Fixed points in reactive distillation

Similarly to the molar fractions, molar flow rates are transformed as suggested by Giittinger (1998), 

These definitions present convenient and simplifying properties ( ) the dimensions of the system are reduced, simplifying the depiction of equilibrium (figure 3.3) (Frey and Stichlmair, 19996 Barbosa and Doherty, 1987a) (ii) they have the same numerical values before and after reaction (m) they sum up to unity (iv) they clearly indicate the presence of reactive azeotropes when Xi =Yf, (v) the nonreactive hmits are well defined (vi) the number of linear independent transformed composition variables coincides with the number of independent variables that describe the chemical equihbrium problem and vii) the lever rule is valid as the chemical reaction no longer impacts the material balance. 

However, the introduction of transformed variables reduces the imderstanding of the phase equilibrium as a whole, hiding eventually some nonreactive azeotropes (Frey and Stichlmair, 19996). 

Fixed-point algorithm is not recommended by Ciric and Gu (1994) and Espinosa et al. (1996) for systems where the reaction kinetics, heat of reaction, liquid hold-up and residence times are the determining design variables. However, Okasinski and Doherty (1998) successfully use it to reproduce the ethylene glycol synthesis example of Ciric and Gu (1994). 

Mahajani and Kolah (1996) successfully extend the approach of Doherty and co-workers to packed columns, where liquid phase backmixing is totally absent. This approach relies on the performance of mass balances over differential elements of length dz in both rectifying and stripping sections, which can be re-written as functions of dimensionless numbers e.g. Da, HTU, dimensionless flux and ratio of characteristic times in the vapor and liquid sides f3 = x x k ))). 

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