## Distillation

To reduce load times, this material is divided into seven files, corresponding to the numbered points below. The present file (distill3.html) contains point 3 only.

## Distillation III: Operating Equations

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For a binary system, we can make a plot of the stage vapor composition vs. stage liquid composition, yn vs. xn. If the points for all the stages are joined, the plot represents the operating path of the system. If the liquid and vapor rates are constant through a section of the column, the operating curve will be a straight line.

In the following analysis, we will assume our column has a single feed and no sidedraw products, but the equations are easily adjusted should this change.

### Rectifying Section

Writing steady state balances over the entire rectifying section, including the accumulator produces:

The latter can be rearranged:
and the total mass balance used to eliminate the vapor rate:
This equation is the operating equation for the rectifying section of the column. Some authors prefer to call this the material balance equation.

### Stripping Section

Next, follow a similar procedure for the stripping section, including the reboiler:

and you have the operating equation for the stripping section.

### Equimolal Overflow

Calculations using these equations are much more convenient if the two operating equations are lines. This is true only if the liquid and vapor flows do not change in a given section of the column. What is required for them to be constant?

Constant Molal Overflow (also called equimolal overflow) is what is needed. This occurs when the molar heat of vaporization of the liquid phase is essentially equal to that of the vapor phase. That is, the heat needed to vaporize one mole of liquid is roughly the same as the heat released when one mole of vapor is condensed. Consequently, any condensation on a stage is balanced out by vaporization and flow rates within the column are changed solely by feed or product streams.

The quickest way to check the validity of an assumption of equimolal overflow is to compare the heats of vaporization of the components. If their ratio is roughly 1:1, the assumption is probably acceptable.

When equimolal overflow is present, and for any given section of the column. (Remember: the only things causing L and V to change are feeds and products). L/V, D/V, and B/V are all constants within a section, so the operating equations are lines:

which can be expressed in terms of the reflux ratio

If you examine the equations, you see that when x=xD that y=xD as well (prove it by substituting xD into the equation). This means that the point (xD, xD) lies on the rectifying line. Thus, if we assume equimolal overflow, the rectifying operating line can be drawn using only this point and the slope.

Similarly, the stripping operating line runs through the point (xB, xB).

If there is a secondary feed or a sidestream product, there are more than two regions in the column; consequently, an additional operating line is required for each sidestream. The forms of the the equations don't change, but the numerical values do. The slope of line through the region will be the (L/V) ratio for that region.

### Feed Line

It would be nice to know where the rectifying line and the stripping line intersect.

The vapor and liquid flow rates will be different in the different sections, so (still assuming equimolal overflow!) the rectifying and stripping section steady state balances are:

Subtracting the stripping balance from the rectifying balance yields
In order to rearrange this to a convenient form, refer to the steady state model for the feed tray, where we showed how the flow rates were changed by the feed:

These can be rearranged to get the flow difference terms we'd like to replace:

and we can substitute them back into the balance equation to get
Similarly, we can substitute the column overall material balance
to obtain

This is a third operating equation. The feed variable q and the feed composition xF are constants so this equation is called the feed line. It can be plotted from xF and q alone. It intersects the diagonal at x=xF.

The slope and position of the feed line depend upon the thermal condition of the feed as described by the parameter q. The line will have positive slope and lie to the right of the vertical for cold feed. The line will be vertical for saturated liquid and horizontal for saturated vapor. For a mixed vapor-liquid feed, it will lie between the horizontal and vertical (negative slope). Superheated feed will produce a line below the horizontal.

The rectifying and stripping lines intersect on the feed line.

If the column has an intermediate feed or product, the same rules apply. A feed/product line, depending only on the composition of the stream and its thermal condition, can be constructed to serve as the set of possible intersections of the operating equations for the regions above and below the sidestream.

R.M. Price
Original: 2/97
Revised: 1/23/98, 1/25/99; 2/14/2003