## LbVb

Degrees of Freedom in Operation of a Distillation Column

With a given feed (F, z and q), and column pressure (P), we have only 2 degrees of freedom in operation of the two-product column in Figure 5, independent of the number of components in the feed. This may be a bit confusing if we think about degrees of freedom as in Gibb's phase rule, but in this context Gibb's rule

Figure 5 An ordinary continuous two-product distillation column.

find the number of theoretical stages for mixtures with constant molar flows. The equilibrium relationship yn = f (xn) (y as a function of x at the stages) may be nonideal. With constant molar flow, L and V are constant within each section and the operating lines (y as a function of x between the stages) are straight. In the top section the net transport of light component w = xDD. Inserted into the material balance eqn [19] we obtain the operating line for the top section, and we have a similar expression for the bottom section:

Figure 5 An ordinary continuous two-product distillation column.

0.6 (mol s-1). This will have a drastic effect on composition. Since the total amount of light component available in the feed is z = 0.5 (mol s-1), at least 0.1 (mol s-1) of the distillate must now be heavy component, so the amount mole faction of light component is now at its best 0.5/0.6 = 0.833. In other words, the amount of heavy component in the distillate will increase at least by a factor of 16.7 (from 1% to 16.7%).

Thus, we generally have that a change in external flows (D/F and B/F) has a large effect on composition, at least for sharp splits, because any significant deviation in D/F from z implies large changes in composition. On the other hand, the effects of changes in the internal flows (L and V) are much smaller.

McCabe-Thiele Diagram (Constant Molar Flows, but any VLE)

The McCabe-Thiele diagram where y is plotted as a function x along the column provides an insightful graphical solution to the combined mass balance (operation line) and VLE (equilibrium line) equations. It is mainly used for binary mixtures. It is often used to

A typical McCabe-Thiele diagram is shown in Figure 6.

The optimal feed stage is at the intersection of the two operating lines and the feed-stage composition (xF, yF) is then equal to the composition of the flashed feed mixture. We have that z = qxF + (1 — q)yF. For q = 1 (liquid feed) we find xF = z and for q = 0 (vapour feed) we find yF = z (in the other cases we must solve the equation together with the VLE). The pinch, which occurs at one side of the feed stage if the feed is not optimally located, is easily understood from the McCabe-Thiele diagram, as shown in Figure 8 (see below).

### Typical Column Profiles - Pinch

An example of a column composition profile is shown in Figure 7 for a column with z = 0.5, a = 1.5, N = 40, NF = 21 (counted from the bottom, including the reboiler), yD = 0.90, xB = 0.002. This is a case where the feed stage is not optimally located, as seen from the presence of a pinch zone (a zone of constant composition) above the stage. The corresponding McCabe-Thiele diagram is shown in Figure 8. We see that the feed stage is not located at the intersection of the two operating lines, and that there is a pinch zone above the feed, but not below.

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