# Trajectory Bundles Under Finite Reflux

To return to Eqs. (2.3) and (2.5) for the rectifying section and to fix xt D and R parameters, we obtain a number of points xij by solving this system from the upper tray.

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Figure 2.5. Trajectory bundles under finite reflux of ace-tone(1)-benzene(2)-chloroform(3) azeotropic mixture for (a) rectifying and (b) stripping section. Solid lines with arrows, trajectories; solid line, a-line; dotty line, separatrix under infinite reflux; big circles, stationary points under infinite reflux; little circles, stationary points under finite reflux.

The concentration profile of the rectifying section under reflux R and an overhead product composition x, D will be represented by broken lines, the lengths of which come through points Xij, which are found by means of solving Eqs. (2.3) and (2.5).

In a similar way, with the help of the Eq. (2.3) and the equation of the material balance:

we create a trajectory for a stripping section.

Just as in the case of the infinite reflux, the broken lines can be replaced by the continuous curves. The distillation trajectories under the finite reflux, first, are different for two column sections and, second, have the composition points of the corresponding product (x, D or Xi B) as parameters as well as reflux ratio or reboil ratio (R or S).

The trajectory bundles of the rectifying and stripping sections for the azeotrope mixture: acetone(1)-benzene(2)-chloroform(3) under R = 2.5, S = 1.4 are illustrated in Figs. 2.5a and 2.5b, respectively, while the product compositions are x1 D = 1, x2d = 0, x3d = 0, and x1b = 0, x2b = 0.85, x3b = 0.15.

The trajectories of Figs. 2.5a and 2.5b were constructed in the following way: in the case of fixed R and xd or S and xb, an arbitrary point in the triangle x was chosen and the calculation was performed from this point to bottom in accordance with Eqs. (2.3) and (2.5) for the rectifying section and from this point to top in accordance with the Eqs. (2.3) and (2.7) for the stripping section.

The trajectory starting in product point xd or xb and ending in the point corresponding to the feed tray is the only one of the whole bundle. It is the profile of concentrations of the column section.

The given trajectory belongs to some trajectory bundle bounded by its fixed points (points N-, Sr and N+ of Fig. 2.5a and points Ss and N+ of Fig. 2.5b), the separatrixes of the saddle points S and the sides of the concentration triangle.

Knowledge about the regularities of the trajectory bundles arrangement under the finite reflux provides an opportunity to develop the reliable and fast-acting algorithm to fulfill design calculations of distillation to determine the required number of trays for each section.