Hydrodynamic dispersion in a tube with diffusive losses through its walls

Advective-diffusive transport of passive or reactive scalars in confined environments (e.g. tubes and channels) is often accompanied by diffusive losses/gains through the confining walls. We present analytical solutions for transport of a reactive solute in a tube, whose walls are impermeable to flow but allow for solute diffusion into the surrounding medium. The solute undergoes advection, diffusion and first-order chemical reaction inside the tube, while diffusing and being consumed in the surrounding medium. These solutions represent a leading-order (in the radius-to-length ratio) approximation, which neglects the longitudinal variability of solute concentration in the surrounding medium. A numerical solution of the full problem is used to demonstrate the accuracy of this approximation for a physically relevant range of model parameters. Our analysis indicates that the solute delivery rate can be quantified by a dimensionless parameter, the ratio of a solute's residence time in a tube to the rate of diffusive losses through the tube's wall.