Shock Wave Science and Technology Reference Library
The new Springer collection, Shock Wave Science and Technology Reference Library, conceived in the style of the famous Handbuch der Physik has as its principal motivation to assemble authoritative, state-of-the-art, archival reference articles by leading scientists and engineers in the field of shock wave research and its applications. A numbered and bounded collection, this reference library will consist of specifically commissioned volumes with internationally renowned experts as editors and contributing authors. Each volume consists of a small collection of extensive, topical and independent surveys and reviews. Typical articles start at an elementary level that is accessible to non-specialists and beginners. The main part of the articles deals with the most recent advances in the field with focus on experiment, instrumentation, theory, and modeling. Finally, prospects and opportunities for new developments are examined. Last but not least, the authors offer expert advice and cautions that are valuable for both the novice and the well-seasoned specialist.
Shock Wave Science and Technology Reference Library
Collection Editors
Hans Gr¨onig Hans Gr¨onig is Professor emeritus at the Shock Wave Laboratory of RWTH Aachen University, Germany. He obtained his Dr. rer. nat. degree in Mechanical Engineering and then worked as postdoctoral fellow at GALCIT, Pasadena, for one year. For more than 50 years he has been engaged in many aspects of mainly experimental shock wave research including hypersonics, gaseous and dust detonations. For about 10 years he was Editor-in-Chief of the journal Shock Waves.
Yasuyuki Horie Professor Yasuyuki (Yuki) Horie is internationally recognized for his contributions in high-pressure shock compression of solids and energetic materials modeling. He is a co-chief editor of the Springer series on Shock Wave and High Pressure Phenomena and the Shock Wave Science and Technology Reference Library, and a Liaison editor of the journal Shock Waves. He is a Fellow of the American Physical Society, and Secretary of the International Institute of Shock Wave Research. His current interests include fundamental understanding of (a) the impact sensitivity of energetic solids and its relation to microstructure attributes such as particle size distribution and interface morphology, and (b) heterogeneous and nonequilibrium effects in shock compression of solids at the mesoscale.
Kazuyoshi Takayama Professor Kazuyoshi Takayama obtained his doctoral degree from Tohoku University in 1970 and was then appointed lecturer at the Institute of High Speed Mechanics, Tohoku University, promoted to associate professor in 1975 and to professor in 1986. He was appointed director of the Shock Wave Research Center at the Institute of High Speed Mechanics in 1988. The Institute of High Speed Mechanics was restructured as the Institute of Fluid Science in 1989. He retired in 2004 and became emeritus professor of Tohoku University. In 1990 he launched Shock Waves, an international journal, taking on the role of managing editor and in 2002 became editorin-chief. He was elected president of the Japan Society for Aeronautical and Space Sciences for one year in 2000 and was chairman of the Japanese Society of Shock Wave Research in 2000. He was appointed president of the International Shock Wave Institute in 2005. His research interests range from fundamental shock wave studies to the interdisciplinary application of shock wave research.
B.W. Asay (Ed.)
Shock Wave Science and Technology Reference Library, Vol. 5
Non-Shock Initiation of Explosives
With 298 Figures and 31 Tables
123
Blaine W. Asay Los Alamos National Laboratory Explosives Applications and Special Projects Los Alamos NM 87545 USA Email:
[email protected] Blaine W. Asay Blaine W. Asay is a Research Scientist and Group Leader in the Explosive Applications and Special Projects group at Los Alamos National Laboratory. His research interests include hypervelocity jet initiation, DDT, novel energetic materials, thermobarics, abnormal initiation, and thermal response. He has published over 50 papers in the refereed literature and presented numerous papers at international conferences, and also holds several patents. Dr. Asay received the Ph.D. in Chemical Engineering for his work in combustion from Brigham Young University in 1982 and worked at Kodak Research Laboratories before joining LANL in 1984.
ISBN 978-3-540-87952-7 e-ISBN 978-3-540-87953-4 DOI 10.1007/978-3-540-87953-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009933123 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: eStudio Calamar/Spain Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Los Alamos National Laboratory is an incredible place. It was conceived and born amidst the most desperate of circumstances. It attracted some of the most brilliant minds, the most innovative entrepreneurs, and the most creative tinkerers of that generation. Out of that milieu emerged physics and engineering that beforehand was either unimagined, or thought to be fantasy. One of the fields essentially invented during those years was the science of precision high explosives. Before 1942, explosives were used in munitions and commercial pursuits that demanded proper chemistry and confinement for the necessary effect, but little else. The needs and requirements of the Manhattan project were of a much more precise and specific nature. Spatial and temporal specifications were reduced from centimeters and milliseconds to micrometers and nanoseconds. New theory and computational tools were required along with a raft of new experimental techniques and novel ways of interpreting the results. Over the next 40 years, the emphasis was on higher energy in smaller packages, more precise initiation schemes, better and safer formulations, and greater accuracy in forecasting performance. Researchers from many institutions began working in the emerging and expanding field. In the midst of all of the work and progress in precision initiation and scientific study, in the early 1960s, papers began to appear detailing the first quantitative studies of the transition from deflagration to detonation (DDT), first in cast, then in pressed explosives, and finally in propellants. Clearly this phenomenon had been observed before, and was undoubtedly the cause of many accidents, but with the improved diagnostic techniques (e.g., imageintensified cameras, rapid oscilloscopes, etc.) workers were finally able to begin to probe this very difficult area of reactive material behavior. These formed the initial studies of what has come to be known as nonshock initiation. This behavior and the reasons it is so difficult to understand are explored in Chap. 1. If an explosive is going to react violently, an ignition event must occur at some location within the volume. That ignition will be at a single point, perhaps in a small portion of a crystal. Whatever its size, to propagate
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the reaction, mass, momentum, and energy must be conveyed (conducted, convected, radiated, or advected) to other locations on a timescale that promotes growth of temperature and pressure. Transport phenomena have been studied for hundreds of years, but in the context of nonshock initiation, we felt that a complete presentation was required to place the common terms into context. This forms the substance of Chap. 2 by Perry. Explosives are metastable molecules – they are always reacting, but the rate is extremely temperature sensitive. Depending on the material and temperature, the reaction rate can double with an increase of only a few degrees centigrade. The physics and chemistry governing this behavior are extremely complex. For example, the decomposition of HMX has been described with over 500 elementary kinetic steps, and it is known that this is not a complete treatment. Trying to use that kind of description to understand the behavior of a reactive system is impossible, even with the latest state-of-the-art computational capabilities. We require a more global approach to the entire decomposition process. The ability to do that requires enormous insight, a fundamental understanding of chemistry, and the capability of reducing processes to their most simple, yet complete, level. But no further. Henson has done just that in Chap. 3. Once initiated, an explosive grain, or hot spot, can either fail, survive for a short amount of time, or progress and grow. This is the problem of criticality, and without knowing how this process occurs, it is impossible to understand or predict the outcome of an initiation event. In Chap. 4, Hill discusses in detail the important issues surrounding the topic of criticality, upon what it depends and how it affects the outcome of an event. If a hotspot survives, the reaction may then proceed through the body of material. This process is called combustion, and involves the conversion of a solid material having chemical energy into gas at much higher temperature and pressure. The rate at which this occurs is governed by the kinetics, but the process is controlled by how the mass, momentum, and energy move. Combustion waves can progress at very slow speeds (fractions of a centimeter per second) or up to supersonic speeds in cracks. Clearly, the mechanism of combustion is central to the concept of nonshock initiation and is covered by Jackson in Chap. 5. HMX is a very powerful explosive, but at atmospheric pressure and no confinement, it burns very slowly and in a laminar fashion. However, allow the pressure to increase somewhat, and provide surface area and confinement, and that lazy laminar combustion wave can easily convert into a detonation wave with pressures of thousands of atmospheres. In Chap. 6, Parker and Rae discuss the damage mechanisms that create the surface area without which the transition to detonation cannot occur in secondary explosives. This is an area of study of relatively recent vintage and which some have been slow to acknowledge. However, our ability to directly observe reacting systems being damaged has increased substantially in the past 10 years, enabling these conclusions to be considered incontrovertible.
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Chapter 7 discusses the topic of cookoff, which comprises each of the topics discussed before. It is a general term that includes explosion as well as detonation. The chapter discusses the small and large-scale tests used to study the field of nonshock initiation and some of its important elements. The deflagration to detonation transition was alluded to in an earlier paragraph and is discussed in other portions of the text, but it needed a chapter of its own to be elucidated more fully. The earliest papers accurately described the overall behavior of explosives, but in many cases missed the quantitative response. In Chap. 8 McAfee provides a complete treatment of the DDT process incorporating all of the advances that have recently occurred, and unifies many of the observations that have been made over the past 50 years. It is possible to drop a relatively large piece of explosive from a substantial distance onto a very hard surface and have no reaction. However, take that same explosive part, and the same hard surface and add a few grains of sand, and an entirely different outcome obtains. The study of frictional initiation has a long history with many elegant experiments having been conducted. Dickson provides a well-reasoned and complete analysis of the topic in Chap. 9. Finally, in Chaps. 10 and 11, Kennedy covers the areas of initiation by shear and impact, followed by spark and laser. These topics are sometimes overlooked because they are not very well characterized, but the first three constitute primary sources of accidental nonshock initiation, and the last is a current topic of study for purposeful ignition of explosives. We hope that readers will see the organization of the text as a natural progression in the study of nonshock initiation. It is by no means an easy field of study, nor one in which all of the major questions have been worked out. Each of the authors of this volume is not only an expert in the field, but someone who has thought deeply about the topic. It took several years to formulate the concepts, reduce them to the written word, and then finally produce something worthy of publication. Each has been a great resource and an important and trusted colleague for many years. I am indebted to each one for his contribution to my understanding of the field. I also acknowledge the work of Phil Howe, without whose tireless and persistent support many of the experiments and analyses discussed in this text would not have occurred. He is an integral and crucial member of the team. I have been incredibly fortunate to have known some of the most influential men and women in the field of explosives research. At Los Alamos, I was instructed in DDT by Wayne Campbell, shock physics by Mac Walsh, detonation physics by Bill Davis and John Bdzil, detonation chemistry by Ray Engelke, propellant behavior and explosives safety by John Ramsay, shaped charge design by Bill Mautz, and the list goes on. I have also had the privilege of meeting pioneer researchers from throughout the world ranging from Kondrikov and Dremin from Russia to Price and Jacobs from White Oak. As funding and support for the kind of work we do erodes, the community is becoming ever smaller. I am grateful to have been involved during some of the high points.
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I want to thank my closest coworkers of many years, Laura Smilowitz, Bryan Henson, Steve Son, John McAfee, and Peter Dickson. The thousands of hours spent in discussion, writing, some arguing, and waiting for that elusive exotherm have been joyous. Finally, I acknowledge the inestimable influence of my four children who have each exceeded every expectation of which I could have ever conceived, and my wife and eternal companion, Patrice, who completes me. Los Alamos, NM, October 2009
Blaine W. Asay
Contents
1 Introduction Blaine W. Asay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What Are We Able to Predict? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 What Are We Unable to Predict? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Motivation for This Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Purpose of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Final Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Transport Phenomena for Nonshock Initiation Processes W. Lee Perry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 An Overview of Transport Theory in Explosives Problems . . . . . . . 2.2 The Origins of Transport Theory in the Field on Nonshock Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Physical Properties Used in Transport Theory Analysis . . . . . . . . . 2.4 Transport Theory Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Selected Examples of Transport Theory Applied to Explosive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Chemical Kinetics of Solid Thermal Explosions Bryan F. Henson and Laura B. Smilowitz . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Chemical Kinetics and Explosives Modeling . . . . . . . . . . . . 3.2.2 Types of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Modeling Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3 5 7 9 11 11 13 14 15 15 19 22 27 30 41 42 45 45 49 49 73 76
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Specific Example: A Global Chemistry Model for HMX . . . . . . . . . 76 3.3.1 HMX Thermal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.2 HMX Thermal Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.3 HMX Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.4 HMX Thermodynamics and Ignition . . . . . . . . . . . . . . . . . . . 103 3.3.5 Global Kinetic Models of PBX 9501 (HMX) . . . . . . . . . . . . 109 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4 Classical Theory of Thermal Criticality Larry G. Hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.1.1 Spontaneous Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.1.2 Thermal versus Physical Explosions . . . . . . . . . . . . . . . . . . . 134 4.1.3 The Three Fundamental Cookoff Safety Questions . . . . . . . 135 4.1.4 The Nature of Preignition and Post-Ignition Cookoff Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.1.5 Historical Development of Thermal Explosion Theory . . . . 137 4.1.6 Definition, Scope, and Modern Relevance of Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.1.7 Emphasis and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.1.8 Additional Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2 Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2.1 Homogeneous and Heterogeneous Chemical Reactions . . . . 142 4.2.2 Spatial Averaging Over Heterogeneous Solid-Phase Reactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2.3 The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.2.4 Arrhenius Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.2.5 Reaction Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.6 Reaction Rate Expressed as Mass Fraction . . . . . . . . . . . . . 147 4.2.7 Reaction Progress Function Behavior for Arbitrary Reaction Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.2.8 Zero-Order Reaction versus Negligible Depletion . . . . . . . . 149 4.2.9 Autocatalytic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2.10 Hot-Spot-Initiated Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.2.11 Reaction Progress Function for an Ideal Hot-Spot-Initiated Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2.12 Modeling Complex Thermal Decomposition Behavior . . . . 162 4.2.13 The Arrhenius Function and its Calibration . . . . . . . . . . . . 163 4.2.14 Activation Energy as a Measure of HE Sensitivity . . . . . . . 165 4.2.15 Thermal Decomposition Parameters for PBX 9501 . . . . . . . 169
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4.3
4.4
4.5
4.6
4.7
4.8
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Semenov Theory of Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.3.1 Volumetric Heat Generation and Surface Heat Loss . . . . . . 171 4.3.2 Heat Flux Balance and the Semenov Graphical Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.3.3 Semenov Criticality Condition . . . . . . . . . . . . . . . . . . . . . . . . 175 4.3.4 Explicit Solutions for the Critical Temperatures . . . . . . . . . 177 4.3.5 Semenov Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.3.6 Semenov Steady-State Solution Space . . . . . . . . . . . . . . . . . . 181 Semenov Criticality Theory with Reactant Depletion . . . . . . . . . . . . 184 4.4.1 The Path to Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.4.2 Generalized Semenov Critical State . . . . . . . . . . . . . . . . . . . . 186 4.4.3 Example: First-Order Reaction with Constant Heating Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.4.4 Behavior of Reaction Runaway Near the Explosion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.4.5 Effect of Reaction Energy Content on the Explosion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.4.6 Sensitivity to Arrhenius Parameters . . . . . . . . . . . . . . . . . . . 192 Reactive Heat Equation and the F-K Approximation . . . . . . . . . . . 192 4.5.1 Heat Equation with Volumetric Reaction . . . . . . . . . . . . . . . 193 4.5.2 F-K Dimensionless Temperature . . . . . . . . . . . . . . . . . . . . . . . 194 Semenov Theory in the F-K Limit: The “SFK” Model . . . . . . . . . . 197 4.6.1 Heat Flux Balance, Graphical Construction, Critical Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.6.2 Criticality Expressed by the Semenov Number . . . . . . . . . . 198 4.6.3 Explicit Solutions for the Critical Temperatures . . . . . . . . . 199 Basic Frank-Kamenetskii Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.7.1 General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.7.2 Critical Temperature and Critical Size . . . . . . . . . . . . . . . . . 201 4.7.3 Critical Condition for the Infinite Slab . . . . . . . . . . . . . . . . . 202 4.7.4 Critical Condition for the Infinite Cylinder . . . . . . . . . . . . . 204 4.7.5 Critical Condition for the Sphere . . . . . . . . . . . . . . . . . . . . . . 205 4.7.6 Critical Temperature Profiles for the Canonical Shapes . . . 207 4.7.7 Relative Critical Temperatures for the Canonical Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.7.8 F-K Critical State for General Shapes: Square Bar Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 F-K Criticality Correlations for General Shapes . . . . . . . . . . . . . . . . 213 4.8.1 Correlation for Bodies of Near-Unity Aspect Ratio . . . . . . . 214 4.8.2 Comparison of F kc Values Obtained by Various Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.8.3 Effective Dimensionality of Near-Unity-Aspect-Ratio Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
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4.8.4
Breakdown of the Shape Correlation for Elongated Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.8.5 On Extending the Shape Correlation to Elongated Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.9 F-K Theory with Generalized Boundary Conditions . . . . . . . . . . . . . 224 4.9.1 Thermal Conductance/Resistance and the Biot Number . . 224 4.9.2 Thomas’ Generalized Boundary Condition . . . . . . . . . . . . . . 225 4.9.3 Infinite Slab: Critical Condition versus Biot Number . . . . . 226 4.9.4 Infinite Cylinder: Critical Condition versus Biot Number . 228 4.9.5 Sphere: Critical Condition versus Biot Number . . . . . . . . . . 230 4.9.6 Critical State versus Biot Number for the Canonical Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.9.7 Quasi-Self-Similar Scaled Criticality Curves vs. Biot Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 4.10 Convergence of the Critical State to the F-K Limit . . . . . . . . . . . . . 235 4.10.1 Canonical Shape Criticality for the Full Arrhenius Rate . . 235 4.10.2 Quasi-Self-Similar Scaled Criticality Curves vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 4.10.3 Critical State Correction for Finite Activation Energy . . . . 237 4.11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 5 Deflagration Phenomena in Energetic Materials: An Overview Scott I. Jackson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5.2 Combustion and Its Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.3 The Conductive Deflagration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.3.1 A Description of the Conductive Flame Structure . . . . . . . 250 5.3.2 Factors Affecting the Burn Rate . . . . . . . . . . . . . . . . . . . . . . . 252 5.3.3 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.3.4 Pressure Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.3.5 Quenching and Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5.3.6 Burning Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 5.4 Convective Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5.4.1 Gas Flow into Porosity and Isolated Voids . . . . . . . . . . . . . 264 5.4.2 Ignition of Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 5.4.3 The Influence of a Melt Layer on the Critical Pressure . . . 272 5.5 Progression of Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5.5.1 Across Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 5.5.2 Spreading of Convective Combustion . . . . . . . . . . . . . . . . . . . 275 5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
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6 Mechanical and Thermal Damage Gary R. Parker and Philip J. Rae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.2 Classification of Pertinent Morphologies . . . . . . . . . . . . . . . . . . . . . . . 294 6.2.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.2.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.3 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.3.1 Issues of Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.3.2 The Representative Elementary Volume . . . . . . . . . . . . . . . . 298 6.3.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 6.4 Pristine Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6.4.1 Energetic Material Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6.4.2 Examination of Defects in Pure Materials . . . . . . . . . . . . . . 310 6.4.3 Examination of Defects in Pressed Materials . . . . . . . . . . . . 316 6.4.4 Examination of Defects in Melt-Cast Materials . . . . . . . . . . 317 6.5 Insult and Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.5.1 Damage From Mechanical Insult . . . . . . . . . . . . . . . . . . . . . . 319 6.5.2 Damage From Thermal Insult . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.6 The Effect of Pore Morphology on Fluid Transport in Damaged EM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 6.6.1 Fundamentals of Fluid Transport . . . . . . . . . . . . . . . . . . . . . . 352 6.6.2 Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 6.6.3 Experiments with EM Measuring Gas Transport in the Preignition Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 6.6.4 Experiments with EM Measuring Gas Transport in the Post-ignition Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 6.7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 7 Cookoff Blaine W. Asay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 7.1 Concept of Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 7.1.1 Semenov’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 7.1.2 Frank-Kamenetskii’s Approach . . . . . . . . . . . . . . . . . . . . . . . . 412 7.1.3 Critical Temperature Determination . . . . . . . . . . . . . . . . . . . 413 7.2 Visual Observation of the Cookoff Event . . . . . . . . . . . . . . . . . . . . . . 418 7.2.1 Mechanically Confined Cookoff Tests . . . . . . . . . . . . . . . . . . 418 7.2.2 Cindy Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 7.3 Small-Scale Instrumented Thermal Ignition Experiments . . . . . . . . 431 7.4 Quantification of Reaction Violence . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 7.4.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 7.4.2 A First Law Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 7.4.3 Violence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 7.4.4 Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
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7.4.5 Violence and P 2 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 7.4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 7.5 Large, Integrated Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 7.5.1 LLNL Large-Scale Cookoff Test . . . . . . . . . . . . . . . . . . . . . . . 460 7.5.2 Scaled Thermal Explosion Experiment (STEX) . . . . . . . . . 460 7.5.3 Large-Scale Annular Cookoff . . . . . . . . . . . . . . . . . . . . . . . . . . 461 7.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 7.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 7.5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 7.6 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 8 The Deflagration-to-Detonation Transition John Michael McAfee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 8.2 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 8.2.1 Solid DDT Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 8.2.2 Solid DDT Experimental Evidence . . . . . . . . . . . . . . . . . . . . 487 8.3 Type I DDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 8.3.1 The Phenomenology of Type I DDT . . . . . . . . . . . . . . . . . . . 489 8.3.2 Experimental Evidence for Type I DDT . . . . . . . . . . . . . . . . 494 8.4 Type II DDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 8.4.1 Type II DDT Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . 508 8.4.2 Experimental Evidence for Type II DDT . . . . . . . . . . . . . . . 511 8.4.3 General Discussion of Type II DDT . . . . . . . . . . . . . . . . . . . 516 8.5 The Change Between Type I and Type II DDT . . . . . . . . . . . . . . . . 518 8.5.1 Experimental Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 8.5.2 Burn Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 8.5.3 Porosity, Permeability, and Particle Size Distribution . . . . 523 8.5.4 Particle Surface Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . 527 8.5.5 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 8.5.6 Igniter Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 8.5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 8.6 DDT in Gases, Vapors, and Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 8.6.1 Gases and Vapors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 8.6.2 Dusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 9 Friction Peter Dickson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 9.1 Introduction to Frictional Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 9.2 Significant Accidents Probably Involving Frictional Processes . . . . 537 9.3 Classical Friction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 9.4 Other Factors Affecting Surface Temperatures . . . . . . . . . . . . . . . . . 540 9.5 Effects of Particulate Contaminants . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 9.6 Frictional Processes Involved in Heating and Ignition of Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
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9.7 9.8 9.9
Friction Testing Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Powder Explosive Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Pressed/Cast Explosive Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 9.9.1 Oblique Impact Testing Revisited . . . . . . . . . . . . . . . . . . . . . 545 9.9.2 LANL Friction Test Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 548 9.10 Drop-Weight Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 9.11 Analysis and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
10 Impact and Shear Ignition By Nonshock Mechanisms James E. Kennedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 10.1 Effects of Shear in Material-Flow Processes . . . . . . . . . . . . . . . . . . . . 555 10.2 Drop-Weight Impact Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 10.2.1 Cavendish Laboratory Glass-Anvil . . . . . . . . . . . . . . . . . . . . . 559 10.2.2 Temperature and Flow Sensors . . . . . . . . . . . . . . . . . . . . . . . . 561 10.3 Large-Scale Crushing Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 10.3.1 IITRI Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 10.3.2 Susan Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 10.3.3 U.S.S. Iowa Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 10.3.4 XDT Response Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 10.3.5 Two-Step Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 10.4 Penetrating Impact of Cased Explosives . . . . . . . . . . . . . . . . . . . . . . . 571 10.5 Shear in Confined Charge Assemblies . . . . . . . . . . . . . . . . . . . . . . . . . 572 10.6 Mechanical Properties of Energetic Materials Relevant to Shear Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 10.6.1 Properties of Unreacted Energetic Materials . . . . . . . . . . . . 575 10.6.2 Mechanical Property Effects on Ignition . . . . . . . . . . . . . . . . 575 10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 11 Spark and Laser Ignition James E. Kennedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11.2 Spark Detonator: A Vehicle for Study of Loose Powder Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 11.3 Initiation Behavior of Secondary Explosive Spark Detonators . . . . 586 11.4 Similarities Among EBW, Laser, and Spark Initiation of Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 11.4.1 EBW Detonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 11.4.2 Laser Detonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 11.5 Sparks as Sources for Ignition of Explosive Powders . . . . . . . . . . . . . 591 11.5.1 Charge Generation and Discharge . . . . . . . . . . . . . . . . . . . . . 591 11.5.2 Electrostatic Discharge (ESD) Sensitivity Testing of Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 11.5.3 Discharges from Human Body or Handling Equipment . . . 595
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11.6 Electrical Discharges Through High-Density Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 11.6.1 Conclusions Concerning High-Density Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 11.7 Other Laser Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
List of Contributors
Blaine W. Asay Los Alamos National Laboratory Los Alamos, NM, 87545
[email protected] Peter M. Dickson Los Alamos National Laboratory Los Alamos, NM, 87545
[email protected] Bryan F. Henson Los Alamos National Laboratory Los Alamos, NM 87545
[email protected] Larry G. Hill Los Alamos National Laboratory Los Alamos, NM, 87545 USA
[email protected] Scott I. Jackson Shock and Detonation Physics Group Dynamic and Energetic Materials Division Los Alamos National Laboratory Los Alamos, NM, 87545 USA
[email protected] James E. Kennedy Hazards and Explosives Research and Education LLC 2710 Via Caballero del sur Santa Fe, NM, 87505 USA
[email protected] John M. McAfee Los Alamos National Laboratory Los Alamos, NM, 87545
[email protected] Gary R. Parker Los Alamos National Laboratory Los Alamos, NM, 87545 USA
[email protected] William Lee Perry Los Alamos National Laboratory Los Alamos, NM, 7545
[email protected] Philip J. Rae Los Alamos National Laboratory Los Alamos, NM, 87545 USA
[email protected] Laura B. Smilowitz Los Alamos National Laboratory Los Alamos, NM, 87545
[email protected] 1 Introduction Blaine W. Asay
Accident 1 On 14 February 2005, two highly trained technicians who had helped write the standard operating procedures were transferring triethoxy-trinitrobenzene (TETNB) from a filter tray into 5 gallon buckets. TETNB is a precursor in the manufacture of triamino-trinitrobenzene (TATB), and each of these compounds is considered to be extremely insensitive explosives as they are very difficult to ignite with shock, friction, or electrostatic discharge. During the transfer, an explosion occurred, killing one of the technicians and critically injuring the other. The technicians were investigating a new approach for synthesizing TATB using a more environmentally friendly method. They had already successfully demonstrated the new technique in 23 kg batches and were scaling up to a 230 kg batch. During the transfer of the TETNB, they encountered an unusual condition, in that the material was caked hard to the bottom and the sides of the filter tray. They proceeded to remove the caked material with a metal scoop and a plastic shovel, whereupon the material exploded. Subsequent investigation revealed that a precursor to TETNB, TNPG, which is extremely impact sensitive, had segregated during the wash step and formed the hard cake. So while the workers thought that they were dealing with TETNB, they were actually forcefully removing TNPG with a metal tool. Accident 2 On 10 August 2004, a truck carrying 17,000 kg of explosives jack-knifed negotiating a tight turn while driving through a canyon. The truck skidded across the highway and started a fire on the mountainside. The fire reached the explosive load, and 3 min later, exploded, leaving a crater 9 m deep and 21 m wide, propelling concrete barriers into the river a 100 m away.
B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 1, c Springer-Verlag Berlin Heidelberg 2010
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Accident 3 On 16 April 1947, the French liberty ship Grandcamp was being loaded with the remainder of its cargo of ammonium nitrate fertilizer. Twenty three hundred tons were already on board. The longshoremen in the hold receiving the 45 kg pallets noticed a small wisp of smoke coming up through the bags. They estimated that the smoke originated about five layers down. Attempts to quench the fire with small quantities of water were unsuccessful. The ship’s captain did not want to use large amounts of water as it would ruin the cargo, so they closed all the hatches and turned on the steam. However, the fire continued and the local fire department was called. The pressure from the steam blew off the hatch covers and large amounts of orange smoke filled the sky. Onlookers began to gather to watch the spectacle. An hour had now passed, but the fire continued to grow. A few minutes later, an enormous explosion occurred that was heard 150 miles away, with a cloud erupting to over 600 m. Two airplanes were knocked out of the sky and ship fragments weighing several tons were found in the surrounding countryside. Hours later, another liberty ship, the High Flyer, containing sulfur and a thousand tons of ammonium nitrate was observed to be on fire. Around 1:00 AM the following morning, the High Flyer also detonated with a blast that was reported to have been even greater than when the Grandcamp was destroyed. The resulting fires burned for days until they were finally extinguished. In all, nearly 600 people perished, with another 3,500 seriously injured. Accident 4 During the development of a new warhead, a series of experiments were conducted to ensure that it would pass the spigot test. This involved dropping a weighted protrusion representing a cleat or other object present during the transfer of warheads between ships, or from ship to shore, onto a warhead. All previous warheads tested had shown a direct relationship between drop height and weapon response. That is, increasing the drop height increased the likelihood of initiation of the charge. During one such test series with the new warhead, it was shown to ignite and burn nonviolently when dropped from 10 m. When the final demonstration was scheduled, the decision was made to reduce the drop distance to ensure no reaction. When dropped from 1.75 m, the charge detonated. No injuries were reported, but significant concern over the ability of the test to predict weapon response resulted. Accident 5 While developing a new super-thermite comprised of 80 nm aluminum formulated in a tetrafluoroethylene-hexafluoropropylene-vinylidene fluoride (THV) polymer, researchers applied the nanocomposite formulation to a fiberglass e-glass substrate, baked off the acetone solvent, and then cooled the 25 cm2
1 Introduction
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sample. The Al/THV mixture had passed all the conventional safety tests including impact, spark, and friction, showing it to be insensitive. The e-glass is inert and showed no chemical incompatibility with the Al/THV. After the sample was cooled, one individual examined the material with bare hands before giving the sample to a coworker to examine. The coworker, wearing nitrile gloves, then manipulated the sample, observing the coating uniformity and other mechanical properties. Approximately 30 s later, the material spontaneously ignited, resulting in second-degree burns to the coworker. These are five examples drawn from thousands that could have been cited. The causes of each are now well understood, but as with many accidents, the response to the stimulus could not have been predicted beforehand. We could continue for many volumes detailing explosives accidents over the past 150 years. In fact, texts have been written doing just that. The frequency of large accidents like those on the USS Forrestal (nine 1,000 lb bombs, 132 perished), Oppau Germany (4,500 tonnes of ammonium sulfate and ammonium nitrate, 500–600 perished), Hanbury England (3,500 tons of bombs, 68 perished) has been greatly reduced because of increased understanding and safer practices. However, accidents continue to occur on a daily basis. Most, if not all, are the result of a phenomenon known as nonshock initiation, the topic of this text.
1.1 Background Defining what constitutes an explosive can be difficult. The DOE Explosives Safety Manual defines an explosive as any chemical compound or mechanical mixture which is designed to function as an explosive, or chemical compound which functions through self-reaction as an explosive, and which, when subjected to heat, impact, friction, shock, or other suitable initiation stimulus, undergoes a very rapid chemical change with the evolution of large volumes of highly heated gases that exert pressures in the surrounding medium. The term applies to materials that either detonate or deflagrate. Explosives can be gas, liquid, or solid. They can be physical mixtures of fuel and oxidizers (e.g., ammonium nitrate and fuel oil) or have the fuel and oxidizer in the same molecule (e.g., RDX). Explosives can be classified as either high or low (e.g., nitroglycerin vs. black powder), primary or secondary (e.g., lead azide vs. HMX), sensitive or insensitive (e.g., CL20 vs. TATB). Their classification can change with the physical state. For example, for shipping purposes, dry HMX is considered a primary and wet HMX is a secondary explosive. PETN can be considered either sensitive or insensitive depending on the particle size. Explosives can be formulated with a binder or used neat (e.g., PBX 9501 vs. TNT), they can be cast or pressed to shape (e.g., Comp B vs. PBX 9404), they can have additives that either increase or decrease the sensitivity (e.g., microballoons vs. wax). Addition of approximately 0.1% carbon
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black can change the spark sensitivity of an explosive from extreme to zero. An explosive component that is quickly heated to 500◦ C may be totally safe and nonviolent while slowly heating the same charge to a much lower temperature might result in a detonation and much damage. Dropping an explosive component from 10 m may result in a benign reaction while dropping the same charge from 1/2 the height might result in a detonation. These facts are cited to illustrate that explosive behavior is complicated. There are a few general rules of thumb that can be quite useful, but for every rule there are a number of exceptions. Having said this, it is possible to get a fundamental and useful understanding of behavior in a relatively short period of time. That is the purpose of this text. Many people think that explosives are dangerous because they have an enormous amount of energy available. While it is true that they can contain considerable energy, when compared to well known non-detonable organic compounds, we see that the total amount is not remarkable. For example, the heat of combustion of kerosene is 43 MJ kg−1 , while HMX, an extremely powerful military explosive, has only 9.9 MJ kg−1 . The difference is attributed to the fact that the oxygen required for detonation is contained in the HMX molecule, while that used by the kerosene is taken from the air. If the oxygen required for complete combustion of the kerosene is included, its energy is reduced to approximately the same as HMX. The relatively weak explosive ammonium perchlorate (AP) has 12.1 MJ kg−1 while lead azide, the very sensitive primary explosive used in detonators, has 2.6 MJ kg−1 . So if it is not the total amount of energy that makes an explosive useful, and in some cases dangerous, what is it? The answer is the rate at which the available energy is released. Typical combustion velocities are in the order of millimeter to centimeter per second. Detonation velocities in condensed phase materials range from 2 to 9 km s−1 . So even a relatively poor explosive releases its energy orders of magnitude faster than nondetonable compounds having relatively high amounts of energy. For example, to provide a context for the numbers that will be used, the following list illustrates the rate of energy release for common processes: – – – – –
Acetylene/air flame: 100 W cm−3 Deflagrating gun propellant: 1 MW cm−3 Detonating explosive: 100 GW cm−3 US generation capacity in 2004 was 1.05 TW Average hurricane represents 1.7 TW
In commercial operations in 2005, according to the International Society of Explosives Engineers, six billion pounds of explosives and 75 million detonators were used in the US and Canada alone. This does not take into account the military explosives or the chemicals with explosive properties that are used to make nonexplosive materials. Two-thirds of these explosives are used in coal mining, and of that, more than 80% is ANFO. These are extremely large numbers. Given the large amount of explosives used, it is remarkable that we do not experience more accidents. This is due in part to the continuous
1 Introduction
5
improvement in handling and processing practices. It is not necessarily due to increased fundamental understanding of the physics and chemistry involved in explosive behavior. Explosives have been studied for nearly 1,000 years, and even longer if Greek Fire is considered to be an explosive. The Chinese described black powder in 1040 ad, and from that point forward, scientific investigations have been carried out to improve the performance, efficiency, and safety of explosives. These investigations began with attempts to purify one of the main ingredients, saltpeter, when it was recognized to have a major effect on behavior. This was followed by modification of the grain structure through corning and moderation, which led to more reproducible burning, elimination of grain disruption, and its attendant uncontrolled increase in surface area and resultant explosion. The destruction of guns and cannon by this explosion process was most likely the earliest observation of the early stages of a process we now call deflagration to detonation transition (DDT). The behavior and performance of black powder were more or less fully understood by the late seventeenth and early eighteenth centuries. Other molecules and mixtures were soon sought out because of the relatively low pressure of black powder, and its inability to effectively fragment granite in mining and tunneling operations. Nitroglycerin, nitrocellulose, picric acid, tetryl, TNT, RDX, and HMX, and many others followed as understanding of chemistry and more specialized needs developed. Many book volumes and literally thousands of research papers have been written on the history, performance, physics, chemistry, and safety of explosives. Most of the early work concentrated on gaseous detonations and combustion. Later came treatises on solid explosives. These include recent texts by Zukas and Walters, Cooper, Davis, Cherest, and others. Some of the most brilliant scientists in the world have made explosives the subject of study. Given the lengthy history and the intellectual effort expended, one would think that prediction of explosives’ response to abnormal events would be a relatively simple exercise, and that certainly another text on the topic would not be needed. What is the problem? The problem is that, even with all of the study and the thousands of years of experience, we still cannot predict with any precision, in general, what will happen to an explosive if we hit it, heat it, drag it, drop it, or do anything else outside of its design envelope.
1.2 What Are We Able to Predict? We actually do know quite a bit about HE response in certain cases. Response is predictable in cases with the following: 1. Strong, planar shocks in well-characterized materials 2. HE that is either extremely sensitive (e.g., NI3 ) or extremely insensitive (e.g., TATB) 3. Large amounts of data in specific systems with specific threats (e.g., fragment impact correlations)
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Sample Detonation Run Distance
Booster
Fig. 1.1. Illustration of wedge test configuration and analysis used for strong shock initiation
Let us address each case individually Case (1). In the case of strong planar shock waves, data from an experiment called the wedge test are used to construct a pressure vs. rundistance plot (Pop plot [1]) as shown in Fig. 1.1. In the wedge test, a planar shock wave is driven into a carefully characterized explosive and the distance required to achieve a steady detonation is recorded. Results from a series of experiments are then compiled and plotted as shown. Different explosives have different slopes and positions. Thus as shown in Fig. 1.1, explosive A will transit to a detonation in a shorter distance than explosive B when presented with a shock of the same strength. These results are functions of temperature, particle size, density, and other parameters. However, once obtained for a specific explosive, the data can be used to make accurate predictions of system response, as long as strong planar shocks are used. Case (2). There are many materials that are either so sensitive or so insensitive that their response to any stimulus is assured, or where the probability of a given event is so great that it does not matter. For example, NI3 will explode at the slightest perturbation. The mere act of touching it, or cooling a crystal, is enough to make it react. On the other hand, TATB and similar highly insensitive explosives require extremely strong shock waves to begin initiation of a detonation wave, and they have never been observed to undergo deflagration-to-detonation transition. For this case, we are able to make predictions of response based on the known extreme sensitivity or insensitivity of the compound. Case (3). In this case, a specific explosive with specific characteristics (e.g., density, particle size distribution, etc.) is tested in a specific environment to measure the response to a specific stimulus, and correlations are compiled. For example, the response of TNT to fragment impact has been measured over a range of fragment velocities and sizes. As long as one interpolates within the acquired data set, relatively precise predictions can be made. However, extrapolation outside the data is not reliable and can oftentimes lead to extremely unwelcome miscalculations. In addition, these types of correlations typically
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yield relatively imprecise predictions of response. For example, a designation of a violent response might be something that ranges from a mild explosion to a full-scale detonation. The effect of either of these outcomes on surrounding structure or personnel might be the same. However, the outcome could arise from completely different mechanisms. Thus the correlation is not useful for deriving an understanding of the behavior, but merely assessing the level of risk involved in a specific application.
1.3 What Are We Unable to Predict? Unfortunately, the cases described above do not constitute a large fraction of those in which accidents happen (e.g., Trident D5 motor detonation, USS Forrestal, Texas City, etc.). Bulk explosives by their nature are designed to be initiated by strong shock waves originating from a detonator. Accidents typically occur when an explosive is subjected to an unplanned stimulus. These stimuli are oftentimes relatively low rate (when compared to shock waves), and relatively low pressure (when compared to the pressure encountered from a detonator output). In fact, accidents involving unplanned thermal ignition occur at atmospheric pressure. In addition, the material is often damaged by the event; that is, it contains cracks or pores that are not part of the original design for the component. Because accidents do not typically occur in the restricted situations described by one of the three cases cited above, our predictive capability is negligible. The major reasons for this observation are as follows: 1. There is no corollary to the Pop Plot for weak shock or low-strain-rate processes 2. Many explosives are of intermediate sensitivity and their response to a given input depends on material and confinement properties, temperaturedependent heat transfer and kinetic parameters, and other complications 3. Sufficient data to produce correlations for even a small fraction of potential accident scenarios have not been acquired, even though many explosives have been subjected to a range of safety tests For example, if an explosive for which a basic set of kinetic data is available is heated, we can predict with fair accuracy when an event will occur, but we cannot say what the event will be (e.g., how violent), or specifically where in the explosive it will begin. Given ignition of an explosive, we cannot predict reaction rates, how the combustion wave will build (or if it will build), or if it will transition to a detonation or merely continue as a low-order deflagration wave. Under mechanical insult, if mechanical and material data are available, we can calculate deformation and heat localization, but we cannot predict how violent the response will be. The reasons that we cannot make accurate predictions are that the problems are (1) multiscale both temporally (microseconds to seconds) and
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spatially (microns to tens-of-centimeters), (2) multidimensional, (3) almost always initially subsonic with the resultant complication that boundaries and surrounding material have a large effect on the outcome, and (4) dependent on the mechanical properties of HE (these may actually dominate). In addition, (5) extremely small amounts of chemical reaction can affect behavior. For example, a chemical energy release of less than 1% of that contained in an initially nonreactive shock wave may be sufficient to build the shock to detonation. Finally, (6) if all we needed to know was whether or not a classic detonation would result, the problem would be almost tractable. However, when the energy is delivered on a relatively short time scale, even modest amounts of reaction can result in disastrous consequences. One example of the complexity of explosive behavior is illustrated in Figs. 1.2 and 1.3, which shows a Viper shaped charge penetrating the liquid explosive nitromethane. Figure 1.2 shows the structure during penetration while the jet penetration velocity is less than that required for prompt ignition. Various mach reflections are evident as well as the (relatively) inert bow shock structure. Figure 1.3 shows penetration at a slightly higher velocity, resulting in unsteady ignition. Ignition began on the lower left hand side of the shock front, failed, and then began again on the lower right side. The internal structure behind the shock front is not visible in these images because of the reacting gases. These images illustrate the immense complexity in a relatively simple liquid explosive system. The complexities and difficulties in heterogeneous systems only become more numerous.
Mach Reflections
Fig. 1.2. Shaped charge penetration of nitromethane. Penetration velocity is subcritical, and so no steady-state initiation is achieved
1 Introduction
2.99 µs
7.0 µs
10.99 µs
14.99 µs
9
Fig. 1.3. Shaped charge penetration of nitromethane. Penetration velocity is slightly higher than that in for Fig. 1.2. Ignition is unsteady
1.4 Numerical Approaches Some in the research community (particularly the funding agencies) hope that numerical simulations are currently, or soon will be, adequate to make up for the deficiency in our understanding. Although it is true that computer modeling can be used to evaluate a large number of situations and examine large portions of parameter space, it is also true that the computer codes are only as good as the physical models used to construct them. These are devised by writing the equations of conservation of mass, momentum, and energy along with the necessary equations of state and closure relations. Because of the complex nature of this set of equations, certain assumptions are employed to make the solution tractable. And because there is no analytic solution to the coupled set of conservation equations, they must be solved numerically. This requires a choice of first or higher order methods, each with its own list of positive and negative attributes, but having in common the inability to solve the equations exactly. Issues with numerical diffusion and viscosity, among many others, must be dealt with. Specifying applicable equations of state is problematic because the material state changes in nonshock processes, so the starting material is usually changed significantly during the course of the event. Thus, its new state must either be calculated (e.g., crack and surface area formation, chemical species, etc.) or assumptions about how the state evolves must be invoked and an ad hoc modification of these properties made.
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Additionally, even with a perfect model, accurate physical, material, and chemical properties are required. Information regarding thermal diffusivity, porosity, density, heat capacity, and a host of other properties must be known as a function of temperature, and to a lesser extent, as a function of pressure. These properties are rarely known to high precision, and while in some cases this does not matter, in others it is a critical deficiency. Also, chemical reaction rate constants, kinetic data, activation energies, etc. are difficult to obtain and evaluate. Because these describe the energy-producing processes, and as indicated earlier, a difference of only 1% in an energy release rate can mean the difference between a detonation and a deflagration, errors in this part of the model can have large effects on the accuracy of the prediction. Other difficulties with computer calculations are the enormous memory and computational requirements for performing high-fidelity calculations in three dimensions. Even with the continuing increase in computational power (0.1–100 Tflops from 1992 to 2006), adaptive mesh refinement (AMR), highly optimized numerical schemes, and increased visualization capabilities, we are still years away from being able to accurately compute generalized HE response in generalized geometries. This would be true even if we did possess completely accurate physical models, which we do not. For example, as computational error is linearly dependent on zone size, and the CPU time required is nonlinear, a 1,000-fold increase in computational speed yields only a tenfold decrease in error for a two-dimensional calculation, and an error reduction of only 5.6 times in three dimensions. In addition, for a generic problem of a planar detonation wave in a 200 mm3 cube with a 1 mm reaction zone, and using AMR to resolve the reaction zone, it would require 185 days of CPU time with a 1,000 processor machine [5]. Increasing the complexity by invoking multistep kinetics, unsteady reactions, material interfaces, and damage only serves to increase the time requirements. The bottom line is that we will not be able to calculate our way out of our problems any time soon. This is not to say that computational tools are not of enormous importance and utility. Even with approximate physical models, estimated material properties, and gross assumptions about material state, we can obtain useful guidance of a general nature. There are many powerful and sophisticated computer codes capable of highly resolved calculations (see, e.g., [4]). As long as the assumptions and difficulties of making such calculations are appreciated, they can be useful. The real danger is in the fact that these approaches tend to be so complicated and involved, requiring tens of thousands of lines of code in some cases, that it is likely impossible to fully understand where they may be misleading or have errors. Thus, if we need specific information about time to explosion, extent of reaction, and violence, we are still dependent on estimates based upon a combination of expert elicitation, correlations, and numerical approaches. Correlations of system response are very useful in some circumstances. However, they typically require a large number of tests. As many processes do not scale, the tests must be run at different sizes, which significantly increases
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the cost. Changing one parameter (e.g., particle size or density) may invalidate a derived relationship. For example, in one set of tests, dropping PBX 9404 vertically from 8 ft results in no reaction. If the same sample is dropped from the same height onto a plate inclined at 45◦ , a violent reaction ensues. The violence of the reaction is a strong function of whether or not loose grit is present and to a lesser extent on surface finish and thermal conductivity. For PBX 9011, the drop height required to produce a violent response at room temperature is 78 ft, while at −29◦ C it is only 4 ft. The types of tests that yield data for correlations are generally deficient, in that they typically do not isolate a single mechanism. For example, in the well-known drop weight test, the sample is undergoing multiaxial and unsteady shear, strain, and compression. This leads to an unknown material state. The only data that are produced from the experiment are whether or not a sound of sufficient magnitude was produced. This information is then compiled to find the drop height that produces this nonspecific diagnostic 50% of the time. Although quite useful for a number of applications, using data of this nature to make predictions about explosive behavior in a general sense is extremely unreliable. The same can be said for every other set of data ranging from differential scanning calorimetry, to Henkin tests, to the one-dimensional time to explosion (ODTX), to fragment impact, to small and large scale gap tests, and beyond.
1.5 Motivation for This Text These observations argue strongly for a sound understanding of the fundamentals of explosive behavior so that informed and reliable decisions can be made in the absence of specific data for a specific situation. If we understand how explosives react, what the mechanisms are for ignition, what the requirements are for propagation, and how violence is determined, then we can more accurately assess the danger of an operation or application, even if the seemingly infinite number of specific details that affect the outcome are unknown or unknowable. For example, a fundamental understanding of thermal ignition and explosives’ kinetic response and the careful application of some significant assumptions were used by Frank-Kamanetskii to formulate a simple approach enabling solutions to problems ranging from autoignition of hay bales to the design of safe shipments of thousands of tons of fertilizers in the holds of ships. However, blind application of this kind of simple approach without a fundamental understanding makes disastrous consequences inevitable. What we present in this text is the basis for that required understanding of explosives in abnormal environments.
1.6 Purpose of This Book There are many books on mining and blasting, detonation physics, explosives engineering, and explosives chemistry. And for every book on detonation,
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there are several that discuss combustion. Given that observation, a legitimate question would be, with all that has been written about explosives over many years, why do we need yet another text on the topic? While there are many books that discuss performance, shock initiation, and a host of engineering topics, none (that we found) assembled into one location a useful summary of all the relevant topics involved in nonshock initiation. For our purposes, shock initiation is defined as prompt initiation of a detonation through shock processes such as detonators, planar impact by plates, etc. Nonshock initiation is defined as arising from bulk heating, fragment impact, spark and friction, or other similar processes. Initiation is not prompt and involves a series of steps that may or may not lead to a steady detonation. Much of the early scientific experimental work (before 1940) on shock initiation was performed with gases because they are more easily studied. Parameters can be quickly varied, optical access is straightforward, and they are homogeneous. However, condensed-phase systems are much more complex with varying material properties, the possibility of damage, and surface area creation, etc. Most of the early theoretical study was performed on idealized one-dimensional systems, and much of this work concentrated not on initiation, but on propagation of steady waves. In the area of nonshock initiation, once again, serious work on DDT began with substances in the gas phase. Later, the problem of DDT in condensed phase materials was addressed at length, with an emphasis on propellants. Much of that work is published in research journals and has been nicely summarized in a few excellent review papers. Some of it appears in internal reports and classified reports and is not easily accessed. A single text that describes DDT, initiation by spark and friction, cookoff, the effects of damage, simplified kinetics, and other important and essential topics is not to be found. As illustrated earlier, it is not possible to completely describe all the physics and chemistry involved in nonshock initiation. Even if they were known, which they are not, that would be too ambitious, even for a series of volumes. This text is meant to provide an introduction to several important topical areas and a foundation in the essential areas; those necessary to understand the relevant details, evaluate situations and procedures, develop models, and to sort out the important facts from those that are not important. This latter point is very important and often overlooked. For example, when considering a scenario of a cased explosive part impacted by a fragment, or heated in a fuel fire, or dropped from some height onto a protrusion, how is the risk evaluated? Or which hazards are important to mitigate and which ones can be ignored? Many have made the mistake of thinking that if they worry about all possible scenarios and hazards, and treat each equally, they will be safer. Nothing could be farther from the truth, because unneeded complexity is just as dangerous, or more so, than not enough rigor. The key is to know what deserves your attention. Do you worry about the temperature, or the confinement, or the velocity, or the material state? And if you worry about the material state, which is more important, the initial pristine mate-
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rial or the damaged material? And before you guess the obvious, it may be important to recognize that the state of the pristine material might actually determine the importance of the damage that might occur, and in some cases, might make the damaged state irrelevant. The knowledge and ability to sort out the important facts from those that are unimportant is critical. This text presents the background and fundamental understanding required to more fully understand non-shock initiation of explosives in abnormal environments. It is not meant as a stand-alone document. Information on detonation and shock physics, explosives formulation, materials science, and other topics is required to fully round out an understanding of the behavior of explosives. References found in each of the chapters can and should be used to focus more attention on a specific topic.
1.7 Final Word of Caution Having said all of this, lest one assume that with sufficient study and acumen he will be able to completely understand this field, we offer a few interesting observations. With respect to understanding and predicting accidents, one early researcher’s view is that: Whatever method we use to predict possible accidents, we have one fundamental problem. The substances can be subjected to so many different situations, each with its own pattern of different kinds of stress, that it is impossible to imagine them all. With more understanding, we can make predictions of risks in more and more situations, but a wholly complete understanding and description of all potential risk situations will never be possible [2] Even though this statement was made in 1976, over 30 years ago, the situation and conclusion have scarcely changed. Smith [3] made the following observations regarding HE sensitivity to initiation by various means: – The sensitivity of an explosive is not a well-defined property of the material, expressible as a single number, say, but is, instead a complex pattern of behavior – Different sensitivity tests, even when intended to measure the same property, will frequently produce different orders of relative sensitivity for a given series of explosives. In other words, there is not even a unique qualitative scale of sensitivity – In most sensitivity tests, the response of an explosive varies in seemingly random fashion over some range of severity of the applied stimulus. That is, there is no sharp threshold above which the explosive will always explode, below which it will never explode
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Given the difficulties described by these authors, we will present as clearly as possible the information required to sort out the conflicting data and outline the approach to making informed decisions.
References 1. Ramsay, J.R., Popolato, A.: Analysis of shock wave and initiation data for solid explosives. In: 4th Symposium (International) on Detonation, pp 233–238. US Naval Ordnance Laboratory, White Oak, 12–15 October 1965 2. Ek, S.: Sensitivity of explosive substances a multivariate approach. In: 6th Symposium (International) on Detonation, pp 272–280. Naval Surface Weapons Center, Coronado, CA, 24–27 August 1976 3. Smith, L.C.: Los Alamos National Laboratory Explosives Orientation Course, Sensitivity and Sensitivity Tests, Los Alamos National Laboratory, LA (1987) 4. Baer, M.R.: Computational modeling of heterogeneous reactive materials at the mesoscale. AIP Conference Proceedings, pp. 27–33. Shock Compression of Condensed Matter: Conference of the American Physical Society Topical Group, Snowbird, UT, USA, 27 June–2 July 1999 5. Aslam, T.D.: Direct numerical simulation of detonation. In: Conference of the American-Physical-Society-Topical-Group on Shock Compression of Condensed Matter, Baltimore, MD, 31 July–5 August 2005. AIP Conference Proceedings, vol. 845, pt.1 and 2, p.931–935 (2006)
2 Transport Phenomena for Nonshock Initiation Processes W. Lee Perry
2.1 An Overview of Transport Theory in Explosives Problems A useful explosive material is, of course, stable under reasonable environmental conditions. It will neither release energy nor produce gas without some kind of thermal stimulus. The possible stimuli include impact, spark, friction, boundary heat, or shock. These processes raise the temperature of the explosive material, either in a localized volume, or throughout its entire volume, to a point where the exothermic, gas-producing reaction becomes self-sustaining (ignition). We refer to the events leading to self-sustaining reaction as the preignition regime, and the behavior after self-sustaining reaction commences as the post-ignition regime. We further generally characterize the post ignition regime by either laminar nonviolent deflagration; violent, convectively driven explosion; or the process may undergo a transition to detonation (DDT). Transport phenomena are inherent to the nonshock initiation process and post-ignition behavior. Specifically, transport phenomena are the movement of heat (energy), molecular species, and momentum under the influence of temperature, species concentration, pressure, and velocity gradients. The theory of these phenomena, together with the production of gas and heat from chemical reactions (kinetics and thermodynamics), in principle fully describe the processes that occur prior to ignition (preignition) and what happens after (post-ignition). In practice, the theory has only been applied to very simple situations due to the complexity of the integrated problem. The stimuli listed above all serve to raise the temperature of the explosive either within a volume small with respect to the charge (impact, spark, friction, shock) or globally (boundary heat). Each of these stimuli will be discussed in detail in other chapters of this text. As these are the initial drivers for the ignition process, we briefly introduce them here. Impact leads to localized heating when deformation becomes concentrated, where the material fails by cracking, or when the yield strength is exceeded and causes shear localization and concentration of energy along shear bands (Chap. 10). Spark energy B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 2, c Springer-Verlag Berlin Heidelberg 2010
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is localized along the spark channel (Chap. 11). Friction (Chap. 9) localizes heat between moving surfaces or at specific locations, where, for example, foreign matter (grit) becomes trapped between the surfaces (the latter has been implicated in accidents.) A shock wave compresses material along its leading edge and causes heat generation directly by adiabatic compression of the material itself, by collapsing void space, or by grain boundary interactions. Boundary heat, as the phrase implies, is heat applied to the outer surfaces of an explosive charge, or its container, and the most common source is fire, and ignition by this method is commonly called “cookoff” (Chap. 7). Chapter 3 discusses reaction kinetics and thermodynamics. We mention them here in the context of heat transport because, of course, exothermic chemical reaction becomes the driving force for the ignition and combustion process once one (or more) of the ignition stimuli meet a critical criterion. To a first order, the general Arrhenius expression provides an illustration of the rate of gas production as a function of temperature and some reactive state, f (X), of the explosive (e.g., reaction extent, density, concentration of reactive species, available surface area, etc.): E ˙ · f (X). N = A · exp − RT
(2.1)
The product of enthalpy change (per mole converted) and the rate of gas production reveal the rate of heat production: Q˙ = N˙ · Δh.
(2.2)
These expressions illustrate the sensitivity of the process to temperature in the exponential term and the dependence of some reactive state of the explosive. When one (or more) of the listed stimuli is applied with sufficient intensity to raise the temperature to a value where significant heat production occurs, ignition may occur. The specific ignition threshold is crossed when the rate of heat production in a volume exceeds the rate of heat removal from that volume by a heat transport process (conduction, convection, or radiation). For combustion to spread, energy must be fed back from the combustion zone to regions of unreacted material at a sufficient rate to maintain combustion. These are central concepts of ignition and propagation theory and will be discussed throughout the book. As described, ignition and reaction spread arise from the specific interplay of heat transport behavior, reaction kinetics, and the thermodynamics of the involved reactions. Heat transport theory has played a central role in determining the factors that lead to ignition. The critical conditions for ignition are determined by applying the principle of energy conservation on the stimulated volume, balancing heat production by chemical reaction and heat loss by conduction or convection. In conjunction with carefully constructed experiments, the application of heat transport theory has also provided a tool to help deduce
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or verify the specific chemical steps that occur in the preignition regime [1]. Heat transport theory is also well-developed in the post-ignition regime for low-order (nonviolent) laminar combustion of simple regressing solid or liquid surfaces where heat feeds back from the combustion zone to the surface liberating the reactive species [2, 3]. However, for cases of more critical interest where violent explosive reaction occurs, heat transport occurs in a very dynamic, multidimensional, multiphase environment. Although the basic processes and theory are fairly well understood, the complexity challenges our best computational tools and this remains a very active area of research. Species transport is the movement of specific chemical constituents throughout a system. Explosives considered in this text generally consist of a single molecular constituent. In the preignition regime, the aforementioned stimuli can cause direct decomposition to combustion products, decomposition to intermediate species, or sublimation/evaporation of the original substance. In each of these cases, species transport phenomena govern the physical movement of the products or sublimated species from a solid or liquid surface into the gas phase. The rates of these processes and any associated exothermicity are again controlled by kinetics, which of course is strongly temperature dependent, but also dependent on species distributions and/or morphology (f (X)). If the explosive is heterogeneous (e.g., HMX powder), species transport governs how the gas phase species will distribute themselves throughout pre-existing void space, or throughout the void space created by decomposition, sublimation, phase changes, etc. (mechanical or thermal damage). Recent research has concluded that the void space (pore structure) and its potential development and spatial distribution prior to ignition has a very significant effect on the post-ignition behavior (Chap. 6) [4]. The distribution of species, whether in homo- or heterogeneous system, determines, in part, f (X) in (2.1) above and therefore affects the preignition heat transport behavior. Recent work has shown this distribution of reactive species affects the temperature distribution [5]. This in turn affects the postignition spread of combustion from the ignition locus as it propagates into a larger volume at a higher temperature and in a more reactive state [6]. Further, combustion spreads more readily into regions previously saturated with combustible species. As with heat transport, the situation during violent explosive reaction becomes exceedingly complex, and to date, no significant computational work has been done, which includes rigorous species transport in the post-ignition regime. In the post-ignition regime, species transport theory has also been applied to the study of laminar flames where convection and diffusion processes carry reactive species into the combustion zone [7, 8]. Heat production, decomposition, combustion, and sublimation/evaporation all potentially lead to pressure gradients, which are the driving force for momentum transport. Momentum transport is responsible for the convective (advective) movement of both heat and species. In the pre-ignition regime, localized heating from impact, spark, or friction leads to local gas formation by decomposition or sublimation, causing a local region of elevated pressure
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relative to the bulk of the explosive. In the case of boundary heating, the evolving temperature profiles will cause a distribution of gas products and a resulting pressure distribution. In both these cases, the pressure gradients cause the transport of momentum, which carry heat and species, affecting preand post-ignition behavior as already described. The rigorous incorporation of momentum transport requires the Navier– Stokes (NS) equations, which for any practical problem require the art of Computational Fluid Dynamics (CFD). The tensorial nature of the NS equations and the presence of multiple phases add a large degree of complexity to an already exceedingly complex problem and have not been applied in the preignition regime. However, the porosity and permeability of some explosives have been determined [9, 10], and it was suggested that Darcy’s law for permeation in porous materials would provide a tractable means to incorporate momentum transport for analysis of the preignition problem [5, 11]. This approximation has provided good insight to the preignition problem and allows models to correctly predict temperature profiles for a boundary-heated system [5]. The Darcy’s law approximation requires low Reynold’s number flow, which of course is not the case for the violent post-ignition regime. In that regime, high flow rates in stationary media occurs in the initial phases; as momentum is transferred to the solid phase and the pressure builds, the solid phase disassembles and is set in motion by drag forces. In this complex multiphase flow regime the full NS system is required. These have not been incorporated into contemporary computational solutions; however, progress has been made in formally formulating the problem and examining tractable limiting cases [9]. For the most violent possible post-ignition outcome, DDT, multiphase momentum transport also governs the formation of compaction waves that build to a shock wave and initiate the shock-to-detonation transition (SDT) [12]. Although our understanding of the mechanism of DDT is fairly mature (Chap. 8) [13], only a limited amount of work has been done to apply formal momentum transport theory to the problem. Hopefully, this brief introduction has given the reader a flavor of the interconnection of the three transport modes through heat and mass production by chemical reaction and the importance of all three modes to the ignition and explosion behavior. In the rest of this chapter, we will provide an overview of the history of the application of heat transport theory to explosive systems. We will then introduce the fundamental equations of heat transport theory as an introduction to other chapters that provide a more complete development. Properties such as heat capacity, thermal conductivity, viscosity, etc. are crucial to accurate transport theory predictions, and so a section will be devoted to that topic. Finally, we will conclude the chapter with an overview of three recent problems that employ formal transport theory in a way that has enhanced our physical understanding of the ignition and combustion process.
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2.2 The Origins of Transport Theory in the Field on Nonshock Initiation In our modern world of advanced computing tools, we may lose sight of the fundamental transport processes involved in the nonshock initiation and combustion of explosives. The early work in this field, prior to the existence of modern computing tools, required very insightful analysis and therefore deserves a brief review. Explosion is merely an extreme form of flame propagation, such that early flame research provides a logical origin of transport theory development for ignition and combustion of explosives. Scientists have known for over 150 years that combustion and flame propagation involved the transport of momentum, chemical species, and heat. This is evident in the lively and engaging compilation of lectures by Michael Faraday entitled The Chemical History of a Candle, published in 1861 [14]. Early theory of flame propagation velocity was published by Ernest-Francios Mallard in 1875 [15], and further developed by both Mallard and the famous physical chemist Henry Le Chatelier [16]. The Nobel Prize winning Dutch physical chemist Jacobus van’t Hoff provided early insight in the late nineteenth century to the nonequilibrium nature of combustion by suggesting that an exothermically reacting system cannot be in equilibrium with its surroundings [17]. The early development of transport theory for combustion and ignition of energetic materials mostly belongs to the Russians. In the 1920s and 1930s, Yakov Zeldovich [18, 19] and Belyaev [20] pioneered the early application of transport theory to the combustion of gasses and solid energetic materials. In the same time period, Todes and Seminov provided the earliest recognized formal heat transport theory of the critical conditions for the initiation of thermal explosion for a boundary-heated system [21, 22]. Seminov’s unsteady treatment is widely cited and discussed today and is distinguished by its simplicity, qualitative nature, and illumination of physical features important to criticality. His analysis showed the conditions for which heat production exceeded heat removal, the condition for ignition, using simplified mathematics. To find an analytical solution, Seminov included an exponential temperaturedependent reaction term, but neglected heat transport within the explosive material, that is, the spatial derivatives, such that all resistance to heat transfer occurred at the material’s surface. However, it was known experimentally at the time that ignition generally occurred at a specific location within the explosive material, implying that an accurate analysis could not neglect internal heat transport. David Frank-Kamenetskii, a student of Seminov, took a different approach from his mentor and developed a steady-state solution that included internal heat conduction (spatial derivatives) and also included the nonlinear reaction term [23]. An essential feature of Frank-Kamenetskii’s analysis was the use of Seminov’s steady-state solution to develop a dimensionless temperature having a value near 1 at the explosion temperature and zero initially. The interplay between the works of these two researchers continued, as it was later recognized that Frank-Kamenetskii’s steady solution could
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provide key parameter information for Seminov’s unsteady solution such that a fairly clear picture of (transient) thermal explosion behavior emerged. When exactly this latter insight came about is unclear, but it is discussed in a text published by Frank-Kamenetskii in 1947 [24]. Their methods of solution and the results are especially clever and insightful considering the lack of computational resources. The work of these early researchers, embodied in the theories of Seminov and Frank-Kamenetskii, are essential for the understanding of energetic material ignition, primarily due to the clear presentation and illumination of salient features. As such, Chaps. 4 and 7 go into much greater detail of the development and predictive qualities of these theories. This early development of ignition theory was concerned with the boundary heating problem; investigation of hot-spot heating followed soon after. It was recognized in the 1940s that the size of a hot spot played a role in the critical conditions for ignition. This idea may have originated from intuition stemming from knowledge of heat transport theory, cognizance of the size effect in the Frank-Kamenetskii/Seminov boundary analyses, or more likely, both. Bowden et al. confirmed this when they observed a size effect in friction experiments, the results of which were reported in 1947 [25]. This size effect was shown theoretically in 1948 by the hot-spot analysis of Rideal and Robertson that examined impact ignition [26]. In that study, the researchers formulated the one-dimensional spherical transient and spatially dependent heat transport problem and presented an analytical solution. The analytical solution was possible because they neglected the nonlinear exothermic reaction term and imposed a temperature, rather than a heat source term. They determined critical hot-spot volume and temperature by equating the rate of heat conduction at a given temperature and spot size to heat evolved from chemical reaction over the same volume. As with the early work on boundaryheated ignition, computational tools did not yet exist to provide numerical solutions to the nonlinear equations that govern the hot-spot problem. In fact, these authors admit: “A more elegant and satisfactory treatment, however, would be to solve the general equation. . .,” referring to the unsteady 1D spherical heat equation with an Arrhenius heat generation term. Apparently, very little work was done on the hot-spot problem until the advent of numerical tools. In 1963, Merzhanov and his coworkers published the results of their numerical study that followed the dimensionless parameterization scheme of Frank-Kamenetskii [27]. They concluded that those parameters were also good indicators of criticality for the hot-spot ignition problem, although with a different functional dependency on temperature. The usefulness of the Frank-Kamenetskii parameters for finding the properties of a critical hot spot arises due to the universal physical basis for criticality. While boundary heating may be generic in nature, specific localized mechanisms cause hot-spot heating. In those cases, an additional criterion is required: localized power dissipation from the heating mechanism must also exceed the rate of local cooling such that the hot spot will reach a critical temperature. A current good review was provided by John Field et al.
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in 1992 [28]. They point out the important early contributions to impact and friction studies of Bowden et al. [25], Eirich and Tabor [29], Rideal and Robertson [26], Bowden and Yoffe [30], Afanas’ev and Bobolev [31], and Field and his coworkers [28]. Many of these studies are mechanistic in nature and some report a temperature rise under the stimulated conditions. From a transport theory perspective, it is most useful to have a rate of energy dissipation (power). Spark initiation studies provide a simpler platform to quantify energy and power from the electrical properties of the spark generator. Langevin and Bicquard published early studies of the spark initiation of primary explosives in 1934 [32]. Wyatt and his colleagues published their studies of spark initiation of primary materials in 1958 [33], and Tucker reported their research on secondary explosives in 1968 [34]. Somewhat surprisingly, these early papers, with some exceptions in spark initiation research, represent the state of the art of hot-spot ignition source research. Although mechanisms are fairly well understood, heat production in the context of transport theory has not been fully investigated and this remains an active area of research. So far, we have reviewed the foundations of ignition theory and laminar (nonviolent) combustion. It was known in the late 1950s that sufficient pressure would cause an explosive deflagrating from its outer surface to transition to a violent thermal explosion as flame penetrated into the bulk via an existing pore network or cracking induced by pressure. It was also known in this period that this phenomenon, usually called “convective burning,” was a critical step for DDT. In terms of transport phenomenon, this physical transition is reflected by the mathematical transition to theory where momentum transport analysis becomes essential for understanding the movement of heat and chemical species. The Englishmen Griffith and Groocock published their early work in 1960 [35], as did the contemporary Russian researchers Andreev and the already-mentioned pioneer Belyaev [36]. In 1962, Taylor published his work on the convective burning mechanism in HMX [37]. The evolution of this research was in part motivated by the potential for solid rocket motors, some of which were propelled by HMX and/or RDX, to explode or detonate under abnormal pressure conditions. As such, the propellant literature from the 1960s and 1970s provide the most complete theoretical transport basis for the convective burning problem and the review text of Kenneth Kuo and Martin Summerfield provides an excellent fundamental overview [38]. The process of DDT begins when the drag forces of moving gases become sufficient to set the solid phase into motion, thus amplifying the complexity of the problem. Bernecker and Price credited the aforementioned work of Griffiths and Groocock [35] in an early review in 1979 as providing the first complete description of DDT in a bed of granular explosive [39]. Perhaps it is not surprising that the theory of explosion originates coincidentally with the understanding of chemical equilibrium. The early contributions of the Russian researchers cannot be overstated, as followed by the English at Cambridge and, most recently, the researchers funded by the American Department of Defense. Over the last 20 years, researchers funded
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by the US Department of Energy have pursued this area of research in order to understand and minimize the consequences of a nonshock initiation event in a nuclear weapon system. While the early research has provided a reasonably clear picture of mechanisms and the appropriate theory, the evolution of computational tools and their ever-increasing ability to incorporate nonlinear, multiphysics, multiphase phenomena at ever-increasing spatial and temporal resolution now drives the advancement of the field. However, one cannot build good models without the basic theory and insight. That is the intent of this book. To follow the evolution of these research topics beyond their origins, the reader may perform a citation search from this section, and the texts of Frank-Kamenetskii and Zeldovich provide a complete theoretical foundation of transport theory in explosive systems. The other chapters of this book will provide the details of the most recent research.
2.3 Physical Properties Used in Transport Theory Analysis Solving the transport equations requires the physical properties including heat capacity, density, thermal conductivity, thermal diffusivity, viscosity, and species diffusion coefficient. These properties depend directly on the thermodynamic state of the system: pressure, temperature, etc. Common high performance explosives and explosive composites have well-characterized heat capacity, density, and thermal conductivity and data are available up to temperatures where decomposition reactions become significant. The reference text of Gibbs and Popalato [40] or the LLNL Explosives Handbook [41] provides these data for a wide range of explosives. For gases or liquids involved in the explosion process, viscosity and the diffusion coefficient also become factors. For tabulated data for all the properties for gases and liquids, except the diffusion coefficient, we refer the reader to general sources such as the CRC handbook [42] and the NIST Chemistry Web Book [43]. The diffusion coefficient depends on factors specific to the specific environment and tabulated data are sparse, but some data are provided in textbooks covering diffusion; See, for example, [44] and the references therein. Here we review the physical basis for each property, general value ranges, and general trends with pressure and temperature. Specific Heat. In conjunction with density and molecular weight, specific heat (on a mole basis) provides a measure of the thermal inertia of a system. That is, it is a measure of the resistance of a system to a change in energy content (temperature.) As such, specific heat affects transient heat conduction, but has no effect on steady-state heat conduction. Specific heat is mathematically defined as c=
δq dT
. path
(2.3)
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In order to make tractable relationships, the path is taken to be one of constant pressure or constant volume. For a constant volume process, specific heat is related to the internal energy: du cV = . (2.4) dT V For a constant pressure process, the specific heat is related to enthalpy: dh cp = . (2.5) dT P For ideal gases, it can be shown that cp − cv = R. And the ratio of the specific heats commonly appears in gas-phase problems: cp γ= . (2.6) cv Due to the relative incompressibility of the solid and liquid states, u = h such that cp ≈ cv . A simple relationship exists for the internal energy stored by a single degree of freedom regardless of the size or shape of the molecule: u=
And cv =
1 RT. 2
du dT
= v
(2.7) n R 2
(2.8)
for n degrees of freedom. For example, monatomic gas molecules translate through three dimensions and therefore have three degrees of freedom, such that 3 cv = R. (2.9) 2 The presence of rotational, vibrational, torsional, etc. modes increases the heat capacity accordingly. The free electrons in conducting solids contribute to the heat capacity at room temperature on the order of 0.03R/2. Not all modes are active at all temperatures and there are other temperature effects for more complex structures. The reader should consult a physical chemistry text or a statistical thermodynamics text for details of specific molecules and temperatures. Monatomic gasses have a specific heat exactly 3R/2 and diatomic gasses such as nitrogen, oxygen, carbon monoxide, etc., at room temperature are also near 3R/2. Liquid water at room temperature is 75.9 J mol−1 K−1 . Solid copper has a specific heat of 24.4 J mol−1 K−1 and the common organic crystalline explosive HMX has a specific heat of 325 J mol−1 K−1 at room temperature.
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Density plays a role in all three transport processes. It affects the transient solution of the heat transport equations and both the transient and steady state solution of the momentum equations. For reasonable temperatures and pressures found in the preignition regime, gas phase density may be computed from the ideal gas law equation of state. In the more nonideal post ignition regime, a more descriptive equation of state, such as the van der Waals equation of state, may be required. In extreme post-ignition conditions, with pressures and temperatures approaching detonation conditions, an equation of state such as the Abel or Becker–Kistiatowsky–Wilson equation of state may be necessary. Solid and liquid phase densities for preignition conditions are almost always measured quantities. Most liquids fall into the range of 800–1,200 kg m−3 . Most inorganic solids have crystal densities in the range of 1,500–3,000 kg m−3 . Most metals are in the range of 2,500–12,000 kg m−3 , with exceptions that exceed 13,000 kg m−3 such as mercury, uranium, plutonium, etc. Common organic crystalline explosives and their binders have densities in the range of 1,500–2,000 kg m−3 . Thermal conductivity is the proportional factor of Fourier’s Law of Heat Conduction, relating heat flux to a temperature gradient. As with density and heat capacity, thermal conductivity affects transient heat conduction. However, it also appears in the steady state solution. Gas thermal conductivity is related to the heat capacity of the particular molecule, the characteristic velocity, and the mean-free path. The kinetic theory of gases reasonably predicts the thermal conductivity of simple gases to 10 atm (or so) [45]: κ3 T 1 , (2.10) k= 2 d π3 m where d is the molecular diameter, k is the Boltzmann’s constant, and m is the mass of the molecule. This general expression shows that pressure has a weak overall effect on thermal conductivity. Thermal conductivity of gases is on the order of 0.01–0.05 W m−1 K−1 , but can increase by a factor of 10 at combustion temperatures. In liquids, thermal conductivity is roughly an order of magnitude greater than gases because of the shorter mean free path. The kinetic theory of gases provides a starting point that leads to Cp ∂P 2/3 2 2/3 2 , (2.11) k ≈ π d vs = π d Cv ∂ρ T where vs is the sound velocity. In most cases for liquids, the ratio of heat capacities is unity and the pressure–density relationship must be found from an appropriate equation of state. As such, liquid thermal conductivity also has a weak dependency on pressure, but no general temperature behavior. It may increase or decrease depending on the equation of state for the particular liquid. The thermal conductivity of liquids is on the order of 0.1–0.5 W m−1 K−1 .
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In solids, phonons and free electrons exchange momentum to cause heat conduction. For metallic solids, the dominant mechanism is the latter, and (2.10) above will provide an estimate and trends when evaluated for the electron gas. For non-metallic solids, phonon interaction (frequency) dominates and thermal conductivity generally increases with temperature. For non-metallic solids, conductivity ranges over 0.5–5, and metallic solids typically are on the order of hundreds of W m−1 K−1 . Common organic explosives and their binders are in the range of 0.1–0.6 W m−1 K−1 . Thermal diffusivity is the ratio of the thermal conductivity to the product of density and heat capacity: k (2.12) α= . ρc It is a measure of the efficacy of conduction of thermal energy relative to the storage of thermal energy. This may be thought of as thermal inertia or the rate of diffusion of heat. Thermal diffusivity always appears in the exponential terms of the transient solution of heat transport analyses; large values reflect short transients. The dimensionless Fourier number provides an estimate of the thermal time constant of a system when its value is near unity: Foc =
ατ ≈ 1, L2
(2.13)
where L is a characteristic length. Simple gases at STP have a thermal diffusivity near 2 × 10−6 m2 s−1 , but can increase to near 1 × 10−3 m2 s−1 at combustion temperatures. Most liquids are on the order of 1 × 10−7 m2 s−1 and solids are on the order of 1–2,000 × 10−7 m2 s−1 , with metallic solids around 2–5 × 10−5 m2 s−1 . Organic solids, including common explosives and their binders, generally fall into the range of 1–2 × 10−7 . Viscosity is the proportionality factor of Newton’s law of viscosity, relating momentum flux (shear stress) to the velocity gradient. It appears in both the transient and steady state solution of the momentum equations, and also appears in Darcy’s Law along with the permeability to relate flow to a pressure gradient. The simple kinetic theory can again be used to qualitatively understand the effect of the state variables on viscosity for gases: √ 2 mκT . (2.14) μ= 3π 3/2 d2 As with thermal conductivity, there is no pressure dependence at normal conditions. Gas viscosities are on the order of 10−5 under normal conditions and can increase by a factor of 10 at combustion temperatures. In the denser liquid phase, in order to translate, molecules must overcome an energy barrier presented by its neighbors. A kinetic theory approach therefore reveals an exponential dependence on temperature relative to the boiling temperature (Tb ): Tb ρNa h exp 3.8 μ= , (2.15) M T
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where h is Planck’s Constant. Examples of liquid viscosity are water: 1.75 × 10−3 ; and engine oil: 3.85 N s m−2 . Kinematic viscosity is the ratio of the viscosity to density: μ ν= . (2.16) ρ Analogous to thermal diffusivity, this parameter may be thought of as the rate of diffusion of momentum. It always appears in the exponential terms of the transient solution of momentum problems; large values reflect short transients. An estimate of a system’s time constant is given by τ≈
L2 . ν
(2.17)
Simple gases at STP have a kinematic viscosity near 1–25 × 10−5 m2 s−1 and the value does not change significantly near combustion temperatures. Liquids have a wide range of values; water at STP is 1.75 × 10−6 . Mass diffusivity is the proportionality constant of Fick’s law of diffusion and relates the flux of a molecular species to the concentration gradient. It appears in the solution to both the steady and unsteady solution of the species mass transport equations. Analogous to kinematic viscosity and thermal diffusivity, the dimensionless mass transfer Fourier number provides an estimate of the time constant of a system when its value is near unity: Fom,c =
Dτ ≈ 1. L2
(2.18)
Kinetic theory tells us that translational velocity increases with temperature, leading to a more rapid redistribution of species, and collision rates increase with increasing pressure, leading to a slower redistribution. The expression for a gas mixture consisting of nearly identical molecules (e.g., isotopes) shows the dependence on pressure and temperature: 3 3/2 κ T 2 . (2.19) DAA = 3 π 3 m P d2 Diffusion gas phase coefficients for simple binary systems generally range from about 1 to 5 × 10−5 m2 s−1 . Similar to viscous momentum transport, diffusion in liquids is an “activated” process where molecules must overcome the barrier presented by nearby neighbors:
2/3 M Tb exp −3.8 . (2.20) ρNa h T In practice, many factors influence the process. In most cases, empirical measurements are required and (2.20) can be used cautiously to extrapolate from known data. Typically, liquid diffusion coefficients are 105 times smaller than gases. D ≈ κT
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2.4 Transport Theory Equations The equations governing transport processes have been developed ad nauseum, and there are many good texts covering each transport phenomenon. In particular, we suggest the text of Bird, Stewart, and Lightfoot as a comprehensive and detailed general text [46], and the aforementioned texts of Frank-Kamenetskii and Zeldovich for developments more specific to combustion and explosions. It is the intent of this text to provide a complete overview of nonshock initiation, and so we include a brief review of the formal development of transport theory. The development of all three transport fields begins with the fundamental concept of the conservation laws: energy, mass, and momentum are conserved within a control volume. Within this very simple framework, and through the straightforward application of differential calculus, we can develop all of the transport equations. It is always useful to begin with a very general form of the transport equations. Simplification is usually straightforward and a very general starting point allows us to see what assumptions must be made to arrive at the most tractable form for a particular situation. A balance on a differential volume equates energy stored and produced in the volume to the energy flux gradient: − ∇φ =
∂h , ∂t
(2.21)
where we have written storage and production in terms of enthalpy. For an Eulerian coordinate system (fixed reference frame) and in terms of temperature and heat capacity, we account for all fluxes (Fourier conduction and convection), storage and generation: ∂T ρcp + (v · ∇T ) = ∇ (k∇T ) + Δhrxn r (T, X) + q, ˙ (2.22) ∂t where Δhrxn r (T, X) is the product of the enthalpy change associated with chemical reaction and the rate of that reaction. The rate, of course, generally depends exponentially on temperature and on some quantity related to the availability of reactants (X); for example, concentration of gas phase reactants or solid phase surface area (see Chap. 3.) q˙ is the heat generated by any other source. In a Lagrangian reference frame (moving with the flow), we invoke the substantial derivative and we can simplify the equation to ρcp
DT = ∇ (k∇T ) + Δhrxn r (T, X) + q. ˙ Dt
(2.23)
The Lagrangian frame might be useful in the post-ignition regime when things begin to move or for the analysis of a propagating flame. Specific enthalpy was used in this derivation. If we were to start the derivation with specific internal energy, the result would depend on the constant
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volume heat capacity and reaction internal energy, rather than the constant pressure heat capacity and reaction enthalpy. The distinction is important as the former generally relates to incompressible media such as solids, liquids, and constant density gas phase processes. The latter generally applies to compressible gas phase processes. In this latter case, one must use caution with respect to reaction energy, as enthalpy is typically reported and the relationship between enthalpy and internal energy must be accounted for. Specifically, ∂ ∂h ∂P ∂v ∂u = (h − P v) = −v −P . ∂t ∂t ∂t ∂t ∂t
(2.24)
In general, all the material and thermodynamic properties can be space-, time-, and temperature-dependent. There are three types of boundary conditions in heat transport analyses. These are the Dirlecht boundary conditions where we impose a surface temperature, Ts ; the Neumann boundary conditions where we specify the heat flux, q˙s , at the surface; and the Robin boundary conditions, which is a weighted hybrid of flux and temperature conditions. Mathematically, the Nuemann condition is (2.25) n · (−k∇T |s ) = q˙s . The Robin boundary condition can take a general form that may account for convection normal (through) the boundary, convection parallel to the boundary, or conduction normal to the boundary: n · (−k∇T|s + ρCp uTs ) = n · [ρCp u ( T∞ − Ts ] + h (T∞ − Ts ) .
(2.26)
In these expressions, the subscript s indicates the boundary coordinate, ∞ indicates the condition far away from the system, and h is the heat transfer coefficient. The parameter h depends on the physical properties and movement (flow) of the medium surrounding the system. The method for computing h comes from boundary layer theory, and that discussion is beyond the scope of this text. However, simple methods to estimate h are readily available in the references provided [44, 46]. We arrive at the species transport equations by a similar route. Balancing the gradient of the mass flux (n˙ i ) with the mass density storage and production (r) of specie i in a differential volume leads to − ∇n˙ i =
∂ρi + ri , ∂t
(2.27)
and we must account for each species present. Including convection transport, diffusion according to Fick’s law and rewriting the equation in terms of the species mole fractions: − v · ∇ρi + ∇ · (ρi Dij ∇xi ) =
∂ρi + ri (T, X) , ∂t
(2.28)
where xi is the mole fraction of species i. This is a very general (and difficult to apply) form of the species mass transport equation with no restriction on
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29
constant density or diffusion coefficient. However, this form is only valid for binary systems, and in general the derivation of equations for more than two components with nonconstant density is difficult. If we make the assumption of constant density and diffusion coefficient, the more familiar diffusion equation results (Eulerian coordinates): ∂Ci + v · ∇Ci = Di,n ∇2 Ci + ri (T, X), ∂t
(2.29)
and in Lagrangian coordinates, DCi = Di,n ∇2 Ci + ri (T, X). Dt
(2.30)
These forms are most valid for dilute solutions at constant temperature and pressure. Of course, those conditions do not describe most of the interesting explosives problems, but there is insight to be gained by the application of simplified theory. Most notably, researchers have applied (2.29) with its heat transport analog (2.22) to describe flame structure [19]. The codependency of these equations of course arises through the heat and species generation term, which depend on temperature and the quantity X that relates to concentration. Two notable assumptions exist to decouple the equations. If the Lewis number, ρCD , (2.31) k is close to unity, the concentration and temperature fields are similar. And the assumption of zero-order chemical reaction allows decoupled solution of the heat and species transport equations. The boundary conditions for the species transport equations are analogous to those for heat transport. We can specify a Dirlecht condition at the boundary (concentration), a Nuemann condition (species flux), or the Robin hybrid condition. The velocity fields of (2.22) and (2.29), if they exist, are computed using the principle of conservation of momentum. As with heat and species transport, momentum transport occurs by convection and by molecular collisions (i.e., conduction or diffusion.) While these two mechanisms cause transport in parallel for heat and species transport, convective momentum transport by fluid motion is orthogonal to transport by molecular collisions. Consequently, the derivation leads to second order tensor expressions. The full derivation of the governing equation falls into the fluid mechanics field of study. Formal application of the full equation is rare in our field of study, but for completeness we provide what is commonly referred to as the Navier–Stokes equation: ∂v 2 + v · ∇v , (2.32) μ∇ v − ∇P = ρ ∂t Le =
where this form requires a constant density and viscosity (μ). Note that the product of the gradient operator vector and the velocity vector and the
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Laplacian operator vector and the velocity vector are both dyadic products and the result is a tensor. The application of the Navier–Stokes equation to nonshock initiation problems requires a level of computing power not generally available. In lieu of those equations, researchers have applied Darcy’s law of permeation in porous media as an approximation for creeping flow conditions (low Reynold’s number, Re < 10): −κ ∇P, (2.33) vs = μ where κ is the permeability and vs is the superficial velocity vector. In the preignition regime, the movement of chemical species under the influence of a pressure gradient has only recently been incorporated into analysis [5], and we will review that work in a later section. Of course, conditions in the postignition regime do not meet the creeping flow requirement and we must apply the Navier–Stokes equations along with the science of Multi-phase Computational Fluid Dynamics to provide meaningful solutions. We see the further interplay of transport processes that occurs through the pressure gradient term in both the Darcy approximation and the Navier– Stokes equation. The pressure gradient must be found from the production of chemical species via coupling with the heat and species transport equations. Strictly speaking, we must simultaneously consider all three transport phenomena and the interplay originates both from chemical reaction heat and species generation and the thermodynamic linkage of pressure, temperature and density. Consideration of fully coupled transport theory remains a significant challenge, mainly due to limitations of computing (and funding) resources.
2.5 Selected Examples of Transport Theory Applied to Explosive Problems We will review three recent cases where transport theory has been applied to problems involving explosives. First, to more fully understand the fundamentals of hot spot initiation, we have chosen a model system for study that uses the internal and localized dissipation of microwave energy to generate hot spots [47]. Next we will review a boundary-heated problem that includes species and momentum transport. Zerkle has invoked Darcy’s Law for a boundary-heated problem and he showed that permeation of decomposition products through the bulk of the explosive was required to more accurately predict temperature profiles and the violence of reaction [5]. Finally, we discuss a post-ignition problem having coupled heat and species transport. The work of Ward et al. used a low activation energy kinetic expression to characterize the combustion of HMX and made assumptions about the similarity between the thermal and species diffusion coefficients [7].
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Theory of microwave-induced hot spots. Organic explosives generally interact weakly with electromagnetic energy in the microwave frequency range. Therefore, the addition of electromagnetically absorbing inclusions provides a tractable system for the study of localized heating and ignition phenomena. The work considered the connections amongst the theories of dielectric mixing, microwave absorption, heat transfer, and thermal ignition theory in order to understand a model system of idealized hot spots; namely, small high thermal conductivity spherical inclusions ( R. However, in the case of microwave hot spot generation, the heat transport properties of a typical high thermal conductivity microwave absorbing inclusion (e.g., silicon carbide) suggest that the low thermal conductivity explosive surrounding the hot spot will limit the rate of heat transport. This is shown formally by evaluating the dimensionless Biot number: Bi =
2hR , ki
(2.35)
where h is the heat transfer coefficient and the subscript i refers to the inclusion. The heat transfer coefficient is found using the dimensionless Nusselt number: 2hR , (2.36) Nu = ke where the subscript e refers to the explosive. For a sphere in a stationary medium, the Nusselt number is exactly 2 and h=
ke . R
(2.37)
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For ke ki and small R, Bi 1 and the transport equation for R < r < ∞ will adequately describe the behavior. Equation (2.22) in spherical coordinates for the region outside the inclusion is ∂T 1 ∂ ρe Δhrxn 2 ∂T [ρcp ]e = ke 2 r(T, X). (2.38) r + ∂t r ∂r ∂r M Let θ=
T − T∞ r ; ξ= , Tig − T∞ R
(2.39)
and we use a Fourier number to define a dimensionless time τ=
αe t . R2
(2.40)
The dimensionless form of the equations is then 1 ∂ ∂θe = 2 ∂τ ξ ∂ξ
ξ2
Ea ∂θe ρΔhrxn R2 + A exp − , ∂ξ M ke (Tig − T∞ ) R (θe (Tig − T∞ ) + T∞ ) (2.41)
where the expression in brackets is the dimensionless Damk¨ohler number, relating the rate of heat conduction to the rate of heat production by chemical reaction. The boundary condition is ωεi E 2 R2 dθ qR ˙ 2 ˙ = = − = Θ. (2.42) 3ke (Tig − T∞ ) 6ke (Tig − T∞ ) dξ ξ=1 Because of microwave skin depth effects, R must be less than ∼100 μm. Even with this abbreviated statement of the problem, a numerical solution is required to find the evolution of the temperature profiles around the hot spot and the time to ignition. However, we can learn salient features about the behavior of the system without the full solution. From our general understanding of hot spot ignition, we know the following physical criteria must be met for ignition to occur: 1. Microwave energy dissipation must exceed heat removal at the ignition temperature. 2. The rate of heat production via exothermic chemistry must exceed the rate of heat transport away from the inclusion at the ignition temperature. Equation (2.42) provides the relationship for condition 1. The dimensionless gradient at the inclusion surface must exceed unity for the hot spot to reach the ignition temperature. To find the ignition temperature, we use the Damk¨ ohler number in (2.41) to find when condition 2 is true and use that information to determine the ignition temperature. By definition, θ = 1 when reaction heat exceeds heat removal by conduction: Ea ρΔhrxn R2 A exp − = 1. (2.43) M ke (Tig − T∞ ) RTig
2 Transport Phenomena for Nonshock Initiation Processes
33
Ignition Temperature [K]
800
750
700
650
600
2
4 68
10–5
2
4 68
10–4
2
10–3
Inclusion Size [m]
Fig. 2.1. The relationship between ignition temperature and hot spot inclusion size for HMX
This transcendental equation must be solved iteratively to find Tig . Figure 2.1 shows the relationship of Tig and hot spot size for values representative of the kinetic, thermochemical, and thermophysical parameters of HMX. We have learned two very important features of this system without solving the governing equations: the ignition temperature and the minimum microwave power dissipation required to reach the ignition temperature. However, we cannot deduce the heating rate or the time required to reach the ignition temperature from this approach. If we do not care about the exact nature of the spatial temperature profiles, we can further simplify the problem by ignoring the spatial dependence and performing an energy balance on the inclusion. The balance leads to a dimensionless equation describing the temperature of the inclusion (recall N u = 2 for a stagnant medium): dθ ˙ − θ) = 3(Θ dτ where τ = and the solution is
ke t ρi Ci R2
˙ [1 − exp (−3τ )] . θ=Θ
(2.44)
(2.45)
(2.46)
The dimensionless time to ignition is τig
1 1 . = − ln 1 − ˙ 3 Θ
(2.47)
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W.L. Perry
This straightforward development illuminates the temporal qualities of the ˙ < 1, a condition idensystem. Mathematically, ignition will never occur for Θ tical to the statement derived from the dimensionless boundary condition of (2.42). Here, we have used R for the characteristic dimension. This most accurately (but still qualitatively) shows the transient behavior prior to ignition ˙ that is, Θ ˙ > 5. For values of Θ ˙ closer to unity, a for larger values of Θ; characteristic length between R and 2R is more accurate. The dielectric loss (εi of (2.34)) of the inclusion depends on its electrical conductivity and the permittivity of the explosive medium. Generally, inclusions embedded in a low permittivity medium that are less than 100 μm in size and that have a conductivity in the semiconductor range (10 ohm) have
a high dielectric loss ∼70 × 10−12 F m−1 , leading to a high rate of energy dissipation [47]. High frequency also favors a high dissipation rate (2.34). Figure 2.2 shows the transient inclusion temperature and Fig. 2.3 shows the dimensionless ignition time over a range of dimensionless microwave power ˙ While this analysis was qualitative, it illuminates the dissipation values (Θ). factors important to microwave hot spot ignition. More importantly, we may apply the analysis for any heating stimulus as long as we can satisfy the stated ˙ assumptions and find an appropriate expression for Θ. Boundary heating problem including species and momentum transport. In the early 2000s, researchers at Los Alamos conducted a series of large-scale (10 kg) PBX 9501 cookoff experiments (the Large Scale Annular Cookoff experiment: LSAC.) The annular charges (8.9 cm i.d., 15.9 cm o.d.) were held at a temperature sufficient to drive sublimation and decomposition reactions (∼180◦C) and the β − δ phase transition such that substantial thermal 1.0 0.8
θ
0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
τ Fig. 2.2. The dimensionless transient inclusion temperature for three values of the dimensionless microwave heating term: 0.5, 1, and 5
2 Transport Phenomena for Nonshock Initiation Processes
35
1.0 0.8
τ ig
0.6 0.4 0.2 0.0 0
2
4
6
8
10
Θ Fig. 2.3. Dimensionless ignition time as a function of dimensionless microwave power dissipation. There is a threshold below which ignition will not occur
damage occurred; that is, significant porosity developed. These experiments were highly instrumented for the primary purpose of calibrating cookoff models that attempt to predict the explosion violence after ignition. The boundary conditions that lead to ignition have already been well established and documented in modern texts [48]. However, ignition theory does not tell us anything about the post-ignition explosive behavior. It was suspected, a priori, that the high degree of porosity generated during the heating phase would facilitate movement of reactive decomposition products through the charge and potentially affect both the pre- and post-ignition behavior. Indeed, it was found that the models could not accurately predict temperature profiles without including the movement of reactive species by momentum transport. Both the temperature field and distribution of reactive species affect the post ignition behavior, making the incorporation of species and momentum transport an important step towards understanding explosion violence. As modeling of the LSAC is an excellent example of the application of heat transport theory coupled with species and momentum transport, we will review the work of David Zerkle towards this end [5]. The model development began by writing the species transport equation (e.g., (2.28)) for a multiphase system: ∂ (φi ρi ) + ∇ · (φi ρi vg ) = r (T, X) , ∂t
(2.48)
where φi is the species volume fraction. Compressible gas phase processes were important to this analysis, and the heat transport equation was derived in terms of internal energy. The convective transport term of (2.22) was written as
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W.L. Perry
Pg ∇ · φg ρg vg ug + , ρg
(2.49)
where the subscript g refers to the gas phase. The specific internal energy is related to temperature through (2.4); pressure and temperature are related by the ideal gas equation of state. These relations lead to the following form of the heat transport equation: ρCv
∂T + ∇ · (φg ρg vg Cpg T ) = ∇ (−k∇T ) + Δurxn r (T, X) . ∂t
(2.50)
In lieu of incorporating the multiphase momentum transport equations, the velocity field was found using Darcy’s law (2.33) where vs = φg vg . The permeability for a range of thermal damage conditions was determined from the small-scale experiments of Parker et al. [49]. The overall effective density, heat capacity, and thermal conductivity were computed from (respectively): φi ρi Cv ; and k = ρ= φi ρi ; Cv = i φi ki . (2.51) i i ρ The reaction term (r) was modeled using the Los Alamos four-step decomposition mechanism that includes the endothermic solid phase transition, the endothermic sublimation of HMX, and the exothermic gas phase reactions [50]. The coupling of momentum, species, and heat transport in this system comes about through the thermodynamic state of the system, where, in this case, the ideal gas law equation of state ties pressure, temperature, and density together and the system is driven by the reactive production (or consumption) of heat and species. The partial differential equation solver, Flex PDE [51], provided the temperature, species concentration, and velocity fields. Figure 2.4 shows the computed temperature profiles, which agree well with the experimentally measured values. Figure 2.5 shows the comparison of temperature profiles with and without including flow, just prior to ignition. The inclusion of flow indicated a much larger volume at a higher temperature and that reactive species were also spread through a large portion of the charge. These two factors mean that the material was primed for faster reaction spreading, hence more violent explosion. Finally, the reader might notice the asymmetry in the results. This comes about from variations in contact resistance imposed on the interface between the explosive and its container. This experiment is discussed more fully in Chap. 7. A laminar flame model. We have examined two preignition scenarios, and now we examine the post-ignition transport phenomena associated with laminar burning of HMX. While we are most interested in violent explosion, the physical processes that occur for laminar burning also occur during violent explosion. Most explosive combustion processes are surface phenomena: combustible gases evolve from a solid surface and burn in the gas phase at a well defined “stand-off” from that surface. Pressure controls the stand-off distance.
2 Transport Phenomena for Nonshock Initiation Processes
37
Fig. 2.4. Calculated temperature profiles just prior to ignition for the annular cookoff shot. The calculation included Darcy flow and agreed well with experimental measurements. The boundaries were held at 180◦ C and the highest temperature contour was 220◦ C. The contour intervals are approximately 2.86◦ C. The inner diameter was 8.9 cm and the outer diameter was 15.9 cm
Fig. 2.5. Comparison between temperatures computed with (upper) and without (lower) species and momentum (Darcy) transport just prior to ignition. The highest temperature contours are 220◦ C. The lowest temperature contour with Darcy flow was 198◦ C and the lowest temperature contour without flow was 191◦ C
The essential difference between laminar burning and violent explosion is that laminar burning occurs over a surface area much larger than any defects, pores, cracks, grains, etc. in that surface and the stand-off is also much larger than
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those surface features. Violent explosive burning occurs when the pressure squeezes the flame stand-off to a dimension on the order of the aforementioned feature sizes, forcing hot combustion products into the cracks and pores, such that combustion occurs on a high surface area within the bulk of the material. Convective (momentum transport driven) processes of course play the dominant role in this regime, but the fundamental process of surface gas evolution and burning near the surface that occur during laminar combustion still occur in the violent explosion regime. Because of the relevance of the laminar analysis to the whole range of ignition response, we will review the work of Ward and Son [7]. Those authors develop the heat and species transport equations for the laminar flame and provide a straightforward and illuminating analytical solution that correctly describes experimentally observed flame structure of burning HMX, which is shown in Fig. 2.6. Their analysis consisted of two parts: development of the steady-state heat transport equation with a high activation energy decomposition reaction for the solid phase and the steady-state coupled heat and mass transport equations with very low activation energy exothermic combustion reactions for the gas phase. One-dimensional cartesian coordinates were used with the origin at the solid–gas interface for both regions. For the solid phase, (2.22) was reduced to mc p
∂T ∂2T = ke 2 + Δhd r (T ), ∂x ∂x
(2.52)
where the subscript d refers to the decomposition reaction HMX → I,
(2.53)
where I is a representative gas phase intermediate. The kinetic expression is 1.2 1.0
Solid
Gas
θ
0.8 0.6 0.4 0.2 –1.0
Stand Off –0.5
0.0
0.5
1.0
ξ
Fig. 2.6. The quantitative laminar flame temperature structure predicted by Ward et al.
2 Transport Phenomena for Nonshock Initiation Processes
Ed r (T ) = ρe Ad exp − . RT
39
(2.54)
The symbol m ˜ is the mass flux and was the variable in the final solution that represented the “feed” of gas phase reactants for the gas phase part of the analysis. It is also the parameter that translates into a burning rate and is thus a parameter of primary interest. A floating boundary Dirlecht boundary condition, Ts , was specified at the solid–gas interface and ambient temperature in the solid bulk (T = T0 @ x = −∞) provided the second boundary condition. Merzhanov and Dubovitskii [52] provided the original solution for the mass flux: Ad RTs2 ke pe · exp (−Ed /RTs ) . (2.55) m 2 = Ed [cp (Ts − T0 ) − Δhd /2] To develop the problem in the gas phase, the authors began with the assumption of equality between the molar heat capacity of the gas and solid phase, and that the dimensionless Lewis number was unity: Le ≡
αg = 1, D
(2.56)
where the subscript g refers to the gas phase. These assumptions are reasonably accurate and they substantially simplify the problem. Expressing the governing equations in terms of dimensionless variables further facilitates the solution; the following relationships were used: E Δhi T ; Δh∗i = ; ; E∗ = T∞ − T−∞ R (T∞ − T−∞ ) Cp (T∞ − T−∞ ) xm ˜ r cp m ˜ ξ= ; and m ˜∗ = , (2.57) kg m ˜r
θ=
where m ˜ r is a reference mass flux. These definitions lead to the following dimensionless and steady forms of (2.22) (heat) and (2.29) (species): m ∗
∂2θ ∂2X ∂θ ∂X = 2 − Dah and m = ∗ + Das , dξ ∂ξ dξ ∂ξ 2
(2.58)
where X is the mass fraction of I and Dah and Das are the heat (h) and species (s) transport Damk¨ ohler numbers. The boundary conditions were θ (0) = θs ; θ (ξ → ∞) = θf ; X (0) = Xs ;
and X (ξ → ∞) = 0.
(2.59)
The similarity of the two equations and the fact that Le = 1 leads to a relationship between the mass fraction and the dimensionless temperature: X=
θf − θ , Δh∗c
(2.60)
where the subscript c refers to the gas-phase combustion reaction. The route leading to this relationship is not entirely straightforward, and we refer the
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W.L. Perry
reader to Zeldovich for the derivation [53]. The gas phase reaction was assumed to be of the form I + X → C + X, (2.61) and the kinetic expression was 2 PM Ec r (T, X) = Ac exp − X. R RT
(2.62)
Using (2.60), the Damk¨ ohler numbers are E∗ Ec∗ , Dah = Da∗ (θf − θ) exp − c ; Das = Da∗ X exp − θ θf − XΔh∗c (2.63) where Da∗ =
kg Ac cp
P ·M m ˜ rR
2 .
(2.64)
They still required a numerical solution for a nonzero E ∗ ; however, an essential feature of this work was the authors’ recognition that a solution using high gas phase activation energy did not qualitatively predict experimentally observed temperatures and they investigated the solution behavior using a very small gas phase activation energy. In the limit, Ec∗ → 0 had the added benefit of allowing an analytical solution. In dimensionless variables the temperature and concentration profiles were ξ X θ − θf = exp , (2.65) Xs θs − θf ξg where ξg the dimensionless flame stand-off distance is 2 . ξg = √ ∗2 m ˜ + 4Da∗ − m ˜∗
(2.66)
The dimensionless form of the mass flux expression (2.55) is m ˜ ∗2 =
Ad θs2 exp (−Ed /θs ) , Ed [θs − θ−∞ − Δh∗d /2]
(2.67)
and the surface temperature was θs = θ−∞ + Δh∗d +
Δh∗c . ξg m ˜∗ +1
(2.68)
The ‘−∞’ subscript in these equations refers to the far upstream (initial) condition. Unfortunately, after systematically following the route of nondimensionalization, the Lewis number assumption, determination of the surface temperature and mass flux required iteration. What the authors found, however, is that the low activation energy assumption indeed resulted in a
2 Transport Phenomena for Nonshock Initiation Processes 2.5x10−3
70 60
2.0
50 1.5
40 30
1.0
20 0.5
Flame Stand Off [m]
Mass Flux [kg/sq.m/s]
41
10 0
6
2
105
4 6
2
4 6
106
2
4
0.0
107
Pressure [Pa]
Fig. 2.7. Mass flux from solid to gas and the flame stand-off distance as a function of pressure according to Ward et al.
fairly accurate prediction of temperature profiles, mass flux, stand-off, and surface temperature. Furthermore, the theory correctly predicts the effect of pressure: m ˜ ∝ P n, (2.69) where n ≈ 0.85, as shown in Fig. 2.7, was also in good agreement with experimental observations. We have known for some time that the burning rate of solid propellants follows this type of pressure relationship, often referred to as Vielle’s law. This relationship, however, was empirically determined, whereas the relationships leading to the theoretically determined behavior depend on physical parameters and thereby provide insight to the factors that affect the overall behavior. The theory also reveals how flame stand-off depends on pressure and this is also shown in Fig. 2.7. This behavior is particularly important; as mentioned above, the flame stand-off distance relative to characteristic surface defect dimensions determines the pressure at which laminar deflagration transits to violent explosion.
2.6 Summary In this chapter, we have discussed the general framework of transport phenomena and their central role as the theory that describes ignition and combustion behavior. The development of this theory has a rich and interesting history. Undoubtedly, in all fields of study, creative and insightful individuals advance the state of the art, and our field is no exception. We have measured or we can compute (or estimate, to a reasonable degree) the physical properties of transport theory, we know the conservation-based equations of
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transport theory, and we can therefore compute solutions to many important problems. However, in the modern era of advanced numerical solutions, we must remain cognizant of the important physical processes that govern ignition and combustion behavior. Progress requires computational advancement while maintaining our intuitive understanding.
References 1. Tarver, C.M., Tran, T.D.: Thermal decomposition models for HMX-based plastic bonded explosives. Combust. Flame 137, 50–62 (2004) 2. Kubota, N.: Survey of rocket propellants and their combustion characteristics. In: Kuo, K.K., Summerfield, M. (eds.) Fundamentals of Solid-Propellant Combustion, Progress in Astronautics and Aeronautics, vol. 90. American Institute of Aeronautics and Astronautics, New York (1984) 3. Prasad, K., Yetter, R.A., Smooke, M.D.: An eigenvalue method for computing the burning rates of HMX propellants. Combust. Flame 115, 406 (1998) 4. Perry, W.L., Dickson, P.M., Parker, G.R., Smilowitz, L.B., Henson, B.F., Asay, B.W.: Factors affecting explosive reaction violence in PBX 9501. In: Proceedings of the 13th International Detonation Symposium (2006) 5. Zerkle, D.K., Luck, L.B.: Modeling cook-off of PBX 9501 with porous flow and contact resistance, Los Alamos National Laboratory document LA-UR-03-8077; or 21st JANNAF Propulsion Systems Hazards Subcommittee Proceedings CD# JSC-CD-24 (2003) 6. Zeldovich, Y.: Flame propagation in a substance reacting at initial temperature. Combust. Flame 39, 219–224 (1980) 7. Ward, M.J., Son, S.F., Brewster, M.Q.: Steady deflagration of HMX with simple kinetics: A gas phase chain reaction model. Combust. Flame 114, 556–568 (1998) 8. Kumar, M., Kuo, K.K.: Flame spreading and ignition transients. In: Kuo, K.K., Summerfield, M. (eds.) Fundamentals of Solid-Propellant Combustion, Progress in Astronautics and Aeronautics, vol. 90. American Institute of Aeronautics and Astronautics, New York 305 (1984) 9. Asay, B.W., Son, S.F., Bdzil, J.B.: The role of gas permeation in convective burning. Int. J. Multiphas. Flow 22(5), 923–952 (1996) 10. Parker, G.R., Peterson, P.D., Asay, B.W., Dickson, P.M., Perry, W.L., Henson, B.F., Smilowitz, L.B., Oldenborg, M.R.: Examination of morphological changes that affect gas permeation through thermally damaged explosives. Propellants, Explosives and Pyrotechnics 29, 274 (2004) 11. Zerkle, D.K., Asay, B.W., Parker, G.R., Dickson, P.M., Smilowitz, L.B., Henson, B.F.: On the permeability of thermally damaged PBX 9501. Propellants, Explosives and Pyrotechnics 32, 251 (2007) 12. Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials. Int. J. Multiphas. Flow 12(6), 861–889 (1986) 13. McAfee, J.M., Asay, B.W., Bdzil, J.B.: Deflagration-to-detonation in granular HMX: ignition, kinetics and shock formation. In: Proceedings of the 10th International Detonation Symposium, 716 (1993)
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14. Faraday, M.: The Chemical History of a Candle. Cherokee Publishing Company, Atlanta, Georgia (1993) 15. Mallard, E.F.: Ann. Mine. 7, 355 (1875) 16. Mallard, E.F., Le Chatelier, H.: Combustion of explosive gas mixtures. Ann. Mine. (1883) 17. van’t Hoff, J.H.: Lectures on Theoretical and Physical Chemistry, Lehfeld, R.A. (trans.). Arnold, London (1899) 18. Zeldovich, Y.B.: On the theory of flame propagation. Zh. Fiz. Khim. 22, 27 (1928) 19. Zeldovich, Y.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions (English Translation). Consultants Bureau, New York (1985); Original Russian text: Nauka Press, Moscow (1980) 20. Belyaev, A.F.: Zh. Fiz. Khim. 12, 93 (1938) 21. Todes, O.M.: Zh. Fiz. Khim. 4, 78 (1933) 22. Seminov, N.N.: Z. Phys. Chem. 48, 571.(1928) 23. Frank-Kamenetskii, D.A.: Zh. Fiz. Khim. 13, 738 (1939) 24. Frank-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics, Appleton, J.P. (trans.). Plenum Press, New York (1969); Original Russian text: Nauka Press, Moscow (1947) 25. Bowden, F.P., Stone, M.A., Tudor, G.K.: Proc. Roy. Soc. Lond. A 188, 329 (1947) 26. Rideal, E.K., Robertson, A.J.B.: Proc. Roy. Soc. Lond. A 195, 135 (1948) 27. Merzhavanov, A.G., Barzykin, V.V., Gontkovskaya, V.T.: A problem of local thermal explosion. Dolk. Akad. Nauk SSSR 148(2), 380 (1963) 28. Field, J.E., Bourne, N.K., Palmer, S.J.P., Wallery, S.M.: Hot-spot ignition mechanisms for explosives and propellants. In: Field, J.E., Grey, P. (eds.) Energetic Materials. The London Royal Society, London (1992) 29. Eirich, F.R., Tabor, D.: Proc. Camb. Phil. Soc. 44, 566 (1948) 30. Bowden, F., Yoffe, A.: Initiation and Growth of Explosion in Liquids and Solids. Cambridge Monographs on Physics, Cambridge University Press, Cambridge (1952) 31. Afanas’ev, G.T., Bobolev, V.K.: Initiation of Solid Explosives by Impact, Jerusalem Israel Program for Scientific Translations (1971) 32. Langevin, A., Bicquard, P.: Memor. Proud. 26, 354 (1934) 33. Wyatt, R.M., Moore, P.W., Adams, G.K., Summer, J.F.: Proc. Roy. Soc. Lond. A 246, 189 (1958) 34. Tucker, T.J.: Spark initiation requirements of a secondary explosive. Ann. New York Acad. Sci. 152, 643–653 (1968) 35. Griffith, N., Groocock, J.M.: The burning to detonation of solid explosives. J. Chem. Soc. Lond. 814, 4154–4162 (1960) 36. Andreev, K.K., Belyaev, A.F.: The Theory of Explosives. Oborongiz, Moscow (1960) 37. Taylor, J.W.: The burning of secondary explosive powders by a convective mechanism. Trans. Faraday Soc. 58, 561 (1962) 38. Kuo, K.K., Summerfield, M. (eds.): Fundamentals of Solid-Propellant Combustion. American Institute of Aeronautics and Astronautics, New York (1984) 39. Bernecker, R.R., Price, D.: Studies in the transition from deflagration to detonation in granular explosives III. Combust. Flame 22, 161–170 (1974)
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40. Gibbs, T.R., Popalato, A.: LASL Explosive Property Data. University of California Press, CA (1981) 41. Dobratz, B.M., Crawford, P.C.: LLNL Explosives Handbook. National Technical Information Service, US Department of Commerce, Springfield, VA (1985) 42. Lee, D.R.: CRC Handbook of Chemistry and Physics, 88th edn. CRC Press, FL (2007) 43. NIST Chemistry WebBook, http://webbook.nist.gov/chemistry/ 44. Incropera, F.P., Dewitt, D.P.: Fundamentals of Heat and Mass Transfer, 5th edn. Wiley, New York (2002) 45. Loeb, L.B.: The Kinetic Theory of Gases, 3rd edn. Dover Publications, Mineola, New York, (2004) 46. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (1960) 47. Perry, W.L., Sewell, T.D., Glover, B.B., Dattelbaum, D.M.: Electromagnetically induced localized ignition in secondary high explosives. J. Appl. Phys. 104(1), 094906-094906-6 (2008) 48. Cooper, P.W.: Explosives Engineering. Wiley-VCH, New York (1996) 49. Parker, G.R., Asay, B.W., Dickson, P.M., Henson, B.F., Smilowitz, L.B.: In: Shock Compression of Condensed Matter, American Physical Society Topical Conference, Portland, OR, 20–25 July 2003 50. Dickson, P.M.: Private Communication 51. FlexPDE 3. Copyright PDE Solutions Inc., Antioch, CA (2003) 52. Merzhanov, A.G., Dubovitskii, F.I.: Proc. USSR Acad. Sci. 129, 153–156 (1959) 53. Zeldovich, Y.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions (English Translation) Consultants Bureau, New York, 98–103 (1985)
3 The Chemical Kinetics of Solid Thermal Explosions Bryan F. Henson and Laura B. Smilowitz
3.1 Introduction In this chapter, we discuss the physics and chemistry of thermal decomposition in energetic materials. We adopt the traditional approach of chemical kinetics, where the chemistry of decomposition and the resulting conversion of chemical potential into heat are captured by systems of differential equations whose variables are the concentrations of original reagent and product chemical species. Rates of transformation are simulated using temperature-dependent equations for the rate constant and the particular mechanisms of chemical transformation are reproduced in the form of the concentration equations. The energy liberated or consumed by reaction is then a function of the calculated concentration change either as a Hess’s Law calculation constrained by real chemical species and their heats of formation or more generally as a free parameter for an unconstrained, unidentified chemistry. These systems of equations then serve as thermal source (sink) terms in the nonsteady heat equation, within which other thermo-physical parameters, such as the average heat capacity or thermal diffusivity, as well as overall geometry and thermal boundary condition, must also be specified. Such models are necessarily global in nature, and given realistic computational resources cannot hope to encompass the full chemical complexity of the decomposition of a material like octahydro-1,3,5,7-tetranitro-1,3,5,7tetrazocine (HMX), an 8 member, 28 atom nitramine ring. Indeed in a paper published more than 20 years ago, Shackleford [1] noted The scientific literature is replete with decomposition studies of the cyclic nitramines but their critical reaction steps and overall chemical mechanisms are still subject to spirited scientific debate. Depending on how one counts them, as many as 8–13 chemical mechanisms have been proposed, and collectively, they eventually identify each type of covalent bond in the HMX molecule as being an important participant in the decomposition process B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 3, c Springer-Verlag Berlin Heidelberg 2010
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On the one hand, it is likely that the authors of these mechanisms, which were generated by sound methods of elemental chemical analysis, were entirely correct in their individual attributions of importance to each type of covalent bond in this molecule. On the other hand, it also seems unlikely that the full suite of possibilities in such a complex decomposition process will be equally important for our goal, which is the construction of models that reproduce a very specific mode of decomposition in an energetic material, that lead to thermal ignition. The fundamental idea in the theory of solid thermal explosions is a simple, albeit highly nonlinear one, and aids in the identification of the subset of the full chemical complexity of a molecule necessary here. We are interested in thermal ignition, a nonlinear process wherein chemical energy is released and temperature increased, with a rate that depends on temperature. We therefore have important criteria for selection of the decomposition chemistry relevant to this process. The reactions we are concerned with must either themselves release or consume energy sufficient to modify the temperature of the mixture, or provide reagents that participate in such reactions. Furthermore, the extremely nonlinear heating and acceleration of decomposition observed prior to a thermal explosion indicates that some reactions must possess an extreme sensitivity of reaction rate on temperature. These criteria require that not just elemental transformation from one species to another be considered but that those transformations which can be shown to ultimately generate heat and accelerate reaction be identified and incorporated into models discussed here. Historically, the earliest attempts to model this problem were based on analytical solutions to the nonsteady heat equation with necessarily very simple source terms, for example, the Semenov and Frank-Kamenetskii treatments discussed in Chaps. 2 and 7. As discussed previously, the analytical solutions generated in this work were of considerable value in elucidating the overall behavior of thermal explosions. Early systems of equations meant to simulate chemical decomposition in the computer modeling of thermal explosions were also very simple but led to important new insight into the relationship between decomposition and thermal explosion, as well as identifying avenues for the introduction of more realistic, and complex, models of decomposition [2, 3]. These simple models serve as the starting point for this chapter. After a discussion of classical kinetics and the construction of the set of differential equations that encodes a decomposition model, we review the early work and discuss the strengths and weaknesses of the general source term models and applications of such models to a number of energetic materials. We discuss the relative advantages and disadvantages of systematically increasing complexity of the source term model used, from simple homogeneous material heating to multiphase considerations, complex solid states of the material, to a final
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consideration of a fully chemical global decomposition model. As an example of such a complex model we present the current state of attempts in our group to produce a chemically accurate global decomposition model for HMX-based explosive formulations. Although we present the modeling considerations here primarily in the context of direct heating and classic thermal explosions, we do not restrict our considerations to a particular regime of temperature or heating rate. Indeed, particularly in the context of the fully chemical global models, the application of the thermal response model is in principle independent of the temperature or rate of heating, and in the limit that the relevant chemistry over the entire range of temperature is included, such models represent a comprehensive thermal response model. We have devoted a considerable amount of effort toward validating this idea as applied to HMX [4–7]. In this spirit, we do not intend to present this work as equivalent to the very complex chemical models constructed for combustion applications [8, 9], or the complex mathematical models used for the deflagration-to-detonation transition [10]. We do intend to explore where the overlap in response to such models exists, and where the solid state thermal response discussed here may compliment or expand upon these other models. Finally, the quote noted above begs a further question. Why pursue more complex models, and in particular, why pursue a more complex model of HMX? The answer is straightforward and has been reiterated throughout this book. Much more is now required of these models than ever before. We are well past a time when the calculation of a time to ignition as a function of temperature sufficed to answer questions of safety regarding explosive systems. Models of decomposition must now be coupled with models of ignition and the subsequent release of energy to fully understand thermal response. Thermal decomposition models must therefore provide more than a spatially resolved source (sink) of heat but must also provide a spatially resolved rate of mass loss to enable calculations of gas phase pressurization. They must also provide the mechanism to couple that mass loss into a model of solid phase morphology in order to address the ability of the material state to support either convective or conductive burning. There has been significant progress recently toward realizing these ambitious goals, and it is useful to note what has been required to realize it. The temperature and kinetics of the β-δ phase transition in HMX have been the subject of study for more than 40 years, yet until recently there was considerable uncertainty even in the equilibrium transition temperature. To fully resolve these difficulties, a means of resolving the relative fraction of δand β-HMX within the solid phase during transformation was required. The discovery of the extreme nonlinear optical susceptibility of HMX and other energetic molecules provided such a probe in the form of the laser spectroscopic technique second harmonic generation (SHG) [11–13]. With this new probe, the separation of nucleation and growth components of the transformation was
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enabled, and the detailed temperature dependence of the transformation was determined. It was immediately recognized that this temperature dependence was consistent with a process governed by melting, albeit at temperatures more than 100 K below the melting point [14], and this led to an entirely new model of solid phase transformation [14–19]. This is not only a new model of HMX transformation or even of energetic material transformation, but an altogether new and general model of solid phase transformation that has already found application to other fields of solid state physics [20]. Further work has revealed a similar thermodynamic constraint in the intermediate steps of the decomposition and ignition of HMX-based materials. The ignition rate was shown to be composed of rates of both sublimation and vaporization, again even though no direct evidence of equilibrium liquid HMX was apparent [4, 7]. Investigations into the complex physical chemistry of liquid phases below the triple point, including the quasiliquid interfacial phase and premelting phenomena, resulted in another general breakthrough in a broader field of physical chemistry, with a new theory of the quasiliquid phase based on the interfacial liquid activity that has resolved decades old questions regarding the extent of the phase as a function of temperature [21]. Work on the final nonlinear acceleration and ignition mechanisms has led to new laser techniques to control and synchronize thermal ignition for the first time [22, 23], and using these techniques, the first radiographic images of the evolution of density within the solid during ignition and the transition to burning have recently been obtained [24, 25]. We will present a suite of global chemistry models for HMX-based materials, which are the results of this progress. Clearly, the significance and scientific breadth of the progress required to produce these models is a testament to the difficulty of these problems. The chapter is organized as follows. We initially discuss the various modeling strategies traditionally applied to the energetic thermal decomposition problem. These span the spectrum of generic source terms in the nonsteady heat equation discussed previously to attempt to seriously constrain the modeling of response by the application of realistic global models of known chemical steps in the decomposition. These strategies are then discussed in the pragmatic context of modeling objectives, where the computational and conceptual cost of more sophisticated models are justified by increased fidelity and additional observables generated as output. We then discuss a specific global modeling effort applied to the HMX plastic-bonded explosive PBX 9501, with particular attention first to the state of understanding of the endoand exothermic decomposition chemistry of HMX and how this understanding constrains the construction of rate constants and kinetic order in the thermal source (sink) model. We then discuss the current state of the resulting global models. We finish with a discussion of the output of these models and how they inform the final, post-ignition component of the problem.
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3.2 General Considerations 3.2.1 Chemical Kinetics and Explosives Modeling The canon of chemical kinetics established over the last two centuries is of course the subject of a voluminous literature and a number of classic texts. In this brief discussion we will review some central ideas that will repeatedly appear as we construct the models of thermal decomposition that are the topic here. In particular, a review of the derivations of the chemical rate constant, the temperature dependence of the chemical rate, and the relationship between kinetic rate and thermodynamics will be of value in Sect. 3.3 as we construct rate constants and kinetic equations based on assumptions of mechanism. While there are a number of classic texts on chemical kinetics, we rely heavily upon [26] and references therein for the particular derivations reviewed here. We refer here, as is traditional, to the rate constant k. The tradition stems from the reality that early kinetic measurements were almost exclusively performed under isothermal conditions, where the temperature dependent rate constant is actually constant, enabling analysis of data in the pre-computer era. In problems of thermal decomposition and ignition, of course some experiments are not isothermal. Also in some instances it is preferable to conduct calorimetric measurements of even very fundamental components of the overall chemistry under nonisothermal (typically linear heating) conditions. We will not change the nomenclature here and will refer to k as the classic rate constant, even in nonisothermal conditions, with the hope that this caveat is sufficient to avoid confusion. Temperature Dependence: The Arrhenius Rate Constant The classic Arrhenius equation for the temperature dependence of the rate constant is used throughout the chemical kinetics literature, and particularly as the generic form for the temperature dependence of the rate of heat generation by decomposition in the nonsteady heat equation discussed in Chaps. 2 and 4. The empirical form of the equation is given by −ΔE , (3.1) k = A exp RT where k is the reaction rate constant, A is a prefactor (integration constant) whose units are determined by the order of the reaction, E is an activation energy, R is the ideal gas constant, and T is the absolute temperature. It is noteworthy here that the original justifications of this equation were derived from consideration of the temperature-dependence of the equilibrium constant. Applying the idea that chemical equilibrium represents a dynamic balance between opposing rates rather than a static condition, van’t Hoff derived (3.1) as the form of the rate constant governing both forward and reverse reactions in equilibrium (k1 and k−1 , with energies E1 and E−1 , in modern
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B.F. Henson and L.B. Smilowitz
nomenclature) [27]. He then showed that the two energies must be such that the difference E1 –E−1 = ΔE, the internal energy difference between the product and reagent species. It was Arrhenius who first proposed that ΔE in (3.1) represented the energy difference between reagent species and an activated form of the reagents in a state intermediate between reagents and products [28]. Subsequent work using similar considerations of chemical equilibrium and utilizing early versions of what would become the Gibbs free energy yielded similar equations [29] −ΔGa k = νa exp , (3.2) RT where νa is again an integration constant that determines the units of the rate constant and ΔGa is now a free energy of activation, which can be decomposed into an activated entropy and enthalpy. Although providing conceptual models of reaction still favored today, there was no theory in classical physics or chemistry with which to deduce the form of the prefactor A, the rate constant at infinite temperature. Conventional transition state theory (TST) provides a theoretical framework for a rigorous understanding of the rate constant. Although the subject of much work and again a voluminous literature [30], the original formulation of TST in its simplest form will suffice here [31, 32]. TST is built upon Arrhenius’ idea of an activated complex, consisting of the combined reagent species and somehow representing a transitional structure on the way to products, and utilizes van’t Hoff’s equilibrium to construct the formula for the rate constant. Four original assumptions concerning the dynamics of transition through the activated complex are required: (1) passage through the transition state is irreversible, (2) the internal energy distribution of the reagent species is at thermal equilibrium, (3) the degree of freedom representing the motion of the complex through the transition (e.g., translational, vibrational) may be separated from the other degrees of freedom of the reagent species, and (4) quantum mechanical tunneling is not involved in passage through the transition state. The derivation of the rate constant begins with a set of equalities derived from the aforementioned assumptions. We review a derivation of the rate constant for bimolecular reaction, as it is of the most utility in the problems discussed in this chapter. The starting point is the expression of equilibrium between the reagent species and the activated complex [26], expressed as a function of concentrations (where A + B form the activated complex Xa ), the classic equilibrium constant, the free energy difference between the two states, and the expression of the equilibrium constant as a function of the molecular partition functions of all species and their internal energy difference [33] −ΔEa −ΔGa [Xa ] [qa ] = Ka = exp exp = . (3.3) [A][B] RT [qA ] [qB ] RT The next step requires specification of the degree of freedom assumed to capture the motion through the transition state. Classic derivations exist based
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on the assumption of both translational [31, 32] and loose vibrational [34] degrees of freedom. Both derivations yield equivalent results. We will review the derivation assuming that a passage through the transition state proceeds via a loose vibrational state of the activated complex. We start with (3.3) and note that the molecular partition functions, qi , are product functions of the translational, rotational, and vibrational degrees of freedom of each species. We make the assumption that one of the degrees of vibrational freedom associated with the activated species represents the coordinate describing the passage through the transition state. The function qa is then a product of one loose degree of freedom and 3 (nA + nB ) − 7 remaining vibrational degrees of freedom (typically assumed to be the uncoupled product states of the two original reagent species), where ni are the number of atoms of each species. The term loose is specifically used to indicate that this degree of freedom ultimately leads to separation of the activated complex into stable product states, and therefore is likely a very low frequency vibration. This assumption allows one to use the molecular partition function for simple harmonic vibration in the zero frequency limit, qv = kB T /hν, where kB and h are Boltzmann’s and Plank’s constants, respectively, and ν is the vibrational frequency. Separating out the single vibrational degree of freedom and assuming qa = qa qv , where qa is the product molecular partition function of the remaining degrees of freedom in the complex yields −ΔEa [Xa ] kB T [qa ] = exp , (3.4) [A][B] hν [qA ] [qB ] RT which can be rearranged to give ν [Xa ] = [A][B]
kB T [qa ] exp h [qA ] [qB ]
−ΔEa RT
.
(3.5)
Finally, ν [Xa ] is the classic rate of reaction, the frequency of passage through the transition state multiplied by the concentration of activated species. The concentration dependence [A][B] is that expected of a second order bimolecular reaction. Equation (3.5) is then the classic second order kinetic equation, with rate constant −ΔEa kB T [qa ] exp k= . (3.6) h [qA ] [qB ] RT Equation (3.6) is a very useful starting point for the application of the Arrhenius equation to problems related to explosives and explosive modeling. The first and most fundamentally satisfying feature is an elucidation of the prefactor in (3.1) and (3.2), particularly the universal frequency factor kB T /h, which establishes the molecular time scale for reaction and enables calculation of activation enthalpy and entropy if temperature-dependent rate data exist. Also satisfying is the identification of the mechanism by which higher order rate constants derive their units. The assumption of equilibrium between reagent and activated species establishes the concentration dependence through the equilibrium constant.
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The assumption of thermal equilibrium is of course of significant practical importance. The exponential temperature dependence of the rate constant ultimately derives from the equilibrium distribution of energy amongst the mechanical degrees of freedom of the reagent species. This is a molecular characterization of equilibrium and should not be confused with the overall thermal equilibrium of the bulk sample. This assumption of molecular equilibrium distinguishes the use of this rate constant from other approaches such as mechano-chemistry or electronically excited chemical pathways initiated by, for instance, a laser. The assumption of the equilibriuim distribution of internal energy, typically achieved on the order of nanoseconds or less, is an excellent one when considering most reactive processes. It can fail in explosive applications, however, when considering processes fast on this time scale, such as applications to detonation or the transition to detonation. However, even in this case, for HMX, we have argued in the past that the assumption of molecular equilibrium appears to be justified [6]. The dependence of the rate constant on pressure can be shown by expansion of the activation enthalpy as an internal energy and work term due to isotropic pressurization of an activation volume, ΔVa , assumed to be the volume necessary to form the activated state T ΔSa − (ΔEa + P ΔVa ) T ΔSa − ΔHa kB T kB T exp exp k= = . h RT h RT (3.7) It is important to properly define the pressure in (3.7), and such definition requires assumptions concerning the mechanism it is intended to reflect. For instance, in solid state processes P is commonly taken to be the isotropic, quasistatic pressure in the sample, and the volume of activation reflects the perturbation to the solid density required to affect the transformation. More sophisticated understanding of the effects of mechanical perturbation to the chemistry might be incorporated, however, by expanding the pressure dependence as a function of the stress tensor of the material. This enables sophisticated considerations of the effects of crystal orientation and nonuniformities in pressurization to be included. Of course, if the mechanism under consideration is a gas phase reaction, then the volume of activation is difficult to interpret, as P and pressure effects are more commonly incorporated in the concentration dependence of the reaction. For models that include both gas and solid phase processes, it is of paramount importance to distinguish between solid stress and gas phase pressurization, which are only equivalent in a truly quasistatic environment such as a fluid supported DAC experiment. The expansion of (3.1) into more fundamental components, (3.6) or (3.7), is not always necessary in the construction of models. It is nonetheless motivated by the additional information carried by the more complex equations, particularly the connection to thermodynamics and thermodynamic phase. For instance, many fundamental kinetic processes in solid decomposition may proceed with rates activated by energies, which are equivalent to changes
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in thermodynamic state, such as when sublimation or vaporization rates are constructed from the latent heats of transformation to the gas phase. In particular for HMX, it was the observation that the rate of growth in the βδ polymorphic phase transformation was thermally activated by an energy equal to the heat of fusion that led to the discovery of a new general class of solid phase transformation [15], and the construction of a very comprehensive specific model of the β-δ transformation in HMX [12, 14], which is the first important step in normal decomposition. This will be discussed further in sections “The Crystalline Phases of HMX and Rates of Transformation” and “The β-δ Transition.” As a brief example, we construct a rate expression for sublimation, which will help to illustrate this increased information content, provide a specific rate expression for use in the next section, and will be used as a step in the global chemistry model of HMX decomposition presented in Sect. 3.3. The process of sublimation is extremely important in solid explosive modeling. Sublimation is a process known relatively well, in particular when compared with covalent bond breaking in large solid explosive molecules, and can be sufficiently endothermic to impact energy balance during decomposition, particularly as the temperature increases to ignition. The rate of sublimation is one of the two competing rates that determine the equilibrium vapor pressure, the other being the rate of collision with the surface by gas phase molecular species. The two rates balance at equilibrium, but are not necessarily equal, as every collision from the gas phase may not lead to incorporation into the solid. The fraction representing this probability of incorporation is the accommodation coefficient [35]. For our purposes, as in some of the earliest applications of TST [34], we consider this coefficient to be equal to one. Experiments on fundamental aspects of sublimation continue and are remarkably recent, with one study in particular demonstrating that molecules leave the surface with a Maxwell–Boltzman (MB) distribution in the velocity reflecting the surface temperature, and resolving the accommodation coefficient as a function of the internal quantum states of the molecule [36]. The collision frequency with a surface follows from the MB distribution of velocities for particles of mass m in the gas phase at temperature T . Briefly, the number of molecules, n, in a volume element dV in the gas phase is given by [37] (3.8) n = ρ dV = ρr2 dr sin θ dθdϕ, where ρ is the gas phase density, r is a ray emanating from the origin of a coordinate system on the surface to that volume element, and the spherical polar volume element is defined using the polar angle θ and azimuthal angle in the surface plane using ϕ. The fraction of the number of molecules in this volume element with velocities between v and v +dv is given by an integration over the MB distribution, f (v)dv. n = f (v) dvρr2 dr sin θ dθdϕ,
(3.9)
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where the MB distribution is m 2/3 −mv 2 exp f (v) = 4π v2 . 2πkT 2kT
(3.10)
Assuming a radially symmetric flux of molecules out of the volume element, we calculate how many will intersect an area A on the surface, centered at the origin of the spherical polar coordinate system. This fraction of the total number of molecules in the volume element is given by the ratio of the projected cross section of the area A, viewed at an angle θ from the volume element dV , over the total solid angle at a distance r from A, or n = ρf (v) dvr2 dr sin θ dθdϕ
A cos θ . 4πr2
(3.11)
The flux of such collisions is calculated by integrating r over velocities sufficiently close to the surface to reach it in 1 s, or integrating over r from zero to v (r = vt, where t = 1 s) as ∂n =ρ ∂A∂t
∞
v
dr [f (v) dv] v=0
r=0
1 4π
π/2
2π
sin θ cos θ dθ θ=0
dϕ.
(3.12)
ϕ=0
The final integration over the MB distribution multiplied by v then gives the mean velocity as ρ ∞ ρ ∂n = vf (v) dv = v, (3.13) ∂A∂t 4 0 4 where
v=
8kT πm
This yields ∂n ρ = ∂A∂t 4
1/2 .
8kT πm
(3.14)
1/2 (3.15)
and using the ideal gas equation ρ = P/kT leads to P ∂n . = √ ∂A∂t 2πmkT
(3.16)
This is the classic Hertz–Knudsen equation for the rate of collision with the surface from the gas phase per unit surface area. A TST rate equation for an equivalent reaction whereby a gas phase molecule becomes incorporated into the solid can be written, beginning with (3.3), as (for a single reagent species) −ΔEa [Xaarea ] [q area ] exp = a , (3.17) ρgas [qgas ] RT
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where the variables have the same assignments as before. The concentration and molecular partition function for the activated state are in units of molecules per unit area and for the reagent gas phase in molecules per unit volume. As before, a frequency of vibration along a reaction coordinate is extracted from the molecular partition function of the activated state and rearranged to give −ΔEa kB T [qaarea ] area exp ν [Xa ] = ρgas . (3.18) h [qgas ] RT At this point a model of the reaction must be provided. If we wish to recover (3.16), a suitable model involves partition functions that describe two- and three-dimensional (2D, 3D) translation for the activated and reagent states, respectively, and a mechanism of incorporation into the solid with no activation barrier, ΔEa = 0. This yields
ν
[Xaarea ]
(2πmkB T ) kB T h2 = ρgas . h (2πmkB T )3/2 h3
(3.19)
Again using ρ = P/kT leads to ν [Xaarea ] =
P ∂n = √ . ∂A∂t 2πmkB T
(3.20)
The assumption of a 2D translational partition function for the activated state evokes a mobile surface phase as a precursor to incorporation of a molecule that has struck the surface. Finally, and equivalently, we construct a TST rate constant for crystalline solid sublimation using −ΔEa [q area ] [Xaarea ] exp = a , (3.21) ρsolid [qsolid ] RT where we again assume a two-dimensional activated state and a threedimensional solid reagent state. For a crystalline solid, which we have carefully defined here, ρsolid is fixed and independent of the amount of solid. If one is dealing with a non-crystalline solid solution or amorphous solid, for which ρsolid is not fixed and is better represented by a true concentration [csolid ], then (3.21) takes the form of a true first order TST rate constant with concentration dependence as −ΔEasolution ] kB T [qaarea exp ν [Xaarea ] = [csolid ] , (3.22) h [qsolution ] RT where [csolid ] denotes a non-crystalline phase of varying concentration. Two important considerations of the reagent phase are important here. A condensed phase at equilibrium with the vapor is at the same chemical potential
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as the vapor. This is the definition of equilibrium. If the solid is a crystalline solid, then, neglecting surface and grain boundary effects, the chemical potential of the solid is constant. No matter how much of the crystalline solid is present, the vapor pressure above it is constant, as is the sublimation rate. The non-crystalline phases of varying concentration are also phases of varying vapor pressure and consequently varying sublimation rate. Proceeding with the assumption of constant crystalline solid density, ρsolid = n/V , as well as writing the dependence on dimension in the solid state molecular partition function leads to n area [qa ] −ΔEa area V Xa = qsolid exp . (3.23) RT V We define the complete solid phase partition function as q=qsolid exp (ΔEa /kT ) in molecular units (substitute k for R), and substitute into (3.23), [qaarea ] q . n We use a series of equalities from Hill [33], q μ = exp − , n kB T μ − μo , P = exp kB T Xaarea =
and for an ideal gas
(2πmkB T )3/2 μo = ln kB T − kB T h3
or exp
μo kB T
=
h3 1 . kB T (2πmkB T )3/2
(3.24)
(3.25) (3.26)
(3.27)
(3.28)
Substituting (3.24) into (3.25) gives Xaarea =
[q area ] a . μ exp − kB T
Rearranging (3.26) and substituting into (3.29) gives μo Xaarea = [qaarea ] exp − P. kB T
(3.29)
(3.30)
Substituting (3.28) into (3.30) gives Xaarea = [qaarea ]
h3 1 P. kT (2πmkB T )3/2
(3.31)
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Making the same assumption of a delocalized 2D activated state at the interface modeled with a 2D translational partition function and extracting the vibrational partition function of the reaction coordinate yields [qaarea ]=
kB T 2πmkB T . hν h2
(3.32)
Substituting (3.32) into (3.31) gives Xaarea =
h3 kB T 2πmkB T 1 P, 2 hν h kB T (2πmkB T )3/2
which reduces to
(3.33)
P ∂n = √ . (3.34) ∂A∂t 2πmkB T We see that once again the assumption of a mobile 2D activated state at the surface is necessary to reconcile the three different derivations of the sublimation flux. Such a precursor could play a significant role in a decomposition mechanism, and we will discuss this possibility in the mechanism of HMX decomposition in Sect. 3.3. The detailed evaluation of these rate expressions for the general process of solid sublimation leads to several important questions concerning realistic chemical modeling of solid thermal decomposition. The identification of a third phase, delocalized on the surface, could have serious ramifications to the mechanism. Does the molecule in this phase exhibit different reactivity than in the gaseous or solid phase, perhaps more resembling the liquid phase? Is there direct decomposition from this phase, especially at elevated temperatures or as the melting point is approached? Does the phase persist long enough, or lead to branches in the decomposition pathway giving rise to different products, such that the latent heat associated with the conversion of a molecule in the crystalline lattice to one in the delocalized surface state needs to be expressed? A framework to discuss the impact of these decisions is illustrated in Fig. 3.1 using the heat of formation and the latent heats for the main thermodynamic phases of HMX. On the left of the figure, the energy levels of the β and δ solid phases, the liquid and the vapor phase are shown plotted on the left ordinate relative to the heat of formation. On the right of the figure an energy diagram of a model trajectory from the δ solid phase to the vapor phase is shown as either a direct path or one composed of two steps and explicitly including the liquid. These trajectories are plotted on the right hand ordinate as the latent heats of each phase, assuming the gas phase potential of zero. Both ways of plotting the energy are equivalent, and differ only in the relative zero of energy. The right hand side is often encountered when dealing with the thermodynamics of a material, and the latent heats of the different phases are more apparent and easier to track. The left hand side is necessary when decomposition and the rearrangement of atoms from reagents to their products are calculated, as the relative zero on the left hand side is the thermodynamic standard where the elements in their pure state are taken to have ν [Xaarea ] =
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Fig. 3.1. Energy levels in the HMX system plotted as a function of a generalized reaction progress from β to δ to liquid to vapor. On the left ordinate, the energies are plotted relative to the heat of formation, and on the right as latents heats of transformation relative to a gas phase energy of zero
zero heat of formation. From the trajectory for sublimation, the assumption that the latent heat of sublimation serves as the energy of activation is intuitive, as is the assumption that condensation proceeds without a barrier. Neither is of course strictly or always true; however, the sense of the approximation is clear. The ramifications of separating out a trajectory that proceeds through the liquid (or delocalized surface) state is also made more clear. If, on the one hand, there is no role for the liquid other than as a transient state in between the solid and the gas phase, then one can consider the overall process as the product of two dependent probabilities, that of transition from solid to liquid, and then from liquid to gas. In this case, the two rates can be compared as ksub = kmelt kvap . Taking the product of the exponentials it is clear that the activation energy is the same. The comparison of the entropy for the overall process as compared to the component processes is more interesting, but we will not pursue that further here. If for some reason the component processes cannot be considered dependent, for instance if the time spent in the liquid state is too long by some measure, then an expression of rates based on the sum of the component rates would clearly not be equivalent to the single overall step.
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The construction of a sublimation rate constant from (3.21), assuming a solid reagent and 2D activated state, also illuminates the particular nature of the crystalline solid as a reagent phase of constant chemical potential. The difference between such a state and one where the chemical potential varies as a function of concentration or morphology, such as a solid solution or amorphous phase, respectively, is apparent from (3.22). In states of variable chemical potential, the loss of reagent to reaction decreases the concentration in a way that can naturally be accommodated by an assumed concentration equation. This is not so for the crystalline solid decomposing by sublimation and vapor phase reaction, for instance. In this later case the flux of molecules to the vapor phase evolves not as the concentration, which is constant, but as the shape or morphology of the solid and the surface area presented to the vapor. This is a fundamentally different type of kinetic order and involves different assumptions. We will discuss this in more detail in the next section on concentration dependence. It is also useful to be familiar with actual rates of the component processes in a model, particularly when they have been directly measured. For instance, again using HMX as an example and the measured vapor pressure as a function of temperature of δ phase HMX in (3.34), we obtain the sublimation flux at 473 K as 9.14 × 1020 molecules m−2 s−1 . This is the areal flux through a surface cross section of 1 m2 in 1 s. Put another way, a square meter of HMX surface exchanges about half a gram per second with the vapor at equilibrium. Given an approximate molecular diameter of 6.5 A, this also corresponds to almost 400 layers a second. These rates are quite important to a number of quantitative considerations, like rates of Ostwald ripening and the evolution of solid morphology over time. They are also important to a number of more qualitative considerations, such as the suitability of the idea of a surface as a dividing plane between the solid and vapor phase. On the time scale of molecular vibration or an elastic collision, say 50 fs, this idea is quite robust. On the other hand, how does this flux affect the diffusion of energy or mass along the surface, which occurs on millisecond timescales? Or more interestingly, how is the idea of an activated molecular state in sublimation, the lifetime of which we have not specified, impacted? A useful expression for comparison to data involves conversion of the flux into a linear regression rate using the molar volume Vm , Avagadro’s number No , and the equation ∂L Vm ∂n = , (3.35) ∂t No ∂A∂t which yields a linear rate of 214 nm s−1 at 473 K. At this rate the sublimation of a typical 100 μm diameter spherical crystal of HMX would take 234 s (three and a half minutes, faster if one includes the change in solid chemical potential as a function of the radius of curvature for diameter less than ∼100 nm through the Kelvin effect). Of course this rate would be difficult to achieve, even under high vacuum, as the high flux from the surface at this temperature
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would likely constitute a hydrodynamic flow condition, rather than a Knudsen regime where ballistic flight from the surface essentially removes material from the surface without return [35]. It is a useful approximation however, and distillation of HMX from hot surfaces in the lab is actually readily achievable on this time scale. A second approximation involves an expression where the specific area per unit mass or volume can be used to generate a volumetric rate. This is useful for decomposition models that are implemented without sophisticated solid morphology models. We have shown that the specific area per unit mass for HMX particles smaller than approximately 100 μm can be modeled as solid oblate cylinders, very nearly spherical [38]. For simplicity, the ratio of the surface area to volume for a sphere is A/V = 6/d, where d is the diameter. For the same 100 μm diameter particle this is ∂A/∂V = 6 × 104 m2 m−3 . Multiplying this surface to volume ratio by the surface flux and normalizing the volumetric rate to the constant molar volume of δ HMX yields the first order rate constant Vm ∂A ∂n . (3.36) k= No ∂V ∂A∂t A calculation of the rate constant at 473 K yields 1.40 × 10−2 s−1 , or a half life for sublimation, independent of the total mass of 100 μm particles, of 50 s. This approximation is of the same order of magnitude, although faster. This volumetric approximation is very simple, however, as the surface to volume ratio will typically change as a function of regression. This will be discussed in more detail in the next section on concentration dependence. Another tempting volumetric approximation involves (3.22) and the assumption that the crystalline solid is a solid solution of concentration equal to the solid density. We assume that the partition functions can be combined with the activation energy to yield the activation free energy of the solid as T ΔSa − ΔHa kB T exp k= . (3.37) h RT If we make the further assumption that the activation free energy is equal to the normal solid free energy change on sublimation, as we do in the derivation of (3.34), ΔGa = ΔGsolid , then the rate constant at 473 K is equal to 1.3 × 107 s−1 . This is far too large to be reasonable and indicates the danger of simply applying variations of (3.2) because one can make a simple association of the mechanism with some known physical process. The details of the mechanism, particularly the dimensionality indicated by the mechanism, are very important. On the other hand, it is possible to make useful approximations in this way, and we will discuss one such volumetric approximation to a sublimation rate for HMX in the section “Comparison of Kinetic and Thermodynamic Rates.” Briefly, using an independent method for approximating the activation entropy for the process, we use (3.37) to describe the rate of sublimation of
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HMX and obtain a rate constant equal to 2.3 × 10−5 s−1 . This is considerably smaller than the first order rate of (3.36), and implies half times for sublimation of hours instead of seconds, but this is expected for this very simple zero order approximation. The last point concerning sublimation is a reminder that unless a loss mechanism for the gas phase species exists, the high rates of sublimation and collision with the surface do not lead to any net transfer of mass from the solid. However, the loss of vapor to leaks or, more likely, to gas phase reactions are loss mechanisms, and if vapor is lost at rates comparable to those discussed above for sublimation, then the overall rate of reaction may be sublimation limited. Finally, the temperature sensitivity of a rate constant is determined by the magnitude of the activation energy in relation to kb T (or RT). This is another consequence of the assumption of internal thermal equilibrium in the reagent molecules. It is useful to examine the simple temperature dependence of a few real rate constants, and this is done in Fig. 3.2. These constants have been taken from the global decomposition chemistry mechanism used to model HMX decomposition and will be described in detail in Sect. 3.3. The general point is the concave down shape of the rates as a function of temperature for most of the rates shown. In the limit of infinite temperature e−1/T goes to
Fig. 3.2. Rate constants plotted as a function of temperature to illustrate the asymptotic approach to the limiting rate at infinite temperature. These rate constants are taken from the global model of HMX decomposition to be discussed in Sect. 3
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1. As the temperature increases in relation to RT the rate approaches this limit, the shape of the curve being determined by Ea , and the limiting rate by A in (3.1), or kB T /h exp (ΔSa /R) in (3.7). Rates k2a and k2b exhibit this behavior. The situation is more complex for highly coupled rate expressions, such as k1 of Fig. 3.2, which represents the overall rate of the β-δ phase transition in HMX. The rate decreases from above and below near the equilibrium transition temperature of 432 K. This is due to the typical observation that the thermodynamic driving force for an equilibrium transformation increases as the difference in free energies of the two phases increases. This difference goes to zero at the equilibrium transition temperature and exactly at that temperature is zero. The mechanism from which the rates shown in Fig. 3.2 are taken reproduces this behavior while accurately reflecting the actual kinetic mechanism of transformation. Rates k3a and k3b are examples of radical reactions that do not typically exhibit a large activation energy, and proceed at very fast rates independent of temperature. We will discuss this in more detail in the next section and in section “The Crystalline Phases of HMX and Rates of Transformation.” Concentration Dependence: Kinetic Order The overall rate of a process is also determined by the dependence on concentration expressed by the mechanism. Kinetic order is a classic term used to describe this concentration dependence as a power law, order being the value of the exponent, as in j ∂[P ] =k [Ci ]oi (3.38) ∂t i=1 for a reaction forming P and involving j reagents of concentration [Ci ]. A particular reaction described by (3.38) is said to be of oi th order in Ci , and the overall order is the sum of the oi . The particular form of (3.38) is often suggested by the mechanism of the reaction, particularly for reactions that proceed in well mixed, homogeneous phases such as liquids or gases, as in the product of concentrations in the binary reaction of (3.5) in section “Temperature Dependence: The Arrhenius Rate Constant,” where the product can be thought of as proportional to a collision probability. In chemical systems that are not homogeneous or well mixed, such as the solid phase reactions typical of the solid explosives under study in this book, the use of (3.38), or even the idea of a dependence on concentration, can be difficult. In such cases, it is of course more correct to use a model describing morphology as well as amount, such that sublimation rate, for instance, may be calculated as a function of the actual surface area available in the sample. There are many reasons, however, that a simplified concentration model might be required instead. Typically the complexity of a full model of morphology is neither obvious from available data nor allowed within the computational capability of a full thermal response simulation.
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Table 3.1. Kinetics of sublimation with various mechanistic approximations Order
Rate equation ∂L Linear = kL ∂t
Rate constant ∂L Vm ∂n = ∂t No ∂A∂t
Units
Integrated rate law
m s−1
c = c0 3 kL t × 1− R
Zero
∂c = −k0 ∂t
k0 =
1 ∂A ∂n No ∂V ∂A∂t
mol m−3 s−1
c = c0 − k0 t
First
∂c = −k1 c ∂t
k1 =
Vm ∂A ∂n No ∂V ∂A∂t
s−1
c = c0 e−k1 t
Second
∂c = −k2 c2 ∂t
k2 =
Vm2 ∂A ∂n No ∂V ∂A∂t
m3 mol−1 s−1
1 1 − = −k2 t c0 c
Mixed
∂c V 2 ∂A ∂n = −kM (c0 − c) kM = m m3 mol−1 s−1 c = c0 ∂t No ∂V ∂A∂t f (1−e−c0 kM t ) ×(f c0 + c) × f +e−c0 kM t
We use the rate of sublimation again to illustrate the use of (3.38), the approximation of a solid phase process using an effective concentration, and to quantitatively compare the resulting rates of loss of the solid. One exact and four approximate schemes for calculating the rate of sublimation of a crystalline solid are shown in Table 3.1. The linear scheme is essentially an exact solution obtained by applying a linear rate based on the Hertz–Knudsen equation, calculated previously as (3.35), to the regression of a sphere of radius R. The rate equation is zero order in the concentration and one-dimensional, in units of meters per second (m s−1 ). The integrated rate equation is cubic in the time, however, and the loss of solid can be expressed as a scaled concentration, c/c0 , going from 1 to 0 and representing the regression to completion of an idealized ensemble of spherical crystals. The zero, first, and second order schemes make use of (3.38), with j = 1 and o1 = 0, 1, or 2, respectively. We approximate a concentration-based rate constant by applying a surface area to volume ratio to the 2D Hertz–Knudsen solution, as discussed earlier. The mixed order scheme is more interesting. In the zero, first, and second order schemes, we have made the approximations necessary to accommodate the 2D nature of the sublimation mechanism in the rate constant only, and used a concentration variable as it would normally be used. Mixed order rate laws using only a single component are used very often to approximate a more complex morphology model using the form of the concentration equation as well. The form of the mixed order rate equation is very general and has been shown to reproduce the behavior of a number of known reaction mechanisms. In this application it is simply a differential equation of the general form
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dx/dt = x(1–x) and could represent, for instance, a nucleation and coalescence type of process. Sublimation in a complex, pressed composite may proceed in this fashion. This equation is often referred to as the Prout–Tompkins equation after the authors of the first application to a kinetic system [39]. By inspection it is apparent that the rate will rise from zero at x = 0 or 1, peak at x = 0.5, and fall again to zero at x = 1 or 0. The integrated rate law is a sigmoidal (step) function going from 0 to 1 where the sharpness of the step is determined by the rate constant. It is also apparent that for a boundary condition reflecting a pure phase, such that x = 0 or 1, there will never be any reaction progress, that is, dx/dt will always be zero, x always 1. Mixed rate equations such as this always require an amount of material to begin progress. This is the reason for the explicit initial concentration dependence, c0 , in the rate equation shown in Table 3.1. This initial concentration of material has been discussed by many authors when using such a mixed order rate equation. For applications involving pure chemical reactivity between stable phases of material, this is straightforward and an equilibrium or pre-existing concentration of product is often not a conceptual difficulty. When using such mixed rate laws to simulate a physical transformation between phases of material, however, the issue of initial concentration is more complex. At temperatures above or below the equilibrium, boundary temperature between the phases one or the other phase is only metastable and therefore cannot simply be considered a pre-existing concentration. Nucleation or some other means to convert the metastable phase must be included in the rate law as well. We have constructed a model of the β-δ phase transition based on such a system of equations and have considered the initial concentration a stable pre-existing concentration [11], a variable concentration based on additional first order components to the overall rate law [14], and as a concentration derived from an independent nucleation rate law [18, 19]. Several important and interesting points with respect to concentration dependence arise in such an application. In addition to initial concentration, the rate equation must reproduce thermodynamic reversibility and the slowing of the transformation rate as the equilibrium boundary temperature is approached. We will discuss this in more detail in the context of the β-δ transformation in HMX in section “The Crystalline Phases of HMX and Rates of Transformation.” We show calculations of reaction progress based on each of the integrated rate laws of Table 3.1 in Fig. 3.3a. In the calculations we use the measured surface to volume ratio of pristine β-HMX based PBX 9501 [38]. The measured specific area was 0.45 m2 g−1 , which we convert to a surface area to volume ratio of 8.39 × 105 m−1 . We have divided the initial concentration from each rate law and plotted reaction progress from 1 to 0. The assumed temperature was 200◦ C, chosen for convenience. The zero order equation provides the fastest overall rate, the linear regression the slowest. Despite the effect of reaction order on the shape of the curves and half times, when plotted as a function of temperature over a large temperature range, this variance is small compared to the temperature dependence, as illustrated in Fig. 3.3b. The mechanisms
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in Fig. 3.3b are ordered, from the fastest to the slowest, as zero, first, second, mixed, and linear. This is an important point and we will return to it in section “Measurements of Rate Constants.” Finally, many other aspects of a thermal explosion kinetic model are dependent on the evolution of concentration produced by the various rate laws. In particular, the evolution of temperature is sensitively dependent on the average heat capacity and thermal diffusivity of the solid or mixture during reaction. These averages must be taken from the values of the individual components and some weighting or mixture rule. This dependence couples with the rate of heat generation to define the nonlinear sensitivity in these models. Classic Kinetic Schemes The rate equations and integrated laws listed in Table 3.1 represent the evolution of one species. The decomposition of a complex (or simple for that matter) molecular species is necessarily modeled by involving subsequent intermediate and product species that represent the entire decomposition. Each of the subsequent species that forms and then reacts requires a rate equation to govern its evolution. The resulting kinetic scheme represents the conceptual and mathematical representation of the chemical decomposition mechanism discussed in the introduction. There are several excellent compilations of simple kinetic schemes, their integration, and mathematical behavior [26, 41, 42]. These compilations are very useful for the development of intuition regarding the typical behavior of a class of kinetic equations, such as the different reaction progress curves shown in Table 3.1 and Fig. 3.3 above. The integral behavior of a coupled series of reactions, each described by some version of (3.38) can be quite complex. In addition, there are a number of classic, named equations that have been applied to particular problems. One such equation is the Avrami–Erofeev equation, which has been shown to reproduce some aspects of a nucleation driven process [43–48]. Another is the Prout–Tompkins equation discussed earlier [39]. We discussed the general form of this equation in terms of replicating a nucleation and growth type behavior. Prout and Tompkins first analyzed the solutions to this equation in the context of the termination of chains in a solid state chain reaction. This illustrates briefly the danger in using such popular, named equations. The equations themselves are quite general and can be shown to reproduce a number of different physical phenomena. They are very useful; however, they carried significantly more importance in the precomputer era when the analysis of kinetic data was very difficult and required many assumptions and approximations to generate a solution for a given kinetic scheme. It is possible now not only to generate sophisticated analytical solutions to even relatively complex kinetic schemes, but also to numerically solve any general scheme. The kinetic model applied to a given system now may be driven by hypotheses concerning the physics and chemistry, and not by the ease or possibility of solving the resulting system of equations. And regardless of the scheme or method of solution, the fit to data by any one of these equations should never be considered sufficient to attribute mechanism.
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Fig. 3.3. Rates of sublimation using the five mechanisms listed in Table 3.1. The mechanisms are labeled in (a). In (a), the progress is shown as a function of time for the complete sublimation of an ensemble of 20 μm HMX particles. In (b), the temperature dependence of the half-time of sublimation for each mechanism is also shown. The mechanisms in (b) are ordered, from the fastest to the slowest, as zero, first, second, mixed, and linear. The free energy of sublimation used was that reported by Lyman et al. [40] and discussed in section “The Crystalline Phases of HMX and Rates of Transformation”
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Summed Component Models We will briefly review a few classic kinetic mechanisms that are used routinely in explosives modeling. The first class of mechanisms we call summed component models and are typified by a sequence of reaction steps or several reactions involving competition for a single reagent species. The summed component models are characterized by regions of rate as a function of temperature, which are dominated by a single step in the mechanism. This is due to the strong dependence of the rate constants on temperature, which results in dramatically different rate dependence on temperature if the activation free energies differ for the series of steps. All mechanisms currently in the literature on energetic material decomposition can be characterized as summed component models. Processes that compete for a single reagent species are common and provide a particularly transparent example of the effects of temperature on the rates of decomposition in a summed component model. A competitive scheme is typically written as j ∂c ki c, =− ∂t i=1
(3.39)
where the ki represent a set of first order processes of decomposition, all of which proceed with a rate proportional to the concentration of the same reagent species. When j = 1 the system is a particularly simple first order rate equation. The half life of a first order rate law is t1/2 = ln(2)/k. Likewise, for the sum of competitive processes in (3.39), it is clear that the sum of rate constants ki is effectively a single global first order rate constant. Furthermore, it follows that the half life of the competitive system is simply t1/2 =
ln(2) , j ki
(3.40)
i=1
which will be dominated by the smallest term in the sum, the term changing as a function of temperature according to the various activation energies associated with each rate. A second common scheme applied to complex reaction systems involves a series of sequential steps, coupled by the successive intermediate concentrations as aA → bB → cC → dD, where a moles of A react to form b moles of B and so on to the formation of D. For a, b, c, and d = 1 the scheme describing such a mechanism involves three coupled first order differential equations
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∂A ∂t ∂B ∂t ∂C ∂t ∂D ∂t
= −ka A,
(3.41)
= ka A − kb B,
(3.42)
= kb B − kc C,
(3.43)
= kc C.
(3.44)
An analytical solution to this system is available and is presented in a number of kinetics and physical chemistry textbooks. For example, the concentration of D with time goes as D=A0 −e−(ka +kb +kc )t fa (ka ) e(ka +kb )t +fb (kb ) e(kb +kc )t − fc (kc ) e(ka +kc )t , (3.45) where A0 is the initial concentration of A and the fi are simple functions of the rate constants ki . The rates appear as sums in exponential functions. The overall rate will therefore again be dominated by individual rates (processes) as a function of temperature if the activation energies for the reactions are significantly different. Significance here is defined by the ability to experimentally resolve the different slopes as a function of temperature. Three examples of the concentration as a function of time for different relative rates are shown in Fig. 3.4. The concentrations are scaled to the initial concentration of A, all other concentrations being zero at t = 0. In Fig. 3.4a, the full sequence of reactions is apparent for ka = kb = kc = 0.005 s−1. In Fig. 3.4b, subsequent reaction rates kb and kc = 0.2 s−1 , 40 times faster than the initial rate, and the two intermediate concentrations are only just apparent on this linear scaling. In Fig. 3.4c, the subsequent rates kb and kc = 5.0 × 10−5 s−1 , 100 times slower than the initial rate, and the two final concentrations are only just apparent on this linear scaling. This example illustrates that even such simple kinetic schemes can generate very complex behavior, and that the experimental work necessary to constrain a complete kinetic scheme is considerable. For instance, if only the loss of reagent A is monitored, then very little is learned about the mechanism, and the three examples are indistinguishable. In Fig. 3.4b if all possible species are observable, one would still conclude that A → D, and the role of B and C in the mechanism would remain unknown. The most significant characteristic of the summed component models, however, remains the dominance of different individual rate constants as a function of temperature. This has been realized for quite some time, and was pointed out as early as 1949 by Eyring et al. in a classic monograph on detonation [49]. The authors cite the example of ignition data from sodium azide [50], which illustrates the dominance of two rate constants over different regimes of temperature, and issue a caution against using low temperature decomposition data to model detonation phenomena.
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Fig. 3.4. Progress as a function of time for a simulated chemical mechanism involving four sequential reactions. By modifying the magnitudes of the rate constants, the system can be shown to exhibit measurable concentrations of either all species (a), only the original reagent and final product (b), or only the original and first intermediate species (c). This illustrates the complexity of observing and determining even a relatively simple mechanism
All models of thermal decomposition and particularly ignition of which we are aware of are the summed component type. This is in part due to the relative lack of data on elemental, solid phase processes in the decomposition of most solid energetic materials. It is also a function of the need for simple models that may be incorporated into larger scale simulations. Of course there are, however, any number of more complex kinetic schemes that can be used to model chemical reactions that are coupled in more complex ways. Coupled Two State Models Another important family of systems involves what we call reversible two-state systems of pure order, which can be written as a family of rate equations as ∂x = k1 (c0 − x)n − k−1 xn , ∂t
(3.46)
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where c0 is an initial concentration of reagent, x is a progress variable denoting the product concentration, the reaction is a reversible two state system and is mass conserved, that is, c0 = x + (c0 –x). The forward-going rate is given by k1 and the reverse rate by k−1 and the reaction is of nth order in the concentration of each state. For n = 1, (3.46) describes the simplest first order equilibrium between two species. The solution of (3.46) for n = 1 is given by k1 (1 − exp (− (k1 + k−1 ) t)) . (3.47) x= k1 + k−1 There are a number of important characteristics of this simple equilibrium system. The system will relax to an equilibrium state where the ratio of the concentrations of the final to the initial state is the ratio of rate constants, k1 /k−1 . This is again a summed component behavior, as can be seen most clearly by solving (3.47) for the half time of the reaction 1 1 k−1 ln t=− . (3.48) 1− k1 + k−1 2 k1 Again we see the summed rate constant that characterizes the separated dependence on temperature. Also note that the reaction time goes to infinity for k1 = k−1 , and that t is only defined for k1 > k−1 . There is therefore a temperature where the direction of flow in the reaction switches from one state to the other. All of these attributes lend themselves to using this scheme to model some aspects of phase transition behavior, although this scheme is not sufficient by itself to model a first order phase transition, as it predicts an equilibrium concentration of the two states. One must go to higher, mixed order so as to mathematically reproduce the first order thermodynamic requirement that one phase exist at a time. These are the considerations that led to the construction of one of the nucleation and growth models of the β-δ phase transition that will be discussed in section “The Crystalline Phases of HMX and Rates of Transformation” below [12, 14]. A very different behavior is observed from the system of (3.46) if n is set to 2. The solution in the progress variable is given by k1 k1 k−1 tanh arctan h − k1 k−1 + c0 k1 k−1 t . (3.49) x= k1 + k−1 Equation (3.49) is more complex than the simple first order case, but essentially produces a sigmoidal shape in x(t), with an induction time determined by the hyperbolic arctangent and a sharpness in t determined by the square root of the product of rates. There is also a singularity for k1 = k−1 . Most important is the reaction half time, given by k1 1 k1 + k−1 1 arctan h − arctan h . (3.50) t=− k−1 2 k1 k−1 c0 k1 k−1
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Now the half time is a function of the product of rate constants, specifically a single new rate constant, which is the exponential average of the two originals. This is a first example of a coupled kinetic scheme, where the individual processes are coupled to the extent that the overall rate is a function of both and are inseparable as a function of temperature. In addition, the particular form of the coupling, the inverse product of the two rate constants, will prove very important to the modeling of HMX ignition in section “Comparison of Kinetic and Thermodynamic Rates.” Another example can be constructed as a strongly coupled sequence of steps that are cyclically coupled, as in ∂y ∂x = k1 y and = −k−1 x, (3.51) ∂t ∂t where x and y represent progress variables in the mechanism. Such systems of equations yield cyclic solutions, the period of which is also a product of the square root of the two rate constants. The overall rate behavior of such systems is not characterized by any single rate constant but rather some function of the constants. Such systems are not currently used in explosive modeling, but some possible advantages to such a formulation will be discussed in section “Comparison of Kinetic and Thermodynamic Rates.” Measurements of Rate Constants The measurement of rate constants and the elucidation of mechanism are closely coupled, and understanding how to extricate one without complete knowledge of the other is the art of the experimental or theoretical chemical kineticist. While much of the work necessary to understand solid thermal explosions ultimately involves the kinetics of gaseous intermediate species, which are relatively easy to observe in isolation and whose interactions constitute a well posed mixing problem, the real difficulties lie in the solid phase chemistry of the problem. Solid phase chemical kinetics have lagged decades, perhaps many decades, behind our understanding of gas phase kinetics precisely because of the difficulties alluded to previously and in sections “Temperature Dependence: The Arrhenius Rate Constant” and “Concentration Dependence: Kinetic Order” involving the complex relationship between state of mixing and the system free energy. We do not pretend to offer a solution to this general difficulty. We will instead offer a number of specific points that are critical in extracting rate constants from data in the literature or by experiment, and finish with a specific example of our work deducing a sub-mechanism of HMX decomposition, the β-δ phase transition. Specific Considerations in the Determination of Rate Constants 1. The null observation of a proposed intermediate species should always be regarded provisionally, and possibly due simply to insufficient sensitivity in detection. This is the straightforward lesson of Fig. 3.4, but the literature is replete with attributions of mechanism based on the lack of observation.
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2. The determination of rate is not the same as the determination of a rate constant. The determination of a rate constant benefits exponentially from the breadth in temperature over which it can be measured. Because a linear regression is performed to deduce energy and entropy of activation, error in the activation energy is exponential in the temperature. Brill has published a wonderful illustration of this in the form of the kinetic compensation effect [51]. Breadth in temperature also renders a decoupling of rate constant from mechanism, as in the illustration of Fig. 3.3a and 3.3b. Despite the factor of two difference in the rate of progress for identical sublimation rates as a function of mechanism illustrated in Fig. 3.3a, one could fix that rate with good precision if the progress at some fixed point (usually x = 0.5) could be measured over the 250◦ C range of Fig. 3.3b. 3. Particularly in work toward ignition models, calorimetry remains the gold standard observable, in that it has an historic use as an observable in chemical kinetics and directly reflects the observable of interest in nonlinear ignition problems. As discussed below, however, as an integral observable it has limitations. 4. Although calorimetry is the gold standard observable, the use of calorimetry is replete with difficulties in interpretation. Probably the most frustrating of such difficulties are debates in the literature regarding the more correct type of experiment, isothermal or nonisothermal, or worse yet, which type of thermal trajectory leads to the more correct model. Of course, unless the rates of heating considered are sufficiently fast to effect molecular equilibrium (and therefore the use of the canonical thermal rate constant as discussed in section “Temperature Dependence: The Arrhenius Rate Constant”) only a model that can accurately predict both types of experiment is correct. We have demonstrated in both the phase transformation models [16, 18] and ignition models [52] that indeed both such types of experiments can be modeled using the same mechanism. This is a standard to which any serious mechanism should be held. A very good recent discussion of this and the general problem of the extraction of rates and mechanisms from calorimetry is provided by Vyazovkin and Wight [53, 54]. Specific Example of Work on the β-δ Phase Transformation Kinetic studies of the phase transformation were conducted using a second harmonic generation (SHG) probe. SHG provides a very sensitive probe that distinguishes the noncentrosymmetric δ phase from the centrosymmetric β phase, allowing a detailed study of the kinetics of the transformation by measuring the conversion efficiency of an input laser wavelength ω into its second harmonic at 2 ω [11, 14]. The SHG probe gives an integral measure of the δ mass fraction. For comparison, differential scanning calorimetry (DSC) studies have been performed, which measure the latent heat during transformation as a progress variable for the reaction. DSC is sensitive to the energy of the phase change, rather than to either the absolute product or reactant mass fractions.
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SHG, being an integral measure of the progress between the phases, is very sensitive and can distinguish the very early onset of the phase transition. It is very difficult, however, to perform an absolute calibration for SHG as the measurement of intensity at 2ω is sensitive to uncontrolled optical properties of the sample through the collection efficiency. DSC is less sensitive, but can be calibrated to provide an absolute rather than a relative progress variable. The kinetics observed by SHG and DSC are in agreement. Another observable often used is Raman spectroscopy with which both the loss of a reactant or the rise of a product species may be monitored. Some discrepancies have been reported between the phase transition kinetics observed by SHG, DSC, and Raman spectroscopy. This is due to the volume of material accessed in a heterogeneous material and how well the total average mass fraction may be infered. DSC studies measure the enthalpy of the transition summed over the entire sample. SHG studies probe an area equal to the size of the laser spot used to generate SHG signal, typically on the order of 0.5–1 cm diameter by several millimeters thick. Raman, on the contrary, probes a sample volume of a focused spot size with a penetration depth determined by the laser wavelength. The heterogeneous nature of the phase transition with multiple nucleation sites and growth from these sites causes the Raman studies to see the kinetics of the growth front propagating across the probe volume [12, 14, 55]. This explains why they appear with a delayed onset and steeper rise time compared to the volumetric nucleation and growth kinetics observed by SHG and DSC [56]. 3.2.2 Types of Models Homogeneous Source Term Models The most common type of models encountered in the literature are those characterized as homogeneous source term models. They are characterized by the use of Arrhenius rate constants and simple, nonchemically specific progress variables to model the dependence on concentration. These terms are then used to determine the rate of heat generation in the nonsteady heat equation. Such models treat only energy transfer and are therefore limited in applicability to calculations where mass transfer are important, such as the observed dependence of time to ignition on the loss of gas from a sample. They must be fit to data, and are therefore interpolative models, rather than extrapolative. The advantages to such an approach include ease of parameterization and computational simplicity. The literature contains many such models for the decomposition of energetic material. Zinn and Mader [2] reported some of the earliest computer simulations of energetic materials using numerical solution of the nonsteady heat equation and Zinn and Rogers [3] presented a reduced single term model for a
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number of materials that captured the time to ignition over the temperature range. The family of One Dimensional Time to Explosion (ODTX) models provide an example where the evolution of homogenous source term models can be followed [57–61]. Dickson et al. reported a four step model that uses a combination of identified chemical steps (the β-δ phase transition) and intermediate species to capture not only the time to ignition, but also the distribution of internal temperatures during the thermal trajectory. In this way, it captures the ignition location as well as time [62]. Zucker et al. have put forth a series of models of this type, including melt steps for explosives such as RDX and PETN which melt prior to thermal ignition to successfully capture ignition times and locations [63]. The strengths of the homogenous source term models is their ease of use. However, neglecting spatially heterogenous conditions such as mass transport limits the applicability of the model. In particular, while condensed phase energy transport may be expected to scale with sample size, mass transport or gaseous transport of sensible heat will depend on the complex morphology of the sample, perhaps determined by the spatially resolved extent of reaction, and therefore must not be expected to be invariant to sample size. Multiphase Models A level of complexity above the homogeneous source term models are multiphase models that distinguish between solid or liquid, and gas. Such models enable the treatment of mass transfer as well as energy transfer and have proven necessary in some instances. These models are significantly more computationally demanding than the homogeneous source term models, requiring finite element meshing of the problem. However, the ability to include mass transport allows the capture of behaviors where mass transport are evident [64]. The Los Alamos Large Scale Annular Cookoff (LSAC) experiment modeling is an example of a case where homogenous models were unable to capture thermal decomposition behavior without the inclusion of gas transport by Darcy flow through the thermally damaged PBX 9501 [65]. Although potentially far superior in fidelity, the computational cost of such models may be prohibitive in large-scale simulation. Complex Material Models The multiphase models used to capture internal temperature distributions in the LSAC experiment on PBX 9501 were parameterized using porosities comparable to those measured for thermally damaged samples [66–68]. To model the evolution of this porosity, more sophisticated consideration of the material is necessary. Such models enable a number of effects known to perturb explosive response to be addressed, and this is the topic of Chap. 6.
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For our purposes here, we note some of the ramifications of morphology on the chemical kinetics. For instance, how are solid and void created/partitioned? What is the best representation of kinetic order of a given process, for example, the kinetic order of a sublimation reaction as discussed in sections “Temperature Dependence: The Arrhenius Rate Constant” and “Concentration Dependence: Kinetic Order.” What are the relative effects of the transport of sensible heat vs. chemical potential. The evolution of these morphological changes can significantly impact the kinetic order or simulated concentration dependence of particular component reactions. The sublimation example described earlier is such a morphologysensitive reaction. The evolving state of the material can have a large impact on the behavior of the material to energy and mass transport through changes in porosity, permeability, on diffusivity, as has been discussed. While clearly preferable, only a few serious modeling efforts have weighted such attention to morphology [69–71]. It is possible that clever modeling strategies, particularly with respect to the simulation of the extensive dependencies like concentration or areal density, may allow inclusion of these effects in model of sufficient simplicity to find application. It is likely that as global chemical modeling is pursued and reasonably accurate mass loss mechanisms are applied, these will enable the calculated loss of mass to generate morphological change. It is certainly clear that as ignition models are used to produce the initial conditions for internal burning in explosions, the state of the material and the ability to support various modes of burning will be required. Global Chemical Models The most sophisticated models are based on a global approximation of the real chemical and physical transformations known to proceed in an explosive. These models incorporate known steps in the decomposition process, which release or transform chemical potential to generate heat. This typically involves a considerable reduction in the chemistry that must be considered compared to the full decomposition mechanism for the molecule. The advantages of such models are that they enable the calculation of many parameters of interest such as mass loss and pressurization. It is also possible to indirectly measure rates and dependence on temperature for component steps in the process. The difficulties of such a modeling strategy, however, include the fact that significant additional constraints on the model are implicit. These constraints arise from the fact that the model is constructed around the transformation of real chemical species, so a host of properties such as reaction enthalpy, thermophysical parameters, and in many cases rate constants, are fixed and cannot (or should not) be modified upon inclusion into the model. For these models, concentration variables are real chemical species rather than unspecified intermediate states. The models require independent measurements of component chemical and physical processes. The use of real species implies constraints
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on the model such as the use of independently measured reaction enthalpies and thermophysical parameters such as thermal conductivity. These independently constrained parameters limit the tunability of the model. The benefit of such constraint, however, is the ability to predict behaviors outside the directly measured space based on the understanding afforded by the model. 3.2.3 Modeling Objectives The level of fidelity required by the model determines which of the modeling schemes described above is necessary. If the only requirement of the model is a prediction of time to ignition at a given temperature, a single term homogenous model will suffice. If, however, in addition to reaction time, the location of the ignition volume is needed, then a multistep homogenous model will be needed, which captures more complex material behavior such as phase transformation. These phase transformations are responsible for the observation of “memory effects” in explosives such as HMX, which once converted to the delta phase, remains more sensitive than its initial beta phase despite being taken back to room temperature. If the model is required to predict the reaction violence of a system response to a thermal condition, more complex models yet are needed. Assumptions such as internal ignition locations and conditions that maximize the volume of heated explosive will contribute to the final reaction violence. Specific information needed to predict the final outcome or violence of a response are pressurization and pressurization rate to determine if an assembly will fail before reaction propagates. In addition, information on how pressure drives internal combustion is needed. This requires knowledge of the material state and pressurization rates at the time when the material transits from thermal decomposition to burning.
3.3 Specific Example: A Global Chemistry Model for HMX We have pursued global chemistry models of HMX decomposition and ignition for the reasons discussed earlier, the inclusion of greater model complexity in order to produce answers to more complex questions with greater reliability. Specifically we will ultimately require the following, in addition to the time to explosion: 1. A spatial map of HMX morphology, that is, regions of β and δ HMX at ignition 2. The rate of generation of gaseous products and the ultimate number of moles of gas generated per HMX molecule at the temperatures generated at ignition 3. Using the temperature trajectory of the simulation, and the spatially resolved conversion of condensed HMX to gaseous products, calculate a
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morphological description of the remaining solid, specifically containing a description of porosity and tortuosity (the length scale and complexity of pathways for gaseous diffusion) 4. Using the morphological model, a conversion of rates of gas generation and ultimate conversion into rates of pressurization and pressures generated by ignition 5. Finally, using the morphological model of tortuosity and pore length scale in concert with a model of pressurization describe whether the calculated ignition will transit to convective burning Such complexity of course leads to many more degrees of freedom in what are typically already grossly underconstrained ignition models. It is for this reason that additional constraints arise from the use of known, albeit simplified, chemical mechanisms in the decomposition mechanism. We will finish this section with our latest examples of such models for HMX. We are not yet ready for (5) and a full safety envelope calculation; however, we are well into (3), and as the rest of this book indicates, we now possess many of the pieces of the puzzle to quantitatively address (5) in the near future. 3.3.1 HMX Thermal Decomposition The literature on the reactions and decomposition pathways of the HMX molecule is considerable, and as the introductory quote by Shackelford [1], indicates often contradictory. The first measurements of the rate of decomposition were conducted by Robertson [72] shortly after the discovery of HMX as a by product of the RDX Bachman synthesis. Work on HMX proceeded over the next several decades, largely as studies incidental to more serious work on the RDX molecule. In 1977, Shaw and Walker compiled various kinetic measurements for what was typically described as HMX gas phase decomposition [73]. The compilation is unfortunately weighted heavily to reports that are more or less difficult to obtain and private communications. The theme of the paper is indicative of the efforts of researchers confronted with the problem of HMX decomposition and energetic response. Shaw and Walker identify eight different possible decomposition pathways, summarized as either identified conceptually or indicated by the measurements of a number of groups up to that time. The rate constants of four of these pathways were approximated using the measured rates (most notably dimethylnitramine) and thermodynamics of small model molecules and it was concluded that N–NO2 covalent bond scission is the rate controlling step in the gas phase decomposition. In the early 1980s, considerable work appeared by Brill et al. [74–82] on the kinetics of transformation among the family of polymorphs of crystalline HMX. In the course of the work, Brill arrived at a conclusion, which we will discuss below as critical in understanding the energetic response of HMX; that the lattice energy of the solid is critical in determining the rate of decomposition of HMX [74]. In the mid 1980s, Shackelford et al. [1] used kinetic isotope
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effects on calorimetrically measured rates of decomposition in the condensed phase to attribute different rate limiting processes depending on the phase of material. In particular, Shackelford et al. identify C–H bond breaking as rate limiting in the solid phase, C–N breaking in the melt, and reiterate Brill’s observation that the lattice, or more particularly its disassembly during decomposition, is a component of the rate limiting process in decomposition. At about the same time much of the compiled Soviet literature on HMX decomposition was recapitulated, and new measurements presented by Maksimov et al. [83]. In this work, the complexities of thermodynamic phase are again encountered, with special kinetic models incorporating mixed phases formulated to explain measurements [83]. Of particular importance here is the observation that there is a characteristic acceleration in the decomposition rate as the melting point is approached. This phenomena was observed to be relatively general as early as 1967 [84, 85] and has since proven to be quite robust [86], although there is no specific mention of HMX. Throughout the 1980s and 1990s, Brill’s group produced a very significant body of work, culminating in the high heating rate temperature jump techniques with which the dominant product distributions were identified and their rates measured [87? –96]. Beginning in the 1990s, Behrens [97, 98] and then Behrens and Belusu [99] published a significant body of work on the decomposition of HMX based upon gravimetric and very sensitive mass spectrometric techniques. With these techniques, many of the pathways of condensed phase decomposition were identified, as well as very important characteristics of the solid morphology of the HMX single crystal during decomposition and their influence on decomposition rates and products. The compilation of data continues to the present, important examples including the extremely low temperature manometric studies of solid HMX decomposition by German et al. [100] and the isoconversional calorimetric techniques applied by Vyazovkin and Wight to HMX [54]. The principle and very significant result of that work was the model-independent determination that HMX condensed phase decomposition is not chemically autocatalytic, as has often been supposed, but rather acceleratory in rate due to morphological effects of the kind previously observed by Behrens and Belusu [99]. We consider that this observation that a single activation energy controls decomposition throughout the reaction progress and over a temperature range of approximately 100 K is also very important and we will discuss our interpretation of the implications of this for the mechanism of ignition below. This brief compilation of the history of HMX decomposition is of course woefully incomplete. In particular we will not include with any justice the body of classic calorimetric work, begun by Rogers [101] and Hall [102] in the seventies, which is extensive and quite valuable. Combined with additional work discussed below, it does however give a flavor for the major bodies of work that have guided our thinking in the HMX problem and so serve as background to the final work of this chapter, which will be to illustrate the current state of our modeling of thermal ignition.
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Rates of HMX Decomposition A compilation of measured rates of HMX decomposition taken from the literature is shown in Fig. 3.5a. The data are either plotted as reported individual measurements or calculated using the reported rate constants and shown over the temperature range reported. The data are also plotted as the inverse rate, or characteristic time of reaction, to facilitate comparison with times to ignition that will be discussed in the next section. The data are plotted against the inverse temperature in order to linearize the exponential behavior expected of a single Arrhenius rate constant. The legend identifies the studies by first author and gives the thermodynamic phase of HMX reported or the solvent used in solution studies. Data from Maksimov et al. The data from Maksimov et al. [83] includes data presented for the first time in that publication, as well as a review of the contemporary Soviet literature. We cite these studies by their reference number in the Maksimov paper, which may be relatively easily obtained in English translation [83], rather than the more difficult original Russian publications, only one of which is an archival journal publication. The authors attribute pure HMX decomposition to gas phase processes in all but one study, that of the solid. In Maksimov et al. [83], a theory is presented of partitioning between gas phase and solid phase components of an experiment with the overall rate observed being a weighted combination of rates from the different phases. It is then supposed that this rationalizes the several order of magnitude discrepencies apparent in the data of Fig. 3.5a. We assume here, and for all the data reported, that the phase of HMX controlling the decomposition is very uncertain. This is not meant to be a criticism of any of the work presented, which together constitute a valuable suite of data for analysis, only a testament to the difficulty of rigorously attributing the phase in a solid-state semi-volatile chemical system below the melting point, or of understanding the importance of extremely rarified solid phases, such as would be found on the surfaces of an experiment where the amount of material used is such that a true bulk condensed phase should not precipitate. We finally mention that in Maksimov [83] it is noted that some of the prior data reported there merits less confidence. For example, the data of Maksimov (Ref. 3) is considered “provisional” (English translation). Solution Studies Data in Fig. 3.5a in grey denotes studies of decomposition conducted on molecular HMX in solution. Data obtained in acetonitrile by FTIR measurement of HMX concentration with time are reported as the single grey square [104], in acetone by chromatographic detection of product species by the grey line [103], and in m-nitrophenol by manometric detection of products by the grey dashed line, Maksimov (ref 5) [83]. In the case of the solution studies, it is
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Fig. 3.5. Compilation of measured rates of decomposition for HMX, plotted as the inverse of the measured rate constant, or the characteristic time of reaction, as a function of the inverse temperature. We have plotted individual points or a calculation based on the reported rate constants and the range of temperature over which the measurements were made. The entire suite of data are plotted in Fig. 3.5a. Measurements reported from the solid phase are plotted as the open diamonds [100], bold solid black line [83], black dashed line [103], and open triangles [1]. Measurements reported from the liquid or mixed phases are plotted as open squares and circles [1] and black dot-dashed line [72]. Measurements reported from the vapor phase are plotted as the solid black lines, and the dashed black line [83]. Measurements reported from solution are plotted as the grey square [104], the grey solid line [103] and the grey dashed line [83]. One calculated result is plotted as the long black dashed line [105]. A subset of the data, discussed in the text, is plotted in Fig. 3.5b, along with a solid black curve added as a guide to the eye
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clear that phase is not an issue, but presumably the solvent environment still presents a profound perturbation to the decomposition rate, exhibited by the four decade variability, and expected due to the strong effect of solvent on precipitate solid phase [106]. In addition to the single measurement of rate at ambient pressure, Naud and Bower [104] determine the pressure dependence of the rate and extract a volume of activation, and Naud and Bower [104] and Oxley [103] both attribute the rate limiting step in their studies to N–NO2 bond scission. Select Sub-Set of the Data The remaining data of Fig. 3.5a for neat HMX decomposition in the solid phase were obtained by Oxley et al. using chromatographic detection of products [103] and are plotted as the short dashed line, by Maksimov (ref 5) [83] using manometric detection of products and plotted as the bold solid black line, and the low temperature solid phase by German et al. [100] also using manometric detection of products and plotted as diamonds. The liquid phase data of Robertson [72] obtained using manometric detection of products are plotted as the single dashed black line. The data from Shackelford et al. [1] were obtained calorimetrically and are reported as representing a heating trajectory through the solid (triangles), mixed liquid/solid (squares), and liquid phases (circles). We also plot the rate constant calculated by Zhang and Troung using ab initio quantum mechanics as the dashed line in Fig. 3.5a [105]. In an attempt to rationalize the trends and differences in these measured rates, we select a subset of the data and plot them again in Fig. 3.5b. In Fig. 3.5b we have removed the solution studies, the “provisional” reference (ref 3) and also (ref 4) in Maksimov [83], and the calculated rates [105] and liquid rates of Robertson [72]. We further plot a guide to the eye through the remaining data set. What remains in Fig. 3.5b looks remarkably similar to the acceleration of rates observed for a number of other energetic materials, commented upon by Maksimov [84, 85] as early as 1961, and recently reviewed [86]. A fairly linear, single Arrhenius slope is observed at low temperatures, until some nonlinearity is exhibited in the German et al. [100] data set at extremely low temperature. This accelerates in a fairly smooth way to an asymptotic form at the HMX melting point, ∼280◦ C. The assignment of phase in the reported measurements is consistent except for the measurement of Maksimov [83], reported as gas phase measurements, which seem more consistent with a condensed phase measurement that approaches the melting point. The data rely heavily for this appearance on the calorimetric measurements by Shackelford et al. [1]. These measurements (reported for normal and deuterated HMX, only the normal HMX are shown here) were deliberately obtained with great temperature control very near the melting point. Extremely high sensitivity to temperature was observed within each phase, with rates changing by factors of two or higher over a three degree range. This is difficult to explain without the asymptotic acceleration that has been observed with many other molecules.
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This is a significant reinterpretation of the data, and at odds with some of the conclusions reported by the authors. It does seem to be the most consistent interpretation of the entire data set however, and provides a rational attribution of solid, liquid, and gas phase decomposition for HMX. Rates of Product Formation T-Jump/FTIR Spectroscopy While decomposition products have been measured since the earliest experiments of Robertson [72], the quantitative measurement of appearance rates and gaseous product distributions from the decomposition of HMX was significantly enhanced by the evolution of fast heating experiments, in conjuction with fast FTIR measurement of gas phase concentration. These experiments began at modest rates of heating, (2 K s−1 ) to temperatures near the melting point of HMX and were used to characterize the IR spectra of the melt phase and identify decomposition products in the melt [94]. Ignition times were subsequently measured [90] and as rates of heating increased to 600 K s−1 , the two well known families of gas phase product species were identified [91, 95, 107], and their rates of appearance measured [92]. Other groups also pursued these types of measurements, with general agreement [108, 109]. Significant discussion of the correct distribution of product species has followed this work, with particular emphasis on the surface of burning HMX [88, 89]. There is now general agreement that two families of products appear during HMX decomposition from 200 to nearly 1,000◦ C and rates of heating from a few degrees to 1,500 K s−1 . We call the first channel the 2a channel and at least three different specific product distributions have been proposed. δ − HMX → 4HCN + 4HONO [93]
ΔH2a = 149 kJ mol−1
(R2a ) −1
δ − HMX → 4HCN + 4NO2 + 4H [93] ΔH2a = 1,460 kJ mol (R2a ) δ − HMX → 4HCN + 2NO2 + 2NO + 2H2 O [110]ΔH2a = 219 kJ mol−1 (R2a) We calculate the heat of reaction, ΔH, for each stoichiometry from the heats of formation tabulated in the NIST Chemistry webbook (positive denoting an endothermic process). While there is disagreement on the product distribution, there is fair agreement between the value of the rate constant, k2a , that governs the appearance of these species [89, 109]. Conversely, there is general agreement on the simple product distribution that describes the second appearance channel, which we call channel 2b, δ−HMX → 4CH2 O+4N2 O [89, 110]
ΔH2b = −220 kJ mol−1
(R2b)
and disagreement on the parameters of the rate constant, k2b [89, 109].
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Fig. 3.6. Measurements of the concentration of two representative gas phase product species in the decomposition of HMX, plotted as the concentration of N2 O over NO2 as a function of the inverse temperature [89]. The solid black line is a fit to an Arrhenius rate constant, with parameters listed in Table 3.2 Table 3.2. Parameters of the fit of (3.1) to the data of Fig. 3.6
ΔHp kJ mol−1 −1 ΔSp J mol K−1
Value
Standard deviation
117.66 72.073
10.4 6.2
The canonical data set regarding these relative rates was obtained by Brill et al. [92] and is expressed as the measured ratio of the two branches, captured by the measurement of the two representative concentrations [N2 O]/[NO2 ]. The data set, taken from [89] is shown in Fig. 3.6, plotted as the concentration ratio as a function of the inverse temperature. Plotted in such a way, linearity implies a functional form similar to the Arrhenius rate constant (3.7), and while such linearity does not imply a single elementary rate constant here, we perform a regression analysis of these data nonetheless and list the parameters in Table 3.2. This set of data and this linear regression parameterization will figure prominently in our attribution of mechanism in subsequent sections.
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Simultaneous Thermogravimetric Modulated Beam Mass Spectrometry (STMBMS) At the same time as Brill and Thynell were pursuing the high heating rate experiments, Behrens [97, 98] and Behrens and Belusu [99] were perfecting the high precision thermogravimetric and mass spectrometric techniques that have come to be called the STMBMS family of techniques. These experiments, designed to achieve electron impact ionization of species for mass spectrometric detection with the minimum possible perturbation [98], significantly clarified the view of product formation at low temperatures and isothermal conditions. While observing a suite of product species in agreement with those being observed at high rates, these experiments also enabled the observation of larger molecular weight, less stable intermediate species as well [97]. The most compelling results of this work have been the focus on the condensed phase and a number of complex mechanisms that have been deduced in the inductive, acceleratory, and late stages of decomposition, particularly the length scales and morphology of the porosity that results from the decomposition [99]. Rates of Reaction for Some Gas Phase Products In the calculation of ignition and thermal response based upon rigorous approximations of known chemistry, some important gas phase reactions among the decomposition products of HMX must be included. We incorporate two approximations to chemical oxidation systems for CH2 O and HCN by NO2 that were worked out in fairly complete detail by the groups of Lin and Melius in the early 1990s. We refer to these two systems by their categorization within detailed chemical combustion mechanisms for the family of energetic nitramine molecules [8, 9, 110]. We refer to the CH2 O + NO2 system therefore as dark zone chemistry, after this reference in these models and because these reactions proceed at the emergence from the solid surface and are not luminous nor sufficiently hot to easily detect by optical emission. We refer to a highly simplified HCN + NO2 approximation as bright zone chemistry after the importance of CN radical formation in the luminous, hot, so-called bright zone of the combustion. With these simple approximations, we hope ultimately to couple the decomposition and ignition models of solid HMX formulations with simple boundary conditions for the burning that follows ignition within a solid explosion. Formaldehyde Oxidation (Combustion Dark Zone Chemistry) It was recognized quite early that the gas phase reaction of CH2 O with NO2 would provide a significant source of exothermicity to the overall decomposition [92] and should play a significant role in gas phase ignition and unconfined burning in particular. The reaction had been studied as early as the 1940s by
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decomposition and ignition studies in the gas phase [111, 112]. As with the two solid phase decomposition pathways, several proposals were put forward regarding the stoichiometry of this reaction. The issue has been settled for all practical purposes by a series of experimental and modeling studies undertaken by Melius and Lin over a temperature range spanning some 1,200 K [113]. These results, in combination with the earlier work of Pollard [111, 112], Feifer and Holmes [114], and He [115] indicate the following overall reaction system. CH2 O + NO2 → HONO + HCO ΔH3 = 49.6 kJ mol−1 NO2 + HCO → CO2 + NO + H ΔH3a = −162 kJ mol−1 NO2 + HCO → HONO + CO
ΔH3b = −264 kJ mol−1
(R3 ) (R3a) (R3b)
The reaction enthalpies have again been determined using the heats of formation of all species taken from the NIST Chemistry Webbook. This system has been shown to be valid from 400 to 1,600 K and in particular is very insensitive at low temperature to a number of radical reactions that presumably proceed concurrently [113]. The reaction has actually been shown to be reversible, with a measured inverse rate several orders of magnitude larger than forward going [116]; however, in the presence of NO2 R(3a) and R(3b) are several orders of magnitude faster still and lead to the completion of this system of reactions. The rate constants determined for the system are plotted in Fig. 3.7 [113]. Data on R(3) is also plotted, with the data of Pollard and Wyatt [112] plotted as open squares, Feifer [114] as the black dashed line, He et al. [115] as the open circles, and Lin [113] as the black filled circles. The reverse rate constant is also plotted as a dashed black line [116]. The radical rate constants have been published both as zero activation energy temperature independent constants; R(3a) as the solid bold black line and R(3b) as the solid bold grey line [115] and with a modest temperature dependence of black and grey bold dashed lines [113]. If one assumes the temperature independent form of the radical rate constants and ignores the reverse reaction, one can sum the system into one overall reaction, limited in rate by R(3) and giving rise to both product branches, weighted by the relative value of the rate constants k3a and k3b , where Fa = k3a / (k3a + k3b ) and fb = k3b / (k3a + k3b )
(3.52)
The summed stoichiometry and heat of reaction are then CH2 O+2NO2 → HONO+fa (CO + NO + H)+fb (HONO + CO2 )
(R3)
and with overall thermochemistry given by a competitive ratio of R(3a) and R(3b) as (3.53) ΔH3summed = ΔH3 + fa ΔH3a + fb ΔH3b . We use this single term as the initial gas phase chemistry in models of thermal ignition of HMX to be presented at the end of this chapter.
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Fig. 3.7. Compilation of data for the oxidation of formaldehyde by nitrous oxide. The rate constant for R(3) is plotted as the solid black line [113]. The data of Pollard and Wyatt [112] are plotted as open squares, of Feifer [114] as the black dashed line, of He et al. [115] as the open circles, and of Lin [113] as the black filled circles. The reverse rate constant is also plotted as a dashed black line [116]. The radical rate constants have been published both as zero activation energy temperature independent constants; R(3a) as the solid bold black line and R(3b) as the solid bold grey line [115] and with a modest temperature dependence black and grey bold dashed lines [113]
Hydrogen Cyanide Oxidation (Combustion Bright Zone Chemistry) We also include a system of reactions, again determined by Lin and Melius, for HCN oxidation based upon the initial isomerization of HCN [117, 118] HCN → HNC, NO2 + HNC → HNCO + NO,
(R4) (R4a)
NO2 + HNCO → HNNO CO2 .
(R4b)
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Fig. 3.8. Calculation of the rate constants for the isomerization of HCN to HNC and subsequent oxidation of HNC by nitrous oxide [118]
The reaction enthalpies must be calculated with respect to the HNC and products of HNC oxidation and are given by Melius [118]. The system is extremely exothermic overall. The rate constants are plotted again as the inverse characteristic temperature in Fig. 3.8 [118]. The system is clearly relevant only at temperatures higher than the dark zone chemistry, and is simply limited by the isomerization reaction R(4). We again use a very simplified approximation of the system, assuming excess NO2 , and write down the single reaction HCN → HNC
(R4)
whose overall enthalpy is quite high. These reactions are not particularly important to ignition calculations in the low temperature ignition regime of HMX, but are expected to be necessary as temperature increases, the approximation of excess NO2 becomes more accurate, and as will be discussed further below, as the highly exothermic dark zone system is less favored by higher temperature branching ratios.
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Fig. 3.9. Time to ignition data for two HMX-based formulations. The data are plotted both linearly (Fig. 3.9a), as the ignition data vs. the inverse temperature, and logarithmically (Fig. 3.9b). Data on PBX 9501 from the ODTX [61] experiment are plotted as open circles, the SITI experiment as solid triangles [119], and from the LARE experiments as solid squares. Data on LX-10 from the ODTX [59] experiment are plotted as open squares and the SITI experiment as solid inverted triangles [119]. The solid black lines in both figures are the calculated using a single Arrhenius rate constant to be discussed below
Summary The gas phase chemical systems discussed here are quite widely used and accepted, and from the standpoint of a global chemical model are the most robust in our models, followed perhaps by the polymorphic phase transformations. We do not routinely include the bright zone chemistry in our models at this time; however, we have shown that such inclusion leads to multiphase ignition processes. We have recently obtained data that may support such phenomena, and if so, this bright zone chemistry may become important for very high fidelity modeling of ignition phenomena. 3.3.2 HMX Thermal Ignition The literature on the ignition behavior of HMX and HMX-based plastic bonded explosive formulations is again considerable. We begin with the confined thermal explosion experiments and early computer modeling first by Zinn and Mader [2] and then by Zinn and Rogers [3]. The work concentrates on the computer solution of the nonsteady heat equation using a thermal
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source term whose rate is exponential in the inverse temperature, thus simulating a simple one reaction chemical decomposition. Using this simulated chemistry generates a particular curve for the time to explosion as a function of boundary temperature for simple geometries. This particular curve may furthermore be expressed in a reduced variable form where the boundary temperature is normalized to the Frank–Kamenetskii critical temperature, Tm , and the time to explosion is normalized to a term containing a dimensional parameter and the material thermal diffusivity (Chap. 4). Zinn and Rogers [3] show that a number of explosives that undergo thermal runaway and explosion in the melt phase obey this simple one term model. They also show that HMX in particular does not, noting that “The behavior of HMX is believed to be essentially different from the rest of the explosives studied, and will be discussed separately.” Thus as early as 1962 it was becoming apparent that the high melting explosive HMX presents a more complex chemical decomposition mechanism than would be admitted by a simple one term source model. More complex computer models were soon developed. Following the lead of Zinn and Rogers [3], Catalano et al. [57] showed that accurate calculation of induction time in HMX-based explosives required more than a single chemical source term. This work marked the introduction of the ODTX. This experiment was subsequently used to generate a number of data sets on many explosives and explosive formulations and has proven to be one of the more valuable experiments and suites of data in the literature. Tarver et al. [60], again using ODTX data, showed that three terms, including an initial endothermic step, were sufficient to accurately reproduce HMX induction times over the range of temperature from 450 to 550 K and McGuire and Tarver [58] subsequently identified these source terms with three possible first order reactions, the rates of which were described by three activation energies and prefactors. The chemistry attributed to these terms included an early endothermic ring opening step, semi-volatile intermediate chemistry and subsequent gas phase reactions [59]. Following the work of Brill [74] and Henson et al. [14], where the initial endothermic step in the decomposition was attributed to the β-δ polymorphic phase transition in HMX, the model has recently been expanded [61] to include the use of binder decomposition models, and early steps in the decomposition have been relabeled to indicate that the β-δ transformation is the initial step, although the comprehensive models that followed on the work of Henson et al. [12–19] have not been incorporated [120]. Next in order of heating rate and ultimate temperature was the T-jump FTIR experiments first of Brill et al. [90] and then of Thynell et al. [108, 109]. Although predominantly experiments were designed to probe distributions and rates of appearance of gas phase products, ignition behavior was observed. Through extensive modeling and measurement, the temperature of the volume of material probed was determined as well. The second type of higher rate ignition experiments conducted since this time have been those related to laser ignition of a surface, primarily in the context of laser sustained combustion. While most of these studies parameterized rate or induction
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time in the laser energy applied, and are thus difficult to interpret without a direct measurement of temperature, some studies were conducted with explicitly modeled temperatures. In particular, the laser ignition study of Lengelle provides a temperature parameterized ignition study in this range [121]. Important for the interpretation of both of these types of experiments is the fact that ignition was generated fast compared to gas phase diffusion rates from the samples, effectively confining ignition to the solid. Throughout the 1990s, work at Los Alamos, associated initially with experiments on mechanical deformation of PBX 9501 by speckle interferometry [122], began to produce ignition times as a function of measured temperatures at higher rates of heating. The first such measurements were conducted with one-color InSb or InGaAs radiometry to measure surface temperatures. These experiments were based on carefully determined IR emmissivities from the constituents of PBX 9501, and rigorous determination of experimental error in the inferred temperature [123]. Using these techniques, ignition times were measured for PBX 9501 subjected to shear impact and high rate frictional heating [5], as well as high rate material flow [7], where very interesting effects of crystalline phase were observed [124]. The boundary temperature applied in these experiments was at times isothermal and at others linear, depending on whether the volume of material was overdriven to ignition during motion or not [7]. The ultimate heating rate experiments considered at this time were actually a reconsideration of an experiment conducted in the 1980s by Tarver and Von Holle, where a run to detonation experiment was imaged radiographically and material temperatures were recorded on the way to ignition [6, 125]. The interpretations of these data at the time centered about the radiographic resolution and hot spot size and the observation of bulk heating. We instead derived (not for the first time) a method of analysis using the rate of heating and observed ignition temperature to determine an effective single Arrhenius rate constant controlling the time to ignition in those experiments and showed that the rate constant determined was consistent with rate constants previously determined at slower rates of heating [6]. Menikoff has since applied this thermal rate constant successfully to the calculation of steady detonation properties of PBX 9501 [126, 127]. In more recent experiments, fast rates of heating resulting from laser irradiation of the central volume of a PBX 9501 during thermal runaway to ignition have been made and used to synchronize thermal explosions to a high energy radiographic source to obtain direct radiography of density evolution during burning within a solid explosion for the first time [23–25]. We have calibrated fast thermocouple measurements of these internal temperatures [22] and shown that observed induction times and temperatures are also consisted with the known rates of thermal decomposition and ignition in HMX [128]. Taken together, the suite of data represents a fairly complete interrogation of HMX ignition behavior over the full temperature range of energetic response, from the minimum critical temperature, Tm , to detonation.
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Small Scale Confined Thermal Explosion Data One Dimensional Time to Explosion (ODTX) The ODTX test was developed at Lawrence Livermore National Laboratory and has served as a benchmark test of energetic response for some time [57, 58, 60]. The experimental configuration utilizes a spherical sample, hence the one-dimensional character of the boundary condition and subsequent analysis, and a relatively high pressure applied to the mechanical confinement. In recent papers, the current status of the experiment, as well as the best data provided for LX-10 [59] and PBX 9501 [61] are reported. ODTX is also a common acronym used to identify the family of thermal decomposition models designed to model these data. Los Alamos Radial Explosion (LARE) The Los Alamos National Laboratory version of a one-dimensional thermal explosion test has been a series of experiments based on solid cylindrical geometry. The earliest versions were designed around a 2:1 length to diameter ratio sufficient to achieve a one-dimensional condition at the midplane. These experiments were designed for internal instrumentation and have been used to measure internal temperature distributions as well as times to explosion. While early versions of the test suffered from a lack of strong confinement and the leaking of gaseous products, later versions have been demonstrated to be sealed. The initial 2:1 aspect ratio experiment was 1 in diameter and was used to construct early versions of thermal decomposition models of PBX 9501 [62]. Later 2:1 versions of 1/2 diameter were used to probe the effects of boundary temperature on reaction violence subsequent to ignition and to field internal radiometry by optical fiber and fast thermocouple measurements of temperature during ignition [22]. Subsequently, laser synchronization was achieved via optical fiber and a laser induced temperature jump at the central volume where thermal runaway occurs [23]. Ultimately this synchronization was successfully used to conduct the first proton radiography of a thermal explosion [25]. The current experimental configuration is a 1:1 aspect ratio 1 diameter version, which enables better radiographic contrast and resolution imaging across the diameter, but defeats the original one-dimensional character of the experiment, offering better agreement with simulation when modeled one-dimensionally as a sphere. Sandia Instrumented Thermal Ignition (SITI) The SITI test is based on the Los Alamos Radial Explosion test. It is a well confined test with internal temperature measurements. Some experiments have been attempted with gaseous pressure measurements. The test has been applied to a broad suite of explosive formulations with good results [119].
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Low Temperature Thermal Ignition Data Set The suite of thermal ignition data for PBX 9501 and LX-10 provided by these techniques is shown in Fig. 3.9. In Fig 3.9a the data are plotted linearly as the time to ignition as a function of the inverse temperature and in Fig 3.9b they are plotted logarithmically. The data from each experiment are labeled in the figure legend with spherical data as open points and cylindrical as the solid. Data on PBX 9501 from the ODTX [61] experiment are plotted as open circles, the SITI experiment as solid triangles [119], and from the LARE experiments as solid squares. Data on LX-10 from the ODTX [59] experiment are plotted as open squares and the SITI experiment as solid inverted triangles [119]. The solid black lines in both figures are calculated using a single Arrhenius rate constant to be discussed below. Note the simple linear relationship in Fig. 3.9b expected for a single rate constant, but inconsistent with the thermal ignition curve of a single rate model presented in Zinn and Rogers [3]. High Heating Rate, Inertially Confined Thermal Explosion Data Temperature Jump Experiments by Hot Wire and Laser The data from Brill and Brush [90] and Lengelle et al. [121] represent a valuable suite of ignition data that are of intermediate rate between the direct heating experiments discussed above and the high rate mechanical impact experiments to follow. The work of both of these groups concentrated on understanding the temperature induced in the material and the subsequent ignition response. In the hot wire experiments of Brill and Brush [90], the extraction of temperature by heating the platinum wire involved considerable modeling support. Laser-induced ignition in the experiments of Lengelle et al. [121] was observed to occur within the solid, below the surface, and temperatures were again constructed with modeling support using the applied laser flux. In both these experimental series, we consider the ignition to be inertially confined, and so comparable to the other series discussed here as will be explained below. Taken together, these experiments represent rates of heating from 300 to 1,500 K s−1 and ultimate temperatures between 500 and 700 K. Los Alamos Mechanical Impact Experiments The mechanical impact experiments were all conducted using light gas guns and projectiles impacting at a few hundreds of m/s. All experiments were confined in steel fixtures and ignition volumes were observed both visibly and by radiometry through sapphire windows. PBX 9501 subject to shear by impacting with a sharp edged plunger at approximately 200 m s−1 yielded temperatures on the order of 900 K over 20 μs and ignition in slightly less than 100 μs [5]. Fast frictional heating by SiC particles at the explosive/sapphire interface, again impacting at approximately 200 m s−1 yielded somewhat lower temperatures with a more linear trajectory and ignition in slightly over 200 μs [5].
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Temperatures in both of these experimental series were obtained by InSb radiometry [123]. Higher velocity impact directly onto the sapphire window was utilized to induce ignition during material flow in PBX 9501 [7]. Velocities on the order of 350 m s−1 yielded temperatures above 1,300 K over approximately 1 μs and led to overdriven ignition during heating in β HMX PBX 9501 and relatively isothermal ignition after a few microseconds in δ HMX PBX 9501 [7]. Temperatures in both of these experiments were again obtained by InSb radiometry. In the most recent series of such experiments, we have also generated very high rate heating in PBX 9501 by application of a 1,064 nm, 100 μs laser pulse to the center of a heated, cylindrical sample. In these experiments, both overdriven and relatively isothermal ignition behavior is observed depending on the rate of heating. In these experiments, we observe heating to 900–1,100 K and ignition after approximately 100 μs. Taken together, these experiments probe ignition over rates of heating from 107 to 109 K s−1 and ultimate temperatures of between 900 and 1,500 K. Inclusion of the run-to-detonation experiments of Tarver and Von Holle do not enhance the range probed, but do introduce yet another means of generating temperature in the material [125]. The Confined, Non-Prepressurized Ignition Data Set A compilation of the suite of ignition data is plotted as the ignition time as a function of the inverse temperature in Fig. 3.10. The ignition time is either determined directly from isothermal like boundary conditions in the experiment or we take the inverse of the heating rate in two nonisothermal experiments as a characteristic time indicating ignition, as discussed later. The data were obtained from experiments on thermal explosion [59, 61] (filled and open circles), fast direct heating [90] (crosses), laser ignition [121] (open inverted triangles), friction and shear ignition [5] (filled squares), fast compression and flow [7] (filled triangles), shock to detonation [125] (open triangles), and laser-induced ignition within a thermal explosion [23] (filled inverted triangles). The data set is compiled subject to the requirement that the experiments be either confined or that ignition proceeds fast compared to material or thermal diffusion. All experiments in this compilation are confined, with the exception of the fast direct heating experiments [90, 121]. In these experiments, laser power and platinum wire current were used such that ignition was observed sufficiently fast that the diffusion of gas phase species away from the heated solid was negligible and the chemical composition of the solid volume element was conserved. Also, all experiments in this compilation were not previously pressurized, although as discussed in a previous publication the pressurization is certainly a component of the higher rate impact experiments [6]. The plotted temperature in each case is either the boundary temperature applied to the experiment, as is the case for the two thermal explosion data sets, or the actual measured temperature of the volume of material observed
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Fig. 3.10. A compilation of ignition data plotted as the ignition time as a function of the inverse temperature. The data were obtained from experiments on thermal explosion [59, 61] (filled and open circles), fast direct heating [90] (crosses), laser ignition [121] (open inverted triangles), friction and shear ignition [5] (filled squares), fast compression and flow [7] (filled triangles), shock to detonation [133] (open triangles), and laser induced ignition within a thermal explosion [23] (filled inverted triangles)
to ignite, as is the case in the laser (external and internal), shear, friction, and compression and flow (on delta PBX 9501) experiments. In these experiments, the temperature is observed to approximate a fast initial heating followed by a relatively constant temperature and are therefore good approximations to an isothermal boundary condition on the material persisting for a long time compared to the final self heating that generates ignition. In the two sets of data representing the fastest rates of heating, the shock initiation [125], and compression and flow (beta PBX 9501) experiments [7], a nonisothermal boundary condition is observed. In these two cases the data are plotted as the inverse rate of heating as a function of the inverse of the temperature at ignition. The analysis that leads to this method of plotting, which accommodates a heating profile where ignition is generated at the temperature where rates of self heating exceed the applied heating rate, is a traditional one and we have presented it previously [6].
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Table 3.3. Parameters of the fit of (3.7) to the data of Fig. 3.10
ΔEign kJ mol−1 ΔSign J mol−1 K−1
Value
Standard deviation
148.54 243.71
1.0 1.8
The HMX formulations presented in this compilation are primarily PBX 9501, although one data set on thermal explosion in LX-10 [59] and one on shock initiation in PBX 9404 are also used [125], as are laser ignition data on neat HMX [121]. The inclusion of data from formulations other than PBX 9501 was necessitated by a desire to provide the most comprehensive coverage in temperature. This may of course introduce complications due to the different binders, but we do not believe this to be the case in the data that we have included here. The LX-10 data is sufficiently close in behavior to PBX 9501 in the thermal explosion regime and PBX 9404 is also sufficiently close in the detonation regime. The principal first observation of the data of Fig. 3.10 is of course the linearity of the data set. We have replotted the data of Fig. 3.10 as the natural log of the ignition time (s) as a function of inverse temperature (K) and performed a linear regression analysis. In the regression we did not weight the times using the error bars plotted in Fig. 3.10 as such error was not established in most of the experiments used here. The resulting parameterization of ΔSign and ΔEign using (3.7) are shown in Table 3.3. Also shown are the standard deviations in each parameter resulting from the linear regression. The calculated ignition time as a function of temperature using (3.7) and the parameters of Table 3.2 is shown as the solid line in Figs. 3.9 and 3.10. Summary The data included in the suite of Fig. 3.10 represent essentially the full range of heating and ultimate temperature applicable to the thermal response of HMX. There is already a considerable conceptual product that arises from the consideration of this HMX thermal ignition data from the point of view of real, global chemical mechanism. These in particular arise due to the linearity of the data set of Fig. 3.10. We list these as follows: 1. The extrapolation of a thermal Arrhenius rate law into the detonation regime suggests the same purely thermal model of chemistry in the detonation reaction zone [6]. If temperature precedes very high pressures in the leading edge of the shock, due to dissipation upon plastic flow or from radiation pre-heating, then thermal chemistry is fast enough to go to completion within the reaction zone defined by the sonic plane. 2. The experiments compiled in Fig. 3.10 represent very many different mechanisms of heating, from direct heating to laser and mechanical impact. The analysis of all these experiments using the single thermal rate
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constant is strong evidence that thermal response in HMX depends solely on the temperature, not on the means of heating [6, 7]. 3. The decomposition remains multiterm and categorized essentially as Tarver did from the outset [60]. We believe there is no single term that exhibits that rate constant in an elemental way. If this is true then a multiterm mechanism cannot be of the summed component type discussed in section “Temperature Dependence: The Arrhenius Rate Constant” [4, 52]. This will be discussed further below. 4. Finally, we note that Menikoff [126, 127] has recently modeled steady detonation in PBX 9501 using a single term kinetic rate published by us previously [4] and equivalent to that shown here in Table 3.3. 3.3.3 HMX Thermodynamics The Crystalline Phases of HMX and Rates of Transformation The solid phase structures and thermodynamics of HMX have been the subject of study for more than 40 years. However, significant deficiencies in the understanding of the crystalline phases have persisted throughout that time, characterized by such questions as the absence of a transition to α-HMX at ambient temperatures [106], or simply the uncertainty in the transition temperature of one of the more common transformations between the β and δ phases [14]. Although the role of the endothermic phase transformation in thermal decomposition and ignition behavior was the subject of early discussion [74], direct verification has come rather late as well [11, 62]. The structures of three stable polymorphs and a fourth hydrate of HMX have been determined under ambient conditions. The typical polymorph observed under ambient conditions is the β form [129]. The lattice structure has been determined to be monoclinic of P21 /c space group by X-ray [130] and neutron [131] diffraction. The molecular conformation involves two NO2 groups above and two below the ring plane. The remaining phases have been generated either by heating or by specific precipitation steps in the synthetic process [106]. The δ phase is the high temperature polymorph most often observed under low pressure heating. The δ lattice structure is of space group P61 (P65 ), again determined by X-ray diffraction [132]. The molecular conformation of δ involves all NO2 groups on one side of the ring plane. There is a 6.7% volume expansion on transition from the β to the δ phase [133]. The remaining polymorph, α, and hydrate, γ, are not considered here, but α HMX is of interest because of the question of stability mentioned above and as a phase sometimes observed upon reversion from δ during cooling. In addition, two more phases have been postulated recently under high pressure conditions [134]. The β-δ phase transition thus involves both a change in lattice symmetry, from a centrosymmetric to a noncentrosymmetric structure, and a change in the molecular conformation. The transition is of the first order
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thermodynamically, with a measured latent heat. Reports of the transition temperature at moderate rates of heating vary by some 50 K, from 140 to 190 K. Brill et al. used the Raman and Fourier transform infrared spectroscopy (FTIR) [76] of the β and δ phases to measure both the kinetics of transformation [74, 79] and the equilibrium pressure–temperature phase diagram [78, 82]. Effects of crystal size were also noted in that work [78, 82]. With respect to crystal morphology, many groups have observed significant cracking of individual crystals upon transition from β to δ, due to the significant molar volume increase [78]. Rates of Transformation The rate of transformation is thermally activated, as shown both in the original work of Brill et al. [74, 79] on HMX crystals and recent work in this laboratory on PBX 9501 at faster rates of heating induced by laser illumination [11]. In the early kinetic work [74, 79, 81], the transition kinetics were assumed first order, and Arrhenius parameters for the rate constant determined. In that work, parameters were determined based on limited extents of transition, as noted by the authors. Subsequent preliminary measurements, based on second harmonic generation [11], showed the full transformation in an ensemble of loose crystals from the β to δ phase. Based on these observations an irreversible second order kinetic rate law was proposed, yielding Arrhenius parameters for the rate constant in fair agreement with those proposed by Brill and Karpowicz [74, 79, 81]. The first step in the thermal decomposition kinetics of HMX is the β-δ transformation. The transition is endothermic with a transition enthalpy of 9.9 kJ mol−1 [14, 74] and a volume increase of 6.7% on converting from β to δ phase [74, 133]. The energy difference for the individual HMX molecules is small (∼30 kJ mol−1 ) [74, 130]. However, the reconstructive nature of the transition means that it is kinetically limited with a high activation energy. The activation barrier for nucleation has been measured [12, 74, 79, 81] and the barrier to growth has been shown to be equal to the heat of fusion [12]. We have published a kinetic model for the β-δ phase transition based upon the data shown in Fig. 3.11. It is a reversible nucleation and growth mechanism that has been described in several previous publications [12–20]. The nucleation is believed to be controlled by defects and leads to significant differences in the observed temperature of transformation, as found in the literature. We have proposed a number of models of nucleation, based on the observation that binder components in HMX formulations such as PBX 9501 may act as solvents at elevated temperatures and tend to reduce the apparent transformation temperature [13, 19]. The rate of growth of the nucleated phase has been shown to be controlled by an activation energy equal to the heat of fusion [12], giving rise to a model for growth occurring by a virtual melt mechanism [15–17]. The virtual melt mechanism explains the fact of melt mediated kinetics for a transition occurring approximately 120 K below the
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Fig. 3.11. Measurements of the β-δ phase transition observed by SHG. The data are plotted as the approximate reaction half-time as a function of the inverse temperature. Data are shown for conversion (open circles) and reversion (open squares) induced by direct heating [12, 14], as well as conversion induced by laser heating (open triangle) [11] and mechanical impact (inverted triangle) [5]. We also show data from Karpowicz et al., which seem to reflect what we model as the nucleation rates [81]. The solid line is a calculation using our reversible model of first order nucleation and second order growth, and the dashed line is a calculation using only the reversible nucleation kinetics [12, 14]
equilibrium melting point. The mechanism is based upon a state of tension induced orthogonal to the direction of propagation of the incoherent phase front in these molecular, reconstructive transformations. This state of tension lowers the effective melting point and enables the formation of a transient liquid layer, which immediately crystallizes to form the new (equilibrium) phase.
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The virtual melt mechanism helps explain many seemingly inconsistent observations regarding the β-δ transformation. In addition to β and δ polymorphs, HMX has a stable α phase, which has the same noncentrosymmetric symmetry as the δ phase, but a volume and thermodynamic stability regime between that of β and δ. It has a 3.8% volume expansion above β, and is thermodynamically stable in the temperature range above 378 K, yet, it is not attainable by direct heating from the β phase [106]. On heating to 432 K, β HMX converts directly to δ. The virtual melt mechanism accounts for the “hidden” α phase. The lattice strains that are induced by the smaller activation volume β-α transformation are not sufficiently high to depress the melt temperature far enough to activate the virtual melt mechanism. We have detailed a number of other observations and inconstancies in the understanding of the HMX polymorph system that are rationalized by this mechanism [16, 17]. Reversibility and Rates of Reversion Reversion from the δ phase to the β phase has been studied using the second harmonic generation (SHG) and Raman spectroscopic probes [12]. Data on reversion obtained using SHG are also shown in Fig. 3.11. DSC as a differential probe is not sensitive enough to capture the enthalpy of reversion due to the extremely slow nature of the transition. While the forward kinetics can be driven to occur on convenient laboratory time scales of seconds to hours (determined by temperature), the reverse transition occurs only on the time scale of hours. It is slowed kinetically due to the fact that the reverse transition is thermodynamically driven only at temperatures below the 432 K δ phase equilibrium temperature, with the highest rates observed at approximately 398 K with reversion to β phase taking tens of minutes at this temperature. SHG is used to follow the kinetics of reversion and Raman spectroscopy is used to confirm the presence of β phase and absence of δ and α phases. Reversion to β phase is found to occur in PBX 9501. For HMX crystals, reversion is observed only when the sample has not been completely transformed to δ phase. The nitroplasticizer is believed to nucleate reversion in PBX 9501. For HMX, the β phase must remain to seed nucleation of the δ to β transition in order for it to occur. Thermally driven transformation from β to α phase has not been observed. It has been observed to occur from δ phase, however, although the details of the kinetics are not well known. The δ has been observed to persist for several weeks, and the transition from δ to α phase has been observed over hours to days at ambient temperature and pressure. Mechanical Damage Conversion of β PBX 9501 to δ PBX 9501 is found to induce significant mechanical damage in the samples that results from the crystallization step in the VM mechanism [15, 19]. Observations include optical micrographs [12, 15] and post-transformation analysis by scanning electron microscopy (SEM) [99]. Thermal expansion and the β-δ transition are predicted to cause an
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expansion of ∼9% on heating from room temperature to 180◦C [133]. Proton radiographic images confirm the filling of void space in cased samples. Allowing extra volume above this allows the sample to expand further during the heating and phase transition with the extra volume incorporated throughout the sample as macroscopic cracks and void volume. Parker et al. [67] performed imaging on unconfined samples heated to 180◦ C and found macroscopic crack formation. The orientation of these cracks was conditioned by the level of confinement of the sample. Radially confined samples with open ends to allow for axial expansion showed cracks oriented parallel to the sample faces. Such features of thermal and mechanical damage are discussed at length in Chap. 4. The β-δ transformation is important in the thermal decomposition and ignition process both due to thermal damage induced throughout the sample, as well as due to the increased sensitivity of the δ phase compared to the β phase. Certain tests such as impact and drop height show that while β HMX is a secondary HE, the sensitivity of δ HMX is comparable to PETN [133]. The ability to revert PBX 9501 from δ back to β phase on cooling without annealing out mechanical damage to the sample caused by the phase transformation has enabled studies to separate out sensitivity due to mechanical damage from sensitivity due to the polymorphic phase of the HMX. A series of impact experiments performed with a single stage gas gun on β PBX 9501, δ PBX 9501, and PBX 9501 reverted from δ back to β phase show an increased sensitivity of the δ PBX 9501 and a decreased sensitivity of the reverted β PBX 9501 [124]. This indicates an increased sensitivity in these experiments due to polymorph phase rather than damage state that is difficult to rationalize simply from the endothermicity of the transformation and may involve the kinetics of transformation in an important way. This is currently an open question. An area of controversy regarding the β-δ phase transition is the ability to retard the formation of δ phase by the application of pressure. As the phase transition involves a volume expansion, the application of pressure could retard the kinetics of the transition according to the volume of activation. However, studies by Renlund et al. [136] using their constant volume hot cell show that the availability of internal volume to accommodate the volume increase of the phase transition is such that even for experiments with fixed volume, the phase transition occurs. They found in this study that the pressure applied to the walls of the cell through the phase transition regime was on the order of only 40 MPa. This pressure is far to low to seriously retard the transition [78, 82]. On lowering the temperature below the δ phase equilibrium temperature with the sample cell in the constant load rather than constant volume mode, the sample showed the volume change indicative of a δ to β phase reversion verifying complete transformation in the experiment. Rates of Sublimation and Vaporization The vapor pressure of HMX is relatively low, and attempts to measure the vapor pressure, and hence sublimation and vaporization free energies, have
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Table 3.4. Condensed phase free energies [40]
ΔH(kJ mol−1 )
ΔS(kJ mol−1 K−1 )
Transition T (K)
β-HMX δ-HMX Liquid PBX 9501a δ-HMX Liquid
184.762 174.930 105.067
279.311 257.942 131.149
(β-δ) 445.67 (m.p.) 558.65
174.930 105.067
256.563 131.149
(β-δ) 432.15 (m.p.) 557.06
a
We have modified the entropy of δ-HMX here to yield the phase transition temperature observed in PBX 9501 [14, 106]. This temperature is likely the true equilibrium temperature (see text)
relied upon techniques to measure the rates of sublimation and vaporization into vacuum from heated cells [137, 138]. These data were then reported as equilibrium vapor pressures as a function of temperature. Maksimov [135] and Lyman et al. [40] have recently used these and other data to review the free energies for the condensed phases, their transition temperatures, and latent heats. The parameters of the free energy, ΔG = ΔH–T ΔS, are shown in Table 3.4, as well as the transition temperatures that arise from setting the free energies of each pair of phases, β-δ and δ-liquid, equal and solving for the temperature. The β-δ phase transition temperature determined using the values of Lyman et al. were chosen by those authors using the temperature determined by Hall [102] for crystalline HMX. We have modified the entropy of the delta phase HMX to yield the equilibrium transition temperature observed by Cady in early synthesis work [106] and by Henson et al. [14] and Smilowitz et al. [12] in studies of the kinetic slowing of the transition rate about the equilibrium temperature. This is likely the true equilibrium transition temperature, as it has been shown that the binder in PBX 9501 acts as a partial solvent for HMX and induces the phase transition with no discernible restriction to nucleation [13, 19]. We show the data of Taylor and Crookes [138] and Rosen and Dickinson [137] in Fig. 3.12 and calculate the vapor pressures above both β and δ HMX using the equation P = Po exp
T ΔS − ΔH , RT
(3.54)
where we have used Po = 760 torr. The data have been plotted the reported pressure (torr) as a function of the temperature. The data of Taylor and Crookes [138] are shown as the solid squares and circles, each set denoting data obtained from cells with different effusion orifices. The data of Rosen and Dickinson [137] are shown as the open circles. The solid and dashed lines are the calculations of vapor pressure over both β and δ HMX.
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Fig. 3.12. Sublimation data from HMX plotted as the reported pressure (torr) as a function of the temperature. The data of Taylor and Crookes [138] are shown as the solid squares and circles, each set denoting data obtained from cells with different effusion orifices. The data of Rosen and Dickinson [137] are shown as the open circles. The solid and dashed lines are the calculations of vapor pressure over both β and δ HMX using the parameters of Table 3.4
Summary We have calculated a simple phase diagram for HMX based on the free energies of each phase and using P ΔV = ΔH – T ΔS to calculate the phase boundaries. We calculate an HMX crystal phase diagram and instead of the free energies of Table 3.4 we use very similar values provided by Karpowicz and Brill [78] for β and δ phases, which results in a β-δ phase boundary of 450 K instead of the equilibrium temperature of 432 K exhibited by PBX 9501. We use values from Cady [106] for the other phases. We also use a more sophisticated calculation of the β liquid coexistence line by Menikoff [139] as a measure of the overall accuracy of this simple diagram. The solid lines are the observed boundaries between the β, δ, vapor, and liquid phases. The dashed lines are the β-α and the metastable β-liquid coexistence lines. This brief summation of the thermodynamics of HMX will be used in the next section. Specifically we will use the free energy differences between the
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phases as free energies of activation in order to rationalize in as simple a way possible the observed rates of transformation and reaction observed in HMX. 3.3.4 HMX Thermodynamics and Ignition The overwhelming consideration in the interpretation of data and the construction of conceptual and quantitative models of HMX decomposition and ignition has been the association of rate limiting processes with the breaking of covalent bonds. Although this is traditional, reasonable, and commonly useful, it is still a hypothesis. It borrows heavily from work in perfectly mixed systems such as the liquid and gas phase, where the relevant chemistry is relatively unrestricted in the mechanisms whereby reagent species become associated. This is not necessarily true in the solid state, and has been challenged in the literature on HMX decomposition [74]. We are going to pose an alternate hypothesis in this section and in the construction of the ignition models to follow. We propose that HMX thermodynamics are principally responsible for controlling the rate of solid decomposition, energy release, and ignition in this material, at least for energetic response in the condensed phase. This thermodynamic rate controlling behavior is manifested as a suite of observed rate constants which, as for the β-δ transformation mechanism, are thermally activated by thermodynamic free energies of transformation. There are at least three reasons why this may be the case: 1. It seems that the dissolution of the lattice is a primary and necessary event before subsequent chemical reactions may take place. We have already encountered this in the reconstructive phase transformations, where the passage between two incommensurate crystalline phases requires essentially randomizing the molecular ensemble such that the new structure might be found. HMX accomplishes this, with help from the stress tensor and the virtual melt mechanism, by forming a transient liquid phase. This necessity of dissolution of the lattice in the subsequent decomposition may be accomplished either by direct sublimation or vaporization from another form of transient liquid. Either process requires that the endothermic latent heat be absorbed, and therefore lost to the process of nonlinear thermal runaway. Finally, either of these processes may be expected to further effect the mechanism of decomposition through the modification of morphology, for example, by the formation of pores, regression of surface, the resulting loss/formation of surface area, and so on. 2. The latent heat that must be absorbed during these transformations is on the same order as the endo/exothermicity expected from covalent bond breaking. The heat of sublimation of δ-HMX, 175 kJ mol−1 (Table 3.4), is as endothermic as the subsequent gas phase dark zone reaction system, 180 kJ mol−1 (ΔH3 summed) on a per molecule basis, and overall exothermicity requires the generation of several reagent species per parent HMX molecule.
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3. If the breaking of covalent bonds is faster than any of these thermodynamic processes, then the true rate limiting phenomena will be thermodynamic, with the expectation that limiting rates may be derived from thermodynamic rates of sublimation, melting, or vaporization. We discussed the construction of rates from thermodynamic free energies in section “Temperature Dependence: The Arrhenius Rate Constant.” Comparison of Kinetic and Thermodynamic Rates This new hypothesis is based on the following observation. We assume the relationship 1 kign = k2a k2b or tign = , (3.55) k2a k2b where kign is the single Arrhenius rate constant determined from the suite of ignition data in section “The Non-Prepressurized, Confined Data Set” and Fig. 3.10 and k2a and k2b are the rates of appearance of the two product channels measured by Brill et al. [89]. We make this assumption without justification at this point. We further assume that the rates of appearance reflect the relative ratio of species expected from a competitive, first order reaction system, as discussed in section “Subsequent Gas Phase Product Decomposition.” The relationship between the observed product concentrations and the rate constants for that system is then k2b t2a [N2 O] = = . (3.56) [NO2 ] k2a t2b Assuming the free energy form of the rate constant given by (3.7), we can rewrite these two equations as ΔGign ΔG2a + ΔG2b exp = exp , (3.57) RT RT
and exp
ΔGp RT
= exp
ΔG2a − ΔG2b RT
.
(3.58)
Using the simple expression for the free energy ΔGi = ΔHi – T ΔSi , where i denotes one of the four free energies in the system, this system of (3.57) and (3.58) can be solved for the activation enthalpy and entropy of reaction for product channels 2a and 2b using the parameters determined by the linear regression of the ignition data, Fig. 3.10 and Table 3.3, and those determined from the product concentration ratio of Fig. 3.6 and Table 3.2. The results are listed in Table 3.5. We compare these activation enthalpies and entropies with the thermodynamic enthalpies and entropies of sublimation and vaporization from Table 3.6.
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Table 3.5. Parameters from the solution of the system of (3.57) and (3.58) Value
σ
From Table 3.2 ΔHp kJ mol−1 ΔSp J mol−1 K−1
72.073 117.66
10.4 6.2
From Table 3.3
ΔHign kJ mol−1 ΔSign J mol−1 K−1
148.54 243.71
1.0 1.8
Solution
ΔH2a kJ mol−1 −1 ΔS2a J mol K−1 ΔH2b kJ mol−1 ΔS2b J mol−1 K−1
184.57 302.54 112.51 184.88
6.2 4.9 6.2 4.9
Table 3.6. Comparison of activation and thermodynamic free energies Thermodynamic free energies from Lyman et al. [40] ΔH(kJ mol−1 )
σ
ΔS(kJ mol−1 K−1 )
β-HMX (sublimation) 184.762 1.5 279.311 δ-HMX (sublimation) 174.930 1.5 257.942 Liquid (vaporization) 105.067 4.2 131.149 Activation free energies for rates of gas phase product appearance 184.57 6.2 302.54 k2a 112.51 6.2 184.88 k2b
σ 3.7 3.7 2.3 4.9 4.9
The enthalpies of sublimation and activation of product channel 2a are equal within experimental error (for β HMX). The enthalpy of vaporization and activation of channel 2b are also equal. It seems that the equality should hold for δ-HMX for channel 2a, and this will be the topic of future work. The entropies of transformation and activation are also only qualitatively equal. However, even given this residual uncertainty, the near quantitative nature of the comparison and association is striking. We therefore attribute the free energy parameters of sublimation with the activation free energy parameters for the formation of product channel 2a, and also for vaporization and the formation of product channel 2b and hypothesize that the appearance of these species is not elementally limited by the rate of covalent bond reorganization, but by the elemental rates of sublimation and vaporization. We use the term “elemental rate” here to indicate that, as discussed in section “Temperature Dependence: The Arrhenius Rate Constant,” we have not yet specified an order or morphology for how evaporation takes place in the experiments we wish to model. And indeed, again as discussed in section “Temperature Dependence: The Arrhenius Rate Constant” such a mechanism for evaporation may not be the same from experiment to experiment. This association, however, is nonetheless the conceptual centerpiece of our strategy for a global chemistry model of ignition for HMX.
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Ramifications for the Modeling of Ignition The ignition rate constant determined in section “The Non-Prepressurized, Confined Data Set” may or may not embody a relatively elemental step in the decomposition, for example, the calculated rate of N–NO2 bond scission that has been reported exhibits a slope similar to ΔHign [105]. We have, however, made a fairly careful survey of the reported rate constants in the nitramine literature [116, 140] and elsewhere [141, 142] and we find no such measured and attributed elemental rate. From this and the observation of section “Comparison of Kinetic and Thermodynamic Rates” above, one may associate this rate constant with a simple functional form comprising measured rates of sublimation and evaporation that lead us to assume that the single Arrhenius rate comes from the geometric mean of the two thermodynamic rates. If one accepts this assumption then certain constraints apply to the models that may be constructed. Comprehensive Thermal Response Model If the rate constant reflects a more complex mechanism and if such a mechanism can be constructed from the two product rates k2a and k2b , then it is possible that such a model will serve as an accurate single model of HMX response over the full range of temperature. This is particularly plausible as the underlying physical processes that exhibit these rates, sublimation and vaporization, are robust physical process that we know are operative over this entire range, perhaps with some increasing complexity as the thermodynamic critical point is reached. In addition, because of the breadth of rates and means of heating described in Sect. 3.3.2 is also possible to argue that this single comprehensive model of response should be applicable for all means of heating the material. This is significant motivation indeed. Sequential or Simple Competitive Models The classic sequential or simple competitive systems that we have termed summed component systems in section “Temperature Dependence: The Arrhenius Rate Constant” and that comprise most if not all current models of energetic material decomposition cannot serve as the mathematical model for such a comprehensive mechanism because of the separate dominance of the individual rate constants in the mechanism; with respect to the specific rate constants here this is shown in Fig. 3.14. We have again plotted the ignition data of Fig. 3.14, along with the calculated ignition rate constant, tign = 1/kign , as the black line. The data are again plotted as a characteristic time as a function of the inverse temperature. We have also plotted the characteristic times of the rates of sublimation and vaporization from section “Comparison of Kinetic and Thermodynamic Rates” as the dashed and dot-dashed lines, respectively. The bold black line is a calculated characteristic time from an analytical two state model where a first and second order process are used to model decomposition as
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Fig. 3.13. Simple phase diagram for the HMX system calculated as discussed in section “Summary”
∂x = k2a (1 − x) + k2b x(1 − x), ∂t
(3.59)
where x is a progress variable going from 0 to 1. Equation (3.59) is of a familiar summed component type. Many decomposition models use systems of equations of this type and our most recent global chemistry model is of this type in the intermediate decomposition steps. These applications of (3.59) are successful at low and intermediate temperatures for the following reason. Equation (3.59) may be solved for the characteristic time (here a solution is obtained for the t when x = 0.5). This solution is given by ⎛ ⎜ arctan ⎝ t = −2
⎞ (k2b − k2a ) 2
⎛
⎟ ⎜ ⎠ + arctan ⎝
⎞ k2a 2
− (k2b − k2a ) − 4k2a k2b − (k2b − k2a ) − 4k2a k2b 2 − (k2b − k2a ) − 4k2a k2b
⎟ ⎠ .
(3.60)
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Fig. 3.14. Ignition data as in Fig. 3.10. In addition, the characteristic times for the rates of sublimation (dashed line) and vaporization (dot-dashed line) are shown. The bold black line is an analytical solution to a common form of summed component model, showing the divergence from the geometric mean ignition behavior observed for HMX
The solution for k2a = k2b is given by 1 2 arctan h 2 . t= 2k1 k−1
(3.61)
This is the functional form assumed at the beginning of this section, (3.55), and is the means by which a summed component model based on sublimation and vaporization rates exhibits the averaged rate behavior observed in Fig. 3.10. The behavior will fail, however, for temperatures away from that which yields k2a = k2b . Incidentally, for k2a and k2b based on activation free energies taken from sublimation and vaporization, the temperature where they are equal is of course the melting point. This may be a factor in the ability to simply model solid decomposition in the family of very complex gas phase combustion models for HMX, where the range of temperatures used is naturally about or slightly above the melting point [8, 9].
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Finally, the black bold line of Fig. 3.14, which overlaps the ignition calculation near the melting point yet diverges to faster time both above and below, according to the two individual rates, is calculated from (3.60) above and from the sublimation and vaporization parameters of Table 3.4. Oscillatory Systems The few systems that we have identified, which might serve to couple the rate constants k2a and k2b throughout the range of energetic response, are the oscillatory systems described in section “Subsequent Gas Phase Product Decomposition,” (3.50) and (3.51). Preliminary calculations using such models are very promising but still in the very early stages of application. We will show some early results, without detail and for comparison with the results of much more mature (and typical) summed component models in the next section. Summary We have shown in this section that an effective conceptual argument can be made for the attribution of rate limiting processes to the thermodynamic rates of sublimation and vaporization. We will further show below that very effective thermal ignition models may be constructed as well. We have not yet acknowledged, however, the role of a vaporization rate, presumably denoting the passage of HMX molecules from the liquid to the gas phase, again at temperatures significantly below the melting point. We find ourselves again with the same problem faced during the construction of β-δ phase transition models with an activation energy equal to the heat of fusion. At the time of this writing this remains an open question. However, as was true of the virtual melt mechanism, which required the use of sophisticated considerations of the incoherent phase front based on the full stress tensor, we are confident that equally sophisticated considerations of the molecular surface under these conditions will yield the answer [21]. 3.3.5 Global Kinetic Models of PBX 9501 (HMX) We present in this section a brief summation of the simplified reaction mechanisms used in our global chemistry models of HMX decomposition and ignition. We will then present our last and most detailed summed component model along with calculations and comparisons with experiment. We will also show some preliminary output from our first and at the time of this writing only very recent coupled model. The chemical decomposition models presented here are based on independently measured rate constants and thermochemical parameters. There are only two degrees of freedom in the models, the kinetic order and one tunable constant, a multiplier of the prefactors for k2a and k2b .
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The calculations represent a simple integration of the differential equations that will be presented and are there homogeneous, model applications. The calculations are not coupled to any larger simulation and therefore also do not strictly conserve energy. These will be the tasks of application when these models are built into more sophisticated computational platforms. The β-δ Transition We have constructed a series of models for the β-δ phase transition in HMX, as discussed in section “The Crystalline Phases of HMX and Rates of Transformation.” These include a single, robust, and well demonstrated model of the growth phase based on the virtual melt, and more speculative models, less well grounded in experiment, for the nucleation step in the transformation. We endeavored throughout the work to create models, which are applicable independent of the rate of heating or type of boundary condition, for example, isothermal or nonisothermal, and the experimental means of determining the progress of the transformation, for example, SHG, IR, or Raman spectroscopy or calorimetry. The model that best reproduces internal temperatures within a solid PBX 9501 sample during heating to ignition remains our original model, based upon a reversible first order nucleation step as well as the reversible second order growth [12, 14]. As we have discussed, although this gives the best reproduction of the data, such a first order nucleation yields a nonzero concentration of the precursor phase after transformation, that is, a residual equilibrium of the two phases is always present. While this is probably unphysical for considerations of the material bulk, if only by consideration of the Gibb’s phase rule, recent interesting observations by a surface nonlinear optical spectroscopy sum frequency generation indicate the presence of a surface phase of δ-HMX at room temperature [143, 144], and serve as a reminder that the real crystalline systems are more complex than an idealized homogeneous bulk phase. Nonetheless, the calculated presence of a minor residual phase is very problematic if subsequent steps in the decomposition involving the loss of δ-HMX are second order and require nucleation mechanisms themselves. Such small concentrations can have a major impact on the kinetics by serving to nucleate the loss of δ-HMX too early. To circumvent this difficulty, we use a single, forward-going mechanism based on the mechanism proposed in Levitas et al. [18] and using a single rate constant that is the sum of the rate constants for both conversion and reversion in the reversible mechanism. We make this modification as it will preserve the forward rate of nucleation, which proceeds via a rate that is the sum of the two rates, as shown in the section “Classic Kinetic Schemes.” Gas Phase Chemistry We utilize the gas phase mechanisms for CH2 O oxidation discussed in section “Classic Kinetic Schemes” essentially without modification, although we make
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use of the simplified application that was discussed. We do not currently employ any bright zone chemistry, but it may be required to fully simulate the temperatures achieved during ignition, as will be discussed below. The Intermediate Decomposition The intermediate decomposition steps, which lead to the conversion of solid δ-HMX to gas phase decomposition products, are the most critical, rate limiting, steps and the focus of our hypothesis concerning the thermodynamic control of rates. Even given full specification of the thermochemical and thermophysical properties by using real chemical species and stoichiometries, and fixing the rates used in the problem by independent measurement, one critical degree of freedom remains to be specified. This is the order of the reactions, expressed as the concentration dependence of the rate equations, and discussed at length in Sect. 3.2.1. The most important point of that discussion was the fact that even a well understood process like sublimation may exhibit many different kinds of concentration dependence, depending upon the solid morphology and the specific observable to be modeled. This important point translates directly into a critical question when composing a set of intermediate steps to reproduce the observed rates of sublimation, vaporization, and their combination to produce the observed geometric mean rate of ignition (3.55). We have published two serious mechanisms, both of the summed component type, which reflect different choices and answers to this critical question [4, 145]. In this final section we present our most recent and likely last summed component model, along with results and comparisons with experiment. In this model we simulate a first order sublimation of δ-HMX using k2a and a second order term in k2b . We have fit a single parameter to the suite of thermal explosion data that will be presented. This parameter is the additional prefactor Q2a =Q2b =0.00048. We treat the gas phase HMX decomposition explicitly, limiting the concentration to the vapor pressure and using the calculated rate constant of Zhang and Troung [105]. We will also present preliminary calculations using a coupled model for comparison. Summed Component Model and Results We present model results from a summed component model. The model is entirely constrained by a global chemistry, and parameterized entirely by independent and more elemental measurements of rate. By this we mean that stoichiometry determines all the enthalpies of reaction, based on measured enthalpies of formation. All thermophysical properties, heat capacity (Cv ), and thermal conductivity (λ), are taken from measurements of both solid phases and measured gas phase properties (all gas phase species are approximated as air). Most importantly, all rate constants used in this model have been determined independently of this model, for example, phase transformation
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rates were measured by SHG in PBX 9501, sublimation rates are taken from Taylor and Crooke [138] and Lyman et al. [40] and modified as discussed in Sect. 3.2.1, gas phase rates were taken from the measurements and modeling of Melius and Lin [113]. We use concentrations to construct the kinetic rate equations, even for the condensed phase components in the model. This modeling strategy has inherent problems, as discussed in section “Temperature Dependence: The Arrhenius Rate Constant,” but it is the strategy adopted here. We use simple mole fraction mixing rules to generate averaged thermophysical properties, as will be shown below. Most importantly this is a homogeneous application and so there is no gas phase diffusion or mass transport. Gas phase “pressures” are calculated using P = [c]RT. In particular, the HMX partial pressure is calculated at each time step and not allowed to exceed the vapor pressure of HMX. The first step in the model is the β-δ phase transformation. We model this using an irreversible nucleation step, first order in the β-HMX concentration, and a reversible growth step, of mixed second order in both β and δ concentrations (the virtual melt model discussed in section “The Crystalline Phases of HMX and Rates of Transformation”). This is followed by the formation of gas phase g-HMX by sublimation, first order in δ HMX, followed by HMX decomposition to form NO2 . Simultaneously, gas phase CH2 O is formed with a rate dependent upon the rate of HMX vaporization. This rate is second order in the δ HMX concentration and presumably proceeds via a melt mediated process similar to the growth mechanism of the β-δ solid–solid transformation. The model ends with the exothermic gas phase reaction mechanism initiated by bimolecular reaction of NO2 and CH2 O discussed in section “Subsequent Gas Phase Product Decomposition.” The identities and stoichiometries of the individual steps are given by the following equations. β − HMX → δ − HMX, δ − HMX ↔ g − HMX,
(R5) (R6)
g − HMX → 2NO2 + 4HCN + 2NO + 2H2 O.
(R7)
R5 is the phase transformation, R6 is the sublimation step, and R7 is the subsequent decomposition of gas phase HMX. Kinetic parameters for the sublimation of δ-HMX are used, modified to incorporate the kB T /h TST prefactor. The kinetic parameters for the gas phase decomposition to NO2 are taken from the ab initio Q.M. calculation of Zhang and Troung [105]. The assumed stoichiometry of R7 is taken from the T-jump experiments and modeling of Thynell’s group [146]. δ − HMX → 4CH2 O + 4N2 O.
(R8)
R8 governs the appearance of gas phase formaldehyde. As discussed earlier, the appearance is governed by a temperature dependent rate proportional to the
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rate of HMX vaporization from the liquid phase. HMX decomposition occurs from the solid, certainly below the melting point of 550 K and it probably occurs from the superheated solid at temperatures above the melting point due to the high rates of heating necessary to drive ignition to such high temperatures. Work is in progress to fully elucidate this mechanism, but we assume that it involves a surface quasi-liquid or premelt phase. This phenomenon has been observed in a number of other systems, and we have worked extensively on models of the equilibrium phase [21]. We currently model this step with a mixed second order mechanism in the δ-HMX concentration, which generates an overall rate Gaussian in the δ-HMX concentration. This form is often used to simulate a pore growth and coalescence process and mimics a dependence on surface area, even though parameterized in the concentration. CH2 O + NO2 → products.
(R9)
R9 is the gas phase decomposition model for the formaldehyde and nitrogen dioxide system discussed in section “Subsequent Gas Phase Product Decomposition.” The first step, R9, is rate limiting, and in the model we approximate the three steps as a single differential equation of modified stoichiometry, where CH2 O + 2NO2 goes to products as discussed. The heats of reaction for each step are calculated from Hess’s Law, ΔHreaction = Σ (ΔHproducts ) –Σ (ΔHreactants ) and taken as independent, with the exception of R9, where a modified equation is used to correct for the simplified stoichiometry. We use the formation enthalpies from the NIST Chemistry Webbook listed in Table 3.7. None of the reaction enthalpies used here exhibits a substantial temperature dependence over the range of interest and so we use the single temperature independent constants. Table 3.7. Heats of formation from the NIST chemistry web page
ΔHformation J mol−1 β-HMX δ-HMX g-HMX CH2 O NO2 HCN NO H2 O N2 O HONO HCO CO CO2 H
74,894 84,694 260,594 −115,900 33,095 135,143 90,291 −241,826 82,050 −76,734 43,514 −110,527 −393,522 217,988
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The set of coupled differential equations describing the evolution of the key chemical species β, δ, g-HMX, NO2 , and CH2 O are given by ∂[β] = −k1nf [β] − (k1gf − k1gr )[β][δ], ∂t
(M1)
∂[δ] = k1nf [β] + (k1gf − k1gr )[β][δ] − k2a [δ] − k2b [δ] [ρ − δ] , (M2a) ∂t ∂ [CH2 O] = k2b [δ] [ρ − δ] − k3 [NO2 ] [CH2 O] , (M2b) ∂t ∂[g] = k2a [δ] − k2c [g] (M2c) ∂t ∂ [NO2 ] = k2c [g] − k3 [NO2 ] [CH2 O] . (M3) ∂t An additional equation is integrated to calculate the total number of unreacted gas phase molecules generated during decomposition, determined by the stoichiometries in R6–R9 above. ∂ [gases] = 4k2b [δ] [ρ − δ] + 7k2c [g] + 3k3 [NO2 ] [CH2 O] ∂t
(M4)
The total pressure is then calculated by adding gas phase concentrations, treating the mixture as ideal and using the equation P = ([gases] + [gHMX] + [CH2 O] + [NO2 ]) RT,
(M5)
where P is the pressure in Pa and R in J mol−1 K−1 . This neglects the few percent free volume present in typical samples and any free volume associated with pressure measurement accessibility. These effects must be considered in any comparison of this calculated pressure with data. And as this is a homogenous calculation, there is no diffusion and the pressures are only really meaningful at times during ignition that are fast compared to diffusion. The rate constants are parameterized using the TST rate constant discussed extensively in section “Temperature Dependence: The Arrhenius Rate Constant,” essentially (3.7) at zero pressure T ΔS ∗ − ΔE ∗ kB T Q exp ki (T ) = (3.62) h RT where ki (T ) is a canonical rate constant in s−1 or m3 mol−1 s−1 , depending on reaction order, T the absolute temperature, kB and h are Boltzman’s and Plank’s constants, respectively, ΔE ∗ and ΔS ∗ are the activation energy and entropy of the activated state, and R is the gas constant. Q is an equilibrium constant relating the concentrations of the activated species to reagents, and ρ is the initial solid density, 5902.2 m3 mol−1 . The other parameter values used in the calculation are given in Table 3.8.
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Table 3.8. Rate constants (3.7) Q (s−1 or m3 mol−1 s−1 ) K1nf K1gf k−1gr K2a K2b K2c K3
1 1 × 10−15 1 × 10−15 0.00048 0.00048 1 1.00 × 10−6
ΔS ∗ (J mol−1 K−1 ) 309.328 138.998 116.238 54.961 −62.704 42.170 −70.000
ΔE ∗ (kJ mol−1 ) 374.329 80.346 70.546 184.57 112.51 155.878 68.000
Table 3.9. Values of the thermophysical parameters of the model
C1 J mol−1 K−1 C2 J mol−1 K−2 λ1 W m−1 K−1 λ1 W m−1 K−2 β-HMX δ-HMX Gases
73.75 104.57 24.0
0.775 0.714 0.038
0.4368 0.7848 0.042
−4.44 × 10−4 −1.39 × 10−3 0
The thermo-physical properties of the model are quite simplified. Both the thermal conductivity (λ) and heat capacity (c) are linearly dependent on temperature using (3.63) C = C1 + C2 T and λ = λ1 + λ2 T,
(3.64) −1
where t is in k, the heat capacity is in units of J mol k−1 , and the thermal conductivity is in w m−1 k−1 . the heat capacity and thermal conductivity of a volume element in the calculation is a mole weighted combination of three states, β-HMX, δ-HMX, and gas products (taken as the sum of gaseous species as in (3.6) for the pressure). The values for the heat capacity for both HMX polymorphs come from a recent compilation of HMX thermodynanmics [40], the thermal conductivity of HMX from recent unpublished measurements here at LANL, and the values for the gases are approximately those of air. In this model we have only addressed questions regarding ignition within the constraints of the ignition data of Fig. 3.10, in other words for confined, unprepressurized systems. High pressure data exist and must eventually be considered and rationalized within this new decomposition and ignition system. Comparison with Data The experimental data presented here for comparison with model calculations were obtained from classic thermal explosion experiments on PBX 9501 samples of solid cylindrical symmetry. These experiments have been described previously [62, 147] and in Chap. 7. They consist of two right circular cylinders of PBX 9501 stacked together and enclosed in a metal (aluminum or copper)
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case. The midplane interface between the halves is outfitted with an array of diagnostics. These experiments, which we call the radial thermal explosion configuration, are typically run for sample sizes with 1/2 diameter and 1 high, 1 diameter and 1 high, and 1 diameter 2 high. A schematic of the experiment and photograph of the midplane diagnostic array are shown in Fig. 10 of Chap. 7. The radial thermal explosion experiments are conducted by heating the explosive directly from the cylinder walls. The heaters are typically nichrome wires wrapped around the metal case. The case is heated through a temperature trajectory at a rate of 5◦ C min−1 with a hold point at 178◦ C to allow for the β-δ phase transition to occur and the PBX 9501 to fully equilibrate before heating to the final hold temperature, which is typically between 185◦C and 210◦C. The internal thermocouples show the time lag for the conductive heating of the PBX 9501 from the outside cylinder wall, which is controlled by the thermal conductivity of the explosive. At the 178◦ C hold point, the endothermic phase transition is evident in the decrease in thermocouple temperatures seen to occur beginning with thermocouples closest to the wall and propagating to the central volume. The shape and duration of the endotherm are determined by the kinetics of the phase transition. After allowing the phase transition to complete and the HE to equilibrate at the boundary temperature, the final 5◦ C min−1 heat step is conducted to the final hold temperature. Figure 3.15 shows the boundary temperature in black and the internal HE temperatures in gray. The bold black line is the model calculation of the central thermocouple for the boundary temperature measured. A comparison of the model predictions for the phase transition region, and time and shape of the final thermal runaway leading to ignition can be seen in the figure. The model also reproduces the spatial temperature gradients. It accurately predicts the ignition location as the central point furthest from the metal boundary, which acts as a heat source until the exothermic chemistry of the explosive becomes the dominant heat source. At this point, the conductive metal boundary, pinned at the final hold temperature, begins acting as a heat sink rather than a heat source and it drives the ignition location to the position furthest away from it, which is the central point of the cylinder. Figure 3.16 shows the temperature traces from Fig. 3.15 plotted in a different way to enhance the details of the final thermal runaway regime leading to ignition. The time axis is now plotted logarithmically with time zero being the ignition event. The origin of the time axis is the start of the experiment, which is approximately 2,000 s from ignition. The abscissa is plotted as time until ignition using a logarithmic scale so that the details of the final second can be seen with microsecond resolution. The grey lines are the measured internal temperatures. The central ignition volume measurement is the top curve and it is recorded with microsecond resolution by directly measuring the voltage generated on the thermocouple into a fast oscilloscope rather than with the standard signal conditioning applied to thermocouple measurements. This increases the error bar on the measurement from the few tenths of degree
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Fig. 3.15. Calculations using the summed component model described in section “Model and Results.” The black and grey curves are data from a solid cylindrical thermal explosion experiment, in this case 1/2 diameter by 1 length PBX 9501 sample held at a final boundary temperature of 478 K. The black curves are the applied boundary temperature and the grey curves the measured internal temperatures. The data are plotted as the measured temperature as a function of time. The bold black curve is the calculated central temperature
accuracy to a few degrees measurement uncertainty. The benefit is that it allows time resolution not limited to the tens of milliseconds available on thermocouple data acquisition systems. The fast temperature record taken in this way shows the final thermal runaway with inflections at the temperature observed in the dark zone region of a surface burn above a burning PBX 9501 surface [8]. Figure 3.16 shows both the final temperature structure in the ignition volume and the large temperature gradient sustained in the PBX 9501 during the final thermal runaway. The grey curves below the highest curve are the temperatures measured millimeters away from the central volume. They demonstrate none of the final thermal runaway but rather end suddenly when the flame front from the central ignition volume sweeps past them. The dashed black curves in Fig. 3.16 are the model calculations for temperatures spaced by ∼10 μm from the central region. The time to ignition and location of the ignition volume are well captured by the model calculations. In addition to calculating internal temperatures, the middle step of the model calculates the solid to gas phase reactions. This can be used to calculate the loss of solid fraction with reaction progress. The solid concentrations as a function of radial position for several different times prior to ignition are shown in Fig. 3.17. As the reaction progresses towards ignition, there is a measurable
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Fig. 3.16. Fast internal temperature measurements and calculations. The black curves are the applied boundary temperature and the grey the measured internal temperatures from experiments described in Fig. 3.15. The data are plotted as the measured temperature as a function of time, with time plotted backwards from ignition and plotted logarithmically. The dashed black curves are internal temperature measurements calculated using the summed component model of section “Model and Results”
loss of solid at the central volume. The final curve in Fig. 3.17 shows a total loss of solid at the central volume with a full-width half-maximum diameter of nearly 500 μm. An experimental measure of the density loss evolution measured using proton radiographic imaging of a radial thermal explosion experiment is shown in Fig. 33, Chap. 7. The diameter of the region where the solid density is decreasing is approximately 1 mm, in fair agreement with the model calculations. The model does not incorporate mass diffusion that is occurring in the actual system. The calculations shown have so far all been for a single isothermal temperature. Figure 3.18 shows the model calculations and comparison to experiments for several different final temperatures. The black lines are the measured boundary temperatures, which for the three experiments shown are 468, 478, and 483 K. The grey lines are the measured internal HE temperatures. The change in the time until ignition over this 15◦ temperature range is approximately a factor of three. The colored curves in Fig. 3.18 are the model calculations for the applied boundary conditions using the summed component model described in section “Model and Results” (shown in bold
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Fig. 3.17. The solid fraction of δ-HMX as a function of the radius of the cylinder in the experiments described in Fig. 3.15. The calculations were done using the summed component model described in section “Model and Results.” The data are plotted as the solid concentration as a function of radius. The times of each calculation begin with the top curve at 2 s from ignition and descend as 1 s, 500 ms, 100 ms, 50 ms, 10 ms, 5 ms, and 1 ms from ignition
black) and the calculations from a preliminary oscillatory model (shown in dashed bold black). What this figure demonstrates is the evolving capability of the global kinetic model to predict both regimes: it captures the details of the internal temperatures and ignition times for the individual experiments, and without tuning parameters, is also able to capture the exponential temperature dependence of the chemistry of thermal explosion.
3.4 Conclusions Global chemical models of thermal ignition represent an approximation of the full chemical complexity of the decomposition of a given material. The selection of the appropriate subset of the full decomposition mechanism of a material was discussed in Sect. 1 and centered on the idea of the extreme temperature sensitivity in the rate of thermal decomposition necessary to drive a nonlinear thermal runaway and explosion. Any realistic model of thermal
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Fig. 3.18. Three experiments at three final boundary temperatures of 468, 478 and 483 K. The black curves are the applied boundary temperature and the grey the measured internal temperatures from experiments described in Fig. 3.15. The data are plotted as the measured temperature as a function of time. The bold black curves are calculations using the summed component model described in section “Model and Results.” The dashed bold black curves are preliminary calculations using a coupled oscillatory model
ignition must deal with this exponential sensitivity, must in some sense find the “correct” description of this temperature dependence. This exponential component of the model sensitivity can and typically does overwhelm the remaining linear sensitivities represented by, for example, the thermal conductivity. A global chemistry model is in a sense the search for this correct rate, justified independently of the model itself. It is also clear from the discussions of Sect. 2 that it is now possible to go quite far past thermal ignition models using the Arrhenius formula as a fitting function and suffering the confusions of kinetic compensation effects in inadequately determined rate constants. The application of sophisticated yet traditional concepts and techniques from chemical kinetics have already led to a full model of ignition in HMX using only parameters determined independently and in more elemental experiments. What we have managed to do using the full suite of kinetics data available for PBX 9501 is put together a global kinetic model that captures the exponential temperature dependence of thermal explosion as well as the detailed internal temperature evolution during thermal decomposition leading to ignition. The current model is a summed component
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model using intermediate steps that are not fully mechanistically constrained, but are based on the independently measured thermodynamics of HMX. Work is ongoing to advance the model towards a fully constrained mechanism for which each differential equation is tied to a physical or chemical step. With this approach we believe that the ultimate product of the pursuit of such global chemistry models, while significantly more demanding and difficult to produce, will be truly predictive not just in calculating the time to ignition, but also the location, pressurization, and other necessary boundary conditions on the subsequent release of energy following ignition.
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4 Classical Theory of Thermal Criticality Larry G. Hill
4.1 Introduction Nomenclature Lower Case Roman Symbol a b c f g h k m q r s t v w x y z
Definition Defined or first arises in Eq... General constant (4.36) General constant (4.36) Specific heat capacity (4.81) Criticality function (4.65) Reaction progress function (4.45) Heat transfer coefficient/area (4.51) Thermal conductivity (4.52) Mass (4.10) Specific heat of thermal decomposition (4.81) Characteristic charge size (4.39) General constant pertaining to slope (4.43) Time (4.15) Speed (4.40) Lambert W function (4.66) Spatial variable within a charge – Spatial variable within a charge – General dummy variable (4.36)
Upper Case Roman Symbol A AR B C E
Definition General constant Aspect Ratio General constant General constant Activation energy
Defined or first arises in Eq... (4.7) (4.160) (4.26) (4.46) (4.2)
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Upper Case Roman Symbol Ei F G H I M N Q R T T∗ W Z
Definition Defined or first arises in Eq... Exponential integral function (4.78) Inverse criticality function (4.65) Function used in depletion-induced explosion limit (4.77) Heat transfer coefficient for entire charge (4.51) Function related to F-K quasi-universality (4.154) Molecular weight (4.10) Number (4.3) Heat of reaction for entire charge (4.50) Universal gas constant (4.2) Temperature (4.2) Activation temperature (4.2) Inverse Lambert W function (4.66) Arrhenius pre-exponential factor (4.2)
Upper Case Roman Script Symbol A B C H K Q R V Z
Definition Defined or first arises in Eq... Total charge surface area (4.51) Incomplete Beta function (4.36) Chemical concentration, expressed as mass fraction (4.1) Critical height in drop weight impact machine – “Specific” reaction rate (4.1) Heat of reaction per unit volume (4.71) “Total” reaction rate (4.1) Total charge volume (4.39) Reaction rate constant for a hot spot reaction (4.41)
Lower Case Greek Symbol α β γ δ ζ η θ κ λ μ ν ξ
Definition Defined or first arises in Eq... Void fraction – Ratio of generalized to aesthetic length scale (4.157) Slope, used in criticality correction function (4.192) Thickness of a plate or reaction layer (4.52) Fractional error, F-K vs. Arrhenius critical temp. (4.195) General dummy variable (4.125) Number density (4.39) Frank-Kamenetskii dimensionless temperature (4.91) Thermal diffusivity (4.82) Charge volume-to-surface area ratio (4.71) General constant (4.43) General function variable (4.125) General dummy variable (4.16) Exponent used in autocatalytic or hot spot reactions (4.23)
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Lower Case Greek (cont.) Symbol ρ σ τ ϕ χ ψ ω
Definition Defined or first arises in Eq... Density (4.81) General constant (4.43) Scaled relaxation function (relates charge shapes) (4.160) General dummy variable (4.137) Dimensionality parameter used in universal scaling (4.154) Mass fraction (product mass fraction if unscripted) (4.11) Function used in spherical criticality problem (4.137) Reaction order (4.1)
Upper Case Greek Symbol Π Υ Ψ Ω
Definition Defined or first arises in Eq... Product of concentrations in law of mass action (4.1) Correction function for finite Biot number (4.190) Inverse of the ψ function (4.141) Correction function for finite activation energy (4.193)
Named Dimensionless Groups Symbol Bi Da Fk Se
Known as... Biot number Damk¨ ohler’s second number Frank-Kamenetskii number Semenov number
Defined or first arises in Eq... (4.161) (4.85) (4.94) (4.102)
Unnamed Dimensionless Groups (Marked by Math Accents) Symbol ˜ H ˆ H q˜ ˜˙ Q tˆ tˇ t˜ ˜ T˜ x ˜
Characterizing... Defined or first arises in Eq... Heat loss compared to generation, entire body (4.55) Heat loss compared to generation, entire body (4.98) Specific heat release (4.85) Total heat release rate Time Time Time Temperature ramp rate Temperature Distance
Subscripts (Miscellaneous Character Types) Symbol a b
Definition “area” “baseline”
(4.54) (4.15) (4.42) (4.74) (4.74) (4.54) (4.112)
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Subscripts (Miscellaneous Character Types) c d e g h i l m n p r s u w A V χ 0 1 2 3
critical depleted edge general hypercritical induction (or equivalently, ignition) limit, explosion maximum near-unity-aspect-ratio shape products reactants surroundings ultimate value of a quantity wall Pertaining to the full charge surface area Pertaining to the full charge volume Pertaining to an arbitrary value of χ Reference or initial value Pertaining to slab geometry Pertaining to cylindrical geometry Pertaining to spherical geometry
4.1.1 Spontaneous Combustion Spontaneous combustion is an unnerving condition whereby a substance suddenly bursts into flames for no apparent reason, that is, in the absence of an obvious ignition source. Some materials are pyrophoric, meaning that they burn spontaneously when exposed to room temperature air. In other words, pyrophoric materials are distinguished by the fact that their auto-ignition temperature is below room temperature. This quality is not a property of the material only, but also depends on its physical form. Some materials (e.g., magnesium) are pyrophoric only when finely powdered such that the surface area to volume ratio is very large. The term hypergolic is used to describe chemical pairs that spontaneously burn upon mutual contact. As such, pyrophoricity is logically a subset of hypergolicity. In practice, however, the latter term is mostly used to describe liquid rocket propellants. The point to be made here is that hypergolic material pairs – one of which is air in the case of a pyrophoric material – are those that spontaneously burn at material interfaces. The subject of this chapter is a different sort of spontaneous combustion, which occurs as the result of volumetric exothermic reaction. The first criterion for volumetric spontaneous combustion is that the material must produce a
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finite amount of heat at ambient temperatures. The second criterion is that the body must be large enough – meaning more precisely that the volume to surface area ratio must be large enough – that heat builds up inside the body faster than it can escape. Because chemical reaction rates are generally exponentially sensitive to temperature, once heat buildup prevails it will generally accelerate to a thermal explosion. The peak temperature attained upon thermal runaway depends on the amount of available energy, and the amount of work performed on the surroundings depends on the volume increase of the reaction products relative to the reactants. In some cases thermal runaway may serve to ignite smoldering reactions, which may in turn ignite flaming combustion. In systems involving high explosives, ignition can lead to violent reaction or detonation. The ultimate example of thermal runaway is star formation, although clearly that system involves several complex processes besides thermal ones. Far from being a mere laboratory curiosity, volumetric thermal runaway poses a serious hazard in industry and agriculture. Some common materials that can spontaneously ignite by volumetric reaction include coal, marshes and peat bogs, haystacks, manure piles, ammonium nitrate fertilizer, cotton bales, tires, mattresses, oil-soaked rags, and – believe it or not – pistachio nuts. Let us examine a few of these systems before launching into the technical discussion. A U.S. Department of Energy bulletin states: “Given the right kind of coal, oxygen, and a certain temperature and moisture content, coal will burn by itself” [1]. The report also quotes J. Dilley, a fireman on the Titanic and one of its survivors, who testified that a coal fire had been burning, and had been continuously fought by firemen, from the time the ship left Southampton until it was struck by the iceberg. Ironically, claimed Mr. Dilley, the largest tear in the Titanic’s hull was under bunker No. 6 where the fire had started, finally supplying ample water for its extinction (and serving to remind us, once again, to be careful what we wish for). In the course of daily living, one is admonished about the hazards of storing oily rags in and around the house, due to the risk of spontaneous combustion. Because most of us have produced numerous oily rags without them bursting into flames, there is an inevitable tendency for us to become desensitized to information that may start to sound like an urban myth. What is usually lost in translation is that the bad actor is most often cotton rags soaked with linseed oil, a combination that demonstrates impressive thermal runaway even in small quantities. Many a home has no doubt been lost to this spontaneously combusting system. Wet haystacks, unprocessed cotton, and manure piles are examples of systems that can spontaneously ignite due to biological exothermic reactions. In an example close to home for this group of Los Alamos authors, the former Santa Fe Downs horse race track had disposed of its manure by heaping it onsite. The local Fire Chief noted that “the manure is mixed with other waste and has repeatedly caught fire over the last 25 years” [26]. (A turnabout
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headline that might have occurred in this instance: “Los Alamos National Laboratory air quality contaminated by smoldering Santa Fe toxic waste dump.”) Little did most of us know that pistachio nuts are quite literally a hot commodity. According to guidance posted by the German Transport Information Service [2], “Self-heating of pistachio nuts is an extremely vigorous process, as the consumption of fatty acids by respiration processes is associated with a considerably greater evolution of heat than is the case with the respiration equation for carbohydrates... The self-heating of pistachio nuts requires only a small seat of moisture, so that within just a few hours heating may occur at moist points for which weeks or months would be required in goods dry on shipment.” The fact that the nuts are spoiled by these same reactions long before they ignite helps to ensure that a large margin of safety is maintained. Each of these common examples is fundamentally the same problem as the thermal runaway phase of a high explosive, and much of the analysis presented in this chapter is as valid for manure as it is for ammonium nitrate. In the post-ignition phase, however, there is really no comparison. As unpleasant as a smoldering manure pile may be, it is in a different class altogether than the Texas City disaster mentioned in Chap. 1, which is generally regarded as the worst industrial accident in American history.1 4.1.2 Thermal versus Physical Explosions A thermal explosion occurs when the temperature of a reacting energetic material spontaneously runs away. A thermal explosion becomes a physical explosion if the temperature runaway also causes the pressure to run away. In response to a large and rapid pressurization, a physically exploding system rapidly expands (often, after failing its containment system) such that a substantial fraction of the internal energy contained in the combustion products is transferred to kinetic energy. The expanding fireball of a physically exploding system displaces the surrounding air, creating a blast shock wave and a following blast wind. (Note that it is not necessary for an energetic material to actually detonate to produce a substantial far-field blast effect.) Most non-shock-initiated explosives require substantial confinement to produce a violent reaction, in which case container fragments are also thrown by the explosion. Air blast and fragments are both important energy transfer mechanisms, which substantially differ in the type of damage they inflict upon targets in their path. Although all non-shock-initiated physical explosions begin as thermal explosions, not all thermal explosions produce physical explosions. A thermal explosion produces a physical explosion only for pressure-building systems, which are those for which the reaction products would seek to occupy a significantly larger volume than the reactants. 1
The Texas City event also resulted in the first ever class action lawsuit against the United States government, on the behalf of 8,485 victims.
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For condensed-phase reactants, a physical explosion can occur only if the combustion products are substantially gaseous, in which case the fully expanded (to atmospheric pressure) reaction product volume is three orders of magnitude greater than the reactant volume. In fact, volume production (which is enhanced by an increase in the number of product moles relative to the number of reactant moles) is the primary attribute that distinguishes explosives from other energetic materials. Indeed, the reaction of some energetic materials produces little or no gas, but rather hot liquid products. For example, heat liberation is an undesired side effect of many two-component epoxies. The thermite reaction, in which a mixture of aluminum and iron oxide powders react to produce molten aluminum oxide and iron, is an example of a system for which highly exothermic condensed-phase reactions are exploited. Among other things, thermite welding (invented in 1895) is widely used to join segments of railroad track. 4.1.3 The Three Fundamental Cookoff Safety Questions A key simplifying observation to be made in connection with the HE cookoff problem is that reaction can be cleanly separated into two distinct regimes. The preignition regime involves the “bootstrapping” buildup of reaction rates. In situations where it occurs, reaction runaway begins very slowly at first and accelerates as the temperature builds. For explosives in particular, the reaction rates are so nonlinear with temperature that there comes a point when the reaction rate appears to suddenly “switch” from a very low level to a very high level. We refer to this apparent switching event as ignition. At the high post-ignition reaction rate level, the reactants are rapidly consumed and/or dispersed. The preignition problem is associated with two closely related practical safety questions. The first is: “Should I flee the scene?” Sometimes the answer to this question seems obvious, but in general it is not. All energetic materials decompose (thereby releasing thermal energy) at a finite rate at room temperature, and yet we still directly handle them. The reason why explosives do not cook off at room temperature is that the heat generation rate is so small that it is lost to the environment without generating a measurable temperature rise. Obviously, materials for which this is not so would not be used for explosives, as they would be very dangerous to handle and would have no shelf life. Under some conditions, reaction-generated thermal energy builds up faster than it can be transferred away, causing the internal temperature to run away to thermal explosion. Whether or not thermal runaway occurs depends on several factors: (1) the reaction pathways of the energetic material in question, (2) the charge size and shape, (3) the temperature of the surroundings, (4) the extent to which the charge is thermally insulated from the surroundings, and (5) the mechanism(s) of heat transfer involved (conduction, convection, and/or radiation). A substantial fraction of the publications in classical thermal explosion theory are devoted to the determination of the critical state
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below which the reaction is stable, and above which the reaction is unstable. That is also the problem we shall consider in this chapter. Assuming that the five influencing factors conspire to suggest that we flee the scene (meaning, more precisely, that we suspect an unacceptable risk of the system becoming supercritical in response to the thermal stimulus), the second preignition safety question is: “How much time do I have to run?” This subject is more technically known as the time-to-ignition problem. A substantial fraction of the publications in classical thermal explosion theory are also devoted to this issue. Unfortunately, space does not allow us to consider timeto-ignition in this chapter. The post-ignition problem addresses the final question, “How far should I run?” To answer this question we need to know both how violent the reaction will be, and how it will be violent (i.e., the partition between blast and fragment damage). Classical thermal explosion theory does not deal with this question because it is not primarily thermal in nature. Rather, post-ignition response is a very complex and multifaceted problem. 4.1.4 The Nature of Preignition and Post-Ignition Cookoff Modeling The preignition regime is relatively tractable because the system is physically static. As such, this problem is primarily one of chemistry and heat transfer. While both of these phenomena can be complicated in detail, they are both well-studied areas that we have experience in dealing with. Another simplifying feature is that explosives tend to be so energetic that in many situations in which a system is heated to criticality, it reaches that condition with very little reactant consumption. In those situations the energetic material can be reasonably considered pristine until ignition. Classical thermal explosion theory usually makes this assumption. On the other hand, the fact that HE products are gaseous means that a small amount of product can distend and otherwise alter the material microstructure. Moreover, many HE’s also experience one or more phase transitions while still in the preignition regime. For example, TNT melts and HMX experiences a solid-to-solid phase transition. Thus, high fidelity preignition calculations should consider the thermal damage state of the explosive (e.g., [34, 58]). There is another important reason for studying preignition thermal damage, which is that the material condition at ignition plays a large role in determining post-ignition violence. For explosives that do not melt prior to ignition, the permeability plays a particularly important role in post-ignition behavior. These effects are considered in detail in Chaps. 5–7. Even though preignition effects can be complicated in detail, making the leap from preignition to post-ignition modeling is much like stepping out of a wading pool and into the ocean. Indeed, the post-ignition problem is sufficiently difficult that there has been some tendency to avoid it. The prevailing research culture with respect to preignition and post-ignition modeling seems
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a bit like the infamous drunk who searches for his lost keys beneath the lamppost because “that’s where the light is.” Although this state of affairs is natural and to an extent inevitable, it becomes dysfunctional when – as sometimes occurs – it is associated with the claim or inference that one has “solved” the cookoff problem by reasonably computing the time to ignition. The unintended consequence of inflated claims of progress is that they tend to retard actual progress. As such, one of the purposes of this book is to unabashedly reveal the many messy and complex details of the post-ignition problem. The ignition event separates the preignition from the post-ignition regime. Ignition occurs at a very distinct time, almost inevitably at a single point within the charge or somewhere along its edges. In numerical computations, the ignition time can be usefully defined as the time at which the heat equation becomes singular, i.e., the time at which a variable computational time step essentially goes to zero. Although this criterion may seem rather arbitrary, the temperature behavior is so nonlinear that the precise definition of what constitutes “singular” is unimportant. The preignition state evaluated at ignition provides the initial condition for the post-ignition problem. The details of the post-ignition problem lie beyond the scope of this chapter; however, many are discussed elsewhere in this book. Here we offer only a brief description of the ways in which the problem is complex. The fact that HE products are gaseous leads to the buildup of pressure, which induces large stresses in a confined system. This in turn leads to substantial material motion and case rupture. It also leads to some manner of complex and self-organizing multiphase behavior, which (at least for HMX systems) often takes the form of burning crack networks (see, e.g., [13] and Chaps. 5–7). Our understanding of the phenomenon by which HMX explosives cook to violent reaction is that burning cracks act as “superhighways” by which reaction is distributed coarsely throughout the system. Provided that the confinement does not fail such that the pressure continues to rise, burning is able to enter even smaller spaces. To a point, such spaces may be generated “on the fly,” by what sometimes appears to be a fractal-like distribution of crack scales. Ultimately, a large surface area resides in the intergranular porous spaces. If burning is able to access that scale, a substantial physical explosion will surely ensue [29]. 4.1.5 Historical Development of Thermal Explosion Theory The mathematical theory of thermal explosions began in Russia in the later 1920’s with the work of pioneers Nikolay N. Semenov2 [51] and O.M. Todes [55]. Semenov developed a criticality model that considered a simple balance between heat generation and loss. His fundamental simplifying assumption was that the HE temperature remains spatially uniform. We present Semenov’s theory in Sect. 4.3. 2
Semenov was awarded the Nobel prize in chemistry in 1956 for his work on the mechanism of chemical transformation. His thermal explosion theory discussed here was only one accomplishment of many.
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About a decade later David A. Frank-Kamenetskii (F-K), who was one of Semenov’s students, performed the same basic criticality problem but allowed the temperature to vary throughout the body [36]. F-K’s problem is mathematically harder than Semenov’s because it requires solution of the heat equation with the highly nonlinear Arrhenius heat generation term. In fact, analytic solutions can only be obtained only if the Arrhenius term is approximated by a simpler function. F-K’s biggest contribution to thermal explosion theory is the introduction and exploitation of this approximation. Prior to the advent of digital computers, workers focused mainly on analytic solutions to the F-K problem for the three canonical shapes (infinite slab, infinite circular cylinder, and sphere). The reason for the emphasis on F-K theory is that Semenov theory is easily analytically solvable, whereas the heat equation with the full Arrhenius rate is analytically insoluble. Thus, the F-K level is intruiging as that which, though often difficult, can sometimes be solved. Later, workers used early analog and digital computers to study criticality problems for non-canonical geometries, at both the F-K and full Arrhenius level. As computing power increased, interest gravitated to more complex systems with complicated shapes, multiple materials, and nonuniform heat loads. More complex chemistry models were also developed. All these advanced elements require a full numerical solution. Much of the current research remains focused on obtaining more accurate chemistry schemes. Such efforts are often very successful; for example, modern HMX models (as discussed in Chap. 3) can predict the temperature runaway with good accuracy up to the time of ignition (e.g., [40]). 4.1.6 Definition, Scope, and Modern Relevance of Classical Theory Although the title of this chapter is classical thermal criticality theory, there is really no universally accepted definition of what is meant by “classical” thermal explosion theory. The dictionary definitions for the word classical: “exemplary standard,” and “traditional and long-established in form or style” aptly describe the work of pioneers Semenov, Frank-Kamenetskii, and a handful of other key contributors. The difficulty with these definitions is that they are ambiguous with respect to the classification of new variations based on the traditional approach. We need a definition that is less rooted in history and more closely tied to technical content. For the purposes of this chapter, we define classical theory as any analysis that treats a non-shock initiation problem by the heat equation with Arrhenius volumetric heat addition, with at most simple mass-fraction-based depletion. It is clear from Sect. 4.1.4 that classical theory mainly applies to the preignition problem. Although it may be formally applied to selected aspects of the post-ignition problem, its applicability in that regime is quite limited. In fact, one might go so far as to say that the application of classical theory to the postignition problem is mainly of mathematical, rather than physical, interest.
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Modern computational capability has advanced to the point where many problems can be computed with substantial real-world detail. This capability may be accompanied by the sentiment that analytic theory is somewhat antiquated – a thing to be used primarily for code validation. Although classical theory is certainly useful for that purpose, it also remains useful in other respects. Firstly, it is important to appreciate that an advanced capability to perform numerical “experiments” on a case-by-case basis does not automatically foster a deep understanding of the overall behavior, any more than an ability to perform real experiments does. Either class of experiment only shows us how the system behaves in specific instances amongst a generally infinite number of possibilities. In either case we seek some greater organizing framework to provide qualitative understanding and intuition. Those analytic theories that are, as Einstein put it, “as simple as possible but no simpler,” provide useful insight into physical competitions, parameter dependence, and limiting behavior. The Semenov and F-K models prove especially useful for heuristic purposes: Semenov for its simplicity and universality, F-K for its ability to show the consequences of adding what is generally the next step in realism. Classical models have a greater overall ability to “hone our intuition” than do complex computer models. Each of us has an intuitive feeling for how heat transfer proceeds in everyday life; however, the highly nonlinear behavior of the Arrhenius reaction-rate equation leads to behavior that can defy that intuition. Classical models retain the basic character of the thermal explosion problem without getting bogged down in details that obfuscate basic dependencies. The study of classical models helps us to re-calibrate our intuition to comply with explosively reactive, rather than inert thermal, behavior. Classical theory also allows one to simply estimate critical states and times to ignition. The fact that sophisticated models running on powerful computers can compute the preignition problem with much greater accuracy than classical theory does not mean that everyone is set up to, or wishes to, run them. In fact, it is rare for an experimentalist to take the time to set up a detailed calculation to compute a critical temperature or a time to ignition. A simple estimate for the purpose of determining the experimental conditions, followed by measurements of real world behavior, is often what is called for. 4.1.7 Emphasis and Outline In this chapter we present the key classical aspects of criticality theory in significant detail, stressing the mathematical and physical aspects in approximately equal measure. For the purposes of presentation simplicity, ease of manipulation and solution, and physical understanding, dimensionless representations are favored. This chapter is not a review article per se, and the review of previous work is by no means exhaustive. Nevertheless, a list of general references and resources is given in Sect. 4.1.8.
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Due to monumental advances in computing power since the time that the seminal exercises were published, we are able in many cases to recreate the pioneering work with greater detail, accuracy, and graphical flair than the original studies.3 Yet one cannot help but be impressed by what the original authors accomplished with the tools they had – hand numerics and analog computers, for example. In Sect. 4.2 we discuss reaction rates as they pertain to classical thermal explosion theory. The goal here is not to duplicate the detailed account of HE chemistry and kinetics presented in Chap. 3; rather, it is to simplify the typically complex energetic material chemistry into a simple form that is digestible by classical theory. In Sect. 4.3 we present Semenov’s original theory of criticality. Through the introduction of the Lambert W special function we develop the theory beyond its normal treatment. This extension allows analytic treatment of aspects of Semenov theory that would otherwise require numerical computation. We also examine and map the structure of the Semenov solution space. The attributes of this space are interesting; moreover, the limited understanding of it has led to occasional confusion and/or obfuscation in the literature. Semenov’s original theory neglects the effects of reactant depletion. In Sect. 4.4 we extend Semenov theory to include these effects, finding an explosion limit of a kind that does not arise in the original theory. This depletion-induced explosion limit represents a significant complication because it depends on the details of the heating history. In Sect. 4.5 we present the reactive heat equation with Arrhenius heat addition. We then derive and discuss F-K’s high activation energy approximation to the Arrhenius term, which leads to a substantial simplification in terms of the scaled dimensionless temperature θ. In Sect. 4.6 we apply F-K’s high activation energy approximation to the Semenov criticality theory presented in Sect. 4.3. Although this approximation is not necessary for the analytic treatment of Semenov’s theory, using it does achieve some convenient simplification, and also allows a more direct comparison to F-K’s theory. In Sect. 4.7 we present the basic F-K criticality problem. We first derive the criticality solutions for the three canonical shapes: the infinite slab, the infinite right circular cylinder, and the sphere. These three symmetric bodies require the solution of an ordinary differential equation. The cylinder can be solved fully analytically, whereas the slab and the sphere must resort to rudimentary numerics. All other shapes require the solution of a partial differential equation. We discuss methods for determining the critical state for general shapes, and present the square bar as a simple example. It would be useful to possess an empirical correlation by which one could determine F-K criticality for arbitrary-shaped bodies without performing numerical computations. Using a combination of integral analysis, dimensional analysis, and logical guesswork, we deduce such a correlation in Sect. 4.8. 3
The calculations in this chapter were performed using Mathematica 7.
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This scaling law is quite accurate for bodies that are not significantly elongated (such as a square in 2D or a cube in 3D). However, the correlation breaks down for bodies that are even modestly elongated. We discuss progress toward generalizing the near-unity aspect ratio correlation to modestly elongated bodies. Highly elongated bodies behave as if they are infinite in the elongated dimension, such that many will obey the near-unity aspect ratio correlation in a reduced dimension. About two decades after F-K developed his theory, P. Thomas unified the Semenov and F-K theories by showing that both variants are special cases of a more general prescription characterized by the Biot number, Bi [54]. Thomas’ theory has quasi-analytic parametric solutions for the canonical shapes. Following Thomas, we show in Sect. 4.9 how Semenov theory (Bi → 0) and F-K theory (Bi → ∞) represent extreme cases of the more general theory. As we show in Sect. 4.5, the fidelity of the F-K approximation depends on how large the critical temperature is compared to the activation temperature, which in turn depends on the charge material and configuration. Following previous authors [16, 48], in Sect. 4.10 we examine the accuracy of the F-K approximation, as a function of this temperature ratio, for the three canonical shapes. We confirm that the F-K approximation becomes exact in the limit of small temperature ratio, and that its error in real situations is minimal. 4.1.8 Additional Resources For a more complete technical history of classical thermal explosion theory, the reader may consult review articles by Merzhanov and Abramov [44] and Gray and Lee [25]. Although these articles are two and four decades old, respectively, at the time of this writing, the fact that only modest attention has been paid to classical theory since the late 1960s means that they are not as dated as one might think. In addition to these review articles, several monographs address thermal explosions, as well as supplementary topics such as high activation energy asymptotics [5, 7, 9, 20, 21, 38, 52, 57, 59]. Other interesting and relevant papers, many of which discuss experiments and more detailed nonclassical models, are sprinkled throughout relevant journals and the proceedings of several conference series, for example: the International Symposium on Combustion (mostly gas phase), the International Detonation Symposium (mostly condensed phase), the APS Topical Conference on Shock Waves in Condensed Matter, the JANNAF Propulsion Systems Hazards Subcommittee Meeting, and the Fraunhofer Institut Chemische Technologie (ICT) Meeting.
4.2 Reaction Rate It is clear from Chap. 3 that the decomposition chemistry of condensedphase explosive molecules typically involves many – perhaps tens or even hundreds – of reactions and intermediate species. Yet for most practical
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purposes, individual reactions are a complication, to be considered only to the extent that we must do so to answer the three basic safety questions concerning a possible physical explosion. Therefore, the best model is usually the simplest one that can adequately answer these questions. Classical theory distills reaction chemistry to a bare minimum, and yet as we saw in Chap. 3 for the case of HMX, the classical Arrhenius rate law is able to fit a wide range of rates remarkably well – at least for that one quintessential explosive. For HMX and any other explosive that exhibits similarly simple rate behavior, classical theory will be applicable so long as the real boundary conditions closely conform to the modeled ones. In practice, boundary conditions are one of the main complicating factors associated with real world problems. As we’ve defined classical theory in Sect. 4.1, the reaction rate law comprises a temperature-dependent Arrhenius term, which is multiplied by a reaction progress function that depends only on the reaction product mass fraction. In general, the reaction progress function expresses a variety of effects, including the relative rates of the various component reactions, as well as any spatially nonuniform reaction structure. Minimally, it attenuates the rate as the reactant becomes depleted. Preignition models usually assume that the explosive is negligibly depleted, in which case the reaction progress function is a constant. We begin the section with examples of how chemical and physical reaction behavior affects the form of the reaction progress function. We then discuss the structure and calibration of the Arrhenius rate function, and explore how one of its parameters (the activation energy) is related to chemical sensitivity. Finally, we list thermal explosion parameters for a sample explosive (Los Alamos’ HMX-based explosive PBX 9501) for use in subsequent examples. 4.2.1 Homogeneous and Heterogeneous Chemical Reactions Chemical reactions can be divided into the two general categories: homogeneous and heterogeneous. Homogeneous reactions proceed spatially uniformly throughout the volume; whereas, heterogeneous reactions occur at interfaces [21]. Not surprisingly, homogeneous reaction theory is simpler and more tractable than heterogeneous reaction theory. For high explosives, cookoff reactions tend to be heterogeneous in the preignition as well as the post-ignition regime. This is true even if, as is often the case, the considered explosive is monomolecular. The reason is that the reactants are solid, while the products are gaseous. Because we define a surface as the boundary between phases, decomposition by definition occurs at surfaces. For slow thermal decomposition, the temperature field is virtually unaffected by reaction heterogeneity. This is because heat generated in localized regions diffuses away much faster than it builds up.4 However, even when the 4
The relative importance of localized heat buildup versus diffusion is characterized by Damk¨ ohler’s second number (see Sect. 4.5).
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temperature is uniform, material strength (or its liquid manifestation, surface tension) causes reactant evaporation to occur at pre-established surfaces, such as intergranular contacts or intragranular flaws. The reason is that, far below the thermodynamic critical point, the creation of new surfaces requires a substantial amount of mechanical work. Thus, even in the slow decomposition limit, reaction is heterogeneous on the scale of flaws and grains. In the post-ignition regime where reaction rates are much higher and are rapidly accelerating, the prevailing physical processes also promote heterogeneous reaction. The reason is that for explosives burning at low pressures, reaction (and hence heat release) occurs in the gas phase. Thus, heat released in gaseous spaces heats adjacent solid surfaces, thereby increasing the reactant evaporation rate. Evaporated reactants burn, produce heat, and evaporate more reactants. Thus, the post-ignition regime develops a sort of small-scale flame structure, wherein the temperature is no longer uniform on the microstructural scale. The difference between the preignition thermal decomposition and the post-ignition flame propagation regimes is important, and the physical distinction just described – including the fact that reaction is heterogeneous in both regimes for different reasons – is often not appreciated. Fortunately, for modeling purposes, ignition is a well-defined event that temporally separates the two behaviors. Sometimes conditions are such that the explosive experiences deflagrationto-detonation transition (DDT). In the detonation regime a third reaction mode occurs, which, depending on the physical structure of the explosive, can be of various degrees of homogeneous or heterogeneous. DDT can occur rather suddenly (see Chap. 8), in which case the detonation reaction regime is temporally separated from the flame propagation regime. 4.2.2 Spatial Averaging Over Heterogeneous Solid-Phase Reactants In classical preignition theory one tracks the bulk temperature and the bulk composition state. We have argued that in the thermal decomposition regime there is negligible temperature variation on the material microscale, despite the fact that reaction is heterogeneous. Consequently, the temperature can be considered to smoothly vary throughout the charge as dictated by continuum theory. This argument does not hold for the composition state which, on the microscale, exists as a heterogeneous mixture of reactants and products. Consequently, treatment of the composition state as a continuum requires additional justification. Specifically, one must perform the same sort of averaging process that is commonly done in connection with two-phase fluid flow. One considers a representative material element containing a mixture of reactants and products. One then considers various element sizes, examining the variation in the product mass fraction as a function of size. For small element sizes this
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property will fluctuate wildly as the size is varied; however, as the element size increases – such that the element is averaged over more and more of the microstructure – these fluctuations will die out. The size at which this occurs is the scale above which the material can be treated as a continuum. A continuum model is justified if the continuum scale is small compared to the domain size. Because it is only necessary to estimate the continuum scale to within an order of magnitude, it is not necessary to actually perform the formal averaging process just described. If, as is virtually always the case, the grain scale is small compared to the charge scale, then the continuum approximation will be satisfied.5 In contrast, observations (e.g., [13]) indicate that post-ignition behavior tends to be heterogeneous on the domain scale, such that continuum modeling is (formally, at least) unjustified. This fundamental observation – that postignition behavior involves a substantial amount of two-phase macrostructure – has caused a fundamental shift in the way one must view the post-ignition problem. 4.2.3 The Law of Mass Action Homogeneous chemical reaction rates are often approximated by the law of mass action: n Ciωi ≡ K Π, (4.1) R=K i=1
where R is the total reaction rate, K is the specific reaction rate, and Ci is the concentration of the ith component of n total components. It is customary to express concentration in units of moles/volume, not considering that this practice wreaks havoc with the physical units. The difficulty is resolved by expressing the Ci factors as mole fractions, thereby rendering each term (as well as the overall product of terms) dimensionless. Then R and K both have simple units of inverse time. But if R is to have units of inverse time, we must express it as the rate of increase of product mole fraction – as opposed to the usual definition, which is the rate of increase of product moles per unit volume [57]. In cases where the Ci -values depend only on the reactants, the reaction is called simple; whereas, if the Ci -values depend on one or more reaction intermediates and/or products, the reaction is called complex [21]. The exponent ωi 5
The situation for explosives that melt before reacting (e.g., TNT) is rather different than for explosives (e.g., HMX) that do not. For molten TNT, reaction products reside in bubbles. Container venting causes nucleate boiling and reactant bubbles. The preignition TNT physical state is thus self-organizing and highly variable. Post-ignition TNT response evidently involves the fragmentation of liquid structures in a manner resembling a boiling liquid expanding vapor explosion (BLEVE).
4 Classical Theory of Thermal Criticality
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sets the relative influence of each contributing constituent, and is called the reaction order with respect to constituent i. Here, we denote the product of the concentration factors by the variable Π. The law of mass action is phenomenological in nature. Roughly, the underlying physical reasoning is that the reaction rate is proportional to the collision rate between appropriate molecules, and the collision rate is proportional to the product of the concentrations. As such, the concentrations in (4.1) must correspond to molecules that actually react on a molecular level [57]. If we consider HMX reaction, for example, then the HMX molecule as a whole does not comprise a Ci -value. Rather the HMX molecule first decomposes into smaller pieces, and those pieces comprise the Ci -values. 4.2.4 Arrhenius Reaction Rate The rate constant K depends mainly on the temperature for virtually all reacting systems [57]. Most often, K is assumed to follow the Arrhenius law: K[T ] = Z e−E/(R T ) = Z e−T
∗
/T
,
(4.2)
where the parameter Z is the pre-exponential factor or frequency factor, E is the activation energy, and R is the universal gas constant. The ratio E/R has dimensions of temperature and is called the activation temperature, T ∗ . Using T ∗ in favor of E/R simplifies the presentation slightly. Note that T and T ∗ (and indeed all other temperatures used in this chapter) are on the absolute scale. Equation (4.2) is derived from the kinetic theory of gases (e.g., [43, 46, 56]), in which E is the minimum molar kinetic energy required for a collision to trigger reaction, and exp[−E/(R T )] is the fraction of the total collisions that result in reaction. This factor multiplied by the collision rate Z gives the reaction rate. Equation (4.2) is actually a slight simplification of the theoret√ ical result, in √ that, like the sound speed, Z is actually proportional to T . Dropping the T factor has minimal mathematical effect because the strong variation of the exponential term dominates it. √ There are two additional practical reasons why it is sensible to drop the T factor for condensed-phase energetic material systems. The first is that the behavior of real chemical systems – especially complex systems, as are condensed-phase explosives – deviates from the ideal theory, such that the √ T term does not necessarily add realism. (In practice, any additional temperature dependence is most often introduced via the kinetic compensation effect [8].) The second is that the exponential term alone (and in particular F-K’s high activation energy √ simplification of it) allows a degree of analytic tractability that is lost if the T term is retained. In any case, (4.2) works well enough that it constitutes the fundamental building block for classical chemical kinetic models. However, in simplifying the theoretical result and using it in a broader context than it was derived, the
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parameters Z and E lose their strict quantitative interpretation. As generally used, they are empirical parameters to be fit to rate data. Nevertheless their approximate physical interpretation endures, and it is sometimes useful to bear in mind the processes that they represent. 4.2.5 Reaction Order Let us first assume that the reactions are simple. We define the concentration of the ith reactant as Cri = Nri /Nr0 , where Nri is the number of reactant moles associated with the ith reaction at any stage of the reaction, and Nr0 is the overall number of reactant moles prior to the start of the reaction. The subscript r reminds us that the concentrations apply to reactants only. The total number of reactant moles remaining at any point in the reaction is m Nri . (4.3) Nr = i=1
Dividing both sides of (4.3) by Nr0 , the total concentration of reactants at any point in the reaction is given by m
Cr =
Cri .
(4.4)
i=1
Prior to the start of reaction, the total reactant concentration Cr0 is unity. It will often be the case that the various reactants are consumed in the same relative proportions throughout the course of the reaction. If this is the case then Nri Nri0 = , (4.5) Nr Nr0 or, equivalently,
Cri = Cri0 , Cr
(4.6)
where Cri0 is the concentration of the ith constituent prior to the start of reaction. Thus Π can be written as Π=
m
m
(Cri0 Cr )ωi =
i=1
ωi Cri0
i=1
where ω=
m
Crωi = A Crω ,
(4.7)
i=1
m
ωi
(4.8)
ωi Cri0 .
(4.9)
i=1
and A=
m i=1
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The parameter ω is called the overall reaction order. Note that the concept of overall order arises as the result of an additional assumption (i.e., 4.5), and as such does not exist in every case for which (4.1) holds. The coefficient A depends on the specific reactions present, but remains constant over the course of the reaction. 4.2.6 Reaction Rate Expressed as Mass Fraction Often in classical thermal explosion theory, and in most detonation reactive burn models, depletion is expressed by mass fraction rather than concentration. Mass has the advantage of being a conserved quantity during reaction; whereas, the number of reactant moles generally differs from the number of product moles. To make the conversion, we begin by writing Cr =
Nr mr /Mr = , Nr0 mr0 /Mr0
(4.10)
where m is mass and M is molecular weight. In Sect. 4.2.3 we noted that the concentrations appearing in the law of mass action (4.1) do not pertain to the HMX molecule as a whole, but rather to the species into which the HMX molecule decomposes, which participate in chemical reactions. Thus, the molecular weights in (4.10) pertain to a mixture of multiple species. In the general case where the species concentrations can vary in an arbitrary way, the corresponding mixture molecular weight is subject to change, such that Mr = Mr0 . However, we have assumed in (4.5) that the relative reactant concentrations are conserved throughout the reaction, such that the mixture on which Mr is based maintains its proportions. Therefore Mr = Mr0 , and Cr =
Nr mr /Mr0 mr = = = χr = 1 − χp ≡ 1 − χ. Nr0 mr0 /Mr0 mr0
(4.11)
For brevity the subscript p, which denotes “product,” is dropped in the final equality. Note that the above analysis has not actually made any assumptions about the number of reactant moles versus the number of product moles. That relationship remains unspecified. Next we must deal with the left hand side of (4.1), and relate the total reaction rate R to the rate of increase of the product mass fraction. Recall from Sect. 4.2.3 that we defined R as the rate of increase of the total product concentration, Cp = Np /Np1 , where Np is the total number of product moles at any phase of the reaction, and Np1 is the final number of product moles when the reaction is complete (χ = 1). Thus, C ≡ Cp =
Np mp Mp1 mp Mp1 mp = × = × = = χp ≡ χ. Np1 Mp mp1 Mp1 mp1 mp1
(4.12)
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Equation (4.12) makes the same assumption as (4.5), but with respect to the products rather than the reactants. That is, we assume that the products accumulate in the same relative proportions throughout the course of the reaction. If this is so, then the same argument that led us to conclude that ˙ where “dot” denotes the Mr = Mr0 tells us that Mp = Mp1 . Finally, C˙ = χ, derivative with respect to time. Assembling the elements derived in Sects. 4.2.4 and 4.2.5 gives R = A Crω Z e−T
∗
/T
.
(4.13)
So far as reaction decomposition measurements are concerned, A and Z are lumped together. Thus, we may assume without loss that A Z → Z. Substituting (4.11) and the time derivative of (4.12) into (4.13) gives the final desired result: ∗ (4.14) χ˙ = (1 − χ)ω Z e−T /T . 4.2.7 Reaction Progress Function Behavior for Arbitrary Reaction Order To characterize the response of the reaction progress function for a range of reaction orders we may integrate (4.14) at constant temperature – as would be appropriate for a body that is reacting slowly, in thermal equilibrium with its environment. To do so we first define a dimensionless time tˆ = Zt e−T
∗
/T
,
(4.15)
which causes (4.14) to reduce to the integral tˆ = 0
χ
dξ . (1 − ξ)ω
(4.16)
For ω > 0 and ω = 1, (4.16) has the implicit solution χ − 1 + (1 − χ)ω tˆ = , ω > 0 and ω = 1. (ω − 1)(1 − χ)ω
(4.17)
For ω = 1, for which (4.17) is singular, (4.16) has the solution: χ = 1 − e −t . ˆ
(4.18)
For ω = 1/2, often called “square root depletion,” the solution to (4.16) can be written: tˆ ˆ χ=t 1− . (4.19) 4 Figure 4.1 plots examples of the product mass fraction versus time for a variety of reaction orders. For 0 ≤ ω < 1, reaction completes in a finite time given by
4 Classical Theory of Thermal Criticality ω = 0 0.25
1.0
0.75
0.50
149
ω=1 1.25
χ
0.8 0.6 0.4 0.2 0.0 0
1
2
3
4
5
t
^
Fig. 4.1. Reaction progress for a constant temperature reaction versus dimensionless time, for various reaction orders
1 , (4.20) 1−ω where the subscript d denotes “depletion”. For ω ≥ 1, the reaction approaches completion asymptotically. tˆd =
4.2.8 Zero-Order Reaction versus Negligible Depletion In Sect. 4.1.4 we noted that one consequence of the high HE energy content is that charges often attain criticality and ignition without suffering a significant amount of reactant depletion. For this reason, it is customary in both criticality and time-to-explosion calculations to set χ = 0. It is often stated that in doing so, one is making a zero-order reaction assumption. However, such an assumption would not be physically reasonable because HE reactions are not zero order. Figure 4.1 clarifies the situation, illustrating that all orders (including zero) behave in the same manner in the limit of small depletion. This attribute may be formally shown by expanding (4.17) in a Taylor series, which yields ω tˆ = χ + χ2 + .... 2
(4.21)
To leading order, tˆ is proportional to χ (independent of ω), and dχ/dtˆ = 1. Transforming this result back to dimensional time via (4.15) gives χ˙ = Z e−T
∗
/T
.
(4.22)
This result, which was obtained for any reaction order and small but finite depletion, is equivalent to the case of precisely zero depletion (obtained by setting χ = 0 in (4.14)) and also to the case of zero-order reaction. The
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main point to be made is that, although the zero-order reaction assumption is mathematically equivalent to the negligible-depletion assumption, the former is physically incorrect while the latter is physically correct. 4.2.9 Autocatalytic Reactions An autocatalytic reaction is one in which the reaction rate increases with an increase in the concentration of one or more reaction products. We can derive this type of behavior by an extension of the methods used in the previous section. Again we start with the law of mass action (4.1), but this time we consider a complex reaction that depends on both reactant and product concentrations (but not intermediate species). The concentration factors can be split into separate products of reactant and product terms as follows: m n Π= Criωi Cpj j . (4.23) i=1
j=1
Having chosen on dimensional grounds to represent concentrations by mole fractions, we must choose molar reference numbers by which to construct the individual species concentrations Cri and Cpj , and also the total concentrations Cr and Cp . Because the number of moles is not generally conserved in the reaction, and because we wish the total concentrations Cr and Cp to vary between zero and one, the only sensible choice is to normalize the reactant and product terms by their own ultimate total values. That is, we choose Cri = Nri /Nr0 and Cr = Nr /Nr0 as before, and Cpj = Npj /Np1 and Cp = Np /Np1 , where Np1 is the total number of product moles associated with complete reaction. Just as in Sect. 4.2.5, we assume that the reactants maintain consistent proportions as they are consumed. Thus, (4.5) applies to the first product in (4.23). Just as in Sect. 4.2.6, we also assume that the analogous situation applies for the products, i.e., that they maintain consistent proportions as they accumulate. This situation is expressed by Npj1 Npj = Np Np1
(4.24)
or, equivalently,
Cpj = Cpj1 , (4.25) Cp where Cpj1 is the concentration of the jth product at the end of reaction. Proceeding as in (4.7), we arrive at the result
(4.26) Π = (A Crω ) B Cp , where A is the same as defined in (4.9), and n B= Cpj1 j . j=1
(4.27)
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151
As in Sect. 4.2.6, the reactant and product mole fractions are equal to the respective mass fractions: Cr = χr , and Cp = χp . Equation (4.26) then becomes Π = A B (1 − χ)ω χ ,
(4.28)
where χ = χp as before. Also absorbing A and B into Z as before, the final form for the autocatalytic reaction rate is χ˙ = (1 − χ)ω χ Z e−T
∗
/T
.
(4.29)
Equation (4.29) is (4.14) multiplied by the term χ . Thus, for any given temperature and reactant concentration: the greater the product concentration the greater the reaction rate. This is the autocatalytic effect. At face value it would also appear that a finite amount of product is necessary to start the reaction – the reason being that if χ = 0, (4.29) dictates that χ˙ = 0. Even though this is so, reaction is indeed self-starting for some values. We can demonstrate this property by examining the reaction progress function as we did in Sect. 4.2.7. We begin by replacing t by tˆ as before, such that dχ = χ (1 − χ)ω . (4.30) dtˆ A variety of behaviors are possible depending on the chosen values of ω and . Figure 4.2 plots selected cases for ω = 1, 0 ≤ < 1. The maximum rate in (4.30) occurs at a value of χ given by χm =
0.7
, ω+
(4.31)
ω=1
− = 0.1 ω
0.6 0.2
χ
0.5
0.3 0.4
0.4
0.5 0.6
0.3
0.7 0.8
− = 0.9 ω
0.2 0.1 0.0
0.0
0.2
0.4
0.6
0.8
1.0
χ
Fig. 4.2. Sample autocatalytic reaction progress functions for ω = 1; 0 < < 1
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L.G. Hill
which occurs at a value of χ˙ given by ω ω χ˙ m = . ω+ ω+
(4.32)
Let us now explore the behavior of (4.30) for small χ, for which dχ = χ , ≥ 0 and χ 1. dtˆ
(4.33)
For the initial condition χ[0] = 0, the solution to (4.33) is χ=
1/(1−) (1 − ) tˆ 0
if 0 ≤ < 1 and χ 1, if ≥ 1.
(4.34)
Equation (4.34) shows that a finite amount of reactant is not necessary to initiate reaction for the parameter range 0 ≤ < 1. The reason is that the slope of (4.30) is infinite at χ = 0, making the starting condition singular. It turns out that this singularity resolves itself in such a way that reaction selfinitiates even though χ˙ = 0 at tˆ = χ = 0. On the other hand, this diminutive initial condition does lead to a very slow reaction runaway at first. We say that there is an induction delay or time-to-ignition, tˆi . Because reaction runaway is a continuous process, there is no unique event that unambiguously marks ignition. On the other hand, one observes that reaction rates are so nonlinear with respect to time that any reasonable criterion we choose will lead to very similar values for tˆi . For the present purposes let us define the ignition time by setting the convenient threshold χ = 1/e in (4.34a). We then solve for tˆ, giving e−(1−) . (4.35) tˆi = 1− Equation (4.35) is plotted in Fig. 4.3, which shows that tˆi → ∞ as → 1. Clearly this calculation of tˆi is a bit artificial in that the temperature is held fixed, which does not actually happen in a real thermal explosion. On the other hand, constant temperature thermal decomposition is maintained in a differential scanning calorimeter, which can be used to directly measure the reaction progress function [35]. This exercise also serves to illustrate that lengthy induction times are caused not only by the slow “bootstrapping” buildup of the temperature. Even if the temperature is fixed, an induction time can arise from the structure of the reaction progress function alone. The full solution to (4.30) can be expressed in terms of the incomplete beta function, which is defined by the integral z ξ a−1 (1 − ξ)b−1 dξ. (4.36) B[z, a, b] = 0
4 Classical Theory of Thermal Criticality
153
104 1000
t^i
100 10 1 0.1
0.0
0.2
0.4
0.6
0.8
1.0
ϖ
Fig. 4.3. Induction time for a constant-temperature autocatalytic reaction (4.30) versus the reaction exponent
1.0
χ
0.8 0.6 0.1
0.6 0.7
ϖ = 0.9
0.8
0.4 0.2 0.0
0
5
10
15
t^
Fig. 4.4. Reaction progress for a constant-temperature autocatalytic reaction versus dimensionless time tˆ, for the cases shown in Fig. 4.2 (ω = 1; 0 < < 1)
In terms of (4.36), the implicit solution to (4.30) is tˆ = B[χ, 1 − , 1 − ω], 0 ≤ < 1.
(4.37)
The reaction progress as a function of dimensionless time (4.37) is plotted in Fig. 4.4 for the same cases (i.e., ω = 1, 0 ≤ < 1, χ[0] = 0) as are plotted in Fig. 4.2. The profiles have the expected sigmoidal shape and closely resemble each other, the primary difference being that each has a different induction delay given by (4.35). Equation (4.34b) indicates that for ≥ 1 reaction is not self-starting; nevertheless, any finite amount of initial product will initiate it. One may
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examine such cases by starting the integration of (4.30) at χ[0] = χ0 > 0, thereby introducing χ0 as an additional parameter. For the purposes of the present discussion it is not necessary to explore the behavior of the many combinations of ω, , and χ0 . Instead, we merely note that letting χ[0] = χ0 results in an extension of (4.35) given by ⎧ −(1−) 1− −χ0 ⎨e if ≥ 0 and = 1, 1− (4.38) tˆi = ⎩−(1 + ln χ ) if = 1. 0
Examination of (4.38) shows that the induction time decreases as χ0 increases, for all values of . In other words, the initial presence of reaction product accelerates the early reaction rate for this autocatalytic reaction progress function. If χ0 = 0, (4.38) reduces to (4.35) if 0 ≤ < 1, but is infinite if ≥ 1. 4.2.10 Hot-Spot-Initiated Reaction The nature and degree of heterogeneity exhibited by energetic material reactions is a very important topic, and also one that is difficult and relatively poorly understood. For example, the shock initiation process of detonating monomolecular explosives shows degrees of homogeneity/heterogeneity depending on the explosive’s chemical sensitivity, its physical structure, and the input shock pressure [32]. No metric currently exists by which one can quantify, much less predict, where a reaction lies on the continuum between homogeneous and heterogeneous behavior. Although this book deals with non-shock initiation mechanisms, the postshock state of a shocked energetic material can mark the beginning of what can amount to a very fast cookoff problem. Suppose that a porous energetic compact has just been dynamically crushed to virtually solid density. The crushing process creates a spatially nonuniform temperature distribution. If a material is shocked hard enough (which may require an overdriven detonation), then all spatial regions will react promptly despite the temperature variation. What constitutes “hard enough”? When a porous material is crushed, some fraction of the mechanical energy goes into collapsing the heterogeneities. Any excess energy goes into compressing and heating the bulk material. Compression-induced reaction will appear homogeneous if the energy that goes into compressing the bulk (what one might call the homogeneous energy) is sufficiently large compared to the energy that goes into removing the heterogeneities (what one might call the heterogeneous energy). For a somewhat weaker stimulus, the material reacts promptly only at the hottest regions, and the reaction spreads from these regions to consume the surrounding material. In the heterogeneous limit, no reaction would occur on a fast time scale other than that which propagates outward from supercritical hot spots.
4 Classical Theory of Thermal Criticality
155
In a hot-spot-dominated reaction, each local temperature maximum has an associated criticality problem that is analogous to those we shall consider for a heated charge as a whole. The extreme state-sensitivity of the reaction rate to temperature causes hot spot propagation to be an all-or-nothing affair. Subcritical hot spots dissipate as if the material was inert. Supercritical hot spots run away with vigor, and as they do so, the amount of reaction energy released soon swamps the amount of mechanical energy that created the ignition spot. By this observation, we conclude that all hot-spot-initiated reaction waves are equivalent except for a time shift. In other words, the details of the initiating hot spot (its shape, size, and peak temperature) are only important in determining criticality. Once that matter is settled, the process quickly forgets its origins – its initial size, which was small, and its initial energy, which was also small. In practice there may be other complicating factors, but this is what a straightforward application of the reactive heat equation has to say about the matter. Whether supercritical hot spots always grow to consume the entire material volume, or whether they can burn for a while and extinguish, has been a matter of some discussion and disagreement. As is often the case, the answer to this question is likely to be regime-dependent. In a typical cookoff problem, a single ignition spot usually occurs, which develops into a solitary reaction wave. In the cookoff regime, the reaction products are gaseous. Consequently, the reaction wave is a hot product-gas bubble that may or may not drive cracking and convective burning ahead of it [13, 28]. The cookoff scenario is much different than the detonation regime where the shocked material is a supercritical fluid. In that case convective burning does not occur, and the density of the reactants is similar to that of the products. Although the detonation regime is simpler than the cookoff regime in many respects, it is significantly complicated by the fact that hot-spotinitiated thermal waves propagate into a material which, as a result of the global detonation reaction zone structure, is rapidly expanding and cooling. One can show that hot-spot propagation could be great slowed by this effect, which would be equivalent to their extinction. We have argued that one supercritical hot spot should behave much like the next. On the other hand, we have noted that detonation initiation experiments suggest that homogeneity/heterogeneity is a matter of degree. Are these two notions compatible? This potential paradox may be resolved by noting that hot-spot-supported detonation requires a cooperative effect between multiple hot spots. Consequently, the number density η of supercritical hot spots must play a large role in determining the degree of homogeneity/heterogeneity. More precisely, the “mix” will depend on a dimensionless parameter of which η is a constituent, such as, the reaction completion time via hot spots, divided by the reaction completion time associated with the spatially averaged temperature. The fundamental difficulty of the hot spot problem is that, although η is surely a key parameter, it is one that is very difficult to measure or
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compute. This problem is marginally within the realm of very large hydrocode calculations configured to model detailed material microstructure (e.g., [4]). Yet for practical purposes, η remains an unknown free parameter. One observes that certain types of processes can produce multiple simultaneously-burning hot spots operating in cooperative effect, while others cannot. We have just described the scenario in which many hot spots are created almost simultaneously by the dynamic crushing of an energetic material. Another scenario that does not involve crushing is self-propagating high-temperature synthesis (SHS) [6], whereby two or more solid reactants react to form a liquid, which then solidifies into a new and hopefully useful material. SHS is complicated by the fact that as a compositionally-heterogeneous material, thermal and mass diffusivity control the process (the ratio of these two effects being characterized by the Lewis number). Nevertheless, much of the generic reasoning concerning hot spots still applies. The thermal runaway of a slow-heated charge is an example of a scenario that does not produce multiple simultaneous hot spots.6 Rather, calculations and experiments agree that the ignition event has a very strong propensity to become highly localized. Localization is driven by an instability caused by the sensitivity of reaction rate to temperature. Even a slow-heated charge bathed in a uniform-temperature environment will experience small temperature variations. State sensitivity causes slightly hotter regions to react disproportionately faster than cooler ones. This disparity continues to be “explosively amplified” until some effect (e.g., reactant depletion or material deconsolidation) acts to mitigate it. A fascinating property of auto-ignition hot spots is that inevitably, it seems, only one develops. It is immediately not obvious that this should be so. Consider, for example, a spherical charge heated in a uniform-temperature bath. The isothermal contours are spherical shells, such that, wherever an ignition spot develops – either within the ball or on its surface – it resides on a spherical isothermal shell upon which every point has an equal probability of ignition. Indeed, a spherically-symmetric thermal calculation would not predict ignition at a point, but rather that the entire isothermal surface ignites simultaneously. Experimentally, however, such surfaces of nominal equal probability rarely result in the occurrence of even two ignition spots before reaction spreads across the charge. Evidently, real systems are so unstable that, even in the absence of substantial global temperature gradients, small perturbations (temperature, mechanical, and compositional in nature) conspire to produce a localized outcome. The single point in the system that leads reaction runaway exponentially outpaces all other points.
6
Note that, because auto-ignition hot spots arise spontaneously, they are by definition supercritical. Thus, the issue of hot spot criticality is only relevant in the context of hot spots created by mechanical work.
4 Classical Theory of Thermal Criticality
157
4.2.11 Reaction Progress Function for an Ideal Hot-Spot-Initiated Reaction Let us now explore the details of the simplest hot-spot-initiated reaction problem. We shall assume that, as the result of some unspecified process, there exists a number N , and number density η, of supercritical hot spots within a cube of energetic material. We shall also assume that N 1, such that the statistics of large numbers assures a unique, reproducible outcome. Our hot spots begin at isolated points, and simultaneously initiate spherically-expanding reaction waves that propagate outward to consume the surrounding material. Each hot spot propagates radially outward at a constant speed v, which we assume is maintained for the entire reaction. In reality, it seems likely that the thermal wave propagation speed will depend on the (dimensionless) product of the total front curvature and the reaction zone thickness, as is the case for curved detonation shocks (e.g., [3]). In the same way that detonation curvature effects are negligible in sufficiently large HE charges, in this problem we are free to choose a number density such that curvature effects are negligible over most of the reaction time. Each spherically-expanding thermal wave is assumed to react through a narrow reaction zone that separates reactants from products. Lastly, we assume that the density of the products is equal to that of the reactants. This last assumption provides a critical simplification, because it leads to strictly thermal waves with no mass motion. Although not reasonable in all cases, it is sensible for the two examples (i.e., detonation and SHS) given above. The objective of this exercise is to demonstrate how one may mathematically model an ensemble of fine-scale heterogeneous reactions as a larger-scale homogeneous reaction. The strategy is to mimic the natural physical averaging process described in Sect. 4.2.2, which causes a granular HE to appear homogeneous on the charge scale. The result is a global reaction progress function of the form used by classical thermal explosion theory. The burn pattern associated with expanding spheres is called progressive burning. More generally, progressive burning refers to net outward burning, which means that the burning surface area increases with time. As the burned spheres expand they start to coalesce, which leads to a more complex burn pattern. As burning continues, only pockets of unburned material remain, until finally the material is fully consumed. The late stages of reaction are regressive in nature. Regressive burning is inward burning (as in, say, a burning fuel droplet). More generally, regressive burning refers to net inward burning, wherein the burning surface area decreases with time. Before launching into details it is useful to consider the overall scaling. Consider a volume V containing N supercritical hot spots. The hot spot number density is η = N/V, and the characteristic distance between hot spots is η −1/3 . For a characteristic hot spot radial spreading speed v, the characteristic time to charge consumption is v −1 η −1/3 , and the characteristic reaction rate is v η 1/3 .
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In the early stages of reaction (before the burned regions have grown enough to coalesce) the spatially averaged reaction progress function can be exactly determined, as follows. Because we have assumed that the product density is equal to the reactant density, the reacted mass fraction is equal to the reacted volume fraction. For a number N of non-overlapping reacted spheres of radius r, the bulk mass fraction is 4πr3 η 4πr3 N = , (4.39) 3V 3 where V is the sample volume and η is the number density of nucleation sites. The reaction rate is: χ˙ = 4πr2 r˙ η = 4πr2 v η, (4.40) where v is the radial propagation speed of the reacting surfaces. Dividing χ˙ by χ2/3 computed from (4.39) gives 1/3 9 π 1/3 9π χ˙ 1/3 2 v η = = Z. (4.41) 2 2 χ2/3 χ=
The chosen characteristic reaction rate Z is a factor two times larger than the characteristic rate identified above. We have introduced this extra factor to define the rate in terms of the simplest reference problem, wherein nucleation sites are spaced on the nodes of a cubic grid. Then 1/Z is the time at which the burned regions touch first. Normalizing the time by Z to define a new dimensionless time tˇ = Z t, (4.41) becomes 1/3 9π dχ = χ2/3 , 0 ≤ χ 1. (4.42) 2 dtˇ Because the exponent in (4.42) is less than one (cf. Sect. 4.2.9), this reaction is self-starting. Considering the frequent emphasis in the preceding discussion on the sensitivity of the reaction rate to temperature, the lack of temperature dependence in (4.42) may seem suspicious. Indeed it should, for there is in fact a temperature dependence, which is buried in v. The form of that dependence has to do with the structure of the reactive wave front. This topic is both interesting and involved. Without considering the details, which are beyond the scope of the present discussion, one can generically say that the reaction zone thickness δ and the reaction wave propagation speed v must depend on the reactant temperature T0 in a way that involves the Arrhenius function. One of the ways that this heterogeneous reactive system differs from a homogeneous one is that, because reaction fronts advance into uniformtemperature reactants, there is no global thermal runaway over time. Instead, thermal runaway occurs only within the isolated reaction zones. Consequently, the global reaction rate increases or decreases with time because of the increase or decrease in the reacting surface area, which depends only on the reaction topology. Given that topology, the overall rate is modulated by Z.
4 Classical Theory of Thermal Criticality
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Let us begin by considering the problem from a numerical viewpoint. Without delving into many details that are discussed elsewhere [33], we randomly place point nucleation sites in a box. These points comprise the center of expanding spherical reaction waves.7 At time t, the radius of each sphere is r = v t. The spheres are grown in a noninteracting way; however, overlapping regions must be removed from consideration because they are already burnt. The problem of discounting overlap regions is fairly straightforward. The total burn fraction at any time is the number of “burnt” computational grid elements in the box divided by the total number of grid elements. The raw computed data set comprises a list of burn-fraction/dimensionless-time pairs. This data can then be recast in the form of an appropriately-scaled reaction progress function. It is easier to visualize the hot spot coalescence problem in 2D, where the burnt regions are circles instead of spheres. There is also useful qualitative information to be gleaned from the 2D problem that carries over to the 3D problem. Snapshots from a representative 2D simulation are plotted in Fig. 4.5.
Fig. 4.5. Snapshots of a 2D hot spot reaction simulation. Nucleation sites are randomly dispersed and grow at a constant rate. White regions are unreacted; black regions are fully reacted 7
The author gratefully acknowledges Dr. Bj¨ orn Zimmermann for programming this problem [33].
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L.G. Hill 1.0
Max of 50 trials Avg of 50 trials Min of 50 trials
χ
0.8 0.6 0.4 0.2 0.0 0. 0
0. 5
1. 0
1. 5 ˘t
2. 0
2. 5
3. 0
Fig. 4.6. Hot spot reaction simulation results for 500 nucleation sites randomly placed in a cubic box. The high and low curves are the extreme results of 50 simulations; the middle curve is the average result of the same simulations
As time and the burnt spot size increase, the white (unburned) areas, which began as the continuous phase, start to lose their connectivity. As reaction continues, the unburned regions morph into a very irregular discrete phase, and the products become the continuous phase. Both the size and the number density of unburned pockets decreases with time. This behavior appears to be rather far from ideal regressive burning, as occurs for the evaporation of distinct droplets, for example. Figure 4.6 plots the results of 3D simulations, which yield the function χ[tˇ]. Each set of numerical experiments tracks 500 randomly-distributed nucleation sites in a cubic box. Each set of calculations was repeated 50 times, each case with a different randomized starting condition. The high- and low-dashed curves are the extreme results of the 50 simulations. The middle solid curve is the average result of the 50 simulations. The small difference between the high and low curves indicates the variability of the bulk effect with respect to detailed nucleation site placement, for the sample size of 500 hot spots. For the case of nucleation sites placed on the nodes of a cubic grid, which we used as a reference to scale the randomized problem, reaction completes when the burnt spherical √ regions meet at the center of the unit cell. This condition occurs at tˇ = 3 ≈ 1.73. The random problem would be expected to finish later because, statistically, there will be some relatively large unburned “holes” that finish late. √ Indeed, Fig. 4.6 shows that reaction is approximately 90% complete at tˇ = 3, and is fully complete by tˇ ≈ 2.5.
4 Classical Theory of Thermal Criticality Eq. 4.44
Eq. 4.42
1.0
161
Eq. 4.43 0.8
χ˘
0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
χ
Fig. 4.7. Reaction progress function for an ideal hot spot reaction. The solid curve is the exact solution (4.43). The dashed curve is the idealized model for pure progressive/regressive burning (4.44). The dotted curve is the exact result for pure progressive burning (4.42)
Although the analytic problem may seem intractable, it actually has a simple solution that may be obtained using the methods of statistical mechanics. We show in [33] that the exact solution to the considered problem is dχ = dtˇ
9π 2
1/3 2/3
(1 − χ) (− ln[1 − χ])
.
(4.43)
Comparison of the integral of Eq. 4.43 with the center curve of Fig. 4.6 shows that the maximum error in the best numerical result is 0.04%. Equation (4.43) is plotted in Fig. 4.7 as the solid curve. The rising portion of the curve represents net progressive burning, and the falling portion represents net regressive burning. In fact, the falling portion of the curve reasonably approximates ideal regressive burning – an attribute that one would not tend to surmise by examining pictures like Fig. 4.5. For a hot spot reaction process that begins with ideal progressive burning and ends with ideal regressive burning, one would expect the reaction progress function to be very close to the following symmetric autocatalytic form: dχ ≈ dtˇ
9π 2
1/3 χ2/3 (1 − χ)2/3 .
(4.44)
By the same means that we derived the properties of ideal progressive burning in (4.39)–(4.42), one may show that the last factor in (4.44) corresponds to ideal regressive burning. By forming the product of progressive and regressive constituent functions, one approximates intermediate regions that are to various degrees globally progressive/regressive, while retaining pure progressive/regressive behavior at the respective ends of the function.
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Equation (4.44) is plotted as the dashed curve in Fig. 4.7. The result is very similar to the exact solution (4.43), especially on the progressive burning side where the two curves become identical as χ → 0. The two curves are also similar on the regressive burning side, although they are readily distinguishable. Being perfectly symmetric about the line χ = 1/2, (4.44) serves to illustrate that (4.43) is skewed slightly to the left. The dotted curve in Fig. 4.7 is the exact progressive burning limit given by (4.42). The regime where this curve closely overlays the other two is that in which, on average, burning is almost ideally progressive. This is true only up to χ ≈ 0.07. The close resemblance of the heterogeneous hot-spot reaction progress function to a homogeneous autocatalytic reaction progress function serves to illustrate that even distinct classes of reactions can exhibit very similar bulk behavior. Consequently the inverse problem, i.e., the inference of governing mechanisms from a bulk result, is ill-posed. 4.2.12 Modeling Complex Thermal Decomposition Behavior Although the decomposition of some explosives follows order-ω depletion behaviour (Sect. 4.2.7), autocatalytic depletion (Sect. 4.2.9), or even perhaps nucleation-and-growth type depletion (Sect. 4.2.11), other explosives (e.g., TATB) exhibit more complex behavior. Certainly, the many HE formulations that comprise a mixture of two or more monomolecular explosives, as well as those that comprise a heterogeneous mixture of fuel and oxidizer, would not be expected to behave so simply. To handle a wider variety of cases, one may generalize (4.14) and (4.29) to the form ∗ (4.45) χ˙ = g[χ]Z e−T /T , where g[χ] is an arbitrary reaction progress function to be determined. The hope – which seems rational given the above analysis – is that by generalizing the form of the reaction progress function, one may model any type of thermal decomposition reaction likely to be encountered, whether homogeneous, heterogeneous, or anywhere in between. This hope is somewhat optimistic, in that (4.45) is not the most general form that the reaction rate can take. Even the notion that the reaction rate depends only on the variables χ and T is an assumption, and (4.45) is not even the most general equation χ˙ = fn[χ, T ] with that dependence. Nevertheless, allowing a general reaction progress function goes a long way towards describing more general reaction behaviors. There is a practical reason for describing reactions by (4.45), which is that one may directly measure the reaction progress function using differential scanning calorimetry; whereas, to chase down all the chemical reaction pathways is generally impractical. Noting that real explosive reaction progress functions are often reminiscent of the elementary forms (or at least some combination of them), J. Janney and R. Rogers [35] have suggested that a
4 Classical Theory of Thermal Criticality
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linear combination of elementary functions should provide a good candidate fitting form: ∗ g[χ] = A + B(1 − χ)ω + Cχ (1 − χ)ω . (4.46) Equation (4.46) is by no means the most general equation that can be formed by this strategy. One may add as many basis functions (elementary or otherwise) as desired, with any combination of exponent values. Janney and Rogers caution us that even nominally-identical explosives may exhibit substantially different reaction progress functions, depending on their purity and physical form. To illustrate this point they compare the reaction rate versus time for three different batches of U.S. production grade HMX. The three curves look substantially different – at least in the temporal plane. By this example, one can appreciate that it is prudent to test a range of representative HE samples before deciding whether a chosen fitting model is adequate. 4.2.13 The Arrhenius Function and its Calibration Given the ubiquitous role of the Arrhenius function in reaction kinetics, we shall now examine its properties in detail. Classical thermal explosion theory is dominated by both the mathematical structure of this function and its calibration to thermal decomposition data. When expressed in terms of the ˜˙ = χ/Z, ˙ (4.2) reduces to the pure dimensionless variables T˜ = T /T ∗ and χ Arrhenius function exp[−1/ξ], as plotted in Fig. 4.8. Although the Arrhenius rate is zero only at absolute zero temperature, it remains very close to zero in a finite region between T˜ = 0 and the temperature corresponding to maximum positive function curvature, which occurs at T˜ ≈ 0.2. The function reaches maximum slope at T˜ = 1/2, after which it ˜˙ = 1. The convergence to the asymptote bends over toward its asymptote at χ ˜ is slow: χ˙ does not reach 0.99 until T˜ ≈ 100. 1.0 0.8
~ χ
0.6 0.4
T = T * (1, 1/e)
0.2
Maximum Slope (1/2, 1/e2) Maximum Curvature (0.202, 0.00715)
0.0 0
1
2
~ T
3
4
5
Fig. 4.8. Universal dimensionless plot of the Arrhenius function
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L.G. Hill
The activation temperature T ∗ is a measure of chemical stability, in the following sense. The larger the activation temperature T ∗ is compared to room temperature, the smaller the nominal decomposition rate is at room temperature, and the more the temperature must be elevated before the decomposition rate becomes significant. The pre-exponential factor Z is the ultimate reaction rate (meaning, the undepleted reaction rate for T T ∗ ). It is the “gain factor” that modulates the reaction for a given value of T relative to T ∗ . Neither T ∗ nor Z alone captures the somewhat different notion of reaction state sensitivity. If one defines the local state sensitivity as dχ/dT ˙ , then state sensitivity involves a combination of the two parameters. The dramatic rise in the function just beyond the point of maximum positive curvature suggests that one might use the condition as a metric for ignition. This simple and apparently logical reasoning turns out to be very much on the wrong track. To understand why this is so illustrates the essential features of the Arrhenius rate law and its calibration. To address this issue, let us identify the properties of a desirable explosive, and deduce the magnitudes of the Arrhenius parameters necessary to achieve them. Firstly, we would like our HE to have a long shelf life at uncontrolled storage temperatures, which we must assume could be on the warm side. Specifically, let us assert that our HE may experience an ambient storage temperature of up to T0 = 100◦ F (38◦ C, or 311 K). Let us further define “long shelf life” as a human life span, 100 years in round numbers. Thus, we would like our HE to experience negligible decomposition (say, less than 0.1% by mass) in 100 years. In round numbers, these conditions correspond to a nominal decomposition rate of χ˙ 0 = O[10−13] s−1 at the temperature T0 = 311 K. Our first condition is therefore χ[311] ˙ = 10−13 s−1 . We would also like our HE to have desirable detonation properties. Detonation performance is a complex subject involving issues beyond the scope of this book; however, we know from experience that desirable detonation wave propagation characteristics in typical charge sizes are associated with reaction zone times of order one nanosecond. Thus, we desire a decomposition rate of about χ˙ † = O[109 ] s−1 under detonation conditions. Detonation reactions in heterogeneous solid explosives are initiated by the hottest shocked regions, which can be T † = O[1000]◦C. Our second condition is therefore χ[1273] ˙ = 109 s−1 . These two conditions determine T ∗ and Z. Evaluating (4.2) at the two specified conditions gives
T∗ χ˙ 0 = Z exp − T0 Solving for T ∗ gives ∗
T =
T∗ and χ˙ = Z exp − † . T
T0 T † T † − T0
†
χ˙ † ln . χ˙ 0
(4.47)
(4.48)
4 Classical Theory of Thermal Criticality
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Solving for Z gives Z = χ˙ 0
χ˙ † χ˙ 0
T† T † −T0
.
(4.49)
Substituting the two specified conditions into (4.48) and (4.49) gives T ∗ ∼ 20,000 K and Z ∼ 1016 s−1 . As we shall see in Sects. 4.2.14 and 4.2.15, these numbers are close to the values that are actually measured for HMX explosive. For T ∗ = 20,000 K, the point of maximum positive curvature in Fig. 4.8 corresponds to about 4,000 K. This value is significantly hotter than the estimated combustion product temperature of HMX. We conclude that not only is the point of maximum positive curvature an inappropriate ignition criterion, but also the entire accessible temperature domain lies well to the left of it, in the region which, on the scale of the entire Arrhenius function, is diminutive. To achieve the high reaction rates observed under thermal explosion and detonation conditions therefore requires an enormous value of the premultiplier Z. Calibration exercises select the diminutive region of the Arrhenius function because that is the region where the function increases most strongly (percentage-wise) with temperature. Only that region can match the extreme reaction state sensitivity exhibited by monomolecular explosives. Conversely, Arrhenius function calibration is our means of quantifying that sensitivity. The origin of the extreme reaction state-sensitivity has much to do with the fact that reaction is tied to the stability of individual molecules. Once these start to decompose, reaction proceeds quickly and easily, because only mass rearrangement on a molecular scale is required to accomplish it. The scale of Fig. 4.8 is much too large to observe the character of the rate equation in the calibrated region. Therefore, in Fig. 4.9 we expand the plot to observe just the calibrated region. Here we show the undepleted reaction rate for the HMX-based explosive PBX 9501. The calibrated rate increases strongly with temperature, and the combustion product temperature associated with slow HMX decomposition is reached by T ≈ 0.1 T ∗. Note that this temperature is only about halfway to the point of maximum positive curvature. 4.2.14 Activation Energy as a Measure of HE Sensitivity In Sect. 4.2.9 we presented the activation temperature as a measure of chemical stability, noting that T ∗ is the scale by which we measure what constitutes a small or large temperature. Larger values of T ∗ relative to a fixed room temperature reference T0 lead to smaller decomposition rates at and near T0 . One may also argue the point on physical grounds. Atomistically, the larger the activation energy E (and the activation temperature T ∗ = E/R), the higher the temperature must be for a significant fraction of collisions to result in reaction.
166
L.G. Hill 0.8
(~2100 C, 1015 s–1)
~ χ(s–1) / 1015
0.6 0.4 0.2 (20 C, 10–20s–1) 0.0 0
500
1000
1500
2000
2500
T (K)
Fig. 4.9. Undepleted reaction rates spanned by PBX 9501 thermal decomposition, from an initial temperature of T0 = 20◦ C to the reaction product temperature of Tp ≈ 2100◦ C
The typical strategy for explosive safety characterization involves the rank ordering of the sensitivity of materials in various categories of stimulus/abuse. This allows one to judge the hazards of a new material by observing where its behavior falls with respect to familiar materials which, by trial and error over a period of many years, we have learned to handle safely. The difficulty is that the rank order of sensitivity can change with the nature of the stimulus. This is because the general initiation problem is more than just a chemical one. Initiation by a mechanical stimulus (friction, impact, shock, etc.) also depends on the mechanical, physical, and microstructural properties of the explosive in question. Consequently, one can only expect activation-energy-based chemical sensitivity metrics (E, T ∗ , T ∗ /T0 ) to be reasonably correlated with mechanical sensitivity metrics for materials within the same class of physical structure (e.g., granular solid, melt-cast solid, liquid, etc.). Table 4.1 shows the level of agreement for several common solid explosives, ranging from very sensitive to very insensitive. The first column gives an abbreviation for each considered explosive, and the second column lists the terms by which each is normally described. HE classification terminology is somewhat imprecise. The term primary explosive is used to describe those that readily burn to detonation (see Chap. 8, DDT). This level of sensitivity makes primary explosives useful as the initial stage of blasting caps and detonators; however, they are too dangerous to use in large quantities. The term secondary explosive is used to describe less sensitive explosives that are used in main charges. Thus, the primary/secondary designation mainly describes how the explosive is used. The designation tertiary explosive is sometimes used to describe very insensitive explosives that are used in large quantities, such as ANFO. Thus, tertiary explosives are initiated by secondary explosives, which are initiated by primary explosives.
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Table 4.1. Chemical sensitivity versus mechanical sensitivity HE* type
Sensitivity classification
H50 P0 (α) E T∗ −1 (cm) [GPa (%)] (kcal mol ) (K)
T ∗ /T0 Data – ref.
MF LA PETN RDX HMX TNT TATB NQ
Primary Primary Borderline P-S Secondary (CHE) Secondary (CHE) Borderline C-IHE Secondary (IHE) Secondary (IHE)
∼3 4 12 22 26 154 >320 >320
50.3 64.5 79.4 79.5 89.0 58.1 101 96.4
– – 1.7 (10) 2.6 (11)a 10 (0.7) 20 (1.6) 20 (2.8) 25 (4.5)
29.8 38.2 47.0 47.1 52.7 34.4 59.9 57.1
15,000 19,230 23,650 23,700 26,540 17,310 30,150 28,740
[18] [7, 17] [22] [22] [22] [22] [22] [22]
*MF Mercury Fulminate, LA Lead Azide, NQ: Nitroguanidine a
Available data are for PBX 9407 (94 wt% RDX, 6 wt% Exon 461 binder)
The terms conventional HE (CHE) and insensitive HE (IHE) are more recent. The IHE designation began as essentially a marketing term to promote the dramatically improved safety features of TATB. The term CHE arose from the need to distinguish explosives that are not IHEs. As the first and only weapons IHE (cf. mining explosives), TATB set the standard for insensitivity. The term IHE is precise (if somewhat arbitrary) because the U.S. Department of Energy maintains a specification that states the official qualifications. Not surprisingly, those qualifications are based on the sensitivity properties of TATB. The term ideal is used to describe explosives for which the detonation speed normal to a curved wave is nearly equal to the plane wave value. This designation is an exaggeration because no explosive is completely ideal. However, for reasons beyond the scope of this chapter, primary explosives come very close – at the expense of safety. The term nonideal is used to describe explosives for which the detonation speed normal to a curved wave varies significantly from the plane wave value. Nonideal explosives are safe to handle but are significantly harder to accurately model. Thus, there are three more or less parallel explosive categorization terminologies within the scientific community: (1) primary/secondary/tertiary, (2) CHE/IHE, and (3) ideal/nonideal. All three designations seek to describe the same basic physical effects; however, the first two classifications focus on safety, while the third focuses on modeling difficulty. Unfortunately, there is no mapping between the systems. To further confuse matters, there are entirely different classification systems (such as that used by the U.S. Department of Transportation) devised for the purpose of controlling storage and transport practice. Devised for legal and regulatory purposes, such classifications are somewhat less ambiguous than the common parlance. The third column of Table 4.1 lists the low-velocity-impact sensitivity, characterized by the critical height on the Los Alamos drop weight impact
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L.G. Hill
machine [22].8 Here H50 is the “50% height” for which half the tests react and half do not. This metric was used to rank order the listed explosives. The fourth column lists shock sensitivity as characterized by a reference point on the Pop plot curve [22]. The quoted input shock pressure, P0 , is that which causes a 3 mm run-to-detonation – the smallest value for which all the data sets are valid. The shock sensitivity varies strongly with void fraction α, which depends on the pressed density. The fairest comparison would be if each HE were pressed to the same α. But as the listed values show, α varies considerably amongst the available data sets. In particular, the PETN, RDX, and Nitroguanidine (NQ) data are for quite low-density pressings (α ∼ 10%), for which the material is significantly more shock-sensitive than for nominal high-density pressings, for which α = 1–2%. Even so, the rank ordering with respect to shock sensitivity agrees with the drop weight impact tests. Columns 5–7 list the chemical sensitivity metrics, E, T ∗ , and inverse dimensionless temperature T ∗ /T0 , where T0 is the room temperature taken to be 20◦ C. These three equivalent metrics consistently rank order with the impact and shock sensitivity metrics in most cases. The two exceptions are TNT and NQ, for which the discrepant numbers are highlighted by bold face type. NQ is only slightly discrepant and is likely consistent to within experimental error. However, the TNT departure is substantial. The chemical sensitivity of TNT falls midway between MF and LA, placing it squarely in the primary explosive category. Yet its impact sensitivity is borderline between CHE and IHE, and its shock sensitivity is the same as TATB. TNT’s markedly different behavior serves to illustrate that there can be substantial exceptions to sensitivity “rules of thumb.” TNT’s departure is by no means fully understood; however, one may venture to guess that it is related to the fact that – in contrast to the other listed explosives and most HE’s in general – TNT melts at a significantly lower temperature than it reacts. In the drop weight impact test, one would expect melting at crystal contacts to have a lubricating and hence desensitizing effect. For shock initiation, the fact that TNT is usually melt-cast rather than pressed leads to a much different (and likely more homogeneous and hence mechanically desensitized) microstructure than a granular compact. In particular, the specialized “creaming” solidification process creates a very homogeneous TNT microstructure and a very insensitive shock initiation behavior. F-K noted that for activation temperatures less than about 10,000 K, energetic materials lack even short-term chemical stability [21]. Such materials spontaneously decompose at room temperature in a short period of time. At T ∗ = 15,000 K, mercury fulminate (MF) is quite spark- and friction-sensitive, 8
The lead azide (LA) result is from the NOL drop weight impact test [19], which is almost identical to the Los Alamos test. The mercury fulminate (MF) result is an estimate, scaled by the LA value using similar data from the Picatinny Arsenal drop weight impact test [17, 18].
4 Classical Theory of Thermal Criticality
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Table 4.2. Approximate sensitivity/stability categorization for pressed granular explosives, arranged by activation temperature T ∗ Range
Typical material behavior and/or categorization
1 for all other shapes of the same dimension. This is so because each canonical shape has the smallest surface area (corresponding to the smallest ra -value for a chosen r-value) of any shape with its same dimension. The dimensionless parameter ϕ is the ratio of the generalized length ra , to the “SMD length,” λ. ϕ may also (and perhaps more usefully) be interpreted as the volume ratio Ara /V, where the numerator Ara provides a crude estimate of the volume of an arbitrary body for which only the surface area
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L.G. Hill
is known.14 The larger a body appears to be from surface area considerations alone, compared to the actual volume V, the greater its ability to shed internally-generated heat. Consequently ϕ represents a sort of dimensionless geometric heat loss factor. We reason that for near-unity-AR bodies, F kac depends on a single dimensionless geometric parameter. We further reason that, due to the dearth of other viable candidates, ϕ is that parameter. For elongated bodies AR arises as a second dimensionless parameter, the dependence of which vanishes if we limit ourselves to cases for which AR ≈ 1. To find the function F kac [ϕ], we may plot values of ϕ and F kac for a variety of candidate shapes, and observe whether there is a consistent trend. If so, we can fit a curve to that trend and thereby determine F kac [ϕ] “experimentally.” Table 4.4 lists the shape factors and F-K critical parameters for six common near-unity-AR objects. Proceeding from left to right, the aesthetic critical F-K number F kc is computed for a scaling based on r. The Semenov dimension λ and the scale factor β determine ϕ, where ϕ = (β r)/λ. For the simple shapes listed in Table 4.4, β and λ are straightforward to compute. However, for arbitrary shapes the determination of these geometric factors is non-trivial. F kac is determined from β and F kc via (4.157). Finally, the critical F-K temperatures for each body are also listed. The six shape correlation points are plotted in Fig. 4.28. There is indeed a good correlation and it is a straight line. The best fit linear equation is F kac = 1.218 ϕ − 0.3397; ϕ ≥ 1.
(4.158)
The negative intercept of (4.158) is not a concern, because negative values are never accessed. Rather, the point corresponding to the infinite slab represents the absolute lower limit of ϕ, and hence the absolute lower limit of F kac . By far the largest deviation from the best-fit line – still a rather modest departure – is the point corresponding to the infinite cylinder. It is curious that one point would deviate from the overall trend much more than the Table 4.4. Shape factors and F-K critical parameters for six common objects Shape description
Length r used in F kc
F kc
λ
β
ϕ
F kac
θmc
∞ Slab ∞ Circular bar ∞ Square bar Sphere Cyl. (AR = 1) Cube
Half width Radius Half edge Radius Radius Half-edge
0.8785 2 1.702 3.322 2.765 2.476
r r/2 r/2 r/3 r/3 r/3
1 1 4/π 1 3/2 6/π
1 2 8/π 3 27/2 54/π
0.8785 2 2.759 3.322 4.148 4.729
1.19 ln[4] 1.37 1.61 1.61 1.61
14
For 1D and 2D cases, the surface area and volume apply to a representative element of the infinite body.
4 Classical Theory of Thermal Criticality 5
217
Cube Cylinder (AR = 1)
4 Fkac
Sphere 3
Infinite Square Bar Infinite Cylinder
2 1 0
Infinite Slab
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
ϕ
Fig. 4.28. Dimensionless shape correlation plot for six near-unity aspect ratio shapes
others. It also seems odd that the point that would do so corresponds to the only shape for which the solution is fully analytic. A logical explanation would be an error in the other (computed) points, except that all are believed to be quite accurate. It is as if the cylindrical case is somehow unique. The function F kac [ϕ] was determined without appeal to (4.154), the primary purpose of which was as a stepping stone to deducing the correct form of the near-unity-AR shape correlation. Having demonstrated via Fig. 4.28 that F kac [ϕ] depends on ϕ only, (4.154) evaluated at criticality tells us that Ic [ϕ] must likewise depend on ϕ only. We may deduce the functional form by combining (4.154) evaluated at criticality with (4.158). The result is Ic = 1.218 −
0.3397 ; ϕ ≥ 1. ϕ
(4.159)
Equation (4.159) is plotted in Fig. 4.29. Its variation with ϕ is rather modest. While the form of Ic [ϕ] is mostly a curiosity, it is interesting that we are able to deduce, by roundabout reasoning, the form of a function whose behavior could not have been easily imagined. Even the parameter(s) that Ic depends upon were non-obvious a priori. 4.8.2 Comparison of F kc Values Obtained by Various Studies We now compare the values of F kc listed in Table 4.4 to those obtained by previous studies. As noted at the beginning of Sect. 4.8, relatively few efforts have been devoted to finding F kc for new shapes, or to confirming the accuracy of F kc for previously considered ones. Table 4.5 tabulates the previous efforts known to the author. The infinite cylinder, for which F kc is exactly 2, helps one to appraise the accuracy of each numerical method. Note that, unlike
218
L.G. Hill 1.4 1.2
Ic
1.0 0.8 0.6 0.4 0.2 0.0
0
5
10
15
20
25
ϕ
Fig. 4.29. The deduced function Ic [ϕ] Table 4.5. Numerical values of F kc for several shapes, obtained by various studies Author/ pub. year
Infinite slab
Infinite cylinder
Infinite sq. bar
Cube
Cylinder (AR = 1)
Sphere
Ref.
F-K/1939 Enig/1956a Enig/1956ba Parks/1960a Hill/2007
0.88 0.87846 0.87983 0.880 0.87846
2.00 – 2.0023 2.01 2.0000
– – – – 1.702
– – – 2.53 2.476
– – – 2.78 2.765
3.32 – 3.3259 3.34 3.3220
[37] [14] [16] [47] [31]
a
These studies used the full Arrhenius heat generation term with a small but finite value of T˜s . These values err slightly on the large side of the F-K limiting values
F kac , comparison of F kc values requires a consistent definition of r for each shape from one study to the next. For these simple shapes, the cleanest choice of r is obvious enough that this was not an issue. F-K determined F kc for the canonical shapes by hand numerical calculations. Later studies have verified that his constants are accurate to the stated precision, which is actually sufficient for most purposes. J. Enig [14] was the first to compute F kc using a computer.15 Enig computed F kc for the slab case (only), which (see (4.123)) is a root-finding exercise. Enig reported his result to five figures. His number is identical to the slab result obtained here. J. Enig et al. [16], and later J. Parks [47], solved the F-K problem for the unapproximated Arrhenius term (the steady-state limit of (4.97)). We previously noted that the F-K high activation energy approximation becomes exact as T˜s → 0. In these two studies, which we shall reproduce in Sect. 4.10, 15
In both 1956 studies listed in Table 4.5, Enig used an IBM 650 “Magnetic Drum Processing Machine.” The IBM 650 was the world’s first mass-produced computer. It was manufactured from 1954 to 1962.
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the intent was to test how the full Arrhenius result converged to the F-K high activation energy result as a function of the surroundings temperature T˜s . The Table 4.5 values of F kc that were taken from [16] and [47] correspond to the smallest values of T˜s computed by the respective authors. These values have nearly converged to their F-K limits; however, to the extent they have not, they would err on the large side. Comparing the values from those studies to the corresponding F-K limit values from the present study, one may verify that such is the case. The largest discrepancy is between Parks’ cubic value and the cubic value from the present study. That discrepancy is still only 2%. 4.8.3 Effective Dimensionality of Near-Unity-Aspect-Ratio Shapes In Sect. 4.8.1 we interpreted the parameter ϕ as a dimensionless geometric heat loss coefficient. Here, we suggest that it can alternatively be interpreted as a measure of the object’s effective dimensionality, in a sense that is at least loosely analogous to the concept of fractal dimension. Fractal geometry is useful for describing objects which, owing to their selfsimilar structure, transcend their topological dimension (the nominal dimension associated with the class of object) so as to partially fill their Euclidean dimension (the dimensionality of the space that the object exists within). B. Mandelbrot posed the enthralling question, “How long is the coastline of Britian?”[41, 42] In that problem the object in question is nominally a curve, which has topological dimension 1. The object exists within a plane, so its Euclidean dimension is 2. However, the object’s effective dimension, also called its fractal dimension, is close to 3/2; that is, the effective dimensionality of the coastline lies about midway between the two bounding dimensions. Clearly there is nothing fractal about the objects considered in Tables 4.4 and 4.5. The main similarity is that, like the fractal dimension, ϕ categorizes a body as having either an integer or fractional effective dimension. In this context, the effective dimension does not characterize the dimension of the shape itself, but rather how the shape builds and sheds heat. The canonical shapes have ϕ values equal to their topological dimension (slab→1, cylinder→2, sphere→3). All other shapes with the same topological dimension have higher ϕ values, which nominally lie between the shape’s topological dimension and the next higher dimension. For example, the square bar has a ϕ-value of 2.55. This measure places the object about halfway between that of a circular bar and a sphere. Such a placement seems sensible in the context of the equilibrium self-heating problem. Likewise at ϕ = 3.67, the unity-AR-cylinder, lies between the effective dimension of the sphere and the next higher integer dimension (i.e., 4). However, at ϕ = 4.15, the cube lies between the effective dimension of the sphere and a dimension slightly greater than 4. It is yet to be seen whether casting ϕ as an effective dimensionality has a deeper meaning that meets the eye, or whether the concept is practically useful. In the meantime this interpretation is at least an interesting curiosity.
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4.8.4 Breakdown of the Shape Correlation for Elongated Shapes Before discussing the criticality of elongated shapes, we must first define what we mean by the term “elongated.” An infinite circular cylinder is extremely elongated; however, this property also ensures that any transverse slice of the body is representative of the whole. Consequently this problem is effectively 2D, and in this reduced effective dimension the body has no elongation whatsoever. Thus, we shall use the term elongated to describe the asymmetry of a body in its effective dimension (i.e., the dimension in which we would naturally analyze it). Bodies having no significant elongation can be relatively simply treated by the shape correlation of (4.158). Likewise, the case of extreme elongation can also be treated by (4.158), after the appropriate reduction of its dimension. The difficulty lies with transitional objects, i.e., those that are modestly elongated in their effective dimension. In 2D, an example of a transitional object would be a rod of elliptical cross-section. In 3D, examples include the prolate spheroid (a football shape) and the oblate spheroid (an “M&M” candy shape). Only in a few applications (e.g., a shaped charge) does one encounter complex HE shapes. Most often, HE charge shapes are elementary. It would be rare to encounter an elliptical HE rod, for example. On the other hand, rectangular blocks are used quite commonly. A rectangular parallelpiped with dimension ratios 1×1.5×2 constitutes transitional elongation in two directions. This is perhaps the most difficult of the common shapes. Let us begin with the simple example of a rectangular bar. The F-K problem is scaled such that the reference dimension is unity. Thus, the other dimension is equal to the aspect ratio, AR, of the cross-section. We need only consider cases for AR ≥ 1, because the cases AR < 1 reproduce the same shapes at a smaller size, and F-K theory is universal with respect to size. Figure 4.30 tests the proposition that the various aspect ratios obey the shape correlation of Fig. 4.28. Obviously, the square bar (AR = 1) does; however, as AR increases from unity, the trend deviates sharply from the unity-AR correlation line. It is clear that for this example – and no doubt for other shapes as well – that elongated bodies exhibit a different behavior. It is interesting to note that even modestly elongated cross-sections (say, AR = 1.5) deviate substantially from the near-unity-AR line. This is so because the departure of the general AR line from the correlation is abrupt. If it had been the case that general-AR curves approach the unity-AR line tangentially as the elongation vanishes (i.e., as AR → 1), the near-unity-AR model would have been more forgiving toward elongation. To put the elongation problem in perspective: from a dimensional analysis viewpoint, AR constitutes an additional parameter. It would be highly optimistic to imagine that this parameter has no influence. We may demonstrate what the approximate nature of this influence is by examining circular cylinders of general aspect ratio. We consider the unity aspect ratio cylinder as
4 Classical Theory of Thermal Criticality 10
AR= 4 Rectangular Bar
8
Fkc
221
AR = 3
AR
6
Cube
1 AR = 2
4
Cylinder (AR = 1) Sphere Infinite Square Bar Infinite Cylinder AR = 1
2 Infinite Slab
0
1.0
1.5
2.0
2.5 ϕ
3.0
3.5
4.0
4.5
Fig. 4.30. Shape correlation for an infinite rectangular bar of variable aspect ratio. As AR increases from unity, the behavior deviates sharply from the near-unity aspect ratio correlation of Fig. 4.28 3.0
Unity-AR Cylinder AR
2.5
Rod
1
Fkc
2.0
Infinite Rod Limit AR
1.5
Tablet
1
1.0 Infinite Slab Limit 0.5 0.0
0
2
4
6
8
10
AR
Fig. 4.31. F kc versus aspect ratio for a finite cylinder. The starting point on the left is the unity-AR case. The upper curve corresponds to elongation in the axial direction to form a rod, and ultimately an infinite cylinder. The lower curve corresponds to elongation in the radial direction to form a tablet, and ultimately an infinite slab
the reference case, and reduce or increase the aspect ratio to consider tablets and bars, the limiting case of which is an infinite slab and an infinite cylinder, respectively. Figure 4.31 plots F kc versus AR for a finite cylinder. The starting point on the left is the unity-AR case, which is one of the objects included in Fig. 4.28. The upper curve corresponds to elongation in the axial direction to form a rod, and ultimately an infinite cylinder. As the cylinder is stretched toward infinite length, the body morphs from a 3D object to a 2D object, and F kc
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drops from the unity-AR value to that of the infinite cylinder. As shown in the embedded schematic, this problem is scaled in the same way as the infinite cylinder, with the radius being unity, such that the half length is AR. This gives the correct limiting value of F kc for the infinite cylinder. The lower curve corresponds to elongation in the radial direction to form a tablet, and ultimately an infinite slab. As the cylinder is stretched toward infinite radius, the body morphs from a 3D object to a 1D object, and F kc drops from the unity-AR value to that of the infinite slab. As shown in the embedded schematic, this problem is scaled in the same way as the infinite slab, with the thickness being unity, such that the radius is AR. This gives the correct limiting value of F kc for the infinite slab. Figure 4.31 makes it clear why the aspect ratio has a large effect. It is because when AR changes, the effective object dimensionality changes and F kc changes accordingly. Sometimes, as in the case of the tablet, the object morphs directly from a 3D object all the way to a 1D object as AR is varied. For both the rod and the tablet, F kc has for all practical purposes converged to the value of the associated infinite body by the point where AR ≈ 10. Thus, it is not necessary for the body to be infinite, nor even that it appear infinite, in order for the asymptotic value of F kc to apply. A good rule of thumb is, “if it looks like a rod then it acts like a rod,” or, “if it looks like a plate then it acts like a plate.” 4.8.5 On Extending the Shape Correlation to Elongated Bodies Aspect ratio as defined in the previous rectangular bar and finite cylinder examples has the same strengths and limitations as did the length scale r. Both are chosen for particular bodies according to simplicity and convenience or, as we have put it, aesthetics. For most bodies of interest, the aesthetic aspect ratio is defined with respect to axes or planes of symmetry. As such, aesthetic AR values for different bodies are not generally comparable. In order to enable such a comparison, it is necessary to generalize the definition of AR in much the same way as we did for r. For some bodies, such as the rectangular bars and cylinders just considered, elongation is characterized by a single value of AR. For other bodies, such as rectangular parallelpipeds, two AR values are required to describe the relative elongation in the three orthogonal directions. But what if the considered charge is a non-descript blob of a moldable explosive like Composition C4 or Semtex, with no axes or planes of symmetry? Happily, the definition of ra works just fine in such cases. But how would one quantify generalized elongation? One possibility is the following. Imagine a line cutting through the considered body, which is normal to the surface at its entrance point (and therefore not generally normal at its exit point). This line is swept over the entire surface in such a way that its entrance point remains normal to the surface. In
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doing so, the line will encounter a maximum span through the body, as well as a minimum span. The ultimate aspect ratio ARu may be defined as the ratio of the maximum to the minimum length. This prescription alone is not sufficient for every body. For example, it can be “fooled” by small protrusions, and is ill-defined at sharp corners. Thus, additional qualifications are needed to handle the general case. It is not our purpose to settle such details here; rather, we merely wish to point out some of the interesting issues that must be considered if one would generalize the near-unity-AR shape correlation to extended bodies. Whether or not a useful general extension exists remains to be seen. In the meantime, we present an interesting and useful partial characterization of the extended body problem. The curves in Fig. 4.31 resemble a relaxation law (i.e., a relationship that describes the temporal evolution of a non-equilibrium state toward an equilibrium one). What if a scaled version of this relaxation-like behavior was widely applicable? In fact, we find that the rectangular bar case plotted in Fig. 4.30, as well as both cylinder cases plotted in Fig. 4.31, closely follow the equation F kc − F kci = AR−9/4 ≡ τ [AR]. F kcn − F kci
(4.160)
Here, F kc is the value we wish to know. F kcn pertains to the near unity-AR F-K number determined from (4.158). F kci is the F-K number associated with the considered shape when extended to infinite dimension. Here, i takes the value 1 or 2 depending on whether the infinite extension is 1D or 2D. Figure 4.32 plots scaled points computed for the three objects. Although the overlay is not perfect, the amount of mutual scatter is minimal. The overlying curve is (4.160). It is interesting that the chosen exponent, although
1.0 0.8
τ
0.6 0.4 0.2 0.0 0
2
4
6
8
10
AR
Fig. 4.32. Aspect ratio correlation for three objects of variable aspect ratio: (1) the infinite rectangular bar, (2) the circular rod, and (3) the circular tablet. As AR increases from unity, the behavior deviates sharply from the value predicted by the near-unity aspect ratio correlation
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close to −2, is clearly distinguishable from it. In fact, the observed exponent is −9/4, or very close to it. The significance of that value is, at this time, unknown.
4.9 F-K Theory with Generalized Boundary Conditions 4.9.1 Thermal Conductance/Resistance and the Biot Number Thermal conductance is defined as the quantity of heat that passes in a unit time through a plate of particular area and thickness, when its opposite faces differ in temperature by 1◦ . For a plate of thermal conductivity k, area A, and thickness δ, the thermal conductance is (kA)/δ. Thermal resistance is its inverse. The heat transfer coefficient H for Newton’s law of cooling (see Sect. 4.3.1) is a generalization of the conductance, for an unspecified system and an arbitrary temperature drop. The thermal conductance for a unit-area plate is k/δ. This quantity has no name other than the heat transfer coefficient. The heat transfer coefficient per unit area in Newton’s law of cooling, h = H/A (see Sect. 4.3.1) is a generalization of this definition, for an unspecified system and an arbitrary temperature drop. Although the term “heat transfer coefficient” has multiple definitions, it is usually clear by the context what definition is meant. The inverse of thermal conductance is called the thermal insulance. The dimensionless Biot number, named for the French physicist Jean-Baptiste Biot, characterizes the thermal insulance of a body in comparison to the thermal insulance of its surroundings. A solid body has a thermal insulance of order r/k, where r is a characteristic dimension and k is the thermal conductivity.16 No such statement can generally be made about the thermal insulance of the surroundings, because they may be essentially infinite in extent (with no identifiable length scale). Moreover, heat transfer may involve convection, in which case it is not characterized by the thermal conductivity. For this reason, we characterize heat transfer to the surroundings by the heat transfer coefficient h, and the thermal insulance by 1/h. The Biot number is the characteristic insulance of the body divided by the characteristic insulance of the surroundings, i.e., Bi =
hr . k
(4.161)
If the Biot number is less than unity, then heat is able to flow through the body interior more readily than it can escape to the surroundings. For Bi 1, the body temperature can be approximated as uniform. This approximation 16
For energetic material systems comprising gases and liquids, in which internal heat transfer is largely convective in nature, the thermal insulance is not characterized by r/k. However, that case is well-approximated by the Semenov model.
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becomes exact as Bi → 0. A good example of a small Biot number problem is the “buried cable” scenario [10]. In stating that the HE temperature is spatially uniform, Semenov theory implicitly assumes that Bi = 0. If the Biot number is greater than unity, then heat is able to flow to the surroundings more readily than it is able to flow through the body interior. For Bi 1 the surroundings maintain a nearly constant temperature, and a heat liberating body experiences internal temperature variation. As Bi → ∞, the body surface temperature becomes “clamped” to the surroundings temperature. In imposing a specified constant edge temperature, F-K theory implicitly assumes that Bi = ∞. From this description it is clear that Semenov theory is not inferior to, nor necessarily less accurate than, F-K theory merely for being simpler. One can imagine scenarios where either model is the better approximation to reality; however, most cases will tend to lie somewhere between these two extremes. For example, in one experimental study A. Merzhenov et al. [45] found Bi to be close to unity for the considered system. Describing the boundary condition in terms of the Biot number has the ˜ in Semenov theory. On same advantages and disadvantages as the use of H the positive side, the description is of great heuristic value, and it is general enough to cover many cases. On the negative side, it is difficult to know what Bi is for real world scenarios. To calculate its value would require a detailed model which, if we had it, would negate the need for a Biot number description. The main caveat is that the boundary condition for real problems may not be spatially uniform, in which case Bi varies over the surface. The Biot number description is not invalid in such situations, but the problem can no longer be solved by the relatively simple extended F-K model presented in this chapter. 4.9.2 Thomas’ Generalized Boundary Condition P. Thomas [54] recognized that the thermal explosion problem could be generalized by expressing the boundary condition in terms of the Biot number. Thomas’ insight and accompanying analysis both extends and unifies the SFK and F-K variants of the theory. Arguably, his contribution is second only to Semenov’s and F-K’s in its importance to the classical theory. Thomas’ generalized boundary condition expresses the fact that heat flow is continuous across a surface. Equating the heat flow on the body side of the boundary to the heat flow on the surroundings side of the boundary gives −k
dT = h(Te − Ts ). dx e
(4.162)
Equation (4.162) is the first instance in which we have had to distinguish between three temperatures: T (the variable temperature within the body),
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Te (the temperature at the charge edge), and Ts (the temperature of the surroundings). For Semenov theory T = Te ≥ Ts , where T is constant throughout the body. For F-K theory T ≥ Te = Ts , where T varies throughout the body. Scaling the distance x by the characteristic body dimension r to form a dimensionless distance x ˜, and substituting the F-K temperature θ for T , gives θ [1] = −Bi θ[1], or θe = −Bi θe ,
(4.163)
where the prime indicates derivatives with respect to x ˜. Note that if Bi → 0, then (4.163) reduces to the condition θ [1] = 0, which is the appropriate boundary condition for the Semenov model. Dividing both sides of (4.163) by Bi and letting Bi → ∞ gives the condition θ[1] = 0, which is the correct boundary condition for the F-K model. In the following three subsections we repeat the F-K criticality problem for each of the three canonical shapes, using Thomas’ generalized edge boundary condition (4.163). The approach is essentially the same as that used by Thomas [54]. Analytic solutions exist in a parametric form; however, the spherical solution is analytic with respect to the numerically-generated function ψ that arose in Sect. 4.7.5 in connection with the basic F-K sphere. 4.9.3 Infinite Slab: Critical Condition versus Biot Number We begin with the slope of the temperature profile prior to the application of the edge boundary condition, which cf. (4.118) is √ (4.164) θ = − 2 F k eθ m − eθ . Next we apply the edge condition (4.163) to give √ Bi θe = 2 F k eθm − eθe .
(4.165)
Repeating the steps in (4.119)–(4.123) (except that in this case θe is nonzero), one obtains √ (4.166) F k = 2 e−θm arcsech2 eθe −θm . Noting that the grouping
F k eθ m (4.167) 2 occurs prominently in the equations, we let ξ be the parameter by which the solutions will be generated. Substituting (4.167) into (4.165) and (4.166) and eliminating θe gives ) * (4.168) Bi (θm + ln sech2 ξ ) = 2 ξ tanhξ. ξ=
In order to find the peak value of F k with respect to θm as in Sect. 4.7.3, we may implicitly differentiate (4.168) and set dF k/dθm = 0. Using the definition
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of (4.167), one may show that this exercise is equivalent to differentiating ξ with respect to θm and setting dξ/dθm = ξ/2. Doing so yields Bi (1 − ξ tanhξ) = ξ(tanhξ + ξ sech2 ξ).
(4.169)
Solving (4.169) for Bi gives the first parametric solution in terms of ξ: Bi[ξ] =
ξ(tanhξ + ξ sech2 ξ) . 1 − ξtanhξ
(4.170)
Substituting (4.170) into (4.168) and solving for θm gives the second parametric solution: ) * θmc [ξ] = 2 − ln sech2 ξ −
2 ξ cosh2 ξ . ξ + coshξ sinhξ
(4.171)
Note that because (4.170) corresponds to the peak (critical) value of F k, θm corresponds to θmc . Solving (4.167) for F k gives F k = 2 ξ 2 e−θm .
(4.172)
Substituting (4.171) into (4.172) gives the third parametric solution: ) * 2 ξ cosh2 ξ F kc [ξ] = 2 ξ 2 exp 2 − ln sech2 ξ − . (4.173) ξ + coshξ sinhξ Examining (4.170), we see that Bi[0] = 0. Thus the minimum value of the parameter ξ is zero. Also examining (4.170), we see that Bi → ∞ when the denominator vanishes, i.e., ξ∞ tanhξ∞ = 1, or ξ∞ = 1.1997.
(4.174)
Thus, the solution domain is 0 ≤ ξ ≤ ξ∞ . Figure 4.33 plots θmc [Bi] (generated parametrically as θmc [ξ] versus Bi[ξ] via (4.170) and (4.171)). The curve asymptotes to the correct F-K value, θmc∞ = θmc [ξ∞ ] = 1.187. Figure 4.34 plots F kc [Bi] (generated parametrically as F kc [ξ] versus Bi[ξ] via (4.170) and (4.173)). This curve also asymptotes to the correct F-K value, F kc∞ = F kc [ξ∞ ] = 0.8785. It is interesting (note the scale of the abscissa on Figs. 4.33 and 4.34) that θmc converges to its asymptotic value at about an order of magnitude smaller value of Bi than does F kc . Gray and Harper [23, 24] presented a stylized three-dimensional plot of the critical temperature profile versus Biot number. A number of workers have found their exercise to be helpful in visualizing how the critical temperature profile varies with Bi. Figure 4.35 plots this surface for the critical slab case. Referring to Fig. 4.33, it is clear that the temperature field on the right hand side of the figure (where Bi = 3) has almost converged to the large-Bi limit value; whereas, F kc is at only about half of its asymptotic value at the point Bi = 3.
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1.2
θmc
1.0 0.8 0.6 0.4 0.2 0.0 0
2
4
6
8
10
Bi
Fig. 4.33. Critical F-K temperature versus Biot number for the infinite slab 1.0
Fkc∞ = 0.8785
0.8
Fk
0.6 0.4 0.2 0.0 0
20
40
60
80
100
Bi
Fig. 4.34. Critical F-K number versus Biot number for the infinite slab
4.9.4 Infinite Cylinder: Critical Condition versus Biot Number The extended criticality analysis of the infinite cylinder proceeds in the same manner as that of the infinite slab (Sect. 4.9.3). We begin with the temperature profile prior to the application of the edge boundary condition, which cf. (4.134) is F k eθ m θ = θm − 2 ln 1 + (4.175) x ˜2 . 8 Next we differentiate (4.175) with respect to x ˜, to obtain θ =
−F k eθm x ˜ . F k eθm x ˜2 2 1+ 8
(4.176)
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Fig. 4.35. A surface plot of the critical F-K temperature profiles versus Biot number for the infinite slab (following Gray and Harper [23, 24])
Evaluating (4.175) and (4.176) at x ˜ = 1 and substituting the result into Thomas’ edge condition (4.163), one obtains 4ζ = Bi (θm − 2 ln[1 + ζ]) . 1+ζ
(4.177)
Equation (4.177) has already incorporated the following substitution for the parametric solution parameter: ζ=
F k eθ m . 8
(4.178)
Incidentally, the parameter ζ is the same as the parameter C in Sect. 4.7.4. To find the peak value of F k with respect to θm , we implicitly differentiate (4.177) as in Sect. 4.9.3, and set dF k/dθm = 0. Using the definition of (4.178), one may show that this exercise is equivalent to differentiating ζ with respect to θm and setting dζ/dθm = ζ. Doing so yields Bi[ζ] =
4ζ , 1 − ζ2
(4.179)
which is the first of the three desired parametric solutions. Substituting (4.179) into (4.177) and solving for θm gives the second parametric solution: θmc [ζ] = 1 − ζ + 2 ln[1 + ζ].
(4.180)
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Note that because (4.179) corresponds to the peak (critical) value of F k, θm corresponds to θmc . Solving (4.178) for F k and substituting in (4.180) gives the third parametric solution: F kc [ζ] =
8 ζ eζ−1 . (1 + ζ)2
(4.181)
Examining (4.179), we see that Bi[0] = 0. Thus, the minimum value of the parameter ζ is zero. Also on examining (4.179), we see that the denominator vanishes (corresponding to Bi → ∞), where ζ∞ = 1. Thus, the solution domain is 0 ≤ ζ ≤ 1. The F-K critical values, corresponding to Bi → ∞ for the present analysis, are correct: θmc∞ = θmc [ζ∞ ] = ln[4], and F kc∞ = F kc [ζ∞ ] = 2. The curves θmc [Bi] and F kc [Bi] for the infinite cylinder are qualitatively similar to the corresponding ones for the infinite slab. We shall defer plotting these curves until Sect. 4.9.6, where we compare them to the corresponding curves for the slab and spherical cases. 4.9.5 Sphere: Critical Condition versus Biot Number The extended criticality analysis of the sphere proceeds in a similar manner as that for the infinite slab (Sect. 4.9.3) and the infinite cylinder (Sect. 4.9.4). We begin with the temperature profile prior to the application of the edge boundary condition, which cf. (4.139) is $√ % θ = θm − ψ F k eθ m x ˜ , (4.182) where ψ is the function defined in (4.138) and plotted in Fig. 4.22. We shall express the results symbolically in terms of ψ, assuming that ψ and its derivatives are known functions. Next we define the parameter √ (4.183) = F k eθ m , which is analogous to those defined in Sects. 4.9.3 and 4.9.4. Differentiating (4.182) with respect to x ˜ gives θ = − ψ [ x ˜].
(4.184)
Substituting (4.182) and (4.184) into Thomas’ edge boundary condition (4.163) gives ψ = Bi (θm − ψ), (4.185) where ψ = ψ[], and the prime now indicates derivatives taken with respect to . To find the peak value of F k with respect to θm , we implicitly differentiate (4.185) as in Sects. 4.9.3 and 4.9.4, and set dF k/dθm = 0. Using the definition
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of (4.183), one may show that this exercise is equivalent to differentiating with respect to θm and setting d/dθm = /2. We do so, while further making the following substitution from (4.138): ψ [] = e−ψ −
2 ψ .
(4.186)
Solving for Bi gives the first of the three desired parametric solutions:
e−ψ − ψ Bi[] = , (4.187) 2 − ψ where, as a reminder, ψ = ψ[] and the prime indicates derivatives taken with respect to . Substituting (4.187) into (4.185) and solving for θm gives the second parametric solution: θmc [] = ψ +
eψ ψ ( ψ − 2) . eψ ψ −
(4.188)
Note that because (4.188) corresponds to the peak value of F k, θm corresponds to θmc . Solving for F k in (4.183) and substituting the result into (4.188) gives the third parametric solution: eψ ψ ( ψ − 2) . (4.189) F kc [] = 2 exp −ψ − eψ ψ − Examining (4.187), we see that Bi[0] = 0. Thus, the minimum value of the parameter is zero. The maximum value of occurs when the denominator of (4.187) vanishes, which corresponds to Bi → ∞. This condition is satisfied if ψ = 2, the solution of which may be determined by inserting the computed function ψ and solving for the root. Doing so gives ∞ = 4.072. Thus, the solution domain is 0 ≤ ≤ 4.072. The F-K critical values are correct: θmc∞ = θmc [∞ ] = 1.607, and F kc∞ = F k[∞ ] = 3.322. The curves θmc [Bi] and F kc [Bi] for the sphere are qualitatively similar to the corresponding curves for the infinite slab and the infinite cylinder. We shall now compare both curves for the three canonical shapes. 4.9.6 Critical State versus Biot Number for the Canonical Shapes Figure 4.36 plots the critical F-K number versus Biot number for the three canonical shapes. The three curves are qualitatively similar to each other, and differ by what appear to be a constant vertical scale factor. Each curve is within about 5% of its asymptotic limiting value at Bi = 30. We noted in Sect. 4.9.2 that the limit Bi → 0 corresponds to the Semenov (or more precisely, the SFK) approximation. Figure 4.36 shows that this limit also corresponds to F kc → 0, for each of the three shapes. Why should the condition F kc [0] = 0 correspond to the Semenov limit? The answer is that
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3.0
Fkc
2.5 2 (infinite cylinder)
2.0 1.5
0.8785 (infinite slab)
1.0 0.5 0.0 0
5
10
15 Bi
20
25
30
Fig. 4.36. Critical F-K number versus Biot number: comparison of the three canonical shapes 1.8 1.607 (sphere) 1.6 ln[4] (infinite cylinder)
θmc
1.4
1.187 (infinite slab)
1.2
1.0 0
2
4
6
8
10
Bi
Fig. 4.37. Critical F-K temperature versus Biot number: comparison of the three canonical shapes
Semenov’s uniform temperature assumption corresponds to the limit of infinite thermal conductivity k. Because k appears in the denominator of the equations defining both Bi (see (4.161)) and F k (see (4.96)), infinite k corresponds to Bi = F k = 0. This is true irrespective of the body shape. Figure 4.37 plots the critical F-K temperature curves for the three canonical shapes. As in Fig. 4.36, the three curves are qualitatively similar to each other. If first subtracted by their minimum value of unity, the curves differ by what appear to be a constant vertical scale factor. As noted in Sect. 4.9.3 in connection with the infinite slab, the temperature curves converge to their F-K limiting values much faster than does the F-K number. Each curve is
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within about a percent of its limiting value at Bi = 10. The Semenov limit, Bi → 0, corresponds to θmc = 1 for all three curves. Recall that this property was shown in Sect. 4.6.1 in connection with SFK theory. 4.9.7 Quasi-Self-Similar Scaled Criticality Curves vs. Biot Number We observed in Sect. 4.9.6 that the curves F kc [Bi] and θmc [Bi] have similar functional forms for each of the three canonical shapes. In this section we put that perception to a quantitative test. Figure 4.38 plots each of the three F kc curves normalized by their respective values of F kc∞ . The three curves overlay almost exactly. The slightly discrepant scaled curves are reversed in order relative to Fig. 4.36. That is, the slab is the highest curve and the sphere is the lowest. Figure 4.39 plots each of the three θmc curves subtracted by one, normalized by their respective F-K limiting values θmc∞ subtracted by one. In this case the overlay is not nearly so tight as in Fig. 4.38. Just as for Fig. 4.38, the scaled curves are reversed in order relative to Fig. 4.37. That is, the slab is the highest curve and the sphere is the lowest. The overlay of the three curves in Fig. 4.38 is tight enough that it is quite reasonable to fit the ensemble with a single analytic curve. Such a curve would allow one to perform a simple correction for finite Biot number. Of course, we have not demonstrated that this correction curve is valid for all body shapes; however, the fact that the scaled curves are nearly identical for the three very different canonical shapes is an encouraging sign. Figure 4.40 plots selected discrete values of the three curves of Fig. 4.38 as “data points.” The scatter in the points indicates the small discrepancy between the three curves. The superimposed curve is a nonlinear least-squares fit of the equation 1.0
Fkc / Fkc∞
0.8 0.6
High curve: infinite slab Middle curve: infinite cylinder Low curve: sphere
0.4 0.2 0.0
0
10
20 Bi
30
40
Fig. 4.38. Critical F-K number normalized by the respective value of F kc∞ , plotted versus Biot number for the three canonical shapes
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(θmc–1)/(θmc•–1)
1.0 0.8
Infinite Slab
0.6
Sphere
Infinite Cylinder
0.4 0.2 0.0
0
2
4
6
8
10
12
14
Bi
Fig. 4.39. Critical F-K temperature minus one, normalized by the respective F-K limiting value minus one, plotted versus Biot number for the three canonical shapes 1.0 0.8
ϒ
0.6 0.4 0.2 0.0
0
5
10
15 Bi
20
25
30
Fig. 4.40. Scaled critical F-K number versus Biot number: comparison of the three canonical forms (points) and an analytic fit of (4.190) (solid curve)
F kc Bib ≡Υ = , F kc∞ a + Bib
(4.190)
which results in the parameters a = 2.411 and b = 1.059. This fit can be considered exact to within the scatter of the three curves. The procedure for correcting F kc for finite Bi is as follows. First, one determines F kc∞ for the considered shape. This may be determined by (4.158) for near-unity aspect ratio shapes, and corrected for elongated shapes in selected cases by the relaxation law given in (4.160). Multiplying F kc∞ by the correction curve Υ [Bi] gives the Bi-corrected value.
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4.10 Convergence of the Critical State to the F-K Limit Historically, the F-K limit was invoked to analytically solve the reactive heat equation in simple geometries. This allowed selected critical conditions to be determined, and other aspects of classical thermal explosion theory (which we have not considered here) to be simplified. Although we have shown that the approximation is theoretically justified in many cases, we have not yet “closed the loop” to show the extent to which the critical state, as characterized by F kc , errs as a function of T˜s . The full Arrhenius problem was first performed by J. Enig [16] for the three canonical shapes, and later by J. Parks [48] for the canonical shapes plus the cube and the unity-aspect-ratio right circular cylinder.17 We shall repeat this exercise for the canonical shapes with the benefit of modern computing tools. 4.10.1 Canonical Shape Criticality for the Full Arrhenius Rate Following the previous authors, we shall numerically compute Fk gc as a function of T˜gs for full Arrhenius heat addition, and observe the convergence to the F-K limit as T˜gs → 0. Here, we have added the subscript g to indicate that F kg and T˜gs apply to the general Arrhenius problem rather than the F-K limit problem. This distinction will become necessary as we compare the two cases. In order to make a comparison to F-K theory, which is formulated in terms of the F-K temperature θ, it is convenient (following Parks) to consider the full Arrhenius heat equation expressed in terms of θ. Thus, we shall consider the steady limit of (4.97), which is & ' θ (4.191) ∇2 θ = −F kg exp , θ [0] = 0, θ[1] = 0. 1 + T˜gs θ Once one makes the leap to numerical solutions – as one must do to compute all non-canonical shapes even in the F-K limit – it becomes as easy to solve the full Arrhenius problem as the F-K problem. In either case, numerical solutions (in contrast to the analytic solutions considered so far) are slightly complicated by the “mixed” boundary conditions. As we have noted, the heat equation reduces from a PDE to an ODE for the three perfectly symmetric canonical shapes. In these cases the general boundary value problem reduces to one condition on the temperature profile at each end of the domain. Such problems (in which the boundary conditions are specified at two locations) may be solved by the shooting method. In the context of the present problem, one assumes a trial value of θ[0] = θm with θ [0] = 0, and integrates (i.e., “shoots”) out to the edge. 17
Enig used an IBM 650 as he did in [14] (see footnote 15). Parks started with an Electronic Associates Inc. PACE analog computer, and later moved on to an IBM 704 digital computer (the first mass-produced computer with floating point arithmetic hardware).
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Compliance with the edge boundary condition is checked. The problem is repeated, iterating on θm , until the correct condition θ[1] = 0 is met. This is the basic method used by Parks and the method used here.18 For some conditions Parks found more than one solution, and, following O. Rice [49], speculated about the solution possibilities. We hope that Figs. 4.10, 4.15, 4.16, and 4.19, together with the companion explanations about the solution space, have sufficiently clarified this issue. The only point to be added is that, because the shooting method tests compliance with a boundary condition, it can access multiple solutions without regard to whether they are stable or unstable. Consequently, spurious solutions are accessed by this technique. For the shooting method described above: if one systematically tracks the trial solutions by constructing a curve θ[1] versus θ[0] ≡ θm , the result has the appearance of an inverted parabola. For subcritical values of F kg , the peak of this curve is positive such that there are two zero crossings. These correspond to the two intersections that a slightly subcritical loss line makes with the production curve – one stable and the other unstable (see Figs. 4.10 and 4.19). As F kg increases, the parabola shifts downward until, at the critical condition, the peak of the parabola just touches the origin. The value of F kg for which that occurs is F kgc . Figure 4.41 plots the function F kgc [T˜gsc ] for the three canonical shapes. For each of the three shapes, F kgc (depicted by the solid lines) increases linearly with T˜gsc . The F-K limiting values F kc are the level dashed lines. The 4 Sphere
Fkgc
3
2
Infinite Cylinder
1 Infinite Slab 0 0.00
0.02
0.04
~ Tgsc
0.06
0.08
0.10
Fig. 4.41. Divergence of the critical state from the infinite activation energy (F-K) limit for the three canonical shapes 18
Enig’s [16] solution technique provides an interesting comparison. He used the dimensionless temperature T˜ rather than θ, and absorbed F k into the dimensionless length scale. He then solved the differential equation by a perturbation technique.
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1.12 1.10 e nit
Fkgc / Fkc
1.08
lin Cy
r de
i Inf
1.06 i Inf
1.04
lab eS nit
e her
Sp
1.02 1.00 0.00
0.02
0.04
0.06
0.08
0.10
~ Tgsc
Fig. 4.42. Scaled divergence of the critical state from the infinite activation energy (F-K) limit for the three canonical forms. Each line F kgc [T˜gsc ] is scaled by its respective F-K limiting value, F kc
deviation of the full Arrhenius problem from the F-K limit problem is indicated by the divergence of the solid lines from the dashed lines. The deviation vanishes at zero temperature and increases linearly with temperature. 4.10.2 Quasi-Self-Similar Scaled Criticality Curves vs. Temperature Figure 4.42 plots the same lines as Fig. 4.41, but with each line normalized by its respective F-K limiting value, F kc . We observe that the three curves nearly overlay, the difference at any point being an order of magnitude less than the mean deviation from the unity reference value. Just as for the Biot number scaling in Sect. 4.9.7, the ordering of the scaled curves is reversed relative to the unscaled curves. That is, the scaled slab curve is the highest and the scaled spherical curve is the lowest. Also, the fact that the scaled behavior nearly overlays for the three very different canonical shapes suggests that the divergence from the F-K limit depends only weakly on the charge shape. Each curve in Fig. 4.42 is of the form F kgc = F kc (1 + γ T˜gsc ),
(4.192)
where γ is the slope of the scaled lines in Fig. 4.42. These values are 1.210, 1.168, and 1.142 for the slab, cylinder, and sphere, respectively. The average value γ¯ = 1.174 is sensible for general estimation. 4.10.3 Critical State Correction for Finite Activation Energy Unfortunately (4.192) is not in a form that tells us what we would mostly like to know, which is: how much error arises in the critical temperature as a result
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in
1.010 Ω
fin In
1.005
b
la te S
ni
Infi
ite
l Cy
r de
re he Sp
1.000 0.00
0.02
0.04
~ Tsc
0.06
0.08
0.10
Fig. 4.43. Correction functions (4.193) by which F-K limit solutions are converted to full Arrhenius solutions for the canonical shapes
of the F-K approximation? Fortunately, it is a fairly simple matter to recast the above results to express this attribute. What one would like is to rearrange (4.192) so as to provide a correction factor function, Ω[T˜sc ] = T˜gsc /T˜sc . We begin by canceling all the terms common to F kgc and F kc (see (4.96)) in (4.192). Further replacing T˜gsc by T˜sc Ω, (4.192) can be written f[T˜sc Ω] = f[T˜sc ](1 + γ T˜sc Ω).
(4.193)
The transcendental (4.193) cannot be further simplified; however, for a given γ-value it is straightforward to solve the equation numerically. The numerical solution to (4.193) is plotted in Fig. 4.43 for the three canonical shapes. The correction is about 1.5% at the very most, and will be significantly less than 1% in many cases. This level of error is generally less than that due to other approximations that arise in the classical theory. Specifically, the error is almost certainly less than the accuracy to which we can know the boundary condition in any real scenario. We therefore conclude that the F-K approximation is close enough for all practical purposes – for the canonical shapes computed in this example at least, and likely (as evidenced by the insensitivity of the result to charge shape in those three cases) for other shapes as well. Although the correction is a small one, we nevertheless provide an analytic fit for a typical case, as characterized by γ¯ = 1.174. Figure 4.44 plots selected discrete values of the three curves of Fig. 4.43 as “data points.” The scatter in the points indicates the small discrepancy between the three curves. The superimposed curve is a nonlinear least-squares fit to the equation σ , Ω = 1 + μ T˜sc
(4.194)
which results in the parameters μ = 2.032 and σ = 2.157. This fit can be considered exact to within the scatter of the three curves.
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1.014 1.012 1.010 Ω
1.008 1.006 1.004 1.002 1.000 0.00
0.02
0.04
0.06
0.08
0.10
~ Tsc
Fig. 4.44. Analytic fit to the three canonical correction functions (4.194)
To correct an F-K critical surroundings temperature T˜sc to an Arrhenius critical surroundings temperature T˜gsc , one simply multiplies T˜sc by the correction function Ω[T˜sc ]. Finally, the fractional error, , is given by =Ω−1=
Tsgc − Tsc σ = μ T˜sc . Tsc
(4.195)
4.11 Concluding Remarks There is no lack of overview-type treatments of Semenov and F-K theory in the literature (e.g., [5, 7, 20, 21, 39, 52, 57, 59]). The world did not need yet another one, just to make a book entitled “non-shock initiation of explosives” seem complete. Because classical thermal explosion theory involves only one main equation and a few ancillary equations, it might seem that the existing basic treatments, together with the few more comprehensive review articles [25, 44], would suffice. If that were the case, nothing more than references to previous publications would be necessary. But as we have seen, the criticality question alone has a surprising number of interesting and often non-intuitive attributes. As such, it has seemed useful to elaborate on the various elements in a way that could not have been done until recently, with the benefit of modern computing tools. In performing the supporting computations for this chapter, it has become clear that the many aspects of even classical thermal explosion theory have not yet been exhausted with respect to useful inquiry. Specifically, the availability of symbolic manipulation combined with advanced numerical computing and plotting capabilities (as brought together in Mathematica, Matlab, Maple, etc.) creates new opportunities in “transitional areas” – those that are analytically tractable only via special functions and implicit equations – which were previously impractical to deal with.
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Although this examination of typically glossed-over details serves a useful purpose, it has not been without opportunity cost. For in pursuing a detailed path, many topics within even the limited scope of classical thermal explosion theory had to be omitted so as to make room for numerous other nonshock initiation subjects. Had this not been the case, several additional topics would have naturally and efficiently flowed from the foundational discussion presented here. These include: 1. 2. 3. 4. 5.
Theory of supercriticality (the induction or time-to-explosion problem) Ignition dynamics and statistics Hot spot criticality and hot spot interaction Supercritical hot spot propagation and thermal waves Cookoff ignition and violence mapping
The general nature and importance of the first four topics has been adequately motivated by our discussion; however, the fifth topic, “cookoff ignition and violence mapping,” requires some elaboration. Here the idea is to generate, for a given HE system subjected to a particular class of thermal insult, an engineering chart that plots both the time to explosion and a chosen violence metric, as a function of key heating parameters. This type of chart, which resembles a thermodynamic chart in both form and function, would allow engineers and safety personnel to estimate the two fundamental safety parameters at a glance. The preignition component of the map (the time to explosion) would be computed from the best available kinetics models (such as Henson et al.’s HMX model described in Chap. 3) and checked against experiments. The post-ignition component of the map (the violence metric) would be guided by elements of the preignition problem, but would primarily rely on direct experimental measurements. Although the idea is only a few years old at the time of this writing, cookoff ignition and violence mapping technique shows substantial promise of becoming a practically useful tool [30]. In addition, some of the included topics should really be investigated more thoroughly. For example, although progress was made toward F-K shape correlations for elongated bodies, no general solution has yet been found. Short of a general solution, the most important uncharacterized shape – the rectangular parallelpiped – should be more fully examined. Other sub-problems in need of further development include the hot spot reaction problem and the depletion-induced explosion limit problem.
References 1. The fire below: Spontaneous combustion in coal. United States Department of Energy Environment, Safety, & Health Bulletin, No. EH-93-4 (1993) 2. Pistachio nuts. German Transport Information Service bulletin, www.tis.gdv.de (2007)
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3. Aslam, T.D., Bdzil, J.B., Stewart, D.S.: Level set methods applied to modeling detonation shock dynamics. J. Comp. Phys. 126(2), 390–409 (1996) 4. Baer, M.R., Hall, C.A.: Isentropic compression studies of the mesoscale response of energetic composites and constituents. In: Proc. 13th Symp. (Int.) on Detonation (2006) 5. Boggs, T.L., Derr, R.L.: Hazard studies for solid propellant rocket motors. Tech. rep., AGARDograph 316, NATO, Brussels (1990) 6. Borisov, A.A., Luca, L.D., Merzhanov, A.: Self-Propagating High-Temperature Synthesis. Taylor & Francis, New York (2002). Translated by Y. B. Scheck 7. Bowden, F.P., Yoffe, A.D.: Fast Reaction in Solids. Butterworths, London (1958) 8. Brill, T., Gongwer, P.E., Williams, G.K.: Thermal decomposition of energetic materials. 66. kinetic compensation effects in HMX, RDX, and NTO. J. Phys. Chem. 98(47), 12,242–12,247 (1994) 9. Buckmaster, J.D., Ludford, G.S.S.: Theory of Laminar Flames. Cambridge University Press, Cambridge (1982) 10. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Oxford Science Publications, Oxford (1986) 11. Chambr´e, P.L.: On the solution of the Poisson-Boltzmann equation with application to the theory of thermal explosions. J. Chem. Phys. 20(11), 1795–1797 (1952) 12. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comp. Math. 5, 329–359 (1996) 13. Dickson, P.M., Asay, B.W., Henson, B.F., Smilowitz, L.B.: Thermal cook-off response of confined PBX 9501. Proc. R. Soc. Lond. A 460, 3447–3455 (2004) 14. Enig, J.W.: The solution of the steady-state heat conduction equation with chemical reaction for the hollow cylinder. Tech. Rep., NAVORD Report 4267 (1956) 15. Enig, J.W.: Critical parameters in the Poisson-Boltzmann equation of steadystate thermal explosion theory. Combustion and Flame 10, 197–198 (1966) 16. Enig, J.W., Shanks, D., Southworth, R.W.: The numerical solution of the heat conduction equation occurring in the theory of thermal explosions. Tech. Rep., NAVORD Report 4377 (1956) 17. Fedoroff, B.T., Aaronson, H.A., Sheffield, O.E., Reese, E.F., Clift, G.D. (eds.): Encyclopedia of Explosives and Related Items, vol. 1. Picatinny Arsenal, Dover, New Jersey (1960) 18. Fedoroff, B.T., Sheffield, O.E. (eds.): Encyclopedia of Explosives and Related Items, vol. 6. Picatinny Arsenal, Dover, New Jersey (1974) 19. Fedoroff, B.T., Sheffield, O.E. (eds.): Encyclopedia of Explosives and Related Items, vol. 7. Picatinny Arsenal, Dover, New Jersey (1975) 20. Frank-Kamenetskii, D.A.: Diffusion and Heat Exchange in Chemical Kinetics. Princeton University Press, Princeton, New Jersey (1955). Translated by N. Thon 21. Frank-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press, New York (1969). Translator: J. P. Appleton 22. Gibbs, T.R., Popolato, A.: LASL Explosive Property Data. University of California Press, Berkeley (1980) 23. Gray, P., Harper, M.J.: Thermal explosions. Part 1: Induction periods and temperature changes before spontaneous ignition. Trans. Faraday Soc. 55, 581–590 (1959)
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24. Gray, P., Harper, M.J.: The thermal theory of induction periods and ignition delays. In: 7th Int. Symp. on Combustion (1959) 25. Gray, P., Lee, P.R.: Thermal explosion theory. In: Oxidation and Combustion Reviews, vol. 2, pp. 3–180. Elsevior, London (1967). Editor: C. F. H. Tipper 26. Grimm, J.A.: Racetrack Hurdles. Santa Fe New Mexican newspaper article, October 8, www.SantaFeNewMexican.com (2004) 27. Henson, B.F., Smilowitz, L., Asay, B.W., Sandstrom, M.M., Romero, J.J.: An ignition law for PBX 9501 from thermal explosion to detonation. In: Proc. 13th Symp. (Int.) on Detonation (2006) 28. Henson, B.F., Smilowitz, L., Romero, J.J., Sandstrom, M.M., Asay, B.W., Schwartz, C., Saunders, A., Merrill, F., Morris, C., Murray, M.M., McNeil, W.V., Marr-Lyon, M., Rightley, P.M.: Burn propagation in a PBX 9501 thermal explosion. In: Shock Compression in Condensed Matter–2007, pp. 825–828 (2007) 29. Hill, L.G.: Burning crack networks and combustion bootstrapping in cookoff explosions. In: Shock Compression in Condensed Matter–2005, pp. 531–534 (2005) 30. Hill, L.G.: On mapping the behavior of energetic material cookoff explosions. In: Proc. 23rd JANNAF Propulsion Hazards Subcommittee Meeting (2006) 31. Hill, L.G.: Critical temperature correlation for a near-unity aspect ratio charge of arbitrary size and shape. In: Shock Compression in Condensed Matter–2007, pp. 829–832 (2007) 32. Hill, L.G., Gustavsen, R.L.: On the characterization and mechanisms of shock initiation in heterogeneous explosives. In: 12th International Symposium on Detonation, pp. 975–984 (2002) 33. Hill, L.G., Zimmermann, B., Nichols, A.L. III: On the burn topology of hot-spotinitiated reactions. In: Shock Compression in Condensed Matter–2009 (2009) 34. Hobbs, M.L., Steyskal, M., Kaneshige, M.J.: Modeling RDX ignition. In: Proc. 24th JANNAF Propulsion Hazards Subcommittee Meeting (2008) 35. Janney, J.L., Rogers, R.N.: Predictive Models for Thermal Hazards. Presented at the 22nd Explosives Safety Seminar, Anaheim, CA. Los Alamos National Laboratory Report LA-UR-86-2925 (1986) 36. Kamenetskii, D.A.: Zhur. Fiz. Khim. 13, 738 (1939) 37. Kamenetskii, D.A.: Acta Physicochimica U.R.S.S. 10, 365 (1939) 38. Kapila, A.K.: Asymptotic Treatment of Chemically Reacting Systems. Pitman Publishing Inc., Marshfield, Massachusetts (1983) 39. Kaye, S.M. (ed.): Encyclopedia of Explosives and Related Items, vol. 9. Picatinny Arsenal, Dover, New Jersey (1980) 40. Konecni, S.: Heat transfer analysis of small PBX 9501 cylinders at LANL. In: Proc. 23rd JANNAF Propulsion Hazards Subcommittee Meeting (2006) 41. Mandelbrot, B.B.: How long is the coast of Britain?: Statistical self-similarity and fractional dimension. Science 156, 636–638 (1967) 42. Mandelbrot, B.B.: Fractal Geometry in Nature. W. H. Freeman, New York (1982) 43. McQuarrie, D.A.: Statistical Mechanics. University Science Books, Sausalito, CA (2000) 44. Merzhanov, A.G., Abramov, V.G.: Thermal explosion of explosives and propellants. a review. Propellants, Explosives, Pyrotechnics 6, 130–148 (1981) 45. Merzhanov, A.G., Abramov, V.G., Dubovitskii, F.I.: Dokl. Akad. Nauk SSSR 128, 889 (1959)
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46. Moore, J.W., Frost, A.A.: Kinetics and Mechanism. John Wiley and Sons, New York (1981) 47. Parks, J.R.: Criticality criteria for various configurations of a self-heating chemical as functions of activation energy and temperature of assembly. J. Chem. Phys. 34(1), 46–50 (1960) 48. Parks, J.R.: Criticality criteria for various configurations of a self-heating chemical as functions of activation energy and temperature of assembly. J. Chem. Phys. 34(1), 46–50 (1961) 49. Rice, O.K.: The role of heat conduction in thermal gaseous explosions. J. Chem. Phys. 8, 727–733 (1940) 50. Rogers, R.N.: Thermochemistry of explosives. Thermochimica Acta 11, 131–139 (1975) 51. Semenov, N.N.: Z. Physik 20, 571 (1928) 52. Semenov, N.N.: Some Problems in Chemical Kinetics and Reactivity (Volumes 1 and 2). Princeton University Press, Princeton, NJ (1958). Translated by M. Boudart 53. Shouman, A.R., Donaldson, A.B., Tsao, H.Y.: Exact solution to the onedimensional stationary energy equation for a self-heating slab. Combustion and Flame 23, 17–28 (1974) 54. Thomas, P.H.: On the thermal conduction equation for self-heating materials with surface cooling. Trans. Faraday Soc. 54, 60–65 (1958) 55. Todes, O.M.: Zhur. Fiz. Khim. 4(78) (1933) 56. Trotman-Dickenson, A.F.: Gas Kinetics. Academic Press, New York (1955) 57. Williams, F.A.: Combustion Theory. Addison-Wesley Co., New York (1985) 58. Zerkle, D.K.: Phase segregation effects on the calculation of ODTX in HMX spheres. Combustion and Flame 117, 657–659 (1999) 59. Zukas, J.A., Walters, W.P.: Explosive Effects and Applications. Springer-Verlag, Heidelberg (1997)
5 Deflagration Phenomena in Energetic Materials: An Overview Scott I. Jackson
5.1 Introduction In this chapter, we review the basic behavior that occurs in deflagrating high explosives. Consideration of the deflagrating characteristics of high explosives is often neglected as they are primarily intended to detonate. However, explosives can exhibit a wide range of deflagrative behavior. Propagation rates can range from slow and steady propellant-like burns at millimeters per second to rapid gas-pressure-driven burns approaching detonation velocities, depending on the burning pressure, the explosive porosity, and the level of confinement. In the most extreme cases, the pressure waves generated by rapid burn rates can induce a detonation in the high explosive. Such a range of behavior is not surprising, as high explosives are closely related to propellants and, in many cases, composed of similar materials. Both are specialized types of energetic materials, a class of compounds or mixtures that contain a large amount of stored chemical energy. In practice, several subcategories of energetic materials exist, including propellants, pyrotechnics, low explosives, and high explosives, which serve to more explicitly define the characteristics of each energetic material. Propellants are optimized for gas generation through controlled deflagration in order to generate thrust. Pyrotechnics are designed to produce large amounts of gas, heat, light, or smoke through deflagration, depending on the application. Low explosives are typically mixtures of fuel and oxidizer that are intended to deflagrate. High explosives stand apart from other energetic materials, in that they are designed to have desirable detonation characteristics, although almost all energetic materials will detonate if given proper initiation conditions. Knowledge of the deflagrative characteristics of high explosives can be quite useful when considering the response of a given explosive system to a specific stimulus. Considerable concern is often devoted to certain issues: Take, for example, the crash of an aircraft or truck carrying high explosive ordinance in a populated area. Fuel tank failure and subsequent ignition submerges the B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 5, c Springer-Verlag Berlin Heidelberg 2010
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ordinance in a liquid pool fire. How much time do rescuers have to extricate any survivors and put out the fuel fire before the ordinance ignites? How violently will the ordinance react upon ignition? Another common worry is the presence of defects or cracks in rocket motors. For sufficiently large cracks, the deflagration that normally drives the rocket motor can enter the crack and increase the burning surface area of the propellant. Depending on the surface area of the crack, this additional release of product gases can increase the pressure of the rocket chamber, leading to improper operation or failure of the rocket motor casing. Defects below a critical size, however, will allow the rocket operation to proceed normally. Thus, proper understanding of the effect of a given defect can avoid undesirable consequences. In each of the above accident scenarios, understanding of the deflagrative characteristics of energetic materials is essential to properly identify the possible outcomes. Even with this knowledge, accurate prediction can often be difficult as the products and reactants in a deflagrating solid explosive span the solid, liquid, and gaseous states of matter. A significant amount of fluidstructure coupling and thermodynamic feedback occurs between each state, which cannot be neglected. Thus, accurate high-explosive deflagration models need to be able to resolve or account for the mechanical, thermodynamic, and chemical responses of each phase to accurately predict burn development. We review the basic phenomena associated with deflagration of high explosives and discuss how such a wide range of combustion propagation modes are possible. Our primary intent is to qualitatively present the deflagrative mechanics of high explosives for those new to the field. This will provide insight into the nonshock ignition mechanisms discussed in other chapters. Even though the term “explosive” is used almost exclusively, the principles discussed herein are applicable to most energetic materials, including propellants. The outline for this chapter is as follows. We discuss deflagrations driven by conductive heating, termed conductive burning. The reaction zone structure is presented and the dependence on burn rate and temperature are reviewed. The phenomenon of extinction is also introduced. In certain situations, the product gases from combustion can accelerate deflagrations by preheating unburned material. This phenomenon, referred to as convective burning, is also discussed. When possible, references are included to manuscripts with more detailed information. Shock-driven detonation phenomena are not discussed in detail, nor is basic deflagrative theory in a theoretical sense. Only phenomena specific to high explosive deflagration are discussed.
Nomenclature English Symbols A An
Area, Constant in reaction rate Andreev number
5 Deflagration Phenomena in Energetic Materials
a ag b C Cp Cs Dc dp Ea F h L Lc Le m ˙ m ˙ in mvoid m ˙ void Nu n P Pc Pr Q Qd Qloss Qpc Qtrans R r Re Sf T T0 Tf Tg Ti Tm Ts Tw T∗ t ug
Burn rate constant Geometrical parameter Temperature coefficient in burn rate law and Vielle’s Law Specific heat capacity Specific heat capacity of gas at constant pressure Specific heat capacity of solid explosive Characteristic grain size Characteristic pore or porosity diameter (or width) Activation energy Arbitrary function Heat transfer coefficient Length of pore or porosity Length of pore over which gas cooling occurs Lewis number Mass flow rate Mass flow rate into pore Mass in void Mass flow rate into void Nusselt number Pressure exponent or combustion index Pressure Critical pressure Prandtl number Heat Heat of decomposition Heat lost from gas to pore wall Heat of phase change Heat transferred from gas to pore wall Specific gas constant Regression rate or conductive burn rate of solid explosive Reynolds number Flame speed Temperature Initial temperature Final product temperature Gas temperature Ignition temperature Melt temperature Surface temperature Wall temperature Ignition temperature Time Velocity of gas phase
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Inflow velocity of gas into pore Width of slot or pore Distance into pore from open end
Greek Symbols α αg αs β δr κ λ λg λs s g ρ ρg ρs σp τ τg τr τs τ∗ ω˙
Thermal diffusivity Thermal diffusivity of gas Thermal diffusivity of solid Intermediate constant Reaction zone thickness Constant in Belyaev’s equation Thermal conductivity Thermal conductivity of gas Thermal conductivity of solid Lengthscale Lengthscale of solid explosive Lengthscale of gas Density Gas-phase density Sold-phase density Temperature sensitivity of burning rate at constant pressure Timescale for heating of pore wall Timescale of gas phase Chemical reaction timescale Solid phase timescale, Time to reach steady-state thermal profile Time to reach ignition temperature Reaction rate
Subscripts k P 1 2
Reaction rate exponent (limiting order of chemical reaction) Constant pressure State 1 State 2
Abbreviations AP DB HE NC NG
Ammonium perchlorate Double base High Explosive Nitrocelluose Nitroglycerin
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5.2 Combustion and Its Modes It is useful to define the different combustion modes, as well as combustion in general, before proceeding. Combustion is an exothermic process characterized by the chemical reaction of a set of atoms or molecules from a reactant state of high chemical potential energy to a product state with lower potential energy. Reactants are composed of either metastable molecules or of heterogeneous mixtures of fuel and oxidizer. This process can take many forms, including the propane–air deflagration in an outdoor grill, the detonation of high explosives, the smoldering of glowing coal embers, and even the cellular respiration process where carbohydrates or lipids are combined with oxygen to release the energy necessary to drive our metabolisms. From the above examples, it is clear that several combustion mechanisms are available to reduce the reactants to products. In practice, most combustible mixtures exist in a stable or metastable state where they will not react until a minimum amount of heat or energy, termed the activation energy, is input to generate the chain-branching radicals necessary for the combustion reactions to occur. This is convenient in that it allows for safe transport, handling, and mixing of unreacted mixtures. The exception is hypergolic propellants, which have low-enough activation energies to allow for spontaneous ignition as soon as a combustible mixture is created. For a useful combustible mixture, the activation energy is much less than the heat released by the exothermic reactions. In reacting systems, energy must be transported from the exothermic reaction zone to activate the unreacted mixture for combustion to propagate. In general, the primary mode of energy transport is either compressive shock heating or thermal diffusion through conductive processes, although convective and radiant processes can also contribute to varying degrees. During shock heating, adiabatic compression from the passage of shock waves through a reactant mixture delivers the activation energy and induces combustion in a reaction zone immediately behind the shock. The energy released in the reaction zone then supports the shock propagation into undisturbed reactants downstream. In practice, this process is called a detonation. By definition, it is a supersonic process, as a shock wave is necessary to drive the reaction. In condensed-phase combustion systems where the chemical reaction is driven by thermal diffusion, a portion of the heat released from the exothermic reaction zone diffuses into the unreacted mixture, heating it to the point of reaction. It is also possible for lower levels of heating to induce chemical reactions that exist purely or partially in the condensed phase through pyrolysis or decomposition. Solid fuels in a gaseous oxidizing environment can also sustain chemical reaction at the solid–gas interface in a process referred to as smoldering. This work deals solely with the mechanism of deflagration.
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5.3 The Conductive Deflagration In conductive burning, energy transport between the reacting material and the unreacted mixture is primarily due to heat conduction. Thus, the thermal diffusivity of the explosive, which is the ratio of the thermal conductivity to the heat capacity, is the dominant parameter in the reaction propagation. In this section, we present the basic flame structure of a conductive deflagration, followed by a review of the dependent parameters and basic theory. Flame quenching and scaling parameters are also discussed. When possible, specific high explosives are used as examples. 5.3.1 A Description of the Conductive Flame Structure For a steady conductive burn, the flame structure is as shown in Fig. 5.1. In general, several different zones are present in the reaction, which can subject the explosive material to multiple phase changes. The explosive begins the process in the solid phase at ambient temperature. Some of the heat released from the gas-phase chemical reaction zone is transmitted throughout the unreacted material, primarily through conduction, although radiation and convection may also play lesser roles. The region of solid explosive that is heated without reaction or phase change is called the preheat zone. In this zone, the explosive temperature steadily rises with increasing proximity to the reaction zone. Deflagrating solid explosives have reaction zone temperatures that are well above the melting and sublimation points of the constitutive materials. In general, reaction zone temperatures are typically around 3,000 K (for HMX and x
Exothermic Flame Zone Tf
Gas-Phase Reaction Zone Ts
Dark Zone Fizz Zone Melt Zone
Surface Reaction Zone Tm
Solid
Preheat Zone T0
T
Fig. 5.1. The reaction zone structure and corresponding temperature profile for a steady conductive burn in a heterogeneous energetic material. Specific zones are described in the text.
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Table 5.1. Approximate melting (M.P.) and decomposition (D.P.) or vaporization (V.P.) point temperatures for common high explosive compounds. Explosive
M.P.
D.P./V.P.
◦
ANFO 170 C/443 K 210◦ C/483 K 280–290◦ C/553–563 K HMX(β-phase) 246◦ C/519 K 280–290◦ C/553–563 K HMX(δ- or γ-phase) 282◦ C/555 K Lead Azide N/O 320◦ C/593 K Lead Styphnate N/O 311◦ C/584 K Mercury Fulminate N/O 165◦ C/438 K ◦ 50◦ C/323 K Nitroglycerin 13 C ◦ 204◦ C/477 K RDX 204 C/477 K ◦ 190◦ C/463 K PETN 141 C/414 K N/O TATB 450◦ C/723 K 160◦ C/433 K TATP 91◦ C/364 K 295◦ C/568 K TNT 81◦ C/354 K N/O denotes that phase change has not been observed.
RDX), while melting points are approximately 500 K [85, 86]. Vaporization or boiling points exist as well, although they are usually less clearly defined as many materials experience significant chemical decomposition near these temperatures. Examples of these values for some common explosives are listed in Table 5.1. All deflagrating solid explosives undergo a phase change to the gaseous state prior to reaching the high-temperature reaction zone. Intermediate phase transitions are also observed in some explosives. HMX, for example, undergoes a solid-to-solid phase transition from the β-phase to the δ-phase near 453 K. In many explosives, a liquid melt layer is also observed prior to boiling. Theoretically, the melt layer should exist for most explosives, but experimentally it is not always observed. Part of the problem may be that the melt layer grows vanishingly small when there is a large temperature gradient near the boiling point. It is also possible for bubbles to be present in the melted explosive in a region termed the fizz zone. This zone exists at the melt–gas layer interface to allow the liquid explosive components to both vaporize and decompose into the gas phase. In general, the melt layer and any fizz zone are collectively referred to as the surface reaction zone due to the occurrence of condensed-phase decomposition reactions in this region. Eventually, the explosive fully decomposes or gasifies and is considered to be in the gas-phase reaction zone. This zone can consist of up to two subzones. The first is an optically transparent layer that is referred to as the “dark zone” due to its lack of luminosity. When present, it exists immediately adjacent to the surface of the condensed (solid or melt) phase and allows gaseous reactants continue to undergo diffusive heating after vaporization. Mixing can also occur in this layer for heterogeneous explosives where the components are not premixed on the molecular scale.
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(a)
(b)
Fig. 5.2. Images from [47] of conductively burning (a) RDX-CMDB (compositemodified double base) and (b) RDX-composite propellant at 2.0 MPa. The dark zone can clearly be seen separating the condensed phase from the luminous flame zone.
The second subzone is an exothermic flame zone that begins immediately after the dark zone, when the reactants have reached a sufficient temperature and decomposed to the point that they can recombine to form products with a lower chemical potential energy. This region is easily located by its luminous nature, which is primarily due to thermal radiation and chemiluminescence. A portion of the heat generated in this region is conducted back into the unreacted mixture to drive the deflagration. Most of the heat, however, is convected away from the reaction zone by the hot product gases, which can also contain solid particulates. In the convective and erosive burning modes, a portion of the convected heat can be passed on to the unreacted material, accelerating the burn rate. These mechanisms are discussed in the following sections. Photographs of conductively burning high explosives are shown in Fig. 5.2. 5.3.2 Factors Affecting the Burn Rate From the above description, it should be clear that a myriad of parameters control the explosive burn rate. As stated, the primary mechanism in propagation of a conductive burn is the diffusion of heat from the reaction zone to the unburned material. The rate of this propagation can be influenced in many ways. Of chief importance in the process is thermal diffusivity
5 Deflagration Phenomena in Energetic Materials
α=
λ , ρC
253
(5.1)
which is a function of the thermal conductivity λ, the material density ρ, and the specific heat capacity C. Note that (5.1) applies throughout all three material phases, although the parameter values will change. Obviously, the heat release or reaction zone temperature will also play a role in the energy transfer as it creates the necessary gradient for conduction to drive heat back to the lower temperature reactants in the preheat zone. These material parameters are specific to each explosive and can vary significantly with temperature. The burn rate is also strongly affected by external conditions, including both the ambient temperature of the explosive and the gas-phase pressure. 5.3.3 Temperature Dependence Increasing the initial temperature of the bulk explosive increases the rate of reaction, as less conductive heating from the reaction zone is required to preheat the reactants to ignition. This is, of course, assuming that the initial temperature of the explosive is low enough not to induce cookoff reactions as discussed in Chap. 7. Unlike the effect of pressure, which is discussed below, the temperature dependence is more difficult to directly back out of theory and is usually measured experimentally by recording the regression rate as a function of initial explosive temperature. This data is then reduced into a coefficient σp termed the temperature sensitivity of burning rate at constant pressure ∂ ln r 1 ∂r = , (5.2) σp = ∂T p r ∂T p which gives the percent change in burn rate per degree change of explosive temperature for a constant pressure. The effect of σp on rocket motor performance is discussed in [47, 48, 80]. In general, values of σp are well tabulated for propellants as they are critical for predicting rocket motor performance, but there is much less data available for explosives. For the typical temperature excursions of 50–100 K of ambient conditions, the explosive regression rate can vary by up to 30% [80]. Elevated burn rates are also observed in geometries where hot product gases pass over a conductively burning surface at high velocities and provide additional heat transfer from convective processes. When the flow of gas is limited to the surface of the explosive, this process is referred to as erosive burning. Additional discussion on this phenomenon is contained in [48, 74, 80]. 5.3.4 Pressure Dependence The conductive burn rate exhibits a strong dependence on the gas-phase pressure. Although varying the gas-phase pressure does little to affect the condensed matter, it does increase the density of the gas phase, which is where
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Fig. 5.3. Images from [47] illustrating the influence of pressure on the reaction zone spacing of conductively burning double-base propellant at (a) 1.0 MPa, (b) 2.0 MPa, and (c) 3.0 MPa.
the primary chemical reactions occur. This serves to cause more frequent molecular collisions and to increase the reaction rate. The elevated reaction rate effectively compresses the width of the gaseous reaction zone (Fig. 5.3), moving it closer to the condensed phase and increasing the rate of heat transfer to the unreacted material. The net effect is to heat the explosive to a gas more rapidly and increase the rate at which the solid explosive is consumed (regression rate). The dependence of the regression rate on the pressure is a topic of significant interest to rocket motor designers and those in the field of internal ballistics. As such, a large amount of data on regression rate versus pressure has been collected for different propellents and explosives. Over limited ranges of pressure, the regression rate has experimentally been found to be well represented by an empirical expression r = a + bP n ,
(5.3)
where a, b, and n are material-specific constants and P is the pressure of the gas phase of the explosive. The pressure exponent n is often referred to as the combustion index. For most high explosives, n is between 0.7 and 1.00, while it is closer to 0.5 for most propellants. It is imperative that propellants have combustion index values that are less than unity in their range of operation, as this ensures stable rocket motor operation. When the pressure exponent exceeds unity, pressure deviations can result in unstable operation and drive the chamber pressure to destructively high values [47].
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For lower-pressure work, (5.3) is often approximated by r = bP n ,
(5.4)
termed Vielle’s Law or Saint Robert’s burn rate equation. In this equation, b is often referred to as the temperature coefficient. This expression is appropriate for rocket motor chamber pressures, which are typically below 100–200 bar. A second approximation, referred to as Muraour’s Law, is common when pressures exceed 1 kbar, r = a + bP, (5.5) as often occurs in gun and artillery munition chambers. It is best to fit the above equations to data obtained in the range of expected operation. Measurements of the conductive burn rate of high explosive materials are given in [57, 60, 61, 78, 79, 84, 85]. Pressure Dependence via Thermal Theory It is possible to study the pressure dependence on the burn rate in more detail using a simplified analysis based on the thermal, laminar-flame theory of Mallard and Le Chˆ atelier [62], who first formally identified that the transport of heat from the reaction zone to the preheat zone was responsible for propagation of the deflagration. Their theory was originally intended for gaseous mixtures, but can also be applied to condensed phase materials. Consider the case of a steady conductive burn in a high explosive. As discussed in Sect. 5.3.1, the heat release in the reaction zone is conducted downstream into the preheat zone where it raises the explosive temperature to the point of decomposition and vaporization. The theory of Mallard and Le Chˆ atelier separates the burning process into two separate zones that steadily travel with the deflagration wavefront as shown in Fig. 5.4. The first zone is a preheat zone, where material enters at the ambient temperature T0 and emerges at an elevated ignition temperature Ti . Originally, Ti was intended to represent the ignition temperature, prior to which no reaction occurred and after which reaction immediately started. Subsequent work has revealed Tf
Melt Layer
r
T0
Gas Preheat Zone
Ti
Reaction Zone
δr
x
Fig. 5.4. A schematic of the flame based on the thermal theory of [62].
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that ignition criteria depend on other parameters besides the temperature, including the heating history, but for the purposes of this model, we can assume that Ti represents the temperature at the boundary between the preheat and reaction zones where significant exothermic chemical reactions begin to occur. Upon exiting the reaction zone of thickness δr , the gas temperature is assumed to be at its final temperature Tf . The energy balance in the preheat zone consists of energy inputs from the inflow of ambient temperature material into the control volume as well as heat flux from the adjacent reaction zone. Approximation of a linear temperature profile allows the heat flux across the interface boundary to be represented with Fourier’s Law as λ(Tf − Ti )A/δr , yielding mC ˙ s T0 + λ
(Tf − Ti ) A + mQ ˙ d = mC ˙ p Ti + mQ ˙ pc, δr
(5.6)
where energy sources to the control volume are on the left and sinks are on the right. Parameters Cs and Cp are the specific heats of the solid explosive and the gaseous reactants, respectively, and A is the cross-sectional area. The heat source Qd represents heat addition from decomposition reactions in the preheat zone and the heat sink Qpc is the energy absorbed by the phase change of the reactants from solid to gas. The term m ˙ is the mass flow rate, which is steady regardless of the material phase. Thus, (5.7) m ˙ = ρs rA = ρg ug A, where ρs and ρg are the densities of the condensed and gaseous explosive leaving the preheat zone. Inflow and outflow velocities to the control volume are r and ug , respectively. The term r is also referred to as the regression rate of the solid explosive. Solving for this term, r=
λ ρs
Tf − Ti Cp Ti − Cs T0 + Qpg − Qd
1 δr
(5.8)
shows the regression rate dependence on temperature, but not pressure. This is because the pressure dependence is hidden in the reaction zone thickness term δr , which is not known a priori. To explore the pressure dependence, a scaling relationship for the reaction zone thickness must be established. Assuming that the rate limiting reactions occur in the gas phase, we can follow the logic of Kuo [53], where the reaction zone thickness is the product of the gas inflow velocity and the time for chemical reaction τr . Additionally, the chemical reaction time scales with the inverse of the reaction rate ω˙ δr = ug τr ∼
ug . ω˙
(5.9)
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The reaction rate can be represented by ω˙ = A e−Ea /RT ω k P k−1 ,
(5.10)
where k is the order of the limiting chemical reaction. Using the ideal gas law along with (5.7), (5.9), and (5.10) to eliminate δr in (5.8) yields λ r ∼ 2 ρc RTi 2
Tf − Ti Cp Ti − Cs T0 + Qpg − Qd
A e−Ea /RTf ω k P k ,
(5.11)
where R is the gas constant for the reactants at the preheat-reaction zone interface. From this expression, the relatively weak dependence of initial temperature T0 on burn rate can be seen. The strong pressure dependence is also clear. In fact, for pressure the major dependence is k
r ∝ P 2,
(5.12)
where k ≈ 2n in Vielle’s Law. The implication of this scaling relationship is that the burn rate of all explosives is pressure dependent, unless their combustion index is zero, which would imply a 0th order limiting reaction. Explosives with pressure indexes near unity are thought to have second-order limiting gas phase reactions, while explosives with pressure indexes near 0.5 have firstorder limiting reactions. The regression rates of explosives with higher pressure exponents are more sensitive to changes in pressure. HMX-based explosives have pressure exponents very close to unity, and so the burn rate scales almost directly with the pressure. This can result in a rapid acceleration of regression rate when the product gases are confined. Note that the pressure and temperature effects discussed above change only the rate of energy release from the explosive. The total amount of chemical potential energy that can be released by combustion does not change. 5.3.5 Quenching and Extinction In a deflagrating high explosive, changes in pressure not only affect the regression rate, but also have the potential to extinguish the reaction zone in severe circumstances. The key issue is a mismatch between the timescales in the solid- and the gas-phase zones. For a purely conductive burn, the length of the solid and the gaseous preheat zones will scale with a few key parameters that are shown in Table 5.2. From dimensional analysis, the timescale associated with the solid preheat zone varies with λs αs = 2 (5.13) τs = 2 ρs Cs r r
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Table 5.2. Relevant scaling parameters and orders of magnitude associated with conductive deflagrations. Variable λ C ρ α r or ug
Dimensions kg m s3 K m2 s2 K kg m3 m2 s m s
Solid
Gas
0.1
0.1
1,000
1,000
1,000
1
1 × 10−7
1×10−4
0.001
1
and the length scaling is s =
αs λs = . ρs Cs r r
(5.14)
Meanwhile, for the gas phase preheat zone, the corresponding parameters are τg =
λg Cs ρg αg τs = 2 λs Cp ρs ug
(5.15)
αg λg = , ρs Cp r ug
(5.16)
and g =
where (5.7) has been used to express (5.15) and (5.16) in terms of the condensed phase flow rate and density, as the gas phase parameters can be more difficult to measure. Using (5.13) and (5.16) with values from Table 5.2 estimates that τs = 100 ms, while τg = 0.1 ms. Thus, the gas-phase zone is able to adjust to any changes in the burn rate much more rapidly than the solid zone. The temperature profile for two different pressures (and burn rates) is shown in Fig. 5.5. At higher pressures, the accelerated burn rates have a high heat flux incident on the solid–gas interface due to the thinner gas-phase zone. This is predicted by Fourier’s law of heat conduction dQ = −λA∇T, dt
(5.17)
as the decreasing reaction zone thickness will result in a higher temperature gradient. The high heat flux delivers energy to the condensed phase more rapidly and creates a steeper temperature gradient in the solid, which is reinforced by the accelerated material flow rate. Decreasing the pressure lowers the burn rate and increases the length of the gas phase zone as shown. This results in a lower heat flux to the solid and a more gradual internal temperature profile, as the slower material flow and energy delivery rates allow for more time for the heat to penetrate the solid.
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Tf P2 > P1 Ts2
Additional Energy Stored by P1 Profile T0
Temperature Profile for P1 Ts1
Temperature Profile for P2
Gas Phase
Fig. 5.5. The effect of different values of heat flux on the solid explosive temperature profile vs. depth. Temperature Ts is the surface temperature of the explosive at the solid–gas interface. Light shading denotes excess energy stored by the P1 curve (solid line), while dark shading is the energy deficit compared to the P2 profile (dashed line).
By comparing the two temperature profiles, it can be seen that the solid temperature profile associated with the lower pressure has additional energy associated with it (difference between light and dark shaded regions) compared to the high pressure profile. Thus, during a pressure increase, the solid material is partially preheated, which results in a temporary overdrive of the burn rate. Pressure drops require the solid to absorb additional energy to adapt to the slower burn rate, resulting in a lower vaporization rate compared to the steady case. Thus, gradual pressure decreases result in marginalized burn rates during this absorption period. For rapid pressure drops, the sudden decreased flow of reactants to the gas phase can extinguish the flame. For some systems, oscillatory behavior may also be observed due to timescale mismatch. 5.3.6 Burning Progression The conductive deflagration mechanism as presented has implicitly assumed that the explosive has a very low level of connected porosity and is sufficiently well formed, such that product gases do not penetrate the preheat zone. During these conditions, the deflagration remains at the surface of the high explosive, and burning proceeds layer-by-layer, so that the solid material regresses normally to the exposed surface in a uniform fashion. This effect is known as Piobert’s Law, which states that all exposed burning surfaces receive heat at an identical rate and thus should burn at the same rate. Typically, this is a reasonable assumption in most pristine explosives. Many explosives are melt cast, such as Comp B, and generally have almost no porosity. Plastic-Bonded Explosives (PBXs), which are composed of binderencapsulated high explosive crystals that are pressed together under temperature, inherently contain some level of porosity, which is usually less than 1%. However, this porosity is generally assumed to exist in an unconnected fashion
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as individual closed voids. Other small cracks or defects may be present in the material as well, due to issues associated with a material’s manufacture, storage, and handling history. A deflagration encountering a small defect will not be able to enter it when the length of the gas-phase reaction zone is larger than the defect width. Presumably, heat losses from the additional wall surface area associated with the defect hinder the reaction. Furthermore, in explosives with melt layers, the liquid can act to fill in or cover any defects, effectively hiding them from the flame. At certain conditions, these effects fail. The gas-phase lengthscale is inversely proportional to the pressure (P −n ) and, at elevated burning pressures, any liquid layer can grow vanishingly small. Thus, for high burning pressures or large defects, the reaction will enter the porosity. For large enough voids, the burning will then proceed normally as per Piobert’s Law and the end result will be an increase in both the rate of regression and of product-gas generation due to the additional burning surface area (which is still an issue of concern to rocket motor designers). In situations where the reaction enters a high-aspect-ratio crack or dense porosity, the product gases may be produced in the crack faster than they are able to escape. This effect can locally increase the pressure and the burn rate to very large values, even causing additional damage to the high explosive. Criteria exist to predict when a deflagration will penetrate an explosive surface containing voids of a given size, resulting in augmentation of the burn rate. When connected porosity is present, sustained burning can exist in this manner and generally occurs at a more rapid rate than simple conductive burning. This mode of deflagration is referred to as convective burning and is discussed below.
5.4 Convective Burning In early work with porous beds composed of explosive material, researchers observed a discontinuous jump in the propagation rate of a deflagration through the bed [19, 21, 81]. Burning rates were recorded that were orders of magnitude faster than the expected conductive rates, propagating on the order of 100 cm s−1 instead of 1 cm s−1 . By filling paper tubes with HMX powders composed of varying grain diameters, Taylor [81], shortly followed by Bobolev et al. [19, 21], showed that, for a given substance, the onset of the burn rate increase would occur only above a critical pressure, as shown in Fig. 5.6. The critical pressure Pc was, in turn, found to be inversely dependent on the characteristic pore diameter dp associated with the bed. Analogous behavior was observed in burning explosives containing cracks or defects [72, 73], where burning was unable to penetrate small cracks until Pc was reached. From this data, it was clear that only a new burn mechanism would explain the discontinuous behavior. An explanation had been proposed in
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60
Apparent Mass Burning Rate (g/(cm2 s))
50
40
(a)
(b) (c) (d)
30
20
10
(e) 0
50
100
150
200
Pressure (atm)
Fig. 5.6. A plot from [81] illustrating the dependence of the onset of convective burning on pressure and pore spacing in tubes filled with HMX particles of diameter (a) 200–600 μm, (b) 104–124 μm, (c) 64–76 μm, (d) 63–64 μm, and (e) 5 μm.
earlier Russian work whereby hot combustion gases were driven through porosity from pressure gradients induced by burning. Bradley and Boggs [22] note that A.F. Belyaev and A.A. Andreev first postulated and quantified the concept that conductive burning would occur only while the defects associated with the explosive porosity were smaller than some critical dimension that reflected the combustion process. Subsequent experimental work continued to explore the onset of the convective burning phenomenon in both cracks and porous systems. Results confirmed the theory that the accelerated propagation rates were a result of flame penetration into the porosity as the flame’s standoff distance (composed of the gas preheat zone and the gas reaction zone) grew small relative to the porosity length scale. Upon accessing the smaller surface defects in the explosive, the flame surface area abruptly increased and, accordingly, so did the burn rate. For materials with high degrees of connected porosity and confinement, continued burn rate acceleration was observed after flame penetration of the porosity. In these cases, burning enhancement was not only due to the deflagration processing additional explosive surface area, but also due to the hot
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product gases being generated in the porosity faster than venting could occur. For porous materials and narrow cracks, the escape of gas from reactions inside the porosity can be constrained due to viscous effects and gas-dynamic choking. With sufficient confinement, the contained product gases locally elevate the porosity pressure and drive the hot products ahead of the deflagration and into unreacted explosive. They may even precede the conductive preheat zone. The degree of convection penetration is currently not fully resolved and, in all likelihood, depends on the porosity of the explosive [6]. In this situation, thermal and frictional losses to the walls quickly slow and cool the product gases and preheat the unreacted explosive in the process, decreasing the amount of heating required by conduction. In some cases, the preheating of the product gases is sufficient to induce ignition by itself. So, in contrast to conductive burning, where propagation of the chemical reaction is primarily due to conduction of heat, the propagation rate of this new mode is augmented by the convection of hot product gases. Thus, it is referred to as convective burning. It is important to note that, in a convective burn, conductive heating is still present and plays a major role in the ongoing reaction of the ignited material, as heat still propagates ahead of the deflagration via thermal conduction. The convection simply enhances the burn rate by providing additional energy transport ahead of the front. The key identifying characteristics of a convective burn are penetration of porosity by the flame and preheating of the porosity by hot gases driven ahead of the flame front. A schematic of the convective burn process is shown in Fig. 5.7. Some authors [22] have gone as far as to classify two modes of convective penetration: “spontaneous” and “forced.” Spontaneous burning occurs where the reaction front penetrates porosity smoothly with a definite velocity. Forced penetration is when the reaction spreads in a highly nonsteady fashion, with fingers of reaction shooting ahead of the main front into pockets of Convective Preheat
Porous Bed
Thermal Diffusion
Final Combustion Products
Multi-Phase Combustion Zone
Burn (Ignition) Front
Fig. 5.7. The convective burn structure based on [6].
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porosity. The spontaneous mode of propagation is most commonly observed in experiments involving porous beds and low rates of pressurization. Forced penetration mode is seen in materials with a defined crack structure and at higher pressurization rates, both of which facilitate localized convective behavior. Experimentally, pressurization rates are often controlled by varying the sample size relative to the test chamber volume. Low pressurization rates are generated in test chambers with large volumes relative to the burning sample and are termed “Crawford bombs” or “strand burners.” These vessels are designed in such a way that product gas production from the burning test material will not significantly vary the chamber pressure during an experiment. Often, the bomb’s pressure is preset and maintained or slowly varied during testing by flowing nitrogen or argon gas into and out of the vessel. This can have the undesired effect of pre-pressurizing any existing porosity with cold, high-pressure gas, which would not be present in a real-world situation. More rapid rates of pressurization are achieved in a “manometric bomb,” which typically has a small chamber volume relative to the explosive sample. During combustion, confinement of the product gases pressurizes the vessel and can force these hot gases into any porosity in the test material. As the porosity is not pre-pressurized with cold gas, manometric-bomb experiments exhibit onset of convective burning at pressures 10–15 times lower than that in Crawford bombs [6]. Subsequent flame spreading in the porosity is also slowed by the presence of high-pressure cold gases. The convective burn mode is an undesirable (and unreliable) mode of combustion for most applications. Studies have shown that the convective burn rate is highly dependent on the amount of connected porosity, a property not generally well known in pristine material and difficult to predict in damaged material. Thus, proper study of the phenomenon requires a high degree of material preparation and characterization as void formation in explosives can occur from many different phenomena. Microscale voids are inherently created in plastic bonded explosives during the manufacturing process, where binderencapsulated explosive crystals are pressed at high pressures and elevated temperatures. This process induces cracking of crystals and spaces persist between pellets that do not bond completely. The pouring process of melt explosives can also create bubbles, while the solidification process can induce cracking from nonuniform cooling of the charge. During storage, aging of the explosive can also form voids due to loss of volatiles, stresses induced by thermal cycling of the material, or from sustained mechanical or thermal damage. Characterizing a convective reaction in a condensed phase explosive material can be challenging. Optical access is usually limited with opaque explosives. Because of the localized propagation nature of the convective mechanism through the connected porosity, pressure and temperature probes embedded in the test material recover only local conditions, requiring either a large amount of embedded diagnostics or a fair degree of subjective analysis. Thus, study of this phenomenon has received significantly less attention than the conductive deflagration and detonation modes. The onset of
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convective burning is considered more of a safety issue, with most work focusing on how to prevent the mode from occurring entirely. Furthermore, for certain porosity ranges and levels of confinement, convective burning has been shown to be a pathway to deflagration-to-detonation transition (DDT) [17, 18, 29, 43, 44] by providing sufficient burn rate acceleration to drive compaction and shock waves into the unreacted explosive (Chap. 8). In spite of these issues, significant effort has been devoted to convective burning. Most early work took place in the USSR in the early 1960s, with efforts shifting to the West in the late 1960–1980s. For the most part, studies are scattered over a wide variety of conditions and lack cohesiveness, which is at least partially due to geopolitical limitations as well as the previously discussed difficulties associated with testing solid explosive materials. Further complication is introduced by the fact that, in practice, connected porosity in high explosives tends to exist as a series of cracks that are irregular in shape and can exhibit high degrees of branching [69]. Most studies, however, have focused on simplified configurations consisting of either single, narrow, and straight channels or statistically porous beds composed of explosive particles. Only a few studies have taken place between these two extreme limits [16, 35, 59]. The end result is the absence of any unified theory governing flame propagating into voids or porosity. Some simplified theories do exist, however, along with several common observations. The general conclusion is that, in order for convective burning to occur, hot product gases must penetrate the voids or porosity present in the explosive, ignition must occur inside these voids, and this ignition must continue to penetrate the void in order to remain ahead of any conductive burn front. Our subsequent discussion is split into these three stages. Section 5.4.1 reviews the conditions necessary for porosity penetration by hot gases, while Sect. 5.4.2 considers the criteria required for ignition of voids. This is followed by a discussion of the effects of the melt layer in Sect. 5.4.3. 5.4.1 Gas Flow into Porosity and Isolated Voids The first stage of the convective burning process requires hot gases to penetrate defects in the explosive. Such a process is primarily fluid dynamic in nature and strongly dependent on many parameters, including the gas pressure, the rate of pressure change, and the gas temperature. The geometry associated with the porosity is also of critical importance, including its characteristic width, and its length for blind pores, or its permeability for open pores. Isolated Voids It is instructive to consider the simplified gas flow into a single blind void of length L as discussed by Bobolev et al. [20] and Belyaev et al. [10]. Consider a single void in a conductively burning explosive as shown in Fig. 5.8. A reaction zone remains above the crack surface. The void is initially filled with cool gases
5 Deflagration Phenomena in Energetic Materials Gaseous Reaction Zone
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P Tg
ρg νg
T0 P
T0 P
(a)
(b)
Fig. 5.8. Gas flow into an idealized explosive void (a) before and (b) during burning.
at temperature T0 , while hot gases from outside the void are at temperature Tg . The gas inflow process is subsonic and quasi-steady, such that both the void and the external gas are at the same pressure P . Several assumptions are necessary to render the analysis tractable. First, the velocity of the product-gas inflow vg must be greater than the regression rate r of the explosive (5.18) vg > r in order for the gas to actually enter the void. The pore is also assumed to be sufficiently narrow such that the internal flow is both isothermal, remaining at T0 , and one-dimensional with no recirculation effects. From these assumptions, the mass inflow to the void is m ˙ in = vg ρg = vg
P A, RTg
(5.19)
where the gas is assumed to be ideal, A is the void’s cross-sectional area, and R is the specific gas constant. Meanwhile, the mass of the gas in the void at any time is P , (5.20) mvoid = ALρ = AL RT0 where ρ is the gas density. The rate of change of mass in the void can be obtained by differentiating (5.20) to yield m ˙ void =
AL dP RT0 dt
(5.21)
with the isothermal assumption applied. Equating (5.19) and (5.21) and solving for vg yields
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vg =
L P
dP dt
Tg T0
,
(5.22)
the dependence of the inflow velocity. It can be seen that the inflow is enhanced for longer channels, lower pressures, higher pressurization rates, hot combustion products, and cooler propellants. Substituting (5.22) and Vielle’s Law (5.4) into (5.18) gives the minimum rate of pressurization dP b T0 n+1 > P (5.23) dt L Tg necessary to drive gas into the channel ahead of the advancing conductive burn. Thus, decreased pressure promotes gas penetration not only by allowing gas to enter the void more easily, but also by slowing the conductive burn rate, resulting in a P n+1 dependence. Equivalent theories exist for the penetration of gas into statistically porous beds as well [22]. 5.4.2 Ignition of Porosity The previous section highlights the necessary conditions for hot product gases to penetrate the explosive porosity. However, gas penetration of the explosive surface is a necessary, but insufficient condition for ignition of convective burning. A separate analysis must be performed to determine if the heat transfer from the products gases will ignite the walls of the porosity. The concept that combustion in an explosive with connected porosity will occur at the conductive rate until the flame is able to penetrate the porosity is first thought to have been proposed by A.F. Belyaev in 1946 in an unavailable doctoral thesis [22]. Belyaev suggested using the pore size normalized by the standoff distance of the gas phase reaction zone as an indicator of when convective burning would occur. Later in 1946, Andreev further quantified this concept for a single pore by using the thermal layer thickness in the solid instead of the gaseous standoff distance. These concepts were first transferred to statistically porous beds (beds with networks of connected porosity) in 1961 by Margolin [63] in theoretical form only. Margolin later developed these ideas [64, 65] with coauthors, as discussed later. In general, explosive material will undergo pyrolysis or gaseous deflagration when heated to an ignition temperature; however, removal of the heat source will lead to quenching unless a sufficient preheat layer is present in the solid. Zeldovich [83] defined criteria for stable ignition to occur, requiring both that the ignition temperature be reached at the surface of the void and that a temperature versus depth profile exists at the void wall that approximates the preheat profile occurring during steady-state burning. Margolin and Chuiko [64] utilized these criteria when deriving simplified and approximate conditions necessary for ignition of pore walls from convective preheating. Their theory assumes that inflowing hot gas at temperature Tg gradually cools to the wall temperature (initially at T0 ) as heat is
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absorbed by the pore wall. The temperature change is assumed to occur in a smooth fashion and the elevated wall temperature Tw is assumed to exist over a length Lc . For a circular pore profile of diameter dp , comparing the heat lost to the wall from the gas to the heat transferred across the gas film yields the heat balance d2p = πdp Lc h (Tg − Tw ) , (5.24) (vg − r) ρg Cp (Tg − Tw ) π 4 where the variable h is the heat transfer coefficient h=
ΔQ A · ΔT · Δt
(5.25)
between the gas and the solid wall. Solving for the length of the cooling section yields (vg − r) d2p ρg Cp dp (vg − r) Lc = = , (5.26) 4h 4N uαg where αg is the thermal diffusivity of the gas (αg = λg /(ρg Cp )). The Nusselt number hdp Nu = (5.27) λg represents the ratio of convective to conductive heat transfer across a surface. Margolin and Chuiko [64] mention that vg and N u can be measured or calculated from the properties of gas, but also note that, for gas inflow to small pores, the gas inflow will initially be laminar. According to Eckert and Drake [27], for laminar flow with the following condition satisfied x/dp > 0.1, ReP r
(5.28)
the Nusselt number can be assumed to be constant. In the above inequality, x is the distance from the start of the pore, Re is the Reynolds number (representing the ratio of inertial forces to viscous forces), and P r is the Prandtl number (representing the ratio of momentum diffusivity to thermal diffusivity). Margolin and Chuiko [64] list calculated values of Lc from (5.26), shown in Table 5.3, which indicate that at small pore diameters, the preheated layer depth is very thin. Their calculated values imply that N u ≈ 3.33, assuming αg = 2.5 × 10−5 as stated in [65]. To apply Zeldovich’s ignition criteria [83], Margolin and Chuiko [64] define several lengthscales and timescales, including the maximum time τ available to heat the pore wall before it is consumed by the advancing conductive burn τ=
Lc . r
(5.29)
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Table 5.3. Convective preheat lengths as calculated by Margolin and Chuiko [64] for vg = 1 m s−1 and P = 1 bar. dp
Lc
1 cm 1 mm 100 μm 10 μm
3 cm 3 mm 30 μm 0.3 μm
After a time τ , the penetration depth of the thermal layer is assumed to be √ = αs τ , (5.30) where αs is the thermal diffusivity of the solid explosive. This definition is conventionally used as the thermal layer depth, defined as the location where the temperature is 90% of (Ts − T0 ). Correspondingly, the time τs to establish a steady-state thermal layer is considered to be s (5.31) τs = . r The variable s is the steady-state thermal layer depth √ s = αs τs , (5.32) which can be combined with (5.31) to yield τs =
αs . r2
(5.33)
Additionally, the time required to heat the pore wall from its initial temperature to the surface ignition temperature T∗ is represented by τ∗ , and can be determined by consideration of the energy flow into the explosive thermal layer. The heat transferred from the gas to the solid can be found by solving (5.25) for ΔQ (5.34) Qloss = hLc dp (Tg − T∗ ) τ∗ and substituting relevant parameters to the problem. Meanwhile, the heat absorbed to the solid preheat layer is √ Qtrans = ρs Cs Lc (T∗ − T0 ) (dp αs τ∗ + ag αs τ∗ ) , (5.35) where ag is a geometrical factor in the calculation of the preheat material volume that is unity for a pore of circular cross-sectional area and zero for a narrow rectangular slit. Equating (5.34) and (5.35) and solving for t∗ (with (5.26)) gives 2 d2p β , (5.36) τ∗ = αs N u − aβ
5 Deflagration Phenomena in Energetic Materials
where β=
λs (T∗ − T0 ) . λg (Tg − T∗ )
269
(5.37)
Expressing the ignition criteria with these scaling parameters gives
and
τ ≥1 τ∗
(5.38)
τ ≥ 1 or ≥ 1. τs s
(5.39)
Equation (5.38) requires that the time to reach the critical surface temperature must be less than the total time available for heating, while (5.39) specifies that the steady-state thermal depth profile must form before the pore is consumed by the conductive burn. The surface temperature inequality can be evaluated using (5.26), (5.29), (5.36), and (5.38) to yield (vg − r) N u r 4
Tg − T∗ T∗ − T0
2
ρg Cp λg ρs Cs λs
2 aβ > 1, 1− Nu
(5.40)
the surface temperature criterion. Equations (5.26), (5.29), (5.33), and (5.39) combine to form (vg − r) d2p r ≥ 1, (5.41) 4N uαg αs the thermal depth criterion. Thus, these two inequalities predict the regime where penetration of the void by hot gas will result in ignition. Margolin and Chuiko [64] sketched these two curves on a plot of pressure versus pore diameter for arbitrary conditions, as shown in Fig. 5.9. By sketching (5.40) as a straight line, it is assumed that vg , N u, and Tg are independent or weakly dependent on dp . Five distinct regimes are identified on the plot, labeled A–E. In region A, both the surface temperature and depth criteria are met, allowing sustained ignition in the pore. In region B, neither requirement is satisfied and no ignition occurs in the pore. Region C has the necessary surface temperature for ignition to occur, but insufficient penetration of heat into the pore wall, leading the authors [64] to conclude that only pyrolysis would occur in this region. The surface temperature in region D is too low for ignition, but the necessary temperature depth profile does exist. Margolin and Chuiko [64] proposed that combustion might propagate through the heated layer of the pore wall in this region via a thermal explosion or cookoff event. Finally, region E exists for high pressures and pore diameters and is cryptically referred to as a regime where “the flow of combustion products becomes turbulent,” violating one of the model assumptions. Eventually, increasing pressures and pore sizes also result in the defect becoming accessible to the conductive burn.
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pth De
Pressure, P
on teri Cri
Region C: Pyrolysis
Region A: Ignition
Region E: Turbulent Gas Combustion
Surface Criterion
Region B: No Ignition (Conductive Burning)
Region D: Thermal Explosion
Pore Diameter, dp
Fig. 5.9. A sketch of the ignition regimes implied by (5.38) and (5.41). Based on Margolin and Chuiko [64].
Figure 5.9 represents only one mechanism of ignition; however, Margolin and Chuiko [64] acknowledged that ignition by other means was possible. In particular, they mentioned that pore wall roughness could allow hotspots to form on any protruding areas of the wall, allowing for earlier initiation than predicted by theory. They also reasoned that adiabatic compression of the gas in the pore could result in high enough temperatures to cause ignition for sufficiently rapid pressurization rates. This effect was observed in subsequent research [50]. Finally, they noted the possibility of the pore filling with partially decomposed explosive products, which could then ignite from either thermal explosion or via a gas-phase deflagration. Any of these effects could result in unsteady or oscillatory ignition of the pore gases and may be sufficient to create the proper thermal depth (in region C) or raise surface temperature (in region D) to a sufficient level that sustained that ignition can occur in the pore. Andreev Criterion The many parameters in (5.40) and (5.41) made for difficult application of the criteria to specific explosives due to the number of physical property measurements required in both the gas and the solid phases. Furthermore, even when assuming that N u was constant, the variables vg and Tg can depend on
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pore diameter and pressure. One year after presenting their ignition criteria, Margolin and Chuiko [65] further simplified their model with a series of assumptions to make it more accessible. They first reasoned that the gas inflow velocity vg was dependent on the conductive flow velocity ρs vg = ug F = rF (5.42) ρg and a function F dp dp Tg Tg Tm , F =F , P r, Le, , , (5.43) Dc T∗ T0 T0 that was dependent on a wide range of experimental conditions. They then assumed that (1) the function F was only weakly dependent on its parameters in the range considered; (2) the steady-state thermal layer thickness of the solid s = αs /r was proportional to that of the gas g = αg /r; (3) the convective burn process was independent of the ratio ρs /ρg ; (4) and the gas inflow velocity to the pore was much greater than the solid regression rate vg r. Applying these assumptions to the thermal depth criterion (5.39), rewritten here as 1 vg − r dp dp ≥ 1, (5.44) 4N u r s g gives the simplified criterion dp ρs Cs rdp = = An ≥ An∗ , (5.45) s λs where the number An is referred to as the Andreev number in honor of A. Andreev, the Soviet scientist who greatly advanced their knowledge of porous charges. Physically, the Andreev number is similar to the Peclet number. Values above unity indicate that convective processes are dominant relative to conductive processes. The onset of convective burning occurs when An exceeds a critical value An∗ . Note that the Andreev criterion requires only the existence of a preheated thermal layer. The expectation that the surface temperature exceeds a specific value (5.40) has been dropped in order to yield a single closed form equation for prediction of the onset of convective burning. Margolin and Chuiko [65] further simplified their An criterion by observing that (λs /Cs ) does not widely vary and assuming a “typical” explosive value of 1.5×10−3 g cm−1 . From a collection of experimental data for various explosive powders [65], they also determined that convective burning occurred at a mean value of ρs rdp = 0.9 × 10−2 g cm−1 s−1 , yielding a critical value of An∗ = 6. More accurate results can also be obtained by experimentally calibrating the model to a given explosive. While the Andreev criterion was initially developed for a single pore, it has been tested more extensively for porous beds, using an effective dp estimated from permeability measurements. Less work has been performed to verify its
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applicability to single pore systems [72, 73], and even fewer attempts have been made to quantitatively test the validity of the An criterion for open or blind end conditions in single pores and porous beds of varying length. Belyaev Criterion A similar closed form criterion was established by Belyaev in 1973 while studying flame penetration into a single blind explosive pore in a small volume test chamber with high pressurization rates [10] P 2n+1 d2p ≥ κ.
(5.46)
This expression, presented by Belyaev without derivation, was found by Bradley and Boggs [22] to result from (5.41), assuming that (1) vg ∝ r; (2) αg ∝ P −1 , an approximation valid at low pressures; (3) vg r; and (4) r = aP n . Like the Andreev number, Belyaev’s equation does not include any surface temperature criteria. However, his data (Fig. 5.10) of critical pressure required for the onset of convective burning in a given pore width appear to show the presence of a temperature criterion, as noted by Bradley and Boggs [22]. 5.4.3 The Influence of a Melt Layer on the Critical Pressure An unresolved issue in convective burning is the exact effect of the melt layer on the gas penetration of the porosity. Conceptually, the idea is that the existence of a liquid melt layer will seal any surface defects to the convective gases. As the melt layer thickness decreases with increasing pressure, it is assumed that gases will penetrate the porosity after a minimum melt thickness is achieved. Thus the melt layer is thought to delay the onset of convective
Threshold Pressure (MPa)
10.0
1.0
0.1 10
100 Crack Width (μm)
1000
Fig. 5.10. Experimental data from Belyaev et al. [10]. The abrupt change in slope towards horizontal may be due to the surface temperature criterion in (5.40).
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burning. Based on this logic, the effectiveness of the melt layer depends on both its thickness and the size of any porosity present, as larger porosity sizes will be less affected by a given melt layer thickness. Taylor [81, 82] proposed that the melt layer has an effect on the resistance of explosives to convective burning. For a given pressure, he reasoned that explosive with a lower melting point would have a thicker melt layer, which would hinder gas penetration of porosity and thus shift the onset of convective burning to a higher pressure. He supported this concept by pointing out that HMX has a high melting point of 278◦ C and a critical pressure of 1.3 MPa, while PETN has a lower melting point of 140◦C and a critical pressure of 2.8 MPa. Andreev [3] felt that Taylor [81, 82] was not fully correct. He [3] agreed that the melting point of an explosive had an effect on the critical pressure, but thought that the boiling point also played a role in both the thickness and the effectiveness of the melt layer. The boiling point of HMX and PETN was given as >340◦ C and 270◦ C, respectively [3]. From this logic, HMX is liquid over a temperature range ΔT of at least 62◦ C while PETN is liquid over a range of 130◦ C. In reality, the thickness of the melt layer will likely depend on both the melting point, the boiling point, and the shape of the temperature gradient (which is a function of pressure for a given explosive). As the temperature gradient in a conductive burn increases more rapidly with distance at higher temperatures, materials with lower melting point should preferentially have thicker melt layers than explosives with high melting points, even if both are liquid over the same temperature range. Even with a lesser melt range, an explosive with lower melting point may have a thicker melt layer than an explosive with a higher melting point and larger melt range. This may explain Taylor’s logic. Belyaev et al. [11] proposed that convective burning can occur only when the melt layer thickness grows smaller than the maximum pore size. This criterion was supported by the work of Andreev and Gorbunov [5], who measured the threshold porosity responsible for the onset of convective burning. Their data supported Belyaev’s statement [11] as the melt layer thickness for each explosive tested was less for those with lower critical pressures. Gorbunov and Andreev [32] further explored this effect with chemically similar explosives that had widely varying melting points and found no correlation, although the effect of the boiling point, viscosity, and other fluid properties were not considered.
5.5 Progression of Combustion The flame spreading process in explosives is analogous to the available burning modes and generally proceeds primarily due to either conductive or convective mechanisms. We review the literature on this topic, discussing first flame spreading over an unconfined flat explosive surface, followed by flame spread through explosive porosity.
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5.5.1 Across Flat Surfaces Flame propagation across explosive surfaces has received very limited attention [79]. More effort has been devoted to flame spread across propellant surfaces due to the applicability to rocket motor applications, but even this field lacks detailed study [52]. The low priority placed on this topic indicates that it is not considered to be a critical factor affecting the outcome in explosive accident scenarios; ignition and conductive burning generally receive the lion’s share of attention here. In rocket motor applications, Kumar and Kuo [52] reason that the flame spread is considered to be a short part of the already short ignition transient and thus worthy of only limited effort. In general, flame propagation proceeds smoothly across flat surfaces in a continuous manner. Son et al. [78] measured the rate of propagation of conductive deflagrations across the surfaces of the HMX-based explosive PBX 9501 as a function of pressure. Care was taken in their experimental setup to keep the hot product gases from flowing across the unburned surfaces, ensuring that burn was predominately conductive. They [78] were able to correlate the results with a rate law (5.47) Sf = 0.259P 0.538, which was a function of pressure, where Sf is the deflagration propagation velocity in cm s−1 . Comparison with their previously determined [14] conductive burn rate for the same explosive r = 0.018P 0.924
(5.48)
showed that the flame spread across the surface more rapidly than the conductive burn rate over the pressure range tested (Fig. 5.11). They also noted that
P
Fig. 5.11. Experimental data from [78] comparing the flame spreading rate to the conductive burn rate for PBX 9501.
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both rate laws would intersect at 110 MPa due to the higher exponent associated with (5.48). After this intersection, they reasoned that the flame spread and burn rate would proceed at an identical rate as it would be physically implausible for the flame spread to propagate at a slower rate than the conductive burn rate. They also concluded that at the high intersection pressure, such consideration was irrelevant anyway as the reaction zone would likely access the inherent porosity present in the pressed explosive and transition to convective burning. More information on flame spreading over propellant surfaces is given by Kumar and Kuo [52], although the discussion is skewed more towards rocket motor applications and involves situations where erosive burning becomes more dominant. In fact, Kumar and Kuo [52] note that most existing flame spread models are better suited to systems dominated by convective heating. This applies to flame spreading both over open surfaces and in cracks. 5.5.2 Spreading of Convective Combustion Convective combustion propagates through penetration of porosity by hot products gases, which preheat the unreacted material. Many of the conditions necessary for the onset of convective combustion (discussed in Sect. 5.4) are also responsible for the spread of the reaction through any connected porosity. Generally, once combustion has penetrated the porosity of an explosive material, it will continue to spread throughout all accessible voids. Early on in the convective burn process, it is possible to quench the reaction by rapidly lowering the external pressure supporting the gas penetration. However, once convective burning has progressed to the point that the porosity pressure exceeds the external pressure, only catastrophic failure of the explosive confinement will stop the convective burning. Thus, after flame penetration, the key question is how fast the reaction will progress through the explosive. As mentioned, this is significant because rapid acceleration of the burn rate, coupled with sufficient levels of confinement, can drive shocks into the unreacted explosive and cause DDT. The constraints required for DDT tend to be restrictive and do not readily occur in practice, but when detonation is accidentally initiated, it tends to be quite memorable. Even without DDT, however, convective combustion is capable of generating pressures in excess of 1 kbar in less than 100 μs [40], which will exceed the limits of most confinement systems. Generally, for damaged materials, the convective combustion process proceeds in the following manner, as illustrated in Fig. 5.12. Temperature-induced pressure gradients force hot gases into any porosity present in the explosive. Ignition then occurs in the void and begins to spread throughout the porosity. In high-aspect ratio voids, the rate of product gas generation inside the porosity exceeds the venting rate, leading to internal pressure buildup. As the pressure in the void rises above the external pressure, gas flow into the porosity will reverse and high temperature gases will start flowing out of the void. Pressurization of the cavity, in turn, will increase the burning rate and pressurize
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(a)
(b)
(c)
(d)
Fig. 5.12. Stages in the development of convective combustion: (a) gas penetration of porosity, (b) ignition of porosity, (c) pressurization of porosity and flow reversal, and (d) continued pressurization resulting in deformation and cracking.
the void even more rapidly. Left unchecked, this positive feedback cycle can drive burn rate and crack pressures towards infinity. However, deformation of the explosive walls (which are generally not mechanically strong) will act to reduce the rate of pressurization. Regression of the crack walls will also mitigate pressurization rate to some degree by increasing the exhaust area available to the product gases. It is the balance of these phenomena that controls the outcome of the convective burn. In cases where the pressure continues to increase, the explosive can be deformed to the point of mechanical failure, resulting in propagation of the crack and possibly crack branching. Depending on the level of explosive confinement, the process will result in one of the three outcomes: the explosive will either (1) experience complete mechanical failure, which will vent the pressure and drop the burn rate; (2) continue to experience a rapid burn rate until all material is consumed; or (3) undergo significant compression from rapid acceleration of the burn rate to the point that DDT results. Prediction of the progression of the convective burn involves accurately resolving the coupling between the gas-dynamic processes in the porosity, the unsteady combustion of the burning walls, and the structural response of the high explosive. Furthermore, the response of most systems is highly dependent on the ignition location, the level of confinement, and the morphology and properties of the explosive. The end result is that no fully predictive theory exists to accurately model the evolution of a convective burn into an arbitrary explosive system, but many fundamental observations do exist. The status of the field is best reviewed by considering specific work devoted to this problem, as is done below. Most testing has been performed on manufactured porosity in pristine explosive, which has the advantage of allowing accurate measurement of the existing void structure. Void formation in these studies has either consisted of well-formed single cracks or statistically porous beds formed of many small particles. However, some studies do exist in explosives that have sustained mechanical or thermal damage.
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Convective Burning Literature The ability of the porosity to control the spreading of convective combustion through confinement of the product gases was shown by Taylor [81], who explored the effect of boundary conditions on the propagation of the flame through porous beds. Three different acrylic tubes were used as shown in Fig. 5.13. The first (a) had an open downstream end and a convergent upstream end. The second (b) had an open upstream and downstream ends, and the third (c) had an open upstream end and a closed downstream end. All were filled with HMX powder and tested under identical conditions. The flame was observed to propagate most rapidly through the tube with the convergent upstream end. The slowest propagation rate was seen in the tube with the closed end. The results indicated that generation and confinement of product gases in the porosity can enhance the burning rate. The influence of porosity boundary effects was not limited to porous systems. Bobolev et al. [20] measured flame penetration in flat cracks in RDX. For cracks with open ends, they measured the flame speed to be Sf, open = 0.15P 1.55 ,
(5.49)
while for blind cracks they found Sf, closed = 0.075P 1.55,
(5.50)
(3) (2)
7.8 cm
(1)
(4)
1.6 cm
(a) Convergent Nozzle
(b) Blank Nozzle
(c) Blank Nozzle and Sealed Lower End
Fig. 5.13. A figure from [81] showing the experimental assemblies used to vary the porosity boundary conditions. (1) All acrylic tubes had a 6-mm inner diameter. (2) Upstream end (top) was fitted with asbestos to prevent widening of the exit during test. (3) For tube (a) the throat diameter was 4.3 mm. (4) Gauze cloth was used to confine explosive in tubes with open downstream ends and still allow for gas flow. Resulting flow velocities are shown in Fig. 5.14.
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Time (sec)
0.4 (c)
0.3
(b) 0.2 (a) 0.1
0
1
2
3
4
5
6
7 Lower end Distance from Top Surface of Powder (cm)
Fig. 5.14. Figure from [81] showing position vs. time of the reaction front for the HMX-filled tubes shown in Fig. 5.13. HMX particle diameter was 200–600 μm and the initial pressure was 2.76 MPa.
where Sf was in cm s−1 and P was in atm. Their results indicated that open and blind cracks have the same pressure dependence on the burn rate, but that flames are able to move twice as fast through open cracks compared to blind ones. Belyaev et al. [12] further studied the development of convective combustion in closed and open flat cracks in NG powder and AP mixtures. They [12] found that pores with open ends were penetrated faster than blind ones. They also measured the pressure inside the pores and observed that narrow pores were able to develop pressures in excess of the external chamber pressure and that the pore pressure increased with the pore aspect ratio. In general, the pore pressure and flame spread rate were found to increase with decreasing pore width. Innovative experiments by Krasnov et al. [46] detected the presence of a convecting flow in a burning channel initially filled with liquid that was allowed to vent at the downstream exit of the channel via a valve. The motion of both the liquid layer and the reaction front was recorded as the flame penetrated the channel. Experiments without liquid were said to yield identical results, showing that the liquid layer did not influence the development of combustion. Their experiments verified that the gas penetration preceded the flame spread into the channel. Early on, researchers were concerned about the ability of convective burning to locally pressurize the porosity to levels capable of damaging the explosive. A theoretical study by Kirsanova and Leipunnskii [42] represents one of the earliest attempts to apply fracture mechanics to the mechanical stability of cracks under these conditions. Their model assumed that cracks
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behaved in an elastobrittle fashion. Subsequent work by Cherepanov [23] extended the previous model [42] using an elastoplastic assumption and quasistatic deformation processes. However, his [23] equations were not solved and so no results were presented. Work by Belyaev et al. [13] reviewed experimental data from previous work for the development of convective burning in a blind slot of unspecified propellant material. He showed that as the slot aspect ratio (L/w) increased, the larger the peak pressures that were measured in the slot, also noting that after a duration of “a few hundredths of a second” the pressure decreased due to “burning away of the pore.” Peak pressures up to 1 kbar were detected in slots with values of L/w = 1,000, prompting him to apply the analytic work of [42] to his experimentally observed results [13] to determine if it was possible for the short duration of local pressurization to cause the crack to grow. He determined that bounded and self-sustaining crack growth were possible depending on the degree of pressurization and the properties of the cohesion coefficient [9] of the substrate, but did not provide any calculated results. More recent studies on the development of convective burning in rigidly confined, extremely narrow slots have also demonstrated the sensitivity of peak slot pressure to aspect ratio. Berghout et al. [14, 15] and Jackson et al. [40] chose aspect ratios similar to those observed in thermally damaged material and found that increasing the aspect ratio from L/w = 500 [14] to values near 2,000 [15, 40] could drive the values of localized pressurization from a few hundred bar to thousands of bar. For smaller aspect ratios, the crack pressures were contained by the experimental cell and decreased after a few milliseconds due to crack wall regression. However, the larger aspect ratio slots developed pressures in excess of 1 kbar in approximately 100 μs (Fig. 5.15), driving the reaction at speeds up to 10 km s−1 [40] and destroying the test cell (Fig. 5.16). Kuo and colleagues also studied the development of convective burning and the local pressurization that can occur in propellants during convective burning. They devoted a multidecade effort of experimental, analytical, and computational effort [45, 49–52, 54–56, 58, 59, 71, 75, 76] to predict the phenomena, culminating with a coupled computational model [59]. During each computational step, the model would solve the gas-dynamic conditions in the crack during pressurization and then pass the computed crack wall loading off to a finite-element analysis code to evaluate the solid mechanics and fracture response of the propellant. Any change in the wall shape from deformation and crack growth was then passed back to the gas-phase part of the code. More recent work has focused on direct measurement of the porosity levels and damage that occurs in mechanically and thermally damaged high explosives. Renewed interest was provided by Dickson et al. [24, 25], who demonstrated that it was possible for extremely violent reactions, including DDT, to occur in small, heated HMX-based samples that were confined and thermally damaged. Images from the study (Fig. 5.17) appeared to show the presence of cracking during violent reaction.
280
S.I. Jackson 1200 P2 1000
P4
P3
Pressure (bar)
Model 800 600 400 P1
200 0 –125 –100 P1
–75
–50 –25 0 Time (μs)
P2
P3
25
50
75
P4
Slot
Fig. 5.15. Pressurization of a narrow slot due to flame penetration. The bottom image is the experimental assembly from [40] with the cover removed to show the 19-cm-long × 80-μm-wide slot between two pieces of PBX 9501. Pressure transducer locations P1–P4 are also shown. The flame entered the slot from the reservoir to the left. The plot shows localized pressurization in the slot (P2–P4) compared to the reservoir pressure P1. The test cell failed at 1 kbar as shown in Fig. 5.16. The “Model” curves are from the theory of Jackson and Hill [38].
Subsequent efforts have been devoted to characterizing the porosity and morphology present in damaged explosives of interest [33, 37, 67–69, 77]. Parker et al. [69], in particular, found that PBX 9501 explosive that had been thermally damaged and then cooled contained cracks with an average width of 4 μm. Many images of the morphology present in thermally damaged high explosive are shown and discussed in the following chapter. The resulting combustion in this damaged material has also been studied. Berghout et al. [16] performed experiments on extensively damaged PBX 9501 using the rate of pressurization as a diagnostic for the type of connected porosity. Explosive pellets, mechanically fractured by quasi-static compression and thermally damaged by heating to 180◦ C, were ignited in a closed-volume bomb, and the resulting pressurization was recorded. The existence of an abrupt change in the pressurization rate was interpreted as marking the onset
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Fig. 5.16. A test cell from [40] before (top) and after (bottom) failure during testing.
0 ms
10 ms
5 ms
15 ms
Fig. 5.17. Images from [25] illustrating the presence significant cracking in a violently reacting, thermally damaged, confined piece of PBX 9501 explosive at 200◦ C. Explosive samples were 25.4 mm in diameter and 2–5 mm thick.
of convective combustion inside the sample. In conjunction with Belyaev’s equation (5.46), this allowed the determination of the pore size associated with the connected porosity. Connected porosity with a length scale of 25 and 4 μm was inferred from the combustion of the mechanically and thermally damaged samples, respectively. These values are within the range of crack widths observed in the literature for these damaged materials.
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Fig. 5.18. Images from the study of [35] showing a portion the copper tube prior to testing (left) and just after the onset of convective burning (right). Deflagration reaction and products can be seen through the tube wall fragments. The ignition source was a hot wire located at the left end of the tube.
Hill et al. [35, 36] and Jackson et al. [41] thermally damaged PBX 9501 pellets that were confined in copper tubes by heating them to temperatures between 150 and 190◦ C. The amount of porosity induced from thermal expansion and the HMX β-to-δ phase transition in the tubes was inferred from tube-wall strain measurements. The resulting combustion was then observed with a framing camera. Imaging of the tube wall deformation allowed determination of the combustion conditions inside the tube. A range of behavior was observed from conductive burning to convective combustion and DDT. An example of a test exhibiting violent convective burning is shown in Fig. 5.18. Dienes and Kershner [26] highlight concerns regarding the inability of current modeling capabilities to predict the effects of this mesoscale damage on the global evolution of the burn reaction. Finite-element codes typically deal with volumes that are too large to accurately model the observed small-scale fracture mechanics, while atomic-scale computer simulations cannot handle systems large enough to encompass a full explosive grain. They [26] conclude that the most efficient path forward is to model the material in an approximate way that captures the main features of the combustion and material failure with generic, statistical material models rather than getting bogged down attempting to accurately resolve the detailed behavior. Such a method would draw statistical data from mechanical and thermal damage studies and requires a generalized approach then to model the evolution of combustion. Current theoretical work on this problem has operated on the principle that some sacrifice of local accuracy is necessary for simplification of global prediction. Hill [34] presented a model of combustion in a confined explosive containing a statistically determined porosity distribution. The numerical results of his simple model agreed well with the behavior observed in the experiments of Dickson et al. [25].
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1000
800 Runaway Reaction
Pressure (bar)
Steady Choking 600
Steady Solution
400
200
0
0
50
100
150
200
250
300
350
L/w
Fig. 5.19. A plot from [39] of pressurization vectors as a function of crack pressure and aspect ratio. In the “steady choking” regime, the crack pressure approaches a finite value. In the “runaway reaction” regime, pressurization vectors increase to infinite values. Belyaev’s equation for a crack with a length of 500 μm and a variable width (w = 1.43 μm at L/w = 350) is overlaid on the plot. The zone where flame penetration is forbidden is hatched.
Analytic work by Jackson and Hill [38] has also developed a simplified model of the internal ballistics occurring in a crack to predict pressurization of a slot based on the occurrence of gas-dynamic choking. Modeling the burn rate with a simple conductive equation (5.3) yields the pressurization rate (and thus steady-state burning pressure) of the crack as a function of only the dimensionless crack aspect ratio L/w (Fig. 5.19). Their predicted pressurization rates compared well with earlier experiments [15, 40] for burning rigidly confined, high-aspect ratio cracks (Fig. 5.15). When coupled with Belyaev’s criterion (5.46) for the onset of flame penetration of a crack, their results [38] implied that any convective burning occurring in cracks with widths and aspect ratios similar to those occurring in thermally damaged PBX 9501 [69] would result in rapid crack pressurization (in some cases towards infinity) that would result in failure of any confinement. Stable convective burning modes were forbidden by Belyaev’s criterion due to the small widths associated with the thermal damage. Jackson and Hill [39] later numerically extended their model to account for regression of the crack surface due to burning and found that crack regression lowered the
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peak pressures that could be achieved, but that the phenomenology of the problem was unchanged and the onset of convective burning at small scales would still generate crack pressures in excess of what could be contained by most casing materials. This concludes the discussion of convective burning literature. Summaries of the works discussed as well as some other unmentioned ones are contained in Tables 5.4–5.6.
5.6 Concluding Remarks In the preceding discussion, we have discussed the possible deflagrative mechanisms available to explosive materials. Chemical properties aside, the dominant factors controlling an explosive burn were seen to be the external pressure and porosity of the material. Increasing pressure always increases the reaction rate. The porosity controls whether or not the increase occurs in a smooth or discontinuous fashion. Predicting the progress of a conductively burning material is well within the limits of current modeling capabilities. However, burn progression in a confined and damaged material can be very difficult to predict due to (1) the lack of a comprehensive theory governing the onset of convective burning for a wide range of explosives and (2) the current inability to accurately resolve the interplay between the multiple phases of reacting matter that control the outcome of any convective burn. Even with ongoing efforts, this issue is sufficiently complex that it will likely remain unresolved for many years to come. Readers interested in pursing more detailed reading on the combustion of energetic materials are encourage to refer to several works. Basic theory on combustion of propellants and explosives is contained in [48]. More indepth theory and data on specific materials are contained in two volumes devoted to the fundamentals [57] and the nonsteady behavior [60] of propellant combustion, respectively. The work by Bradley and Boggs [22] contains a literature review of convective combustion studies performed prior to 1978 with a focus on Russian efforts. Further information must be pursued on a paper-by-paper basis. Tables 5.4–5.6 have been provided to assist with any such effort.
Prentice [72] Taylor [81] Taylor [82]
1962 1962 1962
O-OB hole OC-OB porous solid
Structure O O/D C
Focus
P0 0.1–3.9 MPa 20.0 MPa 20.0 MPa
P˙ ∼ ∼ ∼ NC HMX, PETN, RDX HMX, PETN, RDX
Material 0.25–1.6 mm 5–900 μm 5–900 μm
Dimensions
Melt layer study.
Note
1963 1965 1965 1966
Andreev and Chuiko [4] Bobolev et al. [20] Bobolev et al. [19] Belyaev et al. [11]
O-B porous O-BO slot O-B porous EO-B porous
D O/D O/D O
∼ ∼↑ ↑ ∼↑
0.1–100 MPa >0.1 MPa 0.5–7 MPa 0.1–120 MPa
NC, PETN, Tetryl 5 μm, 200 μm RDX 0.1 mm RDX 50–360 μm PETN, TNT, RDX, 10–550 μm DINA, NC, KClO4 , Picric Acid 50–300 μm 1966 Bobolev et al. [21] O-B porous O/D ∼ 0.05–8 MPa AP, KClO4 , RDX 50–730 μm 1966 Margolin and Chuiko [65] O-B porous O ∼ >0.1 MPa AP, Hg(CNO)2 , HMX, NC, PETN, Picric acid, RDX, Tetryl, TNT 1967 Gorbunov and Andreev [32] O-B porous O ↑ >0.1 MPa DINA, HMX, 50–100 μm TNT, RDX 1969 Belyaev et al. [12] O-BO slot D ∼ >0.1 MPa AP, Nitroglycerin, 0.2–0.6 μm Secondary HE 1969 Margolin and Margulis [66] O-BO hole C ∼ 0.5 MPa Nitroglycerin 1.0 mm “Structure” column denotes porosity type: slot, round hole, porous bed, mechanical (M) or thermal (T) damage, and none (solid); and the prefix denotes condition of upstream and downstream boundary conditions: open (O), blind (B), embedded (E), constricted but open (C). “Focus” column reports whether the paper studied either conductive burning (C), the onset (O) of convective combustion or its development (D). The rate of change of the external or driving pressure at ignition is given by column “P˙ ” and includes steady (∼) or increasing (↑). The initial pressures tested are noted in “P0 .” Explosive components studied are listed under “Material.” The “Dimensions” column contains the particle diameter for porous beds or the critical width for pores, slots, or cracks.
Author
Year
Table 5.4. A partial list of experimental studies on explosive burn development.
5 Deflagration Phenomena in Energetic Materials 285
B-B porous D
1999 Gifford et al. [29]
C
D N/A C C
solid
Andoh [2] B-O hole Asay et al. [6] B-O porous Maienschein and Chandler [61] solid Atwood et al. [7] solid
1993 1996 1998 1999
D O D O D D D
S D O D
D O O >0.1 MPa 5–7 MPa >0.1 MPa >0.1 MPa
∼ ∼ ↑ ∼
Material AP N/A AP
Ballistite AP N/A HMX, PETN, RDX N/A N/A RDX ∼ 0.1–10.0 MPa NC ↑ >0.1 MPa N/A ↑ 0.1 MPa >0.1 MPa 0.05–30.0 MPa 0.1–11 MPa 0.05–10.0 MPa >0.1 MPa
∼ ∼ ∼ ↑ ↑ ∼ ↑ ↑ ↑ ∼ ∼ ∼ ↑
P0
P˙
HMX, TATB PETN HMX HMX HMX HMX, TNT HMX, TATB HMX HMX RDX HMX HMX, RDX HMX
HMX
Material
Table 5.6. Explosive burn studies. (Continued from Table 5.5)
25, 50, 100 μm 1, 2 mm N/A 1 μm 25–200 μm 2–20 μm N/A N/A N/A 80 μm 80 μm N/A N/A N/A N/A
Dimensions
Cookoff study.
DDT study.
Note 5 Deflagration Phenomena in Energetic Materials 287
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65.
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and Aeronautics, pp. 305–360. American Institute of Aeronautics and Astronautics (1984) Kuo, K.K.: Principles of Combustion. Wiley, New York (2005) Kuo, K.K., Chen, A.T., David, T.R.: Convective burning in solid-propellant cracks. AIAA J. 16, 600–607 (1978) Kuo, K.K., Moreci, J.A.: Crack propagation and branching in burning solid propellants. In: 21st International Combustion Symposium, pp. 1933–1941. The Combustion Institute, Philadelphia, PA (1986) Kuo, K.K., Moreci, J.A.: Different modes of crack propagation in burning solid propellants. J. Propul. Power 3(1), 19–25 (1987) Kuo, K.K., Summerfield, M. (eds.): Fundamentals of solid-propellant combustion, vol. 90. Progress in Aeronautics and Aeronautics. American Institute of Aeronautics and Astronautics (1984) Kuo, K.K., Vichnevetsky, R., Summerfield, M.: Theory of flame front propagation in porous propellant charges under confinement. AIAA J. 11(4), 444–451 (1973) Lu, Y.C., Kuo, K.K.: Modeling and numerical simulation of combustion process inside a solid-propellant crack. Propellants, Explosives, Pyrotechnics 19, 217–226 (1994) De Luca, L., Price, E.W., Summerfield, M. (eds.): Nonsteady burning and combustion stability of solid propellants, vol. 143. Progress in Aeronautics and Aeronautics. American Institute of Aeronautics and Astronautics (1992) Maienschein, J.L., Chandler, J.B.: Burn rates of pristine and degraded explosives at elevated temperatures and pressures. In: 11th International Symposium on Detonation, pp. 872–879. Office of Naval Research (1998) Mallard, E., Le Chˆ atelier, H.L.: Thermal model for flame propagation. Annales Des Mines 4(18), 379–568 (1883) Margolin, A.D.: The burning stability of porous high explosives. Doklady Academii Nauk, Phys Chem Sect. 140(1–6), 741–742 (1961) Margolin, A.D., Chuiko, S.V.: Conditions for ignition of the pore walls in the burning of a porous charge. Combustion, Explosion, and Shock Waves 1(3), 15–19 (1965) Margolin, A.D., Chuiko, S.V.: Combustion instability of a porous charge with spontaneous penetration of the combustion products into the pores. Combustion, Explosion, and Shock Waves 2(3), 72–75 (1966) Margolin, A.D., Margulis, V.M.: Penetration of combustion into an isolated pore in an explosive. Combustion, Explosion and Shock Waves 5, 15–16 (1969) Parker, G.R., Asay, B.W., Dickson, P., Perry, W.L.: Closed-bomb combustion of hot, thermally damaged PBX 9501. In: 13th International Detonation Symposium, pp. 284–292. Office of Naval Research (2006) Parker, G.R., Asay, B.W., Dickson, P.M., Henson, B.F., Smilowitz, L.: Effect of thermal damage on the permeability of PBX 9501. In: Shock Compression of Condensed Matter, pp. 1009–1012. American Physical Society (2003) Parker, G.R., Peterson, P.D., Asay, B.W., Dickson, P.M., Perry, W.L., Henson, B.F., Smilowitz, L.B., Oldenborg, M.R.: Examination of morphological changes that affect gas permeation through thermally damaged explosives. Propellants, Explosives, Pyrotechnics 29, 274–281 (2004) Payne, C.E.: Flame propagation in propellant cracks. Technical Report AFRPLTR-69-66, Air Force Rocket Propulsion Laboratory, Edwards Air Force Base, California (1969)
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71. Peretz, A., Kuo, K.K., Caveny, L.H., Summerfield, M.: Starting transient of solid-propellant rocket motors with high internal gas velocities. AIAA J. 11(12), 1720–1727 (1973) 72. Prentice, J.L.: Flashdown in solid propellants. Technical Report NOTS TP 3009, U.S. Naval Ordance Test Station, China Lake, CA (1962) 73. Prentice, J.L.: Combustion in solid propellant grain defects: A study of burning in single- and multi-pore charges. Technical Report NWC TM 3182, U.S. Naval Ordance Test Station, China Lake, CA (1977) 74. Razdan, M.K., Kuo, K.K.: Erosive burning of solid propellants, In: Fundamentals of Solid-Propellant Combustion, vol. 90. Progress in Aeronautics and Aeronautics, pp. 515–598. American Institute of Aeronautics and Astronautics (1984) 75. Siefert, J.G., Kuo, K.K.: Experimental observations of crack propagation in burning solid propellants. In: Proceedings of the 19th JANNAF Combustion Meeting, vol. I, pp. 407–415. CPIA Publication 336, Greenbelt, MD (1982) 76. Siefert, J.G., Kuo, K.K.: Crack Propagation in Burning Solid Propellants, In: Dynamics of Shock Waves, Explosions, and Detonations, vol. 94. Progress in Aeronautics and Aeronautics, pp. 575–595. American Institute of Aeronautics and Astronautics (1985) 77. Smilowitz, L., Henson, B.F., Asay, B.W., Dickson, P.M., Oschwald, D.M., Romero, J.J., Parker, G.R.: Morphology changes during thermal decomposition of PBX 9501. In: Shock Compression of Condensed Matter, pp. 1037–1040. American Physical Society (2003) 78. Son, S.F., Asay, B.W., Whitney, E.M., Berghout, H.L.: Flame spread across surfaces of PBX 9501. In: Proceedings of the Combustion Institute, vol. 31, pp. 2063–2070. The Combustion Institute, Philadelphia, PA (2007) 79. Son, S.F., Berghout, H.L., Bolme, C.A., Chavez, D.E., Naud, D., Hiskey, M.A.: Burn rate measurements of HMX, TATB, DHT, DAAF, and BTATz. In: Proceedings of the Combustion Institute, vol. 28, pp. 919–924. The Combustion Institute, Philadelphia, PA (2000) 80. Sutton, G.P.: Rocket Propulsion Elements. Wiley, New York (1992) 81. Taylor, J.W.: The burning of secondary explosive powders by a convective mechanism. Trans. Faraday Soc. 58, 561–568 (1962) 82. Taylor, J.W.: A melting stage in the burning of solid secondary explosives. Combust. Flame 6, 103–107 (1962) 83. Zeldovich, Y.B.: On the theory of combustion of powders by a convective mechanism. J. Exp. Theor. Phys. 12, 498–524 (1942) 84. Zenin, A.A., Finjackov, S.V.: Response functions of HMX and RDX burning rates with allowance for melting. Combustion, Explosion, and Shock Waves, 43(3), 309–319 (2007) 85. Zenin, A.A., Finjakov, S.V.: Characteristics of RDX combustion zones at different pressures and initial temperatures. Combustion, Explosion, and Shock Waves 5, 521–533 (2006) 86. Zenin, A.A., Puchkov, V.M., Finyakov, S.F.: Characteristics of HMX combustion waves at various pressures and initial temperatures. Combustion, Explosion, and Shock Waves 34(2), 170–176 (1998)
6 Mechanical and Thermal Damage Gary R. Parker and Philip J. Rae
6.1 Introduction The current approach to understanding and predicting the behavior of energetic materials (EM) outside their designed mode of operation is through computational modeling. As stated in the introduction to this text, the problem of response prediction from the countless multitude of starting conditions is a dauntingly complex challenge. Although fitting a model to a controlled data set can lead to reliable predictions within the limits of the parameter space used in the experiment, these models often do not capture behaviors outside the applied experimental conditions (e.g., if one changes the volume or geometry of the explosive, the prediction often fails). Sensitivity to variation is exacerbated by the fact that most nonshock events cause damage to the materials, thus the knowledge of the physical state of a pristine material is often of limited use. If a widely applicable model is the end goal, then a realistic characterization of the physical state of the damaged material is the rational starting point. In collaboration with a modeling effort, the experimentalist is charged with providing this useful morphological and mechanical framework to base the models on and behavioral observations to validate the models against. This chapter is intended to familiarize the reader with the physical state of select types of EM and introduce, through up-to-date research, how nonshockforming actions can characteristically alter the materials in question. Further, we discuss some material properties (i.e., permeability, porosity, surface area, stiffness, etc.) and how they may influence behavior (i.e., time to ignition and burn mode/rate) in ways that are believed to be important for current modeling efforts. We maintain the following premises throughout this chapter: (1) the morphology and mechanical properties of damaged EM influence their sensitivity and response, and (2) a useful model requires realistic assumptions for the physical state of the damaged material.
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We begin in Sect. 6.2 by defining and classifying pertinent morphological features so that we can proceed in a semantically precise and illustrative manner. In Sect. 6.3, we introduce some techniques commonly used to probe morphology and material properties and rank them, where it makes sense to do so, according to spatial resolution. An overriding challenge for this chapter is to answer the question of what constitutes damage. This is not a trivial question especially considering that most pristine energetic materials are not actually pristine; that is, they have defects. While a definition for damage requires comparison with a pristine baseline state and a description of the accumulation of morphological alterations resulting from an insult to the material, this definition first implies a reasonable knowledge of the pristine morphology. Section 6.4 introduces several explosives by type and presents some research describing the varied morphology of the pristine state. In Sect. 6.5, we examine characteristic damage morphologies linked to the insults from which they arose. In Sect. 6.6, we describe how damage affects pre- and post-ignition material behavior, framed as fluid transport problems, and some of the implications for damage in the context of this larger text. Finally, an introduction to engineering terminology is included in Appendix A to aid readers from other backgrounds.
6.2 Classification of Pertinent Morphologies There are many mechanical properties (i.e., strength, stiffness, etc.) and morphological elements (i.e., porosity, surface area, crystal structure, etc.) that influence EM behavior. For this chapter, examination of material morphology and how it changes is a logical starting point because structure often, in turn, determines a material’s mechanical properties and how they can change as a result. 6.2.1 Porosity Porosity is the single most important set of morphological features involved in the study of nonshock initiation. It is defined as the nonsolid volume fraction of a material. It is within porosity that gas phase decomposition chemistry occurs and through which reactive gases are transported. Porosity increases reactive surface area and its accessibility. It can control the mode of combustion (i.e., laminar vs. convective) after the onset of ignition and is useful in hypotheses describing mechanisms for the occurrence of deflagration to detonation transition (DDT). Many theories invoke void collapse mechanisms to explain the formation of hot spots and porosity to explain sensitivity of energetic materials. For these reasons, we must carefully define and classify the different forms of porosity. To begin this classification, we start by discriminating porosity based on its accessibility to fluids (i.e., whether it is open or closed ).
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Open Porosity Open porosity is the void space that is accessible to, and potentially active in, fluid transport processes. To fit this definition, the porosity must access a free surface. Most of the pore features discussed in this chapter fall into this classification, including cracks (both open- and closed-ended), channels, and surface pores. See Fig. 6.1 and Table 6.1 for definitions and illustrations. In real
Fig. 6.1. The corresponding definitions of void morphology can be found in Table 6.1 Table 6.1. Void definition labels. Letter Nomenclature a b
Void, cavity or closed pore Ink bottle pore
c
Dead-end pore, surface pore or blind pore
d
Channel
e
Closed-ended crack
f
Open-ended crack
Description A nonsolid volume that is isolated from the surface or other open porosity A nonsolid volume with a relatively large-volume pore body that is connected to the surface or other open porosity through a narrow constriction or throat A nonsolid volume that is open on one end only and is fully accessible to the surface, though their role in transport is often negligible. Shallow dead-end pores are sometimes called surface roughness features A nonsolid volume formed through erosive or expansive actions of a fluid. Once formed they serve as conduits and will often have a high degree of tortuosity A nonsolid volume that extends in two dimensions, but remains closed to the surface in at least one direction. Cracks are created from application of stress beyond what can be supported by the strength of the material. Closed-ended cracks are the result of a partial or incomplete fracture through a ductile material. The crack tip is the location where the crack ceased to open further; it is the closed end Similar to a closed-ended crack except that open-ended cracks extend completely in two dimensions, resulting in a complete separation
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systems, all these pore morphologies often exist in close spatial proximity and are often joined into complex three-dimensional networks. Closed Porosity Closed porosity is the void space that is isolated from fluid transport processes. Within this category are the intragranular and extragranular pores. Intragranular voids (voids located completely within a single crystal) are formed during crystal growth and include defects such as miniscule lattice vacancies and, often, larger solvent inclusions. In addition to intragranular voids, there can be extragranular, yet closed, voids. These are typically pressing or casting artifacts, where a void is isolated within the binder field or at a grain-to-grain boundary. 6.2.2 Crystal Structure In addition to porosity, crystal growth defects can act as stress concentrators and as nucleation sites for phase changes and chemistry, and therefore, deserve mention in this section on classification (see Fig. 6.2). Crystal Defects, Surface Roughness, and Phase Transitions Crystal growth begins with a solid-phase coalescence (precipitation) of like molecules into a condensation nucleus. It proceeds in a layer-by-layer accretion of solute molecules from a super-saturated solution. The regularly structured arrangement of these molecules is called a crystal lattice.1 Varied surface energies across the growing crystal facets combined with chemical impurities and locally varying concentrations of solute can result in growth irregularities, such as inclusions and lattice dislocations. Inclusions are either caused by impurities (such as solvent) or result from intergrowth of other crystals (e.g., growth twins). Dislocations are lattice displacements resulting from a change in orientation of the molecules on the growth surface. As crystals grow in a layer-by-layer fashion, dislocations may contribute to the surface roughness by initiating the growth of a larger surface irregularity [104]. Further, during mass-production of some energetic materials, small seed crystals are sometimes added to nucleate the growth of larger grains. The external surface of the crystals, as well as the internal boundaries between the seeds and their subsequent accretion layers, often contain many growth irregularities. As is discussed later, these irregularities act as nucleation sites for evaporation, sublimation, and chemical decomposition reactions during thermal insult. 1
Perfect energetic molecular crystals with ideal lattice structure, however, are rare because they are difficult to grow and not required to meet energetic material performance specifications.
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Fig. 6.2. Classification of morphology pertinent to the study of damage in PBX types
Freezing proceeds similar to precipitation. Depending on the cooling rate, a liquid solidifies into large crystals (slow cooling), small crystals (fast cooling), or an amorphous glass (fastest cooling). Here again, defects at grain boundaries are of particular interest.
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6.3 Techniques 6.3.1 Issues of Scale As the goal of damage research is to provide data for physically realistic computational models, it is necessary to describe ma terials with a degree of detail that is useful, yet not so fine that it is computationally cumbersome. This limit of scale, unfortunately, is material- and scenario-dependent. First we describe methods commonly employed for dealing with issues of scale, followed by the practical limitations of experimental techniques and diagnostics. 6.3.2 The Representative Elementary Volume From the study of mechanics and the modeling of shocks in heterogeneous materials, the explosives community has adopted and adapted a classification system for length scale that is related to particle size in a given system. The semantics and classification methodology from the study of mechanics are similar to those used by geohydrologists to address transport through heterogeneous geologic formations. This is convenient for this chapter as the influence of damage in the nonshock initiation regime often reduces to transport phenomena. The primary distinction in this classification system is the characteristic length of an average-sized particle (grain) or pore body diameter. With composite explosives, features shorter than the grain-scale are referred to as microscale or at the shortest extremes as molecular or atomistic. On the longer side of this classification is the macroscale or continuum scale. In between these extremes resides the mesoscale, which is more difficult to define. To do so requires the concept of a representative elementary volume (REV) [16], a.k.a. a representative volume element (RVE). The REV is the smallest ensemble of a material’s constituent elements (i.e., explosive grains, binder, and voids), where the effects of heterogeneity are no longer discernable [13]. The macroscale extends longer than the lengths comprising a side of a cubic REV, where the variations due to effects of smaller material elements are averagedout and the material behaves as if it were homogeneous. The mesoscale is the length between a REV and the length of a grain. As a loose rule of thumb, a typical cubic REV would incorporate the following: 2–5 grain-lengths (and any interstitial elements) along a side for acoustic properties [22], 20 grainlengths for mechanical testing [8], or 15 pore diameters in a pore network for transport measurements [45]. The number of grains needed for a REV for a specific system should be determined by how well-sorted the characteristic element size distribution is. For well-sorted materials (monomodal ) fewer grain-lengths may suffice, and for poorly sorted materials (multimodal ) greater numbers of grain-lengths may be required to capture the necessary material behaviors. For modeling mechanics and acoustics, the mesoscale would need
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to incorporate behaviors such as stress bridging between grains, impedance mismatch between grains and binder or voids, as well as the varied stress– strain properties of each element to name a few. For the study of damage morphology and its effects on the transport of mobile phases, the REV would need to include a distribution of flow-controlling pore throats and features such as voids at the crystal–binder interfaces, or the extent of the binder field across several grain lengths. 6.3.3 Experimental Methods The study of the morphology of energetic materials at the mesoscale and below is challenging due to the small spatial scale of many explosive particles and even the smaller size of the defects. A complete description of energetic material morphology requires the ability to probe features with sizes that range from millimeters to angstroms. Furthermore, as one probes shorter length scales, experimental uncertainties tend to increase and cloud the accuracy of the observations. The challenge has been to find a single technique that can provide qualitative and quantitative information on the microstructure of the materials. Often, multiple techniques must be applied to complete the picture to a level of useful accuracy. A discussion of commonly used techniques and their resolution limits, where sensible, is presented (see Fig. 6.3). It should be noted that this section is not exhaustive and that new techniques and improvements to existing techniques are continually being discovered and/or perfected. Optical Microscopy Using the visible spectrum of light either transmitted through or reflected off the surfaces can allow an investigation of morphology on the length scale of 0.2 μm. However, because of difficulties in attaining definitive images at the
Fig. 6.3. Approximate resolution limits of selected diagnostic probes (adapted from Peterson, LANL)
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extreme of this resolution, the optical techniques discussed further are often not useful below the spatial resolutions of 1 μm. Reflection Through the adoption of reflected-light microscopy methods developed in the field of metallography, great improvements have been made in the study of the microstructure of energetic materials. Particularly, the method of Reflected Parallel Polarized Light microscopy (RPPL), also known as Polarized Light Microscopy (PLM), allows both qualitative and quantitative examination of energetic material morphology. With this technique, polarized light is reflected from the surface of interest and passed though another polarizing filter (oriented in the same plane as the incident light) into magnifying optics through which it is ultimately focused into an eyepiece or image recording device. Specimens are sectioned, vacuum-mounted, and cured in a low-viscosity, transparent epoxy and then polished so that the reflective surface is nominally planar. The reason for this treatment is to maximize the proportion of reflected light that is polarized. Further, using a polarizing filter to minimize diffuse reflection, one images a planar cross-sectional surface of the specimen [114]. The differences in the indices of refraction between compositional elements in the specimen, as well as the stabilizing epoxy, result in contrastrich images where it is possible to resolve and discern microstructural features ranging in size from millimeters to ≈3 μm [97]. Historically, one of the inherent weaknesses of PLM has been the qualitative nature of the results. To extract quantitative details from the PLM images, such as area percentages of solid, void, and crystal size distributions, commercial image-processing software can be employed. These software packages use customizable routines that involve varying levels of color thresholding and geometric constraints to differentiate between microstructural components (see Fig. 6.4). Transmission Light is passed through a specimen into magnifying optics and ultimately focused into an eyepiece or image recording device. Using the material’s optical properties, such as indices of refraction, transmittance, and color, optical transmission microscopy reveals structure through a necessary thickness of material. Because light is passed through the specimen, this technique has some limitations. Viewing of opaque materials is of limited use. Similarly, some types of energetic materials, such as plastic bonded explosives (PBXs), are relatively opaque and, therefore, challenging preparation of thin sections is required [114]. However, with loose powders, this technique can be useful for obtaining particle size information. Another method commonly used to investigate EM is called matched refractive index microscopy. By immersing crystals in a solution with a matched index of refraction, contrast gradients at external particle boundaries are reduced, thus accentuating internal defects such as pores and solvent inclusions.
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Fig. 6.4. PLM image of thermally damaged PBX 9501 (top left) with an example of an image analysis mask identifying crack-like features (top right). Statistics, such as the number of cracks and crack width distributions, can be garnered from this type of analysis (bottom) [96]
The technique of matched refractive index microscopy is limited to materials lacking optical anisotropy because the variance in indices of refraction within a population of anisotropic crystals make it impossible to match with a single immersion medium [21]. As with PLM, transmission microscopy methods are qualitatively valuable but require image analysis routines or laborious handmeasurement to provide quantitative statistics. Electron Microscopy With the ability to resolve many orders of magnitude finer detail than optical microscopes, scanning electron microscopy (SEM) is a valuable, though largely qualitative, tool for investigating the details of microstructure at lengths
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approaching 6 nm. While its resolution is not the best of all available techniques, its relatively large viewing area and wide depth of field provide easily interpretable two-dimensional projections of the actual three-dimensional surface topography. Viewed in conjunction with a PLM internal cross-sectional image, one can reasonably describe a fairly complete three-dimensional structure. With a conventional SEM, nonconductive materials must be coated with a conductive material to prevent charge localization and sample damage under high electron beam voltages [103]. Unfortunately, many energetic materials are not electrically conductive. More versatile SEMs, particularly field emission and environmental scanning electron microscopes (FESEM and ESEM), can image nonconductive materials without application of a conductive coating. FESEM can potentially resolve the microstructure of energetic materials down to 10 nm. In addition, many SEMs have the capability to employ both an energy-dispersive X-ray spectrometer (EDS) and a wavelength-dispersive X-ray spectrometer (WDS) to further identify, quantify, and map elemental compositions. Atomic Force Microscopy and Transmission Electron Microscopy Probing at the finest level of detail, atomic force microscopy (AFM) and transmission electron microscopy (TEM) can resolve the atomic structure of materials (0.1 nm) and reveal fine porosity [78] and crystal defects such as lattice dislocations [110]. AFM provides true three-dimensional topographic information and does not require a conductive coating; however, the viewing area (a few micrometer) and the depth of field are much smaller than the images obtained by SEM. TEM records the diffraction of electrons as they are transmitted through a specimen. This requires the specimen to be very thin or prepared in a thin-section. As with AFM, the field of view can be narrow and, as with SEM, electron beam voltages may cause damage to nonconductive samples. Small-Angle Scattering Small-angle scattering (SAS) techniques, employing neutrons (SANS) or X-rays (SAXS), are used to probe length scales from 1 nm to 1 μm [97]. In a SAS experiment, each microstructural feature (interface, pore, particle, etc.) contributes to the total observed scattering. Additional methods are used to separate these contributions, allowing for quantitative assessment of microstructural features. Because of the penetrating nature of neutrons and X-rays, SAS techniques are highly applicable to the study of porosity and interfacial surface area in pressed and plastic-bonded types, as well as loose EM powders, and allow for correlation between insult and microstructural changes [86].
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X-Ray Tomography X-rays emitted from a source are directed at a specimen through which they penetrate with little refraction and are collected by a detector on the other side. The attenuation of X-ray energy can be correlated to the density of the material being probed, making it possible to convert the X-rays received by the detector into a two-dimensional density map of a slice through the sample (radiograph). To build a three-dimensional map, a series of two-dimensional images are obtained by imaging a specimen sequentially as it is rotated stepwise by known angles. Computer reconstruction of the two-dimensional images can create a three-dimensional map (tomogram) of the relative densities. Current resolution limits are approaching 1 μm, but there is much work going into attaining submicron resolution [125]. The advantages of X-ray tomography are that it is nondestructive, yet able to reveal internal microstructure of materials; in particular, it holds promise for mapping, albeit qualitatively, the extent of interconnectedness of porosity [81]. However, because this method relies on relative density differences within a specimen, there can be difficulty discerning elements of similar densities, such as distinguishing crystals from binder in PBX types. Nonlinear Optical Techniques Some crystalline materials exhibit nonlinear optical properties that result in a phenomenon called second harmonic generation (SHG) or frequency doubling of light. Crystals exhibiting SHG can be probed, for example, with an invisible Nd:YAG laser (wavelength: 1,064 nm) and will conveniently emit visible green light (wavelength: 532 nm) [9]. Research has discovered that HMX behaves nonlinearly after undergoing a polymorphic phase transition from its room temperature β phase to its high temperature-induced δ phase. Coupled with a microscope and a photomultiplier tube, real-time nucleation and growth kinetics have been measured for the phase transition in HMX [116]. This technology presents a rapid probe of crystalline damage resulting from heating [62] (see Fig. 6.5).
Fig. 6.5. SHG of a heated HMX crystal progressing through a polymorphic β − δ phase transition. In addition to alteration of the optical properties, the crystal morphology is altered in the form of cracking [9]
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Imbibition Techniques Imbibition of fluids into porous materials can provide quantitative data regarding the dimensions and size distribution of porosity. The detailed theoretical basis and governing equations for the following techniques can be found in the texts by Adamson [3] and Dullien [45]. Mercury Porosimetry In general, imbibition techniques provide quantitative pore size distributions by measuring the relationship between the pressure applied to a fluid and the volume of the fluid to have intruded into the accessible porosity of the specimen. Ritter and Drake [108] first described this technique based on the fact that the surface tension of a nonwetting fluid (i.e., with a contact angle greater than 90◦ ) causes resistance to the intrusion of the fluid through pore throats (modeled as capillaries). This resistance, however, can be overcome by application of external pressure, P , as described by the following equation: P =
−2σt cos θ , r
(6.1)
where σt is the surface tension, θ is the contact angle, and r is the radius of the capillaries (representing pore throats). As the applied pressure is incrementally increased, the penetrating fluid accesses smaller porosity, assuming all pores are equally accessible to the external surface (see Fig. 6.6).
Fig. 6.6. Mercury porosimetry data showing pressure vs. mercury intrusion and the corresponding calculated pore size distribution (Data courtesy of H. Czerski [39])
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Mercury porosimetry is the most common imbibition technique due to mercury’s nonwetting property and high surface tension and has been shown to be able to resolve porosity on the order of 0.01 μm, at which point experimental uncertainty resulting from deformation of the experimental cell may begin to cloud the results [21]. Specially designed high-pressure mercury porosimeters can determine pore sizes nearing 4 nm [45]. Porosimetry is sensitive to the value used for contact angle and, because the pores are modeled as capillaries, it also leads to intangible and unavoidable errors when the cross section of the real pore throats deviates from circular. As with physical adsorption techniques (see below), this technique is limited to porosity accessible to the surface of the sample, and accuracy also suffers from the occurrence of ink bottle pores and porosity that has convoluted and varying widths, where a narrow pore throat can block access to a larger pore body on the downstream side until the highest pressures are applied. This occurrence results in overrepresentation (by volume) of the smallest pores in the distribution and may be elucidated by measuring the hysteresis effect upon depressurization, where mercury contained in ink bottle pores remains trapped [3]. Physical Adsorption Measurement of the rapid and reversible physical adsorption of inert, pure gases (adsorbates) onto open pore surfaces (adsorbents) has been applied to gain quantitative surface area and pore size distributions of granular and pressed forms of energetic materials. A useful description of a typical adsorption apparatus and experimental technique can be found in the text by Shoemaker et al. [112]. These techniques rely on the measurement of pressure difference, as known quantities (by volume) of adsorbate gas are condensed onto the surfaces of known quantities (by mass) of adsorbent materials when held isothermally at the boiling point (saturation temperature) of the gas. Nitrogen and krypton are the most commonly used adsorbates. The essential steps for nitrogen gas adsorption are as follows: the porous medium is placed into a glass bulb (a calibrated volume) and placed under vacuum, sometimes at high temperatures, though this is not recommended for energetic materials. The bulb is then immersed in a bath of liquid nitrogen. A small volume of nitrogen gas is quantitatively dosed into the bulb and pressure is allowed to equilibrate as condensation occurs on the pore walls. The difference in measured pressure and expected pressure can be used to calculate the volume or moles of gas adsorbed. Stepwise gas dosing proceeds, with each iteration raising the pressure in the bulb, until the saturation pressure is reached and bulk condensation occurs. At first, the condensate adheres (by van der Waals attraction) in a molecular monolayer. However, in most materials, as pressure increases, adsorption becomes multilayer in nature, resulting in an inflection in the isotherm (see Fig. 6.7). In BET (Brunauer, Emmett, and Teller) theory [25], the adsorption of a gas is related to the surface area of the adsorbent through
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Fig. 6.7. Brunaer’s five isotherm types [3]. Type I (a.k.a. Langmuir type) represents monolayer only. Type II is common and is observed typically with powders. Type III is rare. Type IV and V are commonly observed with consolidated porous media where the second inflection and deviation from the Type II and III (respectively) curves is assumed to occur because of capillary condensation. The first inflection on Type II curves has been shown to be indicative of completion of monolayer adsorption [25]
n cx V , = = Vm nm (1 − x)(1 − x + cx)
(6.2)
where V or n is the volume or moles adsorbed, Vm or nm is the volume or moles required to form a complete monolayer on accessible surfaces, x is the relative pressure (i.e., measured pressure, P , divided by saturation pressure, P0 , for the gas), and c is a dimensionless constant, greater than unity and related to temperature. This equation is rearranged to the common linear form of the BET equation, 1 (c − 1)x x = + , (6.3) n(1 − x) nm c nm x) which is convenient because a plot of x/n(1 − x) vs. x (the so-called BET plot, Fig. 6.8) should produce a linear region for 0.05 < x < 0.35, from which nm and c are evaluated from the slope, s, and intercept, i, as follows: nm =
1 , s+i
(6.4)
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Fig. 6.8. A BET plot of arbitrary data. Data are typically linear at x values between 0.05 and 0.35, and so the BET plot, by practice, is limited to this range. The slope and intercept of the linear fit (solid line) are used to calculate nm and c
s c=1+ . i
(6.5)
In turn, nm can be used to calculate surface area, SB , as follows: SB = nm NA Am ,
(6.6)
where NA is Avagadro’s number and Am is the cross-sectional area per molecule of adsorbate (1.62 nm for N2 and 1.95 nm for Kr). The BET equation describes Type I, II, and III isotherms, but fails to capture the second inflection on Type IV and V isotherms. When c is large, BET reduces to the Langmuir equation and fits Type I. Conversely, when c is small, Type III isotherms are predicted. For adsorption of inert gases on polar surfaces, however, c ≈ 100 and results in Type II isotherms [3]. Because it describes three of the five isotherm types well, and some of the extent of the other two types, it is widely used to calculate surface area of solids from regions of isotherms where relative pressure is low. By matching the measured surface area to theoretical calculations of surface area of closest-packed arrangements of regular geometric solids, one can devise a simple geometric model for the surface area of a material [63]. This technique can also be used to compare the accessible surface area of pristine materials to that of damaged materials, gaining an understanding of the relative difference. Further, because physical adsorption is reversible, desorption curves have been shown to be useful as described by BJH (Barrett, Joyner, and Halenda)
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theory, where the adsorption/desorption hysteresis loop allows calculation of a pore size distribution. BET and BJH theory are limited to measurements of surface areas ranging from 1,000 to 0.01 m2 g−1 and pore radii on the length scale of approximately 1 nm, at which point the condensed adsorbate on opposing walls of a pore throat grows thick enough to join (capillary condensation) and prohibit intrusion of gas into downstream pores [3], thus may under-represent porosity in real solids. Another limitation of the physical adsorption method is that it only accounts for open porosity that is accessible to the surface of a sample. Permeametry Measurement, with an instrument called permeameter, of the volumetric flow-rate per unit area of a fluid through a porous medium can provide qualitative information regarding the interconnectedness of porosity throughout the material. Using the empirically based Darcy’s law, macroscale flow rate can be used to quantitatively determine specific permeability (a conductivity property of a porous medium). This can be useful in the study of certain energetic materials where gas transport mechanisms have been proposed to explain reaction behavior. This technique, however, describes only the effects of open porosity on fluid transport and does not account for isolated or closed porosity. Permeametry of damaged explosives is discussed in greater detail in Sect. 6.6 of this chapter. Archimedean Techniques The ISL sink–float experiment is a technique to determine the apparent density of single crystals. This method can also be used to assess an apparent particle density distribution for powders of loose crystals. Crystals are made buoyant in a mixture of toluene and methylene iodide with a known density. Toluene is then titrated into the mixture in small quantities, thus lowering the density of the mixture, until the crystals sink [21]. At this point, the apparent density is calculated and it is a simple exercise of comparison with the material’s theoretical maximum density (TMD) to find the volume contribution of the intragranular porosity. Gravimetric Techniques Thermo-gravimetric analysis (TGA) is a technique where a material’s mass is monitored as it is exposed to controlled thermal environments. Samples can be subjected to varying heating rates to obtain mass loss information as nongaseous phases undergo decomposition or vaporization. As the material is gasified, products vent from the measurement cell and are recorded as mass loss. Another common TGA procedure is to subject the sample to a constant
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(isothermal) temperature soak while mass loss is recorded. Using a variety of heating (ramp) rates, one can deduce decomposition kinetics, while an isothermal soak can provide porosity development measurements. Of course, only gases that escape both the sample and the sample pan are recorded, and so this technique is valuable only for measuring surface-accessible porosity [117]. Acoustic Probes Measurement of the transmission of ultrasonic acoustic waves (velocity and attenuation) through confined explosive specimens can be used to learn about some bulk physical properties (e.g. its elastic moduli) of an explosive composition as it undergoes damage [124]. Factors that affect transmission include density, stiffness, and impedance differences at intermaterial interfaces. In principle, dense, stiff, and homogeneous materials transmit acoustic waves with less inelastic loss and/or dispersion, thus resulting in overall faster sound velocity. In practice, ultrasonic waves are pulsed between two transducers separated by a known length. Time of flight (the time elapsed as the sound waves travel through a sample) is calculated through data analysis and wave velocity is obtained. The advantages of this technique are that it is nondestructive, can be performed on samples in-situ, and observations can be tracked as a function of time and temperature. However, because multiple factors influence wave speed, it can be difficult to deconvolve the causative factors. Further, this is a bulk measurement technique that does not resolve microstructural features and results can be difficult to interpret with dispersive media.
6.4 Pristine Materials Describing a material as being damaged is meaningful only in relation to an undamaged reference; therefore, we must first be able to describe a material’s pristine state. For many pristine energetic materials, qualitative descriptions exist; however, accurate quantitative descriptions are, for the most part, only available for production-controlled military explosive or propellant formulations that are made to meet precise specifications. 6.4.1 Energetic Material Types Energetic materials can be categorized based on their physical characteristics or form. The production processes used to create pure energetic materials rarely produce an explosive or propellant that is safe or useable without further treatment. Often, liquid-phase chemistry is employed to create the energetic molecules that are then precipitated into a solid-phase crystalline form or absorbed into a solid matrix for stabilization. Following purification and drying, many crystalline energetic materials, as loose powders, have undesirably low densities from a performance standpoint, are sensitive to static
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discharge and/or friction, and are difficult to shape for application (the exception is gun propellants, in which a granular form is desirable for combustion performance in internal ballistics applications). To address these issues, energetic materials are often pressed, mixed with desensitizing and/or binding agents, melted and cast, or processed with a combination of the previous treatments. The result is that energetic materials may be classified into many different types, including liquids, matrix stabilized materials (dynamites), slurries and gels, loose powders (granular), pressed pure compounds, melt-castables, and plasticized materials, which are subdivided into plastic bonded explosives (PBXs), putties, and cast-cured extrudables (see Table 6.2). For more in-depth discussion on the types of energetic material, the authors refer the reader to the text by Cooper [34]. Depending on the extent of treatment during production, various energetic material types may exhibit a variety of characteristic defects, yet still be considered pristine; that is, in the state they were intended to be for their given application (a.k.a. their design-mode). For example, in the case of a PBX type, initially, the individual explosive crystals exhibit defects from crystal growth; upon pressing, new defects and alterations may be introduced; further, upon machining even more morphological changes can occur. Despite the accumulation of defects acquired through this pedigree of treatments, the final piece of PBX is considered to be pristine because these defects are inherent in the design mode specifications. In this section, we describe the characteristic defects due to crystallization, pressing, and casting to establish some baseline descriptions. Within the confines of this chapter, it is not necessary, nor practical, to delve into discussion of damage in all the types of energetic materials. For example, most commercially utilized explosives are liquids, slurries, and gels, which by nature, flow and thus cannot crack. We leave it to the readers’ judgment to apply or dismiss descriptions of damage morphology where it is sensible. Additionally, while much of the following discussion may focus on a limited set of types, many of the examples shown should apply to materials with similar physical forms and properties. We focus the discussion of mechanically induced damage in this chapter on three types that encompass the breadth of mechanical stiffness ranging from brittle melt-cast types to ductile high-binder content (>10% binder by weight) PBXs, with low binder content (30%), the induced drag can lead to bed compaction and reduction in length of the permeation zone depth. Bed collapse has important physical implications for a Type I DDT mechanism (see chapter 8). Modeling strategies to predict reaction violence of EMs insulted to the point of ignition can be made to be dependent on pressure. There are three physical characteristics of damaged EMs that have pressure dependence during burning and affect violence: 1. The linear burn rate. 2. The surface area of the pore walls. 3. The critical pressure threshold after which flames can intrude through the pore throats and access this surface area.
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For a pressure-dependent burn model, it is important to be able to experimentally relate these three characteristics to pressure. It is even more telling if the measurements are made in a dynamic pressure environment like that which may exist, albeit post-ignition, in a real accident scenario. The aim of this section is to introduce some techniques that provide pressure-dependent data that can be analyzed to provide this information. The following methods arise from the study of propellant performance in combustion vessels. Therefore, a review of combustion mechanics and classical internal ballistics techniques is useful. One-Dimensional Steady Burn Model for Solids As a surface is heated, decomposition of the solid fuel releases reactive gas phase species that mix with an oxidizer (oxidizer can come from the surrounding atmosphere or from the decomposing molecules themselves). The resulting gas phase oxidation reactions are exothermic and lead to ignition and flame formation. In between the flame sheet and the solid surface is the reaction zone, where the mixing and exothermic chemistry occurs. Some of the heat generated in the reaction zone and from the flame is returned to the decomposing surface via conduction and radiation. In turn, this heat accelerates the kinetics of decomposition and replenishes the fuel to the reaction zone at the expense of solid mass. Because the reaction zone is gaseous, its thickness is susceptible to pressure. As pressure increases, the gases are compressed, the reaction zone thickness decreases and heat is fed back to the surface more efficiently. This feedback loop leads to increasing decomposition kinetics as a function of increasing pressure. If phrased as a conversion of solid to gas, then the one-dimensional rate of surface regression, normal to the surface, is dependent on the pressure at which the burn is occurring. This mode of burn progression is called laminar burning (referring to the layer-by-layer regression of the surface). Mathematically, this pressure effect is reflected in Vielle’s burn rate law [1], x˙ = BP α ,
(6.41)
where x˙ is the linear burn rate (one-dimensional rate of surface regression), B is the material-specific burn rate coefficient, P is pressure, and α is the dimensionless burn rate exponent. As can be seen, if one-dimensional steady burning is occurring, then the linear burn rate is dependent not only on the chemistry of the composition, but also on the pressure. Combustion Vessels Combustion vessel techniques were initially developed to study the gas production rates of burning propellant grains for internal ballistics applications, where it is necessary to control the pressurization rates inside gun breaches
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for launching projectiles. To optimize pressurization rates, propellant grains are manufactured in forms that have various degrees of perforation (increasing the degree of perforation increases the surface area of the grain). Once ignited, flames spread across free surfaces and, as pressure rises, eventually intrude into the perforations, thus accessing this additional surface area. As more surface area is incorporated into the burn, gaseous combustions products are liberated at progressively higher rates, further pressurizing the breach and creating the force needed to launch the projectile. In short, high surface area leads to high pressurization rates. To measure and tune the burn rates and pressurization characteristics resulting from varying the geometric form of the grain, experimenters traditionally use two types of combustion vessels: strand burners and closed bombs [1]. Strand Burners Strand burners are used to measure the linear burn rate of a nonperforated strand of material at controlled pressures and temperatures. Because Vielle’s law assumes that the burning surface is regressing in a planar front normal to the longitudinal axis of the strand, the sides of the strand are coated with flame inhibitor, leaving only one face to burn (see Fig. 6.61a). This practice assures the progression of the burn front down the length of the strand is essentially one-dimensional and, therefore, solely a property of the material composition, not its geometry. Strand burners are typically connected to a large-volume ballast tank (Crawford bomb) so that the pressure is nominally constant during the burn [37]. With traditional Crawford bomb strand burners, flame progression (linear burn rate) is monitored by evenly spaced embedded fuse wires that melt (resulting in a loss of electrical continuity) as the flame arrives. Linear burn rate measurements are repeated across a range of discreet pressures. Burn rate is then plotted against pressure and statistically fit to satisfy the variables in (6.41). Closed Bombs In contrast to the strand burner, the closed bomb is a small-volume vessel intended to simulate the dynamic pressure environment inside a gun breach. In its simplest design, a closed bomb is a high-pressure vessel with a piezoelectric pressure transducer. Propellant grains are sealed within the bomb and ignited and pressure vs. time data are recorded. The closed bomb technique is different from the strand burner technique, in that there is no restraint on propellant geometry and flame spread is not inhibited on any surface. Where the strand burner measures only a linear burn rate (dependent only on chemical composition and pressure), the closed bomb measures the overall rate of sample consumption (dependent on the linear burn rate and surface area). As the initial form and surface area of the propellant grains are usually known and with the following assumptions:
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1. All surfaces are equally and instantaneously accessible to flame and regress at the same rate 2. That the material is homogeneous in composition, one can apply mathematical form functions to the pressure-time data to calculate the linear burn rate [1]. Hybrid Strand Burners In the last couple of decades, research has been performed using so-called hybrid closed bomb-strand burners (hybrid strand burners for short) to determine linear burn rate as well as the burning surface area when a well-behaved laminar burn is not assured [123]. Hybrid strand burners are a type of closed bomb in which strands of EMs are burned with flame inhibitor. As is demonstrated in the following text, they have advantages over traditional strand burners and closed bombs when measuring burn characteristics of damaged explosives. Combustion Vessel Analyses Measuring the Linear Burn Rate of Damaged Explosives Recall the constraints for these traditional techniques: Strand Burner
Closed Bomb
Nonperforated strand
Homogeneous composition Known geometry Instantaneous flame spread
Most propellants meet these criteria while heterogeneous explosives (especially if damaged) do not. If pristine, a traditional strand burner can be used to measure the linear burn rate. A hybrid strand burner, however, has the advantage of providing linear burn rate data over a continuous range of pressures in the course of a single experiment. If damaged, however, laminar burning is usually not observed and the melt wires do not necessarily report in the proper order or at a steady rate, thus making it difficult to know the distance-time progression of the burn front. Attempting to measure the linear burn rate of damaged explosives with a traditional closed bomb is even less reliable because the form of the porosity is often irregular, unknown, and flame spread is not instantaneous. Together with the fact that the compositions of many explosives are heterogeneous, form functions are of no use. The best that can be hoped for is the careful measurement from a pristine strand of material and the assumption that the damaged state has a similar linear burn rate.
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Measuring the Surface Area of the Porosity in Damaged Explosives Only closed bombs or hybrid strand burners can be used to measure surface area. Again, the hybrid strand burner is preferred for damaged explosives due to the practice of inhibiting flame spread on all surfaces except in the direction of flow. As with permeametry, only flow (or flame spread) through open porosity is of interest. If not inhibited, flame spread across all external surfaces would effectively lead to over-representation of the surface area contribution of the pore walls. Again, as with permeametry, closed bomb combustion techniques average out the effects of the multitude of flow controlling structures. Observations are applicable only on a continuum scale, thus care must be taken to use an REV of test material. Calculating burning surface area with closed bomb pressure–time data requires rephrasing Vielle’s law from a linear rate into a volumetric rate, with the assumption that all surfaces (including pore walls) regress at the same linear burn rate of a pristine strand, leading to ˙ V˙b = Sb x,
(6.42)
where V˙b is the volumetric rate of burning material and Sb is the burning surface area. With the solid density, ρ, and molecular weight, Mw , it is possible to relate V˙b to the molar rate of solid burning, n˙b , Mw n˙b . V˙b = ρ
(6.43)
Equations (6.42) and (6.43) can be combined and rearranged to solve for the rate in number of moles of solid burning, n˙b =
˙ Sb xρ . Mw
(6.44)
However, to utilize pressure data obtained in closed bomb experiments, it is necessary to convert this result into a gas-phase number. This can be accomplished if the ratio of moles of product gas to moles of solid burned, n ˜ , is known, leading to the following expression for the rate of product gas formation, n˙g , ˙ n Sb xρ˜ n˙g = n˙b n ˜= . (6.45) Mw Further, with the following assumptions: (1) the combustion products are ideal gases, (2) the measured vessel pressure is the same as the pressure at the burning surface7 , and (3) the vessel volume, Vv , and temperature, T , are constant8 , 7 8
Though this is almost certainly not true due to gas flow choking at pore throats causing local pressure gradients In reality they can vary, but often to a negligible extent in the time for the burn to go to completion, thus the assumption is commonly made
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P˙ Vv . (6.46) RT Combining (6.45) and (6.46) and rearranging, one can solve for absolute burning surface area: P˙ Vv Mw . (6.47) Sb = RT xρ˜ ˙ n n˙g =
Absolute burning surface area is valid only for the sample volume and geometry used during the measurement. For application in a burn model, a more useful number for quantifying burning surface area resulting from damage is the relative surface area per unit area of a pristine strand. Unlike propellants, flame spread into the porosity of damaged explosives cannot be assumed to occur instantaneously, thus reporting surface area per unit volume is not sensible. Proceeding with the following notation convention: single prime for pristine, double prime for damaged, P˙ Vv Mw RT x˙ ρ n ˜ Sb = . Sb RT x˙ ρ n ˜ P˙ Vv Mw
(6.48)
If care is taken to compare damaged and pristine strands that have the same composition, then it can be assumed that x˙ = x˙ , ρ = ρ , Mw = Mw , ˜ . Again it is important that constant temperature and volume and n ˜ = n assumptions are defensible. Further, if the strands have the same initial crosssectional area and flame inhibitor is used, then Sb = Ac and (6.48) reduces to P˙ Sb = . Ac P˙
(6.49)
This is an attractive approach to estimate surface area per unit area from nothing more than measured pressure–time data. If burning surface area was calculated absolutely from (6.47), this would be accomplished by normalizing the burning surface area of the damaged strand by the known cross-sectional area of a pristine strand or more simply by the ratio of pressurization rates (see Fig. 6.57). Measuring Effective Diameter of the Porosity in Damaged Explosives Closed bombs can also reveal other structural information about damaged energetic materials. Pressure–time data from burning packed granular beds, as well as damaged EMs possessing open porosity, often show a sudden increase in pressurization rate when burned inside a closed bomb (see Fig. 6.58, top). This threshold is called the critical pressure and occurs when flames suddenly access new surface area. As pressure rises during initial laminar burning, the gas-phase reaction zone thickness (a.k.a. flame standoff distance) decreases until it is on par with the width of the pore throats (at the critical pressure), after which flames can intrude and access the surfaces comprising the walls
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Fig. 6.57. Raw data (left) from a hybrid strand burner comparing pristine to damaged PBX 9501. The graph on the right shows the increase in burning surface area for the damaged material, using the linear burn rate measured by Son et al. [119]. The critical pressure denoted on the raw data trace allows estimation of an effective diameter of 2.5 μm, using (6.50) and the constant determined by Berghout et al. [18]
Fig. 6.58. An illustration of the reduction of flame standoff distance until critical pressure and the transition from laminar to convective burn modes (top). The frames (bottom), taken with high-speed video through a viewing port on a hybrid strand burner, show the transition to a convective burn mode in thermally damaged PBX 9501. The flame-inhibited strand was ignited on the left side of the image. Luminous flames are seen entering cracks
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of the porosity (see Fig. 6.58, bottom). Below the critical pressure, the flames are not affected by the existence of porosity and the primary mechanisms for heat transfer are conduction and radiation. However, at the critical pressure and above, the porosity becomes accessible and so the burn mode transitions to what is called convective burning. Convective transport is a much more effective heat transfer mechanism than radiation or conduction [57]. Belyaev et al. [18] devised the following expression to relate critical pressure to effective diameter assuming a conduit flow model: Pc(1+2n) d2eff = cB ,
(6.50)
where Pc is the critical pressure, n is the material-specific burn rate exponent (from 6.27), deff is the effective diameter, and cB is a constant with units of (pressure length)−2 . By determining the value for cB through experimentation involving machined slots of known widths, Berghout et al. [18] calibrated this relationship for PBX 9501. Once calibrated to a respective material, it becomes possible to estimate an effective diameter for real damaged specimens that may have a distribution of crack widths. Recall that the effective diameter concept is tied to a bundle of identical capillaries model where the permeability is a function of the number of capillaries and diameter and as such requires some explanation of caveats limiting the assumptions that can be made about the network structure of the real porous medium (see Fig. 6.59). If one pictures a real pore network to be constructed of a number of pore bodies linked by narrower pore throats, then a distribution of pore widths would exist (as is seen in
Fig. 6.59. Diagrammatic illustration of two hypothetical pore spaces drawn with equivalent effective diameters (left), thus when measured with a closed bomb technique would reveal the same critical pressure, Pc (right). Trace A is representative of a bundle of identical capillaries model, while trace B is meant to represent a more realistic pore volume. The sharp transitions on trace A, in contrast with the more gradual transitions on trace B, result from the lack of variability in the pore width distribution. The steeper slope on trace B is due to the greater extent of surface area accessed after critical pressure is exceeded
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image analysis studies, refer back to Fig. 6.4, bottom). Because intrusive flow is limited by constrictions, effective diameter is an averaged representation of flow controlling structures. If the pore width distribution is monomodal, it has been demonstrated that the statistical mean width is close to the effective diameter measured in a closed bomb [96]. Because closed bomb techniques create a dynamic pressure environment, sometimes samples fracture in the course of a burn. When this happens, the distribution of cracks is changing and the critical pressure threshold can become less distinct. Costantino and Ornellas [36] observed that with a brittle plastic bonded explosive (LX-14), there were several regions of uniform burning, each with a steeper slope, thus in effect, several critical pressure thresholds. This was interpreted to be the result of flame intrusion into successive populations of progressively finer, yet similarly wide, fractures that were formed as the sample shattered. This leads us into the next section about interpretation of mechanical response. Interpretation of Mechanical Response of Burning EM from Closed Bomb Analysis Convective burning of damaged EM in hybrid strand burners can provide additional clues about the damaged state of EM. In particular, conclusions can be drawn regarding the mechanical properties of damaged materials. Cole and Fifer [33] studied burning down strands of neat, pressed HMX powders. From their work, they hypothesized a mechanism where convective ignition of downstream particles caused pressurization of the downstream pores. The pore pressure rapidly exceeded the tensile strength of the pressed bed. In turn, this lead to ejection and diminishment of the burning grains in the flame plume and was called “progressive deconsolidation” by the authors. This mechanism is an element of convective burning now widely referred to as “deconsolidative burning” and has been observed frequently in damaged EM (see Fig. 6.60) [85, 94]. When pristine EM is burned in closed bombs, the rapid pressurization inside the bomb has been implicated as a force that mechanically alters the material in the following observations: 1. Deconsolidative burning has been observed to occur more readily in brittle materials (e.g., melt-cast) [77, 84, 127]. 2. Less brittle PBXs with low binder contents resist cracking relative to the melt-cast types, but still exhibit deconsolidative behavior at high pressures [85]. 3. More compliant high-binder content PBX, and extrudable types, seems to resist pressure-induced cracking and often do not transition to convective/deconsolidative burning [77]. When similar materials are thermally insulted before ignition: 1. Melt-cast type materials melt and pre-existing porosity heals. The burns do not exhibit deconsolidative behavior [84].
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Fig. 6.60. The configuration on the left shows a cylinder of PBX 9501 radially confined inside a quartz tube. The copper sleeve was heated to 185◦ C and the HE was damaged for 3 h before ignition by hot wire inside a hybrid closed bomb-strand burner. The high-speed video frames show a so-called “deconsolidative burn,” where luminous flames first penetrate cracks (see frames at 4 and 6 ms) after which pieces of burning HE (labeled “a” and “b”) are ejected into the flame plume [94]
2. Low-binder content PBX types, particularly those containing HMX (which fractures extensively during its polymorphic phase transition), have been observed to transition to violent convective/deconsolidative burning [85, 94]. 3. High-binder content PBXs have been observed not only to resist transition to convective burning, but also to burn more slowly at higher pressures. Speculatively, this has been attributed to the relatively spongy binder compressing under pressure and closing thermally induced porosity before bulk convection can occur [84]. Using Pressurization Rate to Interpret Cohesiveness A convenient way to interpret the cohesiveness of a damaged material burned in a closed bomb is to plot the burning surface area against burn extent (ΔP/Pmax ). Similar analysis can be made by plotting vivacity (which is directly proportional to burning surface) against burn extent [76]. Figure 6.61 shows several geometrically regular shapes (reminiscent of common propellant forms) shown as how they would change throughout the course a burn in a hybrid strand burner. The tube surrounding each shape represents the flame inhibitor, thus surface regression is occurring only on surfaces that are accessible to flame spread in the direction of the longitudinal axis (equivalent to flow through connected porosity). Diagrammatic illustration of raw pressure vs. time data is shown above an accompanying illustration of how the relative burning surface area vs. burn extent plot would appear (taken from the time at critical pressure through to the time at maximum pressure). For the purpose of this device, the samples are considered to be
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Fig. 6.61. Illustration of how geometry affects pressurization in closed bomb combustion
of identical composition (thus the same linear burn rate). With thoughtful interpretation, much can be learned about the physical state of each material. Strand “a” represents a pristine cylinder burning in a laminar, onedimensional progression. The surface area does not change during the burn (horizontal slope on the area plot), thus it is described as a neutral geometry. It does not exhibit a critical pressure, and the increase in pressurization rate is only due to the pressure dependence of a normal, laminar burn. Strands “b,” “c,” and “d” are all intended to represent damaged strands (having open porosity), thus they have a critical pressure. They have been drawn to have equivalent initial effective diameters. Strand “b” represents a bounding case for a damaged cylinder with only a single channel. The burning surface area of the channel increases throughout the burn (positive slope on the area plot), thus it is described as a progressive geometry. It is also cohesive throughout the burn. The increase in pressurization rate is due to the normal pressure dependence for burning and the increasing surface area. Strand “d” is a bed of evenly spaced spherical grains. This represents the other bounding case where, after a short period of rapid flame spread, the surface area of each grain decreases throughout the burn (negative slope on the area plot). In contrast, this is described as a regressive geometry and is a case where there is no cohesion from the start. The last example, strand “c,” represents common burn behavior of damaged explosives (especially if they are heterogeneous formulations, e.g., grains in binder). It is neither fully progressive nor fully regressive. Instead, at some
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point in the burn, the growing voids join and the remaining solid splinters or granulates. On the pressure curve, this occurs at the inflection and correlates to the peak on the surface area plot. From the moment of splintering/granulation, surface area decreases with burn duration. Also, because the splinters/grains become detached, this point can also be interpreted as when deconsolidation occurs. Real damaged materials do not have regular geometry, and flame does not spread down the length of the strand instantaneously, thus splintering/granulation does not occur at the same instant throughout the sample volume as is illustrated in the figure. Instead, flame intrusion into increasingly finer pores is happening simultaneously with splintering and granulation; simply put, there is a competition between the progressive and the regressive phases. Further complicating analysis, flames can continue to intrude into finer voids in the detached grains and pressure-induced fracturing makes surface area a dynamic quantity. So with real media, the peak area does not clearly define the point of splintering/granulation, rather it should only be considered relatively as the point where new surfaces are being accessed at a diminishing rate and the remaining nonporous grains are only regressing. Viewed on an area vs. burn extent plot, this competition is revealed by the skew and broadness of the peak. Skew to the left indicates low cohesiveness. Conversely, skew to the right indicates high cohesiveness. Broadness of the peak correlates with the broadness of distribution of pore widths. Cohesiveness by itself does not tell the whole story. To better grasp the structure of the pore network, one must consider cohesiveness in conjunction with the maximum burning surface area and the effective diameter. Take, for example, the data in Fig. 6.62 where both samples were of the same
Fig. 6.62. Hybrid strand burner data comparing the effects that tightness of confinement has on thermally damaged PBX 9501. The raw pressure–time data (left) is used to determine the critical pressure threshold. The area plot (right) is informative of the relative extent of surface area burning and the cohesiveness of the samples
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composition, initial volume, and damaged at the same temperature for the same duration. They were also both damaged inside an open-ended tube, thus were partially confined (these samples are damaged similarly to the samples examined in Fig. 6.48). The only difference was with the tightness of fit inside the tube. The pressure vs. time curves indicate that the tight fitting sample had a higher critical pressure, thus smaller effective diameter of porosity. The surface area curve for the tightly confined sample is skewed to the left, indicating that it broke into pieces relatively early into its burn. Comparatively, the loose fitting sample is skewed to the right indicating later deconsolidation. When relative surface area is considered, it is clear that the tight fitting sample broke into fewer (necessarily larger) pieces and the loose fitting sample fragmented into more (necessarily smaller) pieces. As a whole, viewed in the frame of the pore network surrounding the solid pieces, one can conclude that the fluid transport in the tight fitting sample occurs through fewer, narrower channels per unit area than the loose fitting sample. Referring back to Fig. 6.48 for microscopy of similarly damaged samples, one can see a qualitative agreement.
6.7 Nomenclature Ac Am B c cB cs d deff E G i I k k0 Kn kp K Kp L La Mw n n˙
Cross-sectional area Molecular cross-sectional area Burn coefficient BET constant Belyaev’s constant Sutherland’s constant Diameter Effective diameter Young’s modulus Shear modulus or conductance Intercept Current Kozeny constant Carman–Kozeny shape factor Knudsen number Specific permeability Bulk modulus Permeability Length Actual length Molecular weight Moles Molar flow rate
6 Mechanical and Thermal Damage n ˜ n˙b n˙g nm NA Nc P P˙ P0 Pa Pc Pmax Pr ∇P Q r R Re s Sb SB Sφ T Ta Tr Tref Ts V V˙ V˙b Vm Vv x x˙ α θ λ μ μref ν ρ σ σtfs σcfs σt φ
Ratio of moles of product gas to moles of solid burned Molar rate of solid burning Molar rate of gas production Moles in monolayer Avagadro’s number Number of conduits/capillaries Pressure or resolved force Rate of pressure change Saturation pressure Atmospheric pressure Critical pressure Maximum pressure Reservoir pressure Pressure gradient Resolved force Radius Universal gas constant Reynold’s number Slope Surface area burning Surface area from BET technique Surface area of porosity Temperature Atmospheric temperature Reservoir temperature Reference temperature Sample temperature Volume, volume fraction or voltage Volumetric flow rate Volumetric rate of burning Volume of monolayer Vessel volume Relative position or constant Linear burn rate Burn rate exponent Strain Contact angle or geometric angle Mean free path Dynamic viscosity Reference viscosity Superficial velocity, Darcys velocity, specific discharge, filter velocity or poisson ratio Density Stress Measured tensile failure stress Compressive tensile failure stress Surface tension Porosity
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Appendix 1: Engineering Terminology Engineering Terminology Stress and Strain In common speech, the words stress and strain are used almost synonymously. In engineering, they have very distinct and different meanings. In simple terms, stress is a measure of load per unit area, while strain is a ratio of the change in a dimension with respect to the starting dimension. Equations (6.51) and (6.52) define these concepts in what is known as uniaxial engineering stress and strain. Later it will be useful to redefine these concepts slightly to allow mathematical and logical consistency. F , A0
(6.51)
X (L − L0 ) = , L0 L0
(6.52)
σ= =
where σ is the stress, F is the applied force, A0 is the original sample area, is the strain, L is the current dimension, L0 is the starting dimension, and X is the change in dimension. Usually mechanical tests are carried under uniaxial loading 9 and the effect is measured. In reality, most structures are biaxially or even triaxially loaded. Therefore, both stress and strain are often described in three dimensions by either matrices or tensors depending on the application. Elasticity In a classical material such as steel, at first the material behaves elastically and following Hooke’s Law (that is upon unloading the object returns to its original dimensions) before undergoing plastic deformation (permanent deformation remains upon elastic unloading). The transition from elastic to plastic deformation is called yielding and the stress at which it occurs, the yield stress (σy ). Increased deformation initially requires increased stress. Eventually, a maximum stress the material can support is reached (historically this is often denoted as σuts for ultimate tensile stress). After this strain, a critical localization (inhomogeneous stress region) is formed that deforms under a lower stress. At some point failure will occur. The uniaxial elastic modulus, also known as the Young’s modulus, is a material constant calculated from the gradient E= 9
Δσ Δ
(6.53)
The load is applied in only one direction on a sample of constant cross-section.
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377
ν = −εx εy
y x
Fig. 6.63. The definition of the Poisson ratio in elasticity
in the elastic region of the stress vs. strain curve. In stiff materials, such as steel (E ≈ 210 GPa), the elastic strains are so small that it is difficult to make accurate mechanical measurements of them. Instead, in many materials, elastic-moduli are measured using ultrasonic techniques that give accurate values. Obviously, if the material does not have a linear stress–strain region, it is not strictly valid to calculate, or quote, the Young’s modulus. Instead a “loading” modulus is defined by fitting a tangent or secant linear approximation to the initial part of the curve. Everyday experience shows that when a rubber band is pulled, it gets longer and thinner. In the elastic region of most materials, material volume is not conserved. Figure 6.63 defines the so-called Poisson ratio (ν), an observed material constant. In most engineering materials, ν varies between 0.25 and 0.35, with the exception of rubbers where ν ≈ 0.49–0.5 (i.e., volume is almost conserved as the material gets longer in one direction and thinner in the other two). Under plastic deformation, volume is often assumed to be conserved. Technically, ν is only an elastic material constant, although occasionally it is quoted incorrectly in the plastic regime. True Stress and True Strain A material is under uniaxial tension when stretched in one direction and under compression when the pushing direction is reversed. Figure 6.64 shows this concept. It is commonly assumed that a material maintains constant volume under plastic deformation. This is not always the case if voids form within the material, particularly under compression (the measure of volume in this case is called dilatation). Nevertheless, constant volume is a common assumption. Examining the tensile and compressive diagrams in Fig. 6.64 reveals a limitation in the strain equation (6.52). Intuition would suggest that stretching a material to twice its original length (an engineering strain of 1.0) should cause the same strain as compressing it to half its length. This is not the case with engineering strain. Additionally, in compression it is not quite possible to reach, or therefore exceed, an engineering strain of 1.0 as the material would
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2L L
1/2 L Original
Tension
Compression a θ
L
Original
Shear
Fig. 6.64. Compressive, tensile, and shear strains
need to have zero or negative thickness. To get around this, and several other problems such as simply summing of sequential straining operations, a true strain under uniaxial stress is defined, thus
L
= L0
L dL = ln . L L0
(6.54)
Clearly, the area over which the applied force is acting will change significantly at large strains. To account for this, a true stress is defined (6.55). In this expression, the current area is used (A) rather than the initial one. If volume conservation is assumed, A may be trivially calculated from the starting area and the current true strain. At small strains (320 98–180 69–120 36 30 100
Type 12a has a sandpaper surface. Adapted from Cooper and Kurowski [2]
the value. Reported values thus must be considered carefully before being used or applied. Serious scientific efforts to measure Tc began in the early part of the last century, but the first easily performed and reliable method was developed by Henkin in 1952 [3]. This involved rapidly immersing a small HE sample (3–25 mg) encased in a copper cylinder into a Wood’s metal bath, being held at a specified temperature. The amount of time required for the sample to explode was then recorded. A range of temperatures was tested and the highest temperature at which no explosion resulted was assumed to be Tc . Because Tc is a function of many different variables, it is impossible to measure for all desired conditions. Therefore, a method of calculation is required. Various theories of thermal decomposition have been presented in earlier chapters, but for solid explosives under the conditions most often encountered in systems of interest, the treatments of Semenov and of Frank-Kamenetskii are normally used. Many treatments of these topics use nondimensionalized variables to make algebraic manipulation more tractable and certain relationships obvious. However, that approach can make it difficult to understand the connections to actual materials and behavior. Here we use physical quantities to enhance application and understanding. For a more quantitative approach, see the discussion by Hill. 7.1.1 Semenov’s Approach An illustration of the behavior of explosives near the critical temperature can be obtained by doing a simple energy balance, assuming no thermal gradients inside the explosive sample. This treatment is attributed to Semenov [4]. The rate of production of heat is q˙1 = mQkr ,
(7.1)
where m is the mass, Q is the heat of decomposition, and kr is the rate constant. For most discussions involving estimations of critical temperatures, the reaction rates are assumed to be of zero order. This is usually a reasonable assumption as typically only very small amounts of explosive are consumed
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during the induction period. This will be illustrated below. The reaction rate constant is assumed to follow the Arrhenius relationship kr = Z e−Ea /RT1 ,
(7.2)
where Z is the frequency factor, R is the universal gas constant, Ea is the activation energy, and T1 is the explosive temperature. Assuming only conduction of heat and no convection, heat loss is represented by q˙2 = hA (T1 − T0 ) ,
(7.3)
where h is the heat transfer coefficient, A is the surface area, and T0 is the surface temperature. Note that the production of heat is exponential while heat loss is a linear process. This difference becomes critical when trying to predict the performance of explosive. Using these equations, taking the values for TNT from Table 7.3 assuming hA of 0.02 cal deg−1 s−1 and a 40 mg sample, we can plot the rates of heat production and loss at different temperatures (see Fig. 7.3). Curve 1 shows heat production rate as a function of explosive temperature. Curves 2, 3, and 4 show heat loss rates at three different bath temperatures. Curve 2 is calculated for a bath temperature of 287◦ C. It crosses curve 1 at two locations. Point A is a stable stationary state. Below that point, the explosive produces more heat than is being lost and the temperature rises. At point A, the loss rate matches the production rate and the temperature remains steady. If more heat is produced, the temperature rises, the loss rate becomes greater than the production rate, the system cools, and the temperature falls back to point A. Point B is also a stationary state, but it is unstable, because any increase in temperature results in an explosion, while a decrease in temperature results in it falling to the value at A. When the bath temperature is 291◦C (curve 3), the heat production curve is tangent to the loss curve at point C. This is the critical temperature, and is the final stationary state before explosion. Curve 4 does not intersect curve 1 at any point. Thus, the heat production rate is everywhere larger than the heat loss rate and an explosion always results. Even though simple kinetics and basic assumptions have been made in this analysis, and in real systems the details will vary, this process of reasoning is most instructive in obtaining a fundamental understanding of the behavior of explosive. Once again, following the simplest of analyses and using what we have previously derived, we can formulate a time-to-explosion relation. dT1 = dt
q˙1 − q˙2 mCp
,
(7.4)
where Cp is the heat capacity. Using 0.3615 cal cm−3 C−1 for Cp for TNT [5], and numerically differentiating the equation, we get the results shown in Fig. 7.4. This shows the effect of temperature on the time to explosion, and
HMX RDX TNT PETN TATB DATB BTF NQ PATO HNS
253–255 215–217 287–289 200–203 331–332 320–323 248–251 200–204 280–282 320–321
Exp.
Tm (◦ C)
253 217 291 196 334 323 275 204 288 316
Calc.
0.033 0.035 0.038 0.034 0.033 0.035 0.033 0.039 0.037 0.037
a(cm) 1.81 1.72 1.57 1.74 1.84 1.74 1.81 1.63 1.70 1.65
ρ(g cm
−1
) 500 500 300 300 600 300 600 500 500 500
Q(cal g−1 )
Values used E(kcal mol−1 ) 52.7 47.1 34.4 47.0 59.9 46.3 37.2 20.9 32.2 30.3
Z(sec−1 ) 5 × 1019 2.02 × 1018 2.51 × 1011 6.3 × 1019 3.18 × 1019 1.17 × 1015 4.11 × 1012 2.84 × 107 1.51 × 1010 1.53 × 109
7.0 2.5 5.0 6.0 10.0 6.0 5.0 5.0 3.0 5.0
λ(104 cal cm−1 sec−1◦ C−1 )
Table 7.3. Comparison between experimental and calculated critical temperatures. Adapted from Rogers [6]
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Fig. 7.3. Rates of heat production and loss as a function of temperature
Fig. 7.4. Effect of temperature on the time to explosion for TNT
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Fig. 7.5. Effect of temperature on rate for TNT
the behavior around the critical temperature. Note the extreme sensitivity to temperature in this regime. Another way to examine the effect of temperature is to examine its effect on rate. Using the values for TNT already given, we plot the results in Fig. 7.5. A change of 100◦ from 300 to 400 K yields a difference of 8 orders of magnitude in reaction rate. 7.1.2 Frank-Kamenetskii’s Approach The Semenov treatment assumes that there are no internal thermal gradients. This represents the condition of a well-stirred reactor. That is, all resistance to heat transfer occurs at the explosive surface. Although these conditions may be obtained in certain situations, in general they are not realistic. If the explosive is considered to have low thermal conductivity, then a different approach is required. A detailed treatment of the theory was given in Chap. 4. Here, we will jump right to the generalized conduction equation with heat addition that Frank-Kamenetskii used [4], which is div λ · ∇T − Cv
dT = −Qf (c) Z e−Ea /RT , dt
(7.5)
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where Ea is the activation energy, R is the gas constant, Z is the preexponential factor, Q is the heat of reaction during the self-heating process, and λ is the thermal conductivity. This equation does not have an analytical solution. At the time it was being considered by Frank-Kamenetskii, no numerical solutions were available either. Thus, researchers were forced to make simplifications while retaining essential elements in order to make the problems tractable and to solve the questions being asked. They gained remarkable insight and their solutions were very close to being exact in many situations. The key assumptions were that the process was at stationary state, the reaction did not depend upon reactant concentration (zeroth order), and exponential terms could be approximated. Additionally, he considered only three shapes: a slab, a cylinder, and a sphere. A reasonable question at this point would be: now that we have very strong numerical capabilities, why bother with simplified treatments. The answer is that great insight can be gained from working through the derivation of these equations, and considering the import and impact of various assumptions. It is very possible (and very often the case) that the equations can be solved numerically and nothing be learned about explosive behavior in the process. Also, following the derivations and the assumptions invoked can be most illuminating as to what is important and what is not. It is one of the best methods to learn the fundamentals of explosive behavior. After several assumptions and approximations, the F-K equations can be reduced to 2 a ρQZEa E = R ln , (7.6) Tc δλTc2 R where a is the effective dimension (i.e., radius of a sphere or cylinder, or half thickness of a slab), ρ is the density, δ is a dimensionless shape factor (i.e., 3.32 for spheres, 2.0 for infinite cylinders, and 0.88 for infinite slabs). Q is the heat of reaction; it is not the heat of combustion, heat of detonation, or heat effect during thermal explosion [6]. Rogers found that to first order, Q averages 500 cal g−1 , and can go as low as 300 cal g−1 or as high as 600 cal g−1 . Thermal conductivities range between 2 and 13 × 10−4 cal cm−1 s−1 ◦ C−1 for most organic explosives. 7.1.3 Critical Temperature Determination Henkin noted that when the reciprocal of the temperature was plotted against the time to explosion, a linear relationship resulted for some of the explosives tested. For others, a bilinear relationship was obtained. He derived Ea and Z values from these data, but found that the activation energies were much lower than those found in the literature. He ascribed the differences to heat absorption by the vapor phase. This can be mitigated somewhat by more careful sample preparation and reduction of head-space. However, in general, time to explosion experiments provide data that are not accurate enough for
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Fig. 7.6. Sample size effect for different explosives
the determination of kinetic constants. As Rogers reported [6], isothermal DSC can be used to obtain data of sufficient quality to make accurate critical temperature predictions. Critical temperature can be easily measured using the apparatus described by Rogers. This provides a value for a sample of specific size, shape, and density. Typically, when accurate kinetic and material constants are known, only two measured values of Tc obtained at two different sizes can be used to check the constants. If the data match the predictions, then the validity of the constants is assured and the F-K relationship can be used to predict behavior under different conditions. For example, Fig. 7.6 shows the effect of sample size for a set of different explosives using (7.6) and the data in Table 7.3. These data demonstrate the extreme dependence on sample size at low values of a. An increase in diameter from 1 to 10 cm results in a reduction of over 100◦ C in some cases. This illustrates the danger of inferring critical temperature from incomplete data. Also shown in the figure is the effect of sample shape, which is not as significant as the effect of diameter. Figure 7.7 shows an example of how the reaction moves from the surface to the center of the charge depending on the heating rate in terms of radius divided by diameter. If the heating is particularly rapid, illustrated here by imposition of a high boundary temperature, owing to the low thermal conductivity of explosives, the reaction will grow near the charge surface. This typically results in a low-violence pressure burst; as the majority of the explosive
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Fig. 7.7. Location of ignition as a function of heating rate. Values at specific temperatures represent times at 98%, 95%, and 90% of the time to explosion
is still at low temperature, significant thermal damage has not occurred, and as a result is not involved in the subsequent reaction. As the reaction moves toward the center, more explosive becomes involved and the violence increases. Depending on the sample characteristics and experimental conditions, it is possible to measure a large range in the values of E and Z. It has been found that these values can be linearly related when E is plotted vs. ln (Z). This is known as the kinetic compensation effect [7]. Each individual critical temperature can be matched with an infinite number of combinations. However, only the correct values of E and Z will predict the correct behavior over a range of particle shapes and sizes. This is illustrated in Fig. 7.8. The curves cross at the correct value of Tc but only one curve reproduces the proper values for Tc across the range of sample sizes. A test that measures critical temperature is very useful in determining a relative measure of thermal stability for a given sample size and shape. This is useful in its own right. It can also be used to determine material compatibility and also to check sample purity. Samples of reagent grade ammonium nitrate have very different thermal behavior than do samples from a commercial source – the critical temperature for the two materials differs by more than 100◦. While this demonstrates that the thermal explosion test can be used for quality control purposes, it actually points to a much more important conclusion; seemingly small or insignificant changes in an explosive can result in drastic changes in thermal response.
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Fig. 7.8. Effects of different E and Z on critical temperature calculations
Fig. 7.9. Effect of additives on critical temperature
One final example of the effects of changes is shown in Fig. 7.9. Lead was added in varying amounts to HMX and the time to explosion was measured. A reduction of 440◦ in the critical temperature was accomplished with the addition of 5% Pb. The difference decreased as more metal was added. We oftentimes attempt to save time and money on the number of tests conducted to determine the safety envelope for a new explosive or formulation.
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One method for doing this is to infer behavior for other concentrations by making measurements at either end of a range. For example, if we wish to add an inert binder to HMX, and if we are interested in concentrations of binder ranging from 0 to 10%, we might determine the drop height, spark, and friction values for pure HMX and HMX with 10% binder, and then infer that the results for any mixture in between will be no worse than that at the endpoints. With a true inert, this is most likely true. However, for materials that are not inert, the behavior can vary in unexpected ways, as demonstrated by the HMX/Pb data. The one-dimensional time to explosion test (ODTX) was a major step forward in quantifying the thermal behavior of explosives and in better enabling a comparison among different materials. The Henkin test, and others of similar construct, suffers from being multi-dimensional and difficult to conduct with precision with respect to heating rates and confinement. As mentioned previously, they are also of little use in abstracting chemical kinetic parameters for accurate thermal modeling. The ODTX experiment was designed by Catalano and colleagues and first described in 1976 [8]. It was the first major advance in standard time to explosion instrumentation and has proven to be very successful. It consists of a sphere of explosive placed between two heated hemispherical anvils. Explosive weight was approximately 2.2 g. In the original design, the anvils closed onto the sample in approximately 0.6 s, and the closing pressure and the confinement pressure could be independently adjusted. The assembly allowed no gas to escape and could be configured to allow a void space or no void, depending on the purpose of the experiment. The principal data obtained are times to explosion, with an explosion being defined in terms of the sound level using microphones. Violence of the event was determined after the fact by examining the anvils. The original device has been extensively modified and has resulted in greatly reduced data scatter. Initial results showed a marked decrease in critical temperature when compared to previous studies using other equipment. The authors concluded that the high, close confinement was responsible, because gas was forced into intimate contact with the explosive, thus accelerating the reaction process. They conclusively demonstrated that the time to explosion can be nearly doubled by merely increasing void space above the explosive, even if no gas is allowed to escape. They also found that using the F-K treatment does a reasonable job of determining the critical temperature, even though it is known that the kinetics are not single step and first order. This is because the critical temperature is determined primarily by the fastest reaction. However, the time to explosion is dominated by the induction time, which is a strong function of the slower reactions, many of which occur in the condensed phase. ODTX can be used to determine relative thermal stability, but it can also provide input for the development of global multistep chemical decomposition models in multiple dimensions. Given the more complicated chemistry,
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these codes are more capable of predicting time to explosion with increased accuracy. It was determined that for many explosives, at least one initial endothermic step was required and a total of 3–4 steps were needed to adequately describe the thermal behavior. Further, when coupled to other codes, and calibrated, relatively accurate approximations could be made of system response. A large number of explosives have been tested using the ODTX apparatus, and kinetic parameters have been determined and reported in the literature [9].
7.2 Visual Observation of the Cookoff Event The details of the deflagration processes occurring after a cookoff ignition were, until quite recently, largely neglected in experimental studies, and for many years it was generally assumed that deflagration proceeds, from an ignition point, in a more or less planar manner as has been observed many times in pristine energetic materials at room temperature. However, in reality, the reaction is progressing through a heated and thermally damaged material, and it has been demonstrated [10–12] that pressurized deflagration through damaged explosive can become very non-laminar. We here describe two tests that were successful in visualizing the internal structure that determines, to a large extent, the system response. 7.2.1 Mechanically Confined Cookoff Tests In a series of experiments designated Mechanically Confined Cookoff tests (MCCO) [13], the spread of reaction after cookoff was investigated directly in one such energetic material (PBX 9501). Small disks of PBX 9501 were heated with a confining glass or sapphire window through which the early stages of the combustion process could be observed directly by high-speed photography. A 25 mm diameter disc of PBX 9501, 5 or 2 mm thick, was confined with a metal ring, a thick steel base, and a thick toughened glass or sapphire top window through which the charge was observed. The strength of the edge confinement was determined by the thickness and the material of the metal ring; copper and mild steel were used in three different thicknesses (6.35, 3.175, and 1.588 mm). The disc of explosive was heated using a combination of a heating plate and/or heating tape either until ignition occurred spontaneously or up to a predetermined temperature, at which point the explosive was ignited at the center by means of a 100 μm thick, coiled, heated nichrome wire. Figure 7.10 shows the construction of the test assembly. Optical images of the event showed the onset and spread of reaction, and strain gauges bonded to the outside of the confining ring measured the early stages of ring expansion. Distinct variations were observed in the response of PBX 9501 to these tests
7 Cookoff 3/8 "
419
3/8 "
steel
toughened glass or sapphire
casing PBX steel steel
spacers 25.4 mm
Fig. 7.10. Test assembly schematic
0 µs
5 µs
10 µs
Fig. 7.11. PBX 9501 ignited at 195◦ C
as the conditions were changed. Some of these variations were reproducible, that is, apparently due to the test configuration, while others were not. Figure 7.11 shows the typical behavior of a 5-mm-thick disc that has been heated to 190◦ C in around 1 h and then ignited by hot wire at the center. The results show that following the ignition of the charge, rapid pressurization at the center causes luminous radial cracks to open and that these are driven outwards at several hundred meters per second. The freshly opened surfaces appear to be ignited as the cracks progress. The dark strip is a thin copper foil conductor that carries electrical current to the top of the igniter wire, which runs perpendicular to the plane of the figure. No failure of the window is evident up to 10 μs. Figure 7.12 shows another longer sequence under similar conditions and lighting. This behavior appears quite reproducible, with typically 3 or 4 cracks that may subsequently bifurcate. Failure of the toughened glass window is visible at 30 μs, spreading fast to obscure the field of view at 35 μs. To investigate the effect of confinement, the tests were repeated with the same thickness (5 mm) of HE, a thick steel ring, and a thin copper ring. Figure 7.13 shows the response with a 6.35 mm steel ring, and Fig. 7.5 with a 1.588 mm copper ring. With the thick steel ring, cracks appear to start as before, but the pressure rises faster and the glass window breaks much earlier.
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0 µs
5 µs
10 µs
15 µs
20 µs
25 µs
30 µs
35 µs
Fig. 7.12. Ignition of PBX 9501 at 200◦ C
Fig. 7.13. Ignition of PBX 9501 at 200◦ C with thicker confinement
Initial fracture of the glass is visible at 5 μs, and the apparent bright cracks at 10 μs are caused by the refraction of light from the reaction by multiple cracks in the glass window. With the weaker confinement (Fig. 7.14), a much more extensive network of cracks and reaction sites appear to develop, and the window does not break within the timeframe of the high-speed images. Further tests were conducted both by heating the samples until self-ignition occurred and by wire-igniting at higher temperatures (∼220◦C). In both cases, the observed response is noticeably different from that observed around 200◦ C. Figure 7.15 shows a sample heated to ignition. It is not clear from the images whether the original ignition is localized to a small region, from which it spreads out, or whether there are multiple ignition sites. Part of the ambiguity may arise because the ignition is almost certainly
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0 µs
10 µs
5 µs
15 µs
421
Fig. 7.14. Ignition of PBX 9501 at 200◦ C with thin copper confinement
Fig. 7.15. Spontaneous ignition of PBX 9501
subsurface. The well-defined cracks observed at lower temperatures seem to be absent, and the glass window breaks while the reaction is apparently still localized around the center of the ring. The earlier time at which the window breaks (15 μs) suggests that the reaction rates are higher than in the lower temperature shots. To try to locate more precisely the original ignition sites, the spontaneous ignition tests were repeated with thinner discs. Two quite distinct outcomes were observed. Figures 7.16–7.18 show the temperature profile, heating rate, and high-speed images from one of these tests. The behavior is at least superficially similar to the self-ignited thicker discs. Reaction is observed over a region of the sample, and this spreads, apparently without the formation of large cracks. This type of response was observed in several samples. In contrast, Fig. 7.19 shows the images from a nominally similar test. After a similar temperature ramp profile, the base temperature was 6◦ C higher
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temperature (C)
200
150
100 base window interface
50 0
1000
2000 time (s)
3000
4000
Fig. 7.16. Temperature profile for spontaneously-ignited 2 mm thick sample 100×10–3 90 80
dT / dt (uCs-1)
70 60 50 40 30 base window interface
20 10 0
40
60
80
100
120 140 160 temperature (uC)
180
200
220
240
Fig. 7.17. Boundary heating rate profile for spontaneously-ignited 2 mm thick sample
Fig. 7.18. Spontaneous ignition of a 2 mm sample
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Fig. 7.19. Spontaneous ignition of a 2 mm sample showing probable detonation
at ignition than in the previous test. The top temperature was not recorded due to the failure of the thermocouple. Reaction appears to start in the same manner, but shortly after the failure of the window at 9 μs, a brief but intense flash of light was observed through the heavily fractured window (18 μs). Inspection of the remains of the test assembly indicated that the sample had at least partially detonated. No violent responses of this kind were observed with the 5 mm thick discs. In the lower temperature shots, it appears that pressurization at the center resulting from ignition, which leads to tensile hoop stresses in the surrounding material, causes the formation of radial cracks. Weaker confinement leads to more cracks, while stronger confinement suppresses crack formation. In the high-speed photographs, we observe these cracks as highly luminous, which indicates that the surfaces of these cracks have been ignited. It is not clear whether this ignition is the result of hot product gases flowing down the cracks, or whether the energy dissipated at the crack tips is sufficient to cause ignition. Calculations [14] suggest that, while we know that at room temperature the propagation of a crack through an explosive crystal or charge does not dissipate enough energy at the crack tip to cause ignition, there may be a critical temperature above which this could occur. As the temperature increases, the activation energy barrier reduces; if that were the only factor, then at temperatures near enough to the ignition point, crack energy would be sufficient. However, we primarily observe the large cracks at temperatures around 190◦C, which is still well below the ignition temperature of HMX. At higher temperatures, both in the wire-ignited and in the self-ignited shots, the well-defined cracks were not apparent. Reaction rates were higher, as might be expected, but the material response was different. Whether that is due to a change in the material properties of the PBX 9501 as the temperature is raised or some other factor(s) is unclear. Allowing the samples to self-ignite does lead to a change in the temperature profile and location of ignition. As the exothermic reactions develop, the base and glass window become heat sinks, and so the temperature rises higher inside the disc of HE, leading to an internal ignition. Figures 7.16 and 7.17 show the heating profile and heating rate as a function of temperature for a self-ignited shot. On the upper surface (HE/glass window interface) record, the endotherm resulting from the β − δ
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phase change is visible, as is the exothermic runaway just before cookoff, while these are not seen on the lower surface record. The reason for this is that the metal base is a better heat sink than the glass or sapphire window. These two temperature records were used as boundary conditions for a numerical calculation of the behavior of the sample using the reduced 4-step Arrhenius decomposition scheme [15]. This indicated that ignition would have occurred around a distance of one-third of the sample thickness below the surface. Subsurface ignition is likely to change the observed spread of reaction to some extent. The widely varying behavior of the thin, self-ignited samples is also difficult to account for, and we noted that these are small samples, which would not normally be expected to detonate as a result of thermal cookoff. We further propose that a DDT mechanism is responsible for the events, and that although it could not occur in a pristine, room-temperature PBX, a combination of factors might allow it to occur in a small, cooked sample of PBX. Those factors may include phase-change and decomposition-induced porosity, arising from a prolonged period at high temperature and leading to the possibility of bed-collapse and compaction/shock wave formation, together with the increased chemical kinetic rates associated with the high temperature. If such conditions occur in the cooked sample, a mechanism such as that previously proposed for DDT in porous beds [16, 17] may be occurring. It was noted that the samples that detonated reached a higher temperature before ignition, and that it may account for the variation in behavior; one possibility is that, in a relatively small sample, the ignition process may be controlled by the presence or absence of large crystals of HMX, which would be expected to cookoff earlier than the smaller ones. PBX 9501 has a bimodal crystal size distribution centered approximately on 25 and 130 μm [18] and those samples that ignited later and hotter may have lacked large particles. The observation of DDT in such small samples supports other observations that both the material state and chemical kinetics are very different after prolonged periods at elevated temperature, and that these changes effectively reduce the insensitivity of the PBX [19–21]. 7.2.2 Cindy Test The MCCO experiments used explosives that were ignited at elevated temperatures. Tests at room temperature have also been performed. The previous experiments elucidated the importance of reactive cracks involved in reaction violence in both thermally ignited experiments and impacted explosives, which stand in contrast to the common classical assumption that combustion in these materials proceeds as a simple wave that propagates uniformly from an ignition point. This section presents work conducted at ambient temperature and demonstrates that the same behavior observed in the MCCO experiments also occurs at room temperature. In accident scenarios, relatively low-speed deflagration processes play a critical role in determining the ultimate reaction violence [22, 23].
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Reaction propagation may begin as conductively driven (normal) combustion, transit to convective combustion (advective energy transport), induce compaction-initiated combustion (mechanical dissipation), and possibly lead to detonation (propagation involving shock waves). Normal deflagration and detonation are relatively well studied. Convective and compaction-driven combustion have received much less study and are not well understood. As the MCCO experiments involved hot PBX 9501, there are two possible mechanisms for reaction in the cracks. One involves crack tip ignition, where the small amount of energy released at the crack tip as the crack opens is sufficient to trigger reaction and thus propagate the combustion along with the crack. Another possibility is that pressure produced by reaction at the center of the disk opens cracks and the existing pressure gradient forces in hot gases, resulting in crack ignition via convective transport. We report here on a set of similar experiments using unheated pristine PBX 9501 as well as heated PBX 9501 that has been allowed to cool before ignition. The sample in each case is at room temperature when ignited and thus the small amount of energy released at the crack tip is insufficient to heat the sample to ignition. The energetic material used in this work is PBX 9501. Samples for these experiments are machined to the required shapes from PBX 9501 molding powder that has been hydrostatically pressed to 1.83 g cm−3 . Figure 7.20 shows the unassembled view of the cell used in the experiments involving ignition of a confined disk of PBX 9501 that we will refer to as the Cindy test. This test uses a 1.00-in. diameter, 0.20-in. thick disk with a 1.0mm hole in the end for sample ignition. The sample is surrounded by a copper ring and is sandwiched between a hardened glass window on one side and a steel flat on the opposite side. The steel flat contains a pressure access port at its center that allows monitoring of the pressure in the ignition hole at the center of the disk. An electric current-heated coiled NiCr wire ignites the sample in the center hole of the disk. Thin pieces of insulated copper foil on each face of the sample disk conduct current to the NiCr wire. Thin polytetrafluoroethylene sheets between the sample and the steel flat on one side and the window on the other produce a seal that allows pressure from initial pyrolysis products to build up sufficiently in the ignition hole to produce sustained combustion. We use several diagnostic tools to monitor the combustion of the disk. Fast framing cameras were used to photograph the event, and pressure sensors monitor the transient pressure in the sample ignition hole. In addition, strain gauges provide ring deformation data. Radial burn experiments known as Cindy tests [11] resemble the mechanically coupled cookoff experiments of Dickson et al. [24], except they are conducted at room temperatures rather than at elevated temperatures and also provide record of the pressure in the sample ignition hole. Before and after pictures of the 0.40-mm-thick copper-ring Cindy test mounted in the cell are shown at the top of Fig. 7.21. A ∼3 A current passed through the coiled NiCr wire in the sample ignition hole pyrolyzes the PBX 9501 in the hole and ignites
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Fig. 7.20. View of disassembled Cindy test cell
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t = 0.815 ms
t = 0.840 ms
t = 0.864 ms
t = 0.889 ms
t = 0.914 ms
t = 0.938 ms
t = 0.963 ms
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Fig. 7.21. Image sequence from a pristine PBX 9501 Cindy test with a 0.40 mm copper ring
the sample. The first four frames in the sequence at the bottom of Fig. 7.21, each separated by 25 μs from the next, show luminous reaction progress growing at the center of the sample. Luminous reaction appears in several cracks in Frame 5 and is fully developed by Frame 6, indicating flame travel velocities of about 500 m s−1. Other experiments with 0.80-mm and 1.60-mm-thick copper rings show similar cracking patterns and luminous reaction propagation rates, the same as those seen in the elevated temperature mechanically coupled cookoff experiments of Dickson et al. [24]. Figure 7.22 shows the pressure record at the ignition hole for the 0.40 mm copper ring experiment. All of the Cindy tests using pristine PBX 9501 experiments show similar traces with maximum pressurization rates of about 107 psi s−1 and peak pressures of about 4–5,000 psi, regardless of ring thickness. Figure 7.23 shows the superimposed pressure transducer and strain gauge outputs for an experiment with 1.6-mm thick copper ring confinement. Pressure rises to a peak that corresponds temporally with the onset of ring deformation. Ring deformation occurs rapidly, resulting in loss of confinement and hence pressure. The consistent pressure performance likely reflects the strength of the PBX 9501 sample sandwiched between the back plate and the window. While the copper ring provides some support for the sample, it appears to quickly yield when pressures are sufficient to damage the PBX 9501 and flames appear in sample cracks.
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4000
Pressure / psi
3000
2000
1000
0 –10x10–3
–8
–6
–4 –2 Time/s
0
2
4
Fig. 7.22. Pressure vs. time for 0.4 mm copper ring Cindy test 6000
10jul_1_CindyTest 1.6mm Copper Ring Ambient external pressure and temperature Maximum pressure = 6182 psi Maximum strain = 792 microstrain
Pressure / psi
400 2000 200
0
0
Ring Strain/microstrain
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Strain Pressure
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–200 –0.5
0.0
0.5
1.0
1.5
–3
2.0x10
Time/s
Fig. 7.23. Superimposed pressure vs. time and strain gauge plot for 1.6-mm copper ring Cindy test
A second set of radial burn experiments was conducted using thermally damaged PBX 9501. Thermally damaged samples of PBX 9501 are prepared by assembling the cell with a pristine sample. The assembled cell with sample is heated to 180◦C for 45 min and then cooled to room temperature before the sample ignition. Figure 7.24 is a micrograph showing typical thermal damage in a piece of PBX 9501 that has been subjected to this treatment
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Fig. 7.24. Thermally damaged PBX 9501 micrograph
[11, 25]. In addition to crystalline damage due to the HMX β -δ phase change, heating PBX 9501 produces changes in the binder [26]. Smaller HMX particles appear to dissolve in the hot binder, producing significant macroscopic changes in PBX 9501 and introducing fairly uniformly distributed porosity at the particle scale. Figure 7.25 shows a sequence of images from a thermally damaged PBX 9501 1.6-mm ring experiment and Fig. 7.26 shows superimposed pressure and strain gauge traces for the same experiment. As in the pristine–sample experiments, pressure rise leads to ring strain, indicating that it is the sample strength that determines peak pressure. Reaction spread velocities are greater than that the video record can resolve, at least 500 m s−1 . The images are qualitatively different than those from the pristine PBX 9501 experiments. The thermally damaged material shows more illuminated cracks and some of the cracks appear to branch. Maximum pressures observed in thermally damaged Cindy tests are roughly half those seen in tests on pristine material and the pressurization rates are more than twice those of the pristine material. These results can be understood in light of the physical changes that occur in the thermally damaged PBX 9501. Because of the changes that occur in the crystalline HMX and the binder, thermally damaged samples are not likely to have the strength of pristine samples [26] and therefore will not support
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t = 0.272 ms
t = 0.296 ms
t = 0.321 ms
t = 0.346 ms
t = 0.370 ms
t = 0.395 ms
t = 0.420 ms
t = 0.444 ms
Fig. 7.25. Image sequence from a pristine PBX 9501 Cindy test on thermally damaged PBX 9501 with a 1.6 mm copper ring 3000
2500
16jul_1_CindyTest Thermally damaged sample, 1.6 mm Copper ring Maximum pressure 2890 psi Maximum strain 1258 microstrain
1000 800 600
1500
400
1000
Strain/microstrain
Strain pressure
2000 Pressure / psi
1200
200 500 0 0 –400
–200
0 Time/s
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400×10–6
Fig. 7.26. Superimposed pressure vs. time and strain gauge plot for 1.6 mm copper ring Cindy test on thermally damaged PBX 9501
the same pressures seen in pristine materials before failing. The increased pressurization rate is likely due primarily to convective burning in the increased surface area, which results from the thermally damaged material’s greater void space.
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Since these experiments involve room temperature PBX 9501, the small amount of energy released at the crack tip is insufficient to heat the sample to ignition. This rules out the possibility that the luminous-reaction-spread rate simply reflects crack propagation rate in the sample. It is likely that the pressure produced by reaction at the center of the disk opens cracks and the existing pressure gradient forces hot gases into the crack, producing convective burning of the crack surface. Previous elevated-temperature experiments reveal luminous combustion propagating through cracks in a confined sample of PBX 9501 at 500 m s−1 . Analogous room temperature experiments with both pristine and thermally damaged PBX 9501 samples produce similar combustion propagation rates. Experiments with thermally damaged samples consistently achieve lower peak pressures than pristine samples before luminous reaction appears. The consistent pressure at which luminous reaction appears in sample cracks likely reflects the strength of the PBX 9501 sample in our experimental configuration. These room temperature experiments show that crack ignition is caused by hot gases entering existing or pressure-induced cracks rather than by energy release at the crack tip during the opening of the crack.
7.3 Small-Scale Instrumented Thermal Ignition Experiments In an effort to predict the reaction violence of PBX 9501 subject to a particular environment (thermal, fragment, shock, etc.), experiments and model development are constantly being refined. The refinement is seen both in additional observables and in the ability of models to demonstrate the phenomena observed as well as in the more precise nature of the questions being asked on subsequent experiments and models. Initial experiments on PBX 9501 yielded time to explosion as a function of boundary temperature with the ability to vary temperature, confinement, and void volume [27, 28]. These experiments used spherical charges and uniform boundary temperature, making them essentially one-dimensional, thus the name one-dimensional time to explosion (ODTX). The time to explosion vs. temperature data they yielded falls on a single line on the Arrhenius-type log time, inverse temperature plot (see Fig. 7.27) [29, 30]. The models generated to capture the ODTX data include the three-step mechanisms [28]. While ODTX and three-step models captured the time to ignition, which was the only observable in these experiments, they did not answer questions about energy release and reaction violence. They did not measure internal temperature distributions or pressurizations, and therefore cannot adequately constrain models. The addition of internal temperature diagnostics was needed to constrain models aimed at determining the location of ignition. The radial thermal explosion studies from LANL and the Sandia Instrumented Thermal
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Fig. 7.27. Time to explosion data placed onto an Arrhenius-type log plot
Ignition (SITI) test from Sandia National Lab are examples of tests designed to measure the internal temperature distributions [15, 31]. These tests use cylinders of explosive encased in a metal confinement sleeve and heated from the outside. The SITI test is 25.4 mm diameter by 25.4 mm long and the LANL radial tests are run at both 25.4 × 25.4 mm2 and 25.4 × 50.8 mm2 sizes. The metal sleeves have been made from both copper and aluminum. The LANL radial tests have been run as approximately two-dimensional tests with insulated endcaps for minimal axial temperature gradients, and three-dimensional tests with uninsulated conductive endcaps that drives and axial gradient forcing the hottest location to the axial midplane to allow for diagnostics to be placed at predetermined hottest location [15]. The SITI experiment has been used to collect hundreds of data points on different explosives at varying temperatures and amounts of void volume [31]. These tests were initially driven by the need to place temperature diagnostics internal to the heating explosive. The suite of diagnostics therefore
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Fig. 7.28. The LANL radial experiment
initially started with boundary temperature and spatially resolved internal temperatures. It has been expanded to add fast temperature measurements, case strain, gas pressure, and spatially resolved solid density. The LANL radial experiment is shown in Fig. 7.28. The left panel shows the outer aluminum casing with partial transparency so that the internal cylinders of PBX 9501 are also visible. The right panel shows the diagnostics placed at the midplane of the HE. These experiments are conducted with 12.7 mm diameter and 12.7 mm tall halves held together by the aluminum casing. There is one joint in the aluminum casing that is sealed with a combination of silicon sealant and high temperature epoxy to minimize gas loss. Success in sealing the experiments is judged by the consistency of the time to ignition from shot to shot and agreement with time to ignition as a function of temperature measured by others [31, 32]. These shots are heated from the outer aluminum casing using resistive heating wire. They are designed with enough ullage at the ends to allow for the difference in thermal expansion between the HE and the aluminum and for the volume expanding β-δ phase change. The thermal profile used is a heating stage at 70◦ C to balance the halves, followed by a 5◦ C min−1 ramp to 178◦ C and a soak at this temperature for 20 min to allow the β-δ phase transition to complete and the material to re-equilibrate at the boundary temperature. After 20 min, there is a final ramp of 5◦ C min−1 to 205◦ C and the boundary is held at this temperature until ignition. During this soak, the PBX 9501 begins generating heat and the aluminum boundary becomes a heat sink driving ignition to the spot furthest from the walls to the center of the piece. The midplane of the piece is outfitted with an array of temperature sensing diagnostics including type K thermocouples and 200 μm core silica fiber optics. Figure 7.29 shows the temperature gradient evolution across the midplane of the HE. Initially, the walls are hottest, but as the exothermic processes begin and the walls remain thermal conductors at a fixed boundary temperature, the hottest point moves towards the center of the HE.
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temperature (C)
250
200
1) Thermal Decomposition: initial endo / exothermic steps: ~ hour 2) Final exothermic steps: ~30 minutes Self heating: ~25 minutes Thermal runaway: ~5 minutes 3) Ignition: ~100 microseconds Burn propagation Case burst
150
100
50 1000
2000
3000
4000 time (s)
5000
6000
Fig. 7.29. The temperature gradient evolution across the midplane of the HE
Having mapped out the internal temperature distribution leading up to thermal explosion, the question can be further refined. The temperature distribution measured with thermocouples sampled by commercial data acquisition systems can be used to model the thermal decomposition steps leading up to thermal ignition, the preignition regime. However, they are not fast enough to study the transition from the preignition to the post-ignition burn propagation regime. The goal for this series of experiments was to study the transition from heating in place to ignition propagation through heated HE. To this end, the thermocouple-based temperature measurements are modified for faster time response, and fiber optics are added for pyrometric temperature measurements. The thermocouple voltage outputs are measured directly using an oscilloscope as an analog to digital converter with a microsecond time base. Laser heating measurements of these thermocouples in air and in water give a response time of between 1.5 and 10 ms depending on the thermal conductivity of the surrounding medium [33, 34]. This range is used to deconvolve the temperature measurements in the ignition volume and back out actual HE temperatures. The fiber optic output is coupled to an InGaAs amplified photodiode with a response time of 100 ns. The gain setting and fiber diameter chosen give us a minimum temperature sensitivity of 800 K. Calibration of the fiber output is performed assuming an emissivity of the thermally damaged PBX 9501 of 0.625 ± 0.375. This uncertainty in emissivity correlates to a temperature uncertainty of ±100 K for peak temperatures measured at 1,500 K as shown in Fig. 7.30. The emissivity of pristine PBX 9501 at room temperature is 0.7. The emissivity certainly changes with the thermal decomposition, but the range of 0.25–1.0 is chosen to bracket the emissivity between that of a metal and that of a perfect blackbody. While a 100 K uncertainty at low
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1600 ε = 0.25 ε = 0.5 ε = 0.75
1400
Temperature (C)
1200
ε=1
1000 800 600 400 delta T(max-min)
200 0.0
0.2
0.4
0.8 0.6 InGaAs signal (V)
1.0
1.2
Fig. 7.30. Uncertainty in emissivity correlates to a temperature uncertainty of ±100 K for peak temperatures measured at 1,500 K
temperatures would limit their utility in pinning down a reaction mechanism, an uncertainty of 100 K in the measured temperature regime of 1,500 K still provides a constraint on reaction rate. Figure 7.31 shows the effects of this temperature uncertainty on constraining the reaction mechanism on an Arrhenius-type log time vs. inverse temperature plot. The temperature error bars are shown as the dashed line above and below the data line. The red point is the measured high temperature. The measured temperatures as the central volume runs to ignition are illustrated in Fig. 7.32. The structure on the tens of microsecond scale is reminiscent of the burn structure set up above a regressing surface, with the surface temperature pinned at approximately 400◦ C and then a dark zone above that, at ∼800◦ C and a bright zone above that [35]. The next observable necessary to fully describe the material response is the generation of gas pressure from the solid decomposition. To measure this, a density sensitive diagnostic is needed. Proton radiography is used to this end. Transmission images were obtained using the 800 MeV proton accelerator at Los Alamos National Laboratory [36]. The proton beam is delivered in beam gates of 800–1 ms duration within which individual proton pulses were delivered with arbitrary duration and inter-pulse time. The proton beam is focused using a quadrupole magnetic configuration to create an image plane. The sample is illuminated in this plane and the spatially resolved transmission through the sample is converted to optical luminescence from a scintillator material and recorded by digital framing cameras. The magnification in these experiments is such that the full field is 50 × 50 mm2 with a spatial resolution of 30 mm. Recorded images were digitally processed to normalize against the proton beam intensity and spatial variability. Images taken with proton radio-
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Fig. 7.31. The effects of this temperature uncertainty on constraining the reaction mechanism
graphy show material density evolution. Figure 7.33 shows the spatial density gradient developed over the final tens of seconds leading the thermal ignition. The internal temperature gradients shown in the top panel of Laura Fig. 7.7 encode a density gradient into the HE. The bottom panel of Fig. 7.33 shows the line profiles of each of the final 15 images, illustrating the conversion from solid to gas at the central ignition volume just prior to ignition. This central ignition volume is seen to be within the central 1 mm diameter both from the fast temperature traces bracketing it and from the spatial density gradients. A series of experiments were performed to probe the effect of the thermal decomposition at elevated temperatures on final explosion violence. In these experiments, radial thermal explosion experiments of 12.7 × 25.4 mm2 are run
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Fig. 7.32. Temperature measured at central volume by thermocouple (line with points) and by fiber optic (solid line). The time axis is plotted in reverse time as time before ignition with ignition at time zero
Fig. 7.33. The spatial density gradient developed over the final tens of seconds leading to the thermal ignition
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with an additional soak time at 180◦ C held for a variable length of time [37]. The observable used to probe reaction violence is the wall velocity at ignition. The results of these experiments are that the ignition location moves from the boundary towards the center of the HE as the soak time increases. The time to ignition initially decreases with soak time, but then begins to increase with soak time again. This is interpreted as initially accelerating reaction due to the onset of thermal decomposition during the soak, but then to decrease the reaction rate due to the slow loss of gases during the thermal decomposition of the soak. The HE response follows from preignition thermal decomposition, to ignition, and then post-ignition burn propagation. So far, the observables have yielded information up to the onset of ignition. To follow the reaction past the ignition event, a diagnostic is needed which can image the event with microsecond time resolution through the complications of burning explosive. Proton radiography is again used to provide details of the burn propagation and case rupture in the radial thermal explosion experiments. The first detail of this experiment that needs to be addressed is the technical difficulty of applying a high energy radiographic technique to a thermal explosion. Proton radiography uses proton pulses generated by an accelerator, which cannot be triggered on demand. This diagnostic needs to provide the master clock for the experiment, rather than the other way around. Until now, the thermal explosion has been allowed to evolve and all diagnostics triggered by the ignition event. For dynamic radiography, a trigger has to be used, which senses when the thermal ignition is imminent. Then a proton pulse triggers the ignition event to occur within the sub-millisecond duration of the proton pulse. A laser temperature jump technique developed at LANL is used to synchronize the ignition event [38]. Laser heating of the central volume is accomplished by fiber coupling the 1064 nm fundamental of a free running Nd:YAG to the central ignition volume via a 200 μm core polyimide-coated fiber optic. The fiber optic is chosen to withstand the 205◦C temperature of the experiment and the 200 μm diameter is chosen as a compromise between good bend diameter for mechanical robustness and large diameter to easily couple in the laser light. The Nd:YAG is used in the free running mode providing a pulse train of approximately 100 pulses, each ∼5 ns wide over the ∼150 μ s envelope of the flashlamp. This allows the central volume to be heated within a 1 ms window without generating an ablation shock in the material, which could perturb the burn propagation. The thermocouples nearest the cleaved fiber end record the temperature rise generated in the material a few millimeter from the ignition volume. The final temperature achieved, while not directly measured, can be determined from the time until ignition breaks out [32]. Fig. 7.34 shows the ignition delay as a function of laser pulse energy. The delay of 350 μ s at 25 mJ corresponds to a central temperature jump of 230◦ C. With a synchronization technique available, dynamic radiography of burn propagation is possible. This diagnostic yields information on how the central ignition event propagates to the case and subsequently bursts the case.
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Fig. 7.34. The ignition delay as a function of laser pulse energy
For dynamic imaging, the central thermocouple is used both to monitor thermal runaway and to provide a trigger for image collection. When the central volume is observed to be in the thermal runaway regime and within minutes of ignition, a request is made for a 1 ms duration proton macropulse. The availability of a proton macropulse sends a TTL pulse to fire the laser, which is coupled to the central ignition volume of the HE. The laser pulse provides a temperature jump at the central, volume which short circuits the thermal runaway and causes ignition to break out within hundreds of microseconds. The central thermocouple is monitored and when ignition is observed, proton micropulses and image collection are triggered. This allows the collection of multiple frames during the burn propagation and case expansion and breakup regimes. These images show cracking from the central ignition volume propagating to the case walls within tens of microseconds after the breakout of ignition (Fig. 7.35) The cracks are propagating through PBX 9501 with a temperature gradient from 230 to 250◦ C to the case that is held at 205◦C for both. In addition to the macroscopic crack formation, an increase in transmission with approximate radial symmetry is observed to propagate from the central volume towards the case with a velocity on the order of 165 m s−1 in the radial direction and 200 m s−1 in the axial direction. This increase in transmission is interpreted as arising from a burn front that is consuming material as it runs, causing the increase in proton transmission. The velocities associated with the burn front imply that it is a convective burn. While the transmission increases as the burn front propagates, the remaining transmission behind the burn front indicates that only a small
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Fig. 7.35. Images that show cracking from the central ignition volume propagating to the case walls within tens of microseconds after the breakout of ignition
fraction of HE is actually consumed by the convective burn. The transmission behind the front increases with time corresponding to a continued material consumption lasting on the order of a 100 μ s after the convective burn front has passed. An interpretation of burn mechanism consistent with these observations is that the convective wave lights the material, but only consumes a small fraction. The rest of the consumption occurs more slowly by a conductive burn. Using the compiled knowledge of burn rates vs. pressure for PBX 9501, the convective velocities observed indicate a pressurization on the order 1 GPa. Assuming this same pressure behind the conductive burn gives a mean particle size to be consumed of 10 μm. Figure 7.35 shows the ratio of proton transmission images collected during the dynamic event to those collected before ignition to enhance the change in HE density due to the burn propagation. The change in transmission is represented in the images by an inverted gray scale with black representing an increase in transmission and white representing a decrease in transmission. These images are taken viewing along the cylindrical axis of the radial thermal explosion experiment. Figure 7.36 shows the direct proton transmission images taken during the dynamic event and viewed with the optical axis normal to the cylindrical axis of the sample. These images show the case response to the HE pressurization. Initially, the
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Fig. 7.36. The direct proton transmission images taken during the dynamic event and viewed with the optical axis normal to the cylindrical axis of the sample
case opens up at the midplane seal and the wall begins to bow out. This is followed by the endcaps bowing and finally punching out plugs from the aluminum when the convective front reaches them. The next refinement of the question of reaction violence is pressurization rate. This will allow random case geometries and confinements to be modeled. Work is ongoing to develop diagnostics for use in the thermal explosion environment to measure both the quasistatic and dynamic pressurization.
7.4 Quantification of Reaction Violence To this point, we have been discussing the mechanisms of thermal explosions. We have also addressed their effects and determined ways in which we could understand how to determine when an explosion might occur. Now we will address the question of how to quantify the violence from a given explosion and how such violence might be controlled. Further details of the work presented
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in this section can be found in [54] and [64]. Any explosive subjected to a given time at temperature will undergo both mechanical and chemical changes, and these changes may substantially modify the performance of the explosive and the behavior during a cookoff event. The explosive behavior may range from a nonviolent deflagration to a violent thermal explosion. In extreme cases, the thermal reaction may transit to detonation (DDT). To effectively mitigate these effects, a more complete understanding of the behavior is required. The ultimate goal of the research presented in this section is to explore the effect of several factors on the violence of a thermally initiated explosive. Three of the factors represent typical cookoff experiment parameters: soak time, soak temperature, and HE mass. Two more factors are mechanical confinement and inertial confinement. These factors are known to influence the violence of a cookoff event and have a strong effect on the likelihood of DDT. Here we discuss the effect of the two important confinement factors on the violence of reaction. We use the word violence to describe the dynamics of an explosive event transferring energy to an adjacent object at a high rate with sustained high pressure. The explosives research community commonly speaks of violence as a metric of energy release rate during an explosive event. This metric is often the subjective observation of an experienced researcher, or may come from measuring fragment velocity during the event and quantifying fragment size after an event. This metric persists even though we have other quantifiable measurements that succinctly reflect explosive performance: power, energy, overpressure, and impulse. We quantify violence by reporting three important quantities: total enthalpy change, energy release rate (power), and the P 2 τ (product of overpressure and impulse) criteria for explosive initiation [39–41]. We know that for a thermal explosion to be violent, the quantity of energy released must raise the pressure inside the confining vessel to a point where its strength is exceeded, and the rate of release must add energy at rate that builds high pressure before the container fragments and surrounding gas can move appreciably. Baker et al. provide in-depth discussion of this phenomenon [42]. In the experiments reported here, the rate condition was always met, and all the shots were considered violent. The combined effects of total enthalpy change and the rate, coupled with the confinement, gives rise to the pressure– time characteristics. We also discuss how the enthalpy change and rate were used together to form a ranking for violence. We also use P 2 τ as a separate ranking for violence and compare the two methods. To examine these effects, we have subjected PBX 9501 explosive samples to a given thermal history under well-controlled laboratory conditions, and then raised the temperature until the samples auto-initiated. This was done in a gun-type apparatus where a small explosive sample was loaded into a chamber sealed with a calibrated rupture disk. The apparatus mimicked a onedimensional configuration where the explosive was sandwiched between a stationary (massive) block and a confining structure. After initiation occurred, pressure rose to a point at which confinement fails and the fragment (projec-
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tile) began to move. During this initial stage of the process, the explosive was mechanically confined. At the failure pressure, the burst disk ruptured and the fragment (projectile) accelerated and the explosive was confined by inertia. Drag effects were not considered. Our experimental apparatus mimicked this situation and the subsequent data reduction determined the P 2 τ delivered to the acceptor (pressure sensor) and the power and enthalpy change of the explosive as a function of mechanical confinement strength and inertia. This study was in the limit of strong confinement where the mechanical confinement was adequate to contain either 9 or 35% of the stored chemical energy (based on the known enthalpy of detonation [43]) in the explosive and the charge-to-projectile-mass ratio was 0.05 or 0.1. The experimental section that follows describes the apparatus in detail. We are also investigating the effect of weak confinement, which has been observed to have a strong effect on violence of reaction. There have been several notable experiments where variable confinement was an experimental feature. Dagley et al. [44] used the “Small-scale Cook-off Bomb” and “Super-small-scale Cook-off Bomb” primarily to study heating rate effects on violence. However, they did explore various confinements and found that confinement affected violence, but their focus was on thermal effects and confinement was not systematically varied. Dickson et al. [13] developed the mechanically confined cookoff (MCCO) experiment that focused on reaction propagation mechanisms, and also examined the effect of the confinement. They briefly concluded that lower confinement led to the formation of many small cracks in the explosive and higher confinement caused the formation of fewer, larger cracks. Collignon et al. [45] performed a study relevant to the current work. They varied confinement until a detonation occurred. This study found Livermore Explosive (LX)-14(N) to have the lowest transition-toviolent-reaction threshold of 1,200 psi of all the HMX-based explosives examined. LX-14(N) is very similar to PBX 9501, containing 4.5% estane and 95.5% HMX. Berghout et al. found a similar transition to violent reaction at 1,334 psi [11]. Chidester et al. noted that confinement played a significant role in cookoff violence and varied confinement in their experiments [46]. However, their experimental program confounded the effects of confinement with other factors. 7.4.1 Experimental Cylindrical samples (0.25 in. diameter) of PBX 9501 were thermally initiated in a radially confining cylindrical chamber. This chamber was sealed at one end by a piezoelectric pressure sensor and sealed at the other by a calibrated burst disk. The empty volume of the breech was 1.6 cc (ullage) and the volume of the explosive was 0.16 cc. The burst disk separated the
explosive from a projectile and gun barrel. Samples were heated rapidly 3◦ C min−1 to a soak temperature, held there for a given soak time, and rapidly heated until an explosion occurred. At the calibrated pressure, the burst disk ruptured and the expanding gas pushed the projectile down the gun barrel. The piezo
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Fig. 7.37. Schematic of the experimental apparatus. All dimensions are in millimeters. The ullage volume was 1.6 cc and the explosive volume was 0.16 cc
80x10–3
160x106
Pressure [Pa]
Press. Data Press. Model
80
Proj. Position
40
40
Position [m]
60
120
20
0 0.0
0.4
0.8
0 1.2×10–3
Time [s]
Fig. 7.38. Typical pressure and position data. Analysis using the raw pressure data was complicated by the pressure ringing and the pressure model shown was used for calculations
pressure sensor provided high-fidelity pressure information, and a microwave interferometer provided position–time data. Figure 7.37 shows a schematic of the system. Figure 7.38 shows typical pressure and position data. The pressure signal showed a structure associated with pressure-wave ringing. To simplify the analysis, a model pressure curve was employed as illustrated by the dotted line. This idealization provided an envelope of the true pressure behavior and provided a consistent method to compare shots under different conditions. To obtain the position data, we used a phase-unwrapping method to convert the quadrature interferometer signals into a continuous record of position vs. time [47].
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The complete experimental program examined the effect of soak time, soak temperature, explosive mass, burst pressure, and projectile mass. We report here the effects of confinement, and only burst pressure (mechanical confinement) and projectile mass (inertial confinement) were varied. Soak time, temperature, and explosive mass were held constant at 1 h, 190◦ C, and 300 mg, respectively. These conditions were the average of the high and low values of the larger study, hence the use of a soak for the confinement study to facilitate the analysis of the entire suite of data at the end of the study. The following sections provide a detailed thermodynamic analysis of the volume in which the explosive reaction occurs (breech) and the development of a violence metric. The thermodynamic analysis requires the input of the observable state variables, pressure, and volume. The outcome of the analysis was enthalpy increase in the breech. The violence metric uses total energy and peak power, and the rationale for our approach is discussed. 7.4.2 A First Law Analysis Our goal was to determine the reaction enthalpy change as a function of observable state variables in the system: pressure and volume (by projectile position). Oscilloscope voltages were converted to pressure and continuous volume vs. time data. We then used this information to calculate the enthalpy change during the explosive reaction and peak power of the process. We did so using a first law analysis on a control volume around the breech volume: Q(t) + W (t) = ΔE(t).
(7.7)
Q is the heat exchanged with the surroundings, W is the work done by the explosive product gases on the projectile, and ΔE is the change in total energy. We wrote these variables as a function of time in order to track the evolution of the reaction. The change in total energy equals the sum of all other forms of energy in the system: ΔE(t) = ΔU (t) + KEg (t).
(7.8)
The expansion process imparts kinetic energy to the gas, KEg , and ΔU is the change in internal energy. The change in internal energy is related to enthalpy, pressure, and volume through the definition ΔU (t) = ΔH(t) − (P (t)V (t) − P1 V1 ) ,
(7.9)
where P and V represent the system pressure and volume; the subscript “1” represents the initial state. ΔH is the change in enthalpy from the initial to current state at time t and was further dissected as follows: ΔH(t) = a(t)ΔHrxn + Cp [N (t)T (t) − N1 T1 ] ,
(7.10)
where a is the reaction progress variable, ΔHrxn is the enthalpy of the reaction, N is the number of moles present, and T is the system temperature. Cp
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is the constant pressure heat capacity, which we assumed was constant for this analysis. The quantity −a (t) ΔHrxn is the enthalpy change during the reaction, RE. We assumed an ideal gas equation of state and wrote the second term on the RHS as Cp [N (t) T (t) − N1 T1 ] =
Cp [P (t) V (t) − P1 V1 ] . R
(7.11)
Substituting (7.11) into (7.10) and then substituting that result into (7.9) gave the following relationship for the change in internal energy: ΔU (t) = a (t) ΔHrxn +
Cp [P (t) V (t) − P1 V1 ] − P (t) V (t) + P1 V1 . R
(7.12)
The ratio of the constant pressure heat capacity to the gas constant is γ / (γ − 1), where γ is the ratio of constant pressure to constant volume specific heats, which we assumed to take a value of 1.4.1 Making these substitutions and rearranging terms: P (t) V (t) − P1 V1 Δ (P (t) V (t)) = a (t) ΔHrxn + . γ−1 γ−1 (7.13) Note that the second term on the RHS is the expression provided by Baker et al. and Brode for the energy stored in a bursting sphere [42, 48], and here we interpreted it to represent the potential energy stored in the compressed reaction products. Inserting this relationship for change in internal energy into (7.8), and the result into the First Law expression (7.7) with heat exchange yielded the following energy balance: ΔU (t) = a (t) ΔHrxn +
Q(t) + W (t) = a (t) ΔHrxn +
Δ (P (t) V (t)) + ΔKEg (t) . γ−1
(7.14)
We computed the work done on the gas in the standard way: V (t)
W (t) = −
P dV.
(7.15)
V1
The negative sign indicated that the system did work on the surroundings. During the process, the projectile gained kinetic energy from the work done by the system. Gas kinetic energy depended on the projectile kinetic energy. We took advantage of this fact to compute the gas kinetic energy in terms of the work done. Assuming a linear gas velocity distribution, KEg was 1
This is the value representative of diatomic species, likely prevalent in reaction products.
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1 2 me up (t) . 6
447
(7.16)
The projectile kinetic energy and work was then written terms of KEg : KEp (t) =
1 mp 2 KEg (t) = −W (t) . mp up (t) = 3 2 me
(7.17)
The negative sign arose because although the work is negative in the control volume, it produced positive kinetic energy in the projectile and hence positive gas kinetic energy. We then wrote KEg in terms of work: KEg = −
1 me W (t). 3 mp
(7.18)
Finally, we estimated heat loss using the familiar relationship Q = −hAΔT (W ), where ΔT was the temperature difference from the expanding gas to the surroundings. It was not possible to obtain an analytic solution for reaction enthalpy change without knowledge of the temperature. We therefore calculated the area from the position data, and estimated the product of h and ΔT . A standard internal flow heat transfer correlation analysis estimated the product of hΔT on the order of 500 MW m−2 using a gas temperature of 2,000 K. Our goal was to determine the evolution of reaction enthalpy (RE) as a function of observable state variables, pressure, and volume. To complete the analysis, we inserted (7.15) and (7.18) into (7.14), applied the relationship for Q described earlier, grouped terms, and solved for a (t) ΔH and RE: RE (t) = −a (t) ΔHrxn =
V(t) 1 me Δ (P (t) V (t)) + AhΔT t. +1 P dV + 3 mp γ−1 V0
(7.19) We measured volume using the interferometer position measurement and pressure directly from the piezo pressure sensor. As mentioned in the “Experimental” section, we used a pressure model fit to the pressure data as shown in Fig. 7.38. The model provides an inclusive envelope of the true behavior, and thus provides, in a sense, limiting case behavior in a way that treats each case consistently. The validity of the model for our purposes was supported by repeated measurements under the same conditions (estimation of error) and by the general consistency of the results, which will become clearer in following sections. 7.4.3 Violence Typically reaction violence is stated in subjective terms or estimated by measuring fragment velocity or by characterizing fragments after an explosive test.
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Almost all the literatures concerning cookoff response discuss the results in terms of violence [13, 49], and comparisons amongst various studies would be simplified if there were a more succinct definition. For the system considered here, we suggest that violence is characterized by the energy release and energy release rate relative to a standard case. In other words, we calculate energy “efficiency” and peak power “efficiency” relative to a standard case. Symbolically, these efficiencies are εE =
E˙ (t)max E and εp = , Eref E˙ max
(7.20)
where the dot over the E represents the time derivative of the energy (=power). This allows us to determine the overall efficiency of the process by taking the product of the two individual efficiencies. It is this product we call Violence for the system under study: Violence = εp × εE .
(7.21)
The rationale behind this definition comes from the fact that a damaging explosive event must have both high energy and high peak power. This in turn produces high pressure and impulse at a high rate. These factors are independent. It is commonly known that an unconfined explosive cookoff would not be violent relative to detonation of the same mass of explosive, assuming no self confinement effects (large charge). Presumably, both examples would release a similar amount of energy, but the cookoff would have significantly lower power. Another example is a laser pulse. Compared to explosives, a laser pulse will generally have much less energy but equivalent or higher power. An experienced observer would likely say that the explosive event was more violent than the laser pulse, unless the pulse was directed at his cornea. This definition for violence has been useful for the study at hand. We would like this definition to eventually have broader application to other geometries and explosive systems, and so we define the reference case as detonating trinitrotoluene (TNT, pressed at 1.63 g cc−1 ), following historical precedent. TNT has an enthalpy of detonation of 5.4 MJ kg−1 [50]. A simple measure of reactive power relevant to the system under study can be determined by assuming a steady axial detonation in a cylinder of TNT. This power is the product of the detonation enthalpy, density, and detonation velocity. The result of this calculation gives a power “flux” across the cross section of the detonating cylinder. Using the same diameter cylinder as used in these experiments, we find a TNT reference power of 1.9 GW. We next present our results for energy, power, violence, and the P 2 τ shock initiation criterion and discuss the details and implications. In this discussion, the burst pressure is referred to as HMC or LMC for high or low mechanical confinement (138 or 34.5 MPa) and the projectile mass are referred to as HIC or LIC for high or low inertial confinement (3.15 or 6.3 g).
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7.4.4 Energy and Power Energy existed in this system in three distinct forms: chemical potential, gas potential, and projectile kinetic. The interplay amongst these forms of energy occurred in a somewhat serial fashion. The reaction energy transformed directly to the potential energy of the gas. The potential energy closely followed the evolution of the reaction energy until the projectile began to move. The projectile then gained energy directly from the gas potential energy. All shots used 300 mg of explosive theoretically capable of producing an enthalpy change of 1,800 J [50], assuming that the reaction products reach equilibrium. Figure 7.39 shows the evolution of reaction enthalpy for the HMC/HIC for several values of the product hΔT . We chose a value for this product such that none of the case analyses resulted in a decrease in enthalpy change with increasing time, which of course was not physically possible. A value of 200 MW m−2 was adequate for the HMC/HIC, HMC/LIC, and LMC/HIC cases. However, the LMC/LIC case required a heat loss of 350 MW m−2 . The rate of heat transfer depends strongly on gas velocity, and this case produced a significantly higher average projectile velocity and gas velocity (because of the early rupture and light projectile), justifying the use of a higher rate of heat loss. Figure 7.40 shows the enthalpy change, potential energy, and work for each of the four cases studied. Included on the graphs are the approximate times of mechanical confinement failure and initial projectile motion. Note that in the first three cases, the explosive was mostly reacted before the projectile gained more than 5% of the total reaction enthalpy change by work interaction, and in the fourth case, the explosive was mostly reacted before the projectile gained more than 10% of the total reaction enthalpy change was converted to work and projectile kinetic energy. The balance was stored in the form of potential
Energy [J]
1200
800 Increasing hΔT 0 100 200 300 2 400 MW/m
400
0
0
200
400
600x10–6
Time [s]
Fig. 7.39. Energy evolution as a function of time for the HMC/HIC case. Five different values for the product hΔT are shown
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Energy [J]
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Time [s] Fig. 7.40. Reaction energy trajectories for the four confinement cases. Crossed circles mark the time of confinement failure and crossed squares mark time at which 5% of reaction energy was converted to work
energy in the form of compressed gas. The compressed gas then applied force to cause projectile motion, meaning the potential energy had the most significant influence on the projectile motion. From a complementary perspective, the projectile mass significantly affected the dissipation of potential energy, but had little influence on the course of the reaction. Another way to state this observation: the explosive rapidly reacts, forming a hot, high-pressure gas before any significant material motion occurs. The motion of the projectile was then governed by the conversion of the potential energy to work and kinetic energy at a rate governed by inertia coupled with the gas equation-of-state. Figure 7.41 shows the total enthalpy change as a function of the confinement conditions. Error bars were determined by performing five repeats of the same experimental conditions and represent the maximum deviations from the average. The values range from 726–944 J. The average value of 824 J was 46% of the theoretical enthalpy change for detonation of 300 mg of explosive [50]. The theoretical enthalpy change assumes product equilibrium, but it was very likely that the system under study did not reach complete equilibrium during the time of observation, leading to the reduced enthalpy change measured. The reaction also may not completely consume the energetic material, but careful screening of gun barrel ejection revealed little or no unreacted material. However, this forms a likely explanation for the scatter in the data.
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12
Reaction Energy Average Power
10
1000
8
800
6
600
4
400 2
200
Reactive Power [MW]
Reaction Energy [J]
1400
451
0 LMC/LIC
LMC/HIC
HMC/LIC
HMC/HIC
0
Fig. 7.41. Enthalpy change and average reactive power for the four confinement cases. Error bars were determined from multiple (5) shots at fixed conditions
Most importantly, the data do not show any clear relationship with the confinement within experimental error. As there was little material motion before complete reaction, the explosive reaction experienced very similar conditions for all cases. The LMC/LIC case provides the notable exception to this statement. In that case, the lower strength of the mechanical confinement allows the explosive to apply pressure to the projectile early in the process. The low mass allows for rapid acceleration. In this case, the projectile has still gained only 10% of the total enthalpy change at the time the reaction was complete. It logically follows that the confinement in this regime (strong confinement) does not significantly affect the process. One could generalize the enthalpy change shown in Fig. 7.41 and approximate that the explosive burns at a steady rate until consumed. This being the case, we report an average energy release rate over the course of the event as shown in Fig. 7.41. The average power values range from 3.9 to 5.7 MW, and average very near to 5 MW. Again, the error was estimated by performing multiple (5) shots at fixed conditions. As with the peak power and total enthalpy change, we find no clear correlation of average power and the state of confinement, within experimental error. Again, this overall insensitivity to this level of confinement results from the fact that the explosive process goes to near completion before the projectile displacement increases appreciably. 7.4.5 Violence and P 2 τ The P 2 τ relationship has been used as a measure of critical shock energy required to initiate an explosive material [39–41]. In this section, we use it as a relative measure of a thermally initiated explosive’s potential to deliver
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energy to an adjacent object. Here this object was the pressure transducer. The P 2 τ relationship is explicitly written as P 2τ = constant. ρ0 U
(7.22)
where P is the peak shock pressure, τ is the duration of the shock, ρ0 is the initial density of the , and U is the shock velocity in the shocked material. Presumably, for this criterion to be valid, the pressure must rise to a maximum at a rate that exceeds some characteristic rate of heat transfer away from hot-spot ignition locations; that is, if the pressure rises too slowly, the energy delivered by the pressure wave will dissipate without a large temperature rise around the hot-spot. Because this criterion has a rate restriction, we feel that it does not adequately rank processes having a slower rate of energy release, allowing energy dissipation from hot spots without a significant temperature rise. For this reason, we have proposed a violence metric that includes the effects of total energy and rate as described earlier. Figure 7.42 shows the behavior of violence and P 2 τ for the four confinement cases. Although we do detect some variation in both metrics, the variation was small when considering the error limits. The error limits account for the combined error in power and energy measurements. As with the power and energy analysis, this insensitivity to confinement results from the explosive reaction progressing to near completion before much material motion occurs. A major concern during a thermal initiation scenario is whether a thermally initiated explosive would cause shock or fragment impact initiation of nearby explosive devices. To determine whether the most intense shot of this series was sufficient to shock initiate another piece of adjacent PBX 9501, we must calculate the shock velocity U at the peak pressure. This velocity 25x10–3
4
2
5
1
0
0 LMC/LIC
10
LMC/HIC
3
HMC/LIC
15
HMC/HIC
Violence
20
Energy Fluence [MJ/m2]
5 Violence P2τ
Fig. 7.42. Violence and P 2 τ (energy fluence) for the four confinement cases. Note that the reported energy fluence approaches the critical level. Error bars were established using the combined error of enthalpy and power
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was approximately 2,700 m s−1 at 150 MPa and the initial density of the PBX is 1.84 g cc−1 . This gives a maximum observed energy fluence of 3.6 MJ m−2 , approaching the reported threshold of 6.3 MJ m−2 for this explosive [43]. This result warrants a cautionary note: the apparatus had considerable ullage. Reducing the ullage significantly increases the violence of the explosion and could perhaps raise the energy fluence above the critical level. By our definition, these shots were on the order of 100 times less violent than detonating TNT, as shown in Fig. 7.42. The interpretation of this, and the general perception of violence, depends strongly on the intended (or inadvertent) receptor of the explosive energy and the distance from the explosive. Qualitatively, an observer would characterize all of these shots as violent. The shots produced a sharp, loud report, and energy was sufficient to imbed the projectile in Lexan. Quantitatively, an observer standing just a few charge lengths away from the explosion would detect a “point source” explosion and blast effects would follow standard scaling laws and the blast wave and damage would not differ from a detonation of the same mass of material (assuming complete reaction). In these cases, it is the dynamic properties of the intervening air that cause the blast wave to evolve into a form that depends only on the stored energy of the explosive and the rate is not important until it drops below a certain threshold that depends on the local sound velocity, size of source region, and the reaction rate [42]. There is therefore a range of events where the rate exceeds a minimum value, up to detonation, that we consider violent in general. This article attempts to attach a specific value to the level of violence above this minimum threshold. In the very near field, rate plays an important and well-known role. A detonating explosive will generate small fragments, severely deform nearby metal, and drive extremely high shock pressures into adjacent objects. A lower rate event, such as a thermal explosion, will produce much larger fragments, metal will not undergo as severe deformation as with a detonation, and the pressure imparted to adjacent objects is much lower than a detonation and persists for a much longer time. It should be noted that in either case, the total kinetic energy of all fragments would not differ significantly and individual fragments in the lower rate/violence event could potentially cause more damage because of their larger mass. These issues are of paramount importance when evaluating the susceptibility of explosive systems to thermal insult where energetic materials are in close proximity with other potentially hazardous or sensitive components. As another point, we know that unconfined explosives generally react nonviolently, but we did not observe nonviolent behavior under these confinement conditions. However, in our on-going work, we have explored the regime of weak confinement and find a significantly reduced violence with decreasing confinement strength. Till now, we have used a value of 1.4 for γ. Although this number allows for accurate computation of trends, a more appropriate value for γ is 1.162, which improves the absolute accuracy of the analysis. This value for
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End of Reaction
Energy [J]
800 600 400 200 0 0
100
200x10 –6
Time [s]
Fig. 7.43. Typical reaction enthalpy data. Violent reaction begins at time zero and ends as indicated
γ can be determined by computing heat capacity for an equimolar mixture of H2 O, CO2 , and N2 at a flame temperature of 2,700 K, the temperature measured by Zenin [51]. Figure 7.43 shows the reaction enthalpy behavior for a typical experiment. Reactive power was determined by measuring the slope between the point of transition to violent reaction and the point of apparent reaction completion. Justification for this relationship comes from the fact that damage level depends on both the total energy of the process and the rate of energy delivery. Critical shock energy provides a specific example [40]. This criterion specifies a threshold of energy fluence required to induce shock-to-detonation transition for a given explosive. The criterion has an implied rate threshold, as energy applied at an insufficient rate will dissipate in the shocked explosive without inducing ignition [52]. Additionally, Price and Wehner concluded that DDT requires both sufficient total pressure (proportional to energy) and pressurization rate (proportional to power) [53]. Figures 7.44–7.47 show the trends in enthalpy, power, violence, and peak pressure with increasing MC strength, which all approach a limiting value. Data were acquired in two sets. In the first, the breech temperature was raised at 5◦ C min−1 until explosion occurred (represented by solid circles). In the second, the explosive was thermally damaged by holding the temperature at 190◦ C for 1 h, and then ignited by increasing the temperature at 5◦ C min−1 (represented by solid triangles) [54]. The reactive power and violence of the temperature-soaked (damaged) explosive exceeded that of the unsoaked (undamaged) explosive, where the damaged samples released the expected quantity of energy within experimental error: 1,809 J 6, 030 J g−1 . The enthalpy change observed for the undamaged samples at the level of confinement investigated was significantly less than the enthalpy change for the damaged samples (∼1,800 J vs. 1,000 J). Incomplete reaction on the relevant timescale
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Enthalpy Fraction 0.00
0.10
0.20
0.30
40
80
120
2
Enthalpy (J)
1000 8 6 4 2
100 8 6
0
Burst Pressure (MPa)
Fig. 7.44. Enthalpy vs. burst pressure. The solid circles represent experiments where the temperature was raised at 5◦ C min−1 , the solid triangles represent experiments where the sample was thermally damaged at 190◦ C for 1 h before being ramped to ignition at 5◦ C min−1 . The dashed line is a fit to the data for undamaged material and is provided as an aid to the eye Enthalpy Fraction 0.00
0.10
0.20
0.30
40
80
120
Power (kW)
104
103
102
101 0
Burst Pressure (MPa)
Fig. 7.45. Power vs. burst pressure. The solid circles represent experiments where the temperature was raised at 5◦ C min−1 , the solid triangles represent experiments where the sample was thermally damaged at 190◦ C for 1 h before being ramped to ignition at 5◦ C min−1 . The dashed line is a fit to the data for undamaged material and is provided as an aid to the eye
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B.W. Asay Enthalpy Fraction 0.00 10–2
0.10
0.20
0.30
40
80
120
Violence
10–3 10–4 10–5 10–6 10–7 0
Burst Pressure (MPa)
Fig. 7.46. Violence vs. burst pressure. The solid circles represent experiments where the temperature was raised at 5◦ C min−1 , the solid triangles represent experiments where the sample was thermally damaged at 190◦ C for 1 h before being ramped to ignition at 5◦ C min−1 . The dashed line is a fit to the data for undamaged material and is provided as an aid to the eye
0.00
Enthalpy Fraction 0.10 0.20 0.30
2
Peak Pressure (MPa)
100 6 4 2
10 6 4 2
1 0
40
80
120
Burst Pressure (MPa)
Fig. 7.47. Peak pressure vs. burst pressure. The solid circles represent experiments where the temperature was raised at 5◦ C min−1 , the solid triangles represent experiments where the sample was thermally damaged at 190◦ C for 1 h before being ramped to ignition at 5◦ C min−1
explains this behavior and presumably, at sufficiently strong confinement levels, the undamaged samples would release the complete enthalpy potential. The physical explanation for this behavior lies in the effect of thermal damage.
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In general, accessible surface area increases with the level of thermal damage, a topic of current study and recent published research [55, 56]. The significant observation stemming from these data was the precipitous decline in all metrics with decreasing mechanical confinement strength at pressures below ca. 20 MPa. In addition, at the lowest level of confinement (a 0.02 mm copper foil), the foil ruptured nonviolently for 4 out of 8 shots. For violent reaction to occur, the pressure must exceed a critical value sufficient to collapse the gas phase reaction zone onto the surface of the explosive and force the hot reacting gas into pores and cracks, creating a cascading process of surface area generation by crack formation and gas penetration into pores [10, 57]. Further, according to Berghout et al. [10], the critical pressure depends on characteristic crack size; a sample with characteristically larger cracks will have a lower critical pressure. If this critical pressure was not exceeded, nonviolent burning occurred. The process of pressure-induced surface area generation and combustion continued and the rate increased until either the reaction was occurring on the surfaces of agglomerations of HMX crystals and the entire charge was consumed, or until the confinement failed. As mentioned earlier, for the highest levels of MC, violent reaction occurred and a significant portion of the reaction was complete before the confinement failed and/or the projectile moved significantly. The trends shown in Figs. 7.44–7.47 are thus explained by the interplay of pressure, confinement strength, surface area generation, and combustion. After MC failure, the reaction continued in an expanding volume defined by the balance of reaction product pressure and projectile inertia. Figure 7.48 shows the behavior of decreasing projectile mass using a constant mechanical
0
2
Mass Ratio 4 6 8
10
Enthalpy (J)
1000
8 6 4 2
100
8 6
0.0
1.0
2.0
3.0
Projectile Mass (g)
Fig. 7.48. Reaction enthalpy behavior as a function of projectile mass. Combustion extinction occurred near 2 g. The enthalpy at the lower masses was the energy required to rupture the burst disk. The dashed lines highlight the extinction behavior
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confinement of 34 MPa. The notable observation was that projectile mass did not affect reaction performance until below a critical mass, at which point reaction quenching was observed. Power, peak pressure, and violence were all similar to values for the 3.15 g projectile data already shown in Figs. 7.45–7.47. At the highest projectile mass, the observed behavior agrees with what we have observed previously [54]: a sufficiently massive projectile does not move significantly during the explosive reaction, and the reaction proceeds as it would in a fixed volume. The reaction failure below the critical projectile mass was most likely caused by dynamic extinction. This phenomenon has been observed for burning propellants, as was reviewed by De Luca [54, 58]. Dynamic extinction theory describes a critical rate of pressure decrease for propellants, below which the reaction will fail due to adiabatic expansion of the reactive gases. This theory was developed in the context of gun and rocket propellant burning, and has been discussed at length in the literature cited in De Luca’s review. Figure 7.49 shows the maximum negative pressure derivative vs. projectile mass that occurred during the explosive reaction for the shots illustrated in Fig. 7.48. The data have considerable scatter due to the inherent variability of the reaction process, but there was a clear trend towards increasingly negative pressure slopes as highlighted by the best-fit line. The reaction for which extinction occurred is indicated on the plot, and produced the most negative pressure derivative. The pressure for this case did not fall below the critical pressure for violent reaction, consistent with the literature consensus that for burning propellants, it is not a critical pressure, but a critical rate of depressurization that led to extinction. De Luca reviewed the extinction mechanism for several combustion systems, but HMX was not among them. A literature review did not reveal any studies specifically detailing extinction in burning HMX.
Pressurization Rate (MPa/s)
50 0 –50 –100 –150 –200 1.5
2.0 2.5 3.0 Projectile Mass (g)
3.5
Fig. 7.49. Pressurization rate vs. projectile mass. Arrow indicates case where extinction occurred
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7.4.6 Conclusion We have determined power and enthalpy change for PBX 9501 as a function of high inertial and strong mechanical confinement. This study was in the limit of strong confinement where the mechanical confinement held off either 9 or 35% of the stored chemical energy and the charge-to-projectile-mass ratio was 0.05 or 0.1. A thermodynamic analysis was employed to calculate the enthalpy change from the explosive reaction by measuring the observable state variables of pressure and volume as a function of time. There was no clear correlation of energy, power, P 2 τ , or violence with confinement in this regime. The explosive reaction caused an enthalpy change approximately 46% of the theoretical enthalpy at an average rate of 5 MW. For comparison, detonating the same diameter cylinder of PBX-9501 releases energy at a rate of 1.9 GW. We conclude that the reason for the insensitivity to these confinement factors stems from the fact that much of the reaction occurs before significant material motion occurs, and the reaction was “ignorant” of the confinement conditions at the studied limit of strong confinement. The logical and intuitive nature of these results, coupled with the reasonable scatter found in estimating the error, supports the use of our pressure model to idealize the true pressure data and minimize the complexity added by the ringing in the true pressure signal. The P 2 τ analysis showed that the potential energy flux to an adjacent object was very near the critical value required to shock initiate secondary explosives in close proximity. The violence analysis found values 100 times lower than detonating TNT. We do claim that the reactions were violent in the general sense, and that a similar free field event would not be distinguishable from a detonation of the same mass of explosive at a few charge lengths from the explosion. Thermally damaged samples produced a higher enthalpy, power, and violence as compared to the undamaged samples. Most importantly, as confinement was decreased, all three of the metrics declined rapidly at pressures below ca. 20 MPa. This approximate pressure has been shown to delineate laminar and convective burning in similar explosive systems. Decreasing IC did not have a significant effect on the reaction metrics until a critical level. This value occurred because of a critical rate of depressurization as described in the literature. The physical mechanism was quenching due to rapid adiabatic expansion of the reactive gases. We hypothesized that the insensitivity to IC above the critical level results from the independence of combustion rate to pressure when the reaction occurred on a surface area of characteristic agglomerates of HMX crystals. The size of the agglomerations were likely controlled by the level of thermal damage, a topic currently under study
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7.5 Large, Integrated Experiments Small-scale experiments as described in earlier sections of this chapter are oftentimes, or nearly always, preferable to large-scale experiments because of the cost and complexity involved when sizes and weights increase. However, large-scale tests are sometimes necessary to put the smaller experiments into context and to examine scaling effects. There are many tests used to evaluate hazards of large devices such as fuel fire tests. However, here we concentrate on those tests that are heavily instrumented in order to obtain as much data as possible. We present three significant examples. The first two will only be highlighted as they have been discussed and published in the literature. 7.5.1 LLNL Large-Scale Cookoff Test The first in the series of large-scale tests was performed by Chidester et al. [46]. The test was designed to provide quantitative data on the violence of an explosive subjected solely to a thermal environment. They conducted 20 experiments using hollow cylinders in which the explosive material was heated at varying rates. X-radiography and other diagnostics were used to determine the response. Comparisons were made to explosive assemblies that were intentionally detonated. They found that in the cookoff tests, the outer cases were accelerated to velocities approaching those produced by a detonation. However, inner cylinder collapse velocities were much less than those produced by detonations. They stated that they did not achieve a deflagration to detonation transition because the residual strength of the case was too low and they did not have enough explosive material. Others, however, have demonstrated DDT in much smaller quantities and with less confinement using different explosives [13]. Chidester et al. performed ODTX calculations and showed very good agreement in the time to explosion predictions and experimental measurements. They also performed hydrodynamic calculations assuming laminar flow and showed good agreement with experimental outcomes. It took over 100 μs to accelerate the outside case to detonation-like velocities. This would imply that approximately the same amount of energy was liberated, but at a lower rate. The inner case did not have the time to accelerate to the high velocities, and also, the geometry of the collapse process inhibits acceleration in the absence of a shock. 7.5.2 Scaled Thermal Explosion Experiment (STEX) In 2002, Wardell and Maienschein published results from a new test designed to quantify the violence of a thermal event and provide data to support calculations by providing a database for validation [59]. It was designed to have extremely well controlled boundary conditions with temperature, confinement,
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and other parameters precisely known. It was also a large-scale, integrated test as opposed to the small-scale tests that were designed to examine a single type of response or mechanism. The test consists of a hardened 4,130 steel cylindrical vessel. This geometry promotes symmetry in response and forces explosive self-heating to the central axis. This makes measurement and calculation of the response much easier. The inner diameter is 5.08 cm and the length is 20.3 cm for an L/D of 4.
Large flanges 152 × 25.4 mm2 are brazed on each end and then an end cap
2 152 × 28.4 mm is bolted onto each flange. Wall thickness can be varied to increase or decrease the structural confinement as desired. Likewise, the explosive can either be press-fit into the cylinder or a gap between the HE and the wall can be used, depending on the desired effect. The explosive is typically heated at 1◦ C h−1 . Internal and external thermocouples are used to control and record the temperature, and strain gauges, pressure gauges, and radar are used to measure system response and wall velocities. While the experiment was designed to provide a quantification of reaction violence, it was found that a single value could not be provided because of the complexity of the response. That is, in some cases the confinement failed nonuniformly and thus the wall velocity measurement at that point was not representative of the entire assembly. The same result occurred for strain gauges that were placed at specific locations along the cylinder. The test has been used, however, to generate integrated data for use in code calculations that report a time to explosion. These models use ODTX and other tests to help generate kinetic parameters. They assume constant burn rates and do not take into account the effects of thermal damage like cracks and porosity that are known to form. However, they do provide useful information as to global effects that may be expected and support the overall understanding of thermal response [60, 61]. 7.5.3 Large-Scale Annular Cookoff Chidester et al. [62] concluded that it was possible that multiple initiation sites occurred during the uniform heating of their explosive samples. This would result in multiple waves propagating throughout the volume of explosive. A series of experiments comprised of eight cookoff tests, generically referred to as the Large-scale annular cookoff tests [63], were designed to determine whether or not a uniform heating profile would result in a volumetric explosion, originating from essentially an infinite number of initiation sites, or whether material properties and thermal fluctuations would mitigate such a response. These were carried out to investigate the symmetry and violence of a cookoff event arising from slow, symmetric heating of a moderately heavily confined annular test assembly containing PBX 9501. Extensive diagnostics were used to measure the internal and external temperature distributions, heat flow, strain, casing motion, and various other parameters.
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The design comprises concentric inner and outer cylinders secured together by an end cap that screws on to the outer cylinder and secured by a locking ring that screws on to the inner cylinder (see Fig. 7.50). Other diagnostics used were heat flow gauges, manganin strain gauges, resistance wire strain gauges, contact wires, accelerometers, fiber optic light probes, X-ray imaging, and VISAR. As a principal diagnostic was observation of the collapse of the inner cylinder, the outer cylinder was made thicker than the inner cylinder. The assembly was heated by a single layer of heating tape wrapped around the outer cylinder. The assembly was suspended horizontally in a custom-built
c
b
3" 3 1/2" 6 1/4" 7 1/4"
7 1/2" 8" 10 3/4" 11 3/4"
Fig. 7.50. (a) Schematic representation of assembly design. (b) Small assembly dimensions. (c) Large assembly dimensions
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vacuum chamber in order to minimize the effect of convection currents in the surrounding air on the temperature distribution. Similar temperature profiles were planned for all the shots: heat from ambient temperature to 150◦ C, hold until bulk HE temperature was close to 150◦ C, heat to 180◦C, hold assembly at 180◦C until the HE self-heated to ignition. Note that “close to 150◦C” was not precisely defined, but in practice was within 2◦ C. The test assembly is not gas-tight by design. Gas may escape through the threaded end caps or via six small holes drilled through the outer casing on the mid plane for the internal thermocouple leads. 7.5.4 Results Now we will discuss a few of the major results from this series of experiments. Because we could not use color plates, the data graphs have been rendered in black and white, thus losing some of their utility. However, the discussion in the text, along with the graphs should provide sufficient detail to permit an understanding of the conclusions that are drawn. Figure 7.51 shows the temperature profile of the shot. The gap in the temperature data arises from a temporary failure in the logging process. The shot 2 assembly was constructed from brass rather than copper, and it appears that conduction of heat to the inner cylinder was reduced from shot 1. The inner cylinder was several degrees cooler in shot 2, relative to the outer cylinder temperature. Figure 7.52 shows the high-speed images of the event. The object in the middle of the field of view is a thermocouple that has become detached from 200
temperature (˚C)
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50
0
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Fig. 7.51. Temperature profile: Shot 2
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Fig. 7.52. High-speed images: Shot 2
the inner cylinder. Asymmetric collapse of the inner cylinder wall, starting from the lower right corner, is visible from 10 μs. At around 20 μs it appears to be overtaken by reaction products. It is believed that these may be escaping at one of the end caps. In any case, the event does not appear to develop any cylindrical symmetry before the view of the inner cylinder becomes completely obscured by gas/debris. Recovery of assembly fragments indicated a violent event, with only small pieces (
