Modeling Explosions and Blast Waves

Preface
Contents
About the Author
1 Basic Concepts and Introduction to Blast Waves and Explosions
	1.1 Noise and Disruption of Objects in an Explosion
		1.1.1 Sound Waves
		1.1.2 Finite Amplitude Waves
		1.1.3 Wave with a Steep Front
		1.1.4 Shock Waves
		1.1.5 Compression Disturbances Intensifying to a Shock Wave
		1.1.6 Expansion Waves
		1.1.7 Role of Compressibility of the Medium
		1.1.8 Wave Propagation and Not Matter Propagation
		1.1.9 Blast Wave and Disruption of Objects
	1.2 Types of Explosions
		1.2.1 Naturally Occurring Explosions
		1.2.2 Intentional Explosions
		1.2.3 Accidental Explosions: Hazard
	1.3 Typical Examples of Accidental Explosions
		1.3.1 Texas City Disaster (April 16, 1947)
		1.3.2 Beirut Explosion (August 4, 2020)
		1.3.3 Explosion of Fuel Tank of Aircraft During Flight
		1.3.4 Largest Man-Made Explosion: Ural Mountains, June 4, 1989
		1.3.5 Fireball and Blast in the Explosion at Crescent City, Illinois: June 2, 1970
		1.3.6 Explosion in a Bakery at Turin Involving Dust
		1.3.7 Explosion in a Copper Smelter at Flin Flon, Canada
		1.3.8 World\'s Worst Industrial Disaster: Bhopal Gas Tragedy (December 2/3, 1984)
		1.3.9 Nuclear Explosions: Chernobyl (April 26, 1986) and Fukushima (March 11, 2011)
	1.4 Classification of Explosions
2 Blast Waves in Air
	2.1 Ideal Blast Wave
		2.1.1 Ideal Blast Trajectory from Dimensional Considerations
	2.2 Modeling of Parameters Across  A Constant Velocity Shock
		2.2.1 Rankine–Hugoniot Equations
		2.2.2 Rayleigh Line and Properties Across a Shock Wave  of Given Velocity S
		2.2.3 Properties Behind a Shock of Mach Number MS
	2.3 Change of Properties in a Blast Wave
		2.3.1 Concentration of Mass at the Wave Front
		2.3.2 Deviation from Wave Phenomenon
		2.3.3 Decay of Blast Waves
		2.3.4 Characteristic Length of Energy Release;  Explosion Length
	2.4 Predictions for Overpressures
		2.4.1 Cranz–Hopkinson Scaling Law for Overpressure
		2.4.2 Overpressure from a Strong Blast Wave
		2.4.3 Smaller Values of Overpressures
	2.5 Non-idealities of Source Influencing  Overpressure
	2.6 Pressure Variations in a Blast Wave: Impulse
		2.6.1 Arrival Time and Mach Number of The  Blast Wave at a Distance L from the Source
		2.6.2 Impulse
		2.6.3 Cranz–Hopkinson Law for Scaling Impulse
	2.7 Missiles, Shrapnels, and Fragments from Blast;  Gurney Constant
	2.8 Salient Features of Blast Waves
3 Interaction of Blast Waves with Rigid and Non-rigid Bodies
	3.1 Reflection of Shock Waves from Non-yielding Surfaces
		3.1.1 Normal Reflection
		3.1.2 Oblique Reflection
	3.2 Reflection and Transmission of Shocks from Yielding Surfaces
		3.2.1 Mechanical Impedance of Medium and Determination of Reflected and Transmitted Waves
		3.2.2 Formation of Expansion Waves
		3.2.3 Spallation
		3.2.4 Crushing of Kidney Stones in Humans
	3.3 Reflection of a Blast Wave from the Ground: Formation of Multiple Shocks and a Mushroom Cloud
		3.3.1 Craters
	3.4 Multiple Pressure Spikes from a Finite Volume Explosion
	3.5 Blast Waves in Water
		3.5.1 Underwater Explosions and Associated Blast Wave
		3.5.2 Explosions over Surface of Water
	3.6 Absorption of Blast Wave Energy in Layered Structures
	3.7 Role of Overpressure and Impulse on Damage from Blast Waves
4 Energy Release and Rate of Energy Release
	4.1 Energy Release
		4.1.1 Heat of Formation
		4.1.2 Chemical Reactions and Energy Release
		4.1.3 Stoichiometry, Fuel-Rich and Fuel-Lean Compositions;  Equivalence Ratio
		4.1.4 Energy Release in a Stoichiometric Mixture  of Fuel Vapor and Air
		4.1.5 Generalized Procedure for Determining  Energy Release
		4.1.6 Influence of Variations in the Temperature And  Pressure of the Reactants on Energy Release
		4.1.7 Energy Release for Fuel-Lean (< 1) And  Fuel-Rich (> 1) Compositions
	4.2 Rate of Energy Release
		4.2.1 Concentration, Law of Mass Action, And  Activation Energy
		4.2.2 Arrhenius Rate Equation
		4.2.3 Rate of Chemical Reactions and Rate of Energy Release
5 Thermal Theory of Explosions
	5.1 Formulation of Theory
		5.1.1 Lumped Mass Assumption
		5.1.2 Variations of Heat Release and Heat Loss Rates
		5.1.3 Stable Temperature and Ignition Temperature
		5.1.4 Critical Temperature and Auto-ignition
		5.1.5 Changes of Ambient Temperature
	5.2 Critical Conditions and Preheat
	5.3 Characteristic Times of Heat Generation  and Heat Loss
		5.3.1 Characteristics of Heat Release From  Chemical Reaction
		5.3.2 Characteristic Time for Energy Release
		5.3.3 Characteristic Heat Loss Time
	5.4 Conditions for Explosion to Occur
	5.5 Ignition and Auto-ignition
	5.6 Induction Times and Nature  Of Chemical Reactions
	5.7 Branched Chain Explosions in Closed Vessels
		5.7.1 First Explosion Limit
		5.7.2 Second Explosion Limit
		5.7.3 Third or Upper Explosion Limit
	5.8 Limitations of Lumped Mass Assumption
6 Propagation of Reaction Front: Detonation, Deflagration and Quasi-Detonation
	6.1 Propagation of One-Dimensional Combustion Waves: Reaction Hugoniot and Rayleigh Line
	6.2 Physically Realizable States on Reaction Hugoniot: Detonations and Deflagrations
		6.2.1 Chapman–Jouguet (CJ) Points
		6.2.2 Detonation Branch of Hugoniot
		6.2.3 Deflagration Branch
		6.2.4 Realizable Combustion Waves
	6.3 Detonations
		6.3.1 Detonation Velocity VCJ and Pressure pCJ at the CJ Point U
		6.3.2 One Dimensional Structure of a Detonation
		6.3.3 ZND Structure of a Detonation
		6.3.4 Detonation Cell and Multi-headed Detonation Front
	6.4 Deflagration and Burning Velocities
		6.4.1 Burning Velocity and Flame Speed
		6.4.2 Thickness of Flame
		6.4.3 Laminar and Turbulent Burning Velocities
		6.4.4 Turbulent Flame Brush
		6.4.5 Pressure Changes Across a Flame
	6.5 Fast Flame at Lower Chapman–Jouguet Point:  Sub CJ or Quasi Detonation
		6.5.1 Velocity and Pressure in a Quasi Detonation
	6.6 Detonations and Flames: Destructive Influence
7 Formation of Flames and Detonations in Gaseous Explosives
	7.1 Initiation of Flame
		7.1.1 Divergence and Loss
		7.1.2 Quenching of Flame
		7.1.3 Minimum Ignition Energy
		7.1.4 Limits of Flammability
		7.1.5 MIE and Flammability Limits
		7.1.6 Low-Pressure Flammability Limits
		7.1.7 Influence of Initial Temperature on Flammability Limits
		7.1.8 Flammability Limit for a Mixture of Gases
		7.1.9 Upward and Downward Flammability Limits
	7.2 Minimum Oxygen Concentration: Maximum Safe Oxygen Concentration
	7.3 Flammability Limits of Vapors From Volatile Liquids
		7.3.1 Formation of Flammable Vapor–Air Mixture from Volatile Liquids
		7.3.2 Flash and Fire Point Temperatures
	7.4 Initiation of Detonation: Detonation Kernel
		7.4.1 Requirement of Strong Shock Wave
		7.4.2 Requirement of a Minimum Kernel for Detonation
		7.4.3 Detonation Kernel in Analogy to Flame Kernel: Energy Required
		7.4.4 Limits of Detonation
	7.5 Transition of Flame to Detonation
8 Condensed Phase Explosions
	8.1 Hydrocarbon Fuels Constituting Condensed Phase Explosives
		8.1.1 Single, Double, and Triple Bonds
		8.1.2 Alkanes, Alkenes, Alkynes, and Alkadienes
		8.1.3 Aromatic Structure: Benzene
		8.1.4 General Classification of Hydrocarbons
	8.2 Explosives from Hydrocarbons
		8.2.1 Nitromethane, Nitroglycerine, and Nitroglycol from Aliphatic Hydrocarbons
		8.2.2 Nitrocellulose from Cellulose
		8.2.3 Penta Erythritol Tetra Nitrate (PETN) from Straight Chain Aliphatic Compound
		8.2.4 RDX and HMX from Cyclo-Aliphatic Hydrocarbons
		8.2.5 Trinitrotoluene (TNT) from Aromatic Benzene Ring
		8.2.6 Picric Acid (PA) from Phenyl
		8.2.7 Tetryl
		8.2.8 TATB
	8.3 Explosives with Radicals of Azide, Fulminate, Acetylide, and Stephnate with Metals
	8.4 Inorganic Explosives: Black Powder
	8.5 Characteristics of Explosive Compositions
		8.5.1 Enhancing Oxygen Content by Addition of Oxygen-Rich Compounds: AN-NM Slurry, ANFO, Gelatine Dynamite
		8.5.2 Reduction in Oxygen Content: Plastic Explosives
	8.6 Volume of Gas Generated from Condensed Explosives: Explosion Severity, Pyrotechnic Compositions, Thermites
	8.7 Deflagration and Detonation of Condensed Explosives
		8.7.1 Deflagration in Confined and Unconfined Spaces
		8.7.2 Detonation in Confined and Unconfined Spaces
		8.7.3 Detonation and Heterogeneity of the Explosive
	8.8 Parameters of Explosive Influencing Detonation; Classification in Four Categories
	8.9 High Values of Activation Energies
	8.10 Ease of Formation of Detonation in Condensed Explosives
	8.11 Low Explosives, Primary Explosives, and Secondary Explosives
	8.12 Overall Classification of Condensed Explosives
9 Unconfined and Confined Gas Phase Explosions
	9.1 Unconfined Explosions
	9.2 Confined Explosions
		9.2.1 Maximum Explosion Pressure
		9.2.2 Violence or Rate of Pressure Rise
	9.3 Methods of Decreasing Maximum Pressure and Maximum Rate of Pressure Rise
		9.3.1 Relief Venting
		9.3.2 Halons; Suppression of Rate of Pressure Rise
	9.4 Maximum Experimental Safety Gap
		9.4.1 Enclosures Joined by Pipes
	9.5 Partial Confinement
	9.6 Sequence of Events in Typical Unconfined and Confined Explosions
		9.6.1 Largest Man-made Unconfined NG Explosion: Ural Mountains
		9.6.2 Propane Vapor Explosion and Fireball at Port Hudson, Missouri
		9.6.3 Semi-confined Explosion at Chemical Plant at Flixborough, England
		9.6.4 Hydrogen Explosion and Rupture of Confinement In Loss of Coolant Accident (LOCA) at Fukushima
		9.6.5 Confined Explosion; Fuel Tank of Aircraft
10 Dust Explosions
	10.1 Organic Dust; Lower and Upper Limits of Concentration
	10.2 Estimation of Concentration
		10.2.1 Gravity Feed
		10.2.2 Forced Feed by Entraining with Air
	10.3 Detonation, Smoldering, And Secondary Explosions
	10.4 Characterization of Dust Explosions
		10.4.1 Severity, KSt and St Classification
	10.5 Ignition Sensitivity, Explosion Severity, and Index of Explosibility
	10.6 Nonvolatile Dusts
		10.6.1 Ignition Sources
11 Physical Explosions and Rupture  of Pressure Vessels
	11.1 Flash Vaporization
		11.1.1 Principle
		11.1.2 Spillage of Volatile Liquids Stored at High Pressures
		11.1.3 General Formulation for Flash Vaporization
		11.1.4 Energy Release in Flash Vaporization
		11.1.5 Flash Vaporization of Water Thrown in A Hot Furnace as an Example
		11.1.6 Rollover and Explosion in Cryogenic Storage Vessels
		11.1.7 Rollover Explosion at Cleveland, Ohio and Intense Damage of Sewage Lines, Homes and Skyrocketing of Manhole Covers
		11.1.8 Condensation Shock and Rupture of Cryogenic Pressure Vessels
	11.2 Burst of Pressure Vessels
		11.2.1 Influence of Specific Heat Ratio
		11.2.2 Heat Addition to a Gas at Constant Pressure
12 TNT Equivalence and Yield from Explosions
	12.1 TNT as an Idealized Energy Source
		12.1.1 Energy Release from TNT
		12.1.2 Energy Liberated at Constant Volume
		12.1.3 Rate of Energy Release in TNT
		12.1.4 Approximation of TNT as a Point Source
		12.1.5 Characterization of Ideal Explosion by TNT
	12.2 Role of Nonidealities and Yield
	12.3 Influence of Source and Surroundings
13 Atmospheric Dispersion of Flammable Gases and Pollutants
	13.1 Dispersion of Gases Released in the Atmosphere
		13.1.1 Temperature Distribution Above the Surface of Earth
		13.1.2 Wind, Air Currents, and Clouds
		13.1.3 Temperature Inversion
	13.2 Escape of Gas Released in the Atmosphere: Atmospheric Stability
		13.2.1 Pasquill\'s Classification of Atmospheric Stability
	13.3 Dispersion
		13.3.1 Diffusion of Gas in One Dimension
		13.3.2 Standard Gaussian Distribution
		13.3.3 Significance of Standard Deviation in the Standard Gaussian Distribution
		13.3.4 Comparison of the Diffusion of Concentration in One Dimension with the Standard Gaussian Distribution
	13.4 Dispersion in Three Dimensions of a Mass  of Gas Release
	13.5 Dispersion of Steady Release: Plume
	13.6 Dispersion Coefficients
	13.7 Uncertainty and Errors in the Gaussian Dispersion Scheme
	13.8 Buoyant and Heavy Gases
	13.9 Major Explosions and Pollutions Involving Atmospheric Dispersion
		13.9.1 Dispersion of Propane in the Explosion at Port Hudson in December 1970
		13.9.2 Dispersion of Pollutant in the Great Smog of London, December 1952
		13.9.3 Dispersion of Poisonous Gases from Runaway Chemical Reaction in a Chemical Plant at Bhopal (December 2–3, 1984)
		13.9.4 Dispersion of Styrene Vapor in Atmosphere in the Visakhapatnam Gas Leak, India (May 7, 2020)
14 Quantification of Damages
	14.1 Probabilistic Nature of Damages
	14.2 Dose Signifying the Stimulus
	14.3 Dose Response Curves
	14.4 Standard Deviations σ, 2σ and 3σ
		14.4.1 Probit
		14.4.2 Form of Probit
Appendix A Dependence of Sound Speed on Temperature
Appendix B Mach Number MS1 Behind a Shock Propagating at Constant Mach Number MS
Appendix C Reflection of a Constant Velocity Shock Wave Normal to a Rigid Surface
Appendix D Shock Reflection and Transmission: Mechanical Impedance and Acoustic Impedance
Appendix E Energy Release in a Chemical Reaction assuming Thermodynamic Equilibrium: Illustration for the Explosion of TNT and a Fuel-rich Gaseous Mixture
E.1  Gibbs Free Energy and Equilibrium
E.2  Equilibrium of Species and Equilibrium Constants
E.3 Chemical Potential and Equilibrium Products  in a Reaction
E.4 Extension to Multiple Species in the Products  of the Reaction
E.5  Equilibrium Constant Kp and Determination of Species
E.6 Example of Using Equilibrium Constants to Determine the Products and Energy Release in the Explosion of TNT
E.6.1  Initial Assumption for Total Number of Moles in Products
E.6.2  Inputs for Solving Equilibrium Relation and Iteration for Temperature of Products
E.7  Example of a Fuel-Rich Mixture of Air and Butane
E.8  Phases of the Species
E.9  Correction for Pressure in the Dense Gases: Fugacity
E.10  Minimization of Gibbs Free Energy
Appendix F Parameters of Chapman–Jouguet Detonation
Appendix G Change of Concentration with Distance and Time: Solution of the Diffusion Equation
Index