# Talk:Fenske equation

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 Definition:  An equation for calculating the minimum number of theoretical plates needed to separate a binary feed stream by a fractionation column operated at total reflux (i.e., meaning that no overhead product is being withdrawn from the column). [d] [e]
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## Wikipedia has similar article

Wikipedia also has an article on the Fenske equation and I contributed to that article very extensively. I created this CZ article essentially by rewriting the WP article, reformatting it and expanding it. I did the rewrite in my CZ sandbox before creating this article. This CZ article is now much better than the one in Wikipedia.

## Equation explanation

It would appear that the equation explanation may be a bit off:

Does this part:

Xd = mole fraction of more volatile component in the overhead distillate Xb = mole fraction of more volatile component in the bottoms product

mean what it says, ie the mole fraction of the same compound, at two locations, or is this a typo from cut and paste? One of the references listed suggests that Xd and Xb are the mole fraction targets (desired level of purify) of the more volatile and the less volatile chemicals, respectively.

Also, I simple graph showing how N varies as Xd (and/or Xb) changes from values near to 1.0 and approaching 0.0 might be a nice graphic for discussion in the article. Or, at what mole fraction is N a minimum? (I suppose 1.0, neat product)

Also, doesn't the mole fraction of the feedstock come into play somehow for # of theoretical plates needed?David E. Volk 19:59, 21 October 2009 (UTC)

David, thanks for your interest in this article. Let me first say that the Fenske equation as now written is correct and Xd and Xb are indeed the mole fractions of the same component (the more volatile component or the one with the lower boiling point) in the distillate (overhead) and bottoms product.
It will take me a bit of time to explain in more detail and I will do that as soon as I can. However, for the time being I noticed that reference 3 (a lecture at Queens University) is no longer online ...so I am replacing it with a reference from a book instead which is also available online in Google books. Milton Beychok 23:30, 21 October 2009 (UTC)
David, as stated just above the common version of the Fenske equation, there are many different forms of the Fenske equation ... in fact, over a dozen. Many of the them use LK (light key) and HK (heavy key). If you just keep in mind, using D for overhead distillate and B for bottoms these equalities for a binary system:
LKD + HKD = 1, ... then HKD = 1 - LKD
LKB + HKB = 1, ... then HKB = 1 - LKB
If we use mol fraction X =LK then mol fraction HK = 1 - X, and we can write:
XD + (1 - XD) = 1
XB + (1 - XB) = 1
In other words, XD and XB are both the light key (LK) or the more volatile component in the distillate and the bottoms, respectively.
With the above equalities in mind, all of the various forms of the Fenske equation can be resolved to the form now in this article.
What I think worried you is that reference 3 (by Schrieber) uses the words "desired product" instead of "more volatile component". If we had a binary mixture of n-butane and n-pentane and wanted to produce say 95 % butane as the overhead product than butane is the "desired product" and XD = mole fraction of butane (LK) in the distillate and XB = mole fraction of butane (LK) in the bottoms. It would have been much better if reference 3 had been written more carefully. Reference 4 uses the same wording as I did, namely "the more volatile component", for both the distillate and the bottoms.
The purpose of the Fenske equation is to determine the theoretical minimum number of trays at essentially total reflux (essentially no overhead product) for the specified distillation at specified design temperature and pressure. Assuming a specific binary and specific design temperature and pressure, plotting a graph to show the various minimum number of trays for various overhead (or bottoms) compositions would require a lot of work and it would be somewhat of no use since it would not be general. Other binary mixtures would have other relative volatilities and hence that specific graph would not be representative of the multitude of other mixures. Also, even the same binary mixture at another design pressure and temperature would have different relative volatilities and hence a different graph (because temperature and pressure affect the vapor pressures of the binary components).
As discussed in the section on shortcut design methods, the designer would use Fenske's equation to find the minimum number of trays attainable and Underwood's equation to find the minimum reflux attainable (at an infinite number of trays) ... and then select a design between those two boundaries as his first pass design.
Finally, the feed composition does not enter into the Fenske equation at all. Milton Beychok 05:15, 22 October 2009 (UTC)
David, I revised the article to include the equivalent forms of Fenske's equation using LK and HK to help avoid any confusion as to the meaning of the "more volatile" component. Milton Beychok 21:33, 24 October 2009 (UTC)