Detonation Phenomena of Condensed Explosives (Shock Wave and High Pressure Phenomena) 9789811953064, 9789811953071, 9811953066

Table of contents :
Preface
Contents
Contributors
1 Shock and Detonation Phenomena
1.1 Sound, Shock, and Detonation Waves
1.2 Types and Properties of Explosives and Target Phenomena
1.3 Overview of the History of Detonation Research
2 Theory of Shock Wave and Detonation
2.1 Shock Wave Physics in Condensed Media
2.1.1 Introduction
2.1.2 Shock Propagation Characteristics
2.1.3 Thermodynamics of Shock Compression
2.2 Detonation Wave Fundamentals in Condensed Phase Explosives
2.2.1 Simple Model for Detonation—C-J Hypothesis
2.2.2 ZND Model
2.2.3 Application of C-J Hypothesis
2.3 Numerical Simulation
2.3.1 Introduction
2.3.2 Finite Difference Scheme for One-Dimensional Lagrangian Code
2.3.3 Computed Results Using One-Dimensional Lagrangian Code
2.3.4 OpenFOAM
References
3 Description of Detonation Phenomena
3.1 Introduction of Equation of State for Explosives
3.2 Equation of State for Detonation Products
3.2.1 Equation of State with Explicit Chemistry
3.2.2 Equation of State Without Explicit Chemistry
3.3 Kihara–Hikita–Tanaka (KHT) Code
3.3.1 Molecular Theory of Detonation by Kihara and Hikita
3.3.2 KH Code
3.3.3 KHT Code
3.3.4 Typical Examples of KHT Code
3.4 Other Models of the Equation of States
3.4.1 Unified EOS for Arbitrary Initial Density
3.5 EOS Model for Unreacted Condensed Explosives
3.5.1 Formulation of the Equation of State from Isothermal Compression Data
3.5.2 Material Functions and Shock Hugoniot
3.5.3 Interpretation of Unreacted Hugoniot Data
References
4 Measurements of Shock and Detonation Phenomena
4.1 Experimental Methods
4.1.1 Production of Planar Shock Wave
4.1.2 Pressure Measurements
4.1.3 Particle Velocity Measurements
4.2 Temperature Measurements of Detonation Phenomena
4.2.1 History of Detonation Temperature Measurements
4.2.2 Temperature Measurements by Time-Resolved Optical Pyrometer
4.2.3 Detonation Temperature of Liquid Explosives
4.2.4 Detonation Temperature of Solid Explosives
4.3 Measurements of Underwater Explosion Phenomena
4.3.1 Description of Underwater Explosion Phenomena
4.3.2 Use and Evaluation of Underwater Explosion Performance
4.3.3 Measurements of Shock Wave and Bubble Pulse
References
5 Shock Initiation
5.1 Introduction
5.2 Overview of the Experiments and Numerical Simulations of Shock Initiation
5.3 Modeling of the Reactive Flow
5.3.1 Introduction
5.3.2 Discussion of Mixture Rule
5.3.3 Simulations of Reactive Flow by Unified EOS
5.4 Gap Test and Its Numerical Simulation
5.4.1 Small-Scale Gap Test
5.4.2 Scale Effect of Sympathetic Detonation
5.5 Laser Initiation
5.5.1 Introduction
5.5.2 Experimental Procedure
5.5.3 Short Summary and Future Prospects
References
6 Ideal and Non-ideal Detonation
6.1 Introduction
6.2 Definition and Overview of Studies for Non-ideal Detonation
6.2.1 Definition of Ideal and Non-ideal Explosive
6.2.2 Overview of Studies for Non-ideal Detonation
6.3 Ammonium Nitrate
6.3.1 Investigation of Non-ideal Behavior of Ammonium Nitrate
6.3.2 Investigation of Non-ideal Behavior of Ammonium Nitrate and Activated Carbon Mixtures
6.3.3 Investigation of Non-ideal Behavior of ANFO Explosive
6.3.4 Diameter Effect of AN-Based Explosive
6.4 Emulsion Explosives
6.4.1 Introduction
6.4.2 Detonation Velocity of Emulsion Explosives
6.4.3 Detonation Pressure of Emulsion Explosives
6.4.4 Detonation Properties of Aluminized Emulsion Explosives
6.4.5 Underwater Explosion Performance of Emulsion Explosives
6.5 Non-ideal Detonation in Aluminized Explosives
6.5.1 Non-ideal Detonation in Gelled Nitromethane/Aluminum Mixtures
6.5.2 Non-ideal Detonation in Packed Aluminum Particles Saturated in Nitromethane
6.5.3 Non-ideal Detonation in Aluminized Solid Explosives
References
7 Future Perspectives on Detonation Research
7.1 Historical Viewpoint
7.2 Future Perspectives

Citation preview

Shock Wave and High Pressure Phenomena

Shiro Kubota   Editor

Detonation Phenomena of Condensed Explosives

Shock Wave and High Pressure Phenomena Founding Editor Robert A. Graham

Honorary Editors Lee Davison, Tijeras, NM, USA Yasuyuki Horie, Santafe, NM, USA Series Editors Frank K. Lu, University of Texas at Arlington, Arlington, TX, USA Naresh Thadhani, Georgia Institute of Technology, Atlanta, GA, USA Akihiro Sasoh, Department of Aerospace Engineering, Nagoya University, Nagoya, Aichi, Japan

Shock Wave and High Pressure Phenomena The Springer book series on Shock Wave and High Pressure Phenomena comprises monographs and multi-author volumes containing either original material or reviews of subjects within the field. All states of matter are covered. Methods and results of theoretical and experimental research and numerical simulations are included, as are applications of these results. The books are intended for graduate-level students, research scientists, mathematicians, and engineers. Subjects of interest include properties of materials at both the continuum and microscopic levels, physics of high rate deformation and flow, chemically reacting flows and detonations, wave propagation and impact phenomena. The following list of subject areas further delineates the purview of the series. In all cases entries in the list are to be interpreted as applying to nonlinear wave propagation and high pressure phenomena. Development of experimental methods is not identified specifically, being regarded as a normal part of research in all areas of interest. Material Properties Equation of state including chemical and phase composition, ionization, etc. Constitutive equations for inelastic deformation Fracture and fragmentation Dielectric and magnetic properties Optical properties and radiation transport Metallurgical effects Spectroscopy Physics of Deformation and Flow Dislocation physics, twinning, and other microscopic deformation mechanisms Shear banding Mesoscale effects in solids Turbulence in fluids Microfracture and cavitation Explosives Detonation of condensed explosives and gases Initiation and growth of reaction Detonation wave structures Explosive materials Wave Propagation and Impact Phenomena in Solids Shock and decompression wave propagation Shock wave structure Penetration mechanics Gasdynamics Chemically Reacting Flows Blast waves Multiphase flow Numerical Simulation and Mathematical Theory Mathematical methods Wave propagation codes Molecular dynamics Applications Material modification and synthesis Military ordnance Geophysics and planetary science Medicine Aerospace and Industrial applications Protective materials and structures Mining

Shiro Kubota Editor

Detonation Phenomena of Condensed Explosives

Editor Shiro Kubota National Institute of Advanced Industrial Science and Technology Tsukuba, Japan

ISSN 2197-9529 ISSN 2197-9537 (electronic) Shock Wave and High Pressure Phenomena ISBN 978-981-19-5306-4 ISBN 978-981-19-5307-1 (eBook) https://doi.org/10.1007/978-981-19-5307-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

In 2019, a workshop was held on shock and detonation physics of condensed-phase explosives in Tsukuba, Japan. Attendees of the workshop were specialists of shock and detonation physics in Japan. There were also two invited speakers: Prof. Yasuyuki Horie and Prof. Kunihito Nagayama. One outcome of the meeting was the consensus view that although there are unique research works on the detonation of condensedphase explosives in Japan, they are not well publicized in the world. After the meeting following up on the assessment, he suggested, being an editor of Springer Nature shock wave series, the publication of a book to document those unique works in Japan, including background and historical materials. In March 2020, the chief organizer of the workshop (Kubota of AIST), liking the suggestion, gathered willing contributors and embarked on the project to assemble a book with focus on the detonation of condensed-phase explosives. Willing contributors are Prof. Nagayama, Dr. Yoshida, Dr. Kato, Dr. Murata, Prof. Miyake, and Dr. Kubota. They are leading scientists and engineer in Japan who have worked in safety research on explosives, development of industrial explosives, and application of explosives. This book presents the fundamental theory of shock and detonation waves as well as detonation related R&D in Japan. Especially, it describes the research in Japan on industrial explosives such as ammonium nitrate-based explosives and liquid explosives such as nitromethane (NM). Some of them are internationally recognized pioneering works. This book is intended as a monograph-style book written by multi-authors, but it emphasized the use of common technical terms and symbols for ease of referencing. Special effort was made to create organic links among the covered topics: detonation phenomena in application of explosives, fundamental theory of detonation wave, methods of measurements, and individual case studies. The book also offers a historical perspective on detonation research in Japan, pedagogical materials for young researchers in detonation physics, and an introduction to EOS works in Japan, which are worthy of attention but very little known internationally. The pedagogical materials are intended to be a primer on the detonation of condensed explosives and

v

vi

Preface

to help and encourage young readers to start their own research on explosives and applications. Condensed phase is either solid or liquid, and commercial explosives are generally in the condensed phase. Dynamite, a solid explosive, was invented in 1866, and the blasting gelatin that allowed dynamite to be used more safely was developed in 1875. Detonation study of explosives, however, was started with gas-phase explosives around 1880. Additionally, phenomena that cannot be understood by the classical model of detonation such as the three-dimensional cell structure of detonation were also discovered in the early stage of gaseous detonation study. This discovery on gas phase is one of the clear differences in the research history between condensed-phase explosives and gas-phase explosives. In addition, the difference in applications is attributed to a huge difference in pressure between the gas- and the condensed-phase detonations. This book mentions gas-phase detonation in the historical context, but the book is primarily concerned with the condensed-phase detonation and explosives. The obvious reason is our own interest in and work on the condensed-phase explosives. Chapter 1 introduces basic phenomena related to detonation and other basic matters regarding condensed-phase explosives. Since condensed-phase explosives are an unreacted material at the beginning and detonation is a special kind of shock wave propagation involving chemical reactions, detonation study requires a good deal of understanding shock wave propagation in the condensed-phase medium with and without chemical reaction. Chapter 2 is devoted to providing fundamentals of shock wave physics of solids as well as detonation wave in condensed explosives. Almost all the key information on the basic materials is covered including shock propagation characteristics, equations of state (EOS) at high pressures and temperatures, Chapman–Jouguet (C-J) hypothesis, and Zeldovich-von Neumann-Doering (ZND) model. In describing the detonation theories, however, even if they were a conceptual diagram, they were drafted using the EOS of condensed explosives. The theory described in Chap. 2 is also the basis for the EOSs described in Chap. 3. Particular utility of the latter is found in detonation simulation. That is, if the EOS of detonation products and the EOS of unreacted solid explosive were known, the detonation phenomenon could be simulated using computational fluid dynamics (CFD) by combining the conservation equations of flow and a reaction rate law. The basics of such a numerical analysis are described at the end of Chap. 2. Chapter 3 describes EOSs for condensed-phase explosives. Two types of EOSs are necessary to express the detonation phenomena involving detonation products and unreacted condensed phase. Since the detonation products reach a high pressure of tens of thousands of atmospheres, ideal-gas EOS cannot be applied to accurately represent the detonation state. The study of the EOS of detonation products using the real gas equation of state began around the 1950s. The EOS developed in Japan and its application are discussed in detail in Chap. 3. For the unreacted condensed explosives, the static compression data base EOS is explained with many experimental data. In Chap. 4, experimental diagnostics are presented. They include pressure and particle velocity gauges, laser velocity interferometer, and optical pyrometer. In

Preface

vii

addition, the chapter contains experimental measurements of detonation properties in Japan, including temperature measurement. Two types of measurements are of special interest. The first is the measurement of detonation temperature, which is very scarce in the open literature. Especially, Sect. 4.2 discusses detonation temperatures measured for various liquid and solid explosives and their comparison with theoretical values calculated using various types of EOSs. The second is the underwater explosion test aimed at accessing the energy content and performance of explosives in a longer time scale than that of detonation. Also included are a discussion of characteristics of underwater explosion phenomena and measurements of underwater shock wave and bubble pulse. Chapter 5 describes an overview of experiments on and numerical simulations of shock initiation and related studies in Japan. These studies are focused on understanding the initiation process from external stimuli and its transition to detonation. These stimuli are typically exposure to high-temperature environment over an extended period of time, collision with a heavy object at a relatively low speed, and shock wave loading resulting from high-speed impact. Included in Chap. 5 is a new observation on “the mixture rule that was obtained through the examination of thermodynamic states in simulation during the shock-todetonation transition. The observation explained why the numerical shock initiation modeling was insensitive to the four assumptions typically used to construct a mixture EOS. This chapter also describes two additional new results. The first is the model of shock initiation of PETN, which can be applied to arbitrary loading density without changing other model parameters. The second is the finding, in the section on “Gap test,” of a new scaling law for sympathetic detonation. It was considered the highlight of the study. The last section of the chapter presents a study on the laser initiation of explosive, which was considered a pioneering work at that time. Chapter 6 is focused on non-ideal detonation and discusses materials that are not often dealt with in standard textbooks. They are the definition/meaning of non-ideal detonation and an overview of studies in the world as well as studies in Japan. Of particular interest in this regard are detonation phenomena that cannot be explained by the C-J hypothesis and have been a subject of intense interest since the 1940s. A typical example is the diameter effect of detonation propagation in a cylindrical charge. That is, for all explosives, detonation velocity is lower than the C-J detonation velocity depending on the charge diameter. Plus, as the diameter is decreased, there is a critical diameter below which the detonation will not propagate. Additional topics covered in this chapter are the studies related to highly nonideal explosive that are based on ammonium nitrate (AN), and a discussion of recent research trends. Also included are the studies of non-ideal detonation behaviors of pure AN, mixture of AN and activated carbon, AN/fuel oil (ANFO), and emulsion explosives (EMX). The effects of Al particle reaction on the non-ideal detonation properties of mixtures consisted of NM and Al particles are also discussed.

viii

Preface

Last but not least, Chap. 7 describes future perspectives on detonation research in Japan. Tsukuba, Japan

Shiro Kubota

Contents

1 Shock and Detonation Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiro Kubota

1

2 Theory of Shock Wave and Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . Kunihito Nagayama, Shiro Kubota, and Masatake Yoshida

7

3 Description of Detonation Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kunihito Nagayama, Shiro Kubota, and Masatake Yoshida

59

4 Measurements of Shock and Detonation Phenomena . . . . . . . . . . . . . . . 103 Yukio Kato and Kenji Murata 5 Shock Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Shiro Kubota and Kunihito Nagayama 6 Ideal and Non-ideal Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Atsumi Miyake, Shiro Kubota, Yukio Kato, and Kenji Murata 7 Future Perspectives on Detonation Research . . . . . . . . . . . . . . . . . . . . . . 287 Atsumi Miyake

ix

Contributors

Yukio Kato Fukushima, Japan Shiro Kubota National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan Atsumi Miyake Yokohama National University, Yokohama, Japan Kenji Murata Akita University, Akita, Japan Kunihito Nagayama Kyushu University, Fukuoka, Japan Masatake Yoshida Explosion Research Institute Inc., Tokyo, Japan

xi

Chapter 1

Shock and Detonation Phenomena Shiro Kubota

Abstract In this chapter, we focus on the detonation of explosives in the condensed phase, and the basic phenomena related to detonation and general matters regarding condensed phase explosives covered in this book are described. Keywords Detonation · Sound wave · Shock wave · Condensed phase · High explosives

1.1 Sound, Shock, and Detonation Waves Pressure wave is a type of wave that appears in a medium, and sound wave is one of the pressure waves. Conversation involves a voice generation mechanism on the transmitting side and a complicated mechanism in the ears on the receiving side, and subtle change of air vibration is recognized as conversation information by the ears. During a conversation, we disregard the speed of sound, however in everyday life, people can hear sounds and make various decisions based on the sounds heard. For example, when a batter hits a ball in an unexpected direction in baseball training, a person who is picking up the ball nearby can avoid being hit by the ball by looking forward the direction of the sound if the person’s reflexes are good. When lightning strikes, some people will judge the danger by the time difference between light and sound. Shock wave is a pressure wave that propagates beyond the speed of sound of the medium through which the wave propagates. Shock waves are generated when the stored energy is momentarily released. The explosion of an explosive, supersonic flying object, rupture of a pressure vessel, and high-speed collision generate shock waves that propagate in the air and condensed media existing in the surrounding. For example, when an accidental explosion occurs, a blast wave is generated and propagates far from the explosion source. Generally, blast is a shock wave that travels

S. Kubota (B) National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8569, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Kubota (ed.), Detonation Phenomena of Condensed Explosives, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-5307-1_1

1

2 Table 1.1 Density and sound velocity of some materials

S. Kubota Material

Density kg/m3

m/s

km/h

Air

1.293

331

1192

Water

1000

1480

5328

Fe

7850

3574

12,866

Trinitrotoluene (TNT)

1624

1934

6962

Sound velocity

through the air. The blast wave exceeds the speed of sound in the air. The sound reaches you after the blast has already hit you and you lose consciousness. The shock pressure in the condensed phase is significantly higher than that in the gas phase. This is due to the difference in acoustic impedance, which is the product of density and speed of sound, between the condensed and gas phases. For water, density is about 775 times larger than that of air, and it is about 1500 m/s. When an explosive is detonated in water, the pressure of the underwater shock wave near the explosion site becomes higher than 1 GPa. Table 1.1 summarizes the density and sound velocity of some materials. Detonation is a form of intense combustion. When a combustion wave through a substance propagates at a velocity beyond that of sound, it is called detonation wave. The detonation wave forms a shock wave at the front, and it is self-supported by the detonation heat, so the detonation wave can be expressed as a combustion wave accompanied by a shock wave. The steady-state detonation is retained as long as the explosive exists. The detonation pressure in the condensed phase is significantly higher than that in the gas phase. This is due to the difference in shock impedance between the condensed and gas phases. In the case of the condensed phase, detonation pressure is in the order of GPa, which greatly exceeds the strength of various materials, whereas in the case of the gas phase detonation, the pressure is in the order of MPa. Another form of intense combustion is deflagration. Deflagration wave propagates at a speed sufficiently lower than that of sound velocity of a substance, such as several tens of meters per second or lower. There is probably no stable deflagration wave propagating at a speed close to the sound velocity. Combustion waves propagating at speed near that of sound velocity are unstable and either transition to detonation or decay to low velocities.

1.2 Types and Properties of Explosives and Target Phenomena There are various types of explosives, and they have a wide range of applications. Generally, explosives are classified as high and low explosives (USA) (propellants (UK)). The former is used mainly for destruction applications using detonation, and

1 Shock and Detonation Phenomena

3

the latter is for propulsion applications using combustion or deflagration. Typical explosives used for propulsion include black powder, smokeless powder, composite propellant, and so on. There is also a category called pyrotechnics, which includes detonators and blasting fuses. In pyrotechnics, both high and low explosives are used. The explosives discussed in this book are mainly high explosives. The materials discussed in this book include ammonium nitrate, which is added to explosives as an oxidant, and explosives with metal powder. Explosives have a wide range of applications in industry, for example, dismantling structures such as dilapidated buildings and bridges with truss structures, as well as for metal processing, material synthesis, and space development. Various types, shapes, and amounts of explosives have been used. The strength of detonation differs greatly depending on the type of explosive. Both the shape and the amount of explosives affect their characteristics. For mining use, such as open pit mining and underground digging, different explosives are used depending on the rock type. The state of detonation also depends on the restraint conditions (confinement). It is necessary to understand the detonation characteristics of explosives and apply them according to their application. The detonation of explosives is caused by a strong external impact. Some explosives explode with a weak impact, but they are usually dangerous and cannot be used for general applications. The process of externally stimulating an explosive to make it detonate is called initiation. The sensitivity of an explosive is the degree to which initiation can be induced, and various sensitivities such as impact sensitivity and friction sensitivity are tested by various methods. The explosives are usually initiated using an electric detonator. Figure 1.1 shows a special detonator for research purposes called a precision detonator. In this detonator, the compressed pentaerythritol tetranitrate (PETN) powder is directly initiated with an explosion of metal wire with precision of 100 ns or less. The shape of precision detonator is the same as that of general electric detonator used for industrial purposes such as blasting. The detonator consists of a metal tube, which contains less than 1 g of high explosive that is relatively sensitive. Therefore, the detonator is not set in the explosive until just before the explosive is detonated. The sensitivity varies depending on the type of explosive, and some explosives cannot be detonated by the detonator alone. Those explosives are detonated by a booster explosive with the detonator. Fig. 1. 1 Precise electric detonator

4

S. Kubota

A word “initiation” which has been used in detonation physics, means the detonation transition process. In the real world, the detonation cannot start without transition time after receiving an external impact. The initiation mechanism is extraordinary important for understanding detonation phenomena. This phenomenon is very complicated non-steady and non-equilibrium phenomenon. The velocity at which a steady detonation wave is propagated is called the detonation velocity, and those of the explosives in the condensed phase range from about 2000 m/s to over 9000 m/s. Even when the explosives have the same composition, when the loading density increases, the detonation velocity increases linearly for many types of explosives. There are also many explosives that have the maximum detonation velocity at a certain loading density. The intensity of detonation can be easily estimated from the product of loading density and the square of detonation velocity, and the higher the density and detonation velocity, the higher the intensity. In addition, considering a classical detonation model and the progress of research on the equation of state for detonation products, the ideal detonation velocity can be predicted, and the predictions close to experimental values can be obtained. Furthermore, when the diameter of the cylindrical sticks of explosives is large, a constant detonation velocity is obtained regardless of the diameter of the explosive. This is called the ideal detonation velocity or the detonation velocity at infinite charge diameter. On the other hand, if the charge diameter is smaller than a certain charge diameter, the detonation velocity will be lower and depend on the charge diameter. This is called the diameter effect. If the charge diameter is made smaller, there is a diameter at which the detonation wave cannot propagate. This charge diameter is called the critical diameter or failure diameter. In general, less powerful industrial explosives such as ammonium-nitrate fuel–oil (ANFO) have a larger critical diameter. In general, the high explosives for military use have high detonation velocity and high detonation pressure and have a wide range of ideal detonation. ANFO explosives, which are widely used in mine blasting, exhibit non-ideal detonation in a wide range of applications. ANFO is a typical non-ideal explosive that not only does not reach the ideal detonation velocity under general conditions of use, but also is greatly affected by confinement. Ammonium nitrate (AN) is used as a fertilizer, but historically, its explosiveness was recognized after the major explosion accidents. When an AN-based emulsion explosive was invented, it replaced dynamite as a result of improved performance through research and development.

1.3 Overview of the History of Detonation Research Around 1880, researchers measuring the burning rate of flammable gases in France discovered a fast burning phenomenon that had never been measured before. This phenomenon was named detonation. The study of detonation was started with the discovery of gas phase detonation. Dynamite, a typical explosive, was invented by Alfred Nobel in 1866. Prior to that, in 1864, Nobel developed an industrial detonator,

1 Shock and Detonation Phenomena

5

so the use of detonation in the condensed phase explosives began before the recognition of the detonation phenomenon. When this phenomenon was discovered, the theory of shock waves had been proposed by Rankine and Hugoniot. The theory was developed for the gas phase. The bold idea of the detonation model and its progress have been completed in a short time because it started from gas phase detonation. Detonation waves have a wave front with discontinuous state changes. Before and after a wave front, the conservation law (jump condition) holds. This model uses the conservation law on shock waves. Flow is one-dimensional, and since the chemical reaction completes in an instant, the reaction process is ignored, and it is assumed that a thermodynamic equilibrium is achieved behind the detonation front. Chapman and Jouguet published their papers on the detonation theory in 1899 and 1906, respectively. Later, in the ZND model that appeared in the 1940s, ideas were developed that included a reaction zone. The C-J hypothesis and the ZND model have several assumptions. Many of the later theories arose by removing those assumptions and expressing the detonation phenomenon. The assumption of the plane wave at the detonation front in the ZND model was first removed by Jones and Eyring et al. in 1940, and Nozzle theory and curved front theory were proposed. In 1951, the most important conference in detonation research, “Conference on The Chemistry & Physics of Detonation”, was held in Washington, D.C. Until the third conference held in 1960, it was a domestic conference in the USA. The Four symposium was held in 1965 as an international conference, “Fourth symposium (international) on detonation”. This symposium is held every four or five years, and recently, the 16th International Detonation Symposium was held in Cambridge, MD, in 2018. People involved in detonation research continue to participate in this symposium. Since this symposium constantly gathers information on advanced detonation research from around the world, the history of this symposium is equivalent to the history of detonation research. The technology of measurement, theory (model), and numerical simulation of the detonation phenomenon have advanced with time, and when an unknown phenomenon was measured, the result was explained by theory and simulation. Of course, there are still many phenomena that are being studied. Detonation research is so extensive that in this section, we only give an overview above. An overview of related detonation research is given in each chapter.

Chapter 2

Theory of Shock Wave and Detonation Kunihito Nagayama, Shiro Kubota, and Masatake Yoshida

Abstract This chapter is devoted to provide fundamentals of shock wave physics for solids as well as detonation waves for condensed explosives. Almost all the key information on the above topics are covered including the shock propagation characteristics, the equation of state at high pressures and temperatures, the C-J hypothesis, ZND model and further application to numerical simulation, etc. Keywords Jump conditions · Rankine–Hugoniot · C-J hypothesis · ZND model · Envelope approximation · Numerical simulation

2.1 Shock Wave Physics in Condensed Media 2.1.1 Introduction The term, ‘solid’ is sometimes used to represent the concept of idealized substances whose dynamical properties is just the opposite side of that of ‘fluid’. Unlike this concept of solids, all the solid materials have a very large but finite rigidity. In other words, all the solid materials are compressible, although their bulk modulus ranges up to more than tens of GPa. Furthermore, as explained below, solid materials can flow under some extreme conditions, and the dynamics should be described by fluid equations. Very large but finite value of the bulk modulus of solid materials indicates that even solid materials can be compressed appreciably, when an instantaneous stress K. Nagayama (B) Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan e-mail: [email protected] S. Kubota National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan e-mail: [email protected] M. Yoshida Explosion Research Institute Inc., Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Kubota (ed.), Detonation Phenomena of Condensed Explosives, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-5307-1_2

7

8

K. Nagayama et al.

or pressure is exerted to solid materials in an extremely fast rate. In exactly the same sense as gases, compressible shock waves can be induced in any solid materials. The most impressive difference of the shock waves in solids from those of gases is the very high pressure necessary to produce shock waves in these substances. Shock waves in any substances are defined as a wave of abrupt jump in both pressure and velocity propagating at the velocity faster than the speed of sound of a medium ahead of the wave front [1, 2]. Concept of high compression of solids by high-pressure shock waves was introduced and established in 1940s. The most comprehensive study of the theory of shock waves in media with arbitrary equation of state was published by Bethe in 1942 [3]. Since solid materials have quite a different equation of state from that of gases, introduction of shock waves in solids adds another wide area of shock wave science and technology [4]. More than thousand solid materials have been studied so far to know the shock wave properties of them. Most part of the shock Hugoniot data have been piled up to a database, which can be accessed or purchased in a form of a book or a Web site [5–7]. As explained, shock waves in solids come always with very high-pressure states of solids. Therefore, shock wave science had a close relationship with high-pressure physics, material synthesis by shock wave loading [8], shock compaction [9], explosive welding of metal plates [10], space debris [11], elastic–plastic behavior of materials [12], high-pressure structural phase transitions [13], pulse laser-induced waves [14], medical applications [15, 16], extremely high-pressure generation [17, 18] and more. Importance of the shock propagation characteristics arises in the fact that shock discontinuity is embedded in the detonation wave structure which will be explained later in this chapter. Shock characteristics in solids is also important in that detonation of high explosives induces the generation of shock wave in solid material adjacent to the explosive. Later in this section, we will see that the differences in the shock wave characteristics in condensed substances stems from the difference in the compression properties of solids. Application of very strong and instantaneous force or very high pressure or stress on solid surface should be required to realize appreciable degree of compression. As a result, very high dynamic pressure and also instantaneous one are required to induce a compression wave in solids. This property is also important to the system of producing shock waves in solids experimentally. Instrumentation of shock generation will be given in Chap. 4.

2.1.2 Shock Propagation Characteristics 2.1.2.1

Shock Jump Conditions

Figure 2.1 shows the schematic illustration to derive the jump conditions representing the conservation equation of mass across shock jump at the wave front. In Fig. 2.1, shock wave is induced by the u p . Piston starts to move at the time t = 0 with velocity u p at x = 0, arrives at the position, x = u p Δt with the time interval Δt. Let us

2 Theory of Shock Wave and Detonation

9

Fig. 2.1 Schematic illustration of shock jump conditions

consider the mass over the interval x = 0 to x = u s Δt with the unit area at t = 0 perpendicular to the piston motion. The mass of this volume should be equal to the mass of the interval x = u p Δt to x = u s Δt with the unit area at t = Δt. The density of the former region at t = 0 is assumed as ρ0 and at t = Δt is now assumed as ρ. Therefore, [ ] (2.1) ρ0 u s Δt = ρ u s Δt − u p Δt , which will finally be the mass conservation expressed as ] [ ρ0 u s = ρ u s − u p .

(2.2)

We will now switch to the effect of momentum flux. During the time interval, t = 0 to t = Δt, the net force p H − p0 per unit area at the shock front during this time interval produces the impulse, [ p H − p0 ]Δt, which is equal to the momentum increase of the mass of the interval at t = 0 from zero velocity to the velocity u p . In this consideration, we have the momentum conservation [ p H − p0 ] Δt = [ρ0 u s Δt] u p ,

(2.3)

p H = p0 + ρ0 u s u p .

(2.4)

which leads to

Similarly, one may consider the energy flow conservation. Piston moves with u p at the net force p H on a unit area which gives the mechanical work, p H u p Δt during the time interval Δt. This work causes the increase in the internal energy and also in the kinetic energy for the mass of the same above interval. This equality is given by [

which leads to

] 1 2 p H u p Δt = ρ0 u s Δt ε H − ε0 + u p , 2

(2.5)

[ ] 1 ρ0 u s ε H − ε0 + u 2p = p H u p , 2

(2.6)

10

K. Nagayama et al.

where u s , u p , ρ, p H and ε denote the shock velocity, the particle velocity, the density, the shock pressure and the specific internal energy, respectively. Suffixes 0 and H denote the value at the initial state and the shock-compressed state, respectively. Equations 2.2, 2.4 and 2.6 expressing the conservation of mass, momentum and energy, respectively, are often called the jump conditions. These conservation equations can also be derived from the general conservation laws in continuum mechanics. Shock-compressed state in solids is assumed here to be in thermally and dynamically equilibrium state. Strictly speaking, we have to adopt the x x-component of stress σx x in the momentum and energy equation, Eqs. 2.4 and 2.6 instead of shock pressure p H . Stress tensor is not equivalent to hydrodynamic pressure but contains some deviatoric stress component due to the finite value of the yield strength even at that high-pressure state. At least, for very high-pressure shock waves or waves in materials with small elastic strength such as organic high explosives, one may safely replace σx x by p H . You can notice from the above discussion that the stress ratio or the pressure ratio, p H / p0 , may not be a good parameter to describe the shock wave propagation in solid materials. This situation is in sharp contrast to that of ideal gases. In fact, the pressure ratio is entirely useless in case of condensed materials. The change in the initial conditions between the atmospheric conditions and the evacuated conditions (vacuum) is physically almost nothing. Generalization of the shock wave theory of gases in terms of pressure ratio fails in this sense. We need another common concept of non-dimensional parameter to represent shock strength in universal terms, although the discussion has not been still completed yet [19].

2.1.2.2

Rankine–Hugoniot Relation and the Shock Hugoniot Curve

By combining Eqs. 2.2, 2.4 and 2.6 with p0 = 0 and eliminating velocity variables, u s and u p , we have another important relationship between the thermodynamic variables, pH (2.7) ε H − ε0 = (v0 − v H ) , 2 where v = 1/ρ denotes the specific volume. This equation is called the Rankine– Hugoniot relationship. One may note that this equation contains only the thermodynamic state variables. The number of variables contained in the shock jump conditions, Eqs. 2.2, 2.4 and 2.6, is five, i.e., the density ρ, the shock velocity u s , the particle velocity u p , the shock pressure p H and the specific internal energy ε H with the known initial variables, the initial density ρ0 , the initial pressure p0 and the initial internal energy ε0 . Among five independent variables, we have three jump conditions. We need two more information to specify all the values of the above variables. We have, however, already additional relationship for the material called as the equation of state in the form F( p, v, ε) = 0. (2.8)

2 Theory of Shock Wave and Detonation

11

Thermodynamically, this equation of state in terms of these three variables does not contain all the thermodynamic information of the material. But the combination of variables is adequate for describing material flows including shock or sound propagation. Since the sound propagation is an adiabatic process, the equation of state can describe the isentropic definition of the sound velocity c derived by the differential equation for the sound wave propagation of infinitesimal amplitude. ( c ≡ 2

∂p ∂ρ

)

( = pv S

2

∂p ∂ε

) v

( +v

2

∂p ∂v

) ε

,

(2.9)

where all the right-hand side terms of Eq. 2.9 can be evaluated by Eq. 2.8. By combining this equation of state, Eq. 2.8 together with the Rankine–Hugoniot relationship, Eq. 2.7, we can derive the relationship between two state variables, for example the shock pressure p H as a function of the specific volume v, etc. This relationship is called the shock Hugoniot compression curve for the material. The function should be unique for the material, but is different from the other thermodynamic compression curve such as an isotherm or an isentrope. The difference is that successive states on the Hugoniot curve are not the one followed by a thermodynamic process, but each state of the Hugoniot curve is the collection of states accessed by a discontinuous shock jump from the initial state. As stated in the Introduction of this chapter, one of the major objectives of the solid shock experiments has been to explore the high-pressure equation of state for the material, since the attained high-pressure state by shock compression, i.e., Hugoniot state, is well defined by Eqs. 2.2, 2.4 and 2.6. By measuring some of the shock wave parameters appearing in the jump conditions, it is possible to specify the attained high-pressure states in terms of parameters appearing in the shock jump conditions. The relationship between two independent shock wave parameters is unique for the material measured and is called the shock Hugoniot curve. One of these Hugoniot curves measured is the shock velocity–particle velocity relationship and is explained in the next section. The combination of variables for the equation of state, Eq. 2.8 is not coincidental, but has clear physical meaning. They are all the mechanical variables, although the internal energy has a thermal aspect as well. In this sense, no explicit information can be obtained on the thermal variables like temperature or entropy by the measurement of the characteristics of shock wave propagation. This is the consequence of the fact that shock wave is a wave which is driven by the break in the mechanical variables, pressure and velocity. Even so, thermal aspects of materials have enough large influence on the propagation behavior implicitly. It is of fundamental importance to discuss thermal contributions to the equation of state for solids at very high pressures and temperatures reached by strong shock compression. Figure 2.2 shows the schematic illustration of shock Hugoniot compression curve p H − v. Increase in the specific internal energy by shock compression, ε H − ε0 is the area of triangle ABC in Fig. 2.2, which is derived by the Rankine–Hugoniot relationship, Eq. 2.7. Energy conservation equation shows that the mechanical work by shock pressure, p H per unit time and unit mass will be

12

K. Nagayama et al.

Fig. 2.2 Pressure–volume Hugoniot

p H u p Δt = p H (v0 − v) ρ0 u s Δt

(2.10)

This equation and the Rankine–Hugoniot relationship show that the mechanical work by piston drives the increase in the specific internal energy and also the increase in the kinetic energy per unit mass u 2p /2, both by the same amount. In Fig. 2.2, the straight line connecting the initial state to the shock-compressed state is called the Rayleigh line. The slope of the line is negative, and the magnitude of the slope is given by pH (2.11) = ρ02 u 2s , v0 − v which shows that the line slope is determined by the propagation velocity of the shock wave and the initial density. 2.1.2.3

Applicability of Jump Conditions and Shock Wave Front Structure

In the derivation process of shock jump condition using Fig. 2.1, we have used only the conservation laws of physical quantities for two equilibrium states at both sides of shock front. One may be worrying about the applicability of the derivation process to any other flow situation. Discontinuous jump in physical variables with infinite spatial derivative cannot happen in reality. Discontinuities should have finite rise time or distance. In this section, we will discuss how the finite rise of the shock front is realized and what is the limitation of the applicability of shock jump conditions. The discussion in this section will be useful to understand the detonation wave structure in the next section.

2 Theory of Shock Wave and Detonation

13

Shock rise time in metals has been measured [20], and it was also discussed theoretically [21]. Actually, the finite but very steep shock rise is understood by the balance of existing energy dissipation mechanisms and nonlinearity of mechanical properties of the material. For example, sound velocity of thermodynamic state is a function of pressure and temperature and increases with increasing pressure. Since sound velocity is determined by the elastic modulus of the material, the elastic modulus itself is a function of the thermodynamic state variables. Due to this mechanical nonlinearity, high-pressure region of some pressure distribution, for example, propagates faster than the lower pressure region; thereby, the spatial derivative of pressure steepens with propagation, while the steeper pressure or other variables makes the viscous or other kinds of energy dissipation increase, thereby makes the pressure rise smoothing. Due to these competing processes of steepening by nonlinearity and of smoothing by energy dissipation, almost discontinuous but finite rise time is realized to form a shock wave front which could be described by the jump conditions. It is important to know that the jump conditions can be applied to the unsteady shock wave as well as to the steady shock wave. And what is the limiting condition of the applicability? Here, we will give an intuitive but probably sufficient criterion of the applicability. Let the rise time of shock front be Δτ . If the shock wave front structure of the physical variables does change infinitesimally by the time interval, Δτ , the shock front propagation can be regarded as steady in the time scale of Δτ . In this condition, one may safely apply the shock jump conditions for this shock profile. One may use the jump conditions for the unsteady shock wave at each time when the change in spatial distribution of physical variables is much slower in the time scale of Δτ . In conclusion, one may compare the time scale of steep shock front and that of ambient slowly varying profile region. If the shock wave front profile is complicated, the jump condition can be applied to the two adjacent states. In some cases, such as that of the detonation wave structure, one may encounter that the jump conditions will be met at arbitrary choice of compressed states and the initial state. These circumstances are realized, since the jump conditions are basically derived by integrating the differential equations of continuum mechanics.

2.1.2.4

Empirical Linear Relation

For many solid substances and over wide ranges of pressures, the following empirical relationship is known [4, 18]. u s = A + Bu p , (2.12) where A and B are constant material parameters. As will be explained in a later section, shock waves in solids will be divided into weak waves, or elastic waves whose shock pressure is lower than the yield strength of the material, or dual waves with

14

K. Nagayama et al.

elastic precursor wave and plastic waves having intermediate pressure, and finally plastic waves of higher pressure. Equation 2.12 is valid for plastic part of shock waves. The parameter A is expected to play a role of sound velocity in a plastic wave region. Experimental values of A, however, does not coincide with those of longitudinal sound velocity. Rather A is related to the hydrostatic compressibility through the relation, K + 43 G K 4G 4 A2 = cb2 = − = = cl2 − ct2 (2.13) ρ ρ 3ρ 3 where K , G, cl and ct denote the adiabatic bulk modulus, the shear modulus, the longitudinal sound velocity and the transverse sound velocity, respectively. This A2 is a square of the velocity of virtual longitudinal sound wave due to the hydrostatic compression. We call this the bulk sound velocity, cb , whose value can be estimated by the above formula or from the data of hydrostatic compression of the material. Parameter A is found to be in good agreement with the bulk sound velocity in many materials. Parameter B represents the nonlinearity of the dynamical properties of materials. The empirical formula, Eq. 2.12, is then considered to be a kind of simple polynomial expansion approximation in terms of particle velocity under the weak shock approximation. Empirical formula is known to hold well over the convergence limit of the weak shock approximation. Various attempts have been published to understand this universal behavior [19]. No sufficient physical explanation of the empirical formula, however, is given. By combining Eq. 2.12 with the shock jump conditions, Eqs. 2.2, 2.4 and 2.6, shock Hugoniot compression curve can be given by pH =

ρ0 A2 η (1 − Bη)2

A2 ε H = ε0 + 2

[

η (1 − Bη)

η =1−

v v0

(2.14) ]2 (2.15)

(2.16)

In this expression, it is assumed that the shock-compressed state is in hydrostatic compression. Functional form of Eq. 2.15 is depicted in Fig. 2.3 for several metals. And several of the parameters A, B, etc., are given in Table 2.1. From Fig. 2.3, one may see that region left to the initial state, v/v0 = 1 corresponds to the shock compression curve, whereas the region right to the initial state is virtual expansion by the negative particle velocity. In the case of particle velocity equal to −A/B, solid medium could be expanded to gaseous state [22]. Apparently, the expansion region of Eq. 2.15 cannot be realized, since the expanded state calculated by Eq. 2.15 has the value of entropy smaller than that of the initial state. Even so, at least in the vicinity of the initial state, this function

2 Theory of Shock Wave and Detonation

15

Fig. 2.3 Hugoniot functions of internal energy both compression and expansion Table 2.1 Rodean’s approximation to the cohesive energy Material name ρ0 (g/cm3 ) Ta W Pt Cu Au Al Pb

16.656 19.235 21.470 8.93 19.280 2.785 11.346

A (km/s) 3.414 4.029 3.598 3.94 3.080 5.328 2.03

B 1.200 1.237 1.544 1.489 1.560 1.338 1.52

( A )2

exp

(km/s)2

−E ξ (km/s)2

4.05 5.30 2.72 3.50 1.95 7.93 0.892

4.320 4.546 2.892 5.324 1.861 11.93 0.9494

1 2

B

has second-order contact on an isentrope both compression isentrope and expansion isentrope. Functions depicted in Fig. 2.3 are considered to be the extended form of the isentrope. Asymptotic value of specific internal energy at very large volume E ξ is calculated by using Eq. 2.15 and is given by 1 − Eξ = 2

( )2 A , B

(2.17)

which is compared with the cohesive energy of each material in Fig. 2.3. One may see that sometimes, Eq. 2.17 gives qualitative measure of the cohesive energy of the material [22]. Table 2.1 shows the comparison of the value calculated by Eq. 2.15 with ex p the experimental data of cohesive energy, −E ξ . Since the information containing the shock compression seems to be the one by atomic repulsive forces, the present

16

K. Nagayama et al.

Fig. 2.4 Periodicity of material parameters, A and B

result is surprising in that the cohesive energy or the energy of sublimation stems principally from the attractive energy or force component of the substituent atoms. Estimated particle velocity of escaping from bulk material calculated by Eq. 2.12 is −A/B. Figure 2.3 and Table 2.1 contain examples only for metals (or elements). Discussions on the chemical compounds like alkali metal halides and for organic high explosive materials will be given in Chap. 3 in the discussions on unreacted EOS for high explosives. Later, one may see Eq. 2.17 is an approximation not for the cohesive energy but for the sublimation energy of the substance. Figure 2.4 shows the periodicity of the parameters A and B for elements. Parameter A, bulk sound velocity shows clear periodicity, while parameter B has the value in a very narrow range. Figure 2.5 shows pressure–density Hugoniot compression curve for selected materials. All of these curves are calculated based on the measured value of A and B together with the initial density of materials. All Hugoniot curves are plotted up to the pressure value, where empirical linear relation cannot apply due to the lack of data or to phase transitions. As is seen from the figure, for the same value of density ratio, higher shock pressure is generated for heavy materials or materials of high shock impedance. Concept of shock impedance is defined in the next section.

2.1.2.5

Reflection and Transmission of Shock Waves at the Material Interface

In contrast to the case of gases, solid materials have a sharp material boundary, and it is a very easy task to put several materials together with a well-defined interface. A shock wave induced in one of the material can propagate into another adjacent material through the interface. This phenomena is used in several of the experimental methods to be explained in Chap. 4. In this section, several concepts concerning the wave reflection and transmission are explained. Schematic of reflection and transmission of shock waves at the material interface is shown in Fig. 2.6. A steady shock wave in material I will propagate into material II through the interface as shown in the figure. In this figure, material II is

2 Theory of Shock Wave and Detonation

17

Fig. 2.5 Pressure–density compression curves

Fig. 2.6 Shock reflection and transmission at material interface

assumed to be a harder, i.e., more incompressible material than material I. In this case, the reflected wave should be a shock wave, whose shock pressure is higher than the incident shock pressure as shown in Fig. 2.6. A transmitted wave is apparently a shock wave, and the pressure and velocity at the material interface are determined such that these two values are equal in both materials. Therefore, wave reflection and transmission is conveniently analyzed by the shock Hugoniot relationship between shock pressure and particle velocity. This kind of

18

K. Nagayama et al.

Fig. 2.7 Pressure–particle velocity Hugoniots

problems can be understood almost by that of sound waves. One of the key parameters of sound wave in a medium is the acoustic impedance, which is defined as the product of density and sound velocity. At the interface of two materials whose acoustic impedance has the same value with each other, sound wave transmits with no reflected waves. This is called impedance matching. As the extended concept of acoustic impedance, shock impedance z is defined as z = ρ0 u s

(2.18)

As in the case of sound waves, shock wave transmits into the other materials without reflection, if the shock impedance of materials is equal. Except for the instant of shock arrival just at the interface, pressure or the x x-component of the stress tensor and particle velocity normal to the interface should have the same value. Change of state by the wave interaction at the material interfaces can be considered by drawing the possible wave reflections to be occurred at the interface. This work should be made in the plane of the particle velocity versus pressure (or stress). Figure 2.7 shows the pressure–particle velocity Hugoniot compression curve for the same materials as those plotted in Fig. 2.3. This plot is useful, if you will estimate the produced shock pressure by the high-speed impact of two materials, whose Hugoniot curve is shown in this plot. You may choose the impactor material, whose Hugoniot curve should be replotted starting at the point of the impact velocity u p = u 0 and p = 0 and plotting the curve to the velocity decreasing direction. Crossing point should be the realized compressed state. Such a plot is invaluable for the experimenter to design the experiment.

2 Theory of Shock Wave and Detonation

19

2.1.3 Thermodynamics of Shock Compression 2.1.3.1

Grüneisen Equation of State for Condensed Media

Besides shock wave parameters arising in the shock jump conditions, other thermodynamic variables are often required to know the values of them. They are temperature, entropy, thermal internal energy, specific heat, etc. The most important parameter in the equation of state for condensed media at high pressure and temperatures is the so-called Grüneisen parameter, which is defined by the following thermodynamic equation [23]. ( ) ∂p (2.19) γ =v ∂ε v Although various efforts have been made to measure the value of Grüneisen parameter at shocked state, precision of the data is not very good. Volume dependence of the eigen frequencies of lattice vibration is, by definition, an equivalent quantity as the Grüneisen parameter, and various microscopic theories and calculations have been published to evaluate the parameter at arbitrary high-pressure conditions. The parameter can be expressed by other measurable thermodynamic quantities as the bulk modulus, the specific heat, the volume expansion coefficient and is given by γ =

vκT β vκ S β = , Cv Cp

(2.20)

where Cv , κT , β denote the specific heat at constant volume, the isothermal bulk modulus and the volume expansion coefficient, respectively, while C p and κ S denote the specific heat at constant pressure and the adiabatic bulk modulus, respectively. Temperature dependence of the Grüneisen parameter was calculated at least at atmospheric pressure conditions. Based on the thermodynamic measurements, the Grüneisen parameter is almost independent on the temperature, γ = γ (v)

(2.21)

to a high temperature near the melting point of the material [24]. Based on this empirical law, together with the definition, Eq. 2.19, pressure– volume–energy relation can be written as p = pc (v) +

γ (v) [ε − εc (v)] , v

(2.22)

where pc (v) and εc (v) are called cold pressure and cold internal energy, which are the values of pressure and internal energy at zero temperature in Kelvin. In this equation, we have three material functions, pc (v), εc (v) and γ (v), which we have to know. Functions of cold pressure and energy are the isotherms at zero Kelvin, and they are also an isentrope at S = 0. Therefore, we have another relation,

20

K. Nagayama et al.

pc (v) = −

dεc (v) dv

(2.23)

By using this relation, we need to know two functions, εc (v) and γ (v). If the Grüneisen parameter is given by a function of volume, whole state surface can be determined by the measurement of only one of the thermodynamic paths, i.e., an isotherm, an isentrope or a Hugoniot. If we have a measured Hugoniot curve, we get p H (v) = pc (v) +

γ (v) [ε H (v) − εc (v)] , v

(2.24)

which is obtained by applying Eq. 2.22 to the Hugoniot state. In this formulation of the equation of state for solids at high pressures, the Grüneisen parameter is a key parameter to determine the entire functional relationship of Eq. 2.22. The Grüneisen parameter is defined also in a microscopic terms as d ln ν j , (2.25) γ =− d ln v where ν j denotes the lattice vibration frequency of the jth mode [18]. Although this depends on the mode itself, it is proved to be an identical definition as Eq. 2.19 under the assumption of Eq. 2.21. Equation 2.25 states that the value of the parameter changes according to the change in volume. Since the lattice vibration frequency should be given by the change in the form of lattice potential function, function γ (v) should be determined by the behavior of εc (v). There have been proposed several model theories for the behavior of γ (v) as a function of lattice potential function.

γ (v) = −

2−m v − 3 2

( ) 2m d 2 pc v 3 (

dv 2

d pc v

2m 3

)

,

(2.26)

dv

where m = 0, 1, 2, which corresponds to the Slater model [25], Dugdale–MacDonald model [26] and Vaschenko–Zubarev model [27] in this order. If the shock wave data is available, functional relations, p H (v), ε H (v) are known. Since relationship between cold components is given by Eq. 2.23, unknown functions contained in Eq. 2.24 are γ (v) and εc (v). By combining Eqs. 2.21 and 2.22 with one of the model theories for γ (v), it is possible to calculate the volume behavior of γ (v). From such calculations, γ (v) is found to decrease with compression. Since experimental determination of function γ (v) was found to be quite difficult, theoretical determination of γ (v) is still very important. For a given set of state variables for a high-pressure state, say ( p, v), internal energy at the state can be calculated by ε = ε H (v) +

p − p H (v) ργ

(2.27)

2 Theory of Shock Wave and Detonation

21

This calculation is extended to the states off the Hugoniot. Since the equation of state contains only p, v, ε, only these variables can be calculated by this method. We need further assumptions to determine other variables like temperature or entropy. In case of extremely high-pressure shock waves in solids, appreciable amount of electronic excitation will have an increasing contribution to the thermal energy. At extreme pressures, it will be the dominant part of the thermal energy. In this case, the so-called three term equation of state is assumed [1, 18]. p = pc (v) + pT (v, T ) + pe (v, T ),

(2.28)

ε = εc (v) + εT (v, T ) + εe (v, T ),

(2.29)

where pT , εT , the second term of Eqs. 2.28 and 2.29 are called thermal pressure and internal energy due to lattice vibration. While the third terms of Eqs. 2.28 and 2.29, pe and εe are the pressure and internal energy due to the electronic excitation. It is also assumed the electronic Grüneisen parameter γe as pe (v, T ) =

γe εe (v, T ) v

(2.30)

Finally, the shocked state at extreme pressures is described by the quantum statistical model of an atom.

2.1.3.2

Shock Temperature Calculation

Since temperature at the shocked state, the shock temperature is a very difficult quantity to measure, theoretical estimation of temperature at the shocked state is still a reliable method to obtain information on the thermal properties of materials under shock compression. Even so, we need to know several quantities except for the shock Hugoniot curve. One is the Grüneisen parameter at high pressure, γ (v), and the other is the specific heat at constant volume, Cv and its temperature and volume dependence. Temperature is calculated by integrating the thermodynamic relationship along the shock Hugoniot curve, since the Hugoniot curve is the most reliable thermodynamic path of high-pressure region known for most materials. For these theoretical calculation to have enough accuracy, the Grüneisen parameter and the specific heat should be known or measured under high pressure, but normally, such a data is quite limited. Rather, the Grüneisen parameter itself is found to be one of the most difficult quantity to measure at high pressures, as stated before. In this sense, high-pressure behavior of these quantities is assumed theoretically. For the specific heat at constant volume, the Debye model is commonly adopted. This model is successful to predict the temperature behavior of specific heat for various materials. This model cannot explain the measured lattice vibration frequency spectrum of materials, but the success of the model may suggest the specific heat value is quite

22

K. Nagayama et al.

insensitive to the detailed frequency mode spectrum. It is assumed that this situation preserves even at high shock pressures. Another assumption is the volume behavior of the Grüneisen parameter. Grüneisen parameter is known to decrease by compression. In many calculation of shock temperature, very simple assumption ργ = ρ0 γ0 = constant

(2.31)

is adopted instead of complicated model theories described by Eq. 2.26. In any case, one cannot derive an analytical expression for shock temperature even if Eq. 2.31 is assumed. Temperature is calculated by using the thermodynamic relationship dT dS dΘ d S = −ργ dv + + = , (2.32) T Cv Θ Cv where Θ denotes the Debye temperature. Nagayama introduced new thermal variables in the framework of the Grüneisen equation of state, which leads to a new simple algorithm of calculating shock temperature and Grüneisen parameter [21]. This calculation procedure of shock temperature contains only the contribution from the lattice vibration. In case of very high-pressure shock waves in solids, appreciable amount of electronic excitation will have an increasing contribution to the thermal energy. At extreme pressures, it will be the dominant part of the thermal energy. In this case, we have to use Eqs. 2.28 and 2.29 instead of Eq. 2.22 as an equation of state. 2.1.3.3

Elastic–Plastic Shock Wave and Constitutive Equation

Flat plate impact or explosive lens introduced in Chap. 4 could realize precisely controlled plane shock compression. In this case, produced shock wave compresses the material with the condition of ideal one-dimensional strain. This condition is exactly the same as that realized by the shock tube instrument for producing gas shock waves. In this sense, shock compression data has a very unique and reliable source of information on the mechanical behavior of materials under compression of one-dimensional strain. Shock compression, however, is special in the following condition; i.e., (i) strain rate is very fast, and (ii) stress–strain curve is complicated due to the effect of high-pressure compression. Above some limit of the value of the deviatoric stress, material yields and stress– strain relation curves downward. Point of deflection of the Hugoniot curve is called Hugoniot elastic limit. The value of stress at the point should be a material parameter to specify the elastic-to-plastic transition [1, 4]. Figure 2.8 shows the schematic illustration of the elastic–plastic Hugoniot compression curve. In this case, elastic wave and plastic wave propagate with different velocities. Apparently, the propagation velocity of elastic wave should be close to the longitudinal sound velocity, which is larger than plastic wave velocity whose peak pressure is less than some definite value. For the elastic shock wave propagating in the x-direction, stress and strain at shock front can be given by

2 Theory of Shock Wave and Detonation

23

Fig. 2.8 Schematic of elastic–plastic shock compression curve

σx x = (λ L + 2μ L )εx x σ yy = σzx = λ L εx x

(2.33)

σzz = σ yz = σzx = 0 where σi j denotes components of the stress tensor and εx x denotes the one-dimensional strain component, and λ L , μ L denotes the Lame’s constant, respectively. If the yielding occurs behind the elastic precursor wave, the shock yield stress Y is given by Y = 2μ L εx x

(2.34)

This result is obtained by assuming either the maximum shear stress or von-Mises criterion for yielding due to the condition of one-dimensional strain. Hydrostatic pressure at the wave front is defined as p=

( ) ) 1( 2 σx x + σ yy + σzz = λ L + μ L εx x 3 3

(2.35)

Departure between the Hugoniot curve and the hydrostatic compression curve is calculated by using Eqs. 2.34 and 2.35 as 4 2 σx x = p + μ L εx x = p + Y 3 3

(2.36)

Yield strength is a rate-dependent parameter and has a pressure dependence as well. It is found that Y is normally an increasing function of hydrostatic pressure. Shock compression process is characterized by the very large strain rate, so that the propagation of the elastic–plastic wave is a very complicated phenomena strongly dependent on materials [12].

24

2.1.3.4

K. Nagayama et al.

Shock Compression of Porous Material

Shock compression whose initial density is lower than the normal or crystalline density of the material is found to be quite different from that of normal density medium. Porous or the lower density materials indicate various forms such as powders, material having pores or particulate aggregates. Early stage of the compression of these substances may be collapsing the pores or any void space. These pore collapsing processes require the pressure or stress of the order of the yield strength of the material. Therefore, much higher pressure compression process than the yield strength can be modeled by the early stage of pore collapse by the yield stress and further volume compression by higher pressure [1, 4]. Figure 2.9 shows the schematic shock compression process of crystalline and porous material. In this figure, v00 denotes the specific volume at the initial porous or lower density state, and shock pressure is assumed to be much higher than the yield stress of the material. In this sense, compression curve from the initial volume, v00 to the continuous state, v0 can be approximated by the straight line at p H = 0. Shock compression curve for the porous material in higher pressure region is approximated by a curve starting at the volume, v0 as schematically shown in Fig. 2.9. In Fig. 2.9, the red curve starting at v0 denotes the shock compression curve of continuous or crystalline material, while the thick brown curve starting at the same point denotes the shock compression curve for the porous material whose original initial state is the point of v00 . What is the difference between these two Hugoniot curves? Why the porous shock Hugoniot pressure is higher than the continuous shock Hugoniot? Blue region in Fig. 2.9 shows the internal energy of cold work, while the pink region shows the increase in thermal internal energy by shock compression of continuous material. One may note that the resultant triangular area, v0 − v − p H 0 composing the blue and pink regions, can be given by Eq. 2.7, and this is the total internal energy change acquired by shock compression of continuous material.

Fig. 2.9 Schematic of shock compression curve for crystalline and porous materials

2 Theory of Shock Wave and Detonation

25

Fig. 2.10 Schematic of shock Hugoniot including phase transition

On the contrary, the corresponding energy change for porous substance up to the same specific volume contains additional area of magenta color in Fig. 2.9. This additional energy increase should also be the thermal energy by shock compression of porous medium. By these discussions, porous shock Hugoniot curve on p − v − ε state surface extends to higher temperature region depending on the initial density or the porosity.

2.1.3.5

Wave Splitting by Elastic–Plastic and High-Pressure Phase Transition

At high pressures, structural phase transition will take place in various materials. Lattice structure changes to more stable configuration for the pressure and temperature realized by strong shock compression. In such cases, shock Hugoniot compression curve can be seen like Fig. 2.10. The similar situation is realized by the elastic–plastic transition as well. According to thermodynamics, isotherm around first-order phase transition in pressure–volume plane has a horizontal line segment at mixed phase region. Shock Hugoniot in this region should have negative slope due to the temperature and entropy increase by decreasing volume. From the conservation equation of continuum mechanics, change of state at shock front rise occurs along the so-called Rayleigh line. Value of shock velocity is proportional to the square root of the slope of the Rayleigh line as explained and shown in Eq. 2.11. If the Hugoniot curve has a shape such as in Fig. 2.10, shock wave splits into two waves, the weaker wave propagates with the velocity which is determined by the slope between the initial point and the deflection point, and the succeeding wave with the velocity determined by the slope between the deflection point and the finally compressed state. In Fig. 2.10, the final state is assumed to be state C. The propagation velocity of the fast (and first) wave, u sB , is given by

26

K. Nagayama et al.

/ u sB = v0

pB v0 − vB

(2.37)

and the propagation velocity of the second wave, u sC is then given by u sC =

/

/ pB (v0 − vB ) + vB

pC − pB , vB − vC

(2.38)

where the first term of Eq. 2.38 denotes the particle velocity of the fast wave. Shock velocity of Eq. 2.38 is the one viewed from outside of the process. Particle velocity is the flow velocity in which the second wave propagates. Such a wave splitting takes place when the condition /

pB > v0 − vB

/

pC − pB vB − vC

(2.39)

is fulfilled. This condition holds for high-pressure phase transition or elastic–plastic transition. This double wave structure disappears, when the final pressure reaches the point D in Fig. 2.10, where the slope from points A to B, and that of A to D are equal. Double wave structure is observed in the shock pressure of the interval ( p B < p < p D ).

2.2 Detonation Wave Fundamentals in Condensed Phase Explosives Many excellent references have been written on explosives including those by Zeldovich and Kompaneets, Ficket and Davis, Mader, Engelke and Sheffield and Cooper [28–32]. This section which treats the basic concept of detonation wave referred these text books. The feature of this section is employing the information of the equation of state for condensed phase substance instead of ideal gas during the explanation of the classical theory. Especially, for description of the ZND model, a whole new approach is introduced. We also elaborated on the classical theory needed in our own research introduction in subsequent chapters. The classical detonation theory is explained from various perspectives, for example, Refs. [33–36].

2.2.1 Simple Model for Detonation—C-J Hypothesis As shown in Fig. 2.11, the first one-dimensional model only considers two boundaries and two substances that are unreacted explosive and its detonation product which is assumed to be completely reacted state. That is, within two slim lines, shock rise and reaction process are included. Since the pressure rise and chemical reaction generated between B1 and B2 are assumed to be completed instantaneously, so B1 and B2 are

2 Theory of Shock Wave and Detonation

27

Fig. 2.11 Conceptual diagram for simple detonation model (define dual boundary separating unreacted and fully reacted substances, which are regarded as two different material)

Fig. 2.12 Conceptual diagram for simple detonation model (jump conditions)

Fig. 2.13 Conceptual diagram for simple detonation model (initial state and Hugoniot)

equivalent to one boundary that defines shock front in the simple theory. The shock front propagates with a constant detonation velocity D. In addition, except for detonation products, any information related to the equation of state is unnecessary. On the unreacted explosive, the variables for initial condition, pressure p0 , density ρ0 and specific internal energy ε0 are only considered. The state

28

K. Nagayama et al.

Fig. 2.14 Conceptual diagram for simple detonation model (C-J condition)

variables for solid explosive will jump from the initial state to gaseous Hugoniot state with detonation heat as shown in Fig. 2.12. The jump condition of shock wave theory explained at Sect. 2.1.2 could be applied. The p − v plane is often used in explanation of the detonation theory, i.e., Chapman–Jouguet (C-J) hypothesis [37]. Figure 2.13 shows initial condition of solid explosive and Hugoniot state for detonation products in p − v plane. The states jump from initial condition (v0 , p0 ) to any point (v, p) at Hugoniot line in Fig. 2.13. The Rayleigh line connects initial point (v0 , p0 ) to any point at Hugoniot by straight line in p − v plane. There are two types for the Rayleigh line. One is a group of lines that intersect Hugoniot at two points, and the other is a tangent line. The necessary condition for specifying the steady-state detonation velocity was determined by the C-J hypothesis. The C-J hypothesis is that the Rayleigh line is tangent to the Hugoniot for detonation products as shown in Fig. 2.14. The tangent point has been called C-J point and defined as detonation pressure. In addition, the Rayleigh line is also tangent to the isentropic line passing through the C-J point for detonation products, and the C-J condition would be expressed as follows, (

∂p ∂v

)

( = H

∂p ∂v

)

( = R

∂p ∂v

) ,

(2.40)

S

where suffices H, R, S denote the Hugoniot, Rayleigh line and isentrope passing through the C-J point. By the same meaning as Eq. 2.40, the following expressions may be common,

2 Theory of Shock Wave and Detonation

29

D − u = c,

(2.41)

where D − u is the flow velocity just behind the detonation front, and c is the local sound velocity of the detonation products. Just behind the detonation front, it is a flow in which the local Mach number is one. The sonic surface of detonation products moves with the detonation front. It will be explained below using the jump condition explained at Sect. 2.1.2. The jump conditions represent the state changes in Figs. 2.12 and 2.13 and could be applied to all points of Hugoniot of detonation products. Equation 2.2 (conservation of mass), Eq. 2.4 (conservation of momentum) are rewritten as follows. ρ0 D = ρ(D − u)

(2.42)

p = ρ0 Du.

(2.43)

Here, u s and u p in Sect. 2.1.2 are replaced by detonation velocity and particle velocity of detonation products. Rayleigh line in p − v plane obtained by eliminating u from Eqs. 2.42 and 2.43 and could be expressed as (

∂p ∂v

) = −(ρ0 D)2 = − R

p v0 − v

(2.44)

At C-J point, (

∂p ∂v

)

( = H

∂p ∂v

) = −(ρ0 D)2 = − R

p . v0 − v

(2.45)

( ) ( ) ( ) ( ) < ∂∂vp , and below the C-J point, ∂∂vp > ∂∂vp . Above the C-J point, ∂∂vp H R H R Note that the inequality sign is inverted when the absolute value of slope is taken. The Rankine–Hugoniot relation, expressed in Eq. 2.7, is also rewritten as follows: ε − ε0 =

p (v0 − v) 2

(2.46)

The discussion will proceed focusing on the C-J point. By differentiating the above equation with v and combining Eq. 2.45, the following relation is obtained only at the C-J point. ( ) ∂ε = −p (2.47) ∂v H

30

K. Nagayama et al.

On the other hand, from the thermodynamic identity, (

∂ε ∂v

) = −p

(2.48)

S

Since Eq. 2.24 satisfies both the Hugoniot and isentrope lines, the next equation is derived. γ (2.49) p H (v) = p S (v) + [ε H (v) − ε S (v)] v By differentiating the above equation with v, the following relation is obtained. (

∂p ∂v

)

] [( ) ( ) ] ( ) d (γ ) γ ∂ε ∂ε ∂p (ε H − ε S ) + = − (2.50) + dv v v ∂v H ∂v S ∂v S [

H

The two state curves pass through at the C-J point, and from Eqs. 2.43 and 2.44, the C-J condition can be confirmed. (It was self-evident because the Hugoniot and isentrope have a second-order contact at the C-J point [38].) (

∂p ∂v

)

( = H

∂p ∂v

) (2.51) S

From the definition of sound velocity, Eqs. 2.40, 2.41 and 2.44, Eq. 2.42 can be obtained. The following relations including a non-dimensional parameter κ are frequently applied to express the state at the C-J point. The definition of parameter κ like adiabatic index is ) ( ) ( ∂ ln p v ∂p v =− ≈ (2.52) κ≡− ∂ ln v S p ∂v S v0 − v If the detonation pressure is high enough comparing with atmospheric pressure, p0 could be ignored. However, in the case of gaseous explosives, p0 must be taken into consideration. Transform Eq. 2.52 or combine it with jump condition to get: v κ = v0 κ +1

(2.53)

p=

ρ0 D 2 κ +1

(2.54)

u=

D κ +1

(2.55)

c=

κD κ +1

(2.56)

2 Theory of Shock Wave and Detonation

31

Fig. 2.15 Conceptual diagram for simple detonation model (when intersecting at two points)

For condensed phase explosives, the value of κ is often around 3. Based on this, the equation, p = (ρ0 D 2 )/4 is the simple prediction formula for detonation pressure. The Hugoniot line is obtained from the isentropic line using Eq. 2.49 and Rankine– Hugoniot, Eq. 2.46. ( ) γ 2v p H (v) = p S (v) + ε S (v) v 2v − γ (v0 − v)

(2.57)

In fact, Hugoniot in figures of this section was drawn based on the JWL equation of state for PETN which will be introduced at Chap. 3. With exception of the C-J point, the Hugoniot and Rayleigh lines have two intersections as shown in Fig. 2.15. At the point 2, the intersection is the solution of the jump condition and is represented by subscript 2. The slope of isentrope line passing through at point 2 is (

∂p ∂v

)

( < S

∂p ∂v

) = −(ρ0 D2 )2 = − R

p2 v0 − v2

(2.58)

where D2 > D j . From this and jump condition, the following equation is c2 > D2 − u 2 .

(2.59)

The local sound velocity immediately behind the detonation front is greater than the flow velocity at the detonation front, so the rarefaction wave overtakes the detonation front. Therefore this is not a solution for a steady-state detonation.

32

is

K. Nagayama et al.

At the point 1 in Fig. 2.15, the slope of isentrope line passing through at point 1 ( ) ( ) ∂p ∂p p1 > = −(ρ0 D1 )2 = − = −(ρ0 D2 )2 (2.60) ∂v S ∂v R v0 − v1

where D1 = D2 > D j . The relationship of local sound velocity and flow velocity is c1 < D1 − u 1

(2.61)

The rarefaction wave behind the detonation front is steadily pulled away from the detonation front. Under the steady-state detonation, the conditions that the local sound velocity is lower than the flow velocity do not exist physically. The point 2 in Fig. 2.15 has been called strong point, and in case of the condensed explosive, it can exist in a limited way for short duration. For instance, higher impact velocity problem between the condensed explosive and metallic flyer is intuitive and easy to understand. After impact, the constant velocity of flyer has to be more than C-J particle velocity. In this case, induced detonation is called overdriven detonation. Overdriven detonation may be possible by placing the high velocity explosive around the low velocity explosive in a cylindrical system. The point 1 in Fig. 2.15 has been called weak point and the weak detonation. The weak detonation has been discussed by several researchers [32, 39].

2.2.1.1

Definition of Specific Internal Energy at C-J Hypothesis

The entropically expanded states from C-J point is expressed by p S (v), which is schematically shown in Fig. 2.16. The specific internal energy, ε in Eq. 2.46 can be written as follows using detonation heat Q. { ε=−

p S dv − Q = ε j − Q

(2.62)

The energy at C-J point of the detonation products, including the detonation heat, can be expressed as follows. ε j − ε0 =

1 p j (v0 − v) + Q 2

Note that Q is often represented by the E 0 symbol [40].

(2.63)

2 Theory of Shock Wave and Detonation

33

Fig. 2.16 Definition of internal energy for detonation products

2.2.2 ZND Model The name of this model comes from names of originators, Zel’dovich, von Neumann and Döring who proposed it in 1940s, independently [28]. This is also onedimensional detonation model, but the compression state of explosive at detonation front and the reaction zone are considered as shown in Fig. 2.17. In the figure, the image of the pressure profile is also shown by the dotted line. There are three boundaries in this model. First boundary (B1) divides the undisturbed explosive and compressed but unreacted explosive and is the front of detonation wave. At the immediately behind the detonation front which is the second boundary (B2), unreacted explosive is compressed and becomes maximum pressure called as Neumann spike. Immediately after B2, reaction of solid explosive starts. The reaction proceeds with pressure degradation and expansion to complete the reaction at third boundary (B3). Just behind B3 is the detonation products, i.e., completely reacted state. The reaction zone is large compared with the width between B1 and B2. Steady profile exists between B1 and B3. Conceptual diagram for ZND model in p − v plane is shown in Fig. 2.18. There are three points, initial state (v0 , p0 ), Neumann spike (vn , pn ) and C-J point (v j , p j ). The Neumann spike is somewhere on the Hugoniot for unreacted solid explosive. If the shock velocity of the unreacted and compressed explosive is same as the detonation velocity D, the Rayleigh line centered at initial state can be expressed as follows. pn (2.64) (ρ0 D)2 = v0 − vn

34

K. Nagayama et al.

Fig. 2.17 Conceptual diagram for ZND model (three boundaries and four types of states) enlarged image near the detonation front Fig. 2.18 Conceptual diagram for ZND model on p − v plane (Initial state and Hugoniots for unreacted solid explosive and its detonation products)

The Rayleigh line at the C-J point can be rewritten as follows using subscript j which corresponds to C-J point, (ρ0 D)2 =

pj . v0 − v j

(2.65)

The state varies as follows. When the detonation front reaches the observation point, the undisturbed state (v0 , p0 ) jump to the Neumann spike (vn , pn ) under the

2 Theory of Shock Wave and Detonation

35

unreacted state. Just after that state, reaction of solid explosive starts, so the state changes the partially reacted state with degradation of pressure and density along the Rayleigh line which has the slope −(ρ0 D)2 to complete the chemical reaction at C-J point (v j , p j ). The states on the Rayleigh line from immediately after the Neumann spike to just before the C-J point are non-equilibrium state existing between the unreacted state surface for solid explosive and the completely reacted state surface for detonation products. This locus satisfies the constant detonation velocity D, even though all states realize during the degradation process from the Neumann spike. We will use the subscript m to express the state variable at arbitrary point in reaction zone. The following equation holds at any point in the reaction zone. (ρ0 D)2 =

pm . v0 − vm

(2.66)

The specific internal energy at the Neumann spike, C-J point and arbitrary point on reaction zone along the Rayleigh line are 1 pn (v0 − vn ) 2

(2.67)

1 p j (v0 − v j ) + Q 2

(2.68)

1 pm (v0 − vm ) + λQ 2

(2.69)

εn = εj = εm =

The variable λ denotes the degree of chemical reaction, changing from 0 for no reaction to 1 for complete reaction.

2.2.2.1

Description of Reaction Zone by ZND Model

The state variables in the reaction zone are determined based on the method often used in the simulation of reactive flow for solid explosives. The feature is that the state during the reaction process is treated as a simple mixed phase of unreacted and completely reacted states [41]. Figure 2.19 shows the reaction path in p − v plane together with Hugoniots for the unreacted and the completely reacted states. In the figure, subscripts u and r denote the unreacted components and the fully reacted components, respectively. When the C-J point and Neumann spike are known in the p − v plane, the pair of (vm , pm ) on the arbitrary point of the Rayleigh line could be calculated by vm = vn + Δv

(2.70)

36

K. Nagayama et al.

Fig. 2.19 Conceptual diagram for description of reaction zone by ZND model ( p − v plane)

p m = pn −

pn − p j Δv v j − vn

(2.71)

Assuming two Hugoniot relations for the unreacted and the completely reacted states, detonation heat Q is known. As mechanical equilibrium is achieved instantly, to determine equations could be employed to determine vu and vr . pu (vu ) = pm (vm ) = pr (vr )

(2.72)

The degree of chemical reaction λ and the specific volumes satisfies the following linear relation (2.73) vm = λvr + (1 − λ)vu Since λ is obtained by Eq. 2.73, the specific internal energy along the Rayleigh line, εm could be calculated by Eq. 2.69. One state in thermodynamic surface vm − pm − εm was specified. Figure 2.20 shows the reaction path in p − ε plane together with Hugoniots for the unreacted and the completely reacted states. The εu and εr are also calculated by Hugoniot relations for both states, and estimated states automatically satisfied the following linear relation. εm = λεr + (1 − λ)εu

(2.74)

2 Theory of Shock Wave and Detonation

37

Fig. 2.20 Conceptual diagram for description of reaction zone by ZND model ( p − ε plane)

Since the target for the simulation of reactive flow is the initiation process of detonation rather than the steady-state detonation, two equations of state for unreacted and completely reacted states are used instead of Hugoniot for two states. In the simulation, λ is calculated by a burn model. The specific volumes and specific internal energies for both components are unknown parameters. The calculation becomes more complicated with four unknowns. The details are described in Sect. 5.3.2 [41].

2.2.2.2

Expression of ZND Model on Thermodynamic Surface

When the ZND model is applied to solid explosives, the initial state and the Neumann spikes are on the unreacted solid surface. On the other hand, since the reaction is completed at C-J point and the solid explosive changes to the detonation products which is thermodynamically equilibrium state, the surface of the detonation products must also be considered. There are above two thermodynamic surfaces, and we call those as the unreacted solid surface and reacted gas surface, respectively here. Figure 2.21 shows the above-mentioned two thermodynamic surfaces. In this section, all figures were drawn using equation of state for penta-erithritol-tetranitrate (PETN). Figure 2.21a corresponds to unreacted solid surface drawn by Hugoniot base Mie Grüneisen form equation of state. The blue line in this figure is the Hugoniot centering (v0 , p0 ). The measurement of unreacted Hugoniot in the high-pressure region is controversial, but it is the optimal equation of state information available currently. Figure 2.21b

38

K. Nagayama et al.

Fig. 2.21 Thermodynamic surfaces for unreacted solid explosive and detonation products

is the reacted gas surface drawn by JWL equation of state. The black line in this figure is the Hugoniot passing through the C-J point. The high-pressure region above C-J point is an extrapolation from the measured value. The detailed discussion of equation of states is described in Chap. 3. Figure 2.21c shows the Hugoniot line for unreacted solid explosive and the surface of reacted gas including the Hugoniot line. If the two thermodynamic surfaces are shown in the same figure, it will be complicated, so in this section, the unreacted solid surface will be omitted except for unreacted Hugoniot line. The expression of ZND model for high explosive in thermodynamic surfaces is shown in Fig. 2.22. In this figure, two Rayleigh lines, one isentropic line and C-J point, have been added comparing with that in Fig. 2.21. In this 3D plot, we will show that the Rayleigh line on p − v plane is seen in a form of different lines, i.e., a line from the initial state to the Neuman point, and another line from the Neuman point to the C-J point as explained below. The path of state variation starts from (A) the initial state, reaches (B) the Neumann spike under the unreacted state and then the reaction proceeds along the Rayleigh line to reach (C) the complete reaction state (thermodynamic equilibrium state) at the C-J point. The Rayleigh lines are not straight on the v − p − ε surface. The internal specific energy along the Rayleigh line pass from (A) to (B) is calculated

2 Theory of Shock Wave and Detonation

39

Fig. 2.22 Expression of ZND model for high explosive in thermodynamic surfaces

Fig. 2.23 Expression of ZND model for high explosive in thermodynamic surfaces from p − v plane direction

by Eq. 2.7. For the pass from (B) to (C), the specific internal energy is estimated by Eq. 2.69. The state along the Rayleigh line changes between two surfaces as a nonequilibrium state with a chemical reaction. The red line corresponds to the isentropic line for detonation products passing through C-J point. On the C-J point, the Hugoniot and isentropic lines for detonation products and Rayleigh line be tangent. This is the C-J condition, so ZND model includes the C-J hypothesis. The Rayleigh line ((A)– (B)) also tangent at the C-J point when it translates in the energy direction with the detonation heat Q. The thermodynamic space with the p − v plane facing forward is shown in Fig. 2.23. As we always see on the p − v plane, the two Rayleigh lines overlap

40

K. Nagayama et al.

Fig. 2.24 Expression of ZND model for high explosive in thermodynamic surfaces from p − ε plane direction

Fig. 2.25 Expression of ZND model for high explosive in thermodynamic surfaces backside view of Fig. 2.24, focusing on the (B)–(C) path

and become one straight line and be tangent at C-J point. In Fig. 2.24, we focus the p − ε surface. Figure 2.25 is a backside view of Fig. 2.24 focusing on the (B)–(C) path. Figure 2.26 shows only the periphery of the path (B)–(C). It is clearly shown that (B) is on the unreacted surface and (C) is on the reacted gas surface, and partially reacted state between them does not belong to two surfaces. Thermodynamic surfaces are effective not only in steady detonation but also in the initiation process, which will be discussed in Chap. 5.

2 Theory of Shock Wave and Detonation

41

Fig. 2.26 Expression of ZND model for high explosive in thermodynamic surfaces focusing on the (B)–(C) path

Fig. 2.27 Conceptual diagram for description of reaction zone by ZND model with partially reacted Hugoniots and its Rayleigh lines, from p − ε plane direction

2.2.2.3

Additional Consideration Using ZND Model

It is difficult to measure the Neumann spike (B), so naturally, there may be also the situations where point (b) in Fig. 2.27 was measured as results. There are two possible paths from (A) → (b). If the path is (A) → (B) → (b), it corresponds to ZND model. The reaction does not start until the peak pressure is reached during compression process. On the other side, if the reaction starts during compression process, the state goes directly from (A) → (b). The base Hugoniot will deflect to a partially reacted Hugoniot before reaching the spike point and passes through a non-equilibrium space. To hold the detonation to be steady, the attaining point (b) is between (B) and (C) on Rayleigh line. If the process from (A) → (b) is very fast and be negligible in building a steady detonation model, it will be a third model that

42

K. Nagayama et al.

could retain the steady-state detonation. It may be considered that there is the case that point (b) very close to (C). Two paths could be considered. One is the ZND model (A) → (B) → (C), another is (A) → (b). The latter is a concept close to C-J hypothesis.

2.2.3 Application of C-J Hypothesis The application of C-J theory is wide, and a typical application is for construction of the equation of state of detonation products. It was specifically shown in this chapter that the equation of state must satisfy the C-J condition. Chapter 3 specifically describes those equation of state. Here, to connect our idea in Chap. 3, envelope approximation by Nagayama [42] and Jones–Stanyukovich–Manson relation [29, 43] are described. The C-J hypothesis predicted that there is a detonation state at one unique point on the Rayleigh line that could be represented by the detonation velocity. The Rayleigh line connects the initial state to C-J state, so the initial state and the C-J state are paired. The detonation characteristics can be expressed by the detonation velocity, pressure, temperature, etc., but the detonation velocity is relatively easy to measure, and that information is actively used in our model in Chaps. 3 and 5.

2.2.3.1

Detonation Velocity Versus Initial Density

It is well known that detonation velocity and loading density of high explosive exhibit linear relationship for various explosives as shown in Fig. 2.28. (Note: There are also many explosives that do not show a linear relationship between detonation velocity and initial density.) The linear relationships are often divided into 2–3 segments with respect to density, and the difference in slope leaves discussions such as the influence of formation of different detonation products [45]. Even if there are multiple segments, the difference may be small. Considering that the effect is small, in such a case, we use one linear relationship that can be expressed by the following equation [46]. (2.75) D[ρ0 ] = j + kρ0 When the difference is large, it may be possible to connect with multiple straight lines.

2.2.3.2

Envelope Approximation

Figure 2.29 shows the Rayleigh lines for PETN. Twenty Rayleigh lines were drawn based on the linear equation in the figure. There is one C-J point corresponding to

2 Theory of Shock Wave and Detonation Fig. 2.28 Relationship between the detonation velocity and initial density for PETN (solid line) [44], RDX (dashed line) [44] and TNT (dotted line) [44]

Fig. 2.29 Rayleigh lines for PETN in p − v plane. From linear relation of detonation velocity and initial density only

43

44

K. Nagayama et al.

Fig. 2.30 Rayleigh lines for PETN in p − v plane together with some C-J point. Black lines drew from linear relation of detonation velocity and initial density. Red lines are drawn by published experimental data

one initial state on one Rayleigh line. If the initial density (specific volume) is moved little by little and countless Rayleigh lines are drawn, the envelope of the group of Rayleigh lines approximately coincide with the locus of C-J points with different initial densities. Figure 2.30 shows the C-J states and Rayleigh lines obtained for the five initial densities in red. It can be imagined that there is a locus of C-J points on the envelope. Hereafter, the points on the envelope are obtained. The formula for the Rayleigh line with the starting point as v0 (= 1/ρ0 ) is p = −(ρ0 D)2 (v − v0 ) = −Z 2 [v0 , ε0 ](v − v0 )

(2.76)

Here, [ ] is introduced to specify the independent variable. The variable Z is a shock impedance. The Rayleigh line starting from v0 + Δv0 that is very close to v0 is p = −Z 2 [v0 + Δv0 , ε0 ] (v − (v0 + Δv0 ))

(

= −Z 2 [v0 , ε0 ](v − v0 ) + Z 2 [v0 , ε0 ]Δv0 − ( +

∂ Z2 ∂v0

) ε0

2)

∂Z ∂v0

ε0

(2.77) Δv0 (v − v0 )

Δv0 Δv0

≃ −Z 2 [v0 , ε0 ](v − v0 ) + Z 2 [v0 , ε0 ]Δv0 −

(

∂ Z2 ∂v0

) ε0

Δv0 (v − v0 )

2 Theory of Shock Wave and Detonation

45

Comparing Eq. 2.76 with Eq. 2.77, the intersection of two adjacent Rayleigh lines satisfies the following condition, ( Z [v0 , ε0 ] = 2

∂ Z2 ∂v0

) Δv0 (v − v0 )

ε0

(2.78)

This is a point on the envelope function and predicts the C-J values, so Z 2 [v0 , ε0 ] v j = v0 + ( 2 ) ∂Z ∂v0

Substitute it into Eq. 2.76,

ε0

Z 4 [v0 , ε0 ] pj = ( 2 ) ∂Z ∂v0

(2.79)

(2.80)

ε0

The specific volume dependence of shock impedance is (

∂Z ∂v0

(

) ε0

=

∂ρ0 D ∂v0

[(

) ε0

= −Zρ0

∂ ln D ∂ ln ρ0

)

] + 1 = −ρ0 Z (Ad + 1)

(2.81)

Here, the very important and dimensionless parameter Ad is introduced. ( Ad = (

∂ Z2 ∂v0

(

) ε0

= 2Z

∂ ln D ∂ ln ρ0 ∂Z ∂v0

) ε0

=

kρ0 D

(2.82)

) ε0

= −2ρ0 Z 2 (Ad + 1)

(2.83)

The C-J value can be predicted only by the initial density dependence of the detonation velocity. Substituting the above equation into Eqs. 2.79 and 2.80, the specific volume and the pressure at the C-J point are v j = v0 − pj =

v0 2( Ad + 1)

ρ0 D 2 2( Ad + 1)

(2.84)

(2.85)

For some explosives, the approximated C-J pressure using envelope and the measured values is compared in Table 2.2. The envelope approximation gives C-J pressure about 2–15% higher than the measured value (Fig. 2.31).

46

K. Nagayama et al.

Table 2.2 Measured C-J pressure ( pexp ) versus envelope approximation ( penv ) for PETN (k = 3.65) Density (cm3 /g) D (m/s) pexp (GPa) penv (GPa) 1.770 1.5 1.26 0.88

8300 7450 6540 5170

33.5 22 14 6.2

34.3 24 15.8 7.3

Fig. 2.31 Measured and predicted C-J pressures for various high explosives

2.2.3.3

Jones–Stanyukovich–Manson Relation

The equation of state for detonation products formula is also in the form of Eq. 2.8, and here, it would be expressed by ε( p, v) and its total derivative is ( dε(v, p) =

∂ε ∂v

)

( dv + p

∂ε ∂p

) dp

(2.86)

v

when both sides are differentiated with v along an isentrope, (

∂ε ∂v

)

( = S

∂ε ∂v

)

( + p

∂ε ∂p

) ( v

∂p ∂v

) (2.87) S

The left-hand side is − p from the thermodynamic identity. The differential coefficient appearing in the second term on the right-hand side relates two parameters that have already been defined. One is the Grüneisen parameter γ defined in Eq. 2.19,

2 Theory of Shock Wave and Detonation

47

and the other is κ defined by Eq. 2.52. Those are expressed as follows, respectively. (

(

∂p ∂v

) S

∂ε ∂p

) v

v =− p

= (

v

(

∂p ∂v

v

) =

∂p ∂ε v

) (

−

S

v γ

p) p = −κ v v

(2.88)

(2.89)

The parameter α is introduced for the first term on the right-hand side of Eq. 2.86, p α ≡ ( ∂ε )

(2.90)

∂v p

Substituting the above relationship into Eq. 2.85 yields: α=

γ κ −γ

(2.91)

The relationship between three thermodynamic parameters related to the slope on the thermodynamic surface holds on all surfaces, including C-J point. Also, at C-J point, Eqs. 2.53 and 2.54 hold. This correlates the state variables at C-J point with the initial state. From here, the relationship between the initial state and the C-J state is considered. The subscript j is added to the variable at the C-J point to emphasize that we are considering the variable at the C-J point. Equations 2.53 and 2.54 are κ=

vj v0 − v j

(2.92)

ρ0 D 2 = p j (κ + 1)

(2.93)

p j v02 = D 2 (v0 − v j )

(2.94)

Rayleigh line is

Differentiation of above equation with initial state as independent variables is dp j v02 + 2 p j v0 dv0 = 2Dd D(v0 − v j ) + D 2 (dv0 − dv)

(2.95)

Dividing both sides by pv02 , dp dv dv0 dD +κ = (κ − 1) +2 pj vj D v0

(2.96)

On the other side, the derivative of Rankine–Hugoniot equation, Eq. 2.41 is

48

K. Nagayama et al.

dε =

1 1 dp j (v0 − v j ) + p j (dv0 − dv) + dε0 2 2

(2.97)

From Eqs. 2.86, 2.88 and 2.89, the above equation is expressed as pj vj 1 1 dv + dp j = dp j (v0 − v j ) + p j (dv0 − dv) + dε0 2 2 α γ

(2.98)

Dividing both sides of the above equation by p j v j and rearranging, dp j dv α +κ = vj 2+α pj

(

dv0 dε0 (κ + 1) + 2κ v0 pjv

) (2.99)

Equations 2.96 and 2.99 are equal, so the derivative of detonation velocity can be expressed as follows with the initial values (specific volume and specific internal energy) as independent variables [47]. dD κ − 1 − α dv0 α(κ + 1)2 dε0 =− + D 2 + α v0 2 + α D2

d D[ρ0 , ε0 ] = D

(

∂ ln D ∂ ln ρ0

=−

) ε0

dρ0 1 + 2 ρ0

(

∂ D2 ∂ε0

) ρ0

(2.100)

dε0 D2

(2.101)

κ − 1 − α dv0 α(κ + 1)2 dε0 + 2 + α v0 2 + α D2

The coefficient of the first term on the right side was defined by Eq. 2.82. (

∂ ln D ∂ ln ρ0

) ε0

=

κ −1−α kρ0 = Ad = 2+α D

(2.102)

The above equation was first introduced by Jones at 1949 [43]. From the above equation and Eq. 2.90, Grüneisen γ could be expressed using Ad and κ, γ =κ

κ − 1 − 2 Ad κ − Ad

(2.103)

We also use this relationship to construct the equation of state for detonation products in Chap. 3 and apply it to the reactive flow simulation in Chap. 5. The initial-state dependence of detonation velocity was discussed in the 1960s by Wood and Fickett [48] and Davis et al. [49]. They showed the initial-state dependence of detonation velocity for solid and liquid explosive as follows.

2 Theory of Shock Wave and Detonation

49

dD dv0 dε0 dp0 = −Ad + Bd 2 − Cd D v0 D ρ0 D 2 where Bd =

(2.104)

α(κ + 1)2 2+α

(2.105)

κ +1 2+α

(2.106)

Cd =

A discussion of the application of this equation continues in Sect. 3.4.1.4 of Chap. 3.

2.3 Numerical Simulation 2.3.1 Introduction Understanding shock and detonation waves in condensed matter is not an easy task. It requires understanding basic theories, learning phenomenology through reading papers, simple calculations, e.g., impedance matching and so on. However if the reader is familiar with some numerical simulation softwares which can handle shock waves in condensed matter and explosive reactions under high pressure, it will become much easier to understand how shock and detonation waves in condensed matter propagate, reflect, attenuate and interact each other. Thanks to the continuous development of computing hardware, we can easily run three-dimensional simulations with personal computers. But we have to take some care with gaps between what we can view beautiful color contour movies and what we can understand from those images. In the first section, we describe a simple one-dimensional Lagrangian example and explain to treat shock discontinuity. A simple one-dimensional Lagrangian example will be shown with partial differential equations and difference equations and how to code equation of state.

2.3.2 Finite Difference Scheme for One-Dimensional Lagrangian Code Lagrangian coordinate, also called material coordinate, is attached to material points and material motion does not affect coordinate values of the material points. It is difficult to use this coordinate in two- or three-dimensional flows which accompany deformation and Eulerian coordinate, attached to space, is commonly used in computational fluid dynamics coding. However, one-dimensional Lagrangian method is suitable to understand propagation of shock and detonation waves. Constructing

50

K. Nagayama et al.

numerical program is easier than Eulerian method as we do not need to consider material motion across the cell boundary.

2.3.2.1

Differential Equations

Now, we describe one-dimensional planar isentropic flow, which does not include shock discontinuity in the flow and then introduce artificial viscosity to the differential equations. One-dimensional isentropic flow can be described as follows: ∂v ∂v = v0 ∂t ∂X

(2.107)

∂u ∂p = −v0 ∂t ∂X

(2.108)

∂ε ∂v = −p ∂t ∂t

(2.109)

Equations 2.107, 2.108 and 2.109 are mass, momentum and energy conservations, respectively. X is Lagrangian coordinate, and t is time. p, v, ε and u are pressure, specific volume, specific internal energy and velocity, respectively, and are function of X and t. Subscript 0 denotes the initial state. Lagrangian coordinate X and Eulerian coordinate x can be converted each other by ∂x v = ∂X v0

(2.110)

Above equations cannot reproduce shock waves as the shock wave front accompanies the entropy increase, and Eq. 2.109 is not correct for the shock wave. von Neumann and Richtmyer [50] introduced a concept of artificial viscosity in 1950. They added an additional term q to p, and Eqs. 2.108 and 2.109 are changed as follows; ∂u ∂( p + q) = −v0 (2.111) ∂t ∂X ∂ε ∂v = −( p + q) ∂t ∂t

(2.112)

q is the artificial viscosity and should be negligibly small except near the shock front and represents dissipative process, like the real viscosity, at the shock front. They proved with the following expression of artificial viscosity, a ∂u q = − ΔX 2 v ∂X

| | | ∂u | | | |∂ X |

(2.113)

2 Theory of Shock Wave and Detonation

51

that Hugoniot equations hold, and it can represent the shock wave thermodynamically, if flow parameters do not change rapidly compared to the thickness of the shock front. The artificial viscosity of the following form, | | | | a1 a2 2 ∂u 2 | ∂u | − ΔX | q = − cΔX ∂X v v ∂X |

(2.114)

is often used to reduce numerical oscillations where c is the sound velocity, a1 and a2 are constants.

2.3.2.2

Pressure and Stress

Before proceeding to finite difference equations, we would like to comment on pressure ( p) and stress (σ ). If we define the term pressure as hydrostatic pressure, stress is defined as (2.115) σ = p + Sx for one-dimensional planar geometry, where S is elastic stress deviator and subscript x denotes x-direction. For gases and liquids, the stress deviator disappears and σ = p, but for solids in many cases, the deviatoric stress is not negligible and play an important role for material behaviors under shock loading. Shock conservation and flow conservation equations should use σ instead of p. However, as this section is an introduction to numerical simulations for those who are not familiar to simulations in shock propagation in condensed matter, we assume that stress deviator is negligible. Sx = 0 and σ = p

2.3.2.3

(2.116)

Finite Difference Equations

Now, we describe finite difference equations for Eqs. 2.107, 2.112, 2.113 and 2.114 with very simple and naive algorithm for easy understanding. The finite difference mesh and time is illustrated in Fig. 2.32. Variables X and u are defined at mesh boundaries, and p, v, ε and q are defined at the center of the mesh. We denote time with superscript and position with subscript. { q n+1 j

=

− van1 c(u nj+1 − u nj ) − j

a2 (u nj+1 v nj

| | | | − u nj ) |u nj+1 − u nj | (u nj+1 u nj ) (u nj+1 ≥ u nj )

0 v n+1 − v nj j Δt

=

u nj+1 − u nj ρ0 ΔX

(2.117)

(2.118)

52

K. Nagayama et al.

Fig. 2.32 Time advancement of variables f and u, where f is p, ε, v or q

u n+1 − u nj j Δt

=−

p nj+1 + q nj+1 − p nj − q nj ρ0 ΔX

− εnj = −( p nj + q nj )(v n+1 − v nj ) εn+1 j j

(2.119) (2.120)

Eulerian positions are updated by x n+1 − x nj = u nj Δt j

(2.121)

With these equations, we can update variables v, q, u, ε and x from values at time t n to new variable values at time t n+1 . Δt should satisfy the Courant–Friedrichs–Lewy (CFL) condition [51], Δx (2.122) Δt = CCFL a CCFL ≤ 1,

(2.123)

where CCFL is the CFL coefficient and a is the local wave speed, a = |u| + c

(2.124)

Δt should be evaluated for all the cells and minimum value will be selected as a time step value. CCFL is usually chosen to be much smaller than unity. The sound velocity is obtained by /( ) ∂p (2.125) c= ∂ρ S

2 Theory of Shock Wave and Detonation

53

Fig. 2.33 Numerical example for a shock tube problem at t = 0.8 ms. The numerical output (blue) was compared with an analytical solution (orange) [52]

Variable p will then be updated from equation of state (EOS) p n+1 = p(v n+1 , εn+1 j j j )

(2.126)

2.3.3 Computed Results Using One-Dimensional Lagrangian Code 2.3.3.1

Shock Tube Problem

Although this code is intended to be used for simulations of shock and detonation propagation in condensed matter, it is useful to check how accurately the shock tube problem is solved and compared with analytical solution. Figure 2.33 is a computed result for shock tube problem compared with an analytical solution calculated according to the procedure in [50]. Numerical and analytical solutions practically agree very well.

2.3.3.2

Detonation Propagation

One of the basic concepts to be understood is that C-J detonation can propagate with stable velocity because it is theoretically stable. There is no need to have a detonation velocity value in the calculation. Detonation will propagate according to the Hugoniot p − v curve of the products at the minimum velocity. We / /propagation ( ) ( ) ∂p ∂p and are used JWL equation with constants listed in Table 2.3. ∂ρ ∂ρ S

H

54

K. Nagayama et al.

Table 2.3 1DL initial conditions for shock tube problem High-pressure section Gas molecular weight (kg) Specific heat ratio (γ ) Pressure (kPa) Temperature (K) Length (m) Number of cells

0.028966 1.403 1000 300 0.5 1000

Low-pressure section 0.028966 1.403 100 300 0.5 1000

Fig. 2.34 JWL Hugoniot and isentrope Table 2.4 JWL EOS constants A B ρ0 1630

3.712e+11 0.03231e+11

C

R1

R2

ω

ε0

0.01045e+11

4.15

0.95

0.30

7.0e+9

plotted in specific volume versus velocity plane in Fig. 2.34 which shows the C-J detonation velocity is about 6930 m/s (Table 2.4). Figure 2.35 shows a pressure profile at 6.24 µs.

2.3.4 OpenFOAM OpenFOAM [53], a free open-source software developed primarily by OpenCFD Ltd., is an object oriented C++ class libraries and utilities to construct and run a solver for partial differential equations. OpenFOAM has many example solvers but in lack

2 Theory of Shock Wave and Detonation

55

Fig. 2.35 Numerical example for steady detonation propagation. (1) Pressure profiles and (2) artificial viscosity profile plotted against time. The maximum position of artificial viscosity is considered to be the position of the detonation front

of solvers which can be used for shock and detonation propagation in condensed phase. Libraries are also necessary which handle equation of state of solids and reaction schemes for high explosives in pressures of tens of GPa. Explosion Research Institute constructed solvers for strong compression under OpenFOAM framework and libraries are also programmed for equation of state of solids and reaction schemes to simulate shock and detonation phenomena in condensed phase. With OpenFOAM classes, we can use a syntax which looks very similar to mathematical expression of partial differential equations. For example, a conservation equation: ∂ρ + ∇ · (φv ρ) = 0 ∂t

(2.127)

is programmed in the solver code as follows. Solver flowchart is shown in Fig. 2.36.

56

K. Nagayama et al.

Fig. 2.36 Flowchart of the code

References 1. Zel’dovich YaB, Raizer YuP (1967) Physics of shock waves and high-temperature hydrodynamic phenomena (English translation), vol 2. Academic Press, New York and London, pp 685–784 2. Davidson L, Shahinpoor M (eds) (1997) High-pressure shock compression of solids I–IV. Springer, New York 3. Bethe H (1942) Theory of shock waves in a medium with arbitrary equation of state. Original paper in report. Republished in: Johnson JN, Cheret R (eds) Classic papers on shock compression science. Springer, London, 1998, pp 421–492 4. McQueen RG, Marsh SP, Taylor JW, Fritz JN, Carter WJ (1970) High velocity impact phenomena. In: Kinslow R (ed), Chap VII. Academic Press, New York, pp 293–417 5. Marsh SP (1981) Los Alamos shock Hugoniot data. University of California, Berkeley 6. van Thiel M (1966) Compendium of shock wave data. University of California Press, Livermore, CA 7. http://www.ihed.ras.ru. Entrance page to shock wave database (2002) 8. Decarli PS, Jamieson JC (1961) Formation of diamond by explosive shock. Science 133:1821– 1822 9. Prümmer R (2006) Explosive compaction of powders and composites. CRC Press, Berlin 10. Cowan GR, Holtzman AH (1963) Flow configuration in colliding plates: explosive bonding. J Appl Phys 34:928–939 11. Christiansen EL (1995) Hypervelocity impact testing above 10 km/s of advanced orbital debris shields. In: Proceedings of APS conference on shock compression of condensed matter, pp 1183–1186 12. Mashimo T (1993) Shock waves in materials science. In: Sawaoka A (ed), Chap 6. SpringerVerlag, Tokyo, pp 113–144 13. Duvall GE, Graham RA (1977) Phase transitions under shock wave loading. Rev Mod Phys 43:523–579 14. Dlott DD (1995) Picosecond dynamics behind shock front. J Phys IV:C4, Suppl. III(5):C4337-1-7 15. Noack J, Vogel A (1998) Single-shot spatially resolved characterization of laser-induced shock waves in water. Appl Opt 37:4092–4099 16. Nagayama K, Mori Y, Motegi Y, Nakahara M (2006) Shock Hugoniot for biological materials. Shock Waves 15:267–275

2 Theory of Shock Wave and Detonation

57

17. Nellis WJ, Moriarty JA, Mitchell AC, Ross M, Dandrea RG, Ashcroft NW, Holms NC, Gathers GR (1988) Metal physics at ultrahigh pressure: aluminum, copper, and lead as prototypes. Phys Rev Lett 60:1414–1417 18. Eliezer S, Ghatak A, Hora H (1986) An introduction to equation of state: theory and applications. Cambridge University Press 19. Nagayama K (1994) New method of calculating shock temperature and entropy of solids based on the Hugoniot data. J Phys Soc Jpn 63:3737–3743 20. Chhabildas LC, Asay JR (1978) Rise-time measurements of shock transitions in aluminum, copper, and steel. J Appl Phys 50:2749–2754 21. Swegle JW, Grady DE (1985) Shock viscosity and the prediction of shock wave rise times. J Appl Phys 58:692–701 22. Rodean HC (1968) Relationship for condensed materials among heat of sublimation, shockwave velocity, and particle velocity. J Chem Phys 49:4117–4127 23. Grüneisen E (1926) In: Greiger H, Scheel K (eds) Handbuch der Physik, 477, vol 10. Springer, Berlin, pp 1–59 24. Steinberg D (1981) The temperature independence of Grüneisen gamma at high temperature. J Appl Phys 52:6415–6417 25. Slater JC (1939) Introduction to chemical physics, Chap XIV. McGraw-Hill, New York, pp 222–240 26. Dugdale JS, MacDonald DKC (1953) The thermal expansion of solids. Phys Rev 89:832–834 27. Vaschenko VYa, Zubarev VN (1963) Concerning the Grüneisen constant. Sov Phys Solid State 5:653–655 28. Zeldovich YaB, Kompaneets AS (1960) Theory of detonation. Academic Press Inc, New York 29. Ficket W, Davis WC (1979) Detonation. University of California Press, Berkeley 30. Mader CL (1979) Numerical modeling of detonations. University of California Press, Berkeley and Los Angeles, California, pp 1–5 31. Cooper PW (1997) Introduction to detonation physics. In: Zukas A, Walters WP (eds) Explosive effects and applications. Springer, Berlin, pp 115–135 32. Engelke R, Sheffield SA (1997) Initiation and propagation of detonation in condensed-phase high explosive. In: Davison L, Shahinpoor M (eds) High-pressure shock compression of solids III. Springer, New York, pp 171–239 33. Dremin AN (1999) Toward detonation theory. Springer-Verlag, New York 34. Nagayama K (2011) Introduction to the Grüneisen EOS and shock thermodynamics, Chap 4. Kindle ebook (Amazon) 35. Forbes JW (2012) Shock wave compression of condensed matter: a primer. Springer-Verlag, Berlin Heidelberg 36. Stanyukovich KP (1960) Unsteady motion of continuous media. Pergamon Press Ltd., Oxford, London, New York, Paris 37. Chapman DL (1899) VI. On the rate of explosion in gases. Lond Edinb Dublin Philos Mag J Sci 47(284):90–104. https://doi.org/10.1080/14786449908621243 38. Zel’dovich YaB, Raizer YuP (1967) Physics of shock waves and high-temperature hydrodynamic phenomena (English translation), vol 1. Academic Press, New York and London, pp 63–67 39. Courant R, Friedrichs KO (1948) Supersonic flow and shock waves. Interscience Publishers Inc, New York, pp 204–235 40. Lee EL, Hornig HC, Kury JW (1968) Adiabatic expansion of high explosive detonation products, UCRL-50422. Lawrence Livermore National Laboratory, Livermore, CA 41. Kubota S, Nagayama K, Saburi T, Ogata Y (2007) State relations for a two-phase mixture of reacting explosives and applications. Combust Flame 151:74–84 42. Nagayama K, Kubota S (2004) Approximate method for predicting Chapman-Jouguet state for condensed explosives. Propellants Explos Pyrotech 29(2):118–123 43. Jones H (1949) The properties of gases at high pressures which can be deduced from explosion experiments. In: 3rd symposium on combustion, flame and explosion phenomena. Williams and Wilkins, Baltimore, Maryland

58

K. Nagayama et al.

44. Dobratz BM (1981) LLNL explosives handbook: properties of chemical explosives and explosive simulants. UCRL-52997 45. Kerley GI (1989) Theoretical model of explosive detonation products; tests and sensitivity studies In: Proceedings of the ninth symposium (international) on detonation, Portland, OR, OCNR 11329-7. Office of the Chief of Naval Research, Arlington, VA, pp 443–451 46. Nagayama K, Kubota S (2003) Equation of state for detonation product gases. J Appl Phys 93:2583–2589 47. Lambourn BD (1985) Derivatives of the Chapman-Jouguet state. In: Proceedings of the 8th symposium (international) on detonation, Albuquerque, NM, 15–19 July 1985, pp 778–784 48. Wood WW, Fickett W (1963) Investigation of the Chapman-Jouguet hypothesis by the “inverse method”. Phys Fluids 6:648–652. https://doi.org/10.1063/1.1706795 49. Davis WC, Craig BG, Ramsay JB (1965) Failure of the Chapman-Jouguet theory for liquid and solid explosives. Phys Fluids 8:2169–2181. https://doi.org/10.1063/1.1761177 50. von Neumann J, Richtmyer RD (1950) A method for the numerical calculation of hydrodynamic shocks. J Appl Phys 21:232–237 51. Courant R, Friedrichs K, Lewy H (1928) Über die partiellen Differenzengleichungen der Mathematischen Physik. Math Ann 100:32–74 (in German) 52. Toro EF (1997) Riemann solvers and numerical methods for fluid dynamics. Springer-Verlag, Berlin Heidelberg New York 53. https://www.openfoam.com

Chapter 3

Description of Detonation Phenomena Kunihito Nagayama, Shiro Kubota, and Masatake Yoshida

Abstract The equations of state (EOS) for condensed phase explosive are described in this chapter. The two types of EOSs that are the EOS for detonation products and for the unreacted condensed explosive are necessary to express the detonation phenomena. For detonation products, the EOS developed in Japan and some case studies in Japan are introduced in detail. For the unreacted condensed explosive, the static compression data base EOS are explained with many experimental data. Keywords Equation of state · Detonation products · Unreacted explosive · Unified EOS

3.1 Introduction of Equation of State for Explosives The equations of state (EOS) for the condensed phase explosive are described in this chapter. Even if the detonation theory is constructed by the conservation laws and C-J condition, the detonation state cannot be expressed quantitatively without equation of state. The condensed explosive becomes the detonation products which are the mixture of the mainly gaseous chemical species after the complete reaction at C-J state. The most important EOS for the explosive is for the detonation products. If the CFD code is equipped with the EOS for detonation products, it is possible to evaluate the steady detonation and its effect on the surroundings. The EOS for the unreacted state is important too. The shock compressed state without chemical reaction exists at the detonation front, called Neumann spike. Since K. Nagayama (B) Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan e-mail: [email protected] S. Kubota National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan e-mail: [email protected] M. Yoshida Explosion Research Institute Inc., Tokyo, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Kubota (ed.), Detonation Phenomena of Condensed Explosives, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-5307-1_3

59

60

K. Nagayama et al.

it is difficult to consider the intermediate chemical species during the reaction, the state in the middle of reaction is expressed as a mixture of the detonation product and the unreacted state. By using EOSs of both for the detonation products and for the unreacted state of explosive, many detonation phenomenon including shock to detonation transition and diameter effect could be estimated quantitatively by numerical simulation with the reaction rate model. The detonation products EOS developed in Japan and some studies in Japan are introduced in detail. For the unreacted condensed explosive, the static compression data base EOS is explained with many including comparisons with experimental data.

3.2 Equation of State for Detonation Products 3.2.1 Equation of State with Explicit Chemistry Solid explosives are solid state before detonation and become gas phase as detonation products after detonation. The gas density just after the reaction is higher than the initial solid state, and the detonation pressure is in the order of 10 GPa. For the solid explosive, the detonation products could not be expressed by the ideal gas EOS. The EOS for detonation products has been an important topic in detonation physics since 1950 and before. At the first detonation symposium held in 1951, Brinkley discussed the EOS for detonation products [1]. Many researchers had employed Abel EOS in which the internal energy is a function of temperature only, i.e., thermodynamically ideal. The generalized Abel EOS in which the covolume is the function of the volume, and the EOS that is the virial expansion with respect to pressure up to the third order had been employed. The Kistiakowsky–Wilson (KW) formula, which is an improved version of Baker’s EOS, was mentioned as the most promising EOS for the detonation products of condensed explosive. Kistiakowsky raised the issue of solid residue from an EOS point of view [2]. Kirkwood argued that the Lennard– Jones–Devonshire (LJD) EOS based on the free-volume theory provides the best compromise [3]. The essential problem for the development of the EOS for detonation products was the lack of experimental information on detonation phenomena. By this time, the relationship between velocity and density was well investigated [4]. However, the following was also pointed out. The EOS is not determined by the information obtained merely from studying detonation velocity as a function of loading density. The different EOSs predict the values of the same order of magnitude in pressure. However, the calculated temperature is quite sensitive to the form of the EOS employed [2]. At this symposium, Price also presented the Kistiakowsky– Wilson EOS developed in Naval Ordinance Laboratory [5]. At that time, there was no computer environment for performing chemical equilibrium calculations considering several chemical species. At the 2nd Detonation Symposium, Fickett and Cowan calculated C-J state using KW EOS considering seven chemical species (H2 , CO2 , CO, H2 O, N2 , NO and graphite) as detonation products [6, 7]. The empirical parameters were selected so

3 Description of Detonation Phenomena

61

as to reproduce the experimental information of the C-J pressure and the relationship between detonation velocity and initial density for RDX, RDX/TNT and TNT. The equation of state developed by LANL was used for graphite. For the C-J pressure, the value measured by Deal at the same symposium was used for comparison. Since many statistical mechanical theories of the EOS of highly dense molecular systems have been proposed, many EOS theories aimed at constructing the EOS of detonation products were studied [8]. For example, the first EOS for detonation products in Japan was proposed in 1952 by Kihara and Hikita (KH EOS), as described in Sect. 3.3 [9, 10]. Jacobs investigated EOS for detonation products at high density using LJD EOS and Monte Carlo method [11]. Reply of Jacobs for Mader’s comment on his presentation in Symposium (International) on Combustion accurately represented the state of detonation research 20–30 years ahead. However, there had not yet been a computer that performs equilibrium calculations taking into account enough chemical species of detonation products. Until the mid-1970s, the development of the Baker–Kistiakowsky–Wilson (BKW) EOS was promoted, and at the same time, the C-J parameters for various explosives were also collected, such as C-J pressure obtained by free surface velocity measurements by Deal [12]. The comparison with the experimental data has increased, and the prediction accuracy of BKW has also improved. During this period, the development of the EOS of detonation products gas promoted over the world. Mader stated in his textbook that BKW’s FORTRAN code became available to research institutes around the world in 1967, including the Department of Reaction Chemistry of the University of Tokyo, Japan [13]. There is a detailed description of the BKW EOS in Mader’s textbook [13]. Original formula is a virial equation of state for mixture of chemical species of detonation products gas [8], pV B C = 1 + + 2 + ··· RT V V

(3.1)

The virial coefficients of B, C, . . . are related to the intermolecular potential, and its relation could be obtained by statistical mechanics. Several similar EOSs had been proposed and used, with different mathematical expressions and the different intermolecular potential models. In the case of BKW, ignoring third-order terms, and using χ = B/ V , pV = 1 + χ (1 + βχ ) (3.2) RT Applying a first approximation, pV = 1 + χ exp[βχ ] RT

(3.3)

Using a repulsive potential of the form U = Ar −1 , where r is the separation distance, the variable χ is expressed below in the form proposed by Cowan and Fickett [7].

62

K. Nagayama et al.

χ=

κB k B V (T + θ )α

κB =

Σ

n i k Bi

(3.4) (3.5)

The original BKW parameters are α, β, κ B and covolumes of detonation products k Bi . The covolume of each component had been obtained from Van der Waals radii. The empirical parameters, α, β, κ B were determined so that calculations using these parameters would reproduce a set of experimental data including both the loading density dependence of detonation velocity and C-J pressure. At the same time, the handling of solid components and the algorithm for chemical equilibrium calculation were established. At the sixth detonation symposium in 1976, Jacobs–Cowperthwaite–Zwisler (JCZ) EOS was proposed by Cowperthwaite and Zwisler with Jacobs comments and had made improvements such as the adoption of Exp 6 potential to the EOS proposed by Jacobs [14, 15]. Similarly, the thermodynamic codes made by existing EOS was developed in each country so that it could be applied to the EOS of detonation products for various explosives. The calculated results by Percus Yevick EOS, Carnahan–Starling EOS, BKW and JCZ3, KH EOS, etc., were compared, and the improvements of thermodynamic codes had been made [16–18]. In Japan, the Kihara–Hikita–Tanaka (KHT) code was developed based on KH EOS, as explained in Sect. 3.3 [19]. The EOS code for evaluating detonation phenomena including the expansion of the detonation products requires a database that stores the EOS information of substances in a high-temperature and high-pressure state. On the other hand, hydrodynamic codes that simulate phenomena involving detonation require various types of EOS information. For example, Sandia National Laboratories (SNL) has developed CTHTIGER by combining the CFD code, CTH and TIGER code. Hobbs et al. introduced brief history of SNL’s EOS work based on Stanford Research Institute’s original TIGER code [20]. As mentioned above, the calculated temperature is very sensitive to the EOS format. On the other hand, measurement is also known to be more difficult than other parameters. Chapter 4 also introduced examples of temperature measurements in Japan.

3.2.2 Equation of State Without Explicit Chemistry The EOS most often used in computational fluid dynamics to simulate detonation phenomena of high explosives in the world is Jones–Wilkins–Lee (JWL) EOS [21– 23]. This EOS was proposed at LLNL in the 1960s, and its parameters are obtained by the cylinder expansion test [24, 25]. In the test, explosives are detonated in a copper pipe, and the expansion process of the pipe is recorded by high-speed streak camera. Figure 3.1 shows an example of the results of cylinder expansion test [26].

3 Description of Detonation Phenomena

63

Fig. 3.1 Example of high-speed photography results of a cylinder expansion test [26]

The parameters are determined so that the test results can be accurately reproduced in the fluid dynamic calculation using the JWL EOS. The isentropic line passing through the C-J point is expressed as follows. p S = Aex p(−R1 V ) + Bex p(−R2 V ) + C V −(ω+1)

(3.6)

where A, B, C, R1 , R2 and ω are the JWL parameters, and V = v/v0 . The specific internal energy along the isentropic line is {v∞ ε S = ε(v∞ ) + v

1 p S dv = ρ0

(

A B C ex p(−R1 V ) + ex p(−R2 V ) + V −ω R1 R2 ω

)

(3.7) Where, attention must be paid to the integral range. This equals to ε j in Eq. 2.62 of Chap. 2. Substituting the isentropic line into Eq. 2.22 of Chap. 2 as a reference line gives the equation of state of the detonation products, i.e., ω (3.8) p = p S + (ε − ε S ) v The variable ω corresponds to the Grüneisen coefficient and is assumed to be a constant in JWL. Substituting 3.6 and 3.7 into 3.8 yields ( ) ) ( ω ω ω ex p(−R1 V ) + B 1 − ex p(−R2 V ) + ρ0 (ε + Q) p = A 1− R1 V R2 V V (3.9) Here, Q is detonation heat as shown in Sect. 2.2.1.1 and had been expressed as a value available chemical energy E 0 in original paper [22]. The JWL parameters should reproduce the results of the cylinder expansion test in the fluid dynamic simulation. In addition, it must satisfy the C-J hypothesis described in Sect. 2.2.1. Isentropic line expressed by Eq. 3.6 has to pass through the C-J point and is tangent with the Rayleigh line in p − v plane. As the expansion progresses and the volume becomes larger, the third term of Eq. 3.9 becomes dominant. For this reason, at the large volume area the JWL EOS considered a condition of κ = ω + 1, so ω is set between 0.2 and 0.4, which is consistent with the heat capacities of ideal gases [22].

64

K. Nagayama et al.

Fig. 3.2 Isentrope passing through C-J point for PETN with initial density 1.77 g/cm3 . The three terms of the JWL EOS are also plotted Table 3.1 JWL parameters for PETN [27] Explosive Density (cm3 /g)

A (Mbar) B (Mbar)

C (Mbar)

ω

R1

R2

PETN

6.17

0.0069

0.25

4.40

1.20

1.770

0.169

Figure 3.2 shows the isentrope passing through C-J point for PETN with initial density 1.77 g/cm3 , and the other parameters in Table 3.1 are used. The three terms of the JWL EOS are also plotted, respectively. The first and second terms are dominant in the pressure range higher than GPa order. The contribution of the third term increases below about 100 MPa. When the third term becomes dominant, it exhibits a linear change behavior on a log-log plot like an ideal gas.

3.3 Kihara–Hikita–Tanaka (KHT) Code 3.3.1 Molecular Theory of Detonation by Kihara and Hikita In 1952, Kihara and Hikita published a paper titled “Molecular Theory of Detonation” in Japanese journal “Kogyo Kayaku” [9]. They assumed that molecular attractive force is negligible at high pressure and considered only repulsive force. Then, intermolecular potential is given by U (r ) = λr −n (λ > 0, n > 3),

(3.10)

where U , r are intermolecular potential, distance of two molecules, and λ, n molecular constant which corresponds to molecular diameter, and a constant corresponding to incompressibility of molecules.

3 Description of Detonation Phenomena

65

Table 3.2 EOS constants obtained by Kihara and Hikita [9] n α a b 9

1.878

0.958

1460 1460

3200 2210

−0.928

1.635

Table 3.3 Computed TNT parameters by Kihara and Hikita [9] Initial density T (K) p (GPa) (kg/cm3 ) Blinkley–Wilson Kihara–Hikita

c

13.3 14.8

Dcalc (m/s)

Dexp (m/s)

6640 6450

6470 6470

Considering that, at extremely high density ( pv ≫ kT ), the internal energy should be sum of static energy of closest packing density and vibrational energy, they expressed kT / pv with a function of x = (λ/ pv)3/n v −1 as follows; kT 1 − αx = , pv 1 + ax + bx 2 + cx 3

(3.11)

so that it can be smoothly extended to low-density region. They theoretically calculated values of constants α, a, b, c for n = 6, 9, 12, 15, 18, ∞ and hand-calculated the detonation velocities of PETN, TNT, Tetryl and NG (nitroglycerin), selecting n = 9 with best detonation velocity values. The determined parameters are listed in Table 3.2. They assumed the composition of detonation products, and as a result, heat of detonation is independent of the initial density of the explosive, and calculations of detonation velocity assumed only one hypothetical average molecule as a product gas to avoid complicated equilibrium computations. Their computed detonation parameters are compared with values computed using Blinkley and Wilson equation of state in Table 3.3. Detonation C-J temperature calculated with Kihara–Hikita equation of state is generally lower than other models and this is also the same trends for the KH code and KHT code explained in the following sections.

3.3.2 KH Code A Ph.D. student Tanaka programmed Kihara–Hikita (KH) code and [28], without assuming product composition and heat of explosion, computed C-J parameters using chemical equilibrium computation. The code, when first completed, was very similar to BKW code of Mader [29]. Yoshida heard from Dr. Tanaka that when Dr. Mader visited Hikita and Tanaka in Japan he taught them the essential theory of BKW code. KH code computes the equilibrium compositions by dG =

Σ

μi dxi = 0,

(3.12)

66

K. Nagayama et al.

where μi is the chemical potential of ith component gas by the method according to White et al. [30]. Equations for thermodynamic variables for gases are as follows: kT 1 − αx = pv 1 + ax + bx 2 + cx 3 = f (x) ≡ 1/g(x) ( x=

λ pv

)3/n

v −1 ,

) 3 ( 0 + RT (g(x) − 1) , E = Σ (E 0 − H 0 ) + H0n n

(3.13) (3.14)

(3.15)

[ ] xi p x cx 2 − RT (c + ab) 2 + μi = (G 0 − H00 )i + H0i0 + R ln p0 a 2a [ ] 1 − αx + RT (α 3 + aα 2 + bα + c) ln + Z (g(x) − 1) − ln g(x) (3.16) i α2 [ ] p S = Σxi (S 0 )i − R Σxi ln xi + ln p0 [ ] 2 x cX |1 − αx| + R (c + αb) 3 + + ln g(x) + (α 3 + aα 2 + bα + c) ln α α2 2α (3.17) Zi ≡ λ

−3 n

√ 3 + λi λ3/n , n 3/n

≡ Σxi λi .

(3.18) (3.19)

KH code used the NASA data [31] at ambient pressure and extended to high pressure using Kihara–Hikita equation of state. As the detonation products include solid carbon, they used the equation of state of graphite by [7] which is based on shock experiments by [32].

3.3.3 KHT Code In 1982, Tanaka [33] revised the Kihara–Hikita equation as follows: kT 1 − αx = pv 1 + ax + bx 2 + cx 3 + d x 4 + ex 5

(3.20)

3 Description of Detonation Phenomena

67

Table 3.4 Equation of state parameters for KHT code/Kihara and Hikita [33] KHT code KH code/Kihara and Hikita 9 1.85 −1.8523 40.245 −235.06 661.49 −670.48

n α a b c d e

9 1.878 0.958 1.635 −9.928

Table 3.5 Molecular parameter λ1/3 (Mbar cc/mol)1/3 of KHT code, KH code and Kihara and Hikita [33] for main product gases KHT code KH code Kihara and Hikita H2 O CO2 CO N2

6.1 14.0 9.8 9.8

6.877 12.837 9.169 9.169

6.665 12.794 9.283 9.283

and re-evaluated parameters to fit Hugoniot data of cryogenic liquids and solid carbon dioxide. The parameters α, a, b, c used by KH code are the same as original values of Kihara and Hikita [9] and those are derived theoretically using n and λ which are determined to reproduce detonation velocity. Tanaka proposed the empirical method for finding such parameters. The parameters of EOS are determined so that the result of the EOS satisfies Hugoniot for gaseous species and so on [33–42]. The equations of state parameters for Kihara and Hikita, KH code and KHT code are summarized in Table 3.4. The parameter λ is listed in Table 3.5. Carbon monoxide is also an important product gas but it lacks experimental data. The values of nitrogen, which has the same molecular weight, were applied by Tanaka.

3.3.4 Typical Examples of KHT Code The KHT code is used in various studies related to explosives applications and has the following functions. They are calculation of Chapman–Jouguet (C-J) detonation characteristics, C-J isentrope characteristics, Hugoniot line, JWL parameters, isochoric isobaric combustion and the power of explosive, rocket performance, blast, chemical equilibrium and maximum calorific value, etc. The above includes the function of fitting the C-J isentrope obtained by KHT code to the JWL form to obtain JWL parameters. It has a thermodynamic database, but users can also add it. Figures 3.3, 3.4 and 3.5 are the calculation results by KHT

68 Fig. 3.3 Pressure–volume relationship for RDX64/TNT36 estimated using KHT code by varying initial density

Fig. 3.4 Relationship between the loading density and the detonation velocity for RDX64/TNT36 estimated by KHT code

Fig. 3.5 Evaluation of the detonation velocity of TNT containing ammonium nitrate. Evaluation when the contained ammonium nitrate partially reacts. The non-reactive part can be treated as inactive

K. Nagayama et al.

3 Description of Detonation Phenomena

69

code. Figure 3.3 shows the C-J isentrope for mixed explosive RDX64/TNT36 which consists of RDX 64% and TNT 36%. Figure 3.4 is the initial density dependence of the detonation velocity for RDX64/TNT36. Figure 3.5 shows non-ideal detonation of the mixed explosive of TNT and ammonium nitrate. The detonation velocity of explosives containing ammonium nitrate is actually lower than calculated. It is believed that this is because ammonium nitrate does not react completely on the detonation front. To estimate such condition, KHT code can define a portion of ammonium nitrate as an inert substance.

3.4 Other Models of the Equation of States 3.4.1 Unified EOS for Arbitrary Initial Density This section presents our method of determining the usable EOS function based on the experimental data on the initial density dependence of detonation velocity and one C-J isentrope line. A simplified method of calculating the Grüneisen parameter function of specific volume is given [43]. In case of liquid and solid explosives, the parameters of EOS for detonation products have been estimated for each initial density. This is because the same explosive does not always produce the same detonation products when the initial density is different. However, it may be considered that highly sensitive explosives such as PETN can generate similar detonation products essentially regardless of the initial density. In fact, in the JWL EOS, the heat of detonation per unit mass of PETN is regarded as constant regardless of the initial density [44]. We have used Jones–Stanyukovich–Manson (JSM) relation described in Sect. 2.2.3.3 [45] to estimate Grüneisen parameter more directly and obtained unique thermodynamic surface of detonation products for PETN which could be applied for all initial density. We have obtained EOS useful for detonation physics the unified form EOS obtained here is applied to the research on the shock to detonation transition in Chap. 5.

3.4.1.1

Formulation

As shown in Sect. 2.2.3.1 it is well known that detonation velocity and loading density of high explosive exhibits linear relationship for various explosives, and we use this relation to find γ . The linear relation is written as follows. D[ρ0 ] = j + kρ0 ,

(3.21)

70

K. Nagayama et al.

where D is detonation velocity, j and k are constants. The state variables defined at C-J point as shown in Sect. 2.2.3.3 are written as follows. γ =

κ(κ − 1 − 2 Ad ) κ − Ad (

γ =v ( κ=−

∂ ln D ∂ ln v (

Ad =

∂p ∂ε

)

) = S

∂ ln D ∂ ln ρ0

(3.22)

(3.23) v

ρ0 D 2 −1 p

(3.24)

ρ0 k D

(3.25)

) ε0

=

They are the JSM relation, Grüneisen parameter, the adiabatic index, and the Jones parameter, respectively. To obtain a functional form of the EOS for the detonation products, it is necessary to have further data set in addition to the D[ρ0 ] data. At least, one additional data is necessary to close the system of equations. We choose to use one isentropic curve determined by the cylinder expansion test to calculate the Grüneisen parameter. We further assume that the Grüneisen parameter defined by Eq. 3.23 is a function of volume, and no dependence on temperature. The fundamental information and assumptions can be listed as follows. (1) D[ρ0 ] data are available. (2) C-J state variables data at theoretical maximum density (TMD) are used and JWL isentrope available. (3) JSM relationship is utilized in every step of calculation, and formulation. (4) Grüneisen parameter should be a function of specific volume alone, γ (v). The Grüneisen-type EOS can be written as follows. p = p S (v) +

γ (v) (ε − ε S (v)) v

(3.26)

In this equation, variables, p S (v), ε S (v) and γ (v) should be determined. If we can obtain an isentrope, from the cylinder expansion test, we know the isentropic functions as p S (v) and ε S (v). Differentiation of both sides of equation with respect to specific volume will lead to the following form. dγ γ (γ + 1) p S (v) − γ = dv v

dps dv

+ ρ02 D 2 p j − ps

(3.27)

To close the differential equation, JSM relation was used. Taking the logarithm on both sides of Eq. 3.22 and differentiating, we obtain a differential equation along the C-J states,

3 Description of Detonation Phenomena

) ( (κ + 1) 2 A2d − 2 Ad κ + κ 2 + Ad − dv0 ) ( = dv κ 3A2d − 2 Ad κ + κ 2

71 (κ−Ad )2 dγ ρ0 (κ+1) dv

(3.28)

Equations 3.26, 3.27 and 3.28 form a closed set of equations, which can be integrated along the C-J state. We can choose the C-J state reached from the initial density of explosive at TMD as a starting point of calculation, although that point is a singular point of the differential equation. One may derive the limiting relationship near the singular point to start the integration. As the information on the isentrope needed to the calculation, we adopt the JWL parameters at the C-J point for PETN with TMD. For calculation of d ps /dv, JWL isentrope for TMD case is used as follows. dps 1 d Ps = v0 d V dv ] 1 [ =− R1 Aex p(−R1 V ) + R2 Bex p(−R2 V ) + (1 + ω)C V −2−ω v0

(3.29)

This is the differentiation of Eq. 3.26. Because the initial density 1.77 cm3 /g is very close to TMD for PETN, this initial density is considered as TMD for PETN in this section.

3.4.1.2

Grüneisen Parameter γ (v) and Thermodynamic Surface for PETN

The Grüneisen parameter, the adiabatic index and the Jones parameter obtained by solving of Eqs. 3.28, 3.29 and 3.30 are shown in Figs. 3.6 and 3.7. By using γ (v) into Eq. 3.28, the unified form EOS of PETN that could applied for all initial density can be constructed. This equation of state is called unified form EOS in this chapter. Since the numerical integration is performed along the C-J point, the relationship between each parameter and the initial density, and the locus of the C-J point were obtained at the same time. Figure 3.8 shows the locus of the C-J point on the p − v plane obtained by solving differential equations, with a solid line. The experimental data by Lee and Hornig and the results of the envelope approximation (red dashed line) described at Sect. 2.2.3.2 are also plotted. The experimental data almost exist between the results of the envelope approximation and the unified form EOS, and disperse, which was also the reason why the experimental locus of the C-J point was not used in our analysis directly. Considering the experimental error in pressure measurements, the envelope approximation may be a good evaluation method for detonation pressure. As shown in Sect. 2.2.3.2 the envelope approximation is the case of which the value of the Grüneisen γ is equal to zero.

72

K. Nagayama et al.

Fig. 3.6 Calculated Grüneisen parameter of PETN by solving differential equations

Fig. 3.7 Grüneisen parameter, the adiabatic index and the Jones parameter obtained by solving of the differential equations

If Eq. 3.22 is rearranged by κ, κ = 2 Ad + 1 + γ

κ − Ad κ

(3.30)

Since the envelope approximation considers γ to be 0, it is an approximation that ignores the third term on the right side of this equation. Based on Fig. 3.8, it can be predicted that the influence is small for detonation pressure. To accurately evaluate this influence and proceed the development of detonation physics, more accurate measurement of detonation states is required. Comparison of the unified form EOS and envelope approximation [46] is shown in Fig. 3.9. The locus of the C-J point in p − v and p − ρ0 planes are plotted. Since the locus of the p j − v j relation exists nearly on the envelope of the Rayleigh lines,

3 Description of Detonation Phenomena

73

Fig. 3.8 Locus of C-J point on p − v plane obtained by unified EOS and envelope approximation, and comparison with experimental data

Fig. 3.9 Comparison of the unified form EOS and envelope approximation (the locus of the C-J point)

all pairs of (v j , p j ) are very close to the envelope curve. On the other hand, the p j − ρ0 relation does not match. This implies that there are differences on the pairs of v j , ρ0 , so the pressure is different at the same initial density. LLNL Handbook lists JWL parameters for PETN with four different initial densities [27]. These parameters were evaluated based on the cylinder expansion test and the experimental values of the C-J state. The value of the C-J volume was calculated by initial density, detonation pressure and adiabatic index κ in the table of the handbook. Table 3.6 shows the experimental results and the calculated results obtained by the unified form EOS and the envelope approximation. For C-J volume, the results of the unified EOS was up to 1.1% smaller than the experimental value, and the difference of the pressure is about 2% as shown in Table 3.7. It is considered that the unified EOS can simulate the experimental results very well. The results of

74

K. Nagayama et al.

Table 3.6 Comparison of the experiments results with the calculated results obtained using unified form EOS and envelope approximation Density Experiments (Exp) Unified EoS (Uni) Envelope functions (g/cm3 ) v j (cm3 /g) P j (GPa) v j (cm3 /g) P j (GPa) v j (cm3 /g) P j (GPa) 1.77 1.5 1.26 0.88

0.4098 0.4907 0.5865 0.8266

33.5 22.0 14.0 6.20

0.4098 0.4866 0.5802 0.8177

33.58 21.73 14.09 6.321

0.4061 0.4759 0.5620 0.7888

Table 3.7 (Continued) Comparison of the experiments versus calculation Density (g/cm3 ) v j Pj vj (Exp)/(Uni) (Exp)/(Uni) (Env)/(Uni) 1.77 1.5 1.26 0.88

1.0000 1.008 1.011 1.011

0.998 1.012 0.993 0.981

0.991 0.978 0.969 0.965

34.38 23.03 15.29 6.895

Pj (Env)/(Uni) 1.024 1.060 1.085 1.091

envelope approximation in this Table 3.6 were calculated during the estimation of the unified EOS with assumption of γ = 0. It is plausible to estimate the accuracy of the envelope approximation result based on the results of the unified EOS. Table 3.7 is the comparison of the envelope approximation and the unified EOS. In the case of envelope approximation, the C-J volume are estimated to be about 3.5% smaller than that estimated by unified EOS, and the pressure is about 9% larger. This is not directly linked to the assumption of zero gamma. Figure 3.10 shows the locus of C-J state estimated by unified EOS and the three C-J points estimated by JWL EOS. Further verification of the effectiveness of our method is a comparison with C-J isentropic lines for each initial density, or a simple simulation of detonation propagation. Since Eq. 3.26 satisfies two isentrope lines on the state surface, arbitrary isentrope line can be obtained using the JWL isentrope line for TMD. By differentiation of Eq. 3.26 with the specific volume and using the thermodynamic identity, (∂ε/∂v)s = − p, the ordinary differential equation could be obtained as follows ( ) dp(v) dps (v) 1 d ln γ (v) (3.31) = + − 1 − γ (v) ( p(v) − ps (v)) dv dv v d ln v The values for unified EOS in Table 3.6 were used as the starting point for numerical integration for each initial density, so the corresponding starting point of JWL isentrope is slightly different. This difference may also have an influence on the comparison of isentropic lines. Figure 3.11 shows the C-J isentropes for PETN with three different initial densities estimated by unified EOS and JWL EOS. Figure 3.12

3 Description of Detonation Phenomena Fig. 3.10 Locus of C-J state estimated by unified EOS (red line) and isentrope line passing through C-J point estimated by JWL EOS (TMD), and three C-J points estimated by JWL EOS. The densities in this figure represent the initial states for the three C-J points

Fig. 3.11 C-J isentropes for PETN with three initial densities, estimated by the unified EOS and JWL EOS

Fig. 3.12 Comparison of the unified EOS and JWL EOS for PETN with three different initial densities (ratio of the two EOSs)

75

76

K. Nagayama et al.

Fig. 3.13 Comparison of the unified EOS and JWL EOS for PETN with three different initial densities (for wide pressure range)

corresponds to the ratio of the two isentrope lines for each initial density. In the case of initial density 1.5 g cm−3 , the difference was relatively large comparing with cases of the other initial density, and it has changed about ±5%. Because the starting point of numerical integration of Eqs. 3.28 and 3.29 is a singular point, the results are affected by the initial values for integration. This means that the validity of the results needs to be confirmed. The most important application of this type of EOS is the reactive flow simulations of detonation phenomena, especially the simulation of shock to detonation transition. Another application is the detonation propagation process, but not for the estimation of blast waves. This means that the relatively high-pressure region as shown in Fig. 3.10 is important for this type of EOS. For the reasons mentioned above, we must validate unified EOS even in a wider pressure range. A solution in which γ tends to decrease smaller than ω as the specific volume increases has also been confirmed, and in such a case, a discrepancy with the JWL EOS is confirmed in the pressure range of the order of MPa, where ω is the JWL parameter that appears in Eq. 3.6. It can be avoided by fixing γ as constant like ω at above a certain specific volume. Figure 3.13 shows the case that γ is set constant for the region where γ is less than ω. Thermodynamic surface for PETN estimated by unified EOS is shown in Fig. 3.14. The isentrope lines passing through the C-J point obtained by JWL EOS are also plotted. In the range of the figure, the isentropic lines were almost located on the state surface.

3 Description of Detonation Phenomena

77

Fig. 3.14 Thermodynamic surface for PETN with some isentrope lines passing through the C-J point. The surface was drawn by unified EOS and isentrope lines were plotted by JWL EOS

3.4.1.3

Numerical Simulation of the Propagation of the Steady State Detonation of PETN Using the Unified EOS

The propagation of steady-state detonation can be simulated with a simple algorithm with the EOS for detonation products. A typical algorithm is C-J volume burn. The principle of it is simple and easy to understand. The variable λ defined at Sect. 2.2.2.1 denotes the degree of chemical reaction, changing from 0 for no reaction to 1 for complete reaction. The model that determines this λ according to the laws such as thermodynamics is called the burn model. The details of burn model will be described in Chap. 5. The C-J volume burn is expressed by the difference formula as follows [13]. )/(v0 − v j ), (3.32) λin+1 = (v0 − v n+1 j where v n+1 is the specific volume defined in ith grid of the calculation field, and the j superscript corresponds to the time step of simulation. The variable v0 is the specific volume of the initial solid explosive, and v j is the specific volume of the detonation products at C-J point. The reaction is completed when the explosive is compressed to C-J state. The equation of state of solid explosives is not required, just the initial state such as density is given to the calculation field. The pressure of the reacted cell is evaluated by the following equation using the EOS of the detonation products, pg (v, ε). (3.33) pin+1 = λin+1 × pg (v, ε)

78

K. Nagayama et al.

Fig. 3.15 Comparison of the simulation results that are employed unified EOS and JWL EOS for PETN with initial density 1.77 g/cm3 PETN

The detonation propagation process obtained by the simulation completely depends on the EOS for detonation products. Below, the calculated results using the unified EOS and JWL EOS were compared [47]. The calculation conditions such as the grid size of 0.05 mm, the artificial viscosity and CFL number were all the same in both calculations. This calculation requires the value of v j . Since this is a known value for each equations of state, the respective values were used. Both simulations can be calculated with the value of the envelope approximation for v j . Although it does not significantly affect the detonation velocity and wave profile, it has a slight effect only near the C-J point at the detonation front. Therefore, we decided to use the respective v j values. The simulations were performed with a one-dimensional Lagrangian code, and the detonation were triggered by impact of 1 mm PETN with the same density at 1 or 0.5 km/s. Figure 3.15 shows the comparison of the simulation results for PETN with initial density 1.77 g/cm3 . The horizontal axis corresponds to position measured from the initial impact point. The detonation velocities are the same with each other, and all pressure distributions overlap and seem as a single line. In the case of the initial density of 0.88, the detonation velocity estimated by the JWL EOS is 1.6% faster than that estimated by unified EOS, so the position of the detonation front shifts with each other. To compare the profiles of the detonation waves, the result of unified EOS was displayed with a shift of 3 mm in the propagation direction. The results are shown in Fig. 3.16. At the position of 10 cm, the pressure difference between the two cases was 4.4%. At least, it can be applied accurately to the numerical simulation of the shock to detonation transition. Currently, the γ adjustments described in the previous section have been implemented and we have achieved higher accuracy. The effect of Grüneisen parameter on the calculated result is examined by simulation of detonation propagation. The detonation propagation processes were compared by the results of simulation. Specifically, the calculated results with constant γ and

3 Description of Detonation Phenomena

79

Fig. 3.16 Comparison of the simulation results that are employed the unified EOS and JWL EOS for PETN with initial density 0.88 g/cm3 PETN

Fig. 3.17 Dependence of the detonation velocity of PETN on the initial density (the sensitivity check of the Grüneisen parameter)

the results employing unified EOS are compared. The base isentrope line of Eq. 3.17 is JWL isentrope with a density of 1.77 g/cm3 . As for the γ constant condition, two conditions, 0.45 and 0.25, were examined. Simulations were performed for the initial densities, 1.7, 1.5. 1.26, 0.88, 0.48 and 0.25 g/cm3 . Figure 3.17 shows the dependance of the detonation velocity of PETN on the initial density. In the case of the initial density of higher than 1.5 g/cm3 , the simulation is not sensitive to the Grüneisen parameter, at least for the detonation velocity and the pressure waveform. For the initial density lower than 1.5 g/cm3 , the contributions of Grüneisen increases as density decreases, particularly when γ = 0.45. Typical results on the difference of the waveform of detonation waves are shown in Fig. 3.18. The results for density range γ = 0.25 tends to have a low detonation velocity in the region near from 0.88 to 1.26 g/cm3 , and for γ = 0.45 have a high detonation velocity at

80

K. Nagayama et al.

Fig. 3.18 Comparison of the simulation results of detonation propagation (unified EOS, and constant γ ; 0.45 and 0.25)

the density range lower than 1.26 g/cm3 , and the waveforms differ greatly. The parameters of the JWL EOS were precisely determined, and it is thought that the isentropic lines of each density passing through C-J point exist in the immediate vicinity of the surface for PETN with initial density 1.77 g/cm3 .

3.4.1.4

Analysis of Temperature Dependence of Detonation Velocity Using Unified Method

At the beginning of this section, γ was calculated as the function of specific volume using the relationship between the initial density dependence of the detonation velocity and JSM relation and was used to construct the thermodynamic surface of PETN. At the same calculation, the dimensionless parameters, γ , κ, and α defined at the C-J point are obtained. The quantity α is defined in Chap. 2 and is written below. p α ≡ ( ∂ε )

(3.34)

∂v p

The equation of the detonation velocity dependence on the initial state in Sect. 2.2.3.3 is shown below. dD dv0 dε0 dp0 = −Ad + Bd 2 − Cd D v0 D ρ0 D 2 κ − 1 − α dv0 α(κ + 1)2 dε0 κ + 1 dp0 =− + − 2 2 + α v0 2 + α ρ0 D 2 2+α D

(3.35)

3 Description of Detonation Phenomena

81

For PETN, the relationships of the dimensionless parameters defined at the C-J point and the C-J state are obtained. We use this information to develop a discussion of the temperature dependence of detonation velocity. It is possible to change the initial pressure if it is a gaseous explosive, but it is generally difficult with liquid or solid explosives. This is because it is almost impossible to maintain specific volume while increasing temperature of the explosives. By selecting the specific volume and the internal specific energy as independent variables and differentiating Eq. 3.35 with respect to initial temperature T , the formula for the temperature dependence of the detonation velocity is derived as follows. (

d D(v0 , ε0 ) dT0

)

( ) ) ∂v0 Bd ∂ε0 + ∂ T0 p0 D ∂ T0 p0 ) Bd ( = −Ad Dβ0 + C p0 − p0 v0 β0 D (

= −Ad Dρ0

where β0 = (

∂ε0 ∂ T0

(

) = p0

1 v0 ∂h 0 ∂ T0

(

) (3.37) p0

)

( C p0 =

∂v0 ∂ T0

( − p0 p0

∂h 0 ∂ T0

(3.36)

∂v0 ∂ T0

) (3.38) p0

) (3.39) p0

The variables β and C p0 are the thermal expansion coefficient and the specific heat at constant pressure, respectively. The first term includes the contribution of the thermal expansion and has negative sign. The initial temperature dependence of the internal energy in the second term consists of the enthalpy change due to heating under isobaric conditions from which the work caused by thermal expansion is removed. We will also consider envelope approximation. Equation 2.78 of Chap. 2 for envelope approximation can be rearranged as D2 v0 − v (2v − v0 )dv0 + d D2 = 0 3 v0 v02

(3.40)

Differentiating with T0 gives the following equation. (

∂D ∂ T0

) =− p0

Dβ0 (κ − 1) 2

(3.41)

The envelope approximation has a simple form and can be estimated from κ. The powder PETN is a mixture of air and PETN crystal, so the changes in the coefficient

82

K. Nagayama et al.

Fig. 3.19 Relationship between the initial temperature derivative of the detonation velocity and the initial density for PETN estimated by unified method [48]

of thermal expansion and constant pressure-specific heat with respect to changes in initial density were ignored. Figure 3.19 shows the relationship between the initial temperature derivative of the detonation velocity and the initial density for PETN. The first and second terms in Eq. 3.36 and the result of envelope approximation were also plotted. The heating causes the thermal expansion. As a result, the density decreases and the detonation velocity decreases. Therefore, the first term takes a negative value. With decreasing initial density, the contribution of the first term decreases linearly. The first term of Eq. 3.36 can be written as the following expressions. (

∂ ln D − Ad Dβ0 = − ∂ ln ρ0

) Dβ0 = −kρ0 β0

(3.42)

p0

where the k is a coefficient of the linear equation of the detonation velocity shown in Eq. 3.21. The first term of Eq. 3.36 is a more accurate approximation compared with the envelope approximation. At high initial densities, the contribution of the thermal expansion increases. In contrast, the contribution of the increment of the internal energy is small. The difference between the approximation by the first term and Eq. 3.36 is only about 3% near the TMD. In the region where the initial density is high, the assumption of (∂ε0 /∂ T0 ) p0 = 0 holds approximately. The formulation can be written as follows for the first-term approximation in the case of a high initial density. (

∂D ∂ T0

(

) p0

∂D ≃− ∂ρ0

) ρ0 β0 = −kρ0 β0

(3.43)

p0

Since the contribution of the thermal expansion becomes small with decreasing initial density, the contribution of the internal energy becomes relatively large. The sign of the temperature derivative changes at an initial density of 0.268 g/cm3 .

3 Description of Detonation Phenomena

83

Fig. 3.20 Relationship between the detonation velocity and initial temperature for various initial density for PETN. D293; detonation velocity at 293 K

Fig. 3.21 Relationship between the relative detonation velocity and initial temperature for various initial density for PETN. D293; detonation velocity at 293 K

The detonation velocity obtained by Eq. 3.21 was regarded as the value at 293 K, and the initial temperature dependence of the detonation velocity is plotted in Figs. 3.20 and 3.21. The temperature range was set from 240 to 360 K. Although it is a narrow range, considering the phase change and melting point, it can be considered as conditions for normal application. Figure 3.21 shows the relative detonation velocity which detonation velocity is divided by its value at 293 K. The detonation velocity changes between only ±1%, the rate of change decreases with the decreases of density, and the slope reverses at the initial density of 0.268 g/cm3 . The accuracy is controversial, it became possible to simulate the detonation phenomena of PETN with different initial temperature. Data for both the initial temperature and initial density derivatives for PBX9404 and LX-04 are given in the LLNL Explosives handbook [27]. Using the experimental data for the initial density derivative, the values of the initial temperature derivative for

84

K. Nagayama et al.

Table 3.8 Measured initial temperature derivative and initial density derivative of detonation velocity and the initial temperature derivative of detonation velocity predicted by the first-term approximation Explosive substance

ΔD/ΔT0 (cm/µs/K)

ρ0 (g/cm3 )

ΔD/Δρ0 (cm/µs cm3 /g)

β0 (µm/K/cm)

1st-term approx. ΔD/ΔT0 (cm/µs/K)

PBX9404

−1.165 × 10−4

1.84

0.36

174

−1.153 × 10−4

LX-04

−1.550 × 10−4

1.86

0.362

228.2

−1.545 × 10−4

these explosives are examined using Eq. 3.43 and are compared with the experimental data in Table 3.8. The difference between the experimental and predicted results are 1% or smaller. It can be concluded that for high-density powdered explosives, the first-term approximation can be accurately applied.

3.5 EOS Model for Unreacted Condensed Explosives 3.5.1 Formulation of the Equation of State from Isothermal Compression Data 3.5.1.1

Compression Curve of High Explosives (Shock Hugoniot and Isothermal Compression Curve)

Shock to detonation transition (SDT) processes for condensed phase high explosive can be simulated by a numerical procedure based on the unreacted and reacted EOS and reaction modeling. Among two EOSs, detonation gas EOS can be constructed by using detonation wave velocity or other measured data [43, 45, 49–53]. In order to reproduce detonation properties as well as wave profile data in high precision, EOS of detonation gas of an explosive has been constructed depending on the value of initial density. We are developing simulation code for explosive with different initial density with different reaction model parameters [43]. Reaction model parameters and/or EOS surface should be unique for one explosive, since these are the basic material properties of the explosive assuming that detonation tends to thermodynamic and chemical equilibrium. Nevertheless, the above attempts have been meaningful for real simulation as an engineering sense. Nagayama and Kubota have published an EOS of detonation gas for PETN which has the measurement data of the dependence of detonation velocity on initial density and that is described in the previous section [43, 54, 55]. The other EOS, that is an EOS for unreacted explosives, has also been reported by various authors [56, 57]. One of possible data source is the direct measurement of shock Hugoniot for the explosive [58]. It is known, however, that data scatter for explosive material is commonly larger than those for inert materials. This is attributed to the possible partial reaction at the shock front. One of an alternative

3 Description of Detonation Phenomena

85

source of information on EOS for unreacted high explosive is the static compression experiments [59–61]. Static isothermal compression data are rather preferable to shock Hugoniot data due to lower possibility of partial reaction during measurement. Efforts to construct an EOS based on static compression data are described in this section.

3.5.1.2

The Grüneisen Equation of State with Volume-Dependent Grüneisen Parameter

The Grüneisen-type equation of state (EOS) of the form p = f (v) +

γ (v) ε v

(3.44)

can only be derived from the fundamental assumption ( γ =v

∂p ∂ε

) v

= γ (v),

(3.45)

where p, v, ε and γ denote the pressure, the specific volume, the specific internal energy and the Grüneisen parameter, respectively. Thermal component of specific internal energy is shown to be written as [54, 62] εT = Θ(v)C(S),

(3.46)

where ⎡ Θ(v) = Θ(v0 ) exp ⎣−

{v v0

⎤ ⎤ ⎡ v { γ (v) ⎦ γ (v) dv = Θ0 exp ⎣− dv ⎦ . v v

(3.47)

v0

Two material functions C(S) and Θ(v) are conjugate with each other such as entropy and temperature and have physical dimension of these thermal variables [54, 62]. Indeed, they satisfy the following relationship T d S = Θ(v)dC(S).

(3.48)

It is noted that these relationships are derived from the assumption, Eq. 3.45. Thus, Eq. 3.44 can be rewritten to [54, 62] ε = εc (v) + Θ(v)C(S),

p = pc (v) +

γ (v) Θ(v)C(S), v

(3.49)

(3.50)

86

K. Nagayama et al.

where suffix c denotes cold part. This equation can also be written as ε = ε(v, S0 ) + Θ(v)Z (S, S0 ), p = p(v, S0 ) +

γ (v) Θ(v)Z (S, S0 ), v

(3.51) (3.52)

where p(v, S0 ) and ε(v, S0 ) denote a reference isentrope with S0 , the value of entropy of uncompressed state. The same or similar expression has been used in the CREST model [63, 64]. Material function Z (S, S0 ) is simply related with C(S) as Z (S, S0 ) = C(S) − C(S0 ), which is used as a measure of entropy in the CREST model [64]. In this way, Eqs. 3.49 or 3.51 provide arbitrary isentrope functions. It is also shown that Θ(v) corresponds to the Debye temperature or Einstein temperature, if these models are adopted, and we call here the characteristic temperature function [54, 62].

3.5.1.3

Procedure of Constructing the Grüneisen-Type Equation of State Based on the Isothermal Compression Data

Under the assumption of Eq. 3.45, it is shown that the specific heat at constant volume should be a function of entropy. It cannot be expressed as a function of temperature alone. This requirement stems from the following thermodynamic identity. dT γ (v) dS dΘ(v) d S =− dv + + = , T v Cv Θ(v) Cv

(3.53)

where Cv denotes the specific heat at constant volume and it must be an entropy function in order for the lhs quantity to be a total differential. For later use, one may derive the relationship derived from Eq. 3.53, x=

T dC(S) = . Θ(v) dS

(3.54)

This x is a function of entropy. Now let us consider the formulation of the Grüneisen-type EOS based on the isothermal compression curve at room temperature, p(v, T0 ). One has to determine the material functions f (v) and γ (v) in Eq. 3.44 to establish the form of the Grüneisen-type EOS. Applying Eq. 3.44 to the isotherm of T = T ∗ , one obtains f (v) = p(v, T ∗ ) −

γ (v) ε(v, T ∗ ). v

(3.55)

3 Description of Detonation Phenomena

87

Differential of the specific internal energy along an isotherm can be written by ∗

[

dε(v, T ) = T

∗

(

∂p ∂T

)

∗

v

]

− p(v · T ) dv,

(3.56)

which can be written by using Eqs. 3.45, 3.47 and 3.54 as dΘ(v) (Cv (x)) − p(v, T ∗ )dv = T ∗ d x − p(v, T ∗ )dv. Θ(v) x (3.57) Note that material function Cv can be a function only of entropy, or of the variable x defined by Eq. 3.54. Equation 3.57 can then be integrated over volume along an isotherm of room temperature T0 to have dε(v, T ∗ ) = −Cv (x)T ∗

T0 { /Θ(v0 )

ε(v, T0 ) = ε(v0 , T0 ) + T0 T0 /Θ(v)

Cv (x) dx − x

{v p(v, T0 )dv.

(3.58)

v0

By inserting Eq. 3.58 to Eq. 3.55, we obtain ⎤ {v γ (v) p(v, T0 )dv ⎦ f (v) = ⎣ p(v, T0 ) + v v0 ⎛ ⎤ T0{ /Θ(v) γ (v) ⎜ Cv (x) ⎥ ε ⎝v0 , T0 ) + dx⎦ T0 − v x ⎡

(3.59)

T0 /Θ(v0 )

Insertion of Eq. 3.59 to Eq. 3.44 gives the Grüneisen-type equation of state based on the isothermal compression curve, p(v, T0 ). Resultant EOS in p − v − ε form is the generalized version of Setchell and Taylor obtained assuming constant Cv , constant γ /v, and Murnaghan form of isotherm [65]. Arbitrary specific heat function and/or Grüneisen function with available isothermal data can be incorporated into Eq. 3.59. One may see from Eq. 3.59 that specific heat term and Grüneisen term can be calculated independently. To use the Grüneisen EOS of the form of Eqs. 3.50 or 3.52, one must calculate the reference isentrope, p(v, S0 ). We will start the following thermodynamic formula for isochoric pressure change ) ( ) ( ) ∂p ∂p ∂ε dS = dS ∂S v ∂ε v ∂ S v γ (v) γ (v) γ (v) = TdS = ΘdC = ΘCv (x)d x v v v (

dp =

(3.60)

88

K. Nagayama et al.

The last expression is derived from the expression derived from Eqs. 3.53 and 3.54 (3.61) dC = xd S = xCv d(ln x) = Cv (x)dx By integrating Eq. 3.60 from isotherm to isentrope along an isochoric, we have

γ (v) p(v, S0 ) − p(v, T0 ) = Θ(v) v =

γ (v) Θ(v) v

T (v,S {0 )/Θ(v)

Cv (x)d x T0 /Θ(v) T0 { /Θ(v0 )

Cv (x)d x,

(3.62)

T0 /Θ(v)

where T (v, S0 ) = T0 Θ(v)/Θ(v0 ) is derived from Eq. 3.53 with d S = 0. With the use of these relationships, the reference isentrope function can be calculated by the specific heat model and the Grüneisen function.

3.5.2 Material Functions and Shock Hugoniot 3.5.2.1

Specific Heat Function at Constant Volume for PETN and HMX

Specific heat at constant volume and at atmospheric pressure can be obtained from the measurements of the temperature dependence of the specific heat at constant pressure, C p (T , p0 ) together with those of volume expansion coefficient β(T , p0 ) and the Grüneisen parameter, γ (v0 ). Those quantities at least for several explosives including PETN and HMX are available [66–68], and Cv can be obtained by the following relationship, Cp Cv = (3.63) 1 + γβT It is shown that the obtained temperature dependence of Cv for these explosives has slight curvature and is approximated as the following several functions. Cv (v0 , T ) = const : Case 1

(3.64)

Cv (v0 , T ) = a + bT : Case 2 Cv (v0 , T ) = a + bT + cT 2 : Case 3

(3.65) (3.66)

Figure 3.22 shows estimated specific heat by using Eq. 3.63 for PETN and HMX together with the fitting functions, i.e., Cases 1–3. HMX data seems almost linear. Best fit for both explosive data is obtained by the function of Case 3, i.e., quadratic function and shows no difference in the figure, although the quadratic function has

3 Description of Detonation Phenomena

89

Fig. 3.22 Estimated specific heat at constant volume for PETN and HMX and several adjusted functions

its own drawback explained later. For Case 1, we use the value at T = 300 K. Linear function in Case 2 may give too large value of Cv at very high temperature under compression or at very large value of x. Quadratic function may give maximum or decrease at very high temperature. This behavior may not be justified on physical basis. It is easily expected that the estimated temperature under shock compression is highest for Case 1, and lowest for Case 2. Since specific heat at constant volume must be a function of entropy and not of temperature, the function Cv (T ) must be read as Cv (T /Θ(v)). This procedure of replacing a variable from T to T /Θ(v) is always possible for arbitrary Cv (T ) functions. Eventually, specific heat at constant volume given in Eqs. 3.64, 3.65 and 3.66 are replaced by an entropy function by using an entropy variable, x as Cv (v0 , T ) = const : Case 1

(3.67)

Cv (v0 , T ) = a + bΘ0 x : Case 2 Cv (v0 , T ) = a + bΘ0 x + cΘ0 2 x 2 : Case 3

(3.68) (3.69)

where we define Θ0 = Θ(v0 ) in Eq. 3.47.

90

K. Nagayama et al.

By inserting Eqs. 3.67, 3.68 and 3.69 to Eq. 3.61 and integrating over x, we obtain C(S) = C(x) = C0 + a [x − x0 ] : Case 1 ] bΘ0 [ 2 x − x0 2 : Case 2 C(S) = C(x) = C0 + a[x − x0 ] + 2 ] bΘ0 [ 2 x − x0 2 C(S) = C(x) = C0 + a[x − x0 ] + 2 ] cΘ0 2 [ 3 + x − x0 3 : Case 3 3

(3.70) (3.71)

(3.72)

These formulae can be used to calculate temperature by giving any values of (v, ε). Given any values of (v, ε), and hence Θ(v) from Eq. 3.47, and then inserting into Eqs. 3.49 or 3.51 to obtain C(S) or Z (S,S0 ). Hence, all other states on the isentrope through (v, ε) can be found. If one knows the value of C(S), value of x can be obtained by solving Eqs. 3.70, 3.71 and 3.72. Finally, with Eq. 3.54, the value of T can be calculated by giving the value of Θ(v). The specific heat term in Eqs. 3.58 and 3.59 for each specific heat function can then be calculated to be T0{ /Θ(v)

Cv (x) Θ(v0 ) d x = T0 a ln x Θ(v)

(3.73)

] [ Cv (x) Θ(v0 ) Θ(v0 ) d x = aT0 ln + bT0 2 −1 x Θ(v) Θ(v)

(3.74)

T0 T0 /Θ(v0 ) T0{ /Θ(v)

T0 T0 /Θ(v0 )

T0{ /Θ(v)

T0 T0 /Θ(v0 )

Cv (x) Θ(v0 ) d x = aT0 ln x Θ(v)

+ bT0

3.5.2.2

[ 2

] ) [( ] Θ(v0 ) T0 3 Θ(v0 ) −1 +c −1 Θ(v) Θ(v) 2

(3.75)

Grüneisen Volume Functions

Almost all of the analyses above are based on the information on the volumedependent Grüneisen parameter. Previous attempts on the studies of shock Hugoniot have been made by using the assumption that γ (v)/v = γ0 /v0 = const. Since no serious evidences have been found whether this assumption is valid or not, we have tried to use several test functions in order to see its contribution to the calculated shock Hugoniot from known isotherm. Here, we have chosen the following four Grüneisen

3 Description of Detonation Phenomena

91

Fig. 3.23 Volume dependence of the assumed Grüneisen functions

functions, i.e., γ (v) = γ (v0 ) = const : Case 4 v γ (v) = γ (v0 ) : Case 5 v0 ( )1.5 v : Case 6 γ (v) = γ (v0 ) v0 ( )2 v : Case 7 γ (v) = γ (v0 ) v0

(3.76) (3.77) (3.78) (3.79)

Decrease in Grüneisen parameter with compression is slower for γ (v) in Eqs. 3.76, 3.77, 3.78 and 3.79 in the cited order. Characteristic temperature function can be calculated by Eq. 3.47 for test functions. Figures 3.23 and 3.24 show test Grüneisen functions and their corresponding characteristic temperature functions.

92

K. Nagayama et al.

Fig. 3.24 Volume dependence of the characteristic temperature function corresponding to the assumed Grüneisen functions

3.5.2.3

Birch–Murnaghan Isothermal Compression Curve for PETN, HMX and the Grüneisen Equation of State

Two data sets for TMD PETN isothermal compression data have been available by Olinger et al. [60] and by Yoo et al. [61]. TMD HMX data have also been included in Yoo et al.’s paper [61]. We will employ the Birch–Murnaghan form as ) 5] v −3 − v0 [( ) 2 ]) ) 3( , v −3 × 1+ K −4 −1 , 4 0 v0

3 p(v, T0 ) = K 0 2 (

[(

v v0

)− 73

(

(3.80)

where K 0 and K 0, denote the bulk modulus and its pressure derivative. These two material parameters for two explosives were given by the above authors. In this case, the first parenthesis in Eq. 3.59 is then calculated by integrating Eq. 3.80 over volume.

3 Description of Detonation Phenomena

γ (v) p(v, T0 ) + v

93

{v p(v, T0 )dv v0

]2 [( ) 2 γ (v) 9 K 0 v −3 = p(v, T0 ) − −1 v 8 ρ0 v0 (( ) 2 [ )] K 0, − 4 v −3 × 1+ −1 2 v0

(3.81)

Using Eq. 3.80 and isothermal term in Eq. 3.59 together with the Grüneisen functions, Eqs. 3.76, 3.77, 3.78 and 3.79 gives the material function f (v) in concrete functional form without no numerical integration. Then, reference isentrope function p(v, S0 ) described in Eq. 3.62 can also be calculated by using Eqs. 3.70, 3.71, 3.72 and 3.80 as ] [ γ (v) aT0 Θ(v0 ) Θ(v) (3.82) 1− p(v, S0 ) = p(v, T0 ) + v Θ(v) Θ(v0 ) ] [ γ (v) aT0 Θ(v0 ) Θ(v) 1− v Θ(v) Θ(v0 ) [ )2 ] ( 2 γ (v) bT0 Θ(v0 ) + Θ(v) 1− v Θ(v) 2Θ(v0 )

p(v, S0 ) = p(v, T0 ) +

] [ γ (v) aT0 Θ(v0 ) Θ(v) 1− v Θ(v) Θ(v0 ) [ )2 ] ( 2 γ (v) bT0 Θ(v0 ) + Θ(v) 1− v Θ(v) 2Θ(v0 ) [ ) ] ( γ (v) cT0 3 Θ(v0 ) 3 + Θ(v) 1− v Θ(v) 3Θ(v0 )

(3.83)

p(v, S0 ) = p(v, T0 ) +

3.5.2.4

(3.84)

Shock Hugoniot for PETN and HMX

We have established the Grüneisen-type EOS using Eq. 3.59 inserting into Eq. 3.44. Specific heat term in Eq. (16) is replaced by Eqs. 3.64, 3.65 and 3.66 and isotherm term calculated using Eq. 3.58 for each Grüneisen functions, Eqs. 3.76, 3.77, 3.78 and 3.79. First, we calculated entropy function C H (v) along shock Hugoniot. Figure 3.25 shows that this material function is almost perfectly independent of both material functions, γ (v) and Cv (S). It is somewhat strange in the sense that entropy function C(S) is

94

K. Nagayama et al.

related with another entropy function Cv (S), and why the different Cv (S) function gives no change in the value on shock Hugoniot. In order only to estimate this entropy function, one can assume any of the material functions. Second, shock Hugoniot compression curve, p H (v) for PETN and those for HMX are shown in Fig. 3.26. From these figures, it is shown that it depends slightly on γ (v), but almost no dependence on Cv (S). This result stems from the fact that an isentrope centering the uncompressed states described by Eqs. 3.82, 3.83 and 3.84 has very little dependence on the specific heat function, although it contains Cv term. In this sense, flow description in terms of mechanical variables and not in terms of thermal variables except for C(S) is not strongly dependent on material functions assumed. Third, Fig. 3.27 shows calculated shock temperature for PETN and also for HMX using different Grüneisen functions. They have almost similar trends as Hugoniots of p H (v) for PETN and HMX. They are insensitive to the Grüneisen functions. Large dependence of shock temperature on specific heat functions are shown in Fig. 3.28. Difference between Cv = const (Case 1) and linear Cv (Case 2) is fairly large. Case of Cv = const produces overestimated value of TH , while Case of linear Cv gives too low value, since Cv increases with temperature and no tendency approaching an asymptotic value. In reality, shock temperature may fall between these two cases. From these discussions, difference in the output values using the Grüneisen EOS with any material functions, γ (v) and Cv (S) is expected to be rather small at least one does not need the estimates of temperature. Even in case of entropy-based reaction modeling like CREST model, the situation is the same. However, temperature calculation is important in simulation, one has to choose an appropriate non-constant specific heat function. As shown in Figs. 3.26 and 3.27, difference between isothermal compression curve and shock Hugoniot is larger in the case of γ (v) = γ0 = const, i.e., Eq. 3.76 irrespective of the explosive. This difference naturally is attributed to the thermal pressure added through irreversible heating by shock compression. From Fig. 3.26, one may see that case 4 γ (v) = γ0 = const gives higher shock pressure with the same specific volume. One may also see the Grüneisen function dependence of shock temperature. Temperature difference is larger than pressure difference for all cases. It is possible to calculate shock velocity–particle velocity Hugoniot from calculated pressure–volume Hugoniot. Bulk sound velocity at uncompressed state can be calculated as follows. By differentiating Eq. 3.62 by volume, we have for the bulk sound velocity as | | | | | | dps | | | = −v 2 | dpt | + γ 2 Cv T0 . cb2 = −v02 || 0| 0 | dv 0 dv |0

(3.85)

One may note that the bulk sound velocity is free for the behavior of Grüneisen functions. Estimated u s − u p Hugoniot for each Grüneisen function is shown in Fig. 3.29. In this figure, published shock Hugoniot data for both crystalline PETN of initial density of 1.773–1.778 g/cm3 and HMX of initial density of around 1.90 g/cm3 are also shown [58]. Data for PETN are seen to be scattered around present calculations,

γ=const : Case 4

TMD PETN

γ=γ (v/v ) : Case 5 0

0

γ=γ (v/v ) 0

1.5

0

: Case 6

2

0

H

γ=γ (v/v ) : Case 7

95 Entropy function C - arbitrary unit

H

Entropy function C - arbitrary unit

3 Description of Detonation Phenomena

0

C =const : Case 1 v

linear C : Case 2 v

quadratic C : Case 3 v

v/v

0

γ=const : Case 4

TMD HMX

γ=γ (v/v ) : Case 5 0

0

γ=γ (v/v ) 0

1.5

0

: Case 6

2

γ=γ (v/v ) : Case 7 0

0

C =const : Case 1 v

linear C : Case 2 v

quadratic C : Case 3 v

v/v

0

Fig. 3.25 Calculated C H − v Hugoniots for PETN and HMX. Functional form of C H has almost no dependence on the specific heat function

Fig. 3.26 Calculated p − v Hugoniots for PETN and HMX

96

K. Nagayama et al.

Fig. 3.27 Calculated T − v Hugoniots for PETN and HMX

while those of HMX are somewhat shifted to higher shock velocities. One may see from this figure that dependence of u s − u p Hugoniot on the functional form of the Grüneisen parameter is also insensitive to the Hugoniot functions. In other words, it is very difficult to obtain information on the Grüneisen parameter by the measurement of the Hugoniot data which has been known for many inert solids. One may note, however, that our calculation does not include the rigidity effects, so that u s − u p Hugoniot will change slightly. Even so, insensitivity of u s − u p Hugoniot to the functional form of specific heat and of the Grüneisen parameter remains valid, if one includes rigidity effects. One may note that Fig. 3.29 contains the estimated Neumann spike point as well as the published C-J point [58]. At least for these two explosives, it is found that the tangent to the initial state of the Hugoniot is directed toward the C-J point. While one may see that the non-linearity of u s − u p Hugoniot will determine the difference between the two points, C-J and the Neumann spike.

3 Description of Detonation Phenomena

Fig. 3.28 Dependence of T − v Hugoniots for PETN on specific heat function

Fig. 3.29 Calculated Us − U p Hugoniots for PETN and HMX

97

98

K. Nagayama et al.

3.5.3 Interpretation of Unreacted Hugoniot Data Starting from the isothermal compression data, we have now arrived at the equation of state of the Grüneisen type capable of calculating the shock Hugoniot curve, etc. We have compared the theoretical prediction with the experimental data for PETN and HMX. Experimental determination of the shock Hugoniot for condensed explosive is not very easy, whose traditional procedures are reviewed in Chap. 4. Furthermore, it seems a possible partial reaction at the generated shock front. Probably due to these situations, the scatter of the available data looks rather large compared with those of inert materials. In order to understand the feasibility of the data which are those with no appreciable reaction at the front, we will investigate the Hugoniot compression curves shown in Fig. 2.5 of Chap. 2. Figure 2.3 depicting the internal energy as a function of specific volume covering the expansion region as well as the shock compression region. Internal energy level at the highly expanded region relative to the initial state is written as (A/B)2 /2, for the empirical linear relation, Eq. 2.12 which is an approximation of cohesive energy of the material. Examples are shown in Table 2.1 The situation for chemical compounds is somewhat different from those of elements. For these materials, the value ( A/B)2 /2 in the empirical linear relation is found to correspond not to the cohesive energy but to the heat of sublimation of the material. Some of the typical examples of the compounds are shown in Table 1 including NaCl and KI. In the table, the value of (A/B)2 /2 are compared with two kinds of ex p energies, i.e., the cohesive energy, −E ξ , and the heat of sublimation, Hsub . One ex p 2 may notice that ( A/B) /2 is much closer to Hsub than to −E ξ . That is, the cohesive energy is much larger than that of sublimation. This result denotes the increase in internal energy acquired by shock compression covered by the linear relation could reach not the complete decomposition but at least the possible sublimation. If the shock data for condensed phase high explosive are the one with no reaction, one may make the same analysis similar to the one above. In the case of high explosives, Fig. 3.29 shows that the u s − u p relationship is not linear but slightly curved. Therefore, the relationship can approximately be given by u s = A + Bu p + Cu 2p ,

(3.86)

where the relationship contains a quadratic term. Although the shock Hugoniot data shown in Fig. 3.29 and the calculated theoretical curves show some curvature, an approximate linear relation for two explosives is derived between the range of the particle velocity 0–3 km/s, and the estimated ex p parameters, A, B and C are used to compare the two energies, −E ξ and Hsub . In order to estimate an approximate value for the cohesive energy from the quadratic equation by Eq. 3.43, we will calculate the value of particle velocity, u p

3 Description of Detonation Phenomena

99

Table 3.9 Rodean’s approximation to the sublimation energy Material A (km/s) B C (km/s)−1 Δε (km/s)2 name NaCl KI PETN HMX

3.40 1.787 2.71 2.70

1.35 1.372 1.93 2.09

0.0 0.0 −0.066 −0.079

3.173 0.8475 0.898 0.756

exp

−E ξ (km/s)2

Hsub (km/s)2

13.08 3.780 10.10 10.50

2.919 0.9379 0.481 0.594

at the shock velocity, u s = 0. This is the condition where specific volume becomes infinite. An averaged quadratic function for two explosives was calculated, and the internal energy value was derived by the Rankine Hugoniot relation Δε =

1 2 u . 2 p

(3.87)

Results for the explosives are included in Table 3.9. The estimated value of Δε for ex p high explosives is again much closer to Hsub than to −E ξ . The agreement of Δε ex p and Hsub is not very good, but these values are far from that of −E ξ . Experimental ex p values of Hsub and −E ξ for high explosives are from [68–70]. The result indicates that the available shock Hugoniot data may not be appreciably influenced by the possible partial reaction at the wave front.

References 1. Brinkley Jr SR (1951) The equation of state for detonation gases. In: Proceedings of conference on the chemistry and physics of detonation (first detonation symposium), Office of Naval Research, Washington, DC, 11–12 Jan 1951, pp 72–78 2. Kistiakowsky GB (1951) Problems and future developments. In: Proceedings of conference on the chemistry and physics of detonation (first detonation symposium), Office of Naval Research, Washington, DC, 11–12 Jan 1951, pp 105–106 3. Kirkwood JG (1951) Theoretical developments in detonation. In: Proceedings of conference on the chemistry and physics of detonation (first detonation symposium), Office of Naval Research, Washington, DC, 11–12 Jan 1951, pp 107–116 4. Campbell AW, Malvin ME, Boyd Jr TJ, Hull JA (1955) Technique for the measurement of detonation velocity. In: Proceedings of second ONR symposium on detonation (second detonation symposium), Office of Naval Research, White Oak, Maryland, 11 Feb 1955, pp 136–150 5. Price D (1951) Recent work at NOL. In: Proceedings of conference on the chemistry and physics of detonation (first detonation symposium), Office of Naval Research, Washington, DC, 11–12 Jan 1951, pp 22–30 6. Fickett W, Cowan RD (1955) Calculation of the detonation properties of solid explosives with the Kistiakowsky-Wilson equation of state. In: Proceedings of second ONR symposium on detonation (second detonation symposium), Office of Naval Research, White Oak, Maryland, 11 Feb 1955, pp 383–403

100

K. Nagayama et al.

7. Cowan RD, Fickett W (1956) Calculation of the detonation properties of solid explosives with the Kistiakowsky Wilson equation of state. J Chem Phys 24(5):932–939 8. Hirschfelder JO, Curtiss CF, Bird RB (1964) The molecular theory of gases and liquids. Wiley, New York 9. Kihara T, Hikita T (1953) Molecular theory of detonation. J Ind Explos Soc (Sci Technol Energetic Mater) 13(2):106–113 (in Japanese) 10. Kihara T (1953) Virial coefficients and models of molecules in gases. Rev Mod Phys 25(4):831– 843 11. Jacobs SJ (1969) On the equation of state for detonation products at high density. In: Twelfth symposium (international) on combustion. The Combustion Institute, pp 501–511 12. Deal WE (1955) Measurement of Chapman-Jouguet pressure for explosives. In: Proceedings of second ONR symposium on detonation (second detonation symposium), Office of Naval Research, White Oak, Maryland, 11 Feb 1955, pp 327–342 13. Mader CL (1979) Numerical modeling of detonations. University of California Press, Berkeley and Los Angeles, California, pp 412–448 14. Cowperthwaite M, Zwisler WH (1976) The JCZ equations of state for detonation products and their incorporation into the TIGER code. In: Sixth international detonation symposium, Coronado, CA. Office of Naval Research, Arlington, VA, pp 162–172 15. Hobbs ML, Baer MR, McGee BC (1998) Extension of the JCZ product species database. In: Proceedings of 11th international detonation symposium, Snowmass, CO. Office of Naval Research, Arlington, VA, pp 958–968 16. Heuze O, Bauer P, Presles HN, Brochet C (1985) The equation of state of detonation products and their incorporation into the QUATUOR code. In: Proceedings of eighth international detonation symposium, Albuquerque, NM. NSWC, White Oak, MD, pp 762–769 17. Chung WK, Lu BC-Y (1985) Calculation of detonation products by means of the CS hardsphere equation of state. In: Proceedings of eighth international detonation symposium, Albuquerque, NM. NSWC, White Oak, MD, pp 805–814 18. Xiong W (1985) Detonation properties of condensed explosives computed with the VLW equation of state. In: Proceedings of eighth international detonation symposium, Albuquerque, NM. NSWC, White Oak, MD, pp 769–804 19. Tanaka K (1985) Detonation properties of high explosives calculated by revised Kihara-Hikita equation of state. In: Proceedings of eighth international detonation symposium, Albuquerque, NM, 15–19 July 1985. NSWC, pp 548–557 20. Hobbs ML, Brundage AL, Yarrington CD (2014) JCZS2i: an improved JCZ database for EOS calculations at high temperature and pressure. SAND2014-15321C 21. Kury JW, Hornig HC, Lee EL, McDonnell JL, Ornellas DL, Finger M, Strange FM, Willkins ML (1965) Metal acceleration by chemical explosives. In: Proceedings of fourth symposium (international) on detonation, Office of Naval Research, White Oak, MD, 12–15 Oct 1965, pp 3–13 22. Lee EL, Hornig HC, Kury JW (1968) Adiabatic expansion of high explosive detonation products. UCRL-50422. Lawrence Livermore National Laboratory, Livermore, CA 23. Lee EL, Hornig HC (1969) Equation of state of detonation product gases. In: Proceedings of 12th symposium combustion (1969), pp 493–499 24. Hornberg H, Volk F (1989) The cylinder test in the context of physical detonation measurement methods. Propellants Explos Pyrotech 14:199–211 25. Souers PC, Haselman Jr LC (1994) Detonation equation of state at LLNL, 1993. Energetic Materials Center, Lawrence Livermore National Laboratory, 1 Mar 1994 26. Kubota S (1996) Characteristics of shock compaction assembly using converged underwater shock waves. Thesis, Kumamoto University, Kumamoto 27. Dobratz BM (1981) LLNL explosives handbook, properties of chemical explosives and explosive simulants. UCRL-52997, Chap 8 8-23 28. Tanaka K, Hikita T (1975) The estimation of detonation properties using Kihara-Hikita equation of state. Kogyo Kayaku (Sci Technol Energetic Mater) 36(4):210–217 (in Japanese)

3 Description of Detonation Phenomena

101

29. Mader CL (1967) Fortran BKW: a code for the detonation properties of explosives. Los Alamos Scientific Laboratory Report LA 3704 30. White WB, Johnson SM, Dantzig GB (1958) Chemical equilibrium in complex mixtures. J Chem Phys 28:751–755 31. Gordon S, Macbride BJ (1971) N71-37775 NASA, No. 273 32. Walsh JM, Christian RH (1955) Equation of state of metals from shock wave measurements. Phys Rev 97(6):1544–1557 33. Tanaka K (1982) The study of detonation properties of high explosives using the intermolecular potential model-I. Derivation of the equation of state of gaseous detonation products. Kogyo Kayaku (Sci Technol Energetic Mater) 43(4):239–248 (in Japanese) 34. Tanaka K (1982) The study of detonation properties of high explosives using the intermolecular potential model-II. Detonation properties of high explosives yielding only gaseous products. Kogyo Kayaku (Sci Technol Energetic Mater) 43(4):335–343 (in Japanese) 35. Tanaka K (1983) The study of detonation properties of high explosives using the intermolecular potential model (III). Equation of state of solid carbon. Kogyo Kayaku (Sci Technol Energetic Mater) 44(1):36–43 (in Japanese) 36. Tanaka K (1983) The study of detonation properties of high explosives using the intermolecular potential model (IV). A repulsive-force coefficient of carbon monoxide. Kogyo Kayaku (Sci Technol Energetic Mater) 44(1):44–47 (in Japanese) 37. Tanaka K (1983) The study of detonation properties of high explosives using the intermolecular potential model (V). Detonation properties of CHNO explosives. Kogyo Kayaku (Sci Technol Energetic Mater) 44(3):134–147 (in Japanese) 38. Tanaka K (1983) The study of detonation properties of high explosives using the intermolecular potential model-VI. The effects of inert additives on liquid explosives. Kogyo Kayaku (Sci Technol Energetic Mater) 44(3):148–153 (in Japanese) 39. Tanaka K (1985) The study of detonation properties of high explosives using the intermolecular potential model-VII. Boron and fluoride explosives. Kogyo Kayaku (Sci Technol Energetic Mater) 46(3):128–134 (in Japanese) 40. Tanaka K (1985) The study of detonation properties of high explosives using the intermolecular potential model-VIII. Aluminum loaded explosives. Kogyo Kayaku (Sci Technol Energetic Mater) 46(3):135–141 (in Japanese) 41. Tanaka K (2003) Detonation properties of fireworks by the revised Kihara-Hikita equation of state. Sci Technol Energetic Mater 64(4):167–174 (in Japanese) 42. Tanaka K (1983) Detonation properties of condensed explosives computed using the KiharaHikita-Tanaka equation of state. National Chemical Laboratory for Industry, Tsukuba Research Center 43. Nagayama K, Kubota S (2003) Equation of state for detonation product gases. J Appl Phys 93:2583–2589 44. Lee, E, Finger M, Collins W (1973) JWL equation of state coefficients for high explosives, UCID-16189. Lawrence Livermore National Lab. (LLNL), Livermore, CA 45. Ficket W, Davis WC (1979) Detonation. University of California Press, Berkeley 46. Nagayama K, Kubota S (2004) Approximate method for predicting Chapman-Jouguet state for condensed explosives. Propellants Explos Pyrotech 29(2):118–123 47. Kubota S, Saburi T, Ogata Y, Nagayama K (2010) Numerical simulations of detonation phenomena in PETN by systematic equation of state for detonation products. Sci Technol Energetic Mater 71–2:44–50 48. Kubota S, Saburi T, Wada Y, Nagayama K (2010) Analysis and prediction of initial state dependence for C-J state using unified EOS. Sci Technol Energetic Mater 77–2:23–27 49. Mader CL (1998) Numerical modeling of explosives and propellants. CRC, Boca Raton, FL 50. Saenz JA, Stewart DS (2008) Modeling deflagration-to-detonation transition in granular explosive pentaerythritol tetranitrate. J Appl Phys 104:043519-1–043519-14 51. Zukas JA, Walters WP (1997) Explosive effects and applications. Springer, Berlin, pp 115–135 52. Sheffield SA, Gustavson RI, Anderson MU (1997) Shock loading of porous high explosives, chap 2. In: High-pressure shock compression of solids IV. Springer, New York, pp 23–61

102

K. Nagayama et al.

53. Kubota S, Saburi T, Ogata Y, Nagayama K (2007) State relations for a two-phase mixture of reacting explosives and applications. Combust Flame 151:74–84 54. Nagayama K (2011) Introduction to the Grüneisen EOS and shock thermodynamics. Kindle ebook, Amazon 55. Kubota S, Saburi T, Ogata Y, Nagayama K (2010) Numerical modeling of shock initiation in PETN by unified EOS. Sci Technol of Energetic Mater 71(4):91–97 56. Sewell TD, Menikoff R (2004) Complete equation of state for β-HMX and implications for initiation. In: Shock compression in condensed matter-2003, CP-706, pp 157–160 57. Wescott BL, Stewart DS, Davis WC (2005) Equation of state and reaction rate for condensedphase explosives. J Appl Phys 98:053514-1–053514-10 58. Marsh SP (ed) (1980) LASL shock Hugoniot data. University of California, Berkeley, London 59. Peiris SM, Piermarini GJ (2008) Static compression of energetic materials, shock wave and high pressure phenomena, chap 3. Springer-Verlag, Berlin Heidelberg 60. Olinger B, Hallack PM, Cady HH (1975) The isothermal linear and volume compression of pentaerythritol tetranitrate (PETN) to 10 GPa (100 kbar) and the calculated shock compression. J Chem Phys 62:4480–4483 61. Yoo C-S, Cynn H, Howard WM, Holmes N (1998) Equations of state of unreacted high explosives at high pressures. In: Proceedings of eleventh symposium (international) on detonation, ONR-33300-5, Snowmass, CO. Office of Naval Research, Arlington, VA, pp 951–957 62. Nagayama K (1994) New thermal variables of condensed matter at high pressures and temperatures. J Phys Soc Jpn 63:3729–3736 63. Lambourn BD, Whitworth NJ, Handley CA, James HR (2006) An improved EOS for nonreacted explosives. In: Shock compression in condensed matter-2005, CP-845, pp 165–168 64. Handley CA (2006) The CREST reactive burn model. In: Proceedings of thirteenth symposium (international) on detonation, ONR-351-07-01, Norfolk, VA. Office of Naval Research, Arlington, VA, pp 864–870 65. Setchell RE, Taylor PA (1988) A refined equation of state for unreacted hexanitrostilbene. J Energetic Mater 6(3–4):157–199 66. Baytos JF (1979) Specific heat and thermal conductivity of explosives, mixtures, and plasticbonded explosives determined experimentally. LASL Preport LA-8034-MS 67. Cady HH (1972) Coefficients of thermal expansion of pentaerithritol tetranitrate and hexahydro1,3,5-trinitri-s-triazine (RDX). J Chem Eng Data 17:369–371 68. Saw CK (2002) Kinetics of HMX and phase transitions: effects of grain size at elevated temperatures. In: Proceedings of the twelfth international detonation symposium, pp 198–203 69. Dobratz BM (1985) LLNL explosive handbook, properties of chemical explosives and explosive simulants. UCRL-52997 70. Rosen JM, Dickinson C (1969) Vapor pressures and heats of sublimation of some high melting organic explosives. NOLTR 69-67

Chapter 4

Measurements of Shock and Detonation Phenomena Yukio Kato and Kenji Murata

Abstract Measurements of shock and detonation phenomena are very difficult task because of destructive nature and extremely short duration of events. The experimental data are very important in verifying the theoretical description of phenomena. The advance in theoretical model and numerical simulation depends strongly on the progress in various experimental methods. During last decades, nanosecond timeresolved measurements of detonation pressure, particle velocity and temperature have become possible owing to the rapid progress of the experimental devices. In this chapter, the experimental diagnostics such as pressure and particle velocity gauges, laser velocity interferometer and optical pyrometer are presented. Actually, the experimental data of detonation temperature are very scarce. Detonation temperatures measured for various liquid and solid explosives are summarized and compared with theoretical values calculated using various types of equation of states. Underwater explosion test is a valuable tool to access the energy content and performance of explosives in longer time scale. Characteristics of the underwater explosion phenomena and measurements of underwater shock wave and bubble pulse are described. Keywords Pressure gauge · Particle velocity gauge · Laser velocity interferometer · Optical pyrometer · Underwater explosion

Y. Kato Fukushima, Japan e-mail: [email protected] K. Murata (B) Akita University, Akita, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Kubota (ed.), Detonation Phenomena of Condensed Explosives, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-5307-1_4

103

104

Y. Kato and K. Murata

4.1 Experimental Methods Because of destructive nature and extremely short duration of events, measurements of shock and detonation phenomena are very difficult task. The current state of knowledge of shock and detonation phenomena is largely dependent on the theoretical model and numerical simulation. The experimental data play a crucial role in verifying and grounding the various theoretical descriptions. The advance in the theoretical model and numerical simulation is direct results of the progress in the experimental methods. During the last decades, several nanoseconds time-resolved measurements of shock and detonation wave properties have become possible owning to the considerable progress in a variety of experimental techniques such as pressure and particle velocity gauges, laser velocity interferometer and optical pyrometer. The experimental results combined with numerical simulation provided detailed description of shock and detonation wave properties. In this section, widely used experimental methods are presented.

4.1.1 Production of Planar Shock Wave To perform precise measurements, it is required to load planar shock wave onto target sample. Most widely used methods to generate planar shock wave are based on explosive plane-wave generator or precisely controlled projectile impact.

4.1.1.1

Explosive Plane-Wave Generator

The explosive plane-wave generator is designed to generate a planar detonation wave so as to produce a planar shock wave in a target sample. The explosive plane-wave generator is composed of a hollow cone of fast explosive of detonation velocity D f which is fitted over a cone of slow explosive of detonation velocity Ds as shown in Fig. 4.1. If the angle of the cone θ is chosen according to the relation cos θ = Ds/D f , a planar detonation wave is generated in a cone of slow explosive. Then, a planar detonation is generated in attached explosive pad by a planar detonation propagating in the cone of slow explosive. When the sample is in direct contact with explosive pad (Fig. 4.1a), shock pressure attainable in the sample depends on shock impedance of the sample material and detonation pressure of explosive pad. Shock pressure of several tens of GPa is generated when explosive pad of high detonation pressure is directly in contact with high shock impedance materials. To obtain higher shock pressure in the sample, explosive plane-wave generator is used to accelerate a flyer plate which impacts a target sample (Fig. 4.1b). This technique can generate shock pressure up to a few hundreds GPa. Graham [1] showed a summary of experimental arrangements of shock loading systems using explosive plane-wave generators to achieve pressure from 3 to 80 GPa.

4 Measurements of Shock and Detonation Phenomena

105

Fig. 4.1 Plane-wave generators (a) and (b)

4.1.1.2

Gun

Smooth bore gun is widely used to accelerate projectile to preselected velocity and impact it upon sample with precise alignment between the impacting surfaces. The impact technique using smooth bore gun has become the standard loading technique for the precise experiments because of its versatility and high precision. Figure 4.2 shows a schematic illustration of gun experiments. The impact takes place very close to the muzzle of gun barrel in an impact chamber which is evacuated before the shot to prevent air from cushioning the impact. The electrical probes measure the tilt at impact and projectile velocity. Three types of guns are used according to velocity range; gas gun, powder gun and two-stage light gas gun. Gas gun operates by sudden release of compressed gas to

Fig. 4.2 Schematic illustration of gun experiments

106

Y. Kato and K. Murata

Fig. 4.3 Schematic of the operation of two-stage light gas gun

accelerate projectile. Impact velocity to about 0.4 km/s is attained with air as driving gas, and helium can produce velocity to over 1 km/s. Shock pressure up to a few tens of GPa can be attained. Gas guns are most widely used because of its safety. Powder gun operates using high pressure gas produced by reaction of gun propellant to propel projectile. Impact velocity to about 2.5 km/s and shock pressure in target sample material to about 100 GPa can be attained in the experiments using powder gun. Two-stage light gas gun is designed to achieve much higher impact velocity than powder gun. To attain extremely high impact velocity, gun propellant is used to accelerate a large piston which compress a reservoir of light gas (hydrogen or helium) to high pressure. Then compressed light gas accelerates the smaller projectile over long acceleration distance. Figure 4.3 shows schematic of the operation of two-stage light gas gun. Impact velocity over 8 km/s can be attained with shock pressure to 1 TPa in high impedance material.

4.1.2 Pressure Measurements The development of pressure gauge has enabled time-resolved measurements of pressure profiles in shock and detonation waves. Two types of material, piezoresistant and piezoelectric material, have been used as sensing elements of pressure gauges.

4.1.2.1

Piezoresistant Pressure Gauge

The electrical resistance of manganin (copper alloy containing manganese and nickel) is very sensitive to pressure. It was demonstrated that the fractional change in resistance ΔR/R0 (ΔR: resistance change, R0 : initial resistance) of manganin increases

4 Measurements of Shock and Detonation Phenomena

107

Fig. 4.4 Structure of typical manganin pressure gauge and pulsed constant current supply circuit

almost linearly with pressure up to the pressure greater than 100 GPa [2], and thus, manganin has been widely used as sensing element of pressure gauge. Manganin pressure gauge is calibrated in controlled planar shock loading by measuring the fractional change in resistance ΔR/R0 as a function of shock pressure. Typical manganin gauge is consisted of a manganin foil of 10 µm thick (effective area: 1 × 4 mm2 , resistance: 50Ω) and four-probe type copper leads formed by photo-etching, which are sandwiched by polyimide film of 12.5 µm thick. A constant current is provided to the gauge by pulsed constant current supply. The circuit of the pulsed constant current supply is selected so as to keep constant current during the measurements of resistance change. Structure of typical manganin pressure gauge and pulsed constant current supply circuit are presented in Fig. 4.4. The measured voltage change, which is proportional to the resistance change of manganin, provides the change of pressure. Manganin pressure gauge has been successfully used for a number of years to measure pressure in inert solid materials. Wackerle et al. [3] and Burrows et al. [4] applied manganin pressure gauge embedded in solid explosive to measure pressure profile of detonation wave. The use of multiple gauges embedded at multiple locations enabled the measurements of pressure profiles in the shock initiation process of solid explosives. The important progress of embedded manganin pressure gauge technique allowed detailed quantitative study of the shock initiation process of detonation, coupled with the numerical simulation using various reaction models.

4.1.2.2

Piezoelectric Pressure Gauge

Piezoelectric materials have been widely used as active elements of pressure gauges. Since the discovery of piezoelectricity in polyvinylidene-fluoride (PVDF) by Kawai [5], PVDF polymer has become most attractive candidate for sensing elements of pressure gauges. The structural and electrical characteristics of PVDF were reviewed by Graham [6]. Bauer [7, 8] developed technology to process PVDF such that its physical properties exhibit reproducibility approaching that of piezoelectric single crystals. PVDF polymer is mechanically stretched into a thin film of typically 25 µm thick

108

Y. Kato and K. Murata

to increase the concentration of polar phase. Then, PVDF film is subjected to special electrical poling procedures to achieve uniform, precisely specified polarization. The piezoelectric response of PVDF pressure gauge was determined by the precisely controlled shock loading experiments. It was demonstrated that the electrical charge Q e generated by shock loading is a continuous function of shock pressure up to 35 GPa [9], and that piezoelectric response is independent of loading path.∑The loading is expressed by V = Q e / C, electrical charge Q e generated by pressure ∑ C is total capacitance of measuring system where V is measured voltage and composed of PVDF pressure gauge, connecting cable and measuring circuit. Then the pressure is determined using the voltage provided by the measuring system. The PVDF pressure gauge has the advantage of self-powered, yielding large voltage signal upon pressure loading. Murata and Kato [10] measured pressure profile of detonation in nitromethane (NM) using PVDF pressure gauge. The PVDF pressure gauge is consisted of a PVDF film of 10 µm thick and 5 mm square, and electrodes made of copper foil, which are sandwiched with polyimide films. Structure of PVDF pressure gauge and equivalent circuit for measurement system are illustrated in Fig. 4.5. The experimental arrangement for detonation pressure measurements is presented in Fig. 4.6. The PVDF pressure gauge was placed on a polymethyl-methacrylate (PMMA) block of 50 mm thick and glued with a PMMA plate of 1 mm thick. PMMA was selected because the shock impedance of PMMA is close to that of detonation products of NM. Sample NM was confined in a steel tube (38 mm inner diameter, 5 mm thick and 150 mm long). Detonation velocity was measured by using optical fiber probes to confirm steady detonation. As the PVDF pressure gauge measured the pressure transmitted into PMMA, detonation pressure was evaluated from the measured pressure by using the impedance match method. Figure 4.7 shows the pressure profile of detonation in NM measured by PVDF pressure gauge and the impedance match method to evaluate detonation pressure. The measured pressure of leading shock wave was much lower than the predicted von Neumann spike pressure, and the pressure profile in the reaction zone was not resolved because the time resolution of PVDF pressure gauge was not sufficient to resolve it. The distinct change in the slope of pressure profile at the pressure corresponding to CJ state and following pressure decay in Taylor wave were observed.

Fig. 4.5 Structure of PVDF pressure gauge and equivalent circuit

4 Measurements of Shock and Detonation Phenomena

109

Fig. 4.6 Experimental arrangement for detonation pressure measurements using PVDF pressure gauge

The measured pressure corresponding to CJ state is on the Hugoniot curve of PMMA. The CJ pressure is on the Rayleigh line of detonation in NM. The CJ pressure is obtained as the intersection between the mirror Rayleigh line which passes the measured pressure corresponding to CJ state and the Rayleigh line of NM detonation. As the Rayleigh line of detonation in NM and the CJ isentrope of detonation products are very close in the proximity of CJ state, the CJ pressure can be evaluated with good precision. The CJ pressure of detonation in NM was measured to be 12.5 GPa, which was in good agreement with the CJ pressures calculated using various types of equation of state (EOS).

4.1.3 Particle Velocity Measurements 4.1.3.1

Electromagnetic Particle Velocity Gauge

Following the pioneering work of Zaitsev et al. [11], Jacobs and Edwards [12] developed electromagnetic particle velocity (EMPV) gauge technique. The principle of EMPV gauge is based on Faraday’s law of electromagnetic induction. Schematic illustration for EMPV gauge measurements is presented in Fig. 4.8. A rectangular loop of thin aluminum foil is embedded in the sample medium to be studied. The base of the loop of length l is the active sensing element which is connected to the leads in the shape of a squared letter U. For a constant magnetic field of flux density B, the induced voltage V in the loop is proportional to the time rate change of flux

110

Y. Kato and K. Murata

Fig. 4.7 Pressure profile of detonation in NM measured by PVDF pressure gauge and the impedance match method to evaluate detonation pressure

within the loop. When the magnetic field is normal to the enclosing area and the change in flux is due to the motion of the base of the loop normal to the magnetic field, the induced voltage is given by V = k × B ×l ×u

(4.1)

where k is constant depending on the resistance of the gauge and termination resistance of the oscilloscope, and u is the velocity of the base. The velocity of the base is the same as the sample medium soon after the passage of shock or detonation wave. Thus, the observed voltage is a direct measure of the particle velocity of the sample medium. The method of generating a constant magnetic field is to mount a pair of Helmholtz coils at either side of sample to be studied. Helmholtz coils used in EMPV gauge produce field strength of 30–100 mT and are powered by regulated DC supply.

Fig. 4.8 Schematic illustration for EMPV gauge measurements

4 Measurements of Shock and Detonation Phenomena

111

Fig. 4.9 Schematic of EMPV gauge package and experimental arrangement

Cowperthwaite and Rosenberg [13] carried out particle velocity measurements of solid explosives using multiple EMPV gauges placed at multiple positions. The measured particle velocity profiles were used for Lagrange analysis to characterize the shock initiation and detonation process in solid explosives. Vorthman et al. [14] developed the EMPV gauge package composed of five EMPV gauges, which allowed the measurement of particle velocity at five locations in the sample explosive. The EMPV gauge package is consisted of patterned aluminum foil gauge elements of 18 µm thick, sandwiched between insulation films of 25 µm thick. The EMPV gauge package is very compact, and its size is smaller than 10 mm square. The EMPV gauge package is inserted into two precisely machined explosive wedge samples at an angle of 30°. Thus, 2 mm separation of gauge pairs in the package gives 1 mm spacing of the gauge elements in the direction of particle motion. Schematic of EMPV gauge package and experimental arrangement are shown in Fig. 4.9. The recent EMPV gauge package [15] is consisted of 10 gauge elements made of 5 µm thick aluminum foil. The important progress of EMPV gauge technique enabled particle velocity measurements with time resolution of about 10 ns. The embedded EMPV gauge technique has been widely used to investigate the shock initiation process of explosives [16, 17]. The measured precise particle velocity profiles have contributed to refine various reaction rate models.

4.1.3.2

Laser Velocity Interferometry

Laser velocity interferometry has been widely used to measure particle velocity in detonation wave and wall expansion velocity in cylinder test. An important advantage of laser velocity interferometry over embedded gauging is that the measurements with extremely high time resolution are possible. Laser velocity interferometry has become most powerful tool in the study of detonation phenomena. Various types of

112

Y. Kato and K. Murata

interferometers are used to make measurements in both inert and energetic materials; VISAR [18, 19], ORVIS [20] and Fabry–Perot [21]. All these interferometers produce interference fringe shifts which are proportional to Doppler shift of the laser light reflected from moving surface. Laser light of wavelength 514.5 nm is usually used. In VISAR interferometer, the laser light reflected from moving surface is split into two beams to form two legs of a Michelson interferometer. The light in one leg is delayed with respect to the other light by passing it through etalons or a pair of lenses. By mixing two beams, the interference light fringes are formed and the fringes shift when the interface is accelerated. The velocity of the moving surface u(t) is related to the fringe count F(t) as a function of time by ( λ0 F(t) τ) ( = u t− 2 2τ (1 + δ) 1 +

Δγ γ

)

(4.2)

where τ is temporal difference between two interfering legs of the interferometer, λ0 is wavelength of laser light, (1 + δ) is a correction term for the frequency dispersion of the etalon and (1 + Δγ /γ ) is a correction factor for the change in index of refraction of the shocked window material. The ORVIS interferometer is essentially similar to VISAR interferometer except that an electronic streak camera is used to record interference fringe motion. Time resolution of VISAR with photomultiplier tubes and digitizers for recording the fringes has a time resolution of 1–2 ns. Time resolution of VISAR and ORVIS using electronic streak camera is estimated to be less than 0.5 ns. In Fabry–Perot interferometer, the Doppler shifted light reflected from moving surface passes through Fabry–Perot plates separated by a precise distance. A Doppler shift in wavelength of the reflected light produces a corresponding shift in fringe ' spacing. The fringe diameter is shifted to a new value d1 from its static value d1. The velocity of the moving surface u is calculated by cλ0 u= 4L

} {( ' ) d12 − d12 ( 2 ) +m d2 − d12

(4.3)

where c is velocity of light, d2 is static fringe diameter of the next fringe and m is the number of fringes shifted at shock arrival time which must be determined from some previous knowledge of the moving surface velocity. Laser velocity interferometry is the only experimental tool that has the temporal resolution required to measure the particle velocity profile in the reaction zone of explosive. The velocity of explosive/window interface is measured to obtain the particle velocity profiles in the reaction zone. A very thin layer (submicron thick) of aluminum is vapor deposited on the diffuse surface of window. As window material, PMMA, fused silica, sapphire and lithium fluoride (LiF) are used depending on the

4 Measurements of Shock and Detonation Phenomena Table 4.1 Reaction zone length determined by particle velocity measurements [23]

Explosive

Fast reaction zone (ns)

113 Slow reaction zone (ns)

PETN

< 10

–

NM

~ 10

50–100 60–80

HMX

~ 20

TNT

~ 80

~ 200

TATB

60–80

240–320

shock impedance of explosives. Sheffield et al. [22] measured the particle velocity of detonating NM/PMMA window interface using VISAR with 1 ns time resolution to obtain the particle velocity profile in the reaction zone of NM. The measured particle velocity profile shows the peak of particle velocity followed by the decay in particle velocity which occurs with both fast (~10 ns) and slow (50–100 ns) components. The time to reach CJ state is estimated to be 50–100 ns. The measured peak particle velocity (~2.2 mm/µs) is much lower than the expected value (~2.7 mm/µs) based on the unreacted shock Hugoniot of NM. This result indicates that even VISAR diagnostics lack the temporal resolution to resolve von Neumann spike. Tarver [23] discussed the particle velocity profiles in the reaction zone for penta-erythritol-tetranitrate (PETN), NM, cyclotetramethylene-tetranitramine (HMX), TNT and triaminotrinitrobenzene (TATB). The reaction zone lengths of these explosives are summarized in Table 4.1. The reaction zone is composed of fast zone corresponding to rapid exothermic reaction, and slow zone corresponding to products equilibration and solid carbon formation. In the fast zone, 80–90% of chemical energy is released, and in the slow zone, 10–20% of chemical energy is released. PETN forms little or no solid carbon in its reaction products and has only fast reaction zone. Even though the interferometers such as VISAR, ORVIS and Fabry–Perot are excellent diagnostics, they have certain disadvantages. These interferometers are very expensive, and require large experimental facility, important data acquisition and analyzing systems, and skilled maintenance. Only a limited number of laboratories can perform the experiments using these interferometers. Recently, Strand et al. [24] invented new type of interferometer Photon-Doppler Velocimeter (PDV). The PVD is a small compact diagnostic that can be used for a broad variety of experiments. The PVD is a displacement interferometer built from high bandwidth telecommunication optical fiber components, high bandwidth detectors and high bandwidth digitizers (20 GHz). The PDV measures the beat frequency of the fringes resulting from mixing the Doppler shifted light from the moving surface with the non-Doppler shifted light from the laser. The frequency of the fringes is the rate at which the surface moves through a distance of λ0 /2. The data are analyzed by using a sliding fast Fourier transform analysis, and the velocity of moving surface u is given by following relation; u = λ0 f /2, where f is frequency of the fringes. Briggs et al. [25] presented principles of PDV and an overview of various experiments carried out using PDV.

114

Y. Kato and K. Murata

4.2 Temperature Measurements of Detonation Phenomena 4.2.1 History of Detonation Temperature Measurements In order to understand fully the detonation phenomena in condensed high explosives, it is necessary to measure the detonation properties such as velocity, pressure and temperature. The important progress of embedded gauges and laser interferometry techniques enabled the precise measurements of particle velocity and pressure. Current phenomenological reactive flow models, when normalized to the measured particle velocity or pressure data, have been able to predict shock initiation process and detonation properties of explosives. However, the temperature is the most important parameter to describe the chemical kinetics in the reaction zone and thermodynamic state of detonation products. Yet, the temperature is the only parameter which is difficult to measure. The measured detonation temperatures are very scarce and for most explosives nonexistent. At present, the detonation properties of condensed explosives are calculated by thermochemical computer codes using various EOSs for detonation products. The calculated detonation temperature is strongly dependent on the choice of EOS. Contrary, the calculated detonation velocity and pressure are much less sensitive to the EOS used. The measurements of detonation temperature are very useful tool as criteria to check the validity of EOS. Time-resolved optical pyrometry is only non-intrusive technique which enables the temperature measurements of detonation phenomena. First attempts to measure the detonation temperatures of high explosives using time-resolved optical pyrometer were performed by Gibson et al. [26] and followed by Voskoboinikov and Apin [27]. Most liquid explosives are transparent in visible and near-infrared range, which enables to record the radiation emitted from propagating detonation front through the unreacted liquid explosive. Dremin and Savrov [28] and Trofimov and Troyan [29] investigated the emissive spectrum of detonation in NM. The spectrum recorded was essentially a continuum, and the distribution of radiation intensity in visible range was very close to that of black body. Thus, the radiation emitted from detonation front in NM was considered to be very close to that of black body. Taking advantage of progress in the opto-electronic technology, Mader [30], Burton et al. [31, 32], Urtiew [33] and Kato et al. [34–38] measured the brightness temperatures of detonation front in various liquid explosives using time-resolved optical pyrometer in visible and near-infrared red range. Temperature measurements of detonation in solid explosives, which are essentially granular and opaque, present various experimental difficulties. Kato et al. [34, 35, 39] carried out temperature measurements of detonation in opaque aluminized NM and observed the temperature increase due to the reaction of aluminum particle in detonation products after the arrival of detonation front at transparent window. These results have demonstrated the possibility to measure the temperature of detonation products through shocked transparent window. Kato et al. [37, 38], Xianchu

4 Measurements of Shock and Detonation Phenomena

115

et al. [40], Huishen et al. [41] and Gogulya et al. [42, 43] measured the brightness temperatures of detonation products of various solid explosives. Recently, Yoo et al. [44, 45], Leal et al. [46] and Leal-Crouzet [47] measured the temperatures in the shock to detonation transition (SDT) and steady-state detonation of NM, tetranitromethane (TNM), PETN single crystal and pressed HMX using timeresolved six wavelength pyrometer in visible and near-infrared range (0.35–0.70 µm and 0.50–1.51 µm) with time resolution of 1–3 ns. Recent very rapid progress of opto-electronics, computer and data processing devices enabled the determination of the true detonation temperature and the emissivity at each wavelength by nonlinear least-squares fitting technique. In order to study the optical properties of shocked NM and detonation products in SDT and steady detonation in NM, Bouyer et al. [48] measured the emissive spectra using fast time-resolved emission spectroscopy with time resolution of 1 ns. The emission spectroscopy measured the radiation intensity at 16 wavelengths between 0.30 and 0.85 µm, with spectral resolution 32 nm. The measured emissive spectra revealed some particularity that could not be recorded by the former pyrometer measurements. To investigate the shock initiation of explosives compressed to relatively low pressure, it is required to measure the temperature below 1000 K. The infrared pyrometer was used by Von Holle et al. [49, 50] to measure the temperatures of shocked solid explosives at explosive/window interface in spectral range 2.0–5.5 µm. Crouzet et al. [51] measured the temperatures of the unreacted NM shocked to the pressure of 7.7 GPa in spectral range 2.5–4.0 µm. Delpuech and Menil [52] applied high speed time-resolved Raman spectrometry to measure the temperatures of NM shocked to the pressure of 2.3–8.5 GPa. The detonation properties of explosives are calculated by thermochemical codes using various EOS such as Becker–Kistiakowsky–Wilson (BKW) EOS [53], Jacobs– Cowperthwaite–Zwisler (JCZ) EOS [54] and Kihara–Hikita–Tanaka (KHT) EOS [55]. BKW-EOS has been extensively used to calculate detonation properties of various solid and liquid explosives. BKW-EOS has empirical constants which are adjusted to fit measured detonation properties. Finger et al. [56] attempted to determine the empirical constants using measured detonation properties and calorimetry data and proposed BKWR-EOS. However, BKWR-EOS as well as original BKWEOS give the CJ temperatures much lower than the measured detonation temperatures. Hobbs and Baer [57, 58] presented BKWS-EOS in which the empirical constants were optimized to fit the measured detonation temperatures and velocities. Fried and Souers [59] optimized the empirical constants based on cylinder test data and proposed BKWC-EOS. The CJ temperatures calculated by BKWCEOS and BKWS-EOS are nearly identical and give reasonable agreement with the measured detonation temperatures. The results of temperature measurements contributed considerably to improve the accuracy of the prediction of the CJ temperature.

116

Y. Kato and K. Murata

4.2.2 Temperature Measurements by Time-Resolved Optical Pyrometer 4.2.2.1

Principle of Temperature Measurements

The principle of temperature measurements by optical pyrometer is consisted of measuring the thermal radiation emitted by detonation phenomena and to convert it into brightness temperature using Plank’s law or into two-color temperature from a set of monochromatic radiance. The monochromatic radiance L(λ0 ) at wavelength λ0 and temperature T is described by Plank’s law when the emissivity is 1: C1 λ−5 ( 0)

L(λ0 , T ) = exp

C2 λ0 T

(4.4)

−1

where C1 and C2 are Plank constant (C1 = 1.19 × 10−16 W m2 sr−1 and C2 = 1.4388 × )10−2 m K). In the experimental conditions considered ( λ0 T < 3.5 × 10−3 m K , Wien’s law can be applied with good precision: L(λ0 , T ) = C1 ×

λ−5 0

× exp

( ) C − λ 2T 0

(4.5)

The radiation energy received by the detector of pyrometer is written in the form: E(λ0 , T ) = S × a × f (λ0 ) × Δλ0 × L(λ0 , T )

(4.6)

where s, a f (λ0 ) and Δλ0 are, respectively, the surface sighted on the sample, view angle of detector, spectral response of detector and bandwidth. The optical pyrometer concentrates the radiation energy E(λ0 , T ) on the detector, and the detector delivers the output voltage V which is proportional to the radiation energy E(λ0 , T ). Thus, the output voltage V is given by V = k × E(λ0 , T ) = k ' × L(λ0 , T ) = k ' × C1 × λ0

(4.7)

where k and k ' are characteristic constant of pyrometer. Using the output voltage V0 measured with black body of known temperature T0 , the brightness temperature Tb at wavelength λ0 is written in the form: 1 1 λ0 − = (lnV0 − lnV ) Tb T0 C2 The true temperature Tt is given by following relation:

(4.8)

4 Measurements of Shock and Detonation Phenomena

1 1 λ0 = + lnε p Tt Tb C2

117

(4.9)

where ε p is the emissivity at wavelength λ0 . To convert the brightness temperatures into true temperature, it is necessary to know the emissivity and hypothesis on its variation with wavelength. Using the brightness temperatures measured at different wavelength, the true temperature and emissivity are deduced by nonlinear least-squares fitting technique assuming linear variation of spectral emissivity with wavelength. The method used for determining two-color temperature is to compare monochromatic radiance at two different wavelengths supposing gray body. The output voltage V1 and V2 delivered by detectors at wavelength λ1 and λ2 are proportional to the monochromatic radiance L(λ1 , T ) and L(λ2 , T ), respectively: V1 = k1 × L(λ1 , T ), V2 = k2 × L(λ2 , T )

(4.10)

where k1 and k2 are characteristic constants of pyrometer. The application of Wien’s law gives the following relation: ( ln

V1 V2

)

( = −5ln

λ1 λ2

) −

) ( ) ( C2 1 k1 1 + ln − T λ1 λ2 k2

(4.11)

The calibration using black body of known temperature T0 provide the output voltage V1 and V2 at wavelength λ1 and λ2 . The two-color temperature Tc is written by the following relation: ( ))/ ( ( )) ( ( ) k1 1 1 1 V1 1 − ln C2 − = ln − Tc To V2 k2 λ1 λ2

4.2.2.2

(4.12)

Description of Multi-wavelength Optical Pyrometer

The experimental set-up to measure the detonation temperature using the timeresolved multi-wavelength pyrometer is shown in Fig. 4.10. The multi-wavelength pyrometer is consisted of optical probe, optical fibers, band-pass filters, detectors, amplifiers and digital recorders. The optical probe collects the radiation emitted from a small area (2–4 mm in diameter) of the sample and forms its image on the head of optical fibers. In the pyrometer which does not use the optical probe, the optical fibers mounted at the back of the transparent window directly collect the radiation emitted from the sample. The optical fibers transmit the radiation to detectors through band-pass filters. Different kinds of detectors allow to cover a large wavelength range (0.4–4.0 µm). Photomultiplier is used in visible range (0.4–0.75 m), and Si PIN photodiode in visible and near-infrared range (0.4–1.1 µm). Ge detector and InGaAs PIN photodiode are used in near-infrared range (1.1–1.8 µm). HgCdTe and InSd detectors

118

Y. Kato and K. Murata

Fig. 4.10 Experimental set-up to measure the detonation temperature using time-resolved multiwavelength pyrometer

are used in mid-infrared range (2.5–4.0 µm), operating at liquid nitrogen temperature (77 K). Band-pass filters, placed in front of the detectors, select each restricted spectral range. The full bandwidth of band-pass filter is 50–100 nm in visible and near-infrared range, and 0.5–1.0 µm in mid-infrared range. The measurable temperature range is 2000–5000 K in visible and near-infrared range, and 700–5000 K in mid-infrared range. The time resolution of the most recent optical pyrometer is 1–3 ns in visible and near-infrared range and less than 10 ns in mid-infrared range. The calibration of the optical pyrometer is performed with a tungsten ribbon lamp (T ~ 2300 K) and a carbon arc (T ~ 3400 K). Recently, black body source is used for calibration of the pyrometer between 850 and 3250 K.

4.2.3 Detonation Temperature of Liquid Explosives 4.2.3.1

Detonation Temperature of Nitromethane

Most of temperature measurements have been carried out for liquid explosives because of its transparency in visible and near-infrared range. The radiation emitted by detonation front can be recorded all along its propagation in sample explosive until the interaction with transparent window. Particularly, NM has been the subject

4 Measurements of Shock and Detonation Phenomena

119

of many investigations because its molecular structure is simple, and its chemical and physical properties are extensively studied. Leal et al. [46] and Leal-Crouzet [47] measured the detonation temperatures of NM using time-resolved six wavelength pyrometers in spectral range 0.5–1.51 µm with time resolution of 3 ns. NM was confined in a polyethylene tube (70 mm diameter, 25 mm long) enclosed by a copper plate on one side, and by a LiF window on the other side. Sample NM was initiated by plate impact method. Figure 4.11 presents the typical record of time-resolved brightness temperature of detonation in NM measured in visible range. The record clearly displays 4 phases. Phase 1–3 correspond, respectively, to SDT, overdriven detonation and steady-state detonation. Phase 4 shows the evolution of the temperature of detonation products after the interaction with transparent window. The signal in phase 3 represents the thermal radiation emitted by detonation front. The signal level in steady-state detonation is kept constant because NM is transparent in the spectral range under 1 µm. The brightness temperatures measured at six wavelengths and the emissivity calculated by least-square fitting technique are shown in Table 4.2. These values represent average over seven experiments with repeatability better than 60 K. The emissivity of detonation front in NM is linear to wavelength and slightly decreases with wavelength. The radiation from detonation front in NM was proved to be very close to black body in nature in the spectral range between 0.5 and 1.51 µm. The detonation temperature of NM is about 3500 K, whereas the shock temperature of unreacted NM is lower than 2000 K at von Neumann spike pressure of about 20 GPa [22, 60]. Therefore, the thermal radiation emitted from detonating NM is substantially stronger than that from shocked but unreacted NM. Under these conditions, the thermal radiation is considered to originate from the reaction zone, although it is measured across the detonation front. Thus, the measured detonation temperature is considered to be identical to CJ temperature of NM. The true detonation temperature of NM is 3600 ± 100 K. Table 4.3 summarizes the various experimental detonation temperatures of NM and the CJ temperatures calculated using various EOSs. The experimental detonation

Fig. 4.11 Typical record of brightness temperature of detonation in NM

120

Y. Kato and K. Murata

Table 4.2 Brightness temperature and emissivity of detonation front in NM [47] Wavelength (µm)

0.50

0.65

0.85

1.10

1.27

1.51

Brightness temp.(K)

3560

3570

3540

3500

3420

3410

Emissivity

0.92

0.91

0.90

0.88

0.86

0.85

temperatures lie in the range 3300–3800 K. Gogulya and Brazhnikov [42] suggested the effect of purity of NM on detonation temperature. The purity of NM is not specified in most of publications. Kato et al. [37] measured detonation temperature in NM of 96% purity using four wavelengths pyrometer between 0.65–0.95 µm with time resolution of 10 ns. Leal-Crouzet [47] measured detonation temperature in NM using high purity NM (>99%) and low purity NM (100 µm), the effect of heat conduction on the measured temperature can be neglected.

4.2.3.2

Detonation Temperature of Nitromethane/Tetranitromethane Mixtures

Detonation products of NM contain large amount of solid carbon, which might be responsible for the black body nature of the detonation front. It is interesting to study the effect of oxygen balance on the radiation properties of detonation products. Kato et al. [36] measured the detonation temperature of NM/TNM mixtures using four wavelengths pyrometer in spectral range 0.650–1.008 µm. The mass fraction of TNM in the mixture was varied from 0.186 to 0.616. Stoichiometric mixture is obtained at TNM mass fraction 0.445. When TNM mass fraction is less than 0.325, detonation products contain solid carbon. The records of brightness temperature of detonation front in NM/TNM mixtures are quite similar to those in NM. The results of brightness temperature measurements are listed in Table 4.4. For all NM/TNM mixtures, it is demonstrated that the measured brightness temperatures show no particular wavelength dependence within

4 Measurements of Shock and Detonation Phenomena

123

Table 4.4 Brightness temperatures of detonation front in NM/TNM mixtures Mass frac.

TMN

0.186

0.278

0.445

0.616

1

Wavelength

0.650 (µm)

3680(K)

3910(K)

4330(K)

3960(K)

2620(K)

Mean temp.

0.783

3710

3920

4330

3940

2640

0.915

3975

–

4470

4100

2660

1.008

3790

4130

4400

4090

2660

–

3740

4020

4380

4020

2650

Fig. 4.14 Comparison between the measured detonation temperatures and the CJ temperatures calculated using various EOSs for NM/TNM mixtures

the accuracy of measurements ±100 K. These results indicate that at the temperature and pressure attained in the detonation of high explosives (2000–5000 K and 10–40 GPa), detonation products have very high optical thickness and radiate like a black body even without solid carbon. The brightness temperature of NM/TNM mixtures increases as the oxygen ratio rises until stoichiometric value and then decreases. The measured brightness temperatures of NM/TNM mixtures are compared with the CJ temperatures calculated using various EOSs in Fig. 4.14. BKWS-EOS provides the CJ temperatures 200–300 K higher than the measured temperatures. BKW-EOS and KHT-EOS give the CJ temperatures 100–800 K lower than the experimental values.

4.2.3.3

Detonation Temperature of Liquid H–N–O Compositions

To investigate the effect of composition on the radiation properties of detonation products, Kato et al. [38] carried out the temperature measurements of detonation in liquid H − N − O compositions using four wavelengths pyrometer described

124

Y. Kato and K. Murata

previously. Liquid explosives studied were hydrazine nitrate (HN)/hydrazine hydrate (HH) and HN/water (H2 O) mixtures. These liquid explosive mixtures were chosen as representative of H − N − O compositions. The compositions of HN/HH and HN/H2 O mixtures are expressed by mass ration of two components. The mixtures were contained in a glass tube to avoid the contact with metal (Fig. 4.15). The initial temperature of HN/H2 O mixtures was maintained at a temperature 5–10 K higher than its fudge point. The records of brightness temperature of detonation front in HN/HH and HN/H2 O mixtures are similar to those in NM. Propagation of steady-state detonation was observed in all mixtures studied. Typical record of brightness temperature measurements for detonation in HN/H2 O mixtures is shown in Fig. 4.16. Temperature decrease caused by localized failure waves is observed for all HN/H2 O mixtures. The measured brightness temperatures of HN/HH and HN/H2 O mixtures are listed in Table 4.5. For all mixtures, the measured brightness temperatures show no particular wavelength dependence within the accuracy of the measurements ±100 K. These results indicate that the detonation products of H − N − O composition are optically thick and radiate as a black body. The measured brightness temperatures of HN/HH and HN/H2 O mixtures are compared with the CJ temperatures calculated by various EOSs in Fig. 4.17. It is shown that the measured brightness temperatures increase linearly with the increase of HN mass ratio. For HN/HH mixtures, the CJ temperatures calculated using BKWS-EOS and KHT-EOS present good agreement with the experimental values. For HN/H2 O mixtures, the CJ temperatures calculated by BKWS-EOS and KHT-EOS are, respectively, about 300 K and 1300 K lower than the experimental values. Urtiew [33] measured the brightness temperatures of detonation front in HN/hydrazine (HY) mixtures. In Table 4.6, the brightness temperatures of detonation front in HN/HY mixtures, TNM, nitroglycerine (NG) and bis-2-fluoro-2,2dinitroethylformal (FEFO) are compared with the CJ temperatures calculated using various EOSs.

Fig. 4.15 Detonation tube used for temperature measurements of HN/HH and HN/H2 O mixtures

4 Measurements of Shock and Detonation Phenomena

125

Fig. 4.16 Brightness temperature of detonation in HN/H2 O (85/15) mixture measured at wavelength 0.85 µm Table 4.5 Brightness temperatures of detonation front in HN/HH and HN/H2 O mixtures 55/45

HN/HH mixture Wavelength

Mean temp

60/40

65/35

70/30

75/25

0.65(µm)

2350(K)

2480(K)

2550(K)

2590(K)

2710(K)

0.75

2300

2450

2490

2580

2660

0.85

2250

2380

2460

2470

2570

0.95

2190

2310

2380

2480

2570

–

2270

2410

2470

2530

2630

HN/H2 O Mixture

80/20

85/15

90/10

95/5

–

Wavelength

0.65(µm)

2650(K)

2740(K)

2790(K)

2900(K)

–

0.75

2630

2730

2780

2860

–

Mean temp

0.85

2580

2700

2760

2870

–

0.95

2350

2560

2620

2690

–

–

2550

2680

2740

2830

–

80

90

Fig. 4.17 Comparison between the measured brightness temperatures and the CJ temperatures calculate using various EOSs for HN/HH and HN/H2 O mixtures

3000

2500

2000

{ {

1500

50

60

70

100

126

Y. Kato and K. Murata

Table 4.6 Detonation temperatures of HN/HY mixtures, TNM, NG and FEFO Theoretical

Experimental Reference HN/HY

Urtiew

[33]

Temp.

Reference

2900(K)

Finger et al. Finger et al.

(79/21)

HN/HY

Urtiew

[33]

2180

(30/70)

TNM

NG

FEFO

Voskboinikov and Apin

[27]

3100

EOS

Temp.CJ

[56]

BKW

1453(K)

[56]

BKWR

2332

Finger et al.

[56]

JCZ3

2695

Tanaka

[63]

KHT

2598

Hobbs and Bear

[57]

BKWS

2700

Finger et al.

[56]

BKW

1200

Finger et al.

[56]

BKWR

1883

Tanaka

[63]

KHT

1610

Hobbs and Baer

[57]

BKWS

2300

Mader

[30]

BKW

1621

Tanaka

[55]

KHT

1645

Mader

[30]

2800

Yoo et al.

[44]

BKWR

2200

Urtiew

[33]

2840

Yoo et al.

[44]

JCZ3

2500

Kato et al.

[36]

2650

Yoo et al.

[44]

BKWC

2830

Yoo et al.

[44]

2950

Hobbs and Baer

[58]

BKWS

2860

Gibson et al.

[26]

4020

Mader

[30]

BKW

3216

Voskoboinikov and Apin

[27]

4000

Hardesty

[64]

JCZ3

4660

Tanaka

[63]

KHT

3871

Mader

[30]

3470

Hobbs and Baer

[58]

BKWR

3750

Burton et al.

[32]

4260

Hobbs and Baer

[58]

BKWS

4550

Gogulya et al.

[65]

4100

Finger et al.

[56]

BKWR

2870

Finger et al.

[56]

BKWR

3510

Finger et al.

[56]

JCZ3

4283

Tanaka

[63]

KHT

3810

Bobbs and Bare

[58]

BKWS

4390

4.2.4 Detonation Temperature of Solid Explosives For homogeneous explosives such as liquid explosives and defect-free single crystals, detonation temperature measurements are more accurate. Yoo et al. [44, 45] carried out the temperature measurements of detonation in PETN single crystals using timeresolved six wavelengths optical pyrometer in the spectral range between 0.4 and 0.7 µm with time resolution of 1 ns. As the unshocked defect-free PETN single crystal (initial density 1.778 g/cm3 ) is essentially transparent in the spectral range studied, the detonation temperature can be determined with high accuracy. PETN

4 Measurements of Shock and Detonation Phenomena

127

single crystal sample was mounted between a sapphire window (3 mm thick) and an aluminum (Al) base plate (2 mm thick). The sample PETN and sapphire window were gently pressed against the base plate to ensure the intimate contact of the sample PETN, window and base plate. The detonation in the sample PETN was initiated by striking an Al impactor (2 mm thick) to the Al base plate. Detonation temperature was determined by fitting the measured radiation emission to a gray body radiation formula. The record of time-resolved detonation temperature in PETN single crystal is very similar to that in liquid explosive such as NM. The record displays four phases; SDT, overdriven detonation, steady-state detonation and the interaction between detonation products and window. The detonation temperature of PETN single crystal is estimated to be higher than 4000 K, whereas the shock temperature of unreacted PETN single crystal is lower than 2000 K. Therefore, the thermal radiation emitted from detonating PETN is substantially stronger than that from shocked but unreacted PETN. The reaction zone of PETN is known to be extremely thin (less than 10 ns or 80 µm). Under these conditions, the thermal radiation is considered to originate from the reaction zone, although it is measured across the detonation front. Thus, the measured detonation temperature of PETN single crystal is considered to be identical to CJ temperature of PETN. The detonation temperature of PETN single crystal was measured to be 4140 K within the precision of 70 K. Actually, the measured detonation temperature of PETN single crystal is the most reliable detonation temperature of solid explosive. The detonation temperature measurements of single crystals of other solid explosives are highly recommended. The measured detonation temperatures of PETN single crystal and PETN with different initial density are compared with the CJ temperatures calculated using various EOSs in Table 4.7. The measured detonation temperature of PETN single crystal is slightly lower than the CJ temperatures calculated using BKWC-EOS and JCZ3-EOS but substantially higher than the CJ temperatures calculated using BKWR-EOS and KHT-EOS. Measurements of detonation temperatures in solid explosives, which are essentially granular and opaque, present various experimental problems. Kato et al. [37, 38] measured detonation temperatures of pressed solid explosives with different initial density using four wavelength pyrometers described previously. The explosives studied were TNT, cyclotrimethylene-trinitramine (RDX), trinitrophenyl-methylnitramine (Tetryl) and PETN. The properties of solid explosives studied are shown in Table 4.8. Sample explosives were pellets of 20 mm in diameter and 10 mm thick. Five pellets of sample explosives were contained in a polyvinyl-chloride (PVC) tube of 50 mm long, 20 mm in diameter and 2 mm thick. The detonation tube had a Pyrex glass window of 10 mm thick at one end and a booster explosive at the other end. The sample explosive pellets were gently pressed against the Pyrex glass window to ensure the intimate contact between them. Figure 4.18 presents typical radiation signal of the detonation in TNT with initial density 1.51 g/cm3 . Because solid explosives are opaque, the optical pyrometer begins to record the radiation emitted from detonation wave when detonation front approaches transparent window. During about 0.3 µs before the interaction between

128

Y. Kato and K. Murata

Table 4.7 Detonation temperatures of PETN Reference Experimental

Voskoboinikov and Apin

[27]

Initial density (g/cm3 )

4200

1.77

Mader

[30]

3400

1.77

Huishen

[41]

4000

Single crystal

Kato et al.

[37]

4300

1.62

Yoo et al.

[45]

4140

Single crystal

Temperature (K)

Initial density (g/cm3 )

EOS

Reference Theoretical

Temperature (K)

Tanaka

[63]

3690

1.77

KHT

Tanaka

[63]

3830

1.60

KHT

Yoo et al.

[44]

3300

1.778

BKWR

Yoo et al.

[44]

4200

1.778

JCZ3

Yoo et al.

[44]

4330

1.778

BKWC

Hobbs and Bare

[57]

3300

1.76

BKWR

Hobbs and Bare

[57]

3520

1.60

BKWR

Hobbs and Bare

[57]

4280

1.76

BKWS

Hobbs and Bare

[57]

4390

1.60

BKWS

Table 4.8 Properties of solid explosives studied

Initial density (g/cm3 )

Composition

PETN

1.62

Cont. 4.7 wt% wax

TNT

1.0, 1.2, 1.4, 1.51

Explosive

RDX

1.0, 1.2, 1.4, 1.66

Cont. 5.4 wt% wax

Tetryl

1.0, 1.2, 1.4, 1.61

Cont. 1.0 wt% graphite

detonation front and transparent window, the radiation increases rapidly as the radiation absorption by the unreacted explosive is decreased. The measured radiation attains its maximum when detonation front arrives at transparent window. The duration of the radiation peak is about 0.1 µs. It is erroneous to deduce the detonation front temperature from this radiation peak because the radiation emitted from detonation front includes that emitted from high temperature hot spots in the vicinity of detonation front. After this radiation peak, the optical pyrometer can record the radiation from detonation products. Figure 4.19 shows the temperature profiles of detonation products measured at wavelength 0.75 and 0.85 µm. The temperature profiles are very similar for all wavelength measured. The scatter between the temperatures measured at different wavelength is less than 300 K. The accuracy of temperature measurements is estimated to be better than 200 K for TNT with initial density 1.51 g/cm3 . To estimate the temperature profile of detonation products without the effects of high temperature hot spots in the vicinity of detonation front, numerical simulations

4 Measurements of Shock and Detonation Phenomena

129

Fig. 4.18 Signal of the radiation emitted from detonation in TNT (initial density 1.51 g/cm3 ) measured at wavelength 0.85 µm

Fig. 4.19 Comparison between the measured and calculated temperatures of detonation products in TNT (initial density 1.51 g/cm3 )

were performed. Detonation propagation in TNT (initial density 1.51 g/cm3 ) was simulated using 2DL computer code with Forest Fire burn model. The CJ state and CJ isentrope of detonation products were calculated using KHT-EOS. The simulations were carried out for the same configuration as experiments. The mesh size and time increment were the same as in the case of detonation in NM. The calculated temperature profile of detonation products, which were initially at the interface and charge axis, was compared with the temperature profiles measured at wavelength

130

Y. Kato and K. Murata

0.75 and 0.85 µm in Fig. 4.19. The dotted lines in the measured temperature profiles correspond to the effects of the radiation emitted from high temperature hot spots in the vicinity of detonation front. The CJ temperature calculated by KHT-EOS was 3460 K. Although the measured temperatures were 50–200 K higher than the calculated temperature, the measured temperature profiles were very similar to the calculated temperature profile except during first 0.1 µs. The detonation front temperature was determined by fitting the calculated temperature profile to the measured profile during first 0.1 µs. In the simulation, the temperature of detonation products was increased 90 K by reflected shock, as the shock impedance of Pyrex glass is slightly higher than that of detonation products. The effect of reflected shock on the measured temperature was estimated to be less than 100 K. The detonation temperature of TNT with initial density 1.51 g/cm3 was measured to be 3750 K within the precision of 200 K. For all solid explosives studied, the measured temperature profiles were similar to that of TNT, although the measured temperatures were 300–800 K higher than the calculated values for RDX and Tetryl. The scatter between the temperatures measured at different wavelength was less than 300 K for all solid explosives studied. The effect of reflected shock or rarefaction waves on the measured temperatures was estimated to be less than 100 K, as the shock impedance of Pyrex glass is close to that of detonation products of solid explosives studied. The same fitting procedure was applied to determine the detonation front temperature for RDX and Tetryl. The detonation temperatures of solid explosives were measured within the precision of 200 K. The measured detonation temperatures of TNT, RDX and Tetryl with different initial density are compared with the CJ temperatures calculated by various EOSs in Figs. 4.20, 4.21 and 4.22. For all explosives investigated, the CJ temperatures calculated by BKWS-EOS and JCZ3-EOS give reasonable agreement with the measured values. The CJ temperatures calculated by BKWR-EOS and KHTEOS are substantially lower than the measured temperatures.

4.3 Measurements of Underwater Explosion Phenomena 4.3.1 Description of Underwater Explosion Phenomena When an explosive charge is detonated in water, the expansion of gaseous detonation products with very high pressure and temperature generates a shock wave which propagates radially outward in the surrounding water. Figure 4.23 presents the typical record of shock wave pressure P measured at 3 m from the explosive charge of 0.25 kg placed at 4 m in depth. Pm is peak shock wave pressure and θs is time constant which is the time for pressure in shock wave to decay from Pm to Pm /e (e is base of natural logarithm). The shock wave pressure decays exponentially over one time constant, then the pressure decays more slowly. Using high speed framing camera, Hantel and Davis [66] measured the decay of the shock wave generated by spherical

4 Measurements of Shock and Detonation Phenomena

131

Fig. 4.20 Comparison between the measured detonation temperatures and the CJ temperatures calculated by various EOSs for TNT

Fig. 4.21 Comparison between the measured detonation temperatures and the CJ temperatures calculated by various EOSs for RDX

charge. The velocity of shock wave near the charge is much higher than the sound velocity in water but approaches the sound velocity rapidly as the wave propagates outward over the distance of about ten charge radii. The other characteristics of the underwater explosion phenomena are subsequent pressure pulses that are observed much later than the shock wave [67, 68]. These pressure pulses are generated by the pulsation of the gas bubble which contains the gaseous detonation products. The high pressure of the gaseous detonation products

132

Y. Kato and K. Murata

Fig. 4.22 Comparison between the measured detonation temperatures and the CJ temperatures calculated by various EOSs for Tetryl

Fig. 4.23 Typical record of shock wave pressure measurements

causes rapid expansion of the gas bubble in the surrounding water. The outward motion of water is stopped only after the gas pressure is fallen sufficiently below the ambient hydrostatic pressure P0 . When the gas bubble reaches its maximum size, the volume of gaseous detonation products is the order of 104 times larger than that of the original charge, and the pressure in gas bubble is less than 10% of the ambient

4 Measurements of Shock and Detonation Phenomena

133

hydrostatic pressure. Then the higher surrounding pressure reverses the water motion, and the gas bubble contracts. When the gas bubble reaches its minimum size, the gas is re-compressed to a very high pressure of several MPa. At this point, strong compression wave called bubble pulse is emitted in the surrounding water, and the whole process is repeated. The gas bubble oscillates in this manner several times until it arrives at water surface. The successive bubble pulses become weaker because the energy of gas bubble is lost during its pulsation process. Figure 4.24 illustrates the typical record of bubble pulses measured at 3 m from the explosive charge of 0.25 kg placed at 4 m in depth. Tb is first bubble period which is the time interval between the shock wave and first bubble pulse. The gas bubble is nearly spherical at its maximum size and kidney shaped at its minimum size. The maximum radial velocity of the bubble surface is about 60 m/s when the bubble approaches its minimum size. The pressure in the gas bubble is much lower than surrounding hydrostatic pressure over most of the cycle. The period of the bubble pulsations is about several hundred times longer than the duration of shock wave. Because of long duration of the bubble pulsation, gravity effects become noticeable. Therefore, the gas bubble has great buoyancy and mitigates upward. However, the gas bubble does not float up like a balloon but jumps up when its size is close to minimum. 5 4 3 2 1 0 0

100

Fig. 4.24 Typical record of bubble pulse measurements

200

300

134

Y. Kato and K. Murata

4.3.2 Use and Evaluation of Underwater Explosion Performance The underwater explosion test is valuable tool to access the energy content and performance of explosives in terms of measured shock and bubble energies [67– 70]. The underwater explosion test has some advantages over other experimental methods. The experimental methods such as pressure, particle velocity measurements and cylinder test are limited to measure the state close to detonation front, and their observation time is limited to the order of ten microseconds. In the underwater explosion test, detonation products are allowed to expand adiabatically from CJ state to a hydrostatic pressure in the water owing to sufficient confinement due to large mass of surrounding water. This particular condition permits to observe the state of detonation products at longer time range. In the case of non-ideal explosives, sufficient confinement realized in the underwater explosion test enables slow rate reaction to complete in detonation products. The underwater explosion phenomena have been studied using various numerical simulation codes. Numerical simulation succeeds to predict accurately shock and bubble energies in the case of ideal explosives but fails to predict shock and bubble energies for non-ideal explosives because of lack of data about slow rate reaction behind CJ state. The underwater explosion test is actually sole method to access shock and bubble energies of non-ideal explosives. In blasting industry, the relative effectiveness of explosives is characterized in terms of shock and bubble energies. It is considered that shock energy is the measure of the explosive’s shattering action in other materials such as rock, and bubble energy is the measure of the heaving action of the explosive. In the engineering applications such as shock forming of metal plates and shock consolidation of metallic and ceramic powder, the measurements of underwater shock wave in high pressure range are required to control shock wave loading on target samples. To understand the destructive effects of underwater explosion phenomena, it is also important to measure precisely bubble pulse as well as shock wave. In the underwater explosion of explosive, the chemical energy of explosive is partitioned to the energy of shock wave transmitted to the water, called shock wave energy, and to the energy of gaseous detonation products, called bubble energy. Shock wave energy is calculated using shock pressure profile measured by pressure gauge placed at a certain distance from the explosive charge. Energy flux density E f (energy per unit time per unit surface area) carried by the shock wave is expressed by {ti Ef = 0

) ( P u2 + ρW uΔ E W + 2 ρW

(4.13)

where ti is the integration time, and ρW , u, E W are, respectively, density, particle velocity, internal energy per unit mass of water. They are all functions of time at a

4 Measurements of Shock and Detonation Phenomena

135

certain location as the shock wave passes through. Time zero is the time of arrival of the shock wave to the gauge. The increment Δ means that the shock wave increases the original energy of water. The increase in internal energy and kinetic energy can be neglected comparing with pressure term. Then, the energy flux density can be calculated by simple equation: {ti Ef =

{ti u(P − P0 )dt=

0

u Pdt

(4.14)

0

The hydrostatic pressure of the water P0 can be neglected comparing to the shock pressure P. From the momentum equation, following expression is derived: {ti

1 P + u= ρW cW ρ W Rs

u Pdt

(4.15)

0

where cW is sound velocity of water and Rs is stand off distance which is the distance from the charge to the measuring point. Neglecting the last term in Eq. 4.15 called after flow, which is small for sufficiently long distance to the charge, particle velocity is given by u = P/ρW cW . Substituting u in Eq. 4.14: 1 Ef = ρW cW

{ti P 2 dt

(4.16)

0

Shock wave energy per unit mass E s is given by 4π Rs2 4π Rs2 Ef = Es = W ρW cW W

{5θs P 2 dt

(4.17)

0

where W is the mass of explosive. The integration time ti is usually taken to be 5θs . Shock wave impulse Is is given by {5θs Is =

Pdt

(4.18)

0

Various parameters related to the shock wave such as peak shock pressure Pm , scaled time constant θs / W 1/3 , scaled shock impulse Is / W 1/3 and scaled energy flux density E f / W 1/3 are expressed in the form of similitude functions [68]:

136

Y. Kato and K. Murata

( Shock wave parameter = K 1

1

W3 Rs

)α (4.19)

Similitude constants K 1 and α are calculated using the shock wave profiles measured at various scaled distance Rs /W 1/3 . Bubble energy is the residual energy of detonation products, and consisted of a sum of three forms of energy; work against hydrostatic pressure, kinetic energy and internal energy. As the bubble expands and contracts, the energy is converted from one form to another. Bubble energy E b is expressed by )( ( ) d Rb 2 4 3 4 3 3 πρw Rb E b = π P0 Rb + + Ei 3 2 3 dt

(4.20)

where Rb is bubble radius and E i is the internal energy per unit mass of detonation products when the bubble radius is Rb . When the bubble reaches a maximum radius Rbm , the velocity at the interface d Rb /dt is zero and the internal energy E i can be neglected. Bubble energy E b is expressed by Eb =

4 3 π P0 Rbm 3

(4.21)

The maximum bubble radius is rarely measured because of its experimental difficulty. Contrary, first bubble period can be measured easily. Bubble energy can be expressed in terms of first bubble period Tb by −3

5

E b = 0.684 · ρw2 · P02 · Tb3

(4.22)

Cole [67] described the theoretical basis for bubble scaling laws. To determine the scaling law for first bubble period and maximum bubble radius, the effects of hydrostatic pressure on the bubble and gas pressure in the bubble must be considered. However, over most of the oscillation cycle, the effects of gas pressure can be neglected as it is relatively small except when the bubble is near its minimum size. According to this approximation, the appropriate scaling factors are hydrostatic pressure and mass of explosive. The scaling laws for first bubble period Tb and maximum bubble radius Rbm are given by 1

−5

Tb = K 2 · W 3 · P0 6 ,

1

−1

Rbm = K 3 · W 3 · P0 3

(4.23)

Constants of scaling law K 2 and K 3 are determined from the experimental data obtained at various experimental conditions. In the underwater explosion tests, the experiments are often carried out in natural pond or sea. The absolute values of measured shock and bubble energies can be influenced by the experimental conditions. In practical applications, the shock and

4 Measurements of Shock and Detonation Phenomena

137

bubble energies relative to those of standard explosive are generally used. Pentolite (PETN/TNT = 50/50 wt%) is used as standard explosive as it is easily detonated with minimum size of initiating material. To determine the total expansion work E t performed by detonation products, it is necessary to evaluate the dissipated energy E d . The dissipated energy is the energy lost as heat in the water when the shock wave travels from the charge to the gauge. The dissipated energy is lost very rapidly within the distance of about five charge radii and more slowly thereafter, so it cannot be measured. Bjarnholt and Holmberg [69] proposed a practical way to estimate the dissipated energy. The total expansion work E t is expressed by Et = Es + Ed + Eb = m Es + Eb

(4.24)

where m is shock loss factor. Then the dissipated energy is proportional to the measured shock energy and factor (m − 1) as following E d = E s (m − 1)

(4.25)

Thermochemical calculations for ideal explosives show that the total expansion work is nearly identical to the detonation heat Q d measured in a detonation calorimeter. Using the relation E t = Q d , Eq. 4.24 gives Q d = m Es + Eb

(4.26)

Then shock loss factor m is calculated using E s , E b and Q d by m=

Q d − Eb Es

(4.27)

The shock loss factor m was determined as a function of detonation pressure for some explosives whose detonation pressures were ranged from 2 to 26 GPa [69].

4.3.3 Measurements of Shock Wave and Bubble Pulse

Pressure gauge using tourmaline crystal as sensing element has been widely used to measure underwater shock wave and bubble pulse. Considering the survivability and reliability of the tourmaline gauge, it is suggested to carry out routine measurements in the pressure range below 50 MPa. In various applications, it is required to measure shock pressure much higher than 100 MPa. Fluoropolymer such as PVDF has been widely used to measure the pressure of shock wave in solids and is a good candidate as sensing element of pressure gauge to measure underwater shock pressure much higher than 100 MPa. Murata et al. [71–73] investigated the performance of

138

Y. Kato and K. Murata

Table 4.9 Physical properties of PVDF, VDF-TFE and tourmaline [73] Piezoelectric material

PVDF

VDF-TFE

Tourmaline

Piezoelectric modules d33 (pC/N)

12

9

2

Dielectric constant ratio ε1 /ε0

12

10

1

Density (g/cm3 )

1.8

1.9

6.0

1.9

Sound velocity (km/sec) Acoustic impedance

(kg/m2

× sec)

3.4 ×

2.4 106

4.5 ×

3.1 106

19 × 106

PVDF; poly-vinylidene fluoride. VDF-TFE; vinylidene fluoride - trifluoroethylene copolymer. d33; Piezoelectric modules of d33 axis. ε1 ; Dielectric constant of piezoelectric materials, ε0 ; Dielectric constant at vacuum.

fluoropolymer as sensing element of pressure gauge. Two kinds of fluoropolymer, PVDF and vinylidene fluoride-trifluoro ethylene copolymer (VDF-TFE; mole ratio = 75/25) were selected as sensing element. Physical properties of PVDF, VDF-TFE and tourmaline are shown in Table 4.9. Structure of fluoropolymer gauge is illustrated in Fig. 4.25. Sensing element is a fluoropolymer sheet of 5 mm square and 50 µm thick. Both sides of the sheet are coated by vapor deposited Al, which are served as electrodes and connected with copper wire leads. Coaxial cable connected with copper wire leads is held by a body made of epoxy resin. Gauge assembly is inserted into polyamide cap of 10 mm inner diameter, 70 mm long and 1 mm thick. The gauge assembly is filled with insulation oil whose shock impedance is very close to that of water. Calibration of pressure gauges was carried out using the oil pressed calibration apparatus (PCB Model 913A02) in the pressure range up to 40 MPa. In order to evaluate the performance of three types of pressure gauge, PVDF gauge, VDF-TFE gauge and tourmaline gauge, underwater explosion tests were carried out in the testing tank of 36 m in diameter and 8 m in depth. Spherical charge of emulsion explosive (EMX) of detonation velocity 3200 m/s and density 1.10 g/cm3 was placed in the center of testing tank at the depth of 4 m. Pressure gauge was also placed at the Fig. 4.25 Structure of fluoropolymer pressure gauge

4 Measurements of Shock and Detonation Phenomena

139

Fig. 4.26 Experimental set-up of underwater explosion tests

depth of 4 m. To cover wide peak shock pressure range, scaled distance was varied between 0.4 and 7 m/kg1/3 . Experimental set-up of underwater explosion tests is shown in Fig. 4.26. Shock wave and bubble pulse were recorded by digital recorder with sampling time of 0.2 and 20 µs. Figure 4.27 presents the shock pressure profiles measured by three types of pressure gauge at a distance of 1 m from a charge of 0.2 kg. The VDF-TFE pressure gauge measured correctly shock pressure profile even when peak shock pressure exceeded 100 MPa. Contrary the PVDF pressure gauge failed to measure shock pressure profile although it could measure peak shock pressure over 100 MPa. The discharge time of PVDF is too short to measure shock pressure profile. The tourmaline pressure gauge was damaged when peak shock pressure exceeded 30 MPa.

Fig. 4.27 Shock pressure profiles measured by three types of pressure gauge

140

Y. Kato and K. Murata

Table 4.10 Properties of sample explosives Sample explosive

Density (g/cm3 )

Det. velocity (m/s)

Composition (wt%) EM

MB

Al powder

EMX

1.15

3390

98.9

1.1

0

Al-EMX

1.22

3360

82.4

0.92

16.7

To confirm the performance of the VDF-TFE pressure gauge, Murata et al. [74, 75] measured shock pressure and bubble pulse in the wide range of scaled distance for two kinds of explosive; EMX and aluminized EMX (Al-EMX). The properties of EMX and Al-EMX are shown in Table 4.10. The composition of base emulsion matrix (EM) was ammonium nitrate (AN)/HN/water/oil and emulsifier = 74.6/10.6/10.6/4.2 in weight ration. Formed polystyrene sphere was used as micro-balloon (MB). Al powder of mean diameter 30 µm was used. Distance between sample explosive and pressure gauge was varied from 0.05 to 3 m, and charge weight was fixed to be 0.1 kg. Scaled distance was varied in wide range from 0.11 to 6.46 m/kg1/3 . Sample explosive in spherical shape of about 2.7 cm in radius was centrally initiated by No.6 electric detonator. The results of shock pressure and bubble pulse measurements are summarized in Table 4.11. Shock wave profiles were successfully measured even in the condition that peak shock pressure was over 400 MPa. In the case of EMX, the measured shock wave energy decreases with the increase of distance within the distance shorter than 0.5 m owing to the loss of shock wave energy by shock heating of surrounding water. However, in the case of Al-EMX, the measured shock energy remains constant within the measured distance range. This fact indicates that the loss of shock wave energy is compensated by the shock wave enhancement due to the late energy release by Al reaction in gas bubble. Figure 4.28 illustrates shock pressure profiles measured for EMX and Al-EMX. The decay of shock pressure in Al-EMX is slower than that in EMX, and time constant of shock wave in Al-EMX is longer than that in EMX. Figures 4.29 and 4.30 present the relation of peak shock pressure and scaled shock impulse to the scaled distance. The peak shock pressure in Al-EMX is lower than that in EMX until the scaled distance shorter than 1 m/kg1/3 and then becomes higher than that in EMX. The scaled shock impulse in Al-EMX is about 25% greater than that in EMX within the measured scaled distance range. Figure 4.31 compares the bubble pulse profiles measured by tourmaline gauge and VDF-TFE gauge. The bubble pulse profile measured by tourmaline gauge presents important shift in base line to negative pressure after the passage of shock wave, which makes the precise measurements of bubble pulse pressure difficult. In contrast, the bubble pulse profile measured by VDF-TFE gauge maintains base line after the passage of shock wave. The enlargement of first bubble pulse profile is shown in Fig. 4.32. The duration of first bubble pulse is in the order of 10 ms, and its pressure rise is very slow compared with shock wave. Bubble pulse impulse Ib is calculated using measured bubble pulse profile Pb (t) by following equation;

4 Measurements of Shock and Detonation Phenomena

141

{t2 Ib =

Pb (t)dt

(4.28)

t1

where t1 and t2 are, respectively, the time when bubble pulse pressure rises and falls to threshold pressure. The threshold pressure is selected as 5% of peak bubble pulse pressure Pbm considering noise level of measurements. Figures 4.33 and 4.34 present the relation between peak bubble pulse pressure Pbm , scaled bubble pulse impulse Ib / W 1/3 and scaled distance. The peak bubble pulse pressure in Al-EMX is about 25% higher than that in EMX, and scaled bubble pulse impulse of AlEMX is about 20% greater than that in EMX within the measured scaled distance range. The experimental results also show that the impulse of first bubble pulse is about 2.5 times greater than the shock wave impulse, although the peak bubble pulse pressure is only 10 to15% of the peak shock pressure. Consequently, bubble pulse gives important dynamic loading to the underwater structures, and the effects of bubble pulse loading as well as shock loading should be considered to study the Table 4.11 Results of shock pressure and bubble pulse measurements Sample Explosive

Rs (m)

Rs/W 1/3 (m/kg 1/3 )

Pm (MPa)

I s (Pa・s /kg1/3)

Es (MJ/kg)

Pbm (MPa)

Ib (Pa・s /kg1/3 )

EMX

0.05

0.11

464.6

21,099

0.924

46.4

80,840

Al-EMX

0.10

0.22

213.4

9999

0.910

20.6

37,713

0.20

0.43

100.1

5207

0.906

9.6

17,594

0.50

1.08

32.0

2643

0.861

4.1

6421

1.00

2.15

17.2

1351

0.887

2.1

2996

2.00

4.31

8.0

724

0.880

1.0

1398

3.00

6.46

5.1

503

0.866

0.7

895

0.05

0.11

383.5

27,879

0.975

58.0

90,927

0.10

0.22

178.5

13,815

1.037

29.4

43,430

0.20

0.43

90.3

7396

0.995

13.5

20,744

0.50

1.08

34.8

3154

1.015

4.7

7811

1.00

2.15

16.9

1656

1.030

2.5

3731

2.00

4.31

8.2

869

1.056

1.4

1782

3.00

6.46

5.4

596

1.018

0.9

1157

142

Y. Kato and K. Murata

Fig. 4.28 Shock pressure profiles measured for EMX and Al-EMX

Fig. 4.29 Relation of peak shock pressure to scaled distance

1000

100

10

1

0.1

1.0

10.0

response of the underwater structures against the underwater explosion of explosives. The measured bubble energies of EMX and Al-EMX are, respectively, 1.8 and 3.0 MJ/kg. It was shown that the addition of Al powder substantially increases the bubble energy.

4 Measurements of Shock and Detonation Phenomena Fig. 4.30 Relation of scaled shock wave impulse to scaled distance

Fig. 4.31 Bubble pulse profiles measured by tourmaline gauge and VDF-TFE gauge

10

5

10

4

10

3

0.1

143

1.0

10.0

144

Y. Kato and K. Murata

Fig. 4.32 Enlargement of first bubble pulse profile

Fig. 4.33 Relation of peak bubble pulse pressure to scaled distance

100

10

1

0.1 0.01

0.1

1

10

4 Measurements of Shock and Detonation Phenomena Fig. 4.34 Relation of scaled bubble pulse impulse to scaled distance

10

10

10

10

145

6

5

4

3

0.01

0.1

1

10

References 1. Graham RA (1993) Chapter 3 in Solids Under high-pressure shock compression Graham RA (ed.), Springer-Verlag 2. Barker LM, Shahinppr M, Chhabildas LC (1993) Chapter 3 in high-pressure shock compression of solids Asay JR, Shahinpoor M (ed.), Springer-Verlag 3. Wackerle J, Johnson JO, Halleck PM (1976) Shock initiation of high density PETN. In: Proceedings 6th international deonation symposium, Coronado, CA, Office of Naval research, Arlington, VA, pp 20–28 4. Burrows K, Chilvers DK, Gyton R, Lambour BD, Wallace AA (1976) Determination of detonation pressure using manganin wire technique. In: Proceedings 6th international detonation symposium, Coronado, CA, Office of Naval Research, Arlington, VA, pp 625–636 5. Kawai H (1969) Piezoelectricity of poly (vinylidene fluoride). Japan J Appl Phys 8:975–981 6. Graham RA (1993) Chapter 5 in solid under high-pressure shock compression Graham RA (ed.), Springer-Verlag. 7. Bauer F (1981) Behaviour of ferroelectric ceramics and PVDF2 polymers under shock loading. AIP Conf Proc 78:251–267 8. Bauer F (2005) Piezoelectric polymer shock gauges. AIP Conf Proc 845:1183–1186 9. Bauer F (2001) PVDF gauge piezoelectric response under two-stage light gas gun impact loading. AIP Conf Proc 620:1149–1152 10. Murata K, Kato Y (2010) Application of PVDF pressure gauge for pressure measurements of non-ideal explosives. Int J Soc Mater Eng Resour 17(2):112–114 11. Zaitsev VM, Pokhil PF, Shedov KK (1960) The electromagnetic method for the measurement of velocities of detonation products. Doklady Acad Sci USSR 132(6):139 12. Jacobs SJ, Edwards DJ (1970) Experimental study of the electromagnetic velocity-gauge technique. In: Proceedings 5th international detonation symposium, Pasadena, CA, Office of Naval Research, Arlington, VA, pp 413–426 13. Cowperthwaite M, Rosenberg JT (1981) Lagrange gauge study in ideal and non-ideal explosives. Proceedings 7th international detonation symposium, Annapolis, MD, Naval Surface Weapon Center, White Oak, MD, pp 1072–1085

146

Y. Kato and K. Murata

14. Vorthman J, Andrews G, Wackerle J (1985) Reaction rates from electromagnetic gauge data. In: Proceedings 8th international detonation symposium, Albuquerque, NM, NSWC, MD, pp 99–110 15. Sheffield SA, Gustavsen RL, Alcon RR (1999) In-situ magnetic gauging technique used at LANL-method and shock information obtained. AIP Conf Proc 505:1043–1048 16. Sheffield SA, Dattelbam DM, Engelke R, Alcon RR, Crouze B, Robbins DL, Stahl DB, Gustavsen RL (2006) Homogeneous shock initiation process in neat and chemically sensitized nitromethane. In: Proceedings 13th international detonation symposium, Norfork, VA, Office of Naval Research, Arlington, VA, pp 401–407 17. Dattelbaum DM, Sheffield SA, Stahl DB, Dattelbaum AM, Trott W, Engelke R (2010) Influence of hot spot features on the initiation characteristics of heterogeneous nitromethane. In: Proceedings 14th international detonation symposium Coeur d’Alene, ID, Office of Naval Research, VA, pp 611–621 18. Baker LM, Hollenbach RE (1972) Laser interferometer for measuring high velocities of any reflecting surface. J Appl Phys 43(11):4669 19. Baker L (1999) The development of the VISAR, and its use in shock compression science. AIP Conf Proc 505:11–17 20. Bloomquist DD, Sheffield SA (1983) Optically recording interferometer for velocity measurements with subnanosecond resolution. J Appl Phys 54(4):1717–1722 21. Dirand M, Leharrangue P, Lalle P, Le Bihan A, Morrian J, Pujols H (1977) Interferometric laser technique for accurate velocity measurement in shock wave physics. Rev Sci Inst 48:275 22. Sheffield SA, Engelke R, Alcon RR, Gustavsen RL, Robins DL, Stahl DB, Stacy HL, Whitehead MC (2002) Particle velocity measurements of the reaction zone in nitromethane. In: Proceedings 12th international detonation symposium, San Diego, CA, Office of Naval Research, Arlington, VA, pp 159–166 23. Tarver CM (2005) Detonation reaction zones in condensed explosives. AIP Conf Proc 845:1026–1029 24. Strand OT, Goosman DR, Martinez C, Whitworth TL, Kuhlow WW (2006) Compact system for high-speed velocimetry using heterodyne techniques. Rev Sci Inst 77:083108 25. Briggs ME, Hill LG, Hull LM, Shinas MA, Dolan DH (2010) Application and principles of photon-doppler velocimetry for explosive testing. In: Proceedings 14th international detonation symposium, Coeur d’Alene, ID, Office of Naval Research, VA, pp 414–424 26. Gibson FC (1958) Use of an electro-optical method to determine detonation temperatures in high explosives. J Appl Phys 29(4):628–632 27. Voskoboinikov IM, Apin AY (1960) Measurement of detonation front temperature for explosives. Dokl Akad Nauk SSSR 130(4):804–806 28. Dremin AN, Savrov SD (1965) Emission spectrum of a detonation wave in nitromethane. Z.P.M.T.F 1:103–105 29. Trofimov VS, Trojan AV (1969) Detonation luminescence spectrum of nitromethane. Fiz Gor i Vzry 5(2):280–282 30. Mader CL (1961) Detonation performance calculations using the Kistiakowsky-Wilson equation of state. Los Alamos Sci Lab Report LA-2613 31. Burton JTA, Hicks JA (1964) Detonation emissivity and temperatures in some liquid explosives. Nature 202:758–759 32. Burton JTA, Hawkins SJ, Hooper G (1981) Detonation temperature of some liquid explosives. In: Proceedings 7th international detonation symposium, Annapolis, MD, NSWC, White Oak, MD, pp 759–767 33. Urtiew PA (1976) Brightness temperature of detonation wave in liquid explosives. Acta Astronaut 3:555–566 34. Kato Y (1978) Contribution a l’etude des detonations des melanges heterogenes de nitromethane et d’aluminium. Doctor Thesis, University of Poitiers 35. Kato Y, Bouriannes R, Brochet C (1978) Mesure de temperature de luminance des detonations d’explosifs transparents et opaques. In: Proceedings HDP symposium CEA, Paris, pp 439–449

4 Measurements of Shock and Detonation Phenomena

147

36. Kato Y, Bauer P, Brochet C, Bouriannes R (1981) Brightness temperature of detonation wave in nitromethane-teranitomethane mixtures and in gaseous mixtures at high initial pressure. In: Proceedings 7th international detonation symposium, Annapolis, MD, NSWC, White Oak, MD, pp 768–774 37. Kato Y, Mori N, Sakai H, Tanaka K, Sakurai T, Hikita T (1985) Detonation temperature of nitromethane and some solid high explosives. In: Proceedings 8th international detonation symposium, Albuquerque, NM, NSWC, White Oak, MD, pp 558–566 38. Kato Y, Mori N, Sakai H, Sakurai T, Hikita T (1989) Detonation temperature of some liquid and solid explosives. In: Proceedings 9th international detonation symposium, Portland, OR, Office of Naval Research, Arlington, MD, pp 939–946 39. Kato Y, Brochet C (1985) Detonation temperatures of nitromethane aluminum gels. Dyn Shock Waves Explosion Detonations Prog Astronaut Aeronaut 94:416–426 40. Xianchu H, Chenbung H, Shufong K (1985) The measurement of detonation temperature of condensed explosives with two colour optical fiber pyrometer. In: Proceedings 8th international detonation symposium, Albuquerque, NM, NSWC, White Oak, MD, pp 567–574 41. Huisheng S, Chengbung H, Shufang K, Lihong H (1989) The studying detonation temperatures of solid high explosives. In: Proceedings 9th international detonation symposium, Portland, OR, Office of Naval Research, Arlington, VA, pp 947–952 42. Gogulya MF, Brazhnikov MA (1993) Radiation of condensed explosives and its interpretation (Temperature measurements). In: Proceedings 10th international detonation symposium, Boston, MA, Office of Naval Research, Arlington, VA, pp 542–548 43. Gogulya MF, Dolgoborodov AY, Brazhnikov MA (1999) Investigation of shock and detonation waves by optical pyrometer. Int J Impact Eng 23:283–293 44. Yoo CS, Holmes NC, Souers PC (1996) Detonation in shocked homogeneous high explosives. Mat Res Soc Symp Proc 418:397–406 45. Yoo CS, Holmes NC, Souers PC, Wu CJ, Ree FH (2000) Anisotropic shock sensitivity and detonation temperature of pentaerythritol tetranitrate single crystal. J Appl Phys 88(1):70–75 46. Leal B, Baudin G, Goutelle JC, Presles HN (1998) An optical pyrometer for time resolved temperature measurements in detonation wave. In: Proceedings 11th international detonation symposium, Snowmass, CO, Office of Naval Research, Arlington, VA, pp 353–361 47. Leal-Crouzet B (1998) Application de la pyrometrie optique a la mesure de la temperature des produits de reaction d’explosifs condenses en regime d’amorcage et de detonation. Doctor Thesis, University of Poitiers 48. Bouyer V, Darbord I, Herve P, Baudin G, Le Gallic C, Clement F, Chavent G (2006) Shockto detonation transition of nitromethane: time-resolved emission spectroscopy measurements. Combust Flame 144:139–150 49. Von Holle WG, Tarver CM (1981) Temperature measurement of shocked explosives by timeresolved infrared radiometry: a new technique to measure shock-induced reaction. In: Proceedings 7th international detonation symposium, Annapolis, MD, NSWC, White Oak, VA, pp 993–1003 50. Von Holle WG, WcWilliams RA (1983) Detonic research infrared radiometer with nanosecond response. Rev Sci Instrum 54(9):1218–1221 51. Crouzet B, Partouche-Sebban D, Carion N (2003) Temperature measurements in shocked nitromethane. AIP Conf Proc 706:1253–1256 52. Delpuech A, Menil A (1983) Raman scattering temperature measurement behind a shock wave. Shock Compression of Condensed Matter, North-Holland 53. Mader CL (1998) Numerical modeling of explosives and propellants. CRC Press, Boca Raton, FL 54. Cowperthwaite M, Zwisler WH (1976) The JCZ equations of stat for detonation products and their corporation into the tiger code. In: Proceedings 6th international detonation symposium, Colonado, CA, Office of Naval Research, Arlington, VA, pp 162–172 55. Tanaka K (1985) Detonation properties of high explosives calculated by revised Kihara-Hikita equation of state. In: Proceedings 8th international detonation symposium, Albuquerque, NM, NSWC, White Oak, MD, pp 548–557

148

Y. Kato and K. Murata

56. Finger M, Lee E, Helm FH, Hays B, Hornig H, McGuire R, Kahara M, Guidry M (1976) The effect of elemental compositon on the detonation behavior of explosives. In: Proceedings 6th international detonation symposium, Colonado, CA, Office of Naval Research, Arlington, VA, pp 710–722 57. Hobbs ML, Baer MR (1992) Nonideal thermoequilibrium calculations using a large product species data base. Shock Waves 2:177–187 58. Hobbs ML, Baer MR (1993) Calibrating the BKW-EOS with a large product species data base and measured C-J properties. In: Proceedings 10th international detonation symposium, Boston, MA, Office of Naval Research, Arlington, VA, pp 409–418 59. Fried L, Souers PC (1996) BKWC: an empirical BKW parametrization based on cylinder test data. Propellant, Explos, Pyrotech 21:215–223 60. Winely JM, Duvall GE, Knudson MD, Gupta YM (2000) Equation of state and temperature measurements for shocked nitromethane. J Chem Phys 113(17):7492–7501 61. Marsh SP (1983) LASL shock Hugoniot data. University of California Press, Berkeley, CA 62. Le Francois A, Baudin G, Le Gallic C, Boyce P, Coudoing JP (2002) Nanometric aluminium powder influence on the detonation efficiency of explosives. In: Proceedings of 12th international detonation symposium, San Diego, CA, Office of Naval Research, Arlington, VA, pp 22–32 63. Tanaka K (1983) Detonation properties of condensed explosives computed using the KihraHikita-Tanaka equation of state. National Chemical Laboratory for Industry, Tsukuba Center, Japan 64. Hardesty DR, Kennedy JE (1977) Thermochemical estimation of explosive energy output. Combust Flame 28:45–59 65. Gogulya MF, Dolgoborodov AY, Brazhnikov MA, Dushenok SA (1999) Shock wave initiation of liquid explosives. AIP Conf Proc 505:903–906 66. Hantel LW, Davis WC (1970) Spherical explosions in water. In: Proceedings 5th international detonation symposium, Pasadena, CA, Office of Naval Research, Arlington, VA, pp 599–604 67. Cole RH (1948) Underwater explosions. Dover Publications, New York 68. Roth J (1983) Underwater explosions in encyclopedia of explosives and related items, vol. 10. US Army Res Develop Comd, Dover, NJ U38–81 69. Bjarnholt G, Holmberg R (1976) Explosive expansion works in underwater detonations. In: Proceedings 6th international detonation symposium, Colonado, CA, Office of Naval Research, Arlington, VA, pp 540–550 70. Bjarnholt G (1980) Suggestions on standards for measurement and data evaluation in the underwater explosion tests. Propellants Explos 5:67–74 71. Murata K (1995) Material chemical research on the application of piezoelectric fluorinepolymer to pressure sensors for underwater explosion tests. Doctor Thesis, Akita University (in Japanese) 72. Murata K, Takahashi K, Kato Y, Murai K (1996) Development of pressure sensors using fluoropolymer for underwater shock wave measurements. Kayaku Gakkaishi (Sci Tech Energ Mater) 57(6):252–262 (in Japanese) 73. Murata K, Takahashi K, Kato Y (1997) Measurement of underwater shock wave by fluoropolymer sensor. In: Proceedings 23rd international pyrotechnics seminar, pp 548–560 74. Murata K, Takahashi K, Kato Y (1999) Precise measurements of underwater explosion phenomena by pressure sensor using fluoropolymer. J Mater Proc Technol 85:39–42 75. Murata K, Takahashi K, Kato Y (2002) Measurements of underwater explosion performances by pressure gauge using fluoropolymer. In: Proceedings 12th international detonation symposium, San Diego, CA, Office of Naval Research, Arlington, VA, pp 336–342

Chapter 5

Shock Initiation Shiro Kubota and Kunihito Nagayama

Abstract This chapter describes the overview of experiments and numerical simulation for shock initiation, and the studies on the shock initiation in Japan. In the “Discussion of Mixture rule,” using thermodynamic state obtained by simulations for shock initiation, these variations during reaction process and relationship between them are discussed. The constructing of the simulation method for shock initiation of PETN which can be applied to arbitrary loading density for PETN without changing the parameters is described. In the section of “Gap test,” the construction of the scale law of the sympathetic detonation is a highlight. The last section presents the study on the laser initiation of explosive, which it may be considered a pioneer study at that time. Keywords Shock-to-detonation transition · Mixture rule · Unified simulation · Gap test · Laser initiation

5.1 Introduction A word, initiation, which has been used in detonation physics, means the initiation to detonation transition process. In the real world, the detonation cannot start without transition time. There are three types of transition process: deflagrationto-detonation transition (DDT), shock-to-detonation transition (SDT), unknown-todetonation transition (XDT). Although DDT and XDT are important topic in detonation physics, we mainly treated SDT in this chapter. The SDT starts from shock entrance to explosive. The mechanism in the shock initiation of explosives differs between homogeneous explosives such as liquid explosives and heterogeneous explosives. In the latter, it is recognized that the behavior of hot spots that are the local

S. Kubota (B) National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan e-mail: [email protected] K. Nagayama Kyushu University, Fukuoka, Japan © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Kubota (ed.), Detonation Phenomena of Condensed Explosives, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-5307-1_5

149

150

S. Kubota and K. Nagayama

high-temperature and high-pressure region in the explosive contributes to the initiation process. The hot spots are generated by the interactions of the shock waves with various density discontinuities such as voids and binders present in explosives. When a shock wave enters the heterogeneous explosive, hot spots are generated, and the explosive begins to decompose. Depending on the degree of inhomogeneity and the strength of the shock wave, hot spots coalesce with nearby hot spots, increasing the reaction rate and reacting region and eventually to reach the steady state detonation. If the incident pressure is low or its duration is short, hot spots will not be created or will be extinguished even if they are created, and detonation transition will not occur. In this chapter, the overview of the experiments and numerical simulations for shock initiation is described. Especially, the concept for basic experiment is explained using simulation results. Other main contents are studies on the shock initiation of condensed explosive carried out in Japan. In the “Discussion of Mixture rule,” using thermodynamic state obtained by simulations for shock initiation under various assumptions. These variations during reaction process and relationship between them are discussed. Next content presents the simulation method for shock initiation of PETN, which can be applied to arbitrary loading density for PETN without changing the parameters. In the section of “Gap test,” the construction of the scale law of the sympathetic detonation is a highlight. The last section describes the results of the study of laser initiation of explosive, and it may be considered a pioneer study at that time.

5.2 Overview of the Experiments and Numerical Simulations of Shock Initiation In 1951, at the First Detonation Symposium, Kistiakowsky [1] mentioned that the initiation mechanism is extraordinary important. Since the third symposium, the shock initiation becomes the main topic of the symposium. It can be clearly understood from Jacob’s invited presentation [2] at third symposium that there are a lot of researches related to shock initiation since 1940s, and that the existence of various phenomena related to shock initiation has been clarified experimentally by 1960. These include research topics to solve, such as the detonation around a corner or around an arc. Unfortunately, we cannot confirm the information of the classified paper at that time at least in the proceedings. Since it is difficult to cover all issues related to the shock initiation in this book, the following describes the studies that are related to the content presented in this book. The first experiment to record the locus of shock-to-detonation transition, which can determine the condition of shock loading to the explosive, is the wedge technique as shown in Fig. 5.1. A lot of researchers employed this type of experimental system, and at the Third Detonation Symposium, Campbell, Davis, Ramsay and Travis and

5 Shock Initiation

151

Seay and Seely presented the studies related to shock initiation using this system [3–6]. Just below an explosive booster-attenuator system, the sample explosive with wedge shape was set to record the locus of shock front by smear camera. Here, the explosive booster-attenuator system corresponds to plane-wave generator system as described in Sect. 4.1.1.1. The incident pressure to the sample explosive can be controlled by the type of explosives, and type and thickness of the attenuator. The Hugoniot for attenuator must be known. The free surface velocity of the attenuator under the same conditions for the wedge technique must also be known before testing. To determine the incident pressure to the sample explosive, the impedance matching method explained in Sect. 2.1.2.5 could be applied. Wedge technique employed large diameter explosive with plane-wave generator to allow one-dimensional assumption. We explain the data analysis using the results of one-dimensional simulation as shown in Fig. 5.2. PMMA is used as the attenuator, and Hugoniot of PMMA is known. The left-hand side corresponds to the locus of shock front which is obtained by the record of smear camera in experiment. An inflection point of the shock locus is the shockto-detonation transition (SDT) point. The distance from origin to SDT point is the run distance to detonation. The initial slope of the shock locus is the shock velocity (U e ) in sample explosive under this condition. The pressure in sample explosive has been estimated as shown in Fig. 5.2. The known variable is the free surface velocity U f , so the particle velocity in PMMA is U f /2 according to free surface approximation [7]. Before the shock wave enters the sample explosive, the state is on the Hugoniot of PMMA (black solid line) and particle velocity is U f /2. After the shock wave enters the sample explosive, the state is on the Rayleigh line (red dashed line). The slope of Rayleigh line is ρU e . The pressure is determined by the intersection of the Rayleigh line and mirror Hugoniot of PMMA. This is an approximation, but it gives accurate results. More strictly, it is necessary to use reflected Hugoniot [8] instead of mirror Hugoniot. Metal attenuator is frequently used and general. Because of high impedance of metal, Hugoniot of attenuator is above the Rayleigh line of sample explosive. In that case, the rarefaction release curve of attenuator has to be used [8]. Since the relation between pressure and particle velocity of sample explosive is obtained, the shock velocity can be determined by Jump condition. Multiple shots with different conditions can determine the Hugoniot for sample explosive. The Fig. 5. 1 Wedge test (plane-wave generator with attenuator and wedge shape sample explosive)

Lens Main explosive Attenuator

To camera

Sample explosive

152

S. Kubota and K. Nagayama PMMA vs PETN 1.72 g cm

2 0.4

Pressure / GPa

Position of shock front / cm

−3

Impedance matching

0.6

SDT point

0.2

Ui = 1.0

PETN 1.72 g

Free surface velocity of PMMA (km

0 0

0.5

Time /µ s

1

PMMA Hugoniot

1

Rayleigh line for PETN

−3

cm s

−1

Reflected Hugoniot for PMMA (Mirror approximation)

)

0

1.5

0

0.2 0.4 −1 Particle velocity / km s

Fig. 5. 2 Locus of shock front along the wedge slope obtained by wedge test (from the onedimensional numerical simulations of wedge tests). Right side is the analysis of wedge test results

relationship between the pressure and the run distance to detonation obtained by the above procedure is plotted by red Δ in Fig. 5.3. In 1965, at the Fourth Detonation Symposium, Ramsey and Poplato have found that there is a linear relationship between incident pressure and run distance to detonation with logarithmic scale [9] as shown in Fig. 5.3. This engineeringly useful relationship is called Pop plot and is still used as calibration data for numerical simulations of shock initiation. They also pointed out the problem of Hugoniot measurement, in which the extension of the measured points on the Us–Up plane predicts the C-J point rather than the spike point. The possibility of the measurement of partially reacted Hugoniot was suggested [9]. By this time, the following was 10 Run distance to detonation / mm

Fig. 5. 3 A linear relationship between incident pressure and distance to detonation (Pop plot)

2 −3

PETN 1.72 g cm

Experimental data by Stirpe et al. 1970 Least square fit to experimental data ( by Cooper 1993)

10

1

0

10 −1 10

0

10 Pressure / GPa

10

1

5 Shock Initiation

153

already known. The existence of the reaction wave that overtakes the shock front for homogeneous explosive was shown by Campbell et al. [10, 11]. The significant energy release behind the shock front in heterogeneous explosive was measured by Craig and Marshall [12]. While the data related to shock initiation had been accumulated, the development of measurement methods such as directly embedding gauges in explosives had been progressed [13, 14]. In 1976, at the Sixth Detonation Symposium, the results of particle velocity and pressure measurements by embedding gauges were presented. These measurements were the first to clarify the reaction growth behavior behind the shock front in the explosive [15–17]. The buildup process can now be understood concretely. Instead of wedge test, the projectile from gas gun was used, and more accurate one-dimensional assumptions were realized. The empirical p 2 τ criterion for shock initiation proposed by Walker and Wasley was discussed by highly professional commentators at this symposium [18]. This critical energy criterion was originally applied to flyers but was subsequently modified to apply to a variety of other impact configurations [19, 20]. Before 1970, the results of numerical studies presented were mainly related to the steady detonation or the shock wave or deformation generated by detonation [21–23]. The simplest propagation model, i.e., C-J volume burn described in Chap. 3 in this book, has already been introduced by Wilkins in 1960 [21]. According to Wilkins’ textbook [24], an early version of Hydrodynamic, Elastic, Magneto, and Plastic (HEMP3D) was developed in the late 1960s. The publications of the simulation results for shock initiation were delayed because it took time to experimentally grasp the shock initiation process and to construct the equation of state at high temperature and pressure ranges. During this period, Mader published some results on the simulation of shock initiation in which BKW EOS and Hugoniot base Mie Gruneisen EOS and Arrhenius rate law were employed [25–28]. He simulated the interaction of shock wave and void in Nitromethane, related to the formation of hot spot. In 1976, Forest Fire burn model has been presented by Mader and Forest [29, 30]. The name is from originator Forest [31]. This initiation model is a semiempirical model based on the concept of single curve buildup [32]. The parameters can be obtained analytically using shock initiation experiment data such as the input pressure and locus of shock-to-detonation transition. This is an epoch-making and excellent model if the experimental results can be reproduced by simulation without any parameter adjustment. In Mader’s textbook [33], it has been described that the model reproduced multiple experiments for shock initiation. In 1980, new burn model has been proposed by Lee and Tarver [34] and accurately reproduced the pressure and particle velocity gauge records for various explosives. This is phenomenological model and consists of two terms which describe hot spot formation and the growth of reaction, respectively. In 1981, at the Seventh Detonation Symposium, many burn models [35–38], including the Forest Fire model [39, 40] and Ignition and growth model [41, 42], have been proposed and reproduced gauge records. In addition, these models were employed in two-dimensional fluid dynamic code and applied to the application problems for the shock initiation, such as the threshold curve of projectile impact, corner turning, gap test, and failure radius.

154

S. Kubota and K. Nagayama

Direct Analysis Generated Modified Arrhenius Rate (DAGMAR) by Anderson et al. [43] and burn model by Partom [44] had also been proposed at this symposium. The above-mentioned research profiles on shock initiation indicated that the understanding of the shock initiation phenomena requires the development of advanced measurement system and the development of the numerical simulation technologies. The development and application of the particle velocity and pressure measurements by embedded gauge in explosive continued to progress [45–52]. Various types of interferometers were developed, VISAR, ORVIS, and Fabry–Pérot as shown in Chap. 4 and were applied in detonation research [38, 53–55]. New photonic Doppler velocimetry (PDV) system was proposed by Strand in 2004 and was modified for the measurements for detonation physics [56–58]. PDV came to be used for the measurement of a large number of detonation studies since around 2014 as a system that can measure particle velocities in the low- to high-velocity ranges relatively easily and high accurately. At the same time, the research on EOSs for unreacted explosive and for detonation products and high-speed computers were progressed. Improvements were added to burn models that had already been developed, and more burn models were proposed. In 1985, Kipp presented model that introduces shear band concept for initiation of granular explosive [59], and Johnson et al. proposed JTF model same year [60, 61]. In 1998, History Variable Reactive Burn (HVRB) model was installed in CTH code by Starkenberg and Dorsey [62]. In the 2000s, the following sophisticated burn models were appeared one after another. There are the physics-based reactive burn model by Horie et al. [63–66], Wescott, Stewart, Davis (WSD) equation of state and reaction rate [67], CREST by Handley [68, 69], and scaled unified reactive front (SURF) model [70], and so on. There are many types of burn model, and many types of EOSs for unreacted phase and for detonation products, and mixture rule, so discussion on the reactive burn models is very difficult. The detailed discussion of development of burn model can be confirmed in paper by Handley et al. [71]. James and Lambourn analyzed the various types of particle velocity gauge records and suggested that shock entropy is the measure of the shock strength which controls the reaction [72]. Its concept is that of CREST. The CREST can reproduce the gauge records for double shock initiation and for short pulse initiation without changing the parameters of burn model [71]. The same parameters could be applied even though changes in the initial state such as density and temperature. We have also proposed a unified method that can reproduce the shock initiation experiments for arbitrary initial density with one parameter set as shown in Sect. 5.3.3.2 [73, 74] In the latter half of the 1990s, mesoscale calculations that took into account particle shape and distribution began to appear. At the same time, an attempt to consider detonation and initiation phenomena from the microscopic behavior of atoms and molecules started. The shock compression characteristics and their equations of state have been determined using techniques such as density functional theory and are being applied to simulations of reactive flow. While simulating explosion phenomena from micro-, meso-, and macroscale perspectives and supporting experiments, the progress and application of explosion simulation technology have

5 Shock Initiation

155

contributed to the safety field, effective use of explosion energy, and elucidation of explosion invention phenomena. However, this chapter does not treat mesoscale and microscale simulations. When applied to complex real phenomena of explosion safety evaluation such as the estimation of the sympathetic detonation of explosive, even now, macroscopic reaction flow simulation is the most effective.

5.3 Modeling of the Reactive Flow 5.3.1 Introduction The simulations of the shock initiation, the failure process of detonation, and the steady detonation may be called as the simulation of reactive flow. The detonation propagation or initiation phenomena is expressed as the flow that has high pressure and temperature distributions and is described by the conservation of mass, momentum, and energy as shown in Chap. 2. The explosives have three different states: unreacted state, completely reacted state, and intermediate state between them. Former two states are described by EOS as shown in Chap. 3. The intermediate state is often treated simple mixture of the former two states as shown in next section. The reaction rate is calculated by burn model and is the main part of the simulation of reactive flow. Other than the burn model, there are various technical problems remained. In particular, the mixture rule is an area that has not been fully discussed and is treated in this section. The very interest relationship of state variables was found and discussed [75–78]. Moreover, the application for gap tests, and the simulations which employed unified form EOS described in Chap. 3 are shown in this section [73, 74].

5.3.2 Discussion of Mixture Rule It is impossible to determine the state in the partially reacted explosive by considering the detailed components of the intermediate products. There are several methods to determine the state variables, and typical one is the two-phase mixture rule [60]. Based on this theory, the reacting explosive is regarded as a simple mixture of phases that correspond to the unreacted explosive and its detonation products. During the simulation, the specific volume v and specific internal energy ε at the explosive part are calculated by the differential equations of the conservation laws. By using the variables in calculation field and the degree of the reaction λ at previous calculation step, the λ at the current step is calculated by solving burn model. The state variables just after the process of reaction must be determined. The unknown independent variables

156

S. Kubota and K. Nagayama

are the specific volume and specific internal energy for unreacted and completely reacted states. There are four independents unknown variables, vU , v R , εU , and ε R . Subscripts U and R indicate the unreacted component and the completely reacted gaseous components, respectively. The internal energy and specific volume of the reacting explosive have been represented by a linear combination of individual internal energies and specific volumes for two phases. Since the specific volume and the internal specific energy are extensive parameters, they are described by linear combination of each component. v = (1 − λ)vU + λv R

(5.1)

ε = (1 − λ)εU + λε R

(5.2)

where the variable λ denotes the degree of chemical reaction, changing from 0 for no reaction to 1 for complete reaction. These equations are the first and second assumptions made to determine the state variables. The equations of state of the two phases are necessary, and in many cases, the form of F( p, v, ε) is adopted. When the EOSs for two phases are known, the third assumption is that of mechanical equilibrium. pU (vU , εU ) = p R (v R , ε R )

(5.3)

To close the equations, another physical assumption is needed, and many assumptions have been adopted. A possible reason that different assumptions have been applied for the closed assumption may be that the numerical results of the shock initiation problems are insensitive to closed assumption. In particular, the thermal equilibrium assumption and the adiabatic behavior of the unreacted component have been used as the closed assumption. There is no evidence that temperature equilibrium is achieved instantaneously. In addition, one of the major reasons for which the assumptions other than thermal equilibrium have been used may be that it has been experimentally or theoretically difficult to obtain accurate information on the temperature of both the reactant and detonation products. In this section, we clarify why the difference of the numerical results caused by difference of the closed assumptions is small. Four types of assumptions were employed as the closed assumption, and the results were examined [76]. The ignition and growth model and the JWL equation of state (EOS) for reactant and products are employed in these hydrodynamic calculations [34]. Kubota et al. [75, 76] found that the relation for the specific volumes of the two components can be related by a single curve of the specific volume of reactant vs that of products. They discussed this relationship and showed how it can be applied, using the results of the numerical simulation.

5 Shock Initiation

5.3.2.1

157

Closed Assumptions

The following relations were examined as the closed assumption: TU = TR

(5.4)

εU = εUi

(5.5)

εU /ε R = εU H /ε Ri

(5.6)

εU = ε R

(5.7)

where T is the temperature, and subscripts H and i indicate the Hugoniot and isentrope line, respectively. We called the above assumptions as (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) the energy-balance model, respectively. Assumption of (III) Hugoniot/isentrope corresponds to HOMSG EOS by Mader [33]. The temperature-dependent JWL EOS [34] was employed for both components to examine the assumption of thermal equilibrium and was expressed as follows: p = A ex p(−R1 V ) + B ex p(−R2 V ) +

ωCv T, V

(5.8)

where Cv is the average heat capacity. The meaning of parameters is the same as in Eq. 3.9 in Sect. 3.2.2. The values of parameters can be confirmed in reference [34]. Since the flow equations do not explicitly include the temperature, the following thermodynamic identity was used to treat the temperature effects. ) ] [ ( ∂p − p dv dε = Cv dT + T ∂T v

(5.9)

Combining this with Eq. 5.8, the function T (ε, v) is obtained. The closed assumption (II) implies the adiabatic behavior of the unreacted component, and under this assumption, the two components are thermally isolated. The equation of state that describes the isentropic state is necessary. The following methods are often adopted. The Grüneisen-type EOS in Eq. 2.27 of Chap. 2 is rewritten as follows. pU = pU H +

γU (εU − εU H ) vU

(5.10)

This equation satisfies both the isentrope and Hugoniot lines for the unreacted component, so,

158

S. Kubota and K. Nagayama

pU i = pU H +

γU (εU i − εU H ) = pU H + aγ (εU i − εU H ) vU

(5.11)

The assumption of aγ = constant is often used in this type of solid-state materials. By differentiating both sides of Eq. 5.11 with vU and rewriting it using the thermodynamic identity (∂εU /∂vU )s = − pU i , the following differential equation is obtained. dpU i + aγ pU i = f (vU ) dvU

(5.12)

As shown in Eqs. 2.14 and 2.15 in Chap. 2, using Hugoniot relations, pU H and εU H are the function of specific volume only, in consequence, the right-hand side of Eq. 5.12 is also the function of specific volume only. The JWL EOS for unreacted explosive is based on the experimental Hugoniot, so energy is adjusted to initial condition. Because the simulation of the phenomena related to shock-to-detonation transition process requires strict condition, before applying it to the simulation, we recommended to confirm whether the necessary conditions for EOSs for both components are satisfied by the p − v diagram and so on. Not limited to self-made software, it is often the case that the initial conditions and C-J conditions are not satisfied depending on the EOS code.

5.3.2.2

Numerical Procedure for SDT Simulation

The governing equations were the mass, momentum, and energy conservation laws and were solved by the finite difference method along with the burn model, the equations of state, and the mixture rule. For burn model, the ignition and growth model [34] was used, which is expressed as dλ = I (1 − λ)2/9 η4 + G(1 − λ)2/9 λ2/3 p z dt

(5.13)

where η = v0 /v − 1. The parameters z, G, and I depend on the explosive properties and are summarized in Table 5.1. The subscript 0 indicates the initial state of a solid explosive. Since the purpose of this study was to investigate the effect of the difference in the closed assumption on the difference in the simulation results, the parameters of the burn model were decided to satisfy the Pop plot under each calculation condition. The JWL EOS for unreacted explosive and detonation products is employed in these simulations of reactive flow. The samples used for this calculation were PBX9404 (HMX/nitrocellulose/tris-β-chloroethylphosphate = 94/3/3 wt%, density; 1842 kg/m3 ), Composition B (RDX/TNT = 60/40 wt%, density; 1712 kg/m3 ) and pentaerythritol tetranitrate (PETN, density; 1750 kg/m3 ). The parameters in the equations of state are the same values as found in Ref. [34] for PBX9404 and PETN, and in Ref. [79] for Comp. B for both components. The mesh size of the simulations in this section was set 0.05 mm. The mesh size, 100 mesh/mm, or more may be recommended to calculate the reaction zone with high resolution. However, regarding the mesh size, it depends on the required evaluation

5 Shock Initiation

159

Table 5.1 Parameters of the ignition and growth model for PBX9404, Comp. B, and PETN, since the purpose of this study was to investigate the effect of the difference in the closed assumption on the difference in the simulation results, the parameters of the burn model were decided to satisfy the Pop plot under each calculation condition Explosives

I

G

z (0.6 km/s)

z (0.8 km/s)

z (1.0 km/s)

PBX9404

44

800

2.26

2.24

2.15

PETN

20

400

1.70

1.45

1.32

Comp. B

44

414

2.24

2.20

2.12

The inside of the parentheses shows the impact velocity in the simulation

accuracy of research subject or the size of the calculation field. We use both finer and coarser meshes, depending on the problem.

5.3.2.3

Influence of the Closed Assumptions for Simulation Results

Fig. 5. 4 Pop plot for PETN, Comp. B, and PBX9404 together with the results of reactive flow simulation which were carried out to determine the parameter of the burn model

Run distance to detonation / mm

The strategy was to carry out a one-dimensional reactive flow simulation of impact problem between the explosive versus the inert material using selected closed assumption and extract the state quantity during the simulation and compare the results. The impact problems were applied to three cases: PBX9404 (1842 kg/m3 ) versus aluminum, Composition B (1712 kg/m3 ) versus copper, and PETN (1750 kg/m3 ) versus PMMA. The impact velocities were 0.6, 0.8 and 1.0 km/s. The simulation was carried out for sustained pulse experiments such as wedge tests. First, the parameters of the ignition and growth models were determined by simulation so that their results would reproduce each Pop plot as shown in Fig. 5.4. The thermal equilibrium was selected as closed assumption for parameters determination. 10

2

10

1

This calculation results Least square fit to experimental data ( by Cooper 1993)

Comp. B PETN 1.75 g/cc

10

0

10

PBX 9404 −1

0

10 10 Pressure / GPa

1

160

S. Kubota and K. Nagayama

thermal equilibrium

2.6, 3.2, 3.8 µ s

60 0.8 Pressure / GPa

Impact velocity 0.6 km/s

40

Pressure Mass fraction

0.4 20

0 0

1 Position / cm

2

Mass fraction of detonation products

Impact problem Aluminum vs. PBX 9404

Fig. 5. 5 Shock-to-detonation transition process obtained by the reactive flow simulation for PBX9404

0

Figure 5.5 shows the SDT process obtained by the simulation for PBX9404. The propagation process of the shock wave in the explosive is shown by the pressure distribution in explosive at three different times. The origin of the horizontal axis is the impact surface. The vertical axis is the pressure in explosive and the mass fraction of detonation products, i.e., the degree of reaction λ. The three selected times are as follows. First one is a time when the reaction has started, but the effect of reaction is still small. The degree of the reaction is confirmed by the vertical axis of the right side. Second is a time when a slight reaction effect at the shock front, and a significant influence of reaction can be confirmed behind the shock front. Near the impact surface which already moves about 1 mm to right side direction, the reaction has already been completed. Third is a time when the reaction wave catches up with the shock front and reaches the steady detonation velocity. Using the same burn model and those parameters, the impact problems were calculated, varying the closing assumption of mixture rule. Similar calculations of impact problems were carried out for Comp. B and PETN with a different projectile. The typical results are shown in Figs. 5.6 and 5.7. Each figure has four lines according to the closed assumption, and it can be seen the difference of the results due to the difference in closed assumption. Taking the case of PBX9404 with 0.6 km/s impact velocity, which has the largest difference in the simulation results, the run distance to detonation is 14 mm and its difference is 1 mm, which is a difference of 7%. Considering the variability of the data in the shock initiation experiment at a relatively low pressure, it is not a considerable difference. In case of the PETN, the difference among the four assumptions is even smaller, and in Comp. B, the four results almost overlap as shown in Fig. 5.7. When the impact velocity was set to 1 km/s, four results overlapped even in the result of PBX9404. The cause was found.

5 Shock Initiation

161

Impact problem Aluminum vs. PBX 9404

Impact problem PMMA vs. PETN 1.75

60 Impact velocity

40

0.6 km/s

Impact velocity

2.6, 3.2, 3.8 µ s thermal equilibrium isentropic Solid Hugoniot/Isentrope energy balance

20

0 0

1 Position / cm

0.6 km/s

3.5, 4.5, 5.5 µ s

Pressure / GPa

Pressure / GPa

60

40

thermal equilibrium isentropic Solid Hugoniot/Isentrope energy balance

20

0 0

2

1 Position / cm

2

Fig. 5. 6 Comparison of the numerical results obtained using four different closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. These figures show the shock propagation process in PBX9404 versus aluminum and PETN versus PMMA, and impact velocity is 0.6 km/s. Left; PBX9404, Right; PETN

Impact problem Copper vs. Comp. B Impact velocity

Impact problem Aluminum vs. PBX 9404 Impact velocity

0.6 km/s

3.0, 4.0, 5.0 µ s thermal equilibrium isentropic Solid Hugoniot/Isentrope

energy balance

20

0 0

1 2 Position / cm

Pressure / GPa

Pressure / GPa

40

60

40

1.0 km/s

0.6, 0.9, 1.2 µ s thermal equilibrium isentropic Solid Hugoniot/Isentrope energy balance

20

0 0

0.2

0.4 Position / cm

0.6

Fig. 5. 7 Comparison of the numerical results obtained using four different closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. These figures show the shock propagation process in Comp. B versus copper with 0.6 km/s impact velocity and PBX9404 versus aluminum with 1.0 km/s impact velocity

5.3.2.4

Relative Evaluation Method of Closed Assumption

Since the specific volumes and specific internal energies for unreacted component (vU , εU ) and detonation products (v R , ε R ) are estimated by Eqs. 5.1 and 5.2 under the assumption of mechanical equilibrium, v and ε for each component and for the

162

S. Kubota and K. Nagayama

reacting explosive are located on the same straight line in the specific internal energy and specific volume (ε − v) plane. The conceptual diagram of the ε −v plane for estimating the mixture rule is shown in Fig. 5.8. In this figure, the three results, i.e., results from thermal equilibrium, isentropic solid, and the energy-balance model are plotted, and we assumed three results give the same results in the numerical simulation of the SDT process. When all results were calculated by same (v, ε, λ) values, the pair of the (v, ε) for both components obtained from each closed assumption are located on the same isobaric lines, as shown in this figure. The two isobaric lines are obtained using the EOS for both components. From Eq. 5.1, v for reacting explosive is satisfied and the following relations hold independent of remaining assumptions for mixture rules: v R − v = (1 − λ)(v R − vU )

(5.14)

v − vU = λ(v R − vU )

(5.15)

This concept is convenient for investigating the influence of the closed assumption. The condition under which all assumptions yield the same results is that in the ε − v plane, the slope of the isobaric line for the unreacted component is equal to that for the detonation products at any pressure level [76]. (

∂εU ∂vU

)

( = p

∂ε R ∂v R

) (5.16) p

Strictly speaking, the condition of Eq. 5.16 does not hold. In the analysis here using JWL EOS for both components, it is considered that there is a situation close to the condition of Eq. 5.16, and we will verify it later. Fig. 5. 8 Conceptual diagram for consideration of mixture rule in ε − v plane

5 Shock Initiation

163

When the slopes of the isobaric lines in the ε − v plane for each component have almost the same values in the range that includes the predicted state variables of each component, all the assumptions yield similar results. In a word, when the shape of the EOS of both components is similar, the difference of the closed assumption almost disappears. The following idea is also plausible. It is considered that a state in which thermal equilibrium is achieved, a state in which both components are completely thermally isolated, and a state between the above two states are physically possible. We define that vU calculated by thermal equilibrium is vU (I) , and vU calculated by isentropic solid is vU (II) . In specific set of (v, ε, λ), the range of specific volume that is plausible as a physical model is vU (II) ≤ vU ≤ vU (I) and v R(I) ≤ v R ≤ vU (II) . Similar relations can be defined for εU and ε R . The energy-balance model is not in this range.

5.3.2.5

Verification of Relative Evaluation Method

Using the state variables extracted from the simulations, the concept of the previous section is verified. The path along the particle in the p − v plane drawn by numerical results will be described. Focusing on the position 3 mm from the initial interface (origin of x-axis) in Fig. 5.5 (or left side of Fig. 5.6 at thermal equilibrium), after the pressure rises to about 4 GP near the initial interface, the pressure rises to about 17 GPa due to the reaction and the reaction completed. The locus of this pressure rise is shown as path A on the p − v plane in Fig. 5.9. In the figure, the isentropic expansion line passing through the C-J point of detonation products and the Hugoniot for unreacted component are also plotted. Paths B and C correspond to the results from the impact velocity of 0.8 and 1.0 km/s, respectively. The observation point is 3 mm from the initial interface. In the paths A and B in Fig. 5.9, after the reaction starts, the process is expansion with pressure rise due to reaction. In contrast, the process along path C is the compression. For the latter case, it is considered at observation point that the influence of the compression by the impact of the aluminum is greater than the influence of the expansion caused by reaction. Figures 5.10, 5.11, and 5.12 correspond to cases where Eq. 5.16 holds approximately. It can be confirmed that the results obtained by the simulation using four different assumptions give almost the same results to the reacting PBX9404. In Fig. 5.11, the upper left figure shows the ε − v relationship among the reacting PBX9404 and two components and extracted at 5, 10, and 20 GPa under the condition of thermal equilibrium. It is a slightly expanding process in which the reaction proceeds and the pressure is increased. However, the two components move toward compression. The remaining three figures are plotted with the extracted state variables at 5, 10, 20 GPa. The simulation result does not depend on the selected closed assumptions. The difference of the specific volume between the two components becomes smaller as the pressure rises. Figure 5.12 shows the result extracted from Path C, where the process is compression.

164

Impact problem Aluminum vs. PBX 9404

Isentrope

PBX 9404 Impact velocity ; 0.6, 0.8, 1.0 km/s

40

Observation point;

Pressure / GPa

Fig. 5. 9 Path along the particle of SDT process in the pressure–specific volume plane obtained by simulation for PBX9404. The observation point was initially 3 mm from the initial impact surface. Path A; 0.6 km/s impact velocity, path B; 0.8 km/s and path C; 1.0 km/s

S. Kubota and K. Nagayama

3.0 mm from initial interface Hugoniot & Isentrope

Path B

20

Path A

Path C Hugoniot

0

0.4

0.6 3

−1

Specific volume / cm g

Fig. 5. 10 Relations of the specific internal energy and specific volume for the reacting PBX9404. The unreacted solid and the detonation products components are also plotted together with the isobaric line for each component. The four results adopt the following closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. The graph on the left extracts the state quantity of 5 GPa from the path A in Fig. 5.9, and the figure on the right extracts the state quantity of 10 GPa

When the slopes of the isobaric lines in the ε − v plane for each component have almost the same values in the range that includes the predicted state variables of each component, all the assumptions yield similar results. In contrast, in the case of PETN, it is confirmed the state where the slopes of the two isobaric lines are different. The right figure of Fig. 5.13 is the SDT process in PETN. Path H in left side figure of Fig. 5.13 is the result of the above-mentioned simulation under the condition of thermal equilibrium. The impact velocity is 0.8 km/s, and the observation point is

5 Shock Initiation

165

Fig. 5. 11 Relations of the specific internal energy and specific volume for the reacting PBX9404. The unreacted solid and the detonation products components are also plotted. The four results adopt the following closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. The graph on the left extracts the state quantity of 5 GPa from the path A in Fig. 5.9, and the figure on the right extracts the state quantity of 10 GPa

1 mm from the initial interface. Each path in Fig. 5.13 will be explained. Even though the velocity is 1.0 km/s and the observation point is only 1 mm from the interface, the sensitivity of PETN is relatively high and the shock impedance of PMMA is small, so path I remains with little change in specific volume during reaction. It is an image that the inertia of PMMA and the expansion due to the reaction are balanced. Path H and Path G are the expansion processes of reacting PETN. Figure 5.14 shows the case where the slope is different between the state variables extracted from path G and path H. When the slopes of the isobaric lines are significantly different, it can be confirmed that the simulation results are different due to the difference in closed assumptions.

166

S. Kubota and K. Nagayama

Fig. 5. 12 Relations of the specific internal energy and specific volume for the reacting PBX9404. The unreacted solid and the detonation products components are also plotted together with the isobaric line for each component. The four results adopt the following closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. The graph on the left extracts the state quantity of 5 GPa from the path A in Fig. 5.9, and the figure on the right extracts the state quantity of 10 GPa

Fig. 5. 13 Path along the particle of SDT process in the pressure–specific volume plane obtained by simulation for PETN (initial density 1.75 g cm−3 ). Path G; 0.6 km/s impact velocity, path B; 0.8 km/s and path C; 1.0 km/s (Left side). Comparison of the numerical results obtained using four different closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. These figures show the shock propagation process in PETN versus PMMA (Right side)

5 Shock Initiation

167

Fig. 5. 14 Relations of the specific internal energy and specific volume for the reacting PETN (initial density 1.75 g cm−3 ). The unreacted solid and the detonation products components are also plotted together with the isobaric line for each component. The four results adopt the following closed assumptions: (I) thermal equilibrium, (II) isentropic solid, (III) Hugoniot/isentrope, and (IV) energy-balance model. The graph on the left extracts the state quantity of 10 GPa from the path G in Fig. 5.13, and the figure on the right extracts the state quantity of 10 GPa from the path H in Fig. 5.13

5.3.2.6

Modeling of the Reaction Zone by vU − v R Relation

In the previous section, the assumption of Hugoniot/isentrope is confirmed as the physically conceivable condition that exists between thermal equilibrium and isentropic solid. We have found very interesting relation between vU and v R that extracted from the simulations of shock initiation under the assumption of Hugoniot/isentrope. vU can be approximately expressed as a monotonically increasing and single-valued function of v R as shown in Fig. 5.15. The simulations are the impact problem for PETN (1750 kg/m3 ) versus PMMA, and the impact velocities are 0.6, 0.8, and 1.0 km/s. The paths G, H, and I in Fig. 5.15 correspond to those in left side of Fig. 5.13. The three vU − v R relations extracted from paths G, H, and I are almost on a single line in vU − v R plane. This suggests that the variables vU and v R can be calculated only by Eq. 5.1 and vU = f (v R ). We will consider the reason why the vU − v R relation exists and its meaning. The simplest reaction path in p −v plane for SDT process is included in the steady detonation. That corresponds to the expansion process from Neumann spike to C-J point. The conceptual diagram for the description of reaction zone by ZND model shown in Fig. 2.19 is rewritten in Fig. 5.16. Hugoniots for unreacted component, completely reacted component, and mixture of both components are plotted together with reaction path, i.e., Rayleigh line. The assumptions are that the unreacted component and completely reacted component exist on Hugoniot for each component. Along the Rayleigh line, all variables

3

Specific volume of unreacted soid component / cm g

Fig. 5. 15 Relationship of the specific volumes between unreacted solid and completely reacted components of partially reacted lines. The colored lines are obtained by reactive flow simulation for the three impact problems with different velocity

S. Kubota and K. Nagayama −1

168

PETN 1.75 g cm

0.5

−3

Closed assumption Hugoniot/isentrope

0.4 from simulations of reactive flow PETN vs PMMA Path Impact velocity

G H I

0.3 0

0.6 km/s 0.8 km/s 1.0 km/s

1

2 3

−1

Specific volume of completely reacted component / cm g

Fig. 5. 16 Conceptual diagram for the description of reaction zone by ZND model in p − v plane

Pn

P(v)

p(vR)

Rayleigh line Hugoniot for solid explosive (unreacted)

Pressure

p(vU)

Hugoniot for detonation products

Pcj Mixture Hugoniot

Specific volume

are calculated by method as shown in Sect. 2.2.2. In the reaction zone of steady detonation, the vU − v R relation can be obtained using the following equation uniquely. pU H (vU ) = p = p R H (v R ).

(5.17)

The assumptions make the vU − v R relation unique. We extend the above concept for arbitrary partially reacted state. We call this model as extended ZND model here after. From the assumptions, ε is dependent variable same as εU and ε R . For specified values of v and λ, the variables of vU and v R that satisfy Eqs. 5.1 and 5.17 are calculated to determine p at the same time. The left side of Fig. 5.17 shows the partially reacted Hugoniot, and the right side is the vU − v R relation obtained from

5 Shock Initiation

169

left side partially reacted states. Nine lines with different λ are drawn and completely overlap each other, so it becomes one thin line. Assumption of Hugoniot/isentrope that was used simulation in Fig. 5.15 corresponds to HOMSG EOS by Mader [33]. This EOS referred an isentrope line for gas component. When the specific volume of the completely reacted component is larger than v0 , the question remains whether it is suitable as Hugoniot state. In addition, the isentropic lines passing through C-J point are very close to Hugoniot passing through C-J point before and after C-J point. Instead of Hugoniot, the isentrope line will be used for completely reacted component in extended ZND model. The assumption is that the v R and vU exist on the isentropic line for completely reacted component and Hugoniot line for unreacted component, respectively. It can be written using internal energies as follows. εU = εU H (vU )

(5.18)

ε R = ε Ri (v R )

(5.19)

At C-J point, the following relation holds, Q = ε Ri (v R )−ε R H (v R )

(5.20)

The advantage lies in its simplicity. It has the following disadvantage, which may limit its use. The variable ε becomes the dependent variable. The left side of Fig. 5.18 shows the state lines of partially reacted explosive obtained by above assumption in p − v plane. The partially reacted state was almost the same compared

Fig. 5. 17 State lines of partially reacted explosive with unreacted and completely reacted components. The left side partially reacted Hugoniot in p − v plane, the right side is the vU − v R relation obtained from left side partially reacted states

170

S. Kubota and K. Nagayama

Fig. 5. 18 State lines of partially reacted explosive obtained by extended ZND model in p − v plane (left side). The pU versus vU and p R versus v R relations for unreacted solid component and for completely reacted component, and the p versus vU and p versus v R relations obtained from the partially reacted states were plotted. The nine lines with different λ are drawn and completely overlap each other, so it becomes one thin black line (right side). From assumption, this is trivial

with Fig. 5.17. The p versus vU and p versus v R relations of the partially reacted explosive are shown in right side of Fig. 5.18. Nine lines with different λ are drawn and completely overlap each other, so it becomes one thin black line. We will consider a closed assumption of Hugoniot/isentrope by analytically. For investigation of the mixture of the reacting explosive, the state variables (v, ε, λ) must be determined. This is obtained naturally from the simulation for SDT process. Here we try to consider more analytical method with EOS only. There are tremendous combination of the (v, ε, λ) for all physical processes of the initiation, so the mixture rule must be able to be applied for arbitrary set of (v, ε, λ). The specific volume v of reacting explosive is determined arbitrary under the condition of vU < v < v R . At the same time, λ is determined arbitrary in the range of 0 < λ < 1. The values of v and λ are arbitrarily specify within the above range. The ε can be defined following relation as thermodynamically possible point. ε = λε Ri (v) + (1 − λ)εU H (v)

(5.21)

The specific volume and the specific internal energy for each component as the independent variables are solved using Eqs. 5.1–5.3 and closed assumption. As a closed assumption, we referred to HOMSG [33]. ε' = λε Ri (v R ) + (1 − λ)εU H (vU )

(5.22)

εU = ε × εU H (vU )/ε'

(5.23)

5 Shock Initiation

171

Fig. 5. 19 State lines of partially reacted explosive with unreacted and completely reacted components in p − v plane. The left side based on the Hugoniot/isentrope, the right side is the Hugoniot/Hugoniot assumptions

ε R = ε × ε Ri (v R )/ε'

(5.24)

After all, dividing Eq. 5.23 by Eq. 5.24 agrees with Eq. 5.6. Based on the set (v, ε, λ), the set of vU , v R , εU and ε R that satisfies the mixture rule already described here is calculated iteratively. If λ is fixed, v is changed within the range to be evaluated, and the state variables are determined and partially reacted state line at λ can be obtained. The lines of partially reacted state in p−v plane are shown in Fig. 5.19. The comparison between the results by the Hugoniot/isentrope and that by the extended ZND for the vU versus v R relation is shown in right side of Fig. 5.19. At the high compression region, the results of the Hugoniot/isentrope and extended ZND is overlap. At least, we predict that both models can be applied with high accuracy to SDT problems caused by high-velocity impact. In the case of the Hugoniot/isentrope, the v R and vU exist very near on the isentropic line for detonation products and on the Hugoniot line for unreacted solid as shown in Fig. 5.20. The mixture rule defines the starting and the ending point and boldly predict the state variables in the midpoint. The relationship between state variables from a basic point were shown. The models described in this section can be applied for the simulation of SDT process and of detonation propagation.

5.3.3 Simulations of Reactive Flow by Unified EOS The shock initiation phenomena depend on the initial state, such as density, temperature, and pressure. In general, there are few applications that use explosives with increased initial pressure. The same is true for temperature. Changes in the initial

172

S. Kubota and K. Nagayama

Fig. 5. 20 pU versus vU and p R versus v R relations for unreacted solid component and for completely reacted component, and the p versus vU and p versus v R relations obtained from the partially reacted states were plotted. The nine lines with different λ are drawn and almost overlap each other, so it becomes one thick black line

pressure and temperature may be due to a special harsh environment or the pressure and temperature due to an unexpected explosion accident. In contrast, the initial density depends on the explosive manufacturing process such as the pressed compression and casting. Since the detonation properties depend on the initial density, the initial density is adjusted for the application. It is natural to encounter various evaluation requirements regarding the detonation phenomena for explosives with the different initial density and same chemical composition. The simulation of reactive flow requires the robust scheme of hydrodynamics, burn model, EOSs for unreacted solid and completely reacted components, and mixture rule. Of these, the EOSs for both components and burn model depend on the initial density. Each requires a lot of parameters, and almost all the parameters need to be reconstructed every time according to the initial density changes. It is usually not easy to get all the data. In fact, the published data related to EOS frequently used for the research related to shock initiation. Therefore, based on a certain theory, we tried to construct a simulation that can simulate the detonation phenomena with one parameter set even if the initial density is changed for the explosives with the same composition. Although it is also important as an application, we have considered that it is one of the necessary approaches when discussing the relationship between the burn model and the initial density. We proposed the numerical modeling of the shock initiation of high explosive using the unified form EOSs for unreacted and completely reacted components [73, 74]. For arbitrary initial density, only one parameter set has been used. Our proposed EOS described in Sect. 3.4.1 was employed for detonation products [80, 81]. For the unreacted solid component, Hugoniot-based Mie Grüneisen-type EOS described in Sect. 2.1.3.1 and the porous model described in Sect. 2.1.3.4 were adopted. PETN was selected as the subject of this study.

5 Shock Initiation

5.3.3.1

173

Numerical Procedure for SDT Simulation by Unified EOS

The isentropic line passing through C-J point is JWL form in Eq. 3.6 and rewritten according to the following, p TMD = A ex p(−R1 VR ) + B ex p(−R2 VR ) + C VR−(ω+1) Ri

(5.25)

{ εTMD =− Ri

p TMD Ri dv

(5.26)

where A, B, C, R1 , R2, and ω are the JWL parameters, and VR = v R /v0 . The subscripts R and i represent the complete reaction component and the isentropic state, respectively. Because the initial density of 1.77 g cm−3 is very close to theoretical maximum density (TMD) for PETN, this initial density was approximately regarded as TMD in this section. Unified form EOS for detonation products is rewritten by, p R = p TMD Ri (v R ) +

) γ R (v R ) ( ε − εTMD Ri (v R ) vR

(5.27)

Here, γ R (v R ) is a Grüneisen function and function of only v R as shown in Fig. 3.6. The porous Hugoniot was shown in Fig. 2.6 εUP H =

1 P p (v00 − vU ) 2 UH

(5.28)

where the variable v00 denotes the initial specific volume of porous state. The subscripts U and H are the unreacted and Hugoniot states. The superscript P denotes the porous state. This relation also satisfies Eq. 2.24, so substituting the above equation gives the following equation, pUP H (vU ) = pUTMD H (vU ) Kγ =

K γ − v0 /vU K γ − v00 /vU

2 γUTMD (vU )

+1

(5.29) (5.30)

The JWL parameter for PETN was referred from [82]. For the unreacted solid component, u s = 2.42 + 1.91u p and γUTMD = 1.15 were used [83]. The ignition and growth model described in Eq. 5.13 was used [34]. The first term of Eq. 5.13 was applied from λ = 0 to the ignition term limit (IGL). Impact problems of PETN and PMMA were solved for PETN with various initial densities. The initial mesh size was set to 50 μm. The strategy of this numerical study is as follows [73]. The parameters of burn model for PETN with initial density 1.72 g cm−3 were determined

174

S. Kubota and K. Nagayama

so that the simulation results reproduce Pop plot. Using determined parameters, it was examined whether the result for arbitrary initial density can be reproduced or not. Since the reaction rate depends on the initial density, the parameters of burn model determined using above method will not be applicable to low initial density cases. Later we will clarify the relationship between the initial density and the appropriate parameter values. In this simulation, we pay attention to growth process. The main reaction growth is expressed using the second term. The main parameters in the second term of Eq. 5.13 are G and z. We select the exponent z as the subject of this numerical study. All parameters are fixed except z and IGL. Finally, it is shown that the detonation problem of arbitrary initial density could be simulated with only one parameter set.

5.3.3.2

Unified Simulation for Reactive Flow

Figure 5.21 is the Pop plot of PETN for three densities, 1.72, 1.6, and 1.4 g cm−3 . The solid symbols and +symbol were obtained using the parameters in Table 5.2. Lines in this figure are fitting lines for wedge test data [6, 84, 85]. Because the parameters were adjusted to PETN with the initial density of 1.72 g cm−3 , there is large discrepancy for cases of the initial densities 1.6 and 1.4 g cm−3 . When the exponent z is decreased as shown with open symbols, the simulation results are close to the experiment. I = 20, G = 8100 IGL = 0.01

Run distance to detonation / mm

Fig. 5. 21 Pop plot for PETN [6, 84, 85] and results of this simulations

10

z = 2.30

Calculaion Initial Density

1

1.72 1.60 1.40 1.60 z = 2.15 1.40 z = 2.05

PETN

10

1.6

0

1.4

10

1.72

0

10

1

Pressure / GPa

Table 5.2 Parameters of ignition and growth model for PETN with the initial density of 1.72 g cm−3

I (μs−1 )

G (μs−1 Mbar−1 )

Z

IGL

20

8100

2.3

0.01

The parameters were adjusted based on the initial density of 1.72 g cm−3 in this study

5 Shock Initiation

175

Figure 5.22 shows the shock locus in PETN. To confirm the effect of reaction on the propagation of shock front, the simulation result without reaction is also plotted by a dotted line. In addition, the locus estimated by Hugoniot (U s = 1.32 + 2.58U p ) which was measured by Stripe et al. [84] is also plotted. Two shock loci without reaction agree well, and this means that porous Hugoniot model is appropriate for the reference line for EOS of unreacted component. The inflection point of the shock locus corresponds to the shock-to-detonation transition (SDT) point. It is the moment when the reaction wave overtakes the shock front. The numerical result with the initiation model shows that before the SDT point, the shock front is accelerated by the reaction behind shock front. The right side in Fig. 5.22 is for the case of initial density 1.4 g cm−3 at which the SDT point is at about 0.2 cm from the initial impact surface as same in left figure. When comparing the condition in which the run distances are almost the same, these results mean that the propagation of shock front in the low initial density case is influenced by the chemical reaction more than that in the high initial density case. The weak point of the porous Hugoniot model is that the model ignores the compression process from the initial density to the theoretical maximum density. Although the compression occurs, the pressure does not rise until TMD. This weak point gives the influence to the calculation of the partially reacted state for the low initial density case. The relative compressibility η is suitable for the ignition term because it differs greatly depending on the initial density. However, when the initial density is very small and no limiter is applied, a large decomposition occurs only in the ignition term, and as the results, the shock wave is accelerated unnaturally. To avoid this situation, the IGL was adjusted in the case of initial density 1.0 g cm−3 . Figure 5.23 shows the effect of the IGL on the velocity of the shock front in PETN of initial density 1.0 g cm−3 . The shock loci were obtained by the numerical simulation without a growth term. In the case of the IGL = 0.22, the velocity of the shock front

Position of shock front / cm

Position of shock front / cm

PETN 1.60 Calculation

0.4

with initiation model without reaction (Porous Hugoniot)

0.2 Us = 1.32 + 2.58 Up

PETN 1.40 Calculation with initiation model

0.4

without reaction ( porous Hugoniot )

0.2

( Under P = 1.95 GPa)

0 0

0.4 0.8 Time / µ s

1.2

0 0

1 Time / µ s

2

Fig. 5. 22 Loci of shock front in PETN obtained by simulations with and without burn model. In left figure, U s ; shock velocity (km/s), U p ; particle velocity (km/s) U s –U p relation [84]

176

S. Kubota and K. Nagayama

agrees with that obtained using the porous Hugoniot without a reaction. By using this IGL value, the exponent z was estimated by the simulation. The results for the initial density 1.0 g cm−3 are plotted in Fig. 5.24 with square symbols. The simulation results simultaneously satisfy the Pop plot and the velocity information of the shock front as shown in Fig. 5.24. Figure 5.25 shows the pressure histories in PETN at various points obtained by numerical simulation of PETN of initial density 1.75 g cm−3 . The results reproduced the pressure gauge records [15, 34]. The parameter z must be adjusted for each initial density. Figure 5.26 shows the relationship between the initial density of PETN and the exponent z in Eq. 5.13. To be 0.8

Fig. 5. 23 Effect of the ignition limit (IGL) into the locus of shock front in PETN. The numerical simulation was carried out without growth term

IGL = 0.22

Position of shock front / cm

PETN 1.0 Ignition term only

IGL = 0.3

0.4 IGL = 0.01

0 0

10

10

1

PETN

10

1.6

1.0

0

1.4

5 Time / µ s

1.72

10

1

PETN 1.0

Run distance to detonation/ mm

Run distance to detonation / mm

I = 20, G = 8100 Calculaion Initial Density 1.72 z = 2.30 1.60 z = 2.15 1.40 z = 2.04 1.0 z = 1.95

IGL = 0.1

Calculation Experiment

0

10

-1

0

10 Pressure / GPa

10

1

10 0 10

10

1

Time of run distance to detonation/ µ s

Fig. 5. 24 Pop plot for PETN [6, 84, 85] and results of the unified simulations for reactive flow. Right side: The relationship between the time of run distance and the run distance to detonation for PETN of initial density 1.0 g cm−3 . Experimental data is Ref. [6]

5 Shock Initiation

177

60

Fig. 5. 25 Calculated pressure history in PETN of initial density 1.75 g cm−3

Obervation points PETN 1.75 0, 1.04, 2.10, 3.79, 4.5, 5.0, and 6 mm

Pressure/ GPa

Sustained shock pulse of 1.90

40

20

0 0

1 Time / µ s

2

able to apply this simple modeling of the shock initiation phenomena for PETN for arbitrary initial density, this relation between the z and initial density is approximated using a function, z = a + b exp(ρ0 /c), where a = 1.928, b = 5.056 × 10−4 , and c = 0.26158.

2.2 Exponent z

Fig. 5. 26 Relationship between the initial density of PETN and the exponent z for initiation model; parameter z was estimated in this numerical study

2

0.8

1.2

1.6

Initial density / g cm

-3

178

S. Kubota and K. Nagayama

5.4 Gap Test and Its Numerical Simulation 5.4.1 Small-Scale Gap Test When the accidental explosion occurs in the explosive magazine, the prevention of sympathetic detonation is important to restrain the expansion of the explosion damage. In Japan, the study to reduce the explosions damage, such as the installation of bulkheads in the explosive magazine had been carried out [86, 87]. The gap test is an impact sensitivity test for sympathetic detonation and is carried out according to the standard. Furthermore, various gap tests have been conducted all over the world because a lot of data on the detonation problem can be obtained by combining with high-speed photography and electric measurements. Unlike the wedge test, the gap test does not allow accurate evaluation of SDT point. The gap test is focused on whether violent reaction occurs or not under given impact conditions. When evaluating the initiation phenomena by simulation of reactive flow, the parameters for burn model that reproduce the wedge test results or the embedded gauge data are obtained. Various phenomena that include the initiation are evaluated by the simulation with above-mentioned parameters. Although the gap test includes the initiation phenomena, whether the above-mentioned simulation procedure could reproduce the gap test results or not is not clear. The finding of the conditions for sympathetic detonation is not the direct purpose of the wedge test. Of course, it also depends on the accuracy required for each evaluation target. For calibration of burn model the wedge test results, embedded gauge data and gap test results have been employed. However, it is easy to see that the cost of the experiments is very high. The understanding of the phenomena of the sympathetic detonation by experiments and the development of direct estimation method by the simulation of reactive flow are important. To obtain the detail information of the sympathetic detonation and to establish the estimation method of the parameters for the simulation of reactive flow, the gap test was carried out combined with the high-speed photography and electric measurement [88, 89]. The simulations have been also carried out and compared with the results of gap test.

5.4.1.1

Experimental Setup for the Small-Scale Gap Test

The schematic illustration and photograph of the gap test are shown in Fig. 5.27. The sample was a Composition C4 filled in PMMA pipe. The inner diameter and thickness of the PMMA pipe were 26 and 2 mm, respectively. The height of donor was 50 mm. Large and the small size PMMA plates were set as gap material. The small plate is for adjusting the gap length, and the large plate serves also as blast shield for the high-speed photography. The height of acceptor charge was mainly 40 mm, in addition, and 100 and 25 mm were also used.

5 Shock Initiation

179

Fig. 5. 27 Experimental setup for the small-scale gap test. The schematic illustration is left side, and photograph is right side

To determine Go/No-go, the witness steel block was used at the bottom of the acceptor. High-speed video made by Shimadzu Corporation Japan was set at its maximum framing rate 1,000,000 f/s. The luminescence at the surface of acceptor holder was also used for Go/No-go decision. In some experiments, the measurements using PVDF gauges (PVF2; Dynasen, INC) were carried out. One gauge was set at the interface of the gap material and acceptor charge, another was set inside the acceptor. For pressure measurements, the acceptor consisted of two blocks. All pressure measurements were done under the No-go condition and were applied to estimate Hugoniot of composition C4. On the other hand, the parameters of burn model were calibrated by the results of the high-speed photography.

5.4.1.2

Numerical Procedure for Multi-material Flow

The governing equations are mass, momentum, and energy conservation law, and the EOS for detonation products is JWL and for unreacted solid is Hugoniot base Mie Gruneisen EOS. The ignition and growth burn model and the mixture rule described in the previous section were employed. The cubic interpolated polynomial (CIP) scheme was adopted for solving the governing equations [90–92]. Since the calculation field includes the explosive, PMMA, air and steel, so the simulation must fit the multi-material flow. The mesh size of this simulation was set 0.5 mm, so the detail structure of the reaction zone was not emphasized. The following equations are used for solving the multi-material flow.

180

S. Kubota and K. Nagayama

( ) K ∂f α + u→ · ∇ f α = f α − 1 ∇ · u→ ∂t Kα

(5.31)

∂ f α ρα + u→ · ∇( f α ρ α ) = − f α ρ α ∇ · u→ ∂t

(5.32)

∂ u→ 1 + u→ · ∇ u→ = − ∇( p + q) ∂t ρ

(5.33)

∂ f α ρ α εα + u→ · ∇( f α ρ α εα ) = − f α ρ α εα ∇ · u→ − m α ( p + q)∇ · u→ ∂t

(5.34)

where α corresponds to each material in calculation field, where u→ is the velocity vector for axial and radial directions, f α , ρ α , and εα are the volume fraction, density, and specific internal energy of the material α. Eq. 5.31 is the advection equation for each material. The advantage of this method is the very simple algorithm as shown below. This method is not suitable for intense blast wave problems with large compression because of no volume control algorithm. The governing equations are solved with two steps in each cycle. Since the same treatment is adopted for each governing equation, only the mass conservation equation will be shown here. The first step is a Lagrangian step and can be expressed as follows for mass conservation. dF α = −F α ∇ · u→ = g ∂t

(5.35)

Here, F α is f α ρ α . The solution of numerical scheme of above equation is the physical value at next time step at Lagrangian coordinate system. In the Lagrange coordinate system, the coordinate moves according to changes in physical variables, and conservation of mass automatically established. In this algorithm, instead of moving the coordinates, Eq. 5.31 and advection equation as shown in Eq. 5.37 were solved. That is both an advantage and a disadvantage. The CIP requires the spatial derivatives of the physical variables in each direction, and the equations obtained by spatial differentiation of the governing equation for each direction should be solved. dFxαi ∂t

= gαxi − u→ xi ∇ · F α

(5.36)

The subscript xi denotes the spatial derivative for the i direction. The second step is an advection step, which can be expressed as follows. ∂ Fα + u→∇ · F α = 0 ∂t

(5.37)

Equation 5.37 can be solved using the spatial profile. Because the spatial derivatives of the physical variables in each direction are used during interpolation, the spatial derivatives are also solved by the equations obtained by the differentiating the

5 Shock Initiation

181

governing equation. After the advection step, the pressure in each cell is determined using the EOS to complete the simulation cycle. In the cell of the Eulerian coordinate system for the multi-materials flow, there is unsolved problem, i.e., how to determine the state variables at the multi-material cell. Dalton’s law can be accurately applied to gases to which the ideal gas can be applied. In contrast, solid versus gas, solid versus solid does not mix each other. For accurate evaluation of the state variables in the multi-material cell, the interfacial capture method may also be used, but here will not be treated. The mixture rule introduced in the previous section is the typical method and was used for reacting explosive here, and the mixed state with other substances adopted the method explained below [93, 94]. The relaxation time is very short compared with the time step in the simulation. Although the volume fraction can change to attain equilibrium, the total volume and mass fraction are fixed. The process is isentropic. The relationship between the pressure in the multi-material cell and the increment in the pressure of each component can be written as p = p α + Δp α = p α −

Δf α α K fα

(5.38)

Because of the assumption that the total volume does not change, ∑

Δf α = 0

(5.39)

Finally, the pressure in the multi-material cell can be expressed as, P=

∑

P α f α /K α ×

(∑

f α /K α

)−1

(5.40)

From the assumption that the process is isentropic, the bulk modulus of each component is K α = −v α

(

∂ Pα ∂v α

)

= ρ α (cα )2 .

(5.41)

S

Here, subscript S denotes the entropy, v is the specific volume, and c is the sound velocity.

5.4.1.3

Experiments and Simulation Results of Small-Scale Gap Test

A total number of the gap test were 20 shots, 11 of which were performed with gap length (L g ) of 22–24 mm. The sympathetic detonation occurred under the condition of L g was shorter than 22.6 mm (Go) and did not occurred under the condition of L g

182

S. Kubota and K. Nagayama

longer than 22.8 mm (No-go) with one exception [89]. High-speed photography of three typical cases is shown in Figs. 5.28, 5.29, and 5.30. When the luminescence at the surface of acceptor was confirmed in high-speed photography as shown in Fig. 5.28, the recovered witness steel block was broken without exception. In such cases, the result is the sympathetic detonation (Go). Distance between the original position of bottom end of gap and the surface point in the acceptor where the luminescence first appears was measured as shown in Fig. 5.28b. When there is no luminescence at the acceptor, no damage on the witness steel block with one exception. The judgment of the results is No-go. Even if the results are No-go, the behavior of the acceptor charge depends on the gap length. In Fig. 5.29, the luminescence is not confirmed, but it would be clearly confirmed the expansion of the acceptor part at time (b) 27 μs and (c) 37 μs. At time 80 μs, it looks like a gas expansion. In contrast, there is no remarkable expansion of the acceptor part and no remarkable gas expansion as shown in Fig. 5.30. Instead of the gas expansion, the PMMA gap was broken downward to cover the acceptor charge. The results of the pressure measurements are shown in Fig. 5.31 and Table 5.3. Since the behavior other than the first peak of the pressure profile cannot explain the physical meaning, only the first peak was considered as the pressure value. The left side of Fig. 5.31 shows the results when the gap length is fixed at 23.7 mm. The right side shows the gauge profiles when the observation point (Op) is fixed 15 mm from the gap end. In this case, the gauge was directly embedded between the

Fig. 5. 28 High-speed photography of the small-scale gap test focus on acceptor. Sample is Composition C4 explosive, 50 mm donor charge, 40 mm acceptor, and gap length 22.0 mm. Times from left; a 16 μs b 19 μs c 20 μs d 21 μs e 44 μs from the initiation

Fig. 5. 29 High-speed photography of the small-scale gap test focus on acceptors. Sample is Composition C4 explosive, 50 mm donor charge, 40 mm acceptor, and gap length 23.0 mm. Times from left; a 17 μs b 27 μs c 37 μs d 80 μs from the initiation

5 Shock Initiation

183

Fig. 5. 30 High-speed photography of the small-scale gap test focus on acceptors. Sample is Composition C4 explosive, 50 mm donor charge, 40 mm acceptor, and gap length 26.0 mm. Times from left; a 19 μs b 29 μs c 39 μs d 80 μs from the initiation

15 mm long PMMA pipe filled with C4 and the 25 mm pipe filled with C4. In all cases of these measurements, there is no sympathetic detonation in acceptor, but in the case of the 23.7 mm gap length, some reaction occurred. By comparison of the results between 23.7 mm and 25.5 mm gap, the effect of reaction can be explained. As shown in Table 5.3, the values of the input pressure in the acceptor are 2.77 and 2.49 GPa, respectively. The difference is only about 10%. At the 15 mm Op, the pressure values are 1.85 GPa for 23.7 mm gap and 0.31 GPa for 25.5 mm gap, respectively. This remarkable difference in pressure is caused from the existence of a reaction in the case of 23.7 mm gap. The average velocities were estimated by the distance between two gauges and those arrival times for each shot. The composition C4 explosive Hugoniot was determined by trial-and-error method. Assuming the linear relationship between the shock and the particle velocities, the simulation of the gap test without reaction was carried out repeatedly varying the variables A and B in Eq. 2.12. The A and B pairs were varied until the simulation results agree well with the average velocity obtained in the experiments. Finally, the parameters for the burn model were examined by trial-and-error method using the gap test results with sympathetic detonation. The parameters of the ignition and growth model were estimated. The whole image of the calculation field is shown in Fig. 5.32 with snapshot just after the sympathetic detonation occurs. The expansion of detonation products gases and PMMA pipe at donor charge and large deformation of PMMA gap are confirmed. As shown in Fig. 5.33, the typical experimental results are well-simulated. The condition of the 23 mm gap length and the 40 mm length acceptor corresponds t the condition of no. sympathetic detonation case. Figure 5.34 is the special condition. Instead of 40 mm, 25 mm length acceptor was used, so the reflection wave at the witness steel block triggered the rapid reaction. After the shock wave arrive at the bottom end of acceptor, the reaction wave propagated from bottom to upper side, like the retonation (not exactly the retonation). In addition to gaining insights into the sympathetic detonation, this study was able to simulate all the typical gap test results obtained in the experiment with one parameter set determined here.

184

S. Kubota and K. Nagayama 3

Gap length : 23.7 mm

6

Observation point 15 mm from gap end Lg ; Gap length Lg : 23.7 mm

Pressure / GPa

Pressure / GPa

Op: 0mm

2 Op: 15 mm Op: 25 mm

1

4

2

Op: 30 mm

Lg : 25.5 mm Op: 45 mm

0 0

10

20 Time / µ s

Lg : 30.0 mm

0 0

30

10

20 Time / µ s

30

Fig. 5. 31 PVDF gauge profiles in small-scale gap test. All measurements were carried out under the condition of no sympathetic detonation in accepter. When the observation point (Op) is 30 mm, the gauge was set at the bottom end of acceptor. Left figure: the gap length fixed and the Op was varied. Right figure: Op was fixed as 15 mm. Only the first peak was considered as the pressure value

Table 5.3 Results of the pressure measurement in Composition C4 by PVDF gauge

L g (mm)

Position 2 (mm)

P1 (GPa)

P2 (GPa)

23.7

15

–

1.85

23.7

25

2.77

1.47

25.5

15

2.49

0.31

30

15

1.65

0.14

L g : gap length; Position 2: distance from the gap end to gauge point P1: pressure at the gap end, P2: pressure at the position 2

Fig. 5. 32 Initial condition and the snapshot of the gap test with density distribution obtained by the simulation. Time is counted from the initiation of donor charge

5 Shock Initiation

185

Fig. 5. 33 Comparison between the snapshot by high-speed photography and that by the reactive flow simulation. The left side is the acceptor part before the shock wave enters the acceptor charge. At the right side, the typical two cases are compared, i.e., Go and No-go results

Fig. 5. 34 Comparison with the snapshot by high-speed photography and by the reactive flow simulation. The left side is the acceptor part before the shock wave enters the acceptor charge. At the right side, the typical two cases are compared, i.e., Go and No-go results

5.4.2 Scale Effect of Sympathetic Detonation Typical purpose of the study of sympathetic detonation is the safe storage of explosive. It is possible to improve the safety of the surrounding area by introducing bulkhead at the explosive storage and taking measures to prevent sympathetic detonation. Generally, it is impossible to verify the safety only with the information obtained from a small-scale experiment, because of tons order of storage amount in explosive magazine. To verify the safety issue of explosive magazine, the understanding the

186

S. Kubota and K. Nagayama

scale effect of the sympathetic detonation is important. This section focuses on the scale effect of sympathetic detonation and introduces the scale effect examined only by simulation, and the study of the gap test with various scales assuming an actual bulkhead in explosive magazine. We investigated the scale effect of the sympathetic detonation by the simulations of gap test. Figure 5.35 is the schematic illustration of the simulation field. The composition B was selected as the donor and acceptor explosive in which the ratio of length and diameter (L/D) was set one for all scale. The gap material is PMMA. The numerical procedure is same as Sect. 5.4.1.2. The parameter of the burn model was calibrated by the Pop plot. The parameters for EOSs were referred for paper by Murphy et al. [79]. The weight of explosive was set to 0.01, 0.036, 0.3, 2.2, and 20 kg, respectively. To judge the sympathetic detonation, the following approach was employed. When SDT occurs before the shock front arrives at the interface of witness plate and acceptor charge, the judgment is sympathetic detonation. This is a reasonable evaluation because the purpose is to investigate the tendency of the scale effect. No emphasis is paid on reproducing actual experiments. The right-hand side of Fig. 5.35 is the simulation results. It was found that the relationship between the critical gap length and the charge weight is almost linear on both log–log scales. This empirical relationship was very helpful in later experiments. The installation of the dividing wall in the explosive magazine is effective in preventing sympathetic detonation. In such case, the dividing wall requires to possess the function of the shock wave attenuation. Sand can be considered as an inexpensive, stable, and environmentally friendly shock absorber material. Since the gap test can investigate both the attenuation effect of shock wave by material and the sympathetic detonation of explosive, the series of the gap test with various scales has been carried out. The left-hand side of Fig. 5.36 shows the gap test configuration for 5 kg explosive

Fig. 5. 35 Simulation model of gap test (left). The condition for sympathetic detonation based on the simulation results (right)

5 Shock Initiation

187

Fig. 5. 36 Typical gap test to estimate scale effect for sympathetic detonation (left). The scale effect for the sympathetic detonation for emulsion explosive (right)

case. The gap part consists of the mortar disk and the sand. The donor and acceptor explosives are the typical industrial explosive, emulsion explosive. The emulsion explosive is formed by 73.2% oxidizer, 10.8% water, 6.5% fuel, 4.6% glass microballoons, and 4.9% others, and the L/D is set to one. The gap tests were conducted on explosive weights of 0.02, 0.16, 1.256, 5, and 36.4 kg. Natural silica sand was used, and loading density was adjusted to 1.6 g cm−3 . The total gap length was defined by summation of the thicknesses of two mortar disks and the sand. The three typical results were obtained and judged based on the condition of the witness plate. One is the condition of detonation in acceptor; i.e., the hole is opened in witness plate (Go). Second is the case of no rapid reaction of the acceptor charge (No-go). Neither of the two witness plates piled up were distorted nor remaining unreacted explosives were discovered on the witness plate. Last one is a marginal (marginal). The distortion of witness plate is small, and no hole was opened; however, no remaining unreacted explosive discovered on the plate. It was not a detonation, but there was a rapid reaction in acceptor. The parameters of the ignition and growth model have been obtained by trialand-error calculations so that these results reproduce all gap test results. A lot of calculations have been done using parallel computer with 128 CPU. Right-hand side of Fig. 5.36 experimental and numerical results was plotted. The marginal line can be evaluated by 71.2W0.407 . In the previous numerical study, we found that the critical gap length and the charge weight approximately have linear relationship on logarithmic scale. Fortunately, this trend was able to use to reduce the number of experiments.

188

S. Kubota and K. Nagayama

In the experiment of hazard evaluation of the explosive’s storage magazine, verification on a full scale is desired. However, it is no easy in Japan to find the place where a ton order explosion experiment can be conducted. Therefore, we have carried out an explosion experiment as close to the real scale as possible and estimated hazard by clarifying the scale law. If the scale law of the subject explosive can be derived in the gap test, it will be possible to evaluate the real size phenomena using a simulation that satisfies the scale law. Preventing the sympathetic detonation is to reduce the explosion damage of surroundings. This can also be reference data for reducing safety distances.

5.5 Laser Initiation 5.5.1 Introduction Nagayama provides a peculiar introduction of the initiation research by a pulse laser. He had a research project longer than 10 years around 2000 starting the experiment of laser–matter interaction especially at the surface of roughened or intentionally roughened surface of transparent medium. The research was stimulated by the experimental discovery that pulsed laser energy is absorbed by the roughened surface, and that laser beam direction to the surface is very important such that enhanced laser energy absorption in the case laser is directed toward the surface from inside the transparent medium [96, 97]. Figure 5.37 shows the illustrative example of laser energy absorption by smooth and intentionally roughened PMMA surface in the laser direction as shown in the figure. The stress wave in PMMA emanated from the roughened surface is apparently stronger than that from the smooth surface. Entirely different phenomenon has been observed in the case metallic or aluminum coating is on the surface. Much stronger shock wave is produced compared with no coating. Instantaneous high-temperature high-pressure state can be achieved on the solid surface by focusing very intense laser beam on it. This results in an explosive burst of plasma and particles from the surface, and the phenomenon is called laser ablation. This is one of the processes by that the laser pulse energy can be converted to other form of energy very efficiently. The phenomena can be applied to various physics experiments as well as engineering and medical applications. Ultrahighpressure states attained by this process are considered promising as a scientific tool to study these high energy density states and its applications. Application of the same phenomena to the initiation of high explosive charge without danger of accidental explosion by electrical noise seems effective [96, 97]. Accurate timing of initiation can be expected by the proper choice of energy deposition method. Various attempts have been made by focusing the laser energy directly

5 Shock Initiation

189

Fig. 5. 37 High-speed imaging of laser focus into air through PMMA plate with smooth (a) and roughened surface (b). Surface of PMMA was roughened by using #800 paper. Delay time of photographs is 0.53 and 0.54 μs, respectively, with a focused laser pulse of 180 mJ. A pressure wave in air can be seen around the laser focal point

on the explosive charge. In this application, infrared and ns duration laser is preferable. Since most of the explosive are not good absorber of light, direct absorption of laser energy by explosives is not very efficient. Laser initiation of condensed phase high explosive is realized in this case by the above procedure that optical fiber with metal coating at the output end. Important specifications for the reliable initiation have to be realized in the following conditions; (i) thickness of metal coating is at least thicker than 0.4 μm, (ii) intentional roughening of the output surface is desired. Roughening treatment is especially effective to avoid the unwanted peel-off of the metal coating. We have tested plastic fiber to observe produced shock wave propagation into the surrounding medium. Figure 5.38 shows the illustrative example of pulse laser shadowgraph of air-shock wave generation emanated from the plastic fiber with roughened and aluminized coating at the output end surface [97]. Dark cloud indicates high-temperature metal plasma produced by laser ablation. Since the laser energy input is almost same for three cases, laser fluence is highest for the most slender fiber and the plasma cloud movement has the highest velocity. These techniques are used in the real design of laser detonator [98].

190

S. Kubota and K. Nagayama

Fig. 5. 38 Pulse laser shadowgraph of air-shock waves from a plastic fiber with roughened and aluminized output end surface. Three shots compare the difference of fiber diameter. Other initial specifications are the same for all fibers

5.5.2 Experimental Procedure PETN powder used was provided by Asahi Chemical Industry, Ltd. Grain size is about 100 μm and is handpressed into the experimental specimen assembly explained later. An important parameter of the sensitivity of explosive is the initial density, which is estimated by the measured mass divided by the calculated volume. Quantity of PETN powder used in an experiment is chosen to be limited to less than 10–15 mg. Due to this small quantity of specimen, estimated initial density may have rather large error of 10–20%. Typical value of the initial density of PETN powder in this experiment is 0.6–0.9 g cm−3 . Absorption of laser pulse energy and its conversion to another form of energy can be achieved here by the following procedure. In practical applications, laser beam energy is provided through the transparent fiber medium. We intentionally roughen and metallize the output surface of the fiber. The most important difference is that the laser fluence should be less than the ablation threshold fluence of the laser transmitting medium, but be larger than the ablation threshold fluence of the roughened and metallized surface layer. Figure 5.39 shows the experimental setup for the present study. PETN powder is filled into 0.5 mm space, one of the wall surface is roughened and aluminized. Pulse laser beam is focused on the rear surface of the aluminized layer to ablate the layer. Explosive sensitivity was tested with varying the focused diameter and laser energy. Although thickness of the explosive layer is very thin, explosion is observed for the laser focus onto less than 2 mm diameter. Explosion is determined by the streak record of self-luminous front propagation, sound, smell, and broken assemblies. In some cases, appreciable retardation of reaction was observed [99]. If minimum

5 Shock Initiation

191

energy density value is necessary to initiate the explosive, the explosive initiation in the present procedure can be scaled independent of the size of explosive medium. It is found that the results cannot be described by the laser fluence. Extremely small amount of explosive used in this experiment is supposed to be less than the limiting size of SDT to steady detonation, although previous data on the impact initiation tests suggests the high sensitivity of powdered PETN. Figure 5.40 shows a typical example of streak photograph of self-emission due to laser-induced reaction of PETN thin layer. As seen in Fig. 5.40, detonation reaction is found to be delayed 200–300 ns after laser ablation. In the photographs, intense flash due to laser ablation is recorded in the streak photograph even through the opaque powder explosive layer. This is due to the extremely large intensity of light flash of ablation. Light intensity of ablation is brighter than the self-emission of detonation. This is attributed to be the extreme high temperature of ablated metal plasma. It is shown in Fig. 5.40 that detonation front proceeds almost at constant velocity. This result is surprising in the sense of existence of limiting diameter of steady detonation. Detonation velocity estimated by the slope of the streak photography is 4 km/s, while the published data of density dependence of detonation velocity in powdery PETN is around 4.8 km/s. Agreement of these two values suggests that the emission observed in this study seems to be due to almost steady detonation wave front even in an extremely thin and very small amount of explosive charge. Delay time of detonation is found to be different depending on the explosive layer thickness. For 0.5 mm thick and 1.0 mm thick assemblies, the difference in delay time is about 300 ns, while the difference in explosive thickness is 0.5 mm. This difference seems Fig. 5. 39 Experimental setup for detonation sensitivity test. Thick PMMA plates of both sides are used to observe the induced high-pressure stress wave in PMMA by explosion

Fig. 5. 40 Streak photograph of delayed detonation of PETN powder of 10 mg. Nd: YAG laser pulse of 180 mJ is focused to about 1.5 mm diameter

192

S. Kubota and K. Nagayama

somewhat too long by considering the propagation velocity of shock wave inside the unreacted powder explosive layer. Further experiments are necessary to study the physics of process realized by plasma-induced explosive initiation process.

5.5.3 Short Summary and Future Prospects GO-NOGO tests with various experimental conditions were published in several papers [99, 100]. It is apparent that for all cases of detonation we observed a delay time from the laser input to flash by detonation. The minimum delay time is found to be around 200 ns. In the course of this project, we experienced several different phenomenon. One of them is the peculiar phenomena in the case of pulse laser ablation of ground glass [100–102]. We have discovered a small particle generation from the ground glass surface upon laser irradiation. Ejected particle cloud have been observed by laser shadowgraphs, and it is found that its velocity is more than 1 km/s. This phenomena was applied to initiate PETN powder attached with ground glass surface but with no metal coating. This high-velocity particle cloud was shown to have the ability of initiate the explosive powder. The delay time of initiation after laser pulse input was shown to be larger than those by the combination of roughened and aluminized surface of PMMA plate. Naturally, it is suggested that the use of glass optical fiber with roughened output end together with the metallic coating may be a promising candidate of short delay initiation of explosive powders. Physical reasoning of particle generation is apparently by the pulse laser interaction with ground glass surface, and simple analysis by optics equations suggests that laser ablation-produced plasma will be more intense for the case of oblique incidence to the transparent surface. This tendency will be most intense for the case of total internal reflection. Another topic is the liquid ablation by oblique incidence of pulse laser to the liquid surface from inside [103]. Something intense phenomena must be expected at the liquid surface in the case of total internal reflection. We observed a liquid jet associated with the flash at the reflected point on the liquid surface. Liquid particles again have a velocity of 1 km/s. This phenomenon will have several applications in different areas. Readers interested in this projects can consult a reference [103].

References 1. Kistlakowsky GB (1951) Problems and future developments. In: Proceedings—conference on the chemistry and physics of detonation (first detonation symposium). Office of Naval Research, Washington D.C. 11–12 January 1951, pp 105–106 2. Jacobs SJ (1960) Non-steady detonation—a review of past work In: Proceedings—third symposium on detonation. Princeton, NJ, Office of Naval Research Department of Navy, ACR-62, pp 784–812

5 Shock Initiation

193

3. Campbell AW, Davis WC, Ramsay JB, Travis JR (1960) Shock initiation of solid explosives. In: Proceedings—third symposium on detonation. Princeton, NJ, Office of Naval Research Department of Navy, OHR Symposium Report ACR-62, pp 499–519 4. Seay GE, Seely Jr LB (1960) Initiation of a low-density PETN pressing by a plane shock wave. In: Proceedings—third symposium on detonation. Princeton, NJ, Office of Naval Research Department of Navy, OHR Symposium Report ACR-62, pp 562–573 5. Campbell AW, Davis WC, Ramsay JB, Travis JR (1960) Shock initiation of solid explosives. Phys Fluids 4:511–521 6. Seay GE, Seely LB Jr (1961) Initiation of a low-density PETN pressing by a plane shock wave. J Appl Phys 32:1092–1097 7. Walsh JM, Christian RH (1955) Equation of state of metals from shock wave measurements. Phys Rev 97(6):1544–1557 8. McQueen RG, Marsh SP, Taylor JW, Fritz JN, Carter WJ (1970) Chapter VII: the equation of state of solids from shock wave studies. In: Kinslow R (ed) High velocity impact phenomena. Academic Press, New York 9. Ramsay JB, Popolato A (1965) Analysis of shock wave and initiation data for solid explosives. In: Fourth symposium (international) on detonation. White Oak, ML, Office of Naval Research, Arlington, VA, ACR-126, pp 233–238 10. Campbell AW, Davis WC, Travis JR (1960) Shock initiation of detonation in liquid explosives. In: Proceedings—third symposium on detonation. Princeton, NJ, Office of Naval Research Department of Navy, OHR Symposium Report ACR-62, pp 469–498 11. Campbell AW, Davis WC, Ramsay JB, Travis JR (1960) Shock initiation of detonation in liquid explosives. Phys Fluids 4(4):491–510 12. Craig BG, Marshall EF (1970) Decomposition of a shocked solid explosive. In: Fifth symposium (international) on detonation. Pasadena, CA, Office of Naval Research, Arlington, VA, ACR-184, pp 321–329 13. Jacobs SJ, Edwards DJ (1970) Experimental study of the electromagnetic velocity-gauge technique. In: Fifth symposium (international) on detonation. Pasadena, CA, Office of Naval Research, Arlington, VA, ACR-184, pp 413–426 14. Cowperthwaite M, Williams RF (1971) Determination of constitutive relationships with multiple gages in non-divergent waves. J Appl Phys 42:456–462 15. Wackerle J, Johnson JO, Halleck PM (1976) Shock initiation of high density PETN. In: Proceedings—sixth symposium (international) on detonation. Colorado, CA, Office of Naval Research, Arlington, VA, ARC-221, pp 20–28 16. Nunziato JW, Kennedy JE, Hardesty DR (1976) Modes of shock wave growth in the initiation of explosives. In: Proceedings—sixth symposium (international) on detonation. Coronado, CA, Office of Naval Research, Arlington, VA, ARC-221, pp 47–61 17. Cowperthwaite M, Rosenberg JT (1976) A multiple Lagrange gage study of the shock initiation process in cast TNT. In: Proceedings—sixth symposium (international) on detonation, Coronado, CA, Office of Naval Research, Arlington, VA, ARC-221, pp 786–793 18. Walker FE (1 Comments), Cowperthwaite M (2 Comments), Fray R, Howe P (3 Comments, 5 Response to C Mader), Mader C (4 Invited discussion of shock initiation mechanism), Fauquignon C (6 Comments) (1976) Discussion on shock initiation and p2 τ. In: Proceedings— sixth symposium (international) on detonation. Coronado, CA, Office of Naval Research, Arlington, VA, ARC-221 p 82–94 19. James HR (1996) An extension to the critical energy criterion used to predict shock initiation thresholds. Propellant Explos Pyrotech 21:8–13 20. James HR, Cook MD, Haskins PJ (1998) The response of homogeneous explosives to projectile attack. In: Proceedings—eleventh symposium (international) on detonation. Coronado, CA, Office of Naval Research, Arlington, VA, ONR 33300-5, pp 581–588 21. Wilkins ML (1960) Detonation and shock review. In: Proceedings—third symposium on detonation. Princeton, NJ, Office of Naval Research Department of Navy, ACR-62, pp 721– 724

194

S. Kubota and K. Nagayama

22. Wilkins ML (1965) The use of one- and two-dimensional hydrodynamic machine calculations in high explosive research. In: Fourth symposium (international) on detonation. White Oak, ML, Office of Naval Research, Arlington, VA, ACR-126, pp 519–526 23. Lambourn BD, Hartley JE (1965) The calculation of hydrodynamic behavior of plane one dimensional explosive/metal systems. In: Fourth symposium (international) on detonation. White Oak, ML, Office of Naval Research, Arlington, VA, ACR-126, pp 538–552 24. Wilkins ML (1999) Computer simulation of dynamic phenomena. Springer-Verlag, Berlin Heidelberg New York 25. Mader CL (1963) Shock and hot spot initiation of homogeneous explosives. Phys Fluids 6:375–381 26. Mader CL (1965) Initiation of detonation by the interaction of shock with density discontinuities In: Fourth symposium (international) on detonation. White Oak, ML, Office of Naval Research, Arlington, VA, ACR-126, pp 394 (Abstract Only) 27. Mader CL (1965) Initiation of detonation by the interaction of shock with density discontinuities. Phys Fluids 8:1811–1816 28. Mader CL (1965) Numerical calculations of detonation failure and shock initiation In: Fifth symposium (international) on detonation. Pasadena, CA, Office of Naval Research, Arlington, VA, ACR-184, pp 177–184 29. Mader CL, Forest CA (1976) Two-dimensional homogeneous and heterogeneous detonation wave propagation. Technical Report LA-6259, Los Alamos Scientific Laboratory 30. Mader CL (1976) Two dimensional homogeneous and heterogeneous detonation wave propagation. In: Sixth symposium (international) on detonation. Coronado, CA, Office of Naval Research, Arlington, VA, ARC-221, pp 405–413 31. Forest CA (1978) Burning and detonation. Technical Report LA-7245, Los Alamos Scientific Laboratory 32. Lindstrom IE (1966) Plane shock initiation of an RDX plastic bonded explosive. J Appl Phys 37:4873–4880 33. Mader CL (1979) Numerical modeling of detonations. University of California press, Berkeley and Los Angeles, California Chapter 4 34. Lee EL, Tarver CM (1980) Phenomenological model of shock initiation in heterogeneous explosives. Phys Fluids 23(12):2362–2372 35. Bahl KL, Vantine HC, Weingart RC (1981) The shock initiation of bare and covered explosives by projectile impact. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 325–335 36. Titov VM, Lobanov VF, Bordzilovsky SA, Karakhanov SM (1981) Investigation of transient processes at initiation of high explosives. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 362–370 37. Khasainov BA, Borisov AA, Ermolaev BS, Korotkov AI (1981) Two-phase visco-plastic model of shock initiation of detonation in high-density pressed explosives. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 435–447 38. Kipp ME, Nunziato JW, Setchell RE, Walsh EK (1981) Hot spot initiation of heterogeneous explosives. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 394–406 39. Forest CA (1981) Burning and detonation. In: Seventh symposium on (international) detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 234–243 40. Bowman AL, Forest CA, Kershner JD, Mader CL, Pimbley GH (1981) Numerical modeling of shock sensitivity experiments. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 479–487 41. Tarver CM, Hallquist JO (1981) Modeling two-dimensional shock initiation and detonation wave phenomena in PBX9404 and LX-17. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 488–497 42. Westmoreland C, Lee EL (1981) Modeling studies of the performance characteristics of composite explosives. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 517–522

5 Shock Initiation

195

43. Anderson AB, Ginsberg MJ, Seitz WL, Wackerle J (1981) Shock initiation of porous TATB. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 385–393 44. Partom Y (1981) A void collapse model for shock initiation. In: Seventh symposium (international) on detonation. Annapolis ML, NSWC, White Oak, ML, MP 82-334, pp 506–516 45. Vorthman J, Andrews G, Wackerle J (1985) Reaction rates from electromagnetic gauge data. In: Proceedings—eighth symposium (international) on detonation. Albuquerque, NM NSWC White Oak, MD MP-86194, pp 99–110 46. Sheffield SA, Engelke R, Alcon RR (1989) In-situ study of the chemically driven flow fields in initiating homogeneous and heterogeneous nitromethane explosives. In: Proceedings— nineth symposium (international) on detonation. Portland, OR, Office of the Chief of Naval Research, Arlington VA, OCNR 11329-7, pp 39–49 47. Sheffield SA, Gustavsen RL, Alcon RR, Graham RA, Anderson MU (1993) Shock initiation studies of low density HMX using electromagnetic particle velocity and PVDF stress gauges. In: Proceedings—tenth symposium (international) on detonation. Boston, MA, Office of Naval Research, Arlington, VA ONR 333-95, pp 166–174 48. Sheffield SA, Gustavsen RL, Hill LG, Alcon RR (1998) Electromagnetic gauge measurements of shock initiating PBX9501 and PBX9502 explosives. In: Proceedings—eleventh symposium (international) on detonation. Snowmass, CO, Office of Naval Research, VA ONR-33300-5, pp 451–458 49. Salisbury DA, Taylor P, Winter RE, Gustavsen RL, Sheffield SA, Alcon RR (2002) Single and double shock initiation of EDC 37. In: Proceedings—twelfth symposium (international) on detonation. San Diego, CA, Office of NAVAL Research, Arlington VA, ONR 333-05-2, pp 271–280 50. Gustavsen RL, Sheffield SA, Alcon RR, Hill LG (2002) Shock initiation of new and aged PBX9501. In: Proceedings—twelfth symposium (international) on detonation. San Diego, CA, Office of NAVAL Research, Arlington VA, ONR 333-05-2, pp 530–537 51. Sheffield SA, Dattelbaum DM, Engelke R, Alcon RR, Crouzet B, Robbins DL, Stahl DB, Gustavsen RL (2006) Homogeneous shock initiation process in neat and chemically sensitized nitromethane. In: Proceedings—thirteenth symposium (international) on detonation. Norfork, VA, Office of Naval Research, Arlington, VA ONR-351-07-01, pp 401–407 52. Gustavsen RL, Gehr RJ, Bucholtz SM, Seitz WL, Sheffield SA, Alcon RR, Robbins DL, Barker BA (2006) Shock initiation of the tri-amino-tri-nitro-benzene based explosive PBX9502 cooled to −55 °C. In: Proceedings—thirteenth symposium (international) on detonation. Norfork, VA, Office of Naval Research, Arlington, VA ONR-351-07-01, pp 970–979 53. Setchell RE (1985) Experimental studies of chemical reactivity during shock initiation of hexanitrostilbene. In: Proceedings—eighth symposium (international) on detonation. Albuquerque, NM NSWC White Oak, MD MP-86194, pp 15–25 54. Tarver CM, Hallquist JO, Erickson LM (1985) Modeling short pulse duration shock initiation of solid explosives. In: Proceedings—eighth symposium (international) on detonation. Albuquerque, NM NSWC White Oak, MD MP-86194, pp 951–961 55. Gustavsen RL, Sheffield SA, Alcon RR (1998) Progress in measuring detonation wave profiles in PBX9501. In: Proceedings—eleventh symposium (international) on detonation. Snowmass, CO, Office of Naval Research, VA ONR-33300-5, pp 821–827 56. Mercier P, Bénier J, Frugier PA, Sollier A, Le Gloahec RM, Lescoute E, Cuq-Lelandais JP, Boustie M, de Rességuier T, Claverie A, Gay E, Berthe L, Nivard M (2009) PDV measurements of ns and fs laser driven shock experiments on solid targets. In: Shock compression of condensed matter—2009; AIP Conf Proc 1195:581–584 57. Hare DE, Holtkamp DB, Strand OT (2010) Embedded fiber optic probes to measure detonation velocities using the photonic doppler velocimeter. In: Proceedings—fourteenth symposium (international) on detonation. Coeur d’Alene, ID, Office of Naval Research, Arlington, VA ONR-351-10-185, pp 401–406

196

S. Kubota and K. Nagayama

58. Briggs ME, Hill LG, Hull LM, Shinas MA, Dolan DH (2010) Applications and principles of photon-doppler velocimetry for explosive testing. In: Proceedings—fourteenth symposium (international) on detonation. Coeur d’Alene, ID, Office of Naval Research, Arlington, VA ONR-351-10-185, pp 414–424 59. Kipp ME (1985) Modeling granular explosive detonations with shear band concepts. In: Proceedings—eighth symposium (international) on detonation. Albuquerque, NM NSWC White Oak, MD MP-86194, pp 35–41 60. Johnson JN, Tang PK, Forest C (1985) Shock-wave initiation of heterogeneous reactive solids. J Appl Phys 57:4323–4334 61. Tang PK, Johnson JN, Forest C (1985) Modeling heterogeneous high explosive burn with an explicit hot-spot process. In: Proceedings—eighth symposium (international) on detonation. Albuquerque, NM NSWC White Oak, MD MP-86194, pp 52–61 62. Starkenberg J, Dorsey TM (1998) An assessment of the performance of the history variable reactive burn explosive initiation model in the CTH code. In: Proceedings—eleventh symposium (international) on detonation. Snowmass, CO, Office of Naval Research, VA ONR-33300-5, pp 621–631 63. Horie Y, Hamate Y, Greening D (2003) Mechanistic reactive burn modeling of solid explosives. Los Alamos National Lab Rep LA 14008:2003 64. Hamate Y, Lu XR, Horie Y (2006) Development of a reactive burn model: effects of particle contact distribution. In: Proceedings—thirteenth symposium (international) on detonation. Norfork, VA, Office of Naval Research, Arlington, VA ONR-351-07-01, pp 1270–275 65. Hamate Y (2007) A computational study of microstructure effects on shock ignition sensitivity of pressed RDX. In: Shock compression of condensed matter—2007; AIP Conf Proc 955:923– 926 66. Lu X, Hamate Y, Horie Y (2007) Physics-based reactive burn model: grain size effects. In: Shock compression of condensed matter—2007; AIP Conf Proc 955:397–400 67. Wescott BL, Stewart DS, Davis WC (2005) Equation of state and reaction rate for condensedphase explosives. J Appl Phys 98(053514):1–10 68. Handley CA (2006) The CREST reactive-burn model. In: Proceedings—thirteenth symposium (international) on detonation. Norfork, VA, Office of Naval Research, Arlington, VA ONR351-07-01, pp 864–870 69. Whitworth NJ (2006) Some issues regarding the hydrocode implementation of the CREST reactive burn model. In: Proceedings—thirteenth symposium (international) on detonation. Norfork, VA, Office of Naval Research, Arlington, VA ONR-351-07-01, pp 871–880 70. Menikoff R, Shawb MS (2010) Reactive burn models and ignition & growth concept. EPJ Web Conf 10:00003 71. Handley CA, Lambourn BD, Whitworth NJ, James HR, Belfield WJ (2018) Understanding the shock and detonation response of high explosives at the continuum and meso scales. Appl Phys Rev 5:011303 72. James HR, Lambourn BD (2006) On the systematics of particle velocity histories in the shock-to-detonation transition regime. J Appl Phys 100(084906):1–12 73. Kubota S, Saburi T, Ogata Y, Nagayama K (2010) Numerical modeling of shock initiation in PETN by unified EOS. Sci Technol Energetic Mater 71–4:91–97 74. Kubota S, Saburi T, Ogata Y, Nagayama K (2012) Numerical simulation of detonation propagation in PETN at arbitrary initial density by simple model. In: Shock compression of condensed matter—2011; AIP Conf Proc 1426:231–234 75. Kubota S, Shimada H, Matsui K, Nagayama K (2001) Modeling of pressure calculation in reaction zone of high explosive (I). Kayaku Gakkaishi (Sci Tech Energetic Mater) 62(4):155– 160 (in Japanese) 76. Kubota S, Nagayama K, Saburi T, Ogata Y (2007) State relations for a two-phase mixture of reacting explosives and applications. Combust Flame 151:74–84 77. Kubota S, Nagayama K, Shimada H, Matsui K (2004) Pressure calculation of reacting explosive by mixture rule. In: Shock compression of condensed matter—2003; AIP Conf Proc 706:359–362

5 Shock Initiation

197

78. Kubota S, Nagayama K, Wada Y, Ogata Y (2006) Simplified pressure calculation of reacting explosives. Sci Technol Energetic Mater 67–5:141–146 79. Murphy MJ, Lee EL, Weston AM, Williams AE (1993) Modeling shock initiation in composition B. In: Proceedings—tenth symposium (international) on detonation. Boston, MA, Office of Naval Research, Arlington, VA ONR 333-95, pp 963–970 80. Nagayama K, Kubota S (2003) Equation of state for detonation product gases. J Appl Phys 93:2583–2589 81. Kubota S, Saburi T, Ogata Y, Nagayama K (2010) Numerical simulations of detonation phenomena in PETN by systematic equation of state for detonation products. Sci Technol Energetic Mater 71–2:44–50 82. Lee E, Finger M, Collins W (1973) JWL equation of state coefficients for high explosives. Lawrence Livermore National Lab (LLNL), Livermore, CA, UCID-16189 83. Dobratz BM (1981) LLNL explosives handbook, properties of chemical explosives and explosive simulants, UCRL-52997 84. Stirpe D, Johnson JO, Wackerle J (1970) Shock initiation of XTX8003 and pressed PETN. J Appl Phys 41:3884–3893 85. Cooper PW (1993) A new look at the run distance correlation and its relationship to other nonsteady-state phenomena. In: Proceedings—tenth symposium (international) on detonation. Boston, MA, Office of Naval Research, Arlington, VA ONR 333-95, pp 690–695 86. Ishikawa K, Abe T, Kubota S, Wakabayashi K, Matsumura T, Nakayama Y, Yoshida M (2006) Study on shock sensitivity of an emulsion explosive by the sand gap test. Sci Tech Energetic Mater 67–6:199–204 87. Kubota S, Ogata Y, Wada Y, Katoh K, Simangunsong G, Nagayama K, Ishikawa K, Wakabayashi K, Homae T, Matsumura T, Nakayama Y, Saburi T, Yoshida M (2006) Numerical study on the scale effect in gap test of emulsion explosive. In: Proceedings—fourteenth symposium (international) on detonation. Coeur d’Alene, ID, Office of Naval Research, Arlington, VA ONR-351-10-185, pp 553–558 88. Kubota S, Ogata Y, Wada Y, Katoh, Saburi T, Yoshida M, Nagayama K (2006) Observation of shock initiation process in gap test. AIP Conf Proc 845:1085 89. Kubota S, Ogata Y, Wada Y, Saburi T, Yoshida M, Nagayama K (2006) Observation and numerical simulation of high explosive under shock loading. In: Proceedings—fourteenth symposium (international) on detonation. Coeur d’Alene, ID, Office of Naval Research, Arlington, VA ONR-351-10-185, pp 940–947 90. Yabe T, Aoki T (1991) A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver. Comput Phys Commun 66(2–3):219–232 91. Kubota S, Liu ZY, Saburi T, Ogata Y, Yoshida M (2005) Simulation of sympathetic detonation by a CIP Eulerian code. In: Computational ballistics II 107, WIT transactions on modelling and simulation, vol 40. WIT Press 92. Saburi T, Kubota S, Wada Y, Yoshida M (2013) Development of numerical code for physical hazard analysis of high-energy materials. Sci Technol Energetic Mater 74–5:124–131 93. Miller GH, Puckett EG (1996) A high-order Godunov method for multiple condensed phases. J Comp Phys 128:134–164 94. Vorobiev OY, Lomov IN (2000) Numerical simulation of gas-solid interfaces with large detonations. LLNL report UCRL-JC-136868 95. Kubota S, Liu ZY, Ohtsuki M, Nakayama Y, Ogata Y, Yoshida M (2004) A numerical study of sympathetic detonation in gap test. Mater Sci Forum 465–466:163–168 96. Nagayama K, Inou K, Murakami K, Kubota S, Nakahara M (2002) PETN powder initiation characteristics by laser ablation induced plasma flow. In: Proceedings—29th IPS seminar at Denver. Colorado, pp 363–367 97. Nakahara M, Nagayama K (1999) High-pressure shock wave flow field around a roughened surface of an optical fiber. In: Proceedings—22nd international symposium on shock and waves, pp 1077–1080 98. Kennedy JE, Thomas KA, Early JW, Garcia IA, Lester CS, Burnside NJ (2002) Mechanisms of exploding bridgewire and direct laser initiation of low-density PETN. In: Proceedings—29th IPS seminar at Denver. Colorado, pp 781–785

198

S. Kubota and K. Nagayama

99. Murakami K, Inou K, Nakahara M, Kubota S, Nagayama K (2002) Pulse laser ignition of a small amount of secondary explosive powder. Kayaku Gakkaishi (Sci Tech Energetic Mater) 63:275–278 100. Nagayama K, Kotsuka Y, Kajiwara T, Nishiyama T, Kubota S, Nakahara M (2007) Pulse laser ablation of ground glass. Shock Waves 17:171–183 101. Nagayama K, Kotsuka Y, Nakahara M, Kubota S (2005) Pulse laser ablation of ground glass surface and initiation of PETN powder. Sci Tech Energetic Mater 66:416–420 102. Nagayama K, Kotsuka Y, Kajiwara T, Nishiyama T, Kubota S, Nakahara M (2007) Pulse laser ablation characteristics of quartz diffusion plate and initiation of PETN. Sci Tech Energetic Mater 68:65–72 103. Nagayama K, Utsunomiya Y, Kajiwara T, Nishiyama T (2011) Pulse laser ablation by reflection of laser pulse at interface of transparent materials (Chapter 7). In: Lasers-applications in science and industry (InTech.), pp 131–150

Chapter 6

Ideal and Non-ideal Detonation Atsumi Miyake, Shiro Kubota, Yukio Kato, and Kenji Murata

Abstract This chapter consists of the definition of non-ideal detonation and the overview of studies in the world and particularly in Japan. Especially, for the studies related to the highly non-ideal explosive based on ammonium nitrate (AN), the trend of recent research was described. The studies of non-ideal detonation behaviors of pure AN, mixture of AN and activated carbon, AN/fuel oil (ANFO), and emulsion explosive (EMX) in Japan are described. In addition, the effects of Al particle reaction on the non-ideal detonation properties of the mixtures consisted of nitromethane (NM) and Al particle are described. Keywords Ammonium nitrate · ANFO · Emulsion explosive · Aluminized explosive · Steel tube test · Diameter effect · Confinement effect · Temperature measurement and pressure measurement

6.1 Introduction This chapter consists of the definition of non-ideal detonation and the overview of studies in the world and particularly in Japan. Section 6.2 describes the explanation of the definition of ideal and non-ideal detonation. It is explained including the ambiguous definition due to the incomplete theory of detonation. In the overview A. Miyake (B) Yokohama National University, Yokohama, Japan e-mail: [email protected] S. Kubota National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan e-mail: [email protected] Y. Kato Fukushima, Japan e-mail: [email protected] K. Murata Akita University, Akita, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 S. Kubota (ed.), Detonation Phenomena of Condensed Explosives, Shock Wave and High Pressure Phenomena, https://doi.org/10.1007/978-981-19-5307-1_6

199

200

A. Miyake et al.

of studies for non-ideal detonation, by touching on the history of early research, we described it so that reader could have image of phenomena of non-ideal detonation and its estimation method. Especially, for the studies related to the highly nonideal explosive based on ammonium nitrate (AN), the trend of recent research was described. Section 6.3 consists of non-ideal detonation behaviors of pure AN, mixture of AN and activated carbon, and ammonium nitrate fuel oil (ANFO) explosives. For pure AN, five types of AN were examined to investigate the effects of particle size distribution and microstructures on the detonation velocity. The results and discussions for a series of steel tube tests were introduced. The maximum inner diameter of steel tube was 300 mm, and a length was 1 m. The measurements of detonation pressure by manganin gauge were also conducted. The small-scale steel tube tests for mixture of AN and activated carbon and for ANFO were carried out with various diameters. The effects of the physical properties of several AN were also discussed. The shape of the detonation front was taken by streak photograph. Section 6.3 is organized by the above data and its discussion. Emulsion explosive (EMX) is widely used in many blasting applications as well as ANFO. The EMX does not contains explosive component and are sensitized by adding hollow microballoon (MB) to base emulsion matrix. The detonation properties of EMX can be widely controlled by changing the amount and size of MB. In Sect. 6.4, the effects of the amount and size of MB on the detonation velocity, pressure, and failure diameter are presented. In the detonation of EMX containing large size MB, large fraction of EMX was reacted in Taylor wave behind CJ state, and highly nonideal detonation behavior was revealed. The underwater explosion performance is closely related to the blasting efficiency of explosive. The effects of the size of MB on the underwater explosion performance of EMX were studied. The underwater explosion performance of EMX-containing aluminum (Al) particle was measured to investigate the effects of the amount and size of Al. To enhance the performance of explosives, Al has been widely used. Still little is understood about the reaction of Al in detonation wave. In Sect. 6.5, the effects of Al reaction on the detonation velocity and temperature are described for the mixtures of nitromethane (NM) and Al particle. The results of temperature measurements showed that the reaction of Al started very close to CJ state, and that large fraction of Al was reacted. The detonation velocity and pressure of the mixtures comprised of packed bed of Al particles saturated with NM were measured. The results of pressure measurements demonstrated the existence of the extended high-pressure zone between leading shock and Taylor wave, which was due to the complex wave interactions between multiple shock waves in Al particles and detonation waves in NM in interstitial voids. These experimental results were shown to agree qualitatively with the results of mesoscale simulations. The pressure increase due to Al reaction was observed.

6 Ideal and Non-ideal Detonation

201

6.2 Definition and Overview of Studies for Non-ideal Detonation 6.2.1 Definition of Ideal and Non-ideal Explosive For the measurements of detonation velocity of high explosive, the cylindrical stick of explosive as shown in Fig. 6.1 has been used. Because the detonation velocity depends on the charge diameter (diameter effect), the diameter must be increased until to obtain constant detonation velocity. Such experiment for research purpose has been called the rate stick experiment. The detonation velocity also depends on the confinement such as steel pipe, and the detonation velocity of confined explosive is higher than that of bare explosive. There are two special diameters related to the detonation phenomena. The first is the minimum diameter d m for ideal detonation. The second is the critical diameter d c below which detonation fails to propagate. The range of ideal detonation is d ≥ d m , and the range of non-ideal detonation is defined by d c < d < d m Similar descriptions can be found in Cook’s textbooks [1]. It may be difficult to experimentally determine d c and d m for some explosives, but this definition is the simplest and clear for ideal and non-ideal detonation. Some distance is required from the initiation point to arrive at steady detonation. For unconfined explosive, it is estimated to be 4 times of the diameter. It is easy to imagine the difficulty of experiments when d m is tens of centimeters. The ideal detonation could be expressed by the ideal theory and equation of state. For examples, both C-J hypothesis and ZND model described in Chap. 2 have the assumption of one-dimensional flow. This assumption holds if an explosive with

Fig. 6.1 Schematic illustration of steady detonation in cylindrical stick of explosive and the relationship between diameter and detonation phenomena. Since both the ideal and non-ideal detonations are steady detonations one of the differences from ideal detonation is that the sound velocity plane and the reaction end plane do not coincide

202

A. Miyake et al.

infinite diameter is detonated in a steady state, and the effect of expansion waves from the side could be ignored clearly. In fact, even with a finite diameter, d m , the effect of expansion waves from the side does not appear in detonation velocity. In such case, the detonation velocity could be expressed by equation of state (EOS) for detonation products only. For non-ideal detonation, the discussion considering the reaction zone is indispensable. The above definition of ideal and non-ideal detonation remains ambiguous in situations where the theory and equation of state of detonation products have not been fully established. If an explosive that exhibits ideal detonation is defined as an ideal explosive, there is no ideal explosive. So, this definition does not fit to the definition of ideal explosive. Strictly speaking, High explosives with a certain density and composition exhibit either ideal detonation or non-ideal detonation depending on their diameter. For example, Mader [2] described that the dividing line between non-ideal and ideal explosives is arbitrary. He defines non-ideal explosive when there are the important differences between the experimental values such as detonation pressure and velocity and the values predicted by Baker-Kistiakowsky-Wilson (BKW) EOS. Specifically, the pressure difference of 5 GPa and the velocity difference of 500 m/s were considered as an index of judgment. Although various judgment methods are considered, as shown in Fig. 6.1 we generally think that it is natural to judge by the size (diameter) of the most common application of the explosive.

6.2.2 Overview of Studies for Non-ideal Detonation ZND model expresses steady-state detonation considering the reaction zone and has the assumption of one-dimensional plane wave front. Jones [3] and Eyring et al. [4] removed the assumption of one-dimensional plane detonation front from ZND model because the diameter effect is due to the expansion wave from the side of the explosive affecting the axis of the detonation (lateral-loss theory). The approach was slightly different between them. Jones considered Prandtl–Meyer expansion waves and established the well-known nozzle theory. On the other hand, Eyring et al. assumed spherical wave front and introduced one-dimensional governing equation using conservation law and Jump condition. Using the Abel equation of state, they examined the diameter effect by the curved front theory. Kistiakowsky asked the authors to explore the effects of charge diameter and finite reaction rates on the velocity of the detonation process [4]. From the experimental data and theoretically predicted information of reaction zone length, Eyring et al. found the following important empirical formula for unconfined explosives. D a = 1 − 0.5 Di R

(6.1)

Here D is detonation velocity of the explosive with radius R and Di is the ideal detonation velocity which corresponds to infinite diameter. The variable a is reaction

6 Ideal and Non-ideal Detonation

203

zone length estimated by curved front theory. This form of formula is still applied to many explosives to better represent the diameter effect. For confined explosives, the diameter effect was expressed by the following equation D W e ( a )2 = 1 − 2.17 Di Wc R

(6.2)

where We and Wc are the mass per unit areas for the explosive and for the confinement. In the 1950s, the artificial viscosity term was proposed by von Neumann and Richtmyer, and the simulation of propagation of shock wave became possible in principle [5]. However, at that time it had not yet reached the level of solving twodimensional fluid equations by computer. Wood and Kirkwood [6, 7] discussed the diameter effect of non-ideal detonation by expressing steady-state plane detonation with two-dimensional axisymmetric flow. This also removes the assumption of spherical wave, allowing the model to consider expansion waves from all directions, which is much closer to the actual detonation phenomenon. The starting equation is twodimensional axisymmetric equation that combines conservation of mass, momentum in the propagation (z) and radial (r) directions, energy, and the reaction rate law. The expression is as follows. ( ) ∂u ∂w w dρ +ρ + +ρ =0 dt ∂z ∂r r

(6.3)

ρ

∂p du + =0 dt ∂z

(6.4)

ρ

dw ∂ p + =0 dt ∂r

(6.5)

∂v dε +p =0 dt ∂t

(6.6)

dλ = Rr dt

(6.7)

where ρ = (1/v), ε and p are density, specific internal energy, and pressure, the u and w are particle velocities for axial and radius direction, respectively. The variable λ denotes the degree of chemical reaction. Because of a steady-state analysis, a coordinate system moving with detonation velocity D was used to stop the flow. Thermicity σ was introduced which gives the pressure change due to reaction as shown as follows. 1 dp dρ = 2 − σρ Rr dt c0 dt

(6.8)

204

A. Miyake et al.

They obtained a generalized ‘Chapman–Jouguet condition’ for detonation in a finite stick, D = U + c0

(6.9)

σ R − 2wr = 0

(6.10)

Mass, momentum (axial direction), and energy conservation equations were integrated into a steady state and only on axis. After a long calculation with some assumptions, they obtained a sufficient number of equations to determine D, the axial values of p, U, ρ, and λ throughout the steady zone. To solve these equations, the explicit information about (a) the equation of state of reaction products and (b) the reaction rate law R was necessary. Eyring et al. eventually encountered a similar problem in the analysis of the curve front theory. Since such information was not available at that time, the following empirical formula was derived assuming a liquid explosions. D/Di = 1 − 3.5ξ ∗ /Sr

(6.11)

The Sr corresponds to the radius of curvature at detonation front, and ξ ∗ is the distance from the shock front to C-J point at the axis. At that time, Cook et al. [8–10] published the papers on the diameter effect for several explosives. Information necessary for the measurement of steady-state detonation is also summarized. For example, L/d of about 4–4.5 is required to measure the steady-state detonation velocity. L is the length of explosive. Therefore, the charge of L/d > 6 was used to measure the detonation velocity or the curvature at the detonation front in their experiments. The relationship between detonation velocity and the inner diameter of some explosive measured by Cook et al. is shown in Fig. 6.2. The confinement was paper of 1 or 2 mm thick. Experiments were conducted on pure AN up to a diameter of 46 cm, and L/d in that case was set to be 4.5. Two explosives reached a steady detonation at diameter of 5–10 cm, and another two explosives did not reach a steady state within the experimental conditions. The latter cases are called a highly non-ideal explosive. Figure 6.3 is obtained by rewriting Fig. 6.2 with the reciprocal of radius as horizontal axis. The diameter effect is often represented by a linear relation in the form of Eq. 6.1. The detonation velocity at zero on the horizontal axis is a simple extrapolation of the detonation velocity to infinite diameter. It should be noted that there are some cases that the extrapolated value is significantly different from the ideal detonation value. Particular attention should be paid to the scale on the horizontal axis. In the figure, the two explosives have a constant detonation velocity near zero on the horizontal axis. Explosives with strong non-ideality do not have this situation in general. This is because the experiments required very large diameters which are difficult to perform. In general sense, the rough classification of ideal and non-ideal explosive is as follows. Except for loose-packed powder cases, cyclotrimethylenetrinitramine

6 Ideal and Non-ideal Detonation

8 70.7/29 Composition B−AN

−1

Detonation velocity / km s

Fig. 6.2 Relationship between detonation velocity and the inner diameter of some explosive [8]

205

6

50/50 TNT−AN (Cast)

4

50/50 TNT−AN (loose packed)

2 pure AN

0 0

10

20

Diameter / cm

−1

8

Detonation velocity / km s

Fig. 6.3 Relationship between detonation velocity and the reciprocal radius of some explosive [8]

6

4

70.7/29 Composition B−AN

2

50/50 TNT−AN (Cast) 50/50 TNT−AN (loose packed)

0

0.4

0.8

1.2

Reciprocal radius / cm

1.6 −1

(RDX), cyclotetramethylene tetranitramine (HMX), pentaerythritol tetranitrate (PETN), trinitrotoluene (TNT) and those mixture Composition B (RDX/TNT = 60/40 wt%), etc., have an intense ideality. When AN is added to some extent, they begin to show non-ideality. AN is frequently added to energetic materials as oxidizer and is very insensitive blasting agents. The diameter effect also depends on the

206

A. Miyake et al.

loading density. As shown in Fig. 6.3, even with the same 50/50 TNT-AN composition, loose-packed TNT-AN shows strong non-ideality. If loose-packed TNT-AN is pressed to dense pack, the density will be the same as that of cast TNT-AN, and the same tendency of diameter effect may be obtained. The study area of non-ideal detonation becomes broad because the diameter effect is defined for a particular density or composition, and applications of explosives are wide range. It is necessary to carefully check the width of the scale on the horizontal axis of figures of the diameter effects. Between 1960 and 1970, studies were conducted on the diameter effects of both ideal and non-ideal explosives. The solid propellants are used for propulsion purposes and usually do not detonate. However, they may detonate when they receive an intense shock wave generated by the detonation of booster explosive. Those are the energetic material that exhibits strong non-ideality. For safety handling, it is important to investigate the diameter effects and the failure diameter. Pandow, Ockert, and Shuey [11, 12] investigated the diameter dependence of detonation velocity in solid composite propellants and calculated the reaction zone thickness. The samples were several plastisol-nitrocellulose composite propellants and were mainly unconfined, and ammonium perchlorate (AP), KCl, RDX, and Al were added, respectively. Composition C-4 with 1.59 g/cc initial density and L/D = 3 was used as booster explosive which had the same diameter as sample. The maximum diameter was up to about 20 cm, and the diameter effects were confirmed by linear relationship. Campbell and Engelke [13] presented the diameter effect in high-density heterogeneous explosives in 1976. Figure 6.4 shows the diameter effect for several explosives. The diameter effect was investigated for 16 explosives in cast, pressed, and liquid state with a density of at least 94% in theoretical maximum density (TMD). The following formula was proposed. D Ar =1− Di R − Rc

(6.12)

where Ar and Rc were described as length parameters. If Rc is equal to 0, Eq. 6.12 is same form as Eyring’s equation Eq. 6.1. This case expresses the linear relation in diameter effect curve in D/Di versus 1/R plane and covers the region of sufficiently large diameter. If Rc has a value greater than 0, then D/Di will be −∞ if R → Rc . R = Rc is the asymptote of the curve of Eq. 6.12. The curvature near the failure diameter depends Ar , and this property of the curve covers a large curvature of the diameter effect curve near the failure diameter. Equation 6.12 has a straight line with a slope of -1 when the horizontal axis is Ar /(R − Rc ), and the experimental data for 10 explosives are on one straight line. The curvatures of detonation front were also measured. For X-2019 (TATB/KelF 800 = 90/10 wt%), the linear relation between the curvature radius of the detonation front and charge radius was confirmed and was used together with Wood–Kirkwood relationship of Eq. 6.11 to estimate the reaction zone length (the distance from detonation front to sonic plane). The reaction zone length

6 Ideal and Non-ideal Detonation

−3

PBX−9404 / 1.846 g cm −3

Comp. A / 1.687 g cm

−1

Detonation velocity / km s

Fig. 6.4 Relationship between detonation velocity and the inverse radius for high-density heterogeneous explosives [13]

207

7.5

−3

Comp. B / 1.70 g cm

Pressed TNT / 1.62 g cm Cast TNT −3 / 1.615 g cm

−3

Nitromethane / 1.128 g cm

−3

5 0

5

10

15

Reciprocal radius / cm

−1

was estimated to be 1.3 mm in Wood–Kirkwood theory and 0.14 mm in curved front theory [13]. In 1980s, the numerical simulations of non-ideal detonation [14] and quantitative evaluations of non-ideal detonation based on Wood–Kirkwood extended theory were carried out [15–17]. For TATB-based explosive, the numerical simulations were carried out using two-dimensional Eulerian code that considered reaction rate, equations of state of both unreacted component and detonation products, and very simple mixing rule. The results of the numerical simulations were compared with the experimental results on the diameter effect. The effect of the size of computational grid was discussed, but at that time, the computational environment was not enough for the simulation of detonation propagation considering the reaction zone. If the evaluation target is high-density heterogeneous explosive, the length of reaction zone is the order of 1 mm or less. It was not easy to evaluate detonation propagation by the numerical simulation with the grid small enough for the detonation phenomena including its reaction zone. An epoch-making evaluation method that extends the Wood–Kirkwood theory to solve this problem has appeared. In 1981, Bdzil [18] proposed an analytical steadystate theory of diameter effect which is a generalization of Wood–Kirkwood analysis. Under the assumption of steady-state and two-dimensional flow, the proposed analysis can estimate the failure diameter, diameter effect, and shock locus at the detonation front. The equations of state and reaction rate law were required for the analysis, and the ideal EOS for both reactants and detonation products and state-dependent rate law was employed. Analysis was applied to nitromethane and PBX9404 (HMX/nitrocellulose/tris-beta-chloroethylphosphate = 94/3/3 wt%), for which reliable experimental data exist. It was shown that the diameter effect and

208

A. Miyake et al.

the curvature at detonation front were in good agreement with the experiments. This approach has been improved by Bdzil and Stewart et al. and used by many researchers in the world [19–24]. For examples, at 9th Detonation Symposium, the updated method was named Detonation Shock Dynamic (DSD) which came from Whitham’s 2D theory for inert shock propagation called Geometrical Shock Dynamics [25, 26]. Many researchers have proposed models that extend Wood–Kirkwood theory by adding or removing the various assumptions and discussing the diameter effect [27]. One example is Matsui’s paper [16] for slurry explosive in which the condition for steady propagation of detonation is evaluated by solving eigenvalue problem numerically without using a small-perturbation approximation. Matsui’s method used the Grunaisen-type EOS for solid component, Abel EOS (validity evaluation by Kihara– Hikita (KH) EOS explained at Sect. 3.3 and Arrhenius reaction rate law to evaluate the diameter effect for slurry explosive. Figure 6.6 presents by Matsui et al. at 8th Detonation Symposium in 1985 is particularly interesting at that time. At the 9th Detonation Symposium, for examples, as the numerical approach, the simulation employing the ignition and growth model was carried for the failure diameter of propellant containing HMX [28]. As an experimental study, Miyake et al. [29] performed the steel tube test for AN, which is described in detail in Sect. 6.3. AN is often used as compositions for explosives, and physical properties such as particle size also influence the detonation characteristics. It is also the subject of this chapter and is often discussed in the study of non-ideal detonation. Below, we focus on the studies for highly non-ideal explosive, such as AN-based industrial explosive [30]. When the generalized C-J condition was proposed, it was shown that the C-J point in which thermodynamic equilibrium is accomplished, and the sonic plane do not always coincide. The definition of C-J condition automatically changed. It may be thought that there is the ambiguity of the expression of the reaction zone. For non-ideal detonation, the sonic plane locates on the partially reacted state and not the end point of reaction. Figure 6.5 shows the conceptual diagram for Hugoniot of partially reacted state of highly non-ideal explosive. The experimentally obtained steady state is the observed C-J state at which the unreacted explosive remains. If the unreacted explosive is assumed to be as inert, the observed C-J state must lie on a partially reacted Hugoniot. Mader estimated the partially reacted state by BKW EOS which assumed the unreacted explosive as the inert. In Fig. 6.5, all red circles are called C-J point. The studies that modified Eyring’s size effect curve were carried out by Souers et al. [31–33]. Souers investigated size effect and detonation front curvature for 26 explosives including emulsion explosive and HANFO in 1997. He defined the average sonic reaction zone length < Xe > which is the length from first reaction to sonic point, and edge lag to characterize the detonation front curvature and showed that the edge lag and the effective reaction zone length are roughly equal [31]. However, there were the differences in the case of emulsion explosive and HANFO. The study on the effect of confinement on detonation velocity was also carried out by Souers et al. [32] and Esen [34] in 2004. In the former, the relational expression between detonation velocities of unconfined and confined explosive was derived based on original concept which combined empirical size effect curve. The hydrocode model,

6 Ideal and Non-ideal Detonation

209

Fig. 6.5 Conceptual diagram for Hugoniot of partially reacted state of highly non-ideal explosive and the C-J point

=0.6

=1

partially reacted Rayleigh line

Pressure

=0.4

C−J point C−J isentrope

Hugoniot for solid

0 Specific volume JWL + + [33] was applied for six explosives. The latter was the construction of the confinement model applicable to the blasting site based on the data analysis of field detonation velocity measurements. The empirical equations of size effect curve including his proposed equation [4, 13, 34] were investigated for their application. Finally, the empirical formula that expresses the detonation velocity of confined condition as a function of the detonation velocity of unconfined condition and rock strength was proposed [34, 35]. In addition, they called the area from the start of the reaction to the sonic plane as the detonation driving zone. The DSD theory was applied to highly non-ideal explosive such as HANFO in 1998 [36]. Before 2001, the series of studies of non-ideal detonation behavior of ANFO were conducted at Los Alamos National laboratory [37, 38]. In 2001, the results of cylinder expansion test for ANFO to obtain JWL EOS by Daivs and Hill [37], and the results of twelve rate sticks experiments which were tested between 77 and 205 mm inner diameter by Catanach and Hill [39] were discussed. In 2002, the DSD calibration for ANFO was developed based on the streak records of these rate sticks data [38]. In Japan, a series of studies on ANFO introduced in this chapter were conducted in early 2000 [40–43]. The study of highly non-ideal explosive, ANFO was carried out in the world. In 2006, James et al. [44] presented the results of experimental and numerical study of ANFO and AN-Al mixture. A simulation using JWL + + was carried out, and the scope of its application was discussed. In 2007, the influence of prill’s porosity and particle size on the detonation velocity of ANFO was investigated experimentally by Zygmunt and Buczkowski [45]. The effect of AN oil absorption and the relation between the prill’s porosity and ANFO density

210

A. Miyake et al.

were also investigated. At the same time, Miyake [46] investigated the detonation velocity of AN/AC mixture for four types of AN, i.e., powder, prill, granular, and phase-stabilized AN to examine the effect of microstructure of AN. In 2009, the study of non-ideal detonation of ANFO comprising new types of AN was presented [47]. In 2010, Jackson et al. [48] presented the effect of the confinement on the detonation wave of ANFO using aluminum tube with 76 mm inner diameter. Detonation velocity, curvature, and leading stress waves were measured with changing the wall thickness and length, and a thorough discussion including DSD was being developed. Salyer et al. [49] carried out the front curvature measurements for detonation wave of ANFO in paper tube with large diameters of 203 or 305 mm and with 14-diameter length. The effects of prill size, microstructure including new type AN, mixture ratio, and charge diameter on detonation velocity were examined. In addition, numerical simulation of the detonation propagation of ANFO in aluminum tube was carried out by Short et al. [50]. One of the main purposes of this simulation was examination of the effect of the precursor elastic wave in confinement on the detonation behaviors. A lot of data on non-ideal detonation of ANFO have been accumulated in the above series of studies. With the support of companies related to blasting, the propagation of detonation waves in ANFO confined by aluminum tube and steel tube were simulated by Schoch et al. [51, 52].

6.3 Ammonium Nitrate 6.3.1 Investigation of Non-ideal Behavior of Ammonium Nitrate Ammonium nitrate (AN) has been widely used as fertilizer or explosive component, but there are still many unknown characteristics that have to be examined. Studies described in this section relate to the increasing need for quantitative data concerning detonation phenomena of AN. From a viewpoint of prevention and control of explosions, it is necessary to know the conditions for initiation as well as the intensity of explosion. Since a great deal of the properties related to these aspects is determined empirically on a laboratory scale, fundamental knowledge of explosion phenomena is indispensable to enable reliable extrapolation of the test results to industrial-scale conditions. It is the purpose of these investigations to obtain the understanding of the explosibility of AN [53]. In this section, the non-ideal detonation behavior is investigated. To prepare for the experimental research, the published data concerning the explosion sensitivity and detonation velocity were collected. It was pointed out that the quantitative and systematic research should be necessary. Generally, the non-ideal detonation behavior is explained by the relatively low decomposition rate of AN, which causes a wide reaction zone, in combination with lateral heat losses and rarefaction waves which extinguish the decomposition reactions. First, the ideal

6 Ideal and Non-ideal Detonation

TIGER code (JCZ3)

6

Velocity

4

10

2

Pressure / GPa

−1

20 Detonation velocity / km s

Fig. 6.6 Detonation properties of ammonium nitrate as function of initial density (by TIGER code with JCZ3)

211

Pressure

0

0.5

1 Initial density / g cm

1.5

0

−3

detonation parameters were calculated by using the thermo-hydrodynamic TIGER code with Jacobs–Cowperthwaite–Zwisler (JCZ) and BKW EOSs, and these were compared with other calculations [54, 55]. Detonation velocities of five types of AN were measured by 4-inch steel tube test which is equivalent to the EC tube test for AN. Although three samples showed stable detonation, the detonation velocities were estimated to be 40–75% of the theoretically predicted values by TIGER-JCZ3. Detonation velocities and pressures of low density prilled AN and micro-prill AN were measured in steel tubes with different wall thickness and charge diameter using the optical fiber system and the piezo-resistive manganin gauge technique, respectively. It was found that the tube diameter has a much larger influence on the detonation velocity and pressure than the confinement. The detonation velocity and pressure of micro-prill AN extrapolated to the infinite diameter were calculated as D = 3.85 km/s and P = 3.1 GPa, which showed a good agreement with the ideal detonation parameters for AN predicted by the TIGER code with the JCZ3 EOS. Based on the results obtained, the adiabatic exponent of the detonation gas products was calculated and changed with the testing condition in the non-ideal detonation region and it could be used as a measure of the non-ideality of the detonation behavior.

6.3.1.1

Evaluation of the Detonation Characteristic of AN by TIGER Code

The detonation characteristics of AN were calculated using TIGER code [56]. The input data of AN are shown in Table 6.1. Seven chemical species (H2 O, N2 , NO, NO2 , O2 , H2 , NH3 ) were considered as the components of detonation products. JCZ3 was selected as EOS. Figure 6.6 shows the calculated detonation velocities and detonation

212

A. Miyake et al.

Table 6.1 Input data of AN for TIGER code

Composition

NH4 NO3

Heat of formation

−87,270 cal/mol

Standard volume

46.4 cc/mol

Standard entropy

36.1 cal/K/mol

Initial pressure

1.0 atm

Table 6.2 Detonation properties of AN of initial density 850 kg/m3 calculated by different EOS Dcj (km/s)

Pcj (GPa)

Tcj (K)

E0 (J/g)

TIGER JCZ3

4.01

3.5

1660

1470

TIGER BKW

4.71

4.9

1080

1480

KHT

4.23

3.9

1290

1480

DTONATE

3.6

2.7

1820

1620

pressures for various initial density. In the calculation assuming ideal detonation, the detonation velocity increases almost linearly as the initial density increases. If AN with a large charge diameter can be detonated under certain conditions, it will approach the ideal value. In this section, AN such as prill and powder was filled in steel tube in an uncompressed state, and its detonation characteristics were investigated. Table 6.2 shows the results of calculation of the characteristic values of ideal detonation with four types of EOS, taking 850 kg/m3 as a typical initial density.

6.3.1.2

Steel Tube Test for AN

Two series of steel tube tests, 4-inch steel tube test [57], and test to investigate the effects of wall thickness and charge diameter on detonation characteristics were conducted. We call latter as large—diameter steel tube test. The purpose of the first series was to clarify the detonation characteristics of the five types of AN [58]. The second series revealed the effect of steel pipe thickness and charge diameter on detonation characteristics of AN [59]. Table 6.3 shows the physical properties of the five types of ammonium nitrate whose detonation characteristics were examined by 4-inch steel pipe test. The samples were powder AN, high-purity crystal AN, granular AN relatively low density ANFO quality prill (D), and low density and small diameter ANFO quality prill (E). Here, we call D as low density prill AN, and E as micro-prill AN. The experimental setup of 4-inch steel tube test is shown in Fig. 6.7. The outer diameter, the thickness t, and the length of the tube were 114 mm, 5 ≤ t ≤ 6.5 mm, and 1005±2 mm, respectively. Booster explosives were 500 g PETN/Oil (88/12, 1.5 g/cc, D = 7.5 km/s) or 600 g GX-1 dynamite (1.55 g/cc, 7.0 km/s). The former was initiated by No.6 electric detonator (ED), and later by No.8 ED. Pressure-sensitive velocity probe was set along the axis of the steel tube as shown in Fig. 6.7 It was consisted

6 Ideal and Non-ideal Detonation

213

Table 6.3 Physical properties of five types of ammonium nitrate A

B

C

D

E

Loading density

kg/m3

910–950

960–1050

920–940

800–850

800–840

Type of AN

–

Powder

High-purity crystal

granular

Low density prill

Micro-prill

Sample

Purity

wt%

> 99.0

> 99.8

> 99.0

> 99.0

> 99.0

Water content

wt%

–

–

0.06

0.07

0.09

Nitrogen content

wt%

34.1

34.3

34

33.7

33.8

Particle distribution

wt%

–

–

–

–

–

< 125

μm

52

–

–

–

–

125–210

–

38

2

–

–

–

210–350

–

10

6

–

–

1

350–600

–

–

27

–

–

13

600–850

–

–

57

–

–

32

850–1000

–

–

8

5

–

52

1000–1400

–

–

–

23

2

2

1400–1700

–

–

–

46

11

–

1700–2000

–

–

–

18

21

–

2000–2400

–

–

–

18

51

–

> 2400

–

–

–

–

15

–

of very thin stainless-steel tube with 1 mm inner diameter as shown in Fig. 6.8. The bare resistance wire coated by nylon insulation was inserted in the stainless tube. The change of resistance caused by propagation of the shock wave was measured. Instead of the above method, four optical fibers were installed at intervals of 50 mm from the tube end to avoid the influence of electrical noise, and the detonation velocity was measured from the arrival time and the distance between installations, when performing pressure measurement. The experimental setup of large-diameter steel tube test is shown in Fig. 6.9. The low density prill AN and micro-prill AN, i.e., ANFO quality AN prill in Table 6.3, were selected for this test. Seamless steel tube (JIS G 3445, equivalent to STKM13A) was used. For steel tube diameter of about 100 mm, the wall thickness was changed in the range of 5–30 mm, and the influence of wall thickness was examined. The wall thickness was fixed at 10 mm, and steel tube diameter was changed from 50 to 300 mm to investigate the diameter effect. For almost cases, the length of steel tube was about 1 m. When the booster explosive was initiated by one electric detonator, the influence of curvature of detonation wave in booster to the curvature of detonation wave in sample AN increases as the charge diameter increases.

214

A. Miyake et al.

Fig. 6.7 Experimental setup of 4 inches steel tube test. Length: 1005 mm ±2, outer diameter: 113–115 mm, thickness: 5–6.5 mm

Fig. 6.8 Pressure-sensitive velocity probe

Fig. 6.9 Experimental setup of the large-diameter steel tube test

In order to reduce this effect, in the case of diameter larger than 200 mm, we used a special initiation method for detonating a booster explosive using seven detonating cords as shown in Fig. 6.10. A manganin gauge was installed between a 10 mm thick polymethyl-methacrylate (PMMA) plate and a 50 mm PMMA block. The manganin gauge assembly was put at the bottom of a 1 m steel tube to perform pressure measurement. The shock wave propagation in the PMMA plate and block put at the bottom of steel tube was recorded by streak photograph [53]. As a result, it was found that the velocity was constant up to 25 mm from the contact surface with the steel tube. The detonation pressure

6 Ideal and Non-ideal Detonation

215

Fig. 6.10 Booster charge with seven-point initiation

can be estimated by considering the impedance matching between the detonation products and PMMA. It was observed that pressure decay in Tayler wave was small due to the effect of the tube length of 1 m, and detonation wave without large pressure decrease in the reaction zone was obtained. Figure 6.11 shows typical example of the time-volt output of a constant shock wave velocity measured with pressure-sensitive velocity probe. The shock wave reaches the probe at 100 μs, and the resistance decay is confirmed. Detonation wave is overdriven in the vicinity of booster, and the transition to a steady detonation in which the detonation velocity decays but gradually reaches at a constant velocity is observed.

6.3.1.3

Results of Steel Tube Test for AN

The results of 4-inch steel tube test for five types of AN are shown in Table 6.4. When steady detonation was not measured, the two types of results was observed after shot. In the case of failure, the remained sample was confirmed after shot. In the case of decay, no remained sample was confirmed after shot but the shock

216

A. Miyake et al.

Fig. 6.11 Example of the time-volt output of a constant shock wave velocity measured with pressure-sensitive velocity probe

wave decay was measured. For granular AN, all results were failure. For high-purity crystal AN, all results were failure or decay. For powder AN, low density prill AN and micro-prill AN, the steady-state detonation was confirmed. In the case of the results of low density prill notified as ‘decay,’ the charge diameter may be close to critical diameter under the condition of 5 mm wall thickness. It was considered that the shock sensitivity of low density prill was lower than that of powder and micro-prill. Figure 6.12 shows the relationship between the initial density and detonation velocity for three types of AN. Both low density prill and micro-prill were used as prill AN for ANFO, and the difference is particle size. The low density prill has variation in density, but the loading densities were almost the same. Under this experimental condition, the difference in detonation characteristics due to the difference in particle size of prill AN was confirmed, and the detonation velocity of micro-prill was 1.6 times faster than that of low density prill. In the case of powder AN, detonation velocity was similar to that of micro-prill AN. The loading density of powder AN varied slightly wider than the range shown in Table 6.3, but loading density was similar to that of granular AN. In addition, steady detonation was confirmed even at loading density close to that of high-purity crystal AN. From the results of the 4-inch test, the order of the detonability for five types AN is as follows. Micro-prill > Powder > Low density prill > Granular > High-purity crystal. Figure 6.13 shows the microscope photographs of five types of AN samples. The ascending order of density for five types of AN is Micro-prill ≤ Low density prill < Powder < Granular < High-purity crystal. The ascending order of particle size is Powder < High-purity crystal < Micro-prill < Granular ≤ low density prill. If the particle size is almost the same as for low density prill and granular AN, lower loading density AN had higher detonability. This is the effect of the porosity of the particle. The hot spots generate more easily in high porosity particle because of the shock wave interaction in substance and voids. The micro-prill AN and the low density prill AN have close loading density and the surface condition. In such

6 Ideal and Non-ideal Detonation

217

Table 6.4 Detonation velocities of five types of ammonium nitrate in 4-inch steel tube test. Length:1005 mm ±2, outer diameter: 113–115 mm, thickness: 5–6.5 mm

B; high-purity crystal

C; granular

D; low density prill

E; micro-prill

Vob (km/s)

Vcal (km/s)

Vob/Vcal

910

2.45

4.2

0.58

920

2.7

4.24

0.64

955

2.8

4.35

0.64

965

2.4

4.39

0.55

985

2.55

4.46

0.57

960

Failure

4.37

–

990

Failure

4.47

–

1050

Decay

4.69

–

1050

Decay

4.69

–

925

Failure

4.25

–

930

Failure

4.27

–

935

Failure

4.29

–

800

Decay

3.85

–

825

1.8

3.92

0.46

850

Decay

4.01

855

1.7

4.03

0.42

855

1.75

4.03

0.43

820

2.8

3.91

0.72

825

2.85

3.92

0.73

830

2.7

3.94

0.69

Fig. 6.12 Detonation velocities of three types of ammonium nitrate in 4 inches steel tube test

−1

A; powder

Loading density (kg/m3 )

Detonation velocity / km s

Sample

4

4 inches steel tube tests Powder Low density prill Micro−prill

2

0

0.8

0.9

1

−3

Initial denisity / g cm

1.1

218

A. Miyake et al.

Fig. 6.13 Microscopic photographs of five types of AN samples (a reduced copy of a photo from Miyake’s thesis). Micro-prill (× 100) Powder (× 350) Low density prill (× 30) granular (× 50) High-purity crystal (× 650)

situation, the smaller particle size distribution showed better detonability. In case of the high-purity crystal AN, the detonability increases with the loading density increases. This tendency is unique comparing with other types of AN. The large-diameter steel tube test revealed the effect of steel pipe thickness and charge diameter on detonation characteristics of AN. The micro-prill AN and the low density prill AN were selected as sample. The results of the large-diameter steel tube test for micro-prill were shown in Table 6.6. The results for low density prill were shown in reference [60]. The effects of wall thickness on detonation velocity are shown in Fig. 6.14. The left-hand side corresponds to the results for low density prill AN, right to the results for micro-prill AN. In 4-inch steel tube test, under the condition of 5 mm thick steel tube, it was predicted that 100 mm diameter may be close to the critical diameter for low density prill AN. Therefore, it was expected result that the measured detonation velocity increases as the wall thickness increases from 5 mm that was expected to some extent. Although the detonation velocity increases with the increased wall thickness, the increment became smaller. When wall thickness is larger than 20 mm, detonation velocity is nearly constant. In the case of micro-prill, although the increase of detonation velocity with increasing wall thickness was small comparing with the case of low density prill, the same tendency as low density prill was confirmed. The comparison of the results for both samples was shown in Fig. 6.15. At the 5 mm wall thickness, the average detonation velocity was 1.75 km/s for low density prill and was 2.8 km/s for micro-prill. The detonation velocity for low density prill was about 63% of that for micro-prill. With the increase of wall thickness, the velocity difference between two cases became smaller. At wall thickness of 30 mm, the average detonation velocity was 2.98 km/s for low density prill and was 3.3 km/s for micro-prill. The detonation velocity for low density prill became about 90% of that for micro-prill. The difference between the two samples was the difference in particle size distribution as shown in Table 6.3. This differences in particle size distribution appear as a difference in the state of detonation products in the reaction zone due to the difference in the formation and growth of hot spots and reaction. Because the thermodynamic state also becomes different, the deference of the influence of rarefaction wave due to the expansion of steel tube becomes important in the reaction zone.

6 Ideal and Non-ideal Detonation

219

Table 6.5 Detonation velocities of sample E (micro-prill) obtained by steel tube test I.D

wall thickness

Tube length

Booster

Loading density

D

Di

mm

mm

mm

g

kg/m3

km/s

km/s

100

5

1000

600

810

2.85

3.88

100

5

1000

600

820

2.8

3.91

100

5

1000

600

830

2.7

3.94

100

5

1000

600

830

2.85

3.94

98

8

1000

600

830

2.9

3.94

98

8

1000

600

840

2.9

3.98

100

10

1000

600

830

2.85

3.94

100

10

1000

600

830

2.9

3.94

100

10

1000

600

830

3.1

3.94

100

10

1000

600

840

2.8

3.98

100

10

1000

600

840

3

3.98

100

10

1000

600

840

3

3.98

100

10

1000

600

840

3

3.98

100

15

1000

600

810

3.2

3.88

100

15

1000

600

810

3.2

3.88

100

15

1000

600

820

3.2

3.91

100

15

1000

600

820

3.2

3.91

100

15

1000

600

830

3.1

3.94

100

15

1000

600

830

3.2

3.94

100

20

1000

600

790

3.15

3.81

100

20

1000

600

800

3.15

3.85

100

20

1000

600

810

3.2

3.88

100

20

1000

600

810

3.35

3.88

100

20

1000

600

830

3.3

3.94

100

20

1000

600

850

3.25

4.01

100

30

1000

600

800

3.2

3.85

100

30

1000

600

800

3.4

3.85

100

30

1000

600

810

3.3

3.88

100

30

1000

600

810

3.3

3.88

These differences are reflected in the difference of detonation velocity, which is remarkably different at the wall thickness of 5 mm. The influence of the difference of the particle size distribution is reduced significantly at the wall thickness of 30 mm. However, when the charge diameter is 100 mm, the influence of the difference in particle size distribution remains in two samples. Under the condition of 10 mm wall thickness of steel tube, the diameter effect for AN sample is shown in Fig. 6.16. The left-hand side corresponds to the low density

A. Miyake et al.

JCZ3 EOS −1

4

Detonation velocity / km s

Detonation velocity / km s

−1

220

2

Steel tube test for ammonium nitrate Low density prill

0 0

10

20

30

4

JCZ3 EOS

2

Steel tube test for ammonium nitrate Micro−prill

0 0

Wall thickness / mm

10

20

30

Wall thickness / mm

Fig. 6.14 Effect of wall thickness on detonation velocity. Diameter; 100 mm. left; low density prill AN. right; micro-prill AN. The dashed line indicates the theoretically predicted value by TIGER-JCZ3 at loading density of 850 kg/m3 Table 6.6 (Continued) Detonation velocities of sample E (micro-prill) obtained by steel tube test I.D

wall thickness

Tube length

Booster

Loading density

D

Di

mm

mm

mm

g

kg/m3

km/s

km/s

50

10

1000

150

790

2.2

3.81

50

10

1000

150

810

2.35

3.88

50

10

1000

150

810

2.45

3.88

50

10

1000

150

840

2.35

3.98

148

10

1000

1350

820

3.25

3.91

148

10

1000

1350

820

3.4

3.91

199

10

1000

2400

810

3.35

3.88

199

10

1000

2400

820

3.45

3.91

199

10

1000

2400

820

3.5

3.91

199

10

1000

2400

820

3.6

3.91

247

10

1000

3650

800

3.45

3.85

247

10

1000

3650

800

3.45

3.85

247

10

1000

3650

810

3.5

3.88

299

10

1000

5400

820

3.55

3.91

299

10

1000

5400

820

3.55

3.91

299

10

1000

5400

820

3.6

3.91

299

10

1000

5400

820

3.7

3.91

Detonation velocity / km s

Fig. 6.15 Effect of wall thickness on detonation velocity. Comparison of the results for low density prill AN with those for micro-prill AN. Diameter; 100 mm

221

−1

6 Ideal and Non-ideal Detonation

4

JCZ3 EOS

2 Steel tube test for ammonium nitrate Low density prill micro−prill

0 0

10

20

30

Wall thickness / mm

prill case. One shot was carried out under the condition of the inner diameter of 50 mm, but steady detonation was not obtained, and the result was ‘decay.’ The socalled non-ideal detonation velocity, in which the detonation velocity increases with increasing diameter, was measured. The large-diameter experiments were carried out, and in the case of inner diameter 303 mm, measured detonation velocity reached 99% of the predicted ideal detonation velocity. The right-hand side of Fig. 6.16 corresponds to the micro-prill case. The detonability of micro-prill AN was good, and steady detonation velocities were stably measured even at a charge diameter of 50 mm. In the case of 299 mm, average detonation velocity reached 92% of the predicted ideal detonation velocity. The comparison of the results for low density prill AN with those for micro-prill AN was shown in Fig. 6.17. The low density prill has lower detonation velocity up to diameter of 150 mm. At diameter of 200 mm, there is almost no difference between the two samples. At diameter larger than 200 mm, low density prill had slightly higher detonation velocity. Two AN sample are chemically same substance. Although the particle size distribution is different, the bulk density is same. The diameter effect was shown in Figs. 6.18 and 6.19. In the range where data exist, the diameter effect could be approximated linearly when the reciprocal charge radius was smaller than 0.02. The absolute value of the slope is larger for low density prill AN. When charge diameter is larger than 200 mm, the influence of fine structure disappears. The explained charge diameter effect is shown by the solid line in Fig. 6.19. The detonation velocity at infinite charge diameter by linear approximation was 4.27 km/s. The pressure measurement by manganin gauge was carried out [61]. The gauge was set between a 10 mm thick PMMA plate and a 50 mm thick PMMA block. The

A. Miyake et al.

JCZ3 EOS

−1

4

Detonation velocity / km s

Detonation velocity / km s

−1

222

2

Steel tube test for ammonium nitrate Low density prill

0 0

100

200

300

400

4

JCZ3 EOS

2

Steel tube test for ammonium nitrate micro prill

0 0

Diameter / mm

100

200

300

400

Diameter / mm

Detonation velocity / km s

Fig. 6.17 Effect of wall thickness on detonation velocity. Comparison of the results for low density prill AN with that for the micro-prill AN. Wall thickness; 10 mm

−1

Fig. 6.16 Effect of wall thickness on detonation velocity. Wall thickness; 10 mm. left; low density prill AN. right; micro-prill AN. The dotted line indicates the theoretically predicted value by TIGERJCZ3 at a density of 850 kg/m3

4

JCZ3 EOS

2 Steel tube test for ammonium nitrate Low density prill micro−prill

0 0

100

200

300

400

Diameter / mm

gauge assembly was set just below the steel tube. Because the no decay of shock wave generated by detonation of AN in PMMA was confirmed by high-speed photography, the pressure decay was ignored during pressure measurement; i.e., the impedance matching method was directly applied. The method to estimate the detonation pressure of AN is explained in Fig. 6.20. It is based on the impedance matching explained in Sect. 2.1.2.5. The measured pressure in PMMA is on the Hugoniot line of PMMA. On the other hand, the detonation

−1

223

4

Detonation velocity / km s

Detonation velocity / km s

−1

6 Ideal and Non-ideal Detonation

2 Steel tube test for ammonium nitrate Low density prill

0 0

0.02

2 Steel tube test for ammonium nitrate micro prill

0 0

0.04

Reciprocal radius / mm

4

−1

0.02

0.04

Reciprocal radius / mm

−1

Fig. 6.18 Relation between detonation velocities and reciprocal charge diameter. Wall thickness; 10 mm. left; low density prill AN. right; micro-prill AN

−3

4

Loading density / kg m

Detonation velocity / km s

−1

1000

2 Steel tube test for ammonium nitrate Low density prill micro−prill

0 0

0.02

0.04

Reciprocal radius / mm

−1

800

Steel tube test for ammonium nitrate Low density prill micro−prill

600 0

0.02 0.04 −1 Reciprocal radius / mm

Fig. 6.19 Relation between detonation velocities and reciprocal charge diameter. Comparison of the results for low density prill AN with and micro-prill AN. Wall thickness; 10 mm. Right figure is the relation between loading density and reciprocal charge diameter

pressure of AN is on the Rayleigh line of detonation products of AN in which the slope is the product of the initial density and the detonation velocity. Because the shock impedance of PMMA is greater than that of the detonation products of AN, the condition is ‘reflect.’ Unfortunately, no information of the partially reacted Hugoniot for AN is available. Instead of the reflected Hugoniot, mirror of Rayleigh line is used as approximate evaluation as shown in Fig. 6.20. It can be written as follows [62]. pAN = 0.5 × (ρ0 D/(ρm0 Um0 ) + 1) pm

(6.13)

224

A. Miyake et al. Detonation wave for AN vs PMMA

Fig. 6.20 Estimation of detonation pressure of AN from the measured pressure in PMMA using the impedance matching method

6

AN

PMMA Hugoniot (Deal 1965)

Pressure /GPa

Loading density -3 0.82 g cm -1 D=2.85 km s

Us = 2.88 + 1.38 up

4 2.6 GPa

2

estimated 2.0 GPa detonation pressure Slope; 0D Rayleigh line

0 0

measured pressure

Mirror

1

2 -1

Particle velocity / km s

where pm is the measure pressure in PMMA, ρm0 , Um0 are the initial density and shock velocity in PMMA. The detonation velocity must be obtained by measurement. The effect of the wall thickness to the detonation pressure is shown in left side of Fig. 6.21, and the diameter effect is shown in left side. It was found that the effect of wall thickness saturates at wall thickness about 15 mm, while the effect of the diameter tends to increase linearly at the diameter more than 100 mm. From the ideal theory, the detonation pressure is proportional to the square of the detonation velocity, so the relatively small change of detonation velocity is reflected as the relatively large change in pressure. The tendency can be confirmed in the diameter effect in pressure.

6.3.2 Investigation of Non-ideal Behavior of Ammonium Nitrate and Activated Carbon Mixtures By adding fuel to ammonium nitrate (AN), the characteristics as energetic material can be improved. Detonation characteristics are improved, and it is necessary to understand their characteristics for safe application. Here, we use activated carbon (AC) as fuel and investigate the detonation characteristics of the AN/AC mixtures [63–67].

6 Ideal and Non-ideal Detonation

225 4

Pressure / GPa

Pressure / GPa

4

2

2

Steel tube test for ammonium nitrate micro prill

Steel tube test for ammonium nitrate micro prill

0 0

10

20

0 0

30

100

200

300

400

Diameter / mm

Wall thickness / mm

Fig. 6.21 Effect of wall thickness and charge diameter on detonation pressure. Diameter: 100 mm. (Left) and wall thickness: 10 mm. (right) sample is micro-prill AN

6.3.2.1

Steel Tube Test for AN/AC Mixture

The influence of types of AN on the detonation velocity of AN/AC with various mixing ratio was investigated using four types of AN, i.e., powder, prill, granular, and phase-stabilized (PSAN). PSAN contained 10 wt% of potassium nitrate to prevent phase transitions below 373.15 K. Properties of AN and AC used in this study are shown in Tables 6.7 and 6.8, and the photographs of four types AN are shown in Figs. 6.22 and 6.23. The stoichiometric composition of the AN/AC and PSAN/AC mixtures is 93.0/7.0 and 92.0/8.0 wt%, respectively. Photographs of steel tube and protective pipe used in the experiment are shown in Fig. 6.24. The AN/AC mixture was prepared by putting a predetermined amount of each of powders of both components into a plastic box and shaking the container for 10 min to mix both powders. The sample mixture was filled in a steel tube by Table 6.7 Properties of four types of ammonium nitrate Powder Loading density

[g/cm3 ]

Moisture [%]

PSAM

Prill

Granular

0.93

0.9

0.732

1

0.1

0.11

0.094

–

Purity [%]

> 99

~ 90

> 99

> 99

average particle diameter [mm]

0.14

0.13

1.15

1.5

Table 6.8 Properties for activated carbon (AC)

Density [g/cm3 ]

Average particle diameter [μm]

Spec. surface area [m2 /g]

0.26

3.41

1433

226

A. Miyake et al.

(a) Powder

(c) Prill

(b) PSAN

(d) granular

Fig. 6.22 Photograph of four types of AN

Fig. 6.23 Photograph of activated carbon (AC)

tamping, and the booster explosive was set on the top of steel tube. The detonation breaks the steel tube and makes many high-speed fragments. Protective pipe was used to protect the walls of the explosion pit, and it was destroyed only in less than 5 shots. By installing cardboard on the inner wall of

6 Ideal and Non-ideal Detonation

227

(a) Before setting

(b) After setting

(c) After shot Fig. 6.24 Photographs of steel tube and protective pipe

protective pipe, the resistance against explosion was greatly improved, and the fragments could be easily collected. The photograph of the inside of the protective pipe after shot is shown in Fig. 6.24c. When the cardboard was not used, the fragments partially adhere to the inner wall of protective pipe as if it were explosively welded. Figure 6.25 is the schematic illustration of typical experimental setup of the steel tube test. The length of steel tube was 350 mm, and 10 types of steel tubes with different diameters were used. The diameters and thicknesses of steel tubes used are shown in Table 6.9. Four to six ionization probes were mounted with 50 mm interval from the bottom of the steel tube, and one probe was set between the booster and sample explosive. The shock velocity was evaluated from the relationship between the arrival time and position of each probe. The detonation velocity was estimated, and if deceleration was confirmed, it was judged as detonation failure. In addition, the pressure

228

A. Miyake et al. No.6 electric detonator Composition C-4 in PVC tube (50 g)

180

Steel tube (Inner diameter : 35.5) AN/AC (93/7 wt.%)

350

Ionization probe

20 50

3-5

Pulse generator Digital oscilloscope

PMMA PVDF gauge PMMA

[Unit mm]

Fig. 6.25 Experimental setup of the steel tube test

Table 6.9 Inner diameter and thickness of the steel tube; φ: diameter (mm), t: thickness (mm) φ

5.7

7.8

10.9

15.3

20.4

t

2.4

3.0

3.2

3.2

3.4

φ

27.2

35.5

41.2

52.7

65.9

t

3.4

3.6

3.7

3.9

5.2

measurements with a manganin gauge or a polyvinylidene-fluoride (PVDF) gauge (Dynasen Inc. PVF2 -11-(0.125)-EK) attached between PMMA blocks were also carried out on several shots. The booster explosive consisting of 50 g composition C4 (density: 1.4 g/cm3 ) loaded into a polyvinyl chloride (PVC) tube was used to generate steady-state detonation. It was initiated with an electric detonator. First, by using the steel tube of inner diameter of 35.5 mm, the detonability of four types of AN and AN/AC mixture was examined. As the results, powder AN was selected as the component of AN/AC mixture. For powder AN, the number of shot for each combination of the inner diameter of steel tube and mixing ratio of activated carbon is shown in Table 6.10. The results of steel tube tests for powder AN, prill AN, and PSAN are as follows. In case of powder AN, the fragments were relatively large and only a part of steel tube was broken as shown in Fig. 6.26a. A small amount of remained AN without reaction were recovered after the shot. For PSAN, the same result was confirmed. The results clearly indicate the detonation failure. In case of the prill AN, although the fragments were relatively large, the whole tube was greatly deformed and broken as shown in Fig. 6.26b. There was no remained AN after the shot. It was considered

6 Ideal and Non-ideal Detonation

229

Table 6.10 Number of shots for each combination of the inner diameter of steel tube and mixing ratio of activated carbon for the powdered AN/AC mixture AC %

7.8 mm

10.9 mm

15.3 mm

20.4 mm

27.2 mm

0.1

–

–

–

–

2(failure)

0.5

–

–

–

1(failure)

2

1

–

3

2

7

4

5

–

–

–

1

1

7

1

2

2

3

5

10

–

–

–

1

2

AC%

35.5 mm

41.2 mm

52.7 mm

65.9 mm

–

0

2(failure)

3

4

3

–

0.1

5

4

5

–

–

0.3

1

–

–

–

–

0.5

2

2

2

–

–

1

5

6

7

–

–

3

1

–

–

–

–

5

5

2

3

–

–

7

6

11

3

–

–

10

2

2

1

–

–

13

2

–

–

–

–

15

1

–

–

–

–

20

1

–

–

–

–

that all sample was reacted. The judgment of the steady-state detonation was based on detonation velocity measurements using ionization probes. Figure 6.27 shows the results of detonation velocity measurements for prill AN. The measured shock wave had not reached steady state, so the judgment was detonation decay. It may be considered that the type of AN which have the highest sensitivity among four samples is prill AN. This result is consistent with the result in the previous section.

6.3.2.2

Influence of Type of AN on the Detonation Velocity of AN/AC Mixture

Tables 6.11, 6.12, and 6.13 and Fig. 6.28 show the relationship between detonation velocity and mixing ratio of AC for AN/AC mixture. The mixture of granular AN and AC did not give stable detonation at any mixing ratio and unreacted AN and AC were confirmed after shot. This shows that granular AN was clearly insensitive compared with other three types of AN. In case of powder AN/AC mixture, even when the mixing ratio of AC is only 0.1 wt%, the steady detonation was measured. Up to the mixing ratio of 5 wt% of

230

A. Miyake et al.

(a) powder

(b) Prill

Distance from uper end of steel tube / mm

Fig. 6.26 Fragments and a part of broken steel tube after shot (I.D.; 35.5 mm) Fig. 6.27 Results of steel tube test (rill AN, inner diameter of steel tube; 35.5 mm)

300

200

100

0 0

100

200

Time /

s

Table 6.11 Results of steel tube test for the powder AN/AC mixture, tube diameter; 35.5 mm AC (%)

0

0.1

0.3

0.5

1

3

ρ (g/cm3 )

1.01

1.01

1.01

1.01

0.965

0.961

D (km/s)

Failure

1.25

1.6

1.75

2.05

2.95

AC (%)

5

7

10

13

15

20

ρ (g/cm3 )

0.915

0.919

0.895

0.84

0.839

0.745

D (km/s)

3.4

3.3

3.3

3.35

2.95

2.6

6 Ideal and Non-ideal Detonation

231

Table 6.12 Results of steel tube test for the prill AN/AC mixture, tube diameter; 35.5 mm AC%

0

1

3

5

7

10

20

ρ (g/cm3 )

0.835

0.84

0.88

0.875

0.865

0.815

0.725

D (km/s)

Failure

Failure

Failure

3.15

3.1

2.85

Failure

Table 6.13 Results of steel tube test for the PSAN/AC mixture, tube diameter; 35.5 mm AC%

0

0.1

0.5

ρ

(g/cm3 )

1

3

1.1

1.04

1.06

0.983

1

D (km/s)

Failure

Failure

1.25

1.6

2.1

AC%

5

8

10

20

–

ρ

1

1

887

717

–

2.3

2.35

2.85

1.95

–

Fig. 6.28 Relationship between detonation velocity and AN/AC mixing ratio. (Three types of AN powder, prill, and PSAN; inner diameter; 35.5 mm)

4

Inner diameter of steel tube ; 35.5 mm

-1

D (km/s)

Detonation velocity / km s

(g/cm3 )

2

Type of AN

0 0

Powder prill PSAN

10 Mixing ratio of AC / wt.%

20

AC, the detonation velocity was increased as the mixing ratio increased. When the mixing ratio of AC is increased for 5–13 wt%, the detonation velocities are almost the same and those are near the maximum values. Regardless of the mixing ratio, the powder AN/AC mixture always has higher detonation velocity compared to other types of mixture. The combination of the powder AN and AC gives the most effective reaction of mixture. In case of prill AN/AC mixture, when the detonation occurs near the stoichiometric mixing ratio, its velocity is close to that of the powder AN/AC mixture. More

232

A. Miyake et al.

specifically, the range of mixing ratio is around 5–7 wt%. In the other range, lower detonation velocity or failure is confirmed. On the other hand, in case of PSAN/AC mixture, when the mixing ratio is 0.5 wt%, the steady detonation is observed. PSAN shows higher sensitivity than prill AN. It shows the highest detonation velocity of 2.85 km/s at AN/AC composition of 90.0/10.0 wt%, which is higher in AC content than the stoichiometric composition of 92.0/8.0 wt%. In the experiment of ammonium nitrate alone, the type of AN which has the highest sensitivity among four samples is powder AN. Therefore, the above results may also be related to how particles of different sizes are mixed. One of the related factors is the relative particle size of the AN. The average particle diameter of powder AN and PSAN is 0.14 and 0.13 mm, about 40 times diameter of that of AC particle. On the other hand, the prill AN and granular AN particle have diameter about 10 times larger than that of powder AN. This may be disadvantage for prill AN. However, the prill AN has rough surface, so it has advantage with respect to granular AN on the adhesion of particles.

6.3.2.3

Diameter Effect of Powder AN/AC Mixture

The results of the steel tube test include the effects of both diameter and confinement. The effect of confinement was investigated by Souers et al. [32] and by Esen [34]. However, in the case of thin wall, it is considered that the accurate estimation of the effect of the wall thickness is still difficult. Fortunately, we have knowledge on the effect of the wall thickness as shown in Fig. 6.14. The detonability of the micro-prill is higher than that of low density prill. The difference of the detonation velocity due to the difference of the wall thickness for micro-prill is small compared with that for low density prill AN. This means that as the detonability increases the effect on the detonation velocity due to wall thickness becomes small. It is proved that even if the mixture ratio of AC is 0.1 wt%, the detonability of the mixture of AN/AC is higher than that of pure AN (micro-prill). This suggests that the effect of the difference of the wall thickness on the detonation velocity is small. The results of the steel tube tests are summarized in Table 6.14. The measured detonation velocities are average values which are estimated by more than two shots in all cases. The relations of measured detonation velocities with loading density together with the corresponding ideal detonation velocity calculated by thermodynamic equilibrium code are plotted in Fig. 6.29. Cheetah code was used for estimation of ideal detonation velocity. The loading density mainly varied from about 0.82–1.05 g/cc during the preparation. The ratio of measured and ideal detonation velocity as function of loading density is shown in Fig. 6.30. The results for four different diameters are included. Although the density of the AC particle is greater than that of the AN, the bulk density of the AC is very smaller than that of the AN. Therefore, the bulk density of the AN/AC mixture decreases with increasing the mixing ratio of AC.

6 Ideal and Non-ideal Detonation

233

Table 6.14 Detonation velocities (km/s) obtained by the steel tube test for AN/AC mixture. First and fourth lines correspond to the diameter of steel tube (mm) and first column is wt% for the activated carbon Mixing ratio wt%

Inner diameter (mm) 5.7(mm)

7.8

10.9

15.3

20.4

1

–

Failure

1.525

1.575

1.7

7

Failure

1.55

1.95

2.55

2.95

Mixing ratio wt.%

Inner diameter (mm) 27.2(mm)

35.5

41.2

52.7

65.9

0

–

Failure

1.3

1.475

1.775

0.1

Failure

1.3

1.45

1.675

–

1.85

2.15

2.2

2.475

–

3.15

3.5

3.55

3.7

–

Detonation velocity / km s

Fig. 6.29 Detonation velocity of AN/AC mixture as function of loading density measured by steel tube test. Ideal detonation velocity estimated by Cheetah code is also plotted by triangle symbol. Mixing ratio of AC; from 0 to 20 wt %, inner diameter; from 7.8 to 65.9 mm

-1

1 7

4

2

0.7

0.8

0.9

1

Loading denisity / g cm

1.1 -3

In addition, in the range shown in the figure, larger mixing ratio brings the high detonation velocity, so the slope of the data distribution is negative. The range of loading density variation is about 0.1 g/cm3 under all conditions. The diameter effect for AN and AN/AC mixture is shown in Figs. 6.31 and 6.32. The left-hand side of Fig. 6.31 shows the relationship of measured detonation velocity with reciprocal radius. The mixing ratio of AC is 0.1, 0.5, 1.0, and 5.0 wt%, respectively, and the inner diameter is 20.4–65.9 mm. The result for all mixing ratio shows the similar diameter effect and may be able to fit by linear relation with the same slope. In the right figure, the vertical axis is D/Di, and the horizontal axis is wide

234

A. Miyake et al.

Fig. 6.30 Relative detonation velocity is function of loading density obtained for AN/AC of six mixing ratio and by pure AN

AN/AC mixture

D/Di

0.8

0.4

Steel tube

10 wt% AC 7 wt% AC 5 wt% AC 1 wt% AC 0.5 wt% AC 0.1 wt% AC 0 wt% AC Diameter

52.7 mm 35.5 mm 27.2 mm 20.4 mm

0

0.6

0.8

1

1.2

Loading denisity / g cm

-3

to visualize all results. At the small diameter region, the above-mentioned linear relations are broken. All data for each mixing ratio must attain D/Di = 1 at 1/R = 0. However, the above-mentioned linear relations do not attain D/Di = 1. It is just local linear relations and cannot predict the detonation velocity at infinite diameter. For the larger charge diameter, the larger wall thickness should be set. However, as mentioned at the beginning of this section, the effect of the difference in wall thickness is considered to be small, so we think that is not the only reason why it does not attain D/Di = 1. There is no doubt that it is a combined effect. 4

D/Di

Detonation velocity / km s

−1

1

2

0.5

AN/AC mixture

0 0

5wt.% AC

1wt.% AC

0.5wt.% AC

0.1wt.% AC

0.04

powder AN

0

0.08

Reciprocal radius / mm

−1

AN/AC mixture

7wt.% AC

5wt.% AC

1wt.% AC

0.5wt.% AC

0.1wt.% AC

0.1

powder AN

0.2

Reciprocal radius / mm

−1

Fig. 6.31 Diameter effect for AN and AN/AC mixture (as the function of reciprocal radius)

6 Ideal and Non-ideal Detonation

235 1

1 AN/AC mixture

5wt.% AC

AN

AN/AC mixture

1wt.% AC 0.5wt.% AC 0.1wt.% AC

D/Di

D/Di

0.6

0.4

0.4

0.2 0

0.1wt.% AC

0.8

0.8

0.6

1wt.% AC

0.1

0.2

0.3

0.2 0

0.2

0.4

0.6

Fig. 6.32 Diameter effect curve for AN and AN/AC mixture (horizontal axis includes the effect of wall thickness.) Left side; pure; AN, 0.5, 1.0 and 5 wt% AC, right side 0.1 and 1 wt% AC

Because the ratio between the wall thickness and inner diameter of steel tube is different for each inner diameter, so strictly speaking, the confinement effect is also different for each inner diameter. To deduce this effect, Eq. 6.2 was employed as shown in Fig. 6.32. The relation between the detonation velocity and the charge radius can be shown as follows. ( ) ρe 1 D = Dx − k (6.14) Di ρc t R This empirical equation has a non-dimensional parameter Dx at 1/R = 0. Except for the tube with 20.4 mm inner diameter, all the experimental results approximately satisfy Eq. 6.14 with k equal about one. It must be noted that this relationship hold in a narrow range as shown here. For a mixing ratio of 1.0 wt% of AC, as shown in Fig. 6.32, the discrepancy between the prediction dashed line and the experimental data is confirmed for small diameters. Corresponding diameters are 20.4 and 15.3 mm. Both walls’ thickness is 3.2 mm. When the inner diameter is larger than 27.2 mm, the ratio of wall thickness to diameter is about 0.1. In contrast, when the inner diameter is 20.4 mm, the ratio is about 0.17. It can be judged that the wall thickness is thicker than the other conditions, relatively. Since the condition is small inner diameter and the wall thickness is thicker against the charge weight, the deformation becomes smaller than in other conditions and the rigidity of the tube also contributes to the detonation velocity. Figure 6.33 shows the results of the pressure measurements by PVDF gauge. Except for the thickness of the PMMA plate, the experimental conditions are same for each shot. The inner tube diameter is 35.5 mm, and the tube thickness is 3.6 mm. The thicknesses of the PMMA plate are 3, 4, and 5 mm, and the loading densities of AN/AC mixture are 0.951, 0.943, and 0.927 g/cm3 , respectively. The corresponding

236

A. Miyake et al.

4

Fig. 6.33 Results of the pressure measurements by PVDF gauge

Pressure / GPa

3.23 GPa

2

AN/AC mixture 93/7 wt % Thickness of PMMA 3 mm 4 mm 5 mm

0 -2

0 2 Time ( s)

4

measured detonation velocities are 3.24, 3.22, and 3.20 km/s, respectively. Although the peak pressure measured in the case of PMMA of 5 mm thickness is the greater than other cases, the gradient of the pressure attenuation behind the detonation front is almost the same in all cases. The 15% variation was also confirmed in the pressure measurement in the previous section. We also have adopted the following assumptions. The pressure damping is gradual and the pressure damping from the contact surface between the steel tube and PMMA plate to the gauge can be ignored. The detonation pressure is estimated by impedance matching method described. The peak pressures at detonation front were estimated 3.3 and 2.9 GPa, respectively.

6.3.3 Investigation of Non-ideal Behavior of ANFO Explosive The most common explosive in blasting site is the mixture of AN and fuel oil (FO) called ANFO which is consisted of 64 wt% AN and 6 wt% FO. ANFO has also been known as typical non-ideal explosive. In order to investigate its non-ideal detonation behavior, detonation velocities of ANFO prepared with different kinds of ammonium nitrate (AN) were measured in steel tube test [40–43].

6 Ideal and Non-ideal Detonation

6.3.3.1

237

Effects of Physical Properties of ANFO on Detonation Velocity

In Sect. 6.3.1, the detonation characteristics of pure AN were examined to investigate the effect of physical properties of AN. In this section, some kinds of ANFO which have different physical properties were examined. The experimental setup was similar to that described in Sect. 6.3.2.1. The inner diameter of the steel tube is 35.5, and the thickness is 3.6 mm. The length of the steel tube is 400 mm. Emulsion explosive of 50 g is used as the booster explosive to initiate ANFO explosive. Table 6.15 shows the physical properties of AN which were used in steel tube test for ANFO. Six kinds of AN were used to investigate the effects of the pore size and the particle diameter on the detonation velocity of ANFO. Samples A, B, and C have similar particle diameter but the different pore size, and samples C, E, and F have same pore size but different particle size. The porosimeter with the mercury intrusion (pressurization) method with the assumption that the pore in the particle was cylindrical shape was used to estimate mode pore diameter. The mode pore diameter means the diameter in which the relative frequency shows the maximum in the diameter versus differential volume distribution. All results obtained by steel tube test are shown in Table 6.16. Figure 6.34 shows the influence of the mode pore diameter of AN on the detonation velocity of ANFO. The measured detonation velocity was divided by ideal detonation velocity calculated by Cheetah code with JCZ3-EOS. The detonation velocity increased with the mode pore diameter decreased in this experimental condition. The pore in the particle is considered to act as the hot spot when the shock wave enters in, and it generates the local high temperature and high pressure, so that it accelerates the chemical reaction. The formation of hot spot is influenced by the size and volume of the pore which exists after mixing with FO. If the pore is relatively small, the oil is restrained from permeating into the particle because of the surface tension, and then the rest of the voids could be hot spots. It is possible to consider that the role played by the pore is different depending on its size, and in this experimental range, smaller pore has a strong influence on the detonation propagation. Table 6.15 Physical properties of AN used in steel tube test for ANFO AN

A

Bulk density

Mode pore diameter

Total pore volume

Total Pore area

Average particle diameter

Specific surface area

[g cc−1 ]

[μm]

[cc g−1 ]

[cm2 g−1 ]

[mm]

[cm2 cm−3 ]

0.717

15.1

0.227

9

1.46

20.5

B

0.732

8.2

0.1855

10.4

1.4

21.4

C

0.744

4.5

0.177

10.8

1.35

22.2

D

0.733

7.6

0.1913

8.1

> 1.4

19.1

E

0.754

4.6

0.171

9.7

1.0 - 1.18

26.3

F

0.76

4.6

0.1774

9.2

0.85 >

42

238

A. Miyake et al.

Table 6.16 Results of steel tube test for ANFO with calculated ideal detonation velocity Bulk density

D

Di

D/Di

AN

kg cm−3

km s−1

km s−1

–

A-1

843

2.84

4.76

0.60

A-2

844

2.93

4.77

0.61

B-1

868

3.33

4.86

0.69 0.66

B-2

869

3.20

4.86

C-1

849

3.31

4.79

0.69

C-2

850

3.50

4.79

0.73

D-1

839

2.90

4.75

0.61

D-2

840

2.92

4.75

0.61

E-1

850

3.39

4.79

0.71

E-2

849

3.47

4.79

0.73

F-1

860

3.75

4.83

0.78

F-2

866

3.85

4.85

0.79

0.8

D/Di

D/Di

0.8

0.4

0 0

0.4 Steel tube test for ANFO

Steel tube test for ANFO

Length : 400 mm Inner diameter:35.5 mm Thickness : 3.6 mm

Length : 400 mm Inner diameter:35.5 mm Thickness : 3.6 mm

5 10 15 Mode pore diameter /

20 m

0 0

20 40 60 2 −3 Specific surface area / cm cm

Fig. 6.34 Results of the steel tube test for ANFO. Left: normalized detonation velocity as a function of mode pore diameter. Right: normalized detonation velocity as a function of specific surface area. Normalized detonation: measured detonation velocity/ calculated ideal detonation velocity

To study the detonation mechanism of ANFO, it is necessary to understand the reaction mechanism between AN and FO and formation of hot spot at the pore. As the distribution of the pore dimensions may also play an important role, the effective pore size for the detonation propagation and its spatial distribution should be subject to further investigation. The right-hand side of Fig. 6.34 shows the influence of the specific surface area on the detonation velocity of ANFO. Since samples (D-F in Table 6.15) were prepared with the same batch of AN, the physical properties were considered as the same

6 Ideal and Non-ideal Detonation

239

in each ANFO except particle diameter. The detonation velocity increased with the increases of specific surface area. This means that the detonation velocity of ANFO with smaller particle diameter prills is faster than with larger diameter prills, which is also consistent with the result of pure ammonium nitrate. The information provided here gives material for studying the hot spot formation and subsequent growth of chemical reaction.

6.3.3.2

Steel Tube Test for ANFO and Its Numerical Simulation

In this section, the results of the experiments and numerical simulation for non-ideal behavior of ANFO are described [68]. Table 6.17 shows the physical property of AN. The sample AN was prepared separately from the AN shown in the previous section. The sample AN was sieved, and its particle size was reduced. The experimental setup was same as previous section. The inner diameters of steel tubes were 41.6, 35.8, and 27.2 mm, and thickness are 3.5, 3.5, and 3.2 mm, respectively. Table 6.18 shows the results of the detonation velocity measurement. For numerical simulation, we adopted the simulation method proposed by Souers [33]. The EOS for an unreacted explosive and the detonation products are the Murnaghan equation, and the JWL isentrope, respectively. The mixture rule for the unreacted explosive and reaction product is the following equation, p = (1 − λ) pU + λpR

(6.15)

where p and λ are pressure and mass fraction of detonation products, subscripts U and R denote unreacted explosive and detonation products, respectively. Table 6.17 Physical properties of AN which was used in steel tube tests with three different diameters AN Bulk density [g Mode pore Total pore Average particle Specific surface cc−1 ] diameter [μm] volume [cc g−1 ] diameter [mm] area [cm2 cm−3 ] K

0.818

7.55

0.1226

0.85 >

42

Table 6.18 Results of steel tube tests for ANFO with the calculated ideal detonation velocity Inner diameter mm ‘-shot no.

Bulk density kg cm−3

D km s−1

Di km s−1

D/Di

27.2–1

908

3.45

5.01

0.69

27.2–2

904

3.43

4.99

0.69

35.8–1

938

3.67

5.12

0.72

35.8–2

925

3.63

5.07

0.72

41.6–1

907

3.66

5.01

0.73

41.6–2

901

3.78

4.98

0.76

240

A. Miyake et al.

The reaction rate of explosive is represented by the following equation, dλ = G( p + q)b (1 − λ) dt

(6.16)

Detonation velocity / km s

Fig. 6.35 Comparison of the diameter effect of ANFO obtained by the experiment and the simulation

−1

Variable q is artificial viscosity, G and b are reaction rate parameters. The numerical method described in Sect. 5.4.1.2 was used. The governing equations are twodimensional axisymmetric Eulerian conservation law. A high-density ANFO was set in the calculation field as a booster, and the plane initiation was simulated. Before the simulation results, we show the definition of sonic point. The variables c and u are the local sound velocity and particle velocity. The sonic point is defined as the position where the value of D−(c + u) is 0, and the detonation driving zone length is defined as the distance between the reaction start plane to sonic plane. Only the driving zone in the reaction zone contributes to detonation propagation. Figure 6.35 shows comparison of the diameter effect of ANFO obtained by the experiment and the simulation. We examined the influence of parameters G and b in Eq. 6.16. The detonation velocity increases as the value of b decreases, and it increases as the value of G increases. When the reaction rate parameters b and G were set to 2.0 and 400, respectively, the simulated detonation velocities fitted to the experimental values for all charge diameters. The distributions of mass fraction of detonation products near the detonation front are shown in Fig. 6.36. The left-hand side is the case of inner diameter 42 mm, and the comparison of the distribution at the axis with that at inner wall of the tube. The horizontal axis shows the distance from the initial contact surface between booster and ANFO. Under the condition of plane initiation, it is expected that the shape of the

4

2

Steel tube test ANFO (Diameter: 42, 38.5, and 27 mm) Simulation

0 0

0.04

0.08

Reciprocal radius / mm

−1

241 Mass fraction of detonation products

Mass fraction of detonation products

6 Ideal and Non-ideal Detonation Detonation driving zone in axis

7.5 mm

0.8

Steel tube test

ANFO Diameter 42 mm

0.4

Position from an axis 0 mm 21 mm

0 0.37

0.38

0.39

Distance from initiation point / m

0.8

Simulation of Steel tube test ANFO

0.4

Diameter (mm) 42 27

0 0.37

0.38

0.39

Distance from initiation point / m

Fig. 6.36 Distributions of mass fraction of detonation products near the detonation front. Left: inner diameter 42 mm, and the comparison of the distribution along the axis and inner wall. Right: the comparison with the cases of diameter 42 and 27 mm for the distribution along the axis

detonation front will have hardly curved. In this case, the lengths of the detonation driving zone are about 7.5 mm for both at axis and near the inner wall. The ratio D/Di is about 0.7 for this experimental condition; in contrast, the mass fraction of detonation products at sonic point obtained numerical simulation was predicted about 0.9. It should be noted that the evaluation of the simulation is very effective, but at the same time, the problem of how to determine the pressure and the sound velocity at the driving zone remains.

6.3.3.3

Measurements of Detonation Front Curvature in Confined ANFO

The curvature of detonation front in confined ANFO was measured by high-speed photography. The Cordin model 119 was used to record by smear camera, and the PMMA and steel tubes were used as confinement. Total number of curvature measurements is thirteen, and only six shots are represented. The prill AN with average particle diameter 1.36 mm and bulk density 753 kg/m3 was used and mixed with FO. Figure 6.37 shows the experimental setup for PMMA and steel tube confinement. Top of the experimental setup is an explosive lens and consists of nitromethane and mixture of nitric acid and hydrazine, hydrazine and water. The explosive lens was initiated by PETN/Silicon explosive. The bottom of the test tube was covered with a 1 mm thick PMMA container with a 100 μm thick space. The mirror was installed under the test tube at an angle of 45° to record the detonation wave that arrived at bottom of the test tube. The 100 μm slit was set between the mirror and smear camera. The shape of detonation front in ANFO in PMMA tube is shown in Fig. 6.38. In case of inner diameter 40 mm, the decay of measured detonation/shock velocity

242

A. Miyake et al.

Fig. 6.37 Experimental setup for measurement of curvature at detonation font of ANFO plane wave generator was used. Left; PMMA tube, right; steel tube

was confirmed by another experiments. It may be possible that detonation failure occurs at about L = 300 mm. All results in Fig. 6.38 correspond to non-steady state. Figure 6.39 shows the shape of detonation front in ANFO in steel tube. In case of inner diameter 27 mm, the propagation velocity was over 3 km and had no decay near the L = 300 mm. This means that the detonation wave reached the steady state. Within the measurement range, no significant curvature was observed.

6.3.4 Diameter Effect of AN-Based Explosive Figure 6.40 shows the relation between detonation velocity and reciprocal radius for pure AN (left side) and for ANFO and the mixture of AN/AC (right side). The results for confined and unconfined (paper tube) cases are compared. The figure of left-hand side corresponds to pure AN. As shown in Fig. 6.19, loading density of AN for Miyake’s experiments is almost distributed 0.8–0.85 g/cm3 [28, 60]. On the other hand, the loading densities for Cook’s experiments are 0.95 and 1.05 g/cm3 [8]. These results relate to the fact that the detonation velocity increases with increasing of loading density. The detonation velocity for Cook’s data at infinite diameter may be slightly higher than that for Miyake’s data. In the figure of right-hand side, detonation velocity of unconfined data referred from Catanach and Hill [39], and Salyer et al. [49] are plotted. The loading density for former is 0.88–0.92 g/cm−3 ,

6 Ideal and Non-ideal Detonation

243

Fig. 6.38 Shape of detonation front in ANFO in PMMA tube. D: inner diameter, L: length of PMMA tube

(PMMA, D:40 mm, L:75 mm)

(PMMA, D:40 mm, L:150 mm)

(PMMA, D:40 mm, L:200 mm) Fig. 6.39 Shape of detonation front in ANFO in steel tube. D: inner diameter, L: length of steel tube

(Steel, D:27 mm, L:300 mm)

(Steel, D:36 mm, L:300 mm)

(Steel, D:42 mm, L:300 mm)

244

A. Miyake et al.

4

Miyake et al.

Powder

Pure AN

2

Cook et al.

failed

−1

fragmax

Detonation velocity / km s

Detonation velocity / km s

−1

6 Steel tube test for ammonium nitrate Low density prill Miyake micro−prill

Mader's book

0.02

0.04

Reciprocal radius / mm−1

Zygmunt and Buczkowski

4

2 Catanach and Hill

Esen D=5.057−0.212/d

ANFO in paper tube

Salyer et al.

Jackson et al.

Unconined (paper tube)

0 0

Miyake et al AN+7wt% AC in steel tube ANFO in steel tube ANFO with confinment

0 0

ANFO in aluminum tube

0.02

0.04

ANFO in paper tube

0.06

0.08

Reciprocal radius / mm −1

Fig. 6.40 Detonation velocity vs. reciprocal radius for pure AN (left side) and for ANFO and the mixture of AN/AC (right side)

and for latter are 0.793–0.903 g/cm3 and 1.17 g/cm3 (Fragmax made by Dyno Novel). The data plotted by Symbol * were picked up from Mader’s textbook [2]. There are two types of data. The detonation velocity of ANFO at 0.88 g/cc increases from about 3500 m/sec in 3.5-cm-diameter charges in rock or steel to the ideal velocity of 5500 m/sec in 26.8-cm-diameter charges in rock [2]. Another data were obtained by Finger’s cylinder expansion test with copper tube. The ANFO density from Jackson et al. experiments was 0.86–0.90 g/cm3 , and thickness of aluminum tubes was 6.35, 12.7 and 25.4 mm, and charge diameter was 76.2 mm [48]. Zygmunt and Buczkowski measured the detonation velocity of ANFO in steel tube of 36 mm inner diameter and the prill’s size are varied [45]. The loading densities were 1.01, 1.02, 0.98, and 0.86 g/cm3 . The smaller the density, the higher the detonation velocity. The dotted line corresponds to fitting curve by Eyring type equation for ANFO for low density ANFO at a density of 0.71 g/cm3 [34].

6.4 Emulsion Explosives 6.4.1 Introduction The use of commercial explosives was drastically changed in the mid-1950s with the development of slurry and emulsion explosives, which are water-resistant and safe to handle. Prof. Cook of the University of Utah developed first water-resistant slurry explosives consisting of AN, water and Al powder in 1956. Since then, many explosive companies have developed slurry and emulsion explosives with various formulations. The early slurry explosives were sensitized with Al and/or TNT powders. In 1962, first emulsion explosive containing neither Al nor TNT powders was developed. Slurry and emulsion explosives expanded rapidly their range of use and replaced dynamite by the end of 1980s. At present time, more than 90% of commercial explosives are consumed using bulk methods. Slurry and emulsion explosives

6 Ideal and Non-ideal Detonation

245

are delivered in cartridge or in bulk, pumpable form. In Japan, they have been delivered mainly in small size cartridge form for tunnel construction and underground coal and metal mining. The emulsion explosive (EMX) is typically a water-in-oil type of emulsion consisting of fine droplets (smaller than a few μm) of a highly concentrated aqueous solution of oxidizer surrounded by continuous oil phase. The emulsion matrix (EM) is consisted of oxidizer (80–85 wt%), water (10–15 wt%), and fuel oil with emulsifier (about 5 wt%). In typical commercial EMXs, the main oxidizer is AN and small amount of sodium nitrate (SN) or calcium nitrate (CN) is added to optimize the oxygen balance in order to minimize the amount of poisonous CO and NOX gases in detonation products. The EM does not contain explosive components and requires the sensitizer such as hollow microballoon (MB) or gas-forming chemical agent to make emulsion explosive detonable. Both capsensitive and non-capsensitive EMXs can be produced by varying the amount of MB added. The EMXs often contain Al powder to increase its energy density and blasting performance. The aqueous solution of oxidizer, fuel oil, and emulsifier is heated to high temperature of 353–373 K, and mixed together using a high-speed mixer. The size of the oxidizer solution droplets is determined by the speed and time for the mixing. To sensitize the mixture, MB or gas-forming agent is added to EM with a low-speed mixer.

6.4.2 Detonation Velocity of Emulsion Explosives The EMXs exhibit non-ideal detonation behavior in which the detonation velocity can be considerably lower than the value predicted by thermochemical calculation. As EMXs are sensitized by hollow MB, their detonation properties can be widely controlled by changing the amount and size of MB added. Hattori et al. [69], Yoshida et al. [70], Lee et al. [71, 72], and Chaudhri and Almgren [73] investigated the effects of the amount and size of MB on detonation velocity and failure diameter of EMXs sensitized by glass microballoon (GMB) smaller than 0.15 mm in diameter. Hirosaki et al. [74–80] carried out an extensive study of the effects of the amount and size of plastic microballoon (PMB) on the detonation properties such as detonation velocity, pressure, and failure diameter. The EMXs were sensitized by PMBs ranging from 0.05 to 2.42 mm in diameter. The detonation velocity of an explosive is governed by the competing rate of chemical reaction behind shock front and lateral expansion of detonation products which reduces the temperature and pressure in reaction zone. As EMXs may have long reaction zone, their detonation velocities depend strongly on charge diameter for weakly confined charge. Hirosaki et al. [74, 75, 79, 80] measured the detonation velocity of EMXs with various density sensitized by PMBs of different size. The properties of PMBs are shown in Table 6.19. Figure 6.41 presents microscopic photographs of PMB of mono-cell structure and multi-cell structure, and compared with that of GMB. The composition of the EM is AN/SN/water/oil and emulsifier = 77.7/4.7/11.2/5.4 in weight ratio. The EM is oxygen balanced, and its density is

246

A. Miyake et al.

1.39 g/cm3 . The density of the EMXs was varied by changing the amount of PMB added. Sample EMXs of various density were loaded into plastic film tubes of 20, 30, 40, and 50 mm in diameter and 300 mm long to measure detonation velocity by ionization probes. Figure 6.42 shows the dependence of detonation velocity on the inverse charge diameter for the EMXs with various density sensitized by PMB of mean diameter 0.05 and 1.73 mm. The relation between the detonation velocity and the inverse charge diameter is expressed by the following linear function, ( ) A D = Di 1 − d

(6.17)

where D and Di are detonation velocities at a given diameter d and infinite diameter, and A is fitting constant related to the reaction zone length of the explosive [4]. It is shown in Fig. 6.42 that the diameter effect lines become steeper with the increase of density of EMX, which corresponds to the increase of constant A and failure diameter. Figure 6.43 presents the relation between the constant A and charge density for EMXs loaded with PMB of different size. The constant A increases as the charge density increases for all PMB size. Figure 6.44 presents the relation between the constant A and failure diameter. It is shown that the failure diameter increases linearly with the increase of constant A. According to the classification of explosives by Price [81], the EMX is typical of Group 2 explosive in which the failure diameter increases with the increase of charge density. Table 6.19 Properties of plastic microballoons Mean dia (mm) Std. dev (mm) Bulk density (g/cm3 ) Structure

Material

0.05

0.02

0.027

Mono-cell Acrylonitrile/Vinylidene Chloride

0.47

0.06

0.051

Multi-cell

Polystyrene

0.80

0.13

0.077

Multi-cell

Polystyrene

1.73

0.27

0.032

Multi-cell

Polystyrene

2.42

0.40

0.064

Multi-cell

Polystyrene

100 100

Mono-cell stricture PMB

500

Multi-cell stricture PMB

GMB

Fig. 6.41 Microscopic photographs of PMB of mono-cell structure and multi-cell structure and compared with that of GMB

6 Ideal and Non-ideal Detonation

247

Fig. 6.42 Dependence of detonation velocity on the inverse charge diameter for the EMXs with various density sensitized by PMB of mean diameter 0.05 and 1.73 mm

Hirosaki et al. [74, 77] presented a linear relation between the failure diameter and average spacing between PMBs (Fig. 6.45). Lee and Persson [72] also observed the same linear relation between the failure diameter and average spacing between GMBs for the EMXs containing GMB smaller than 0.15 mm in diameter. These results showed that the size of MBs and spacing between MBs are of major importance in affecting the failure diameter. Figure 6.46 presents the effects of the density of EMX on detonation velocity at charge diameter 50 mm for EMXs sensitized by PMB of different size. Dotted line shows theoretical detonation velocity which is calculated by KHT thermochemical Fig. 6.43 Relation between the constant A and density of EMXs sensitized with PMBs of different size

248

A. Miyake et al.

Fig. 6.44 Relation between the constant A and failure diameter

Fig. 6.45 Linear relation between the failure diameter and average spacing between PMBs

code [82] supposing EMX completely reacted at CJ state. Detonation velocity was calculated varying the fraction of EMX reacted at CJ state using KHT code. The fraction of reacted EMX was estimated so as to fit the calculated detonation velocity to the observed value. Figure 6.47 shows the variation of the fraction of EMX reacted at CJ state with the density of EMX at charge diameter 50 mm for EMXs containing PMB of different size. Many of the detonation properties of EMX can be explained by the role of MB on the detonation propagation. Since each MB acts as a hot spot, the number and size of MB have strong influences on the detonation propagation. In the EMXs containing the smallest PMB of 0.05 mm in diameter, the detonation velocity increases with the increasing density until charge density about 1.25 g/cm3 . The fraction of reacted EMX increases slightly up to the density about 1.10 g/cm3 ,

6 Ideal and Non-ideal Detonation

249

Fig. 6.46 Effects of the density of EMX on the detonation velocity at charge diameter 50 mm for EMXs sensitized by PMBs of different size

Fig. 6.47 Variation of the fraction of EMX reacted at C-J state with the density of EMX at charge diameter 50 mm for EMXs containing PMBs of different size

and then it decreases. As the number of hot spots is sufficiently large, the fraction of reacted EMX is maintained at high level (0.8–0.9). The effect of the density on the detonation velocity dominates and the detonation velocity increases with the increasing density. As the density of EMX approaches to that of EM, the number of hot spots decreases rapidly and the fraction of reacted EMX decreases drastically.

250

A. Miyake et al.

Then the detonation velocity decreases rapidly and finally fails. This property is also observed in the EMXs sensitized by GMB which is generally smaller than 0.15 mm in diameter. In the case of EMXs sensitized by PMB larger than 1.72 mm in diameter, the detonation velocity is nearly constant at 2000–3000 m/s in the density ranging from 0.50 to1.10 g/cm3 . As the number of hot spots decreases drastically with increasing density, the fraction of reacted EMX decreases from 0.6 to 0.1. The effect of increasing density on the detonation velocity is estimated to be canceled by the decrease of the fraction of reacted EMX. At density 1.10 g/cm3 , only 10–20% of EMX supports steady detonation propagation. For EMXs sensitized by PMB of 0.47 and 0.80 mm, the detonation velocity increases up to the density about 1.0 g/cm3 , and then it decreases because of the balance between the effect of the increasing density and the decreasing number of hot spot. It is speculated that very low fraction of reacted EMX in the EMXs loaded with large PMB is due to rarefaction waves not only from the lateral expansion, but also from the large PMB itself. The detonation velocity increases with the decrease of PMB size at all charge density. Engelke et al. [83] commented that there is no direct relation between the constant A and reaction zone length since the A value depends strongly on the confinement of the charge. They stated that for two closely related explosives with the same confinement, the ratio of the A value is a measure of the ratio of the reaction zone length. However, this relation between the constant A and reaction zone length cannot be applied to the EMXs in which large fraction of explosive reacts behind C-J state.

6.4.3 Detonation Pressure of Emulsion Explosives Hirosaki et al. [74, 78–80] measured the detonation pressure of the EMXs sensitized by PMBs of different size using polyvinylidene-fluoride (PVDF) pressure gauge. The detail of PVDF pressure gauge is explained in Chap. 4. The experimental setup for detonation pressure measurements is illustrated in Fig. 6.48. The PVDF pressure gauge was placed on a block of 50 mm thick polymethyl-methacrylate (PMMA) and glued with a PMMA plate of 1 mm thick. Sample EMX was loaded into polyvinyl chloride (PVC) tube of 51 mm in diameter, 4.5 mm thick, and 200 mm long. As the PVDF pressure gauge measured the pressure transmitted into PMMA plate of 1 mm thick, detonation pressure was calculated using the impedance match method. Detonation velocity was measured with ionization probes or optical fibers. Figure 6.49 presents the measured pressure profiles for the EMXs with density 1.05 g/cm3 sensitized by PMBs of different size. The pressure rises sharply to its peak pressure, and pressure decrease in the reaction zone behind leading shock and following pressure decay in Taylor wave can be observed in the EMXs containing PMB of mean diameter 0.05 and 0.47 mm. In these EMXs, CJ state is estimated to be the start point of pressure decay in Taylor wave. Whereas the EMXs sensitized by PMBs larger than 1.73 mm require very long time to reach its peak pressure which is considerably lower than that of the EMXs containing PMBs smaller than 0.47 mm.

6 Ideal and Non-ideal Detonation

251

Fig. 6.48 Experimental setup for detonation pressure measurements

Electric detonator Booster explosive Sample explosive confined in PVC or steel tube

30

Ionization probes or optical fiber

140 200

PMMA plate (1mm)

50

PVDF pressure gauge 100

PMMA block (50mm) To Osilloscope

This is due to the important irregularity of detonation front of the EMXs containing large PMBs, which was observed in the photographic records [76]. For the EMXs containing PMBs larger than 0.80 mm, the position of CJ state is difficult to identify in the measured pressure profile. The position of CJ state was estimated to be the measured pressure peak, which is identical to the head of Taylor wave. Detonation pressure was calculated varying the fraction of EMX reacted at CJ state using KHT code. The fraction of reacted EMX was estimated so as to fit the calculated detonation pressure to the measured value. Table 6.20 shows the effects of PMB size on the detonation properties of EMXs with density 1.05 g/cm3 . The measured detonation pressure and velocity considerably decrease with increasing PMB size. The fraction of EMX reacted at CJ state decreases with increasing PMB size. The fraction of EMX reacted at CJ state is as high as about 0.85 for the EMXs sensitized by PMB of 0.05 mm. On the other hand, the fraction of EMX reacted at CJ state is only about 0.3 for the EMXs containing large PMB of 2.42 mm. This result is due to the reduction of the number of PMBs which act as hot spots. The fraction of EMX reacted at CJ state evaluated from the detonation pressure agrees well with that estimated from the detonation velocity, which justifies the hypothesis on the position of CJ state. For the EMX containing PMB of 0.05 mm, the distance between leading shock and CJ state is estimated to be about 2 mm, which agrees with the constant A value. However, for the EMX sensitized by PMB of 2.42 mm, the distance is about 5 mm, which is considerably shorter than the A value. The detonation pressures of the EMXs sensitized by PMBs of 0.05 and 0.47 mm were measured at charge density 0.90, 1.05, and 1.20 g/cm3 . The measured pressure profiles are shown in Figs. 6.50 and 6.51. Table 6.21 presents the effects of the density of EMX on the detonation properties of EMXs sensitized with PMB of 0.05

252

A. Miyake et al.

Fig. 6.49 Measured pressure profiles for the EMXs with density 1.05 g/cm3 sensitized by PMBs of different size

Table 6.20 Effects of PMB size on the detonation properties of EMXs with density 1.05 g/cm3 PMB size (mm)

0.05

0.47

0.80

1.73

2.42

Detonation velocity (m/s)

5230

4480

3510

3360

2960

Measured C-J pressure (GPa)

6.4

5.1

2.8

2.5

2.1

Calculated C-J pressure* (GPa)

8.2

8.2

8.2

8.2

8.2

Fraction of EMX reacted at C-J state estimated from detonation pressure

0.84

0.69

0.42

0.39

0.33

Fraction of EMX reacted at C-J state estimated from detonation velocity

0.87

0.68

0.43

0.39

0.30

*

Al is fully reacted at C-J state

and 0.47 mm. For the EMXs containing PMB of 0.05 mm, the fraction of EMX reacted at CJ state is as high as 0.8–0.9, and the detonation pressure increases with increasing density. For the EMXs sensitized by PMB of 0.47 mm, the detonation pressure is nearly constant. Since the number of hot spots is reduced with increasing density, the fraction of EMX reacted at CJ state decreases and the effect of increasing density is canceled.

6.4.4 Detonation Properties of Aluminized Emulsion Explosives The EMX has been widely used in variety of blasting applications because of its safety in handling, high performance, and low cost. Al powder has been used to improve the blasting performance of EMX. However, there are few fundamental studies on the effects of the addition of Al powder to the detonation properties of EMX.

6 Ideal and Non-ideal Detonation

253

Fig. 6.50 Measured pressure profiles for the EMXs with density 0.90, 1.05, and 1.20 g/cm3 sensitized by PMB of 0.05 mm

Fig. 6.51 Measured pressure profiles for the EMXs with density 0.90, 1.05, and 1.20 g/cm3 sensitized by PMB of 0.47 mm

Table 6.21 Effects of the density of EMX on the detonation properties of EMXs sensitized with PMB of 0.05 and 0.47 mm PMB size (mm) EMX density

(g/cm3 )

0.05

0.05

0.05

0.47

0.47

0.47

0.90

1.05

1.20

0.90

1.05

1.20

Detonation velocity (m/s)

4720 5230 5890 4260 4480 4300

Measured C-J pressure (GPa)

4.2

6.4

10.0

4.2

5.3

5.0

Calculated CJ pressure* (GPa)

5.9

8.2

11.1

5.9

8.2

10.9

Fraction of EMX reacted at C-J state estimated from 0.79 detonation pressure

0.83

0.92

0.78

0.71

0.50

Fraction of EMX reacted at C-J state estimated from 0.82 detonation velocity

0.87

0.94

0.78

0.68

0.48

* Al is fully reacted at C-J state

254

A. Miyake et al.

Kato et al. [84] measured the detonation velocity and pressure of EMXs loaded with 10–40 wt% of 5.7 μm size Al powder to study the effects of the addition of Al. The composition of EM was AN/SN/water/oil and emulsifier = 78.5/4.8/11.3/5.4 in weight ratio. The EM was oxygen balanced, and its density was 1.39 g/cm3 . The initial density of EMX was adjusted to 1.21 g/cm3 by adding mono-cell type PMB of 0.05 mm in diameter to EM. For detonation velocity measurements, sample explosives were confined in steel tubes of different inner diameters 12.7, 16.2, 21.2, and 38.7 mm, 250 mm long, and 5.1 mm thick. Detonation velocity was measured by 4 optical probes placed at 50 mm interval. First probe was set at 90 mm from booster explosive to assure steady detonation propagation. Detonation pressure was measured using PVDF pressure gauge. The experimental arrangement for detonation pressure measurements is shown in Fig. 6.48. Sample explosives were contained in steel tube of inner diameter 38.7 mm, 150 mm long, and 5.1 mm thick for detonation pressure measurements. The relation between detonation velocity and the inverse charge diameter is presented in Fig. 6.52 for base EMX and aluminized EMXs (Al-EMXs). Measured detonation velocity is shown to decrease linearly with the increase of the inverse charge diameter. The charge diameter effect on detonation velocity is expressed by Eq. 6.17. The charge diameter effect becomes considerably important at Al content higher than 30 wt%, and detonation fails at charge diameter 12.7 mm when Al content is 40 wt.%. The results of detonation velocity measurements are summarized in Table 6.22. The constant A value is almost constant up to Al content 20 wt% and increases considerably at Al content higher than 30 wt%. The variation of Di and D (d = 38.7 mm) with Al content is presented in Fig. 6.53. Detonation velocities calculated by KHT thermochemical code supposing Al reactive and inert are also presented for comparison. Di agrees well with detonation velocity calculated supposing Al inert. For D (d = 38.7 mm), velocity deficit becomes very large when Al content is higher than 30 wt.% even with thick steel confinement. According to the results of Hirosaki et al. [74, 78–80], the fraction of EMX reacted at C-J state is estimated to be higher than 0.9 for base EMX and Al-EMXs with Al content up to 20 wt% at d = 38.7mm. In the case of Al-EMXs with Al content up to 20 wt%, it is estimated that the effects of increasing density and decreasing number of hot spots by dilution with unreacted Al powder are balanced to maintain high fraction of EMX reacted at C-J state. For Al-EMXs with Al content higher than 30 wt%, it is estimated that the effects of decreasing number of hot spots become dominant and fraction of reacted EMX at C-J state is decreased. Figure. 6.54 shows the pressure profiles measured by PVDF pressure gauge for detonation waves in base EMX and Al-EMXs. As PVDF pressure gauge measured detonation pressure transmitted into PMMA plate of 1 mm thick, detonation pressure was calculated by impedance match method. In the case of detonation in base EMX, and Al-EMX containing 20 wt% of Al, leading shock wave, steep pressure decrease in reaction zone and following pressure decay in Taylor rarefaction wave are observed. Abrupt pressure decrease due to side rarefaction wave is observed at about 4 μs behind leading shock wave. For base EMX, the measured detonation pressure of 10. GPa is lower than C-J pressure of 11.2 GPa calculated by KHT thermochemical code

6 Ideal and Non-ideal Detonation

255

Fig. 6.52 Relation between detonation velocity and the inverse charge diameter for base EMX and Al-EMXs

Table 6.22 Results of detonation velocity measurements for base EMX and Al-EMXs Al mass fraction (%)

Density (g/cm3 )

D (d = 38.7 mm) (m/s)

Di (m/s)

A (mm)

0

1.21

5860

6150

1.81

10

1.28

5830

6100

1.71

20

1.36

5610

5910

1.99

30

1.46

5240

5700

3.12

40

1.56

4660

5500

5.92

Fig. 6.53 Variation of Di and D (d = 38.7 mm) with Al content

256

A. Miyake et al.

supposing EMX fully reacted, which indicates that the fraction of EMX reacted at C-J state is about 0.9. The length between leading shock and C-J state is 2–3 mm, that is close to the A value 1.81 mm. The pressure profile of Al-EMX loaded with 20 wt% of Al is similar to that of base EMX except that leading shock pressure is lower and the length between leading shock wave and C-J state is about 50% longer. The measured detonation pressure of 10.7 GPa is very close to C-J pressure of 10.8 GPa calculated supposing Al inert and EMX fully reacted. This result shows that the reaction of EMX is nearly completed at C-J state. The extended high-pressure zone behind leading shock wave suggests the effects of the multiple interactions between leading shock and Al particles. The rate of pressure decay in Taylor wave of Al-EMX containing 20 wt% of Al is very similar to that of base EMX. In the pressure profile of Al-EMX containing 20 wt% of Al, the effect of Al reaction on the pressure of detonation products is not observed in the time frame of pressure measurements of about 4 μs. Lefrancois et al. [85] carried out the pressure measurements of detonation products of Al-EMX loaded with 14 wt% of fine Al powder (particle size was not specified) using carbon resistor pressure gauge which was directly implanted at the back of sample explosive charge. The initial density and detonation velocity of sample AlEMX were, respectively, 1.28 g/cm3 and about 5000 m/s. The effect of Al reaction on the pressure of detonation products was not observed during about 4 μs after the interaction of detonation front with pressure gauge. For Al-EMX loaded with 40 wt% of Al, the measured pressure profile is completely different. The pressure of leading shock is lowered to 7.5 GPa, and pressure continues to increase until CJ state. The measured detonation pressure of 8.8 GPa is lower than CJ pressure of 10.1 GPa calculated supposing Al inert and EMX fully reacted, which shows that the fraction of EMX reacted at C-J state is much lower than 0.9. The length between leading shock wave and C-J state is about 6 mm, which is close to the A value 5.92 mm. The rate of pressure decay in Taylor wave is much lower than that of base EMX and Al-EMX loaded with 20 wt% of

Fig. 6.54 Measured pressure profiles for detonations in base EMX and Al-EMXs

6 Ideal and Non-ideal Detonation

257

Al, which suggests the effects of the energy released by Al reaction as well as EMX reaction in Taylor wave.

6.4.5 Underwater Explosion Performance of Emulsion Explosives The underwater explosion test is a valuable tool to characterize the performance of explosives in terms of measured shock wave and bubble energies. For commercial explosives, both the rock-breaking capability and the work done by the expanding detonation products are important depending on the particular blasting application. In blasting industry, the relative efficiency of explosives is characterized by measured shock wave and bubble energies. It is considered that shock wave energy is the measure of the rock-breaking capability and bubble energy is the measure of the heaving action of the expanding detonation products. The underwater explosion phenomena have been extensively investigated since several decades [86–88]. However, the correlation between the detonation properties and underwater explosion performance has not been studied systematically. As EMXs are sensitized by MB, their detonation properties can be widely controlled by changing the amount and size of MB. Kato et al. [89] studied the correlation between the detonation velocity and underwater explosion performance of EMXs whose detonation velocity was varied in the range of 2500–5500 m/s maintaining the same chemical energy and initial density. The composition of EM was AN/hydrazine nitrate (HN)/water/oil and emulsifier = 74.6/10.6/10.6/4.2 in weight ratio. Three types of PMB were used to control detonation velocity of sample EMXs, and their properties are shown in Table 6.23. The compositions of sample EMXs are presented in Table 6.24 with detonation velocities measured at charge diameter 50 mm. The initial density of sample EMXs was kept constant 1.10 g/cm3 . Detonation velocity of sample EMXs was measured using paper cartridge of 50 mm in diameter and 200 mm long by ionization probes. For sample explosives EMX-1–6, the detonation velocity was changed in the range of 2520–5360 m/s at charge diameter 50 mm. The detonation velocity of sample EMXs calculated by KHT thermochemical code supposing EMX fully reacted at C-J state is 5908 m/s. The fraction of EMX reacted at C-J state is estimated to be about 0.9 for EMX-1 and about 0.3 for EMX-6. Table 6.23 Properties of three types of microballoon Microballoon

Material

Particle size (mm)

Bulk density (g/cm3 )

A

Resin

0.015–0.04

0.02

B

Polystyrene

0.4–0.5

0.05

C

Polystyrene

1.5–2.5

0.03

258

A. Miyake et al.

Table 6.24 Composition of emulsion explosives with density 1.10 g/cm3 EMX

Emulsion matrix (wts%)

Microballoon (wt%) A

B

Detonation velocity (m/s) C

EMX-1

99.60

0.40

–

–

5360

EMX-2

99.10

0.16

0.74

–

4680

EMX-3

98.96

–

1.04

–

4200

EMX-4

99.07

–

0.74

0.19

3790

EMX-5

99.24

–

0.32

0.44

2750

EMX-6

99.37

–

–

0.63

2520

The experimental setup for the underwater explosion tests is shown in Fig. 6.55. Shock wave profile P(t) and bubble period Tb were measured by tourmaline pressure gauge (PCB type 138A10). Sample explosive and tourmaline pressure gauge were placed at the depth of 4 m, and the distance between sample explosive and tourmaline pressure gauge Rs was 3 m. Shock wave profile was recorded by digital oscilloscope (DC ~ 20 MHz), and bubble period was recorded by tape recorder (DC ~ 40 kHz). Sample explosive was cylindrical shape of 50 mm in diameter and charge weight W was 0.25 kg. Shock wave energy E s and bubble energy E b per unit mass are calculated by the following equation: {5θs

4π Rs2 Es = ρw Cw W

P(t)2 dt

(6.18)

0

where ρw , cw , and θs are density of water, sound velocity of water, and characteristic time of shock wave. 5

3

E b = 0.684 P02 Tb3 /ρw2 W

(6.19)

where P0 is hydrostatic pressure at charge depth. The correlation between shock wave energy and detonation velocity is presented in Fig. 6.56. Shock wave energy is almost constant when detonation velocity is varied from 2520 to 5360 m/s. The peak pressure of shock wave transmitted in water at the charge interface is strongly dependent on detonation velocity and pressure. In contrary, the measured peak shock wave pressure and shock wave energy are shown to be practically constant and independent of detonation velocity and pressure. These results explain that more shock wave energy is dissipated as heat in the surrounding water when detonation velocity is higher. The correlation between bubble energy and detonation velocity is shown in Fig. 6.57. The measured bubble energy is decreased from 2.04 to 1.77 MJ/kg when detonation velocity is increased from 2520 to 5360 m/s. In the underwater explosion of explosive, the chemical energy liberated by reaction of explosive is partitioned into various forms of energy when detonation products

6 Ideal and Non-ideal Detonation

259

expand and do work; shock wave energy E s , shock wave energy dissipated as heat E d , bubble energy E b and internal energy of detonation products E i when bubble is at its first maximum. Total expansion work E t done by the explosive is given by Et = Es + Ed + Eb = m Es + Eb

(6.20)

where m is shock loss factor. Then dissipated energy Ed is obtained by E d = (m − 1)E s

(6.21)

Bjarnholt and Holmberg [88] determined shock loss factor m as a function of detonation pressure. The dissipated energy of sample EMXs was calculated using the measured shock wave energy and shock loss factor estimated from detonation pressure. Detonation pressure was calculated by the simple equation P = ρ0 D 2 /4, where ρ0 is the initial density of explosive and D is measured detonation velocity. Table 6.25 gives the summary of detonation pressure, shock loss factor, shock wave energy, dissipated energy, bubble energy, and total expansion work for sample EMXs. The dissipated energy is increased from 0.16 to 0.55 MJ/kg when detonation velocity is increased from 2520 to 5360 m/s. The total expansion work of sample EMXs is nearly equal to the detonation heat of 3.23 MJ/kg calculated by KHT thermochemical code. As detonation products expand adiabatically to the pressure close to atmospheric pressure in the underwater explosion of explosive, the total expansion work is nearly equal to the detonation heat with the difference corresponding to the internal energy of expanding detonation products. The addition of Al enhances various performances of explosives. However, the reaction of Al is relatively slow and is not completed during the detonation regime. The effects of the addition of Al cannot be evaluated by the measurements of detonation velocity and pressure. As sufficient confinement realized in the underwater explosion test enables Al reaction to complete in detonation products, the underwater explosion test can evaluate the effects of the addition of Al. The underwater explosion phenomena have been widely studied using various numerical simulation codes. Table 6.25 Summary of detonation pressure, shock loss factor, shock wave energy, dissipated energy, bubble energy, and total expansion work EMX

Detonation pressure (GPa)

Shock loss factor

Shock energy (MJ/kg)

Dissipated energy (MJ/kg)

Bubble energy (MJ/kg)

Total expan. work (MJ/kg)

EMX-1

7.90

1.7

0.79

0.55

1.77

3.11

EMX-2

6.02

1.6

0.80

0.48

1.89

3.17

EMX-3

4.85

1.5

0.84

0.42

1.96

3.22

EMX-4

3.95

1.4

0.75

0.30

1.99

3.04

EMX-5

2.08

1.25

0.80

0.20

2.02

3.02

EMX-6

1.75

1.2

0.80

0.16

2.04

3.00

260

A. Miyake et al.

Numerical simulation has succeeded to predict the shock wave and bubble energies for ideal explosives. However, numerical simulation cannot predict the shock wave and bubble energies for aluminized explosives because of lack of data on Al reaction. Actually, the underwater explosion test is sole method to evaluate the shock wave and bubble energies of aluminized explosives. Kato et al. [90, 91] measured shock wave and bubble energies as well as detonation velocity for aluminized EMXs (Al-EMXs) loaded with different size and amount of Al powder. The EMX was selected as sample explosive because its detonation products are composed of mainly H2 O, CO2 , N2 , gases and contain no carbon. Aluminum powder with three different particle sizes 5.7, 32, and 108 μm (named Al-1, Al-2, and Al-3) was used, and mass fraction of Al powder was increased up to 55 wt%. The composition of EM was AN/HN/water/oil and emulsifier = 74.6/10.6/10.6/4.2 in weight ratio. Base EMX was consisted of 98.96 wt% of EM and 1.04 wt% of PMB of 0.4–0.5 mm in diameter. The initial density of base EMX was 1.10 g/cm3 . Detonation velocity was measured using paper cartridge of 50 mm in diameter and 200 mm long by ionization probes. The experimental setup of the underwater explosion tests is shown in Fig. 6.55. Sample explosive and tourmaline pressure gauge were placed at the depth of 4 m, and distance between sample explosive and tourmaline pressure gauge was 3 m. Sample explosive was cylindrical shape of 50 mm in diameter, and charge weight was 0.25 kg. Shock wave and bubble energies were calculated by Eqs. 6.18 and 6.19 using measured shock wave profile and bubble period. Results of detonation velocity measurements are shown in Fig. 6.58. Detonation velocity of base EMX is 4200 m/s which is 1700 m/s lower than that calculated by KHT thermochemical code supposing EMX fully reacted at C-J state. The fraction of EMX reacted at C-J state is estimated to be about 0.7 for base EMX. For Al-EMXs containing Al-1, detonation velocity is slightly higher than that of base EMX when Al content is 9 wt%. It is considered that the effect of higher density increased the fraction of EMX reacted at C-J state and compensated the decrease of EMX content. Then detonation velocity of Al-EMX containing Al-1 is decreased with the increase of Al content. Detonation velocity of Al-EMX loaded with Al-3 is decreased linearly with the increase of Al content.

(a) Experimental arrangements

(b) Underwater explosion testing tank

(A) Charge, (B) Sensor, (Rs) Stand off distance

Fig. 6.55 Experimental setup for the underwater explosion tests

6 Ideal and Non-ideal Detonation Fig. 6.56 Correlation between shock wave energy and detonation velocity

Fig. 6.57 Correlation between bubble energy and detonation velocity

Fig. 6.58 Effects of Al mass fraction and particle size on detonation velocity of Al-EMXs

261

262

A. Miyake et al.

Fig. 6.59 Effects of Al mass fraction and particle size on shock wave energy of Al-EMXs

The effects of Al content and particle size on shock wave energy are presented in Fig. 6.59. The variation of shock wave energy with Al content is nearly identical for Al-EMX loaded with Al-1 and Al-2. Shock wave energy is maximum at Al content 33 wt%, which is 60% higher than that of base EMX. Then, it decreases rapidly with the increase of Al content. For Al-EMX containing Al-3, shock wave energy is maximum at Al content 33 wt%, but its value is only 15% higher than that of base EMX. According to the result of calculation by KHT thermochemical code, detonation products of 1 kg of base EMX are composed of 30.06 mol of H2 O, 3.47 mol of CO2 , 10.86 mol of N2 , and negligibly small quantity of other gases. Beckstead [92] presented the summary on the characteristics of Al reaction based on almost 400 data performed in various experimental conditions and stated that the burning time of Al with CO2 is twice as long as that with H2 O. If Al reacts primarily with H2 O, all H2 O is consumed to form Al2 O3 at Al content 35 wt%. The experimental results suggest that for Al-EMX loaded with Al-1 and Al-2, Al was fully reacted with H2 O up to Al content 33 wt.%, and unreacted Al remained at higher Al content in the timescale of shock wave (~400 μs). For Al-EMX containing Al-3, it was estimated that only about 50% of Al was reacted at Al content 33 wt% because of longer burning time of Al-3 particle. The effects of Al content and particle size on bubble energy are shown in Fig. 6.60. The variation of bubble energy with Al content is nearly identical for Al-EMX containing Al-1 and Al-2. Bubble energy of Al-EMX loaded with Al-3 is slightly lower. Bubble energy is increased up to Al content 50 wt% for all Al-EMXs. Bubble energy of Al-EMX containing Al-1 and Al-2 is 2.4 times higher than that of base EMX at Al content 50 wt.%. It is considered that Al is fully reacted in Al-EMX containing Al-1 and Al-2, and small amount of Al was remained unreacted in AlEMX loaded with Al-3. If Al is supposed to react with H2 O and CO2 , all H2 O and

6 Ideal and Non-ideal Detonation

263

Fig. 6.60 Effects of Al mass fraction and particle size on bubble energy of Al-EMXs

CO2 are consumed to form Al2 O3 at Al content 38 wt%. However, bubble energy is continued to increase up to Al content 50 wt%. This result suggests the reaction of Al with N2 to form AlN. The total expansion work of Al-EMXs was calculated using measured shock wave and bubble energy. The dissipated energy was calculated from measured shock wave energy, and shock loss factor was determined from detonation pressure. Detonation pressure of Al-EMXs was in the range of 4.8–5.6 GPa, and shock loss factor was fixed to be 1.5. In Fig. 6.61, the total expansion work of Al-EMX loaded with Al-1 and Al-2 is compared with the detonation heat calculated by KHT thermochemical code supposing Al fully reacted at C-J state. The total expansion work increases up to Al content 50 wt%. The detonation heat calculated by KHT code attains its maximum at Al content about 40 wt% because KHT code does not consider the reaction of Al with N2 , and detonation products contain unreacted Al at Al content higher than 40 wt%. For base EMX, total expansion work is nearly equal to detonation heat. For Al-EMXs, the difference between total expansion work and detonation heat increases with the increase of Al content. This result indicates that the temperature of detonation products is very high and the internal energy is still important at maximum bubble radius for aluminized explosives.

6.5 Non-ideal Detonation in Aluminized Explosives Reaction of Al with oxidizing gases produces a very large amount of energy comparing with the energy of detonation of explosives. For nearly 100 years, Al has been utilized in explosive formulations to enhance their performance in many

264

A. Miyake et al.

Fig. 6.61 Comparison of total expansion work with calculated detonation heat for Al-EMXs

practical applications. Large number of experimental and theoretical works have been carried out to study the effects of Al reaction in various stages of a detonation event; reaction in the zone close to detonation front (< μs); reaction in the post-detonation early expansion phase (1–20 μs); and late reaction to contribute to blast effects and underwater explosion performance (100’s μs–100’s ms). Aluminized explosives are typical non-ideal explosives with long sequential reaction zones that may depend on the amount and size of Al particle and other factors, which makes them difficult to model and interpret. There are a large number of studies devoted to the reaction of Al particle in low-pressure combustion experiments. However, still little is understood about the reaction kinetics of Al particle at high temperature and pressure conditions behind detonation front.

6.5.1 Non-ideal Detonation in Gelled Nitromethane/Aluminum Mixtures Because of the extensive research carried out on the detonation properties of pure nitromethane (NM), the heterogeneous mixtures of liquid NM and Al particle are particularly convenient systems to study the various effects of the addition of Al particle in detonation processes. Kato and Brochet [93, 94] conducted a series of experiments to determine the characteristics of aluminized NM comprised of Al particle of mean diameter 10 μm suspended in gelled NM, which is a mixture of

6 Ideal and Non-ideal Detonation

265

97 wt% commercial grade NM (purity 96%) and 3 wt% PMMA added as a gelling agent. They observed detonation propagation in NM and gelled NM/Al mixtures containing 5–15 wt% of Al particle using high-speed framing camera (106 frames/s, exposure time 0.3 μs) [93]. The sample mixtures were confined in a 100 mm long, rectangular (20 × 20 mm) tube composed of 1.5 mm thick upper and lower brass plates and 6 mm thick PMMA side plates. Figure 6.62 shows the photographic records of detonation waves in NM and gelled NM/Al mixtures observed through a side PMMA plate. For detonation in NM, only the narrow bright zone corresponding to the high-temperature zone very close to reaction zone is recorded because the sensitivity of photographic film is strongly dependent on the intensity of light. In the case of detonations in gelled NM/Al mixtures, very long bright zone is recorded because of high temperature of detonation products due to the exothermic Al particle reaction. Very fine regular crisscross pattern composed of dark lines and bright zones is characteristic of these records. Bright zones corresponding to high-temperature detonation products due to the Al reaction are observed only about 2 mm behind detonation front, which suggests that the reaction delay time of Al particle is shorter than 0.3 μs. These bright zones are observed until side PMMA plate loses its initial transparency, which indicates that the burn time of 10 μm size Al particle is much longer than several μs. Persson and Bjarnholt [95] succeeded to demonstrate the long-life failure and re-initiation process in NM detonation confined in glass tube using the open-shutter aquarium technique. For gelled NM/Al mixtures, it was identified that dark lines are traces of failure waves generated continuously at side PMMA plate and bright zones correspond to high-temperature zones due to Al reaction, and that Al reaction is initiated after the re-initiation of NM behind failure wave. The failure waves exist only at the perimeter of side PMMA plate. The regular repetition of failure and re-initiation process was revealed owning to Al reaction. In gelled NM/Al mixture loaded with 5 wt% of Al, the reaction induction time in shocked NM behind failure wave is longer, and failure wave can propagate over a longer distance before the re-initiation wave catchup the failure wave. Contrary, in gelled NM/Al mixture containing 15 wt% of Al, because of very short reaction induction time in shocked NM, the re-initiation wave catch-up the failure wave much sooner. These facts show that the increase of Al concentration increases the sensitivity of gelled NM/Al mixture and decreases the reaction induction time in shocked NM. Recently, particle velocity profiles in shock initiation process of gelled NM containing solid silica beads of 1–4 and 40 μm size with 6 wt% were measured by Dattelbaum et al. [96] using in situ electromagnetic gauges. They found a balance between thermal explosion-driven and hot spot-driven initiation depending on the size and number density of the particles. It was shown that at low number density of hot spots and low shock pressure, the initiation mechanism is dominated by thermal explosion, and that at high number density of hot spots and high shock pressure, hot spot-driven initiation becomes dominant. Kato and Brochet [93] measured detonation velocity of gelled NM/Al mixtures loaded with 5–15 wt% of Al particle of 10 μm size as well as gelled NM. Sample mixture was confined in brass tube (290 mm long, 0.5 mm thick) of different inner

266

A. Miyake et al.

Fig. 6.62 Photographic records of detonation waves in gelled NM/Al mixtures (106 frames/s, exposure time 0.3 μs); I-shaped mark in photograph shows the length of 10 mm printed on side PMMA plate

diameter (11, 13, 18, 25, and 31 mm) with 5 ionization probes accurately set inside polypropylene holders. The relation between detonation velocity and the inverse charge diameter is presented in Fig. 6.63. The results of detonation velocity measurements for gelled NM/Al mixtures loaded with 30 and 45 wt% of Al particle of 16 μm size confined in PVC tube (inner diameters 13, 16, 20, 31 mm, 250 mm long) are also shown in Fig. 6.63. The detonation velocities at infinite diameter and at 31 mm diameter are shown with the parameter A value in Table 6.26. The failure diameter of gelled NM in thin brass tube was 11 mm, and that of gelled NM/Al mixtures was smaller than 8 mm. The parameter A value for gelled NM was nearly identical to that for neat NM, and the parameter A value for gelled NM/Al mixtures was reduced to less than half of that for gelled NM. Engelke et al. [83] have given the experimental evidence that the ratio of A value for closely related pairs of explosives, fired in the same confinement, is a measure of the two explosive’s relative chemical reaction zone length. The reduction of A value for gelled NM/Al mixtures suggests the reduction of reaction zone length of NM in gelled NM/Al mixtures. Engelke [97] measured detonation velocity of gelled NM containing 6 wt% of silica beads of mean diameter 15 μm confined in Pyrex tube. The failure diameter of gelled NM/silica beads mixture was 9.6 mm, and that for neat NM was 16.2 mm. The A value for gelled NM/silica beads mixture was less than half of that for neat and gelled NM. The detonation velocity of gelled NM/Al mixtures was shown to decrease linearly with the increase of Al particle concentration up to 45 wt%. Hence, Al particles are effectively inert within the reaction zone of NM.

6 Ideal and Non-ideal Detonation

267

Fig. 6.63 Relation between detonation velocity and the inverse charge diameter for gelled NM and gelled NM/Al mixtures

Table 6.26 Detonation velocity Di , D(d = 31 mm) and parameter A value

Di (m/s)

D(d = 31 mm) (m/s)

A (mm)

gelled NM*

6290

6265

0.143

gelled NM/Al (95/5)*

6220

6206

0.064

gelled NM/Al (90/10)*

6163

6152

0.058

gelled NM/Al (85/15)*

6104

6091

0.046

gelled NM/Al (70/30)**

5940

5930

0.093

gelled NM/Al (55/45)**

5795

5784

0.101

*

in brass tube, ** in PVC tube

Kato and Brochet [94] carried out the temperature measurements of detonation in gelled NM/Al mixtures containing 5–40 wt% of Al particle of 10 μm size using timeresolved optical pyrometer (see Chapter 4). To simulate the behavior of inert Al, the detonation temperatures of gelled NM loaded with 15 and 30 wt% of lithium fluoride (LiF) particle of 44 μm size were also measured. LiF particle was used as an inert substitute for Al particle because the density and sock Hugoniot of LiF are similar to those of Al. Sample mixtures were confined in brass tube of 18 mm in inner diameter, 1 mm thick, and 100 mm long. The brass tube had 10 mm thick glass window on one end and solid booster explosive on the other end. As the glass window under shock loading is transparent until shattered by tension waves, the duration of temperature

268

A. Miyake et al.

measurements is about 2.3 μs in the case of glass window of 10 mm thick. The detonation temperatures of gelled NM/Al (85/15) mixture were measured by four wavelength pyrometer. Hence, the scatter between the brightness temperatures measured at four wavelength was less than 150 K, and most of temperature measurements were performed by monochromatic pyrometer (wavelength 0.657 μm). Because the emissivity of detonation front in NM is 0.97 at wavelength 0.65 μm [98], the measured brightness temperature of detonation front is nearly identical to C-J temperature. Typical records of detonation temperature measurements for gelled NM and gelled NM/Al (85/15) are presented in Fig. 6.64. For detonation in gelled NM, the record shows three phases: overdriven detonation after initiation (part A), steady-state detonation propagation (part B), and the expansion of detonation products in Taylor wave after the interaction with glass window (part C). As gelled NM is transparent, the radiation emitted from detonation front is recorded through unreacted gelled NM in parts A and B. After the interaction between detonation front and glass window, the radiation from detonation products is recorded in part C. Because gelled NM/Al mixtures are opaque, only the radiation emitted from detonation products is recorded after the interaction of detonation front with glass window. The records for gelled NM/Al mixtures show the increase of radiation which is characteristic of Al reaction in detonation products. Figure 6.65 presents the temperature profiles of detonation products in gelled NM and gelled NM/Al mixtures loaded with 5–40 wt% of Al particle of 10 μm size. For gelled NM, the temperature of detonation products decreases from 3450 K at detonation front to about 2400 K in 2 μs due to the pressure decay in Taylor wave. For gelled NM/Al mixtures loaded with 5–15 wt% of Al particle, the temperature of detonation products decreases during first 0.2–0.3 μs, and then it tends to increase Fig. 6.64 Typical records of temperature measurements for gelled NM and gelled NM/Al mixture (85/15)

6 Ideal and Non-ideal Detonation

269

due to the exothermic reaction of Al particle. This fact indicates that the reaction delay time of Al particle is shorter than 0.3 μs. Yoshinaka et al. [99] revealed by shock recovery experiments that thin oxide layer over Al particle was damaged and removed under shock loading and bare Al was exposed. The reaction of Al particle starts immediately when the surface temperature of Al particle arrives at melting point of Al. This fact supports the very short reaction delay time of Al particle observed in the photographic observations and temperature measurements. For gelled NM/Al mixtures containing 15–40 wt% of Al particle. The temperature of detonation products increases very rapidly during first 1 μs, and it attains its maximum temperature. Then it remains nearly constant during the next μs. The maximum temperature of detonation products increases with the increase of Al content up to 30 wt%, and then it decreases with the increase of Al content. Baudin et al. [100] carried out the measurements of detonation temperature in gelled NM/Al mixtures loaded with 20 and 40 wt% of Al particle of 5 μm size and obtained quite similar results. Kato et al. [101] compared the temperature of gelled NM/Al mixtures with that of gelled NM/LiF mixtures, respectively, containing 15 and 30 wt% of particulate to study the reaction of Al particle with detonation products of NM within the experimental time frame of 2 μs. Hence, gelled NM/LiF mixtures are semitransparent, the records of the temperature measurements are similar to those of neat NM gelled NM. The detonation front temperature of gelled NM/LiF mixture can be measured, which is considered to be nearly identical to CJ temperature. In Table 6.27, the measured detonation front temperatures of gelled NM/LiF mixtures as well as those of neat NM and gelled NM are presented and compared with CJ temperatures of NM/Al mixtures calculated using KHT thermochemical code. The measured detonation front temperature of gelled NM loaded with 30 wt% of LiF is only 100 K lower than that of gelled NM. Whereas, CJ temperature of NM loaded with 30 wt% of Al calculated

Fig. 6.65 Temperature profiles of detonation products in gelled NM and gelled NM/Al mixtures containing 5–40 wt% of Al particle

270

A. Miyake et al.

Table 6.27 Comparison between the measured detonation front temperatures of gelled NM/LiF mixtures and calculated C-J temperatures of NM/Al mixtures

Meas. detonat. front temp. (K)

NM

Gelled NM

3550

3450

NM

Gelled NM/LiF (85/15)

Gelled NM/LiF (70/30)

3440

3350

NM/Al(85/15)

NM/Al(70/30)

Calcul. CJ temp. (Al inert) (K)

3262

2926

2613

Calcul. CJ temp. (Al reacted) (K)

3262

4154

4982

supposing inert Al in thermal equilibrium with detonation products of NM is 649 K lower than that of NM. Baudin et al. [102] calculated CJ temperature of NM loaded with 37 wt% of inert Al in thermal non-equilibrium with detonation products of NM. It is shown that the temperature of detonation products of NM and inert Al particle are, respectively, 3454 and 1798 K. It is also shown that CJ temperature of NM is 3445 K. These results clearly indicate that Al particle is inert and in thermal non-equilibrium with detonation products of NM at CJ state (Table 6.27). The temperature profiles of detonation products in gelled NM/Al and gelled NM/LiF mixtures containing, respectively, 15 and 30 wt% of particulate are presented in Fig. 6.66. The temperatures of detonation front in gelled NM/Al mixtures are 300– 400 K lower than those of gelled NM/LiF mixtures during first 0.1–0.2 μs and do not represent the real temperature of detonation front because the unreacted Al particle partially shuts the radiation emitted by detonation front. The measured temperatures of detonation products also do not represent the real temperature before the initiation of Al reaction by the same reason. After the initiation of Al reaction, reaction products of Al such as Al2 O3 attain immediately thermal equilibrium with other detonation products, and the measured temperature represents the real temperature, even if the temperature of the unreacted core of Al particle is lower. The difference of temperatures of detonation products between gelled NM/Al and gelled NM/LiF mixtures loaded with the same amount of particulate represents the increment of temperature due to the exothermic reaction of Al in detonation products. Figure 6.67 shows the increment of temperatures due to Al reaction for gelled NM/Al mixtures loaded with 15 and 30 wt% of Al particle. At 2 μs behind detonation front, the increase of temperature due to Al reaction is about 900 and 1600 K for NM/Al mixtures, respectively, loaded with 15 and 30 wt% of Al particle. These results indicate that the majority of Al particle is reacted in first few μs. The rate of Al reaction is very fast during first μs, and then it is rapidly slowed down. This rapid slowdown of the Al reaction rate is caused by the pressure decrease, the decrease of oxidizer concentration, and the decrease of Al mass depletion rate. The pressure of detonation products is decreased by Taylor wave during 2 μs, and then the pressure decrease is accelerated by side rarefaction wave due to tube expansion after 2 μs. Large amount of oxidizer gas is consumed, and the concentration of oxidizer gas is decreased considerably during first 2 μs. The mass depletion rate of Al particle is proportional to square of Al particle diameter and radial burning rate, if the reaction

6 Ideal and Non-ideal Detonation

271

Fig. 6.66 Temperature profiles of detonation products in gelled NM/Al and gelled NM/LiF mixtures containing 15 and 30 wt% of particulate

of Al is surface reaction. The mass depletion rate is slowed down rapidly with the decrease of Al particle diameter. In consequence, the Al reaction takes longer time to complete in the expanding detonation products after 2 μs behind detonation front. This fact suggests the long total burning time of Al particle estimated from the results of cylinder tests. Lefrancois et al. [85] measured the detonation temperatures of EMX and aluminized EMX loaded with 14 wt% of fine Al particle (particle size was not specified) using large spectral range optical pyrometer (0.8–1.7 μm). The initial density of EMX and aluminized EMX were, respectively, 1.20 and 1.28 g/cm3 . Their detonation velocity was about 5000 m/s. During first 0.5 μs after the interaction of detonation front with PMMA window of 70 mm thick, the pyrometer detected high temperature of hot spots, and then it recorded the real temperature of detonation products. The detonation front temperature of EMX is estimated to be 2200–2500 K. The temperature of detonation products of aluminized EMX initially decreases and then begins Fig. 6.67 Temperature increase due to Al reaction for gelled NM/Al mixtures loaded with 15 and 30 wt% of Al particle

272

A. Miyake et al.

to increase due to Al reaction within less than 2 μs. It attains its maximum value at about 4 μs behind detonation front. These results show that the reaction delay time of Al particle in aluminized EMX is much longer than that in gelled NM/Al mixture (shorter than 0.3 μs) because of low temperature of detonation products of EMX (2200–2500 K) compared to high temperature of detonation products of NM (~3600 K). The temperature increase due to Al reaction is estimated to be about 800 K at 4 μs behind detonation front.

6.5.2 Non-ideal Detonation in Packed Aluminum Particles Saturated in Nitromethane There has been growing interest in detonation of heterogeneous system comprising a condensed phase explosive and a large quantity of reactive metal particle in an attempt to exploit the high energy content of the particles through their rapid reaction. A large number of experimental works and numerical simulations have been carried out to study the propagation mechanism of detonation in such system during last decades. The detonation in such system presents very complex interaction between detonation front and metal particles, and its propagation mechanism is not well understood. The heterogeneous system comprised of packed bed of Al particles saturated with neat NM (packed Al/NM) is convenient system to investigate the propagation mechanism of detonation in the explosives containing a large amount of Al particles. Kato et al. [103, 104] conducted a series of experimental work to study the characteristics of detonations in packed Al/NMs containing Al particle of seven different mean diameter from 5.7 to 300 μm. The properties of packed Al/NM mixtures composed of Al particle of 7 different sizes are shown in Table 6.28. Ripley et al. [105] gave the normalized interparticle spacings investigated with the corresponding Al volume fraction, mass fraction, and density of the mixture for packed Al/NM composed of spherical Al particles of uniform size. In the real packed Al/NM mixtures, the volume and mass fraction of Al particle is much less than that of ideal closely packed bed of Al particles because Al particle has a size distribution. Particularly, the volume and mass fraction of Al particle are smaller for Al particle smaller than 9.4 μm. Detonation velocity measurements were carried out for packed Al/NM mixtures confined in 250 mm long, 3.5 mm thick PVC tube of different inner diameters 13, 16, 20, and 31 mm using optical fibers placed at 50 mm interval. Figure 6.68 presents the relation between the failure diameter and Al particle size for packed Al/NM confined in PVC tube, which shows U-shaped curve. The failure diameter of neat NM confined in PVC tube is about 15 mm. The failure diameter is smaller than 13 mm for packed Al/NM mixtures composed of Al particle of 32 and 108 μm size. Contrary, the failure diameter is larger than 20 mm for packed Al/NM mixtures containing Al particle smaller than 9.4 μm and larger than 300 μm. Frost et al. [106] showed the similar U-shaped curve for packed bed of glass beads saturated with chemically sensitized

6 Ideal and Non-ideal Detonation Table 6.28 Properties of packed Al/NMs composed of Al particle of 7 different size

273

Mean diameter of Al (μm)

Packed Al/NM density (g/cm3 )

Mass fraction of Al (%)

Volume fraction of Al (%)

5.7

1.76

60

38

9.4

1.76

60

38

16

1.91

69

48

32

1.97

72

51

66

2.04

75

55

108

1.83

64

42

300

1.83

64

42

NM confined in glass tube. It is known that the reaction zone length of chemically sensitized NM is about half of that of neat NM. The failure diameter of packed bed of solid particles saturated with neat and sensitized NM does not depend on the reaction zone length of NM, and it is controlled by various parameters such as hot spot size and density, and desensitization due to the momentum and heat transfer. The relation between detonation velocity and the inverse charge diameter for packed Al/NM mixtures is shown in Fig. 6.69. Linear relation between detonation velocity and the inverse charge diameter was obtained only for packed Al/NM mixtures composed of Al particle of 32 and 66 μm. The detonation velocity of packed Al/NM mixture loaded with Al particle of 108 μm is lowered about 500 m/s when the charge diameter is decreased from 16 to 13 mm. The detonation velocity of packed Al/NM containing Al particle of 300 μm size is more than 500 m/s lower than that of other mixtures at charge diameter 31 mm, which suggests that the detonation in this mixture propagates in quite different mode. Figure 6.70 presents the variation of detonation velocity with Al mass fraction at charge diameter 31 mm for gelled NM/Al and packed Al/NM mixtures. Detonation velocity deficit is represented by single linear function of Al mass fraction independent of Al particle size. This fact Fig. 6.68 Relation between the failure diameter and Al particle size for packed Al/NM mixtures confined in PVC tube

274

A. Miyake et al.

indicates that Al particle is completely inert in the reaction zone of NM, and detonation velocity is controlled by chemical energy reduction and momentum and heat transfer. Detonation pressures in NM and packed Al/NM mixtures were measured using PVDF pressure gauge [103, 104]. The detail of PVDF pressure gauge is described in Chap. 4. The experimental setup for detonation pressure measurements is shown in Fig. 6.48. Sample explosive was confined in PVC tube of 31 mm inner diameter, 3.5 mm thick, and 150 mm long. As the PVDF pressure gauge measured the pressure transmitted into PMMA plate of 1 mm thick, detonation pressure was calculated using the impedance match method. Detonation velocity was measured by optical fibers to confirm steady detonation. Figure 6.71 presents the measured pressure profiles for detonations in NM and packed Al/NM mixtures composed of Al particles of different size. For detonation in NM, the measured pressure of leading shock wave is much lower than predicted von Neumann spike pressure, and the reaction zone of NM is not resolved because of very short reaction zone length. Sheffield et al. [107] measured the reaction zone length of NM to be 300 μm (~50 ns) using interferometric techniques. The pressure profile in NM shows the abrupt change of pressure gradient at CJ state and following pressure decay in Taylor wave. The pressure decrease owning to side rarefaction waves is observed at about 4 μs behind leading shock. In the case of detonations in packed Al/NM mixtures, the measured pressure profiles present strong dependence on Al particle size and show the extended high-pressure zone between leading shock and Taylor wave. The results of pressure measurements demonstrate the existence of the extended high-pressure zone due to the complex wave interactions between multiple shock waves in Al particles and detonation waves in NM in interstitial voids, and the strong dependence of pressure profile on Al particle size.

Fig. 6.69 Relation between detonation velocity and the inverse charge diameter for packed Al/NM mixtures

6 Ideal and Non-ideal Detonation

275

6200

6000

5800

5600

5400

0

20

40

60

80

100

Fig. 6.70 Variation of detonation velocity with Al mass fraction at charge diameter 31 mm for gelled NM/Al and packed Al/NM mixtures

Fig. 6.71 Measured pressure profiles for detonation in NM and packed Al/NM mixtures composed of Al particles of different size

Milne [108] and Zhang et al. [109] carried out two- and three-dimensional mesoscale simulations to study the shock and detonation front interaction with solid particles in liquid NM. Ripley et al. [105] conducted three-dimensional mesoscale simulations of detonation propagation in packed Al/NM mixtures under the various packing configurations, interparticle spacings and ratio of particle diameter to reaction zone length. The Al particles are assumed to be inert within the reaction zone of NM because of the greater timescale associated with Al particle ignition. The reaction zone length of neat NM and particle size is assumed to be 1 μm. The results

276

A. Miyake et al.

of mesoscale simulations demonstrate that the pressure of leading shock in NM in interstitial void is much higher than von Neumann pressure in neat NM, and it decays rapidly. The calculated pressure profiles in NM in interstitial void show the quasisteady high-pressure zone between leading shock and Taylor wave over a distance of 6 particle diameter, in which pressure oscillates around a mean value of approximately 17.5 GPa. The measured peak pressure is very close to the mean value of pressure oscillation in the quasi-steady high-pressure zone. It is also shown that the length of the quasi-steady high-pressure zone is much longer than the reaction zone of NM, and that it scales with Al particle size. The reaction zone length of NM in the void is decreased from 1 μm to about 0.3 μm because of the high temperature resulting from strong shock focusing and formation of hot spots in void. From these facts, the observed extended high-pressure zone corresponds to the quasi-steady high-pressure zone demonstrated by the results of mesoscale simulation. The pressure profiles of detonations in packed NM/Al mixtures containing Al particle smaller than 108 μm are similar except for the pressure increase due to Al reaction. The pressure increase due to Al reaction is observed in packed Al/NM mixtures containing Al particle smaller than 16 μm. The rate and amount of pressure increase due to Al reaction depends strongly on Al particle size. The pressure increase is observed to start at 2.0–3.5 μs behind leading shock in Taylor wave for packed Al/NM mixtures containing Al particle of 5.7, 9.4, and 16 μm size. The pressure increase due to Al reaction is not observed in packed Al/NM mixtures composed of Al particle larger than 32 μm in the time frame of pressure measurements. The pressure decrease owning to side rarefaction wave is observed at about 6 μs behind leading shock. The pressure profile of detonation in packed Al/NM mixture loaded with Al particle of 300 μm is completely different from those of other mixtures, which suggests that the detonation in this mixture propagates in quite different mode. To study the effects of Al mass fraction on pressure profile, the detonation pressure of gelled NM/Al loaded with 30 wt% of 5.7 μm size Al was measured. In Fig. 6.72, the measured pressure profile for detonation in gelled NM/Al containing 30 wt% of Al is compared with that for detonation in packed Al/NM composed of 60 wt% of Al of the same size. It is shown that the length of the extended high-pressure zone increases with the increase of Al mass fraction, and that the rate and extent of pressure increase depend strongly on Al mass fraction. The particle velocities of detonations in gelled NM/Al mixtures loaded with 5–15 wt% of 10 μm size Al particle were measured using electromagnetic particle velocity gauge (cited in [103]). The results of particle velocity measurements present the increase of particle velocity due to Al reaction at 0.5–1.2 μs behind leading shock. These results support the observed pressure increase due to Al particle reaction. Recently, Loiseau et al. [110] studied the flyer plate accelerating ability of gelled NM/Al mixtures loaded with 15 wt% of Al particle of mean diameter 3.5, 12, 55, and 108 μm. The prompt increase in flyer velocity over the baseline gelled NM indicated that Al reaction starts very close to CJ state. Ripley et al. [111] derived the model for momentum and heat exchange from the results of three-dimensional mesoscale simulation of detonations in packed Al/NM mixtures. In this model, the momentum and heat exchange are expressed in terms of

6 Ideal and Non-ideal Detonation

277

Fig. 6.72 Comparison of the measured pressure profile for detonation in gelled NM/Al containing 30 wt% of Al with that for detonation in packed Al/NM composed of 60 wt% of Al

velocity and temperature transmission factors, which are a function of the metal-toexplosive density ratio, solid volume fraction, and ratio of particle size to explosive reaction zone length. Ripley et al. [112] incorporated these models as source terms in the governing equations for continuum two-phase flow, and applied them to macroscopic simulation of detonation in packed Al/NM mixtures composed of Al particle of different size. The calculated detonation features such as velocity deficit, extended high-pressure zone, and critical diameter effects are compared with the experimental results. The reaction zone length in neat NM is assumed to be 300 μm. The pressure profiles calculated by macroscopic simulation are consistent with the experimental results obtained by Kato et al. [93, 94] and show the existence of the observed extended high-pressure zone. As the individual particle response is homogenized in macroscopic simulation, the pressure oscillation in quasi-steady zone is exhibited as the extended high-pressure zone observed in the experiments. Frost and Zhang [113] gave the review of the recent experimental works and numerical simulations on the detonations in heterogeneous mixtures consisted of liquid NM and solid particles.

6.5.3 Non-ideal Detonation in Aluminized Solid Explosives During last decades, the effects of Al particle reaction in the detonation wave and early post-detonation expansion of detonation products have been studied extensively by cylinder tests. The cylinder tests give a clear picture of how the post-detonation energy release due to Al reaction contributes to accelerate wall velocity on timescale from first 1 μs to cylinder breakup. Finger et al. [114] carried out the cylinder tests to investigate the rate and extent of Al reaction in the post-detonation expansion for explosive formulations composed of cyclotetramethylene-tetranitramine (HMX), binder and 4–19wt% of 5 μm size Al particle. They showed that about 50% of 5 μm size Al particle was reacted within 20 μs in all explosive mixtures. Recently, Manner et al. [115] conducted a series of cylinder tests with mixtures of HMX and Al or LiF. Sample explosive formulations are composed of HMX, binder and Al (69/16/15 wt%) (HMX/Al) and

278

A. Miyake et al.

HMX, binder and LiF (69/16/15 wt%) (HMX/LiF). Mean particle size of Al was 3.2 μm and particle size of LiF was smaller than 5 μm. The sample mixture was confined in copper tube of 12.7 mm inner diameter, 2.5 mm thick, and 127 mm long. Wall velocity was measured using photonic Doppler velocimetry (PDV) probes. The ratio of wall velocity for HMX/Al relative to that for HMX/LiF from 1 to 20 μs was compared. The wall velocity for HMX/Al and that for HMX/LiF are identical at 1 μs, and the wall velocity for HMX/Al begins to increase relative to that for HMX/LiF between 1 and 2 μs. The wall velocity for HMX/Al is 13% larger than that for HMX/LiF at 2 μs, and 20% larger at 7 μs (continuing to 20 μs). The ratio of wall velocity for HMX/Al relative to that for HMX/LiF stabilizes after 7 μs. These results suggest that the reaction of Al particle of 3.2 μm is completed within 20 μs. The ratio of HMX/Al velocity relative to HMX/LiF is considered to be a measure of the extent of energy liberated by Al particle reaction. These results suggest that the majority of Al reaction occur in first few μs. These results also support the results of the temperature measurements for gelled NM/Al mixtures [93, 94]. In most explosives, H2 O is prevalent oxidizer as product gases for post-detonation Al oxidation. To investigate the reaction of Al with CO2 in post-detonation expansion, Tappan et al. [116] carried out cylinder tests with mixtures of benzotrifluroxan (BTF) and 15 wt% of Al or LiF. BTF is a hydrogen-free explosive that selectively forms CO2 as major oxidizing species. They compared the wall velocity for BTF/Al with that for BTF/LiF. The results of cylinder tests showed that the wall velocity of BTF/Al exceeds that of BTF/LiF even as early as 1 μs, and that Al reaction is completed before cylinder breakup (