Concentration simplexes elements

As far as c-lines cannot cross each other and boundary elements of concentration simplex are filled with their c-lines bundles, c-lines cannot pass from the internal space of the simplex to its boundary element. Therefore, the distillation trajectories at the infinite reflux can lie completely inside the concentration simplex or inside its boundary elements. 

The main difference between the azeotropic mixtures (and also nonideal zeo-tropic mixtures) and the ideal ones are that, to determine possible sphts of an azeotropic mixture, special analysis is required. The availability of a few distillation regions under the infinite reflux Reg can result in sharp separation becoming completely impossible or in a decrease in sharp splits number. Let s note that for ideal mixtures the fine of possible products compositions atR = ooandiV=oo and set feed composition goes partially inside the concentration simplex and partially along its boundary elements. For azeotropic mixtures, this line can go along the boundary elements of the distillation region (Fig. 3.6a, line 2). 

For the sake of briefness, we call the first of these conditions the term of connectedness. It has general nature - it can be applied to mixtures with any number of components and azeotropes. Moreover, the term of connectedness embraces not only sharp splits, when the product points lie in the boundary elements of the concentration simplex, but also the semisharp and nonsharp splits, when the product points lie in the boundary elements of the distillation region. 

At R = oo and N = oo, distillation trajectories bundles fill up distillation regions Reg°° in concentration simplex limited by node and saddle stationary points (points of components and azeotropes) and by boundary elements of various dimensionality, part of which are located at boundary elements of concentration simplex and part of which are located inside it. 

As one can see in Fig. 4.2, the trajectory of each section at sharp reversible distillation consists of two parts the part, located inside the (n -1) component boundary element C i of concentration simplex, lying between the product point Xd or xb and the tear-off point of the trajectory from this boundary element x[, and the part located inside concentration simplex C , lying between the tear-off point of the trajectory and the feed point xp. Only the second part should be located inside a region of reversible distillation Reg y orRegJ g, and product point Xd or xb can lie outside this region. 

Unlike trajectories of distillation at infinite reflux, which may come off the boundary elements of the concentration simplex in the saddle points S only, reversible distillation trajectories come off in ordinary points x[. ... 

Let s examine the tear-off points of the trajectories of reversible distillation from the boundary elements of the concentration simplex (Fig. 4.9). These points are points of branching one branch of the trajectory is being torn off from the boundary element and goes inside the concentration simplex, and the second branch stays inside the boundary element. Conditions [Eqs. (4.11) or (4.13)] should be... 

If product point Xd or xp belongs to the possible product point region Reg or Regg, the condition [Eq. (4.19) or (4.20)] is valid in one or two points along the trajectory of reversible distillation located at (n - 1) component boundary element C i of the concentration simplex (i.e., there is one tear-off point xj v of the trajectory or there are two x )- In the last case, right side of the expression [Eq. (4.19) or (4.20)] should have an extremum. 

Therefore, Eqs. (4.19) and (4.20) allow determination of the boundaries of the possible product composition region Reg, or Reg at sharp reversible distillation in (n - l)-component boundary elements C i of the concentration simplex. 

It follows from the aforesaid that sharp separation in a reversible distillation column is feasible only if the liquid-vapor tie-line of feeding is directed to the possible product composition region Reg at the boundary element C i of the concentration simplex and from region Reg at other boundary element... 

We previously examined the process of reversible distillation for a given feed point. Below we examine trajectories of reversible distillation sections for given product points located at any -component boundary elements Q of the concentration simplex (xd e C or xg e Q). If / < (n - 1), then in the general case such trajectories should consist of two parts the part located in the same -component boundary element where the product point lies and the part located at some (k+ l)-component boundary element adjacent to it. Along with that, the product point should belong to the possible product composition region Reg or Reg for the examined ( )-component boundary element, and the boundaries of this region can be defined with the help of Eqs. (4.19) and (4.20). 

The above-described way of definition of the possible composition region contour in face 2-3-4 has the most general nature. It can be applied for any (n - 1)-component boundary elements of the concentration simplex of n-component mix-... 

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. 

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). 

At D = Dpr and at i = R in both sections, there are two zones of constant concentrations - in the feed point Xf and in the trajectory tear-off points of sections x from the boundary elements of concentration simplex. For a three-component mixture there is a transition from the first class of fractioning right away into the third class, omitting the second class. At further increase of reflux number, the product compositions do not change any more. 

AtD < Dpr and R = in the top section, there are two zones of constant concentrations in feed point xp and in trajectory tear-off point from the boundary element of concentration simplex and in the bottom section there is one zone in feed point xp. At D > Dpr and R = on the contrary, in the bottom section there are two zones of constant concentration and in the top the section there is one zone. In both cases there is a transition from the first class of fractioning to the second one (i.e., in one of the sections, zone of constant concentrations in feed cross-section disappears, and in the other section, the zone is preserved, but the composition in it starts to change with the change of R). 

At nonsharp separation, the stationary points of section working regions, except the stable node N+, are located outside the concentration simplex (the direction of trajectory from the product is accepted). At sharp separation, other stationary points - trajectory tear-off points x from the boundary elements of concentration simplex - are added to the stable node. These are the saddle points S and, besides that, if the product point coincides with the vertex corresponding to the lightest or to the heaviest component, then this point becomes an unstable node N. ... 

If the problem is stated in this way, it is necessary to determine what product compositions xd and xb of sharp separation are feasible at distillation of nonideal zeotropic and azeotropic mixtures. The theory of distillation trajectory tear-off from the boundary elements of concentration simplex answers this question. 

Let s examine two constituent parts of section distillation trajectory at the example of sharp preferable split of three-component ideal mixture (Fig. 5.6a) the part located in the boundary element (the side of concentration triangle), and the part located inside concentration simplex (triangle). There is a trajectory tear-off point from the boundary element x between these two parts. 

Therefore, it follows from the inequality x f > xf points X from the boundary elements of concentration simplex, that ... 

Inequalities (Eqs. [5.9] and [5.10]) for the components absent in the product and in the boundary element are valid inside concentration simplex not only in the vicinity of trajectory tear-off points x from the boundary elements, but also in other trajectory points that are not stationary. 

Equations (5.9), (5.10), (5.15), and (5.16) are necessary and sufficient conditions of trajectory tear-off from the boundary element of concentration simplex. Equations (5.9) and (5.10) can be called operating ones because they depend on separation mode, and Eqs. (5.15) and (5.16) can be called structural ones because they depend only on the structure of the field of phase equilibrium coefficients. 

In trajectory tear-off points of the top section x( phase equilibrium coefficients of the components present in the product Kj should be greater than those of the components absent in the product Kj, and vice versa in the bottom section. Therefore, tear-off of trajectories from the boundary elements of concentration simplex is feasible only if in the vicinity of this boundary elements there are component-... 

Figure 5.8. Tear-off conditions from boundary elements of the concentration simplex for the section trajectories (a) rectifying section, and (b) stripping section.

Product points can belong only to those boundary elements of concentration simplex that contain trajectory tear-off regions. Along with that, product points should be located at these boundary elements within the limits of some region,... 

It follows from Eqs. (5.15) and (5.16) that distillation trajectory tear-off at finite reflux from A -component product boundary element inside concentration simplex is feasible in that case, if in tear-off point x conditions of tear-off into all the (k + l)-component boundary elements, adjacent with the product boundary element are valid. 

Besides splits without distributed components, we also discuss splits with one distributed component l,2,.../c-l,/c /c, /c- -l,...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]). 

In the mode of minimum reflux, i min at sharp distillation without distributed components traj ectory of the top (bottom) section goes from the product point xd xb) to the trajectory tear-off point Sj Sj) into the boundary element, containing one additional component referring to product components, that is the closest one by phase equilibrium coefficient, then it goes from point S (Sj) to the point of trajectory tear-off S (S ) inside concentration simplex, then it goes from point to point X/ I (xf) in the feed cross-section of the column. Along with that, material balance should be valid in the feed cross-section. 

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... 

Theoretical analysis of the separation of azeotropic mixtures with the help of extractive distillation was carried out in the works (Levy Doherty, 1986 Knight Doherty, 1989 Knapp Doherty, 1990 Knapp Doherty, 1992 Wahnschafft Westerberg, 1993 Knapp Doherty, 1994 Wahnschafft, Kohler, Westerberg, 1994 Bauer Stichlmair, 1995 Rooks, Malone, Doherty, 1996 Stichlmair Fair, 1998 Doherty Malone, 2001). Characteristic peculiarities of the process of extractive distillation of binary azeotropic mixtures were investigated in these works. More general conception of the processes of extractive and autoextractive distillation on the basis of the theory of intermediate section trajectory tear-off from boundary elements of concentration simplex was introduced in the works (Petlyuk, 1984 Petlyuk Danilov, 1999). Trajectory bundles of intermediate section for multicomponent mixtures were examined in the latter work. 

This allows us to actively influence the location of the pseudoproduct point of the intermediate section in order to maintain sharp separation (i.e., separation at which the intermediate section trajectory ends at some boundary element of the concentration simplex). This is feasible in the case when inside concentration simplex there is one trajectory of reversible distillation for pseudoproduct point x ) that ends at mentioned boundary element, and there is the second trajectory inside this boundary element. To maintain these conditions, pseudoproduct point x j) of the intermediate section should be located at the continuation of the mentioned boundary element, because only in this case can liquid-vapor tie-hues in points of reversible distillation trajectory located in this boundary element he at the lines passing through the pseudoproduct point x jy. We discuss these conditions in Chapter 4. It was shown that in reversible distillation trajectory tear-off point x[ev e from the boundary element the component absent in it should be intermediate at the value of the phase equUibrium coefficient between the components of the top product and of the entrainer rev,D > Kevj > Kev.s)- This condition is the structural condition of reversible distillation trajectory tear-off for the intermediate section. Mode condition of tear-off as for other kinds of sections consists of the fact that in tear-off point the value of the parameter (LfV) should be equal to the value of phase equilibrium coefficient of the component absent at the boundary element in tear-off point of reversible distillation trajectory ((L/V)m =... 

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). 

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