Aggregation Kinetics in Sedimentation: Effect of Diffusion of Particles

N. V. Brilliantov; Бриллиантов Н. В.; R. R. Zagidullin; Загидуллин Р. Р.; S. A. Matveev; Матвеев С. А.; A. P. Smirnov; Смирнов А. П.

doi:10.31857/S0044466923040051

Aggregation Kinetics in Sedimentation: Effect of Diffusion of Particles

Authors: Brilliantov N.V.¹^,2, Zagidullin R.R.¹^,3, Matveev S.A.³^,4, Smirnov A.P.³
Affiliations:
1. Skolkovo Institute of Science and Technology
2. University of Leicester
3. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University
4. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences
Issue: Vol 63, No 4 (2023)
Pages: 629-638
Section: Mathematical physics
URL: https://rjonco.com/0044-4669/article/view/664864
DOI: https://doi.org/10.31857/S0044466923040051
EDN: https://elibrary.ru/IPBWKP
ID: 664864

Cite item

Full Text

Open Access
Restricted Access

Access granted
Restricted Access

Subscription Access

Abstract

The aggregation kinetics of settling particles is studied theoretically and numerically using the advection–diffusion equation. Agglomeration caused by these mechanisms (diffusion and advection) is important for both small particles (e.g., primary ash or soot particles in the atmosphere) and large particles of identical or close size, where the spatial inhomogeneity is less pronounced. Analytical results can be obtained for small and large Péclet numbers, which determine the relative importance of diffusion and advection. For small numbers (spatial inhomogeneity is mainly due to diffusion), an expression for the aggregation rate is obtained using an expansion in terms of Péclet numbers. For large Péclet numbers, when advection is the main source of spatial inhomogeneity, the aggregation rate is derived from ballistic coefficients. Combining these results yields a rational approximation for the whole range of Péclet numbers. The aggregation rates are also estimated by numerically solving the advection–diffusion equation. The numerical results agree well with the analytical theory for a wide range of Péclet numbers (extending over four orders of magnitude).

Keywords

coagulation kernel, spatial inhomogeneity, Péclet number

References

Vowinckel B., Withers J., Luzzatto-Fegiz P., Meiburg E. Settling of cohesive sediment: particle-resolved simulations // J. Fluid Mech. 2019. V. 858. P. 5–44.
Fischer A., Chatterjee A., Speck T. Aggregation and sedimentation of active Brownian particles at constant affinity // J. Chem. Phys. 2019. V. 150. № 6. P. 064910.
Yang Y.-J., Kelkar A.V., Corti D.S., Franses E.I. Effect of Interparticle Interactions on Agglomeration and Sedimentation Rates of Colloidal Silica Microspheres // Langmuir 2016. V. 32. № 20 P. 5111–5123.
Hongsheng Ch., Wenwei L., Zhiwei Ch., Zhong Zh. A numerical study on the sedimentation of adhesive particles in viscous fluids using LBM-LES-DEM // Powder Technol. 2021. V. 391. P. 467–478.
Whitmer J.K., Luijten E. Sedimentation of aggregating colloids // J. Chem. Phys. 2011. V. 134. P. 034510.
Pinsky M., Khain A., Shapiro M. Collision Efficiency of Drops in a Wide Range of Reynolds Numbers: Effects of Pressure on Spectrum Evolution // J. Atmos. Sci. 2001. V. 58. P. 742–766.
Khodzher T.V., et al. Study of Aerosol Nano-and Submicron Particle Compositions in the Atmosphere of Lake Baikal During Natural Fire Events and Their Interaction with Water Surface // Wat. Air and Soil Poll. 2021. V. 232. P. 266.
Zhamsueva G., et al. Studies of the Dispersed Composition of Atmospheric Aerosol and Its Relationship with Small Gas Impurities in the Near-Water Layer of Lake Baikal Based on the Results of Ship Measurements in the Summer of 2020 // Atmosphere. 2022. V. 13. P. 139.
Shahad H.A.K. An experimental investigation of soot particle size inside the combustion chamber of a diesel engine // Energy Convers. Manag. 1989. V. 29. P. 141–149.
Krapivsky P.L., Redner A., Ben-Naim E. A Kinetic View of Statistical Physics. Cambridge: Cambridge University Press, 2010.
Leyvraz F. Scaling theory and exactly solved models in the kinetics of irreversible aggregation // Phys. Rep. 2003. V. 383. P. 95–212.
Smoluchowski M.V. Attempt for a mathematical theory of kinetic coagulation of colloid solutions // Z. Phys. Chem. 1917. V. 92. P. 265–271.
Higashitani K., Ogawa R., Hosokawa G., Matsuno Y. Kinetic Theory of Shear Coagulation for Particles in a Viscous Fluid // J. Chem. Eng. Japan. 1982. V. 15. № 4. P. 299–304.
Saffman P.F., Turner N.F. On the collision of drops in turbulent clouds // J. Fluid Mech. 1956. V. 1. № 1. P. 16–30.
Falkovich G., Fouxon A., Stepanov, M. Acceleration of rain initiation by cloud turbulence // Nature. 2002. V. 419. P. 151.
Falkovich G., Stepanov M.G., Vucelja M. Rain Initiation Time in Turbulent Warm Clouds // J. Appl. Meteor. Climatol. 2006. V. 45. P. 591.
van de Ven T.G.M., Mason S.G. The microrheology of colloidal dispersions VIII. Effect of shear on perikinetic doublet formation // Colloid Polym. Sci. 1977. V. 255. № 8. P. 794–804.
Melik D.H., Fogler H.S. Effect of gravity on Brownian flocculation // J. Colloid Interface Sci. 1984. V. 101. № 1. P. 84–97.
Feke D.L., Scjowalter W.R. The effect of Brownian diffusion on shear-induced coagulation of colloidal dispersions // J. Fluid Mech. 1983. V. 133. P. 17–35.
van Kampen N. Stochastic Processes in Physics and Chemistry. Amsterdam: Elsevier, 1992.
Tikhonov A.N., Samarsky A.A. Equations of mathematical physics. New York: Dover Publications, 2013.
Andrianov I., Shatrov A. Padé Approximants, Their Properties, and Applications to Hydrodynamic Problems // Symmetry. 2021. V. 13. P. 1869.
Brezinski C. History of Continued Fractions and PadГ© Approximants. Berlin: Springer, 1991.
Reed C.C., Anderson J.L. Hindered settling of a suspension at low Reynolds number // AIChE J. 1980. V. 26. № 5. P. 816–827.
Samarskii A.A., Vabishchevich P.N Computational heat transfer. New York: John Wiley and Sons, 1995.
Davis T.A. Algorithm 832 // ACM Trans. Math. Softw. 2004 V. 30. № 2 P. 196–199.