Intrinsically linear models

The models described in Equations (7.70) and (7.71) are non-linear. It is possible to transform some non-linear models into linear models these types of models are referred to as intrinsically linear. It is important to note that the disturbance term is also transformed when transforming a model. We present two examples of models that can be transformed and are thereby intrinsically linear. For catalytic reactions, the reaction rate may contain an inhibition term in the denominator. The reaction rate at constant temperature can, for some cases, be expressed as 

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With the exception of the Tucker3 model, the models discussed here are intrinsically linear models, and a straightforward application thus assumes linear interactions and behavior of the samples. While many of the systems of interest to chemists contain nonlinearities that violate the assumptions of the models, the PARAFAC model forms an excellent starting point from which many subsidiary methods are... 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

Some attempts to exploit sensor dynamics for concentration prediction were carried out in the past. Davide et al. approached the problem using dynamic system theory, applying non-linear Volterra series to the modelling of Thickness Shear Mode Resonator (TSMR) sensors [4], This approach gave rise to non-linear models where the difficulty to discriminate the intrinsic sensor properties from those of the gas delivery systems limited the efficiency of the approach. 

The square model is simpler than the linear model. Since all the sites are identical, in a weak sense, we have only one (first) intrinsic binding constant defined by... 

Figure 6.4 shows the structures of benzene-1,2,3,4- and 1,2,4,5- and 1,2,3,5-tetracarboxylic acids. The first is the analogue of the linear model discussed in Section 6.3. Here, we expect two different intrinsic binding constants, (for the... 

With this chosen model, the adjustable parameters are obtained by simnltaneously solving the two equations that resnlt from Equation (3.28). If the chosen model is intrinsically linear, it mnst first be transformed to the form of Equation (3.29) before proceeding. Defining the following terms (SS denotes sums of squares ) simplifies further analysis [10] ... 

Cumene is cracked in a recycle reactor over commercial H-ZSM5 extrudates. A Thiele modulus approach is used to determine the diffusion coefficient and the intrinsic rate constant. The results are compared to those obtained from pulse experiments. A linear model for diffusion, adsorption and reaction rate is applied for reactants and products. In contrast to literature it is argued that if the Thiele modulus is greater than five, the system becomes over parameterised. If additionally adsorption dynamics are negligible, only one lumped parameter can be extracted, which is the apparent reaction constant found from steady state experiments. The pulse experiment of cumene is strongly diffusion limited showing no adsorption dynamics of cumene. However, benzene adsorbed strongly on the zeolite and could be used to extract transient model parameters which are compared to steady state parameters. 

When using logarithms or, more generally, other transformations, it is important to note that the weights for the dilferent measuring points will differ. The two examples of models given here can be transformed and are therefore intrinsically Unear. However, non-linear models cannot in most cases be transformed these are discussed in Section 7.7.2. 

The effect of molecular orientation on the linear CTEs has been modeled (31-32) in terms of intrinsic linear CTEs of individual chain segments using an approach similar to that used to account for the efrect of orientation on polarizabilities (33). Assuming qrlindrical symmetry about the chain axis, only two intrinsic linear CTEs, Oj parallel and Oj normal to the chain axis, are needed to account for the thermal expansion. The volumetric CTE, /3, is related to the linear CTEs in the three principle directions, a, and a, and the two intrinsic linear CTEs by the expression... 

A striking feature of the effect of current on the CO oxidation oscillations is shown in Fig. 8.33. It can be seen that the frequency of oscillations is a linear function of the applied current. This holds not only for intrinsically oscillatory states but also for those which do not exhibit oscillations under open-circuit conditions, such as the ones shown on Fig. 8.31. This behaviour is consistent with earlier models developed to describe the oscillatory behaviour of Pt-catalyzed oxidations under atmospheric pressure conditions which are due to surface Pt02 formation35 as analyzed in detail elsewhere.33... 

The molar mass dependence of the intrinsic viscosity of rigid chain polymers cannot be described by a simple scaling relation in the form of Equation (36) with molar mass independent of K and a. over a broad molar mass range. Starting from the worm-like chain model, Bohdanecky proposed [29] the linearizing equation... 

Anionic polymerization techniques were also critical for the synthesis of a model cyclic triblock terpolymer [cyclic(S-fo-I-fr-MMA)] [196]. The linear cctw-amino acid precursor S-fr-I-fr-MMA was synthesized by the sequential anionic polymerization of St, I and MMA with 2,2,5,5-tetramethyl-l-(3-lithiopropyl)-l-aza-2,5-disilacyclopentane as the initiator and amine generator, and 4-bromo-l,l,l-trimethoxybutane as a terminator and carboxylic acid generator. Characterization studies of the intermediate materials as well as of the final cyclic terpolymer revealed high molecular and compositional homogeneity. Additional proof for the formation of the cyclic structure was provided by the lower intrinsic viscosity found for the cyclic terpolymer compared to that of the precursor. 

The term F2/CsRT is obtained from the constant capacitance model (Chapter 3.7). Fig. 4.6 gives a plot of the linear free energy relation between the rate constants for water exchange and the intrinsic adsorption rate constant, kads. 

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