Quasiequilibrium model of the kinetics of coupled modes of CO2 molecules

The correct allowance for the influence of anharmonicity in the vibrational spectrum of CO₂ on the level distribution of molecules under nonequilibrium conditions, when the vibrational temperature departs significantly from the gas temperature, has become especially urgent in connection with obtaining generation on a number of long-wavelength transitions of CO₂ molecules [1, 2]. The shifts in the levels of coupled modes (symmetric and deformation) are due mainly to Fermi resonance and can reach a considerable value, comparable with the gas temperature even for low levels. In [3] the main features of the quasisteady level distribution of coupled modes were clarified within the framework of the Treanor model of vibrational kinetics. The influence of the ascending flux of quanta, excited by VV exchange under nonequilibrium conditions, on the vibrational distribution was considered in [4–6]. In the present paper we propose a quasiequilibrium model of CO₂ kinetics, obtained without presuming quasisteadiness of the ascending flux of quanta, and making it possible, in contrast to [3–6] to describe the dynamics of the variation of the distribution of molecules among multiplets as a result of processes of VV exchange and VT relaxation between multiplets, with allowance for possible processes of pumping by outside sources. With a Boltzmann population distribution within the multiplets, having the translational temperature of the gas, the problem of studying relaxation in coupled modes is reduced to the equations for an effective anharmonic oscillator with levels corresponding to the multiplets of CO₂ molecules. In this case the levels of the effective oscillator are degenerate with a multiplicity equal to the number of levels in the corresponding multiplet, and they have an anharmonicity constant dependent on the gas temperature. The population distribution of the effective oscillator can be studied by methods developed for the investigation cf a one-mode anharmonic oscillator. The proposed quasiequilibrium model was used for a numerical calculations of the temporal evolution of the distribution function of CO₂ molecules over the levels of coupled modes under the conditions of an extremely maintained discharge.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 16–22, May–June, 1986.     

Formation of inversion in jet of CO2-H2O-N2 gas mixture expanding through a slot

A calculation is made of the kinetics of vibrational relaxation of CO₂ molecules in a CO₂ -H₂O-N₂ mixture escaping into a vacuum from a slot. The examination of vibrational relaxation led to a solution of the kinetic equations corresponding to the most important channels of energy exchange in vibration-vibration and vibration-translation processes. It proved possible to consider the dynamics of a nonequilibrium gas in an approximation of the adiabatic motion of a medium with an effective adiabatic index corresponding to a certain degree of freezing in of the vibrational component of the heat capacity of the gas. The calculated values of the gain index agree well with experimental data. The gain index was calculated with allowance for Doppler and Lorentz mechanisms of line broadening. The results of the calculation were analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 25–31, November–December, 1973.The authors thank A. S. Biryukov and N. N. Sobolev for advice and comments and É. A. Ashratov and G. K. Bunin for conducting the gasdynamical calculation on an electronic computer.     

Study of vibrational deactivation of molecules of carbon dioxide gas during cooling of stream in a supersonic nozzle

The vibrational temperature of the antisymmetrical type of vibrations (v ₃) of the CO₂ molecule at the exit of a supersonic nozzle is measured in the present work using the method of recording the infrared emission. Freezing in of thev ₃-type vibrations was observed during the flow of undiluted carbon dioxide in a nozzle. In this case the vibrational temperature T₃ considerably exceeded the translational temperature. On the basis of a comparison of the experimental results with calculation it can be concluded that vibrational deactivation of CO₂ molecules occurs three to five times faster than the excitation of the vibrations during heating in a shock wave. All the experiments were conducted under the following conditions: maximum expansion of gas in nozzle A/A* = 115, temperature range 1900–2400 °K, pressure range 1–17.5 atm.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 32–40, November–December, 1973.The authors are grateful to U. G. Pirumov and É. A. Ashratov for the calculation of the nozzle profile and the distribution of streamlines as well as for a discussion of the results.     

Flow of gas mixtures with vibrational energy relaxation in plane and axisymmetric nozzles

A method is described for the calculation of plane and axisymmetric flows of gas mixtures with vibrational energy relaxation in the subsonic, transonic, and supersonic regions of the nozzle. The method is based on numerical solution of the inverse problem of nozzle theory. Results are given for the flow of a C0₂-N₂-H₂O-He mixture with vibrational relaxation and compared with the results of one-dimensional calculations. It is found that vibrational-energy relaxation has a significant effect on the gasdynamic parameters of flow in nozzles with large, relative expansion and therefore in choosing a nozzle shape, especially in the supersonic region, it is necessary to calculate the nonequilibrium flow. It is shown that the geometry of the transonic and supersonic regions of the nozzle has a considerable effect on the distribution of the inverse population of the level and the amplification factor.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 125–131, September–October, 1977.     

Calculation of the characteristics of a gas-dynamic laser

The dependence of the radiated power on the characteristics of optical cavities in the case of flow systems has been investigated in a number of papers [1–3], in which it is assumed that population inversion of the laser levels is obtained until entry into the cavity. The operation of a cavity is analyzed in [1] in the geometric-optical approximation with allowance for vibrational relaxation in the gas flow. A simplified system of relaxation equations is solved under steady-state lasing conditions and an expression derived for the laser output power on the assumption of constant temperature, density, and flow speed. The vibrational relaxation processes in the cavity itself are ignored in [2, 3]. It is shown in those studies that the solution has a singularity at the cavity input within the context of the model used. In the present article the performance characteristics of a CO₂-N₂-He gas-dynamic laser with a plane cavity are calculated. A set of equations describing the processes in the cavity is analyzed and solved numerically. Population inversion of the CO₂ laser levels is created by pre-expansion of the given mixture through a flat hyperbolic nozzle. The dependence of the output power on the reflectivities of the mirrors, the cavity length, the pressure, and the composition of the active gas medium is determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi FiziM, No. 5, pp. 33–40, September–October, 1972.     

Investigation of the inverted medium of a quasi-stationary CO2 laser with pulsed excitation

The results of an investigation of the inverted medium of a quasi-stationary CO₂ laser is presented. The medium is distinguished by the fact that the time of flight of individual molecules through the discharge gaps is less than the relaxation time of the 00 °1 CO₂ laser level. The emitted power, the gain, the saturation intensity, and the gas temperature are measured. Using the experimental data, the distribution of the molecules in the vibrational and rotational states of the inverted medium is calculated. The maximum power density attained in this experimental model is 25 W/cm³. For comparison, the characteristics of a model in which cold CO₂ is added to the flow of excited nitrogen are investigated. It is shown that in this case the output power level is determined by the efficiency with which the jets are mixed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 23–29, January–February, 1973.The authors thank V. M. Fedorov for useful discussions and A. A. Borynyak for his help with the experiments.     

Theory of two-component gasdynamic laser

In the present paper a numerical calculation is made of the vibrational relaxation of a binary mixture of molecular nitrogen and carbon dioxide gas. The calculation is performed for the entire range of variation of the concentrations of the components and over a wide range of mixture temperatures and pressures for various geometries of the supersonic part of the nozzle (throat dimensions, degree of expansion). It is shown that population inversion of the CO₂ molecules exists within a certain range of variation of the parameters of the mixture and the nozzle. The population inversion of the vibrational levels and the gain of the gaseous mixture are calculated as functions of these parameters and of distance measured from the critical cross section of the nozzle. The energy characteristics of the two-component gasdynamic laser are optimized.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 23–30, May–June, 1974.     

Nonequilibrium hypersonic flow of carbon dioxide gas over blunt bodies

The Navier-Stokes equations are used to investigate hypersonic flow of carbon dioxide gas over blunt bodies with allowance for nonequilibrium development of chemical reactions and vibrational relaxation of the CO₂ molecules. The problem is solved by the method of stabilization by means of an implicit difference scheme that includes the use of Newton's iterative process. The results are given of calculations of the flow field, the convective heat flux, and the frictional stresses on the surface of blunt cones with spherical noses. The influence of admixtures on the flow field and the heat fluxes is investigated. The results of the calculations are compared with the locally self-similar solution for the neighborhood of the front stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1979.     

Inverted population of Co2 molecules in expanding gas streams

The system of equations of hydrodynamics, which describes the process of escape of the mixtures CO₂ + N₂ + He, H₂O from a nozzle, is solved numerically in conjunction with the equations of the kinetics of the excitation of the vibrational degrees of freedom of the molecules. It is found that an inverted population of the CO₂ molecules with respect to the transition [00 °1] – [10 °0], is produced under certain conditions at the exit from the nozzle. The magnitude of the inversion depends both on the nozzle configuration and on the initial values of the gas temperature and pressure. It is shown that for a specified nozzle configuration there exist optimal values of these parameters, at which the inverted population of the CO₂ molecules reaches approximately 10¹⁵ cm^–3.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 24–34, September–October, 1971.     

Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO3–CoFe2O4 composite wedge with an interface crack

The antiplane analysis is made for a bimaterial BaTiO₃–CoFe₂O₄ composite wedge containing an interface crack. The coupled magneto-electro-elastic field is induced by the piezoelectric/piezomagnetic BaTiO₃–CoFe₂O₄ composite materials. For the crack problems, the intensity factors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field at crack tips are derived analytically. Also, the energy density criterion is applied to predict the fracture behavior of the interface crack. The numerical results also show that the energy release rate for a crack in a single wedge is negative.     

Influence of the initial spatially inhomogeneous temperature perturbations on vibrational relaxation dynamics

The vibrational relaxation of a nonequilibrium molecular gas (T_V>T) plays an important role in the physics of gas lasers, laser chemistry [1], and plasma chemistry [2]. This paper is devoted to an analysis of the dynamics of V-T relaxation with spatially inhomogeneous perturbations of the translational temperature taken into account.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 6, pp. 77–80, November–December, 1984.     

Experimental Investigation of Vibrational Deactivation of CO Molecules in a Supersonic Gas Flow

The vibrational temperature and vibrational deactivation time of CO molecules in collisions with hydrogen atoms are measured using the broadband version of the coherent anti-Stokes Raman scattering technique (CARS). Carbon monoxide with hydrogen-containing admixtures (H₂, H₂O) heated in a reflected shock wave up to temperatures 2900–5100 K escaped through a supersonic wedge-shaped nozzle. The measurements demonstrate the high efficiency of hydrogen atoms in the vibrational deactivation of CO. A difference in the measured temperature dependences of the vibrational excitation and deactivation times of CO molecules in collisions with H atoms, which seems to be associated with a difference in the mechanisms of CO-H complex formation, is noted.     

Vibrational-translational relaxation of anharmonic oscillators at low temperatures

An approximate analytic solution is found to the problem of the vibrational-translational relaxation of anharmonic oscillators at translational temperatures which are small compared with the energy difference between adjacent levels of the oscillator. The deviation of the obtained distribution from the Boltzmann distribution in the relaxation process is analyzed. A study is made of the behavior of the vibrational energy near equilibrium at temperatures such that dissociation has only a small effect on the rate of vibrational relaxation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 3–8, March–April, 1976.The author wishes to thank M. B. Zheleznyak and A. Kh. Mnatsakanyan for a useful discussion of the work.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Nonequilibrium dissociation of diatomic molecules in flows with convective and diffusive particle transport

The nonequilibrium effects in the kinetics of the thermal dissociation which occurs in a streaming gas of diatomic molecules are investigated. Expressions are obtained for the macroscopic reaction rate and the vibrational energy distribution of the molecules, taking into account the influence of the gas motion. Cases of flows with convective and diffusive particle transport are considered. The dissociating molecules are simulated by cutoff harmonic oscillators. The vibrational kinetics is described in the framework of the so-called diffusion approximation.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 147–153, November–December, 1985.     

A Knudsen layer in a flow with two-temperature relaxation

The boundary conditions are considered for the hydrodynamic equation of [1, 2] for highly nonequilibrium diatomic gas with vibrational relaxation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 38–43, November–December, 1972.     

Laser diagnostics of molecular states in nonequilibrium gas flows

Laser diagnostics techniques are developed, and some typical results are presented on the vibrational CO₂ molecule level population measurements under highly nonequilibrium conditions for two types of flows: (i) in a conventional gasdynamic lasers and (ii) in a flow mixing gasdynamic laser. The measuring procedure is based on recording spectral gain coefficient distributions at several resolved rotational transitions of different vibrational bands. The laser optical system with spatial selection of single lines is described. The system allows fast (about 10^–5–10^–4 s) line tuning, thus providing a variable and arbitrary choice of the selected line sequences. The optimum choice of the laser generation spectrum is discussed, and the sources of measuring uncertainties are analyzed. Typical illustrations and results are given and discussed.     

Problems of the theory of monomolecular dissociation of a one-component gas and dissociation constant of CO2 at high temperatures

Equations determining the temperature of vibrations and dissociation constant of polyatomic molecules with consideration of fast exchange of vibrational quanta are formulated. The equations are simplified considerably if different groups of oscillators have similar temperatures of vibrations. In the case of practical interest, it is sufficient to know the vibrational relaxation time and monomolecular dissociation constant at high densities for solving the problem in a harmonic approximation. Quantitative results are obtained for carbon monoxide.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskaya Fiziki, No. 3, pp. 46–52, May–June, 1972.The author thanks S. A. Losev and N. A. Generalov for additional measurement data and I. S. Zaslonko for useful discussions.     

Numerical investigation of the propagation of a pulse of radiation with λ=10.6 μm through absorbing media

The passage of electromagnetic radiation through gaseous media is of special interest when reasonantly absorbing impurities are present in the gas. The interaction of radiation with such a medium can lead, for example, to a temporal decrease of the gas temperature or to its strong heating [1-3]. At the same time the index of refraction in the channel of the light beam is altered, which leads to a deviation of the light rays from the initial direction. The main characteristics of such thermal selfaction within the framework of linear absorption theory for steady and nonsteady processes have been discussed in [4-12]. Nonequilibrium processes in the medium upon absorption of resonant radiation were not taken into account. The effect of the kinetics of vibrational energy exchange on the state of a medium upon the propagation of radiation through a mixture of CO₂ and N₂ gases was first considered in [2, 13, 14]. However, the simplest models of vibrational energy exchange were used, and saturation of the absorbing transition P20 [10°0 00°1] in the CO₂ molecule was not taken into account. Thus linearized equations of vibrational kinetics were used in [13], and only one channel of relaxation of asymmetric vibrations of CO₂ and excited nitrogen was considered in [14]. The propagation of a pulse of radiation with =10.6 m through an absorbing medium is investigated and the influence of the saturation effect and nonlinear processes of vibrational energy exchange on the self-action of light beams of Gaussian profile is studied in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 14–19, May–June, 1984.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].