A theoretical study of continuous counter-current chromatography for adsorption isotherms with inflection points
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 -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:
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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.
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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).
Section snippets
The principle of continuous counter-current chromatographic processes
Continuous counter-current chromatographic processes can be well described by the true moving bed (TMB) model Rajendran et al., 2009, Ruthven and Ching, 1989. To illustrate the principle and to establish the notation used in this paper we provide below a mathematical formulation of an equilibrium stage model for the classical -zone continuous counter-current chromatographic process.
In a chromatographic process the separation is based on a different adsorption behavior of the components and
Isotherms based on statistical thermodynamics
A main aspect of modeling a chromatographic separation process is to find a suitable description of the relation between the liquid-phase concentrations and the corresponding solid-phase concentrations for equilibrium conditions. A simple way to model such a relation is to assume a linear relationship between the liquid-phase concentration of a component and its concentration in the solid-phase,where stands for the Henry constant of the th component that
Separation regions
An important question during the design of continuous counter-current chromatographic processes is to find suitable values of the dimensionless flow-rate ratios . Significant contributions in this direction have been made for linear and Langmuir isotherms (Mazzotti, 2006, Storti et al., 1993). For (i) linear isotherms, (ii) complete separation () and (iii) an infinite number of plates, Storti et al. (1993) developed the so-called triangle theory: For given values of and the
A computational study for second-order isotherms by a scanning technique
In this section we will apply at first a conventional scanning technique to identify the separation regions of continuous counter-current chromatographic processes with second-order isotherms as a function in three variables, namely the number of plates per zone, the purity requirements and the feed concentrations . For this, we consider three series of symmetric -zone TMB test instances:
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Test series TS1 is based on a TMB process with a relative small number of plates per zone (
Alternative approach: proving infeasibility regions via linear relaxations
In this section we will apply a relaxation technique that is based on ideas from global optimization(e.g., see Haus et al., 2007, McCormick, 1976, Tawarmalani and Sahinidis, 2004). In contrast to the scanning technique the alternative relaxation approach does not focus on the feasibility of a certain point but on the infeasibility of subdomains. Therefore, this approach will lead to the negative image of the results obtained by the scanning technique described above. This already indicates one
Conclusion
We investigated continuous counter-current chromatographic processes with second-order isotherms possessing inflection points. A parametric study was carried out using a TMB model and considering different number of plates per zone, different purity requirements and different feed concentrations. Compared to the separation regions valid for Langmuir isotherms having no inflection points, the shapes of the separation regions identified in this work are more elaborate.
The separation regions were
Acknowledgments
This work was supported by grant FOR 468 of the German Science Foundation (DFG). The first author was financially supported by the Research Focus “Dynamical Systems in Biomedicine and Process Engineering” of the Saxony-Anhalt Ministry of Education, Germany.
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