## L

We can plot this line in our x-y plot, as shown in Figure 11B. It is a straight line with slope L/V which intersects the diagonal line at xB. Starting from the

Figure 10 Mass balance on the rectifying section.

These expressions can be combined to give an equation which relates the vapour entering and liquid flows leaving stage n:

On an x-y diagram for component i, this is a straight line starting at the distillate composition with slope R/(R + 1) and which intersects the diagonal line at xD,i.

Starting at the distillate composition xD in Figure 10, a horizontal line across to the equilibrium line takes us to the composition of the vapour in equilibrium with the distillate, y1. A vertical step down takes us to the liquid composition leaving stage 1, x1. Another horizontal line across to the equilibrium line gives us the composition of the vapour leaving stage 2, y2. A vertical line to the operating line gives us the composition of the liquid leaving stage 2, x2, and so on. Thus, as we step between the operating line and equilibrium line in Figure 10, we follow the change in vapour and liquid composition through the rectifying section of the column.

Mass balance on the stripping section.

Mass balance on the stripping section.

bottom composition, xB, a vertical line to the equilibrium line gives the composition of the vapour leaving the reboiler, yB. A horizontal line across to the operating line gives the composition of the liquid leaving stage N, xN. A vertical line to the equilibrium line then gives the vapour leaving stage N, yN, and so on.

Let us now bring together the rectifying and stripping sections at the feed stage. Consider the point of intersection of the operating lines for the rectifying and stripping sections. From eqns [22] and [27]:

heat required to vaporize 1 mol of feed molar latent heat of vaporization of feed

Combining eqns [32], [34] and [35] gives a relationship between the compositions of the feed and the where yi and xi are the intersection of the operating lines. Subtracting eqns [29] and [30] gives:

(Vn +1 — Vm + i)yi = (Ln - Lm)Xi + Dx^i # BxB,i [31]

Substituting the overall mass balance, eqn [20], gives:

Now we need to know how the vapour and liquid flow rates change at the feed stage.

What happens here depends on the condition of the feed, whether it is sub-cooled, saturated liquid, partially vaporized, saturated vapour or superheated vapour. To define the condition of the feed, we introduce the variable q, defined as:

vapour and liquid leaving the feed tray:

For a saturated liquid feed q = 1 and for a saturated vapour feed q = 0. The flow rate of feed entering the column as liquid is q • F. The flow rate of feed entering the column as vapour is (1 — q) • F. Figure 12A shows a schematic representation of the feed stage. An overall mass balance on the feed stage for the vapour gives:

An overall mass balance for the liquid on the feed stage gives:

This equation is known as the q-line. On the x-y plot, it is a straight line with slope q/(q — 1) and intersects the diagonal line at zi. It is plotted in Figure 12B for various values of q.

We are now in a position to bring together the mass balance for the rectifying and stripping sections. Figure 13 shows the complete design. The construction is started by plotting the operating lines for the rectifying and stripping sections. The q-line intersects the operating lines at their intersection. The intersection of the operating lines is the correct feed stage, i.e. the feed stage necessary to minimize the overall number of theoretical stages. The construction steps off between the operating lines and the equilibrium lines. The construction can be started either from the overhead composition working down or from the bottom composition working up. The q

Feed

xB 2 xD

Figure 13 Combining the rectifying and stripping sections.

xB 2 xD

Figure 13 Combining the rectifying and stripping sections.

vapour is refluxed and all of the bottom liquid re-boiled. Figure 14A also shows total reflux on an x-y plot. This corresponds with the smallest number of stages required for the separation. The other limiting case, shown in Figure 14B, is where the reflux ratio is chosen such that the operating lines intersect at the equilibrium line. As this stepping procedure approaches the g-line from both ends, an infinite number of steps are required to approach the g-line. This is the minimum reflux condition, and we term the condition at the feed stage to be a pinch.

This method of design for binary distillation is known as the McCabe-Thiele method. It is restricted in its application because it only applies to binary systems and involves the simplifying assumption of constant molar overflow. However, it is an important method to understand as it gives important conceptual insights into distillation, which cannot be obtained in any other way.

stepping procedure changes from one operating line to the other at the intersection with the g-line. We should also note that a partial reboiler represents a separation stage and a partial condenser (as opposed to a total condenser) also represents a separation stage.

There are two important limits that we need to consider for distillation. The first is illustrated in Figure 14A. This is total reflux in which no products are taken and there is no feed. All of the overhead

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