Abstract
The chemical engineering community began the first efforts that can be associated with the concepts of the population balance in the early 1960s.
Keywords
- Population Balance Equation (PBE)
- Daughter Bubbles
- Daughter Particles
- Source Term Closure
- Collision Density
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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- 1.
Smoluchowski [121], Williams and Loyalka [136], Friedlander [27] and Kolev [48] (p. 170), among others, considered very small particles and interpreted the cross sectional area in a slightly different way representing an extrapolation of the concepts in kinetic theory of gases. In these analyzes the traveling particle is treated as a point in space without cross sectional area, thus the effective area becomes equal to the cross sectional area of the stagnant particle. This alternative cross sectional area approximation becomes: \(\sigma '_{A_T} = \frac{1}{4} \pi d^2\).
- 2.
The collision tube concept is familiar from the kinetic theory of gases. Consider a particle in such a cube, moving with a relative speed with respect to the other particles which are fixed. The particle in the tube sweeps a volume per unit time (\(\mathrm{m}^3 \mathrm{s}^{-1}\)). Venneker [132] named the rate of volume swept by the particle for the effective swept volume rate. Henceforth this name is used referring to this quantity.
- 3.
- 4.
Luo and Svendsen [80] did not distinguish between the experimentally determined relation (9.32) and the Kolmogorov structure function (9.15). In their work the theoretical parameter value \(C\) was calculated as \(C=(3/5)\varGamma (1/3)C_k\approx (3/5)\times 2.6789 \times 1.5 = 2.41\). It follows that their mean droplet velocity estimate is \(\bar{v}_\text {drops} \approx (\frac{8 \overline{\delta v^2 (d)}}{3\pi })^{1/2} = \sqrt{8 \times C/(3\pi )} (\varepsilon d)^{1/3} =\sqrt{2.046}(\varepsilon d)^{1/3}\).
- 5.
Politano et al. [98] adopted the kernel functionality of Luo and Svendsen but modified the definition of the effective collision cross-sectional area, \(\sigma _{A_T} = \frac{\pi }{4} \left( \frac{d_i + \lambda }{2}\right) ^2\), in accordance with other work in nuclear engineering (e.g., Kolev [48], p. 168).
- 6.
Luo and Svendsen [80] did not distinguish between the value of the theoretical parameter \(C\) in the structure function relation (9.15) and the empirical parameter \(\beta = 2.0\) in the droplet rms velocity relation (9.32) when calculating the mean eddy velocity from (9.39). To reproduce the parameter value in the eddy number density relation used by Luo and Svendsen, the value of the \(\beta \) parameter is computed using the structure function relation (9.15) instead of the empirical relation (9.32). That is, they let \(\beta = \frac{8 C}{3\pi }\), and \(C=\frac{3}{5}\varGamma (1/3)C_k \approx \frac{3}{5}\times 2.6789\times 1.5=2.41\). The parameter in the eddy number density of eddies was thus approximated as: \(\frac{9 C_k}{C 2^{5/3} \pi ^{2/3}} = 0.822\).
- 7.
Luo and Svendsen used the parameter values \(C=\frac{3}{5}\varGamma (1/3)C_k = 2.41\), and \((8C/3\pi )\approx 2.045\), hence their parameter value in (9.45) becomes \(\frac{\pi }{4} (0.822) \sqrt{2.045} \approx 0.923\).
- 8.
Notice that this efficiency function is not necessary volume or mass conservative as Luo and Svendsen [80] did consider the breakage efficiency function being equal to the kinetic energy distribution function. It would probably be better to consider the breakage distribution function purely proportional to the empirical kinetic energy distribution function, and determine the probability constant by requiring bubble volume or mass conservation within the breakage process.
- 9.
Luo [79] employed (9.16) with \(C\approx 2.41\) and obtained \(\bar{e}(d_i,\lambda ) = \rho _c \frac{\pi }{6} \lambda ^3 \frac{\bar{v}^2_\lambda }{2} = \rho _c \frac{C \pi }{12} \lambda ^{11/3} \varepsilon ^{2/3} = \rho _c \frac{C\pi }{12} \xi ^{11/3} d_i^{11/3} \varepsilon ^{2/3}\). Moreover, with this parameter value and bubble velocity estimate the critical energy ratio becomes \(\chi _{cr} \approx e_s (d_i,d_j)/\bar{e}(d_i,\lambda ) = 12C_f\sigma _I/(C \rho _c \varepsilon ^{2/3} d_i^{5/3} \xi ^{11/3})\).
- 10.
- 11.
A local space dependency of the \(P_B\)-function is retained as the integration is over a differential microscopic volume \(dV_{r'}\), thus the integrated \(P_B\)-function varies locally in space \(\mathbf {r}\) but the \(P_B\)-function does not differ over the microscopic differential volume \(dV_{r'}\) of the different particles.
- 12.
A local space dependency of the \(\kappa \)-function is retained as the integration is over a differential microscopic volume \(dV_{r'}\), thus the \(\kappa \)-function varies locally in space \(\mathbf {r}\) but the \(\kappa \)-function does not differ over the microscopic differential volume \(dV_{r'}\) of the different particles.
References
Angelidou C, Psimopoulos M, Jameson GJ (1979) Size distribution functions of dispersions. Chem Eng Sci 34(5):671–676
Azbel D (1981) Two-phase flows in chemical engineering. Cambridge Univerity Press, Cambride
Azbel D, Athanasios IL (1983) A mechanism of liquid entrainment. In: Cheremisinoff N (ed) Handbook of fluids in motion. Ann Arbor Science Publishers, Ann Arbor, p 473
Barrow JD (1981) Coagulation with fragmentation. J Phys A: Math Gen 14:729–733
Batchelor GK (1953) The theory of homogeneous turbulence. Cambridge University Press, Cambridge
Batchelor GK (1956) The theory of homogeneous turbulence. Cambridge University Press, Cambridge
Batchelor GK (1982) The theory of homogeneous turbulence. Cambridge University Press, Cambridge
Bertola F, Grundseth J, Hagesaether L, Dorao C, Luo H, Hjarbo KW, Svendsen HF, Vanni M, Baldi G, Jakobsen HA (2005) Numerical analysis and experimental validation of bubble size distribution in two-phase bubble column reactors. Multiph Sci Technol 17(1–2):123–145
Brenn G, Braeske H, Durst F (2002) Investigation of the unsteady two-phase flow with small bubbles in a model bubble column using phase-doppler anemometry. Chem Eng Sci 57(24):5143–5159
Buwa VV, Ranade VV (2002) Dynamics of gas-liquid flow in a rectangular bubble column: experimental and single/multigroup CFD simulations. Chem Eng Sci 57(22–23):4715–4736
Carrica PM, Drew D, Bonetto F, Lahey RT Jr (1999) A polydisperse model for bubbly two-phase flow around a surface ship. Int J Multiph Flow 25(2):257–305
Chen P, Dudukovic MP, Sanyal J (2005) Three-dimensional simulation of bubble column flows with bubble coalescence and breakup. AIChE J 51(3):696–712
Chen P, Sanyal J, Dudukovic MP (2005) Numerical simulation of bubble columns: effect of different breakup and coalescence closures. Chem Eng Sci 60:1085–1101
Cheng J, Yang C, Mao Z-S, Zhao C (2009) CFD modeling of nucleation, growth, aggregation, and breakage in continuous precipitation of barium sulfate in a stirred tank. Ind Eng Chem Res 48:6992–7003
Chester AK (1991) The modelling of coalescence processes in fluid-fluid dispersions: a review of current understanding. Trans IchemE 69(A):259–270
Colella D, Vinci D, Bagatin R, Masi M, Bakr EA (1999) A study on coalescence and breakage mechanisms in three different bubble columns. Chem Eng Sci 54(21):4767–4777
Coulaloglou CA, Tavlarides LL (1977) Description of interaction processes in agitated liquid–liquid dispersions. Chem Eng Sci 32(11):1289–1297
Danckwerts PV (1953) Continuous flow systems: distribution of residence times. Chem Eng Sci 2(1):1–18
Dorao CA (2006) High order methods for the solution of the population balance equation with applications to bubbly flows. Dr ing thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim
Dorao CA, Lucas D, Jakobsen HA (2007) Prediction of the evolution of the dispersed phase in bubbly flow problems. Appl Math Model, Accepted for publication
Doubliez L (1991) The drainage and rupture of a non-foaming liquid film formed upon bubble impact with a free surface. Int J Multiph Flow 17(6):783–803
Drazin PG, Reid WH (1981) Hydrodynamic stability. Cambridge Univerity Press, Cambridge
Eastwood CD, Armi L, Lasheras JC (2004) The breakup of immersible fluids in turbulent flows. J Fluid Mech 502:309–333
Fleischer C, Bierdel M, Eigenberger G (1994) Prediction of bubble size distributions in G/L-contactors with population balances. In: Proceedings of 3rd German/Japanese symposium on bubble columns, Schwerte, Germany, June 13–15, pp 229–235
Falola A, Borissova A, Wang XZ (2013) Extended method of moment for general population balance models including size dependent growth rate, aggregation and breakage kernels. Comput Chem Eng 56:1–11
Frank T, Zwart PJ, Shi J-M, Krepper E, Lucas D, Rohde U (2005) Inhomogeneous MUSIG model—a population balance approach for polydispersed bubbly flows. In: International conference on nuclear energy for new Europe 2005, Bled, Slovenia, 5–8 September
Friedlander SK (2000) Smoke, dust, and haze: fundamentals of aerosol dynamics, 2nd edn. Oxford University Press, New York
Guido Lavalle G, Carrica PM, Clausse A, Qazi MK (1994) A bubble number density constitutive equation. Nucl Eng Des 152:213–224
Hagesaether L, Jakobsen HA, Svendsen HF (1999) Theoretical analysis of fluid particle collisions in turbulent flow. Chem Eng Sci 54(21):4749–4755
Hagesaether L, Jakobsen HA, Hjarbo K, Svendsen HF (2000) A coalescence and breakup module for implementation in CFD-codes. Comput Aided Chem Eng 8:367–372
Hagesaether L, Jakobsen HA, Svendsen HF (2002) A model for turbulent binary breakup of dispersed fluid particles. Chem Eng Sci 57(16):3251–3267
Hagesaether L, Jakobsen HA, Svendsen HF (2002) Modeling of the dispersed-phase size distribution in bubble columns. Ind Eng Chem Res 41(10):2560–2570
Hagesaether L (2002) Coalescence and break-up of drops and bubbles. Dr ing thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim
Havelka P, Gotaas C, Jakobsen HA, Svendsen HF (2004) Droplet formation and interactions under normal and high pressures. In: Proceedings at the 5th international conference on multiphase flow, ICMF’04, Yokohama, Japan, May 30–June 4
Hesketh RP, Etchells AW, Russell TWF (1991) Experimental observations of bubble breakage in turbulent flows. Ind Eng Chem Res 30(5):835–841
Hesketh RP, Etchells AW, Russell TWF (1991) Bubble breakage in pipeline flows. Chem Eng Sci 46(1):1–9
Himmelblau DM, Bischoff KB (1968) Process analysis and simulation: deterministic systems. Wiley, New York
Hinze JO (1955) Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J 1(3):289–295
Hulburt HM, Katz S (1964) Some problems in particle technology: a statistical mechanical formulation. Chem Eng Sci 19(8):555–574
Human HJ, Enkevort WJP, Bennema P (1982) In: Jancic SJ, deJong EJ (eds) Industrial crystallization 81. North Holland Publishing Co., Amesterdam, p 387
Jakobsen HA, Lindborg H, Dorao CA (2005) Modeling of bubble column reactors: progress and limitations. Ind Eng Chem Res 44:5107–5151
Kamp AM, Chesters AK, Colin C, Fabre J (2001) Bubble coalescence in turbulent flows: a mechanistic model for turbulence-induced coalescence applied to microgravity bubbly pipe flow. Int J Multiph Flow 27(8):1363–1396
Klaseboer E, Chevallier JP, Masbernat O, Gourdon C (1998) Drainage of the liquid film between drops colliding at constant approach velocity. In: Proceedings of the 3rd international conference on multiphase flow, ICMF98, Lyon, France, 8–12 June
Klaseboer E, Chevallier JP, Gourdon C, Masbernat O (2000) Film drainage between colliding drops at constant approach velocity: experimental and modeling. J Colloide Interface Sci 229(1):274–285
Kocamustafaogullari G, Ishii M (1995) Foundation of the interfacial area transport equation and its closure relations. Int J Heat Mass Transfer 38(3):481–493
Konno M, Aoki M, Saito S (1980) Simulations model for break-up process in an agitated tank. J Chem Eng Jpn 13:67–73
Kolev NI (1993) Fragmentation and coalescence dynamics in multi-phase flows. Exp Thermal Fluid Sci 6(3):211–251
Kolev NI (2002) Multiphase flow dynamics 2: mechanical and thermal interactions. Springer, Berlin
Kolmogorov AN (1941) Local structure of turbulence in incompressible viscous fluid for very large reynolds number. Dokl Akad Nauk SSSR 30:301–306
Kolmogorov AN (1949) On the breakage of drops in a turbulent flow. Dokl Akad Navk SSSR 66:825–828
Krishna R, Urseanu MI, van Baten JM, Ellenberger J (1999) Influence of scale on the hydrodynamics of bubble columns operating in the churn-turbulent regime: experiments vs Eulerian simulations. Chem Eng Sci 54(21):4903–4911
Krishna R, van Baten JM, Urseanu MI (2000) Three-phase Eulerian simulations of bubble column reactors operating in the churn-turbulent regime: a scale up strategy. Chem Eng Sci 55(16):3275–3286
Krishna R, van Baten JM (2001) Scaling up bubble column reactors with the aid of CFD. Inst Chem Eng Trans IChemE 79(A3):283–309
Krishna R, van Baten JM (2001) Eulerian simulations of bubble columns operating at elevated pressures in the churn turbulent flow regime. Chem Eng Sci 56(21–22):6249–6258
Kuboi R, Komasawa I, Otake T (1972) Behavior of dispersed particles in turbulent liquid flow. J Chem Eng Jpn 5:349–355
Kuboi R, Komasawa I, Otake T (1972) Collision and coalescence of dispersed drops in turbulent liquid flow. J Chem Eng Jpn 5:423–424
Laakkonen M, Moilanen P, Alopaeus V, Aittamaa J (2007) Modelling local bubble size distributions in agitated vessels. Chem Eng Sci 62:721–740
Laari A, Turunen I (2003) Experimental determination of bubble coalescence and break-up rates in a bubble column reactor. Can J Chem Eng 81(3–4):395–401
Lafi AY, Reyes JN (1994). General particle transport equations. Final Report OSU-NE-9409. Department of Nuclear Engineering, Oregon State University
Lasheras JC, Eastwood C, Martínez-Bazán C, Montañés JL (2002) A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow. Int J Multiph Flow 28(2):247–278
Lathouwers D, Bellan J (2000), Modeling of dense gas–solid reactive mixtures applied to biomass pyrolysis in a fluidized bed. In: Proceedings of the 2000 U.S. DOE hydrogen program, Review. NREL/CP-570-28890
Lathouwers D, Bellan J (2000) Modeling and simulation of bubbling fluidized beds containing particle mixtures. Proc Combust Inst 28:2297–2304
Lathouwers D, Bellan J (2001) Modeling of biomass pyrolysis for hydrogen production: the fluidized bed reactor. In: Proceedings of the 2001 U.S. DOE hydrogen program, Review. NREL/CP-570-30535
Lathouwers D, Bellan J (2001) Yield optimization and scaling of fluidized beds for tar production from biomass. Energy Fuels 15:1247–1262
Lathouwers D, Bellan J (2001) Modeling of dense gas–solid reactive mixtures applied to biomass pyrolysis in a fluidized bed. Int J Multiph Flow 27:2155–2187
Laurent F, Massot M (2001) Multi-fluid modeling of laminar polydisperse spray flames: origin, assumptions and comparison of sectional and sampling methods. Combust Theor Model 5:537–572
Laurencot P, Mischler S (2002) The continuous coagulation–fragmentation equations with diffusion. Arch Ration Mech Anal 162:45–99
Laurencot P, Mischler S (2004) Modeling and computational methods for kinetic equations. Birkhäuser, Boston
Lee C-K, Erickson LE, Glasgow LA (1987) Bubble breakup and coalescence in turbulent gas–liquid dispersions. Chem Eng Comm 59(1–6):65–84
Lehr F, Mewes D (2001) A transport equation for the interfacial area density applied to bubble columns. Chem Eng Sci 56(3):1159–1166
Lehr F, Millies M, Mewes D (2002) Bubble-size distributions and flow fields in bubble columns. AIChE J 48(11):2426–2443
Levich VG (1962) Physicochemical hydrodynamics. Prentice Hall, Englewood Cliffs
Liao Y, Lucas D (2009) A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem Eng Sci 64:3389–3406
Liao Y, Lucas D (2010) A literature review on mechanisms and models for the coalescence process of fluid particles. Chem Eng Sci 65:2851–2864
Litster JD, Smit DJ, Hounslow MJ (1995) Adjustable discretized population balance for growth and aggregation. AIChE J 41:591–603
Lo S (1996) Application of population balance to CFD modelling of bubbly flow via the MUSIG model. AEA Technology, AEAT-1096
Lo S (2000) Some recent developments and applications of CFD to multiphase flows in stirred reactors. In: Proceedings of AMIF-ESF workshop: computing methods for two-phase flow. Aussois, France, 12–14 January
Lo S (2000) Application of population balance to CFD modeling of gas–liquid reactors. In: Proceedings of trends in numerical and physical modelling for industrial multiphase flows, Corse, 27–29 September
Luo H (1993) Coalescence, break-up and liquid circulation in bubble column reactors. Dr ing Thesis, The Norwegian Institute of Technology, Trondheim
Luo H, Svendsen HF (1996) Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J 42(5):1225–1233
Marchisio DL, Vigil RD, Fox RO (2003) Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems. Chem Eng Sci 58(15):3337–3351
Marchisio DL, Vigil RD, Fox RO (2003) Quadrature method of moments for aggregation-breakage processes. J Colloid Interface Sci 258(2):322–334
Marrucci G (1969) A theory of Coalescence. Chem Eng Sci 24(6):975–985
Martínez-Bazán C, Montañés JL, Lasheras JC (1999) On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency. J Fluid Mech 401:157–182
Martínez-Bazán C, Montañés JL, Lasheras JC (1999) 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 401:183–207
Martínez-Bazán C, Rodrtíguez-Rodrtíguez J, Deane GB, Montañés JL, Lasheras JC (2010) Considerations on bubble fragmentation models. J Fluid Mech 661:159–177
Melzak ZA (1957) A scalar transport equation. Trans Am Math Soc 85:547–560
Melzak ZA (1957) A scalar transport equation II. Mich Math J 4(3):193–206
Millies M, Mewes D (1999) Interfacial area density in bubbly flow. Chem Eng Process 38(4–6):307–319
Mitre JF, Takahashi RSM, Ribeiro CP Jr, Lage PLC (2010) Analysis of breakage and coalescence models for bubble columns. Chem Eng Sci 65:6089–6100
Nayak AK, Borka Z, Petruno LE, Sporleder F, Jakobsen HA, Dorao CA, (2011) A combined multifluid-population balance model for vertical gas–liquid bubble driven flows considering bubble column operating conditions. Ind Eng Chem Res 50:1786–1798
Olmos E, Gentric C, Vial C, Wild G, Midoux N (2001) Numerical simulation of multiphase flow in bubble column reactors. Influence of bubble coalescence and breakup. Chem Eng Sci 56(21–22):6359–6365
Olmos E, Gentric C, Midoux N (2003) Numerical description of flow regime transitions in bubble column reactors by multiple gas phase model. Chem Eng Sci 58(10):2113–2121
Oolman TO, Blanch HW (1986) Bubble coalescence in stagnant liquid. Chem Eng Commun 43(4–6):237–261
Orme M (1997) Experiments on droplet collisions, bounce, coalescence and disruption. Prog Energy Combust Sci 23:65–79
Patruno LE (2010) Experimental and numerical investigations of liquid fragmentation and droplet generation for gas processing at high pressures. PhD thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim
Pilon L, Fedorov AG, Ramkrishna D, Viskanta R (2004) Bubble transport in three-dimensional laminar gravity-driven flow—mathematical formulation. J Non-Cryst Solids 336(2):71–83
Politano MS, Carrica PM, Baliño JL (2003) About bubble breakup models to predict bubble size distributions in homogeneous flows. Chem Eng Comm 190(3):299–321
Pope SB (2001) Turbulent flows. Cambridge University Press, Cambridge
Prasher CL (1987) Crushing and grinding process handbook. Wiley, Chichester
Present RD (1958) Kinetic theory of gases. McGraw-Hill, New York
Prince MJ, Blanch HW (1990) Bubble coalescence and break-up in air-sparged bubble columns. AIChE J 36(10):1485–1499
Ramkrishna D (1985) The status of population balances. Revs Chem Eng 3:49–97
Ramkrishna D (2000) Population balances: theory and applications to particulate systems in engineering. Academic Press, San Diego
Randolph AD (1964) A population balance for countable entities. Can J Chem Eng 42(6):280
Randolph AD, Larson MA (1988) Theory of particulate processes: analysis and techniques of continuous crystallization, 2nd edn. Academic Press Inc, Harcourt Brace Jovanovich, Publishers, San Diego
Reyes Jr JN (1989) Statistically derived conservation equations for fluid particle flows. Proceedings of ANS Winter Meeting. Nuclear Thermal Hydraulics, 5th Winter Meeting
Population balance modelling of polydispersed particles in reactive flows. Prog Energy Combust Sci 36:412–443
Risso F, Fabre J (1998) Oscillations and breakup of a bubble immersed in a turbulent field. J Fluid Mech 372:323–355
Rodríguez-Rodríguez J, Martínez-Bazán C, Montañes JL (2003) A novel particle tracking and break-up detection algorithm: application to the turbulent break-up of bubbles. Meas Sci Technol 14(8):1328–1340
Ross SL, Curl RL (1973) Measurement and models of the dispersed phase mixing process. Proceedings of 4th joint chemical engineering conference, Paper 29b, Symposium Series 139, AIChE Press, Vancouver, Canada, 9–12 September
Saboni A, Gourdon C, Chesters AK (1999) The influence of inter-phase mass transfer on the drainage of partially-mobile liquid films between drops undergoing a constant interaction force. Chem Eng Sci 54(4):461–473
Saboni A, Alexandrova S, Gourdon C, Chesters AK (2002) Inter-drop coalescence with mass transfer: comparison of the approximate drainage models with numerical results. Chem Eng J 88(1–3):127–139
Sha Z, Laari A, Turunen I (2004) Implementation of population balance into multiphase-model in CFD simulation of bubble column. Proceedings of the 16th International Congress of Chemical Engineering, Praha, Czech Republic (paper E3.2)
Sha Z, Laari A, Turunen I (2006) Multi-phase-multi-size-group model for the inclusion of population balances into the CFD simulation of gas–liquid bubbly flows. Chem Eng Technol 29(5):550–559
Shah YT, Kelkar BG, Godbole SP, Deckwer W-D (1982) Design parameter estimations for bubble column reactors. AIChE J 28(3):353–379
Shi J, Zwart P, Frank T, Rohde U, Prasser H, (2004). Development of a multiple velocity multiple size group model for poly-dispersed multiphase flows. Annual Report. Institute of Safety Research. Forschungszentrum Rossendorf, Germany
Simonin O (1996) Combustion and turbulence in two-phase flows. von Karman Lecture Series 1996–02, von Karman Institute for Fluid Dynamics
Smoluchowski M (1916) Drei vortrage uber diffusion, Brownsche molekularbewegung und koagulation von kolloidteilchen. Phys Z 17:557–585
Smoluchowski M (1917) Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen. Z Physik Chem 92:129–168
Smoluchowski M (1918) Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen. Z Physik Chem, Leipzig Band XCII, pp 129–168
Solsvik J, Tangen S, Jakobsen HA (2013) On the evaluation of breakage kernels for fluid particles. Accepted for publication in Rev Chem Eng
Solsvik J, Jakobsen HA (2013) On the solution of the population balance equation for bubbly flows using the high-order least squares method: implementation issues
Sporleder F (2011) Simulation of chemical reactors using the least-squares spectral element method. PhD thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim
Sporleder F, Borka Z, Solsvik J, Jakobsen HA (2012) On the population balance equation. Rev Chem Eng 28:149–169
Stewart CW (1995) Bubble interaction in low-viscosity liquids. Int J Multiphase Flow 21(6):1037–1046
Thomas RM (1981) Bubble coalescence in turbulent flows. Int J Multiphase Flow 7(6): 709–717
Tsouris C, Tavlarides LL (1994) Breakage and coalescence models for drops in turbulent dispersions. AIChE J 40:395–406
Valentas KJ, Bilous O, Amundson NR (1966) Analysis of breakage in dispersed phase systems. I & EC Fundam 5(2):271–279
Valentas KJ, Amundson NR (1966) Breakage and coalescence in dispersed phase systems. I & EC Fundam 5(4):533–542
Vanni M (2000) Approximate population balance equation for aggregation-brakage processes. J Colloid Interface Sci 221(2):143–160
Venneker BCH, Derksen JJ, van den Akker HEA (2002) Population balance modeling of aerated stirred vessels based on CFD. AIChE J 48(4):673–685
Wang T, Wang J, Jin Y (2003) A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow. Chem Eng Sci 58(20):4629–4637
Wang T, Wang J, Jin Y (2005) Population balance model for gas–liquid flows: influence of bubble coalescence and breakup models. Ind Eng Chem Res 44:7540–7549
Williams FA (1985) Combustion theory:The fundamental theory of chemically reacting flow systems, 2nd edn. Benjamin/Cummings, Menlo Park
Williams MMR, Loyalka SK (1991) Aerosol science theory and practice: with special applications to the nuclear industry. Pergamon Press, Oxford
Zhu Z (2009) The least-squares spectral element method solution of the gas-liquid multi-fluid model coupled with the population balance equation. Dr ing thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim
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Jakobsen, H.A. (2014). The Population Balance Equation. In: Chemical Reactor Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-05092-8_9
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