Besides splits without distributed components, we also discuss splits with one distributed component 1, 2,...k _ 1, k: k, k + 1,...n. The significance of these splits is conditioned, first, by the fact that they can be realized for zeotropic mixtures at any product compositions, while at two or more distributed components only product compositions, belonging to some unknown regions of boundary elements of concentration simplex, are feasible. Let's note that for ideal mixtures product composition regions at distribution of several components between products can be determined with the help of the Underwood equation system (see, e.g., Fig. 5.4). This method can be used approximately for nonideal mixtures. From the practical point of view, splits with one distributed component in a number of cases maintain economy of energy consumption and capital costs (e.g., so-called "Pet-lyuk columns," and separation of some azeotropic mixtures [Petlyuk & Danilov, 2000]).

The analysis of dimensionality of sections trajectory separatrix bundles shows that for splits with one distributed component trajectory of only one section in the mode of minimum reflux goes through corresponding stationary point S2 or S2 (there is one exception to this rule, it is discussed below). The dimensionality of bundle S2 _ N+ is equal to k _ 2, that of bundle S _ N+ is equal to n _ k _ 1. The total dimensionality is equal to n _ 3. Therefore, points xf _1 and xf cannot belong simultaneously to minimum reflux bundles at any value of (L/V)r. If only one of the composition points at the plate above or below the feed cross-section belongs to bundle S2 _ N + and the second point belongs to bundle S1 _ S2 _ N+, then the total dimensionality of these bundles will become equal n _ 2; therefore, such location becomes feasible at unique value of (L/V)r (i.e., in the mode of minimum reflux).

At quasisharp separation with one distributed component in the mode of minimum reflux zone of constant concentrations is available only in one of the sections (in that, trajectory of which goes through point S2).

The following cases of location of composition points at plates above and below feed cross-section x f_1 and xf: (1) point x f_1 lies in rectifying minimum reflux bundle S2 _ Nr+, and point xf lies inside the working trajectory bundle of the bottom section (at nonsharp separation) or in separatrix bundle

S1 _ S2 _ N+ (at sharp separation) - Fig. 5.35a (xf_1 e Regm^, xf e RegfpR);

(2) point xf lies in stripping minimum reflux bundle S2 _ N+, and point xf_1 lies inside the working trajectory bundle of the top section (at nonsharp

Figure 5.35. The joining of section trajectories under minimum reflux for the split 1,2 : 2,3,4 of

the ideal mixture with K1 > K2 > K3 > K4: (a) case a when xf-1 e S^ ^ N+ = Reg™'R, xf e

Si ^ N+ = Reg&R, (b) case a when xf = N+ = Reg^R, xf-1 e S ^ S, ^ N+ = Reg^R.,

Figure 5.35. The joining of section trajectories under minimum reflux for the split 1,2 : 2,3,4 of

the ideal mixture with K1 > K2 > K3 > K4: (a) case a when xf-1 e S^ ^ N+ = Reg™'R, xf e

Si ^ N+ = Reg&R, (b) case a when xf = N+ = Reg^R, xf-1 e S ^ S, ^ N+ = Reg^R.,

(L/ V)min on xD 2. Separatrix sharp split region for rectifying section RegS^, shaded.

(c) case a when xf-1 e S, ^ N+ = Reg™'R and xf = N+ = Regmp'R, and (d) dependence on

separation) or in separatrix bundle S, - S2 - N+ (at sharp separation) - Fig. 5.35b

At some ratio of amounts of the distributed component in the separation products, there is a transitional split between above-mentioned ones: both points x f-1 and xf belong correspondingly to minimum reflux bundles Sr2 - Nr+ and

S2 - N+(xf-1 e Reg^ and xf = N+ = Reg^pR; Fig. 5.35c). In contrast to the

general case, for this split the trajectories of both sections go through the corresponding points S2. When designing columns with one distributed component, one of the tasks is to find out this distribution coefficient because the smallest value of the parameter (L/ V)mm corresponds to it (Fig. 5.35d).

Different types of joining of section trajectories at different component distribution coefficients reflect the fact that split with one distributed component 1, 2,...k _ 1, k: k, k + 1,...n occupies intermediate position between two splits without distributed components: 1,2,... k _ 1 : k, k + 1,... n and 1,2,... k _ 1, k : k + 1,... n. Number and location of stationary points for rectifying minimum reflux bundle S2 _ N+ for the split with one distributed component is the same as for split 1,2,... k _ 1, k : k + 1,... n and, for stripping minimum reflux bundle S2 _ N+, it is the same as for split 1,2,... k _ 1 : k, k + 1,... n.

At relatively small content of the distributed component k in top product, joining of section trajectories goes on at type, characteristic for splits 1,2,... k _ 1 : k, k + 1,... n (i.e., point xf lies in bundle S2 _ N+) and at big content joining goes on at type, characteristic for split 1,2, ... k _ 1, k : k + 1, ... n, (and point xf_1 lies in bundle S2 _ N+).

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

The algorithm of calculation of minimum reflux mode for splits with distributed component includes the same stages as for intermediate splits without distributed components.

The value of (L/ V)Jnin, at which there is intersection of linearized bundles S2 _ N+ and S1 _ S2 _ N+ or S2 _ N+ and S1 _ S2 _ N+ (i.e., the smallest value of [L/V]r, at which there is intersection of bundles S} _ $ _ N+ and S1 _ S2 _ Ns+, is determined at the first stage). The point of intersection can be located both inside bundles S2 _ N+ and S/ _ S2 _ N+, and inside bundles S2 _ N+ and Si _ S2 _ N+, which determines the type of joining of sections trajectories in the mode of minimum reflux (see Fig. 5.35a,b).

The coordinates of points x f _1 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 xf_1 lies in bundle S2 _ N+ and point xf lies in bundle S1 _ S| _ N+, then point xf_1 can be found as intersection point of linear manifolds S2 _ N+ and xF _ Si _ S2 _ N+ and point xf can be found as intersection point of linear manifolds S/ _ Ss2 _ N+ and xF _ Sf _ N+. In other respects, the second stage of search for (L/ V)}™ for splits with distributed components remains the same as for splits without distributed components.

Nonlinearity of separatrix trajectory bundles is taken into consideration only at the third stage, if it is necessary to determine precisely the value of (L/ V)}™. Usually to solve practical tasks, it is sufficient to confine oneself to the first two stages of the algorithm.

Figure 5.35 is carried out according to the results of calculation of (L/ V)nin for equimolar mixture pentane(1)-hexane(2)-heptane(3)-octane(4) were made at separation of it with distributed component at split 1,2: 2,3,4 at different distribution coefficients of component 2 between products. This figure shows the location of rectifying plane Si _ S2 _ N+ and of bottom section trajectory in minimum reflux mode at several characteristic values of distribution coefficient of compo-nent2: (1) at joining "atthe type of direct split" (1:2,3,4) (Fig.5.35b; xD2 = 0.1, xf = Ns+, zone of constant concentrations is located in feed cross-section in bottom section); (2) at joining "at the type of intermediate split without distributed components" 1,2 : 3,4 (Fig. 5.35a; xD = 0.4, Xf-1 lies at line S2 - N+, zone of constant concentrations is located in the middle part of the top section); (3) at joining with optimal distribution of component 2 between products (Fig. 5.35c; xD2 = 0, 268, xf = Ns+, xf-1 lies at line S2 - N+, zones of constant concentrations are available in both sections: in the middle part of the top section and in feed cross-section in the bottom section, [L/ V]Jnin is the smallest comparing with any splits at items 1 and 2).

Figure 5.35d shows for this example the change of the value of (L/ V)mm, depending on the parameter xD2 (the first two stages of general algorithm were used for calculations). The column trajectories, the calculation direction taken into consideration, may be presented in the following brief form:

D ^ Sr1(2) ^ Sr2(3) ^ xf, xf ^ Ss1(3) ^ xB (Fig. 5.35a)

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