Before we try to combine the use of the rectifying and stripping operating equations and analyze how many stages a given separation should take, we must first look at how the feed affects our calculations:

There are five possibilities for how the feed could impact the column flows:

- Feed is a saturated liquid (so it simply increases L')
- Feed is a saturated vapor (so it simply increase V)
- Feed is a subcooled liquid (so it
*condenses*some of V and increases L' significantly) - Feed is a superheated vapor (so it
*evaporates*some of L' and increases V significantly) - Feed is a mixture of vapor and liquid (both L' and V increase)

Performing a balance on the feed stage, we can see how these options affect column operation:

F + V' + L = V + L'

Recalling that the rectifying and stripping sections meet at the feed stage, we can envision either plotting the lines and finding where they intersect (i.e., where $x_{rectifying} = x_{stripping}$ and $y_{rectifying} = y_{stripping}$) or thinking of the column as a continuum and doing out balance on a tiny slice of column right where the feed stage would go (but currently is not, yet). This would give us for the stripping section:

yV' = xL' - x_{B}B

and for the rectifying section:

yV = xL + x_{D}D

We can subtract the operating equation for the stripping section from the rectifying section, to get:

y(V-V') = x(L-L') + x_{D}D + x_{B}B

From an overall column balance we have:

zF = x_{D}D + x_{B}B

so:

y(V-V') = x(L-L') + zF

which is essentially a component balance on the feed "slice" discussed above. Similarly, a total balance on this slice yields:

(V-V') = (L-L') + F

The **feed quality, q** is the fraction of the feed that is
liquid (just like it was for our flash calculations), or:

q = (L'-L)/F

Rearranging our total balance on the feed stage/slice and dividing by F, we get:

F/F = 1 = (L'-L)/F + (V-V')/F

or

1-q = (V-V')/F

Dividing our component feed stage/slice balance also by F, and using this relation for (V-V')/F, we get

y(1-q) = -xq + z

solving for y, gives us the feed equation as:

y = xq/(q-1) - z/(q-1)

Returning to our five feed types we see:

- If feed is a saturated liquid, q=1 (i.e., the difference in liquid flows is purely due to the feed: L'-L=F ), "slope" of feed equation is infinite
- If feed is a saturated vapor, q=0 (i.e., the difference in vapor flows is purely due to the feed: V-V'=F ), "slope" of feed equation is zero
- If feed is a subcooled liquid, q>1 (i.e., L' augmented by vapor condensation), "slope" of feed equation is positive
- If feed is a superheated vapor, q<0 (i.e., L' decreased evaporation), "slope" of feed equation is positive
- If feed is a mixture of vapor and liquid, q is between 0 and 1, "slope" of feed equation is negative

Combining this feed information with our equations for the rectifying and stripping sections allows us to analytically determine the number of stages necessary to achieve a particular outlet concentration.

Calculate the feed quality and determine its effect on flowrates

Determine the number of stages required to achieve a separation, using the Lewis method