A simplified mathematical model is derived that is useful for studying the effects of vibration-dissociation coupling in fluid flows. The derivation is based on energy-moment procedure for simplifying the master equations. To obtain the model equations it is assumed that the vibrational energy can be approximated by the introduction of two vibrational temperatures. The effects of molecular anharmonicity are also accounted for in an approximate manner. The parameters contained within the equations are evaluated by making comparisons with experimental data. It is shown that the model contains the minimum required structure allowing favorable agreement with existing experimental data. Numerical solutions are given for the quasi-steady zone behind a normal shock wave, for the complete structure of a shock wave, and for nozzle flow. The results provide the appropriate pre-exponential temperature dependence of the effective dissociation rate, yield and induction time before dissociation is observed, and, in the case of expanding flow, yield one-fourth less effective relaxation time than the Landau-Teller theory. The thermodynamic quantities for the vibrational mode (partition function, internal energy, and specific heat) agree accurately with like quantities evaluated from spectroscopic data. By the introduction of appropriate assumptions it is shown that the equations reduce to a form identical to the Marrone-Treanor model except for a "truncation factor". When the vibrational temperatures are not large, the model is identical to that of Landau and Teller. The numerical procedure used to integrate the system of rate and flow equations is also described.

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The terms in the master equation for vibrational relaxation of anharmonic oscillators are ordered according to the rates of the relaxation processes (vibrational exchange, vibrational-energy transfer to translation). A perturbation procedure is developed by making use of the fact that the vibration-translation energy exchange transitions are much slower than the vibration-vibration transitions. The population distributions in the master equation are expanded about their values when the vibration-vibration mechanism is the only one present. To zeroth order, the master equation is satisfied by a distribution derived previously by Treanor for anharmonic oscillators. This distribution is maintained by vibration-vibration exchange, and is only Boltzmann during relaxation in the limit of vanishing anharmonicity. The perturbation on this distribution, created by vibration energy transfer to translation, is calculated to first order for a special case. The relaxation of the first moment of the zeroth-order distribution function is also investigated. The population of the lowest vibrational states can be considerably lower than the population predicted by the Landau-Teller model for the relaxation process. Furthermore, since the inversion tends to weight those states having fast vibration-to-translation energy transfer rates, the overall energy relaxation rate can be accelerated. (Author).