Hodograph transform

For a simple wave, apphcation of the method of characteristics (hodograph transformation) gives... 

Rearrange the problem. Do not get fixed ideas on which variables are dependent and which independent. The use of parametric representations (see The Use of Parametric Representations in Chapter 3) and the hodograph transformation come under this rubric. Even more radically, the shift to a different type of model (e.g., the wave model of Westerterp see also General Observations and Forming the Model in Chapter 1) is a possibility. 

There is a dearth of competitive adsorption data, in a large part because they are difficult to measme, but also because little interest has been devoted to them, as, until recently, there were few problems of importance whose solution depended on their understanding. Besides the static methods, which are extremely long and tedious and require a large amoimt of material, the main methods of measurement of competitive isotherms use column chromatography. Frontal analysis can be extended to competitive binary isotherms [14,73,93-99], as well as pulse techniques [100-104]. The hodograph transform is a powerful method that permits an approach similar to FACP for competitive binary isotherms [105,106]. 

Figure 4.31 Experimental determination of competitive isotherms by the method of the hodograph transform, (a) Individual elution profiles of the two components of a binary mixture, (b) Hodograph transform of the profiles in (a). Reproduced with permission from Z. Mfl and G. Guiochon,. Chromatogr. 603 (1992) 13 (Fig. 3).

Figure 4.32 Hodograph transformation obtained from a rectangular injection, (a) Increasing concentration of first component at constant second component, (b) Increasing concentration of both components, (e) Increasing concentration of second component at constant concentration of first component. Reproduced with permission from Z. Ma, A. Katti, B. Lin and G. Guiochon, J. Phys. Chem., 94 (1990) 6911 (Fig. 6a, 6b, and 6c), 1990 American Chemical Society.

The system of Eqs. 8.1a and 8.1b is the classical system of reducible, quasihnear, first-order partial differential equations of the ideal model of chromatography [1, 2,4r-6,9-17]. The properties of these equations have been studied in detail [4,9,10, 18-24], We discuss here those properties that are important for the xmderstanding of the solutions of the ideal model in the case of elution or displacement of a binary mixture. They are the existence of characteristic fines, called characteristics, the coherence condition, and the properties of the hodograph transform. 

Figure 8.2 Hodograph transform of the elution profile of a wide rectangular injection pulse of a 1 2 binary mixture.

The theory of the hodograph transform and the relationship derived between the equations of the two lines given by this transform in the case of a binary mixture and those of the competitive equilibrium isotherms were briefly presented in Section 8.1.2. The theory is easily extended to multicomponent mixtures, although in this case we must represent the hodograph transform in an n-dimensional coordinate system, Ci, C2, , C , or in its planar projections. If the solution presents a constant state (Figure 8.1), it is a simple wave solution, and there is a relationship between the concentrations of the different components in the eluent at the column exit (Figure 8.2). This result is valid for any convex-upward isotherm. In the particular case in which the competitive Langmuir isotherm apphes, these relationships are linear. 

The hodograph transform is valid only within the framework of the ideal model. It has been shown, however, that the hodograph plots derived from actual chromatograms are very similar to those predicted by the ideal model [18]. If the column efficiency exceeds 100 to 200 theoretical plates, there is no significant difference between the hodograph plot obtained with the ideal model and the plot derived from the profiles calculated with the equilibrium-dispersive model, except very near the axes of coordinates (Figure 8.13). Figure 8.14a compares the... 

Figure 8.15 Elution of a wide injection band of a ternary mixture, (a) Experimental chromatogram and profile predicted by the model, (b) Trajectories of the concentration shocks and simple wave regions, (c) Hodograph transform in the Cj,C2 plane, (d) Hodograph transform in the C2, C3 plane, (e) Hodograph transform in the Ci,C3 plane. Reproduced with permission from R. Zenhdusem, Ph.D. Thesis, Eidgenosische Technische Hochschule, Zurich, Switzerland, 1993.

There is an abimdant literature on the comparison between experimental and calculated band profiles for binary mixtures. The most popular methods used have been the forward-backward finite difference scheme and the OCFE method. The former lends itself readily to numerical calculations in many cases representative of the present preoccupations in preparative chromatography. We present first a comparison between the band profiles obtained with the ideal and the equilibrium-dispersive model to illustrate the dispersive influence of the column efficiency. Related to the comparison between these two models is the issue of the use of the hodograph transform of experimental results discussed in Section 11.2.2. Computer experiments are easy to carry out and most instructive because it is possible to show e effects of the change of a single parameter at a time. Some... 

When a wide rectangular injection pulse is injected in a column and the width is such that the plateau is not completely eroded when it is eluted, the solution of the system of equations of the ideal model (Eqs. 8.1a and 8.1b) includes a constant state, followed by a simple wave, as shown by the theory of partial differential equations [12,13]. The importance of this result is due to the existence of a relationship between the concentrations of the two components of the binary mixture in the simple wave region. This relationship is independent of the position of the band along the column. We have discussed the properties of the hodograph transform in the case of the ideal model (Oiapter 8, Sections 8.1.2 and 8.8). In the case of the equilibrium-dispersive model (Eqs. 11.1 and 11.2), this result is no longer valid. However, the plots of Ci versus C2 are often close to the simple wave solu-... 

Figure 11.8 Illustration of the hodograph transform applied to actual chromatograms. Numerical solutions of the equilibrium-dispersive model for a 310 (Top left) and a 120 (Top center) theoretical plate column, a = 1.8, = 0.7. (Bottom left and center) Hodograph...

However, in practice, the deviation between the solutions of the ideal and the equilibrium-dispersive models is small along most of the profiles. The deviations of the plot of the experimental values of C2 versus C from the hodograph transform of the ideal model solution are also small in most parts of the graph, see Figure 11.8, top and bottom left [5,14,15]. The only exceptions are the parts of this plot that correspond to the beginning of the second shock layer and to the... 

Using the hodograph transform, Rhee and Amundson [3] have also shown that a plot of Q versus Q+i is a straight line (solid line in Figure 16.3), provided that these two components have the same axial dispersion coefficient (D ) and mass transfer rate constant (fcy), in addition to the competitive Langmuir isotherm behavior. The equation of this straight line is... 

Figure 16.3 Hodograph transform of the concentration profiles of two components in the shock layer between their zones. Solid line, competitive Langmuir isotherms, Di i = = fc/,2- Dot-...

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