Page 85 - Distillation theory

P1: JPJ/FFX P2: FCH/FFX QC: VINOD/IYP T1: FCH
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3.5 Feasible Splits at R =∞ and N =∞ 59

and (2) for the mixture i-propanol(1)-benzene(2)-cyclohexane(3)-n-butanol(4)
of composition x F (0.15, 0.40, 0.15, 0.30). In addition, the location of points x D and
x B at direct and indirect split is shown [for mixture 1 at direct split x D ≡ 12(0.785,
0.215, 0.0, 0.0), x B (0.414, 0.036, 0.412, 0.138), at indirect split x D (0.687, 0.168, 0.089,
0.056), x B ≡ 13(0.344, 0.0, 0.656, 0.0); for mixture 2 at direct split x D ≡ 123(0.376,
0.169, 0.455, 0.0), x B (0.047, 0.510, 0.0, 0.443), at indirect split x D (0.215, 0.570, 0.215,
0.0), x B ≡ 4(0.0, 0.0, 0.0, 1.0)].
However, the above-described method is unfit for splits other than the direct
and indirect ones, and the number of such splits grows dramatically with the
increaseofn.Thegeneralmethodsofsplitssetdeterminationarebasedontheusage
of structural matrix and of method of product simplex for distillation subregions
(Petlyuk, Kievskii, & Serafimov, 1979).

3.5.1. Method of Product Simplex for Distillation Subregions (m = n)
Let’s return to the notion of distillation subregion Reg sub (Reg sub ∈ Reg ). It is
∞
a polygon, a polyhedron, or a hyperpolyhedron, the vertexes (components and
azeotropes) of one bonds chain connecting nodes of distillation region Reg and
∞
including all the components of a mixture. The types of boundary elements of
distillation subregion are the following: (1) parts of boundary elements of concen-
tration simplex coincident with boundary elements of distillation region Reg ∞ ;
bound
(2) boundary elements of distillation region Reg ∞ , separating it from other
bound
distillation regions; (3) straight lines, planes, or hyperplanes connecting nodes
of distillation region and separating the subregion under consideration from the
other subregions inside one distillation region.
The term distillation subregion was introduced in a number of works (Petlyuk
et al., 1975a; Petlyuk et al., 1977; Petlyuk & Serafimov, 1983). In contrast to that of
distillation region, the notion of distillation subregion includes not only location
of c-lines bundles, but also definite conditions of material balance. In Safrit &
Westerberg (1997), for distillation subregion the terminology the region of con-
tinuous distillation in contrast to the terminology region of batch distillation was
used.
Let’s show that if product points belong to the first and second types boundary
elements of distillation subregion Reg sub (x D ∈ Reg sub and x B ∈ Reg sub ), then these
product points meet the conditions of connectedness (e.g., all these splits are feasi-
ble; at R =∞ and N =∞, product points should lie on these boundary elements).
Let distillation subregion correspond to the following chain of bonds: A 1 →
A 2 → A 3 →· · · → A m−1 → A m , where A 1 , A 2 ,... A m are stationary points, the
set of which includes all the n components of the mixture, m ≥ n. (One should
bear in mind that we call bonds chain the sequence of bonds for which the end of
one is the beginning of the next one.)
Let’s examine the case when m = n. In this case the distillation subregion
is a simplex (Reg ∞ ≡ Reg simp ), the amount of vertexes of which is equal to the
sub
amountofcomponents.Fromthepracticalpointofview,itisfrequentlyconvenient