Nested factors

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

However, process chemistry has several features that distinguish it from other DOE applications. As noted, time is always a critical factor. Time can also be a response, as in time to reaction completion, but its primary role is as a factor with carefully chosen levels. Generally, time is considered a nested factor, the levels selected depend on the level of other factors in the experimental run. Reactions proceed at very different rates depending on the levels of the other factors, especially temperature, thus the time points providing the most information will differ from experiment to experiment. One can think of time as a factor that is subsampled from within the other factor combinations in the design. In fact, the design is usually built in a sequential fashion. First, the nontime factors (whole plot factors) are combined into a design, then the levels of time are determined separately for each experimental run (whole plot combination). In order to facilitate comparisons, one may choose to select some time points that are common across the experiments, but the key criteria is to choose time points that build toward quality estimates of the main responses—the reaction rates. 

The team has brainstormed the factors and developed a prioritized list of the key ones. For each factor, ranges were established that were well outside of the anticipated operating range in order to maximize the effect size (but not so large as to change the fundamental reaction chemistry or overly stress the process facilities). Table 5 summarizes the final list of factors and their ranges that are to be included in the first DOE. Time, of course, will also be a (nested) factor, but, as previously discussed, its levels will be determined after the whole plot design is established. 

Multivariate calibration Calibration in which multiple independent variables are used to establish the calibration model. (Section 5.2) Nested factor In multiway ANOYA a factor that is varied separately for each level of another factor. (Section 4.8)... 

Appleyard, J.R. and L.M. Chesire, "Nested Factorization" paper SPE 12264 presented at the 1983 SPE Symposium on Reservoir Simulation, San Francisco, CA(1983). 

The vertical component of flow can be determined if a well is screened at two different elevations as shown in Figure 1. Frequently, nested wells are used instead of a single well and multiple screenings to determine the vertical component of flow (2). Nested wells must be situated close enough to one another so horizontal gradients do not become a factor. 

Gener Block Factor Fraction Respon SpHt-pl Nested Taguc Num Emphasis Approxim Exampl Tec AppHcatio... 

In fact, the air was not vented. The 1 -inch vent line on the air supply was choked by a wasp s nest. Although this example primarily illustrates a wrong assumption, a second factor was the inadequate indication of the state of the oxygen valve by the panel light. A similar error was a major contributor to the Three Mile Island nuclear accident. 

A complete list of the reaction conditions tested for this response surface design can be found in [76], The center point reaction condition was repeated six times. This was done to measure the variability of the reaction system. Also, the space velocity is kept constant, as it was the least important factor predicted by screening design, for all the reaction conditions. The purpose of this nested response surface design was to develop an empirical model in the form of Eqn (5) to relate the five reaction condition variables and the three catalyst composition variables to the observed catalytic performance. 

Impact 2002 This model provides close to 1,000 characterization factors for the midpoint categories human toxicity, aquatic ecotoxicity, and terrestrial ecotoxicity according to the LCIA methodology. The model is parameterized in a nonspatial and a spatial European model nested in a nonspatial world model, as well as a complete world model... 

Also, due to the nature of the physical factors involved, the values of some of the factors may not be under the experimenter s control. Thus, some factors may be nested, while others may not be. 

The last of the major varieties of experimental design are the nested designs, where the levels of one factor are nested within (or are subsamples of) another factor. That is, each subfactor is evaluated only within the limits of its single larger factor. 

We can only marvel that such lifestyles ever arose and are successfully maintained. One significant factor in their continuation is that a large portion of caterpillars survive once they reach the nests of appropriate ants. A butterfly population can persist even when 95 percent of the caterpillars are adopted by the wrong ants and so are destined to die. The successfully adopted caterpillars high survival rate also helps justify the wasps risky behavior in laying its... 

However, because measurements are kinetically determined, this is a less accurate form of the equation. Very often it is observed that the measured shift factors, defined for different properties, are independent of the measured property. In addition, if for every polymer system, a different reference temperature is chosen, and ap is expressed as a function of T — rj, then ap turns out to be nearly universal for all polymers. Williams, Landel and Ferry believed that the universality of the shift factor was due to a dependence of relaxation rates on free volume. Although the relationship has no free volume basis, the constants and may be given significance in terms of free volume theory (Ratner, 1987). Measurements of shift factors have been carried out on crosslinked polymer electrolyte networks by measuring mechanical loss tangents (Cheradame and Le Nest, 1987). Fig. 6.3 shows values of log ap for... 

The analyst should avoid creating impossible factor combinations. This occurs, for instance, when choosing both the batch number and the manufacturer of the capillary as factors in a robustness test by means of a two-level design. It is impossible to define two unique batch numbers that exist for both manufacturers. The way to examine both factors is by using nested designs. ... 

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