Page 249 - Distillation theory

P1: JPJ/FFX P2: JMT/FFX QC: FCH/FFX T1: FCH
0521820928c07 CB644-Petlyuk-v1 June 11, 2004 20:18

7.2 Distillation Trajectories of Finite Columns 223

2
1
+
Because the dimensionality of region Reg sh,R (S − S − N ) is larger by one
sep,r
r
r
r
2
than that of its boundary element Reg min,R (S − N ), the dimensionality of the
+
sep,r r r
sh
set of points {x f −1 } at given reflux larger than minimum should be larger by
lin
one than the dimensionality at minimum reflux. As far as at minimum reflux, the
sh
dimensionality of {x f −1 } is equal to zero then at reflux larger than minimum
lin
sh sh
the dimensionality of {x f −1 } is equal to one; that is, the set of points {x f −1 }
lin lin
2
1
is a segment lying in linear region Reg sh,R (S − S − N ). Similarly, the set of
+
sep,r r r r
sh sh,R 1 2
+
points {x f } is a segment lying in linear region Reg (S − S − N ). There is
lin sep,s s s s
a correspondence between each point of segment [x f −1 ] sh and certain point of
lin
segment [x f ] sh that is connected with the first one by Eq. (5.18). The greater the
lin
sh
length of segments [x f −1 ] sh and [x f ] , the greater the reflux number.
lin lin
One of the ends of segment [x f −1 ] sh that we designate (x ∞ ) sh should lie at
lin f −1 lin
2
linear boundary element Reg min,R (S − N ) lin . The section trajectory starting in
+
sep,r r r
1
2
point (x ∞ ) sh should pass through two stationary points S and S , that is, for
f −1 lin r r
point (x ∞ ) sh the top section is infinite not only at sharp, but also at quasisharp
f −1 lin
separation (Fig. 7.1b). The other end of the segment [x f −1 ] sh that we designate
lin
2
1
)
+
(x min sh should lie inside region Reg sh,R (S − S − N ) lin , the farthest possible
f −1 lin sep,r r r r
2
from boundary element Reg min,R (S − N ) lin . The section trajectory starting in
+
sep,r r r
1
point (x min sh should pass only through stationary point S , the farthest one from
)
f −1 lin r
2
stationary point S . Therefore, in this case, at quasisharp separation the top section
r
has the smallest number of trays.
∞ sh
Similarly, for the bottom section, the ends of segment [x f ] sh are (x ) and
lin f lin
∞ sh
sh
) .Point(x
) correspondstopoint(x
(x min sh min sh ∞ ) andpoint(x ) corresponds
f lin f lin f −1 lin f lin
to point (x min sh (Fig. 7.1c). In the first case at quasisharp separation, there is an
)
f −1 lin
infinite number of trays in the top section and the smallest number in the bottom
section, and vice versa in the second case.
The smallest summary number of trays of two sections at quasisharp sepa-
sh
ration corresponds to some middle location of points (x f −1 ) sh and (x f ) . Such
lin lin
compositions in the feed cross-section are optimal.
7.2.2. Possible Compositions in Feed Cross Section
The coordinates of segments [x f −1 ] sh and [x f ] sh can be determined from purely
lin lin
geometric considerations from the known coordinates of the stationary points and
of point x F .
While solving this task, we act the way we did when we determined points x f −1
and x f in the mode of minimum reflux (see Section 5.6).
It follows from the condition of material balance in the feed cross-section (Eq.
[5.18]) that segments [x f −1 ] sh and [x f ] sh should be parallel to each other and to
lin lin
1
2
the line of intersection of surfaces or hypersurfaces Reg sh,R (S − S − N ) lin and
+
r
sep,r
r
r
2
1
+
) , (x )
Reg sh,R (S − S − N ) lin . We examine points (x min sh ∞ sh and x F for which
sep,s s s s f −1 lin f lin
∞ sh
Eq. (5.18) should be valid. Therefore, straight line (x min sh − (x ) − x F should
)
f −1 lin f lin
1
2
+
be the intersection line for linear manifolds Reg sh,R ≡ (S − S − N ) lin and
sep,r
r
r
r