Fenske Equation

The equation that is used for the calculation of the minimum required number of the theoretical plates for separating the binary feed stream by a fractionation column which is controlled at the total reflux is called the Fenske equation. Merrell Fenske derived this equation in the year 1932. He worked at the Pennsylvania State University as the head of the Chemical Engineering department between 1959 and 1969.

N = minimum number of theoretical plates required at total reflux (of which the reboiler is one)

X_d = mole fraction of more volatile component in the overhead distillate

X_b = mole fraction of more volatile component in the bottoms

α_avg = average relative volatility of more volatile component to less volatile component

The light key (LK) and the heavy key (HK) in the Fenske equation are the expressions which are used for the more volatile and the less volatile components.

From the column top till the column bottom if the relative volatility of light key to heavy key is maintained constant then the  α_avg will be just simply α. In contrary, from the column top to the column bottom if the relative volatility is not maintained constant then an approximation is used as used in the following:

α_avg = √(α_{avg) (}α_avg)

α_t= relative volatility of light key to heavy key at top of column

α_b= relative volatility of light key to heavy key at bottom of column

Gap between the Volatilities:

There are two important things to be noted in the Fenske equation. The first thing is that there might a wide gap between the volatilities of the heavies and the lights. In such case there is a possibility for all the lights to be stored into one pseudo component with the properties of the lights averaged on the basis of mole-fraction and then being used as the light key. Similarly, all the heavies can be stored into a pseudo-component and then it can be used as one heavy key.

Assumption:

The second thing to be kept in mind is that in assumption that the relative volatility of the light and the heavy keys is constant over the length of the column is not that appropriate where the bubble point at the top of the column is quiet lower than that of the bubble point which is in the bottom. So, it is not a reasonable approach to average the two values of the alpha. In the so-called stabilizer columns after a flash drum the residual lights are purged using the columns. It is quite evident through the difference in volatilities that the separation is quite easy to acquire.

Questions:

• What is Fenske equation?
• What are the two things to be noted in the Fenske equation?