Approximate solutions for developing shear dispersion with exchange between phases

We consider the transport of a tracer substance in Poiseuille flow through a pipe lined with a thin, fixed wall layer in which the tracer is soluble. A formal solution is given for the variation of concentration with time at a fixed downstream position following an initial release of tracer. Asymptotic approximations are derived assuming that: (i) the Péclet number is large; (ii) the time scale for diffusion across the wall layer is much larger than that for diffusion across the fluid phase and (iii) the dimensionless distance downstream of the point of release, z, is large. This means that the transverse concentration variation is small within the fluid phase, so that transport is dominated by the exchange of tracer between the phases and radial diffusion within the wall layer. The character of the concentration transient is found to be determined by two dimensionless numbers, an absorption parameter κ and an effective wall layer thickness ν (both rescaled to take account of the ratio of diffusivities in the two phases); by assumption (ii), ν is large. Several different regimes are possible, according to the values of κ, ν and z. At sufficiently large distances, a Gaussian approximation, analogous to Taylor's solution, is applicable. At intermediate distances, provided κ is not too large, a highly skewed transient is predicted. If κ is small, there exists another region further upstream where the effect of the wall is negligible, and Taylor's Gaussian approximation applies. More complicated behaviour occurs in the zones of transition between these three regions. The behaviour described is expected to be typical of a range of similar systems. In particular, it may be shown that the basic form of the skewed approximation is insensitive to the geometry of the system, and also applies when the Péclet number is of order unity.