Branch distribution model

Pladis, P., Kiparissides, C., 1988. A comprehensive model for the calculation of molecular weight long-chain branching distribution in Ifee-radical polymerizations. Chem. Eng. Sci. 53, 3315-3333. 

The steady states which are unstable using the above-discussed static analysis are always unstable. However, steady states that are stable from a static point of view may prove to be unstable when the full dynamic analysis is performed. In other words branch 2 in Figure 7.8 is always unstable, whereas branehes 1, 3, and 4 can be stable or unstable depending on the dynamic stabiUty analysis of the system. As mentioned earher, the analysis for the CSTR presented here is mathematically equivalent to that of a catalyst pellet using lumped models or a distributed model made discrete by a technique such as the orthogonal collocation technique (see Appendix E). However, in the latter case, the system dimensionality will increase considerably, with n dimensions for eaeh state variable, where n is the number of internal eolloeation points. 

Amer and van Reenen [39] fractionated isotactic polypropylenes by TREE to get fractions with different molar masses but similar tacticities. The DSC results of the fractions indicated that the crystallization behaviour is strongly affected by the configuration (tacticity) and the molar mass of the PP. Soares et al. [40] proposed a new approach for identifying the number of active catalyst sites and the polymer chain microstructural parameters produced at each active site for ethylene/l-olefin copolymers synthesized with multiple-site catalysts. This method is based on the simultaneous deconvolution of bivariate MMD/CCD, which can be obtained by cross-fractionation techniques like SEC/TREE or TREE/SEC. The proposed approach was validated successfully with model ethylene/1-butene and ethylene/ 1-octene copolymers. Alamo and co-workers [41] studied the effects of molar mass and branching distribution on mechanical properties of ethylene/1-hexene copolymer film grade resins produced by a metallocene catalyst Molar mass fractions were obtained by solvent/non-solvent techniques while P-TREE was used for fractionation according to the 1-hexene content. 

This implies that under the condition of a maximum of one TDB per chain, the set of population balance equations of the TDB branching moment variant of the model is solvable without requiring any additional closure assumption. The results obtained with the pseudo-distribution model are identical to those obtained with the classes model shown before (see Figure 9.6). 

Copolymerization of ADMET EP monomers with 1,9-decadiene, thereby forming linear EP copolymers with random branch distribution, has also been accomplished (Sworen et al., 2003). In this study it was again found that as the branch content increased, overall crystallinity as well as the melting temperatures and enthalpies decreased. In the cases of the highest amount of branch incorporation the random materials exhibited a broad, ill-defined melting behavior in contrast to the sharp melting endotherm observed for the precise models with similar branch content. This drastic difference in the behavior between precise and random models punctuates the effect of precise branch placement (Sworen et al., 2003 Smith et al, 2000). 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

Various PIB architectures with aromatic finks are ideal model polymers for branching analysis, since they can be disassembled by selective link destmction (see Figure 7.7). For example, a monodisperse star would yield linear PIB arms of nearly equal MW, while polydisperse stars will yield linear arms with a polydispersity similar to the original star. Both a monodisperse and polydisperse randomly branched stmcture would yield linear PIB with the most-probable distribution of M jM = 2, provided the branches have the most-probable distribution. Indeed, this is what we found after selective link destruction of various DlBs with narrow and broad distribution. Recently we synthesized various PIB architectures for branching analysis. 

For acute releases, the fault tree analysis is a convenient tool for organizing the quantitative data needed for model selection and implementation. The fault tree represents a heirarchy of events that precede the release of concern. This heirarchy grows like the branches of a tree as we track back through one cause built upon another (hence the name, "fault tree"). Each level of the tree identifies each antecedent event, and the branches are characterized by probabilities attached to each causal link in the sequence. The model appiications are needed to describe the environmental consequences of each type of impulsive release of pollutants. Thus, combining the probability of each event with its quantitative consequences supplied by the model, one is led to the expected value of ambient concentrations in the environment. This distribution, in turn, can be used to generate a profile of exposure and risk. 

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