Modeling of Vertical Bubble-Driven Flows

Industrial and Engineering Chemistry Research

Volumn 36, Issue 10, 1997, Pages 4052-4074

  Jakobsen, Hugo A ^a   Sannæs, Bente H ^b   Grevskott, Sverre ^b   Svendsen, Hallvard F ^b  

^a NORWEGIAN METEOROLOGICAL INSTITUTE   (Norway)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

[No Author keywords available]

Indexed keywords

BUBBLES (IN FLUIDS); FLOW OF FLUIDS; FLUID MECHANICS; MATHEMATICAL MODELS; NUMERICAL METHODS;

BUBBLE-DRIVEN FLOW; STEADY-STATE MODELS;

TWO PHASE FLOW;

COMPUTATIONAL FLUID DYNAMICS; FLOW; MODELING;

BUBBLE-DRIVEN FLOW; BUBBLES; FLOW MODELLING;

EID: 0031254713     PISSN: 08885885     EISSN: None     Source Type: Journal    
DOI: 10.1021/ie970276o     Document Type: Review

References (181)

1. Abou-Arab, T. W. Turbulence models for two-phase models. Gas-Liq. Flows 1986, 3, 863-907.
2. Abou-El-Hassan, M. E. Correlations for bubble rise in Gas-Liquid systems. Gas-Liq. Flows 1986, 3, 110-120.
3. Anderson, H. I.; Kristoffersen, R. Numerical simulation of unsteady viscous flow. Arch. Mech. 1989, 41, (2-3), 207-223.
4. Antal, S. P.; Lahey, R. T., Jr.; Flaherty, J. E. Analysis of phase distribution in fully developed laminar bubbly two-phase flow. Int. J. Multiphase Flow 1991, 17, 635.
5. Arakawa, A.; Lamb, V. R. Computational design of the basic dynamical processes of the UCLA general circulation model. Meth. Comput. Phys. 1977, 17, 173-265.
6. Auton, T. R. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 1981.
7. Auton, T. R. The lift force on a spherical body in a rotational flow. J. Fluid Mech. 1987, 183, 199-218.
8. Becker, S.; Sokolichin, A.; Eigenberger, G. Gas-Liquid Flow in Bubble Columns and Loop Reactors: Part II. Comparison of Detailed Experiments and Flow Simulations. Chem. Eng. Sci. 1994, 49 (24B), 5747-5762.
9. Berge, E.; Jakobsen, H. A. A new regional scale multi-layer model for the calculation of long term transport and deposition of air pollution in Europe. Submitted to Tellus 1997.
10. Beyerlein, S. W.; Cossmann, R. K.; Richter, H. J. Prediction of bubble concentration profiles in vertical turbulent two-phase flow. Int. J. Multiphase Flow 1985, 11, 629.
11. Biesheuvel, A.; van Wijngaarden, L. Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid. Mech. 1984, 148, 301-318,
12. Boisson, N.; Malin, M. R. Numerical prediction of two-phase flow in bubble columns. Int. J. Num. Meth. Fluids 1996, 23, 1289-1310.
13. Bott, A. A positive definite advection scheme obtained by non-linear renormalization of the advective fluxes. Mon. Weather Rev. 1989, 117, 1006-1015.
14. Celik, I.; Wang, Y.-Z. Numerical simulation of liquid circulation in a bubbly column. Num. Meth. Multiphase Flow, ASME 90 1990, 91, 19.
15. Celik, I.; Wang, Y. Z. Numerical-Simulation of Circulation in Gas-Liquid Column Reactors-Isothermal, Bubbly, Laminar-Flow. Int. J. Multiphase Flow 1994, 20(6), 1053-1070. Note: English Article.
16. CHAM. PHOENICS Training Course Notes, TR/300, 1990.
17. Chao, W. C. Formulation of an explicit-multiple-time-step integration method for use by a global primitive equation grid model. Mon. Weather Rev. 1982, 110, 1603-1617.
18. Chen, J. J. J.; Jamialahmadi, M.; Li, S. M. Effect of Liquid Depth on Circulation in Bubble Columns. Chem. Eng. Res. Des. 1989, 67, 203-207.
19. Chen, R. C.; Reese, J.; Fan, F.-S. Flow structure in a Three-Dimensional Bubble Column and a Three-Phase Fluidized Bed. AIChE J. 1994, 40 (7), 1093-1104.
20. Chesters, A. K. The modelling of coalescence processes in fluid-liquid dispersions: A review of current understanding. Trans. Inst. Chem. Eng. 1991, 69, 259.
21. Class, G.; Meyder, R.; Sengpiel, W. Measurement of spatial gas distribution and turbulence structure in developing bubbly two-phase flow in vertical channels. Presented at the International Conference on Multiphase Flows, Tsukuba, Japan, 1991.
22. Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic: San Diego, CA, 1978.
23. Cook, T. L.; Harlow, F. H. Vortices in Bubbly Two-Phase Flow. Int. J. Multiphase Flow 1986, 12 (1), 35.
24. Cross, M.; Marcatos, N. C.; Aldham, C. In Control'84 Mineral/Metallurgical Processing; Herbst, J., Ed.; Society of Mining Engineers of the American Institute of Mining, Metallurgical and Petroleum Eng.: New York, 1984; pp 291-297.
25. Dandy, D. S.; Dwyer, H. A. A. A sphere in shear flow at finite Reynolds number: effect of shear on particle lift, drag, and heat transfer. J. Fluid Mech. 1990, 216, 381.
26. Dardub, D.; Seinfeld, J. H. Numerical advective schemes used in air quality models. Sequential and parallel implementation. Atmos. Environ. 1994, 28 (20), 3369-3385.
27. Deardorff, J. W. Numerical investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. 1972, 29, 91-115.
28. Delnoij, E.; Lammers, F. A.; Kuipers, J. A. M.; van Swaaij, W. P. M. Dynamic simulation of dispersed gas-liquid two-phase flow using a discrete bubble model, Chem. Eng. Sci. 1997, 52 (9), 1429-1458.
29. De Vahl Davis, G.; Mallison, G. An evaluation of upwind and central differences approximations by a study of recirculating flow. Comput. Fluids 1976, 4, 29-43.
30. Devanathan, N. Investigation of liquid hydrodynamics in bubble columns via a computer automated radioactive particle tracking (CARPT) facility. D.Sc. Thesis, Washington University, St. Louis, MO, 1991.
31. Devanathan, N.; Moslemian, D.; Dudukovic, M. P. Flow mapping in bubble columns using CARPT. Chem. Eng. Sci. 1990, 45 (8), 2285-2291.
32. Drazin, P. G.; Reid, W. H. Hydrodynamic stability, Cambridge University: Cambridge, 1981.
33. Drew, D. A. Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 1983, 15, 261.
34. Drew, D. A.; Lahey, R. T., Jr. The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int. J. Multiphase Flow 1987, 13, 113.
35. Drew, D.; Cheng, L.; Lahey, R. T., Jr. The analysis of virtual mass effects in two-phase flow. Int. J. Multiphase Flow 1979, 5, 233.
36. Dudukovic, M. P. Problem 4 Presented at the Engineering Foundation Conference, Computational Fluid Dynamics in Chemical Reaction Engineering, San Diego, CA, Oct. 1996.
37. Dudukovic, M. P.; Devanathan, N.; Holub, R. Multiphase reactors: Models and experimental verification. Rev. Inst. Fr. Petrol. 1991, 46 (4), 439-465.
38. Durst, F.; Taylor, A. M. K. P.; Whitelaw, J. H. Experimental and Numerical Investigation of Bubble Driven Laminar Flow in an Axi-symmetric Vessel. Int. J. Multiphase Flow 1984, 10, 557-569.
39. Elghobashi, S. E.; Abou-Arab, T. W. A two-equation turbulence model for two-phase flows. Phys. Fluids 1983, 26, 931-938.
40. Ervin, E. Computations of Bubbles and Drops in Shear Flow. Ph.D Thesis, The University of Michigan, 1993.
41. Esmaeeli, A.; Ervin, E.; Tryggvason, G. Numerical Simulations of Rising Bubbles. In Bubble Dynamics and Interface Phenomena. Proc. IUTAM Symp. Birmingham, U.K., 6-9 September, Blake, J. R., Boulton-Stone, J. M., Thomas, N. H., Eds.; Kluwer Academic: Dordrecht, 1993; pp 247-255.
42. Esmaeeli, A.; Ervin, E.; Tryggvason, G. Numerical Simulations of Rising Bubbles. In Bubble Dynamics and Interface Phenomena; Blake, J. R., Ed.; Kluwer Academic: Dordrecht, 1994.
43. Esmaeeli, A.; Ervin, E.; Tryggvason, G. An inverse energy cascade in two-dimensional low Reynolds number bubbly flows. J. Fluid. Mech. 1996, 314, 315-330.
44. Ferziger, J. H.; Leslie, D. C. Large eddy simulation-A predictive approach to turbulent flow computation. Amar. Inst. Aeronaut. Astronaut., Pap. 1979, 79-1441.
45. Franz, K.; Börner, T.; Kantorek, H. J.; Buchholz, R. Strömungsstruktur in Blasensäulen. Chem.-Ing.-Tech. 1984, 56 (2), 154-155.
46. Gashe, H.-E.; Edinger, C.; Kömpel, H.; Hofmann, H. A Fluid-Dynamically based model of Bubble Column Reactors. Chem. Eng. Technol. 1990, 13, 341-349.
47. Geary, N. W.; Rice, R. G. Circulation in bubble columns: Corrections for distorted bubble shape. AIChE J. 1991, 37, 1593.
48. Gresho, P. M.; Lee, R. L. Don't suppress the wiggles-They're telling you something. Comput. Fluids 1981, 9, 223-253.
49. Grevet, J. H.; Szekely, J.; El-Kaddah, N. An experimental and theoretical study of gas bubble driven circulation systems. Int. J. Heat Mass Transfer 1982, 25, 487-497.
50. Grevskott, S.; Sannaes, B. H.; Dudukovic, M. P.; Hjarbo, K. W.; Svendsen, H. F. Liquid Circulation, Bubble Size Distribution, and Solids Movement in Two- and Three-Phase Bubble Columns. Chem. Eng. Sci. 1996, 51 (10), 1703-1713.
51. Grienberger, J.; Hofmann, H. Investigations and modelling of bubble columns. Chem. Eng. Sci. 1992, 47, 2215.
52. Groen, J. S.; Oldeman, R. G. C.; Mudde, R. F.; Van Den Akker, H. E. A. Coherent structures and axial dispersion in bubble column reactors. Chem. Eng. Sci. 1996, 51 (10), 2511-2520.
53. Haltimer, G. J.; Williams, R. T. Numerical prediction and dynamic meteorology, 2nd ed.; John Wiley & Sons: New York, 1980.
54. Harper, J. F. On Bubbles Rising In-Line at Large Reynolds Numbers. J. Fluid. Mech. 1970, 41, 751-758.
55. Harten, A. High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 1983, 49, 357-393.
56. Harten, A.; Osher, S. Uniformly high-order accurate non-oscillatory schemes. SIAM J. Num. Anal. 1987, 24, 279-309.
57. Herring, J. R. Subgrid scale modelling - An introduction and overview. Turb. Shear Flows 1979, 1, 347-352.
58. Hidy, G. M.; Broch, J. R. The Dynamics of Aerocolloidal Systems; Pergamon: Oxford, 1970.
59. Hillmer, G.; Weismantel, L.; Hofmann, H. Investigations and modelling of slurry bubble columns. Chem. Eng. Sci. 1994, 49 (6), 837-843.
60. Hills, J. H. Radial non-uniformity of velocity and voidage in a bubble column. Trans. Inst. Chem. Eng. 1974, 52, 1-9.
61. Hills, J. H. The rise of a large bubble through a swarm of smaller ones. Trans. Inst. Chem. Eng. 1975, 53, 224.
62. Hills, J. H.; Darton, R. C. The rising velocity of a large bubble in a bubble swarm. Trans. Inst. Chem. Eng. 1976, 54, 258-264.
63. Hjertager, B. H.; Morud, K. E. Computational Fluid Dynamics Simulation of Bioreactors. Model. Identif. Control 1995, 16 (4), 177-191.
64. Hwang, S. T.; Soliman, H. M.; Lahey, R. T., Jr. Phase separation in dividing two-phase flows. Int. J. Multiphase Flow 1988, 24, 439.
65. Ishii, M. Thermo-Fluid Dynamic Theory of Two-Phase Flow; Eyrolles: Paris, 1975.
66. Ishii, M.; Zuber, N. Drag Coefficient and Relative Velocity in Bubbly, Droplet or Particulate Flows. AIChE J. 1979, 25 (5), 843.
67. Jakobsen, H. A. On the Modelling and Simulation of Bubble Column Reactors using Two-Fluid Model. Dr. Ing. Thesis 97, NTH, Trondheim, 1993.
68. Jakobsen, H. A.; Svendsen, H. F.; Hjarbo, K. W. On the prediction of local flow structures in internal loop and bubble column reactors using a two-fluid model. Comp. Chem. Eng. 1993, 17S, S531-S536.
69. Jakobsen, H. A.; Berge, E.; Iversen, T.; Skålin, R. Status of the development of the multi-layer Eulerian model, a) Model description, b) A new method of calculating mixing heights. c) Model results for sulphur transport in Europe for 1992 in the 50 km grid. EMEP/MSC-W Note 3/95; The Norwegian Meteorological Institute (DNMI), Oslo, Norway, Research Report 21, 1995.
70. Jakobsen, H. A.; Jonson, J. E.; Berge, E. Transport and deposition calculations of sulphur and nitrogen compounds in Europe for 1992 in the 50 km grid by use of the multi-layer Eulerian model. EMEP/MSC-W Note 2/96; The Norwegian Meteorological Institute (DNMI), Oslo, Norway, Research Report 34, 1996.
71. Jeelani, S. A. K.; Hartland, S. Collision of oscillating liquid drops. Chem. Eng. Sci. 1991a, 46, 1807.
72. Jeelani, S. A. K.; Hartland, S. Effect of approach velocity on binary and interfacial coalescence. Trans. Inst. Chem. Eng. 1991b, 69, 271.
73. Johansen, S. T. On the Modelling of Disperse Two Phase Flows. Dr. Techn. Thesis, NTH, Trondheim, 1990.
74. Johansen, S. T.; Boysan, F. Fluid Dynamics in Bubble Stirred Ladles: Part II. Mathematical Modeling. Metallurg. Trans. B 1988, 19B, 755.
75. Johansen, S. T.; Boysan, F.; Ayers, W. H. Mathematical modelling of bubble driven flows in metallurgical process. Appl. Sci. Res. 1987, 44, 197-207.
76. Kariyasaki, A. Behavior of a single gas bubble in a liquid flow with a linear velocity profile. ASME-JSME Thermal Engineering Joint Conference, 1987, p 261.
77. Kasahara, A. Computational aspects of numerical models for weather prediction and climate simulations. Methods in Computational Physics; Academic: New York, 1977; Vol. 177, pp 1-66.
78. Krahn, E. Negative Magnus Force. J. Aeronaut. Sci. 1956, 23, 377-378.
79. Kumar, S.; Vanderheyden, W. B.; Devanathan, N.; Padial, N. T. Numerical Simulation and Experimental Verification of the Gas-Liquid Flow in Bubble Columns. AIChE Symp. Ser. 1995, 305, 11-19.
80. Lahey, R. T., Jr. Turbulence and phase distribution phenomena in two-phase flow. Invited Lecture, ICHMT International Seminar, 1987, p 139.
81. Lahey, R. T., Jr. Turbulence and Phase Distribution Phenomena in Two-Phase Flow. Transient Phenomena in Multiphase Flow, ICHMT International Seminar, 1988.
82. Lahey, R. T., Jr. The Analysis of Phase Separation and Phase Distribution Phenomena using Two-Fluid Models. Nucl. Eng. Des. 1990, 122, 17-40.
83. Lahey, R. T., Jr. Void wave propagation phenomena in two-phase flow (Kern Award Lecture). AIChE J. 1991, 37 (1), 123-135.
84. Lahey, R. T., Jr.; Drew, D. A. Analysis of Phase Distribution in Fully Developed Two-Phase Flows. 2nd Multiphase Flow Heat Transf. Symp. 1979, 1, 343.
85. Lahey, R. T., Jr.; Drew, D. A. The three-dimensional time and volume averaged conservation equations of two-phase flow. Adv. Nucl. Sci. Technol. 1988, 20, 1.
86. Lahey, R. T., Jr.; Drew, D. A. On the development of multidimensional two-fluid models for vapor/liquid two-phase flows. Chem. Eng. Commun. 1992, 118, 125-139.
87. Lahey, R. T., Jr.; Lee, S. J. Phase distribution and two-phase turbulence for bubbly flows in pipes. Multiphase Sci. Technol. 1991, 6, 303.
88. Lai, K. Y. M.; Salcudean, M. Computer analysis of multi-dimensional, turbulent, buoyancy-induced, two-phase flows in gas-agitated-liquid reactors. Comput. Fluids 1987, 15, 281.
89. Lapin, A.; Lübbert, A. Numerical simulation of the dynamics of two-phase gas-liquid flows in bubble columns. Chem. Eng. Sci. 1994a, 49 (21), 3661-3674.
90. Lapin, A.; Lübbert, A. Dynamic Simulation of the Two-Phase Flow Mixing in Bubble Columns. Inst. Chem. Eng., Symp. Ser. 1994b, 136, 365-373.
91. Lin, T.-J.; Reese, J.; Hong, T.; Fan, L.-S Quantitative Analysis and Computation of Two-Dimensional Bubble Columns. AIChE J. 1996, 42(2), 301-318.
92. Lopez de Bertodano, M.; Lee, S.-J.; Lahey, R. T., Jr.; Drew, D. A. The prediction of two-phase distribution phenomena using a Reynolds stress model. J. Fluid Eng. 1990, 112, 107-113.
93. Lübbert, A.; Lapin, A. Dynamic Numerical Experiments with 3-D Bubble Columns. Analysis of some Key Mixing Mechanisms. Rev. Inst. Fr. Petrol. 1996, 51 (2), 269-277.
94. Maccoll, J. W. J. R. Aeronaut. Soc. 1928, 32, 777-798.
95. Magnus, G. Ueber die Abweichung der Geschosse, und: Ueber eine auffallende Erscheinung bei rotirenden Körpern. Anal. Phys. Chem. 1853, 88 (1), 1-29.
96. Marchuk, G. I. Numerical methods in weather prediction; Academic: New York and London, 1974.
97. Maryon, R. H.; Ryall, D. B.; Saltbones, J.; Bartnicki, J.; Jakobsen, H. A.; Berge, E. An intercomparison of three long range dispersion models developed for the UK meteorological office, DNMI and EMEP. Report series of UK Meteorological Office, Turbulence and Diffusion Note 234, 1996.
98. Marrucci, G. Rising velocity of a swarm of spherical bubbles. Ind. Eng. Chem. Fundam. 1965, 4, 224.
99. Maxey, M. R.; Riley, J. J. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 1983, 26, 883.
100. Mednikov, E. P. Theory of turbulent transport of suspended particles flowing in dispersed media. Theor. Found. Chem. Eng. 1986, 20, 237-244.
101. McRae, G. J.; Goodin, W. R.; Seinfeld, J. H. Numerical solution of the atmosphere diffusion equation for chemically reacting flows. J. Comp. Phys. 1982, 45, 1-42.
102. Menzel, T.; in der Weide, T.; Staudacher, O.; Wein, O.; Onken, U. Reynolds Shear Stress for Modeling of Bubble Column Reactors. Ind. Eng. Chem. Res. 1990, 29, 988.
103. Millies, M.; Mewes, D. Liquid circulation in columns due to bubble drift. Chem. Eng. Technol. 1992, 15, 258.
104. Millies, M.; Mewes, D. Circulatory Flow and Mixing of the Continuous-Phase in Bubble-Columns. Chem.-Ing.-Tech. 1994, 66 (3), 332-342.
105. Millies, M.; Mewes, D. Back-mixing of the continuous phase in bubble columns. Chem. Eng. Sci. 1995a, 50 (13), 2107-2115.
106. Millies, M.; Mewes, D. Calculation of circulating flows in bubble columns. Chem. Eng. Sci. 1995b, 50 (13), 2093-2106.
107. Miyauchi, T.; Shyu, C.-N. Flow of fluid in bubble columns. Kagaku Kogaku 1970, 34, 958 (in Japanese).
108. Mokeyev, Yu. G. Effect of Particle Concentration on their Drag and Induced Mass. Fluid. Mech. Sov. Res. 1977, 6, 161.
109. Mollinger, A. M.; Nieuwstadt, F. T. M. Measurement of Lift Force on a Particle Fixed to the Wall in the Viscous Sublayer of a Fully developed Turbulent Boundary Flow. J. Fluid Mech. 1996, 316, 285-306.
110. Morud, K. E. Turbulent Two-Phase Flow in Bubble Columns and Stirred Fermentors. Dr. Ing. Thesis, NTH, TMIH, Porsgrunn, 1994.
111. Morud, K. E.; Hjertager, B. H. LDA Measurements and CFD Modelling of Gas-Liquid Flow in Stirred Vessel. Chem. Eng. Sci. 1996, 51 (2), 233-249.
112. Moslemian, D.; Devanathan, N.; Dudukovic, M. P. Liquid circulation in bubble columns via the CARPT facility. Effects of gas superficial velocity and column diameter. In Advances in gas-liquid flows; Kim, J. H., S., Upendra, Rohatgi, Hashemi, A., Eds.; Papers presented at the International Symposium on Gas-Liquid Two-Phase Flows; ASME: 1990, HTD 155, FED 99, pp 441-451.
113. Mudde, R. F.; Lee, D. J.; Reese, J.; Fan, L.-S. Role of coherent structures on Reynolds stresses in a 2-D bubble column. AIChE J. 1997, 43 (4), 913-926.
114. Nigmatulin, R. I.; Lahey, R. T., Jr.; Drew, D. A. On the Different Forms of Momentum Equations and on the Intra- and Interphase Interaction in the Hydromechanics of a Monodispersed Mixture. Chem. Eng. Commun. 1996, 141-142, 287-302.
115. Oliver, D. R. Influence of particle rotation on radial migration in the poiseuille flow of suspensions. Nature 1962, 194, 1269-1271.
116. Ostiguy, L.; Laprise, J. P. R. On the positivity of mass in common used numerical transport schemes. Atm. Ocean 1990, 28 (2), 147-161.
117. Pavlov, V. P. Liquid Circulation in Batch Bubbler. Khim. Promst. 1965, 9, 698-700 (in Russian).
118. Pourahmadi, F. Turbulence Modelling of Single and Two-Phase Curved Channel Flow. Ph.D. Thesis, University of California, Berkeley, 1982.
119. Raithby, G. A critical evaluation of upstream differencing applied to problems involving fluid flow. Comp. Meth. Appl. Mech. Eng. 1976, 9, 75-103.
120. Ranade, V. V. Flow in bubble columns: Some numerical experiments. Chem. Eng. Sci. 1992a, 47, 1857.
121. Ranade, V. V. Numerical simulation of dispersed gas-liquid flows. Sadhana 1992b, 17, 237.
122. Ranade, V. V. Modelling of Turbulent Flow in a Bubble Column Reactor. Trans. Inst. Chem. Eng. 1997, 75A, 14-23.
123. Reese, J.; Mudde, R. F.; Lee, D. J.; Fan, L.-S. Analysis of multiphase systems through Particle Image Velocimetry. AIChE Symp. Ser. 1996, 92 (310), 161-167.
124. Reynolds, O. On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. R. Soc. 1895, A186, 123.
125. Rietema, K.; Ottengraf, S. P. P. Liminar liquid circulation and bubble street formation in a gas-liquid system. Trans. Inst. Chem. Eng. 1970, 48, 54.
126. Rood, R. B. Numerical Advection Algorithms and Their Role in Atmospheric Transport and Chemistry Models. Rev. Geophys. 1987, 25(1), 71-100.
127. Rubinow, S. I.; Keller, J. B. The transverse force on a spinning Sphere moving in a viscous fluid. Fluid Mech. 1961, 11, 447.
128. Saffman, P. G. The lift on a small sphere in a slow shear flow. J. Fluid Mech. 1965, 22, 385.
129. Saffman, P. G. Corrigendum, J. J. Fluid Mech. 1968, 31, 624.
130. Sannaes, B. H. Solids movement and concentration profiles in column slurry reactors. Dr. Ing. Thesis 1997:42, NTNU, Trondheim, June 1997.
131. Schwarz, M. P.; Turner, W. J. Applicability of the standard k-e turbulence model to gas-stirred baths. Appl. Math. Model. 1988, 12, 273.
132. Segré, G.; Silberberg, A. Radial particle displacements in poiseuille flow of suspensions. Nature 1961, 189, 209.
133. Segré, G.; Silberberg, A. Behavior of macroscopic rigid spheres in Poiseuille flow. Part 1 and 2. J. Fluid Mech. 1962, 14, 115-157.
134. Sekoguchi, K.; Sato, T.; Honda, T. Two Phase Flow (first Report). Trans. Jpn. Soc. Mech. Eng. 1974, 40 (333), 1395-1403 (in Japanese).
135. Serizawa, A.; Kataoka, I. Phase distribution in two-phase flow. Proc. ICHMT Conf. Transport. 1987, p 179.
136. Serizawa, A.; Kataoka, L; Michiyoshi, I. Turbulence structure of air-water bubbly flow. Part 1,2 and 3. Int. J. Multiphase Flow 1975, 2, 221-259.
137. Serizawa, A.; Zun, I.; Michiyoshi, I. Analysis of Bubbly Flow in Vertical Tube, II, Phase Distribution, Proc. 4th Two-Phase Flow Symp. Jpn. 1982, pp 13-16.
138. Serizawa, A.; Kataoka, I.; Gofuku, A.; Takahashi, O.; Kawara, Z. Effect of initial bubble size on bubbly flow structure. Proc. Int. Conf. Multiphase Flows 1991, 547.
139. Simonin, O.; Balzer, G.; Deutsch, E.; Dalsecco, S.; Flour, I.; He, J. Numerical Modeling of Turbulent Reactive Two-Phase Flows with a Dispersed Phase. CFD Applied to process engineering, IFP, Solaize, March 1993a.
140. Simonin, O.; Deutsch, E.; Minier, J. P. Eulerian Prediction of the Fluid Particle Correlated Motion in Turbulent Two Phase Flows. Appl. Sci. Res. 1993b, 51 (1-2), 275-283.
141. Smolarkiewicz, P. K. A fully multi-dimensional positive definite advection transport algorithm with small implicit diffusion. J. Comput. Phys. 1984, 54 (2), 325-362.
142. Sokolichin, A.; Eigenberger, G. Gas-Liquid Flow in Bubble Columns and Loop Reactors: Part 1. Detailed Modelling and Numerical Simulation. Chem. Eng. Sci. 1994, 49, 5735-5746.
143. Sokolichin, A.; Becker, S.; Eigenberger, G. Modellierung und Numeriche Simulation von G/L-Blasen-Stromungen. Chem.-Ing.-Tech. 1994, 66 (4), 505-510.
144. Sokolichin, A.; Eigenberger, G.; Lapin, A.; Lübbert, A. Dynamical numerical simulation of gas-liquid two-phase flows. Euler/Euler versus Euler/Lagrange. Chem. Eng. Sci. 1997, 52, 611-626.
145. Spalding, D. B. A guide to the PHOENICS input language. TR/100. Technical Report, CHAM, 1992.
146. Stewart, H. B.; Wendroff, B. Two-phase flow: Models and methods. J. Comput. Phys. 1984, 56, 363.
147. Stuhmiller, J. H. The Influence of Interfacial Pressure Forces on the Character of Two-Phase Flow Model Equations. Int. J. Multiphase Flow 1977, 3, 551-560.
148. Subbotin, V. I.; Ibragimov, M. Kh.; Bobkov, V. P.; Tychinskii, N. A. Turbulent channel-flow characteristics of gas-water mixtures. Sov. Phys.-Dokl. 1971, 16 (3), 192-194.
149. Svendsen, H. F.; Jakobsen, H. A.; Torvik, R. Local flow structures in internal loop and bubble column reactors. Chem. Eng. Sci. 1992, 47 (13/14), 3297-3304.
150. Swanson, W. M. The Magnus effect: A summary of investigations to date. J. Basic Eng. 1961, 461.
151. Taneda, S. Negative Magnus Effect. Rep. Res. Inst. Appl. Mech. 1957, 5, 123.
152. Taneda, S. Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106. J. Fluid Mech. 1978, 85, 187.
153. Thomas, N. H.; Auton, T. R.; Sene, K.; Hunt, J. C. R. Entrapment and transport of bubbles by transient large eddies in multiphase turbulent shear flows. Phys. Model. Multiphase Flow 1983, 169.
154. Tomiyama, A.; Zun, I.; Sou, A.; Sakaguchi, T. Numerical analysis of bubble motion with VOF method. Seminar on Two-Phase Flow Dynamics, 1992, p 67.
155. Tomiyama, A.; Sou, A.; Zun, I.; Kanami, N.; Sakaguchi, T. Effect of Eötvös Number and Dimensionless Liquid Volumetric Flux on Lateral Motion of a Bubble in a Laminar Duct Flow. Proc. 2nd Int. Conf. Multiphase Flow, Kyoto 1995, PD1, 11-18.
156. Torvik, R. Investigations of Two- and Three-Phase Bubble Columns. Dr. Ing. Thesis 51:1990, NTH, Trondheim, 1990.
157. Torvik, R.; Svendsen, H. F. Modelling of slurry reactors. A fundamental approach. Chem. Eng. Sci. 1990, 45, 2325.
158. Trapp, J. A. The mean flow character of two-phase flow equations. Int. J. Multiphase Flow 1986, 12, 263-276.
159. Tuann, S.; Olson, M. Review of computing methods for recirculating flows. J. Comp. Phys. 1978, 29, 1-19.
160. Tzeng, J.-W.; Chen, R. C.; Fan, L.-S. Visualization of flow characteristics in a 2-D bubble column and three-phase fluidized bed. AIChE J. 1993, 39 (5), 733.
161. Van Thai, D.; Minier, J. P.; Simonin, O.; Freydier, P.; Olive, J. Multidimensional Two-Fluid Model Computation of Turbulent Dispersed Two-Phase Flows. Numerical Method in Multiphase Flows ASME, Fluids Eng. Div. FED 185 1994, 277-291.
162. Viollet, P. L.; Simonin, O.; Olive, J.; Minier, J. P. Modelling Turbulent Two-Phase Flows in Industrial Equipment. In Computational Methods in Applied Sciences; Hirsh, Ch., Ed.; Elsevier Science: New York, 1992; pp 213-224.
163. Van Wijngaarden, L. Hydrodynamic Interaction between Gas Bubbles in Liquid. J. Fluid Mech. 1976, 77, 27-44.
164. Wachi, S.; Yates, J. G. Comments on modelling of slurry reactors. A fundamental approach. Chem. Eng. Sci. 1991a, 46, 1528.
165. Wachi, S.; Yates, J. G. Downward lean powder flow in a vertical tube. Chem. Eng. Sci. 1991b, 46, 929.
166. Walter, J. G.; Blanch, H. W. Liquid Circulation Patterns and their Effect on Gas Holdup and Axial Mixing in Bubble Columns. Chem. Eng. Commun. 1983, 19, 243-246.
167. Wang, S. K.; Lee, S. J.; Jones, O. C., Jr.; Lahey, R. T., Jr. Three-dimensional turbulence structure and phase distribution measurements in bubbly two-phase flows. Int. J. Multiphase Flow 1987, 13 (3), 327-343.
168. Webb, C.; Que, F.; Senior, P. R. Dynamic Simulation of gas Liquid Dispersion Behavior in a 2-D Bubble Column Using a Graphics Minicomputer. Chem. Eng. Sci. 1992, 47 (13/14), 3305-3312.
169. Yang, H. Q.; Przekwas, A. J. A comparative study of advanced shock-capturing schemes applied to Burgers' equation. J. Comp. Phys. 1992, 102, 139-159.
170. Yang, Y. B.; Devanathan, N.; Dudukovic, M. P. Liquid backmixing in bubble columns via Computer-Automated Radioactive Particle Tracking (CARPT). Exp. Fluids 1993, 16, 1-9.
171. Yao, B. P.; Zheng, C.; Gasche, H. E.; Hofman, H. Bubble behavior and flow structure of bubble columns. Chem. Eng. Process. 1991, 29, 65.
172. Yu, Y. H.; Kim, S. D. Bubble properties and local liquid velocity in the radial direction of co-current gas-liquid flow. Chem. Eng. Sci. 1991, 46, 313.
173. Zun, I. The transverse migration of bubbles influenced by walls in vertical bubbly flow. Int. J. Multiphase Flow 1980, 6, 583.
174. Zun, I. The analysis of bubble relative velocity and void fraction radial profiles in co-current upward bubbly flow. Proc. 2nd. World Cong. Chem. Eng. Montreal 1981, 290.
175. Zun, I. The role of void peaking in vertical two-phase bubbly flow. Int. Conf. Multiphase Flow 1985, 2, 127.
176. Zun, I. The non-rectilinear motion of bubbles rising through a stagnant and disturbed liquid. World Cong. Chem. Eng. 1986, 3, 214.
177. Zun, I. Pressure drop prediction for bubbly flow based on local turbulent characteristics. Particul. Phenom. Multiphase Transp. 1988a, 5, 207.
178. Zun, I. Transition from wall void peaking to core void peaking in turbulent bubbly flow. Trans. Phenom. Multiple Flow 1988b, 225.
179. Zun, I. The mechanism of bubble non-homogeneous distribution in two-phase shear flow. Nucl. Eng. Des. 1990, 118, 155.
180. Zun, I.; Kljenak, L; Moze, S. Space-time evolution of bubble non-homogeneous distribution in upward flow. Proc. Int. Conf. Multiphase Flows 1991, 463.
181. Zun, I.; Kljenak, I.; Moze, S. Space-time evolution of the non-homogeneous bubble distribution in upward flow. Int. J. Multiphase Flow 1993, 19 (1), 151-172.