Coagulation and fragmentation: Universal steady‐state particle‐size distribution

A population balance model presented describes simultaneous coagulation and fragmentation during shear-induced flocculation. Given sufficient time, a floc-size distribution reaches steady state that reflects the balance between coagulation and fragmentation. The model agrees with experimental data for the evolution of the average floc size. Higher shear shifts the steady-state size distribution to smaller sizes. When the steady-state size distributions obtained at various shear rates are scaled with the average floc size, however, they collapse onto a single line. This indicates that the steady-state floc-size distribution is self-preserving with respect to fluid shear. This distribution is universal for the employed coagulation and fragmentation rates provided that less than 5% (by number) of the particles remain unflocculated. This result is supported with experimental data on shear-induced flocculation of polystyrene particles, although a detailed quantitative comparison is limited by the irregular structure of the flocs.

[1]  K. Higashitani,et al.  Kinetic theory of shear coagulation for particles in a viscous fluid. , 1982 .

[2]  J. D. Boadway.  Dynamics of Growth and Breakage of Alum Floc in Presence of Fluid Shear , 1978 .

[3]  Steven G. Thoma,et al.  Ultrasonic fragmentation of agglomerate powders , 1993 .

[4]  J. C. Godfrey,et al.  Measuring drop size in continuous liquid-liquid mixers , 1989 .

[5]  J. Berg,et al.  Simulation of particle size distribution in an aggregation-breakup process , 1990 .

[6]  Meakin,et al.  Kinetics of coagulation with fragmentation: Scaling behavior and fluctuations. , 1986, Physical review letters.

[7]  P. J. Blatz,et al.  Note on the Kinetics of Systems Manifesting Simultaneous Polymerization-Depolymerization Phenomena , 1945 .

[8]  W. Lick,et al.  The flocculation of fine‐grained sediments in estuarine waters , 1989 .

[9]  A. Golovin.  The Solution of the Coagulation Equation for Raindrops. Taking Condensation into Account , 1963 .

[10]  L. Spielman,et al.  Floc breakage in agitated suspensions: Theory and data processing strategy , 1982 .

[11]  S. Saito,et al.  Scale effect on breakup process in liquid-liquid agitated tanks. , 1983 .

[12]  Mark R. Wiesner,et al.  Kinetics of aggregate formation in rapid mix , 1992 .

[13]  勝美 横井,et al.  nippon kagaku kaishi , 1978 .

[14]  N. Tambo.  Physical aspect of flocculation process—I: Fundamental treatise , 1979 .

[15]  Sotiris E. Pratsinis,et al.  Time-Lag for Attainment of the Self-Preserving Particle Size Distribution by Coagulation , 1994 .

[16]  R. D. Vold,et al.  Flocculation-Deflocculation in Agitated Suspensions. 1. Carbon and Ferric Oxide in Water , 1959 .

[17]  Richard A. Williams,et al.  Direct Measurement of Floc Breakage in Flowing Suspensions , 1994 .

[18]  M. Williams.  An Exact Solution of the Fragmentation Equation , 1990 .

[19]  Ruben D. Cohen.  Steady-state cluster size distribution in stirred suspensions , 1990 .

[20]  L. Spielman,et al.  Kinetics of floc breakage and aggregation in agitated liquid suspensions , 1985 .

[21]  Leonard G. Austin,et al.  Introduction to the mathematical description of grinding as a rate process , 1971 .

[22]  David Jenkins,et al.  Floc Breakup in Turbulent Flocculation Processes , 1972 .

[23]  K. Danov,et al.  Kinetic Model for the Simultaneous Processes of Flocculation and Coalescence in Emulsion Systems , 1994 .

[24]  L. Spielman.  Viscous interactions in Brownian coagulation , 1970 .

[25]  Robert M. Ziff,et al.  On the stability of coagulation--fragmentation population balances , 1989 .

[26]  R. Probstein,et al.  The Effect of Coalescence on the Average Drop Size in Liquid-Liquid Dispersions, , 1976 .

[27]  F. B. Sprow.  Drop size distributions in strongly coalescing agitated liquid‐liquid systems , 1967 .

[28]  M. Hounslow,et al.  A discretized population balance for nucleation, growth, and aggregation , 1988 .

[29]  V. Oles.  Shear-induced aggregation and breakup of polystyrene latex particles , 1992 .

[30]  Karel Antonius Kusters,et al.  The influence of turbulence on aggregation of small particles in agitated vessels , 1991 .

[31]  Sotiris E. Pratsinis,et al.  A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime , 1990 .

[32]  J. Alvarez,et al.  A population balance approach for the description of particle size distribution in suspension polymerization reactors , 1994 .

[33]  PROTEIN PRECIPITATION-ANALYSIS OF PARTICLE SIZE DISTRIBUTION AND KINETICS , 1981 .

[34]  P. C. Kapur,et al.  Self-preserving size spectra of comminuted particles , 1972 .

[35]  David G. Thomas.  Turbulent disruption of flocs in small particle size suspensions , 1964 .

[36]  K. Higashitani,et al.  Turbulent coagulation of particles dispersed in a viscous fluid. , 1983 .

[37]  F. A. Holland,et al.  Liquid Mixing and Processing in Stirred Tanks , 1966 .

[38]  Doraiswami Ramkrishna,et al.  Factors affecting coalescence frequency of droplets in a stirred liquid-liquid dispersion , 1994 .

[39]  Desmond F. Lawler,et al.  The (Relative) Insignificance of G in Flocculation , 1992 .

[40]  Ruben D. Cohen.  The self-similar cluster size distribution in random coagulation and breakup , 1992 .

[41]  L. A. Cutter.  Flow and turbulence in a stirred tank , 1966 .

[42]  P. Saffman,et al.  On the collision of drops in turbulent clouds , 1956, Journal of Fluid Mechanics.

[43]  S. Friedlander,et al.  The coagulation of hydrosols by brownian motion and laminar shear flow , 1964 .

[44]  Lawrence L. Tavlarides,et al.  Description of interaction processes in agitated liquid-liquid dispersions , 1977 .

[45]  L. Glasgow,et al.  Simulation of aggregate growth and breakage in stirred tanks , 1987 .

[46]  A. Rushton,et al.  Floc breakage: The dynamic response of the particle size distribution in a flocculated suspension to a step change in turbulent energy dissipation , 1987 .

[47]  N. Titchener-Hooker,et al.  Chapter 1 – Turbulent aggregation and breakup of particles in liquids in stirred vessels , 1993 .