Triangle theory

Currently, the most successful methodology for the optimization of an SMB s performance is the so-called triangle theory, which was recently also applied to the SMBR [158]. The analysis was based on a mathematical model describing the esterification of acetic acid and ethanol into ethyl acetate and water in a fixed-bed chromatographic reactor [159]. A mixture of ethanol and acetic acid is intro-... 

Keywords Preparative chromatography, Simulated moving bed chromatography, Continuous separation technique, Triangle theory, Bioseparation... 

For the simulation of SMB-separations efficient software packages,based on the Triangle-Theory, are commercially available. The number of columns, the column dimensions, the theoretical number of plates in the columns, the feed concentration, the bi-Langmuir adsorption isotherm parameters and the number of cycles need to be defined by the user. Then the separation is simulated and values for the flow rate ratios, the flow rates, the switching time and the quality of the separation, purity and yield, are calculated. Based on these values an actual separation can be performed. However, some optimization/further development is usually necessary, since the simulations are based on an ideal model and the derived parameters and results therefore can only be taken as indications for the test runs. 

The triangle theory is acknowledged as one of the most effective design tool for SMB. Although this theory neglects mass transfer resistance and axial dispersion, it provides a useful approximation of the real SMB behavior. 

This section presents a systematic strategy for the optimization of operating parameters, based on the transport-dispersive SMB model (Chapter 6.7). A first approximation is performed by applying the triangle theory described above. Notably, the basic strategy of this procedure is not limited to a model-based design but can also be applied for the experimental optimization of a running separation process. 

Here Vsoiid is the apparent solid flow rate, HA and ffB describe the slopes of the adsorption isotherm, which are calculated in the nonlinear case by linearization of the adsorption isotherm for the feed concentration Cfeed, . The transformation reflects the fact that, in a counter-current process, it is not the net flow rates that are important but rather their values relative to the apparent solid movement. For this reason, Morbidelli et al. introduced the m factors in their graphical design (known as the triangle theory) (Biressi et al., 2000 and Mazzotti et al., 1997c). 

In Chapter 17, we discussed the optimization of the flow rate ratios in the four zones of the SMB process and that of the switching time. The triangle theory allows the determination of the optimum conditions for maximum production rate and minimum eluent consumption. Due to the complexity of the simulated moving bed process, most current studies limit studies on the optimization of an SMB unit operation to investigating the influence of these parameters. Few data are available on the optimization of many other experimental parameters e.g., pressure drop, product purity) and column design conditions e.g., column length, particle size, efficiency) or on that of the column configuration (optimum number of columns in the individual zones). 

Based on the triangle theory for TMB reactors Lode et al. (2002, 2003) as well as Fricke et al (2003, 2005) developed short cut methods for the prelirninary design of SMB reactors. They derived analytical solutions for different types of reactions by taking into account the conditions for different subdivisions of the separation region. 

Before Varicol, PowerFeed, or Modicon is taken into account for process design make sure that appropriate optimization tools are at your disposal. In contrast to SMB, no shortcut methods such as triangle theory are available for these processes. 

It is obvious that optimal separation conditions can hardly be attained by trial and error methods. Thus it is necessary to use methods that permit the simulation of the process. In recent years TMB and SMB process models, respectively, in combination with the triangle theory after Storti, et al. [6] became accepted. An overview of the basic methods was given by Ruthven and Ching [7]. 

Pedeferri et al. [23] separated the enantiomers of Troger s base in an SMB plant with eight columns (250 x 4.6 mm). By variation of the process parameter the separation behavior of the SMB plant was analyzed against the background of the triangle theory. 

The starting point for the SMB modeling is the modeling of a chromatographic separation on a single column. The triangle theory , which can be used to determine a suitable set of parameters for the operation of an SMB process with some simplifying assumptions, will be described in detail. 

Because of the analogy between the TMB and the SMB process, this model can be applied for the determination of the operating parameters of an SMB process in the case of certain adsorption isotherms. In SMB-HPLC this so-called triangle theory is established for the generation of operating parameters because it enables the estimation of operating points by means of simple equations with analytical solutions. This method, and its expansion to the conditions of SFC, will be shown in more detail in Section 9.4.2.4. 

If the method of the triangle theory is applied in this case, the region of complete separation can be determined in the same way as described before. However, the shape of the region of complete separation changes. 

If the separation regimes for the gradient operation are compared to the one for the isobaric operation with respect to the purities of the products, the separation regions described for the isobaric case can be found. Notable are the new regions 7 to 9, in which one or both components accumulate in the apparatus. The assumptions of the triangle theory of linear isotherms and infinite solubility of the components lead to a nonsensical prediction of a permanent accumulation in the apparatus. In reality, the accumulating component will break through after a certain time. 

For modeling of real SMB-SFC processes the results of the triangle theory are applicable only in a limited way. But the model provides a boundary case for the validation of the dynamic SMB-SFC model. Furthermore, the application of the triangle theory enables the derivation of parameters for the evaluation of the quality of products and the separation process. 

Before analyzing the effect of operating variables on the process performance, the model for the dynamic simulation of the SMB-SFC process has to be verified. For validation of the SMB-SFC model the triangle theory illustrated in Section... 

For comparison of the separation regions predicted by the triangle theory with the results of the dynamic simulation, the mass flow rates in zones 2 and 3 are varied systematically. A low feed concentration of 0.05ml/min was chosen in order to operate in the linear range. The purities resulted by the simulation will be recorded. A product is assumed to be pure if it has a purity of more than 99.9%. 

In Fig. 9.13 a comparison of the separation regions predicted by the triangle theory with those resulting from dynamic simulations is shown. The agreement between simulation and triangle theory is excellent despite the restriction to 500 theoretical stages per column. 

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