Skip to main content
SpringerLink
Log in

Multiphase Flow Kinetic Theory, Constitutive Equations, and Experimental Validation

  • Chapter
  • First Online:
Transport Phenomena in Multiphase Systems

Part of the book series: Mechanical Engineering Series ((MES))

  • 1245 Accesses

Abstract

The kinetic theory of gases is extended to particulate flow where the interaction between particles is not conserved. Granular temperature and the equation of state for the particle phase is developed. Granular temperature is defined as the average of the random kinetic energy, with the conversion factor of the Boltzmann constant. For dense flows, in addition to the kinetic and collisional stresses using the kinetic theory approach, a model account for the frictional stresses based on soil mechanics principles is presented. Gas/particle flows are inherently oscillatory and they manifest in non-homogeneous structures. Therefore, the drag force for different regimes of non-homogeneous gas/solid flow systems is discussed. Fluid/particle systems are composed of particles of different properties, in which transfer of momentum and segregation by size or density occurs during the flow, thus the kinetic theory was extended to modeling of multitype particle flow using the kinetic theory approach. Finally, heat and mass transfer equations for gas/solid flow systems are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Chapman S, Cowling TG (1960) The mathematical theory of non-uniform gases. Cambridge at the University Press, Cambridge, UK

    MATH  Google Scholar 

  2. Savage SB, Jeffrey DJ (1981) The stress tensor in a granular flow at high shear rates. J Fluid Mech 110:255–272

    Article  MATH  Google Scholar 

  3. Jenkins J, Savage SB (1983) Theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J Fluid Mech 130(1):187–202

    Article  MATH  Google Scholar 

  4. Lun CKK, Savage SB, Jeffrey DJ, Chepurniy N (1984) Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J Fluid Mech 140:223–256

    Article  MATH  Google Scholar 

  5. Jenkins JT, Richman MW (1985) Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys Fluids 28(12):3485–494

    Article  MATH  Google Scholar 

  6. Gidaspow D (1994) Multiphase flow and fluidization: continuum and kinetic theory description. Academic Press, San Diego, California, USA

    MATH  Google Scholar 

  7. Kim H, Arastoopour H (2002) Extension of kinetic theory to cohesive particle flow. Powder Technol 122(1):83–94

    Article  Google Scholar 

  8. Iddir H, Arastoopour H (2005) Modeling of multitype particle flow using the kinetic theory approach. AIChE J 51(6):1620–1632

    Article  Google Scholar 

  9. Gidaspow D, Jung J, Singh RK (2004) Hydrodaynamics of fluidization using kinetic theory: an emerging paradigm: 2002 Fluor-Daniel plenary lecture. Powder Technol 148:123–141

    Article  Google Scholar 

  10. Arastoopour H (2001) Numerical simulation and experimental analysis of gas/solid flow systems: 1999 Fluor-Daniel plenary lecture. Powder Technol 119(2–3):59–67

    Article  Google Scholar 

  11. Songprawat S, Gidaspow D (2010) Multiphase flow with unequal granular temperatures. Chem Eng Sci 65:1134–1143

    Article  Google Scholar 

  12. Shuai W, Zhenhua H, Huilin L, Guodong L, Jiaxing W, Pengfei X (2012) A bubbling fluidization model using kinetic theory of rough spheres. AIChE J 58:440–455

    Article  Google Scholar 

  13. Strumendo M, Gidaspow D, Canu P (2005) Method of moments for gas-solid flows: application to the riser. In: Cen K (ed) Circulating fluidized bed technology VIII. New York: Intern. Acad. Pub., New York, pp 936–942

    Google Scholar 

  14. Strumendo M, Arastoopour H (2010) Solution of population balance equations by the finite size domain complete set of trial functions method of moments (FCMOM) for inhomogeneous systems. Ind Eng Chem Res 49:5222–5230

    Article  Google Scholar 

  15. Gidaspow D, Huilin L (1998) Equation of state and radial distribution functions of FCC particles in a CFB. AIChE J 44(2):279–291

    Article  Google Scholar 

  16. Gidaspow D, Jiradilok V (2009). Computational techniques: the multiphase CFD approach to fluidization and green energy technologies. Nova Science Publishers, New York USA

    Google Scholar 

  17. Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge, UK

    Book  MATH  Google Scholar 

  18. Jiradilok V, Gidaspow D, Breault RW (2007) Computation of gas and solids dispersion coefficients in turbulent risers and bubbling beds. Chem Eng Sci 62(13):3397–3409

    Article  Google Scholar 

  19. Kashyap M, Gidaspow D (2012). Dispersion and mass transfer coefficients in fluidized beds. Lap Lambert Academic Publishing, Saarbruchen, Germany

    Google Scholar 

  20. Miller A, Gidaspow D (1992) Dense vertical gas-solid flow in a pipe. AIChE J 38:1801–1815

    Article  Google Scholar 

  21. Johnson PC, Jackson R (1987) Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J Fluid Mech 176:67–93

    Article  Google Scholar 

  22. Benyahia S, Syamlal M, O’Brien TJ (2005) Evaluation of boundary conditions used to model dilute, turbulent gas/solids flows in a pipe. Powder Technol 156(2):62–72

    Article  Google Scholar 

  23. Syamlal M (1985) Multiphase hydrodynamics of gas-solids flow. PhD thesis, Illinois Institute of Technology, Chicago, IL

    Google Scholar 

  24. Tartan M, Gidaspow D (2004) Measurement of granular temperature and stresses in risers. AIChE J 50:1760–1775

    Article  Google Scholar 

  25. Schlichting HT (1960) Boundary-layer theory, 4th ed. McGraw Hill, New York, USA

    MATH  Google Scholar 

  26. Kim J, Moin P, Moser R (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–166

    Article  MATH  Google Scholar 

  27. Benyahia S, Syamlal M, O’Brein TJ (2007) Study of ability of multiphase continuum models to predict core-annulus flow. AIChE J 53(10):2549–256

    Article  Google Scholar 

  28. Berruti F, Chaouki J, Godfroy L, Pugsley TS, Patience GS (1995) Hydrodynamics of circulating fluidized bed risers: a review. Canadian J Chem Eng 73(5):579–602

    Article  Google Scholar 

  29. Gidaspow D, Chandra V (2014) Unequal granular temperature model for motion of platelets to the wall and red blood cells to the center. Chem Eng Sci 117:107–113

    Article  Google Scholar 

  30. ANSYS, Inc., FLUENT user’s guide. Canonsburg, PA

    Google Scholar 

  31. Syamlal, M, Rogers W, O’Brien TJ (1993) MFIX documentation theory guide: technical note. DOE/METC-94/1004, NTIS/DE94000087, U.S. Department of Energy, Office of Fossil Energy, Morgantown Energy Technology Center Morgantown, WV, National Technical Information Service, Springfield, VA

    Google Scholar 

  32. Kashyap M, Gidaspow D, Koves WJ (2011) Circulation of Geldart D type particles: part I- high flux solids measurements and computation under solids slugging conditions. Chem Eng Sci 66:183–206

    Article  Google Scholar 

  33. Wei F, Lin H, Chang Y, Wang Z, Jin Y (1998) Profiles of particle velocity and solids faction in a high density riser. Powder Technol 100:183–189

    Article  Google Scholar 

  34. Li J, Kwauk M (1994) Particle-fluid two-phase flow. Metallurgical Industry Press, Beijing, China

    Google Scholar 

  35. Jung J, Gidaspow D, Gamwo IK (2005) Measurement of two kinds of granular temperatures, stresses and dispersion in bubbling beds. Ind Eng Chem Res 44(5):1329–1341

    Article  Google Scholar 

  36. Lyczkowski RW, Gidaspow D, Solbrig W (1982) Multiphase flow-models for nuclear, fossil and biomass energy conversion. In: Mujumdar AS, Mashelkar RA (eds) Advances in transport processes (1982), Wiley-Eastern Publisher, New York, pp 198–351

    Google Scholar 

  37. Arastoopour H, Gidaspow D, Abbasi E (2017) Computational transport phenomena of fluid-particle systems. Springer, Cham, Switzerland

    Book  MATH  Google Scholar 

  38. Jung J, Gidaspow D (2002) Fluidization of nano-size particles. J Nanoparticle Research 4:483–497

    Article  Google Scholar 

  39. Gelderbloom S, Gidaspow D, Lyczkowski RW (2003) CFD simulation of bubbling and collapsing fluidized bed experiments for three Geldart groups. AIChE 49:844-858

    Article  Google Scholar 

  40. Matsen JM (2000) Drift flux representation of gas-particle flow. Powder Technol 111(1-2):25–33

    Article  Google Scholar 

  41. Tsuo YP, Gidaspow D (1990) Computation of flow patterns in circulating fluidized beds. AIChE J 36:885–896

    Article  Google Scholar 

  42. Sun B, Gidaspow D (1999) Computation of circulating fluidized-bed riser flow for the fluidization VIII benchmark test. Ind Eng Chem Res 38:787–792

    Article  Google Scholar 

  43. Driscoll MC (2007). A Study of the fluidization of FCC and nanoparticles in a rectangular bed and a riser. PhD thesis, Illinois Institute of Technology, Chicago

    Google Scholar 

  44. Gidaspow, D, Driscoll M (2007) Wave propagation and granular temperature in fluidized beds of nanoparticles. AIChE J 53(7):1718–1726

    Article  Google Scholar 

  45. Roy R, Davidson JF, Tuponogov VG (1990) The velocity of sound in fluidised beds. Chem Eng Sci 45(11):3233–3245

    Article  Google Scholar 

  46. Polashenski W, Chen JC (1999) Measurement of particle stresses in fast fluidized beds. Ind Eng Chem Res 38:705–713

    Article  Google Scholar 

  47. Gidaspow D, Mostofi R (2003) Maximum carrying capacity and granular temperature of A, B, C particles. AIChE J 49(4):831–843

    Article  Google Scholar 

  48. Makkawi Y, Ocone R (2005) Modelling the particle stress at the dilute-intermediate-dense flow regimes: a review. KONA Powder and Particle J 23(0):49–63

    Article  Google Scholar 

  49. Ocone R, Sundaresan S, Jackson R (1993) Gas-particle flow in a duct of arbitrary inclination with particle-particle interactions. AIChE J 39(8):1261–1271

    Article  Google Scholar 

  50. Laux H (1998) Modeling of dilute and dense dispersed fluid-particle flow. Trondheim, Norway, Norwegian University of Science and Technology

    Google Scholar 

  51. Savage SB (1998) Analyses of slow high - concentrations flows of granular materials. J Fluid Mech 377:1–26

    Google Scholar 

  52. Schaeffer DG (1987) Instability in the evolution equations describing incompressible granular flow. J Diff Eq:19–50

    Google Scholar 

  53. Dartevelle S (2003) Numerical and granulometric approaches to geophysical granular flows. Houghton, MI, Michigan Technological University

    Google Scholar 

  54. Srivastava A, Sundaresan S (2003) Analysis of a frictional-kinetic model for gas-particle flow. Powder Technol:72–85

    Google Scholar 

  55. Atkinson, JH, Bransby PL (1978) The mechanics of soils: an introduction to critical state soil mechanics. McGraw-Hill Book Company, London, New York

    Google Scholar 

  56. Tardos GI (1997) A fluid mechanistic approach to slow, frictional flow of powders. Powder Technol 92(1):61–74

    Article  Google Scholar 

  57. Prakash JR, Rao KK (1988) Steady compressible flow of granular materials through a wedgeshaped hopper: the smooth wall, radial gravity problem. Chem Eng Sci 43(3)

    Google Scholar 

  58. O’Brien TJ, Mahalatkar K, Kuhlman J (2010) Multiphase CFD simulations of chemical looping reactors for CO2 capture. 7th International Conference on Multiphase Flow, ICMF 2010, Tampa, FL USA

    Google Scholar 

  59. Nikolopoulos A, Nikolopoulos N, Charitos A, Grammelis P, Kakaras E, Bidwe AR, Varela G (2013) High-resolution 3-D full-loop simulation of a CFB carbonator cold model. Chem Eng Sci 90:137–150

    Article  Google Scholar 

  60. Abbasi E, Abbasian J, Arastoopour H (2015) CFD–PBE numerical simulation of CO2 capture using MgO-based sorbent. Powder Technol 286:616–628

    Article  Google Scholar 

  61. Nikolopoulos A, Nikolopoulos N, Varveris N, Karellas S, Grammelis P, Kakaras E (2012) Investigation of proper modeling of very dense granular flows in the recirculation system of CFBs. Particuology 10(6):699–709

    Article  Google Scholar 

  62. Abbasi E, Arastoopour H (2011) CFD simulation of CO2 sorption in a circulating fluidized bed using deactivation kinetic model. In: Knowlton TM (ed) Proceeding of the Tenth International Conference on Circulating Fluidized Beds and Fluidization technology, CFB-10, ECI, New York, pp 736–743

    Google Scholar 

  63. Ghadirian E, Arastoopour H (2017) Numerical analysis of frictional behavior of dense gas–solid systems. Particuology, 32:178–190

    Article  Google Scholar 

  64. Das BM (1997) Advanced soil mechanics (2nd ed). Taylor & Francis, Washington DC

    Google Scholar 

  65. Jyotsna R, Rao KK (1997) A frictional-kinetic model for the flow of granular materials through a wedge-shaped hopper. Journal Fluid Mech 346:239–270

    Article  MATH  Google Scholar 

  66. Igci Y, Pannala S, Benyahia S, Sundaresan S (2008) Validation studies on filtered model equations for gas-particle flows in risers. Ind Eng Chem Res 51(4):2094–2103

    Article  Google Scholar 

  67. Benyahia S (2012) Fine-grid simulations of gas-solids flow in a circulating fluidized bed. AIChE J 58(11):3589–3592

    Article  Google Scholar 

  68. Nikolopoulos A, Atsonios K, Nikolopoulos N, Grammelis P, Kakaras E (2010) An advanced EMMS scheme for the prediction of drag coefficient under a 1.2 MW CFBC isothermal flow, part II: numerical implementation. Chem Eng Sci 65(13):4089–4099

    Article  Google Scholar 

  69. Jang J, Rosa C, Arastoopour H (2010) CFD simulation of pharmaceutical particle drying in a bubbling fluidized bed reactor. In: Kim S.D. et al. (ed) Fluidization XIII, ECI, New York, pp 853–860

    Google Scholar 

  70. Arastoopour H, Pakdel P, Adewumi M (1990) Hydrodynamic analysis of dilute gas-solids flow in a vertical pipe. Powder Technol 62:163–170

    Article  Google Scholar 

  71. Syamlal M, O’Brien TJ (2003). Fluid dynamic simulation of O3 decomposition in a bubbling fluidized bed. AIChE Journal 49(11):2793–2801

    Article  Google Scholar 

  72. Wen CY, Yu YH (1966) Mechanics of fluidization. Chem Eng Prog Symp Series 62: 100–111

    Google Scholar 

  73. Benyahia S (2009) On the effect of subgrid drag closures. Ind Eng Chem Res 49(11): 5122-5131

    Article  Google Scholar 

  74. Sarkar A, Xin S, Sundaresan S (2014) Verification of sub-grid filtered drag models for gas-particle fluidized beds with immersed cylinder arrays. Chem Eng Sci 114:144–154

    Article  Google Scholar 

  75. Ghadirian E, Arastoopour H (2016) CFD simulation of a fluidized bed using the EMMS approach for the gas-solid drag force. Powder Technol 288:35-44

    Article  Google Scholar 

  76. Arastoopour H, Gidaspow D (1979) Analysis of IGT pneumatic conveying data and fast fluidization using a thermohydrodynamic model. Powder Technol 22(1):77–87

    Article  Google Scholar 

  77. Milioli CC, Milioli FE, Holloway W, Agrawal K, Sundaresan S (2013) Filtered two-fluid models of fluidized gas-particle flows: new constitutive relations. AIChE J 59(9):3265–3275

    Article  Google Scholar 

  78. Benyahia S, Sundaresan S (2012) Do we need sub-grid scale corrections for both continuum and discrete gas-particle flow models? Powder Technol 220:2–6

    Article  Google Scholar 

  79. Li J, Kwauk M (1994) Particle-fluid two-phase flow: the energy-minimization multi-scale method. Metallurgy, Industry Press, Beijing

    Google Scholar 

  80. Wang W, Li J (2007) Simulation of gas-solid two-phase flow by a multi-scale CFD approach: extension of the EMMS model to the sub-grid level. Chem Eng Sci 62(1):208–231

    Article  Google Scholar 

  81. Benyahia S (2012) Analysis of model parameters affecting the pressure profile in a circulating fluidized bed. AIChE J 58(2):427–439

    Article  Google Scholar 

  82. Krishna BSVSR (2013) Predicting the bed height in expanded bed adsorption column using RZ correlation. Bonfring Int J Ind Eng and Mgmt Sci 3(4):107

    Google Scholar 

  83. Lu B, Wang W, Li J (2009) Searching for a mesh-independent sub-grid model for CFD simulation of gas-solid riser flows. Chem Eng Sci 64(15):3437–3447

    Article  Google Scholar 

  84. Arastoopour H, Lin SC, Weil, SA (1982) Analysis of vertical pneumatic conveying of solids using multiphase flow models. AIChE J 28(3):467–473

    Article  Google Scholar 

  85. Cutchin, J, Arastoopour H (1985) Measurement and analysis of particle interaction in a concurrent pneumatic conveying system. Chem Eng Sci 40(7):1134–1143

    Google Scholar 

  86. Arastoopour H, Wang CH, Weil SA (1982) Particle-particle interaction force in a dilute gas-solid system. Chem Eng Sci 37(9):1379–1386

    Article  Google Scholar 

  87. Savage SB, Sayed M (1984) Stresses developed by dry cohesionless granular materials sheared in an annular shear cell. J Fluid Mech 142:391–430

    Article  Google Scholar 

  88. Jenkins JT, Mancini F (1987) Balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic, circular disks. J App Mech 54(1):2734

    Article  MATH  Google Scholar 

  89. Jenkins JT, Mancini F (1989) Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Phys Fluids A: Fluid Dynamics (1989-1993) 1(12):2050–2057

    Google Scholar 

  90. Alam M, Willits JT, Arnarson BÖ, Luding S (2002) Kinetic theory of a binary mixture of nearly elastic disks with size and mass disparity. Phys Fluids (1994-present) 14(11):4085–4087

    Article  MATH  Google Scholar 

  91. Willits JT, Arnarson BÖ (1999) Kinetic theory of a binary mixture of nearly elastic disks. Phys Fluids (1994-present) 11(10):3116–3122

    Article  MATH  Google Scholar 

  92. Zamankhan P (1995) Kinetic theory of multicomponent dense mixtures of slightly inelastic spherical particles. Phys Rev E 52(5):4877

    Article  Google Scholar 

  93. Wildman RD, Parker DJ (2002) Coexistence of two granular temperatures in binary vibrofluidized beds. Phys Rev Lett 88(6):064301

    Article  Google Scholar 

  94. Feitosa K, Menon N (2002) Breakdown of energy equipartition in a 2D binary vibrated granular gas. Phys Rev Lett 88(19):198301

    Article  Google Scholar 

  95. Huilin L, Gidaspow D, Manger E (2001). Kinetic theory of fluidized binary granular mixtures. Phys Rev E 64(6):061301

    Article  Google Scholar 

  96. Huilin L, Wenti L, Rushan B, Lidan Y, Gidaspow D (2000) Kinetic theory of fluidized binary granular mixtures with unequal granular temperature. Physica A: Statistical Mechanics and its Applications 284(1):265–276

    Article  Google Scholar 

  97. Garzó V, Dufty JW (2002) Hydrodynamics for a granular binary mixture at low density. Phys Fluids (1994-present), 14(4):1476–1490

    Article  MATH  Google Scholar 

  98. Iddir H, Arastoopour H, Hrenya CM (2005) Analysis of binary and ternary granular mixtures behavior using the kinetic theory approach. Powder Technol 151(1):117–125

    Article  Google Scholar 

  99. Benyahia S (2008) Verification and validation study of some polydisperse kinetic theories. Chem Eng Sci 63(23):5672–5680

    Article  Google Scholar 

  100. Ferziger JH, Kaper HG (1972) Mathematical theory of transport processes in gases. North Holland Publishing Company

    Google Scholar 

  101. Lebowitz JL (1964) Exact solution of generalized Percus-Yevick equation for a mixture of hard spheres. Phys Rev 133(4A):A895

    Article  MATH  Google Scholar 

  102. Alder BJ, Wainwright TE (1967) Velocity autocorrelations for hard spheres. Phys Rev Let 18(23):988

    Article  Google Scholar 

  103. Khotari KA (1967) M.S. Chem. Engineering, Illinois Institute of Technology, Chicago

    Google Scholar 

  104. Ranz WE, Marshall WR (1952) Evaporation from drops. Chem Eng Prog 48(3):141–146

    Google Scholar 

  105. Ranz WE (1952) Friction and transfer coefficients for single particles and packed beds. Chem Eng Prog 48(5):247–253

    Google Scholar 

  106. Gunn, DJ (1978) Transfer of heat or mass to particles in fixed and fluidised beds. Int J Heat Mass Transf 21(4):467–476

    Article  Google Scholar 

  107. Buist, KA, Backx BJGH, Deen NG, Kuipers JAM (2017) A combined experimental and simulation study of fluid-particle heat transfer in dense arrays of stationary particles. Chem Eng Sci 169:310–320

    Article  Google Scholar 

  108. Chaiwang P, Gidaspow D, Chalermsinsuwan B, Piumsomboon P (2014) CFD design of a sorber for CO2 capture with 75 and 375 micron particles. Chem Eng Sci 105:32–45

    Article  Google Scholar 

  109. Fogler HS (1999) Elements of chemical reaction engineering. Prentice Hall, New Jersey

    Google Scholar 

  110. Levenspiel O (1999) Chemical reaction engineering. John Wiley and Sons, New York

    Google Scholar 

  111. Bolland O, Nicolai R (2001) Describing mass transfer in circulating fluidized beds by ozone decomposition. Chem Eng Comm 187:1–21

    Article  Google Scholar 

  112. Welty, JR, Wicks CE, Wilson RE, Rorrer G (2001) Fundamentals of momentum, heat and mass transfer. John Wiley and Sons, New York

    Google Scholar 

  113. Chalermsinsuwan B, Piumsomboon P, Gidaspow D (2008a) Kinetic theory based computation of PSRI riser – Part 1. Estimate of mass transfer coefficient. Chem Eng Sci 64:1195–1211

    Article  Google Scholar 

  114. Chalermsinsuwan B, Piumsomboon P, Gidaspow D (2008b) Kinetic theory based computation of PSRI riser – Part II. Computation of mass transfer coefficient with chemical reaction. Chem Eng Sci 64:1212–1222

    Article  Google Scholar 

  115. Kato K, Kubota H, Wen CY (1970) Mass transfer in fixed and fluidized beds. Chem Eng Prog Symp Series 105:87–99

    Google Scholar 

  116. Smith, JM (1970) Chemical engineering kinetics. McGraw-Hill Book Company, New York

    Google Scholar 

  117. Williams, FA (2018) Combustion theory, 2nd edn. CRC Press: Taylor & Francis Group, Boca Raton, Florida

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical and Biological Engineering and Department of Mechanical, Materials, and Aerospace Engineering, Wanger Institute for Sustainable Energy Research (WISER), Illinois Institute of Technology, Chicago, IL, USA

    Hamid Arastoopour

  2. Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, IL, USA

    Dimitri Gidaspow

  3. Darien, IL, USA

    Robert W. Lyczkowski

Authors
  1. Hamid Arastoopour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitri Gidaspow
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert W. Lyczkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Arastoopour, H., Gidaspow, D., Lyczkowski, R.W. (2022). Multiphase Flow Kinetic Theory, Constitutive Equations, and Experimental Validation. In: Transport Phenomena in Multiphase Systems. Mechanical Engineering Series. Springer, Cham. https://doi.org/10.1007/978-3-030-68578-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-68578-2_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-68577-5

  • Online ISBN: 978-3-030-68578-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation