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column that is being operated at total reflux (i.e., which means that no overhead product distillate is being withdrawn from the column).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}}
where:  
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
\alpha_{avg} = 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 \alpha_{avg.} is simply \alpha. 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)}
where:
\alpha_t = relative volatility of light key to heavy key at top of column
\alpha_b = relative volatility of light key to heavy key at bottom of column

Two comments should be made regarding the above discussion. First, in multi-component distillation, there may be a wide gap between the volatilities of the "lights" and the "heavies." In this case, all the lights can be grouped into one pseudo-component, with their properties being averaged on a mole-fraction basis, and used as the light key. Likewise, all the heavies can be grouped into another pseudo-component and used as the heavy key. Second, in columns in which the bubble point at the top of the column is much lower than the bubble point at the bottom, the assumption that the relative volatility of the heavy and light keys is constant over the length of the column is not appropriate, and hence, averaging the two values of alpha is not a reasonable approach. This is the case, for example, in so-called "stabilizer" columns used to purge residual lights after a flash drum. In such cases, however, the wide difference in volatilities means that separation is in fact quite easy, few stages are required, and hence the final design of the column is fairly insensitive, from the perspective of cost, to error in the value of alpha used. Hence, the smaller value (i.e. the value at the bottom of the column) can safely be used.[6]

The above Fenske equation can be modified for use in the total reflux distillation of multi-component feeds.[3] The Fenske Equation is also helpful in solving liquid-liquid extraction problems, because an extraction system can also be represented as a series of equilibrium stages, and rather than relative volatility, relative solubility can be substituted.

Contents

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:

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

See also

References

  1. ^ Fenske, M.R. (1932). Ind.Eng. Chem., Vol. 24: 482.
  2. ^ a b Distillation notes (Loren Schreiber, Florida State University)
  3. ^ a b Lecture 13: Fenske Equation (Queens University, Canada)
  4. ^ Tutorial 6: Separation Processes (J. Skilling, University of Edinburgh, Scotland)
  5. ^ Maxwell, J.B. (1950). Data Book on Hydrocarbons (1st ed.). D. Van Nostrand. 
  6. ^ Douglas, James M. (1988). Conceptual Design of Chemical Processes (1st ed.). McGraw Hill. 

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