Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T01:11:45.120Z Has data issue: false hasContentIssue false
  • English
  • Français

On the foundations of generalized Taylor dispersion theory

Published online by Cambridge University Press:  26 April 2006

I. Frankel  and
H. Brenner
I. Frankel
Affiliation:
Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel
H. Brenner
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalized Taylor dispersion theory extends the basic long-time, asymptotic scheme of Taylor and Aris greatly beyond the class of rectilinear duct and channel flow dispersion problems originally addressed by them. This feature has rendered it indispensable for studying flow and dispersion phenomena in porous media, chromatographic separation processes, heat transfer in cellular media, sedimentation of non-spherical Brownian particles, and transport of flexible clusters of interacting Brownian particles, to mention just a few examples of the broad class of non-unidirectional transport phenomena encompassed by this scheme. Moreover, generalized Taylor dispersion theory enjoys the attractive feature of conferring a unified paradigmatic structure upon the analysis of such apparently disparate physical problems. For each of the problems thus treated it provides an asymptotic, macroscale description of the original microscale transport process, being based upon a convective-diffusive ‘model’ problem characterized by a set of constant (position- and time-independent) phenomenological coefficients.

The present contribution formally substantiates the scheme. This is accomplished by demonstrating that the coarse-grained (macroscale) transport ‘model’ equation leads to a solution which accords asymptotically with the leading-order behaviour of the comparable solution of the exact (microscale) convective–diffusive problem underlying the transport process. It is also shown, contrary to current belief, that no systematic improvement in the asymptotic order of approximation is possible through the incorporation of higher-order gradient terms into the model constitutive equation for the coarse-grained flux. Moreover, the inherent difference between the present rigorous asymptotic scheme and the dispersion models resulting from Gill–Subramanian moment-gradient expansions is illuminated, thereby conclusively resolving a long-standing puzzle in longitudinal dispersion theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 97 - 119
Copyright
© 1989 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A., 1968 Handbook of Mathematical Functions. Dover.
Adler, P. M. & Brenner, H., 1984 Transport processes in spatially periodic capillary networks II. Taylor dispersion with mixing vertices. PhysicoChem. Hydrodyn. 5, 269–285.Google Scholar
Aris, R.: 1956 On the dispersion of solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A A235, 67–77.Google Scholar
Brenner, H.: 1979 Taylor dispersion in systems of sedimenting nonspherical Brownian particles: I. Homogeneous, centrosymmetric, axisymmetric particles. J. Colloid Interface Sci. 71, 189–208.Google Scholar
Brenner, H.: 1980a A general theory of Taylor dispersion phenomena. PhysicoChem. Hydrodyn. 1, 91–123.Google Scholar
Brenner, H.: 1980b Dispersion resulting from flow through spatially periodic porous media. Phil. Trans. R. Soc. Lond. A 297, 81–133.Google Scholar
Brenner, H.: 1981 Taylor dispersion in systems of sedimenting nonspherical Brownian particles II. Homogeneous ellipsoidal particles. J. Colloid Interface Sci. 80, 548–588.Google Scholar
Brenner, H.: 1982a A general theory of Taylor dispersion phenomena II. An extension. PhysicoChem. Hydrodyn. 3, 139–157.Google Scholar
Brenner, H.: 1982b A general theory of Taylor dispersion phenomena IV. Direct coupling effects. Chem. Engng Commun. 18, 355–379.Google Scholar
Brenner, H. & Adler, P. M., 1982 Dispersion resulting from flow through spatially periodic porous media II. Surface and intraparticle transport. Phil. Trans. R. Soc. Lond. A A307, 149–200.Google Scholar
Brenner, H. & Gaydos, L. J., 1977 The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius: Models of the diffusive and convective transport of solute molecules in membranes and porous media. J. Colloid Interface Sci. 58, 312–356.Google Scholar
Brenner, H., Nadim, A. & Haber, S., 1987 Long-time molecular diffusion, sedimentation and Taylor dispersion of a fluctuating cluster of interacting Brownian particles. J. Fluid Mech. 183, 511–542.Google Scholar
Chapman, S. & Cowling, T. G., 1961 The Mathematical Theory of Non-Uniform Gases, 2nd edn. Cambridge University Press.
Chatwin, P. C.: 1970 The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 43, 321–357.Google Scholar
Chatwin, P. C.: 1972 The cumulants of the distribution of concentration of a solute dispersing in solvent flowing through a tube. J. Fluid Mech. 51, 63–67.Google Scholar
Chatwin, P. C. & Sullivan, P. J., 1982 The effect of aspect ratio on longitudinal diffusivity in rectangular channels. J. Fluid Mech. 120, 347–358.Google Scholar
DeGance, A. E. & Johns, L. E., 1978a The theory of dispersion of chemically active solutes in a rectilinear flow field. Appl. Sci. Res. 34, 189–225.Google Scholar
DeGance, A. E. & Johns, L. E., 1978b On the dispersion coefficients for Poiseuille flow in a circular cylinder. Appl. Sci Res. 34, 227–258.Google Scholar
DeGance, A. E. & Johns, L. E., 1985 The theory of dispersion of chemically active solutes in a rectilinear flow field: The vector problem. Appl. Sci. Res. 42, 55–88.Google Scholar
Dill, L. H. & Brenner, H., 1982a A general theory of Taylor dispersion phenomena III. Surface transport. J. Colloid Interface Sci. 85, 101–117.Google Scholar
Dill, L. H. & Brenner, H., 1982b A general theory of Taylor dispersion phenomena V. Time-periodic convection. PhysicoChem. Hydrodyn. 3, 267–292.Google Scholar
Dill, L. H. & Brenner, H., 1983a Dispersion resulting from flow through spatially periodic porous media III. Time-periodic processes. PhysicoChem. Hydrodyn. 4, 279–302.Google Scholar
Dill, L. H. & Brenner, H., 1983b A general theory of Taylor dispersion phenomena VI. Langevin methods. J. Colloid Interface Sci. 93, 343–365.Google Scholar
Doshi, M. R., Daiya, P. M. & Gill, W. N., 1978 Three-dimensional laminar dispersion in open. and closed rectangular conduits. Chem. Engng Sci. 33, 795–804.Google Scholar
Dungan, S. R. & Brenner, H., 1988 Sedimentation and dispersion of nonneutrally buoyant Brownian particles in cellular circulatory flows simulating local fluid agitation. Phys. Rev. A 38, 3601–3608.Google Scholar
Gajdos, L. J. & Brenner, H., 1978 Flow field fractionation: Extensions to non-spherical particles and wall effects. Sep. Sci. Technol. 13, 215–240.
Gill, W. N. & Sankarasubramanian, R., 1970 Exact analysis of unsteady convective-diffusion. Proc. R. Soc. Lond. A A316, 341–350.Google Scholar
Gill, W. N. & Sankarasubramanian, R., 1971 Dispersion of a nonuniform slug in time-dependent flow. Proc. R. Soc. Lond. A A322, 101–117.Google Scholar
Mauri, R. & Brenner, H., 1989 Rheological properties of suspensions determined by momentum tracer methods (to be published).
Mavrouniotis, G. M. & Brenner, H., 1988 Dispersion of Brownian spheres in cylindrical pores. J. Colloid Interface Sci. 124, 269–283.Google Scholar
Nadim, H. & Brenner, H., 1989 Longtime nonpreaveraged diffusion and sedimentation properties of flexible Brownian dumbbells. Physicochem. Hydrodyn., (in press).Google Scholar
Nadim, A., Cox, R. G. & Brenner, H., 1986a Taylor dispersion in concentrated suspensions of rotating cylinders. J. Fluid Mech. 164, 185–215.Google Scholar
Nadim, A., Pagitsas, M. & Brenner, H., 1986b Higher-order moments in macrotransport processes. J. Chem. Phys. 85, 5238–5245.Google Scholar
Pawula, R. F.: 1967 Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Phys. Rev. 162, 186–188.Google Scholar
Reichl, L. E.: 1980 A Modern Course in Statistical Physics, p. 143. University of Texas Press.
Sankarasubramanian, R. & Gill, W. N., 1973 Unsteady convective-diffusion with interphase mass transfer. Proc. R. Soc. Lond. A A333, 115–132.Google Scholar
Shapiro, M. & Brenner, H., 1986 Taylor dispersion of chemically reactive species: Irreversible first-order reactions in bulk and on boundaries. Chem. Engng Sci. 41, 1417–1433.Google Scholar
Shapiro, M. & Brenner, H., 1987a Chemically reactive generalized Taylor dispersion phenomena. AIChE J. 33, 1155–1167.Google Scholar
Shapiro, M. & Brenner, H., 1987b Green's function formalism in generalized Taylor dispersion theory. Chem. Engng Commun. 55, 19–27.Google Scholar
Shapiro, M. & Brenner, H., 1988 Dispersion of a chemically reactive solute in a spatially periodic model of a porous medium. Chem. Engng. Sci. 43, 551–571.Google Scholar
Shapiro, M. & Brenner, H., 1989 Dispersion/reaction model of aerosol filtration by porous filters. J. Aerosol Sci. (submitted).Google Scholar
Shapiro, M., Oron, A. & Gutfinger, C., 1989 Electrostatic precipitation of charged particles from a turbulent flow between coaxial cylinders. J. Colloid Interface Sci. (in press).Google Scholar
Smith, R.: 1981a A delay-diffusion description for contaminant dispersion. J. Fluid Mech. 105, 469–486.Google Scholar
Smith, R.: 1981b The importance of discharge siting upon contaminant dispersion in narrow rivers and estuaries. J. Fluid Mech. 108, 43–53.Google Scholar
Smith, R.: 1985 When and where to put a discharge in an oscillatory flow. J. Fluid Mech. 153, 479–499.Google Scholar
Smith, R.: 1987a Diffusion in shear flows made easy: the Taylor limit. J. Fluid Mech. 175, 201–214.Google Scholar
Smith, R.: 1987b Shear dispersion looked at from a new angle. J. Fluid Mech. 182, 447–466.Google Scholar
Taylor, G. I.: 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A A219, 186–203.Google Scholar
Taylor, G. I.: 1954 Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. R. Soc. Lond. A A225, 473–477.Google Scholar