The Mathematical Theory of Combustion and Explosions

vi The amalgamation of individual approaches and results from various schools into a comprehensive scientific theory, which can be generally appreciated throughout the international scientific com munity, is oftendifficult and time consuming. We believe that one of the best ways to give a complete and clear presentation of a theoryis to include a review of the developmental history of that theory. We are convinced that explaining a theory in a historical perspectiveis essential for a proper understanding ofits present state and for a sound choice of future developments. So we have endeavored to present a complete picture of investi gations performed in both Western and Soviet nations. We understand that Soviet investigations are less familiar to English-speaking readers due to the languagebarrier and the obvious sad circumstances of the interruption of scientific connections before and after World War II, because of this there is an emphasis on Soviet publications in the bibliography. Our attempt to present a comprehensive picture has made our book rather large becauseit has had to include some fundamentals of thermochemistry and kinetics as well as self-ignition and flame propagation invarious conditions. We have also included stability problems in some detail but we have had to leave out the problems of combustion of solid propellants and detonation. We hope that our bookwill be useful to the reader wishing to learn about both the present state of combustion theory and howit originated due to the efforts of many people from different countries. Ya.B.Z. 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)