Modeling of the SMB process

This chapter deals essentially with the apphcations of the theory of chromatography to the calculation of solutions of the SMB model in different cases of general interest. The theoretical tools required are a general model of the SMB process and a model for its columns. The former is an integral mass balance that is easy to write. The possible column models were described in the previous chapters. Finally, an accurate model of the competitive isotherms of the feed components is necessary. 

For the PDF model of the SMB process, full discretization was used, that is, both temporal and spatial variables were discretized leading to a huge system of algebraic equations. The standard SMB optimization problem has 33 997 decision variables and 33 992 equality constraints while the superstructure SMB optimization problem has 34 102 decision variables and 34 017 equality constraints. Note that there are many more degrees of freedom in the superstructure formulation (altogether 85) than in the standard SMB formulation (5 degrees of freedom). 

The models suitable for modeling of the SMB process (Fig. 9.8) can be classified accordingly if either a true countercurrent flow (TMB) for simplification is as-... 

In the recent years Simulated Moving Bed (SMB) technology has become more and more attractive for complex separation tasks. To ensure the compliance with product specifications, a robust control is required. In this work a new optimization bas adaptive control strategy for the SMB is proposed A linearized reduced order model, which accounts for the periodic nature of the SMB process is used for online optimization and control purposes. Concentration measurements at the raffinate and extract outlets are used as the feedback information together with a periodic Kalman filter to remove model errors and to handle disturbances. The state estimate from the periodic Kalman filter is then used for the prediction of the outlet concentrations over a pre-defined time horizon. Predicted outlet concentrations constitute the basis for the calculation of the optimal input adjustments, which maximize the productivity and minimize the desorbent consumption subject to constraints on product purities. 

In this study we identify an SMB process using the subspace identification method. The well-known input/output data-based prediction model is also used to obtain a prediction equation which is indispensable for the design of a predictive controller. The discrete variables such as the switching time are kept constant to construct the artificial continuous input-output mapping. With the proposed predictive controller we perform simulation studies for the control of the SMB process and demonstrate that the controller performs quite satisfactorily for both the disturbance rejection and the setpoint tracking. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

The overall model of an SMB process is developed by linking the models of individual chromatographic columns (Section 6.2). As with the chromatographic batch process, the plant set-up of Fig. 6.35 is converted into a simulation flowsheet. Figure 6.36 shows the SMB column model. 

Strube (1996) has shown that an increasing interaction of components or a decreasing number of columns per section results in significant deviations of both the calculated concentration profiles and purities. Model based optimal design requires a correct description of the dynamic behavior of the SMB process. Therefore, Diinnebier et al. (2000a) recommend the use of the detailed SMB model. These considerations are also valid for SMBR processes. Additionally, Lode et al. (2003a) have shown that the residence time calculated with the TMBR model differs from that in the SMBR model and, in consequence, different conversion rates are calculated. 

Kniep, H., Bliimel, C., Seidel-Morgenstem, A. Efficient design of the SMB process based on a perturbation method to measure adsorption isotherms and on a rapid solution of the dispersion model, Oral presentation at SPICA 98, Strasbourg, August 23-25th, 1998. 

There are simple algebraic solutions for the linear ideal model of chromatography for the two main coimter-current continuous separation processes. Simulated Moving Bed (SMB) and True Moving Bed (TMB) chromatography. Exphcit algebraic expressions are obtained for the concentration profiles of the raffinate and the extract in the columns and for their concentration histories in the two system effluents. The transition of the SMB process toward steady state can be studied in detail with these equations. A constant concentration pattern can be reached very early for both components in colimm III. In contrast, a periodic steady state can be reached only in an asymptotic sense in colunms II and IV and in the effluents. The algebraic solution allows the exact calculation of these limits. This result can be used to estimate a measure of the distance from steady state rmder nonideal conditions. 

Figure 6.36 Simulation flow sheet of the SMB process ( SMB column model ...

From this optimization with a detailed SMB model another very important aspect can be observed. Due to the simplifying assumption made to determine the operating diagram (Section 7.4.1.1), the real optimum of the SMB process does not coincide with the predicted theoretical optimum, as can be seen from Figure 7.26. 

Figure 7.36a and b proves the applicability of shortcut calculations based on the ideal equilibrium model for the estimation of process conditions. The results of rigorous process simulation based on the transport-dispersive model are in very good agreement with the shortcut calculation for isocratic (a) as well as nonisocratic (b) SMB processes. Expectedly safety margins have to be taken into account when the process conditions of an SMB process are estimated by shortcut calculation. The scattering of the numerical data results from an increased grid size for the numerical calculations that has been chosen in order to reduce computer time. The model parameters coincide with the data for the protein separation presented in Section 6.6.2.2.3 the separation quality of the SMB process was set to 99.9% purity. 

For the dynamic simulation of the SMB-SFC process a plug-flow model with axial dispersion and linear mass-transfer resistance was used. The solution of the resulting mass-balance equations was performed with a finite difference method first developed by Rouchon et al. [69] and adapted to the conditions of the SMB process by Kniep et al. [70]. The pressure drop in the columns is calculated with the Darcy equation. The equation of state from Span and Wagner [60] is used to calculate the mobile phase density. The density of the mobile phase is considered variable. 

The first modeling software which allowed for the optimization of nonlinear separations by SMB was presented in the early 1990s [46]. Today, numerous publications from academia allows one to have a better understanding of the SMB system [47-51]. Industry now has the practical tools for modeling SMB for quick and efficient process optimization [41, 52]. 

An SMB process is identified by using the subspace identification method. The input/output data-based prediction model is used to obtain the prediction model. The identified model exhibits an excellent prediction performance. The input/output data-based predictive controller based on the identified model is designed and applied to MIMO control problems for the SMB process under the presence of the input and output constraints. The simulation results demonstrate that the controller proposed in diis study shows an excellent control performance not only for the disturbance rejection but also for the setpoint tracking. 

Fig. 6 shows performance predictions obtained with the equilibrium-dispersive model for such single-column recycling with and without ideal solvent removal (TSR). The same requirements were used as in section 3. The process is basically infeasible without ISR. Also shown is the steady state performance of an SMB-based process (6 columns, ISR cf Fig. 3a). As is often found, the SMB process achieves a lower productivity, but at the same time allows for significantly lower solvent consumption.

Other control studies, such as robustness and other control strategies, will be carried out in next works. Although the SMB control was carried out in simulation based on a realistic model of the process, the application of these control strategies to a real SMB for validation purposes should be done. 

The benefits of model-based control strategies for the operation of SMB processes are demonstrated in Chapter 9. This is a rather new concept as, in today s industrial practice, SMB processes are still controlled" manually, based on the experience of the operators. A nonlinear model predictive (NMP) controller is described that can deal with the complex hybrid (continuous/discrete) dynamics of the SMB plant and takes hard process constraints (e.g. the maximal allowable pressure drop) and the purity requirements into account. The NMP controller employs a rigorous process model, the parameters of which are re-estimated online during plant operation, thus changes or drifting of the process parameters can be detected and compensated. The efficiency of this novel control concept is proven by an experimental study. 

Chromatographic separations with product purities exceeding 99% and high separation costs require precise prediction of the optimal process design. This demands carefully validated models, especially in the separation sections of the SMB plant. Consequently, methods are needed to increase the number of measured data points. 

Subsequently, methods for the model-based design and optimization of batch and SMB (Simulated Moving Bed) processes are introduced. For this purpose dynamic simulations of an experimentally validated model are used. When a model-based approach is not affordable, as for example in the design and optimization of complex SMB processes, a short-cut design and optimization method for SMB is introduced. Finally, the performances of both batch and SMB chromatography are compared. 

Due to the simplifying assumptions made for the TMB approaches, accurate design and optimization of SMB processes is not possible. Several approaches based on SMB models have been suggested to improve the prediction and optimization of the SMB operation. Zhong and Guiochon (1996) have presented an analytical solution for an ideal SMB model and linear isotherms. The results of this ideal model are... 

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