Response constant, contours

Figure 12.15 Contours of constant response as functions of jr, and for the response surface of Figure 12.14.

Figure 12.17 Upper left panel contours of constant response in two-dimensional factor space. Upper right panel a subset of the contours of constant response. Lower left panel canonical axes translated to stationary point of response surface. Lower right panel canonical axes rotated to coincide with principal axes of response surface.

However, it is not possible to add °C and min In a normalized factor space the factors are unitless and there is no difficulty with calculating distances. Coded rotatable designs do produce contours of constant response in the uncoded factor space, but in the uncoded factor space the contours are usually elliptical, not circular. 

Figure 11.17a. Contours of constant response in two-dimensional factor space.

Point M in the figure is the optimum that by one definition of the research problem objective should be determined. Each intersection line in the plane is a line of constant response values and is called contour lines-contour diagram. 

We can obtain a two-dimensional representation of the modeled surface by drawing its response contour lines, that is, lines of constant response. The contour lines of a plane are line segments. For example, if we let y = 70 in Eq. (6.3) we arrive at the expression... 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

Figures 5 and 6 show the response surfaces plotted for Property A and Property B, respectively. Note that two variables are plotted at once, with the values of the other variables fixed at levels chosen by the experimenter. The contours in the graph represent constant levels of the response. Fortunately, the computer allows rapid replotting for various levels of the fixed variables, as well as changing the identities of the fixed and floating variables, so that the entire design space can be investigated.

When the response surface has an extreme, then all coefficients of a canonic equation have the same signs and the center of the figure is close to the center of experiment. A saddle-type surface has a canonic equation where all coefficients have different signs. In a crest-type surface some canonic equation coefficients are insignificant and the center of the figure is far away from the center of experiment. To obtain a surface approximated by a second-order model for two factors, it is possible to get four kinds of contour curves-graphs of constant values ... 

Because the generator electrodes must have a significant voltage applied across them to produce a constant current, the placement of the indicator electrodes (especially if a potentiometric detection system is to be used) is critical to avoid induced responses from the generator electrodes. Their placement should be adjusted such that both the indicator electrode and the reference electrode occupy positions on an equal potential contour. When dual-polarized amperometric electrodes are used, similar care is desirable in their placement to avoid interference from the electrolysis electrodes. These two considerations have prompted the use of visual or spectrophotometric endpoint detection in some applications of coulometric titrations. 

Univariate optimization is a common way of optimizing simple processes, which are affected by a series of mutually independent parameters. For two parameters such a simple problem is illustrated in figure 5.3a. In this figure a contour plot corresponding to the three-dimensional response surface is shown. The independence of the parameters leads to circular contour lines. If the value of x is first optimized at some constant value of y (line 1) and if y is subsequently optimized at the optimum value observed for x, the true optimum is found in a straightforward way, regardless of the initial choice for the constant value of y. For this kind of optimization problem univariate optimization clearly is an attractive method. 

The entire relationships between reaction factors and response can be better understood by examining the planned series of contour plots (Fig. 9.2) generated from the predicted model (equation 2) by holding constant the enzyme amount (20, 30, and 40%, weight of canola oil) and substrate molar... 

A good way to visualise the data is via contours in a mixture triangle, allowing three components to vary and constraining the fourth to be constant. Using a step size of 0.05, calculate tire estimated responses from tire model in question 2 when... 

A complete picture of how the independent variables affect Tgl can be obtained from examining a contour plot of the predicted response surface of T. i at constant DEGM level DEGM was held constant because it had little effect on Tgl (Figure 9). It is evident from the contours that Tgi is increased not only by increased epoxy prereaction time but also by increased initiator concentration. The lowest values for Tgl are found with low epoxy prereaction times and low initiator concentrations, independent of DEGDM concentration. 

Figure 4.3 shows some contours of constant error density p(e 27) for systems with two responses. In all four cases the responses are linearly independent, but in the last two they are statistically correlated because the off-diagonal elements of 27 are nonzero. 

In this section, we look at methods of obtaining a mathematical model that can be used for qualitative predictions of a response over the whole of the experimental domain. If the model depends on two factors, the response may be considered a topographical surface, drawn as contours or in 3D (Fig. 4). For more factors, we can visualize the surface by taking slices at constant values of all but two factors. These methods allow both process and formulation optimization. 

Figure 7.3. Typical sustained-load crack growth response, showing incubation, transient (non-steady-state) and steady-state crack growth, under constant load (where K remained constant, with crack growth, through specimen contouring) [3].

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