Modelling and state observation of Simulated Moving Bed processes based on an explicit functional wave form description
Introduction
The Simulated Moving Bed (SMB) is a continuous chromatographic separation process for mixtures of two components. It is based on the counterflow principle between the solvent and the adsorbent. The counterflow is simulated by the discrete shifting of the solvent and feed inlet ports and the product outlet ports on a circle of chromatographic separation columns. The two components of the mixture, which are separated by the SMB plant, are distributed over the circle of separation columns. These distributions are represented by concentration profiles which have the form of waves. Their spatial range is limited by one front and one rear wave front. Because of the circulating solvent stream, the concentration profiles perform a cyclic movement through the circle of separation columns.
With respect to unit operation, it is the aim to recover the products with a pollution that is below given thresholds. These values are usually prescribed by the recipe of the production line, in which the SMB process is included. Therefore, the inlet and outlet ports have to be switched in accordance with the position of the wave fronts. The knowledge of the current wave front positions is crucial for the SMB process control. This paper concerns their determination from discrete measurement values.
The physical model of the process comprises a set of convection–diffusion equations which are coupled by cyclic boundary conditions. The initial conditions show switching behaviour [9]. The concentration profiles, which are the solution of the physical model, are time-varying because of their spatial propagation. The pollution of the product streams is determined by the position and movement of the wave fronts of the concentration profiles. If the form, the movement and the position of the wave fronts are available, e.g. by a state observation, the control of the product pollution can be performed. Because of the limited possibility to measure the concentrations in an SMB plant, a simple process model is necessary for the solution of the observation task. Such a model can be obtained from the application of the explicit functional form approach. However, this yields the problem of the derivation of an analytic solution of physical SMB process model, which is a complex task [5], [8], [13]. Butkovskiy and Pustyl’nikov [1] proposes explicit solutions to chromatographic models with cyclic boundary conditions. However, these solutions are very difficult to handle. A means for simplifying this task is to consider the approximation of the True Moving Bed (TMB) process (cf. Section 2) by the SMB process. This is possible because the concentration profiles of the SMB process and the TMB process are similar, if the SMB process has at least eight separation columns and, thus, is a good approximation of the TMB.
In several publications on observation and control of SMB processes, the similarity of the SMB and the TMB is used to represent the SMB process behaviour [6], [12], [19]. Other publications encounter the mixed continuous and discrete dynamics of the SMB process [3], [22]. However, all the model-based state observers lead to complex algorithms because the observers are derived from the complex process models.
In this contribution, a new approach to SMB process modelling and state observation is presented. It uses the aforementioned similarity of the TMB and the SMB concentration profiles and leads to a strongly simplified model. Considering the convection–diffusion equation as a physical process model, the stationary solution of the TMB model can be explicitly determined [10]. The presented approach to SMB modelling uses the stationary TMB solution to describe the characteristic form of the SMB wave fronts. A spatial shift is applied to model the propagation of the SMB concentration profiles that takes place between two port switching events. A wave front model is obtained which encounters the functional form description, the continuous spatial shift and the switching of the profiles in the moment of port switching. In this simplified version, the form, propagation, and position of the wave fronts is determined by three parameters. It is possible to derive a dynamical model of these parameters which encounters linear time-varying discrete-time dynamics. Based on this model, a state observer for the wave fronts of an SMB process is proposed that uses selected measurements of the SMB concentration curves.
This paper is structured as follows. The SMB process and the TMB process and their characteristic behaviour are described in Section 2. In Section 3, the problem of the state observation and the control of SMB processes is addressed. A common physical model for the two processes is described in Section 4. In Section 5, the explicit solution of the stationary TMB process model is derived. It is used to present an approximation of the stationary SMB process behaviour. The derivation of the wave front model is described in detail in Section 6. Section 7 presents the wave front observer. Simulation studies described in Section 8 show the applicability of the prepared observer to an example SMB plant. In Section 9, a brief investigation of the modelling error is presented.
Section snippets
Simulated Moving Bed principle
The Simulated Moving Bed principle is widely applied to the preparative chromatographic separation of binary mixtures because it allows for the continuous separation of two fractions. It is based on the counterflow between the adsorbent and the solvent. The true counterflow cannot be realised in practice. For an implementation on a technical scale, the counterflow has to be simulated by moving the input and output positions of the mixture and the products on a circle of separation columns.
State observation and control of SMB processes
With the SMB process, the continuous separation of binary mixtures into two fractions is performed. The main aim is to keep the pollution of the product streams below a prescribed value. SMB processes exhibit the following properties:
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The operation point is very sensitive to disturbances.
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The time constants lie in the range of hours, which leads to a large delay of the perturbation impact.
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The options of concentration measurements are very limited.
Because of these points, it is desirable to
Physical modelling of TMB and SMB processes
The physical modelling of chromatographic separation processes is based on the mass balances of infinitesimally small volume elements of the separation column. The balances yield fluid dynamic models describing the mass transport. Several publications treat the modelling of chromatographic processes [2], [4], [7], [18], [20], [21]. For the derivation of a chromatographic process model, the combination of different kinds of mass transport and adsorption mechanisms in the solvent, the adsorbent
Approximate solution of the SMB model
This section proposes a new approach to the description of the SMB concentration curves by considering the stationary solution of a physical TMB model. The basic idea is to use the stationary state solution of the TMB model to approximate the form of the concentration profiles. The solution is extended later on to describe the movement of the SMB concentration curves.
SMB wave front modelling task
The wave fronts should be represented in an accuracy that is appropriate to the solution of the wave front observation task. Therefore, a wave front model shall
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represent the form, movement and position of the wave fronts , , for all , , each over the range of the SMB separation column through which the wave front propagates,
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encounter the variation of the wave front form, movement and position during the dynamic transition of the SMB process,
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represent the
Determination of the wave fronts by state observation
With the wave front model at hand, it is possible to solve the wave front observation task. As the linear wave front model has the same structure for each of the four SMB wave fronts, the wave front observer derived in this section applies to each of these wave fronts and the index i is omitted.
Application example
To show the capability of the proposed wave front observer, the principle is applied to the SMB process shown in Fig 5. This SMB process has eight separation columns and two columns per SMB section. The operation point was chosen such that the purity of the the product streams are larger than 99.9% and the switching time was . The measurement positions are chosen as shown in Fig. 9 and the measurement times are selected as follows:
Modelling error
The derivation of the functional description of the SMB wave fronts is based on the assumption of a good approximation of the TMB by the SMB. It was assumed that the chromatographic process in the SMB and also in the TMB is represented by the convection–diffusion equation. Furthermore, for the derivation of the wave front model, Assumption 6.1, Assumption 6.3 were made. Because these are assumptions, which are not met exactly in practical applications, there will always be a deviation between
Conclusion
The concern of this contribution is the modelling and state observation of Simulated Moving Bed processes. With respect to physical process modelling, the SMB is described by a set of convection–diffusion equations with switching initial conditions and cyclic boundary conditions. Because of the complexity of the process and the limited options of concentration measurements, it is desirable to implement a closed-loop control of the SMB operation based on a state observer. The herein presented
Acknowledgements
The authors would like to thank Prof. Dr. Ing.-G. Brunner and Dipl.-Ing. S. Peper (Institute for Thermal Separation Processes, Technical University Hamburg–Harburg) for providing all necessary data for the numerical simulation of the example SMB process. Special thanks applies to Prof. Thomas Kriecherbauer (Institute for Applied Mathematics, Ruhr-Universität Bochum) for the discussions and his help in deriving the explicit stationary solution of the TMB process model.
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