On the population balance equation

Reviews in Chemical Engineering

Volumn 28, Issue 2-3, 2012, Pages 149-169

  Sporleder, Federico ^a   Borka, Zsolt ^a   Solsvik, Jannike ^a   Jakobsen, Hugo A ^a  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Mathematical modeling; Population balance; Population balance equation (PBE)

Indexed keywords

POPULATION BALANCE EQUATION; POPULATION BALANCES; THEORETICAL FOUNDATIONS;

INDUSTRIAL ENGINEERING; MATHEMATICAL MODELS;

ENGINEERING;

EID: 84867822513     PISSN: 01678299     EISSN: None     Source Type: Journal    
DOI: 10.1515/revce-2011-0013     Document Type: Review

References (90)

1. Alopaeus V, Koskinen J, Keskinen KI. Simulation of the population balance 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.
2. Alopaeus V, Koskinen J, Keskinen KI, 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.
3. Argyriou DT, List HL, Shinnar R. Bubble growth by coalescence in gas fluidized beds. AIChE J 1971; 17: 122-130.
4. Azizi F, Al Taweel AM. Algorithm for the accurate numerical solution of PBE for drop breakup and coalescence under high shear rates. Chem Eng Sci 2010; 65: 6112-6127.
5. Baldyga J, Podgorska W. Drop break-up in intermittent turbulence: maximum stable and transient sizes of drops. Can J Chem Eng 1998; 76: 457-470.
6. Buffo A, Vanni M, Marchisio DL. Multidimensional population balance model for the simulation of turbulent gas-liquid systems in stirred tank reactors. Chem Eng Res 2012; 70: 31-44.
7. Carrica PM, Drew D, Bonetto F, Lahey Jr RT. A polydisperse model for bubbly two-phase flow around a surface ship. Int J Multiphase Flow 1999; 25: 257-305.
8. Chapman S, Cowlings TG. The mathematical theory of non-uniform gases. Cambridge University Press, Cambridge, 1970.
9. Chen P, Dudukovic MP, Sanyal J. Three-dimensional simulation of bubble column flows with bubble coalescence and breakup. AIChE J 2005; 51: 696-712.
10. Coulaloglou CA, Tavlarides LL. Description of interaction processes in agitated liquid-liquid dispersions. Chem Eng Sci 1977; 32: 1289-1297.
11. Diemer RB, Olson JH. A momentmethodology for coagulation and breakage problems: part 3 - generalized daughter distribution functions. Chem Eng Sci 2002; 57: 4187-4198.
12. Dorao CA. High order methods for the solution of the population balance equation with applications to bubbly flows. PhD thesis, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 2006.
13. Dorao CA, Lucas D, Jakobsen HA. Prediction of the evolution of the dispersed phase in bubbly flow problems. Appl Math Modelling 2008; 32: 1813-1833.
14. Doubliez L. The drainage and rupture of an on-foaming liquid film formed upon bubble impact with a free surface. Int J Multiphase Flow 1991; 17: 783-803.
15. Duineveld PC. Bouncing and coalescence of two bubbles in water. PhD thesis, University of Twente, Twente, The Netherlands, 1994.
16. Ekambara K, Nandakumar K, Joshi JB. CFD simulation of bubble column reactor using population balance. Ind Eng Chem Res 2008; 47: 8505-8516.
17. Fan LS, Tsuchiya K. Bubble wake dynamics in liquids and liquidsolid suspensions (Butterworth-Heinemann series in chemical engineering). Butterworth-Heinemann, Boston, 1990.
18. Frank T, Zwart PJ, Shi JM, Krepper E, Lucas D, Rohde U. Inhomogeneous music model - population balance approach for polydispersed bubbly flows. 2005 Int Conf Nuclear Energy for New Europe, Bled, Slovenia, 2005.
19. Greenberg JB, Silverman I, Tambour Y. On the origins of spray sectional conservation equations. Combust Flame 1993; 93: 90-96.
20. Guido-Lavalle G, Clausse A, Converti J. A contribution to a kinetic formulation of gas-liquid flows. Chem Eng Commun 1993; 120: 99-109.
21. Guido-Lavalle G, Carrica P, Clausse A, Qazi MK. A bubble number density constitutive equation. Nucl Eng Des 1994; 152: 213-224.
22. Herrero VA, Guido-Lavalle G, Clausse A. Non-equilibrium effects in void fraction distribution. Nucl Eng Des 1995; 154: 183-192.
23. Hesketh RP. Bubble breakage in pipeline flow. Chem Eng Sci 1991; 46: 1-9.
24. Hill PJ, Ng KM. Statistics of multiple particle breakage. AIChE J 1996; 1: 289-295.
25. Hirschfelder JO, Curtiss CF, Bird RB. Molecular theory of gases and liquids. John Wiley & Sons, New York, 1964.
26. Howarth WJ. Coalescence of drops in a turbulent flow field. Chem Eng Sci 1964; 19: 33-38.
27. Hsia MA, Tavlarides LL. Simulation analysis of drop breakage, coalescence and micromixing in liquid-liquid stirred tanks. Chem Eng J 1983; 26: 189-199.
28. Hulburt HM, Katz S. Some problems in particle technology - a statistical mechanical formulation. Chem Eng Sci 1964; 19: 555-574.
29. Jakobsen HA. Chemical reactor modeling: multiphase reactive flows. Springer, 2008.
30. Jakobsen HA, Lindborg H, Dorao CA. Modeling of bubble column reactors: progress and limitations. Ind Eng Chem Res 2005; 44: 5107-5151.
31. Kim JW, Chang JH, Lee WK. Inhibition of bubble coalescence by the electrolytes. Korean J Chem Eng 1990; 7: 100-108.
32. Kocamustafaogullari G, Ishii M. Foundation of the interracial area transport equation and its closure relations. Int J Heat Mass Transfer 1995; 38: 481-493.
33. Konno M, Aoki M, Saito S. Scale effect on breakup process in liquidliquid agitated tanks. J Chem Eng Jpn 1983; 16: 312-319.
34. LafiAY, Reyes Jr JN. General particle transport equation. Technical Report DOE/ER/13034-2, OSU-NE-9409. Oregon State University, Corvallis, OR, 1994.
35. Lasheras JC, Eastwood C, Martinez-Bazan C, nes JLM. 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.
36. Laurent F, Massot M. Multi-fluid modelling of laminar polydisperse spray flames: origin, assumptions and comparison of sectional and sampling methods. Combust Theor Model 2001; 5: 537-572.
37. Lehr F, Mewes D. A transport equation for the interfacial area density applied to bubble columns. Chem Eng Sci 2001; 56: 1159-1166.
38. Lehr F, Millies M, Mewes D. Bubble-size distributions and flow fields in bubble columns. AIChE J 2002; 48: 2426-2443.
39. 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.
40. 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.
41. Luo H, Svendsen HF. Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J 1996; 42: 1225-1233.
42. MaaßS, Kraume M. Determination of breakage rates using single drop experiments. Chem Eng Sci 2012; 70: 146-164.
43. Marchetti JM, Patruno LE, Jakobsen HA, Svendsen HF. Mathematical framework for higher order breakage scenarios. Chem Eng Sci 2010; 65: 5881-5886.
44. Marchisio DL, Vigil RD, Fox RO. Quadrature method of moments for aggregationbreakage processes. J Colloid Interface Sci 2003; 258: 322-334.
45. Martinez-Bazan C, Montaes JL, Lasheras JC. On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency. J Fluid Mech 1999a; 401: 157-182.
46. Martinez-Bazan C, Montaes JL, Lasheras JC. On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size pdf of the resulting daughter bubbles. J Fluid Mech 1999b; 401: 183-207.
47. Martinez-Bazan C, Rodriguez-Rodriguez J, Deane GB, Montaes JL, Lasheras JC. Considerations on bubble fragmentation models. J Fluid Mech 2010; 661: 159-177.
48. Millies M, Mewes D. Interfacial area density in bubbly flow. Chem Eng Process 1999; 38: 307-319.
49. Mitre JF, Takahashi RSM, Ribeiro CP, Lage PLC. Analysis of breakage and coalescence models for bubble columns. Chem Eng Sci 2010; 65: 6089-6100.
50. Morel C, Ruyer P, Seiler N, Laviéville JM. Comparison of several models for multi-size bubbly flows on an adiabatic experiment. Int J Multiphase Flow 2010; 36: 25-39.
51. Nayak AK, Borka Z, Patruno LE, Sporleder F, Dorao CA, Jakobsen HA. A combined multifluid-population balance model for vertical gas-liquid bubble driven flows considering bubble column operating conditions. Ind Eng Chem Res 2011; 50: 1786-1798.
52. Patruno LE. Experimental and numerical investigations of liquid fragmentation and droplet generation for gas processing at high pressures. PhD thesis, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 2010.
53. Patruno LE, Dorao CA, Dupuy PM, Svendsen HF, Jakobsen HA. Identification of droplet breakage kernel for population balance modelling. Chem Eng Sci 2009; 64: 638-645.
54. Prince MJ, Blanch HW. Bubble coalescence and break-up in airsparged bubble columns. AIChE J 1990; 36: 1485-1499.
55. Ramkrishna D. Analysis of population balance - IV. The precise connection between Monte Carlo simulation and population balances. Chem Eng Sci 1981; 36: 1203-1209.
56. Ramkrishna D. The status of population balances. Rev Chem Eng 1985; 3: 49-95.
57. Ramkrishna D. Population balances, theory and applications to particulate systems in engineering. Academic Press, San Diego, 2000.
58. Ramkrishna D. On aggregating populations. Ind Eng Chem Res 2004; 43: 441-448.
59. Ramkrishna D, Borwanker JD. A puristic analysis of population balance - I. Chem Eng Sci 1973; 28: 1423-1435.
60. Ramkrishna D, Borwanker JD. A puristic analysis of population balance - II. Chem Eng Sci 1974; 29: 1711-1721.
61. Ramkrishna D, Mahoney AW. Population balance modeling. Promise for the future. Chem Eng Sci 2002; 57: 595-606.
62. Ramkrishna D, Shah BH, Borwanker JD. Analysis of population balance - III. Agglomerating populations. Chem Eng Sci 1976; 31: 435-442.
63. Randolph AD. A population balance for countable entities. Can J Chem Eng 1964; 42: 280-281.
64. Randolph AD, White ET. Modeling size dispersion in the prediction of crystal-size distribution. Chem Eng Sci 1977; 32: 1067-1076.
65. Randolph AD, Larson M.A. Theory of particulate processes - analysis and techniques of continuous crystallization, 2nd ed. Academic Press, San Diego, 1988.
66. Reeks MW. On the dispersion of small particles suspended in an isotropic turbulent fluid. J Fluid Mech 1977; 83: 529-546.
67. Reeks MW. The transport of small discrete particles in inhomogeneous turbulence. J Aerosol Sci 1983; 6: 729-739.
68. Reeks MW. On a kinetic equation for the transport of particles in turbulent flows. Phys Fluids 1991; A3: 446-456.
69. Reeks MW. On the continuum equations for dispersed particles in nonuniform flows. Phys Fluids 1992; A4: 1290-1303.
70. Reeks MW. On the constitutive relations for dispersed particles in nonuniform flows. I: dispersion in a simple shear flow. Phys Fluids 1993; 3: 750-761.
71. Richardson J, Zaki W. Sedimentation and fluidisation: part I. Trans Inst Chem Eng 1954; 32(Suppl. 1): 35-53.
72. Rigopoulos S. Population balance modelling of polydispersed particles in reactive flows. Prog Energy Combust Sci 2010; 36: 412-443.
73. Roghair I, Lau YM, Deen NG, Slagter HM, Baltussen MW, Annaland MVS, Kuipers JAM. On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers. Chem Eng Sci 2011; 66: 3204-3211.
74. Sampson KJ, Ramkrishna D. Particle size correlations in brownian agglomeration closure hypothesis for product density equations. J Coll Int Sci 1986; 43: 410-423.
75. Sha Z, Laari A, Turunen I. Implementation of population balances into multiphase-model in CFD simulation of bubble column. 16th Int Cong Chem Eng, Praha, Czech Republic, Paper E3.2, 2004.
76. Sha Z, Laari A, Turunen I. Multi-phase-multi-size-group model for the inclusion of population balances into the CFD simulation of gas-liquid bubbly flows. Chem Eng Technol 2006; 29: 550-559.
77. Shi J, Zwart P, Frank T, Rohde U, Passer H. Development of a multiple velocity multiple size group model for poly-dispersed multiphase flows. Technical Report Annual Report 2004. Institute of Safety Research, Forschungszentrum Rossendorf, Germany, Dresden, Germany, 2004.
78. Shinnar R, Church JM. Predicting particle size in agitated dispersions. Ind Eng Chem 1960; 52: 253-256.
79. Smoluchowski MV. Drei Vorträgeüber Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Z Angew Phys 1916; 17: 557-571.
80. Sporleder F. Simulation of chemical reactors using the least-squares spectral element method. PhD thesis, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 2011.
81. Sporleder F, Dorao CA, Jakobsen HA. Model based on population balance for the simulation of bubble columns using methods of the least square type. Chem Eng Sci 2011; 66: 3133-3144.
82. Tambour Y. Vaporization of polydisperse fuel sprays in a laminar boundary layer flow: a sectional approach. Combust Flame 1984; 58: 103-114.
83. Tomiyama A. Struggle with computational bubble dynamics. Multiphase Sci Technol 1998; 10: 369-405.
84. Tsouris C, Tavlarides LL. Breakage and coalescence models for drops in turbulent dispersions. AIChE J 1994; 40: 395-406.
85. Valentas KJ, Bilous O, Amundson NR. Analysis of breakage in dispersed phase systems. Ind Eng Chem Fundam 1966; 5: 271-279.
86. Venneker BCH, Derksen JJ, Van den Akker HEA. Population balance modeling of aerated stirred vessels based on CFD. AIChE J 2002; 48: 673-685.
87. White FM. Fluid mechanics. McGraw-Hill, Boston, 1979.
88. Williams FA. Spray combustion and atomization. Phys Fluids 1958; 1: 541-545.
89. Yao BP, Zheng C, Gasche HE, Hofmann H. Bubble behavior and flow structure of bubble columns. Chem Eng Process 1991; 29: 65-75.
90. Zhu Z. The least-squares spectral element method solution of the gas-liquid multi-fluid model coupled with the population balance equation. PhD thesis, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 2009.