any two components in the mixture. The same limitations as to the constancy of relative volatility would apply as for the binary mixtures* Feed-plate Location. The criterion for the optimum location of the feed plate is that the relative enrichment of the key components should be a maximum. As with binary mixtures, the feed plate corresponds to the step that passes from one operating line to the other. The change from one operating line to the other should be made just as soon as it will give a greater enrichment than continuing on the same operating line. In coming up from the still, the feed plate is the last step on the lower operating line; calling this the nth plate, and the key component vapors entering, yik(n-i) and yhHn-i>, for the light, or more volatile, and heavy, or less volatile, components, respectively, the foregoing criterion states that if the nth plate is the optimum position for the feed, then the x ratio on this plate should be greater when calculated by the lower operating line from the yn-i values than by the upper operating line, or

(1) + (W/Vm)xwik > y'lKn-1) — (P/V^XPlh /g^ \Xhk)n Vmn-1) + (W/Vm)Xwhk ~~ y'hhin-1) ~ (D/Vn)xDhk where yn~ 1 = mol fraction in vapor from plate n — 1 when feed is added to plate n 2/n-i = mol fraction in vapor from plate n — 1 when feed is added to plate n — 1 For a binary mixture, yn-1, would equal y'n_lf assuming that the bottoms concentration is kept constant. However, the two values are not equal for multicomponent mixtures because of the presence of the components lighter and heavier than the key components. Owing to the presence of these components, the total vapor is not available for fractionating the key components. In order to reduce the interference of the components lighter than the light key component, it is desirable to utilize the lower operating line to a higher value of the ratio of the key components than would be the case for yn-1 = y'n-v Owing to the interference of the heavy components, it would be desirable to change to the upper operating line at a lower ratio of the key components. However, as a first approximation, it will be assumed that yn„i sa yrn_x and a correction for this assumption will be made later. Combining Eq. (9-4) with

Also, by this criterion the x ratio on the (n + l)th plate should be greater when calculated by the upper than by the lower operating line:

Vnlk - (D/Vn)XPlh ^ Vnlk + (W/Vm)Xwik ^v Vnhk ~ (D/Vn)XDhk ~~ Vnhk + {W/Vm)Xwhh where yn = mol fraction in vapor from plate n when feed is added to plate n y'n = mol fraction in vapor from pla/te n when feed is added to plate n + 1 Again using yn = y'n gives i^lli) ^ z™ + (W/V»)(P + Vx™ /9.7s

The right-hand sides of Eqs. (9-5) and (9-7) are equivalent and equal to (xik/xhk)i as given by the intersection of the operating line; thus, since n is now the optimum feed plate, the subscript may be changed, and the criterion for the feed-plate step becomes

(xiA ^ ZFik + {W/Vm){y + 1 )xwik = (xik\ ^ fxu\ \XhkJ/+1 ~ zrhk + (W/Vm)(p + l)xwhk \Xhk)i \Xhk/f where (xik/xhk)i is the ratio of the key components as given by the intersections of the operating lines. However, it should be emphasized that the feed plate does not necessarily step across the intersection of the operating lines, as it does for a binary, but simply that the ratio of the keys for the optimum feed-plate step passes over the ratio of the values given by the operating-line intersections. The absolute value of both key components may be several times the values given at the intersection, provided the ratio satisfies Eq. (9-8).

The derivations of Eqs. (9-5) and (9-7) neglected the effect of the changing concentrations of the light and heavy components. The light components have a relative constant concentration above the feed plate, and this fact can be used to calculate their value. Thus,

and, assuming

giving

As shown on page 228, the concentration of a light component changes approximately by a factor of Om/VmKi per plate below the feed plate. Thus the change in the concentration of a light component per plate at the feed plate is *

and for most cases Dxdi = Fzfi-

Let Ai equal the sum of the changes per plate for all light components, then

where \ is the sum of the term for all components more volatile than Ikir the light key components.

In a similar manner the sum of the changes in the vapor concentrations for the heavy components per plate at the feed plate, Ah, can be calculated.

where } is the summation for all terms less volatile than the heavy hk-\-

key components.

The corrections due to these changes can be utilized (1) to calculate terms to be added to the intersection ratio {xik)/xhk)i or (2) to modify the expression for the intersection ratio. The relations can be formulated so that both methods give essentially the same optimum ratio for the key components at the feed plate. The latter method is believed to be the more convenient, and using </> as the optimum ratio, an approximate expression is

Fzfiu +

FzFhk +

WXwik

WXwhk

In using this expression, it is recommended that Ki and Kk be calculated as (ai/aik) and (ah/aik), respectively. This expression is similar to that for the intersection ratio, but if A* is large, <j> will be larger than (xik)/(xhk)i\ if A* is large, <f> will be smaller. Using these corrections, the optimum feed-plate location is such that

Optimum Feed-plate Location. The use of Eq. (9-12) will be illustrated by the examples already considered.

1. Benzene-Toluene-Xylene example

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