## Introduction

Equilibrium theory is a powerful tool for understanding, designing and controlling chromatographic processes. It is based on an idealized model assuming thermodynamic equilibrium between the fluid and the solid phase. It represents a system of first order quasilinear partial differential equations and admits an analytical approach using the method of characteristics. Thermodynamic equilibrium is described by the adsorption isotherm, which represents the essential nonlinearity of the system. Previous approaches have been limited to explicit adsorption isotherms including Langmuir, Bi-Langmuir and generalized Langmuir isotherms. The focus in this project is on processes with implicit adsorption isotherms, which are often required for an accurate process description. Prominent examples are stoichiometric ion exchange and ideal adsorbed solution theory (IAST).

## Results

We succeeded in extending the equilibrium theory for classical ion exchange to full chromatographic cycles and pulse patterns, which are of major interest in chromatography [1]. Analytical solutions obtained from the equilibrium theory approach were compared to numerical solutions. For an efficient numerical simulation of chromatographic processes with implicit adsorption isotherms a clever reformulation strategy was developed and combined with a method of lines approach and established DAE (differential algebraic equation) solvers. It was possible to prove that hyperbolicity of the model equations implies a differential index of one, which greatly alleviates the numerical solution. This is a relatively ’mild’ condition, which should be always satisfied from the thermodynamic point of view.

The numerical approach was also extended to the widely used IAST, which makes it possible to infer multicomponent sorption behavior from single component adsorption isotherms. It was shown that the novel solution approach outperforms all other available approaches reported in the literature [2] and is therefore also a promising approach for computationally more expensive rigorous mathematical process optimization.

The numerical and analytical approaches for ion exchange chromatography have also been extended to account for variable solution normality and steric hindrance [3,4]. An example is given in Fig. 1. Both are very important in many bioseparations including, for example, protein separations. This opens new ways of rational design of bioseparations based on analytical insight.

This project is supported by the International Max Planck Research School for Advanced Methods in Process and Systems Engineering (IMPRS ProEng). Cooperation partners are Prof. Kaspereit (FAU Erlangen-Nürnberg) and PCF group.

## Publications

[1] M. Fechtner and A. Kienle. Efficient simulation and equilibrium theory for adsorption processes with implicit adsorption isotherm - Mass action equilibria. Chem. Eng. Sci., 171, 471–480, 2017.

[2] M. Fechtner and A. Kienle. Efficient simulation and equilibrium theory for adsorption processes with implicit adsorption isotherm - Ideal adsorbed solution theory. Chem. Eng. Sci., 177, 284–292, 2018.

[3] M. Fechtner, M. Kaspereit, and A. Kienle. Efficient simulation of ion exchange chromatography with application to bioseparations. Computer Aided Chem. Eng. 43, 295-300, 2018.

[4] M. Fechtner and A. Kienle. Equilibrium theory for ion exchange with steric hindrance and variable ionic strength of the fluid phase. Manuscript submitted to Chem. Eng. Sci..

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