On the rate of convergence to steady state in a linear chromatography model

TL;DR

This paper analyzes the convergence rate to steady state in a linear chromatography model, deriving eigenvalues and asymptotic profiles using analytical and numerical methods, with applications to a case study.

Contribution

It provides a rigorous eigenvalue characterization of convergence rates and combines analytical and numerical techniques for asymptotic analysis in a complex chromatography model.

Findings

Convergence rate determined by a dominant eigenvalue.

Constructed a characteristic function for eigenvalues.

Applied results to separation of omeprazole enantiomers.

Abstract

We study the rate of convergence to the steady state in the True Moving Bed model of linear chromatography, as a function of the six parameters that appear in the model. The model is a system of eight linear partial differential equations of hyperbolic type, coupled through the equations themselves and also through boundary conditions. We prove that the rate of convergence is given by a dominant eigenvalue, whose existence we prove by means of the Krein-Rutman Theorem, and by comparison arguments. We show how to construct a (not at all simple) characteristic function, whose roots are the eigenvalues. We also study the asymptotic profile of the solutions for large times, although this part is not purely analytical, but a combination of analytical and numerical techniques. Beyond the theoretical results, these models also offer explicit quantitative information: we apply all our results to a Case Study, namely the separation of omeprazole enantiomers. Finally, we consider a simpler limit case, where all the calculations become explicit.