Partial factorial design

Because the experimental expenditure increases strongly with the increasing number of influence factors, fractional factorial design FFD (partial factorial design) is applied in such cases. It is not possible to evaluate all the interactions by FFDs but only the main effects. 

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. 

For the complete vapor-phase oxidation of methane over a palladium alumina catalyst, conversion-space-time data were taken at 350°C and 1 atm total pressure the fractional factorial design of Table X (Hll) specified the settings of the feed partial pressures of the reacting species. 

For a partial separation situation after screening, the organic modifier content and temperature are decreased according to a 2 full factorial design. When baseline separation is obtained, the retention factor can be further optimized by changing the... 

The experiments depicted in Figs. 1 and 4 did not determine true optima. In the study of carbon and nitrogen interactions, the optima appeared to lie in the range of 2.4 to 4.0 g/1 for carbon and 0.084 to 0.14 gA nitrogen. One point fell in this range, and it was the maximum for this series of experiments but it is not necessarily an optimum. Likewise, in the partial factorial design depicted in Fig. 4, all of the maxima occurred at star points rather than within the matrix, so it is apparent that the optimum or optima lie somewhere outside the limits of the experimental design. Despite these limitations, several useful inferences can be drawn from these data. 

The disadvantage of full factorial designs is that the number of experiments increases rapidly when the number of factors increases. For 6 factors 64 experiments are required and for 7 factors 128. In practice it is usually not possible to perform such a large number of experiments in a reasonable span of time. For this reason often only a fraction of a full factorial design is performed. This kind of design is called fractional (or partial) factorial design. 

The analysis of variance lends itself best to balanced factorial designs, whether complete, partially replicated, or otherwise modified. The concept of balance simplifies the calculations tremendously. There are ways of coping with missing data, unequal replication under various conditions, and even some lack of orthogonality in the design, but these methods seem to involve more calculation than the data may deserve. The analysis of variance is a procedure which makes it possible to compare the effects of the variables being studied, first independently of the effects of all other variables, and second in all possible combinations with one another. Sometimes the effect of a variable within a given level of another variable... 

Alias in a partial factorial design, certain treatments cannot be distinguished. These ate known as aliases ... 

When some of the possible data points are omitted in factorial designs they are known as fractionally or partially replicated designs. The choice of points to be omitted is of considerable importance. There is no single fractional replicate which is best for any given complete factorial design. Usually the experimenter will have some idea as to the expected effects. When this sort of intuition is available, a particular design may be developed to fit a particular problem. 

The development of the design is illustrated in Table 22. Such designs can also be denoted /-level partial factorial designs. Note that a full five-level factorial design for eight compounds would require 58 or 390625 experiments, so there has been a dramatic reduction in the number of experiments required. 

Table 2.29 Development of a multilevel partial factorial design.

In most cases, it is best to use a full factorial design for the factorial points, but if the number of factors is large, it is legitimate to reduce this and use a partial factorial design. There are almost always 2k axial points. 

Three operating variables have been analyzed hydrogen partial pressure, temperature and reaction time. A factorial design of experiments was used to choose the conditions of each run. The levels of the three factors used during the runs are shown in Table I In the same table we can see the expressions employed to normalize the factors. 

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. 

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... 

Use some parametric search algorithm, like factorial designs [126] or a partial or unsaturated factorial design [127] to reduce the experimratal space in which to find the optimum, or... 

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