Page 311 - Distillation theory

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8.4 Multicomponent Azeotropic Mixtures: Presynthesis 285

j j
boundaryelementsofwhichReg bound,D andReg bound,B arecurvilinearandvertexes
i i
are ends of segments at edges of the concentration simplex. j j
(k) (k)
To determine the location of possible product regions Reg D and Reg ,itis
B
i i
necessary to determine the coordinates of the ends of these segments. The algo-
rithm described in the previous section should be used for this purpose, taking
into consideration that we are interested now not in the possible product seg-
j j
(2) (2)
ments Reg and Reg , but in a segments whether are boundary elements of
D B
i i j j
some possible product region Reg bound,D or Reg bound,B .
i i
We examine what conditions should be satisfied by the possible product point
x D orx B belongingtothek-componenthyperfaceC (k) oftheconcentrationsimplex
of an n-component mixture C n and lying at edge i 1 − i 2 of this hyperface (x D ∈
(k)
[i 1 − i 2 ] ∈ C (k) or x B ∈ [i 1 − i 2 ] ∈ C ). Components j 1 , j 2 ... j n-k do not enter into
the number of components of the k-component hyperface under consideration
(k)
(k)
(k)
( j 1 /∈ C , j 2 /∈ C ,... j n−k /∈ C ). Point x D or x B can be product point of the
distillation of the n-component mixture if it is a possible product point x D or x B of
the distillation for all the three-component mixtures C (3) composed of i 1 , i 2 , j 1 ; i 1 ,
j j
i 2 , j 2 ;...; i 1 , i 2 , j n−k . Therefore, possible product segment Reg bound,D or Reg bound,B
i i
at edge i 1 − i 2 being part of the k-component hyperface C (k) of the concentration
j j
(2) (2)
simplex C n is a common part of all possible product segment Reg or Reg at
D B
i i
edge i 1 − i 2 at separation of all ternary mixtures C (3) composed of i 1 , i 2 , j; that is,
j 1 ÷ j n−k j 1 j 2 j n−k
Reg bound,D = Reg D • Reg D ,..., • Reg D or
i 1 ,i 2 i 1 ,i 2 i 1 ,i 2 i 1 ,i 2
j 1 ÷ j n−k j 1 j 2 j n−k
Reg bound,B = Reg B • Reg B ,..., • Reg B .
i 1 ,i 2 i 1 ,i 2 i 1 ,i 2 i 1 ,i 2
The difference between the algorithms of determination of the ends of possible
j j
(2) (2)
product segments Reg or Reg and of segments that are boundary elements of
D B
i i
j j
possible product regions Reg bound,D or Reg bound,B in the k-component hyperface
i i
C (k) consists of the fact that, in the first case, all the components except i 1 and i 2
enter into the number of components j and, in the second case, all components,
except the components of the k-component hyperface C (k) under consideration,
enter into this number.
Thus, the algorithm of determination of coordinates of vertexes of possible
j j
(k) (k)
product regions Reg and Reg (for example of points A 1 ÷ A 7 in Fig. 8.10)
D B
i i
includes the following steps:
1. Determinationofphaseequilibriumcoefficientsofallcomponentsinpoints
of each edge, taken with the set step.