Two-Level Designs

FIGURE 3.3 Composite designs, two-level factorial plus star design, for (a) two factors and (b) three factors. 

Figure )-11 gives an ovemew of the building hloc ks of the ELECTRAS system ELECTRAS was designed for two levels of user experience. The novice part offers a guided data analysis for inexperienced users. Experienced users can analyze their data fast and directly using the expert mode. 

Two-Level Factorial Design with Three Variables... 

The eight fl ials in this design have included evety combination of the variables, with each variable at one of the two levels in each trial, there being four trials with each variable at one of the two levels. Needless to say, the results of the analysis will be improved if each trial is repeated (replicated) but for an initial study this is usually not necessary. 

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... 

The BSD can either shut down the entire facility, or it can be designed for two levels of shutdown. The first level shuts down equipment such as compressors, lean oil pumps, and direct fired heaters, and either shuts in the process or diverts flow around the process by closing inlet/outlet block valves and opening bypass valves. The second level shuts down the remaining utilities and support facilities, including generators and electrical feeds. 

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. 

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. 

The final structure of resins produced depends on the reaction condition. Formaldehyde to phenol (F/P) and hydroxyl to phenol (OH/P) molar ratios as well as ruction temperahne were the most important parameters in synthesis of resols. In this study, the effect of F/P and OH/P wt%, and reaction temperature on the chemical structure (mono-, di- and trisubstitution of methyrol group, methylene bridge, phenolic hemiformals, etc.) was studied utilizing a two-level full factorial experimental design. The result obtained may be applied to control the physical and chemical properties of pre-polymer. 

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. 

A two level full factorial experimental design with three variables, F/P molar ratio, OH/P wt %, and reaction temperature was implemented to analyses the effect of variables on the synthesis reaction of PF resol resin. Based on the composition of 16 components of 10 samples, the effect of three independent variables on the chemical structure was anal3 ed by using 3 way ANOVA of SPSS. The present study provides that experimental design is a very valuable and capable tool for evaluating multiple variables in resin production. 

To extract and evalnate the color pigments from cochineals Dactylopius coccus Costa), a simple method was developed. The procednre is based on the solvent extraction of insect samples nsing methanol and water (65 35, v/v) and a two-level factorial design to optimize the solvent extraction parameters temperature, time, methanol concentration in mixtnre, and yield. For hydrophilic colorants that are more sensitive to temperatnre, water is the solvent of choice. For example, de-aerated water extraction at low temperatnre was applied to separate yellow saffrole and carthamine from saffron (Carthamus tinctorius) florets that contain about 1% yellow saffrole and 0.3% red carthamine. ... 

Based on the experimental data kinetic parameters (reaction orders, activation energies, and preexponential factors) as well as heats of reaction can be estimated. As the kinetic models might not be strictly related to the true reaction mechanism, an optimum found will probably not be the same as the real optimum. Therefore, an iterative procedure, i.e. optimization-model updating-optimization, is used, which lets us approach the real process optimum reasonably well. To provide the initial set of data, two-level factorial design can be used. 

It is interesting to note that threshold problems are quite common in practice and although they do not have a process pinch, pinches are introduced into the design when multiple utilities are added. Figure 16.13a shows composite curves similar to the composite curves from Figure 16.10 but with two levels of cold utility used instead of one. In this case, the second cold utility is steam generation. The introduction of this second utility causes a pinch. This is known as a utility pinch since it is caused by the introduction of an additional utility4. 

Table 3. Conversion, selectivity, yield, and reaction rate at various conditions based on two-level factorial design.

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

Table 5.9. Design matrix for three factors at two levels (+ and — stand for +1 and —1) ...

Table 5.10. Plackett-Burman design matrix lor N 8 experiments and consequently m = 7 factors (including dummy variables) at two levels... 

If nonlinear effects are expected the variables must be varied at more than two levels. A screening plan comparable to the Plackett Burman design but on three levels is that of Box and Behnken [I960]. 

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