(37a) Delta-Shock, a New Type of Composition Front in Nonlinear Chromatography
AIChE Annual Meeting
2009
2009 Annual Meeting
Engineering Sciences and Fundamentals
Mathematical Modeling of Transport Processes
Monday, November 9, 2009 - 8:30am to 8:48am
The local equilibrium theory of chromatography is applied to study binary systems subject to the mixed generalized Langmuir isotherm:
(with H2 > H1), where component 1 and 2 are anti-Langmuir-like and Langmuir-like, respectively. Riemann problems, where a stream of a given composition is fed to a column initially saturated at a different composition, are analyzed. Within the hyperbolic region of the hodograph plane close to the origin it is shown that such Riemann problems admit not only classical solutions, consisting of constant states separated by simple wave and shock wave transitions, but also non-classical solutions, namely including a continuous non-simple wave transition and a delta-shock [1]. The latter can be viewed as a travelling spike of growing strength superimposed to the discontinuity separating the initial and the feed states. Exact conditions for the occurrence of classical and non-classical solutions are derived. Thus, this work adds two new types of composition fronts to the classical ones already known in nonlinear chromatography.
In the case of the delta-shock, the most striking new transition, it is possible to derive exact relationships for its properties, including the slope of its image in the physical plane, i.e. the reciprocal of its propagation speed:
where the symbol [∙] denotes the jump across the discontinuity of the quantity enclosed. These analytical expressions match the results obtained through numerical simulations. The physical legitimacy of the model considered in this work is demonstrated theoretically, and is further supported by the experimental evidence of the existence of delta-shocks that has been recently provided [2].
[1] Mazzotti M., Non-classical composition fronts in nonlinear chromatography - Delta-shock, Ind. Eng. Chem. Res. (2009) submitted.
[2] Mazzotti M., Tarafder A., Gritti F., Guiochon G., Experimental evidence of a delta-shock in nonlinear chromatography, J. Chromatogr. A (2009) submitted.