Page 113 - Distillation theory

P1: FCH/FFX P2: FCH/FFX QC: FCH/FFX T1: FCH
0521832772c04 CB644-Petlyuk-v1 June 11, 2004 17:49

4.3 Trajectory Bundles of Sharp Reversible Distillation 87

valid in the tear-off point of the trajectory for the component that is absent in the
product and conditions [Eqs. (4.6) or (4.8)] should be valid for the rest of compo-
t
nents. After simple transformations of these equations for the tear-off point x ,
rev
we get (Petlyuk & Serafimov, 1983)
t
t
t
x Di = x t (K − K )/(1 − K ) (for top section) (4.17)
rev,i i n n
t
t
t
x Bi = x t (K − K )/(K − 1) (for bottom section) (4.18)
rev,i 1 i 1
In Eq. (4.17), the component n is the heaviest one and, in Eq. (4.18), the
component 1 is the lightest one.
Equalities [Eqs. (4.17) and (4.18)] can be written as follows:
t
t
t
x t = x Di (1 − K )/(K − K ) (i = 1, 2,... n − 1) (4.19)
rev,i n i n
t
t
t
x t = x Bi (K − 1)/(K − K ) (i = 2, 3,... n) (4.20)
rev,i i 1 i
The location of reversible distillation trajectories in the concentration simplex
at sharp separation may be presented in the following brief form:
x D → x t → x F and x B → x t → x F (x F ∈ C n , x D ∈ C n−1 , x B ∈ C n−1 )
rev,r rev,s
4.3.3. Possible Product Composition Regions
If product point x D or x B belongs to the possible product point region Reg or
D
t(n−1)
Reg , the condition [Eq. (4.19) or (4.20)] is valid in one or two points x along
B i
the trajectory of reversible distillation located at (n − 1) component boundary
element C n−1 of the concentration simplex (i.e., there is one tear-off point x t of
rev
t
the trajectory or there are two x ). In the last case, right side of the expression
rev
[Eq. (4.19) or (4.20)] should have an extremum.
If the product point x D or x B does not belong to the possible product point
region (x D /∈ Reg D or x B /∈ Reg B ), then the condition [Eq. (4.19) or (4.20)] is not
valid anywhere (i.e., tear-off points of the trajectory are absent).
Therefore, Eqs. (4.19) and (4.20) allow determination of the boundaries of
the possible product composition region Reg or Reg at sharp reversible dis-
D
B
tillation in (n − 1)-component boundary elements C n−1 of the concentration
simplex.
In Fig. 4.9b, there are two tear-off points x t of the reversible distillation trajec-
rev
(2)
tory for any point x B of the possible product composition segment Reg B at side
t1
1
2-3. The trajectory goes from one of these points x rev to N rev ≡ 1 and from the other
↑ → t t2 → N 2
rev rev
1
t1
t2
2
one x rev the trajectory goes to N rev ≡ 13 ( x B → t rev → N rev ). For
Reg B Reg t rev,s 1 13
the point x B located at the end of the possible product composition segment, both
t2
these points coincide with each other (x t1 = x ) (i.e., two trajectories go inside
rev rev
t
from this common tear-off point x ). The set of trajectories tear-off points for
rev
any points x B at Fig. 4.9b fills up the tear-off segment Reg t(2) of side 2-3 from
rev,s
vertex 2 up to point α 13 .