Branching probability

The event" list, across the top of the event tree, specifies events for which the probability of failure (or success) must be specified to obtain the branching probabilities of the event tree. Events that are the failure of a complex system may require fault tree or equivalent methods to calculate the branching probability using component probabilities. In some cases, the branching probability may be obtained directly from failure rate data suitably conditioned for applicability, environment and system interactions. 

The branching probability at a node is determined by either fault tree analysis of the event system or by data from operating experience. 

If the PSA were constructed in the ZIP format, these common cause effects would be incorporated by changing the event tree branching probabilities as a function of water depth a.s the flood fails the various systems. 

The initiators are separated into two classes those for which the event iree/fauJt tree analysis is appropriate and those for which it is not. The former are called internal initiators and the latter, external initiators (externalities). If dependencies are accounted for by modifying the branching probabilities of the event tree, both internal and external initiators can be accounted for in the same event tree. 

The left side connects with the Initialor, the right side with the damage stale the lop defines the operations, branching probabilities ere inserted at the nodes... 

The branching probability a equals pb, the fraction of B groups reacted. Since Pb = p/(/— 1) where p is written for pa, the fraction of the A groups which have reacted... 

With somewhat greater generality we may let a represent the actual branching probability determined by the extent of condensation irrespective of whether or not the critical point has been exceeded, while a is the lowest root of Eq. (20) for the same value of If then a = and according to Eqs. (44) and (45) Ws = l and Wg O. If a>acj then a>a and In the trifunctional case, for example,... 

Characteristic function of the branching probability a (Eq. IX-20) or of the cross-linking index y (Eq. IX-48). 

FIGURE 9.14 Chain branching probability. Co-catalyst with 0.15% wt. Pt. 

The values of branching probability with cobalt are in general more than two times higher in the beginning of an experiment than at steady state. It is concluded that initially the spatial constraints on the growth sites are lower than at steady state. This also indicates a change in the nature of the growth sites. 

In amylopectin and glycogen p =p is the branching probability, which does not depend on the molar mass of the branched polysaccharides. 

In a real free-radical polymerization, the probability that a given monomer unit in the polymer bears a branch is not constant, but increases with time. Mullikin and Mortimer (tOt) have extended their model to take into account the time-dependence of branching probability and the distribution of residence times in a continuous reactor. They assume that the branching probability b is given by ... 

Fig. 30. Zimm-plot and Kratky-plot of glycogen from edible mussels. Neutron small-angle measurements in deuterated water. The dotted line describes the scattering behavior of the ABC model with ac = aAp, and a branching probability of p = 0.2590,174)...

Fig. 31. Molecular weight dependence of (S2) for the ABC model. The numbers indicate the branching probability p in Oq = aAp. The value 1/p gives the number of monomeric units between two branching points104 ...

Another type of behavior is obtained in the polymerization of a multifunctional monomer constructed so that it has one functional group of the type A, and /-1 functional groups of the type B, with the only type of reaction being between A and B. This multifunctional monomer can be written as A-R-By i. The reaction produces highly branched structures, but without gelation. 9 For this system, the branching probability is... 

Fig. 3 above Chain growth probability in dependence of carbon number at different transient episodes of catalyst transformation with the synthesis gases H2/CO2 = 3 and H2/CO = 2.3 below Branching probability (For further exp. data see Fig. 1)... 

Carreira et al.86) studied the Raman spectrum of 1,3-cyclohexadiene vapor and observed a series of sharp Q branches probably due to Av = 2 transitions of the ringtwisting vibration (Fig. 4.17). The double-minimum potential function derived by fitting the data with the two-parameter quartic-quadratic Hamiltonian is shown in Fig. 4.18. The barrier height is 1099 50 cm-1, about twice as high as the highest... 

Each individual branch probability can be approximated in a similar way [recall Eq. (6.76)]. 

If a resonance Is not narrow, then the partial widths may still be defined in terms of the residue at the pole, but Equation 4 is no longer valid (92,93,107). Branching probabilities are now defined by... 

Let us define a as the branching probability, that is, the probability that any functional group from a branch point leads via bifunctional units to another branch point rather than to a bifunctional non-brartch unit. Then (l-a) is the probability that the functional group from the branch point leads to a nonbranch unit. 

Considering an zth envelope from the randomly selected chain section in envelope 1 in Fig. 5.8, suppose there are Yi branch points on the ith envelope. If all chain sections arising from these branch points ended in branch points on the (i + l)th envelope, then there would be 2Yi branch points on the (i + l)th envelope. However, because of the branching probability a, defined above, the expected number (li+i) of branch points on the [i + l)th envelope will be 2Y a. The criterion for gelation or continuous expansion of the network is that the number of chain sections emanating from the [i + l)th envelope, i.e., 2Yi+i, be greater than the number of chain sections emanating from the ith envelope, i.e., 2Yi. In other words, the criterion is... 

---