Engineering
Fenske Minimum Distillation Stages (N_min) & Relative Volatility
Estimate binary total-reflux minimum equilibrium stages from mole fractions and supplied relative volatility, with input checks and explicit stage-counting assumptions.
Choose a binary total-reflux model
B2 and B3 are mole fractions of the same more-volatile component in the distillate and bottoms. For a binary mixture, the other component has mole fraction 1−x. This complement shortcut is not generally valid for multicomponent mixtures.
B4 is a supplied constant relative volatility of that component relative to the less-volatile component. Values 2.5 and 1.15 below are illustrative inputs, not property data for named chemicals. Obtain a suitable volatility model for the actual temperature, pressure and composition range.
Fenske estimates the minimum equilibrium contacts at total reflux. A single average volatility is an approximation if volatility varies along the column. The result does not directly give an operating tray count at finite reflux.
Compute the minimum without rounding intermediate values
E2 gives the separation factor [xD/(1−xD)]/ [xB/(1−xB)]. E3 evaluates its logarithm as a sum and difference of logarithms, avoiding the intermediate product in the stage calculation.
For xD=0.98, xB=0.02 and alpha=2.5, the separation factor is 2401 and Nmin is approximately 8.494728. At alpha=1.15, Nmin is approximately 55.692132. Display fewer decimal places without replacing the stored result.
=IF(COUNT(B2:B4)<>3,"Enter numeric fractions and alpha",IF(OR(B3<=0,B2>=1,B2<B3),"Require 0<xB<=xD<1",IF(B4<=1,"Require alpha>1; check component order and model",(LN(B2)-LN(1-B2)+LN(1-B3)-LN(B3))/LN(B4))))Interpret whole stages and the equipment boundary
E4 rounds the raw nonnegative minimum upward. The examples require 9 and 56 whole equilibrium contacts on this basis. Do not round Nmin to two decimals before taking the ceiling.
For the total-condenser/partial-reboiler arrangement used here, the partial reboiler counts as one equilibrium contact and the total condenser does not. Thus 9 contacts correspond to 8 ideal column stages plus that reboiler at total reflux. This is not a specification for 8 real trays: finite reflux and actual stage efficiency require further design.
=IF(ISNUMBER(E3),ROUNDUP(E3,0),E3)Reject invalid specifications without diagnosing an azeotrope
Require numeric 0<xB<=xD<1 and alpha>1. Equal product compositions return zero as the no-enrichment mathematical limit; they do not imply a functioning zero-stage column. Exact zero and one compositions are excluded from the logarithmic formula.
For a nonzero separation, Nmin rises as alpha approaches one from above. Alpha<=1 is rejected under the selected component-order convention. A value below one can mean reversed component labels; this single input does not prove azeotropy or universal inseparability.
Try changing alpha, or reducing the target xD. Use a vapor–liquid equilibrium model to assess feasibility for nonideal mixtures; the guard is an input check, not an azeotrope detector.
=IF(COUNT(B2:B3)<>2,"Enter numeric mole fractions",IF(OR(B3<=0,B2>=1,B2<B3),"Require 0<xB<=xD<1",(B2/(1-B2))*((1-B3)/B3)))Try the example
Use the original values or change them to test your understanding.