  We develop a mathematical model for adsorption based on averaging the flow
around, and diffusion inside, adsorbent particles in a column. The model
involves three coupled partial differential equations for the contaminant
concentration both in the carrier fluid and within the particle as well as the
adsorption rate. The adsorption rate is modelled using the Sips equation, which
is suitable for describing both physical and chemical adsorption mechanisms.
Non-dimensionalisation is used to determine the controlling parameter groups as
well as to determine negligible terms and so reduce the system complexity. The
inclusion of intra-particle diffusion introduces new dimensionless parameters
to those found in standard works, including a form of internal Damk\"ohler
number and a new characteristic time scale. We provide a numerical method for
the full model and show how in certain situations a travelling wave approach
can be utilized to find analytical solutions. The model is validated against
available experimental data for the removal of Mercury(II) and CO$_\text{2}$.
The results show excellent agreement with measurements of column outlet
contaminant concentration and provide insights into the underlying chemical
reactions.
