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8.4 Multicomponent Azeotropic Mixtures: Presynthesis 287

and number of product components is equal. Figure 8.9 shows divide of possible
composition regions into simplexes (see dotted lines).
To check whether product point with coordinates (x 1 , x 2 ... x k ) belongs to the
(k) (k)
simplex of possible product compositions x D ∈ Reg or x B ∈ Reg , coor-
simp,D simp,B
2
k
2
1
k
1
2
1
k
dinates of vertexes of which are (x , x ... x ), (x , x ... x ),...(x , x ... x ), it is
1 2 k 1 2 k 1 2 k
sufficient to solve the following system of linear equations:
1
1
x 1 = a 1 x + a 2 x +· · · + a k x 1
1 2 k
2
2
x 2 = a 1 x + a 2 x +· · · + a k x k 2 (8.12)
2
1
..............................................
k
k
x k = a 1 x + a 2 x +· · · + a k x k
1 2 k
In this system of equations, the unknown parameters a 1 , a 2 ,..., a m are pro-
portional to the distance from point (x 1 , x 2 ,... x k ) to the corresponding vertexes
of possible product composition simplex. If all parameters a 1 , a 2 ,..., a k obtained
from Eq. (8.12) turn out to be positive, the potential product point being checked
belongs to the possible product composition simplex under consideration; other-
wise, it does not belong to it. A similar method was used before to determine feasi-
ble splits at infinite reflux (Petlyuk, Kievskii, & Serafimov, 1979) (see Chapter 3).
Besides splits without distributed component, splits with one distributed com-
ponent can be of great practical importance. Therefore, it is necessary to check
which splits of this type are feasible.
The check-up is realized in the same way as for the splits without distributed
components, taking into consideration the fact that coordinates of the product
pointsdependonthedistributioncoefficient.Therefore,thecheck-upisperformed
for a values 0 and 1 of this coefficient.
8.4.4. Possible Sharp Splits in Columns with Two Feeds
To determine possible splits in columns with two feedings, it is necessary to find
j
t(2)
trajectory tear-off segments of the intermediate section Reg e at the edges of the
i
concentration simplex, while using various autoentrainers. As shown in Chapter
6, the following order of components at decreased phase equilibrium coefficients
should be valid for these segments: first comes the group of components of the top
product, and last comes the group of components of entrainer, and between them
there is the group of the rest of components that are absent at the edge under
consideration. j
t(k)
The similar condition should be valid for the trajectory tear-off regions Reg e
i
at the faces and hyperfaces of the concentration simplex. Like possible product
composition regions, trajectory tear-off regions of the intermediate section are
polygons, polyhedrons, and hyperpolyhedrons, the vertexes of which are ends of
j
segments Reg t at edges.
bound,e
i
Therefore, to identify the trajectory tear-off segments and regions, it is neces-
sary at the beginning to calculate the values of phase equilibrium coefficients of