Order parameter numerical values

In order to obtain numeric values for the optimized parameters, it is necessary to define a given separation and the equipment by which the sample is to be analyzed. 

In a similar manner to the optimization of an LC column, in order to obtain numeric values for the optimized parameters, it is necessary to define a given separation and the equipment and materials by which the sample is to be analyzed. The data given in Table 2 are for a general GC separation using an open tubular column. 

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

As equation 2.4.8 indicates, the equilibrium constant for a reaction is determined by the temperature and the standard Gibbs free energy change (AG°) for the process. The latter quantity in turn depends on temperature, the definitions of the standard states of the various components, and the stoichiometric coefficients of these species. Consequently, in assigning a numerical value to an equilibrium constant, one must be careful to specify the three parameters mentioned above in order to give meaning to this value. Once one has thus specified the point of reference, this value may be used to calculate the equilibrium composition of the mixture in the manner described in Sections 2.6 to 2.9. 

Assessing the dependence of rate on concentration from the point of view of the rate law involves determining values, from experimental data, of the concentration parameters in equation 4.1-3 the order of reaction with respect to each reactant and the rate constant at a particular temperature. Some experimental methods have been described in Chapter 3, along with some consequences for various orders. In this section, we consider these determinations further, treating different orders in turn to obtain numerical values, as illustrated by examples. 

This particular type of transfer function is called a first-order lag. It tells us how the input affects the output C/, both dynamically and at steadystate. The form of the transfer function (polynomial of degree one in the denominator, i.e., one pole), and the numerical values of the parameters (steadystate gain and time constant) give a complete picture of the system in a very compact and usable form. The transfer function is property of the system only and is applicable for any input. 

Figure 5.55 Mutual dependence of Q i and Q d order parameters. In the upper part of the figure is outlined the T dependence of substitutional disorder Qod for different values of Qdi and, in the lower part, the T dependence of the displacive disorder parameter Qdt for different values of The heavy lines on the surface of local curves represent the solution for thermal equilibrium. From E. Salje and B. Kuscholke, Thermodynamics of sodium feldspar II experimental results and numerical calculations. Physics and Chemistry of Minerals, 12, 99-107, figures 5-8, copyright 1985 by Springer Verlag. Reprinted with the permission of Springer-Verlag GmbH Co. KG.

We have adjusted the parameters X, X, t, tg in order to obtain the best fit with the experimental points (the numerical values are given in Ref (6). In Figure 2-b, one can see that both hypothesis seem to be in a reasonable agreement with the experiments. The phenomenological analysis is unable to give an indication of the microscopic mechanisms involved. Further information on the structure is necessary. Some is provided by the melting behaviour of the gels. 

The traditional criteria approach is to identify specific performance parameters and to assign numeric values to these. These numeric values represent cutoff or threshold values the method parameters must meet in order for the method to be acceptable. The alternative approach is focused on fitness for purpose and MU. In this fitness-for-purpose approach, the overall MU is estimated as a function of the analyte concentration (see Section 8.2.2). 

The zero and first-order phase correction parameters may be modified manually either by entering numerical values in a dialog box (Numerical button), or in an interactive way, using the Zero Order and First Order buttons and their corresponding slider box. With the Automatic button a fast and rough phase correction is performed which speeds up the subsequent manual fine adjustments. This automatic phase correction is not the same as the fully automatic (and more time consuming) phase correction mentioned above. 

As indicated, a single mean-field parameter amf is included as the proportionality factor. It is noteworthy that the numerical value of amf is unimportant to the condensation phenomenon itself, so that even an infinitesimally small value (e.g., of order 10-6 a.u.), is sufficient to reward thermodynamic condensation and yield an alternative phase of greatly reduced V under appropriate conditions of temperature and pressure. 

The evaluation of the sublimation pressure is a problem since most of the compounds to be extracted with the supercritical fluids exhibit sublimation pressures of the order of 10 14 bar, and as a consequence these data cannot be determined experimentally. The sublimation pressure is thus usually estimated by empirical correlations, which are often developed only for hydrocarbon compounds. In the correlation of solubility data this problem can be solved empirically by considering the pure component parameters as fitting-parameters. Better results are obviously obtained [61], but the physical significance of the numerical values of the parameters obtained is doubtful. For example, different pure component properties can be obtained for the same solute using solubility data for different binary mixtures. 

The simple two-dimensional phase-field simulations in Figs. 18.4 and 18.5 were obtained by numerically solving the Cahn-Hilliard (Eq. 18.25) and the Allen-Cahn equations (Eq. 18.26). Each simulation s initial conditions consisted of unstable order-parameter values from the top of the hump in Fig. 18.1 with a small spatial... 

When i and are both heteroatoms, we can take / I( = h h) p. The recommended values for X = O, N, F, Cl, Br and Me are given in Table 2.2. The exact numerical values of these parameters are not crucially important but it is essential that values of appear in the correct order of electronegativity and / I( in the correct order of bond strength.14... 

The solution of the above equation in order to obtain W(y) requires an appropriate form of the spreading function and the numerical values of its parameters. Furthermore, to convert W(y) into a size distribution requires a relationship between the mean retention volume y and the particle diameter D (i.e., a calibration curve). 

Figure 2 illustrates the character of temperature dependence of hydrogen solubility in the numerical value of energetic constants, the order parameter value and the hydrogen atoms activity, which can be evaluated from independent experimental data. The knowledge of these values permits to evaluate numerically the hydrogen solubility at each temperature in the differ from the respective maximum value. 

Now by the use of the selected values of the Ti = Ti, T2 ordering temperatures (22), (23) and with the determination of numerical values of order parameters for all temperatures by the curve plots of Fig. 3, the temperature dependence of configurational heat capacity can be elucidated by formula (16) and the peculiarities of this dependence can be established over the region of scl feel phase transition. 

In order to conduct model measurements in only one model device, the numerical values of the pi-numbers in question (i.e. the fixation of the operational process point of the system) must be adjusted by the appropriate selection of the numerical values of the process parameters and/or of the model material system. If this is not feasible, then the process characteristic has to be evaluated from model experiments in devices of different scales or the operational process point has to be extrapolated from measurements in differently sized model devices. 

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