The TMB Model

The chromatographic resolution of bi-naphthol enantiomers was considered for simulation purposes [18]. The chiral stationary phase is 3,5-dinitrobenzoyl phenyl-glycine bonded to silica gel and a mixture of 72 28 (v/v) heptane/isopropanol was used as eluent. The adsorption equilibrium isotherms, measured at 25 °C, are of bi-Langmuir type and were proposed by the Separex group  

The operating conditions and model parameters used in simulation for the TMB approach are presented in Table 9-1. The feed concentration of each enantiomer is 2.9 g L and columns were 2.6 cm wide and 10.5 cm long. The section length was 

Solid flow rate Recycling flow rate Eluent flow rate Extract flow rate Eeed flow rate Raffinate flow rate  

Solid/fluid volumes, (1 - e)/e = 1.5 Ratio between fluid and solid velocities = 6.65 r =4.23 

---

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 ... 

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 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). 

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... 

Due to the analogy between TMB and SMB the operating parameters of the processes can be converted into each other. Thus, from the solution of the TMB model operating parameter of an SMB process can be obtained. The rules for calculation are given by Mazzotti [57] ... 

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... 

The solution of the TMB model is given by a set of inequalities, where the complete separation of both components is required as a further condition. For several usual isotherm equations analytical terms for these inequalities exist. In the simplest case of linear isotherms these equations are ... 

---