Page 258 - Distillation theory

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

232 Trajectories of the Finite Columns and Their Design Calculation

and the bottom product contains impurity component k):

x D,k+1 = 1 − η D (7.8)
(7.9)
x B,k = 1 − η B
Given the product compositions at nonsharp separation, one can determine
3
2
2
3
+
stationary points of section trajectory bundles S , S ... N and S , S ... N for
+
r r r s s s
2
2
any values R. Points S and S will belong to the same boundary elements of the
s
r
concentration simplex as at sharp separation, but they will be shifted relatively to
2
2
points S and S for sharp separation. Apparently, this shift will be the bigger the
r
s
smaller is the purity of products η D and η B .
3
3
Points S ... N and S ... N will be located inside the concentration simplex
+
+
r
r
s
s
at some distance from those of its boundary elements to which they belong at
sharp separation. This distance is the bigger the smaller is purity of products
1
1
η D and η B . Points S and S are located outside the concentration simplex at
r s
some distance from its boundary elements, which they belong to at sharp separa-
tion.
Coordinates of these points cannot be determined precisely for a nonideal
mixture because mathematic models used to describe phase equilibrium are de-
termined only inside the concentration simplex. However, there is no necessity of
that because only composition in the rest of the stationary points is significant for
determination of the value R min and coordinates of the ends of segments [x f −1 ] lin
and [x f ] lin . The most rigorous variant of the algorithm requires that the composi-
1
1
tion in all stationary points, besides points S and S , are determined for nonsharp
r s
separation. In other respects, the modified algorithm of design calculation does
not differ from the one described above, taking into consideration the fact that
segments [x f −1 ] lin ,[x f ] lin and points (x f − 1 ) lin ,(x f ) lin are determined instead of
sh
sh
sh
segments [x f −1 ] , [x f ] sh and points (x f −1 ) , (x f ) .
lin lin lin lin
We note, however, that the main algorithm described above can be used in
the majority of cases. It will be used in the examples given below. The modified
algorithm is necessary only for the modes close to the mode of minimum reflux.
We now illustrate the algorithm described at the example of an ideal four-
component mixture (K 1 > K 2 > K 3 > K 4 ) for split 1,2 : 3,4. Feed composition x F,i
and purity of products η D and η B are set.
Component 4 is the non-key impurity component in the top product, and com-
ponent 1 is the non-key impurity component in the bottom product. Trial cal-
culations of the top section are carried out until the summary concentration of
components 3 and 4, which is equal to the concentration of these components in a
sh
chosen point (x f −1 ) sh of segment [x f −1 ] , is achieved at some tray. Similarly, trial
lin lin
calculations of the bottom section are performed until the summary concentration
of components 1 and 2, which is equal to the concentration of these components
sh
in a chosen point (x f ) , is achieved at some tray.
lin
The little concentrations of non-key impurity components in products x D,4
and x B,1 are determined during the search process with the help of the described
algorithm, and the concentrations of the rest of the components in the products are