Substituting Eqs. (2.25) and (2.26) in Eq. (2.24), and dividing both sides by {q - 1)F

Equation (2.27) represents the locus of the points at which the rectifying section component balance line intersects the stripping section component balance line. This equation is called the q-line equation. The q-Iine is illustrated later in Fig. 2.96.

Intersection of the online with the 45° diagonal. If xt = z, then Eq. (2.27) gives y, - jc, = z. Therefore, the g-line intersects the 45° diagonal line at the point (z,z).

Slope of the q-llne. The slope is qKq - 1), per Eq. (2.27). Equation (2.25), which defines q, can be rewritten as

From this equation, the quantity q is the fraction of the feed that is liquid. The product qF is the quantity of liquid contained in the feed. This quantity joins the liquid descending from the rectifying section to provide the liquid for the stripping section. Similarly, (1 - q)F is the quantity of vapor in the feed; this vapor joins the vapor ascending from the stripping section to provide the rectifying section vapor flow. Table 2.2 summarizes the relationship between q, the thermal condition of the feed, slope of the <7-line, and column flows. Figure 2.8a illustrates the slope of the <?-line for each of these conditions. Figure 2.86 illustrates the effect of the slope on the component balance line, assuming the rectifying section component balance line (and therefore the reflux ratio) is fixed.

Summary. In order to draw a straight line on an x-y diagram, the slope of the line and one point on the line need to be determined. The derivations above enable the determination of the slope and one point on the following lines:

1. The rectifying section component balance (operating) line.

2. The stripping section component balance (operating) line.

In each case the point defined is the intersection of the line with the 45° diagonal line. The slopes and intersection points of each of these lines are summarized in Table 2.3. In addition, it has been shown that the rectifying section component balance line and the stripping section component balance line meet on the qr-line.

2.2.4 McCabe-Thiele diagrams: construction

Example 2.1 It is required to separate 200 lb-mole/h of a 40% benzene and €0% toluene mixture into a top product containing 95% benzene and a bottom stream containing 90% toluene. The feed mixture is 25 percent vaporized. The reflux ratio is 3:1, and a total condenser is to be used, (a) How many theoretical stages are required? (ft) At what stage should the feed be introduced?

solution step 1 Obtain an overall material balance for the column. (Refer to Figs. 2.6c and 2.9a.)

1. Given F = 200 lb-mole/h, xF = 0.4, xD = 0.95, xB = 0.1

3. Overall component balance on benzene, Eq. (2.15)

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