The Mathematical Theory of Combustion and Explosions

1 Introduction. Foundations of the Science of Combustion: Basic Physical Concepts.- 1. Some Results from Chemical Kinetics and Thermochemistry.- Rates of chemical reactions. Reaction rate constants.- Chemical equilibrium. The equilibrium constant.- The heat of reaction.- The molecular energetics of combustion.- The adiabatic combustion temperature.- Complex reactions. The method of quasi-stationary concentrations.- The reaction of hydrogen with chlorine as an example of a nonbranching chain reaction.- The oxidation of nitrogen during combustion. A reversible reaction at variable temperatures.- 2. Self-Accelerating Chemical Reactions and Critical Phenomena — Explosions and Self-Ignition. Semenov’s Theory.- The properties of explosive reactions.- Chain self-ignition. The oxidation of hydrogen.- Thermal self-ignition. Adiabatic thermal explosions.- The Frank-Kamenetskii transformation.- Semenov’s theory of thermal explosions.- The induction period of a thermal explosion.- Interpretation of the theory of thermal explosions on the phase plane.- 3. The Homogeneous Ideally-Mixing Chemical Reactor.- Methods of producting combustion in flows.- Stationary combustion in a homogeneous chemical reactor.- The heat release rate from combustion in an ideally-mixing chemical reactor.- The effect of heat losses.- 4. Laminar Flames.- Chemical reaction waves. Why do they propagate through a fuel mixture.- Chemical transformation in flames.- The similarity of the temperature and concentration distributions in flames.- A formula for the normal propagation velocity of a flame.- Some consequences of the flame velocity formula.- The heat release rate of chemical reactions in flames.- Limits on the propagation of combustion.- Isothermal flames. The decisive role of diffusion of active centers.- Are flames always plane and stationary? Flame stability.- References.- 2 The Time-Independent Theory of Thermal Explosions.- 1. Basic Assumptions of the Theory.- 2. The Conditions for a Thermal Explosion.- Thermal explosions in flat reaction vessels.- Comparison with Semenov’s theory of thermal explosions.- Thermal explosions in cylindrical vessels.- Thermal explosions in spherical vessels.- Thermal explosions with Newtonian heat exchange at the vessel wall.- 3. The Stability of the Solutions in the Time-Independent Theory of Thermal Explosions.- 4. Some General Properties of Solutions from the Time-Independent Theory of Thermal Explosions for Vessels of Arbitrary Shape.- 5. Numerical Studies of Thermal Explosions. Experiments.- 6. Application of the Idea of Thermal Explosions to the Mechanics of Polymers.- References.- 3 The Initiation of Chemical Reaction Waves in Fuel Mixtures: Time-Dependent Statements of the Problem.- 1. Ignition of a Fuel Mixture by Heated Surfaces.- The theory of ignition by a heated plane surface.- Generalization to chemical reaction rates with an arbitrary temperature dependence.- Calculating the ignition conditions in specific instances.- Accounting for reagent consumption near a heated surface.- The effect of surface curvature on the ignition conditions. Cylindrical surfaces.- Ignition of a fuel mixture flowing around a hot object.- 2. The Time-Independent Theory of Ignition by Hot Surfaces as an Intermediate Asymptote of a Time-Dependent Process.- The time evolution of ignition. The transition from a thermal explosion to ignition.- The effect of reagent consumption on nonstationary ignition.- 3. The Initiation of Chemical Reactions by Active Centers.- References.- 4 Laminar Flames.- 1. The Equations of the Theory of Flame Propagation.- Basic ideas and simplifications of the theory.- Propagation of laminar flames at constant velocity. The structure of a flame front; some general properties.- 2. Existence and Uniqueness of the Steady-State Solution of the System of Flame Propagation Equations.- The case Le = 1: the similarity of the temperature and concentration distributions.- Existence of a solution for an arbitrary constant Lewis number.- Uniqueness of the solution for 0 ? Le(z) < 1.- Calculating the flame propagation velocity estimation by the “trial and error” methods.- 3. The velocity and Structure of a Flame Front for Reactions with Large Activation Energies.- The range of variation of the flame propagation velocity.- An asymptotic formula for the flame propagation velocity.- The structure of a laminary flame front.- An appropriate method of deriving a formula for the flame propagation velocity.- Matching asymptotic expansions in the theory of normal flame propagation.- 4. The Spectrum of Flame Propagation Velocities for a Nonzero Reaction Rate near and in the Initial State, The Propagation of Chain Isothermal Flames.- The Kolmogorov-Petrovskii-Piskunov problem.- Summary of results on the existence and uniqueness of solutions to the flame propagation velocity problem.- Flame propagation in a mixture that reacts at the initial temperature.- Spontaneous propagation of chemical reactions. The influence of initial conditions.- 5. Formation of a Stationary Laminar Flame Front.- Review of analytic studies.- Application of numerical methods to the transition to a stationary combustion regime.- Stationary combustion waves as self-similar solutions of the second kind.- 6. The Diffusional-Thermal Stability of Laminar Flames.- One-dimensional stability of flame propagation for Le = 1.- Flame propagation in media with a weakly perturbed initial temperature.- Effects of diffusion and heat conduction on the stability of flames with respect to spatial perturbations.- Diffusional combustion in gaseous mixtures.- 7. Flames in Flows with a Velocity Gradient. Flame “Stretching”.- 8. Self-Igniting Mixtures.- Exothermic chemical reaction regimes in gaseous flows.- Analysis of the limiting cases.- Self-igniting combustion of condensed systems.- References.- 5 Complex and Chain Reactions in Flames.- 1. The Theory of Flames with Multi-step Reactions: Purpose and Methods.- Basic system of equations; assumptions.- An approximate method for calculations involving hot flames with complex and chain reactions.- Application of asymptotic methods to flames with complex structures.- 2. Flames with Unbranched Chain Reactions.- The structure and propagation velocity of flames with unbranched chain reactions involving the combination of two substances.- Calculation of the flame velocity in H2-Cl2 mixtures and comparison with experimental data.- 3. Flames with Branching Chain Reactions.- Properties of flames with branching chain reactions; the simplest chain branching scheme in hot flames.- Flames in hydrogen-oxygen fuel mixtures. Simplest model.- 4. Simple Bulk-Schemes for Multi-step Chemical Reactions in Flames.- Consecutive chemical reactions in flames; “merging,” “control,” and “separation” regimes.- Parallel chemical reactions in flames.- 5. Cool (Isothermal) Flames.- The simplest model of an isothermal flame.- The necessary conditions for existence of an isothermal flame in a multi-component gas mixture. Interaction of chains.- Numerical calculations of the propagation velocity of cool flames in CS2-O2 mixtures and of the critical conditions for propagation; comparison with experiment.- References.- 6 The Gas Dynamics of Combustion.- 1. Motion of Flames in Gas Flows. The Condition for Stationary Combustion.- Propagation of a flame front in a given gas flow field. Huygens’ principle.- The condition for stationary combustion. The Mach angle. The concept of retention points.- The flame on a Bunsen burner.- Propagation of a flame front in a horizontal tube.- Turbulent combustion (general concepts).- Flames as surfaces of gasdynamic discontinuity. Conservation conditions at a flame front. The Hugoniot adiabat.- Inclined plane flame fronts.- Combustion in a fast flow in a tube. Ideal mixing.- Rapid combustion in a tube. Nonuniform flow of combustion products.- Flame fronts as sources of vertical perturbations.- 2. Combustion in Closed Vessels. The Mach Effect.- The difference between combustion at constant volume and combustion at constant pressure. The average temperature and pressure in a closed vessel.- Layered combustion in closed vessels. The Mach effect.- The Mach effect in burned gases. The case of a fuel gas and combustion products with constant and equal adiabatic indices.- The temperature distribution in a spherical vessel with central ignition. Computation of P(t) and rf(t) diagrams.- Experimental manifestations of the Mach effect. Basic approximations of the theory and the conditions for their realization.- Production of nitric oxide during combustion in closed vessels.- 3. Hydrodynamic Instability of Flames.- Statement and solution of Landau’s problem.- Discussion of the results. Physical interpretation of the solution.- Stabilizing effects on plane flame fronts. Markstein’s solution.- The effect of acceleration on the hydrodynamic instability of flames.- The effect of transport processes on the hydro-dynamic instability of flames.- A Laplace transform study of hydrodynamic instability.- The effect of the form of the initial perturbation on the temporal development of hydrodynamic instability. The relationship between Landau’s model and the Laplace transform solution.- Stationary combustion following loss of stability by a plane flame front.- Combustion of liquid explosives. The effect of surface tension on the stability of combustion.- The propagation of spherical flames. Why spherical flames attract the attention of researchers.- Changes in the hydrodynamic stability problem for a spherical flame.- Stability of spherical flames with respect to higher harmonics.- Discussion of the theoretical results and comparison with experimental observations of spherical flames.- 4. Flame Acceleration and Detonation in Tubes.- Experimental observations of detonation onset in tubes.- The breakup of an arbitrary discontinuity in a fuel mixture.- Propagation of a flame from the closed end of a tube. Production of compression waves.- Why a flame develops into a detonation wave. Shchelkin’s explanation.- References.- 7 Diffusional Combustion of Gases.- 1. The General Properties of Diffusion Flames. The Combustion Surface.- 2. The Burke-Schumann Problem.- 3. The Limit of Diffusional Combustion in Unmixed Gases.- 4. Diffusion Flames in Opposed Jets of Oxidant and Fuel.- References.