Feed Line (q-Line)

The feed enters somewhere in the middle, and how hot it is changes what the column has to do. A cold feed condenses vapour rising past it and swells the liquid flow below; a vaporised feed does the reverse. The q-line is how that thermal condition gets onto the diagram, and it is the third line of the McCabe-Thiele construction.

The quantity $q$ has two equivalent definitions and it is worth carrying both. Physically, it is the moles of liquid added to the stripping section per mole of feed. Thermally, it is the heat needed to bring one mole of feed to a saturated vapour divided by the molar latent heat. The second makes the five cases fall out immediately: $q > 1$ subcooled liquid, $q = 1$ saturated liquid, $0 < q < 1$ partially vaporised, $q = 0$ saturated vapour, and $q < 0$ superheated vapour.

The slope $q/(q-1)$ behaves strangely on purpose, and the strangeness is the useful part. At $q = 1$ it is infinite and the line stands straight up; at $q = 0$ it is zero and the line lies flat. In between, for a partially vaporised feed, it is negative, which is why a flashing feed draws a line that leans backward across the diagram. Every case pivots about the same point, $(z_F, z_F)$ on the 45-degree diagonal, so the line is drawn by putting a pencil there and setting the slope.

What it decides is the feed stage. Both operating lines must intersect on the q-line, and stepping the diagram off so that you switch from the rectifying line to the stripping line at that intersection gives the fewest total stages. Feeding a tray or two away from the optimum costs real trays, and feeding badly wrong costs many. This is also where preheating gets evaluated honestly: heating the feed moves work from the reboiler to the condenser and shifts the intersection, and whether that is worth doing depends on the relative cost of hot and cold utility at your site, not on a general rule.