Optimization linear isotherm

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... 

Abel, S., Erdem, G., Mazzotti, M., Morari, M., Morbidelli, M. Optimizing control of simulated moving beds - linear isotherm, J. Chromatogr. A, 2004b, 1033, 229-239. 

Optimization of an SMB System with a Linear Isotherm, Using the Safety Margin... 

Optimization of the SMB Process with a Linear Isotherm Using the Triangle... 

Application of concentration-independent HETP coefficients from the linear isotherm region in calculating the number of stages in a nonlinear region is only a formalism. However, since feed concentration is not varied during the optimization, the calculated number of stages can nevertheless be used further as it is a characteristic value for the specific chromatographic separation. Scale-up of the... 

Optimization Strategy for Linear Isotherms Figure 7.23 shows the internal axial concentration profiles at the end of a switching interval for a system of linear isotherms and no competitive interaction of the two components. After switching all ports downstream in the direction of the liquid flow, the initial concentration profile is represented by the upper gray line. In this case the extract will be polluted with component B because the desorption front of B violates point 2 (Figure 7.18). 

Thus, knowing the particle characteristics, R and e, and the slope of the linear isotherm, we can readily determine the effective diffusivity D. This method can be conveniently carried out with a simple linear plot, without recourse to any numerical optimization procedures (provided, of course, that the capacity ratio B is known beforehand). 

Because of the extremely low frequencies of the base isolated building, the gas compression might be close to isothermal conditions. A subsequent state space optimization quickly renders the optimal natural frequencies slightly lowered and the optimal linearized damping coefficients reduced. The increase in effective... 

The optimization of gas chromatographic temperature programs is similar to optimization of isothermal temperature or carrier-gas linear velocity criteria that... 

A. Gentilini, C. Migliorini, M. Mazzotti and M. Morbidelli, Optimal operation of simulated moving-bed units for non-linear cliromatograpliic separ ations. II. Bi-Langmuir isotherm , 7. Chromatogr. 805 37-44 (1998). 

Gentilini A., Migliorini C., Mazzotti M., Morbidelli M. (1998) Optimal Operation of Simulated Moving-Bed Units for Non-Linear Chromatographie Separations. II. Bi-Langmuir Isotherm, J. Chromatogr. A 805 37-44. 

The main difference between the chromatographic process carried out in the linear and the nonlinear range of the adsorption isotherm is the fact that in the latter case, due to the skewed shapes of the concentration profiles of the analytes involved, separation performance of a chromatographic system considerably drops, i.e., the number of theoretical plates (N) of a chromatographic system indisputably lowers. In these circumstances, all quantitative models, along with semiquantitative and nonquantitative rules, successfully applied to optimization of the linear adsorption TLC show a considerably worse applicability. 

To optimize a given preparative chromatographic process (highest productivity, lowest mobile phase consumption, and product dilution) the separation has to be performed with the highest product concentrations still compatible with the system. One immediate consequence of this is that each column has to be operated in the non-linear range of the adsorption isotherm. 

The overloading of the stationary phase is related to the maximum solute concentration. Cm, at which the sorption isotherm associated with equilibrium distribution underlying chromatographic retention ceases to be linear. That deviation results in a broadening and deformation of the peak profile. Since this review deals with chromatographic phenomena and optimization we consider thermodynamics as beyond its scope. 

A similar argument holds for the influence of the peak shape on the separation criterion. In the non-linear part of the distribution isotherm, the shape of the peak will be a function of the injected quantity. Hence, once again, the location of the optimum may be affected by the composition of the sample. Also, the effect of column dimensions on the peak shape may be hard to predict, and the peak shape may to a large extent be determined by the characteristics of the instrument, rather than of the column. Therefore, if the composition (or the concentration) of the sample can be expected to vary considerably, and if it is desirable that the result of an optimization process can be extrapolated to different columns (of the same type) and to different instruments, then it is advisable to use criteria that are not affected by the relative peak areas, nor by the shape of the peaks. 

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

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