A Gas-Dynamic Model for Coupled Vibrational and Radiative Nonequilibrium in CO2—with Application to the Spectrophone

Abstract

Macroscopic equations are formulated for the nonequilibrium interaction of vibrational and radiative rate processes in CO₂. A model for CO₂ is used that assumes that the energy levels of each vibrational mode are evenly spaced and that each mode can be assigned a vibrational temperature. The bending and symmetric‐stretching modes are, however, taken to be in mutual equilibrium so that they have the same vibrational temperature. Collisional rate equations for the V—T and V—V processes are derived on a phenomenological basis. The resulting phenomenological coefficients are interpreted, with the help of basic knowledge of the microscopic physics, in terms of characteristic relaxation times and parameters measuring the relative amounts of energy exchanged by the various modes during V—V transitions. Radiative transfer equations are developed for the three strongest infrared bands, located in the spectrum at 15, 4.3, and 2.7 μ, by appropriately summing a previously derived microscopic transfer equation. It is found that the absorption coefficient is a function of the kinetic and the vibrational temperatures in a manner that depends on the band in question. The source function, which is also different for each band, depends strongly on the temperature of the vibrational modes involved in the transition and weakly on the kinetic temperature. The rate and transfer equations are then incorporated with the conservation equations of gas dynamics for application to linearized nonequilibrium flows of carbon dioxide gas. As a particularly simple example, a solution for the spectrophone is presented, including expressions for the phase lag in the pressure signal that results from radiative excitation via each of the bands individually. One of these expressions can be used in conjunction with experimental results from other sources to draw conclusions about predominant V—V transitions in CO₂. The results of doing this using the 4.3‐μ band have been reported previously. The utility of a spectrophone can also be extended by exploiting the possibility of selectively exciting CO₂ through the different bands and making use of all the phase‐lag expressions. All equations were derived without restriction to room temperature, allowing for possible use of spectrophones at higher temperatures. The approach used to derive the macroscopic rate equations can equally well be applied to other polyatomic molecules.