Shock formation in binary systems with nonlinear characteristic curves

The study of a typical chromatographic cycle in the presence of a bi-Langmuir isotherm and of Langmuir isotherm with temperature dependent coefficients in an adiabatic column is carried out. In both cases, the relevant feature is represented by the nonlinearity of the ΓΓ characteristic curves in the hodograph plane. Under these conditions, peculiar solutions can be obtained, which are not possible in the case of Langmuir isotherms, for which the characteristic curves are straight lines. In particular, the solution found in the hodograph plane can be severely deformed compared to the corresponding Langmuir case in isothermal columns, due to the nonlinearity of the characteristic curves and to the fact that, under these conditions, these are no more superimposed to the corresponding curves describing the shocks. The most relevant consequence is that, when a shock starts interacting with a simple wave of the same type, the typical cancellation effect observed for Langmuir isotherms comes together with the production of a new simple wave of different type. This study can be applied to a multitude of other problems. In particular, the study of the adiabatic columns can be applied to the study of gradient elution chromatography in general, for which the role of temperature is played by the so-called modifier.