Experimental data extrapolation

We confine our attention to the limit of zero ionic strength (i.e., an extremely dilute solution of reactants) and make comparison with experimental data extrapolated to zero ionic strength (10, 11). Furthermore, a very simple primitive... 

Equation 3.37, known as the Einstein function, is tabulated for various X-values (see, for instance, Kieffer, 1985). In the Einstein function, the characteristic frequency o), (and the corresponding characteristic temperature see, for instance, eq. 3.40) has an arbitrary value that optimizes equation 3.35 on the basis of high-T experimental data. Extrapolation of equation 3.35 at low temperature results in notable discrepancies from experimental values. These discrepancies found a reasonable explanation after the studies of Debye (1912) and Born and Von Kar-man (1913). 

Based on the interaction parameters for the mixing rules obtained from the binary subsystems, the ternary phase behavior has been modeled by inter- and extrapolation of the interaction parameters. Because of lack of suflicient experimental data, extrapolation is considered to be the only option in some cases, although it obviously wiU introduce errors. Table 14.7 shows the interaction parameters of the various mixing rules used for the prediction of the ternary phase behavior, whereas the resulting partition coefficients per isothermal series... 

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

This parameter should also be extrapolated to C2 = 0, so the amount of experimental data required in this approach is not significantly less than in the method... 

Until now we have looked at various aspects of light scattering under several limiting conditions, specifically, C2 = 0, 0 = 0, or both. Actual measurements, however, are made at finite values of both C2 and 6. In the next section we shall consider a method of treating experimental data that consolidates all of the various extrapolations into one graphical procedure. 

This has the advantage that the expressions for the adsotbed-phase concentration ate simple and expHcit, and, as in the Langmuir expression, the effect of competition between sorbates is accounted for. However, the expression does not reduce to Henry s law in the low concentration limit and therefore violates the requirements of thermodynamic consistency. Whereas it may be useful as a basis for the correlation of experimental data, it should be treated with caution and should not be used as a basis for extrapolation beyond the experimental range. 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

Mathematical Consistency. Consistency requirements based on the property of exact differentials can be apphed to smooth and extrapolate experimental data (2,3). An example is the use of the Gibbs-Duhem coexistence equation to estimate vapor mole fractions from total pressure versus Hquid mole fraction data for a binary mixture. 

Numerous assessments of the rehabiUty of UNIFAC for various appHcations can be found in the Hterature. Extrapolating a confidence level for some new problem is ill-advised because accuracy is estimated by comparing UNIFAC predictions to experimental data. In some cases, the data are the same as that used to generate the UNIFAC interaction parameters in the first place. Extrapolating a confidence level for a new problem requires an assumption that the nature of the new problem is similar to that of the UNIEAC test systems previously considered. With no more than stmctural information, such an assumption may not be vaHd. 

When liquid-phase resistance is important, particular care should be taken in employing any given set of experimental data to ensure that the equilibrium data used conform with those employed by the original author in calculating values of fci or Hi. Extrapolation to widely different couceutratiou ranges or operating conditions should... 

On occasion one will find that heat-transfer-rate data are available for a system in which mass-transfer-rate data are not readily available. The Chilton-Colburn analogy provides a procedure for developing estimates of the mass-transfer rates based on heat-transfer data. Extrapolation of experimental or Jh data obtained with gases to predict hquid systems (and vice versa) should be approached with caution, however. When pressure-drop or friction-factor data are available, one may be able to place an upper bound on the rates of heat and mass transfer, according to Eq. (5-308). 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

FiOyRE 5.57 Results of alternative extrapolation models for the same experimental data. (Reprinted with permission from Risk Assessment in the Federal Government. Copyright 1983 by the National Academy of Sciences. Courtesy of the National Academy Press, Washington, DC.)... 

The solid lines in Figure 4.5 represent extrapolations of experimental data to full-scale vessel bursts on the basis of dimensional arguments. Attendant overpressures were computed by the similarity solution for the gas dynamics generated by steady flames according to Kuhl et al. (1973). Overpressure effects in the environment were determined assuming acoustic decay. The dimensional arguments used to scale up the turbulent flame speed, based on an expression by Damkohler (1940), are, however, questionable. 

Table 1. Calculated properties of ratile, anatase, brookite, and columbite phases. Relative deviation from experimental values is shown in brackets. Structural experimental data are from 19,20,21,9 respectively. Bulk modulus of ratile extrapolated to 0 K is from 2.

Finally, the total preexponential factor includes the stoichimetry deviation represented by c°(, or c° so an extrapolated Arrhenius plot will show an intercept which is very sensitive to composition. Experimental data will be hard to reproduce both because of stoichiometry variations and because of the slow approach to thermal equilibrium. 

Here a complicated problem arises for finding a correct modulus, since it depends on the amplitude of deformation. There is no generally accepted method of extrapolating the dependence of the modulus on the amplitude of A to A - 0. Therefore, here during the discussion of experimental data there may always be disagreements. 

A word of caution should be added with regard to the calculation of the burn-out flux for a pressure intermediate to the main pressure groups that have been correlated this calculation must not be done by taking intermediate y values from Table II. The recommended procedure is to estimate the burn-out flux for the required conditions for the main pressure groups above and below the required pressure, and then to interpolate linearly. It must be also emphasized that while the above correlations can be used with confidence within the experimental ranges of the data, extrapolation outside these ranges should not be taken very far without allowing for a possible reduction in the accuracy obtained. 

With the currently available information, the largest uncertainty is in the oxygen-potential model and the parameter values within the model. A recent assessment of the Pu/0 system (42) has indicated that the values of the parameters used in the Blackburn model yield slightly smaller oxygen potentials than those of Alexander (22), those of Tetenbaum (22-42) and those extrapolated from the data of Woodley (43). A reevaluation of the model parameters would allow a better fit to these experimental data ... 

Because it is very difficult to measure the flow characteristics of a material at very low shear rates, behaviour at zero shear rate can often only be assessed by extrapolation of experimental data obtained over a limited range of shear rates. This extrapolation can be difficult, if not impossible. From Example 3.10 in Section 3.4.7, it can be seen that it is sometimes possible to approximate the behaviour of a fluid over the range of shear rates for which experimental results are available, either by a power-law or by a Bingham-plastic equation. 

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