Page 252 - Distillation theory

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226 Trajectories of the Finite Columns and Their Design Calculation

7.3. Design Calculation of Two-Section Columns

Purity of the products is the set (specified) parameter at designing, and number
of trays in each section n r and n s and reflux number L/D are the parameters that
have to be determined.
Knowledge of the general regularities of location of separatrix trajectory bun-
dles of the sections and of possible composition segments in the feed cross-section
develops reliable and fast algorithms of calculation of the necessary tray num-
bers. These algorithms include (1) the determination of coordinates of the sta-
tionary points of sections trajectory bundles at different values of parameter
(L/V) r (i.e., calculation of reversible distillation trajectories for the set product
points at sharp separation); (2) the obtaining of linear equation systems describ-
2
2
1
1
2
2
+
ing planes and hyperplanes S − N , S − S − N , S − N , and S − S − N +
+
+
r r r r r s s s s s
by the coordinates of the stationary points through which these planes and hy-
perplanes pass (we remember that the number of the stationary points of bundles
2
2
+
Reg min,R (S − N ) and Reg min,R (S − N ) is equal to the number of components
+
sep,r r r sep,s s s
absent in the top and bottom products, respectively, and the number of the sta-
2
1
1
2
+
tionary points of bundles Reg sh,R (S − S − N ) and Reg sh,R (S − S − N )is
+
s
r
sep,r
r
sep,s
r
s
s
larger by one); (3) obtaining linear equation systems describing planes and hy-
2
2
+
+
perplanes x F − S − N and x F − S − N by the coordinates of point x F and of
s
s
r
r
the corresponding stationary points; (4) the determination of the coordinates of
sh
) , (x
) , and (x )
points (x min sh ∞ ) , (x min sh ∞ sh with the help of the algorithm de-
f −1 lin f −1 lin f lin f lin
scribed in Section 7.2; and (5) the calculation of sections trajectories by method
“tray by tray” and the determination of necessary tray numbers n r and n s in
the sections at the set value of parameter (L/V) r ([L/V] r > [L/V] min ) and at
r
different coordinates of points (x f −1 ) sh and (x f ) sh at segments [x f −1 ] sh and
lin lin lin
sh
[x f ] .
lin
The first four items of this algorithm are of general nature and do not depend
on the split. But the efficiency of the choice of the initial point and of the direction
of calculation by method “tray by tray” depends to a great extent on the accepted
split. In some cases, it is easy to calculate the whole column in one direction (the
direct and the indirect splits). It is considerably more complicated to perform
calculation at intermediate splits and at splits with one distributed component.
It is shown in the next section that for these most general splits the calculation
of each section trajectory should be performed from the end of the column. We
examine all the listed cases.
7.3.1. Direct and Indirect Splits of Mixtures with Any Number of Components
At direct split 1 : 2, 3 ... n and at set small concentrations of impurities in products
(1 − η D ) and (1 − η B ), one can quite precisely set the bottom product composition
x B : x B1 = 1 − η B , x B2 = x F2 /(1 − D/F) − D/F(1 − η D )(1 − D/F), Bx Bi =
Fx Fi for i = 3, 4 ... n. Components are arranged in the order of decreasing phase
equilibrium coefficients. For split 1 : 2,3,4, it is important that component 1 is
the lightest one in all the points of both section trajectories (both trajectories are