Fenske equation

TheFenske equationin continuousfractional distillationis anequationused for calculating the minimum number oftheoretical platesrequired for the separation of a binary feed stream by afractionation columnthat is being operated at totalreflux(i.e., which means that no overhead product distillate is being withdrawn from the column).

The equation was derived in 1932 by Merrell Fenske,^[1]a professor who served as the head of thechemical engineeringdepartment at thePennsylvania State Universityfrom 1959 to 1969.^[2]

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.

Common versions of the Fenske equation

This is one of the many different but equivalent versions of the Fenske equation valid only for binary mixtures:^[3]^[4]^[5]^[6]^[7]

\ N={\frac {\log \,{\bigg [}{\Big (}{\frac {X_{d}}{1-X_{d}}}{\Big )}{\Big (}{\frac {1-X_{b}}{X_{b}}}{\Big )}{\bigg ]}}{\log \,\ Alpha _{avg}}}

where:

$N$ is the minimum number of theoretical plates required at total reflux (of which the reboiler is one),
$X_{d}$ is themole fractionof lessvolatilecomponent in the overhead distillate,
$X_{b}$ is the mole fraction of more volatile component in the bottoms,
$\ Alpha _{avg}$ is the averagerelative volatilityof 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 thelight key(LK) and theheavy key(HK), respectively. Using that terminology, the above equation may be expressed as:^[4]

\ N={\frac {\log \,{\bigg [}{\Big (}{\frac {LK_{d}}{HK_{d}}}{\Big )}{\Big (}{\frac {HK_{b}}{LK_{b}}}{\Big )}{\bigg ]}}{\log \,\ Alpha _{avg}}}

or also:

\ N={\frac {\log \,{\bigg [}{\Big (}{\frac {LK_{d}}{1-LK_{d}}}{\Big )}{\Big (}{\frac {1-LK_{b}}{LK_{b}}}{\Big )}{\bigg ]}}{\log \,\ Alpha _{avg}}}

If therelative volatilityof the light key to the heavy key is constant from the column top to the column bottom, then $\ Alpha _{avg.}$ is simply $\ Alpha$ .If the relative volatility is not constant from top to bottom of the column, then the following approximation may be used:^[3]

\ Alpha _{avg.}={\sqrt {(\ Alpha _{t})(\ Alpha _{b})}}

where:

$\ Alpha _{t}$ is the relative volatility of light key to heavy key at top of column,
$\ Alpha _{b}$ 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.^[6]It is also helpful in solvingliquid–liquid extractionproblems, 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 theU.S. Naval Academy's web site. UsingRaoult's lawandDalton's Lawfor a series of condensation and evaporation cycles (i.e.,equilibrium stages), the following form of the Fenske equation is obtained:

\ {\frac {Z_{a}}{Z_{b}}}={\frac {X_{a}}{X_{b}}}\left({\frac {P_{a}^{0}}{P_{b}^{0}}}\right)^{N}

where:

$N$ is the number of equilibrium stages,
$Z_{n}$ is the mole fraction of component n in the vapor phase,
$X_{n}$ is the mole fraction of component n in the liquid phase,
${P_{n}^{0}}$ is thevapor pressureof pure component n.

References

^"Fenske's helices".
^M.R. Fenske (1932).Ind.Eng. Chem.,Vol. 24:482.
^^a ^bChapter 4 Archived2014-05-31 at theWayback Machine,(From the website of Cal Poly Pomona in California. Lecture notes of Professor Thuan Ke Nguyen for the course entitledCHE313, Mass Transfer.See page 4-42.)
^^a ^bJones, David S.J.; P. Pujado, Peter, eds. (2006).Handbook of Petroleum Processing(1st ed.). Springer.ISBN 1-4020-2819-9.(See page 200.)
^Henry Kister (1992).Distillation Design(1st ed.). McGraw-Hill.ISBN 0-07-034909-6.(See equations 3.4 and 3.5 on page 106.)
^^a ^bA. Kayode Coker (2010).Ludwig's Applied Process Design for Chemical and Petrochemical Plants, Volume 2(4th ed.). Gulf Professional Publishers.ISBN 978-0-7506-8366-1.
^Binay K. Dutta (2007).Principles of Mass Transfer and Separation Processes.Prentice Hall of India.ISBN 978-81-203-2990-4.(See equation 7.88 on page 375.)

External links

Lecture Notes Archived2012-05-02 at theWayback Machine(R.M. Price,Christian Brothers University,Tennessee)
Studies in Chemical Process Design and SynthesisArchived2020-04-11 at theWayback Machine,Y. A. Liu, T.E. Quantrille, and S. Chengt, Ind. Eng. Chem. Res., Volume 29, 1990
Multi-component Distillation^{[permanent dead link‍]}(M.B. Jennings,San Jose State University)

[1] "Fenske's helices".

[2] M.R. Fenske (1932).Ind.Eng. Chem.,Vol. 24:482.

[Nguyen-3] Chapter 4 Archived2014-05-31 at theWayback Machine,(From the website of Cal Poly Pomona in California. Lecture notes of Professor Thuan Ke Nguyen for the course entitledCHE313, Mass Transfer.See page 4-42.)

[Jones-4] Jones, David S.J.; P. Pujado, Peter, eds. (2006).Handbook of Petroleum Processing(1st ed.). Springer.ISBN 1-4020-2819-9.(See page 200.)

[5] Henry Kister (1992).Distillation Design(1st ed.). McGraw-Hill.ISBN 0-07-034909-6.(See equations 3.4 and 3.5 on page 106.)

[Coker-6] A. Kayode Coker (2010).Ludwig's Applied Process Design for Chemical and Petrochemical Plants, Volume 2(4th ed.). Gulf Professional Publishers.ISBN 978-0-7506-8366-1.

[7] Binay K. Dutta (2007).Principles of Mass Transfer and Separation Processes.Prentice Hall of India.ISBN 978-81-203-2990-4.(See equation 7.88 on page 375.)

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Fenske equation

Contents

Common versions of the Fenske equation

Another form of the Fenske equation

See also

References

External links