When designing large-scale, continuous industrial distillation towers, it is very useful to first calculate the minimum number of theoretical plates required to obtain the desired overhead product composition.
is the mole fraction of more volatile component in the bottoms,
is the average relative volatility of the more volatile component to the less volatile component.
For a multi-component mixture the following formula holds.
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. Using that terminology, the above equation may be expressed as:
If the relative volatility of the light key to the heavy key is constant from the column top to the column bottom, then is simply . If the relative volatility is not constant from top to bottom of the column, then the following approximation may be used:
is the relative volatility of light key to heavy key at top of column,
is the relative volatility of light key to heavy key at bottom of column.
The above forms of the Fenske equation can be modified for use in the total reflux distillation of multi-component feeds. It is also helpful in solving liquid–liquid extraction problems, because an extraction system can also be represented as a series of equilibrium stages and relative solubility can be substituted for relative volatility.
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:
is the number of equilibrium stages,
is the mole fraction of component n in the vapor phase,
is the mole fraction of component n in the liquid phase,
^ abChapter 4Archived 2014-05-31 at the Wayback Machine, (From the website of Cal Poly Pomona in California. Lecture notes of Professor Thuan Ke Nguyen for the course entitled CHE313, Mass Transfer. See page 4-42.)
^ abDavid S.J. Jones and Peter P. Pujado (Editors) (2006). Handbook of Petroleum Processing (1st ed.). Springer. ISBN1-4020-2819-9.CS1 maint: extra text: authors list (link) (See page 200.)
^Henry Kister (1992). Distillation Design (1st ed.). McGraw-Hill. ISBN0-07-034909-6. (See equations 3.4 and 3.5 on page 106.)
^ abA. Kayode Coker (2010). Ludwig's Applied Process Design for Chemical and Petrochemical Plants, Volume 2 (4th ed.). Gulf Professional Publishers. ISBN978-0-7506-8366-1.
^Binay K. Dutta (2007). Principles of Mass Transfer and Separation Processes. Prentice Hall of India. ISBN978-81-203-2990-4. (See equation 7.88 on page 375.)