## T4iMrb

Solution of Example 1-6 With the aid of the above equations, the number of plates required to effect the specified separation may be determined. To plot the operating line [Eq. (1-37)] for the rectifying section, the y intercept (DXD/Vr) is computed in the following manner. Since Vr = Lr + D, and Lt = Lr, it follows that

Since y2 = XD (for a total condenser), the point (y2, X D) lies on the 45° diagonal. The y intercept and the point (y2, X D) locate the operating line for the rectifying section as shown in Fig. 1-9.

When xl = X is substituted in Eq. (1-42), the result yl = X is obtained, and hence the q line passes through point (X, X) which in this case is the point (0.5, 0.5). Since q = 0.6, the y intercept of the q line [Eq. (1-42)] is computed as follows

Since the operating line for the stripping section [Eq. (1-38)] passes through the point (xBi xB) = (0.05, 0.05) and the intersection of the q line with the operating line for the rectifying section, it may be constructed by connecting these two points as shown in Fig. 1-9.

The number of perfect plates required to effect the specified separation may be determined graphically as indicated in Fig. 1-11. It is readily confirmed that the construction shown in Fig. 1-11 gives the desired solution. Since y2 = XD = x{ (for a total condenser) and since y2 is in equilibrium with x2, the desired value of x2 is determined by the point of intersection of line 1 and the equilibrium curve as shown in Fig. 1-9. Line 1 also represents plate 1. When x2 is substituted into Eq. (1-37), the value of y3 is obtained. Since (x2, y3) lies on the operating line for the rectifying section, this point is located by passing a vertical line through (x2, y2). The ordinate y3 obtained is displayed graphically in Fig. 1-9. When the first opportunity to change operating lines is taken, the minimum number of total plates needed to effect the specified separation at the specified operating conditions is obtained. When the feed is introduced on stage number 8, a total of 14 stages are required, 12 plates plus the reboiler and a total condenser (see Fig. 1-9).

It should be noted mat if the operating line for the rectifying section is used indefinitely instead of changing to the operating line for the stripping section, the specified value of xB = 0.05 can never be attained even though infinitely many plates are employed.

### Minimum Reflux Ratio

As the specified value of the reflux ratio (Ll/D) is decreased, the intersection of the two operating lines moves closer to the equilibrium curve and the minimum number of plates required to effect the specified separation (xB = 0.05, XD = 0.96) increases. On the other hand, as L1/D is decreased, the condenser and reboiler duties decrease. The minimum reflux ratio is the smallest one which can be used to effect the specified separation. This reflux ratio requires infinitely many plates in each section as demonstrated in Fig. 1-10. It should be noted that for this case, the plates at and adjacent to the feed plate have the same composition. (In the case of multicomponent systems, these limiting conditions do not necessarily occur at and adjacent to the feed plate as discussed in Chap. 11). From the standpoint of construction costs, this reflux ratio is unacceptable because infinitely many plates are required, which demands a column of infinite height.

### Total Reflux

At total reflux, the operating lines coincide with the 45° line. This gives the smallest number of plates needed to effect the separation. As pointed out by Robinson and Gilliland.12 two physical interpretations of total reflux are possible. From a laboratory or plant operational point of view, total reflux is attained by introducing an appropriate quantity of feed to the column and then

x ,Mole fraction of benzene in liquid

Figure 1-10 At the minimum reflux ratio (LxjD\ infinitely many plates are required to effect the specified separation (XD, xB).

x ,Mole fraction of benzene in liquid

Figure 1-10 At the minimum reflux ratio (LxjD\ infinitely many plates are required to effect the specified separation (XD, xB).

operating so that F = D = B = 0. From the standpoint of design, total reflux can be thought of as a column of infinite diameter operating at infinitely large vapor and liquid rates, and with a feed that enters at a finite rate F and with distillate and bottoms that leave at the rates D and B, where F = D + B. Thus, infinite condenser and reboiler duties are required as well as a column having an infinitely large diameter. At total reflux, six plates, a total condenser, and a reboiler are required to effect the specified separation as shown in Fig. 1-11. A graph of the total costs per year versus the reflux rate Lx at a fixed set of specifications is shown in Fig. 1-12 for reflux rates over the range from minimum to total reflux.

The set of equations required to describe a distillation column in the process of separating a binary mixture is merely an extension of the sets stated previously for the boiling-point diagram [Eq. (1-3)], bubble-point and dew-point temperatures [Eq. 1-12)], and the flash process [Eq. (1-26)]. The complete set of

' 0.0 XB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 XD 1.0

Figure 1-11 Determination of the total number of plates required to effect the specified separation at total reflux.

' 0.0 XB 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 XD 1.0

x, Mole fraction of benzene in liquid

Figure 1-11 Determination of the total number of plates required to effect the specified separation at total reflux.

equations solved above by the McCabe-Thiele method are as follows

Equilibrium relationships y jî Kji xji

Material balances

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