Modelling fractionation

Note that it is not possible in this case to carry out the three-site analysis on the whole-polymer HMR data alone. There are only eight known values (the eight pentads) and seven parameters. Mathematically, it is therefore difficult to apply the three-site model. Fractionation data have to be used. 

Albarede and Bottinga, 1972), where t is time since the beginning of the process (at constant accretion or dissolution rates), is the trace element concentration measured at distance x from the interface and at time t, and L (in the case of fractional crystallization) is half the mean distance between centers of accreting crystals less half the mean thickness of the crystals. We will see later the effects of erroneous evaluation of and K when modeling fractional crystallization processes. 

Fig. 2.24. Model of a poorly agitated continuous stirred-tank reactor (a) Flow model fraction / of flow v by-passes only a fraction w of tank volume V is well-stirred (b) Equivalent C, curves for pulse input...

Figure 10.15. Quantitative Description of the MWC Model. Fractional activity, 7, is the fraction of active sites bound to substrate and is directly proportional to reaction velocity a is the ratio of [S] to the dissociation constant of S with the enzyme in the R state L is the ratio of the concentration of enzyme in the T state to that in the R state. The binding of the regulators ATP and CTP to ATCase changes the value of L and thus the response to substrate concentration. 

Eased on two main lines of evidence, Niu et al. (1997) concluded that abyssal peridotites are the end products of melt extraction followed by variable amounts of olivine crystallization. First, in their set of reconstructed compositions they found that model fractional and batch melt extraction trends could not reproduce major and minor element variations in their data set. Most importantly, they found that melt extraction models failed to account for the strong positive correlation between FeO and MgO, as well as incompatible minor-element concentrations. Specifically, at a given Na20 or Ti02 content, abyssal peridotites are enriched in MgO relative to model melt extraction residues. Niu et al. (1997) showed that these compositional anomalies can be reconciled by a model of melt extraction followed by olivine crystallization, with more MgO-enriched samples having more accumulated olivine. If correct, this model has important implications for understanding melt extraction at oceanic ridges, and it has recently been the focus of re-evaluation. 

Keywords Multistage modeling. Fractional extraction. Enantiomer... 

The behavior of the dielectric spectra for the two-rotational-degree-of-ffeedom (needle) model is similar but not identical to that for fixed-axis rotators (one-rotational-degree-of-fireedom model). Here, the two- and one-rotational-degree-of-freedom models (fractional or normal) can predict dielectric parameters, which may considerably differ from each other. The differences in the results predicted by these two models are summarized in Table I. It is apparent that the model of rotational Brownian motion of a fixed-axis rotator treated in Section IV.B only qualitatively reproduces the principal features (return to optical transparency, etc.) of dielectric relaxation of dipolar molecules in space for example, the dielectric relaxation time obtained in the context of these models differs by a factor 2. 

Vapor flow model Liquid flow model Fraction of flooding System factor... 

One of the important issues is the possibility to reveal the specific mechanisms of subdiflFusion. The nonlinear time dependence of mean square displacements appears in different mathematical models, for example, in continuous-time random walk models, fractional Brownian motion, and diffusion on fractals. Sometimes, subdiffusion is a combination of different mechanisms. The more thorough investigation of subdiffusion mechanisms, subdiffusion-diffiision crossover times, diffusion coefficients, and activation energies is the subject of future works. 

Fractional Kelvin model Fractional Maxwell model ... 

For the conformational statistics we rely on a model developed by Williams, Flory, and Mark [116, 117, 118] which assumes three states (trans t (0°), gauche g (-+-120°) and g (—120°)). Note that in PVC, pseudoasymmetric carbons are present, and the special convention for the sense of rotation of the dihedral angles must be obeyed. A recent MD simulation [119] is in good agreement with the rotational isomeric state model (fraction of trans states, characteristic ratio). 

AR Bhaskar. Supercritical fluid processing of milk fat Modelling fractionation and applications. PhD Dissertation, Cornell Univ, Ithaca, NY, 1997. 

While this chapter has focused extensively on the requirements for modeling fractionation systems, we can use the same techniques in the context of modeling refinery reaction process as well. We illustrate this process in Chapters 4 through 6 of this text It is possible to obtain good predictive results for fractionation systems provided that we make reasonable choices for the thermodynamic models and physical properties of the pseudocomponents involved. 

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