Experimental response surface

Optimization can be simplified by employing the predictive capabilities of an artificial neural network (ANN). This multivariate approach has been shown to require minimal number of experiments that allow construction of an accurate experimental response surface (5, 6). The apposite model created from an experimental design should effectively relate the experimental parameters to the output values, which can be used to create an ANN with a strong predictive capacity for any conditions defined within the experimental space (4). 

To develop an empirical model for a response surface, it is necessary to collect the right data using an appropriate experimental design. Two popular experimental designs are considered in the following sections. 

In optimizing a method, we seek to find the combination of experimental parameters producing the best result or response. We can visualize this process as being similar to finding the highest point on a mountain, in which the mountain s topography, called a response surface, is a plot of the system s response as a function of the factors under our control. 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

Note These equations are from Doming, S. N. Morgan, S. L. Experimental Design A Chemometric Approach. Elsevier Amsterdam, 1987, and pseudo-three-dimensional plots of the response surfaces can be found in their figures 11.4, 11.5, and 11.14. The response surface for problem (a) also is shown in Color Plate 13. 

After the dominant independent variables have been brought under control, many small and poorly characterized ones remain that limit further improvement in modeling the response surface when going to full-scale production, control of experimental conditions drops behind what is possible in laboratory-scale work (e.g., temperature gradients across vessels), but this is where, in the long term, the real data is acquired. Chemistry abounds with examples of complex interactions among the many compounds found in a simple synthesis step,... 

A Box-Behnken design was employed to investigate statistically the main and interactive effects of four process variables (reaction time, enzyme to substrate ratio, surfactant addition, and substrate pretreatment) on enzymatic conversion of waste office paper to sugars. A response surface model relating sugar yield to the four variables was developed on the basis of the experimental results. The model could be successfully used to identify the most efficient combination of the four variables for maximizing the extent of sugar production. 

Interpretive methods Involve modeling the retention surface (as opposed to the response surface) on the basis of experimental retention time data [478-480,485,525,541]. The model for the retention surface may be graphical or algebraic and based on mathematical or statistical theories. The retention surface is generally much simpler than the response surface and can be describe by an accurate model on the basis of a small number of experiments, typically 7 to 10. Solute recognition in all chromatograms is essential, however, and the accuracy of any predictions is dependent on the quality of the model. 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

For this last stage, the one-at-a-time procedure may be a very poor choice. At Union Carbide, use of the one-at-a-time method increased the yield in one plant from 80 to 83% in 3 years. When one of the techniques, to be discussed later, was used in just 15 runs the yield was increased to 94%. To see why this might happen, consider a plug flow reactor where the only variables that can be manipulated are temperature and pressure. A possible response surface for this reactor is given in Figure 14-1. The response is the yield, which is also the objective function. It is plotted as a function of the two independent variables, temperature and pressure. The designer does not know the response surface. Often all he knows is the yield at point A. He wants to determine the optimum yield. The only way he usually has to obtain more information is to pick some combinations of temperature and pressure and then have a laboratory or pilot plant experimentally determine the yields at those conditions. 

This model is capable of estimating both linear and non-linear effects observed experimentally. Hence, it can also be used for optimization of the desired response with respect to the variables of the system. Two popular response surface designs are central composite designs and Box-Behnken designs. Box-Behnken designs were not employed in the experimental research described here and will therefore not be discussed further, but more information on Box-Behnken designs can be obtained from reference [15]. 

Table 11.6. Experimental levels of reaction conditions which were investigated for response surface study of NOx storage and reduction catalysts...

Figures 5 and 6 show the response surfaces plotted for Property A and Property B, respectively. Note that two variables are plotted at once, with the values of the other variables fixed at levels chosen by the experimenter. The contours in the graph represent constant levels of the response. Fortunately, the computer allows rapid replotting for various levels of the fixed variables, as well as changing the identities of the fixed and floating variables, so that the entire design space can be investigated.

When from initial experiments, conditions that indicate the enantioselectivity of the system towards a given enantiomer pair or towards a limited series of substances are known, one might optimize their separation. To obtain optimal conditions, the different chemometric techniques used for method optimization in classic chromatographic or electrophoretic separations can also be applied for the chiral ones. Different experimental design approaches, using both screening and response surface designs can be In Reference 331, for... 

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