Abstract
A modification of Aris's method of moments solution to the heat/mass dispersion equation has been developed. In addition to taking axial moments of concentration, the present approach also defines mean cross-sectionally weighted values of the concentration. This leads to a set of ordinary differential equations rather than the partial differential equations generated by Aris's method. Approximate solutions have been developed for the cases of steady and oscillatory flows in a flat channel and values of the dispersion coefficient have been in good agreement with exact analytical predictions calculated by Dewey and Sullivan and by Watson.
Zusammenfassung
Eine Modifaktion der Momenten-Methode von Aris für die Lösung der Dispersionsgleichung für Wärme/Masse ist entwickelt worden. Die neue Methode benützt nicht nur axiale Momente der Konzentration, sondern definiert zusätzlich im Querschnitt gewogene Durchschnittswerte der Konzentration. Dieses führt zu einem Satz gewöhnlicher Differentialgleichungen, an Stelle der partiellen Differentialgleichung nach der Methode von Aris. Angenäherte Lösungen sind entwickelt worden für die Fälle von stationären und oszillatorischen Strömungen in einem ebenen Kanal. Ergebnisse für Dispersions-Koeffizienten stimmen gut überein mit den exakten analytischen Werten, die von Dewey und Sullivan sowohl wie von Watson berechnet worden sind.
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References
R. Aris,On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Soc.235 A, 67–77 (1956).
R. E. Babcock, D. W. Green, R. M. Perry,Longitudinal dispersion mechanisms in packed beds. A. I. Ch. E. J.12(5), 922–927 (1966).
P. C. Chatwin,On the longitudinal dispersion of a passive contaminant in oscillatory flow in tubes. J. Fluid Mech.71, 513–527 (1975).
R. Dewey and P. J. Sullivan,Longitudinal dispersion in flows that are homogeneous in the streamwise direction. Z. angew. Math. Phys.30, 601–613 (1979).
H. B. Fischer,Mixing and dispersion in estuaries. Ann. Rev. Fluid Mech.5, 59–78 (1973).
E. R. Holley, D. R. F. Harleman and H. B. Fischer,Dispersion in homogeneous estuary flow. Proc. A. S. C. E. J. Hydraul. Div.96 (8), 1691–1709 (1970).
C. H. Joshi, R. D. Kamm, J. M. Drazen and A. S. Slutsky,An experimental study of gas exchange in laminar oscillatory flow. J. Fluid Mech.133, 245–254 (1983).
J. C. Schuster,Canal discharge measurements with radioisotopes. Proc. A. S. C. E. J. Hydraul. Div.91 (2), 101–124 (1965).
R. Smith,The importance of discharge siting upon contaminant dispersion in narrow rivers and estuaries. J. Fluid Mech.108, 54–63 (1981).
R. Smith,Contaminant dispersion in oscillatory flows. J. Fluid Mech.114, 379–398 (1982).
I. J. Sobey,The effect of small dispersion on two phase operations. I. M. A. J. Appl. Math.27, 237–249 (1981).
J. W. Stairmand,Dispersion in oscillatory flows. University of Oxford, D. Phil. thesis (1983).
J. W. Stairmand,Dispersion coefficients for large Schmidt number oscillatory flows. Submitted to Chem. Eng. Sci. (1984).
G. I. Taylor,Dispersion of soluble matter in a solvent flowing slowly through a tube. Proc. Roy. Soc.219 A, 186–203 (1953).
G. I. Taylor,Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. Roy. Soc.225 A, 473–477 (1954).
E. J. Watson,Diffusion in oscillatory pipe flow. J. Fluid Mech.133, 233–244 (1983).
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Stairmand, J.W. A modified moments analysis of dispersion in steady and oscillatory flows. Z. angew. Math. Phys. 36, 658–668 (1985). https://doi.org/10.1007/BF00960378
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DOI: https://doi.org/10.1007/BF00960378