Factorial designs with response surface

Factorial design and response surface techniques were used in combination with modeling and simulation to design and optimize an... 

Based on the obtained response surface, a second roimd of optimization follows, using the steepest ascent method where the direction of the steepest slope indicates the position of the optimum. Alternatively, a quadratic model can be fitted around a region known to contain the optimum somewhere in the middle. This so-called central composite design contains an imbedded factorial design with centre... 

In the introduction to this chapter, it was said that complete three-level factorial designs with more than two variables would give too many runs to be convenient for response surface modelling. It is, however, possible to select a limited number of runs from such designs to obtain incomplete 3 designs which can be used to fit quadratic models. 

Assume that you have run experiments by a factorial design (with Np runs) with a view to assessing the significance of the experimental variables fi om estimates, hj, of the coefficients in a linear response surface model. Assume also that you have made Nq repeated runs of one experiment to obtain an estimate of the experimental error standard deviation. From the average response, J, in repeated runs, an estimate of the experimental error standard deviation, Sq, with (Nq - 1) degrees of freedom is obtained as... 

The experimental optimisation step was performed by both a two-level full factorial design and a Box-Behnken design combined with response surface methodology. 

O. L. Davies and co-workers. The Design andAna/ysis of Industria/Experiments, 2nd ed., Hafner, New York, 1956 reprinted by Longman, New York, 1987. This book, which is a sequel to the authors basic text Statistica/Methods in Eesearch and Production, is directed at industrial situations and chemical appHcations. Three chapters are devoted to factorial experiments and one chapter to fractional factorial plans. A lengthy chapter (84 pp.) discusses the deterrnination of optimum conditions and response surface designs, which are associated with the name of G. Box, one of the seven co-authors. Theoretical material is presented in chapter appendices. 

Initial screens can be distinguished between methods that are used to determine what factors are most important, and follow-up screens that allow optimization and improvement of crystal quality (Table 14.1). In experimental design, this is known as the Box-Wilson strategy (Box et al., 1978). The first group of screens is generally based on a so-called factorial plan which determines the polynomial coefficients of a function with k variables (factors) fitted to the response surface. It can be shown that the number of necessary experiments n increases with 2 if all interactions are taken into account. Instead of running an unrealistic, large number of initial experiments, the full factorial matrix can... 

One limitation of two-level factorial designs is the assumption of linearity of the effects. If it is possible that the effect of one or more of the factors is nonlinear, a response surface design may be selected. A central composite response surface design is a full factorial or fractional factorial design that is supplemented with additional trials to allow for estimation of curvature from the factors of interest. For each factor of interest to be studied for curvature, two additional trials are performed (1) one trial with all of the factors at their middle level except for... 

Steven Gilmour is Professor of Statistics in the School of Mathematical Sciences at Queen Mary, University of London. His interests are in the design and analysis of experiments with complex treatment structures, including supersaturated designs, fractional factorial designs, response surface methodology, nonlinear models, and random treatment effects. 

This is the design most often used for response surfaces. It is a combination of a factorial with an axial design with experiments at a distance of a along each axis (thus, the name). It requires a relatively large... 

TJ apid entrainment carbonization of powdered coal under pressure in a partial hydrogen atmosphere was investigated as a means of producing low sulfur char for use as a power plant fuel. Specific objectives of the research were to determine if an acceptable product could be made and to establish the relationship between yields and chemical properties of the char, with special emphasis on type and amount of sulfur compound in the product. The experiments were conducted with a 4-inch diameter by 18-inch high carbonizer according to a composite factorial design (1, 2). Results of the experiments are expressed by empirical mathematical models and are illustrated by the application of response surface analysis. 

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