Reversible Distillation Trajectories

The main peculiarity of thermodynamically reversible distillation process consists of the fact that flows of two different phases (vapor and liquid) found in any cross-section are in equilibrium, and flows found in the feed cross-section are of the same composition as feed flows. 

Using the equation of material balance and of phase equilibrium for an arbitrary cross-section of the upper section, we get 

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). 

Sharp and Nonsharp Reversible Distillation of Ideal Mixtures 

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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. 

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[. ... 

The region of reversible distillation can contain one or several reversible distillation trajectory bundles (lines of stationarity). Some of these bundles can be true some of them can be fictitious. Fictitious bundles always have two node points and true ones have one node point or no node point. 

Condition in Tear-Off Points of the Reversible Distillation Trajectories... 

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

In Fig. 4.9b, there are two tear-off points xA of the reversible distillation trajectory for any point xp of the possible product composition segment Regg at side... 

In other words, for sharp separation in a two-section column it is necessary that the feed point belongs to the intersection of true bundles of reversible distillation trajectories of the two sections Xf e (Regrev.r Reg v. ) (Fig. 4.10). The reversible... 

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). 

Let s examine the analysis of structure of reversible distillation trajectory bundles at the concrete example of four-component mixture acetone(l)-benzene(2)-chloroform(3)-toluene(4). At the beginning, the segments of the components order Regff at the edges of the concentration tetrahedron are defined by means of scanning and calculation of the values Ki (Fig. 4.13a). The corresponding regions of components order Reg in the tetrahedron are shown in Fig. 4.13b and in its faces - in Fig. 4.14. The whole face 1-2-3, where the component 4 that is absent... 

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. 

The approximate calculation method of minimum reflux mode (Koehler, Aguirre, Blass, 1991) - the method of the smallest angle, which holds good for mixtures with any component numbers and for any sphts, including frequently found at azeotropic mixtures separation cases of tangential pinch, is based on the calculation of reversible distillation trajectories for the given product compositions. 

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. 

Another deviation appears in the case of availability ofazi-line inside concentration triangle. This line is one of the branches of reversible distillation trajectory for the product point xm = 1. At the increase of the parameter L/F first trajectory bundle appears N S Point S moves along side 1-2 to 23 - point, and point moves along side 1-3. Second trajectory bundle appears when point S coincides with a 23- point on side 1-2 S N y Then point S moves along... 

L/V < (L/y) g point xd at side 1-2 cannot be the distillation top product point. Along with that, at side 1 -2 there is a stable node (Fig. 5.14a). At (L/ E) = (L/ y), the bifurcation goes on. Distillation trajectory tear-off from side 1-2 inside concentration triangle and the distillation process becomes feasible. The stable node at side 1-2 turns into a saddle in point x[. The trajectory bundle appears with saddle point S at side 1-2 and with the stable node A+ inside concentration triangle at reversible distillation trajectory in the vicinity of point x[. At further increase... 

Deviations from the above evolution of distillation trajectory found at separation of nonideal zeotropic and, especially, of azeotropic mixtures are connected with the peculiarities of location of reversible distillation trajectories and with nonmonotony of change of the parameter L/V along these trajectories (see Chapter 4). 

The most important pecuUarities of location of reversible distillation trajectories of azeotropic mixtures, influencing the evolution of distillation trajectories at the change of the parameter L/V, are limitedness of trajectory tear-off segment... 

It is expedient to discuss the influence of these pecuharities on the evolution of distillation trajectories at the concrete example of azeotropic mixture, such as acetone(l)-benzene(2)-chloroform(3). At side 2-3, there is reversible distillation trajectory tear-off segment of the bottom section from vertex 2 to 0 13-point... 

The analogous situation arises in the case if reversible distillation trajectory bundle does not have node points. For the mixture under consideration, such reversible distillation trajectories are available for the top product points, located... 

Figure 5.19. (L/V)rev as functions X2 on the reversible-distillation trajectories in rectifying section for the split 1,3 2 (for given xo in the tangential-pinch region Regu g) of the acetone(l)-...

Figure 5.20. The trajectories of rectifying section for quasisharp separation of the ideal mixture. (L/Vjs = 1, (L/V)2 > (L/V)2 > (L/V)i, the region between the reversible-distillation trajectory and the distillation trajectory underinflnite reflux is shaded. qS, quasistationary point.

Let s note, that distillation trajectory of three-component mixture for the set product point is located between trajectory at infinite reflux (ie., at L/V = 1, and reversible distillation trajectory) (Kiva, 1976 Petlyuk Serafimov, 1983 Wahn-schafft et al, 1992 Castillo, Thong, Towler, 1998). 

For bottom product points Xb, located in this region, distillation trajectory at some values of the parameter V/L is directed to vertex 1 and at other values, bigger than them, it is directed to vertex 3 (Fig. 5.21b). It is connected with the fact that for these product points reversible distillation trajectory has two branches, one of the branches goes to vertex 1 and another branch goes to vertex 3. The point iV+ small values of the parameter V/L is located at one of the branches, and at bigger values it is located at other branches. 

At small values of the parameter L/V, the stable node A+ that at the increase of the parameter L/V moves away from the product point xd in the direction to vertex 2 appears at edge 1-2 (Fig. 5.24a). After this node reaches reversible distillation trajectory tear-off point x into face 1-2-4, it turns into the saddle with one trajectory going out (Fig. 5.24b). After reaching reversible distillation trajectory tear-off point into face 1-2-3, it turns into the saddle with two... 

If the bottom product point lies on the edge 2-4 (Fig. 5.28b), the point 5, as the value of V/L parameter is increased, first goes along the reversible distillation trajectory within the face 2-3-4 until it meets the an-line (x = ), then along the reversible distillation trajectory within the an-surface up to azeotrope 13 (at V/L = 1). Simultaneously, in the face 2-3-4, in the trajectory of reversible distillation, after point x / a stable node arises, and the point engenders a separatrix 5 - that divides the whole separatrix bundle Reg into two separate trajectory bundles. j... 

To check conditions that possible product point at some k-component boundary element(Cj j, ) should meet, it is necessary (1) for the product point x or xy under exanunation to construct reversible distillation trajectory inside the product boundary element and (2) to define all the first and second (if they are) reversible trajectory tear-off points x[] and x from the product boundary element into all the adjacent (k + l)-component boundary elements. If there is only one reversible distillation trajectory tear-off point into each adjacent boundary element, the point under examination is possible product point and part of reversible distillation trajectory from the most remote from it tear-off point... 

If there are two points of reversible distillation trajectory tear-off into, at least, one of the adjacent boundary elements and if there is segment of reversible distillation trajectory, limited by the most distant from the product point under... 

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