Factorial Variable

Factorial designs are usually specified in terms of factorial variables, designated in this section by x and defined as... 

One class of designs directly applicable to computer experiments is orthogonal designs [17,27,35]. An example of such a design for three independent factorial variables (xi, X2, and X3) is shown in Table 1. Each row of this table represents a computer experiment—the hrst column designates its sequential number (1 through 15), the third to hfth columns, labeled Independent variables, list the factorial variable values held in this experiment, and the last column reports the computed response obtained in this computer run. Several different responses could be obtained in a single computer experiment. 

The use of polynomial chaos expansions for the generation of response surfaces is based on the spectral uncertainty method introduced for combustion models in Reagan et al. (2003, 2004, 2005) and Najm et al. (2009) which was extended to an RSM in, e.g.. Sheen et al. (2009). Here an uncertainty factor m, is first assigned to each input variable. Note that this uncertainty factor m, is related to uncertainty parameter/to be discussed in Sect. 5.6.1 by Mj= 10. Taking the example of rate coefficients, they are then normalised into factorial variables x as follows ... 

Factorial design methods cannot always be applied to QSAR-type studies. For example, i may not be practically possible to make any compounds at all with certain combination of factor values (in contrast to the situation where the factojs are physical properties sucl as temperature or pH, which can be easily varied). Under these circumstances, one woul( like to know which compounds from those that are available should be chosen to give well-balanced set with a wide spread of values in the variable space. D-optimal design i one technique that can be used for such a selection. This technique chooses subsets o... 

An important purpose of a designed experiment is to obtain information about interactions among the primary variables. This is accompbshed by varying factors simultaneously rather than one at a time. Thus in Figure 2, each of the two preparations would be mn at both low and high temperatures using, for example, a full factorial experiment. 

Two-Level Factorial Design with Three Variables... 

When a reaction has many participants, which may be the case even of apparently simple processes like pyrolysis of ethane or synthesis of methanol, a factorial or other experimental design can be made and the data subjected to a re.spon.se. suiface analysis (Davies, Design and Analysis of Industrial Experiments, Oliver Boyd, 1954). A quadratic of this type for the variables X, Xo, and X3 is... 

Labelling die variables as T, S and M, widi reference symbols, a, b, c respectively in trials involving a low (1) and high (h) levels of each variable, each trial having die result jc, die factorial procedure would produce die design code as follows ... 

This is a novel feature of factorial design when compared with the classical laboratory procedure which excludes indications of the interaction of tire variable. The method of airalysis of the data, due to Yates, which is commonly used to evaluate these effects, requires tlrat tire uials are conducted in the sequence shown above, and proceeds as follows. 

The table of results is laid out in a column, and a second column is constructed in which in the hrst four rows the results would be added sequentially in pahs, e. g. Xi + X2, xj, + X4, x + jcg etc., and the lower four rows are calculated by subuacting the second value from dre preceding value thus, JC2 — JCi, JC4 — JC3 etc., a thh d column is prepared from these results by canying out the same sequence of operations. The process is continued until there are as many columns as the number of variables. Thus in the present tluee-variable, two level-study the process is repeated tluee times (Table 15.1), and in the general -variable, two-level case it is repeated n times. (The general description of uials of this kind where tlrere are n variables and two levels, is 2 factorial uials ). 

As there now exists a large body of laboratory studies on each of the variable systems, for example the effect of die lime/silica ratio in the slag on the desulphurization of liquid iron, the most appropriate phase compositions can be foreseen to some extent from these laboratory studies when attempting to optimize the complex indusuial process. The factorial uials are not therefore a shot in the dark , but should be designed to take into account die laboratory information. Any qualitative difference between die results of a factorial uial, and the expectations predicted from physico-chemical analysis might suggest the presence of a variable which is important, but which was not included in the nials. 

This resulted in a 2 =16 factorial experiments. To these were added 8 outlayers and 3 repeated centerpoint altogether 27 experiments. The levels of variable are shown on the table in Figure 6.4.1. 

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... 

Because all the variables that influence the properties of the final product are known, one can use a statistical design (known as a one-half factorial) to optimize the properties of the GPC/SEC gels. Factorial experiments are described in detail by Hafner (10). For example, four variables at two levels can be examined in eight observations. From these observations the significance of each variable as related to the performance of the gel can be determined. An example of a one-half factorial experiment applied to the production of GPC/SEC gel is set up in Table 5.2. The four variables are the type of DVB, amount of dodecane, type of methocel, and rate of stirring. 

Other variables in the factorial experiment also have an impact on the character of the final product. The amount of nonsolvent is a very important variable to examine as the pore size of the gel depends on the amount of it present in the formulation. The stabilizer acts as a suspending agent and influences the particle size of the GPC/SEC gel. Lower viscosity suspending agents... 

Another difference between utilities and factories is that most industrial facilities tend to operate boiler plants with a lower quality (and often variable) FW compared to power generators. Although boiler heat flux is usually lower, this practice nevertheless adds an additional water chemistry control burden, especially because most factories do not employ chemists with specific water chemistry duties. 

Additionally, because most factories do not operate steady-state processes, the efforts to match steam output to variable production department steam demands may further tax the boiler plant s generation capacity and manpower resources. 

Factorial design One method of experimental design that allows interactions between factors to be investigated, i.e. whether changing one experimental variable changes the optimum value of another. 

---