Asymptotic Behavior of Two Interreacting Chemicals in a Chromatography Reactor
Daniel N. Ostrov · SIAM Journal on Mathematical Analysis · 1996
The chromatographic separation of two chemical species ($c_1 $ and $c_2 $) that transform into each other with first-order kinetics as they pass through a Langmuir isotherm reactor is governed by the following system of nonlinear hyperbolic conservation equations: \[\begin{gathered} \frac{{\partial c_1 }}{{\partial x}} + \frac{\partial }{{\partial t}}\left( {\frac{{c_1 }}{{1 + c_1 + c_2 }}} \right) = - kc_1 + k'c_2 \hfill \\ {\text{and}}\quad \frac{{\partial c_2 }}{{\partial x}} + \frac{\partial }{{\partial t}}\left( {\frac{{\gamma c_2 }}{{1 + c_1 + c_2 }}} \right) = \gamma \left( {kc_1 + k'c_2 } \right), \hfill \\ {\text{where }}t \in ( - \infty ,\infty ). \hfill \\ \end{gathered} \] An analysis is presented of the two species’ asymptotic behavior as they progress down a semiinfinite (i.e., $x \in [ {0,\infty } )$) separation reactor with cyclic (periodic) entering feed concentrations. First it is shown that the method of generalized characteristics can be extended to describe the above system of equations. Then generalized characteristics are applied to show that the $\omega $-limit set for the species concentrations is comprised of a single determined point on the curve of chemical equilibrium and that this point is approached at an exponential rate.