The Mathematics of Diffusion by John Crank

By John Crank

Although it contains a lot new fabric, this new version preserves the overall personality of the publication in supplying a suite of ideas of the equations of diffusion and describing how those recommendations should be received.

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The solution is correct to four significant figures when MJM^ < § for desorption. 18) also applies to diffusion along a cylindrical rod or tube of length /, with one end and its surface sealed and the other end maintained at a constant concentration. Talbot and Kitchener also obtained a solution for a slightly tapering tube. If dx and d2 are the diameters of the two ends, the degree of taper is specified by \i = (d2 — dl)/dl9 where dl is the closed end, and may be positive or negative. g. around 0-01, we have effectively radial diffusion and a solution is M, 6 2 ^ 2 where the an are the roots of tana = —a///, and, to the first order in a 2 = (n — \)2n2 + 2jj..

5. Sorption curve for a surface evaporation condition in a semi-infinite medium. These fundamental properties hold in general in semi-infinite media, provided the initial concentration is uniform and the surface concentration remains constant. They also hold for point and line sources on infinite surfaces or in infinite media, and also for the case of diffusion in an infinite medium where the diffusing substance is initially confined to the region x < 0. Clearly they do not apply to cases where parameters other than x/2yJ(Dt) are involved, such as the width of an extended source or the rate of change of surface concentration, etc.

The spherical and cylindrical sources are likely to be of practical interest. 8) where Co is the uniform concentration in the sphere initially. 8) may easily be written in terms of the total amount of diffusing substance M, since :0. ^ \ « / • * - - «- / rC I Jo •W^»^ >2 y » / i l I rr' , M ^ V , ~ t t . °\ INFINITE AND SEMI-INFINITE MEDIA 30 where Io is the modified Bessel function of the first kind of order zero. Tables of / 0 are available. 10) becomes C = C0{l-exp(-a2/4Dt)}. 11) These expressions may be applied, for example, to the diffusion of a sphere or cylinder of solute into a large volume of solvent.

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