Section Trajectories Joining

As we saw in the previous sections, at the increase of the parameter (LlV)r in top section and of the parameter (V/L)s in bottom section trajectory bundles of sections Reg and Reg increase, filling up bigger and bigger parts of concentration simple. Along with that the increase of the parameter L/V)r leads to the certain increase of the parameter (V/L)s in accordance with the equations of material and thermal balance of the column at given xd and xb. 

At some value of parameter (L/ trajectory bundles of sections Reg and Regi adjoin each other by their boundary elements - separatrix min-reflux regions Reg , s (5 = A+, shortly 5 - A+) and Re = (S N, shortly - N+), mostly remote from product points xd and xb, if one uses for determination of (L/ y) the model in Fig. 5.29b, or if validity of condition (Eq. [5.18]) is achieved between some points of these boundary elements, if one uses the model in Fig. 5.29a. At this value of the parameter (L/F)J , the distillation process becomes feasible in infinite column at set product compositions. Such distillation mode is called the mode of minimum reflux. It follows from the analysis of bundle dimensionality 5 - 1V+ and that, at separation without distributed 

5 -A/is equal (k -1) (see section 5.5). Therefore, total dimensionality of those bundles is equal to (n - 2) at dimensionality of concentration simplex (n - 1). 

Therefore, points x/ i and Xf cannot lie in bundles 5 - and 5 - A + at arbitrary values of the parameter (LfV)r, but only at one definite value of this parameter. 

The question about location of points Xf-i and x/ for splits with distributed component is discussed below. 

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Numerous works (Levy, Van Dongen, Doherty, 1985 Levy Doherty, 1986 Julka Doherty, 1990) in which distillation trajectory bundles of three-and four-component mixtures for two sections of distillation column were used at hxed product compositions and at different values of reflux (vapor) number, are of great importance. They defined the conditions of two section trajectories joining in the feed cross-sections of the column in the mode of minimum reflux, and they developed the methods of this mode calculation for some splits. 

To overcome these defects, it was necessary to apply the conception of sharp separation and to develop the theory of distillation trajectory tear-off from the boundary elements of concentration simplex at sharp separation (Petlyuk, Vinogradova, Serafimov, 1984 Petl50ik, 1998) and also to develop the geometric theory of section trajectories joining in feed cross-section in the mode of minimum reflux that does not contain simplifications and embraces mixtures with any number of components and any splits (Petlyuk Danilov, 1998 Petlyuk Danilov, 1999b Petlyuk Danilov, 2001a Petlyuk Danilov, 2001b). 

Conditions of Section Trajectories Joining and Methods of Minimum Reflux Calculating... 

So the distillation process in two-section column may be feasible, it is necessary that sections trajectories are joined with each other (i.e., that there is material balance between sections flows at the plates above and below feed cross-section). 

Figure 5.30. The joining of section trajectories under minimum reflux for the direct spht of (a) the acetone(l)-benzene(2)-chloroform(3) mixture, and (b) the acetone(l)-benzene(2)-chloroform(3)-toluene(4) mixture. The attraction region RegJ is...

Figure 5.33. The joining of section trajectories under minimum reflux (thick hues with arrows) for the intermediate split of the ideal mixture. A progressive change in...

At some intermediate ( boundary ) content of the component A in top product joining of section trajectories goes on simultaneously at two mentioned types. 

The coordinates of points x/ i and Xf are defined at the second stage in accordance with determined at the first stage type of joining of sections trajectories. If, for example, point x/ i hes in bundle — N+ and point Xf lies in bundle 5/ - - A/, then point x/ i can be found as intersection point of linear man-... 

Figure 5.35 is carried out according to the results of calculation of (L/y) " for equimolar mixture pentane(l)-hexane(2)-heptane(3)-octane(4) were made at separation of it with distributed component at spht 1,2 23,4 at different distribution coefficients of component 2 between products. This figure shows the location of rectifying plane S - S - and of bottom section trajectory in minimum reflux mode at several characteristic values of distribution coefficient of component 2 (1) at joining at the type of direct spht (1 2,3,4) (Fig. 5.35b X02 = 0.1, x/ = Aj+, zone of constant concentrations is located in feed cross-section in bottom...

But we saw in Section 5.4 that the values of parameters L/V and V/L and the sizes of trajectory bundles of adiabatic colunms sections are limited, if for product point there are two reversible distillation trajectory tear-off points. Therefore, necessary conditions of separability in adiabatic columns can be insufficient if for one or for both product points there are two reversible distillation trajectory tear-off points from boundary elements to which points 5 belong. In these cases, to check separabiUty it is necessary to verify whether corresponding separatrix sections trajectory bundles join at the maximum possible value of the parameter L/V or V/L. 

We now examine the conditions of joining of sections trajectories at a set flow rate of entrainer (i.e., at set value of the parameter E/D) for a three-component mixture in the mode of minimum reflux. Each of two feeds can be the control one, and the intermediate section trajectory in the mode of minimum reflux in both cases should pass through the saddle point Sm because this trajectory passes through the node point not only in the mode of minimum reflux, but also at reflux bigger than minimum (point arises at the boundary element of the concentration simplex because the extractive distillation under consideration is sharp). 

Figwe 6.10. Joining of the stripping, intermediate, and rectifying section trajectories of extractive distillation of the acetone(l)-water(2)-methanol(3) azeotropic mixture at minimum reflux (bottom feed is the control one) xf+e, total composition of inital feed F and entrainer E,... 

Therefore, the joining of trajectories of the bottom and the intermediate sections in the mode of minimum reflux for the case when the bottom feed is the control one is similar to that of section trajectories of two-section column at direct split (see Section 5.6). In this mode, zones of constant concentration arise in the bottom and in the intermediate sections. The column trajectory may be put in brief as follows ... 

We now examine the four-component mixture for = 2 (Fig. 6.9a). The conditions of joining of section trajectories at a set flow rate of the entrainer in the mode of minimum reflux in the cases of top or bottom control feed do not differ from the conditions for three-component mixtures discussed above. In the case of bottom... 

For four-component mixtures at nim = 3 and at two components in the bottom product (Fig. 6.9b), the conditions of joining in the case of bottom control feed are defined by the dimensionality of trajectory bundles N - Sm(d = 1) and (d = 1) and are similar to those of joining of sections trajectories of two-section column in the mode of minimum reflux at intermediate split (see Section 5.6). Point X/-1 should lie on the separatrix min-reflux region Re (N - Sm) and point Xf should lie on the separatrix min-reflux region Re ( - N ). 

It is shown below that such a joining variant of sections trajectories at arbitrary compositions of the top product and of the pseudoproduct is unfeasible. Joining is feasible if point Xe-i belongs to working trajectory bundle of top section Reg. (Sr - N+). In other words, the top feed cannot be the control one. 

If the control feed is the top one, then the trajectory bundle of the top section should join the stationary point (i.e., the bundle of zero dimensionality) in the concentration space of dimensionality trim - 1- Therefore, joining is feasible at some value of the parameter (L/E) " if the dimensionality of the trajectory bundle of the top section Reg, . is equal to trim - 2. 

In particular, for the most widespread spht with one-component entrainer and one-component top product nim = 2, wzr = 1), the joining of intermediate section trajectories with the trajectories of the top and the bottom sections goes on the way it is at direct split in two-section colunms. This uses the simplest modification of the algorithm of calculation of the minimum reflux mode. 

The theory of trajectory tear-off from boundary elements of concentration simplex and the theory of joining of section trajectory bundles find possible product... 

In this chapter, we turn from infinite columns to real finite columns. On the basis of the analysis performed previously for infinite columns, we determine the regularities of location of finite columns trajectories in the concentration simplex and, in particular, the regularities of joining of finite column section trajectories. This will allow us to develop simple and reliable algorithms of distillation design calculation. 

In complex columns and distillation complexes, geometric conditions of joining of section trajectories are similar to those for simple columns. 

In connection with it, we examine in detail conditions of the joining of section trajectories and the algorithms of design calculation of simple colunms at various splits, and then on this basis we examine these questions for complex columns. 

Design calculation for the set value of a = R/Rmin comes to the determination of tray numbers in the sections of the column Nr and Ns at which section trajectories are joined (i.e., the componentwise material balance in the feed cross-section is valid). The distillation trajectory may be put as follows ... 

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