Inadequate second-order model

There is another solution to movement to optimum, i.e. when interaction effects are included in an inadequate regression model and this movement is done by an incomplete second-order model. In that case, the second-order model is analyzed and the gradient direction changed from point to point. 

If the optimum region is close by, the research by this model ends and we switch to constructing the design of experiments for the second-order model. Fig. 2.38 shows the block diagram of searching for an optimum for an inadequate linear model. 

The regression model is inadequate. The second-order model may be built up to an incomplete third-order model by including design point No. 7. By using the expression (3.27) we get ... 

In practice, augmentation can be performed after the experimenter has completed a full factorial design and found a linear model to be inadequate. A possible reason is that the true response function may be second order. Instead of starting a completely new set of experiments, we can use the results of the previous design and perform an additional set of measurements at points having one or more zero coordinates. All of the data collected can be used to tit a second-order model. 

Higher-order models are rarely applied. In many cases, the true response surface can be sufficiently well approximated by the second-order model. Occasionally, higher-order models can be used when quadratic models are clearly inadequate, for example, when a sigmoid-like relation between the response and a variable is observed (7). Then, either a third-order model, an appropriate transformation, a mechanistic physical model, nonlinear modeling techniques, or neural networks can be applied (1,7). 

We conclude that the regression is highly significant. Using the reduced cubic model does not improve the regression and the second-order model could be used for response-surface analysis. The first-order model is seen to be inadequate. 

Screening designs are mainly used in the intial exploratory phase to identify the most important variables governing the system performance. Once all the important parameters have been identified and it is anticipated that the linear model in Eqn (2) is inadequate to model the experimental data, then second-order polynomials are commonly used to extend the linear model. These models take the form of Eqn (3), where (3j are the coefficients for the squared terms in the model and 3-way and higher-order interactions are excluded. 

The second order aqueous reaction A + B R + Sis run in a large tank reactor (V = 6 m ) and for an equimolar feed stream (C o = Qo) conversion of reactants is 60%. Unfortunately, agitation in our reactor is rather inadequate and tracer tests of the flow within the reactor give the flow model sketched in Fig. P12.ll. What size of mixed flow reactor will equal the performance of our present unit ... 

On the other hand, it may be that the first-order model is inadequate for describing the system within the domain or that it was decided from the beginning to postulate a more complex model. If only the vertices are taken, the number of experiments will probably be insufficient or the design will be of poor quality for determining the second-order or the reduced cubic model. In these cases, it is usual to add further experiments. 

There are several cases where NMR spectroscopy has been used to investigate copolymers which deviate from the terminal model for copolymerisation (see also chapter 3). For example, Hill and co-workers [23, 24] have examined sequence distributions in a number of low conversion styrene/acrylonitrile (S/A) copolymers using carbon-13 NMR spectroscopy. Previous studies on this copolymer system, based on examination of the variation of copolymer composition with monomer feed ratio, indicated significant deviation from the terminal model. In order to explain this deviation, propagation conforming to the penultimate (second-order Markov) and antepenultimate (third-order Markov) models had been proposed [25-27]. Others had invoked the complex participation model as the cause of deviation [28]. From their own copolymer/comonomer composition data. Hill et al [23] obtained best-fit reactivity ratios for the terminal, penultimate, and the complex participation models using non-linear methods. After application of the statistical F-test, they rejected the terminal model as an inadequate description of the data in comparison to the other two models. However, they were unable to discriminate between the penultimate and complex participation models. Attention was therefore turned to the sequence distribution of the polymer. 

As mentioned earlier, the dispersion model, like other models, is subject to two errors inadequate representation of the RTD and improper allowance for the extent of micromixing. We can evaluate the first error for a specific case by using the dispersion model to obtain an alternate solution for Example 6-5. The second error does not exist for first-order kinetics. However, the maximum value of this error for second- and halforder kinetics was indicated in Sec. 6-8. 

Where does all this leave the model developed in this chapter Clearly, basically correct (if one is prepared to accept that the wrong placement of ligand orbital energy levels is unimportant) but unable to account for fine details such as unpaired spin distributions (the total uneven spin density on each carbon in [Cr(CN)6] amounts to about 0.1 electron). How can the model be improved The explanation given for the observation of an uneven spin density depended on, loosely, how one electron behaved consequent on the behaviour of another. The model presented in this chapter is a one-electron model electrons were talked of as individuals. In order to explain fine details this one-electron model is inadequate, two-electron correlations have to be included in our treatment, the second time in this chapter that this conclusion has been deduced. This is not the last time that we shall find a need to explicitly include electron correlation. At this point all that needs... 

---