Page 140 - Distillation theory

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114 Distillation Trajectories and Conditions of Mixture Separability

In a more general case, when there are several distributed components, it is
necessary to obtain from Eq. (5.3) the common roots for two sections. After the
substitution of each of these roots into Eq. (5.1) or (5.2), we obtain the system
of linear equations relatively to d i and V min or b i and V min , the solution of which
r s
determines separation product compositions and internal vapor and liquid flows
in the column sections. In addition, one can find the compositions of equilibrium
phases in the cross-sections of constant concentration zones (i.e., stationary points
of sections trajectories bundles).
The main problem in solving the Underwood equation system, as it was shown
in Shiras et al. (1950), is the correct determination of the list of distributed com-
ponents at two specified parameters. At the wrong setting of this list, the solution
of the equation system leads to unreal values of d i and b i for some components
(d i > f i or d i < 0).
In this case, it is necessary to correct the list of distributed components, re-
ferring those components, for which unreal values of d i or b i were obtained, to
undistributed ones.

5.2.2. Evolution of Separation Product Compositions of One-Section Columns
at Set Feed Composition

The use of the Underwood equation system allows for examination of the evolu-
tion of separation product compositions of one- and two-section columns at set
feed composition and at variable reflux number. Figure 5.2 shows such evolution
for one-section rectifying column (Shafir et al., 1984). This figure also shows tra-
jectory bundles N ⇒ N (N → S → N ) in accordance with the notion of
+
−
+
−
the bundle introduced by Serafimov et al. (1973a). The trajectory of distillation
section is a line in concentration space that connects the points in which the set
of equations of distillation at given product point and reflux is satisfied. This line
obtains by means of “tray by tray” method from any point of concentration space.
The trajectory bundle at given finite reflux R is a set of trajectories with the same
+
−
initial and final stationary points (unstable N and stable N nodes) at the same
product point x D or x B . At the small reflux numbers (the first class of fractioning),
the stationary point of trajectory bundle (composition point in the zone of con-
stant concentrations) coincides with point x F (equilibrium to the feed point y F )
and the product point x D , as at reversible distillation, lies at the continuation of
the liquid–vapor tie-line of feeding (Fig. 5.2a). The stationary point x F is a stable
node N of rectifying section bundle (region Reg R ). At the increase of reflux
+
w,r
number, the top product point x D moves along the straight line, passing through
liquid–vapor tie-line of feeding, moving away from the feed point and coming
nearer to side 1-2.
1
In conclusion, at the limit (boundary) value of reflux number R , the product
lim
point x D approaches side 1-2 (sharp separation, the second class of fractioning; Fig.
5.2b). At the same time, the saddle stationary point S (trajectory tear-off point
t
x from side 1-2) appears at side 1-2. Therefore, at boundary reflux number in