The SMB Model

In the TMB model, the adsorbent is assumed to move in plug flow in the opposite direction of the fluid, while the inlet and outlet lines remain fixed. As a consequence, each column plays the same function, depending on its location. An equivalence between the TMB and the SMB models can be made by keeping constant the liquid velocity relative to the solid velocity, i.e., the liquid velocity in the TMB is  

Introducing the dimensionless variables x = t/L. and 9 = t/r, with = Lj/u = where is the solid space time in a section of a TMB unit, L. is the length of a TMB section, and is the number of columns per section in a SiWb unit, the model equations become  

The initial and boundary conditions are the same presented before and, for x = 0, Equation (20) becomes  

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Mathematically, the SMB model is achieved by connecting the boundary conditions of each column model, including nodes represented by material balances of splitting or mixing models. These so-called node models (Ruthven and Ching, 1989) are given for a component i in the sections I-IV by ... 

Two different modeling approaches are used for simulated moving bed reactors. The first approach combines the model of several batch columns with the mass balances for the external inlet and outlet streams. By periodically changing the boundary conditions the transient behavior of the process is taken into account. The model is based on the SMB model introduced in Chapter 6 and is, therefore, referred to as the SMBR model. The second approach assumes a true counter-current flow of the solid and the liquid phase like the TMBR. Therefore, this approach is called the TMBR model. 

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. 

Beste et al. [104] compared the results obtained with the SMB and the TMB models, using numerical solutions. All the models used assumed axially dispersed plug flow, the linear driving force model for the mass transfer kinetics, and non-linear competitive isotherms. The coupled partial differential equations of the SMB model were transformed with the method of lines [105] into a set of ordinary differential equations. This system of equations was solved with a conventional set of initial and boundary conditions, using the commercially available solver SPEEDUP. Eor the TMB model, the method of orthogonal collocation was used to transfer the differential equations and the boimdary conditions into a set of non-linear algebraic equations which were solved numerically with the Newton-Raphson algorithm. 

The SMB model consists of partial differential equations (PDFs) for the concentrations of chemical components, restrictions for the connections between different columns and cyclic steady-state constraints (Kawajiri and Biegler, 2006b). Previously, the SMB processes have been usually optimized with respect to one objective only. Recently, multi-objective optimization has been applied in periodic separation processes (Ko and Moon, 2002), in gas separation and in SMB processes (Subramani et al., 2003). Ko and Moon used a modified sum of weighted objective functions to obtain a representation of the Pareto optimal set. Their approach is valid for two objective functions only. On the other hand, Subramani et al. applied EMO to a problem where they had two or three objective functions. 

We can say that interactive methods have not been used to optimize SMB processes and, usually, only one or two objective functions have been considered. The advantages of interactive multi-objective optimization in SMB processes has been demonstrated in Hakanen et al. (2008, 2007) for the separation of fructose and glucose (the values of the parameters in the SMB model used come from Hashimoto et al. (1983) Kawajiri and Biegler (2006b)). In Hakanen et al. (2008, 2007), the problem formulation consists of four objective functions maximize throughput (T, m/h ), minimize consumption of solvent in the desorbent stream (D, m/h ), maximize product... 

Considering an SMB design problem with four objective functions was a novel approach because, previously, only two or three objective functions had been considered. This enabled full utilization of the properties of the SMB model without any unnecessary simplifications. In addition, the DM obtained more thorough understanding of the interrelationships of different objectives considered and, thus, learned more about the problem. 

The examples for experimental validation of the SMB model are based on the extended model (Figure 6.37) that takes into account the fluid dynamic effect of piping, especially recycle lines and other peripheral equipment such as measurement devices. From point of process simulation these are additional elements of the plant that have to be regarded within the overall flow sheet. 

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. 

Qj volume flow rate in zone j for the TMB model QjSmB volume flow rate in zone j for the SMB model Qs volume flow rate of the solid... 

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