Page 33 - Distillation theory

P1: FCH
0521820928c01 CB644-Petlyuk-v1 June 11, 2004 17:45

1.4 Phase Diagrams of Three-Component Mixtures 7

increases in the opposite direction of this point). The rest of stationary points are
called saddles (Fig. 1.4).
A stationary point type is defined by the proper values of Yakobian from Eq.
(1.11). For a stable node, both proper values are negative, λ 1 < 0 and λ 2 < 0; for
an unstable node, both proper values are positive, λ 1 > 0 and λ 2 > 0; and for a
saddle, one proper value is negative, λ 1 < 0, and the second is positive, λ 2 > 0.
Foradistillationprocessnotonlythestationarypointtype,butalsothebehavior
of the residue curve in the vicinity of the stationary point is of special importance.
If the residue curves in the vicinity of the specific point are tangent to any straight
line (singular line) (Fig. 1.4a, b, d, e, g, h), the location of this straight line is of great
importance. A special point type and behavior of residue curves in its vicinity are
called stationary point local characteristics.
The whole concentration space can be filled with one or more residue curve
bundles. Each residue curve bundle has its own initial point (unstable node) and
its own final point (stable node). Various bundles differ from each other by initial
or final points.
The boundaries separating one bundle from another are specific residue curves
that are called the separatrixes of saddle stationary points. In contrast to the other
residue curves, the separatrixes begin or come to an end, not in the node points but
in the saddle points. A characteristic feature of a separatrix is that in any vicinity
of its every point, no matter how small it is, there are points belonging to two
different bundles of residue curves. The concentration space for ideal mixtures
is filled with one bundle of residue curves. Various types of azeotropic mixtures
differ from each other by a set of stationary points of various types and by the
various sequence of boiling temperatures in the stationary points.
The first topological equation that connects a possible number of stationary
points of various types for three-component mixtures (N, node; S, saddle; upper
index is the number of components in a stationary point) was deduced (Gurikov,
1958):
2
2
1
3
3
2(N − S ) + N − S + N = 2 (1.12)
Figure1.5showsmainlyphysicallyvaluabletypesofthree-componentazeotropic
mixtures deduced by Gurikov (1958) by means of systematic application of Eq.
(1.12). In Fig. 1.5, one and the same structure cover a certain type of mix-
ture and an antipodal type in which stable nodes are replaced by unstable
ones and vice versa (i.e., the direction of residue curves is opposite). Besides
that, the separatrixes are shown by the straight lines. Let’s note that the later
classifications of three-component mixture types (Matsuyama & Nishimura, 1977;
Doherty & Caldarola, 1985) contain considerably greater number of types, but
many of these types are not different in principle because these classifications
assume light, medium, and heavy volatile components to be the fixed vertexes of
the concentration triangle.
Types of azeotropic mixture and separatrixes arrangements are also called mix-
ture nonlocal characteristics.
The part of the concentration space filled with one residue curve bundle is called
a distillation region Reg (Schreinemakers, 1901). A distillation region Reg ∞ has
∞
(3)