Many high-temperature hydrocarbon conversion processes are based on or contain an element of thermal reaction chemistry. The essential challenge in modeling these pyrolysis reactions stems from the complexity of the hydrocarbon feed. These are often multicomponent mixtures of molecules undergoing the elementary steps of bond fission, hydrogen transfer, @-scission and radical recombination/disproportionation. Mixture components will interact through the bimolecular steps of H-transfer and radical recombination/ disproportionation. Quantitative modeling of these processes requires not only the kinetics of the elementary steps but also a formalism for their assembly into an analytical rate law or the calculation of a numerical scheme.Toward this end, the elementary steps noted above can be organized into a general Rice-Herzfeld chain process comprising initiation, propagation, and termination components, as shown in Figure 1. In deriving a rate law for reactions such as shown in Figure 1, the nonlinearities arising from the bimolecularity (in radical concentration) of the termination step can present a serious mathematical burden when the combinatorial complexities of a mixture need be addressed. However, the chemistry will be such that frequently the disappearance of reactant is predominantly by the propagation cycle of Figure 1. Gavalas (1966) demonstrated the reduction of mathematical complexity that occurs under such a "long kinetic chain" condition. Essentially, the nonlinear problem becomes a linear one.The appeal of this long-chain approximation (LCA), especially for modeling the reaction of mixtures, motivated the present scrutiny of its range of applicability. Limitations were expected because the LCA is more system-specific than, for example, the steady-state approximation, as the structure of the reacting molecule provides the critical balance of propagation vs. initiation rates. Thus, the application of the LCA appeared to require molecule-by-molecule, a-priori inspection. Moreover, because of the high activation energies of initiation, it seemed reasonable to suspect that a given system could shift away from long-chain conditions with increases in temperature. At extremely high temperatures, the highly activated initiation step could be expected to dominate.
Approximate Rate LawThis line of reasoning suggested that the simple sum of the initiation plus the long-chain rates might approximate the exact kinetics of RH chains well (Eq. 1). Limiting-case accuracy seemed assured. A reactant with a highly unfavorable 0-scission pathway would have a negligible contribution from the "long-chain rate," and may be well modeled by a simple initiation rate. Conversely, a reactant with a long chain would see a negligible contribution from the initiation rate. It is the goal of the present communication to show that the simple sum of Eq. 1 can be applied to predict the disappearance kinetics of a range of molecules without a-priori knowledge of the chain length. The value of this is that the exact solution