A theoretical study of continuous counter-current chromatography for adsorption isotherms with inflection points

Abstract

Continuous counter-current chromatographic processes have been increasingly applied in the last years. For chromatographic systems characterized by linear or Langmuir adsorption equilibria, there are nowadays reliable design rules available to decide whether a separation is feasible and which operating parameters should be used. For more complex equilibrium functions, theoretical methods are less developed. More realistic adsorption isotherms are frequently characterized by inflection points in their courses. A flexible model capable to quantify single solute and competitive adsorption isotherms is provided by statistical thermodynamics. In this work, second-order truncations of a statistical model involving inflection points are investigated. The classical scanning technique is used to identify the region of applicable operating parameters. Then an alternative approach is suggested, which verifies the existence and shape of the suitable parameter region by infeasibility certificates. Using the True Moving Bed model, both approaches are compared for different feed concentrations, purity requirements and column efficiencies.

Introduction

Among the different unit operations available, column chromatography has been established as one of the methods of choice for the isolation and purification of valuable products. Due to the costs of the stationary phases and solvents involved, it is an expensive technique. Increasing the effectiveness of chromatographic processes in terms of stationary phase usage and solvent consumption can help reducing these expenses. An important step in this direction was the introduction of the continuous simulated moving bed (SMB) technique by Broughton and Gerhold (1961). Utilizing a counter-current movement of liquid and solid phases, emulated by a periodic switching of port or column positions in a multicolumn arrangement, a binary mixture can be split continuously into two product streams. In its classical implementation, SMB chromatography can meet high purity requirements together with high productivities and comparably low solvent consumption. The initial impact of this technology outside of petrochemical applications was limited. In the 1990s, the process experienced a renaissance. One of the driving forces behind this development was the realization that the well-known equilibrium theory of chromatography can be used as a valuable theoretical design tool for this specific process (Rhee, Aris, & Amundson, 1986/1989). A driving force for further developing SMB processes were the first successful applications to separate enantiomers, which was recently reviewed by Rajendran, Paredes, and Mazzotti (2009). During the last two decades, new and refined derivatives of the original SMB approach were developed, e.g. the FAST process for sugar separations, VariCol or ModiCon. Several of these (and other) process modifications have in common that a chosen operating parameter, kept constant in the original process, is varied dynamically during separation. In a different class of derived processes, the permanent feeding and withdrawal of streams realized in the classical SMB process was replaced by more sophisticated regimes that introduced on/off characteristics to either inlet or outlet ports. An overview regarding modern operating regimes of SMB-chromatography was given recently by Seidel-Morgenstern, Keßler, and Kaspereit (2008).

A crucial part in the analysis of SMB processes is to determine appropriate isotherms that reflect the relation between the concentration of the components in the solid and in the liquid-phase. In practice, linear, Langmuir and bi-Langmuir isotherms are widely used for large classes of separation processes and well-studied in the literature, e.g., Guiochon, Felinger, Shirazi, and Katti (2006). For many applications, these isotherms are, however, not sufficient since they do not allow to model inflection points. One type of isotherms that allows to model inflection points are isotherms based on statistical isotherms of an order that is equal or larger than two (see Hill, 1960).

In this paper we will study the classical $4$-zone continuous counter-current chromatographic process with second-order adsorption isotherms using the simplifying true moving bed (TMB) model (e.g. Ruthven & Ching, 1989). It assumes a true counter-current movement of the solid-phase and reaches a real steady state allowing to model the process using the equilibrium cell model with algebraic equations. In our study we will vary the number of stages or plates (i.e. the column efficiency), the purity requirements and the feed concentrations. In particular, we will compare the shapes of feasible separation regions in terms of dimensionless flow-rate ratios. This requires for each situation the determination of the feasible region which is a non-trivial task as the underlying mathematical model is highly non-convex. Hence, a corresponding optimization problem is very difficult to solve, globally.

We will approximate the shapes of the feasible separation regions from two sides:

• First, we apply a conventional scanning technique over a grid of the flow-rate ratios where we evaluate each grid point w.r.t. feasibility (i.e. the potential of the continuous counter-current chromatography to fulfill the process goal for given operating parameters). In this way, a set of feasible points is identified.

• In a second step, an alternative concept is introduced to solve the same problem. A family of linear relaxations for the underlying nonlinear models is constructed and solved. If a relaxed model is infeasible over a certain subdomain of operating parameters then the original TMB model is infeasible over this subdomain, as well. Thereby, we can mark this subregion definitely as infeasible, i.e. the goal of the separation process is not reachable.

This two-step procedure is similar to the procedure that has been applied by Haus, Michaels, Seidel-Morgenstern, and Weismantel (2007) to evaluate linear continuous counter-current chromatographic processes for the separation of different mixtures of sugars w.r.t. feasibility. Here, we extend the previous study to more complicated adsorption isotherms.

This paper is structured as follows: In Section 2 we give a brief overview of the main ideas of continuous counter-current chromatographic processes. Then we introduce several ways for modeling isotherms and discuss their advantages and disadvantages in Section 3, and summarize the theory regarding the corresponding separation regions in Section 4. We will point out that there is currently no theory available to predict separation regions for isotherms that posses inflection points. In order to deduce the behavior of processes with such isotherms we will apply first a classical scanning technique in Section 5 where we investigate the influence of the number of separation stages per zone, purity requirements and feed concentrations on the shape of the separation region by considering a numerical example. In Section 6 an alternative relaxation technique is presented which was found to be very useful in the field of global optimization (Tawarmalani & Sahinidis, 2002).