The Steady State TMB Model

Increasing the switch time interval is equivalent to decrease the solid flow rate and the net fluxes of components in all sections of the TMB unit will be pushed in the same direction of the liquid phase. This implies that, first, the more retained species will move upwards in section III and will contaminate the raffinate stream and the less retained species will move upwards in section IV, will be recycled to section I, and will contaminate also the extract stream. The decrease of the switch time interval will have similar consequences. The equivalent solid flow rate will increase and the net fluxes of component in all four sections of the TMB unit will be pushed in the opposite direction of the liquid phase. This implies that, first, the less-retained species will move downwards in section II and will contaminate the extract stream and the more retained component will also move downwards in section I, will be recycled with the solid to the section IV, and will contaminate the raffinate stream. It is possible to obtain simultaneously high purities and recoveries in a SMB, but the tuning must be carefully carried out. 

The influence of the mass transfer resistance on the purity and on the steady state internal concentration profiles are shown in Figs. 9-11 and 9-12. A higher value for the mass transfer coefficient corresponds to a situation where mass transfer resistance is less important, and a better performance of the SMB will be obtained with sharper internal concentration profiles. 

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The steady state TMB model equations are obtained from the transient TMB model equations by setting the time derivatives equal to zero in Equations (25) and (26). The steady state TMB model was solved numerically by using the COLNEW software [29]. This package solves a general class of mixed-order systems of boundary value ordinary differential equations and is a modification of the COLSYS package developed by Ascher et al. [30, 31]. 

The steady state TMB package was used to compare the theoretical and experimental internal concentration profiles in Fig. 9-19. Figure 9-20 shows the transient evolution on the concentration of both species in the raffinate. Average concentrations over a full cycle were evaluated experimentally for cycles 3, 6, 9, 12, 15, and 18. Also shown are the corresponding SMB model predictions. The agreement between them is good and the cyclic steady-state, in terms of raffinate concentrations, is obtained after 10 full cycles. 

Since the TMB and SMB configurations are nearly equivalent, i.e., since they achieve the same separation performance provided geometric and kinematic conversion rules are fulfilled, the simpler model of the equivalent TMB unit can be used to predict the steady state separation performance of SMB units. The conversion rules are given by the following relationships ... 

If an SMB process is discretized by an increasing number of columns in the functional zones, the concentration profile converges to that of the TMB model. Thus, the TMB model represents a boundary case of the simulated moving-bed process. If, additionally, only the solution in the steady state of the system is considered, the balance equations in the formulation of the stage model can be simplified in a way that only one nonlinear system of equations has to be solved. Such... 

The equivalent TMB operating conditions and model parameters for the reference case were given in Table 9-1 and Fig. 9-9 presents the corresponding steady state internal concentration profdes obtained with the simulation package. The extract and raffinate purities were 97.6 % and 99.3 %, respectively the recoveries were 99.3 % and 97.6 % for the extract and raffinate streams. The solvent consumption was 1.19 L g and the productivity was 68.2 g/day - L of bed. 

With small interactions between different components and a sufficiently large number of columns in each section a TMB model can be used to approximate the mean concentration profile of a periodic steady state SMB separation process. 

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. 

If the number of columns is increased, the process characteristics gradually approach to the characteristics of a hypothetical process, in which the solid and liquid phases are moving continuously in countercurrent directions. This hypothetical process is designated as a true moving bed (TMB) process. It reaches a real steady state that can be described mathematically much easily compared to the approach described above based on exploiting a dynamic model (Liapis and Rip-pin, 1979 Ruthven and Ching, 1989 Barker and Ganetsos, 1993). 

The main difference between the TMB and SMB approaches is related to the stationary regime. The time dependence of the boundary conditions in the SMB leads to a cyclic steady state instead of a real steady state as occurs in the TMB model. The cyclic steady state is reached after a certain number of cycles, but the system states are shll varying over time because of the periodic movement of the inlet and outlet ports along the columns (Fig. 3.4-11). 

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