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[[Image:Total Reflux.png|thumb|right|275px|{{#ifexist:Template:Total Reflux.png/credit|{{Total Reflux.png/credit}}<br/>|}}Industrial fractionation column operating at total reflux]]
The '''Fenske equation''' is used for calculating the minimum number of [[theoretical plate]]s required for the separation of a binary feed stream by a [[Fractional distillation|fractionation column]] that is being operated at total [[reflux]] (i.e., which means that no overhead product is being withdrawn from the column). The derivation of the Fenske equation assumes that the [[relative volatility]] is constant in the fractionation column.


Theoretical plates are also often referred to as theoretical plates or [[equilibrium stages]].
The equation was derived by Merrell Fenske in 1932 <ref>Fenske, M.R. (1932). ''Industrial Engineering Chemistry'', '''Vol. 24''': 482.</ref>, a professor who served as the head of the [[chemical engineering]] department at the [[Pennsylvania State University]] from 1959 to 1969.
==A common version of the Fenske equation==
This is one of the many different but equivalent versions of the Fenske equation:<ref name=Schreiber>[http://www.eng.fsu.edu/~schreiber/uol/exp300/ Distillation notes] (Loren Schreiber, [[Florida State University]])</ref><ref name=Queens>[http://www.chemeng.queensu.ca/courses/CHEE317/documents/Lecture_13_2007.pdf Lecture 13: Fenske Equation] ([[Queens University]], Canada)</ref><ref>[http://www.chemeng.ed.ac.uk/~jskillin/teaching/sepprocs/2004-05/tutorials/Tut6.pdf Tutorial 6: Separation Processes] (J. Skilling, [[University of Edinburgh]], Scotland)</ref><ref>{{cite book|author=Maxwell, J.B.|title=Data Book on Hydrocarbons|edition=1st Edition|publisher=D. Van Nostrand|year=1950|id=}}</ref>
:<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>
where:
{| border="0" cellpadding="2"
|-
!align=right|<math>N</math>
|align=left|= minimum number of theoretical plates required at total reflux (of which the reboiler is one)
|-
!align=right|<math>X_d</math>
|align=left|= [[mole fraction]] of more [[Volatility (chemistry)|volatile]] component in the overhead distillate
|-
!align=right|<math>X_b</math>
|align=left|= mole fraction of more volatile component in the bottoms product
|-
!align=right|<math>\alpha_{avg}</math>
|align=left|= 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 <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=Schreiber/>
:<math>\alpha_{avg.} = \sqrt {(\alpha_t)(\alpha_b)}</math>
{| border="0" cellpadding="2"
|-
|align=right|where:
|
|-
!align=right|<math>\alpha_t</math>
|align=left|= relative volatility of light key to heavy key at the top of the column
|-
!align=right|<math>\alpha_b</math>
|align=left|= relative volatility of light key to heavy key at the bottom of the column
|}
The above form of the Fenske equation can be modified for use in the total reflux distillation of multi-component feeds.<ref name=Queens/><br><br>
==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|U.S. Naval Academy's]] web site.<ref>[http://chemistry.usna.edu/IntegratedLabs/SC263/200603Distillation_06.pdf Fenske Equation] (U.S. Naval Academy)</ref> 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:
:<math>\ \frac{Z_a}{Z_b} = \frac{X_a}{X_b} \left (\frac{P^0_a}{P^0_b} \right) ^N </math>
{| border="0" cellpadding="2"
|-
|align=right|where:
|&nbsp;
|-
!align=right|<math>N</math>
|align=left|= number of equilibrium stages
|-
!align=right|<math>Z_n</math>
|align=left|= mole fraction of component n in the vapor phase
|-
!align=right|<math>X_n</math>
|align=left|= mole fraction of component n in the liquid phase
|-
!align=right|<math>{P^0_n}</math>
|align=left|= [[vapor pressure]] of pure component n
|}
==Shortcut calculations for designing fractionation columns==
There are many so-called ''shortcut'' calculation methods for designing industrial fractionation columns. The most commonly used one is the Fenske-Underwood-Gilliland method.
The Fenske equation estimates the minimum number of theoretical plates or equilibrium stages at total reflux. The [[Underwood equation]]<ref>A.J.V. Underwood (1948). ''Chemical Engineering Progress'', '''Vol. 4''':603.</ref> estimates the minimum reflux for an infinite number of theoretical equilibrium stages. The [[Gilliland method]]<ref>E.R. Gilliland (1940). ''Industrial Engineering Chemistry'', '''Vol.32''':1220.</ref> then uses Fenske's minimum plates and Underwood's minimum reflux to estimate the theoretical plates for a given distillation at a chosen reflux.
The estimates such as provided by the Fenske-Underwood-Gilliland shortcut calculations are most effective when used to obtain a preliminary design before following up with the use of distillation simulation software which utilize much more rigorous calculation methods.
==References==
{{reflist}}

Revision as of 14:35, 17 February 2008

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