On Laminar Dispersion for Flow Through Round Tubes

Abstract

A general method for the solution of the axially symmetric transient diffusion-convection equation for laminar dispersion in round tubes subject to arbitrary square-integrable initial conditions is analytically developed. The solution representing the local concentration is expressed by a series in terms of the zeroth-order Bessel function, and the order of approximation (equal to the number of terms in the series) required at a given value of the dimensionless time τ for flow with a specified Peclet number Pe is clearly established. It is shown that the approximation used by Gill, et al. [5–8], is a special case of the present analysis under certain conditional assumptions. For the case of fundamental interest with an initial input concentrated at a section of the tube, the mean concentration as a function of the axial distance measured from the origin of a coordinate moving with the average flow velocity determined by the present method at given values of the Peclet number and the dimensionless time is compared with those by Taylor [1], Lighthill [4], Chatwin [9], Gill, et al. [7], and Hunt [23]. The comparison of the concentration profiles shows that Lighthill’s solution is perhaps valid as τ → 0, Hunt’s solution obtained by first-order perturbation approximation yields too large a dispersion by molecular diffusion even at small times, and the other solutions are asymptotically correct at large values of time for flow with high Peclet numbers.