Pseudo-distribution Approach

We conclude, however, that the classes approach in this case produces useful information on the multiradical issue, even when employing a limited number of classes. 

We will introduce this approach in the case of the 2D CLD/DBD computation for mixed-metallocene polymerization of ethylene. Subsequently, we present applications of the approach to the 3D problems of radical polymerization of vinyl acetate (CLD/DBD/number of terminal double bonds distribution), AB radical copolymerization (CLD/comonomer composition distribution/sequence length distribution), and finally the 2D problem of radical polymerization of polyethylene, where random scission is a complicating factor. 

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We conclude that the pseudo-distribution approach can be appUed successfiilly, provided a good approximation can be made for the branching distribution at given chain length from the branching moments. The method is valid for batch reactors as well, in contrast to, for example, the pgf-cascade method (Section 9.7), which is restricted to steady state reactors. It is to be preferred over classes methods in cases, like the metallocene one, where the second distribution dimension may assume high values as well. 

TDB Pseudo-distribution Approach for More than one TDB per Chain... 

The rate-based models usually use the two-film theory and comprise the material and energy balances of a differential element of the two-phase volume in the packing (148). The classical two-film model shown in Figure 13 is extended here to consider the catalyst phase (Figure 33). A pseudo-homogeneous approach is chosen for the catalyzed reaction (see also Section 2.1), and the corresponding overall reaction kinetics is determined by fixed-bed experiments (34). This macroscopic kinetics includes the influence of the liquid distribution and mass transfer resistances at the liquid-solid interface as well as dififusional transport phenomena inside the porous catalyst. 

The solution of Equations 12.38 and 12.39 (which is a linear algebraic system for the p/s) provides the instantaneous radical-type distribution. Once a procedure to explicitly calculate this distribution has been made available, it is possible to apply the pseudo-homopolymer approach by performing the following steps ... 

Choudhuiy and Nimbalkar (2005) computed the seismie passive earth pressure distribution using pseudo-d5mamie approach. The total seismie passive earth pressure (P ) ean be defined as... 

This problem has been introduced in the discussion of the classes approach. For reaction equations and a full set of population balances, see Tables 9.5 and 9.6. Here, we address the more general problem of more than one TDB per chain [9]. This occurs as a consequence of insertion of TDB chains created by disproportionation or of recombination termination. We start with the full 3D set of Table PVAc2 and then reduce it to a ID formulation by developing the TDB and branching moment expressions. The (N, M)th branching-TDB moments or pseudo distributions for living and dead chains are defined by ... 

In PAMPA measurements each well is usually a one-point-in-time (single-timepoint) sample. By contrast, in the conventional multitimepoint Caco-2 assay, the acceptor solution is frequently replaced with fresh buffer solution so that the solution in contact with the membrane contains no more than a few percent of the total sample concentration at any time. This condition can be called a physically maintained sink. Under pseudo-steady state (when a practically linear solute concentration gradient is established in the membrane phase see Chapter 2), lipophilic molecules will distribute into the cell monolayer in accordance with the effective membrane-buffer partition coefficient, even when the acceptor solution contains nearly zero sample concentration (due to the physical sink). If the physical sink is maintained indefinitely, then eventually, all of the sample will be depleted from both the donor and membrane compartments, as the flux approaches zero (Chapter 2). In conventional Caco-2 data analysis, a very simple equation [Eq. (7.10) or (7.11)] is used to calculate the permeability coefficient. But when combinatorial (i.e., lipophilic) compounds are screened, this equation is often invalid, since a considerable portion of the molecules partitions into the membrane phase during the multitimepoint measurements. 

Figure 5.10 Representation of the formation of the lone pair in the PF3 molecule, (a) An isolated P3 + ion consisting of a P5+ core surrounded by two nonbonding electrons in a spherical distribution, (b) Three approaching F ions distort the distribution of the two valence shell electrons pushing them to one side of the P5+ core, (c) When the F ligands reach their equilibrium positions, the two nonbonding electrons are localized into a lone pair, which acts as a pseudo-ligand giving the PF3 molecule its pyramidal geometry.

The overall gain of the multiphase mixture model approach above is that the two-phase flow is still considered, but the simulations have only to solve pseudo-one-phase equations. Problems can arise if the equations are not averaged correctly. Also, the pseudo-one-phase treatment may not allow for pore-size distribution and mixed wettability effects to be considered. Furthermore, the multiphase mixture model predicts much lower saturations than those of Natarajan and Nguyen - and Weber and Newman even though the limiting current densities are comparable. However, without good experimental data on relative permeabilities and the like, one cannot say which approach is more valid. 

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