The composition of the vapor was calculated on the basis of (#26)1 because it is believed that the equilibrium data are more reliable in the high benzene region than in the high water region. If 26 theoretical plates were employed and two liquid layers were refluxed to match #27, then plate 26 would have two liquid layers present. Usually only one liquid layer is refluxed.
The tables of data and the liquid compositions plotted on Fig. 10-13 illustrate the factors involved. Starting at the bottom of tower 1, the system behaves like a mixture of benzene and alcohol, and the benzene concentration increases rapidly. The relative volatility of water is low and does not increase significantly until the benzene concentration has built up enough to increase the volatility of the water.
1 These calculations are not exact, because the solubility data given in Fig. 10-13 are for 25°C., while the temperature on plate 26 is about 66°C. The solubilities are somewhat different at the two temperatures, but the 25°C. data are used to illustrate the principle.
The water concentration then increases rapidly until the feed plate is reached. Above the feed plate the heavy key component (alcohol) decreases rapidly, and the benzene and water attain values that result in two-layer formation. It will be noted that the liquid compositions are heading for ternary azeotrope composition as reported by Young (Ref. 7). The composition of the mixed vapors to the condenser is a point on the tie line through xrt and xr2, and the relative distances from this composition to xr and xr2 are inversely as Orx and Or2.
The limiting conditions for azeotropic conditions are not easily expressed in analytical equation, but they can be evaluated for each specific case.
Minimum Number of Theoretical Plates at Total Reflux. Owing to the wide variation of the relative volatility, equations of the type of (7-53) are not applicable. The number of theoretical plates required is calculated best by the plate-to-plate method using y = x as the operating line for each component. For the benzene-alcohol-water system considered in the preceding section, this plate-to-plate method indicates that between 12 and 13 theoretical plates are required at total reflux.
Minimum Reflux Ratio. This limit corresponds to a pinched-in position, or positions, in the tower. Because of the wide variation in relative volatilities with composition, this limit frequently corresponds to a tangent condition of the operating lines and equilibrium values rather than an intersection. In such cases it is difficult to calculate the exact tangent condition, and each system is essentially a new problem. However, in a number of cases, the minimum reflux ratio is determined by intersections of the operating lines and the equilibrium values, and these often occur near the feed plate, because the mixture to be treated is usually a binary and the azeotrope agent is approximately constant above and below the feed plate. For these cases, the general principles employed for multicomponent mixtures can be applied. As an example, consider the benzene-alcohol-water system already studied, which has this type of limiting condition. The asymptotic concentrations below the feed plate are given by equations of the type of (9-15). Solving for the values between water and benzene,
The concentration of benzene in the bottoms, Xwb, is much smaller than the asymptotic value, xB, and the last term of the denominator will be neglected. The value of xs is much larger than the numerator of the right-hand side of the equation, and this necessitates aB being essentially equal to an, Thus, for this case where xwh and xwb are very small, the pinched-in condition corresponds to the relative volatility of benzene to water being unity; i.e., <xbh « 1.0. A study of Fig. 10-15 indicates that <xBs = 1 for only a limited concentration range for benzene. Above the feed plate the net alcohol and benzene removals are very small, and the same type of analysis leads to the conclusion that otb « a*. The conditions abh = 1 below the feed plate and cxab = 1.0
above the feed plate can be used to evaluate the minimum reflux ratio. One approximation for this limit can be obtained by equating the concentration ratio of alcohol to water for <xB 5=8 a a to the feed ratio. The composition for this condition can be obtained by drawing a line through xa = 0.89, xh = 0.11, and the benzene corner of the diagram. Where this line cuts the aH = as line gives the desired values. This construction has been carried out in Fig. 10-16, and the intersection gives xa « 0.55, xm = 0.07, and xB » 0.38. From Figs. 10-14 and 10-15, - 0.54 and a** - 1.0. By Eq. (9-15),
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