Mathematical analysis of a Sips-based model for column adsorption

We investigate the dynamics of a contaminated fluid flowing through an adsorption column. We derive a one dimensional advection–diffusion equation coupled to a sink term that accounts for the contaminant adsorption. The adsorption rate is modelled by the Sips equation, where the order of the exponents is obtained through analysing the chemical reaction (as opposed to data fitting). By applying a travelling wave substitution the governing equations are reduced to two coupled ordinary differential equations. After non-dimensionalising and imposing typical values for the operating parameters we are able to identify negligible terms and so reduce the system to one where explicit solutions can be obtained. Distinct solution forms are provided for a range of Sips exponents. The approximate solutions are verified against numerics and experimental data. It is shown that if the breakthrough data is plotted in the form suggested by the analytical solutions then the result is a straight line when the correct Sips exponents are employed. This provides a clear indicator of the correct adsorption model. The straight line form permits a trivial optimisation procedure to determine the adsorption rate. This is the first time that analytical solutions have been obtained for sink terms beyond the standard Langmuir and Linear Driving Force models.