Statistical methods factorial experiments

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... 

Factorial design of experiments, combined with statistical methods of data analysis, offers wider and more differentiated information on the system, while conclusions are of greater usability. The results of all the eight runs in the analyzed example serve for determining the factor effects, with seven trials being independent possibilities of testing the effects and one serving for their comparison with the chosen fixed values. Three out of seven independently determined factor effects serve for... 

The last twenty years of the last millennium are characterized by complex automatization of industrial plants. Complex automatization of industrial plants means a switch to factories, automatons, robots and self adaptive optimization systems. The mentioned processes can be intensified by introducing mathematical methods into all physical and chemical processes. By being acquainted with the mathematical model of a process it is possible to control it, maintain it at an optimal level, provide maximal yield of the product, and obtain the product at a minimal cost. Statistical methods in mathematical modeling of a process should not be opposed to traditional theoretical methods of complete theoretical studies of a phenomenon. The higher the theoretical level of knowledge the more efficient is the application of statistical methods like design of experiment (DOE). 

Brenneman (2000) found that Harvey s method could underestimate the dispersion effect of factor j if that factor was left out of the location model. This result led Brenneman and Nair (2001) to propose a modified version of Harvey s method for two-level factorial experiments that is based on the results of Bergman and Hynen (1997). In the modified version, the dispersion statistic for factor j is computed from residuals from an expanded location model that includes the effect of factor j and all its interactions with other effects in the location model. For two-level designs, the modified Harvey s statistic for factor j is then... 

The first method to be described was an incomplete factorial approach (Carter and Carter, 1979). This is basically a method that, given a matrix of compositional components and their concentrations, defines how to sample the variables with a minimum number of experiments. Using statistical methods to analyze the results, it is possible to identify variables that are correlated, and in later stages to concentrate on their variation to optimize crystallization conditions. This method has been... 

If xi and X2 are varied one at a time, then the method is known as a univariate search and is the same as carrying out successive line searches. If the step length is determined so as to find the minimum with respect to the variable searched, then the calculation steps toward the optimum, as shown in Figure 1.15a. This method is simple to implement, but can be very slow to converge. Other direct methods include pattern searches such as the factorial designs used in statistical design of experiments (see, for example, Montgomery, 2001), the EVOP method (Box, 1957) and the sequential simplex method (Spendley et ah, 1962). 

Burner tests are complex, time-consuming, and expensive experiments. In the case of this category of experiments, one should always put a strong emphasis on the planning supported by statistical methods. Should the examined value be influenced by a low number of factors (< 3) and should the dependence between the values be known as linear, it is sufficient to use the so called factorial plan [29] for fhis fype of experiment, or a fractional factorial plan [29]. Nevertheless, most experiments are supposed to have a response variable without linear dependence on some of the factors, and it... 

Statistical methods have been used for a long time, especially in chemical industry, to reduce the number of experiments when a multiparameter problem has to be solved. Design of experiments (DoEs) or factorial design is the most frequently used method in high-throughput catalysis. Examples are NO oxidation over supported Pt influence of synthesis variables (Pt precursor, support, loading, calcination gas and temperature two levels each) [14]... 

Methods for the production of rubber bonded components have to be established and firmly founded within strict limits of the many parameters for the control of quality. It is in the initial stages of the development of the production process that the use of suitable bond tests is vital. The test values will allow the manufacturer to discover the operational limits for all process variables and ensure that the set conditions for production do not allow a knife-edge situation where small changes can produce large variations in the quality of the bond. This is best achieved through the use of factorial experiment design and statistical regression analysis of the results. 

Crystal Structures.—Crystallization, a pre-requisite for diffraction studies, is a notoriously and unpredictably difficult exercise with new proteins. Carter and Carter have introduced a method which searches a large number of experimental variables (e.g., buffer components and pH) that can influence rates of crystallization. Random combinations of the variables are used to attempt crystallization and the resulting precipitated protein scored on an arbitrary crystallinity scale. After applying the appropriate statistic, a complete factorial experiment is set up using the conditions which promoted crystallization in the first experiment. 

Consider a chemical engineer who is studying the yield of a chemical process. There are 2 variables of interest reaction time and reaction temperature. Since there is some uncertainty regarding the appropriateness of a linear model, a single unreplicated 2 -factorial experiment was performed with 5 centre point replicates. The results of the experiment are shown in Table 4.9. Based on the provided results, analyse the model using the methods provided in the above discussion to determine the best model for the process. Be certain to analyse the residuals to determine the adequacy of the model Data taken from Montgomery, Applied Statistics and Probability for Engineers, 4th edn.). 

The task of finding, for a given set of potential catalytic materials, a subset of representatives conveying the required information about the whole set has for nearly a century been addressed by methods of statistical design of experiments (DOE). A very simple kind called factorial design is used if the required information represents the impact of any combination of possible values of some n independent factors. If... 

A factorial design is a statistical method in which all possible factor combinations are considered, allowing the calculation of the single effects of each factor and any factor interactions. The number of experiments required can be calculated using the following equation ... 

This study shows that the optimization of process conditions could be achieved rapidly by a judicious use of statistics and parallel reactors. A two-level factorial method with two center points was used to limit the total number of experiments to ten. Using two identical high-pressure reactors in parallel further shortened the time required to conduct these experiments. For the model reaction of phenol hydrogenation over a commercially available Pd/C, it was experimentally determined that the optimal yield was 73% at 135 °C, 22.5 bar, and 615 ppm w/w NaOH... 

Some authors [29,34] present the statistical interpretation method as an ANOVA table. A general example for a 2 full factorial design is given in Table 3.19. The sums of squares (55x) are obtained with the effect values (Ex) and the number of experiments in the design (N). The mean square... 

DOE is a methodical statistics approach to studying the qualitative effects of process variables. Variables of interest are given a number of values based on the expected relationships [8]. For example, if the relationship is expected to be linear over a range, two variations can be used to approximate the effect of the variable. For effects that are expected to be quadratic, three variations may be needed. These variations are then matrixed to create a set of trials that differentiate and quantify the effect of each variable. If the number of variables is small, then the experiments can be designed as a full factorial. An example of a full two factorial design of an experiment for three variables is shown in Table 15.1. 

This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. There are two main disadvantages of the complete factorial designs. First, when many factors were defined or when each factor has many levels, a large number of experiments is required. Remember the expression number of experiments = replicates x Oevels) " (e.g. with 2 replicates, 3 levels for each factor and 3 factors we would need 2 x 3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

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