Mathematical Programming in Refining

Possibly the chemical industry does not have as much need for mathematical models in process simulation as does the petroleum refining industry. The operating conditions for most chemical plants do not seem subject to as broad a choice, nor do they seem to require frequent reappraisals. However, this is a matter which must be settled on the basis of individual circumstances. Sometimes the initial selection of operating conditions for a new plant is sufficiently complex to justify development of a mathematical model. Gee, Linton, Maire, and Raines describe a situation of this sort in which a mathematical model was developed for an industrial reactor (Gl). Beutler describes the subsequent programming of this model on the large-scale MIT Whirlwind computer (B6). These two papers seem to be the most complete technical account of model development available. However, the model should not necessarily be thought typical since it relies more on theory, and less on empiricisms, than do many other process models. 

A critical point in this process is the evaluation of the model, as the model should only be altered if a change improves its quality. There are several mathematical approaches to define a function which is assumed to possess a minimum for the best possible model in the world of small molecules (typically less than 2(X) independent atoms) the least-squares approach is by far the most common method, while for protein structures other methods Ufce maximum likelihood have also been employed. The program SHELXL, on which this book focuses, is predominantly a program for small-molecule structures and the least-squares refinement is the only method on which we need to concentrate. The concept is simple by means of Fourier transformation, a complete set of structure factors is calculated from the atomic model. The calculated intensities are then compared with the measured intensities, and the best model is that which minimizes M ... 

For the refinement of an obverse/reverse twin SHELXL needs a special reflection file in HKLF 5 format and the refinement is not possible with a single TWIN command (see the two examples in 7.8.3 and 7.8.4). This restriction is unnecessary and will be removed if and when there is a new release of the program. After producing the HKLF5 format file further merging of equivalent reflections is not possible. Therefore the data should be merged before producing this file. Otherwise all data would be treated as independent, which leads to mathematically incorrect standard uncertainties. 

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