Fenske equation

The Fenske equation is used to estimate the number of minimum required theoretical trays in the rectification of a binary chemical mixture under the condition of complete return. The column is operated without a feed and without removing top and bottom product. The calculated number is therefore always lower than the number of trays actually required for separation columns in production.

Determining equation

Distillation column as considered for the Fenske equation

In the Fenske equation, the compositions of the top and bottom products are related. The proportionality factor is the repeated evaporation and condensation (on the theoretical trays), which is described by the relative volatility . The different volatility of the components ensures that the low boiler accumulates in the steam and the high boiler in the liquid. The number of soils is the exponent of the relative volatility:

${\displaystyle {\frac {x_{{\text{Kopf}},L}}{x_{{\text{Kopf}},S}}}=\alpha _{L,S}^{N_{\text{min}}}{\frac {x_{{\text{Sumpf}},L}}{x_{{\text{Sumpf}},S}}}}$

With

${\displaystyle L}$ = Low boiler

${\displaystyle S}$ = High boiler

${\displaystyle x}$= Mole fraction (mole fraction)

${\displaystyle \alpha _{L,S}}$ = Relative volatility of the low boiler in relation to the high boiler in the composition of the feed

${\displaystyle N_{\text{min}}}$ = Minimum number of floors

This equation can be rearranged so that the theoretical number of trays can be determined directly:

${\displaystyle N_{\text{min}}={\frac {\log \left[{\frac {x_{{\text{Kopf}},L}}{x_{{\text{Kopf}},S}}}\cdot {\frac {x_{{\text{Sumpf}},S}}{x_{{\text{Sumpf}},L}}}\right]}{\log \alpha _{L,S}}}}$

However, the relative volatility is not constant over the entire separating column, since it depends both on the composition of the mixture and is temperature and pressure dependent; Therefore, an approximation is usually made using the geometric mean of the relative volatilities of the components in the top and bottom of the column:

${\displaystyle \alpha _{\text{Mittel}}={\sqrt {\alpha _{L,S}^{\text{Kopf}}\cdot \alpha _{L,S}^{\text{Sumpf}}}}}$

Sample calculation

With the arbitrarily set values

${\displaystyle \alpha _{L,S}=1{,}1}$ (assumed here as constant)

${\displaystyle x_{{\text{Kopf}},L}=0{,}98}$

${\displaystyle x_{{\text{Kopf}},S}=0{,}02}$

${\displaystyle x_{{\text{Sumpf}},L}=0{,}05}$

${\displaystyle x_{{\text{Sumpf}},S}=0{,}95}$

follows

${\displaystyle N_{\text{min}}={\frac {\log \left[{\frac {0{,}98}{0{,}02}}\cdot {\frac {0{,}95}{0{,}05}}\right]}{\log 1{,}1}}}$

${\displaystyle N_{\text{min}}=71{,}7}$

The number of calculated theoretical trays increases with lower relative volatility and with higher degrees of purity for the low boiler in the top and for the high boiler in the bottom of the column.

credentials

1. ↑ Fenske, MR (1932). Ind. Eng. Chem., Vol. 24, page 482.