Two-factor system

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... 

In Chapter 2 it was seen that a response surface for a one-factor system can be represented by a line, either straight or curved, existing in the plane of two-dimensional experiment space (one factor dimension and one response dimension). In two-factor systems, a response surface can be represented by a true surface, either flat or curved, existing in the volume of three-dimensional experiment space (two factor dimensions and one response dimension). By extension, a response surface associated with three- or higher-dimensional factor space can be thought of as a hypersurface existing in the hypervolume of four- or higher-dimensional experiment space. 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

Fig. 11. The response surface of a two-factor system. The lines represent equi-response lines. Optimization by varying one factor at a time. From P. J. Golden and S. N. Deming, Laboratory Microcomputer 3, 44 (1984). Reproduced by permission of Science Technology Letters, England...

Figure 3.6. A Simplex optimization of a two-factor system. Numbers give the order of experiments. The first three experiments (points 1, 2, and 3) define the initial Simplex, and dashed-line arrows indicate which points are dropped in favor of new factor values.

A two-factor system is thought to be described by the model yu = fi(l + /3lxu + fi2x2i + ru over the domain 0 < xl < 10, 0 < x2 < 10 (see Figure 2.16). What is the minimum number of experiments that will allow the model to be fit What are some of the ways you would not place these experiments in the domain of xl and x2 [Hint calculate (X X) 1 for questionable designs.]... 

The Box-Wilson experimental designs are a general series of experiments that have been developed to efficiently serve as a basis for deriving the mathematical model of a physical process. Their usefulness is enhanced in the study of industrial applications because most physical situations can usually be approximated by a quadratic function over a reasonable range of the factors. For a two-factor system, the generally used form of this model is... 

Before we proceed to discuss energy changes in detail it is first necessary to be clear that two factors determine the stability of a chemical system—stability here meaning not undergoing any chemical change. These two factors are the energy factor and the kinetic factor,... 

Many systems that cannot be represented by a first-order empirical model can be described by a full second-order polynomial equation, such as that for two factors. 

A kinetic study of the mercuric acetate oxidation of l-alkyl-3,5-dimethyl-piperidines (81) and 3-alkyl-3-azabicyclo[3.3.1]nonanes (82) was made to evaluate the effect of the N-alkyl group on the rate of oxidation and to contrast these two ring systems (70). The maximum factor in the piperidine... 

Risk is defined as tlie product of two factors (1) tlie probability of an undesirable event and (2) tlie measured consequences of the undesirable event. Measured consequences may be stated in terms of financial loss, injuries, deatlis, or Ollier variables. Failure represents an inability to perform some required function. Reliability is the probability that a system or one of its components will perform its intended function mider certain conditions for a specified period. Tlie reliability of a system and its probability of failure are complementary in tlie sense tliat the sum of these two probabilities is unity. This cluipler considers basic concepts and llieorenis of probability tliat find application in tlie estimation of risk and reliability. 

The choice of the anion ultimately intended to be an element of the ionic liquid is of particular importance. Perhaps more than any other single factor, it appears that the anion of the ionic liquid exercises a significant degree of control over the molecular solvents (water, ether, etc.) with which the IL will form two-phase systems. Nitrate salts, for example, are typically water-miscible while those of hexaflu-orophosphate are not those of tetrafluoroborate may or may not be, depending on the nature of the cation. Certain anions such as hexafluorophosphate are subject to hydrolysis at higher temperatures, while those such as bis(trifluoromethane)sulfonamide are not, but are extremely expensive. Additionally, the cation of the salt used to perform any anion metathesis is important. While salts of potassium, sodium, and silver are routinely used for this purpose, the use of ammonium salts in acetone is frequently the most convenient and least expensive approach. 

The world production of plastics in 1995 is projected at 76 million metric tons (mT) with an annual growth rate (AGR) of 3.7%. The expected AGR of PBAs is 12% and that of composites 16%. In 1987, 21% of polymers were used in blends and 29% in composites and filled plastics [56]. If this trend continues, by 1995 all manufactured resins will be used in multiphase polymeric systems. Two factors moderating the tendency are ... 

As discussed in the preceding section, filter bags must be periodically cleaned to prevent excessive build-up of dust and to maintain an acceptable pressure drop across the filters. Two of the three designs discussed, reverse-flow and reverse-pulse, depend on an adequate supply of clean air or gas to provide this periodic cleaning. Two factors are critical in these systems the clean-gas supply and the proper cleaning frequency. 

The ideal design is one in which ail parts can be operated satisfactorily with water flowing with the least turbulence and aeration, and at a rate of flow within the limits that the materials involved can securely withstand. These limits, with regard to flow-rate limitations, vary with the material, as described in Section 1.2, but turbulence, aeration or presence of suspended particulates can lower these limits considerably, and designs that eliminate these two factors go a long way towards preventing impingement attack, which can be the major cause of failures in sea-water systems. (See also Sections 1.6 and 2.1.)... 

The study of the two-electron systems was greatly simplified by the fact that the total wave function could be factorized into a space part and a spin part according to Eq. III. I. ForiV = 3, 4,. . , such a separation of space and spin is no longer possible, and an explicit treatment of the spin is actually needed in considering correlation effects. This question of the connection between space and spin in an antisymmetric spin function is a rather complicated problem, which has been brought to a simple solution first during the last few years. 

---