Calculate minimum theoretical stages at total reflux or relative volatility for binary and multicomponent distillation from product compositions.

Total reflux; constant average relative volatility. Mole fractions are of the whole product stream.

Illustrative pair

Rough examples; use VLE data for your temperature, pressure and composition.

Related Calculators

Fenske Equation Formula

The Fenske equation is used to estimate the minimum number of theoretical stages (equilibrium stages) required in a distillation column at total reflux to achieve a specified separation between a light key (LK) and heavy key (HK). For a binary system written in terms of the LK mole fractions in the distillate and bottoms, a common form is:

Nโ‚˜in = log((xD / (1 - xD)) ร— ((1 - xB) / xB)) / log(ฮฑ)

Variables:

  • Nmin is the minimum equilibrium-stage count at total reflux. This calculator includes an equilibrium partial reboiler or partial condenser and excludes a total condenser; upward rounding gives a whole-stage lower bound, not an actual tray count.
  • xD is the distillate mole fraction of the light key (LK)
  • xB is the bottoms mole fraction of the light key (LK)
  • ฮฑ is the average relative volatility of the light key relative to the heavy key (ฮฑLK/HK > 1)

Specify LK composition targets and an appropriate average relative volatility under the column conditions. Feed composition is not required for this total-reflux estimate. In multicomponent mode, use LK and HK fractions of each whole product stream; each pair must sum to 1 or less, and the distillate LK/HK ratio must exceed the bottoms ratio.

What is the Fenske Equation?

The Fenske Equation is a fundamental equation in the field of distillation that provides an estimate of the minimum number of theoretical stages required to achieve a given key-component separation. It assumes equilibrium stages, total reflux, and (in its most common use) an average relative volatility that is treated as constant over the column. The equation is particularly useful during the early design phase of a distillation column; for operation at finite reflux, additional methods (e.g., Underwood and Gilliland correlations) are typically used.

How to Calculate the Number of Theoretical Plates?

The following steps outline how to calculate the minimum number of theoretical stages using the Fenske equation.

  1. Select the light key (LK) and heavy key (HK) components for the desired separation.
  2. Determine the LK mole fraction in the distillate, xD.
  3. Determine the LK mole fraction in the bottoms, xB.
  4. Estimate an average relative volatility, ฮฑLK/HK, appropriate for the expected temperature/pressure/composition range.
  5. Use the Fenske equation: Nmin = log( (xD/(1 โˆ’ xD)) ยท ((1 โˆ’ xB)/xB) ) / log(ฮฑ).
  6. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Average relative volatility (ฮฑ) = 2.50

Distillate mole fraction of LK (xD) = 0.95

Bottoms mole fraction of LK (xB) = 0.05 (gives Nmin โ‰ˆ 6.42 theoretical stages at total reflux; in practice stage counts are typically rounded up)