  A mathematical model describing the erosion or leaching of a solid material
by a flowing fluid in a column is developed. This involves an
advection-diffusion equation coupled to a linear kinetic reaction describing
the mass transfer between the solid and fluid. Two specific cases are analysed,
the first where the extracted material has the same saturation solubility and
rate of mass transfer throughout the process, the second where the solubility
switches after a certain amount of erosion. In the first case there are only
two model unknowns, the solubility and mass transfer coefficient, in the second
there is a third unknown, the second solubility. Exploiting the fact that
erosion is a slow process (relative to the flow rate) a perturbation solution
based on the smallness of the amount removed is developed to describe the
concentration and radius throughout the column. From this an analytical
expression for the extracted fraction is obtained. The extracted fraction has a
large linear section which results in a simple calculation to estimate the
initial solubility from a very few or even a single data point. The remaining
unknowns may also be easily calculated from the formula and later data points.
A numerical solution, using finite differences, is developed to verify the
perturbation solution. The analytical solution is also verified against
experimental data for the removal of lanolin from wool fibres with a
supercritical CO$_\text{2}$/ethanol solvent. Values for the mass transfer rate
and two solubilities are obtained for different pressures and shown to provide
excellent agreement with a series of experimental results for the extracted
fraction.
