Multicomponent Langmuir isotherm

However, when complexation experiments are performed with both d- and L-enan-tiomers (Fig. 5-18), this leads to selectivity values between 1.4 and 1.9. It was shown that complexation by enantioselective micelles can effectively be described using straightforward multicomponent Langmuir isotherms [74]. 

Fig. 5-18. Selectivity for D,L-phenylalanine, fitted with a multicomponent Langmuir isotherm [73],...

For constant-separation factor systems, the /(-I rails formal ion of Helfferich and Klein (gen. refs.) or the method of Rhee et al. [AlChE J., 28, 423 (1982)] can be used [see also Helfferich, Chem. Eng. Sci., 46, 3320 (1991)]. The equations that follow are adapted from Frenz and Horvath [AlChE ]., 31, 400 (1985)] and are based on the h I ransiomialion. They refer to the separation of a mixture of M — 1 components with a displacer (component 1) that is more strongly adsorbed than any of the feed solutes. The multicomponent Langmuir isotherm [Eq. (16-39)] is assumed valid with equal monolayer capacities, and components are ranked numerically in order of decreasing affinity for the stationary phase (i.e., Ki > K2 > Km). 

Fig. 5.9. Pathgrid of the eigenvectors of the transformed equilibrium function of a reactive chromatographic process with a reaction of type 2 A B + C in the liquid phase and a mixture with competitive multicomponent Langmuir isotherms (schematic)...

If the adsorption and desorption Damkohler numbers (Da 1 and Daf) are sufficiently large i.e., if the adsorption and desorption rates, and k i, are large), the partial differential term in the LHS of Eq. 15.53 can be set to zero and the equation reduces to the multicomponent Langmuir isotherm if the saturation capacities of all the components are the same [43], with... 

In the multicomponent Langmuir isotherm 2.15, each denominator term b,p is proportional to the number of sites occupied by the respective species i, scaled so that the leading "1" is proportional to the number of unoccupied sites. 

The multicomponent Langmuir isotherm based on Equation 2.53 is shown in Equation 2.58 (Charton and Nicoud, 1995) ... 

To explain these effects Figure 2.20 depicts elution profiles that have been theoretically calculated based on the transport dispersive model (Chapter 6). The isotherms are described by a multicomponent Langmuir isotherm and the mass ratio of the feed concentration is varied from 1 9 to 9 1. 

Figure 2.19 Single component Langmuir isotherms and multicomponent Langmuir isotherm for different mixture compositions.

The terms ijsat,i represent again the maximum possible loadings for each component. Interaction between the different components is considered by the summation over all N<-omp components. Equation 6.31 is the nonequilibrium form of the multicomponent Langmuir isotherm (Equation 2.57). 

Helfferich (1997) and Rhee, Aris, and Amundson (1970, 1986, 1989), who derived instructive analytical solutions of the first-order system of nonlinear partial differential equations given by Equation 6.43. Most of the solutions are available for multicomponent Langmuir isotherms. Interested readers will find in the excellent treatment of Rhee et al. the basics of the method of characteristics applied to derive the solutions and to explain the wave phenomena that take place in chromatographic columns. 

The corresponding equilibrium data for pure components and different mixtures are represented in Figure 6.22 by filled rhombuses and triangles. Also shown are the results of different isotherm equations (solid and dashed lines) that have been fitted to the experimental data. In this case the modified multicomponent Langmuir isotherm (Equation 2.58)... 

Figure 6.31 Comparison of the simulated profiles for the modified multicomponent Langmuir isotherm and the IAS equation (Figure 6.27) (Cfeed = 4.4g 10 ml min ...

We have already enployed superposition in some exanples. In Exartple 18-2 we solved a linear chromatography system for the separation of anthracene and naphthalene. This solution was derived by obtaining the solution for anthracene and the solution for naphthalene and then superinposing these two solutions fFigure 18-61. We inherently assumed that the two adsorbates are independent. This type of procedure cannot usually be applied to nonlinear systems since the solutes interact [e.g., as shown by the multicomponent Langmuir isotherm, Eq. (18=8)]. 

To model the adsorption of copper cyanide complexes, the multicomponent Langmuir isotherm can be modified to correlate the aqueous phase speciation and the basicity of the ICAA. The available protonated sites on the ligands attached to the surface also can be assumed to be a function of pH. 

The adsorption of trivalent copper cyanide from solution can be neglected when the pH is lower than 8 because it exists in very small amounts. If we assume that all the protonated sites can adsorb copper cyanide species, and the effect of ionic strength can be neglected in the adsorption processes, the modified multicomponent Langmuir isotherm can be written for total amount adsorbed q as follows ... 

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