Fenske equation

 
The Fenske equation in continuous fractional distillation is an equation used for calculating the minimum number of theoretical plates required for the separation of a binary feed stream by a fractionation column that is being operated at total reflux (i.e., which means that no overhead product distillate is being withdrawn from the column).
The equation was derived in 1932 by Merrell Fenske, a professor who served as the head of the chemical engineering department at the Pennsylvania State University from 1959 to 1969.
When designing large-scale, continuous industrial distillation towers, it is very useful to first calculate the minimum number of theoretical plates required to obtain the desired overhead product composition.
Common versions of the Fenske equation.
This is one of the many different but equivalent versions of the Fenske equation valid only for binary mixtures:
where:
For a multi-component mixture the following formula holds.
For ease of expression, the more volatile and the less volatile components are commonly referred to as the light key (LK) and the heavy key (HK), respectively. Using that terminology, the above equation may be expressed as:
or also:
If the relative volatility of the light key to the heavy key is constant from the column top to the column bottom, then formula_8 is simply formula_9. If the relative volatility is not constant from top to bottom of the column, then the following approximation may be used:
where:
The above forms of the Fenske equation can be modified for use in the total reflux distillation of multi-component feeds. It is also helpful in solving liquid–liquid extraction problems, because an extraction system can also be represented as a series of equilibrium stages and relative solubility can be substituted for relative volatility.
Another form of the Fenske equation.
A derivation of another form of the Fenske equation for use in gas chromatography is available on the U.S. Naval Academy's web site. Using Raoult's law and Dalton's Law for a series of condensation and evaporation cycles (i.e., equilibrium stages), the following form of the Fenske equation is obtained:
where: