Yield curve fitting simultaneously

The theoretical model contains four parameters, (3, F K, and K2, whose values are to be obtained from the best fit of the experimental data. Note that all 11 curves in Figure 5.2 are fitted simultaneously." In other words, the parameters (3, F K, and K2 are the same for all curves. The value of F, obtained from the best fit of the data in Figure 5.2, corresponds to 1/ F = 31 A. The respective value of is 82.2 mVmol, which in view of Equation 5.49 gives a standard free energy of surfactant adsorption = 12.3 kT per DS- ion, that is, 30.0 kJ/mol. The determined value of K2 is 8.8 X 10 mVmol, which after substitution in Equation 5.49 yields a standard free energy of counterion binding = 1.9 kT per Na" " ion, that is, 4.7 kJ/mol. The value of the parameter P is positive, 2p TJkT = h-2.89, which indicates attraction between the hydrocarbon tails of the adsorbed surfactant molecules. However, this attraction is too weak to cause two-dimensional phase transition. The van der Waals isotherm predicts such transition for l TJkT> 6.75. 

By lifting the simplifying restrictions, the kinetic observations can be examined in more detail over much wider concentration ranges of the reactants than those relevant to pseudo-first-order conditions. It should be added that sometimes a composite kinetic trace is more revealing with respect to the mechanism than the conventional concentration and pH dependencies of the pseudo-first-order rate constants. Simultaneous evaluation of the kinetic curves obtained with different experimental methods, and recorded under different conditions, is based on fitting the proposed kinetic models directly to the primary data. This method yields more accurate estimates for the rate constants than conventional procedures. Such an approach has been used sporadically in previous studies, but it is expected to be applied more widely and gain significance in the near future. 

The simultaneous solution of the equations for ai, 02, and K will yield an a versus X curve if all the underlying parameters were known. To this end, Futerko and Hsing fitted the numerical solutions of these simultaneous equations to the experimental points on the above-discussed water vapor uptake isotherms of Hinatsu et al. This determined the best fit values of x and X was first assumed to be constant, and in improved calculations, y was assumed to have a linear dependence on 02, which slightly improved the results in terms of estimated data fitting errors. The authors also describe methods for deriving the temperature dependences of x and K using the experimental data of other workers. 

Figure 20 shows the result for a flow curve, where a small positive separation parameter was necessary to fit the flow curve and the linear viscoelastic moduli simultaneously. The data are compatible with the (ideal) concept of a yield stress, but fall below the fit curves for very small shear rates. This indicates the existence of an additional decay mechanism neglected in the present approach [32, 33]. Again, the A-formula describes the experimental data correctly for approximately four decades. For higher shear rates, an effective Herschel-Bulkley law... 

It is found that the shear stress and yield stress of the PAG suspension are higher than those of the pure PANI suspension at the equal electric field strength. Furthermore, under electric fields, the shear stress of the PAG suspension shows a decline as a function of shear rate to a minimum value after the appearance of yield stress. The widely accepted flow model for ER suspensions, i.e., the Bingham fluid model cannot fit well the flow curves of the PAG suspension, especially in the low shear rate region (see Figure 14.8a). However, the flow curves of the pure PANI suspension maintain a relatively stable level, which can be fitted by the Bingham fluid model (see Figure 14.8b). This different flow behavior reflects that the PAG sheets possess a different ER response from the pure PANI particles under the simultaneous effect of both electrical and mechanical fields. In addition. 

The collisional preexcitation has the drawback, that several levels are simultaneously excited, which may feed by cascading fluorescence transitions the level k) whose lifetime is to be measured. These cascades alter the time profile IpL(t) of the level i> and falsify the real lifetime rj (Fig. 11.41b). This problem can be solved by a special measurement cycle For each position x the fluorescence is measured alternately with and without laser excitation (Fig. 11.41c). The difference of both counting rates yields the LIF without cascade contributions. In order to eliminate fluctuations of the laser intensity or the ion-beam intensity a second detector is installed at the fixed position Xq (Fig. 11.42). The normalized ratios S(x)/S(xq) are then fed into a computer which fits them to a theoretical decay curve [11.104]. 

As discussed above, the I-V relation of a PEN element depends on material properties and electrode structures as well as on operating parameters such as gas composition, pressure, and temperature. Using a simple first-order electrochemical model and the potential balance, Eq. (7), combined with simplified expressions for the various polarisation contributions such as Eqs. (8a), (10b), and (ll)-(14b), I-V curves can be predicted. These predicted curves may be used to fit experimental I-V data and deduce, from an optimal fit, certain material and structure properties such as Ri and ioa, which cannot be measured directly. In cells with sizable electrode area, which tend to have appreciable fuel and oxidant utilisation, temperature and gas partial pressures are local quantities dependent on the extent of the electrochemical and chemical conversion (i.e., the fuel and oxidant utilisations). The electrochemical model predicting the I—V curve simultaneously yields the current distribution, temperature distribution, and other quantities of interest. 

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