Page 247 - Distillation theory

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0521820928c07 CB644-Petlyuk-v1 June 11, 2004 20:18

7.2 Distillation Trajectories of Finite Columns 221

product and components 1, 2 ... k are impurity ones; key components are k and
k + 2.
In Chapter 5, we examined in detail the structure and evolution of sec-
tion trajectory bundles for various sharp splits. In this section, we examine
in detail the peculiarities of location of section trajectories at quasisharp and
nonsharp separation and at given reflux R bigger than minimum. Along with
that, we are interested in the location of trajectories with respect to separa-
1
2
2
1
trix sharp split bundles (regions) Reg sh,R (S − S − N ) and Reg sh,r (S − S −
+
sep,s
r
s
sep,r
r
s
r
2
1
2
2
+
N ), to the boundary elements of these bundles S − N , S − S , S − N ,
+
+
s
s
r
r
r
s
r
1
2
S − S and to the product boundary elements of the concentration simplex
s
s
(i.e., with respect to the boundary elements in the vicinity of which the prod-
uct points are located). We note that at quasisharp separation stationary points
1
1
S and S are absent inside the concentration simplex (they are located out-
s
r
side it close to the product boundary elements), but there are separatrix bundles
1
2
2
1
Reg sh,R (S − S − N ) and Reg sh,R (S − S − N ). These separatrix bundles iso-
+
+
sep,r r r r sep,s s s s
late the working trajectory bundles Reg R and Reg R to which section trajectory
w,r w,s
belongs from other bundles of dimensionality n − 1.
At quasisharp distillation and at reflux bigger than minimum composition
points at the first trays above x f −1 and below x f feed cross-section are very close
to these separatrix bundles. In their turn, these separatrix bundles nearly coincide
with the separatrix bundles at sharp separation. While decreasing, the sharpness
of separation the compositions at the first trays above and below the feed cross-
section move away from the trajectory separatrix bundles deep into the working
bundles. Therefore, at quasisharp separation, the part of the trajectory of each sec-
2
1
tion passes in the small vicinity of separatrix bundle of this section S − S − N +
and, at nonsharp separation, the whole trajectory of the section passes far from
the separatrix bundle. The mentioned regularities are of great importance for the
development of the general algorithm of design calculation. This algorithm should
include calculation “tray by tray.” Any other algorithms, in particular, those based
on componentwise grouping of distillation equations, do not take into consider-
ation the structure of trajectory bundles. The choice of the initial point of the
calculation and its direction plays the key role in the calculation by method “tray
by tray.” The calculation from one of the ends of the column is efficient only at
direct or indirect splits, because at these splits there is one impurity component in
one of the products, which sets the composition of this product with high precision.
Besides that, the structure of section trajectory bundles promotes the execution
of calculation from this product.
In general, at intermediate splits and splits with a distributed component, the
calculation from one of the ends of the column for such splits encounters large dif-
ficulties. Determination of possible compositions in the feed cross-section of the
column is of great importance for overcoming these difficulties. To estimate cor-
rectly the limits of change of component concentrations at the trays above and
below feed cross-section, this limits have to be determined at sharp separation
sh
sh
([x f −1 ] and [x f ] ).
At minimum reflux for the splits without distributed components, there is only
one composition point at the first tray above the feed cross-section x f −1 and only