I don't use extent of reaction in reactor models. I prefer the mass balance to be more explicit and therefore I use mass fractions as the composition state variables. However, I do use extent of reactions for two purposes...(1) computing the equilibrium curve for a single reaction and (2) computing initial guesses of the molar composition when simultaneous equations are involved (e.g. minimization of Gibbs free energy). The next post shows an example of the first use, so I need to define some terms.
Two kinds of extent of reaction
In the image below, two kinds of extent of reaction are defined. The most commonly used is called the extent of reaction, which has the symbol X below. I use a fractional extent of reaction with the Greek letter xi as the symbol. In the literature, that symbol is commonly used for the extent of reaction. The easy way to determine which is being used is by the dimensions...the extent is molar and the fractional extent is dimensionless.
Definitions and Formulas
The example below demonstrates the differences between the two kinds of extent of reaction. Both examples show the values of the respective extents needed to obtain full conversion of the limiting reactant.
The feature I like about the fractional extent is that it is bounded by 0 and 1. For the molar extent, the upper value depends upon the feed composition and is unbounded.
Using extents in reactor models
The molar extent is probably more conveniently used in models requiring differential equations. This is probably the reason that fractional extent of reaction is not as common in the literature.
An early reference to fractional extent of reaction
I don't know the origin of the fractional extent, but Boudart used it for single reactions.
Boudart, M., "Kinetics of Chemical Processes", Prentice-Hall, 1968.