Experimental domains, selection

The presented block diagrams link the factor-fixing accuracy, range of response change and response-surface curvature with the width of factor-variation interval. When selecting a factor variation interval one should, if possible, account for the number of factor variation levels in the experimental domain. Depending on the number of these levels, are the experiment range and optimization efficiency. 

In cases where several variables have to be optimized, one often uses experimental design. An experimented design is a predefined experimental set-up in which a given number of variables are examined with a given number of experiments. The experiments are chosen such that the experimental domain is mapped (covered) in a systematic way. The experimental design selected depends on the goals of the study that is carried out. For instance, some experimental designs make it possible to estimate the effect or the influence of the variables (often also called factors) on the considered response(s). 

Another possibility to deal with irregularly shaped experimental domains is to use so-called uniform mapping algorithms such as the algorithm of Kennard and Stone. They have the advantage that the number of experiments can be sequentially increased. It ensures that the experiments cover the space as uniformly as possible and that they are situated as far as possible from each other. It consists of maximizing the minimal distance between a newly selected point and those previously selected. The distance is the Euclidean distance, given by... 

However, in order to use these criteria in the selection of an experimental design from a set of possible candidate experiments, certain assumptions must be fullfilled (1) The mathematical form of the model to be fitted is perfectly known. (2) The region in which the experiments can be run, i.e. the experimental domain is known, and it is excluded that experiments can be run outside this region. 

It is possible to select the number of center point experiments so that dj is fairly constant in the whole experimental domain. This will give almost the same precision of the predicted response, for all possible settings of the experimental variables in the explored domain. Such a design is called a uniform precision design.[4] It is evident that this is a desirable property if the model is to be used for simulations. 

The number of experiments N in response surface designs is larger than the number of b-coefficients that needs to be estimated. The obtained model then can be used to predict the response for given experimental conditions. It should be emphasized that only predictions within the examined experimental domain are recommended. Extrapolations should be avoided because the model may not be correct anymore and the prediction error will increase (7). However, most frequently the model is used to determine the optimum, and this is selected from the graphical representation (Figure 2.18), rather than using the model for predictive purposes. 

An improvement is brought by the sequential approaches. They select the best variable first, then the best pair formed by the first and second, and so on in a forward or backward progression. A more sophisticated approach applies a look back from the progression to reassess previous selections. The problem with these approaches is that only a very small part of the experimental domain is explored and that the number of models to be tested becomes very high in case of highly dimensional data sets, such as spectral data sets. For instance, with 1000 wavelengths, 1000 models are needed for the first cycle (selection or removal of the first variable), 999 for the second cycle, 998 for the third cycle, and so on. 

The number of factors studied is usually between 2 and 5 and in any case should not exceed 7. Thus RSM is often preceded by one or more screening and factor-effect studies, enabling the most important factors to be selected and isolated. These also allow adjustment and modification of the experimental domain, recentring the design, if necessary, in a more interesting region. 

In this case, both actions were taken. A more restricted experimental domain was selected, eliminating regions which, according to the 2 factor study, would result in a high turbidity. Alcohol concentration was omitted as a variable (it was fixed at 2%) and a 3 factor second-order model was postulated. 

The possible experimental domain is not always known at the beginning of a study. On the one hand, there is a risk of choosing too extensive an experimental region, and then the responses may vary over so wide a range that it becomes impossible to fit the simple quadratic model to them. On the other hand, the response surface may show discontinuities, such as in a tableting experiment, where there may be whole regions where tablets cannot be obtained because of incompatibility with the machine, or lack of cohesion. To avoid this problem a much smaller domain may be selected, with the result that the optimum may then be discovered to be outside it. 

Not only solids, but many liquid systems also, have discontinuities in their properties over the factor space. Figure 10.1 shows a ternary mixture of water, an oil, and a surfactant. According to the proportions of the components, completely different phases are obtained and the methods of experimental design that we have seen so far, along with the polynomial or related models developed for then-analysis, are not suitable for analysis of the total system. For the experimental domain, it would be necessary to select a portion of the factor space with the same phase or phases over all of it. 

This calculation helped selection of the experimental domain gas flow reactor,... 

The thermodynamic study has permitted selection of the experimental domain, and a good correlation with experimental results has been obtained (thermodynamic conversion = 30%, experimental conversion = 26 %). 

The influence of the residence time on the selectivities is studied at 527 °C. Figure 4 shows that benzaldehyde is favoured at short residence times and 1,2-diphenylethane at higher residence times. The highest selectivity for the 1,2-diphenylethane, obtained in the experimental domain, is 0.61. 

The variables selected for use in optimization of the process were reaction ten erature and the molar ratio of dimethylamine to 4-chloroacetophenone. The experiments were run in sealed reactors. The initial design contained seven experiments (rotatable hexagonal design with center point). Six experiments were equally spaced around the center point that was set at 3 equivalents of dimethylamine and 230 C. The unit variation in the equivalents of dimethylamine was set to 1 molar equivalent, and the unit variation in temperature was set to 20 °C. At the conclusion of this series of experiments, it was determined that the optimum conditions were not within the experimental domain. Consequently, the original study was augmented with additional experiments. 

Oqce the variables have been selected one needs to define the boundaries of the experimental domain, i.e., the ranges... 

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