Experimental centered

Figure 4. Experimental (center) and best-fit simulated (right middle and left middle) H-NMR spectra of DABES on silica from a 2% treating solution (approximately monolayer coverage). On the right-hand side, the fractions of rigid and mobile components were varied from 30% rigid (upper) to 10% rigid (lower). On the left-hand side, the anisotropic rotational rates for the faster rotation

In this way the basic experiment is defined for the linear model, and the gradient that indicates the direction of the fastest response increase or decrease is obtained. When a response maximum or minimum is searched for, the experimental center is moved that way and a new experiment for the linear model performed. The procedure is repeated until moving along the gradient has an effect. When this has no effect, it means we are close to the optimum. Polynomials of higher order, mostly the second, are used in the optimum region. 

This experimental center was suggested based on previous information xio=2500 min 1 x2o=100 °C x30=45 min... 

Conditions (2.76) and (2.77) define independence of the design from rotation of coordinates. When selecting the null/centerpoints points (points in experimental center) take into consideration a check of lack of fit of the model, an estimate of experimental error and conditions of uniformity [37]. Centerpoints are created by setting all factors at their midpoints. In coded form, centerpoints fall at the all-zero level. The centerpoints act as a barometer of the variability in the system. All the necessary data for constructing the rotatable design matrix for k<7 are in Table 2.137. This kind of designing is called central, because all experimental points are symmetrical with reference to the experimental center. This is shown graphically for k=2 and k=3 in Fig. 2.40. 

To obtain the second-order regression model, second-order CCRD has been used. The number of design points (trials) for k=5 was 32. The design core has corresponded to half-replica 2s 1 with this generating ratio Xs=XrX2X3X4. The value and number of design points in the experimental center n0=6 are determined from Table 2.137. The design of experiments with outcomes is shown in Table 2.145. 

Based on the outcomes in the experimental center, these reproducibility variances have been determined Sp =4.466 with this degree of freedom f=n0-l=5. Regression coefficients have these values ... 

Values for all variation levels are shown in Table 2.154. Select FUFE 23 as a basic design of experiment. Determine the linear regression model from experimental outcomes, Table 2.155. Assume that the obtained linear model is inadequate and that there is curvature of the response surface. To check these assumptions, additional design points were done in the experimental center so that their average is y0=0.1097 (y0—estimate of free member in linear regression, i.e. y0 — 30). Since h0 — y0 = 3 is the measure... 

The basic levels and variation intervals are given in Table 2.170. It is known from previous design points that the optimum is within the studied factor space. Orthogonal design has therefore been done to obtain the regression model, Table 2.171. Reproducibility variance of the experiment is determined from four additional design points in the experimental center (y01=61.8 y02=59.3 y03=58.7 and y04=69.0). 

Six design points in the experimental center had to be done due to Table 2.164. For the sake of economy only one design point in the experimental center was set in this example, since the reproducibility variance was obtained in the basic experiment. By processing all 15 design points, the following regression coefficients for second-order model were obtained ... 

The experimental error is by analogy determined also for second-order designs with no replications in null point or experimental center (n(l=0 or n(l=1), for orthogonal B4, Hartley s designs, etc. 

CCRD for k=3 has been analyzed in Example 2.44. The design included twenty trials, six of which in null point (six replications under identical conditions in the experimental center), eight FUFE and six trials in starlikC points (Table 2.139). The reproducibility error or reproducibility variance was determined from replications in the design center. 

In applying this method one should also account for the effect of time, since between the first and series of trials a lot of time may pass. A suggestion in such situations is to systematically replicate trials in experimental center of the basic design of experiments as well as those when moving to optimum, but in a completely random sequence. This approach makes a check of hypothesis on existence of time effects possible. Situations given in Fig. 2.44 are possible when moving to optimum. 

This is taken into account in an engineering sketch drawn by R. Erdmenger himself in 1953 of a chemical screw process (Fig. 2.18). From today s point of view, the last two illustrations display the historical patina and archaic simplicity of engineering times long past. The same also applies for Fig. 2.19 which shows a somewhat rustic experimental center from 1957. 

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