Factorial model experimental design

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Gonzalez, A. G., TWo Level Factorial Experimental Designs Based on Multiple Linear Regression Models A Tutorial Digest Illustrated by Case Studies, Analytica Chimica Acta 360, 1998, 227-241. 

A two levels of full factorial experimental design with three independent variables were generated with one center point, which was repeated[3]. In this design, F/P molar ratio, Oh/P wt%, and reaction temperature were defined as independent variables, all receiving two values, a high and a low value. A cube like model was formed, with eight comers. One center point (repeated twice) was added to improve accuracy of the design. Every analysis results were treated as a dependent result in the statistical study. 

The form of the mathematical model fitted to the consensus factorial table must be reassessed after the hierarchical tree Is pruned and the experimental design has been revised by the statistician. 

Rapsomanikis and coworkers observed methylation of tin(II) and lead(II) in sediments by carbanion donors74. A factorial experimental design determined separate and combined effects of MeCoBi2 (methylcobalamin) and Me2Co(N4)+ (a methylcobalamin model) (6) on methylation of Sn(II) in sediment matrices. Experimental results for methylation of tin are shown in Table 11. 

Because variables in models are often highly correlated, when experimental data are collected, the xrx matrix in Equation 2.9 can be badly conditioned (see Appendix A), and thus the estimates of the values of the coefficients in a model can have considerable associated uncertainty. The method of factorial experimental design forces the data to be orthogonal and avoids this problem. This method allows you to determine the relative importance of each input variable and thus to develop a parsimonious model, one that includes only the most important variables and effects. Factorial experiments also represent efficient experimentation. You systematically plan and conduct experiments in which all of the variables are changed simultaneously rather than one at a time, thus reducing the number of experiments needed. 

In the following discussion, we shall again separate the terms of a hyperbolic model and identify two parameters Cl and C2. As before, each of these two parameters will be a collection of terms, one of which is multiplied by conversion and one not multiplied by conversion. In previous formulations, however, we have oriented the discussion toward a familiar type of experimental design in kinetics conversion versus space-time data at several pressure levels. Consequently, the parameters Cx and C2 were defined to exploit this data feature. Another type of design that is becoming more common is a factorial design in the feed-component partial pressures. 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

This section has given an overview of some of the experimental designs that are suitable for collecting data to estimate the coefficients of the first-order and second-order model. Many of these designs are based on factorial and fractional factorial designs. 

Designing a stability study is based on a factorial design of experiments where a systemic procedure is used to determine the effect on the response variable of various factors and factor combinations. A linear model is used to represent the relationship between the factors and factor combinations with the response variable. Once the experimental design is established, the assays are conducted and stability data are saved to finally estimate the shelf life period. 

The fundamentals and some case studies of experimental design on the basis of different factorial plans and the following empirical modeling were described by KOSC1EL-NIAK and PARCZEWSKI [1983 1985],... 

Symmetric experimental designs for mixture+process factor spaces are the cross products of symmetric designs for process variables and mixture variables. Figure 8.12 shows an experimental design in mixture+process factor space for a model where both types of variables are of the first order. In both of the examples shown in Figure 8.12, the process variables are described by a two-level full factorial... 

Sukigara et al. [78] designed a factorial experiment by using two factors (electric field and concentration). For a quadratic model, experimentsmust be performed for at least three levels of each factor. These levels are best chosen equally spaced. The two factors (silk concentration and electric field) and three levels resulted in nine possible combinations of factor settings. A schematic of the experimental design is shown in Figure 28(A) and (B). 

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