# Multiplestage Separation Of Binary Mixtures

Although all of the separation problems involving binary mixtures may be solved by use of the general methods presented in subsequent chapters for multi-component mixtures, it is, nevertheless, rewarding to consider the special case of the separation of binary mixtures because this separation may be represented graphically in two-dimensional space. Many of the concepts of distillation may be illustrated by the graphical method of design proposed by McCabe and Thiele.9

### The McCabe-Thiele Method

In the description of this process, the following symbols are used in addition to those explained above. The mole fraction of the most volatile component in the feed is represented by X, in the distillate by Xand in the bottoms by xB. The subscript j is used as the counting integer for the number of the stages. Since the distillate is withdrawn from the accumulator (j = 1) and the bottoms is withdrawn from the reboiler (j = AT), the mole fractions in the distillate and bottoms have double representation j that is, X ยก^ = Xn (for a column having a total condenser) and xBi = xAl. For the case where the column has a partial condenser (D is withdrawn as a vapor), X Di = yu.

The rectifying section consists of the partial or total condenser and all plates down to the feed plate. The stripping section consists of the feed plate and all plates below it including the reboiler. When the total molar flow rates do not vary from plate to plate within each section of the column, they are denoted by Vr (vapor) and Lr (liquid), in the rectifying section and by Vs and Ls in the stripping section. The feed rate F, distillate rate D, bottoms rate and reflux rate Lx are all expressed on a molar basis.

The design method of McCabe and Thiele9 is best described by solving the following numerical example.

Example 1-6 It is desired to find the minimum number of perfect plates required to separate an equal molar mixture of benzene and toluene into a distillate product containing 96 percent benzene (XD = 0.96) and a bottom product containing no more than 5 percent benzene (xB = 0.05) at the following operating conditions: (1) the column pressure is 1 atm, and a total condenser is to be used (D is a liquid), (2) the thermal condition of the feed is such that the rate Ls at which liquid leaves the feed plate is given by Ls = Lr -I 0.6F, and (3) a reflux ratio Lt/D = 2.2 is to be employed. The equilibrium sets {xA, yA} of benzene used to construct the equilibrium curve shown in Fig. 1-9 were found by solving Prob. 1-1.

This set of specifications fixes the system; that is, the number of independent equations that describe the system is equal to the number of unknowns. Before solving this problem, the equations needed are developed. First, the equilibrium

x, Mole fraction of benzene in liquid

Figure 1-9 Graphical solution of Example 1-6 by the McCabe-Thiele method.

x, Mole fraction of benzene in liquid

Figure 1-9 Graphical solution of Example 1-6 by the McCabe-Thiele method.

pairs {x, y} satisfying the equilibrium relationship y = Kx may be read from a boiling-point diagram (see part (a) of Prob. 1-1) and plotted in the form of y versus x to give the equilibrium curve; see Fig. 1-9. Observe that the equilibrium pairs {x, y} are those mole fractions connected by the tie lines of the boiling-point diagram; see Fig. 1-6.

A component-material balance enclosing the top of the column and plate j (see Fig. 1-5) is given by

Similarly, for the stripping section, the component-material balance (see Fig. 1-5) is given by

The component-material balance enclosing the entire column is given by

The total molar flow rates within each section of the column are related by the following defining equation for q, namely

By means of a total material balance enclosing plates /- 1 and / it is readily shown through the use of Eq. (1-40) that

By means of energy balances, it can be shown that q is approximately equal to the heat required to vaporize one mole of feed divided by the latent heat of vaporization of the feed (see Prob. 1-22).

Since Eqs. (1-37) and (1-38) are straight lines, they intersect at some point (*/> yd* provided of course they are not parallel. When the point of intersection is substituted into Eqs. (1-37) and (1-38) and Lr, Vr9 Ls, Vs, xB, and XD are eliminated by use of Eqs. (1-37) through (1-41), the following equation for the q line is obtained yi