Stage calculations computer methods

For systems with more than two components, there are two approaches to stage calculations approximate methods, which can be done quickly by hand, and rigorous methods, which require a computer. The approximate methods will be discussed first. 

For the first time through a liqmd-liquid extrac tion problem, the right-triangular graphical method may be preferred because it is completely rigorous for a ternary system and reasonably easy to understand. However, the shortcut methods with the Bancroft coordinates and the Kremser equations become valuable time-savers for repetitive calculations and for data reduction from experimental runs. The calculation of pseudo inlet compositions and the use of the McCabe-Thiele type of stage calculations lend themselves readily to programmable calculator or computer routines with a simple correlation of equilibrium data. 

The classic papers by Lewis and Matheson [Ind. Eng. Chem., 24, 496 (1932)] and Thiele and Geddes [Ind. Eng. Chem., 25, 290 (1933)] represent the first attempts at solving the MESH equations for multicomponent systems numerically (the graphical methods for binary systems discussed earlier had already been developed by Pon-chon, by Savarit, and by McCabe and Thiele). At that time the computer had yet to be invented, and since modeling a column could require hundreds, possibly thousands, of equations, it was necessary to divide the MESH equations into smaller subsets if hand calculations were to be feasible. Despite their essential simplicity and appeal, stage-to-stage calculation procedures are not used now as often as they used to be. 

The extreme form of the estimation procedures is ab initio calculation from theory. There is every reason to hope that improvements in theory and in computational methods will eventually reach the stage where such data can be used confidently in models, and even at its present level of accuracy such data are useful for checking experimental results. However, further progress in theory will depend on validation by experimental kinetics data, drawn from a wide range of sources. 

Density functional methods is an attractive alternative to Hartree Fock method due to its computational efficiency. There are density functional methods with different correlation functions [42-48]. Recently several calculations [49-53] were done on zeolites of varying cluster models with density functional methods. Periodic density functional method similar to periodic Hartree Fock is also being used recently [54]. Thus the quantum chemical technique has been at the center stage among the other computational methods in improving our understanding of stmeture and reactivity of zeolites. We apply this powerful technique to study the mechanism of zeolite synthesis. 

An extension of the Underwood method for distillation columns with multiple feeds is given by Barnes, Hanson, and King. Exact computer methods for determining minimum reflux are available. For making rigorous distillation calculations at actual reflux conditions by the computer methods of Chapter 15, knowledge of the minimum reflux is not essential but the minimum number of equilibrium stages is very useful. 

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