Page 40 - Distillation theory

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0521820928c01 CB644-Petlyuk-v1 June 11, 2004 17:45

14 Phase Equilibrium and Its Geometric Presentation

K
a)
K 2
K 3
1
K 1
x 2
0 1
K
b) K 2 Figure 1.11. Dependences K − x on the sides of the concen-
tration triangle for mixture in Fig. 1.9a: (a) side 1–2, (b) side
1 2–3, (c) side 1–3. Thick lines, K − x for present on side com-
α
K 3
13 ponents; dotted lines, K − x for absent on side components
K 1
x 2 (K ∞ − x).
0 1
K
c)
K 2
K 3 K 1
1
K 1 13 K 3
x 1
0 1

Indeed, if α ij = K i /K j = 1, then y i /y j = x i /x j (i.e., points [x i , x j ] and [y i , y j ] lie
on the straight line that passes through vertex k [k = i, k = j]). For example,
in the points of α 13 -line in Fig. 1.9a, the liquid–vapor tie-lines are directed to
vertex 2.
In the concentration tetrahedron, all points of α-surfaces are characterized by
the property that the liquid–vapor tie-lines in these points are directed along the
straight lines passing through that edge of the concentration tetrahedron, which
connects the vertexes whose numbers are missing in the index of α-surface. For
example, in the points of α 13 -surface in Fig. 1.9b, the liquid–vapor tie-lines are
directed to edge 2–4.
In the concentration tetrahedron, the ternary azeotrope gives rise not only to
three α-surfaces, but also to one specific α-line in the points of which not two but
three components of the phase equilibrium coefficients are equal to each other.
We will call the line a three-index α-line. For example, in Fig. 1.10b, the ternary
azeotrope 123 gives rise to the α 123 -line, which crosses the face 1–3–4 in the α 123 -
point (it isn’t shown).
It is characteristic of all points of the three-index α-line that the liquid–vapor
tie-lines in these points are directed along the straight lines passing through that
vertex of the concentration tetrahedron, the number of which is missing in the
index of α-line.
For example, in Fig. 1.10a in the points of the α 123 -line, the liquid–vapor tie-lines
are directed to vertex 4. Let’s note that the α 123 -line is a line of intersection of all
three α-surfaces (α 12 , α 13 , and α 23 ).
The quaternary azeotrope gives rise to six α-surfaces in the concentration tetra-
hedron (the number of combinations is every two from four). Each α-surface gives