Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T14:55:49.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Description of mixing with diffusion and reaction in terms of the concept of material surfaces

Published online by Cambridge University Press:  20 April 2006

J. M. Ottino
J. M. Ottino
Affiliation:
Department of Chemical Engineering, Goessmann Laboratory, University of Massachusetts, Amherst
Get access Rights & Permissions [Opens in a new window]

Abstract

The essence of fluid-mechanical mixing of diffusing and reacting fluids can be traced back to kinematics, connectedness of material volumes and transport processes occurring across deforming material surfaces. Descriptions based on kinematics of homoeomorphic deforming material surfaces (tracers) are restricted solely to continuous motions and conveniently analysed by transport equations in Lagrangian frames.

Connectedness of material volumes restricts the mixing topology and generates bicontinuous structures characterized by intermaterial-area and striation-thickness distributions. Upper bounds for area generation and material-line elongation are related to mean values of viscous dissipation and govern the average reaction rate in diffusion-controlled reactions. Two concepts are introduced: micromixing, related to local flows, rate of stretching and local viscous dissipation, and macromixing, associated with connectedness of isoconcentration surfaces, vorticity and average viscous dissipation.

Several small-scale flows can be used to typify the interplay between fluid mechanics, mass and energy transport, and chemical reactions: elliptically symmetrical stagnation flows, vortex decay, and swirling flow with uniform stretching. It is proposed that complex fluid motions might be interpreted in terms of integrated behaviour of populations of small-scale flows distributed in space and time to simulate mixing behaviour.

The objective of this work is to present the foundations of a continuum mixing description making reference to earlier approaches to demonstrate computational applicability and practical significance.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 114 , January 1982 , pp. 83 - 103
Copyright
© 1972 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 1960 Transport Phenomena. Wiley.
Bowen, R. M. 1976 Theory of mixtures. In Continuum Physics (ed. A. C. Eringen), vol. 3, pp. 1–127. Academic.
Brodkey, R. S. 1975 Turbulence in Mixing Operations. Academic.
Chan, W. C. & Scriven, L. E. 1970 Absorption into irrotational stagnation flow. Ind. Eng. Chem. Fundam. 9, 114–120.Google Scholar
Damköhler, G. 1940 Influence of turbulence on the velocity of flame in gas mixtures. Z. Electrochem. 45, 601–626. (English translation: N.A.C.A. Tech. Memo. no. 1112, 1947.)Google Scholar
Fisher, D. A. 1968 A model for fast reactions in turbulently mixed liquids. M.Sci. thesis, University of Minnesota, Minneapolis.
Fisher, D. A. 1974 Development and applications of a model for fast reactions in turbulently mixed liquids. Ph.D. thesis, University of Minnesota, Minneapolis.
Gibson, C. H. & Libby, P. A. 1972 On turbulent flows with fast chemical reactions. Part II. The distribution of reactants and products near a reacting surface. Comb. Sci. & Tech. 6, 29–35.Google Scholar
Hardy, G. H., Littlewood, J. E. & Pólya, G. 1973 Inequalities, 2nd edn. Cambridge University Press.
Hawthorne, W. R. 1948 Mixing and combusion in turbulent gas jets. In Proc. 3rd. Symp. on Combustion, Flame, and Explosion Phenomena, pp. 266–288. Williams & Wilkens.
Hill, J. C. 1976 Homogeneous turbulent mixing with chemical reaction. Ann. Rev. Fluid Mech. 8, 135–161.Google Scholar
Hottel, H. C. 1952 Burning in laminar and turbulent fuel jets. In Proc. 4th Symp. (Int.) on Combustion, pp. 97–133. Williams & Wilkens.
Karlovitz, B., Denniston, D. W. & Wells, F. E. 1951 Investigations of turbulent flames. J. Chem. Phys. 19, 541–547.Google Scholar
Mao, K. W. & Toor, H. L. 1970 A diffusion model for reactions with turbulent mixing. A.I.Ch.E. J. 16, 49–52.Google Scholar
Marble, F. E. & Broadwell, J. E. 1977 The coherent flame model for turbulent chemical reactions. Project SQUID Tech. Rep. TRW-9-PU.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. MIT Press.
Murthy, S. N. B. 1975 Turbulent Mixing in Non-Reactive and Reactive Flows. Plenum.
Ottino, J. M. 1979 Development and applications of a lamellar model for analysis of liquid-liquid mixing. Ph.D. thesis, University of Minnesota, Minneapolis.
Ottino, J. M. 1980 Lamellar mixing model for structured chemical reactions and their relationship to statistical models, macro- and micro-mixing and the problem of averages. Chem. Engng Sci. 35, 1377–1391.Google Scholar
Ottino, J. M. 1981 Efficiency of mixing from data on fast reactions in multi-jet reactors and stirred tanks. A.I.Ch.E. J. 27, 184–192.Google Scholar
Ottino, J. M., Ranz, W. E. & Macosko, C. W. 1979 A lamellar model for analysis of liquid-liquid mixing. Chem. Engng Sci. 34, 877–890.Google Scholar
Ranz, W. E. 1979a Application of a stretch model to mixing, diffusion, and reaction in laminar and turbulent flows. A.I.Ch.E. J. 25, 41–47.Google Scholar
Ranz, W. E. 1979b Fast reactions in a vortex. Conf. on Mixing, Engineering Foundation Conferences, New Hampshire.
Roshko, A. 1975 Progress and problems in understanding turbulent shear flows. In Turbulent Mixing in Non-Reactive and Reactive Flows (ed. S. N. B. Murthy). Plenum.
Serrin, J. B. 1960 In Encyclopedia of Physics (ed. S. Flugge), vol. VIII/1. Springer.
Shearer, C. J. 1973 Mixing of highly viscous liquids: flow geometries for streamline subline subdivision and redistribution. Chem. Engng Sci. 28, 1091–1098.Google Scholar
Spalding, D. B. 1976a The ESCIMO theory of turbulent combustion. Imperial College Mech. Engng Dept Rep. No. HTS/76/13.Google Scholar
Spalding, D. B. 1976b Mathematical models of turbulent flames: a review. Comb. Sci. & Tech. 13, 1–25.Google Scholar
Spalding, D. B. 1976c A general theory of turbulent combustion; the Lagrangian aspects. Imperial College Mech. Engng Dept Rep. no. HTS/76/14.Google Scholar
Spalding, D. B. 1978a Chemical reactions in turbulent fluids. In Physico-Chemical Hydrodynamics; Proc. Levich 60th Birthday Conf. Advance Publications.
Spalding, D. B. 1978b The influence of laminar transport and chemical kinetics on the time-mean reaction rate in a turbulent flame. In Proc. 17th Symp. (Int.) on Combustion, pp. 431–440. The Combustion Institute.
Sperb, R. P. 1979 Nonlinear diffusion-reaction problems with time-dependent diffusion coefficient. Z. angew. Math. Phys. 30, 663–675.Google Scholar
Uberoi, M. S. 1979 Mechanisms of decay of laminar and turbulent vortices. J. Fluid Mech. 90, 241–255.Google Scholar
Wohl, K., Shore, L., von Rosenberg, H. & Weil, C. W. 1952 Burning velocity of turbulent flames. In Proc. 4th Symp. (Int.) on Combustion, pp. 620–635. Williams & Wilkens.
Yamazaki, T. & Tsuji, H. 1960 An experimental investigation of the stability of turbulent burner flames. In Proc. 8th Symp. (Int) on Combustion, pp. 543–553. Williams & Wilkens.