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A q-line construction for a partially flashed feed is given in Fig. 13-31. It is easily shown that the q line must intersect the diagonal at xF. The slope of the q line is q/(q - 1). All five q-line cases are shown in Fig. 13-32.

The derivation of Eq. (13-28) assumes a single-feed column and no sidestream. However, the same result is obtained for other column configurations. Typical q-line constructions for sidestream stages are shown in Fig. 13-33. Note that the q line for a sidestream must always intersect the diagonal at the composition (y1 or x1) of the sidestream.

Figure 13-33 also shows the intersections of the operating lines with the diagonal construction line. The top operating line must always intersect the diagonal at the overhead-product composition xD. This

Top Operating Line
FIG. 13-31 Typical intersection of the two operating lines at the q line for a feed stage. The q line shown is for a partially flashed feed.
Line For Saturated Liquid
FIG. 13-32 All five cases of q lines; (1) superheated-vapor feed, (2) saturated-vapor feed, (3) partially vaporized feed, (4) saturated-liquid feed, and (5) sub-cooled-liquid feed. Slope of q line = q/(q - 1).

can be shown by substituting y = x in Eq. (13-21) and using V - L = D to reduce the resulting equation to x = xD. Similarly (except for columns in which open steam is introduced at the bottom), the bottom operating line must always intersect the diagonal at the bottom-product composition xB.

Equilibrium-Stage Construction The alternate use of the equilibrium curve and the operating line to "step off" equilibrium stages is illustrated in Fig. 13-34. The plotted portions of the equilibrium curve (curved) and the operating line (straight) cover the composition range existing in the column section shown in the lower right-hand corner. If yn and xn represent the compositions (in terms of the more volatile component) of the equilibrium vapor and liquid leaving stage n, then point (yn, xn) on the equilibrium curve must represent the equilibrium stage n. The operating line is the locus for compositions of all possible pairs of passing streams within the section and therefore a horizontal line (dotted) at yn must pass through the point (yn, xn +1) on the operating line since yn and xn +1 represent passing streams. Likewise, a verticalline (dashed) at xn must intersect the operating line at point (yn _ x, xn). The equilibrium stages above and below stage n can be located by a vertical line through (yn, xn +1) to find (yn +!, xn +1) and a horizontal line through (yn -!, xn) to find (yn -1, xn -1). It can be seen that one can work upward or downward through the column by alternating the use of equilibrium and operating lines.

Total-Column Construction The graphical construction for an entire column is shown in Fig. 13-35. The process, pictured in the lower right-hand corner of the diagram, is an existing column with a number of actual trays equivalent to eight equilibrium stages. A partial reboiler (equivalent to an equilibrium stage) and a total condenser are used. This column configuration was analyzed earlier (see Fig. 13-24) and shown to have C + 2N + 9 design variables (degrees of freedom) which must be specified to define one unique operation. These may be used as follows as the basis for a graphical solution:

Specifications

Stage pressures (including reboiler)

Condenser pressure

Stage heat leaks (except reboiler)

Pressure and heat leak in reflux divider

Feed stream

Feed-stage location

Total number of stages N

One overhead purity

Reflux temperature

External-reflux ratio

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