Problem Of Thermal Explosion Research Articles

Approximate critical conditions in thermal explosion theory for a two-step kinetic model

Approximate critical conditions for a thermal explosion problem is developed for a two-step reactions based on theories of Semenov and Frank-Kamenetskii. The aim is to examine the contributions of the radical termination step and the temperature dependent pre-exponential factor on the critical parameters within the framework of classical stationary and non-stationary theories. In the non-stationary case, a more general expression for the critical Semenov parameter (Ψ cr ) and critical temperature (θ cr ) were obtained by asymptotic procedure. In the stationary case, numerical estimates for the critical Frank-Kamenetskii parameter (δ cr ) and the critical temperature (θ cr ) were obtained by variational method technique. It was observed that the Semenov and Frank-Kamenetskii parameters are greatly influenced by the termination step and the variable pre-exponential factor. Apart from elucidating hitherto unknown features in the theory of thermal explosion, the results are more general as some known results are easily recovered.

Ignition characteristics, conditions of criticality and disappearance of criticality of cumene hydroperoxide reaction by modeling approach

The thermal explosion problem of cumene hydroperoxide exothermic reaction which is used in chemical industries for production of some chemical materials is investigated. The analytical solutions of the problem to determine the margin between ignition and non-ignition systems are presented. The solution offers different analytical expressions which relate between the critical parameters for both steady and unsteady-states in different planes of solutions for different cases. The numerical solutions in different planes offer different trajectories of solution as sub-critical (non-ignition) and supercritical (ignition). Also from the numerical solution the relations between the critical parameters are presented. The critical behaviors from both analytical and numerical solutions are concise and pertained the same results.

Application of heat-balance integral method to conjugate thermal explosion

Conjugate thermal explosion is an extension of the classical theory, proposed and studied recently by the author. The paper reports application of heat-balance integral method for developing phase portraits for systems undergoing conjugate thermal explosion. The heat-balance integral method is used as an averaging method reducing partical differential equation problem to the set of first-order ordinary differential equations. The latter reduced problem allows natural interpretation in appropriately chosen phase space. It is shown that, with the help of heat-balance integral technique, conjugate thermal explosion problem can be described with a good accuracy by the set of non-linear first-order differential equations involving complex error function. Phase trajectories are presented for typical regimes emerging in conjugate thermal explosion. Use of heat-balance integral as a spatial averaging method allows efficient description of system evolution to be developed.

Critical conditions for conjugate thermal explosion

The thermal explosion problem is considered within the two thermally interacting regions with different thermal and kinetic properties. Thermal exchange with the surroundings is assumed to obey Newton's law. Critical conditions for such a problem are established for the planar, cylindrical and spherical geometries. The major conclusion of the present paper is that critical conditions in the two-media system differ both quantitatively and qualitatively from that of the single-media system. In particular, thermal runaway may occur in the composite system at the parameters that are subcritical for both regions, considered as a single media. On the other hand, under certain conditions where a single-media system would experience thermal explosion, conjugate thermal explosion is preventable by adjusting the properties of the neighboring material and the rate of heat exchange with the surroundings. Different scenarios of thermal explosion development are presented for a typical (cylindrical geometry) case by way of unsteady numerical simulations.

Solving the multidimensional problem of site-origin thermal explosion by the perturbation method

Solving the multidimensional problem of site-origin thermal explosion by the perturbation method

Thermal Explosion in a Flammable Gas Containing Fuel Droplets: Asymptotic Analysis

The problem of thermal explosion in a flammable gas mixture with addition of volatile fuel droplets is studied based on the asymptotic method of integral manifolds. The model for the radiative heating of droplets takes into account the semitransparency of droplets. A simplified model for droplet heat-up is used. The results of the analysis are applied to the modelling of thermal explosion in diesel engines. Two distinct dynamical situations have been considered, depending on the initial droplet concentration. These are ‘far zone’ (small initial liquid volume fraction and small droplet radii) and ‘near zone’ (large initial liquid volume fraction and large droplet radii). The conditions of the first zone are typical for the areas in the combustion chamber which are far from the fuel injectors, while the conditions of the second zone are typical for the areas in the combustion chamber which are relatively close to the fuel injectors. It has been pointed out that small droplets’ heating and evaporation time in the far zone is smaller than the chemical ignition delay of the fuel vapor/air mixture. The total ignition delay decreases with increasing initial gas temperature. In the near zone for large droplets, the process starts with the initial gas cooling and slight heating of droplets. This is followed by a relatively slow heating of gas due to the chemical reaction, and further droplet heating. The total ignition delay in the near zone is larger than in the far zone. It is expected that before thermal explosion in the near zone takes place, the droplets break up and are removed from this zone. In optically thick gas effects of thermal radiation are negligible for small droplets but are noticeable for large droplets.

A conjugate problem of thermal explosion of a hollow cylinder

A conjugated problem of heat conduction for two hollow cylinders that are in close contact, with one consisting of a reagent, has been solved analytically. The problem has been solved by the backward method under boundary conditions of the first kind. The critical conditions of thermal explosion were investigated as functions of the parameters of a reagent and of an inert body. An engineering method for determining the critical conditions of a thermal explosion is suggested.

Delayed thermal explosion in flammable gas containing fuel droplets: Asymptotic analysis

The problem of thermal explosion in a flammable gas mixture with addition of volatile fuel droplets is studied based on the asymptotic method of integral manifolds. The model for the radiative heating of droplets takes into account the semitransparency of droplets. A simplified model for droplet heat-up is used. The results of the analysis are applied to the modelling of thermal explosion in diesel engines. Two distinct dynamical situations have been considered, depending on the initial droplet concentration. These are ‘far zone’ (small initial liquid volume fraction and small droplet radii) and ‘near zone’ (large initial liquid volume fraction and large droplet radii). The conditions of the first zone are typical for the areas in the combustion chamber which are far from the fuel injectors, while the conditions of the second zone are typical for the areas in the combustion chamber which are relatively close to the fuel injectors. It has been pointed out that small droplets’ heating and evaporation time in the far zone is smaller than the chemical ignition delay of the fuel vapor/air mixture. The total ignition delay decreases with increasing initial gas temperature. In the near zone for large droplets, the process starts with the initial gas cooling and slight heating of droplets. This is followed by a relatively slow heating of gas due to the chemical reaction, and further droplet heating. The total ignition delay in the near zone is larger than in the far zone. It is expected that before thermal explosion in the near zone takes place, the droplets break up and are removed from this zone. In optically thick gas effects of thermal radiation are negligible for small droplets but are noticeable for large droplets.

Postinduction Processes During Thermal Explosion in the Systems “Porous Material–Reactive Gas–Solid Product”

A complete solution of the unsteady‐state filtration problem of thermal explosion incorporating the postinduction period is given for the first time. The paper describes a study of the temperature‐field dynamics, pore‐gas pressure, and the degree of condensed‐phase conversion versus the reactive‐gas deficiency in a reactive porous material. Focus is on the formation and propagation of frontal regimes of exothermic chemical reactions (their number, direction, and velocity of propagation, degree of condensed‐phase conversion at the front). The study revealed “double self‐ignition” phenomena and combustion‐wave propagation regimes with incomplete conversion at the front. A surface regime of thermal explosion limited by gas filtration from the outside was considered. The regularities in the dynamics of the exothermic chemical reaction found in the present study allow one to qualitatively control high‐temperature synthesis under thermal explosion conditions.

On the Theory of Interphase Interaction in a Mixture of Reacting Metal Particles

A mathematical model of interphase interaction in a mixture of powders is constructed on the basis of the principles of mechanics of multiphase, multispecies, and heterogeneous media, as applied to SHS processes. The evolution of structural transformations, chemical reactions, and phase transitions are considered at the level of a mesocell of the mixture. Analytical solutions are presented for conjugate thermodiffusion problems on single particles and in the mesocell, which close the equations of conservation laws. By an example of a single-stage chemical reaction in a mixture of nickel and aluminum powders (Ni + Al = NiAl), the problem of the structure of an SHS heat wave propagating over a finite-thickness semi-infinite rod is solved numerically. It is shown that the wave structure is strictly nonstationary and contains inflections and isothermal sectors at the melting point of the components, which vary periodically in time. To solve the problem of thermal explosion in a mixture of metal powders in a three-dimensional domain of complex geometry, a numerical algorithm with the use of the method of dummy domains was developed. Key words: high-temperature synthesis, intermetallides, thermal explosion, mathematical simulation.

Exact solutions of nonlinear heat- and mass-transfer equations

The method of generalized separation of variables for solving nonlinear steady and unsteady heat- and mass-transfer equations is outlined. New exact solutions of one-, two-, and three-dimensional heat equations are obtained. Anisotropic media with a nonlinear heat source of general form are considered for the case in which the main thermal diffusivities show a power or an exponential dependence on the spatial coordinates. Equations with a logarithmic heat source are analyzed in detail. The results obtained are applied to the problem of thermal explosion in an anisotropic medium.

Solution to the dusty gas explosion problem with reactant consumption part I: the adiabatic case

The solution to the adiabatic thermal explosion problem of a dusty gas mixture is presented. The same definition of the critical state used for the homogeneous solid is used for the gas mixture. The critical trajectory is defined as the trajectory containing only one inflection point before the maximum temperature is reached in the temperature–concentration domain. In the case of the solid, the inflection point occurred before the maximum temperature. In the case of the dusty gas mixture it was found that the inflection point could occur as well at the maximum temperature. The presence of the gas produces a second critical temperature, and the limiting cases of no gas present or infinite amount of gas present lead to the classical adiabatic solid case or the case of the solid losing heat to the gas as its environment. The solution led to the discovery of three different singular solutions that help in understanding the various trajectories in the subcritical, critical, or supercritical state.

Accounting for reactant consumption in the thermal explosion problem. part IV. numerical solution of the arrhenius problem

The Marching Taylor expansion method is used to determine the critical conditions for the thermal explosion problem with the reaction governed by the Arrhenius reaction kinetic equation. The contrast between the behavior with high θ ad and high θ a, low θ ad and low θ a as well as high θ ad and low θ a is demonstrated. The effect of the degree of reaction is examined. The criticality boundaries with different degrees of reaction are established as a function of the ambient temperature. Comparison with results existing in the literature is shown. Finally an approximate method for determining the critical state is presented. This reduces the computation time needed for establishing accurately the critical states.

Conjugate problem of thermal explosion

The problem of thermal explosion of a reagent in the form a plane layer placed between two plane-parallel inert walls is solved analytically under boundary conditions of the third kind. Critical conditions for thermal explosion for the general case and particular cases are investigated. Results of calculations are compared with previously published data. An engineering method for evaluation of the critical conditions for thermal explosion is proposed.

A Particular Solution Trefftz Method for Non-linear Poisson Problems in Heat and Mass Transfer

This paper proposes a computational procedure based on the Trefftz method for the solution of non-linear Poisson problems. The problem is solved by finding an approximate particular solution to the Poisson equation and using boundary collocation to solve the resulting Laplace equation. This method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions. The solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency. A quasi-Newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme. The efficacy of the new formulation has been demonstrated for two classes of problems, viz. the thermal explosion problem and the diffusion–reaction problem in a partially wetted catalyst pellet. A brief error analysis of the method, as well as future research directions, is presented.

A very simple yet accurate solution to the thermal explosion problem

A very simple solution to the thermal explosion problem for one-dimensional solids is presented. The solution produces analytical expressions for the criticality condition. Two different solutions are used. One solution gives higher answers than those known from exact solutions and the second solution gives lower answers. This allows bracketing the expected answer with a determined tolerance. The results obtained compare quite favorably with those obtained from known exact numerical solutions. This allows the utilization of the present method for many a practical problem with an acceptable tolerances.

Thermal explosion of an unbounded plate under third-kind asymmetric boundary conditions

An analytical solution is given for the problem of a thermal explosion of a plane reagent layer under third-kind asymmetric boundary conditions. The critical Frank-Kamenetskii parameters are determined for the general and particular cases. Calculation results are compared with the literature data. It is shown that the critical parameters, determined by direct and inverse methods, do not coincide for small values of the Biot criterium and depend on the scale temperature. The particular feature of the approach is the application of the inverse method to the solution of the thermal-explosion problem.

The influence of the flow of the reacting gas on the conditions for a thermal explosion

The classical problem of thermal explosion is modified so that the chemically active gas is not at rest but is flowing in a long cylindrical pipe. Up to a certain section the heat-conducting walls of the pipe are held at low temperature so that the reaction rate is small and there is no heat release; at that section the ambient temperature is increased and an exothermic reaction begins. The question is whether a slow reaction regime will be established or a thermal explosion will occur. The mathematical formulation of the problem is presented. It is shown that when the pipe radius is larger than a critical value, the solution of the new problem exists only up to a certain distance along the axis. The critical radius is determined by conditions in a problem with a uniform axial temperature. The loss of existence is interpreted as a thermal explosion; the critical distance is the safe reactor's length. Both laminar and developed turbulent flow regimes are considered. In a computational experiment the loss of the existence appears as a divergence of a numerical procedure; numerical calculations reveal asymptotic scaling laws with simple powers for the critical distance.

Accounting for reactant consumption in the thermal explosion problem part II: A direct solution with application to the Frank-Kamenetskii problem

The thermal explosion problem with reactant consumption is investigated. A direct solution for the problem using a marching Taylor series expansion was obtained. The derivatives in the Taylor series are obtained from the original equations. The states defining criticality are accurately determined for various degrees of reaction. The results obtained are compared with those in the literature by other authors. An approximate method for determining the flammability limits as well as the critical states is presented.

Criterion for thermal explosion with reactant consumption in a dusty gas

The thermal explosion problem in a dusty gas is investigated. Dynamical regimes of the system are classified: slow regimes, thermal explosion with delay, thermal explosion (without delay). The critical transition conditions for the different dynamical regimes are analysed. We emphasize that the critical condition for transition between slow regimes and explosion with delay is a thermal explosion limit. The thermal explosion limit is described in the phase space by a so-called duck-trajectory. (The notion of duck-trajectory was introduced by Cartier for detailed investigation of relaxation oscillations.)