Four-Component Models

In all the above three-component models as well as in the four-component models presented next, an Arrhenius-type temperature dependence is assumed for all the kinetic parameters. Namely each parameter k, is of the form A,erJc (-El/RT). 

---

Fig. 7.83 Mossbauer transmission spectra of Au/Fe multilayer systems with varying Au-layer thickness, measured at 16 K and fitted by a four-component model, including magnetic hyperfine interaction at the Au layer atoms (from [437])...

Preliminary work showed that first order reaction models are adequate for the description of these phenomena even though the actual reaction mechanisms are extremely complex and hence difficult to determine. This simplification is a desired feature of the models since such simple models are to be used in numerical simulators of in situ combustion processes. The bitumen is divided into five major pseudo-components coke (COK), asphaltene (ASP), heavy oil (HO), light oil (LO) and gas (GAS). These pseudo-components were lumped together as needed to produce two, three and four component models. Two, three and four-component models were considered to describe these complicated reactions (Hanson and Ka-logerakis, 1984). 

We consider the following four-component models. Model N is depicted schematically in Figure 18.10 and the corresponding mathematical model is given by the following three ODEs ... 

Four component models were found very difficult or impossible to converge. Models K, M and O are more complicated and have more reaction paths compared to models 1 or N. Whenever the parameter with the highest variance was eliminated in any of these three models, it would revert back to the simpler ones Model I or N. Model N was the only four pseudo-component model that converged. This model also provides an estimate of the HO/LO split. This model together with model 1 were recommended for use in situ combustion simulators (Hanson and Kalogerakis, 1984). Typical results are presented next for model I. 

Predicted versus known concentration of pyrene, using a four component model and the wavelengths 330, 335, 340 and 345 nm (uncentred)... 

A decline in the urinary excretion of uric acid to a level below the rate of production leads to hyperuricemia and an increased miscible pool of sodium urate. Almost all the urate in plasma is freely filtered across the glomerulus. The concentration of uric acid appearing in the urine is determined by multiple renal tubular transport processes in addition to the filtered load. Evidence favors a four-component model including glomerular filtration, tubular reabsorption, tubular secretion, and postsecretory reabsorption. ... 

Figure 7.10. Result of split-half analysis of a four- and a five-component PARAFAC model of fluorescence data. In the left-most plots, the emission loadings are shown and in the right-most the excitation loadings. Upper plots show the results from the four-component model and the lower ones for the five-component model. The profiles from set A and B are shown superimposed. It is seen that in the upper plots, the profiles of set A and B coincide quite closely whereas this is not the case in the lower plots.

In Figure 10.11, a plot similar to Figure 10.9 is shown for the data where the low excitations have been removed. It is quite clear now that there are no more than four components in the data. After four components, the number of iterations increases considerably and the fit of the four-component model is close to 100 % as expected. 

Figure 10.14. Emission and excitation loading plots from a four-component model with (thin solid lines) and without (thick dashed lines) samples 2, 3, and 4.

CoMFA results peak at three PLS components for 5FfT1A, in agreement with ALMOND after FFD selection (Table 7). A one-component and a four-component model for D2 antagonism are illustrated for CoMFA with respect to cross-validated predictivity (see Figs. 13 and 14). Both plots indicate a clear separation between active... 

The PLS model becomes identical to the MLR when the number of latent variables of a PLS derived model becomes equal to the number of actual independent variables, something that rarely happens as a consequence of model validation. The regression coefficients of the MLR model are straightforward to interpret while the PLS latent variables need to be retransformed into original variable space to be interpreted in a similar manner. This also means that the PLS regression coefficients are dimensional dependent, that is, they depend on how many latent variables (PLS components) are used. However, since each PLS component explains a decreasing amount of variance it is usually not that important if a PLS model is based on three or four components, which also means that the PLS regression coefficients will not differ very much between the three- and four-component models. 

The use of statistical models to interpret (and to rationalize) NMR tacticity and sequence data is well established (97,98). In this volume the enantiomorphic-site model has been used by Segre et al. in their studies of polypropylene at high fields (55). A two-site model has been en loyed by Shimozawa et al. to observe the effects of internal donors m propylene polymerization (56). Other models for polyolefins have been reported in the literature, e.g., the multi-site model (99), the dual catalytic-site/chain-end model (100), the perturbed model (101), the consecutive two-site model (102), the four-component model (103), and the chain end model (104). 

Problems of pH in gastric juice may be overcome by bringing the samples to pH 6.2—7.2 with sodium hydroxide prior to ionised calcium analysis. In this way, calcium ion-selective electrode studies [248] have shown that the appearance of calcium in gastric juice involves both secretory and non-parietal components. A four-component model was developed to account for the observed calcium... 

Figure 3.49. Comparison of experimental data with peak resolution results for a four-component model system.

Morals (also sometimes called morality ) generally refer to behavior that conforms to codes of cOTiduct that are held to be authoritative in matter of right and wrong. There are four component models of morality developed by Rest (in Fowler, Zeidler, Sadler, 2009) ... 

According to the assumptions of the four-component model the state of the system is characterised by ... 

The results have been evaluated using the leave-one-out method (see Example 8.9.1). The first block in the printout shows that using this method of cross-validation the number of components required to model Cj is four. The third block in the table gives the reason for this choice it shows that value of the PRESS is lowest for a four-component model, taking the value 0.0052733. (This, incidentally, is higher than it was for the PCR model.) Note that the predictive value of the model, as measured by the PRESS value, decreases if more components are added. The first column of the last block in the table gives the coefficients in the equation for this model. So the regression equation is... 

Serum uric acid concentration in man is the result of complex biosynthetic, catabolic, and excretory processes. Recent investigations of the urate transport system in the human kidney have provided evidence for a four-component model glomerular filtration, presecretory reabsorption, tubular secretion, and postsecretory reabsorption. ... 

According to the four-component model for the renal handling of urate, renal uric acid wasting might result from defects either of of presecretory or postsecretory reabsorption, or enhaced secretion of urate in the tubule. Future studies are required in order to delineate the tubular phases that may account for the uricosuric action of calcitonin. 

Ettouney et al. (1995) compared two-component and four-component models for the analysis of natural gas well enrichment. In the flow system evaluated, gas from the well is recycled through a permeation unit and back to the well. Removal of a CO2 and H2S-rich permeate results in an increase in purity of the well contents with lime. Results of the study showed that proper design of this system requires that the models take into consideration flow patterns, the presence of more than one species, and permeability rate functions that include the effects of both composition and pressure. The use of a simplified two-component model with constant permeabilities gave large deviations from the more detailed models. 

Bebeau, M. J. (2002). The defining issues test and the four component model Contributions to professional education. Journal of Moral Education, 31(3), 271-295. 

---