where: xdA and xnA are the top and pinch compositions of the light key component, xdB and xnB are the top and pinch compositions of the heavy key component, and aAB is the volatility of the light key relative to the heavy key component.
The difficulty in using this equation is that the values of xnA and xnB are known only in the special case where the pinch coincides with the feed composition. Colburn(39) has suggested that an approximate value for xnA is given by:
rf where: ry is the estimated ratio of the key components on the feed plate. For an all liquid feed at its boiling point, ry equals the ratio of the key components in the feed. Otherwise ry is the ratio of the key components in the liquid part of the feed.
xyh is the mole fraction of each component in the liquid portion of feed heavier than the heavy key, and a is the volatility of the component relative to the heavy key.
Using this approximate value for Rm, equation 11.109 may be rearranged to give the concentrations of all the light components in the upper pinch as:
The concentration of the heavy key in the upper pinch is then obtained by difference, after obtaining the values for all the light components. The second term in the denominator is usually negligible, as the concentration of the heavy key in the top product is small.
A similar condition occurs in the stripping section, and the concentration of all components heavier than the light key is given by:
where: xm and are the compositions of a given heavy component at the pinch and in the bottoms, xmA and are the compositions of the light key component at the pinch and in the bottoms,
Lm/ W is the molar ratio of the liquid in the stripping section to the bottom product, aAB is the volatility of the light key relative to the heavy key, and a is the volatility of the component relative to the heavy key.
Again, the second term in the denominator may usually be neglected.
The essence of Colburn's method is that an empirical relation between the compositions at the two pinches for the condition of minimum reflux is provided. This enables the assumed value of Rm to be checked. This relation may be written as:
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