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Dispersion in fixed beds

Published online by Cambridge University Press:  20 April 2006

Donald L. Koch  and
John F. Brady
Donald L. Koch
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
John F. Brady
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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Abstract

A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Péclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Péclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Péclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Péclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Péclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 154 , May 1985 , pp. 399 - 427
Copyright
© 1985 Cambridge University Press

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References

Acrivos, A., Hinch, E. J. & Jeffrey, D. J. 1980 Heat transfer to a slowly moving fluid from a dilute bed of heated spheres. J. Fluid Mech. 101, 403–421.Google Scholar
Acrivos, A. & Taylor, T. D. 1962 Heat and mass transfer from a single sphere in Stokes flow. Phys. Fluids 5, 387–394.Google Scholar
Adler, P. M. 1984 Transport processes in fractals. III. Taylor dispersion in two examples of fractal capillary networks. Intl J. Multiphase Flow (to appear).
Adler, P. M. & Brenner, H. 1984 Transport processes in spatially periodic capillary networks, II. Taylor dispersion with mixing vertices. Physico Chem. Hydrodyn. 5, 269.Google Scholar
Batchelor, G. K. 1974 Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech. 6, 227–255.Google Scholar
Batchelor, G. K. & O'Brien, R. W.1977 Thermal or electrical conduction through a granular material. Proc. R. Soc. Lond. A 355, 313–333.Google Scholar
Brenner, H. 1980 Dispersion resulting from flow through spatially periodic porous media. Phil. Trans. R. Soc. Lond. A 297, 81–133.Google Scholar
Carbonell, R. G. & Whitaker, S. 1983 Dispersion in Pulsed Systems-II: theoretical developments for passive dispersion in porous media. Chem. Engng Sci. 38, 1795–1802.CrossRefGoogle Scholar
Carton, J. P., DuBois-Violette, E. & Prost, K. 1983 Brownian diffusion of a small particle in a suspension. II. Hydrodynamic effect in a random fixed bed. Physica 119A, 307–316.Google Scholar
Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. M. & Ranger, K. B.1976 Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644.Google Scholar
Eidsath, A., Carbonell, R. G., Whitaker, S. & Herrman, L. R. 1983 Dispersion in pulsed systems - III: comparison between theory and experiments for packed beds. Chem. Engng Sci. 38, 1803–1816.CrossRefGoogle Scholar
Fried, J. J. & Combarnous, M. A. 1971 Dispersion in porous media. Adv. Hydrosci. 7, 169–282.Google Scholar
Gavalas, G. R. & Kim, S. 1981 Periodic capillary models of diffusion in porous solids. Chem. Engng Sci. 38, 1111–1122.Google Scholar
Gunn, D. J. & Pryce, C. 1969 Dispersion in packed beds. Trans. Inst. Chem. Engrs 47, T341–T359.Google Scholar
Han, N., Bhakta, J. & Carbonell, R. G. 1983 Longitudinal and lateral dispersion in packed beds: effects of column length and particle size distribution. Submitted to AIChE J.
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720.Google Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 335, 335–367.Google Scholar
Kesten, H. & Papanicolaou, G. C. 1979 A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128.Google Scholar
Maxwell, J. C. 1873 Electricity and Magnetism. Clarendon.
Reis, J. F. G., Lightfoot, E. N., Noble, P. T. & Chiang, A. S. 1979 Chromatography in a bed of spheres. Sep. Sci. Tech. 14, 367–394.Google Scholar
Saffman, P. G. 1959 A theory of dispersion in porous media. J. Fluid Mech. 6, 321–349.Google Scholar
Saffman, P. G. 1973 On the settling speed of free and fixed suspensions. Stud. Appl. Maths 52 115–127.Google Scholar
Sahimi, M., Hughes, B. D., Scriven, L. E. & Davis, H. T. 1983 Stochastic transport in disordered systems. J. Chem. Phys. 6849–6864.Google Scholar
Sangani, A. S. & Acrivos, A. 1983 The effective conductivity of a periodic array of spheres. Proc. R. Soc. Lond. A 386, 263–275.Google Scholar
Sih, P. H. & Newman, J. 1967 Mass transfer to the rear of a sphere in Stokes flow. J. Heat Mass Transfer 10, 1749–1756.Google Scholar