Fixed factor levels

Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. 

The estimation of purely experimental uncertainty is essential for testing the adequacy of a model. The material in Chapter 3 and especially in Figure 3.1 suggests one of the important principles of experimental design the purely experimental uncertainty can be obtained only by setting all of the controlled factors at fixed levels and replicating the experiment. 

To optimize the process of isomerization of sulphanylamide from Problem 2.6, a screening experiment has been done by the random balance method. Factors X1 X2 and X3 have been selected for this experiment. Optimization of the process is done by the given three factors at fixed values of other factors. To obtain the second-order model, a central composite rotatable design has been set up. Factor-variation levels are shown in Table 2.148. The design of the experiment and the outcomes of design points are in Table 2.149. 

Factors are fixed if the values of the levels are given or do not vary. Factors are random if the value is, for pragmatic purposes, randomly selected from a population of all values. To illuminate in an example if a small lab has two analysts that will always be running the assays then the factor analyst is considered fixed with two levels (one level for analyst 1, the other level for analyst 2) however, if the lab is large and/or there is a high turnover in analysts then for each run of an assay the analyst could be regarded as selected at random among a population of trained analysts. 

One object of this exercise is to select factors that are probably active and to get a preliminary idea as to their effects. Another object is to eliminate as many of the factors as possible. The factor is not retained in the remainder of the study if it appears to have no effect. This usually means that from then on the factor is fixed at one of the levels used in the screening study. 

If each of the 3 factors is fixed at 2 levels, and we do experiments at all possible combinations of those levels, the result is a 2 full factorial design, of 8 experiments. The design, plan and results (turbidity measurements only) are listed in table 3.5a, in the standard order. 

The curves show how the responses vary with each factor, maintaining the levels of the other factors fixed at the specified values. In the second graph of the first column, for example, we see that Young s modulus is practically unaffected by a variation in t, as we have already discovered in Section 6.3. The crucial factor for the determination of the optimum point is C, the oxidant concentration, which is the factor with the most pronounced slopes. These slopes are all instructive, because they provide an idea of how safe it is to vary the factor levels around the optimized values. The plot of the overall desirability as a function of t shows that... 

In the final phase of the optimization study for earplug production (Sato, 2002) a 3 factorial design was carried out to study the dependencies of the three responses, apparent density (AD), equilibrium force (EF) and retention time (RT), on the fiow rate of the reticulate solution (A) and the quantity of water in the polyol solution (B). Only these two factors were found to be important, after the execution of a 2 " fractional factorial and a blocked 2 factorial design (see case studies in Sections 3A.9 and 4A.8). The other factors were fixed at levels chosen to provide the most desirable apparent densities, EFs and RTs, as well as convenient operational conditions. Table 6A.7 contains the results of the three responses for the three-level factorial experiments, which were performed in duplicate. 

The response surface representing the model can be visualised graphically by means of 2D contour plots and/or 3D response surface plots. In a 2D contour plot, the isoresponse lines are represented as a function of levels of two factors, while in a 3D plot the response is represented on a third dimension, as a function of the factor levels (see Figure 3.24). When more than two factors are examined and modelled, all but two factors need to be fixed at a given level to draw both plots. The optimal or acceptable experimental conditions can be derived from the graphical representation of the model or by mathematical analysis of its equation. 

However, in the case of entropy the identity of the particles is a factor. In section 17.2 we assumed that we could tell the difference between individual particles that is, we assumed they were distinguishable. In fact, at the atomic level we cannot distinguish between individual, identical particles atoms and molecules are macroscopically indistinguishable. This means that we are overcounting the total number of possible distributions for El. The factor that fixes this overcounting is a factor of M in the denominator of El. (That is, there are 1/M times fewer distributions for indistinguishable particles than for distinguishable particles.) When this factor is considered, the equations become... 

Design values are calculated trough a representative value (called characteristic value) and a set of partial factors. Through partial factors the designer assigns a certain level of reliability to the structure. The partial factors are fixed values, calibrated using probabilistic methods. 

The X-ray instrumentation requires a commercial small angle X-ray camera, a standard fine structure X-ray generator and a sample manipulator if scanning is requested. The essential signal is the relative difference between the refraction level Ir and the absorption level Ia. Both levels are measured simultaneously by two scintillation detectors. At fixed angles of deflection this signal depends solely on the inner surface density factor C and thickness d of the sample [2] ... 

The classical experiment tracks the off-gas composition as a function of temperature at fixed residence time and oxidant level. Treating feed disappearance as first order, the pre-exponential factor and activation energy, E, in the Arrhenius expression (eq. 35) can be obtained. These studies tend to confirm large activation energies typical of the bond mpture mechanism assumed earlier. However, an accelerating effect of the oxidant is also evident in some results, so that the thermal mpture mechanism probably overestimates the time requirement by as much as several orders of magnitude (39). Measurements at several levels of oxidant concentration are useful for determining how important it is to maintain spatial uniformity of oxidant concentration in the incinerator. 

More details of other factors that affect the critical pitting potential have been discussed by Uhlig and his co-workers" . They indicated that for stainless steel the critical pitting potential decreased with increasing concentration of chloride ion. At a fixed chloride level, passivating ions in solution, such as sulphate and nitrate, etc., cause the pitting potential to become more positive at a sufficient concentration these ions totally inhibited pitting, as shown in Fig. 19.40 for SO and CIO . 

Note the close analogy with the Lineweaver-Burk form of the simple Michaelis-Menten equation. In a diagram representing MV against MX one obtains a line which has the same intercept as in the simple case. The slope, however, is larger by a factor (1 + YIK-) as shown in Fig. 39.17b. Usually, one first determines and in the absence of a competitive inhibitor (F = 0), as described above. Subsequently, one obtains A" from a new set of experiments in which the initial rate V is determined for various levels of X in the presence of a fixed amount of inhibitor Y. The slope of the new line can be obtained by means of robust regression. 

---