Mixing deviation parameters

Polymer gel Polymer cohesive energy density P (cal cm ) Deviation from geometric mean mixing rule parameter z Non-Gaussian elasticity parameter, N Curve fit crosslink density (10s mol cm-3) Experimental crosslink density (10s mol cm-3)... 

The two-parameter van Laar equation cannot represent maxima or minima in the activity coefficients, nor can it represent the mixed deviations from ideality exemplified by the benzene-hexaflnorobenzene system of Fig. 1.4-1. However, it is suparior to the Margules equation for some extremely nonideal systems such as alcohol-hydrocarbon mixtures, for example, the ethanol-n-haptane system of Fig. 1.4-1. For such mixtures the two-parameter Margules equation often incorrectly predicts liquid-liquid phase splitting. Higher order Margules equations can sometimes be u red for these systems, but at the expense of many additional parameters. 

The method can be extended to include nonpherical, nonpolar species (such as the lower molecular weight alkanes) by introduction of a third parameter in the equation of state, namely the Prigogine factor for chain-type molecules (9). This modified hard-sphere equation of state accurately describes VE(T, x) for liquefied natural gas mixtures at low pressures. Ternary and higher mixture VE values are accurately predicted using only binary mixing rule deviation parameters. 

There are two mixing rule deviation parameters (fey and /y) which must be evaluated for each pair of species in a mixture. In the present investigation, only binary mixture VE data were used in the evaluation of these parameters. 

Using Equation 17 instead of Equation 16 in the mixing rules for the GIB model did not greatly change the ability of the equation to fit the VE data. There were only slight changes in deviation parameters and standard deviations for the N2 -f CH4 and Ar -f C2H6 systems. 

Flow Pattern Ideality. A straightforward interpretation of the observed kinetics can only be made if the flow pattern in the reactor used corresponds to an ideal flow pattern. In particular for plug flow reactors, deviations from the ideal reactor behavior can be encountered. For perfectly mixed reactors such as a batch reactor and a continuous stirred tank reactor, the rotation speed of the stirrer is the key parameter that needs to be set sufficiently high to ensure complete mixing. Deviations from the ideal plug flow pattern can, for example, be caused by a less-dense packing of the catalyst pellets near the reactor wall, by a too high dilution of the catalyst bed with inert pellets or by the importance of effective axial diffusion compared to convection (15). 

Real reactors deviate more or less from these ideal behaviors. Deviations may be detected with re.sidence time distributions (RTD) obtained with the aid of tracer tests. In other cases a mechanism may be postulated and its parameters checked against test data. The commonest models are combinations of CSTRs and PFRs in series and/or parallel. Thus, a stirred tank may be assumed completely mixed in the vicinity of the impeller and in plug flow near the outlet. 

Although the above treatment is based on the assumption of perfect mixing, it was found experimentally that deviations from this ideal situation can be taken into account by introducing the additional parameters of Wolf and Resnick (W5) into the kernel (253). 

The reason is that classical thermodynamics tells us nothing about the atomic or molecular state of a system. We use thermodynamic results to infer molecular properties, but the evidence is circumstantial. For example, we can infer why a (hydrocarbon + alkanol) mixture shows large positive deviations from ideal solution behavior, in terms of the breaking of hydrogen bonds during mixing, but our description cannot be backed up by thermodynamic equations that involve molecular parameters. 

The major contribution to the components of the D tensor as well as the deviations of the g values from 2.0023 arises from the mixing of ligand field states by SOC other contributions to D result from direct spin-spin coupling, which mixes states of the same spin S. The D tensor and the g matrix both carry chemical information as they are related to the strength and symmetry of the LF, which is competing and counteracting to the effects of SOC. Details on the chemical interpretation of the parameters by quantum chemical means is found in Chap. 5. 

Next, the VLE was calculated using these parameters and the results together with the experimental data are shown in Figure 14.4. The erroneous phase behavior has been suppressed. However, the deviations between the experimental data and the EoS-based calculated phase behavior are excessively large. In this case, the overall fit is judged to be unacceptable and one should proceed and search for more suitable mixing rules. Schwartzentruber et al. (1987) also modeled this system and encountered the same problem. 

It was shown by Englezos et al. (1998) that use of the entire database can be a stringent test of the correlational ability of the EoS and/or the mixing rules. An additional benefit of using all types of phase equilibrium data in the parameter estimation database is the fact that the statistical properties of the estimated parameter values are usually improved in terms of their standard deviation. 

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

Immiscibility phenomena in silicate melts imply positive deviations from ideality in the mixing process. Ghiorso et al. (1983) developed a mixing model applicable to natural magmas adopting the components listed in table 6.12. Because all components have the same standard state (i.e., pure melt component at the T and P of interest) and the interaction parameters used do not vary with T, we are dealing with a regular mixture of the Zeroth principle (cf sections 2.1 and 3.8.4) ... 

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