Page 309 - Distillation theory

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8.4 Multicomponent Azeotropic Mixtures: Presynthesis 283

3,4,5
t
Reg ≡ [min x 1−2−5 − max x 1−2−3 ] (common part of segments [0-1], [0 – max
r t2 t2
1,2
3,4,5 3 4 5
1−2−3 1−2−5 t t t t
x ], and [min x – 1]) (i.e., Reg = Reg • Reg • Reg ). For each
t2 t2 r r r r
1,2 1,2 1,2 1,2
tear-off point at segment [min x 1−2−5 − max x 1−2−3 ], there are three prod-
t2 t2
uct points x 1−2−3 , x 1−2−4 , and x 1−2−5 at separation of mixtures 1,2,3; 1,2,4, and
D2 D2 D2
3,4,5
1,2,5, respectively. Possible product segment Reg D at adiabatic distillation of
1,2
a five-component mixture is a common part of possible product segments at
adiabatic distillation of mixtures 1,2,3; 1,2,4, and 1,2,5 for tear-off points at seg-
3,4,5 3
1−2−5
t
ment Reg ≡ [min x t2 − max x 1−2−3 ] (i.e., Reg D is function of K 1 , K 2 , and K 3
t2
r
1,2 1,2
3,4,5 4
t
at segment Reg [see for example Fig. 4.11], Reg D is function of K 1 , K 2 , and
r
1,2 3,4,5 5 1,2 3,4,5
t
t
K 4 at segment Reg , Reg D is function of K 1 , K 2 , and K 5 at segment Reg and
r r
1,2 1,2 1,2
3,4,5 3 4 5 3,4,5
Reg D = Reg D • Reg D • Reg D ). This segment Reg D is located between point x D2 =
1,2 1,2 1,2 1,2 1,2
0 and point x D2 = min[max x 1−2−3 , max x 1−2−4 , max x 1−2−5 ]. In the example un-
D2
D2
D2
Reg t Reg t Reg t
r r r
3,4,5
der consideration, possible top product segment is Reg D = [0, max x 1−2−5 ].
D2
1,2 Reg t
r
In the general case, if components of edge are components i 1 and i 2 and the
j 1 ÷ j k
other components are j 1 , j 2 ,... j k , then possible product segments are Reg D =
i 1 ,i 2
j 1 j 2 j k j 1 ÷ j k j 1 j 2 j k
Reg D • Reg D ... • Reg D = [0, max x D ]or Reg B = Reg B • Reg B ... • Reg B = [0,
i 1 ,i 2 i 1 ,i 2 i 1 ,i 2 i 1 ,i 2 i 1 ,i 2 i 1 ,i 2 i 1 ,i 2
max x B ], where the ends of the segments are:
i 1 −i 2 − j
max x D = min[max x D ] (8.10)
j Reg t
r
i 1 −i 2 − j
max x B = min[max x B ] (8.11)
j Reg t
r
8.4.2. Possible Product Regions at the Boundary Elements of
Concentration Simplex
j j
(k) (k) (k)
Possible product regions Reg and Reg at the faces and hyperfaces (C ) of the
D B
i i j
concentration simplex have its edges as their boundary elements Reg bound,D and
j i
Reg bound,B . Figures 8.9 and 8.10 show possible location of these regions in two-
i j j
(3) (4)
dimensionalfacesReg andthree-dimensionalhyperfacesReg ofconcentration
D D
i i
simplex, respectively. As one can see in these figures, possible product regions
j j
(k) (k)
Reg and Reg are polygons, polyhedrons, or hyperpolyhedrons, part of the
D B
i i