Section trajectory bundles

Such an approach on the basis of product points will be necessary at the analysis of the location of adiabatic sections trajectories bundles (at finite reflux) which products consist less (n - 1) components (see Chapter 5). 

Calculation investigations (Petlyuk, 1978 Petlyuk Vinogradova, 1980 Shafir et al., 1984) determined the conditions under which saddle and saddle-node stationary points of sections trajectory bundles at finite reflux arise inside the concentration simplex, but not only at its boundary elements, promoted the development of this trajectory bundles theory. 

As far as stationary points of section trajectory bundles should be located at the trajectories of reversible distillation, the systematic analysis of these trajectories locations was of great importance (see Chapter 4). 

In a more general case, when there are several distributed components, it is necessary to obtain from Eq. (5.3) the common roots for two sections. After the substitution of each of these roots into Eq. (5.1) or (5.2), we obtain the system of hnear equations relatively to di and or bi and the solution of which determines separation product compositions and internal vapor and liquid flows in the column sections. In addition, one can find the compositions of equilibrium phases in the cross-sections of constant concentration zones (i.e., stationary points of sections trajectories bundles). 

Structure and Evolution of Section Trajectory Bundles for Three-Component Mixtures... 

To understand the structure of section trajectory bundles for multicomponent mixtures and their evolution with the increase of reflux number, let s examine first three-component mixtures, basing on the regularities of distillation trajectory tear-off at finite reflux and the regularities of location of reversible distillation trajectories. 

Let s note that the top product point cannot be located at sides 2-3 and 1-3 because Eq. (5.15) is not vahd for these sides (i.e., Xo i [2-3] and xd [1-3]). Let s also note that ihc separatrix sharp split region of section trajectories bundle... 

General regularities of the evolutions of sections trajectory bundles, discussed in the previous section for three-component mixtures, are valid also for the mixtures with bugger number of components. Figure 5.23 shows evolution of top section trajectory bundle at separation of four-component ideal mixture, when the product is pure component (i.e., at direct split) Ki > K2 > >... 

Stationary points 5, 5, and iV+ (5 = A ) move along the edges of concentration tetrahedron. The section trajectory bundle may be presented in the following brief form (the bundle s direction is indicated by the double arrow, its stationary points around it) ... 

Let s turn now to the mixtures with any number of components, and let s discuss general conditions of existence of sections trajectory bundles and their structure. 

So far, discussing distillation trajectories and their bundles, we proceeded from the fact, that separation stages are equilibrium ( theoretical plates). In real separation process at plates of distillation columns equilibrium is not achieved and the degree of nonequilibrium is different for different components. That leads to decrease of difference between compositions at neighboring plates and to change of curvature of distillation trajectories (Castillo Towler, 1998), but does not influence the location of stationary points of distillation trajectory bundles because in the vicinity of stationary points equilibrium and nonequilibrium trajectories behave equally. Therefore, implemented above analysis of the structure and of evolution of section trajectory bundles is also valid for nonequilibrium trajectory bundles. 

Section trajectory bundle in its general form may be put in brief as follows ... 

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. 

For the intermediate spfits, it is possible to take into consideration in the best way the regularities of location of section trajectory bundles, using two- or three-stage algorithm of search for (L/y) with gradual precise of the value of this parameter. 

The example of tangential pinch for four-component mixture is quasisharp separation of azeotropic mixture acetone (l)-benzene (2)-chloroform (3)-toluol (4) of composition Xf (0,350 0,250 0,150 0,250) at intermediate split 1,3(2) 2,4(3) (admixture components heavy and light key are in brackets correspondingly) at the following composition the products xd (0,699 0,001 0,300,0) and xb (0 0,500 10 0,500). The same top product composition, as in the previous example (Fig. 5.18b) of separation of three-component mixture in the top section, is accepted for convenience of analysis. In this case, the boundary elements of top section trajectory bundle, located in face 1-2-3, completely coincides with top section trajectory bundle at separation of previously mentioned three-component mixture. 

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. 

However, in some cases, even usage of nonadiabatic columns does not maintain separability. These are the cases, when reversible distillation trajectories for both product points do not have node points. In these cases, section trajectory bundles not only of adiabatic, but also of nonadiabatic columns, are limited because reversible distillation trajectories at which section trajectory bundles stationary point lie are located in limited parts of concentration space, adjacent to product boundary elements (see Fig. 5.17b for xd). To check whether it is possible to separate the mixture of this kind into a set product, it is necessary to examine the bundles 5 - N/r and 5 - A(+ for points 5 and S, the most remote along... 

In Chapter 5, to develop a general algorithm of calculation of minimum reflux mode for columns with one feed, we had to understand the location of reversible distillation trajectories and the structure of top and bottom section trajectory bundles. 

In Chapter 5, we saw that the distillation process in a column section is feasible only if there are reversible distillation trajectories inside concentration simplex and/or at several of its boundary elements, because only in this case a section trajectory bundle with stationary points lying at these trajectories of reversible distillation arises in concentration simplex. This condition of feasibility of the process in the section has general nature and refers not only to the top and the bottom, but also to intermediate sections. Therefore, pseudoproduct points can... 

This develops the general algorithm of calculation of minimum reflux mode for the columns with two feed inputs at distillation of nonideal zeotropic and azeotropic mixtures with any number of components. The same way as for the columns with one feed, the coordinates of stationary points of three-section trajectory bundles are defined at the beginning at different values of the parameter (L/V)r. Besides that, for the intermediate section proper values of the system of distillation differential equations are determined for both stationary points from the values of phase equihbrium coefficients. From these proper values, one finds which of the stationary points is the saddle one Sm, and states the direction of proper vectors for the saddle point. The directions of the proper vectors obtain linear equations describing linearized boundary elements of the working trajectory bundle of the intermediate section. We note that, for sharp separation in the top and bottom sections, there is no necessity to determine the proper vectors of stationary points in order to obtain linear equations describing boundary elements of their trajectory bundles, because to obtain these linear equations it is sufficient to have... 

We now examine the structure of intermediate section trajectory bundle for four-component mixtures at the example of ideal mixture (ATi > K2 > > K4). 

The dimensionality of intermediate section trajectory bundle is equal to n -m + 1, where n is total number of components, and m is summary number of components of top product and entrainer. Pseudoproduct point is located at the continuation of the boundary element, formed by all the components of the top product and entrainer. Reversible distillation trajectories and the stationary points are located at the mentioned pseudoproduct boundary element and at all boundary elements whose dimensionality is bigger by one (at m = n — 1, they are located inside concentration simplex). 

This uses the previously stated general theory of section trajectory bundles in columns with one and two feeds for analysis of mixtures separation in the complexes of heteroazeotropic and heteroextractive distillation. 

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

Given the product compositions at nonsharp separation, one can determine stationary points of section trajectory bundles 5, 5. .. A(+ and S, Sf... N+ for any values R. Points S and will belong to the same boundary elements of the concentration simplex as at sharp separation, but they will be shifted relatively to points S and for sharp separation. Apparently, this shift will be the bigger the smaller is the purity of products rjo and rjB. 

The general approach to design calculation of extractive distillation columns is similar to the approach applied for two-section columns. We use our notions about the structure of intermediate section trajectory bundles (see Sections 6.4 6.6), about possible compositions at the trays adjacent to the feed cross-section from above and below (see Section 7.2), and about possible directions of calculation... 

Separatrix element of section trajectories bundle (Reg, p, or Re gp, or Reg ep,e) separatrix element that belongs to both working and unworking bundles of section trajectories Reg = Regf Reg". ... 

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