Tear-off points

Figure 2.7. A location of product points and trajectories under minimum reflux for given three-component feed xp (a) first class of fractionation, (b) second class of fractionation, (c) third class of fractionation. Ri < R2 < R3 < R4 < Rs < Re < = 00 sphts xo(i) xb(i) at Ri, xo(2) xb(2) aiR2,XD(i)-XB(3) atiis = 7, x i(4) xb(4) at R4, xd(5) xb(S) at R = at R(, and R-j = 00, x and xl — tear-off points of rectifying and stripping section trajectories.

In the case of = R3 (boundary mode of the first fractionation class), point Xd reaches side 1-2. At this time, the trajectory of distillation of the rectifying section (Fig. 2.7a) is situated along side 1-2 from point xd up to the tear off point X, and later it comes inside the concentration triangle up to point Xi Under these conditions, the trajectory of the stripping section is located completely inside the concentration triangle. The zones of constant concentrations of the column are given in Fig. 2.8b. 

In the case of further R increase (the third fractionation class), the compositions xd and Xg do not change and the tear-off points x and x travel along sides 1-2 and 2-3 toward vertex 2 until they join in this vertex (Fig. 2.7c) atR = Rt = oo. 

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. 

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

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... 

Figure 4.9. Reversible section trajectories of acetone (l)-benzene(2)-chlorofonn(3) mixtnre forgiven product points (a) rectifying section, and (b) bottom section Xij(i), xd(2). xoQ), xb, product points x. , tear-off points. 

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. 

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 both cases, the trajectory tear-off point of sharp reversible distillation in the intermediate extractive section should lie at side 1-3 and the trajectory of intermediate section is a line, which is a geometric locus of points where the straight lines passing through a given point of pseudoproduct are tangent to residue curves. This trajectory reaches side 1-3 at the tear-off point and vertex 2 is the node... 

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. 

In conclusion, at the limit (boundary) value of reflux number the product point Xd approaches side 1 -2 (sharp separation, the second class of fractioning Fig. 5.2b). At the same time, the saddle stationary point S (trajectory tear-off point x from side 1-2) appears at side 1-2. Therefore, at boundary reflux number in... 

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 further increase of R at direct separation, top product point xd begins to move along side 1-2 to vertex 1 till component 1 will be completely in top product. After that, further movement of product points xd and xb is stopped (i.e., the third class of fractioning ensues). At indirect separation, bottom product point Xb moves to vertex 3 till component 3 will be entirely in bottom product. At the second class of fractioning, trajectory tear-off point x of one of the sections is not changed and, for mixtures with constant relative volatilities, part of trajectory of this section x s S Ai+ is also not changed (Stichlmair et al., 1993). 

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. ... 

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. 

It is necessary for the distillation trajectory to be able to tear off from the boundary element that certain conditions be valid in the tear-off point. ... 

Trajectory tear-off points x are a special kind of stationary points that can be called pseudostationary ones because, in the vicinity of these points, the concentrations of components, absent in the boundary element, along the distillation trajectory in the direction to the product decrease monotonously at infinite number of separation stages > xf, where is a separation stage closer to the... 

Therefore, in the vicinity of tear-off points, x concentrations of components present and absent in the product behave differently at neighboring plates concentrations of components present in the product are constant, and those of... 

We see that these conditions differ from those in other stationary points (Eq. [5.6]). The difference in change of concentrations ratio of the components at two neighboring plates in the stationary points and of the components absent in the boundary element in trajectory tear-off points x is shown for the top section in Fig. 5.7. 

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. 

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-... 

As far as Eqs. (5.15) and (5.16) should be valid for the points of trajectory located inside concentration simplex, the whole trajectory of the section from tear-off point to point of junction with the trajectory of the second section should be located in one region of section sharp split Reg or Reg. For each sharp split region, Reg or Reg consist of quite definite component-order regions... 

In the tear-off point of the first of this trajectories the value of the parameter V/L (for the bottom section, it plays the same role as the parameter L/Ffor the top section) equals and in tear-off point of the second... 

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... 

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. 

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... 

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