Multiphase Flow Analysis Using Population Balance Modeling: Bubbles, Drops and Particles

Multiphase Flow Analysis Using Population Balance Modeling: Bubbles, Drops and Particles

Volumn , Issue , 2014, Pages 1-365

  Yeoh, Guan Heng ^a   Cheung, Chi Pok ^a   Tu, Jiyuan ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84988234069     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1016/C2011-0-05568-0     Document Type: Book

References (360)

1. Agrawal K., Loezos P.N., Syamlal M., Sundaresan S. The role of mesoscale structures in rapid gas-solid flows. J. Fluid Mech. 2001, 445:151-185.
2. Fredrickson A.G., Ramkrishna D., Tsuchiya H.M. Statistics and dynamics of prokaryotic cell populations. Math. Biosci. 1967, 1:327-374.
3. Friedlander S.K. Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics 2000, Oxford University Press, Oxford. second ed.
4. Fujita M., Yamaguchi Y. Multiscale simulation method for self-organization of nanoparticles in dense suspension. J. Comp. Phys. 2007, 223:108-120.
5. Hidy G.M., Brock J.R. The Dynamics of Aerocolloidal Systems 1970, Pergamon, Oxford.
6. Hulburt H.M., Katz S. Some problems in particle technology: a statistical mechanical formulation. Chem. Eng. Sci. 1964, 19:55-574.
7. Lu J., Tryggvason G. Effect of bubble deformability in turbulent bubbly upflow in a vertical channel. Phys. Fluids 2008, 20-040701.
8. Marshall J.S. Discrete-element modeling of particulate aerosol flows. J. Comp. Phys. 2009, 228:1541-1561.
9. Pandis S.N., Seinfeld J.H. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change 1998, John Wiley & Sons, New York.
10. Quan S.P., Lou J., Schmidt D.P. Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations. J. Comp. Phys. 2009, 228:2660-2675.
11. Ramkrishna D., Borwanker J.D. A puristic analysis of population balance. Chem. Eng. Sci. 1973, 28:1423-1435.
12. Ramkrishna D. Statistical models of cell populations. Adv. Biochem. Eng. 1979, 11:1-47.
13. Ramkrishna D. The status of population balances. Rev. Chem. Eng. 1985, 3:49-95.
14. Ramkrishna D. Population Balances. Theory and Applications to Particulate Systems in Engineering 2000, Academic Press, San Diego.
15. Randolph A.D., Larson M.A. A population balance for countable entities. Can. J. Chem. Eng. 1964, 42:280-281.
16. Serizawa A. Some remarks on mechanisms of phase distribution in an adiabatic bubbly pipe flow. Multiphase Sci. Tech. 2003, 15:79-88.
17. Adeniji-Fashola A., Chen C.P. Modeling of confined turbulent fluid-particle flows using Eulerian and Lagrangian schemes. Int. J. Heat Mass Transf. 1990, 33:691-701.
18. Agee L.J., Banerjee S., Duffey R.B., Hughes E.D. Some Aspects of Two-Fluid Models and Their Numerical Solutions, Second OECD/NEA Specialists Meeting on Transient Two-Phase Flow 1978, CEA, France.
19. Bagchi P., Balachandar S. Effect on free rotation on the motion of a solid sphere in linear shear flow at moderate Re. J. Fluid Mech. 2002, 473:379-388.
20. Banerjee S., Chan A.M.C. Separated flow model I. Analysis of the averaged and local instantaneous formulations. Int. J. Multiphase Flow 1980, 6:1-24.
21. Besnard D.C., Harlow F.H. Turbulence in multiphase flow. Int. J. Multiphase Flow 1988, 14:679-699.
22. Brackbill J.U., Johnson U.N., Kashiwa B.A., Vander-Heyden W.B. Multiphase Flows and Particle methods 1997, CFD Annual Conference of Computational Fluid Dynamics, Victoria, Canada.
23. Burry D., Bergeles G. Dispersion of particles in anisotropic turbulent flows. Int. J. Heat Mass Transf. 1993, 4:651-664.
24. Chen P.P., Crowe C.T. On the Monte-Carlo method for modeling particle dispersion in turbulence. ASME FED 1984, 10:37-42.
25. Clift R., Grace J.R., Weber M.E. Bubbles, Drops and Particles 1978, Dover Publications, New York.
26. Crowe C.T., Sharma M.P., Stock D.E. Particle-source-in cell (PSI-Cell) model for gas-droplet flows. J. Fluid Eng. 1998, 99:325-332.
27. Delhaye J.M., Achard J.L. On the averaging operators introduced in two-phase flow modeling 1976, Proceedings of CSNI Specialists Meeting on Transient Two-phase Flow, Toronto, Canada.
28. Drew D.A., Passman S.L. Theory of Multicomponent Fluids 1999, Springer-Verlag, Berlin.
29. Drew D.A. Mathematical modeling of two-phase flow. Ann. Rev. Fluid Mech. 1983, 15:261-291.
30. Elghobashi S.E. On predicting particle-laden turbulent flows. Appl. Sci. Res. 1994, 52:309-329.
31. Faeth G.M. Recent advances in modeling particle transport properties and dispersion in turbulent flow. Proceedings of ASME-JSME Thermal Engineering Conference 1983, vol. 2:517-534.
32. Fede P., Simonin O. Numerical study of the subgrid fluid turbulence effects on the statistics of heavy colliding particles. Phys. Fluids 2006, 18:045103.
33. Gosman A.D., Iosnnides E. Aspects of computer simulation of liquid-fueled combustors. J. Energy 1983, 7:482-490.
34. Gouesbet G., Berlemont A. Eulerian and Lagrangian approaches for predicting the behaviour of discrete particles in turbulent flows. Prog. Energy Combust. Sci. 1999, 25:133-159.
35. Hoomans B.P.B., Kuipers J.A.M., Briels W.J., Swaaij W.P. Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidized bed: a hard-sphere approach. Chem. Eng. Sci. 1996, 51:99-118.
36. Ishii M., Hibiki T. Thermo-Fluid Dynamics of Two-Phase Flow 2006, Springer-Verlag, Berlin.
37. Joseph D.D., Lundgren T.S., Jackson R., Saville D.A. Ensemble averaged and mixture theory equations for incompressible fluid-particle suspensions. Int. J. Multiphase Flow 1990, 16:35-42.
38. Kallio G.A., Stock D.E. Turbulent particle dispersion: a comparison between Lagrangian and Eulerian modeling approaches. ASME FED 1986, 35:23-34.
39. Kashima B.A., Rauenzahn R.M. A multimaterial formalism 1994, ASME Fluids Engineering Summer Meeting, New York, USA.
40. Kolev N.I. Multiphase Flow Dynamics 1: Fundamentals 2005, Springer-Verlag, Berlin. second ed.
41. Lhuillier D. The macroscopic modelling of multi-phase mixtures. Flow of Particles in Suspensions 1996, Springer-Verlag, Wien, New York. U. Schaflinger (Ed.).
42. Li A., Ahmadi G. Dispersion and deposition of spherical particles from point sources in a turbulent channel flow. Aerosol Sci. Technol. 1992, 16:209-226.
43. Milojevic D. Lagrangian stochastic-deterministic (LSD) predictions of particle dispersion in turbulence. Part. Part. Syst. Charact. 1990, 7:181-190.
44. Mostafa A.A., Mongia H.C. On the modeling of turbulent evaporating sprays: Eulerian versus Lagrangian approach. Int. J. Heat Mass Transf. 1987, 30:2583-2593.
45. Mostafa A.A., Mongia H.C. On the interaction of particles and turbulent fluid flow. Int. J. Heat Mass Transf. 1988, 31:2063-2075.
46. Panton R.J. Flow properties for the continuum viewpoint of a non-equilibrium gas particle mixture. J. Fluid Mech. 1968, 31:273-304.
47. Rivero M., Magnaudet J., Fabre J. Quelques Résultats Nouveaux Concernant les Forces Hydrodynamiques sur une Sphère Solide ou une Bulle Sphérique. C. R. Acad. Sci. Paris Sér. II 1991, 314:1499-1506.
48. Saffman P.G. The lift on a small sphere in a slow shear flow. J. Fluid Mech. 1965, 22:385-400.
49. Shirolkar J.S., Coimbra C.F.M., McQuay M.Q. Fundamental aspects of modeling turbulent particle dispersion in dilute flows. Prog. Energy Combust. Sci. 1996, 22:363-399.
50. Shuen J.S., Chen L.D., Faeth G.M. Evaluation of a stochastic model of particle dispersion in a turbulent round jet. Chem. Eng. J. 1983, 29:167-170.
51. Vernier P., Delhaye J.M. General two-phase flow equation applied to the thermohydrodynamics of boiling nuclear reactors. Acta Tech. Belg. Energie Primaire 1968, 4:3-43.
52. Yadigaroglu G., Lahey R.T. On the various forms of the conservation equations in two-phase flow. Int. J. Multiphase Flow 1976, 2:477-494.
53. Cheung S.C.P., Yeoh G.H., Tu J.Y. On the modeling of population balance in isothermal vertical bubbly flows - average bubble number density approach. Chem. Eng. Proc. 2007, 46:742-756.
54. Cheung S.C.P., Yeoh G.H., Tu J.Y. On the numerical study of isothermal bubbly flow using two population balance approaches. Chem. Eng. Sci. 2007, 31:164-1072.
55. Cheung S.C.P., Yeoh G.H., Qi F.S., Tu J.Y. Classification of bubbles in vertical gas-liquid flow: Part 2 - a model evaluation. Int. J. Multiphase Flow 2012, 39:135-147.
56. Friedlander S.K. Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics 2000, Oxford University Press, Oxford. second ed.
57. Jakobsen H. Chemical reactor Modelling 2008, Chapter 9, 807-865, Springer Berlin Heidelberg.
58. Kocamustafaogullari G., Ishii M. Foundation of the interfacial area transport equations and its closure relations. Int. J. Heat Mass Transf. 1995, 38:481-493.
59. Luo H., Svendsen H. Theoretical model for drop and bubble break-up in turbulent dispersions. AIChE J. 1996, 42:1225-1233.
60. Luo H. Coalescence, break-up and liquid recirculation in bubble column reactors 1993, PhD dissertation, Norwegian Institute of Technology, Norway.
61. Prince M.J., Blanch H.W. Bubble coalescence and break-up in air-sparged bubble column. AIChE J. 1990, 36:1485-1499.
62. Qi F.S., Yeoh G.H., Cheung S.C.P., Tu J.Y., Krepper E., Lucas D. Classification of bubbles in vertical gas-liquid flow: Part 1 - an analysis of experimental data. Int. J. Multiphase Flow 2012, 39:121-134.
63. Ramkrishna D. Population Balances. Theory and Applications to Particulate Systems in Engineering 2000, Academic Press, San Diego.
64. Tsouris C., Tavlarides L.L. Breakage and coalescence models for drops in turbulent dispersions. AIChE J. 1994, 40:395-406.
65. Angelidou C., Psimopoulos M., Jameson G.J. Size distribution functions of dispersions. Chem. Eng. Sci. 1979, 34:671-676.
66. Azbel D., Athanasios I.L. A mechanism of liquid entrainment. Handbook of Fluids in Motion 1983, 473. Ann Arbor Science Publishers, Ann Arbor, U.S.A. N. Cheremisinoff (Ed.).
67. Batchelor G.K. The Theory of Homogeneous Turbulence 1956, University Press, Cambridge.
68. Bilicki Z., Kestin J. Transition criteria for two-phase flow patterns in vertical upward flow. Int. J. Multiphase Flow 1987, 13:283-294.
69. Chatzi E.G., Gavrielides A.D., Kiparissides C. Generalized model for prediction of the steady-state drop size distributions in batch stirred vessels. Ind. Eng. Chem. Res. 1989, 28:1704-1711.
70. Chesters A.K. The applicability of dynamic-similarity criteria to isothermal, liquid-gas, two-phase flows without mass transfer. Int. J. Multiphase Flow 1975, 2. 191-121.
71. Chesters A.K., Hoffman G. Bubble coalescence in pure liquids. Appl. Sci. Res. 1982, 38:353-361.
72. Chesters A.K. The modeling of coalescence processes in fluid-liquid dispersion: a review of current understanding. Trans. Chem. Eng. 1991, 69A:259-270.
73. Clift R., Grace J.R., Weber M.E. Bubbles, Drops and Particles 1978, Dover Publications, New York.
74. Colella D., Vinci D., Bagatin R., Masi M., Bakr E.A. A study on coalescence and breakage mechanisms in three different bubble columns. Chem. Eng. Sci. 1999, 54:4767-4777.
75. Colin C., Riou X., Fabre J. Turbulence and Shear-induced Coalescence in Gas-Liquid Pipe flows, Fifth Int. Conf. Multiphase Flow, ICMF'04 2004, Yokohama, Japan.
76. Coulaloglou C.A., Tavlarides L.L. Description of interaction processes in agitated liquid-liquid dispersions. Chem. Eng. Sci. 1977, 32:1289-1297.
77. Coulaloglou C.A. Dispersed phase interactions in an agitated flow vessel 1975, PhD dissertation, Illinois Institute of Technology, Chicago.
78. Davis R.H., Schonberg J.A., Rallison J.M. The lubrication force between two viscous drops. Phys. Fluids A 1989, 1:77-81.
79. de Nevers N., Wu J.L. Bubble coalescence in viscous fluids. AIChE J. 1971, 17:182-186.
80. Doubliez L. The drainage and rupture of a non-foaming liquid film formed upon bubble impact with a free surface. Int. J. Multiphase Flow 1991, 17:783-803.
81. Duineveld P.C. Bouncing and coalescence of two bubbles in water 1994, PhD dissertation, University of Twente, Netherlands.
82. Fan L.S., Tsuchiya K. Bubble Wake Dynamics in Liquids and Liquid-Solid Suspensions 1990, Butterworth-Heinemann, Stoneham, MA.
83. Friedlander S.K. Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics 1977, Wiley, New York.
84. Fu X.Y., Ishii M. Two-group interfacial area transport in vertical air-water flow. I. Mechanistic model. Nucl. Eng. Des. 2002, 219:143-168.
85. Grace H.P. Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. Chem. Eng. Comm. 1982, 14:225-277.
86. Harmathy T.Z. Velocity of large drops and bubbles in media of infinite or restricted extent. AIChE J. 1960, 6. 281-288.
87. Hesketh R.P., Fraser Russell T.W., Etchells A.W. Bubble size in horizontal pipelines. AIChE J. 1987, 33:663-667.
88. Hibiki T., Ishii M. One-group interfacial area transport of bubbly flows in vertical round tubes. Int. J. Heat Mass Transf. 2000, 43:2711-2726.
89. Hibiki T., Ishii M. Two-group interfacial area transport equations at bubbly-to-slug flow transition. Nucl. Eng. Des. 2000, 202:39-76.
90. Hibiki T., Ishii M. Interfacial area transport equations for gas-liquid flow. J. Comp. Multiphase Flows 1, Transf. 2009, 44:1-22.
91. Howarth W.J. Coalescence of drops in a turbulent flow field. Chem. Eng. Sci. 1964, 19:33-38.
92. Howarth W.J. Measurement of coalescence frequency in an agitated tank. AIChE J. 1967, 13:1007-1013.
93. Ishii M., Kim S., Uhle J. Interfacial area transport: model development and benchmark experiments. Int. J. Heat Mass Transf. 2002, 45:3111-3123.
94. Jeelani S.A.K., Hartland S. Effect of approach velocity on binary and interfacial coalescence. Chem. Eng. Res. Des. 1991, 69:271-281.
95. Kalkach-Navarro S., Drew D.A., Lahey R.T. Analysis of the bubbly/slug flow regime transition. Nucl. Eng. Des. 1994, 151:15-39.
96. Kamp A.M., Chesters A.K. Bubble coalescence in turbulent flows: a mechanistic model for turbulence-induced coalescence applied to microgravity bubbly pipe flow. Int. J. Multiphase Flow 2001, 27:1363-1396.
97. Kennard E.H. Kinetic Theory of Gases 1938, McGraw-Hill, New York.
98. Kocamustafaogullari G., Ishii M. Foundation of the interfacial area transport equation and its closure relations. Int. J. Heat Mass Transf. 1995, 38:481-493.
99. Kolmogorov A.N. The local structure of turbulence in incompressible viscous fluids at very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 1941, 30:299-303.
100. Kuboi R., Komasawa I., Otake T. Collision and coalescence of dispersed drops in turbulent liquid flow. J. Chem. Eng. Jpn. 1972, 5:423-424.
101. Lafi A.Y., Reyes J.N. General Particle Transport Equation 1994, Report OSU-NE-9409, Department of Nuclear Engineering, Oregon State University, Oregon.
102. Lasheras J.C., Eastwood C., Martinez-Bazan C., Montanes J.L. A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow. Int. J. Multiphase Flow 2002, 28:247-278.
103. Lee C.-H., Erickson L.E., Glasgow L.A. Bubble breakup and coalescence in turbulent gas-liquid dispersions. Chem. Eng. Comm. 1987, 59:65-84.
104. Lehr F., Mewes D. A transport equation for the interfacial area density applied to bubble columns. Chem. Eng. Sci. 1999, 56:1159-1166.
105. Lehr F., Mewes D. A transport equation for the interfacial area density applied to bubble columns. Chem. Eng. Sci. 2001, 56:1159-1166.
106. Lehr F., Mewes D., Millies M. Bubble-size distribution and flow fields in bubble columns. AIChE J. 2002, 42:1225-1233.
107. Levich V.G. Physicochemical Hydrodynamics 1962, Prentice Hall, Englewood Cliffs.
108. Liao Y., Lucas D. A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem. Eng. Sci. 2009, 64:3389-3406.
109. Liao Y., Lucas D. A literature review on mechanisms and models for the coalescence process of fluid particles. Chem. Eng. Sci. 2010, 65:2851-2864.
110. Lo S., Zhang D.S. Modelling of break-up and coalescence in bubbly two-phase flows. J. Comp. Multiphase Flows 2009, 1:22-38.
111. Luo H., Svendsen H. Theoretical model for drop and bubble break-up in turbulent dispersions. AIChE J. 1996, 42:1225-1233.
112. Luo H. Coalescence, break-up and liquid recirculation in bubble column reactors 1993, PhD dissertation, Norwegian Institute of Technology, Norway.
113. Martinez-Bazan C., Montanes J.L., Lasheras J.C. On the breakup of an air bubble injected into fully developed turbulent flow. Part 1. Breakup frequency. J. Fluid Mech. 1999, 401:157-182.
114. Martinez-Bazan C., Montanes J.L., Lasheras J.C. On the breakup of an air bubble injected into fully developed turbulent flow. Part 2. Size PDF of the resulting daughter bubbles. J. Fluid Mech. 1999, 401:183-207.
115. Miyahara T., Tsuchiya K., Fan L.S. Effect of turbulent wake on bubble-bubble interactions in a gas-liquid-solid fluidized bed. Chem. Eng. Sci. 1991, 46:2368-2373.
116. Nambiar D.K.R., Kumar R., Das T.R., Gandhi K.S. A new model for the breakage frequency of drops in turbulent stirred dispersions. Chem. Eng. Sci. 1992, Vol. 47:2989-3002.
117. Narsimhan G., Gupta J.P., Ramkrishna D. A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersions. Chem. Eng. Sci. 1979, 34:257-265.
118. Oolman T.O., Blanch H.W. Bubble coalescence in stagnant liquids. Chem. Eng. Comm. 1986, 43:237-261.
119. Park J.Y., Blair L.M. The effect of coalescence on drop size distribution in an agitated liquid-liquid dispersion. Chem. Eng. Sci. 1975, 30:1057-1064.
120. Prince M.J., Blanch H.W. Bubble coalescence and break-up in air-sparged bubble column. AIChE J. 1990, 36:1485-1499.
121. Richardson J.F., Zaki W.M. Sedimentation and fluidization, part I. Trans. Chem. Eng. 1954, 32:35-53.
122. Risso F., Fabre J. Oscillations and breakup of a bubble immersed in a turbulent field. J. Fluid Mech. 1998, 372:323-355.
123. Ross S.L. Measurements and models of the dispersed phase mixing process 1971, PhD dissertation, The University of Michigan, Ann Arbor.
124. Ruckenstein E., Jain R.K. Spontaneous rupture of thin liquid films. J. Chem. Soc. Faraday Trans. 1974, 2(Vol. 70):132-147.
125. 2 bubbles in water. Can. J. Chem. Eng. 1976, 54:392-398.
126. Schlichting H. Boundary Layer Theory 1979, McGraw-Hill, New York. seventh ed.
127. Shinnar R., Church J.M. Predicting particle size in agitated dispersions. Ind. Eng. Chem. 1960, 52:253-256.
128. Simon M. Koaleszenz von Tropfen und Tropfenschwärmen 2004, PhD dissertation, die Teschinschen Universtität Kaiserslautern.
129. Sovova H. Breakage and coalescence of drops in a batch stirred vessel-II. Comparison of model and experiments. Chem. Eng. Sci. 1981, 36:1567-1573.
130. Stewart C.W. Bubble interaction in low-viscosity liquids. Int. J. Multiphase Flow 1995, 21. 1037-1010.
131. Sun X., Kim S., Ishii M., Beus S.G. Modeling of bubble coalescence and disintegration in confined upward two-phase flow. Nucl. Eng. Des. 2004, 230:3-26.
132. Tennekes H., Lumley J.L. A First Course in Turbulence 1972, MIT Press, Cambridge, MA.
133. Tsouris C., Tavlarides L.L. Breakage and coalescence models for drops in turbulent dispersions. AIChE J. 1994, 40:395-406.
134. Wang T.F., Wang J.F., Jin Y. A novel theoretical breakup Kernel function for bubbles/droplets in a turbulent flow. Chem. Eng. Sci. 2003, 58:4629-4637.
135. Wang L., Marchisio D.L., Virgil R.D., Fox R.O. CFD simulation of aggregation and breakage processes in laminar Taylor-Coutte flow. J. Colloid Interface Sci. 2005, 282:380-396.
136. Wang T.F., Wang J.F., Jin Y. Theoretical prediction of flow regime transition in bubble columns by the population balance model. Chem. Eng. Sci. 2005, 60:6199-6209.
137. Wieringa J.A., Dieren F., van, Janssen J.J., Agterof W.G.M. Droplet breakup mechanics during emulsification in colloid mills at high dispersed volume fractions. Trans. Inst. Chem. Eng. 1996, 74-A:554-562.
138. Williams M.M.R., Loyalka S.K. Aerosol Science Theory and Practice: With Special Applications to the Nuclear Industry 1991, Pergamon Press, Oxford.
139. Wu Q., Kim S., Ishii M., Beus S.G. One-group interfacial area transport in vertical bubbly flow. Int. J. Heat Mass Transf. 1998, 41:1103-1112.
140. Zhao H., Ge W. A Theoretical bubble breakup model for slurry beds or three-phase fluidized beds under high pressure. Chem. Eng. Sci. 2007, 62:109-115.
141. Abrahamson J. Collision rates of small particles in a vigorously turbulent fluid. Chem. Eng. Sci. 1975, 30:1371-1379.
142. Aoki K.M., Akiyama T. Simulation studies of pressure and density wave propagations in vertically vibrated beds of granules. Phys. Rev. E 1995, 52:3288-3291.
143. Austin L., Shoji K., Bhatia V., Jindal V., Savage K., Klimpel R. Some results on the description of size reduction as a rate process in various mills. Ind. Eng. Chem. Process Des. Dev. 1976, 15:187-196.
144. Cheng J., Yang C., Mao Z.S., Zhao C. CFD modeling of nucleation, growth, aggregation and breakage in continuous precipitation of barium sulfate in a stirred tank. Ind. Eng. Chem. Res. 2009, 48:6992-7003.
145. Cundall P.A., Strack O.D.L. A discrete numerical model for granular assemblies. Geotechnique 1979, 29:47-65.
146. Davis R.H., Serayssol J.M., Hinch E.J. The elastohydrodynamic collision of 2 spheres. J. Fluid Mech. 1986, 163:479-497.
147. Derjaguin B.V., Muller V.M., Toporov Y.P. Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci. 1975, 53:314-326.
148. Diemer R.B., Olson J.H. A moment methodology for coagulation and breakage problems: Part 3-generalized daughter distribution function. Chem. Eng. Sci. 2002, 57:4187-4198.
149. Firth B.A., Hunter R.J. Flow properties of coagulated colloidal suspensions I. Energy dissipation in the flow units. J. Colloid Interface Sci. 1976, 57:248-256.
150. Flesch J.C., Spicer P.T., Pratsinis S.E. Laminar and turbulent shear-induced flocculation of fractal aggregates. AIChE J. 1999, 45:1114-1124.
151. Greenwood J.A. Adhesion of elastic spheres. Proc. R. Soc. Lond. A 1997, 453:1277-1297.
152. Hertz H. Ueber die Berührung fester elastischer Koerper. J. Reine Angewandte Mathematik 1882, 92:156-171.
153. Hill P.J., Ng K.M. New discretization procedure for the breakage equation. AIChE J. 1995, 41:1204-1216.
154. Hsia M.A., Tavlarides L.L. Simulation analysis of drop breakage, coalescence and micromixing in liquid-liquid stirred tanks. Chem. Eng. J. 1983, 26:189-199.
155. Johnson K.L., Kendall K., Roberts A.D. Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. A 1971, 324:301-313.
156. Kargulewicz M., Iordanoff I., Marrero V., Tichy J. Modeling of magnetorheological fluids by the discrete element method. J. Tribology Trans. ASME 2012, 134:031706.
157. Kostoglou M., Dovas S., Karabelas A.J. On the steady-state size distribution of dispersions in breakage processes. Chem. Eng. Sci. 1997, 52:1285-1299.
158. Kruggel-Emden H., Simsek E., Rickelt S., Wirtz S., Scherer V. Review and extension of normal force models for the discrete element method. Powder Tech. 2007, 130:157-173.
159. Kruggel-Emden H., Wirtz S., Scherer V. A study on tangential force laws applicable to the discrete element method (DEM) for materials with viscoelastic or plastic behaviour. Chem. Eng. Sci. 2008, 63:1523-1541.
160. Kusters K.A., Wijers J.G., Thoenes D. Aggregation kinetics of small particles in agitated vessels. Chem. Eng. Sci. 1997, 52:107-121.
161. Kuwabara G., Kono K. Restitution coefficient in collision between two spheres. Jpn. J. Appl. Phys. 1987, 26:1230-1233.
162. Langston P.A., Tüzün U., Heyes D.M. Continuous potential discrete particle simulations of stress and velocity fields in hoppers: transition from fluid to granular flow. Chem. Eng. Sci. 1994, 49:1259-1275.
163. Lee J., Herrmann H.J. Angle of repose and angle of marginal stability-molecular-dynamics of granular particles. J. Phys. A, Math. Gen. 1993, 26:373-383.
164. Li S.Q., Marshall J.S., Liu G.Q., Yao Q. Adhesive particulate flow: the discrete-element method and its application in energy and environmental engineering. Prog. Energy Combust. Sci. 2011, 37:633-668.
165. Lian G., Thornton C., Adams M.J. A theoretical study of the liquid bridge forces between two rigid spherical bodies. J. Colloid Interface Sci. 1993, 161:138-147.
166. Liu G.Q., Li S.Q., Yao Q. On the applicability of different adhesion models in adhesive particulate flows. Front. Energy Power Eng. China 2009, 4:280-286.
167. Loyalka S.K. Thermophoretic force on a single particle-I. Numerical solution of the linearized Boltzmann equation. J. Aerosol Sci. 1992, 23:291-300.
168. Luo H., Svendsen H. Theoretical model for drop and bubble break-up in turbulent dispersions. AIChE J. 1996, 42:1225-1233.
169. Marchisio D.L., Vigil D.R., Fox R.O. Quadrature Method of Moments for Aggregation-Breakage Processes. J. Colloid Interface Science 2003, Vol. 258:322-324.
170. Markatou P., Wang H., Frenklach M. A computational study of sooting limits in laminar premixed flames of ethane, ethylene and acetylene. Combust. Flame 1993, 93:467-482.
171. Marshall J.S. Discrete-element modeling of particulate aerosol flows. J. Comp. Phys. 2009, 228:1541-1561.
172. Matthewson M.J. Adhesion of spheres by thin liquid films. Philos. Mag. A 1988, 57:207-216.
173. Maugis D. Adherence of elastomers: fracture mechanics aspects. J. Adhes. Sci. Tech. 1987, 1:105-134.
174. McCoy B.J., Wang M. Continuous-mixture fragmentation kinetics: particle size reduction and molecular cracking. Chem. Eng. Sci. 1994, 49:3773-3785.
175. Mindlin R.D., Deresiewicz H. Elastic spheres in contact under varying oblique forces. J. Appl. Mech. Trans. ASME 1953, 20:327-344.
176. Nicolson M.M. Interaction between floating particles. Proc. Camb. Philol. Soc. 1949, 45:288-295.
177. Peng S.J., Williams R.A. Direct measurement of floc breakage in flowing suspension. J. Colloid Interface Sci. 1994, 166:321-332.
178. Pitois O., Moucheront P., Chateau X. Liquid bridge between two moving spheres: an experimental study of viscosity effects. J. Colloid Interface Sci. 2000, 231:26-31.
179. Ramkrishna D. Population Balances. Theory and Applications to Particulate Systems in Engineering 2000, Academic Press, San Diego.
180. Sadd M.H., Tai Q.M., Shukla X. A contact law effects on wave-propagation in particulate materials using distinct element modeling. Int. J. Non-linear Mech. 1993, 28:251-265.
181. Saffman P.G., Turner J.S. On the collision of drops in turbulent clouds. J. Fluid Mech. 1956, 1:16-30.
182. Schafer J., Dippel S., Wolf D.E. Force schemes in simulations of granular materials. J. Physique 1996, 6:5-20.
183. Serayssol J.M., Davis R.H. The influence of surface interactions on the elastohydrodynamic collision of two spheres. J. Colloid Interface Sci. 1986, 114:54-66.
184. Serra T., Casamitjana X. Effect of the shear and volume fraction on the aggregation and breakup of particles. AIChE J. 1998, 44:1724-1730.
185. Smoluchowski M.V. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 1917, 92:129-168.
186. Tabor D. Surface forces and surface interactions. J. Colloid Interface Sci. 1977, 58:2-13.
187. Talbot L., Cheng R.K., Schefer R.W., Willis D.R. Thermophoresis of particles in a heated boundary layer. J. Fluid Mech. 1980, Vol. 101:737-758.
188. Thornton C. Coefficient of restitution for collinear collisions of elastic perfectly plastic spheres. J. Appl. Mech. Trans. ASME 1997, 64:383-386.
189. Tomas J. Mechanics of nanoparticle adhesion-a continuum approach. Particles on Surfaces 8: Detection Adhesion and Removal 2003, 183-229. VSP, Utrecht. K.L. Mittal (Ed.).
190. Tsuji Y., Tanaka T., Ishida T. Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Tech. 1992, 71:239-250.
191. Tsuji Y., Kawaguchi T., Tanaka T. Discrete particle simulation of two dimensional fluidized bed. Powder Tech. 1993, 77:79-87.
192. Tyndall J. On dust and disease. Proc. R. Inst. 1870, 6:1-14.
193. Verlet L. Computer "experiments" on classical fluids, I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 1967, 159:98-103.
194. Vigil R.D., Ziff R.M. On the stability of coagulation-fragmentation population balances. J. Colloid Interface Sci. 1989, 133:257-264.
195. Vu-Quoc L., Zhang X. An accurate and efficient tangential force-displacement model for elastic frictional contact in particle-flow simulations. Mech. Math. 1999, 31:235-269.
196. Waldmann L., Schmitt K.H. Thermophoresis and diffusiophoresis of aerosols. Aerosol Science 1966, 137-162. Academic Press, New York. C.N. Davies (Ed.).
197. Walton O.R., Braun R.L. Viscosity, granular temperature and stress calculations for shearing assemblies of inelastic, frictional disks. J. Rheology 1986, 30:949-980.
198. Wang L., Marchisio D.L., Virgil R.D., Fox R.O. CFD simulation of aggregation and breakage processes in laminar Taylor-Coutte flow. J. Colloid Interface Sci. 2005, 282:380-396.
199. Yamamoto K., Ishihara Y. Thermophoresis of a spherical particle in a rarefied gas of a transition regime. Phys. Fluids 1988, 31:3618-3624.
200. Aldeniji-Fashola A., Chen C.P. Modeling of confined turbulent fluid-particle flows using Eulerian and Lagrangian schemes. Int. J. Heat Mass Transf. 1990, 33:691-701.
201. Baldyga J., Bourne J. Turbulent Mixing and Chemical Reactions 1999, Wiley, New York.
202. Barth T.J., Jespersen D.C. The Design and Application of Upwind Schemes on Unstructured Meshes. AIAA Paper 1989, 89-366.
203. Barrett J.C., Webb N.A. A comparison of some approximate methods for solving the aerosol general dynamics equation. J. Aerosol Sci. 1998, 29:31-39.
204. Boussinesq J. Théorie Analytique de la Chaleur 1903, vol. 2. Gauther-Villars, Paris, pp. 154-176.
205. Bove S. Computational fluid dynamics of gas-liquid flows including bubble population balance 2005, PhD thesis, Aalborg University Esbjerg, Denmark.
206. Cheung S.C.P., Yeoh G.H., Tu J.Y. On the modeling of population balance in isothermal vertical bubbly flows - average bubble number density approach. Chem. Eng. Proc. 2007, 46:742-756.
207. Cheung S.C.P., Yeoh G.H., Tu J.Y. On the numerical study of isothermal bubbly flow using two population balance approaches. Chem. Eng. Sci. 2007, 62:4659-4674.
208. Darwish M., Moukalled F. TVD Schemes for Unstructured Grids. International Journal of Heat and Mass Transfer 2003, Vol. 46:599-611.
209. Deardoff J.W. A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 1970, 41:4 53-480.
210. Debry E., Sportisse B., Jourdain B. A stochastic approach to numerical simulation of the general equation for aerosis. J. Comp. Phys. 2003, 184:649-669.
211. Domilovskii E.R., Lushnikov A.A., Piskunov V.N. A Monte Carlo simulation of coagulation processes. Izvestkya Akademi Nauk SSSR, Fizika Atmosfery I Okeana 1979, 15:194-201.
212. Dorao C.A., Jakobsen H.A. Numerical calculation of the moments of the population balance equation. J. Comp. Appl. Math. 2006, 196:619-633.
213. Dorao C., Jakobsen H. A least squares method for the solution of population balance problems. Comp. Chem. Eng. 2006, 30:535-547.
214. Dorao C., Jakobsen H. Least-squares spectral method for solving advective population balance problems. J. Comp. Appl. Math. 2007, 201:247-257.
215. Dorao C., Jakobsen H. Time-space-property least squares spectral method for population balance problems. Chem. Eng. Sci. 2007, 62:1323-1333.
216. Dorao C.A., Lucas D., Jakobsen H.A. Prediction of the evolution of the dispersed phase in bubbly flow problems. Appl. Math. Model. 2008, 32:1813-1833.
217. Fletcher C.A.J. Computational Techniques for Fluid Dynamics 1991, vols I and II. Springer-Verlag, Berlin.
218. Frenklach M. Dynamics of discrete distribution for Smoluchowski coagulation model. J. Colloid Interface Sci. 1985, 108:237-242.
219. Frenklach M. Method of moments with interpolative closure. Chem. Eng. Sci. 2002, 57:2229-2239.
220. Fulgosi M., Lakehal D., Banerjee S., De Angelis V. Direct numerical simulation of turbulence in a sheared air-water flow with a deformable interface. J. Fluid Mech. 2003, 482:319-345.
221. Germano M.U., Piomelli U., Moin P., Cabot W.H. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 1991, 3:1760-1765.
222. Ghosal S., Lund T.S., Moin P., Akselvoll K. A dynamic localization model for large-eddy simulation of turbulent flows. J. Fluid Mech. 1995, 286:229-255.
223. Gordon R.G. Error bounds in equilibrium statistical mechanics. J. Math. Phys. 1968, 9:655-672.
224. Grosch R., Briesen H., Marquardt W., Wulkow M. Generalization and numerical investigation of QMOM. AIChE J. 2007, 53:207-227.
225. Heinz S. Statistical Mechanics of Turbulent Flows 2003, Springer, Berlin.
226. Hulburt H.M., Katz S. Some problems in particle technology: a statistical mechanical formulation. Chem. Eng. Sci. 1964, 19:55-574.
227. Jayatilleke C.L.V. The influence of prandtl number and surface roughness on the resistance of the laminar sublayer to momentum and heat transfer. Prog. Heat Mass Transf. 1969, 1:193-321.
228. John V., Angelov I., Oncul A.A., Thevenin D. Techniques for the reconstruction from a finite number of its moments. Chem. Eng. Sci. 2007, 62:2890-2904.
229. Kader B. Temperature and concentration profiles in fully turbulent boundary layers. Int. J. Heat Mass Transf. 1993, 24:1541-1544.
230. Karema H., Lo S.M. Efficiency on interphase coupling algorithms in fluidized bed conditions. Comp. Fluids 1999, 28:323-360.
231. Kim S.E., Choudhury D. A near-wall treatment using wall functions sensitized to pressure gradient. ASME FED 1995, 217. (Separated and Complex Flows, ASME).
232. Krepper E., Lucas D., Prasser H. On the modeling of bubbly flow in vertical pipes. Nucl. Eng. Des. 2005, 235:597-611.
233. Krepper E., Frank T., Lucas D., Prasser H.M., Zwart P.J. Inhomogeneous MUSIG model - a population balance approach for polydispersed bubbly flows 2007, Proceedings of the Sixth International Conference on Multiphase Flow, Leipzig, Germany.
234. Lahey R.T., Drew D.A. The analysis of two-phase flow and heat transfer using multidimensional, four field, two-fluidxe "fluid" model. Nucl. Eng. Des. 2001, 204:29-44.
235. Lambin P., Gaspard J.P. Continued-fraction technique for tight-binding systems. A generalized-moments method. Phys. Rev. B 1982, 26:4356-4368.
236. Launder B.E., Spalding D.B. The numerical computation of turbulent flows. Comp. Meth. Appl. Mech. Eng. 1974, 3:269-289.
237. Launder B.E. Second-moment closures: present and future?. Int. J. Heat Fluid Flow 1989, 10:282-300.
238. Leonard B.P. A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Eng. 1979, 19:59-98.
239. Li S.Q., Marshall J.S., Liu G.Q., Yao Q. Adhesive particulate flow: the discrete-element method and its application in energy and environmental engineering. Prog. Energy Combust. Sci. 2011, 37:633-668.
240. Liffman K. A direct simulation Monte Carlo method for cluster coagulation. J. Comp. Phys. 1992, 100:116-127.
241. Lilly D.K. The representation of small-scale turbulence in numerical simulation experiments 1966, NCAR Manuscript No. 281.
242. Lilly D.K. The representation of small-scale turbulence in numerical simulation experiments. Proc. IBM Sci. Comput.Syrup. on Environmental Science 1967, 195-210. Thomas J. Watson Research Center, Yorktown Heights, N.Y., 14-16 November 1966. IBM Form 320-1951.6. IBM Form 320-1951.
243. Lilly D.K. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids 1992, 4:633-635.
244. Lo S.M. Mathematical Basis of a Multiphase flow Model, United Kingdom Atomic Energy Authority, Computational fluid Dynamics Section 1989, Report AERE R 13432.
245. Lo S.M. Multiphase Flow Model in the Harwell-FLOW3D Computer Code 1990, AEA Industrial Technology, Report AEA-InTech-0062.
246. Lo S.M. Application of Population Balance to CFD Modeling of Bubbly Flow via the MUSIG Model 1996, AEAT-1096, AEA Technology.
247. Lopez de Bertodano M., Lahey R.T., Jones O.C. Development of a k model for bubbly two-phase flow. J. Fluid Eng. 1994, 116:128-134.
248. Lopez de Bertodano M., Lahey R.T., Jones O.C. Phase distribution in bubbly two-phase flow in vertical ducts. Int. J. Multiphase Flow 1994, 20:805-818.
249. Maisels A., Kruis F.E., Fissan H. Direct simulation Monte Carlo for simulation nucleation, coagulation and surface growth in dispersed systems. Chem. Eng. Sci. 2004, 59:2231-2239.
250. Marchisio D.L., Fox R.O. Solution of population balance equations using the direct quadrature method of moments. J. Aerosol Sci. 2005, 36:43-73.
251. Marchisio D.L., Vigil D.R., Fox R.O. Quadrature method of moments for aggregation-breakage processes. J. Colloid Interface Sci. 2003, 258:322-324.
252. Marchisio D.L., Pikturna J.T., Fox R.O., Vigil R.D. Quadrature method of moments for population-balance equations. AIChE J. 2003, 49:1266-1276.
253. Marchisio D.L., Vigil D.R., Fox R.O. Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems. Chem. Eng. Sci. 2003, 58:3337-3351.
254. McCoy B.J., Madras G. Analytic solution for a population balance equation with aggregation and fragmentation. Chem. Eng. Sci. 2003, 58:3049-3051.
255. McGraw E., Wright D.L. Chemically resolved aerosol dynamics for internal mixtures by the quadrature method of moments. J. Aerosol Sci. 2003, 34:189-209.
256. McGraw R. Description of aerosol dynamics by the quadrature method of moments. Aerosol Sci. Technol. 1997, 27:255-265.
257. Menter F.R. Zonal Two Equation kω Turbulence Models for Aerodynamics flows 1993, AIAA paper 93-2906.
258. Menter F.R. A comparison of some recent eddy-viscosity turbulence models. J. Fluids Eng. 1996, 118:514-519.
259. Mueller M.E., Blanquart G., Pitsch H. A joint volume-surface model of soot aggregation with the method of moment. Proc. Combust. Inst. 2009, 32:785-792.
260. Mueller M.E., Blanquart G., Pitsch H. Hybrid method of moments for modeling soot formation and growth. Combust. Flame 2009, 156:1143-1155.
261. Olmos E., Gentric C., Vial Ch., Wild G., Midoux N. Numerical simulation of multiphase flow in bubble column reactors. Influence of bubble coalescence and break-up. Chem. Eng. Sci. 2001, 56:6359-6365.
262. Patankar S.V., Spalding D.B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transf. 1972, 15:1787-1806.
263. Patankar S.V. Numerical Heat Transfer and Fluid Flow 1980, Hemisphere Publishing Corporation, Taylor & Francis Group, New York.
264. Piomelli U., Liu J. Large-eddy simulation of rotating channel flows using a localized dynamics model. Phys. Fluids 1995, 7:839-848.
265. Piomelli U., Ferziger J.H., Moin P. Models for Large Eddy Simulation of Turbulent Channel Flows Including Transpiration 1987, Technical Report, Rep. TF-32, Dept. Mech. Eng., Stanford University.
266. Pochorecki R., Moniuk W., Bielski P., Zdrojkwoski A. Modeling of the coalescence/redispersion processes in bubble columns. Chem. Eng. Sci. 2001, 56:6157-6616.
267. Rai M.M., Moin P. Direct simulation of turbulent flow using finite-difference schemes. J. Comp. Phys. 1991, 96:15-53.
268. Rhie C.M., Chow W.L. A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA J. 1983, 21:1525-1532.
269. Rodi W. Turbulence Models and Their Application in Hydraulics 1993, Balkema, Rotterdam.
270. Saad Y., Schultz M. Conjugate gradient-like algorithms for solving nonsymmetric linear systems. SIAM J. 1985, 44:417-424.
271. Sato Y., Sadatomi M., Sekoguchi K. Momentum and heat transfer in two-phase bubbly flow-I. Int. J. Multiphase Flow 1981, 7:167-178.
272. Scott W.T. Analytic studies of cloud droplet coalescence. J. Atm. Sci. 1968, 25:54-65.
273. Shi J.M., Zwart P.J., Frank T., Rohde U., Prasser H.M. Development of a Multiple velocity Multiple size Group Model for Poly-dispersed Multiphase flows 2004, Annual Report of Institute of Safety Research, Forschungszentrum Rossendorf, Germany.
274. Smagorinsky J. General circulation experiment with the primitive equations: part I. The basic experiment. Mon. Weather Rev. 1963, 91:99-164.
275. Spalding D.B. A novel finite-difference formulation for differential expressions involving both first and second derivatives. Int. J. Num. Meth. Eng. 1972, 4:551-559.
276. Tennekes H., Lumley J.L. A First Course in Turbulence 1976, MIT Press, Cambridge, MA.
277. Tersoff J. Modeling solid-state chemistry: interatomic potentials for multicomponent systems. Phys. Rev. B 1989, 39:5566-5568.
278. Thomas J.L., Diskin B., Brandt A. Textbook multigrid efficiency for fluid simulations. Ann. Rev. Fluid Mech. 2003, 35:317-340.
279. Timmermann G. A cascadic multigrid algorithm for semilinear elliptic problems. Numerische Mathematik 2000, 86:717-731.
280. Tokoro C., Okaya K., Sadaki J. A Fast Algorithm for the Discrete Element Method by Contact Force Prediction,. Kona 2005, 23:182-193.
281. Tu J.Y., Fletcher C.A.J. Numerical computation of turbulent gas-particle flow in a 90° Bend. AIChE J. 1995, 41:2187-2197.
282. Van der Host H.A. BICGSTAB: a fast and smoothly converging variant of the Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. 1992, 13:631-644.
283. Van Driest E.R. On turbulent flow near a wall. J. Aerosol Sci. 1956, 23:1007-1011.
284. van Gunsteren W.F., Berendsen H.J.C. Algorithms for brownian dynamics. Mol. Phys. 1982, Vol. 45:637-647.
285. Verlet L. Computer "experiments" on classical fluids, I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 1967, 159:98-103.
286. Versteeg H.K., Malalasekera W. An Introduction to Computational Fluid Dynamics - The Finite Volume Method 1995, Prentice Hall, Pearson Education Ltd., England.
287. Vreman A.W. An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 2004, 16:3670-3681.
288. Wesseling P. Introduction to Multi-Grid Methods 1995, CR - 195045 ICASE 95-11, NASA.
289. Wilcox D.C. Turbulence Modeling for CFD 1998, DCW Industries, Inc.
290. Yeoh G.H., Tu J.Y. Thermal-hydrodynamic modelling of bubbly flows with heat and mass transfer. AIChE J. 2005, 51:8-27.
291. Yeoh G.H., Tu J.Y. Computational Techniques for Multiphase Flows: Basics and Applications 2009, Butterworth-Heinemann, Elsevier, UK.
292. Zhu Z.J., Dorao C.A., Lucas D., Jakobsen H.A. On the coupled solution of a combined population balance model using the least-squares spectral element method. Ind. Eng. Chem. Res. 2009, 48:7994-8006.
293. Alcamo R., Micale G., Grisafi F., Brucato A., Ciofalo M. Large-eddy simulation flow in an unbaffled stirred tank driven by a Rushton turbine. Chem. Eng. Sci. 2005, 60:2303-2316.
294. Alopaeous V., Koskinen J., Keskinen K.I. Simulation of the population balances for liquid-liquid systems in a nonideal stirred tank. Part 1-description and qualitative validation of the model. Chem. Eng. Sci. 1999, 54:5887-5899.
295. Alopaeous V., Koskinen J., Keskinen K.I., Majander J. Simulation of the population balances for liquid-liquid systems in a nonideal stirred tank. Part 2-parameter fitting and the use of the multiblock model for dense dispersions. Chem. Eng. Sci. 2002, 57:1815-1825.
296. Antal S.P., Lahey R.T., Flaherty J.E. Analysis of phase distribution and turbulence in dispersed particle/liquid flows. Chem. Eng. Comm. 1991, 174:85-113.
297. Bapat P.M., Tavlarides L.L. Mass transfer in a liquid-liquid CFSTR. AIChE J. 1985, 31:659-666.
298. Bothe D., Schmidtke M., Warnecke H.J. VOF-simulation of the lift force for single bubbles in a simple shear flow. Chem. Eng. Technol. 2006, 29:1048-1053.
299. Bourne J.R., Yu S. Investigation of micromixing in stirred tank reactors using parallel reactions. Ind. Eng. Chem. Res. 1994, 33:41-55.
300. Burns A.D., Frank T., Hamill I., Shi J. The Favre averaged drag model for turbulent dispersion in Eulerian multiphase flow 2004, Proc. Fifth Int. Multiphase Flow, ICMF-2004, Yokohama, Japan.
301. Cheung S.C.P., Yeoh G.H., Tu J.Y. On the modeling of population balance in isothermal vertical bubbly flows - average bubble number density approach. Chem. Eng. Proc. 2007, 46:742-756.
302. Cheung S.C.P., Yeoh G.H., Qi F.S., Tu J.Y. Classification of bubbles in vertical gas-liquid flow: Part 2 - a model evaluation. Int. J. Multiphase Flow 2012, 39:135-147.
303. Coulaloglou C.A., Tavlarides L.L. Description of interaction processes in agitated liquid-liquid dispersions. Chem. Eng. Sci. 1977, 32:1289-1297.
304. Drew D.A., Lahey R.T. Application of General Constitutive Principles to the Derivation of Multi-dimensional Two-Phase Flow Equation. Int. J. Multiphase Flow 1979, Vol. 5:243-264.
305. Fan R., Marchisio D.L., Fox R.O. Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds. Powder Tech. 2004, 139:7-20.
306. Frank T., Shi J., Burns F.A.D. Validation of Eulerian multiphase flow models for nuclear safety application 2004, Proc. Third Symp., Two-Phase Modeling and Experimentation, Pisa, Italy.
307. Gordon R.G. Error bounds in equilibrium statistical mechanics. J. Math. Phys. 1968, 9:655-672.
308. Hibiki T., Ishii M. One-group interfacial area transport of bubbly flows in vertical round tubes. Int. J. Heat Mass Transf. 2000, 43:2711-2726.
309. Hibiki T., Ishii M. Development of one-group interfacial area transport equation in bubbly flow systems. Int. J. Heat Mass Transf. 2002, 45:2351-2372.
310. Hibiki T., Ishii M. Interfacial area transport equations for gas-liquid flow. J. Comp. Multiphase Flows 2009, 1:1-22.
311. Hibiki T., Ishii M., Xiao Z. Axial interfacial area transport of vertical bubbly flows. Int. J. Heat Mass Transf. 2001, 44:1869-1888.
312. Hsia M.A., Tavlarides L.L. A simulation model for homogeneous dispersion in stirred tanks. Chem. Eng. J. 1980, 20:225-236.
313. Ishii M., Chawla T.C. Local Drag Laws in Dispersed Two-Phase Flow, Technical Report ANL-79-105 1979, Argonne National Laboratory, Chicago IL, 1979.
314. Ishii M., Zuber N. Drag coefficient and relative velocityxe "velocity" in bubbly, droplet or particulate flows. AIChE J. 1979, 25:843-885.
315. Ishii M., Kim S., Uhle J. Interfacial area transport: model development and benchmark experiments. Int. J. Heat Mass Transf. 2002, 45:3111-3123.
316. Krepper E., Lucas D., Prasser H. On the modeling of bubbly flow in vertical pipes. Nucl. Eng. Des. 2005, 235:597-611.
317. Lebowitz J.L. Exact solution of generalized Percus-Yevick equation for a mixture of hard spheres. Phys. Rev. A 1964, 133:895-899.
318. Liu T.J., Bankoff S.G. Structure of air-water bubbly flow in a vertical pipe - I. Liquid mean velocity and turbulence measurements. Int. J. Heat Mass Transf. 1993, 36:1049-1060.
319. Liu T.J., Bankoff S.G. Structure of air-water bubbly flow in a vertical pipe - II. Void fraction, bubble velocity and bubble size distribution. J. Heat Mass Transf. 1993, 36:1061-1072.
320. Lopez de Bertodano M. Turbulent bubbly two-phase flow in a triangular duct 1992, PhD dissertation, Rensselaer Polytechnic Institute, New York.
321. Lopez de Bertodano M. Two-fluid model for two-phase turbulent jet. Nucl. Eng. Des. 1998, 179:65-74.
322. Lucas D., Krepper E., Prasser H.M. Use of models for lift, wall and turbulent dispersion forces acting on bubbles for poly-disperse flows. Chem. Eng. Sci. 2007, 62:4146-4157.
323. Luo H., Svendsen H. Theoretical model for drop and bubble break-up in turbulent dispersions. AIChE J. 1996, 42:1225-1233.
324. Menter F.R. Zonal Two Equation kω Turbulence Models for Aerodynamics flows 1993, AIAA paper 93-2906.
325. Moraga F.J., Larreteguy A.E., Drew D.A., Lahey R.T. Assessment of turbulent dispersion models for bubbly flows in the low stokes number limit. Int. J. Multiphase Flow 2003, 29:655-673.
326. Politano M.S., Carrica P.M., Converti J. A model for turbulent polydisperse two-phase flow in vertical channels. Int. J. Multiphase Flow 2003, 29:1153-1182.
327. Prasser H.M., Beyer M., Carl H., Gregor S., Lucas D., Pietruske H., Schutz P., Weiss F.P. Evolution of the structure of a gas-liquid two-phase flow in a large vertical pipe. Nucl. Eng. Des. 2007, 237:1848-1861.
328. Prince M.J., Blanch H.W. Bubble coalescence and break-up in air-sparged bubble column. AIChE J. 1990, 36:1485-1499.
329. Serizawa A., Kataoka I. Phase distribution in two-phase flow. Transient Phenomena in Multiphase Flow 1988, 179-224. Hemisphere, New York. N.H. Afgan (Ed.).
330. Shen X.Z., Mishima K., Nakamura H. Two-phase phase distribution in a vertical large diameter pipe. International Journal of Heat and Mass Transfer 2005, 48:211-225.
331. Smoluchowski M.V. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 1917, 92:129-168.
332. Syamlal M., Rogers W., O'Brien T.J. MFIX Documentation Theory Guide 1993, US Department of Energy, Morgantown, West Virginia.
333. Taitel Y., Bornea D., Dukler A.E. Modeling Flow Pattern Transitions for Steady Upward Gas-liquid in Vertical Tubes. Aiche Journal 1980, 26:345-354.
334. Takagi S., Matsumoto Y. Numerical Study on the Forces Acting on a Bubble and Particle 1998, Proc. Third Int. Conference on Multiphase Flow, Lyon, France.
335. Taitel Y., Bornea D., Dukler A.E. Modeling Flow Pattern Transitions for Steady Upward Gas-liquid in Vertical Tubes. Aiche Journal 1980, 26:345-354.
336. Tomiyama A. Struggle with Computational Bubble Dynamics 1998, Proc. Third Int. Conf. Multiphase Flow, Lyon, France.
337. Vermeulen T., Williams G.M., Langlosi G.E. Interfacial area in liquid-liquid and gas-liquid agitation. Chem. Eng. Prog. 1955, 51:85-F-94-F.
338. Wallis G.B. One-Dimensional Two-Phase Flow 1969, McGraw-Hill, New York.
339. Wang S.K., Lee S.J., Lahey R.T., Jones O.C. 3-D turbulence structure and phase distribution measurements in bubbly two-phase flows. Int. J. Multiphase Flow 1987, 13:327-343.
340. Wang L., Marchisio D.L., Virgil R.D., Fox R.O. CFD simulation of aggregation and breakage processes in laminar Taylor-Coutte flow. J. Colloid Interface Sci. 2005, 282:380-396.
341. Wellek R.M., Agrawal A.K., Skelland A.H.P. Shapes of liquid drops moving in liquid media. AIChE J. 1966, 12:854-862.
342. Wu Q., Kim S., Ishii M., Beus S.G. One-group interfacial area transport in vertical bubbly flow. Int. J. Heat Mass Transf. 1998, 41:1103-1112.
343. Yao W., Morel C. Volumetric interfacial area prediction in upwards bubbly two-phase flow. Int. J. Heat Mass Transf. 2004, 47:307-328.
344. Zuber N. On the dispersed two-phase flow in laminar flow regime. Chem. Eng. Sci. 1964, 19:897-917.
345. Akroyd J., Smith A.J., Laurence R.S., McGlashan R., Kraft M. A coupled CFD-population balance approach for nanoparticle synthesis in turbulent reacting flows. Chem. Eng. Res. Des. 2011, 66:3792-3805.
346. Amara N., Ratsimba B., Wilhelm A., Delmas H. Growth rate of potash alum crystals: comparison of silent and ultrasonic conditions. Ultrason. Sonochem. 2004, 11:17-21.
347. Briesen H. Two-dimensional population balance modeling for shape dependent crystal attrition. Chem. Eng. Sci. 2009, 64:661-672.
348. Brown M.E. The growth of brown trout (Salmo trutta Linn.). J. Exp. Biol. 1957, 28:4473-4491.
349. Christofides P.D., Li M., Madler L. Control of particulate processes: recent results and future challenges. Powder Tech. 2007, 175:1-7.
350. Fredrickson A.G. Population balance equations for cell and microbial cultures and revisited. AIChE J. 2003, 49:1050-1059.
351. Gadewar S.B., Doherty M.F. A dynamic model for evolution of crystal shape. J. Cryst. Growth 2004, 267:239-250.
352. Liu Y.H., Bi J.X., Zeng A.P., Yuan J.Q. A population balance model describing the cell cycle dynamics of myeloma cell cultivation. Biotechnol. Prog. 2007, 23:1198-1209.
353. Liu J.J., Ma C.Y., Hu Y.D., Wang X.Z. Modelling protein crystallisation using morphological population balance models. Chem. Eng. Res. Des. 2010, 88:437-446.
354. Ma C.Y., Wang X.Z., Roberts K.J. Morphological population balance for modeling crystal growth in face directions. AIChE J. 2008, 54:209-222.
355. McMillan J., Briens C., Berutti F., Chan E. Particle attrition mechanism with a sonic gas jet injected into a fluidized bed. Chem. Eng. Sci. 2007, 62:3809-3820.
356. Ouchiyama N., Rough S.L., Bridgwater J. A population balance approach, to describing bulk attrition. Chem. Eng. Sci. 2005, 60:1429-1440.
357. 2 partial pressure conditions. Chem. Eng. Sci. 2010, 65:550-555.
358. Thompson R.W., Cauley D.A. A population balance model for fish population dynamics. J. Theor. Biol. 1979, 81:289-307.
359. Tsantilis S., Kammler H.K., Pratsinis S.E. Population balance modeling of flame synthesis of titania nanoparticles. Chem. Eng. Sci. 2002, 57:2139-2156.
360. Villermaux J., Devillon J.C. Représentation de la coalescence et de la redispersion des domaines de ségrégation dans unfluide par un modèle d'interaction phénoménologique 1972, Second Int. Symp. Chem. React. Eng, Elsevier, New York, pp. 1-13.