Six-parameter model

The interdependence of the Gibbs energy of adsorption and the molecular interaction parameter was recently discussed in detail by Karol-czak, who used a six-parameter model. Contrary to the rather general Damaskin model, no relation between the molecular interaction parameter A and AGads was assumed. It was suggested that this is an arbitrary relation dependent on the theoretical model used in fitting experimental data within acceptable experimental errors. 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

Two-Region Models. Recognizing that the bubbling bed consists of two rather distinct zones, the bubble phase and the emulsion phase, experimenters spent much effort in developing models based on this fact. Since such models contain six parameters, see Fig. 20.7, many simplifications and special cases have been explored (eight by 1962,15 by 1972, and over two dozen to date), and even the complete six-parameter model of Fig. 20.7 has been used. The users of this model. 

Before looking at the results we mention that, as an alternative to the Fourier transforms just described, one may take advantage of the fact that both the classical line shape, Gc (correlation function, Cci(t), may be represented very closely by an expression as in Eq. 5.110 [70]. The parameters ti T4, e and S of these functions are adjusted to match the classical line shape. These six parameter model functions have Fourier transforms that may be expressed in closed form so that the inverse and forward transforms are obtained directly in closed form. We note that the use of transfer functions is merely a convenience, certainly not a necessity as the above discussion has shown. 

As discussed herein, the model could not adequately describe the particular resin system studied and the use of an expanded six-parameter model is described. 

The creep-compliance data on the mayonnaises were modeled by means of the six parameter model ... 

Although any of the parameters in a substitution model might prove critical for a given data set, the best model is not always the one with the most parameters. To the contrary, the fewer the parameters, the better. This is because every parameter estimate has an associated variance. As additional parametric dimensions are introduced, the overall variance increases, sometimes prohibitively (see Li, 1997, p. 84, Table 4.1). For a given DNA sequence comparison, a two-parameter model will require that the summed base differences be sorted into two categories and into six for a six-parameter model. Obviously, the number of sites sampled in each of the six categories would be much smaller (and perhaps too small) to give a reliable estimate. 

We have fitted 18 observed bands with a six parameter model, and obtained a fit to the observed band centres good to a few cm In every case. The calculated band centres are shown alongside the observed bands that appear In Figures 9 to 12, and the parameters obtained from the least squares fit are shown In Figure 13. Figure 14 shows the vibrational hamiltonlan matrices for V=2, in the normal mode basis functions, by way of example. Note that the Fermi Interacting levels cross over between V=1 and V=3. At V=l, around 3000 cm i, and are more than 100 cm-i above 2 2 V=3, around 8700 cm, the [3,0]. ... 

In order to circumvent some deficiencies of the Kraus model, namely the fact that the whole set of model parameters has to be reconsidered if the frequency is changed. Lion et al. proposed a interesting phenomenological theory which leads also to a six-parameter model for the DSS effect. Both the frequency and the amplitude are taken into account with this model and by interpreting the observed history and recovery effects on the elastic and viscous moduli as manifestations of thixotropy, a so-called... 

Thus only six parameters are required to determine the potentials entering our model for handling the R-T effect in asymmehic tetraatomic molecules... 

Determination of the paiameters entering the model Hamiltonian for handling the R-T effect (quadratic force constant for the mean potential and the Renner paiameters) was carried out by fitting special forms of the functions [Eqs. (75) and (77)], as described above, and using not more than 10 electronic energies for each of the X H component states, computed at cis- and toans-planai geometries. This procedure led to the above mentioned six parameters... 

Although two-parameter models are rather restrictive, three-parameter models of the intermolecular potential have been developed which provide reasonable descriptions of the thermodynamic behavior of solids. Examples include the Morse potential, the exponential-six potential, and, more recently, a form proposed by Rose et al. (1984) for metals. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

For a comparison, to obtain multiple values of each parameters six different models have been applied CAESAR, Toxtree, T.E.S.T., Lazar, and ECOSAR, which are freely available, and ToxSuite (ACD/Labs) [55],... 

This model can have as many as six parameters for its characterization Kbe, Pe Pec, and ratios of volumes of regions, of solid in the regions, and of fluid in the regions. The number can be reduced by assumptions such as PF for the bubble region (Pe, - ), all solid in the emulsion, and all fluid entering in the bubble region. Even with the reduction to three parameters, the model remains essentially empirical, and doesn t take more detailed knowledge of fluidized-bed behavior into account. 

At least six distinctly different factor combinations (/ = 6) are required to fit the six parameters of this model (p = 6). To provide three degrees of freedom for lack of fit, /must be increased to 9. To provide three degrees of freedom for purely experimental uncertainty, n must be increased to 12. 

Again, certain terms are turned on or turned off to correspond to a particular factor combination ij. Notice that Equation 15.39 has eight parameters Equation 15.32 has only six parameters. It would appear that the two models are not equivalent... 

The activity coefficients can be calculated using any of the existing models if the binary parameters for all combinations of binary pairs are known. These parameters are obtained by fitting to experimental data. For ternary systems, one can either simultaneously fit all six parameters or first determine the parameters using binary data for those binary systems that have a phase separation and the rest of the parameters from ternary data. 

Models vary in complexity. One-parameter models seem adequate to represent packed beds or tubular vessels. On the other hand models involving up to six parameters have been proposed to represent fluidized beds. 

Fig. 26. General two-region model of a fluidized bed. Fluid is in dispersed plug flow in both regions. The six parameters of this model are m, x, Vi, vi, Di, and Dz (L13).

The model for the adsorption of the surfactant mixtures includes six parameters Tioo, F onoo Py l i> 2 and fCnon. which can be obtained by regression analysis. 

We now have a total of six parameters four from the autonomous system (p, r0, and the desorption rate constants k, and k2) and two from the forcing (rf and co). The main point of interest here is the influence of the imposed forcing on the natural oscillations. Thus, we will take just one set of the autonomous parameters and then vary rf and co. Specifically, we take p = 0.019, r0 = 0.028, fq = 0.001, and k2 = 0.002. For these values the unforced model has a unique unstable stationary state surrounded by a stable limit cycle. The natural oscillation of the system has a period t0 = 911.98, corresponding to a natural frequency of co0 = 0.006 889 6. 

Conformational characteristics of PTFE chains are studied in detail, based upon ab initio electronic structure calculations on perfluorobutane, perfluoropentane, and perfluorohexane. The found conformational characteristics are fully represented by a six-state RIS model. This six-state model, with no adjustment of the geometric or energy parameters as determined from the ab initio calculations, predicts the unperturbed chain dimensions, and the fraction of gauche bonds as a function of temperature, in good agreement with available experimental values. 

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