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# 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column that is being operated at total reflux (i.e., which means that no overhead product distillate is being withdrawn from the column).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 by Merrell Fenske in 1932 [1], a professor who served as the head of the chemical engineering department at the Pennsylvania State University from 1959 to 1969.

This is one of the many different but equivalent versions of the Fenske equation:[2][3][4][5]

$\ N = \frac{\log \, \bigg[ \Big(\frac{X_d}{1-X_d}\Big)\Big(\frac{1-X_b}{X_b} \Big) \bigg]}{\log \, \alpha_{avg}}$
N where: = minimum number of theoretical plates required at total reflux (of which the reboiler is one) = mole fraction of more volatile component in the overhead distillate = mole fraction of more volatile component in the bottoms = average relative volatility of more volatile component to less volatile component

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.

If the relative volatility of the light key to the heavy key is constant from the column top to the column bottom, then αavg. is simply α. If the relative volatility is not constant from top to bottom of the column, then the following approximation may be used:[2]

$\alpha_{avg.} = \sqrt {(\alpha_t)(\alpha_b)}$
αt where: = relative volatility of light key to heavy key at top of column = relative volatility of light key to heavy key at bottom of column

The above Fenske equation can be modified for use in the total reflux distillation of multi-component feeds.[3]

## 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.[6] 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:

$\ \frac{Z_a}{Z_b} = \frac{X_a}{X_b} \left (\frac{P^0_a}{P^0_b} \right) ^N$
N where: = number of equilibrium stages = mole fraction of component n in the vapor phase = mole fraction of component n in the liquid phase = vapor pressure of pure component n