Factorial design mathematical model

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

Agenda 6 The last agenda consists of a team review and approval of a write-up that documents the final test design The documentation must Include the consensus factorial table, hierarchical tree, and mathematical model used to fit the predicted values In addition, the documentation must Include all basic arguments and considerations, even if these considerations do not appear in explicit form in the final design The specific reasons for excluding certain test... 

Factorial tables, hierarchical trees, and associated mathematical models are elementary tools used to guide the efforts of the design team ... 

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. 

This is rarely the case in engineering. Most of the time we do have some form of a mathematical model (simple or complex) that has several unknown parameters that we wish to estimate. In these cases the above designs are very straightforward to implement however, the information may be inadequate if the mathematical model is nonlinear and comprised of several unknown parameters. In such cases, multilevel factorial designs (for example, 3k or 4k designs) may be more appropriate. 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

Because earlier experimental results and data analyses (3-10) had led us to anticipate the inadequacy of the simple approach considered above, we also planned and carried out (2) a second order factorial design of experiments and related data analysis. Mathematical analysis (of the results of 11 experiments) based on the second order model showed that all of these results could be represented satisfactorily by an equation of the form... 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

Historically, factorial designs were introduced by Sir R. A. Fisher to counter the then prevalent idea that if one were to discover the effect of a factor, all other factors must be held constant and only the factor of interest could be varied. Fisher showed that all factors of interest could be varied simultaneously, and the individual factor effects and their interactions could be estimated by proper mathematical treatment. The Yates algorithm and its variations are often used to obtain these estimates, but the use of least squares fitting of linear models gives essentially identical results. 

To obtain the mathematical model of the process, 1/4-replica of a full factorial experiment of type 2s has been realized. Design points-trials have been done in a completely random order. The Table 2.129 shows conditions and outcomes of doing a 26 2 fractional factorial experiment. 

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). 

Many methods have been used to size relief systems area/volume scaling, mathematical modeling using reaction parameters and flow theory, and empirical methods by the Factory Insurance Association (FIA). The Design Institute for Emergency Relief Systems (DIERS) of the AIChE has performed studies of sizing reactors undergoing runaway reactions. Intricate laboratory instruments as described earlier have resulted in better vent sizes. 

Andres, T. H. and Hajas, W.C. (1993). Using iterated fractional factorial design to screen parameters in sensitivity analysis of a probabilistic risk assessment model. Proceedings of the Joint International Conference on Mathematical Models and Supercomputing in... 

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. 

Two level factorial designs are primarily useful for exploratory purposes and calibration designs have special uses in areas such as multivariate calibration where we often expect an independent linear response from each component in a mixture. It is often important, though, to provide a more detailed model of a system. There are two prime reasons. The first is for optimisation - to find the conditions that result in a maximum or minimum as appropriate. An example is when improving die yield of synthetic reaction, or a chromatographic resolution. The second is to produce a detailed quantitative model to predict mathematically how a response relates to die values of various factors. An example may be how the near-infrared spectrum of a manufactured product relates to the nature of the material and processing employed in manufacturing. 

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. 

The experimental distribution catalyst to be tested, reaction temperature, inlet gas composition NO concentration, propane concentration, presence of water vapour and presence of CO and the measured NO conversion values are collated in Table 3. A simple direct correlation from the NO conversion results presented in Table 3 with the experimental variables was not obvious. However, by application of the factorial design the effects and interactions between different variables may be defined by a mathematical model. 

General forms of full factorial designs and their mathematical models... 

We will begin by demonstrating the form and meaning of the mathematical models used in factor-influence studies, using a simple 2 factor example. We will go on to look at the most widely used designs, mainly factorial and fractional factorial designs at 2 levels, but also Rechtschaffner and %-factorial designs and... 

The coefficients in the model equation 3.4 may be estimated as before, as linear combinations or contrasts of the experimental results, taking the columns of the effects matrix as described in section III.A.5 of chapter 2. Alternatively, they may be estimated by multi-linear regression (see chapter 4). The latter method is more usual, but in the case of factorial designs both methods are mathematically equivalent. 

E. General Forms of Full Factorial Designs and their Mathematical Models 1, The synergistic model... 

Additional experiments are needed for there to be enough data (N > p) for a statistical analysis. We add an experiment to the design at the centre of the domain, which is the point furthest from the positions of the experiments of the factorial design. This will allow us to verify, at least partially, the mathematical model s validity. Therefore the solubility was determined in a mixed micelle containing... 

In the last item it has been cited that a factorial design was applied. The mathematical model has been used, as it was the real plant. The basic idea is to obtain an analytical expression, statically representative of the plant, to foresee plant outputs. Chosen the desired output with its respective inputs influences, a cluster of 16 data has been collected according to the following schema. 

The symbol + mean that the output result of the studied variable has been considered for an upper reference, while is the opposite, the variable is calculated with it lower reference. The lower and upper limits are fixed according to the interest area, because this range of values will determine the region that it is possible to apply the model. Equation (1) represents the commented factorial design mathematical model. 

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