A Linear Model of the Motion of a Low-Concentration Suspension of Monodisperse Stokes Particles in a Flat Channel

In the framework of the convection-diffusion approach to the monodisperse low-concentration solid phase sedimentation in a suspension moving in a flat horizontal channel, we obtain a linear boundary value problem for a parabolic equation on the local concentration of particles. We impose boundary conditions of the third kind using the condition that the flux of particles on the wetted surface is proportional to their concentration at the wall. Integral transformations yield an analytical solution of the stated boundary value problem, which we use to find the relations to determine the thickness of sediment on the bottom and top walls of the channel. Simulations show that the kinetics of the solid phase settling from a flowing suspension, as well as the sediment formation rate and its distribution on the botton and top walls of the flat channel, substantially depend on the degree of mixing of the dispersion medium and the absorption capacity of wet surfaces. We establish that for walls with low absorption capacity a decrease in the mixing intensity reduces the rate of particle sedimentation on the walls, but increases it in the case of high absorption capacity.

Keywords

convection-diffusion equation; boundary value problem; Laplace transform; analytical solution; sediment thickness; sedimentation; degree of mixing; wall absorption capacity.

References

1. Einstein A. Eine neue Bestimmung der Molekuldimensionen. Annalen der Physik, 1906, vol. 19, pp. 289-306. DOI: 10.1002/andp.19063240204
2. Cunningham E. On the Velocity of Steady Fall of Spherical Particles Through Fluid Medium. Proc. R. Soc. Lond. A, 1910, vol. 83, pp. 357-365. DOI: 10.1098/rspa.1910.0024
3. Smoluchowski M. 'Uber die Wechselwirkung von Kugeln, die sich in einer z'ahen Fl'ussigkeit bewegen. Bull. Int. Acad. Sci. Cracovie. Ser. 1A, 1911, pp. 28-39.
4. Einstein H.A., Chien N. Effects of Heavy Concentration Near the Bed on the Velocity and Sediment Distribution. MRD Sediment Series 8, University of California, Berkeley, 1955.
5. Pokrovskiy V.P. Statisticheskaya mehanika razbavlennyh suspenziy [Statistical Mechanics of Dilute Suspensions]. Moscow, Nauka, 1978. 135 p.
6. Junke G., Pierre Y.J. Turbulent Velocity Profiles in Sediment-Laden Flows. Journal of Hydraulic Research, 2001, vol. 39, no. 1, pp. 11-23. DOI: 10.1080/00221680109499798
7. Venturu A.F., Garcia A.P., Ferreira P.J., Rasteiro M.G. Modelling the Turbulent Flow of Pulp Suspensions. Ind. Eng. Chem. Res., 2011, vol. 50, issue 16, pp. 9735-9742. DOI: 10.1021/ie101500h
8. Shanliang Z., Jianzhong L., Weifeng Z. Numerical Research on the Fiber Suspensions in a Turbulent T-Shaped Branching Channel Flow. Chinese Journal of Chemical Engineering, 2007, vol. 15, no. 1, pp. 30-38. DOI: 10.1016/S1004-9541(07)60030-5
9. Gus'kov S.B. [Self-consistent Field Method is Applied to the Dynamics of Viscous Suspensions]. Prikladnaja Matematika i Mehanika [Applied Mathematics and Mechanics], 2013, vol. 77, no. 4, pp. 557-572. (in Russian).
10. Nigmatulin R.N. Osnovy mehaniki geterogennyh sred [Fundamentals of Mechanics of Heterogeneous Media]. Moscow, Nauka, 1978. 336 p.
11. D.A. Baranov, A.V. Vjaz'min, A.A. Tuhman Processy i apparaty himicheskoy tehnologii. Javleniya perenosa, makrokinetika, podobie, modelirovanie, proektirovanie. T. 1. Osnovy teorii processov himicheskoy tehnologii [Processes and Devices of Chemical Technology. Transport Phenomena, Macrokinetics, Likeness, Modelling, Design. Fundamentals of the Theory of Chemical Technology Processes]. Moscow, Logos, 2000. 480 p.
12. Harin V.M., Ryazhskih V.I. [On the Theory of Deposition]. Teoreticheskie osnovy himicheskoy tehnologii [Theoretical Foundations of Chemical Engineering], 1989, vol. 23, no. 5, pp. 651-658. (in Russian)
13. Cvetkov F.F., Grigor'ev B.A. Teplomassoobmen [Heat and Mass Transfer]. Moscow, Izd-vo MJeI, 2005. 550 p.
14. D'otsch G. Anleitung zum praktischen gebrauch der Laplace-transformation und der z-transformation, dritte Auflage. Wien, 1967.
15. Belyaev N.M., Ryadno A.A. Metody teorii teploprovodnosti [Methods of the Teory of Heat Conduction. Ch. 1]. Moscow, Vysshaya shkola, 1982. 327 p.