On the method of moments for solute dispersion

A cloud of solute injected into a pipe or channel is known to spread out by a dispersion process based on cross-sectional diffusion across a velocity shear. The original description of the process is due to Taylor (1953, 1954), and an important subsequent contribution was by Aris (1956), who framed and partially solved equations for the integral moments of the cloud of contaminant. The present work resolves some technical difficulties that occur when Aris’ solution method (separation of variables) is pursued in depth. In particular, it is shown that Aris’ technique has to be modified to give the moments at short and moderate times after the injection of solute into the flow. The paper is concerned with dispersion in those parallel flows for which an associated eigenvalue problem has a discrete spectrum of eigenvalues; fortunately, this case appears to be the rule rather than the exception. Expressions are obtained for the second and third moment about the mean, and the theory is applied to three cases of interest.

[1]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[2]  G. Taylor.  Dispersion of soluble matter in solvent flowing slowly through a tube , 1953, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  Geoffrey Ingram Taylor,et al.  The dispersion of matter in turbulent flow through a pipe , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[4]  J. W. Elder.  The dispersion of marked fluid in turbulent shear flow , 1959, Journal of Fluid Mechanics.

[5]  R. Aris,et al.  On the dispersion of a solute by diffusion, convection and exchange between phases , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  P. C. Chatwin,et al.  The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe , 1970, Journal of Fluid Mechanics.

[7]  Edward A. Mason,et al.  Gaseous Diffusion Coefficients , 1972 .

[8]  P. Chatwin.  A CALCULATION ILLUSTRATING EFFECTS OF THE VISCOUS SUB-LAYER ON LONGITUDINAL DISPERSION , 1973 .

[9]  Ian R. Wood,et al.  Longitudinal Dispersion with Dead Zones , 1977 .

[10]  The initial dispersion of soluble matter in three-dimensional flow , 1978, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[11]  Carleton J. Howard,et al.  Kinetic measurements using flow tubes , 1979 .

[12]  Ronald Smith.  The early stages of contaminant dispersion in shear flows , 1981, Journal of Fluid Mechanics.

[13]  B. Andersson,et al.  Dispersion in laminar flow through a circular tube , 1981, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.