On the theory of stationary velocity of propagation of an exothermic reaction front in a condensed medium

It has been demonstrated experimentally that in the combustion of many explosives and powders in the condensed phase (k-phase) an exothermic chemical reaction occurs. Although the heat release in the k-phase is usually small in comparison with the calorific value, it may play an important role in the multistage reaction in the combustion zone.Analysis of the heat balance of the k-phase reveals that in a number of cases heating of the substance before gasification is primarily due to self-heating. According to the thermocouple measurements made by A. A. Zenin, the heat release in the k-phase during combustion of nitroglycerine N powder is more than 80% of the total quantity of heat in the heated layer of the k-phase (pressure 50 atm). This makes it possible to speak of the propagation of the exothermic reaction front in a condensed medium as the first stage in the combustion of condensed systems. Cases are also known where the propagation of the reaction front is maintained only by self-heating (flameless combustion [1]), and there are cases when such propagation is not accompanied by gasification (combustion of thermites, sometimes the polymerization process). Theoretical investigations of stationary propagation of a reaction front in a condensed medium were made in [2–6]. We note that this problem is also of interest in relation to the study of various nonstationary phenomena associated with the combustion of powders [7–9]. One of the principal theoretical problems is the derivation of a formula for the velocity of propagation of the reaction front in the k-phase. The Zel'dovich-Frank-Kamenetskii method [10] was used in [2–5] in the solution of this problem.This paper is an investigation of the applicability of the Zel'dovich-Frank-Kamenetskii method to the case of propagation of a zero-order reaction front in the k phase. A method is proposed for deriving a formula for the propagation velocity of the front leading in the case of a zero-order reaction to a formula identical to that obtained using the Zel'dovich-Frank-Kamenetskii method, and this method is then used to derive a formula for the propagation velocity of a first-order reaction front in the k-phase. The upper and lower limits of the velocity given by this formula are investigated.     

Investigation of the stability of a stationary exothermic reaction front in a condensed phase

A model system of combustion theory describing an exothermic reaction in a condensed phase is investigated. The Arrhenius dependence of reaction rate on temperature is replaced by a piecewise-constant dependence. This makes it possible to find the distribution of the quantities in the stationary wave and to investigate the stability of the solution with respect to one-dimensional perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 115–113, January–February, 1985.The author is grateful to G. G. Chernyi for his constant interest in the work.     

Two-dimensional stability of the combustion of condensed systems

The question of the combustion stability of condensed systems relative to curvature of the front is investigated in a linear approximation. Two of the simplest combustion models are examined, a gasless system and a model of flameless combustion of a solid fuel. In the first case, the combustion products are condensed, just as are the initial materials, and in the second the solid fuel is converted into a gas in which no chemical reactions occur. Boundaries of the stability of the stationary combustion mode are found. It is shown that gasless systems are less stable with respect to two-dimensional perturbations than to one-dimensional perturbations. For the flameless combustion model the result depends on the relationship between the thermophysical constants of the initial material and the products. The question of the influence of heat emission on the one-dimensional stability of the gasless composites is considered. An increase in the heat emission diminishes the stable combustion region, where a one-dimensional instability originates earlier than collapse of combustion occurs because of strong heat emission to the wall.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 51–59, September–October, 1971.     

Calculation of one transient mode of combustion of a condensed system

A one-dimensional process representing the combustion of a powder sample on a metal substrate at constant pressure is considered on the basis of a two-phase model of the thermal decomposition of a condensed system. The results of numerical computer calculations are presented. Qualitative comparison is made with experiment.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 23–30, January–February, 1972.     

Asymptotic analysis of stationary distribution of the front of a two-stage consecutive exothermic reaction in a condensed medium

An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in the article. The method of joined asymptotic expansions is used in constructing the solutions. The ratio of the sum of the activation energies of the reactions to the final adiabatic combustion temperature is a parameter of the expansion. The characteristic limiting states of the stationary distribution of the wave corresponding to different values of the parameters figuring in the problem are shown. Approximate analytical expressions for the wave velocity and distribution of concentrations are obtained for each of the states.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 75–87, January–February, 1973.     

Multizone combustion of condensed systems

In the general case the combustion of condensed systems is of a stage-wise character and the combustion front is multizone [1, 2]. Following the investigation of two-zone models [3–5] it became clear that, during multizone combustion, one of the zones of heat evolution is the controlling zone. The velocity of the front is equal to the velocity of the controlling zone; however, with a change in the parameters of the system, there is the possibility of a transition of the controlling role from one zone to another, as well as of the coalescence and splitting of zones. This paper discusses a generalization of the two-zone problem which makes it possible to go over to the analysis of a complex, multizone front and shows that, for a front with two reactions (in the condensed phase and in the gas) and with dispersion, there are in all three possible arrangements of the zones of heat evolution (two three-zone variants and one two-zone variant). All possible types of dependence of the combustion rate on the depth of the dispersion are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 99–105, November–December, 1972.     

Theory of the combustion of small drops of metal

The article discusses the combustion of small drops of metal. It is postulated that the formation of an oxide in the liquid phase starts with the origin of a condensed phase and continues as the result of a reaction between the vapors of the drop and the oxidizer at the surface of the forming particles of the condensed phase. It is shown that the process of the formation of particles of condensed oxide in the gas, for very small drops, has an essentially unsteady-state character. Under these circumstances, a considerable fraction of the vaporization products of a drop does not succeed in condensing after the complete gasification of the drop and remains in the gaseous state.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 47–53, March–April, 1976.     

On the question of the stability of powder combustion in a half-closed space

The influence of gas temperature perturbations on the stability of powder combustion in a rocket chamber is investigated theoretically on the basis of the Zel'dovich-Novozhilov theory of powder combustion. The influence of the bow space adjacent to the burning channel is also examined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 74–79, November–December, 1971.     

Role of the Knudsen layer in the problem of droplet evaporation

A study is made of the part played by the Knudsen layer in the problem of weak unsteady evaporation of a spherical droplet in its own vapor. It is shown that use of the classical Hertz—Knudsen formula may lead to appreciable errors, in particular, in the determination of the time required by the droplet temperature to relax to the state corresponding to steady evaporation.Translated from Izvestiya Akademii Nauk SSSR; Mekhanika Zhidkosti i Gaza, No. 1, pp. 127–131, January–February, 1984.     

Estimates of the normal velocities of propagation of laminar and very small-scaled turbulent flames

On the most general assumptions (taking account of the Lewis-Semenov number, thermal expansion, variability of thermophysical parameters, etc.), analytical estimates are obtained for the normal velocities of combustion of laminar and turbulent flames. In the case of an Arrhenius dependence of the reaction velocity on the temperature, the combustion velocity is represented by an asymptotic series with respect to the Frank-Kamenetskii dimensionless temperature; for turbulent flames, with respect to a parameter of the relative scale of turbulence. The final results over a wide range of change of parameters are compared with a numerical calculation on a computer of the exact equations and with the relations obtained by the method of combined asymptotic expansions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 28–37, May–June, 1976.     

Solid propellant combustion in the presence of photoirradiation

Previous studies of the nonsteady processes associated with the irradiation of propellants with light have chiefly been devoted to the question of ignition [1–3]. It is also important to consider the effect of such an easily controlled influence as light on the propellant combustion process. We have attempted to estimate the dependence of the propellant burning rate on the intensity of the luminous radiation. Cases of steady-state combustion and combustion in the presence of a light flux varying harmonically with time are considered. It is assumed that the incident light flux is absorbed in the solid phase in accordance with the Bouguer-Lambert exponential law with constant transparency index. Steady-state combustion is considered within the framework of the Zel'dovich theory [4]. It is shown that in the steady state irradiation is equivalent to a certain increase in the initial temperature of the propellant. In the case of combustion with irradiation this makes it possible to use the data on steady-state combustion without irradiation. Nonsteady combustion in the presence of a periodically varying light flux is described with the aid of the Novozhilov model [5]. A correction to the mean burning rate (u °), proportional to the square of the light flux amplitude, is obtained. In the case of an exponential dependence of the burning rate on initial temperature the correction u ° is negative. The effect of irradiation on the stability of the steady-state propellant combustion mode is discussed.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicneskoi Fiziki, No. 5, pp. 70–77, September–October, 1971.In conclusion the authors thank O. I. Leipunskii and V. B. Librovich for their valuable advice.     

Theory of nonsteady propellant combustion. Combustion in the presence of harmonically varying pressure

It is shown that in order to construct a theory of nonsteady propellant combustion it is necessary to know the steady-state dependences of the burning rate u₀ °, surface temperature T_s °, and flame temperature T_F ° on the external parameters and the initial temperature of the propellant. The combustion processes in an unbounded space, when one of the external parameters varies according to a harmonic law, are examined within the framework of such a theory.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 60–69, September–October, 1971.     

Radiative and convective heat transfer during blowing into a compressed hypersonic layer of two-phase products from the ablation of a covering

The study examines the screening of the radiative heat flux in conditions of hypersonic flow around blunt bodies with ablated carbon-based coverings. In contrast to the studies already known [1–3], allowance is made for the presence of condensed microscopic particles in the products of ablation. In [4] the problem of radiative transfer is considered in a layer of two-phase ablation products with parametrically prescribed dimensions, particle temperature, and layer thickness. The present study uses a closed system of equations which describes the processes of heat and mass transfer. This gives rise to considerable differences in the numerical results, according to the degree of screening.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–166, November–December 1985.deceased     

Study of the extinction of gunpowder in the combustion model with a variable surface temperature

The results of a calculation of the rate of transient combustion of gunpowder during a fall in pressure are presented; these are obtained by the numerical integration of the equations of transient-combustion theory, allowing for the variable surface temperature of the k phase. For rapid and severe pressure drops extinction always occurs, no introduction of special extinction conditions being required. The change in the rate of burning during the extinction process is of a smooth nature.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 92–100, May–June, 1973.     

Analytical calculation of the parameters of a molecular gas on a surface in the Smoluchowski problem

An analytical solution of the classical Smoluchowski problem on the temperature jump in molecular (monatomic, diatomic, and polyatomic) gases is presented. The gas occupies a half–space above a flat wall, with a constant temperature gradient and evaporation rate from the gas—condensed phase interface set far from this wall. The distribution function is explicitly constructed both in the half–space and at its boundary. Formulas for the concentration and temperature at the interface are derived; in the case of diatomic and polyatomic gases, formulas for temperatures determined by translational and rotational degrees of freedom of molecules are obtained. Numerical calculations are performed.     

Measurement of the Combustion Temperature of a Solid Propellant by the Cars Method

To measure the solidpropellant combustion temperature, a procedure was used based on examination of the temperature dependence of the Qbranch of the coherent antiStokes Raman scattering (CARS) spectrum due to nitrogen contained in combustion products. The measurements were carried out at a pressure of 4 MPa, under which the spectrum demonstrated a substantial overlap between spectral lines. CARS intensities at two frequencies were registered; their ratio was used to determine the combustion temperature of a stoichiometric ammonium dinitramide–polycaprolacton mixture.     

Calculation of the flow of a two-phase mixture in a Laval nozzle,allowing for the turbulent diffusion of the particles

The flow of a mixture of gas and condensed particles in an axisymmetric Laval nozzle is considered. The motion of the particles is calculated in a specified field of gas flow, with due allowance for their turbulent diffusion. The results of calculations indicating the necessity of allowing for this phenomenon when considering the motion of particles toward the wall of a profiled nozzle are presented.Translated from Izvestiya Akademii Nauk SSSR, No. 2, pp. 161–165, March–April, 1973.     

Effect of condensed powder phase reactions on the stability of steady-state combustion processes

A study has been made of the low-frequency stability of powder combustion in a semiclosed chamber, working within the framework of a linear theory with account taken of condensed-phase (k-phase) inertia and evolution of thermal energy. The case treated is that of the first-order reaction. It is shown that k-phase exothermic chemical decomposition increases the stability of the combustion process. The results of numerical computations are interpreted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 102–111, September–October, 1973.     

Calculation of hydrogen-air combustion behind detached shock wave for supersonic flow past a sphere

Many of the published theoretical studies of quasi-one-dimensional flows with combustion have been devoted to combustion in a nozzle, wake, or streamtube behind a normal shock wave [1–6].Recently, considerable interest has developed in the study of two-dimensional problems, specifically, the effective combustion of fuel in a supersonic air stream.In connection with experimental studies of the motion of bodies in combustible gas mixtures using ballistic facilities [7–9], the requirement has arisen for computer calculations of two-dimensional supersonic gas flow past bodies in the presence of combustion.In preceding studies [10–12] the present author has solved the steady-state problem under very simple assumptions concerning the structure of the combustion zone in a detonation wave.In the present paper we obtain a numerical solution of the problem of supersonic hydrogen-air flow past a sphere with account for the nonequilibrium nature of eight chemical reactions. The computations encompass only the subsonic and transonic flow regions.The author thanks G. G. Chernyi for valuable comments during discussion of the article.