Two-factor response surface

Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. 

A two-factor response surface is the graph of a system output or objective function plotted against the system s two inputs. It is assumed that all other controllable factors are held constant, each at a specified level. Again, it is important that this assumption be true otherwise, as will be seen in Section 12.2, the response surface might appear to change shape or to be excessively noisy. 

An alternative approach is the use of response surface or mixture designs to determine the transfer function. A schematic of a two-factor response surface design is given in Fig. 15B. This is accomplished in nine experimental runs and evaluates main effects, as well as factor interactions. If the variability of the inputs is known, then one can model the predicted... 

A Two-Factor Response Surface Methodology Case Study 3—Dissolution and Residual Solvent Control in... 

Figure 7.5 The contour diagram for a two-factor response surface.

One of the most effective ways to think about optimization is to visualize how a system s response changes when we increase or decrease the levels of one or more of its factors. A plot of the system s response as a function of the factor levels is called a response surface. The simplest response surface is for a system with only one factor. In this case the response surface is a straight or curved line in two dimensions. A calibration curve, such as that shown in Figure 14.1, is an example of a one-factor response surface in which the response (absorbance) is plotted on the y-axis versus the factor level (concentration of analyte) on the x-axis. Response surfaces can also be expressed mathematically. The response surface in Figure 14.1, for example, is... 

Suppose you leave your home, go to the grocery store, visit a friend, stop at the drug store, and then return home. Draw a response surface of your round trip showing relative north-south distance (y,) as a function of relative east-west distance (Xj). What does the term hysteresis mean and how does it apply to your round trip that you have described Give examples of other systems and response surfaces that exhibit hysteresis. Why does there appear to be two different response surfaces in systems that exhibit hysteresis Is there a factor that is not being considered Is y, really a response Is x, really a factor ... 

Thus there are two factors responsible for the maximum in the solvent entropy and its deviation from the pzc. Those parameters are the different configurations the monomers and dimers are able to assume on the surface of the electrode, and that, as we discussed above, depend on the free energy associated with each configuration [Fig. 6.85(a)]. The second parameter is related to the entropy of libration, i.e., how the water molecules oscillate and how these oscillations arc affected by the electrode charge [Fig. 6.85(b)]. The vibrational movements of the molecule do notgivatly affect the position of the maximum in the entropy-charge curve. 

From this expression, the two factors responsible for an enhanced Yield for NEA surfaces are evident. First, the ratio apE(hv)/a(hv) equals unity because all absorption processes that lead to final states at and above the conduction band minimum can, in principle, contribute to the Yield, provided they reach the surface. Second, the electron mean free path increases as the electron affinity decreases since, on average, each electron can lose more energy to inelastic scattering and still escape. Hence, the secondary electron yield will, in general, increase with decreasing electron affinity. 

In general there are two factors capable of bringing about the reduction in chemical potential of the adsorbate, which is responsible for capillary condensation the proximity of the solid surface on the one hand (adsorption effect) and the curvature of the liquid meniscus on the other (Kelvin effect). From considerations advanced in Chapter 1 the adsorption effect should be limited to a distance of a few molecular diameters from the surface of the solid. Only at distances in excess of this would the film acquire the completely liquid-like properties which would enable its angle of contact with the bulk liquid to become zero thinner films would differ in structure from the bulk liquid and should therefore display a finite angle of contact with it. 

For a two-factor system, such as the quantitative analysis for vanadium described earlier, the response surface is a flat or curved plane plotted in three dimensions. For example. Figure 14.2a shows the response surface for a system obeying the equation... 

Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as... 

The initial simplex is determined by choosing a starting point on the response surface and selecting step sizes for each factor. Ideally the step sizes for each factor should produce an approximately equal change in the response. For two factors a convenient set of factor levels is (a, b), a + s, h), and (a + 0.5sa, h + 0.87sb), where sa and sb are the step sizes for factors A and B. Optimization is achieved using the following set of rules ... 

Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical relationships between the response and the factors. In spectrophotometry, for example, Beer s law is a theoretical model relating a substance s absorbance. A, to its concentration, Ca... 

Empirical Models of the Response Surface In many cases the underlying theoretical relationship between the response and its factors is unknown, making impossible a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response surface. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors... 

A problem with the simplex-guided experiment (right panel) is that it does not take advantage of the natural factor levels, e.g., molar ratios of 1 0.5, 1 1, 1 2, but would prescribe seemingly arbitrary factor combinations, even such ones that would chemically make no sense, but the optimum is rapidly approached. If the system can be modeled, simulation might help. The dashed lines indicate ridges on the complex response surface. The two figures are schematic. 

For the PS case, a three-variable Box-Behnken response surface methodology (RSM) design using formulation variables has been carried out. For the RF system, an eight-variable fractional-factorial screening study was done first to select significant factors, and this was followed by two RSM s which were similar in design to the one done for PS. The results have led directly to substantial improvements in both materials. 

On the basis of the grid experiments a mathematical function y = f(xi, 2, 3, ), called the response surface, is estimated that characterizes the response as a function of the factors. In case of only two factors the response surface can be visualized by plots like that in Fig. 5.5. 

Fig. 5.7. Simplex optimization within a response surface of two factors X and x2 with simplexes of invariable size... 

In reference 69, results were analyzed by drawing response surfaces. However, the data set only allows obtaining flat or twisted surfaces because the factors were only examined at two levels. Curvature cannot be modeled. An alternative is to calculate main and interaction effects with Equation (3), and to interpret the estimated effects statistically, for instance, with error estimates from negligible effects (Equation (8)) or from the algorithm of Dong (Equations (9), (12), and (13)). Eor the error estimation from negligible effects, not only two-factor interactions but also three- and four-factor interactions could be used to calculate (SE)e. 

To analyze response surface designs, a model is fitted to the data for each response. Usually the results are visualized in response surface plots, showing the change in response as a function of two factors. " These plots allow deciding on the optimal conditions. However, as already mentioned in Section IV, these response surfaces seem not so useful when only small variations around the nominal conditions are examined. 

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