Page 173 - Distillation theory

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5.5 Trajectory Bundles for Four- and Multicomponent Mixtures 147

1
−
If the top product point coincides with vertex 1 (x D1 = 1) (i.e., N ≡ S ≡ x D1 )
2
(Fig. 5.28a), then, as the value of L/V parameter is increased, the point S first
moves along the edge 1-2 up to the α 23 -point, then along the α 23 -line in the face
1-2-3, then along the edge 1-3 up to azeotrope 13 (at L/V = 1).
If the top product point lies on the edge 1-3, it is the case of tangential pinch,
which we are going to consider in the next section (not shown in Fig. 5.28).
2
If the bottom product point lies on the edge 2-4 (Fig. 5.28b), the point S ,as
the value of V/L parameter is increased, first goes along the reversible distilla-
tion trajectory within the face 2-3-4 until it meets the α 13 -line (x branch ), then along
rev
the reversible distillation trajectory within the α 13 -surface up to azeotrope 13 (at
V/L = 1). Simultaneously, in the face 2-3-4, in the trajectory of reversible distil-
lation, after point x branch a stable node N +(2) arises, and the point S 2(α) engenders
rev
1
a separatrix S − S 2(α) , that divides the whole separatrix bundle Reg sh,R into two
sep,s
separate trajectory bundles. j
(3)
If the bottom product point lies in the possible composition region Reg in
B
i
2
+
the face 2-3-4 (Fig. 5.28c), the point N ≡ S moves, as the value of V/L goes up,
along the reversible distillation trajectory inside the concentration tetrahedron up
to vertex 1 (V/L < (V/L) max ).
5.5.2. Mixtures with Any Number of Components

Let’s turn now to the mixtures with any number of components, and let’s dis-
cuss general conditions of existence of sections trajectory bundles and their
structure.
It follows from Eqs. (5.15) and (5.16) that distillation trajectory tear-off at finite
reflux from k-component product boundary element inside concentration simplex
t
is feasible in that case, if in tear-off point x conditions of tear-off into all the
(k + 1)-component boundary elements, adjacent with the product boundary ele-
ment are valid.
To check conditions that possible product point at some k-component bound-
(k) (k)
ary element(C ) should meet, it is necessary: (1) for the product point x
bound D
(k)
or x under examination to construct reversible distillation trajectory inside the
B
product boundary element; and (2) to define all the first and second (if they are)
t2
reversible trajectory tear-off points x t1 and x rev from the product boundary el-
rev
ement into all the adjacent (k + 1)-component boundary elements. If there is
only one reversible distillation trajectory tear-off point x t1 into each adjacent
rev
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
(k)
max x t1 to the end of reversible distillation trajectory N rev is trajectory tear-off
rev
j
(k) t1 (k)
segment Reg t ≡ [max x , N rev ].
rev
i
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 dis-
tillation trajectory, limited by the most distant from the product point under