Feed Section Operating Line (q-line)

See the Figure below for a feed tray and consider F moles/hr of feed, with fraction q of liquid; thus

liquid
flow = q F moles/hr

vapour flow = (1-q) F moles/hr

Overall material balance:

L' = L + q F

V = V' + (1-q) F

Component balance for the more volatile component - see the Figure below:

Rectifying section : V y = L x + D xD

Stripping section : V' y = L' x - B xB

At the feed point where the two lines operating lines intersect:

( V - V' ) y = ( L - L' ) x + D xD + B xB

we have:

V - V' = ( 1 - q ) F

L - L' = - q F

In addition, from component
balance around the entire column:

F xF = D xD + B xB

Thus, ( 1 - q ) F y = - q F x + F xF

Re-arranging in the form y = f(x), we have:

For a given feed condition, xF and q are fixed, therefore the q-line is a straight line with slope -q / (1-q) and intercept xF / (1-q).

If x = xF
, then y = xF.

i.e. the q-line passes through
the point (xF, xF)
on the 45o diagonal.

Different values of q will result in different slope of the q-line. See the Figure below.

Note that the q-line passes through the point (xF, xF) on the 45o diagonal for all values of q.