Page 91 - Distillation theory

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3.5 Feasible Splits at R =∞ and N =∞ 65

being carried out by means of solution of the following linear equation system for
each product simplex:

1 2 n
x F1 = x a 1 + x a 2 +· · · + x a n
1 1 1
1 2 n
x F2 = x a 1 + x a 2 +· · · + x a n (3.8)
2 2 2
······························
1 2 n
x Fn = x a 1 + x a 2 +· · · + x a n
n
n
n
j
where x is concentration of component i in stationary point A j of the product
i
simplex Reg simp .
The system [Eq. (3.8)] is an expression for center of gravity of the product
simplex, when gravity is applied only to its vertexes (stationary points). For all this,
the center of gravity coincides with feed point x F and relative distances of vertexes
of the simplex A j from feed point are equal to the corresponding coefficients a j .
If the feed point belongs to product simplex Reg simp being examined, then solu-
tion of the system [Eq. (3.8)] relative to coefficients a j should answer inequalities
0 < a j < 1. If part of coefficients a j does not answer these inequalities, then feed
point is located out of the product simplex being examined. This method of deter-
mination of whether the feed point belongs to one or to another product simplex
was proposed in Petlyuk et al. (1979).
If it is ascertained by means of such analyses that feed point belongs to several
product simplexes, then the mixture at R =∞ and N =∞ can be separated into
pseudocomponents at any border between two adjacent key stationary points of
each product simplex if the rule connectedness is not broken. The splits with one
pseudocomponent being distributed between products is also feasible.
The described approach allows ascertainment of a complete set of splits of fixed
initial mixture x F at R =∞ and N =∞ independently on the number of com-
ponents and azeotropes. Thus, this method can be applied for synthesis of sep-
aration flowsheets of polyazeotropic mixtures. Let’s show an application of
product simplex method at obvious case of four-component mixture separation
acetone(1)-benzene(2)-chloroform(3)-toluene(4)ofcompositionx F1 =0.25,x F2 =
0.30, x F3 = 0.20, and x F4 = 0.25. The mixture has one binary azeotrope 13 (x az , 1 =
0.66; x az , 2 = 0.34). The boiling temperatures of the components and of the
azeotrope are T 1 = 56.5 C, T 2 = 80.1 C, T 3 = 61.2 C, and T 13 = 66 C. Bonds
◦
◦
◦
◦
between stationary points are shown in the concentration tetrahedron (Fig. 3.18a)
and in the structural matrix (Figs. 3.18b,c). It is seen from the structural matrix that
there are only two bonds chains: 1 → 13 → 2 → 4(1 ⇒ 4, m = n) and 3 → 13 →
2 → 4(3 ⇒ 4, m = n). These two bonds chains corresponds to two distillation
regions Reg , which differ from each other by their unstable nodes (points 1 and
∞
3). Each distillation region contains one product simplex Reg simp and feed point
gets into product simplex Reg simp ≡ 1 → 13 → 2 → 4(x F ∈ Reg simp − it is seen
even without solution of the system [Eq. (3.8)]). Therefore, feasible splits at R =
∞ and N =∞ without distributed components or pseudocomponents: (1) 1 :
13,2,4; (2) 1,13 : 2,4; (3)1,13,2 : 4.