Hodograph

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

Fig. 17-15. Hodograph showing variation of wind speed and direction with height above ground. SFC = surface wind.

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. 

In the multidimensional case (i.e. N > 1), wave solutions are constructed in the phase space of the concentrations also called the hodograph space. For simplicity, the procedure will be demonstrated for a two-dimensional problem (i.e. N = 2). However, the same concepts apply to higher dimensional problems [16, 34, 38]. 

Fig. 5.6. Construction of wave solutions in the hodograph space corresponding to the scenario in Fig. 5.1.

Expression (15.208) repre.sents a simple algorithm of diffraction tran.sforma-tion, based on summation of the time section along the hodographs of the scattered wavefield (Tirnoshin, 1978). 

This means that SuC E is a constant for a homogeneous rock and that the accumulated metal mass is a linear function of time. The relationship is represented as a geoelectrochemical hodograph (Fig. 2-21, branch I) in which the angle of inclination of the curve to the time axis, t, depends on the concentration, C. ... 

Fig. 2-21. Geoelectrochemical hodograph (metal accumulation, m, versus time, x) for a two-layer medium.

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.

When the second component front appears, the concentration of the second component jumps from 0 to C, while at the same time the concentration of the first component falls from to C°. This is a simultaneous concentration discontinuity. It corresponds to the segment AF in the hodograph plot (Figure 8.2). Afterward, the concentrations of both components in the eluent remain constant and equal to their concentrations in the feed until the end of the injection plateau and the beginning of the diffuse rear profiles. This second front shock moves at the velocity associated with the concentration discontinuity AC2 = in the presence of a concentration of the first component (see Figure 8.1, conditions at the rear of the second shock). From Eq. 8.15 and since Aq2 is equal to 2(Cj, C ), this velocity is... 

This equation gives the ratio of the concentrations of the first component on both sides of the shock. Since the point C, remains on the same continuous line of the hodograph plot, r in Eqs. 8.41a and 8.43 is still the solution of Eq. 8.12 with C . Combination of Eqs. 8.41a and 8.43 gives [14] ... 

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.

Figure 9.7 Operating lines when the isotherms of two feed components intersect. (a) Isotherms of A, B, and the displacer (solid lines) and two possible operating lines (dashed lines), (b) Corresponding hodograph plots showing tie lines (dashed lines) and the unit selectivity line. The operating and tie lines demarcate the space into three operating regions (see text). Reproduced with permission from F. Antia, Cs. Horvdth, J. Chromatogr., 446 (1991) 119 (Fig. 3).

---