Page 35 - Distillation theory

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1.5 Residue Curve Bundles of Four-Component Mixtures 9

a) b) c)

d) e) f)

Figure 1.6. Types of saddle points of four-component mixtures: (a) one-
component saddle, (b, c) two-component saddle, (d, e) three-component sad-
dle, and (f) four-component saddle. Separatrix surfaces are shaded. Arrows,
direction of residium curves; dotty lines, separatrixes.

number of four-component mixture structures makes their overall sorting out
practically impossible. However, a topological equation for four-component mix-
tures similar to Eq. (1.12) was obtained (Zharov & Serafimov, 1975).
To understand the peculiarities of location of residue curve bundles of four-
component mixtures, let’s consider their behavior in the vicinity of saddle points
(Fig. 1.6) and the nonlocal characteristics of the residue curve bundles using sepa-
rate examples of the four-component mixture structures (Fig. 1.7). In Fig. 1.7, the
separating surfaces of the residue curve bundles representing the two-dimensional
bundles Reg ∞ are shaded. Considering the nonlocal characteristics of the residue
(3)
curve bundles, the simplest of such characteristics refers to each pair of stationary
points.Apairofstationarypointscanbeconnectedornotconnectedbytheresidue
curve. To be brief, let’s call the line of distillation that connects a pair of stationary
points a bond (link) – it will be designated by the arrow (→) that is directed
toward the side of the temperature increase (Petlyuk, Kievskii, & Serafimov,
1975a, 1975b, 1977, 1979). For example, in Fig. 1.7a, 12 → 23. In the same fig-
ure, points 1 and 2 are not bonded.
The totality of all bonds characterizes the mixture’s structure. The bond serves
as the elementary nonlocal characteristic of the residue curve bundle structure.
Bondsformbondchains.Thebondchainsofmaximumlengthconnecttheunstable
∞
−
node N and the stable node N of the distillation region Reg . Let’s call a
+
polyhedron formed by all stationary points of one maximum-length bond chain
and containing all components of the mixture a distillation subregion Reg .
sub
The distillation region Reg ∞ is a polyhedron formed by all stationary points
of the totality of all maximum-length bond chains connecting the same unstable
node of the composition space with the same stable node (it will be designated ⇒).
The examples of distillation regions Reg ∞ are 12 ⇒ 4, 12 ⇒ 2 (at Fig. 1.7a),