Process & Water Chemistry · The column balance
Two balances, and nothing else
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Two balances, and nothing else

Before a single tray is counted, the split is already decided. Draw one box around the whole column and write two balances: total flow, F=D+BF = D + B, and light key, FzF=DxD+BxBF z_F = D x_D + B x_B. Solve them together and everything else falls out:

D=FzFxBxDxBD = F\,\dfrac{z_F - x_B}{x_D - x_B} — read aloud D equals F, z-F minus x-B, over x-D minus x-B.

Every letter, in words. FF is the feed flow, DD the distillate leaving overhead and BB the bottoms leaving the base, all three in kmol/h. zFz_F, xDx_D and xBx_B are the mole fractions of the light key in those same three streams — the subscript names the stream, and the letter z is reserved for a feed that may be part vapour, while x is a liquid. All three fractions are bare numbers between 0 and 1, and the feed's must sit between the two products, because a column cannot make both ends purer than what it was given.

Look at the shape and you have a lever rule. Lay the three compositions on a line: xBx_B at one end, xDx_D at the other, zFz_F somewhere between. The fraction of the feed that goes overhead is exactly how far along zFz_F sits. A feed close to the distillate specification means most of it leaves out the top.

Now notice what the answer did not need. No reflux ratio. No stage count. No relative volatility. No column. That independence is why this is always the first calculation on the pad: if the split it returns is commercially wrong, no quantity of trays or reflux will rescue it, and the specification itself has to change. One warning attached — the same arithmetic works on a mole basis or a mass basis, but not on both at once. Mole fractions against mass flows is the classic error, and it hands back a number that looks entirely reasonable.