Continuous factors

When significant effects are found on the response(s) describing the quantitative aspect of the method, the results from the robusmess test can be used to set restrictions on the levels of significant continuous factors. When factor X has a significant effect, the initially examined interval is reduced and the levels can be estimated, where the effect is eliminated. The nonsignificance interval limits are estimated as follows ... 

Figure 2.2 is a response surface showing a system response, y, plotted against one of the system factors, x,. If there is no uncertainty associated with the continuous response, and if the response is known for all values of the continuous factor, then the response surface might be described by some continuous mathematical model M that relates the response y, to the factor x. ... 

A continuous factor is a factor that can take on any value within a given domain. Similarly, a continuous response is a response that can take on any value within a given range. Examples of continuous factors are pressure, volume, weight, distance, time, current, flow rate, and reagent concentration. Examples of continuous responses are yield, profit, efficiency, effectiveness, impurity concentration, sensitivity, selectivity, and rate. 

Figure 2.10 Response surface showing an inherently discrete response (number of lines in the Lyman series) as a function of an inherently continuous factor (excitation energy).

The discrete factor solvent number is recognized as a simple bookkeeping designation. We can replace it with the continuous factor dipole moment expressed on a ratio scale and obtain, finally, the response surface shown in Figure 2.13. A special note of caution is in order. Even when data such as that shown in Figure 2.13 is obtained, the suspected property might not be responsible for the observed effect it may well be that a different, correlated property is the true cause (see Section 1.2 on masquerading factors). 

Natural equality constraints exist in many real systems. For example, consider a chemical reaction in which a binary mixed solvent is to be used (see Figure 2.15). We might specify two continuous factors, the amount of one solvent (represented by X,) and the amount of the other solvent ( 2). These are clearly continuous factors and each has only a natural lower bound. However, each of these factors probably should have an externally imposed upper bound, simply to avoid adding more total solvent than the reaction vessel can hold. If the reaction vessel is to contain 10 liters, we might specify the inequality constraints... 

In the following example, the measured chemical quantities are fixed by four continuous factors age, treatment A, treatment B and analytical error. Treatment B has been applied to only one sample of the class. Age and treatment A produce variations in composition that cannot be interpreted as deviations from the model they are the inner factors. On the other hand, treatment B cannot be identified as an inner factor because of the lack of information, and its effects fall, with the analytical error, in the outer space. Besides, if the effects of treatment B are noticeably greater than those of the analytical error, the B-treated object can be identified as an outlier, as it really is. 

The discrete factor solvent number is recognized as a simple bookkeeping designation. We can replace it with the continuous factor dipole moment and obtain, finally, the response surface shown in Figure 2.13. 

The method becomes difficult with four independent continuous factors, and for five or more variables, the method is totally impracticable despite its... 

For systems with three independent continuous factors, we could in principle perform experiments corresponding to any point within the cube of Fig. 7.2a. A study of the variation of a reaction 3deld with x = time, X2 = temperature and X3 = pressure, for example, would be a typical case. If the system is a mixture of three components, however, it would have to obey the constraint X1+X2+X3 = 1, which defines an equilateral triangle inscribed in the cube, also illustrated in Fig. 7.2a. 

Fig. 7.2. (a) The experimental space for processes with three continuous factors includes all the points within the cube. The experimental space for mixtures of three components is limited to points on the triangle, (b) A response surface for all possible mixtures of components 1-3. (c) Its contour curves. 

These are factors whose different levels can take numerical values within a well-defined domain of variation representing the domain of variation of the factor consido ed. It is theoretically possible to choose in this domain any state (or level) for the factor. Although it often occurs that only a few discrete values can be chosen in this domain of variation, we generally consider that a quantitative factor is also a continuous factor. Examples are temperature of reaction, duration of addition, concentration of a reactant, and ccrniposition of an emulsion. 

Discrete or continuous factor discrete factor continuous factor continuous factor... 

The real values, which lie between these limits, presumably derive from complicated phase morphology, and this can be described by a continuity factor/. The continuity factor is normalized such that/ = 0 for a completely discontinuous hard phase and a completely continuous soft phase. On the other hand, / = 1 for a totally continuous hard phase and a totally discontinuous soft phase. The discontinuity factor is/ = 1/2 when hard and soft phases are continuous or discontinuous to about the same amount. The shear modulus of the total system is then... 

The continuity factor/must vary with the hard- and soft-phase volume fractions. The greatest change is expected when about the same fractions of hard and soft phases are present, since then, a phase reversal occurs. The following can be given for this change ... 

Equation (35-9) describes the experimental data of a large number of systems quite well (Figure 35-11). Low values of the parameter, n, lead to high continuity factors that only vary slightly with the volume fraction of the hard phase (Table 35-6). High values of n, on the other hand, give continuity factors that decrease sharply with diminishing hard-phase volume fraction. 

Yet another approach to tackle the conditionally convergent Coulomb stun is used for MMM. Instead of defining the summation order, one can also multiply each summand by a continuous factor c, nj, Ukim) such that the sum is absolutely convergent for > 0, but c(0,., .)= . The energy is then defined as the limit 0 of the sum, i.e., is an artificial convergence parameter. For a convergence factor of the limit is the same as the spherical limit, and... 

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