Chem. Phys., 79 (1983), 926
  123. Horn H. W. et al., J. Chem. Phys., 120 (2004), 9665
  124. Abascal J. L. F., Vega C., J. Chem. Phys., 123 (2005), 234505
  125. Mitchell P. J., Fincham D., J. Phys., 5 (1993), 1031
  126. Ponder J. W., Case D. A., Adv. Protein Chem., 66 (2003), 27
  127. Wang J. et al., J. Comput. Chem., 25 (2004), 1157
  128. MacKerell A. D. (Jr.) et al., J. Phys. Chem. B, 102 (1998), 3586
  129. Jorgensen W. L., Maxwell D. S., Tirado-Rives J., J. Am. Chem. Soc., 118 (1996), 11225
  130. Allen W., Rowley R. L., J. Chem. Phys., 106 (1997), 10273
  131. Liu H. et al., Ind. Eng. Chem. Res., 51 (2012), 7242
  132. Siu S. W. I., Pluhackova K., Böckmann R. A., J. Chem. Theory Comput., 8 (2012), 1459
  133. Dodda L. S. et al., J. Phys. Chem. B, 121 (2017), 3864
  134. Martin M. G., Siepmann J. I., J. Phys. Chem. B, 102 (1998), 2569
  135. Moore J. D. et al., J. Chem. Phys., 113 (2000), 8833
  136. Dysthe D. K., Fuchs A. H., Rousseau B., J. Chem. Phys., 112 (2000), 7581
  137. Nieto-Draghi C., Ungerer P., Rousseau B., J. Chem. Phys., 125 (2006), 044517
  138. Ungerer P. et al., J. Chem. Phys., 112 (2000), 5499
  139. Nieto-Draghi C. et al., Mol. Simul., 34 (2008), 211
  140. Sun H., J. Phys. Chem. B, 102 (1998), 7338
  141. Kondratyuk N. D., Pisarev V. V., Fluid Phase Equilib., 498 (2019), 151
  142. Kondratyuk N., Lenev D., Pisarev V., J. Chem. Phys., 152 (2020), 191104
  143. Kondratyuk N. D., Pisarev V. V., Fluid Phase Equilib., 544–545 (2021), 113100
  144. Orekhov N., Ostroumova G., Stegailov V., Carbon, 170 (2020), 606
  145. Car R., Parrinello M., Phys. Rev. Lett., 55 (1985), 2471
  146. Кон В., УФН, 172 (2002), 336
  147. Levashov P. R. et al., J. Phys., 22 (2010), 505501
  148. French M. et al., Atrophys. J. Suppl., 202 (2012), 5
  149. Wang W. Y. et al., Acta Mater., 97 (2015), 75
  150. Weber H. et al., Phys. Rev. B, 96 (2017), 054204
  151. Rong Z. et al., Renewable Energy, 163 (2021), 579
  152. Harada A., Shimojo F., Hoshino K., J. Phys. Soc. Jpn., 74 (2005), 2017
  153. Ohmura S., Shimojo F., Phys. Rev. B, 80 (2009), 020202
  154. Ohmura S., Shimojo F., Phys. Rev. B, 84 (2011), 224202
  155. Koura A., Ohmura S., Shimojo F., J. Chem. Phys., 138 (2013), 134504
  156. Ohmura S., Shimojo F., Phys. Rev. B, 83 (2011), 134206
  157. Lopanitsyna N., Ben Mahmoud C., Ceriotti M., Phys. Rev. Mater., 5 (2021), 043802
  158. Kamaeva L. V. et al., J. Phys., 32 (2020), 224003
  159. Ryltsev R., Chtchelkatchev N., J. Mol. Liq., 2022, accepted
  160. Behler J., Parrinello M., Phys. Rev. Lett., 98 (2007), 146401
  161. Wang H. et al., Comput. Phys. Commun., 228 (2018), 178
  162. Yeh I.-C., Hummer G., J. Phys. Chem. B, 108 (2004), 15873
  163. Green M. S., J. Chem. Phys., 22 (1954), 398
  164. Kubo R., J. Phys. Soc. Jpn., 12 (1957), 570
  165. Helfand E., Phys. Rev., 119 (1960), 1
  166. Alder B. J., Wainwright T. E., Phys. Rev. A, 1 (1970), 18
  167. Anikeenko A. V., Malenkov G. G., Naberukhin Yu. I., J. Chem. Phys., 148 (2018), 094508
  168. Orekhov M. A., J. Mol. Liq., 322 (2021), 114554
  169. Moultos O. A. et al., J. Chem. Phys., 145 (2016), 074109
  170. Волков Н. А., Посысоев М. В., Щeкин А. К., Коллоидный журн., 80 (2018), 264
  171. Jamali S. H. et al., J. Chem. Theor. Comput., 14 (2018), 2667
  172. Tazi S. et al., J. Phys. Condens. Matter, 24 (2012), 284117
  173. Humphrey W., Dalke A., Schulten K., J. Mol. Graph., 14 (1996), 33
  174. Lee A. L., Ellington R. T., J. Chem. Eng. Data, 10 (1965), 101
  175. Belonoshko A. B. et al., Nat. Commun., 10 (2019), 2483
  176. Adjaoud O., Steinle-Neumann G., Jahn S., Earth Planet. Sci. Lett., 312 (2011), 463
  177. Gordon P. A., Ind. Eng. Chem. Res., 44 (2005), 5828
  178. Kondratyuk N. D., Orekhov M. A., J. Phys. Conf. Ser., 1556 (2020), 012048
  179. Бражкин В. В., Ляпин А. Г., УФН, 170 (2000), 535
  180. Храпак С. А., Храпак А. Г., Письма в ЖЭТФ, 114 (2021), 615
  181. Viscardy S., Servantie J., Gaspard P., J. Chem. Phys., 126 (2007), 184512
  182. Allen M. P., Brown D., Masters A. J., Phys. Rev. E, 49 (1994), 2488
  183. Nevins D., Spera F. J., Mol. Simul., 33 (2007), 1261
  184. Fomin Yu. D., Brazhkin V. V., Ryzhov V. N., Письма в ЖЭТФ, 95 (2012), 349
  185. Fomin Yu. D., Brazhkin V. V., Ryzhov V. N., Phys. Rev. E, 86 (2012), 011503
  186. Rudyak V. Ya., Krasnolutskii S. L., Phys. Lett. A, 378 (2014), 1845
  187. Рудяк В. Я., Краснолуцкий С. Л., ЖТФ, 85:6 (2015), 9
  188. Brazhkin V. V. et al., Physica A, 509 (2018), 690
  189. Zhang Y., Otani A., Maginn E. J., J. Chem. Theory Comput., 11 (2015), 3537
  190. Maginn E. J. et al., Living J. Comput. Mol. Sci., 1 (2019), 6324
  191. Hess B., J. Chem. Phys., 116 (2002), 209
  192. Rey-Castro C., Vega L. F., J. Phys. Chem. B, 110 (2006), 14426
  193. Kondratyuk N., J. Chem. Phys., 151 (2019), 074502
  194. Zhang Y. et al., J. Phys. Chem. B, 119 (2015), 14934
  195. Messerly R. A. et al., Fluid Phase Equilib., 495 (2019), 76
  196. Kondratyuk N., Lenev D., Pisarev V., J. Chem. Phys., 152 (2020), 191104
  197. Liesen N. T. et al., J. Chem. Phys., 153 (2020), 024502
  198. Heyes D. M., Smith E. R., Dini D., J. Chem. Phys., 150 (2019), 174504
  199. Heyes D. M., Dini D., Smith E. R., J. Chem. Phys., 154 (2021), 074503
  200. Müller-Plathe F., Phys. Rev. E, 59 (1999), 4894
  201. Bordat P., Müller-Plathe F., J. Chem. Phys., 116 (2002), 3362
  202. Evans D. J., Morriss G. P., Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London, 1990
  203. Tuckerman M. E. et al., J. Chem. Phys., 106 (1997), 5615
  204. Daivis P. J., Todd B. D., J. Chem. Phys., 124 (2006), 194103
  205. Ewen J. P., Spikes H. A., Dini D., Tribol. Lett., 69 (2021), 24
  206. McCabe C. et al., J. Chem. Phys., 114 (2001), 1887
  207. Jadhao V., Robbins M. O., Proc. Natl. Acad. Sci. USA, 114 (2017), 7952
  208. Jadhao V., Robbins M. O., Tribol. Lett., 67 (2019), 66
  209. Liu P. et al., J. Chem. Phys., 147 (2017), 084904
  210. Carreau P. J., Trans. Soc. Rheology, 16 (1972), 99
  211. Prentice I. J. et al., J. Chem. Phys., 152 (2020), 074504
  212. Lee S. H., Cummings P. T., Mol. Simulat., 16 (1996), 229
  213. Travis K. P., Searles D. J., Evans D. J., Mol. Phys., 95 (1998), 195
  214. Kelkar M. S. et al., Fluid Phase Equilib., 260 (2007), 218
  215. Galvani Cunha M. A., Robbins M. O., Fluid Phase Equilib., 495 (2019), 28
  216. Bair S., McCabe C., Cummings P. T., Phys. Rev. Lett., 88 (2002), 058302
  217. Chen T., Smit B., Bell A. T., J. Chem. Phys., 131 (2009), 246101
  218. Simonnin P. et al., J. Chem. Theory Comput., 13 (2017), 2881
  219. Bair S., Martinie L., Vergne P., Tribol. Lett., 63 (2016), 37
  220. Bair S., Fluid Phase Equilib., 488 (2019), 9
  221. Zhang J., Spikes H., Tribol. Lett., 68 (2020), 42
  222. Тропин Т. В., Шмельцер Ю. В. П., Аксенов В. Л., УФН, 186 (2016), 47
  223. “Industrial Fluid Properties Simulation Challenge”, Industrial Fluid Properties Simulation Collective, Accessed: 2021-03-24
  224. Weiss H. et al., Annu. Rev. Chem. Biomol. Eng., 7 (2016), 65
  225. Lowitz D. A. et al., J. Chem. Phys., 30 (1959), 73
  226. Benchmarks for the 11th Challenge, Industrial Fluid Properties Simulation Collective
  227. Stegailov V. et al., Int. J. High Perform. Comput. Appl., 33 (2019), 507
  228. Kostenetskiy P. S., Chulkevich R. A., Kozyrev V. I., J. Phys. Conf. Ser., 1740 (2021), 012050

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