Concentration simplexes

Fig. 8.15 Concentration simplex of the La-B-C-N system. Combinations include one quaternary system, four ternary systems, and six binary systems.

Figure 1.1. Concentration simplexes (a) for binary mixtures, (b, c) for three-component mixtnres. and (d) for four-component mixtnres. xi,X2,xs,X4, concentrations of components.

Lines, surfaces, and hypersurfaces of equal phase equilibrium coefficients of two components split the concentration simplex into regions of identical order of components Reg (A) >Kj > Kk) that define the possible causes of separation under the finite reflux mode. 

In the case of the reflux ratio alteration and conservation of the product composition, the stationary points of trajectory bundle are traveling along the reversible distillation trajectories built for a given product, so the trajectories may be called lines of stationarity. Thus, the analysis of the reversible distillation trajectory arrangement in the concentration simplex is decisive in general geometric theory of distillation. 

Obviously, at a flnite number of stages, the distillation trajectory under the infinite reflux should he in one of the c-lines and cannot pass through a stationary point of the concentration simplex, start or end in it. At the infinite number of... 

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). 

One of three parts of trajectory can transform into nodal or saddle point of concentration simplex. 

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. 

For an n-component mixture, the rule of connectedness in some cases can be used without graphic interpretation of the concentration simplex and without application of structural matrix (Petlyuk et al., 1977 Knight Doherty, 1990). Such elementary cases are the following (1) = Ng, and (2) and Ng are... 

It follows from the structural matrix that concentration simplex contains three distillation regions Reg (l 6, 1 = 8, and 1 9), with common unstable node corresponding to the point of acetaldehyde—1. The points corresponding to ethanol-6, water-8, and diethylketone-9 are stable nodes. 

Azeotropic mixtures can almost never be separated completely into pure components in the sequence of columns without recycles at R = oo and N = oo. The set of products of such a system of columns almost always contains not only pure components, but also azeotropes (pseudocomponents). Mixtures, for which concentration simplex contains only one distillation region, are an exception. For three-component azeotropic mixtures, the only phase diagrams of such type are the diagram shown at Fig. 3.10b and antipodal it. Such a mixture can be separated into two columns and into pure components. Two variants of flowsheet with direct 1 2,3 or indirect 1,2 3 split in the first column are feasible. 

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. 

Product points at fixed feed composition and at various values of parameter D/F fill up some line in concentration simplex connecting nodes of distillation region Reg°° with the feed point. 

Where within the concentration simplex can the product points of distillation at... 

It follows from Eq. (4.6) that the points of the upper section Xi, y, yoi lie in one straight line in the concentration simplex. 

In addition, the product points should lie on the straight hne passing through the liquid-vapor tie-line of feeding. Hence, it follows that the maximum length of reversible distillation trajectory is achieved at the intersection of this straight line with the hyperfaces of concentration simplex that (hyperfaces) correspond to (n - 1)-component constituents C i of the initial mixture (sharp separation). 

For the ideal mixture, one and the same order of increase and decrease of phase equilibrium coefficients is consistent throughout the whole concentration simplex ... 

Feasible sharp reversible distillation split of ideal mixtures can be presented as follows 1, 2,... (n - 1) 2,3... n. Therefore, at the reversible distillation, components 2, 3,... (n - 1) are distributed among the top and the bottom products. At nonsharp and semisharp reversible distillation, both products contain all the components or one of the products does not contain the lightest or the heaviest component. At nonsharp reversible distillation, product points lie in the same straight line as at sharp distillation but at some distance from the hyperfaces of the concentration simplex. 

Therefore, at at = const, the sharp reversible trajectory of the upper (lower) section goes from the feed point to hyperface of concentration simplex along secant, passing through the vertex of the simplex corresponding to the heaviest (the lightest) component (Fig. 4.2) (i.e., to the components absent in the section products). 

It is clear from Fig. 4.1 that the location of a reversible distillation section trajectory is determined only by the location of its product point (i.e., it does not depend on any parameter). One component is absent at sharp reversible distillation [i.e., the product point of the section is located at some (n - 1) component edge, face, or hyperface of the concentration simplex]. That is why, in all the points of a section trajectory at sharp reversible distillation, liquid-vapor tie-lines should be directed to this edge, face, or hyperface or from it [i.e., one and the same component should... 

Therefore, one or several trajectory bundles, filUng up that region inside the concentration simplex, where one and the same component is the heaviest (for the top section) or the Ughtest (for the bottom section), will correspond to all the product points located at one and the same (n — 1) component edge, face, or hyperface of concentration simplex. We call this region the region of reversible... 

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[. ... 

Therefore, the concentration simplex can contain one or several regions of reversible distillation Reg v.r or for each section. 

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... 

The location of reversible distillation trajectories in the concentration simplex at sharp separation may be presented in the following brief form ... 

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). 

Besides the location of reversible distillation trajectories in the concentration simplex, the character of the liquid and vapor flow rates changing is of great importance. In accordance with the formulas [Eqs. (4.11) and (4.13)], the ratio of liquid and vapor flow rates in each cross-section in the top section should be equal to the phase equilibrium coefficient of the heaviest component and in the bottom section to that of the lightest component. For ideal mixtures, these phase equilibrium coefficients should change monotonously along the sections trajectories, which leads to maximum liquid and vapor flow rates in the feed cross-section (see Figs. 4.3 and 4.6). 

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-... 

It is possible to formulate a general structural condition that should be valid in the tear-off point of the extractive reversible distillation trajectory from a (n - 1)-component face or hyperface of the concentration simplex of any multicomponent azeotropic mixture the phase equilibrium coefficient of the component that is absent in this face or hyperface and does not rank among the top product components and of the entrainer should be smaller than that of the top product components and bigger than that of the entrainer components. 

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