Model validation, response surface designs

The response surface design results are analysed by building, interpreting and validating an empirical model describing the relationship between responses and the studied factors. A second-order polynomial model is selected usually because frequently only two or three important factors are optimised, but for more factors the models would be similar. Equations (3.11) and (3.12) present the models for two x and x ) and three factors (xi, X2 and x ), respectively. 

The latest DoE was focussing on the axial and swirl stream temperature (Tswiri) the rotation speed (n) in a face-centred central composite (CCF) response surface design with those three factors (/=3) on three levels. Levels were set linearly as mentioned in section Improved Experimental Setup so that N = 2 +2/+1 = 15 experiments were required for this model. The centre point was repeated five times to ensure reproducibility and reasonable model validity. Particle size, span, particle shape, surface roughness, flowabDity and BET surface area were chosen as responses to evaluate the significant effects of the factors on these particle properties [34, 35]. 

Furthermore, optimal design theory assumes that the model is true within the region defined by the candidate design points, since the designs are optimal in terms of minimizing variance as opposed to bias due to lack-of-fit of the model. In reality, the response surface model is only assumed to be a locally adequate polynomial approximation to the truth it is not assumed to be the truth. Consequently, the experimental design chosen should reflect doubt in the validity of the model by allowing for model lack-of-fit to be tested. 

Response Surface Methodology (RSM) is a well-known statistical technique (1-3) used to define the relationships of one or more process output variables (responses) to one or more process input variables (factors) when the mechanism underlying the process is either not well understood or is too complicated to allow an exact predictive model to be formulated from theory. This is a necessity in process validation, where limits must be set on the input variables of a process to assure that the product will meet predetermined specifications and quality characteristics. Response data are collected from the process under designed operating conditions, or specified settings of one or more factors, and an empirical mathematical function (model) is fitted to the data to define the relationships between process inputs and outputs. This empirical model is then used to predict the optimum ranges of the response variables and to determine the set of operating conditions which will attain that optimum. Several examples listed in Table 1 exhibit the applications of RSM to processes, factors, and responses in process validation situations. 

Statistical design of experiments (DOE) is often used in the early stages of process optimization. This is followed by a validation of the predictive model using actual plant production data. The response surface model described below captures the process performance window and shows the effect of changing composition and extruder screw speed on blend properties. A validation step can be easily implemented. 

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