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| [[Image:Total Reflux.png|frame|right|Fractionation at total reflux]]
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| The '''Fenske equation''' in continuous [[fractional distillation]] is an [[equation]] used for calculating the minimum number of [[theoretical plate]]s required for the separation of a binary feed stream by a [[Fractionating column|fractionation column]] that is being operated at total [[reflux]] (i.e., which means that no overhead product distillate is being withdrawn from the column).
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| The equation was derived in 1932 by Merrell Fenske, a professor who served as the head of the [[chemical engineering]] department at the [[Pennsylvania State University]] from 1959 to 1969.<ref>M.R. Fenske (1932). ''Ind.Eng. Chem.'', '''Vol. 24''': 482.</ref>
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| 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.
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| ==Common versions of the Fenske equation==
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| This is one of the many different but equivalent versions of the Fenske equation:<ref name=Nguyen>[http://www.csupomona.edu/~tknguyen/che313/pdf/chap4-4d.pdf Chapter 4], (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.)</ref><ref name=Jones>{{cite book|author=David S.J. Jones and Peter P. Pujado (Editors)|title=Handbook of Petroleum Processing|edition=1st|publisher=Springer|year=2006|isbn=1-4020-2819-9}} (See page 200.)</ref><ref>{{cite book|author=Henry Kister|title=Distillation Design|edition=1st|publisher=McGraw-Hill|year=1992|isbn=0-07-034909-6}} (See equations 3.4 and 3.5 on page 106.)</ref><ref name=Coker>{{cite book|author=A. Kayode Coker|title=Ludwig's Applied Process Design for Chemical and Petrochemical Plants, Volume 2|edition=4th|publisher=Gulf Professional Publishers|year=2010|ISBN=978-7506-8366-1}}</ref><ref>{{cite book|author=Binay K. Dutta|title=Principles of Mass Transfer and Separation Processes|edition=|publisher=Prentice Hall of India|year=2007|isbn=978-81-203-2990-4}} (See equation 7.88 on page 375.)</ref>
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| :<math>\ N = \frac{\log \, \bigg[ \Big(\frac{X_d}{1-X_d}\Big)\Big(\frac{1-X_b}{X_b} \Big) \bigg]}{\log \, \alpha_{avg}} </math>
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| where:
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| * <math>N</math> is the minimum number of theoretical plates required at total reflux (of which the reboiler is one)
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| * <math>X_d</math>  is the [[mole fraction]] of more [[Volatility (chemistry)|volatile]] component in the overhead distillate
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| * <math>X_b</math>  is the mole fraction of more volatile component in the bottoms
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| * <math>\alpha_{avg}</math>  is the average [[relative volatility]] of the more volatile component to the less volatile component
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| 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:<ref name=Jones/>
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| :<math>\ N = \frac{\log \, \bigg[ \Big(\frac{LK_d}{HK_d}\Big)\Big(\frac{HK_b}{LK_b} \Big) \bigg]}{\log \, \alpha_{avg}} </math>
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| or also:
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| :<math>\ N = \frac{\log \, \bigg[ \Big(\frac{LK_d}{1-LK_d}\Big)\Big(\frac{1-LK_b}{LK_b} \Big) \bigg]}{\log \, \alpha_{avg}} </math>
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| If the [[relative volatility]] of the light key to the heavy key is constant from the column top to the column bottom, then <math>\alpha_{avg.}</math> is simply <math>\alpha</math>. If the relative volatility is not constant from top to bottom of the column, then the following approximation may be used:<ref name=Nguyen/>
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| :<math>\alpha_{avg.} = \sqrt {(\alpha_t)(\alpha_b)}</math>
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| where:
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| * <math>\alpha_t</math> is the relative volatility of light key to heavy key at top of column;
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| * <math>\alpha_b</math> is the relative volatility of light key to heavy key at bottom of column.
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| The above forms of the Fenske equation can be modified for use in the total reflux distillation of multi-component feeds.<ref name=Coker/> 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.
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| ==Another form of the Fenske equation==
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| A derivation of another form of the Fenske equation for use in gas chromatography is available on the [[U.S. Naval Academy|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 stage]]s), the following form of the Fenske equation is obtained:
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| :<math>\ \frac{Z_a}{Z_b} = \frac{X_a}{X_b} \left (\frac{P^0_a}{P^0_b} \right) ^N </math> | |
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| where:
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| * <math>N</math>  is the number of equilibrium stages
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| * <math>Z_n</math>  is the mole fraction of component n in the vapor phase
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| * <math>X_n</math>  is the mole fraction of component n in the liquid phase
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| * <math>{P^0_n}</math>  is the [[vapor pressure]] of pure component n
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| ==See also==
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| *[[Continuous distillation]]
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| *[[Distillation]]
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| *[[Fractional distillation]]
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| *[[Fractionating column]]
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| *[[McCabe-Thiele method]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.cbu.edu/~rprice/lectures/distill4.html Lecture Notes] (R.M. Price, [[Christian Brothers University]], Tennessee)
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| *[http://www.design.che.vt.edu/publication/74_LiuQuantrilleCheng1993df.pdf ''Studies in Chemical Process Design and Synthesis''], Y. A. Liu, T.E. Quantrille, and S. Chengt, Ind. Eng. Chem. Res., Volume 29, 1990
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| *[http://www.engr.sjsu.edu/jennimi/Classes/CHE160B/Notes/25%20-%20MULTICOMPONENT%20DISTILLATION%20CONCEPTS%20.ppt Multi-component Distillation] (M.B. Jennings, [[San Jose State University]])
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| {{Distillation}}
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| [[Category:Distillation]]
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| [[Category:Equations]]
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