One-temperature Approximation Research Articles

Numerical Research of Spontaneous Condensation of Boron Oxide Vapor in Flat Nozzles

A model of spontaneous condensation of boron oxide vapor in chemically reacting gas mixtures is proposed, based on the classical theory of nucleation and one-speed and one-temperature approximation for the equations of two-phase mixture flow. The model considers nucleation, condensational growth, coagulation, and gas-phase chemical reactions of molten boron oxide droplets. Numerical research of the spontaneous condensation of boron oxide in flat nozzles of different sizes and geometries was carried out using the model. The condensation shock wave is shown to occur immediately behind the nozzle throat. The highest nucleation rates (up to ) are characteristic of a zone adjacent to the sidewall of the nozzle, where the rate of expansion of combustion products is maximum. In nozzles with similar geometries, the position of the front of the condensation shock wave does not depend on the linear dimensions of the nozzle; and only the Mach number in front of the condensation shock wave determines its position. In flat nozzles with geometrically similar inlet regions, the front of the condensation shock wave position does not depend on the nozzle linear dimensions because the condensation shock wave is formed in the field that is independent of the expansion angle of the nozzle. The effect of gas-phase chemical reactions on the spontaneous condensation of boron oxide vapor development is estimated. This effect is weak and tangible only in large nozzles.

Numerical simulation of a CO2, CO, O2, O, C mixture: Validation through comparisons with results obtained in a ground-based facility and thermochemical effects

Various issues of numerical simulation of high-enthalpy non-equilibrium flows of a gas mixture of CO2, CO, O2, O, and C are discussed. Navier–Stokes equations are considered within the framework of one-, two-, and three-temperature approximations with allowance for chemical reactions. In the one-temperature approximation, chemical reactions are modelled by the Arrhenius law. In the multi-temperature approximations, simulations are performed with the use of the Treanor–Marrone model for carbon dioxide dissociation and the Park model for dissociation of diatomic gas molecules, which take into account vibrational non-equilibrium of molecules. The Arrhenius law is also used for calculating recombination reactions and also forward and backward exchange reactions. Translational-vibrational and vibrational-vibrational energy exchange is simulated by a simplified Landau–Teller model. Some examples of flows with the use of the above-mentioned approximations are calculated. It is demonstrated that the one-temperature approach ensures an adequate reproduction of experimental data on the shock standoff distance, heat flux, and pressure coefficient on the cone, which are obtained in a high-enthalpy wind tunnel. The calculation of the flow around a cylinder under Martian re-entry conditions reveals a significant effect of thermochemical non-equilibrium.

Rotational Energy Relaxation Time for Vibrationally Excited Molecules

The effect of the vibrational level of a molecule on the relaxation time of its rotational energy is studied within the state-to-state kinetic theory approach. The rotational levels of molecules are described by the non-rigid rotator model, while the interaction between molecules is described by the variable soft sphere model. This model is used to calculate the N2-N, O2-O, and NO-O collision cross sections for different vibrational and rotational levels of molecules. The rotational energy relaxation time is introduced for each vibrational level using the methods of the kinetic theory of nonequilibrium processes. The relaxation times are numerically calculated within a broad temperature range and compared with the relaxation time determined by the well-known Parker formula. The effect of various multi-quantum rotational transitions on the accuracy of the rotational relaxation time calculation is analyzed, and the convergence of the solution with an increase in the maximally possible number of quanta transmitted in the course of transition is demonstrated. It has been established that the vibrational state of a molecule has an appreciable effect on the rotational energy relaxation time in the state-to-state approach, and using the Parker formula may lead to a noticeable error in the calculation of state-to-state transport coefficients. The Parker formula provides a satisfactory agreement with the results obtained via the averaging of state-resolved relaxation times with a Boltzmann vibrational energy distribution in the one-temperature approximation at moderate temperatures.

Vibrational-Chemical Coupling in mixtures CO2/CO/O and CO2/CO/O2/O/C

The paper deals with modeling of vibration-chemical coupling in mixtures containing CO2 molecules. The temporal vibrational and chemical relaxation in the five-component and three-component space-homogeneous mixtures CO2/CO/O2/C/O and CO2/CO/O is studied taking into account vibrational energy transitions, dissociation, recombination and exchange chemical reactions. The vibrational-chemical coupling in considered mixtures is studied on the basis of three-temperature, two-temperature and one-temperature vibrational CO2 distributions. Governing equations for macroscopic mixture parameters in the three-temperature, two-temperature and one-temperature approximations are solved numerically for different initial conditions. Comparison of the results obtained in three approaches showed the in uence of non-equilibrium vibrational distributions on chemical reaction rates and on the temporal variation of the gas temperature, mixture composition and vibrational temperatures in two considered mixtures. The role of exchange chemical reactions in relaxation processes is also discussed in the paper.

Thermodynamic parameters of heterogeneous materials under shock-wave loading in presentation of equilibrium model

The results of numerical experiments on modeling of shock wave loading of solid and porous heterogeneous materials on the example of molybdenum and some alloys included molybdenum as a component are presented. A thermodynamically equilibrium model is applied to describe the behavior of solid and porous materials. This model ensures good compliance with the experiment in a wide range of pressures. The gas in pores, which is a component of the medium, is taken into account in this model. The equation of state of the Mie-Grüneisen type with allowance for the dependence of the Grüneisen coefficient on temperature is used for condensed phases. The applied model allows the behavior of the molybdenum with porosity from 1 to 3 to be calculated under shock-wave loading at pressures above 5 GPa in the one-velocity and one-temperature approximations, as well as on the assumption of equal pressures for all the phases. Computational results are compared with the well-known experimental results obtained by different authors. The model permits the shock-wave loading of solid and porous alloys with molybdenum in their composition to be described reliably solely by using species parameters.

Mathematical models of water nucleation process for the Direct Simulation Monte Carlo method

Two models of the nucleation process are developed on the basis of (a) size corrected classical nucleation theory and (b) kinetic approach within the Direct Simulation Monte Carlo method. Expressions for probabilities of cluster formation/decomposition reactions are proposed using the models for corresponding rate constants. Both models are employed for the direct simulation of stationary nucleation process in a spatially homogeneous water vapor, where the monomer parameters are kept constant. The relaxation of size distribution function of water clusters is computed for moderate supersaturation values and temperature of 320 K. Numerical analysis shows sufficient difference in results obtained within above mentioned models. The difference arises due to the incorrect description of the internal cluster temperatures caused by the limitation of the classical nucleation theory consisting in one-temperature approximation, which does not describe the details of the energy relaxation. The impact of the dimer formation mechanism (by triple and binary collisions) on cluster size distribution is also analyzed.

Modeling shock loading of multicomponent materials including bismuth

The behavior of multicomponent mixtures and alloys, including bismuth, under high dynamic loads is described by the thermodynamically equilibrium (TEC) model. For condensed phases, the Mie–Gruneisen-type equation of state with regard to the Gruneisen coefficient depending on temperature is used, and gas in pores is among the main environmental components. The model used makes it possible to calculate the behavior of bismuth and materials based on this element (mixtures and alloys) for pressures higher than 6 GPa in one-velocity and one-temperature approximations on the assumption that the pressure is identical for all phases. The calculation results have been compared with the known experimental data and the model calculations performed by different researchers for porosity values varying from 1 to 3. It has been indicated that the model reliably describes shock loading of solid and porous bismuth as well as multicomponent alloys containing bismuth.

Plasma heating to super-hot temperatures (>30 MK) in the August 9, 2011 solar flare

We investigate the August 9, 2011 solar flare of X-ray class X6.9, the “hottest” flare from 2000 to 2012, with a peak plasma temperature according to GOES data of ≈32.5 MK. Our goal is to determine the cause of such an anomalously high plasma temperature and to investigate the energy balance in the flare region with allowance made for the presence of a super-hot plasma (>30 MK). We analyze the RHESSI, GOES, AIA/SDO, and EVE/SDO data and discuss the spatial structure of the flare region and the results of our spectral analysis of its X-ray emission. Our analysis of the RHESSI X-ray spectra is performed in the one-temperature and two-temperature approximations by taking into account the emission of hot (∼20 MK) and super-hot (∼45 MK) plasmas. The hard X-ray spectrum in both models is fitted by power laws. The observed peculiarities of the flare are shown to be better explained in terms of the two-temperature model, in which the super-hot plasma is located at the flare loop tops (or in the magnetic cusp region). The formation of the super-hot plasma can be associated with its heating through primary energy release and with the suppression of thermal conduction. The anomalously high temperature (33 MK according to GOES) is most likely to be an artefact of the method for calculating the temperature based on two-channel GOES measurements in the one-temperature approximation applied to the emission of a multi-temperature flare plasma with a minor contribution from the low-temperature part of the differential emission measure.

Model of the behavior of aluminum and aluminum-based mixtures under shock-wave loading

A thermodynamically equilibrium model is applied to describe the behavior of solid and porous materials. This model ensures good compliance with the experiment in a wide range of pressures. The gas in pores, which is a component of the medium, is taken into account in this model. The equation of state of the Mie-Gruneisen type with allowance for the dependence of the Gruneisen coefficient on temperature is used for condensed phases. The applied model allows the behavior of the aluminum with a porosity from 1 to 8 to be calculated under shock-wave loading at pressures above 5 GPa in the one-velocity and one-temperature approximations, as well as on the assumption of equal pressures for all the phases. Computational results are compared with the well-known experimental results obtained by different authors (shock adiabats, double compression by shock waves, and temperature estimation). The model permits the shock-wave loading of solid and porous mixtures with aluminum in their composition to be described reliably solely by using species parameters.

The boundary layer in compact binaries

Disk accretion onto stars leads to the formation of a Boundary Layer (BL) near the stellar surface where the disk makes contact with the star. Albeit a large fraction of the total luminosity of the system originates from this tiny layer connecting the accretion disk and the accreting object, its structure has not been fully understood yet. It is the aim of this work, to obtain more insight into the Boundary Layer around the white dwarf in compact binary systems. There are still many uncertainties concerning the extent and temperature of the BL and the rotation rate of the white dwarf. We perform numerical hydrodynamical simulations, where the problem is treated in a one-dimensional, radial approximation (slim disk). The turbulence is described by the alpha parameter viscosity. We include both cooling from the disk surfaces and radial radiation transport. The radiation energy is treated in the one-temperature approximation. For a given M_dot our results show a strong dependence on the stellar mass and rotation rate. Both, the midplane and the effective temperature rise considerably with increasing stellar mass or decreasing stellar rotation rate. Our simulations further show, that the radiation energy and pressure are indeed important in the BL. However, some models show a low optical depth in the BL, making it necessary to find a better representation for optically thin regions. The combination of a high mass and a small radius, such as in white dwarfs, can lead to an enormous energy release in the BL, provided the WD rotates slowly. Since the radial extent of BLs is typically very small (about 0.02 to 0.05 R_star), this leads to surface temperatures of a few hundred thousand Kelvin. All of our models showed subsonic infall velocities with Mach numbers of < 0.4 at most.

Dimensional Analysis, Thermodynamics and Conservation Laws in a Problem of Radiation Processes Simulation

The mathematical modelling of radiation processes, based on experimental data of modern physics and traditional approaches of dimension theory, thermodynamics and gas dynamics is considered. Dimensional analysis allows predicting important characteristics of radiation processes (mass of particles-bosons radiation, the magnitude of the radiation pressure, the characteristic linear dimensions), and also indicate the possibility of superluminal motion (without violating the principle of causality). Methods of thermodynamics lead to the refined modelling of equilibrium radiation, the equation of state of radiation protection, determine the heat capacity and the value of radiation pressure in the source. Gas dynamics approaches to formulate a closed system is thermodynamically consistent conservation laws for radiation heat-conducting gas. In this direction we present one-velocity two-temperature and one-temperature approximation. Demonstrate some practical applications of the methods of modelling.

Simple model for calculating shock adiabats of powder mixtures

A model is proposed for calculating the behavior of porous powder mixtures under shock-wave loading in the one-velocity and one-temperature approximations and also under the assumption of identical pressures of all phases. The model takes into account the presence of the gas in pores. The calculated results are compared with available experimental data for solid and porous media (shock adiabats, double compression by shock waves, and adiabatic unloading). The calculated results are in good agreement with the experimental data, including those for two- and three-species (in terms of condensed phases) mixtures.

Peculiarities of Thermoelectric Cooling in p–n Structures

The basic physical peculiarities of the phenomenon of thermoelectric cooling in p–n structures by an electric current are investigated in the linear approach. It is proved that, for an adequate research of such structures in the general case, a multi-temperature approximation is required. The multi-temperature approximation means that each of the quasi-particle subsystems (electrons, holes, phonons) is characterized by its own temperature. Also, a criterion is found for applying the one-temperature approximation. The most interesting particular cases are studied, explaining the conditions for each case.

Two-temperatures approximation theory of thermo-emf in bipolar semiconductors

For the first time, the thermo-emf is calculated in the two-temperatures approximation and taking into account non-equilibrium charge carriers. The results strongly differ from the results obtained in the one-temperature approximation. To emphasize the role played by these non-equilibrium charge carriers, it was considered a very thin semiconductor sample, so recombination processes and energy exchange processes are not present. The thermo-emf generated depends only on electron's parameters and Peltier's resistance appears.

Criteria for thermal stability of parabolically oxidable heterogeneous systems

A system of equations that interaction of an isolated particle or a dense ensemble of particles with an oxidizing agent in the case of parabolic growth of the product layer is analytically treated in the one-temperature approximation. It is shown that a jumplike transition from the lower reacting mode (regime with a temperature maximum) to the upper reacting mode (regime with a monotonically increasing temperature) occurs with a continuously changing ignition parameter. The critical ignition parameter obtained differs from its numerically predicted value by 7%.

Analytical study of the combustion waves propagation under filtration of methane–air mixture in a packed bed

The study of combustion waves during the filtration of lean methane–air mixtures in inert porous media is carried out using the one-temperature approximation in a semi-infinite canal. The analytical solution is built in three different regions, the pre-heating region, the reaction region and the region occupied by the combustion products. By means of the solution, the temperature and mass fraction profiles of the methane are built for three regions, as well as the longitudinal extension of the reaction region, the ignition temperature of the mixture and the combustion wave propagation velocity in the system. The results obtained are validated by means of comparisons with the known numerical and analytical solutions and experimental results found in literature. Finally, an analysis of the built analytical solution in terms of five dimensionless parameters is made, which define heat and mass transport in mathematical model.

Two-dimensional modeling of filtration combustion in porous charges in a one-temperature approximation

Filtration combustion with a stationary front in porous charges is modeled within the framework of a one-temperature approximation. The results of numerical calculations are compared with an analytical evaluation for the quasi-one-dimensional case of a tube of variable cross section.

Model of the physico-chemical kinetics behind the front of a very intense shock wave in air

A model of the physico-chemical kinetics of the reactions taking place behind the front of an intense shock wave propagating in air with a speed of 9–14 km/s is proposed. The problem of describing the chemical reactions, namely, molecular dissociation and exchange reactions involving vibrationally excited molecules in the absence of vibrational equilibrium, is solved. The vital role of the vibrational excitation delay in the dissociation of oxygen and nitrogen is established. The rate of the exchange reaction between nitrogen molecules and oxygen atoms in the shock wave depends only slightly on the vibrational excitation level. It is demonstrated that the rate constants for thermally nonequilibrium dissociation reactions can be represented within the framework of the one-temperature approximation at constant vibrational temperatures of the dissociating species satisfying quasi-stationary conditions.

Standing waves of gaseous chemical reactions in inert porous media

Standing waves of gaseous, exothermic, chemical reactions in inert porous media are investigated in the two-temperature and one-temperature approximations. A “semiinfinite” reaction zone method is developed to solve the problem analytically. It is shown that the wave solutions obtained by this method form a large set. Traditional solutions with strong heat transfer at the end of the wave make up but a small fraction of this set.

Entrainment of dispersed impurities into the atmosphere by a rising thermal

The ascent of a dusty turbulent thermal in a nonuniform compressible atmosphere is numerically investigated. The dispersed phase is described in the one-velocity and one-temperature approximation. A comparative calculation of the entrainment of active and passive impurities into the stratosphere is made, the vertical aerosol density distribution in the hanging cloud is obtained, and the maximum amount of active impurity that can be carried beyond the tropopause by a thermal of given energy is determined. The ascent of a thermal formed in an air blast is contrasted with that of a thermal formed in a surface blast, and the results of the calculations are compared with previously published data.