Three-parameter model

Three Parameter Models. Most fluids deviate from the predicted corresponding states values. Thus the acentric factor, CO, was introduced to account for asymmetry in molecular stmcture (79). The acentric factor is defined as the deviation of reduced vapor pressure from 0.1, measured at a reduced temperature of 0.7. In equation form this becomes ... 

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. 

With a three-parameter model of the intermolecular potential, the theoretical spall strength is not simply a constant times the bulk modulus. Although the slightly greater accuracy obtained is not critical to the present investigation, an energy balance is revealed in the analysis which is not immediately transparent in the Orowan approach. 

A useful method of weighting is through the use of an iterative reweighted least squares algorithm. The first step in this process is to fit the data to an unweighted model. Table 11.7 shows a set of responses to a range of concentrations of an agonist in a functional assay. The data is fit to a three-parameter model of the form... 

Excel calculates the following parameters for the three-parameter model... 

In Table 3.1 the experimentally determined pressures together with the calculated ones using the two models are shown. As seen the three-parameter model represents the data better than the two-parameter one. 

Temperature (K) Experimental Pressure (MPa) Calculated Pressure with three-parameter model (MPa) Calculated Pressure with two-parameter model (MPa)... 

Wee and Kalogerakis (1989) have also considered the simple three-parameter model given next... 

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

Although this appears to have four parameters, it is really a three-parameter model, because the combination r/0/k2p is a single parameter, along with the two parameters p and r]. ... 

By means of Equations 5 and 6, the adsorption process can be described in terms of the parameters and Ks. Since the effective area of solid surface available to polymer adsorption is not generally known in practice, Ng is a third parameter which must be fitted from experiment and Equations 5 and 6 define a three-parameter model for the process. 

A more detailed analysis using multivariable regression of the ibuprofen data demonstrated that a three-parameter model accurately fit the data (Table 7). The Bonding Index and the Heywood shape factor, a, alone explained 86% of the variation, while the best three-variable model, described in what follows, explained 97% of the variation and included the Bonding Index, the Heywood shape factor, and the powder bed density. All three parameters were statistically significant, as seen in Table 7. Furthermore, the coefficients are qualitatively as... 

Table XII would indicate that the effect of water is not described adequately by the model. Utilizing the principle of parsimonious parameterization, one can consider both water and carbon dioxide to be adsorbed and oxygen to be nonadsorbed, resulting in the three-parameter model 4. The residuals in Table XII for model 4, however, are correlated with the oxygen level. Hence model 5 would perhaps be preferable, for it likewise contains only three parameters while allowing adsorbed oxygen. The random residuals of Table XII for model 5 indicate that this model cannot be rejected using the...

It is possible to fit zero-, one-, two-, and three-parameter models to the data shown... 

What information could be obtained if all ten experiments were carried out at only three levels of pH Three levels of x, (f= 3) provides sufficient factor combinations for being able to fit a three-parameter model, but leaves no degrees of freedom for estimating lack of fit / - p = 3 - 3 = 0. Because one of our objectives was to determine if a parabolic relationship provides an adequate model for the observed rate, we must be able to estimate the variance due to lack of fit of the model the number of factor combinations (levels of x, in this example) must therefore be greater than three. 

Figure 2 shows the profile of the 27-29 ppm spectral region of three polymers which served as models (1J ) for ethylene propylene rubber. The better agreement between the observed spectrum and the five-parameter model strongly suggests the three-parameter model is less realistic as an explanation for the polymerization mechanism. Table VII compares the observed profiles of EPDM rubbers made with a Ziegler catalyst system. The ratio of... 

Three-parameter models such as the VFT equation (4.14) can be lit to data by first placing the equation in linear form ... 

At the lowest temperature where the para-H2 and ortho-H2 concentrations are in thermal equilibrium, the rotational ground state and the lowest excited state (J = 0 and 1) are about equally populated, hence the comparable line intensities at 354 and 587 cm-1 at 77 K. With increasing temperature, the J = 1 state is more highly populated, and states with J > 1 are increasingly populated as well, at the expense of the J = 0 ground state, so that the So(l) line shows up much more prominently than So(0) at the higher temperatures. Profiles obtained at temperatures T > 100 K may similarly be fitted by simple three-parameter model profiles if one accounts for the higher So(J) and Qo(J) lines, J > 1, as well. Very satisfactory fits of the laboratory data have resulted [15]. The profiles of the individual lines vary with temperature. Fairly accurate empirical spectra may be constructed, even at temperatures for which no measurements exist, when the empirical temperature dependences of the three BC parameters are known, see Chapter 5 below. 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

In designing the accessories and choosing an injection and extrusion installation for low-pressure moulding1, it is important to evaluate the design procedure for determining the channel resistance to non-Newtonian liquid flow. Practically simple and convenient design procedures for pressure losses and hydraulic resistance of non-round channels for power liquids and liquids described by the three-parameter model may be found, e.g. in 68,69). 

The Ellis model (41), is a three-parameter model, in which the non-Newtonian viscosity is a function of the absolute value of the shear stress tensor, x,... 

A number of other power law equations have been proposed that predict an approach to constant viscosity at low shear rates. Cross (1965, 1968) formulated a three-parameter model ... 

The potency of compounds derived from concentration response assays is expressed most commonly as IC50 or EC50 defined as the compound concentration that produces half maximum response. A common model is the four-parameter Hill-slope equation (Table 14.1). A three-parameter model can be used if a maximum or minimum asymptote is not available because compound potency falls outside the concentration range. One recommendation is to fit the logarithm (loglO) of IC50 or EC50 instead of the untransformed concentration because the concentration response errors are normally... 

If one has intrinsic and apparent reaction kinetics available, then Equation 10 may be viewed as a three-parameter model (t)ce, b wo> B -d f°r prediction of isothermal trickle-bed reactor performance. However, Biwo depends on two mass transfer coefficients and a priori model parameter evaluation is no simpler than before. 

Here F, denotes the tensile strength of the material, Fc is the compressive strength, and Ybc represents equal biaxial compression strength of the material. The reader is directed to Palko25 where the specific forms of B and A are presented. This model is termed a three-parameter model, referring to the three material strength parameters (F FC, and Ybc) used to characterize the model. Failure is defined when / = 0, and the multiaxial criterion is completely defined in the six-dimensional stress space. 

The other interpretation of surface activity coefficients supposes that the interactions between the pairs of the exchanging and exchanged ions (Me, - Mex, Me2-Me2, and Mex - Me2, respectively) are not equivalent, so the surface activity is necessarily the function of surface composition (Hogfeldt 1984, 1988 Konya et al. 1989). This is the Hogfeldt three-parameter model, in which the selectivity coefficient depends on the surface molar factions of the ions and their interactions, as follows ... 

K6nya J., N. M. Nagy, and E. Hogfeldt. 1989. Zinc-calcium exchange on bentonite Application of a simple three-parameter model. Acta Chim. Scand. 43 612-614. 

Hence, the effectiveness factor is a function of three parameters a, e and a Thiele modulus. Therefore we speak of a three-parameter model. 

Thus, the effectiveness factor is given by two parameters and a Thiele modulus. So, a two-parameter model is obtained if assumption 7.3 holds. The error we introduce by using a two- instead of three-parameter model is... 

Table 7.1 Value of the maximum relative errror <5 that will be made if the two-instead of the three-parameter model is used, for some typical processes and, hence, for some typical values e and a (data from Hlavacek and Kubicek [1])...

From now on the two-parameter model is used because it is almost as accurate as the three-parameter model and it gives a better insight. For example, the curves which were drawn by Weisz and Hicks [2] for different values of a and s (Figure 6.4) reduce to one. This is illustrated in Figure 7.1 where the effectiveness factor is plotted versus An0 for several values of C, and for a first-order reaction occurring in a slab. Notice that all the curves in Figure 7.1 coincide in the low ij region, since ij is plotted versus An0. The formulae used for Ana now follow. 

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