Process Modeling with Single-Response Data

The statistical investigation of a model begins with the estimation of its parameters from observations. Chapters 4 and 5 give some background for this step. For single-response observations with independent normal error distributions and given relative precisions, Bayes theorem leads to the famous method of least squares. Multiresponse observations need more detailed treatment, to be discussed in Chapter 7. 

This chapter uses Gauss 1809 treatment of nonlinear least squares (submitted in 1806, but delayed by the publisher s demand that it be translated into Latin). Gauss weighted the observations according to their precision, as we do in Sections 6.1 and 6.2. He provided normal equations for parameter estimation, as we do in Section 6.3, with iteration for models nonlinear in the parameters. He gave efficient algorithms for the parameter 

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Chapter 6 Process Modeling with Single-Response Data... 

Process Modeling with Single-Response Data Insertion of Eq. (6.4-3) gives... 

In the resolution of any multicomponent system, the main goal is to transform the raw experimental measurements into useful information. By doing so, we aim to obtain a clear description of the contribution of each of the components present in the mixture or the process from the overall measured variation in our chemical data. Despite the diverse nature of multicomponent systems, the variation in then-related experimental measurements can, in many cases, be expressed as a simple composition-weighted linear additive model of pure responses, with a single term per component contribution. Although such a model is often known to be followed because of the nature of the instrumental responses measured (e.g., in the case of spectroscopic measurements), the information related to the individual contributions involved cannot be derived in a straightforward way from the raw measurements. The common purpose of all multivariate resolution methods is to fill in this gap and provide a linear model of individual component contributions using solely the raw experimental measurements. Resolution methods are powerful approaches that do not require a lot of prior information because neither the number nor the nature of the pure components in a system need to be known beforehand. Any information available about the system may be used, but it is not required. Actually, the only mandatory prerequisite is the inner linear structure of the data set. The mild requirements needed have promoted the use of resolution methods to tackle many chemical problems that could not be solved otherwise. 

However, the argument that the cyclic nature of the perturbation ehminates the intrusion of heat effects must be treated with caution. For both p-xylene and 2-butyne in silicalite Shen and Rees [31,32] observed a bimodal response spectrum and they interpreted the two peaks as indicative of two different transport processes corresponding to diffusion through the straight and sinusoidal channels. There is some NMR evidence to support the view that such molecules cannot easily reorient themselves at the channel intersections, and for silicalite-2, which contains only straight channels of similar dimensions, only a single response peak is observed so this hypothesis is certainly plausible. However, Sun and Bourdin [34] have shown that an alternative explanation is also possible. If the heat balance equations are included in the theoretical model, the predicted response assumes a bimodal form and the heat-transfer parameter required to match the experimental data appears to be quite reasonable. 

A single selected experiment with glass and a single experiment with alumina, both conducted at mild thermal conditions, were used to adjust the collision frequency pre-factor Tcoii- In this way, the value of Fcoii = 10m (/coll = 1.6 s ) was obtained for glass, and Fcoii = 45 m (/cou = 4.1 s ) for alumina. Once this parameter was fitted, it was used without further change for comparison with all other data gained with the respective material. A deeper discussion on the effect of the number of collisions on model response, more details about equipment and material properties, and a full documentation of the experimental results can be found in Terrazas-Velarde (2010). Here, just a few comparisons with measured data are presented to show that the model can reliably describe the influence of process parameters. 

Another advantage of nonlinear impedance analysis is that measurement of several harmonics may facilitate extraction of kinetic parameters at a single DC "offset" potential V [8] not available from small-amplitude fundamental-frequency impedance measurement. NLEIS can be used to calculate all the harmonics of the current response to a sinusoidal potential perturbation -V+ V sin(( >t) and derive the nonlinear impedance. Results from a simulation study can be compared with experimental NLEIS data, leading to more accurate quantification and modeling of the impedance data and better interpretation of the electrochemical kinetic processes [8,9,10,11,12,13]. 

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