Process branching

There are also branching processes in dris system such as those found in hydrogen oxidation, e.g. 

Auslaufer, m. runner offshoot, branch process (Bot.) stolon (iSpec2.) attendant line, satellite. 

As a result of these branching processes, the chain produces an increasingly larger number of radicals that can take part in even more branching steps. The reaction rate increases rapidly, and an explosion typical of many combustion reactions may occur (Fig. 13.20). 

The basis of model calculations for copolymerization, branching and cross-linking processes is the stochastic theory of Flory and Stockmayer (1-3). This classical method was generalized by Gordon and coworkers with the more powerful method of probability generating functions with cascade substitution for describing branching processes (4-6). With this method it is possible to treat much more complicated reactions and systems (7-9). 

In this study computational results are presented for a six-component, three-stage process of copolymerization and network formation, based on the stochastic theory of branching processes using probability generating functions and cascade substitutions (11,12). 

In the stochastic theory of branching processes the reactivity of the functional groups is assumed to be independent of the size of the copolymer. In addition, cyclization is postulated not to occur in the sol fraction, so that all reactions in the sol fraction are intermolecular. Bonds once formed are assumed to remain stable, so that no randomization reactions such as trans-esterification are incorporated. In our opinion this model is only approximate because of the necessary simplifying assumptions. The numbers obtained will be of limited value in an absolute sense, but very useful to show patterns, sensitivities and trends. 

POLYMQ is similar to POLYM, but with the additional tetrafunc-tional 0 monomers in stage 1. These two programs contain the formulae derived with the stochastic theory of branching processes which are also specified elsewhere (12). 

Depending on the values of the probability parameters of the branching process two paths of evolution for every population are possible either it will degenerate in some generation or it will infinitely grow. The probability of... 

A general theory of the equilibrium polycondensation of an arbitrary mixture of monomers, described by the FSSE model, has been developed [75]. Proceeding from rigorous thermodynamic considerations a branching process has been indicated which describes the chemical structure of condensation polymers and expressions have been derived which relate the probability parameters of this stochastic process to the thermodynamic parameters of the FSSE model. 

Relationships (61)—(63) admit simple probabilistic interpretation in terms of the branching process. To the reproducing particles of this process the reacted functional groups correspond distinguished by color i and label r. Integer i characterizes the type S, of monomeric unit to which a given group was attached at the moment r of its formation. 

Once the particular branching process that specifies the probability measure on the set of macromolecules of a polymer specimen has been identified, the statistical method provides the possibility to determine any statistical characteristic of the chemical structure of this specimen. In particular, the dependence of the weight fraction of a sol on conversion can be calculated by formulas [extending those (55)] which are obtainable from (61) provided the value of dummy variable s is put unity ... 

Harris TE (1963) The theory of branching processes. Springer, Berlin Heidelberg New York... 

Chain reactions can lead to thermal explosions when the energy liberated by the reaction cannot be transferred to the surroundings at a sufficiently fast rate. An explosion may also occur when chain branching processes cause a rapid increase in the number of chains being propagated. This section treats the branched chain reactions that can lead to nonthermal explosions and the physical phenomena that are responsible for both branched chain and thermal explosions. 

If the pressure in the reaction vessel is now decreased, the rate at which chains are broken will also decrease. Eventually the pressure will reach a point at which the rates of the termination processes will become equal to the rate at which the radical concentration is increasing because of the chain branching process that is... 

The sulfochlorination chain reaction is not a branching process, but the chain can be very long as the reactions of the indivudual steps indicated in the following are all presumably exothermic ... 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

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