Model discrimination for

Model discrimination, for example by experimental investigation of the dependence of polymerization rate versus pressure, is required if one wants to use the model over a wide pressure range. 

If the terminal model adequately explains the copolymer composition, as is often the case, the terminal model is usually assumed to apply. Even where statistical tests show that the penultimate model does not provide a significantly better fit to experimental data than the tenninal model, this should not be construed as evidence that penultimate unit effects are unimportant. It is necessary to test for model discrimination, rather than merely for fit to a given model. In this context, it is important to remember that composition data are of very low power when it comes to model discrimination. For MMA-S copolymerization, even though experimental precision is high, the penultimate model confidence intervals are quite large 0.4[Pg.348]

Model discrimination for reactions with stop-effect... 

It is clear that in the future, advanced experiments and high-accuracy analytical equipment will further increase the ability to carry out the model discrimination for radical copolymerization. 

Thullie, J. and A. Renken, Model discrimination for reactions with a stop-effect. Chem. Eng. Sci., 1993 48 3921-3925. 

Golay, S., O. Wolfrath, R. Doepper, and A. Renken, Model Discrimination for Reactims with Stop-effect, in Dynamics of surface and reaction kinetics in heterogeneous catalysis, G.F. Froment and K.C. Waugh, Editors. 1997, Elsevier Amsterdam. Studies in Surface Science and Catalysis, 109, 295-304. 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

Model discrimination is a procedure for developing a suitable description of the unit performance. The techniques are drawn from the mathematics hterature where the goodness-of-fit of various proposed models are compared. Unfortunately, the various proposed models will usually describe a unit s performance equally well. Model discrimination is better accomplished when raw or adjusted measurements from many, unique operating conditions provide the foundation for the comparisons. 

While many data are suggestive of chain length dependence, the data are not usually suitable for or have not been tested with respect to model discrimination. Values of ,u have been determined for a variety of small monomeric radicals to be ca I09 M s 1.4 Taking kt0 as Jk,lj and a as 1.0 in the geometric expression yields values of ,iJ as shown in Figure 5.4a.49 Use of the Smoluchowski mean or the harmonic mean approximation prediets a shallower dependence of k 1 on the chain length (Figure 5.4b). All expressions yield the same dependence for j=i. 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

It is also possible to process copolymer composition data to obtain reactivity ratios for higher order models (e.g. penultimate model or complex participation, etc.). However, composition data have low power in model discrimination (Sections 7.3.1.2 and 7.3.1.3). There has been much published on the subject of the design of experiments for reactivity ratio determination and model discrimination.49 "8 136 137 Attention must be paid to the information that is required the optimal design for obtaining terminal model reactivity ratios may not be ideal for model discrimination.49... 

The important aspect of this problem is that while the penultimate model involves a four dimensional parameter space, the model discrimination problem can be reduced to a two dimensional space by dealing with functions of the original parameters. This approach requires that probabilities for array locations in the four dimensional r, r/, rj, rj ) space be mapped to array locations in the ((ri-r/), (rj-rj )) space. 

The positive values obtained in practically all cases indicate that all these models may be plausible representations of the data and indeed, the correlation coefBcients, R, are greater than 0.9. Thus, statistical compliance is not a sufficient basis for model discrimination. Specifically, the thermodynamic consistency of the estimates, as proposed by Boudart et al. [3], is appropriate further scrutinizing criterion during kinetic modelling and has been gainfully employed in other reactions [4-6]. 

Kinetic Model Discrimination. To discriminate between the kinetic models, semibatch reactors were set up for the measurement of reaction rates. The semi-batch terminology is used because hydrogen is fed to a batch reactor to maintain a constant hydrogen pressme. This kind of semi-batch reactor can be treated as a bateh reactor with a constant hydrogen pressme. The governing equations for a bateh reactor, using the product formation rate for three possible scenarios, were derived, as described in reference (12) with the following results ... 

Some data fitting results are displayed in Figures 12.1 and 12.3. The general conclusion is that both models describe the behaviours of the main components, lactose and lactitol very well, both for sponge nickel and ruthenium catalysts. In this respect, no real model discrimination is possible. Both models also describe equally well the behaviour of lactobionic acid (D), including its concentration maximum when the reversible step is included (ks) (Figure 12.3). 

The sequential experimental design can be made either for precise parameter estimation or for model discrimination purposes. 

Selection of Optimal Sampling Interval and Initial State for Model Discrimination... 

If instead of precise parameter estimation, we are designing experiments for model discrimination, the best grid point of the operability region is chosen by maximizing the overall divergence, defined for dynamic systems as... 

Determination of Optimal Inputs for Precise Parameter Estimation and Model Discrimination... 

The use of time stages of varying lengths in iterative dynamic programming (Luus, 2000) may indeed provide a computationally acceptable solution. Actually, such an approach may prove to be feasible particularly for model discrimination purposes. In model discrimination we seek the optimal inputs, u(t), that will maximize the overall divergence among r rival models given by Equation 12.23. 

As a third example let us consider the growth kinetics in a chemostat used by Kalogerakis (1984) to evaluate sequential design procedures for model discrimination in dynamic systems. We consider the following four kinetic models for biomass growth and substrate utilization in the continuous baker s yeast fermentation. 

Buzzi-Ferraris, G., P. Forzatti, G. Emig and H. Hofmann, "Sequential Experimental Design for Model Discrimination in the Case of Multiple Responses", Chem. Eng. Sci., 39(1), 81-85 (1984). 

With this book the reader can expect to learn how to formulate and solve parameter estimation problems, compute the statistical properties of the parameters, perform model adequacy tests, and design experiments for parameter estimation or model discrimination. 

Procedures on how to make inferences on the parameters and the response variables are introduced in Chapter 11. The design of experiments has a direct impact on the quality of the estimated parameters and is presented in Chapter 12. The emphasis is on sequential experimental design for parameter estimation and for model discrimination. Recursive least squares estimation, used for on-line data analysis, is briefly covered in Chapter 13. 

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