Factorial designs center points

Factorial design (center point) Bransonic 52 ultrasonic bath (output power of 120 W frequency of 35 KHz) ... 

The central composite design (CCD) is based on the full quadratic polynomial. Hence it is composed of 2 factorial design, center points and 2k axial portion of design. 

Full two-level factorial design. Star design. Center point. 

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. 

Full factorial 23-design with center point... 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is... 

In Figure 13.9, instead of carrying out four replicate experiments at the center point (as in Figure 13.2), the four replicates are carried out such that one experiment is moved to each of the existing four factorial points. The experimental design matrix is... 

The reactions were conducted according to a two factorial design with three variables, which contains experimental points at the edges and the center of a face-centered cube leading to 9 different experiments. Typically, the experiment at the center point is conducted at least 3 times to add degrees of freedom that allow the estimation of experimental error. Hence a total of 11 experiments are needed to predict the reaction rate within the parameter space. The parameter space for the catalysts to be prepared is shown in columns 2-4 in Table 1. 

In general the cube portion might be replicated times and the star portion might be replicated times. Also, it might be possible to use a fractional factorial design of resolution less than V if the experimenter is prepared to assume that certain interactions are negligible. A central composite design in four variables is shown in Table 2.6. In this table, runs 1-16 are the cube portion, runs 17-24 are the star portion, and runs 25-27 are the center points. 

A 20 run 2 fractional factorial design with four replicate center points was carried out to assess whether it was possible to optimize the current formulation for scale-up or if major reformation would be necessary. Table 3 lists the formulation variables that were evaluated. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

The full factorial central composite design includes factorial points, star points, and center points. The corresponding model is the complete quadratic surface between the response and the factors, as given by Eq. 1 ... 

In this model, the regression coefficients of the pure quadratic terms (the fijj) are not estimable because the typical screening design has all factors at only two levels. However, the experimenter should be alert to the possibility that the second-order model is required. By adding center points to the basic 2f factorial design we can obtain a formal test for second-order curvature, that is, a test of the null hypothesis... 

The center points consist of nc replicates run at the design point Xj = 0, j = 1,2,...,/, when all of the design factors are quantitative. Let yF be the average of the response values at the nF factorial design points and yc be the average response at the center points. The f-statistic for testing the null hypothesis in (3) is... 

A five-level-five-factor CCRD was employed in this study, requiring 32 experiments (Cochran and Cox, 1992). The fractional factorial design consisted of 16 factorial points, 10 axial points (two axial points on the axis of each design variable at a distance of 2 from the design center), and 6 center points. The variables and their levels selected for the study of biodiesel synthesis were reaction time (4-20 h) temperature (25-65 °C) enzyme amount (10%-50% weight of canola oil, 0.1-0.5g) substrate molar ratio (2 1—5 1 methanol canola oil) and amount of added water (0-20%, by weight of canola oil). Table 9.5 shows the independent factors (X,), levels and experimental design coded and uncoded. Thirty-two runs were performed in a totally random order. 

A five-factor (buffer concentration, pH, IPR and organic solvent concentrations, and column temperature) two-level fractional factorial design with four center points was performed. The center points of the design were the midpoints of the range for each factor. The response from the design did not focus on bare retention but on resolution, and particularly on the separation of potential impurities around the main peak. Peak tailing, run time and backpressure were also considered chromatographic responses [76]. 

Fig. 4 Central composite design for three factors. The factorial points are shaded, the axial points unshaded, and the center point(s) filled.

---