Single-parameter models

Figure 1 Experimental B-factors of a-carbon atoms (thin curve) compared with those predicted using their single-parameter model for subunit A of endodeoxyribonuclease I complexed with actm (thick curve). Calculations were performed using both subunits A and D comprising 633 residues. Their parameter is adjusted such that the area under the predicted curve equals the area under the experimental curve. (From Ref. 46.)...

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter /z, the viscosity of the fluid. No single parameter model will describe non-Newtonian behaviour and models involving two or even more parameters only approximate to the characteristics of real fluids, and can be used only over a limited range of shear rates. 

The dispersion and stirred tank models of reactor behavior are in essence single parameter models. The literature contains an abundance of more complex multiparameter models. For an introduction to such models, consult the review article by Levenspiel and Bischoff (3) and the texts by these individuals (2, 4). The texts also contain discussions of the means by which residence time distribution curves may be used to diagnose the presence of flow maldistribution and stagnant region effects in operating equipment. 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

Thus, for a single-parameter model such as y,j = p + r,j, the estimated variance-covariance matrix contains no covariance elements the square root of the single variance element corresponds to the standard uncertainty of the single parameter estimate. 

The two single-parameter models give RTD s which are somewhat different from each other, although at this low dispersion the differences are small. Neither of the two models fits the RTD obtained Irum the tracer response curve. 

Reaction engineering texts provide two simple single-parameter models that represent two extremes of flow. These can be used to obtain clues to which flow regimes are occurring in the vessel. These two equations also can be used to predict the retention-time distribution of this vessel, within the limiting assumptions. 

In Section 6.2, the standard uncertainty of the parameter estimate b0 was obtained by taking the square root of the product of the purely experimental uncertainty variance estimate, sand the (X X) 1 matrix (see Equation 6.3). A single number was obtained because the single-parameter model being considered (yu = /30 4 r1() produced alxl (.Y A T 1 matrix. 

Section 9.6 shows how to test the significance of a set of parameters in a model. This set could contain just one parameter. Is it possible to fit a large, multiparameter model, and then test each parameter in turn, eliminating those parameters that do not have a highly significant effect, until a concise, best model is obtained Is it possible to start with a small, single-parameter model, and then add... 

The assumption that the fraction (1-Hce) of external catalyst surface is dry, as used by some other investigators (11), results in a very large B1q which cannot explain or even match the observed experimental results. Dryout of a catalyst surface appears possible only when much larger temperature gradients are present. On the other hand the assumption of riCE= 1 everywhere leads to unrealistic dependence of mass transfer coefficients on liquid velocity. Matching the data with a single parameter model (an overall mass transfer coefficient) results in too high an effect of velocity on such a parameter and in the loss of model predictive ability for different solvents. 

To demonstrate the procedures to be used to obtain probability parameters from crystallization data, let us consider the distribution which occurs if we have a single parameter model. Under these conditions the equilibrium distribution is,... 

A number of limiting cases can now be considered. If catalyst particles are assumed to be completely wetted by actively flowing liquid, then riCE = 1 so that one obtains a single parameter model where Biwo is the only parameter ... 

All of these models given by Equations 13-14 are single parameter models and are limiting cases of Equation 12. 

There is no single parameter model for calculating the diethylether and chloroform distribution coefficients, so you need to use the multiple parameter model given in Eq. (2) and Tables 1-3. 

A single parameter model that forces the data to go through the origin can be written as... 

Consider two single-parameter model classes with the likelihood functions shown in Figure 6.1 for the same set of measurement. This figure shows also the log-likeHhood functions. In this case, p(T> 6, C2) = 4p(V 0, Ci) so In p(T>j0, C2) = 2In 2 - - In p(T> 6, C ). With the same prior distribution, the posterior PDFs and uncertainty of the parameter of these two model classes are identical since the difference between the log-likelihood functions is a constant. However, model class C2 provides better fitting to the data as its maximum Ukelihood value is four times of that for Cp. 

The idea of electronegativity has always interested chemists and there have been many scales published, which can be categorized into one- and two-parameter models. The two-parameter models assume equalization of electronegativities, i.e., charge is transferred between the atoms until they have the same electronegativity. 3.2 3,2 7 single-parameter models use other devices (including hybridization,... 

The procedure that was used to obtain the parameters Pi and P2, is similar to the one that was used for the single parameter model in the previous section, which implies that the parameters are fitted again to experimental tensions. This yields the results presented in Table 2. 

In case of reaction control or ash layer diffusion-control, SCM is a single parameter model, and Treaction or rash as characteristic time. The above equations show that, to achieve any conversion of Xb, for the diffusion of gas film, r oc °, when R increases, the index decreases, and for ash diffusion con-... 

This paper has shown that energy analysis can allow transparent understanding of adhesion phenomena. In the first place, the adhesion equilibrium can be established under certain circumstances for elastic materials with smooth, clean surfaces and low adhesion. This equilibrium can be defined by a single parameter model based on the work of adhesion V/. Adhesion forces can then be predicted for many different geometries and elasticities. 

Among the approaches proposed so far, we recall here single-parameter models [102-111, 115, 118-120, 122, 123, 125, 126, 129], and multi-parametric correlation equations (either based on the combination of two or more existing scales or on the use of specific parameters to account for distinct types of effects) [112, 113, 116, 117, 121, 124]. Additional popular models are the Abraham s scales of solute hydrogen-bond acidity and solute hydrogen-bond basicity [127, 128], and the Catalan et al. solvatochromic scales [130,132, 133]. Methods based on quantitative stmcture-property relationships (QSPR) with solvent descriptors derived from the molecular structure [131, 134], and on principal component analysis (PCA) [135, 136] have been also proposed. An exhaustive review concerning the quantification of the solvent polarity has been recently published [138-140], including a detailed list of solvent scales, interrelations between parameters and statistical approaches. 

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