Clarification of a Common Misunderstanding of Collision Frequencies in the Smoluchowski Equation
Publication: Journal of Environmental Engineering
Volume 137, Issue 4
Abstract
Flocculation is universally modeled on the basis of the seminal work of M. von Smoluchowski. The key equation is a particle number balance that expresses the net rate of formation of particles of a given type as the difference between the rate at which such particles are formed by collisions of smaller particles and the rate at which they are destroyed by collisions with other particles to form larger ones. The equation is commonly presented in a concise form in which the rates of formation and destruction of particles of a given size are each represented by a single summation containing the term , in which = collision frequency function, and and = number concentrations of type- and type- particles, respectively. Although the product has been interpreted as the rate of collisions between and particles in several important publications, that interpretation is correct only for collisions between unlike particles; the collision rate between like particles is (i.e., ). This fact alters the interpretation of both the formation and loss terms in the Smoluchowski equation for collisions between like particles. This technical note clarifies the basis for those terms.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The writer acknowledges the helpful technical input of Peter Mackenzie in the preparation of this paper.
References
Adler, P. M. (1981). “Heterocoagulation in shear flow.” J. Colloid Interface Sci., 83(1), 106–115.
Camp, T., and Stein, P. C. (1943). “Velocity gradients and internal work in fluid motion.” J. Boston Soc. Civ. Eng., 30, 219–237.
Clark, M. M. (1996). Transport modeling for environmental engineers and scientists, Wiley-Interscience, New York.
Crittenden, J. C., Trussell, R. R., Hand, D. W., Howe, K. J., and Tchobanoglous, G. (2005). Water treatment—Principles and design, 2nd Ed., Wiley, New York.
Delichatsios, M. A., and Probstein, R. F. (1975). “Coagulation in turbulent flow: Theory and experiment.” J. Colloid Interface Sci., 51(3), 394–405.
Friedlander, S. K. (2000). Smoke, dust, and haze: Fundamentals of aerosol dynamics, 2nd Ed., Oxford Univ. Press, New York.
Han, M., and Lawler, D. F. (1992). “The (relative) insignificance of in flocculation.” J. Am. Water Works Assoc., 84(10), 79–91.
Hounslow, M. J., Ryall, R. I., and Marshall, V. R. (1988). “A discretized population balance for nucleation, growth, and aggregation.” AIChE J., 34(11), 1821–1832.
Ives, K. J. (1978). “Rate theories.” The scientific basis of flocculation, K. J. Ives, ed., Sijthoff & Noordhoff, Alphen aan den Rijn (The Netherlands), 37–61.
Kumar, S., and Ramkrishna, D. (1996). “On the solution of population balance equations by discretization—I. A fixed pivot technique.” Chem. Eng. Sci., 51(8), 1311–1332.
Lawler, D. F., O’Melia, C. R., and Tobiason, J. E. (1980). “Integral water treatment plant design: From particle size to plant performance.” Chapter 16, Particulates in water, advances in chemistry series 189, M. C. Kavanaugh and J. O. Leckie, eds., American Chemical Society, Washington, DC.
Letterman, R., Amirtharajah, A., and O’Melia, C. R. (1998). “Coagulation processes: Destabilization, mixing, and flocculation.” Chapter 6, Water quality and treatment, 5th Ed., American Water Works Association, ed., McGraw-Hill, New York, 6.1–6.66.
O’Melia, C. R. (1989). “Particle-particle interactions in aquatic systems.” Colloids Surf., 39(1-3), 255–271.
Pierce, R. (2010). “Combinations and permutations.” MathsIsFun, 〈http://www.mathsisfun.com/combinatorics/combinations-permutations.html〉 (Sep. 2010).
Tambo, N., and Watanabe, Y. (1979). “Physical aspect of flocculation process I. Fundamental treatise.” Water Res., 13(5), 429–439.
von Smoluchowski, M. (1917). “Versuch einer mathematischen Theorie der koagulationskinetik Kolloider Lösungen.” Z. Phys. Chem., 92, 129–168 (in German).
Weber, W. J., Jr., and DiGiano, F. A. (1996). Process dynamics in environmental systems, Wiley, New York.
Information & Authors
Information
Published In
Journal of Environmental Engineering
Volume 137 • Issue 4 • April 2011
Pages: 297 - 300
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Apr 2, 2010
Accepted: Oct 17, 2010
Published online: Nov 11, 2010
Published in print: Apr 1, 2011
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.