Time-dependent transport by convection and diffusion with exchange between two phases

We consider the transport of a tracer substance through a system consisting of a tube containing flowing fluid surrounded by a wall layer in which the tracer is soluble. The fluid moves with either a Poiseuille or a uniform flow profile, and the outer boundary of the wall layer is either impermeable to tracer or absorbs it perfectly. The development of dispersive transport following the injection of tracer is described in terms of three time-dependent effective transport coefficients, viz. the fraction of tracer remaining in the system, the apparent convection velocity and the dispersion coefficient ; the last two are defined in terms of the rates of change of the mean and variance of the axial tracer distribution. We assume that the timescale for tracer diffusion across the wall layer is much larger than that for diffusion across the flowing phase, and derive an asymptotic approximation corresponding to each timescale. Numerical results are given to illustrate sensitivity to the physical parameters of the system. It is shown that if the coefficients are based on tracer concentration in the fluid phase alone, as in previous work, paradoxical behaviour, such as negative apparent convection velocities, can result ; we therefore base our results on averages of concentration over both phases. On the shorter timescale (the same timescale over which Taylor dispersion develops) at leading order it is found that the influence of the wall layer can be characterized by a single dimensionless parameter, and that conditions at the outer boundary have no effect. In many cases transport is also rather insensitive to the form of the flow profile. On the longer timescale, at leading order the influence of the wall layer is characterized by another dimensionless parameter, and unless uptake is very small diffusion within the layer is the rate-determining process ; consequently transport is independent of the form of the flow profile. A further important conclusion is that the usual effective convection and dispersion coefficients, based on spatial moments, are of little use in predicting the time-varying concentration at a fixed position, because the spatial concentration profile becomes Gaussian only over the longer timescale.