Components and Separable Algebras

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

This number can be reduced to 12 by transforming the coordinates [20] so that terms in the momentum components of the center of mass are separated. As they must remain constant, these terms can be ignored. If account were also taken of the constancy of total angular momentum, the number of equations could be decreased still further, but the algebra required is formidable and conservation of angular momentum throughout the calculated trajectory can be used to check the accuracy of the computation. 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

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. 

Compute Mixing. This corresponds to the ideal flow pattern in Fig, 20,6-2e, Practically, this flow pattern rarely is realized except in a flat or spiral-wound separator operated at low cuts and low feed rales. Nevertheless, it is useful mainly doe to the algebraic simplicity of the corresponding model equations. Stem and Walawendei have reviewed threa different methods for modeling this flow pattern.u The eqontions derived by Weller and Steiner for a binary-component feed are summarized below.13... 

We neglect the energy balance. We chose the recycle as tear stream, so that the computational sequence is Mixer-Reactor-Separator. The tear stream has been cut explicitly in two parts, the streams 4 and 5. The convergence is obtained when the difference in component flow rates between the streams 4 and 5 is less than a prescribed tolerance. Let us denote the molar flow rates of the components. 4, fi, C in the stream 5 by, Fg, F(-. The convergence condition leads to the following algebraic equations ... 

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