We assume that the flows of the species are approximately constant across each section of the column:

${y}_{n}=\frac{L}{V}{x}_{n.-1}+\frac{D}{V}{x}_{\mathrm{D}}$

${y}_{m}=\frac{L\prime}{V\prime}{x}_{m-1}+\frac{W}{V\prime}{x}_{\mathrm{W}}$

The feed to the column is partially liquid. Thus we define the liquid and vapor fraction of the feed as ${\mathit{\Phi}}_{\mathrm{L}}$ and ${\mathit{\Phi}}_{\mathrm{V}}$, respectively:

${\mathit{\Phi}}_{\mathrm{L}}=\frac{L\prime -L}{F}{\mathit{\Phi}}_{\mathrm{V}}=\frac{V-V\prime}{F}$

Thus the *q* line can be written as follows:

$q=\frac{{\mathit{\Phi}}_{\mathrm{V}}-1}{{\mathit{\Phi}}_{\mathrm{V}}}{x}_{n.}+\frac{1}{{\mathit{\Phi}}_{\mathrm{V}}}{x}_{\mathrm{f}}$

We now ...