Assessment of Safety and Risk with a
Microscopic Model of Detonation
Front picture: Precursors in detonation shown by a wrong color technique from X-ray isodensitographs,see page 360, and Figure XVI-2, center (page 362)
Back picture: Crater of Toulouse Ammonium Nitrate explosion 21.09.2001, see page 273, Figure Xlll-4.
Assessment of Safety and Risk with a
Microscopic Model of Detonation
Carl-Otto Leiber
by
Brigitta Dobratz
2003
ELSEVIER AMSTERDAM - BOSTON - LONDON -NEW YORK - OXFORD - PARIS SAN DIEGO - SAN FRANCISCO SINGAPORE - SYDNEY - TOKYO ~
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DEDICATION Personal dedication and thanks to my wife Barbara Leiber for all that that only she knows! To my children and grandchildren Ortrun, Christoph, Tabea, Jonas and Simon.
Special dedication to the hture colleagues, fbrthering our safety and detonation science for the world’s benefit. This text was written for them!
ACKNOWLEDGEMENTS Thanks to Brigitta Dobratz. Without her perseverance this treatise would not have been written and without the courage of Charon Duermeijer and Linda Versteeg, Elsevier, never published. Birgitte Blom, Elsevier, cared for the quality in print. Usually persons are addressed by name. In this case, however, I had a list of over 50 people that seemed to grow daily. Even when I thought the list was finished, additional important names came to me that night. Thanks to colleagues and fi-iendswho supported all this by active discussions. Special thanks and respect I owe to persons who at a very early stage of the work have seen positively developing new aspects, and had the courage to express this openly. Also there are several contributors to individual chapters whom I wish to acknowledge individually: Siegfi-iedHaussiihl, Rolf Hessentnuller, Boris Kondrikov, Eric Lesleighter, Siegfi-iedWinkler.
This Page Intentionally Left Blank
ASSESSMENT OF SAFETY AND RISK WITH A
MICROSCOPIC MODEL OF DETONATION
PROLOGUE ............................................................................................................
.............1 References ...............................................................................................................
I
SHORTCOMINGS
THE MACROSCOPIC a A N E - W A V E MODEL OF
...................................................................................................
DETONATION of
Wave
5
................................................................................ ......................................... ................
Discussion.............................................................................................................. References .............................................................................................................
7 7
11 11
...................13
IMPEDANCE MIRRORPHOTOGRAPHY OF H. DEANMALLORY of ............................ of ....................... ........................................................................ ..........................................................
15
........................... Evaluation of the Organizing Pressure Spots ................. Pluto - Flagstaff Technique........................................................... A B Stereographic Technique ... ........................................................ a Two Projector Methods ....................... b Inclined Mirror Technique .......................................... ........................................................ c Wrong Color Technique ..
on References ......
I11
............................... ..................... ........................................................
26
26
28 31
.....................................................
PRESSURE GENERATING MECHANISMS of
on
33 ..................34 ................................................................................. 36 by ..................................... . 3 7 vii
...
Vlll
Content
................................. ..........................
......................
..................................
39
................................................................................................ ...........................................................................
41 41
of
References
IV
EQUATIONS
A - ACOUSTIC QUANTITIES
...........................................
.........
..........................................
42
........................................
................ ......................................................
................................................................
B - SPHEMCAL WAVES EQUATION
47 .............................................. 47 .................... ......... 48 ..................................................... ..................................... 49 c - COMPARISON OF THE PROPERTIES OF PLANE AND SPHERICAL WAVES 49 ...................................................................... .................. 49 .......................................................................................... .50 ....................................... D - GENERAL SPHERICAL WAVEEQUATION of
....
..........
.................................
.............................................. RADIAL
.......................................................................................
SOLUTIONS
55
..................................
....................................................... 56 ..................................................................... ...................................... ..........................
57
58
Particle Velocities. In Stokes-Rayleigh terms ................................................................................. 59 In Stenzel or Riccati-Bessel terms ................................................................... 59 In terms of the spherical Bessel functions ................................... Correspondencies (identical expressions) .................................................... 62 Radiation Impedance ...................................... ................................. .64
.........................................
E - SCATTERING OF A PLANEWAVEAT A SPHERE of ................................................. on ............................................
64
66
ix Content
............................................................................. ....................................................
F - CYLINDRICAL WAVES G - WAVES IN ELASTIC ISOTROPICMEDIA ...................................... ................... ..................................................................... Bulk c o ............................................................... .... ......................................
69 70 72
73
of Aelotropic Medium .................................... .......................................... Infinite Isotropic Elastic Medium ................ Semi-infinite Isotropic Elastic Medium ................................................................... Surface (Rayleigh) Waves ...................... .......................................................... Estimate of Dominant Wave Types ......... .......................................................... Additional Wave Types .............................................................................
............................................
V
.74 75 75 76 76 77
.........................................................79
PRESSURE SOURCES FOR MODELING
.... ............................................... .................................................................. Pressure ............................. .......................................... Particle velocity....................................................................
79 80
................................................................. Pressure .................................................. ................................... Particle velocity ...................................................................
................................................................................................ Phase Control .................................................................. ..... Stochastic Source Strength ................................................................. Stochastic Source Phase ....... ........................................................................
................................................................................ m l A T 1 O N OF A SHORT MOTION SOURCE
....................................................
Short Constant Source Expansion ............ Pressure ........................................................................
83
.84 86
86 86
...................................
88
a: Constant distance r at variable time t... ................................ b: Constant time t and variable distance r .............................. Source Expansion Pulse
89
................................
90
A:
References .............................................................................
VI
....................................................................93
RAYLELGH'S BUBBLE MODEL ...................................................................................
..................................................................
............................................................................................
DEVIN'S BUBBLE 's
.................................................................
93 95 97 .97
X
....................... .... ..................... ............................................................. ................................................................... m = a ................................................................................... W O ........................................................... < ~ ....................... of ‘s ................................. ................................................................... &, > 2 ..................................................................... = 2 ..................................................................... RELATIONS OF ENERGY AND POWER
98 99 100
100 102
..........
.................................................................... 103 Experimental Determination of the Loss 6 ............................................................ 104 of
...............................................................................
.......................................................
V11
LOSSES BY VOLUME VARIATIONS Loss
.......................................
..... &,d
of
....................................... ......... ...............................................................
Thermal LASS &, ................................................................................... .................................................... On the ‘adiabatic hot-spot’ ..................................................................... Viscous Loss 6, ..................
............................................ ..................... Other Loss producing Mechanisms ....................................................
107 107 109
112 114 1 14
115
VARIETY OF INITIATION MODESBY BUBBLES.....................................123 .................. 123 ............................ SOFTEXCITATIONS .................... 124
VTIl
‘Soft‘
..............................................
Probability of Resonance... How to Avoid Spontaneous
.................................................................
125
.................................................................
125
............................ .............................................. FREQUENCY EXCITATION MODES............................................................... by ) ..................................................... A B C D
Stiffness Activation .................................................................. Damping Activation ................................................. Mass Activation ..................... Dynamic Pressure Activation ................... .............................
(NM)
HOTSPOTS
128 130 131 131 132 135
(NG) ............... 136 137 .............................................................................. 137
.................................................................................................. ....................................................................................
Cvntent
.............................................. ...................
of SHAPEOF PRESSURE PULSES
138
............................................................,..........140
by
........................................ .................
qf References ...
IX
X
...................................................................
144
...
VARIOUS APPROACHES TO DESCRIBE BUBBLE DYNAMIC PHENOMENA145 SCATTERING CROSS-SECTIONS 145 QUASICONWNUUM APPROACH TO SLOW-AND LOW-VELOCITY DETONATION 148 .......................... 148 ,151 of .................................... 151 .. 153 ........ ... ............................................. 157 ............... ........................................................ ... 157 ............. ................... 159 .................................................. 160
.................................................................... ...........
..............................................................................
SENSITIVITY TESTING ..........................................................................
161 161 .................. 163 ............................................. 164 ..................... 165
.......................
ON PREDICTIONS BASEDON G O N O G
............................................
166
170
..
Sensitivity Testing (50% values) ................... ...................... 172 Risk Assessment ............................................................................................. 172 Recognizable Risk (Long Practice - Judgement of Experts) ....................... .I73 Risky Operations ............ ....................................... 173 ....... 174 Producer and User Risks (Ri
.............................................
EXPERIMENTS ON MECHANICAL SENSITIVrTY .................................................... Influence of Bubble-Size Roth’s Experiments .......
.................
........................................
174 75 177
xii Content
Influence of Bubble-Content on Sensitivity ...........................................................
178
......................................................................... Initiation by Entrainment caused by an Explosive Blast (NM)......
179
..................................................................... of .................................................... ..................................... .............................................. ............................................................................................................ ........................................................ .......................................... nces ....................................................................................................
X1
179 180 181 181 181
Low- ( L W ) AND SLOW-VELOCITY DETONATION ( S m ) OF LIQUID EXPLOSIVES. 185 LVD governs Safety ........................................................ LVD Stimuli .. Anomalous Hu L ..................................................... First Observations of an LVD in Liquids SVD (3rd LVD SVD LVD Transitions ................................... 2L ..............................
Joyner’s last Experiments T N M M - p~v~/D’LvDRelationship .......................
......
..........................
................................
...........................
..201 ...................202
........................................................ 204 Aging Properties ................................................................ ........ 206 GENERAL RESULTS ............ .................................................................. 206
....................................... .................................................................................
EXPLOSION TESTSOF THE US BUREAU OF MINES ................................................................................. PHYSICAL EXPLOSION
207 207 209
.................................................... ,210 By Eric J Lesleighter, Cooma NSW, Australia Dartmouth Dam Project - Modeling and Field Measuremen ............................................................... 214 Conclusions. References ..........................................................................
...
Xlll
Content
XI1
.........................
LOW VELOCITY DETONATION OF SOLID EXPLOSIVES
Stability of LVD and C LVD of Pressurized Porous Explosiv
219 ............................................ 219 .................. 219 ...220 .................................... ............................ 221
.................................................... Strength of Confinement... ExplosiveKonfncment Interactions and LVD Luminosity .................................... Detonation Transfer .............................................................................................
223 224 224
................................. .. 226 ................................. .................... 226 Experiments on Double Explosions and Jump Phenomena .................................... 226 ........................................ ............................. 228 Initiation and Ignition Experiments ................................................. 229 LVD Spin Detonations in Solids.......................... ................................ 230 Further Transients ................................................................ .............230 LVD OF PVROTECHNICSAND PROPELLANTS 230 ...............
of
...............................................
....................................... ............... 231 Dead Pressing ................................................................................................. .................. 231 ............................................ Less-known Problems with Pyrotcchnic Delays..........
......................................... .................. ,232 ...................................
..........................
TESTING FOR LVD WSKS..................................... ............................................................................................. 233 GAPTESTS 236 1 ....................................................................................... 240 .......................................................... 240 ............................................................. TECHNICAL APPLICATIONS 244 .... 1 Double Explosions ..................... 2 Flash-over of pyrotechnic D .................... ,244 Cap and Fuze Train Failures ..................................... 3
....................................,.............................................................. .........................................................................
4 5 6
7
Irregular Response of Device Set Backs ....................................... Premature Detonations _ _ _ _ . Breech Blows ....................
..................
247
....................................
......... References ...............................
...............................................................................
253 .253 DOUBLE EXPLOSIONS ..................................................................................... EXPLOSIONS OF PURE CHLORATES 258 CASES W T H AMMONIUM NITRATE(AN) 261
XI11
CASE HISTORWS..
..........................................
.............................................................. .....................................................
x1v
Content
.....................................................
..................................................................... ,263 ........................................ .266 270 .......................................................... .270
.................................................................................
ACCIDENT HISTORIES
BASF, Oppau, Germany, 21.09.1921 .................................... SS GRAND CAMP, Texas City, Texas, USA, 16.04.1947 __.. SS HIGH FLYER, Texas City, Texas, USA, 17.04.1947 ................................. Consequences and Lessons of both Explosions ........... Tests performed thereafter ...... Benign Burning of AN, Texas Cit SS OCEAN LIBERTY, Brest, France, 28.07.1947 .........
272
1.5 1.6 ............... 274 ANFO Explosions, Kansas City, USA, 29.11.1988 ......................................... 274 Herlong, CA, USA, 07.08.1978 ............... NM Tank Car Explosion, Niagara Falls, U ............................. 275 NM Tank Car Explosion Mt. Pulaski, Ill. USA, 01.06.1958 ........................... 276 Non-Explosions of NM ................................................. ...............278 Truck / Train Collis onticello, Arkansas, 22.03.1962 xplosions 278 Apparent safety in Drag Racing ... .............................................................. 278 1. ..................................... ........... 278 MMAN Tank Car Explosion, Wenatchee, Wash., USA, 06.08.1974 ............... 278
.......
..............................
.................................. Explosions and Detonations of Films .............. Fires and Explosions in Covered Rifle Ranges .................................................
279 2x0 280
................................................
of
Port Chicago, CA, USA, 17.0 Damages .........................
..............................................................
283
CASE HISTORIES OF NON-CHEMICAL BASEDEXPLOSIONS (PHYSICAL EXPLOSIONS) .............................................,...............................284 .......................................................................................................... 285 Tunguska-Event Meteor Impacts................... ................................... 285 ~
Barringer Arizona Metcor Cratcr Energies of Crater Formation. Energy of TNT-Equivalents ... Comet Shoemaker-Levy 9 Impacts on Jupiter ... Volcanic Explosions ......................................... Crater Lake Formations. .............
PHYSICAL EXPLOSIONS IN THE KIT Vapor Explosions ..................................................................
................................... 286 ................................... 286 .................. .................. ................................... 289 FE...........................
290
..........291
xv Content
Vapor Explosions and Associated Chemical-Room Explosions ... .291 Vapor Explosions with small Amounts of Water (Entrapment Explosions)............ 292 Bubble Resonance Explosions .............................................................................. 292 Bubble Resonance Explosions (Superheated Water) .................. Sorption-DesorptioliResonance Explosions ...............................
.......................................
SAFETY AND INDUSTRIAL PHYSICAL EXPLOSIONS. 293 293 Nuclear Reactor Estimates of Explosion Risks.. .............................. Vapor Explosions ........... ............................................ 294 ..................,294 Quebec Foundry Accidcnt ....................................... Reynolds Metal Co., McCook, USA, 1958 ...................................................... 294 Various othcr Cases of Steam Explosions {Fuel Liquid Interactions (FLI)) ...... 294 Corc Melting (Explosions) in Nuclear Reactors ........................ Technical Use of Vapor Explosions ...........................................
............................................................................
HYDRAULIC TRANSIENTS
Water-Hammer Effects ......................................................... Joukowski Shocks ..... Tarbela Dam Pakistan,
EVAPORATIONS - ‘FLASH’-EVAPORATIONS
......................
.................................................
296 298
299
............................................ 300 Tank Ruptures and Explosions ...................... 300 Ammonia Tank Car Crash, Explosion of liquid Carbon-Dioxide Tanks .................................... Dimethyl ether Explosion, Ludwigshafen, 28.07.1948 .................................... 302 Shocks in Silos........................................................ Grain Silo Explosion, Blaye, France, 20.08.1997 “Boiling Liquid Expanding Vapor Explosions” (BLEVE) .................................. ,304 Crescent City, Propane Jumbo Tank Car Explosions 21.06.1970 ............... 304 Challenger Accident 28.01.1986 ............................................... 305 ................................................................. 305 ........................................ 306 306 Waverley, Tern., dclayed Propane Tank Car Explosion 24.02.1978 ................ 306 Terrorist Attack on Tank Farm S. Dorligo, Triest, 04.08.1972 ........................ 306 Aspects of Theory, Countermeasures against BLEVE or Rolling Over Events .......307 up.. ................... Gushing of Beer ................................. Blow-Down Systems - Safety Disks .... Degassing of Solids .. ....................... ............................................... 310 Degassing of Natural ................................. 310 ................................. by .........
.......................................... References ...........................................................................................................
x1v
DIPOLE SCATTERING
..........................................................................
................... A B
.................................. ............................
in Tiny Particle in an inviscid Medium.. .... Tiny Particle in a viscid Medium .......................
312 3 14
321 321
.................................. 322 ...................... 323
xvi Content
Particle of arbitrary Size in an inviscid Medium ......................................... Arbitrarily sized Particle in a compressible, viscid Medium ........................
C D
325 327
.................... 330 ......................... 331 ......................... ...333
of
Hydrodynamic Pair Formati
.....................................
............................
334
.........................
336
SELECTIVITY ...............................................................................................331 .............................. 338 .......................................................................... 339 Theory of Surface Coatings.. . ............................ .................... 339 .........................................................................
CASCADED COLLISIONS .................................................................................. ............................ 341 ... 343 References ...................................................................................... XV
FINITESHOCKRISE............................................................................... ........
XVI
........................
.............................................................................
VOIDPRECURSORS .............................
.........................
346
359 359
......................... ..................................... ............................................................. 361 Nitromethane (NM) .................................................... ............................ 363 NitromethaneiAcetone Mixture 84i 16 ...................................................... ...364 NitromethaneiAcetone Mixture 75/25 ................................................................. 364 . . TNT Liquid ............................................................
.................................. Nitroglycerine (NG) .................................... Rule of Thumb ..............................................
............................
365
xvii Content
............ ,.............................................. 367 by ...................... 367 References ........................................................................................................... 368 of
................................
ALTERATIONS OF HUGONIOTS BY BUBBLE FLOW
XVII
................................
371
Carbon Tetrachloride.......... ...................................... .................. 372 Other Liquids............................................. ................................... Qualitative Explanation of Hugoniot Effects .................... .......................... 374
ENERGY BARRIE ........................................................................................... ............................. Model of an Expanding Rayleigh Bubble .............. .........
380
.......................
by ..........
..............................
APPENDIX: SUMMARY OF EXPERIMENTAL HUGONIOTS Referenccs ..................................... .............................................................. XVJII
385 390
CRITICAL DIMENSIONS ......................................................................
.......................
................
394
................396 397
.........................
.................398
of
.........................
C
Fresnel Approximation.................... of
..................... ..................................................
Application to Detonation Problems ..........................
401 401
.............................
401
....405
CRlTICAL DIMENSION PHENOMENA NEAR SOURCES IN A LINEAR ARRAY
Further Description of the Harmonica1 Spherical Source.......... ................... 405 Transition from a Linear Array to an Areal Monopole-Cosine Source ................................. ..................................... 408
......................
.................409
........... 1. 2. 3.
Triggered Source Array of Consta Source Array of Constant Dynamics, Variation of Geometrical Dimensions 41 4 Is there any Linearity or Similarity?....... ................................... 416
......................................................................
417
xviii Content
.............................................................. 417 ........................... 417 Refcrences ........................................................................................................... 422 of
XIX
CRlTICAL DIAMETER@) OF NITROMETHANE (NM)
............................
425
.......................................................
........................................................ on
....................................... .....................................
VARIATION OF THE CRITICAL
428 429 433 434 434 435 435
............
DIAMETER@) OF NM BY ADDITIVES
of of of
............................................................. .................................................... ..........................................................
on on
.
437
............................................
.....................
443
.................................................................. References ..................
xx
. 445
SMOOTH AND ROUGHPRESSURE FRONTS. DARKWAVES AND DDT
a) b) c)
....
447 ................................................................................................. 447 of ........................................................................ 447 Monopole Source. m = 0 .......................................................................... 448
Monopole-Cosine Source Dipole Source. m = 1 ................................................................................
448 449
449 452 ....................................................................... 453 of .................................................................. 457 CELL AND HERRINGBONE STRUCTURES IN DETONATION ............................. 458 ...................................................................................... 458 ............................. ........................... 458 ............................................... 462 .......................................................................
Example ............................................................................................................... 463
..................................................................
PROPAGATION OF DETONATION
References ...........................................................................................................
XXI
.....................................................................................
SHOCK TUBES .......................................................................................
467 470
473 473
.............................................................
474
..... Suggested Physical Model of NONEL . Equidistant Sourccs on a Circular Ring Critical-Dimension Phenomena ....................................................................... Scaling Parameters .......................................................................................... Effects of the Source Distribution in the Circular Area
on
........................................................................
477 479
483
by SURFACE DETONATIONS
.......
...................... on Sufa on Sufbce ............................................................................ References ..........................
XXII
DETONATION PHENOMENA IN CHARGES WITH AN AXIALCAVITY
486 486
.......487
.................................................................................
487
....................... .................................................................................. References .........................
491
...........493 493 A: PROPERTIES OF SINGLE CRYSTALS ......................................................... 495
XXlII
MICROSCOPIC AND MACROSCOPIC PROPERTIES OF SOLIDS
With Siegfried Haussiihl .................................................................................
................................................................................................. .. ............................................................... Ax ............................................................... .......... ............................................................... ................................ .................................. Stress Components ...................................................................................
495 495 496 497 498 499
Interconversion between the Constants Physical Significance of the Quantities
.................................................................................
................................................................................. of
Triclinic 21 Constants in c and s, 6 in Thermal Expansion .................................. Monoclinic 13 Constants in c and s, 4 in Thermal Expansion .............................. Orthorhombic 9 Constants in c and s, 3 in Thermal Expansion ............................ Trigonal, 7 Constants in c and s, 2 in Thermal Expansion ................................... Trigonal, 6 Constants in c and s, 2 in Thermal Expansion ................................... Tetragonal, 7 Constants in c and s, 2 in Thermal Expansion ..................... Tetragonal, 6 Constants in c and s, 2 in Thermal Expansion ............................... Hexagonal, Constants in c and s, 2 in Thermal Expansion ................................ Cubic, 3 Constants in c and s, 1 in Thermal Expansion ....................................... Isotropic, 2 Constants in c and s, 1 in Thermal Expansion ...................................
500 501
502 502
502 503 503 503 504 504 504 504
Conten1
..........................................................
505
Hydrodynamic State .............................................................................................
................................................ .................................................................... ..................................................... ..................................................
505
506 507 507 509 . 509
.................................................................... 509 ELASTIC PROPERTIES OF THE POLYCRYSTALLINE AGGREGATE 511 Body ......... 511 Necrfeld Mean Average........................................................................................ 511 ................................................................. 513 Rule of Thumb ..................................................................................................... 513 ....................... 514 ................................ .......... 515 ................................................ 515 ........................ 516 Measures of Microscopic Anisotropies ................................................................. 516 rk U s i k .for Some 51 7
..................
Orthorhombic 9 Constants in c and s, 3 in Thermal Expansion Trigonal, 6 Constants in c and s, 2 in Thermal Expansion ....... Tetragonal, 6 Constants in c and s, 2 in Thermal Expansion ................................. Hexagonal, 5 Constants in c and s, 2 in Thermal Expansion .................................. Cubic, 3 Constants in c and s, 1 in Thermal Expansion Isotropic, 2 Constants in c and s, I in Thermal Expansi
517 518
ELASTIC DATAOF PETN, RDX, E-HMX AND AP ....................................... 519 ................................................................................... 520 Tensor invariants in the dimensions of Haussuhl ............................................ 520 Voigt/Reuss averages of the Elastic Properties and wave velocities ..................520 Thermal expansion ........................................................................................ 520 ..................................................... 521 Measures of Anisotropy of PETN
Rotated General (Shear modulus ..................................................... Thermal expansion ........................ Selected Sound Veloci Longitudinal (compression) waves .................................................................. .......................................... Transversal (equivoluminal) waves ..... Longitudinal and transverse waves in combination ........................................
..........................................................................
521
523 523 523
524
Tensor invariants in the dimensions of Haussiihl .......... ......524 Voigb'Reuss averages of the Elastic Properties and wave velocities .................. 524 Thermal expansion (at room temperature) ....................................................... 524 Measures of Anisotropy of RDX (20°C) ........................................................ 525 Rotated relative uniaxial compressibilty ......................................................... 525
XXI
Content
Rotated Young's modulus Rotated general Shear modulus .........................
................................
,525
26 Tcnsor invariants in the dimensions of Haussuhl ............... VoigtiReuss averages of the Elastic Properties and wave ve
Rotated Young's modulus
................. 528 ...................................................
............................................. ..........................
528 528
................................... 530 Possible longitudinal (compression) waves ....... ................ 530 Possible transversal (equivoluminal) waves ........................ Possible combination ..................... 531 ..................................... Tensor invariants in the dimensions of Haussuhl ............................................ 531 VoigtiReuss averages of the Elastic Properties and wave velocities..................531 Measures of Anisotropy of R-HMX (1 07°C)......................... Rotated rclativc uniaxial compressibilty ............................. Rotated Young's modulus ....................... ............................................ 532 ......................... 532 Rotated general Shear modulus ................................ Thermal expansion .......................................... nic B-HMX (107"C), (Pure Selected Sound Veloci ........................... 533 Possible longitudinal (compression) waves .................... Possible transversal (equivoluminal) waves .................................................... 533 ........
...................................
534
Tensor invariants in the dimensions of Haussiihl ........................................ 534 Voigt/Reuss averages of the Elastic Properties and wave velocities..................534 ................. 534 Thermal expansion .......................................................... Measures of Anisotropy of (Haussuhl) ..................................................... 535 ................................. 535 Rotated relative uniaxial compressibilty ......... ...................535
............................................ ........................
Measures of h i
535
537
.............................. ote: a and b axes wr
..................................
538
Rotated Young's modulus ................................................... Rotated general Shear modulus ..................................... ............... Anisotropy of thermal expansion ...................
MACROSCOPIC APPEARANCE OF THE MESOSC ESTIMATES OF POLYCRYSTALLINE PROPERTIES
CTER
..................539
..........................................
.............................................
539
... 539
xxii Content
........................................................... .539 .................. 541 ....
of of
...............................
.........................................................
....
.........................................
545
.............................................. ................... 546 ...................... ..................................................................... 547 CONCLUSIONS FROM YSTAL DATA 548 ............... ............... ....................................... 548 by
..............................................
...............
............................................. ................... 550 .......................................................... ,550 ..................................................................... ............. Referenccs .....................................................................................
XXIV A
FRACTURE DYNAMICS OF INITIATION........ on ...................
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SUBJECT INDEX
ASSESSMENT OF SAFETY AND RISK WlTH A
MICROSCOPIC MODEL OF DETONATION
PROLOGUE Classical Theories of Detonation, a Description of the State of the Art The principles of detonation theory are based on Schuster’s [ 11 interpretation of Riemann’s [2] ideas for the behavior of shock waves in detonating gases, and have been later applied also to condensed explosives. These principles are: Macroscopic quasicontinuum laminar flow is approached. Plane-wave considerations and corresponding impedances are established. The jump discontinuity is independent of time, whereas the flow may be time dependent. Chemical reaction occurs completely and instantaneously within the detonation jump. Quantities are in thermodynamic equilibrium.
In the more recent past, developments in detonation theory have been accomplished by Becker [3], Zel’dovich [4], J. von Neumann and Doring [6]. When speaking of this ZND-theory I refer to a steady state theory. It is usually assumed that the origin of detonations is thermal. This means that shock compression induces a temperature increase in the volume, which results in a thermal decomposition that would drive the shock. Many assumptions have been based on this classical ZND-theory, where the originators are not quite clear. This is best evidenced in the paper of W. C. Davis [7]: The basic idea should be “their key insight was to take a shocked material’s temperature and pressure as givens and to concentrate on how chemical reactions in the shocked state
Prologue
proceeded". Doring [8] commented that this had never been his idea. He feels that this idea would be strange and misleading. It must be noted that before 1940 physical detonation theories had been considered to be of only academic interest; no practical need for such a tool was seen because everything had been solved experimentally. In these early days, the science of explosives was concerned only with inventing and developing new ever-more powerful explosives. This process had been entirely in the hands of able chemists, who approached the synthesis problems through application of their own insight. Therefore, many crude chemical and physical ideas and tests, as well as some sophisticated ones, are still in use and can present dangerous situations. For example, Schmidt [9] used the thermohydrodynamic detonation model to calculate detonation properties in the 1920s. According to the chemist's viewpoint, he believed in thermochemical limits of explosion properties. Even though bubble sensitization of explosive materials leading to detonation was known at that time, the German authority Chemisch-Technische Reichsanstalt (CTR) denied such a possibility [lo]. The most serious error seems to be that explosives behavior is taken to be only a matter of chemical reactions (kinetics). The chemists view any hazards in the context of chemical reactions, totally ignoring physical explosions. As we have seen fiom the Oppau-accident in 1921, which was not an 'allowed' explosion, we lack even today the tools to predict or avoid such risks, other than prior experience. After this catastrophe the government reported to the Reichstag as naively as did their American colleagues to their government after the Texas City catastrophe in 1947. The first practical need for a physical detonation theory resulted from the development of atomic bombs, where knowledge of precise high velocity processes of explosively driven materials are essential. Using conservation principles in plane-wave terms it became possible to calculate with this theory the materials' behavior under (regular) shock attack. So shock wave physics was born in terms of excellent engineering estimation methods. The classical detonation theories are described in several monographs [l l-161, and many computer codes have been developed since that time. In the following chapters I will outline some ideas in a rudimentary fashion. For the original goal of shock and detonation research, there had been no need to consider so-called 'pathological' situations in explosives behavior though there are many. Some unsolved problems of detonation are: 1)
Plane-wave concepts are used, although plane and wavy detonation fronts exist, along with dark waves, in the center of a charge.
2)
Classical theory only addresses High Velocity Detonation (HVD), in which the detonation velocity exceeds 6 W s , in spite of the fact that the first detonation velocities observed in nitroglycerine (NG) were < 2 k d s . This detonation process in NG, called Low Velocity
3 State of the Art
Detonation (LVD), is not well understood, and its existence in solids was denied for a long time. Auniversal detonation velocity gap between 2 - 2.5 and 6 km/s exists for condensed explosives of normal high densities. This phenomenon is neither understood nor generally recognized. Deflagration to Detonation Transition (DDT) does exist. Working codes have been developed using simple assumptions. The physical origin of DDT is not clear, partly because in plane-wave terms always and invariably one-half of the energy is in compression one-half in flow. Detonation is considered to be a shock wave accompanied by a chemical decomposition, which means that chemical processes drive hydrodynamic events. This coupling process has been known since Becker’s days (the 1920s), but no explanation has been advanced up to the present. If there is a reliable theory, one expects that a univalent detonation pressure should be determined from different and very precisely determined experimental data. This is not the case. Von Neumann [6] suggested in his classical paper for the substantiation of the ZND-theory that particle velocities would probably be much more effective for initiation than the shock heating process. No discussion ever followed this suggestion.
No guidelines exist at present to explain either physical explosions or unexpected chemical explosions and their causes. Based on the fact that so many irregularities of detonation are assumed to be Kamlet‘s [ 171 statement characterizes the state of the art:
I’
This demonstrates that detonation was often treated as a ‘chef3 art’, where the ‘chef’ decided about truth. This is probably best evidenced by the different interpretations of the same facts at different institutions in even a single country. This situation has not improved since the pioneering years of detonation development. Another old (ignored) question has arisen in Becker‘s original work on detonation physics for the German academic degree of ‘Habilitation‘. Nernst, as a winner of the Nobel-prize for chemistry in 1920 and an explosives authority, considered the volume
4
Prologue
heating process to be inadequate. Instead he pointed out the importance of hot-spots. No detonation model has yet been developed from this initiation mechanism. Therefore it is the intent of the following considerations, to suggest principles of a microscopic detonation model, where the macroscopic (classical) models get asymptotic solutions. Also, shortcomings of the macroscopic view will be evaluated. So it appears that the formerly “pathological” phenomena out to be natural results, which can be explained in acoustic terms. This rationale is counter to the theory of shock phenomena. But the first leading term in each shock expansion is based on acoustics. So something is missing whenever acoustical results are neglected. A hrther argument is that the impedances are linear acoustic terms, therefore there should be no linear shock impedances present - but they are. In the end, my personal conviction is: Even in linear acoustics everything is so complicated that one must simplify the theory in order to develop the model. Some analogies to the difficulties may be understood by considering the history of the development of the gas laws based on the kinetic gas molecules as opposed to the thermodynamic version. The ideas presented here are not all new but were developed over centuries. In the sense of Newton: “If I have seen further, it is because I have stood on the shoulders of giants”. Remember, for example, the first physical application of a resonator in the Neolithic period about 2400 B. C. in the Hypogeum of Malta, see Figure 1, and the recognition of the physical effects of particles (in the atmosphere) by Derham, 1708 [18]! Figure 1. Oracle of Hypogeum of Malta. The Helmholtz-resonator (Dimensions: 180 1 volume, neck 0.2 mz, length 5 cm, resonance frequency 91 Hz). The voice is augmented and distorted due to the transformation of the energy of flow to acoustic waves. (Kind permission of Briiel & Kjaer, Kopenhagen).
5
Art
References [ 11 A. Schuster, Note to H. B. Dixon, Bakerian Lecture: On the Rate of Explosion in Gases,
Phil. Trans. Roy. Soc. London A 84 (1893), p. 1521154. [2] B. Riemann, iiber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Gotting. Nachr. (1859), p. 1921196; Gotting. Abh. VIE, p 243/265. [3] R. Becker, StoRwelle und Detonation, Z. Physik 8 (1922), p. 321/362. R. Becker, Physikalisches uber feste und gasfdrmige Sprengstoffe, Z. t e c h . Physik (1922), p. 253 ff. [4] Ya. B. Zel'dovich, Collected Scientific Papers, Nauka, Moscow, 1984. [5] J. von Neumann, Theory of Detonation Waves, OSRD-Rept. No 549, 1942. [6] W. Doring, Der Dmckverlauf in den Schwaden und im umgebenden Medium bei der Detonation, in Probleme der Detonation, Berlin, Deutsche Akademie der Luftfahrtforschung, 1940/1941. W. Doring, iiber den Detonationsvorgangin Gasen, Ann. Physik 5. Folge 43 (1943), p. 421/436. [7] W. C. Davis. The Detonation ofExplosives, Scientific American (1987)5, p. 981105, in German: Die Detonation von Sprengstoffen, Spektmm der Wissenschaft (1987)7, p. 70/77. [8] W. Doring, Letter from 06.10.1987 to Dr. Rudi Schall. [9] A. Schmidt, Beitrage zur thermodynamischenBehandlung explosibler Vorgange, Z. ges. SchieR- und Sprengstoffwes. 24 (1929)2,3,4,p. 41/46; 90/93; 97/101; 144/148, and later contributions. [lo] J. F. Roth, Personal communication. 1975. [ 1 11 Ya. B. Zel'dovich, and A. S. Kompaneets: Theory of Detonation, Academic Press, New
York and London, 1960. [12] Ya. B. Zel'dovich, and Yu. P. Raizer: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. I and 11, Academic Press, New York and London, 1966. [13] H. D. Gruschka, and F. Wecken: Gasdynamic Theory of Detonation, Gordon & Breach, London, 1972. [14] W. Fickett and W. C. Davis: Detonation, University of California Press, Berkeley, Los Angeles, London, 1979. [ 151 Ch. L. Mader, Numerical Modeling of Detonations, University of California Press, Berkeley, Los Angeles, London, 1979. [ 161 M. A. Nettleton, Gaseous Detonations, their Nature, Effects and Control, Chapman and Hall, London, New York, 1987. [ 171 M. J. Kamlet, Lectures on Detonation Chemistry, No 1, 9/1986
[ 181 W. D. Derham, Experimenta & Observationes de Soni Motu, aliisque ad id attinentibus, factae, Phil. Trans, Roy. SOC.London 26 (1708), p. 2/36.
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I
SHORTCOMINGS IN THE MACROSCOPIC PLANE-WAVE MODELOF DETONATION
Description of the Physics The plane-wave detonation model considers the impedance and therefore the sound velocity in the Chapman-Jouguet (CJ) compressed state to be independent of the shape of the pressure wave and of the diameter of the charge. A plane piston in motion leads to a plane detonation wave. To test this assumption, linear acoustics is applied to the piston model of detonation in the CJ compressed state. This cannot be accomplished by classical exact solutions of the Huygens-Rayleigh integral for the piston as a plane membrane, because the pressure and impedance over the diameter depend on geometry and dynamics, which in this case means a wavelength h (Fig. 1-2 to 1-4). The impedance becomes complex. In the case of high dynamics, the real plane-wave impedance is asymptoticallyobtained, but large pressure variations result fiom the location of the membrane. By contrast, smooth pressure fronts are obtained for low dynamics, where the impedance disappears. Thus, a plane piston in motion will not produce a plane pressure wave. The classical macroscopic approach of the piston model of detonation as laminar flow of a homogeneous reactive medium is not consistent in itself, because if each element of the piston radiates pressure waves, then no plane wave can be expected by the application of Huygens' principle on the elements of the radiating membrane.
Plane-Wave Detonation Starting from the experimental demonstration that plane detonation fronts exist, then the plane piston model of detonation has brought much progress. Its main advantage is that engineering approximations of detonation can be obtained by application of the conservation equations of mass, momentum and energy, assuming a homogeneous laminar flow and ignoring any phase transitions. The conservation laws result in: p , , ~= p ( -~u p ) and
8
-
=
Also, the Chapman-Jouguet (CJ) condition for stability is used in the form (1-3) c is the sound velocity in the Chapman-Jouguet (CJ) compressed state. This is linked with the Equation of State (EOS) of the reaction products, which is the result of the conservation of energy. A detonation proceeds through an explosive mixture with a velocity D. Initially the medium (density p,,) is at rest (particle velocity is zero) and has a pressure p,,, which is small in comparison to the detonation pressure p. In the reacted explosive the density has increased to p and the particle velocity is up, see Figure 1-1.
P1Ston
Detmstlon "l'0"l
Figure 1-1 Classical steady state piston (diameter 2a) model of detonation The plane piston (membrane) is driven by the CJ-particle velocity leading to a plane pressure wave propagation with detonation velocity D
- _
Rearrangement of Equations 1-1 to 1-3 leads to an expression for the shock impedance Z, which corresponds to the acoustic impedance ( P C ) ~of, the shock-compressed medium in the CJ-state. This is the well-known plane wave impedance of acoustics, which depends neither on the diameter of the piston nor on the wave shape, i. e., it is always constant.
Plane-wave assumptions and this shock impedance are the key tools of classical shockand detonation physics to calculate shock reflection and transmission, for example to evaluate the detonation pressure {Goranson, see [I-l]}. It is assumed in the classical detonation model that a plane piston of diameter 2a in motion with the appropriate CJ particle velocity up drives a plane pressure front propagating with velocity D (Figure 1-1). The pressure and corresponding particle velocity are linked by the plane-wave impedance, which is determined from the original density po and detonation velocity D. On the other hand, the local sound velocity c is
9 plane
not known by direct measurements, but is inferred from the equation of state (EOS). Practically, engineering approximations are obtained by adjusting this EOS to actual calibration measurements of an explosive. Therefore, such solutions are only valid for the appropriate range of composition and geometry for which the EOS has been determined. It is therefore hardly surprising that any conclusions drawn from this model have similar limited validity. For example, determination of the detonation pressure for a single explosive by different experimental methods leads to a wide range of values: 195 kbar [I-21, 268, 275, 289, 292, and 312 kbar [I-31. The spread of values (1 17 kbar) is much greater than the statistical uncertainty in the measurements (ca. 5 kbar).
Acoustics of a Piston Again we assume a piston of diameter 2a and velocity up, but now postulate that the velocity varies harmonically with a frequency f, so that the wave vector k = 2 7cUc = = 27cIh. Any area element of this piston is a pressure-radiating element. At a given point of observation the total pressure may be obtained by Huygens' principle. The added parts from these area elements vary as the point of observation varies. If this point moves out of the axis of symmetry, the pressure contributions change accordingly. According to Rayleigh [I-41, Backhaus [I-5], Stenzel [I-61, and StenzeVBrosze [I-71 one gets:
or in integral form [I-8,I-9] f
~ = p I-c
1
Y-
1
1-t2 exp(-2ikut)dt
where J1() and HI( ) are the first order Bessel and Struve functions, respectively. In Figure 1-2 and 1-3, the impedance values are plotted in two different coordinates for various ka values. It is inferred from these figures that a plane-wave impedance, Re Z = pc, is realized for large values of ka, and Im Z = 0 is realized asymptotically for ka (for large arguments J1() 0, and HI( ) 3 2h). Contrary to the plane-wave assumption in detonation theory, this impedance is diameter dependent in that this diameter is related to the dynamic quantity h. The exact calculation of this pressure distribution over the surface of the membrane we owe to Stenzel, and StenzelIBrosze [I-6,1-71, see Figure 1-4. The result is that a very approximate 'plane' wave at low pressure amplitudes is obtained only for low ka values,
10
Chapter
where the impedances are complex as shown in Figures 1-2 and 1-3. Large pressure variations at the location on the membrane result for larger ka-values.
0
0.5 ReZlpc
-
1.0
Figure 1-2 Impedance of the piston of diameter 2a of Figure 1-1 in usual Gaussian terms of complex quantities.
t
0.01
0.1
-
1.0
ko
rr
10
100
Figure 1-3 Impedance of the piston of diameter 2a of Fig. 1-1 in polar notation. The modulus of the impedance is the radius vector to the appropriate diameter ka = 2 x d h .
Figure 1-4 Pressure distribution across the diameter of the piston. x is the distance &om the center of the piston (StenzeliBrosze with kind permission).
Because the impedance of a "real" plane wave does not depend on the shape of the pressure or particle velocity function, we can not for another non-harmonic function. Apparently one never gets plane-wave behavior [Im Z = 0, Re Z = pc] a plane pressure wave propagation by a plane motion of the piston
11
Discussion The plane-wave detonation model assumes the impedance and therefore the sound velocity in the CJ-compressed state to be independent of the shape of the pressure wave and the diameter of the charge. Acoustic considerations show that this can not be true, even for a loss-free medium. This "incredible" result in applied acoustics was accepted only after direct experimental demonstration [I-lo], [I-1 11 B. Furthermore, if losses and rate effects of the chemical reaction in the detonation zone are taken into account, then the corresponding complex sound velocities also come into play [I-121. If these acoustical considerationshold, then they indicate that plane pressure waves can not exist, even though they may be readily observed experimentally. We believe that the considerations do hold, and therefore the plane wave model is at best a rough approximation.
References [I-11 R. E. and E. Houston, Measurement of the Chapman-Jouguet Pressure and Reaction Zone Length in a Detonating High Explosive, J. Chem. Phys. 23 (1955), p. 1268/1273. [I-21 G. Hollenberg, H.-R. KleinhanR and G. Reiling, Messung des Chapman-Jouguet-Druckes mit Rontgen-Absorption,Z. Naturforsch. 36a (1981), p. 437/442. [I-31 W. C. Davis, and D. Venable, Pressure Measurements for Composition B-3, Proc. 5th Symp. (Int.) Detonation, 1970, p. 13/21, Office ofNaval Research, ACR-184. [I-41 J. W. S. Rayleigh, The Theory of Sound, Vol. 11, $302, Dover, New York, reprint 1945 (1 896). H. Backhaus, Das Schallfeld der kreisfdnnigen Kolbenmembran, Ann. Physik 5. Folge, 5 (1930), p. 1/35. [I-61 H. Stenzel, iiber die akustische Strahlung von Membranen, Ann. Physik, 5. Folge, (1930), p. 947/982. [I-71 StenzeVBrosze, Leitfaden zur Berechnung von Schallvorgangen,2ndedition, 1958, Springer, Berlin, Gottingen, Heidelberg.
[I-81 L. Huber, Personal communication, 1982. [I-91 M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Appl. Math. Ser. 55, 1970. [I-101 0. Brosze, Personal communication, 1985 .
Publication of these results had been refused by six esteemed journals in the 80ies due to apparent absurdness. But the well-known acoustician Otto Brosze remembered the same controversies in the 20ies. Finally to settle the argument a 6-m plane membrane was attached at the walls of the Reichstelegrafenamtin Berlin, and no plane pressure field could be noted! Even after the measurements, the controversies continued for many years, and were finally forgotten!
12
Chapter I
[I-1 11 H. Backhaus, and F. Trendelenburg, ijber die Richtwirkung von Kolbenmembranen, Z. techn. Physik (1926), p. 630/635. [I-121 A. Einstein, Schallausbreitungin teilweise dissoziierten Gasen, Sitzungsber. Preuss. Akad. Wiss. (1920), p. 380/385.
I1
IMPEDANCE MIRRORPHOTOGRAPHY OF H. DEANMALLORY
Photographs of detonation fronts, the development of detonation, and their side views may be obtained by using the impedance mirror streak and framing camera technique of H. Dean Mallory [11-11, [11-21. The principle of this technique is shown in Figure 11-1.
-
Direction of Detonation
Figure 11-1 Arrangements of the impedance mirror technique of Mallory to obtain the structure of a detonation front. .
This technique is based on the fact that Plexiglas shows some shock elasticity; and thus will not break instantaneously in a shock attack. Therefore, an impinging structured, inhomogeneous pressure stem produces characteristic fingerprints, which may be observed on the mirror-coated Plexiglas.
Interpretation of Impedance Mirror Photographs If a constant plane-wave impedance exists, pressure and particle velocities are in phase and Mallory's original interpretation fully applies. However, if the impedance varies, as will be described below, pressure and particle velocities behave differently. In contrast to the pressure imprint on the Plexiglas, which may change with time, the particle velocity may erode the Plexiglas surface irreversibly. This was discussed with H. Dean Mallory, and he remembered his investigations of the detonating mixture of hydrazine mononitratelhydrazine, where only one picture had been published up to now [Ref. 11-21. This detonating mixture shows no detonation luminosity, and the detonation products are optically transparent.
13
14
Chapter
Figure 11-2 A series of detonating 18-molal hydrazine mononitrate/hydrazine looking through the reaction gases at a backlight. The grid lines are on the back surface of the tank. The numbers give multiples of the inter-frame distances of 1.5 bs.
Figure 11-3 Wrong color photo The common structures of the picture pairs appear in white, The engraved structure seems to remain when one compares figures 11-3a and fig. 11-3b. Fig. 11-3a. The blue picture 10 (not shown in Figure 11-2) is copied with the red picture 15.
Fig. 11-3b. From Figure 11-2 the blue picture 5 is copied with the red picture 15.
Side-on framing camera snap shots at an exposure time of 0.6 ps and an interframe distance of 1.5 ps were taken with equipment shown schematically in Figure 11-lb without mirror. Some of the photographs are shown in Figure 11-2.
TOfind erosion patterns, if present, a blue (early state) and a red (later stage) film copy has been made from each exposure of Figure 11-2. This enabled us to compare photos taken at different time intervals by copying the different color films together. The f d l addition of the colors produced white images, whereas the earlier situation appears in blue, and the later in red. In Figure 11-3 this is shown for two different time intervals. Note that from Figure 11-3 the early engraved structures do not disappear. Most probably the erosion of the particle velocity remains forever, so that quantity variations at later times cannot be detected. The Plexiglas erosions are captured on film [II-31.
15
Impedance Mirror Photographs of Detonating Nitromethane Nitromethane (NM) is the classical liquid explosive used to test detonation ideas. It was found that volume homogeneous thermal explosion describes the detonation behavior. In the classical view, a volume homogeneous shock heating process leads to a uniform plane-wave detonation. Diluted NM will not change the mechanism, provided that detonation is still possible. In Figure 11-4, a previously unpublished series of photographs by the late Dr. Mallory are shown of the side views of the NM-detonation with increasing acetone content. Close examination of these photographs shows that the fronts roughen with increasing acetone content. It is extremely difficult to understand how such behavior can result from a uniform isotropic shock heating process. By head-on photography of the detonation front, Mallory [11-41 has shown the transition from smooth to rough fronts with increasing dilution of NM with acetone (Fig. 11-5). Figure 11-6 shows this problem even more dramatically: An NM/acetone = 75/25 vol% charge of 5.75-in. diameter and 6-in. length has been initiated by a plane-wave generator. As is evident from the streak camera record, detonation started at the left side of the charge in approximately the middle, and the lower part of the charge followed somewhat later when a second detonation started there. Since boundary effects and rarefactions cannot affect the apparently inert regions between the two detonations, we cannot 'explain' this result in usual plane-wave terms. These findings are also supported by many Russian experiments [11-51, showing that the detonation front roughens with increasing dilution. Typically, the front of a cm with a detonating neat high explosive exhibited a front roughness of about periodicity of about 5 x cm. Pimbley, Mader and Bowman [II-6] also investigated plane-wave generators both theoretically and experimentally. They concluded that significant gradients appear in compression behind the front across the diameter even though the wave-front is plane. All these results indicate that the assumption of a plane-wave detonation model is an inadequate approximation for more sophisticated problems, especially on the microscale. The paradox may be removed if we consider not a plane- but a spherical-wave model. This will be discussed in detail later in this book.
Chapter
Figure 11-4 Impedance mirror streak camera views. All shots were plane-wave initiated by a 10.2 cm diameter booster and a 2.54 cm thick Comp. B pad. (Unpublished result of Dr. H. D. Mallory.)
a) 100 % NM, 6.35-cm 0,45.7-cm length, reaction time = 22 3 ns.
b) 90/10 vol.% NM/acetone, 7.6cm 0, 30.5-cm length.
c) 85/15 vol.% NM/acetone, 9.5-cm 0 , 30.5-cm length.
d) 80/20 vol.% NM/acetone, 7.6-cm 0,30.5-cm length.
e) 75/25 vol.% NM/acetone, 7.6-cm 0, 30.5-cm length.
17
Figure 11-5 Framing camera views into the detonation fionts ofNM with varying dilution with acetone @lanewave initiated) fiom H. D. Mallory [II-4]. X has a real height of 3.2 mm. Exposure time 0.75 ps.
The reflecting light from a detonating liquid should be specular for a plane detonation, but in diluted detonating NM it was only partially specular, and showed some diffuse components arising from a small roughness, according to Zel'dovich, Kormer, Krishkevich and Yushko [I1-7]. Using the setup shown in Figure 11-7, a streak photograph of detonating NWacetone 45/55 vol.% is shown in Figure 11-8 that shows considerable roughness of the detonation front, but the same experiments on inert liquids (like hexane, water or
18
Chapter I1
methylene iodide) showed no patterns. The period of intense activity lasts only for about 1 ps [II-41.
19 Impedance Mirror Photography: Nitromethane
Figure 11-9. Impedance mirror photographs of detonating NIWacetone = 75125 vol.%. Top Fig. II-9a..: Tank 7-cm I. D., 3.8-cm long, booster P-40 + 2.5 cm Comp. B. Center: The same. Right: The same, but tank length was 15.2 cm.
Bottom Fig. 11-9b. Same as above, but 7.6-cm tank length. Center: Tank 10.2-an d. 7.6-cm long, booster: P-40 + 12.75 mm Comp B. Right: Tank 5.7-cm 1. D., 7.6-cm long, boosted with PA0 + 2.5 cm Comp. B.
20 Chapter
Figure 11-11. Impedance mirror photograph of cast baratol on a polished 1.22-mm Cu-foil into Alhexane tank. (Unpublished result ofH. Dean Mallory.) Left: Cast baratol with subsurface bubbles, where the left one made a breakthrough, and the rings below the X are rekaction patterns of another bubble, The height of X indicates 3.2 mm. Right: Magnification of the interesting part.
Comparing all these results, one may puzzle on the many, widely varying manifestations of detonation phenomena. Six camera records of detonating NMIacetone 75/25 vol.% are shown in Figure 11-9 with very different behavior: The initiating pressures ranged from strong overdrive (120 kbar) down to 83 kbar, which is the steady state detonation pressure of this composition. By an overdrive the detonation becomes 'more normal'. The system behaves similar to liquid propellants, which
21
exhibit large critical diameters. Mallory and Graham [II-81 suggested the use of an overdrive to investigate such systems on a smaller scale (e. g. postage-stamp-size, very small-scale tests). These photos are partly unpublished results of Dr. H. Dean Mallory. The comments are given in the legend of Figure 11-9. We note an exotic characteristic here: Diluting NM with carbon tetrachloride instead of acetone does not change the critical diameter even in large dilutions {Kusakabe and Fujiwara [II-91). Their suggestion that there should be a relationship between the critical diameter and the reaction time means that the reaction time for NM/CC14 systems should be independent up to 45 vol % CC14 dilution. Impedance mirror lo], proved this assumption approximately correct. But the photographs of Mallory mechanism of the diluent is not understood.
Solid Explosives Such structures are also found for solid explosives. Figure 11-10 shows impedancemirror photographs of detonating TNT as the detonation 'front' impinges the impedance mirror. More interesting is Figure 11-11, where subsurface bubbles in detonating cast baratol
22
Figure 71-12. Dark waves and interaction patterns on the detonation front in undiluted NM; 22-mm0 glass tube with 1.6-mm walls, and length of 12.7 cm. Exposure time 0.8 ps. Unpublished result ofH. Dean Mallory. A similar set was published in [II-13]. The time sequence is upper left to right, and the same for the lower sequence.
Figure 11-13 shows a smear camera record of detonating NM1acetone = 80120, where initiation occurred at the left lower part by a bubble. Convergent and divergent waves resulted. More details can be found in the original paper [11-131. Figure 11-14 shows the consecutive detonation fi-ontsafter a run of 7.5 cm [11-131. Smooth and rough detonation areas, separated by a dark wave, can be seen clearly. Chapter XX (Figure XX-6) shows models of the smoothhough transition and a dark wave.
23 Impedance Mirror Photography: Solid ExplosivedDark Waves
Figure 11-13. Smear camera record (covering about 15 ps) of detonating NMiacetone = 80120 vol% in a 55-mm i. d. glass tube. The detonation stopped after a run of 10 cm. Initiation occurred by a bubble at the left side, bottom [11-13].
There is often the belief that dark waves are typical for small dimensions andor explosive compositions that tend toward chemical inertness. However, dark waves can probably be produced in pure nitromethane in very long tubes of 50 m andor more with diameters of 15 to 20 cm. Events are observed that indicate a weakening of detonation or even a transition to burning of nitromethane. A reinitiation was also observed.
Reaction and Pressure Centers We discussed with Dr. Mallory the old question, whether pressure waves are really coupled with any chemical reaction. To find an answer, he developed an impedancemirror technique that made possible simultaneous observation of flame fronts. He used the system sketched in Figure 11-15: In a Plexiglas prism one side has been mirror coated and the other remained optically transparent, so that, using an angular mirror, both sides could be simultaneously observed by a framing camera. However, it was necessary to illuminate the impedance mirror side with an argon flash. Figure 11-16 shows the photograph of this Plexiglas container filled with ball powder. This arrangement was ignited at one side with a squib. This squib opened after 25 ps, and first luminous events were observed after 1,500 ps. At this time the framing camera started, so that 25 pictures could be taken each 10 ps, with an exposure time of 7 ps for each photo. In this way the events between 1,500 and 1,750 ps after the ignition of the squib were covered. Some results are shown in Figure 11-17. The numbers indicate the time, so that the actual snap-shot time of each picture is (1,500 ps + n x 10 ps) aRer ignition. The
24
Chapter
Figure 11-14. Detonating NM/acetone = 80120 vol.% in a 55-nun inside diameter glass tube after 7.5 cm wave run. These consecutive pictures had exposure times of 0.8 ps. The X was 3.2-mm high [11-13].
impedance-mirror picture is seen in the upper part of Fig. 11-17, and direct optical observation of burning in the lower part. No macroscopic pressure front is visible in the initial stages, but singular pressure points appear that increase in number and density until a macroscopic pressure plug appears, which was agglomerated from the many cooperating single pressure sources. But, curiously, these pressure sources appear at the other end of ignition. The burning proceeds relatively slowly and continuously, but is well separated fiom the pressure points. Development of the cooperating pressure points can be seen, but any evaluation of the development to the macroscopic pressure plug cannot be easily described.
25 Mirror
Reaction/Pressure
Argmflash to Kluminade the Impedance Mirror (only)
Mirror Impedance Mrar Rexiglass
ridi in
upper side Mirror C o a t i Lower side: optical transparent Hrror
Figure 11-15. Test arrangement.
Figure 11-16. Explosion view of the Plexiglas Prism.
Evaluation of the Organizing Pressure Spots [II-14] Evaluation of small changes in the location of the pressure points from picture to picture presents problems. In a suggestion from the US-Geological Survey, Flagstaff, AZ, this set was first considered with stereographical tools, and it appeared that the single pressure spots contribute towards a macroscopic wave.
26 Chapter
-
Technique
In Flagstaff, the planet Pluto was discovered by comparing two pictures against the same background but at different times. These two pictures were projected from two projectors consecutively shadowed by a Maltese cross, so that these two pictures could be seen alternately on the screen. Thus it became possible to find the moving point. This moving point was Pluto at that time. This technique was also applied to Mallory's picture series in a video. The mobility of the pressure points in the first 16 pictures appears to be quite stochastic, but with increasing time more and more organization becomes visible. Whereas the last four single pictures apparently show no information, this technique demonstrated that these last events are most important: The prism is widened laterally due to the macroscopic internal pressure.
B
Stereographic Technique
Another option exists for visualization of the displacement of the pressure points. Stereographic pictures are taken fkom two different points that correspond to the distance between the eyes. One can assume that the displacement in time corresponds to the geometrical displacement, like the eye distance. Then, by comparing consecutive pictures, one gets 3-D images for the pressure points that have moved. This can be done by the following methods: A
Two PROJECTOR METHODS
Color stereographic system, where the consecutive pictures are projected by 2 projectors with appropriate color filters. A 3-D image is obtained by viewing this projection with glasses with the corresponding color filters. The original color of the photographs is lost with this method. Original color is seen when the original slides are projected by polarized light from the two projectors. The angle of the planes of polarization of these two projectors is 90". Using the corresponding polarization glasses, one gets 3-D images using an aluminized projection screen. This was by far the optimum presentation. B
INCLINED MIRROR TECHNIQUE
Two consecutive pictures in original colors are simultaneously observed by inclined mirrors. The eye must adapt so that the two different pictures are merged into one. This method has the advantage that the pictures may be printed in original color; but an inclined mirror or an angular prism is necessary for observations.
c WRONGCOLOR TECHNIQUE The above techniques are not very suitable for printing purposes, so a method appropriate to obtain a printed visualization follows: The consecutive pictures are projected on a screen in coincidence of the fields, the two projectors having
27 Impedance Mirror Photography: ReactionIPressure Centers
Figure 11-17. Collection of some simultaneous impedance mirror photographs (upper side), and flame photographs (lower side) kom H. Dean Mallory. The exposure time of a single set was 7 ps. The time interval between the consecutive numbers was 10 ps. The first photo (not shown) started 1,500 ps after ignition ofthe squib, when the ball powder started to burn. The time of the numbered sequences is therefore (1,500 + nx 10) ps after the opening of the squib.
28
filters in complementary colors. Locations that overlap (coincide) appear in white, and parts that change within the time interval appear in the appropriate color. Figure 11-18 demonstrates this technique for the consecutive pictures 718, 10/11, and 14/15, where blue is used for the earlier picture and red for the later one. The red dots appeared new, the white areas showed no change with respect to the former picture, but the blue stripes indicated local shifts or the disappearance of former positions. Such effects are seen more clearly when one compares pictures of larger time intervals, as shown in Figure 11-19. No classical explanation has been found for these phenomena. Also, it is not clear fiom the classical viewpoint why regions of burning and pressure buildup are SO far apart. Therefore, more physical insight into detonation phenomena is desirable.
Initiation Experiments on NG Films Mallory's pictures show a bubbly structure of detonation, which indicates that spherical reaction sites are at work in the microscale. For substantiation of this observation, we note in some of the initiation pictures of Bowden and McOnie [11-151 for a sparkignited NG-film that flames break through into the cavitation zone, where then an explosion results, see Figure 11-20. Therefore, we conclude that spherical sites are reaction centers that also emit pressure waves.
29 Mirror Photography: Reaction/Pressure Center~s
Figure 11-18. Demonstration of the wrong color technique for the consecutive pictures 7/8, loill, and 14/15, where blue is used for the earlier picture and red for the later one. The red dots appeared new, the white areas (of overlap) showed no change with respect to the former picture, but the blue stripes indicate local shifts or the disappearance of former situations.
30 Chapter
Figure 11-19. Demonstration ofthe wrong color technique for pictures with larger differences in time 10112, 12/15, and 11\15, where blue is used for the earlier picture, and red for the later one.
31 Impedance
Photography: Reaction/Pressure Centers
Figure 11-20. Growth o f burning to detonation in a NG film after Bowden and McOnie [II-I 51 (Kind permission of Prof Field)
References [11-11 H. D. Mallory, and W. S. McEvan: Transparency of Glass and Certain Plastics under Shock Attack, J. Appl. Phys. 32 (1961)11, p. 2421/2424. [II-2] H. D. Mallory, Evidence of Turbulence in the Reaction Zone of Detonating Explosives, J. Appl. Phys. 37 (1966)13, p. 479814803. [II-3] R. Hessenmiiller, and C. 0. Leiber: Der Detonation ins Gesicht geschaut, - Mallory's - , Sprengstoffe, Pyrotechnik Schonebeck 27 (199 1)i, p. 10/13.
32
[&4] H. D. Mallory, Turbulent Effects in Detonation Flow: Diluted Nitromethane, J. Appl. Phys. 38 (1967), p. 530215306.
[II-5] A. N. Dremin, S . D. Savrov, V. S . Trofimov, and K. K. Shvedov: Detonation Waves in Condensed Media, Nauka, Moscow 1970. English version: FTD-HT-23-1889-7 1. [II-6] G. H. Pimbley, Ch. L. Mader, and A. L. Bowman: Plane Wave Generator Calculations, LANL-Rept. LA-91 19, UC-45 (1 982). [LI-7] Ya. B. Zel’dovich, S . B. Kormer, G. V. Krishkevich and K. B. Yushko: Smoothness of the Detonation Front in a Liquid Explosive, Sov. Physics Doklady 11 (1967), p. 936/939. [ I N ] H. D. Mallory, and K. J. Graham: The Postage Stamp Very-Small-Scale Test for Propellant Sensitivity, Proc. 1986 JANNAF Propulsion Systems Hazards Subcommittee Meeting, Monterey, California, March 1986.Laurel , Maryland, Chemical Propulsion Infomation Agency, June 1986.(ADD600-974, CPIA Publication 446, Vol. 1. [II-9] M. Kusakabe, and S . Fujiwara: Effects of Liquid Diluents on Detonation Propagation in Nitromethane, Proc. 6Ih Symp. (Int.) Detonation, ACR-221, Oftice of Naval Res., Arlington, 1976, p. 1331147. [II-lo] H. D. Mallory, Personal Communication from 26. 11. 1980. [II-l I ] A. W. Campbell, T. E. Holland, M. E. Malin and T. P. Cotter, jr.: Detonation Phenomena in Homogeneous Explosives, Nature 178 (l956), p. 381 .
[II-12] A. N. Dremin, 0 . K. Rozanov, and V. S. Trofimov: On the Detonation of Nitromethane, Combustion and Flame 7 (1963), p. 1531162. [El31 H. D. Mallory, and G. A. Greene: Luminosity and Pressure Aberrations in Detonating Nitromethane Solutions, J. Appl. Phys. 40 (1969)12,p. 493314938. [ E l 4 1 R. Hessenmuller, First Steps of DDT Observed by a Combined Optical and Impedance Mirror Technique by Dean Mallory, Proc. MWDDEA-AF-7 1-F/G-7304 Physics of Explosives, Bad Reichenhall, 23,126. June 1992, Vol. I, p. 41/50. [II-l5] F. P. Bowden, and M. P. McOnie: Formation of Cavities and Microjets in Liquids and their Role in Initiation and Growth of Explosion, Proc. Roy. SOC. A 298 (1967), p. 38/50.
111
PRESSURE GENERATING MECHANISMS
It is apparent from the examples in explosion or detonation that plane pressure waves cannot exist independently, but an assembly of spherical pressure sources does constitute a plane-wave source. Therefore we have to search for basic pressuregenerating mechanisms. Externally and internally generated waves exist in a medium that produces pressure waves into the surrounding medium. Most pleasant sources are those from music; other widely known sources are those from sirens and bells. Noises from technical processes, such as air hammers and the noise of supersonic planes are annoying. In addition, there are less well-known sound sources, driven thermoacoustically, where mostly resonating systems are encountered. Whenever explosion or detonation occurs, pressure waves are radiated into the surrounding medium. We should know the physical origin of the pressure sources to understand the phenomena. A variety of mechanisms is known, which in burning processes leads to acoustic sensations. Very often we do not know for sure whether these processes pose risks or whether they are benign but annoying. A typical example is that of internal engine knocking, which most people believe is dangerous, but no motor has ever been destroyed by engine knock. On the other hand, the speed of combustion in internal engines increases with the number of revolutions, which would be impossible without acoustic interactions. Therefore, such combustion rates are not observable except in a motor. The acoustic effectiveness of sound sources is usually poor - at the upper end flutes have 1 to lo%, organs 0.3% and blowing bubbles in water about 0.01%. Effectiveness can be increased by an order of magnitude or more by using resonators. Whenever this occurs, we get powerful apparent pressure augmentation even for the same energetic event. This is important for safety considerations. In daily life, we usually attribute a definite pressure-wave radiation (of 'forgotten' low effectiveness) to a set of given energetics. But whenever mechanisms are at work to increase this effectiveness even a little, we are surprised about the pressure augmentation, and wonder about the role of energy conservation. This knowledge leads to careful examination of acoustic effects in energetic systems, since unexpected scale-ups lead to explosions under certain conditions, transitions of a pulsating combustion to deflagration is an example.
33
34
Generation of Pressure Waves in Gases and Condensed Systems Whereas in condensed detonics the ambient static pressure po can be ignored, this is not the case in gaseous systems. In gaseous combustion processes the ambient pressure po - assumed to be constant usually exceeds any acoustic pressures. The generation mechanisms of acoustical instabilities are found by the pressure differentials of the acoustic pressure pa, 11. These mechanisms are represented by the equations: ~
p = PO+ pa,
as pressure, with
dp = dp,,,
p = PO+ pa,
as density, with
dp = dp,,,
u = uo + u,,
as streaming velocity, with
du = du,,.
(111-1)
The first assumption is the adiabatic equation of state p = p( p), so that the pressure p is a unique function of density p. This assumption only holds if u is small compared to the molecular velocity, and the medium's cinematic viscosity q/po is negligible. The system also should be equilibrated and the temperature meaningful. Such assumptions hold best in gas phase burning processes such as in the internal ballistic of guns and rockets and in the onset ofdeflagrations. A Taylor series o f p = p(p) leads to:
(111-2) 2
6p = c 6p , where c2 = - is the square of the sound velocity . '
aP
Another assumption is that the mass m of identical molecules in a unit volume v is constant, m = pv. So one gets p-
+
=0
ap
B
Using the ideal adiabatic gas law
v
or -= -- . ap
=
P
where y is the pressure-dependent ratio of the
where v is the volume given by the ratio of the molecular weight M and the density p, R the gas constant, and T the absolute temperature. As an estimate of the sound velocity one gets in this
E
case ~ ~ 9 1 . 1 8 3
As the temperature derivative of
c=
dc
1 dT for ideal gases. 2 T
one gets finally -= -c
For real gases one should use the appropriate gas law, but it is important to know that y is pressure dependent, since C, and Cv have different pressure dependencies. This effect is most pronounced for pressurized argon (argon-flash) [III-21.
The compressibility is defined by
K=
Therefore, one gets for the square of the sound velocity ap - -ap c2 =dp
dp
1 =1 p
(111-3)
pK
and the total pressure is p = p o + ~ =pp o +c2 ~ p = p +o c 2 (PO + ~ P ) - c2 PO p = c2 p +
7
(111-4)
This latter equation (111-4) is equation of state of the homogeneous medium for small amplitudes under the given assumptions and restrictions. The time derivative offers some of the keys to acoustic instabilities: p = 2 p c i + c 2 p = 2 p c --
2
i
p-
(111-5)
This equation indicates the following pressure-generating mechanisms: Differences in temperature with time lead to acoustic instabilities,where some phase relations can also appear when AT is positive or negative. Such temperature variations can arise by thermal transfer or by direct combustion processes. Another externally or internally controlled mechanism controls the injection of a fuel into a chamber - or that the flame pulsates internally by any means. Pressure wave augmentation or attenuation can occur when these variations are in phase with natural fkequencies of a resonating system. The mechanisms 1. and 2. are physically manifested in historically well-known thermoacousticdevices, such as Higgins gas harmonica (singing flames), Sondhauss’ sounds, the gauze organ-like sounds of Rijke and RieS. Raun et al. [III-3] surveyed these phenomena comprehensively, giving references to the original papers, their descriptions are summarized in Table 111-1. Another criterion for sound generation was given along the lines of Rayleigh [IlI-4, In-51 by Putnam Dennis [111-61 by the integral:
which must be positive for sound generation by the addition of heat or cold h(t).
36
Table 111-1: Survey of classical thermoacoustical devices ~
A
on
An
of
No of
5)
Equ. (111-5) the injection of mass into the unit volume
appears as an additional
pressure-generating effect, which can be realized by both physical and chemical means: The physical effects of inertial volume variations are nearly absent in gases caused by their low density, but become most important in solids and liquids. In solids such volume variations occur by the opening of cracks or closures. Analog void formations or their collapse both lead to physical explosions in liquids. Onset of a chemical reaction is induced by the loss of the volume variations, which blows up these centers like balloons. This greatly alters the volume, density, and compressibility of the matrix. Static pressure normally compresses a volume element corresponding to normal compressibility. Compressibility turns negative when a volume blow-up happens, and an increase in pressure as well as an increase of the volume element occur. 6)
One may question the view that the thermally driven oscillations described above constitute a mass injection into the unit volume due to thermal expansions.
Questions on Resonance 1
Obviously the thermally driven mechanisms, described in the previous section, are greatly amplified by resonating systems such as tubes (organ pipes) or vessels. For closed and open-ended tubes the first harmonics are in different phases, but in an open tube the first where h is the wave length. Such systems can be realized in harmonic is about to rocket motors, in the interior ballistics of guns, in internal combustion engines, and also in pyrotechnic whistles.
31
Generation
In cases where the resonator sizes compare with the wave lengths, the generation of a limited number of low-frequencyresonances by proper engineering arrangements is relatively easy. Unwanted resonances can often be avoided by proper devices like baffles, absorbers, or by special shaping of the resonator. Such tools are applicable only in the dimension ranges cited above.
However, resonances are always generated when high-frequency sources of fiequency f occur in a large volume V. According to Weyl [III-7] the number Am of eigenffequencies in a frequency range Aflf becomes asymptotically (111-7) so that the probability of resonance increases with the total volume and decreasing sound velocity c, of the volume’s content. Because pressure waves can become greatly amplified by resonances, it is necessary to consider both pressure-wave generating mechanisms and resonance conditions for safe technical applications. Finally, real hazards (explosions) can result by escalations of occurrences. The applicability Equation (111-7) to safety considerations had been established by the comparison of spontaneous tank-car explosions of several kinds of liquid gases. Indeed jumbo tank cars with a volume of 113.5 m3are more than twice as vulnerable to explosions than normal tank cars (volume 41.6 m3) [III-8]. by
(111-7)
by works
not
Contrary to the low-frequencyresonances, high-frequencyresonances are practically unavoidable if they occur unintentionally because of the many nonlinear resonant conditions. The only chance to eliminate these sounds is to damp them away by proper attenuation mechanisms.
Pressure-Wave Generation by Non-Thermal Means Several basic mechanisms are identified with pressure sources of varying effectiveness. The Monopole Source, (above described under 5.) was first described by Lord Rayleigh [III-4,111-5], who stated that the injection of a mass rate into the unit volume Vo leads to a pressure generation. If p is the density, and V the volume variation with time, then the quantity pV/Vo characterizes a monopole pressure source in terms of masshime per unit volume (opening or closing cracks or voids). The mechanisms driving such volume variations are not important here. As may be seen from the V -term, the dynamics becomes most important, whereas in the classical detonation approaches no typical unique dynamic events are involved.
38 Chapter
Dipole Source: Another mechanism is the momentum injection into the unit volume, such as is produced by playing stringed instruments, and the Quadrupole Source: is produced by a curl injected into the unit volume. (Supersonic Flow). The latter two mechanisms are described by Lighthill [ I S 9 to 111-111 in the 1950’s. Lord Rayleigh’s work in the 1880’s was oriented toward the acoustics of music, which had monopole sources responsible for harmonic high-fidelity sounds. But the modem acoustic stimuli are cacaphonic roars and bangs in mostly non-linear terms. Jet noises are only understandable in Lighthill’s terms. A review of technical important noises has been given by Heck1 [111-121. Now one might pose the question: which mechanism will become or is most important in detonics?
Acoustic Power The effectiveness of these sources depends on the particle velocity u, which is expressed in terms of the Mach number Ma = u/c (c being the sound velocity). Rough estimates of the acoustic power radiation can be made in terms of Monopole source
prop. Ma’ x u3
prop. u
Dipole source
prop. Ma3 x u3
prop.
Quadrupole source
prop. Ma’
x
u3
4 9
6 2
8
prop. u .
We can conclude that the monopole terms are dominant in domains where u 5 c or Ma 1. This is further substantiated by the fact that cracks and voids have been known for a long time to be of utmost importance in explosives behavior. In cases, where Ma approaches 1, dipole influences also come into play, which must explained separately. Such dipole forces are of great importance in High Velocity Detonation (HVD) events (see the dipole scattered monopole source).
39 Pressure wave Generation
The Radiating Single Pressure Source Figure 111-1 shows the collapse of a laser-induced spherical cavitation bubble water, taken with an image converter camera from Lauterborn and Timm [111-131 at one million fiames per second. The shock wave radiated upon collapse (seen in Frames 3 and 4). This is an ideal situation, usually not present.
Figure 111-1. Shock radiating cavitation bubble after its collapse. The width of one frame is 2.7 mm. (Courtesy of Lauterborn [III-13]).
Auxiliary Branches of Science When the above Mach relationship applies, we have to use results from other crossdisciplinary science branches, such as cavitation research, multiphase behavior, fracture mechanics and dynamics, and materials science. The links between cavitation and fiacture and the onset of chemical reactivity are important. Therefore the steps from single crystals toward the macroscopic continuum are essential. We have to choose a proper single pressure source and its radiation to solve the problem of a microscopic detonation model with reaction at one single hot spot, or of a shocked porous material. We use acoustic models, because we are testing the idea of a mechanism without specified properties. This acoustical model will be idealized by assuming that radial symmetrical sources are present. Since these are complicated and unspecified, we make hrther simplifications by using harmonic point or finite sized sources or another specified simple motion. In this process we gather and use solutions from many references that may apply to different problems of a microscopic detonation model. But comparison of the same solutions, often written in different notations for the same problem, is obviously difficult. Therefore, we present a review of different solutions. I.
The acoustic quantities in current use, valid for homogeneous materials, are specified, where especially the shortcomings are outlined when these results are applied to twophase systems.
40
Chapter III
2.
By the continuity of mass and the concept of mass injection into the unit volume as pressure generating mechanism, the spherical wave equation for the central symmetrical case is obtained. The solutions have general applicability in an otherwise homogeneous medium.
3.
Again the differences between plane-wave and spherical-wave descriptions are demonstrated. The shortcomings of the description of detonation in plane-wave terms are again demonstrated.
4.
Current harmonic solutions of the more generalized wave equations of different authors are reported and compared for further use.
5.
The complications of a polycrystalline matrix are addressed.
References [111-I] E. Skudrzyk, Die Grundlagen der Akustik, Springer, Wien 1954. [111-21 C. 0. Leiber, Approximative quantitative Aspects of a Hot-Spot, Part 11. Initiation, Factors of safe Handling, Reliability and Effects of hydrostatic Pressure on Initiation, J. Haz. Materials 13 (1986), p. 3 11/328. [I1131 R. L. Raun, M. W. Beckstead, J. C. Finlinson, K. P. Brooks, A Review of Rijke Tubes, Rijke Burners and related Devices, Prog. Energy Combust. Sci. 19 (1993), p. 3 131364. [III-4] J. W. S. Lord Rayleigh, The Theory of Sound, Vol. 2, Znd edition, Dover, New York, 1945, Reprint of the 1896 edition. [III-5] J. W. Strutt, Baron Rayleigh, Die Theorie des Schalles, autorisierte Deutsche Ausgabe, translated by Fr. Neesen, 2. Vol., Vieweg, Braunschweig, 1880. [1II-6] A. A. Putnam and W. R. Dennis, Burner Oscillations of the gauze-tone type, J. Acoust. SOC.Amer. 26(1954), p. 716/725. [III-7] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte he a re r partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), p. 441/479. [111-81 C. 0. Leiber, Approximative quantitative Aspects of a Hot-Spot, J. Haz. Materials 12 (1985), p. 43/64. [III-9] M. J. Lighthill, On Sound Generated Aerodynamically, I. General Theory, Proc. Roy. SOC.London A, 211 (1952), p. 564/587. [III-101 M. J. Lighthill, On Sound Generated Aerodynamically, 11. Turbulence As A Source Of Sound, Roy. SOC.London A, 222 (1954), p. 1/32.
[111-111 M. J. Lighthill, The Bakerian Lecture, 1961. Sound Generated Aerodynamically, Proc. Roy. SOC.London A, 267 (1962), p. 147/182. [111-121 M. Heckl, Stromungsgerausche, Fortschritts Ber. VDI-Z. Reihe 7, Nr. 20, 1969. [III-13] W. Lauterborn, and R. Timm, Bubble Collapse Studies at a Million Frames per Second, in W. Lauterborn, Cavitation and Inhomogeneities in Underwater Acoustics, Springer Series in Electrophysics 4, Springer, Berlin, Heidelberg, New York, 1980, p. 42/46.
IV
EQUATIONS
QUANTITIES A - ACOUSTIC Equation of State In acoustics as well as shock-wave physics and detonics, a primary relation between the pressure p, density p, and internal energy E exists: = f (I%
(IV-1)
Using the specific volume V=vP
(IV-2)
9
the compression or dilatation A, defined by A = (V
-
vo)/vo, so that v =
(1 + A)
,
(IV-3)
and the condensation s s=
- po)/po,
so that p = p,(l+
s).
(IV-4)
Consequently (1 +
(I + A ) = I
(IV-5)
should hold and only depend on pressure for constant energy. This is true for a homogeneous medium outside a dynamically activated void or crack, but such an assumption does not hold for the cross behavior of dynamically activated two-phase media. This is also the reason why neither the bulk modulus K nor the compressibility K nor the corresponding wave velocity c, are characteristics of compressed dynamically activated two-phase media. K and K are for homogeneous materials, only.
(IV-6)
41
42 Chapter
(IV-7)
(IV-8) or the local sound velocity c
(IV-9)
Euler Equation The application of Newton's law (force = mass x acceleration) to a fixed volume in a flow results in the Euler equation. Therefore, convective terms are present in a greater than one-dimensional flow. These terms are ignored (by linearization)both in acoustics and current shock relations, though they take into account that particles are moved into domains of a different flow velocity u. Omission of the convective terms will probably have serious consequences in two-phase systems.
Acoustic Energy Density Since a pressure wave shows a pressure p leading to condensation or compression, there also exists a potential energy of compression. The flow velocity u has also a kinetic energy. These questions to determine the proper energies are even in acoustics very difficult, because second-order terms must be applied {see References [IV-11, [IV-31, [IV- 1O]}. Energy densities E are considered to avoid consideration of dependence on volume. The potential energy density is -
(IV-10)
&pot.- -
2P c2
and the density of kinetic energy 1
1
(IV-11)
=-Pu2=-p(grad(D)2, 2 2
where the total energy density is the sum of both. The ratio plane waves, as will be discussed later.
/&kin. =
1 holds only for
43
Time Averages [IV-101 By definition the time average over z of the fhction f(t) is given by (IV- 12)
The time average would be zero if f(t) were a harmonic hnction, so the root mean squared average is now used (IV- 13)
f r m s = J 7 j
where for a cosine this averaged quantity is A / f i , if A is the maximum amplitude. The harmonic solutions for pressure and particle velocities are complex, but the "interesting" solutions are usually the real part. So there is a need for effective (root mean squared) quantities governing the energy relations. (IV-14) where p is the complex pressure, and p* the conjugate complex. So (IV- 15) where IpJ=
is the modulus of the pressure. The same holds for the
particle velocities. Therefore, in the following we only consider the moduli of the quantities, but split the impedance into its real and imaginary parts because the real part controls the energy partition.
The Velocity Potential As we will see later, solutions of wave equations are primarily in terms of the velocity potential Harmonic solutions may also be solved in terms of a (harmonic) pressure ; the reason for this will be outlined below. But there is the disadvantage that there is scarcely an interpretation of the physical significance of this velocity potential @. Skudrzyk [IV-11, [IV-21 gives an interpretation as a momentum per unit mass
44
Chapter IV
(IV- 16) but Lamb [IV-31 offers an explanation in terms of a condensation integral f
(IV- 17)
@=c2 0
where s is the condensation. The velocity potential is mainly an auxiliary function to obtain other quantities by differentiation. So the pressure p is:
(IV- 18)
p=*p-
If harmonic velocity potentials are used, then immediately
- =pikG
(IV- 19)
= pi@
holds. Conversely, one can use directly the harmonic pressure potential {see the above Equation (IV-19)).
p
instead of the velocity
In general the (different) particle velocities are obtained by the process .
u=T
(IV-20)
There is a convention that if Equation (IV-18) is positive, then Equation (IV-20) shows a negative sign, and the reverse. For a plane wave we have the particle velocity up=--.
dx
(IV-21)
but the general spherical wave equation is a fimction of the radial distance r, polar angle 8, and azimuth cp (see the notations in Figure IV-1). Therefore corresponding particle velocities exist, and we get:
ur=--, dr
(IV-22)
45
Figure IV-1. Sketch and notations for the spherical wave equation in general spherical coordinates. {Taken from SkUdrVk [IV-I]}.
(IV-23) (IV-24)
Monopole Cosine Sources There is now a specific situation in detonics. Experimental shock and detonation phenomena are described in classical plane-wave terms, using the plane-wave particle velocity up. But spherical pressure sources with the Yrue’ particle velocity u, have been applied. These are observed in terms of a plane-wave expectation in an assumed uniaxial direction, where the other component of u, is ignored. However, the pressure is isotropic. This situation is sketched in Figure 1V-2. The observed components of the real velocities under assumed plane-wave conditions are: up=
and
U ~ C O S ~
1 a@
1
ut=------~,osine=ursin~. r
(IV-25) (IV-26)
This means that the velocity potential @ is used for the pressure, and x cos 8 is used to determine the particle velocity. We call this condition a monopole cosine source.
46
Chapter IV
Figure IV-2. Monopole pressure sources. a) Monopole source, isotropically pulsating with surface velocity u,
h
a
b) Monopole cosine source, isotropically pulsating with surface velocity u,. According to plane-wave observations, only the components up in the direction of plane-wave propagation are monitored, and the other component ut is ignored (see the left parts). 6 is the angle between the direction of observation and the direction of the assumed plane-wave propagation.
c) Dipole source pulsating to and ko in axial direction of propagation. In the case of 'plane wave' observation, the appropriate quantities are also sketched by the arrows.
It appears that ut is out of phase to the pressure, and corresponds to the near field of the particle velocity up, in cases b) and c) of Figure TV-2. If one likes to avoid the imaginary i in harmonic solutions, it is possible to introduce a constant phase shift x / 2 into all the velocity potentials Because exp(-ix/2) = - i the quantity i can be removed.
Impedances As usual the impedances Z are calculated as a quotient of the pressure p and the appropriate particle velocity ux. Thus many different impedances are possible.
zx=G
(IV-27)
AS we will see later, the plane-wave impedance pc is a constant characteristic of the medium. We also use relative impedances Zx/pc to characterize the deviations fiom the often useful plane-wave assumptions.
41 Acoustic Equations
COMMENT: In terms of a wrong physics we expect quantities to apply, that, however, do not really apply. Example: Epicycles had been expected by Ptolemaeus!
Radiation Impedance One often encounters the term radiation impedance. The above-defined impedances are properties of the waves at any point in the wave field as function of the distance r (for spherical waves), but radiation impedance is the impedance only on the surface of the source. For example, if this source is a sphere of radius R, then one replaces the distance r of the above impedance by R, and gets the radiation impedance included in Table IV-1. Only the impedances for vibrations are interesting, not those in angles. Non-planar waves have real and imaginary parts, with very different relevance. The real part addresses wave radiation, the imaginary part kinetic energy in the nearfield, which is of no importance to the rachation process.
Power of Radiation The power of radiation N per unit area is given by 2
(IV-28)
N =(ReZ)K. 2
B - SPHERICAL WAVESEQUATION Continuity of Mass, Pressure Wave Generation Using the basic concept of mass injection, pV , into the unit volume at a specified local point as a pressure generating mechanism, it will be demonstrated that a spherical wave equation results by continuity of mass, according to Skudrzyk [IV-11, “-21 (see Figure 1V-3). The excess of fluid mass entering and leaving the shell in time dt is determined, where u, is the velocity of the sphere‘s surface.
Chapter
Figure Model used for derivation of the spherical wave equation in central symmetry.
471 y 2 p(r)u,.(r)dt - 471(r + dr)2 p ( r + dr) ur (r + dr) dt = -471 W d r d t . dr
(IV-29)
The variation of mass within the shell can also be formulated in terms of density differences. dP 6p 4nr2dr = 4xy2-dtdr
4xr2 = -c2 at
dt dr
(IV-30)
Central Symmetric Spherical Wave Equation Equating Equations (IV-29) and (IV-30), one gets the final wave equation in terms of a velocity potential which is defined as u, = - grad (IV-3 1)
Equation (IV-3 1) appears as a plane-wave equation, if rCD is replaced by radial distance r by the distance
and the
Equ. (IV-3 1) holds for the centrally symmetric case whithout angles, so that the solutions depend on time t and radial distance r only. The general solutions are: f i = f , ( c t - r ) + f 2 ( c t+ r )
,
(IV-32) (IV-33)
where the first term represents a diverging and the second term a converging wave, which travels into its origm.
49
Further, there is, see Equ. (IV-17) above, the result that from Equ. (IV-32) results crs =
(ct - r )
+
f12
(ct
+ r ).
(IV-34)
crs travels unchanged, or the condensation s decreases as l/r with increasing distance r.
Plane Waves Plane waves will be approximated asymptotically at a great distance r. The plane wave equation is, at a distance x and with a velocity potential @: (IV-35)
with the corresponding solutions of diverging and converging waves (IV-36)
@
C - COMPARISON OF THE PROPERTIES OF PLANE AND SPHERICAL WAVES Plane Waves Using the solution Equation (IV-36) for diverging plane waves one gets: (IV-37)
Z=-=pc
(IV-39)
The plane wave impedance is always constant and independent fi-om any geometry and wave shape, as assumed in plane-wave detonation theory. Furthermore, the particle velocity up corresponds to the usual particle velocity in plane-wave terms.
50
Chapter IV
Spherical Waves Using the solution Equation (IV-33) for a diverging spherical wave one gets:
af(4
a@ P p=p-=---=-. at r du a t
pcf'
' af
Since f ' = - =af(u) --
au
(IV-40)
r
and f
one gets for the particle
=
cat
velocity urrwhich is the radial velocity and different fiom the plane-wave velocity u p
a@
ur=-grad@=--=ar
1
I
r
y2
=-+pc
'I
pr
pdt.
(IV-41)
And the impedance Z becomes PC
1 +f y
.
(IV-42)
.f
The expression for the radial velocity shows two terms. The first corresponds directly to the pressure, making it decisive for the energy flow by pressure waves, the second drops with respect to the pressure with l/r much faster with distance r. In other words, flr disappears more quickly than f, so that the plane-wave impedance is obtained at large(r) distances. The quantity Z/pc is a convenient parameter to characterize the actual deviations fi-om plane-wave assumptions. These deviations disappear as Z/pc 3 1. The second term in Equ. (IV-41) is important only near the source. This is the reason of using the word "near-field term" (NF), in contrast to the first, the "far-field term" (FF). Since the near-field term is characterized by a pressure/time integral, it shows no similarity with the original pressure for arbitrary pressure functions. Due to the different dependencies o f f and f ' in Equ. (IV-41) fi-om the distance r, the shape and the amplitude of the near-field term vary with the distance r in the vicinity of a source. The particle-wave shapes become a function of the distance! Only the derivatives of harmonic functions retain the shape of the original function. Consideration of the time dependence is avoided by using these harmonic functions. That is why harmonic equations are general and simple to apply. summary: Only spherical waves have a near-field term, plane waves do not! Ignoring the near-field near a source is a most serious fault. Therefore, considerations near a hot spot in terms of plane waves are of no significance.
51
Table IV-1. Comparison of plane and spherical harmonica1 Waves (of a Monopole-Source)
I
QUANTITY
1
PLANE WAVES
SPHERICAL WAVES
I
Solution pressure p=
-ipm@ = -ipkc@
particle velocities up,
-ik@ - @/r
-ik@
=
Farfield term (FF) ~~
@(I,t) = B exp[i(kr-wt)]/r
@(x,t) = A exp[i(kx-ot)]
-ik@
~
Nearfield Term (NF)
~
1
~
I
0
Density of potential energy
1
density of kinetic energy (FF)
1
density of kinetic energy (NF)
- @/r
0
1
Impedance Z = Power of radiation per unit area N=
PC
pc-
u2
2
In the following, we focus our interest on energy partition problems to understand the differences between plane and spherical waves. The energy of hot spots divides into pressure wave emissions (compression) and into kmetic energy of flow. Therefore, the ratio of these terms determines whether more compression or more flow is obtained. Here the problem of the Deflagration Detonation Transition fits in, where energy release may be more in flow than in compression.
52
Chapter Table IV-2. Comparison of arbitrary diverging plane and spherical Waves ~
SPHERICAL DIVERGING WAVE
PLANE DIVERGING WAVE do = p af(u) du p = p- = pcf ' du
p df(u) au
p=p--=---r du
at
- pc-
r
af(4
with u = c t - r onegets f ' = - = - du
1 af c at
and f = [cf 'dt = [%it PC
1
=-
f'+,
r
*
'arbipary spherical wave -
zarbitrav plane wave
rf '
.f' + f
-1
1
f
r
r+m
We use the Equs. (IV-10) and (IV-11) to obtain the ratio of the energy densities
(IV-43) and get the square of the relative impedance Zlpc. This is always unity for plane waves, and a function of these sources and the distances for spherical waves. The reason a quadratic relative impedance appears in Equ. (IV-43) is found in the timeaveraging process. The differences between plane- and spherical-wave descriptions are not easily seen in general terms. Therefore in Table IV- 1 we compare the corresponding harmonica1 plane and spherical wave expressions, and in Table IV-2 the general diverging waves. The most important result is that for plane waves the ratio of the potential energy (coupled with pressure wave emission), and kinetic energy of flow is always 1. But for
a spherical wave at distance r = R at the surface of a monopole source (bubble) the ratio is (due to the near (or lunetic) field} is a variable both of the medium and the size of the source compared with the wave length =
(IV-44)
The ratio of Equ. (IV-44) also governs the power of radiation per unit area of this source. This power is controlled by the impedance at the source surface. Re Zlpc compares therefore the powers of pressure-wave emission of a spherical source with an assumed plane-wave radiator. The values in table IV-3 demonstrate that, depending on the source size, there are considerablemagnitudes in varying the ability of pressure wave radiation. Therefore a dramatic transformation process in pressure-wave radiation is made possible by alterations of the source size.
0.05
1.00 2.00 10.00
0.0001 0.0025 0.0099 0.0588 0.2000 0.3600 0.5000 0.8000 0.9901
Table 1V-3. Ability of the pressure wave radiation as function of the source size kR.
The Scattering Cross Section Qs The scattering cross-section Qs is an average summary quantity that compares the power of a source with an incident intensity of a plane wave. From Equ. (IV-28) we get the power of radiation for the unit area. The power of radiation of the whole source is given over its surface area 4nRz as:
(IV-45)
54 Chapter
D - GENERAL SPHERICAL WAVEEQUATION Despite the fact that the central symmetrical sources are found in Chapter XX, we will consider the general solution in harmonic waves in all three coordinates r, the polar angle 0 and the azimuth cp (see Figure IV-1). The main reason is that the dependencies may be separated, so that pure solutions of the central symmetrical case result. Many solutions exist in different terms, which are outlined and compared below. The derivation is identical to that of the central symmetrical case {see References [IV- 11 [IV-71, and [IV-101).
-
(IV-46)
One obtains the above radial symmetric form of wave equation (IV-3 1) by dropping 0 and cp. Equation (IV-46) shows infinite numbers of solutions and of special groups of solutions. Again, the solution is the velocity potential @. The pressure is determined as usual. However, several different velocities may also be derived {see Equations (IV-20) to (IV-24)).
Harmonic Solutions The solutions of Equation (IV-46) may be harmonic standing waves: =
R(r) U(0) V(cp) T(t) .
(IV-47)
Inserting this solution into Equation (IV-46) gives, by division with R x U x V
The left-hand side depends on r, 8 and cp, the right-hand side on t. So we get Time solutions T(t), Polar solutions U(0), Azimuth solutions V(cp) and their combinations with U(0), - and the Radial solutions of R(r).
x
T:
55
Time Solutions Equation (IV-48) will be satisfied if the right-hand side is constant. -k2is used as a constant. One obtains standing or progressive waves for negative values, but a positive constant leads to exponential decay. 1
__---
-
(IV-49)
k2
shows the solution
+ 5 e-imr , where
F =2
E = O when
= kc
.
(IV-50)
iscomplex.
Further Intermediate Procedures Setting the left-hand side of Equation (IV-48) to -k2and multiplying with r2, one gets two independent equations
[-The left bracket depends on r and the right on 0 and cp. These brackets must be constant. For some reasons the integer arbitrary constant n(n + 1) is used {see References [IV-1] and [IV-2]}. One gets the radial part of the equation
(IV-52)
r
ar2
and
-sine - [ s i naO ~ ] + ~ , ? p ' = 1- n ( n i l ) s i n 2 e
u
(IV-53)
RADIAL SOLUTIONS Stokes-RayleighRadial Solution [IV-11, [IV-5], "-61 Equation (IV-52) can be condensed to
(IV-54)
56
IV
(IV-55) The central part of Equation (IV-54) can be ignored for large r. Then one gets the solutions of the form =
e-ikr
+
j j ecikr ,
(IV-56)
are constants that depend on n. For n > 0 they become functions of r where 2 and for small distances, and describe the nearfield. Finally one gets as solution of Equation (IV-54) an expression in the Stokes functions -
= e-jkr
(ik)+
(-
(IV-57)
Using Equation (IV- 19), this fbnction may be written directly as a
(IV-58) The first parts of Equations (IV-57) and (IV-58) characterize a diverging wave, the second parts a converging wave; f,(ikr) describes the nearfield. Setting z = kr
+ 1)
f n ( i z ) = l + y
+
(n -I)(.)(.
+
2x 4x
......+
+ 2) + ........... (IV-59)
l x 2 x 3....x 2n
2 x 4 x 6..... x 2n x
'
Equation (IV-59) is the Stokes-(Rayleigh) h c t i o n , where the following asymptotic expressions hold: For z - 0
fn(iz)= 1 x 3 x . . . ~(2n - l)/(iz)" and
(IV-60)
for z
fn(iz)= 1.
(IV-61)
For n = 0 to 3 the functions f,(iz) and F,(iz) are:
SphericuUPlune Waves
I
Stokes (Rayleigh) functions n
fn
0
1
1
1+-
2
3 1+-+-
3
1+-+-+-
(iz>
Fn(iz) 1
1
6
2
3
15
9 +4++
9 ~
27 +7++
15
60 ~
60 +-
Particle Velocity According to Equation (IV-20), u = - grad one gets from Equation (IV-58)
=i
grad pkpc. For the particle velocity
(IV-62) Forz=kr F,(iz)
=
(1 + iz) fn(iz) - iz fi(iz)
(IV-63)
holds. The means differentiationwith respect to the argument iz. One comes ftom diverging to converging waves by changing the sign of iz. Standing waves are obtained by the superposition of both types, or by the separation of e-ikrfn(ikr)/rinto its real and imaginary parts.
In summmary, the components for a diverging wave are: Pressure:
(IV-64) Particle velocity:
(IV-65) If kr 3 00 then f,(ikr)
=
1 and F,(ikr)
=
ikr.
(IV-66)
Chapter
Bessel Solution of the Radial Equation With the Lommel transformation Y ( x ) =x a ~ p ( p x )(see Skudrzyk [IV-11, [IV-21, one obtains another form of the radial wave equation:
(IV-67) The current solutions in use are:
(IV-68)
(IV-69)
(IV-72) A
(kr).
(IV-73)
These solutions are spherical Bessel and Neumann (spherical Bessel function of the second kind) functions, and their combinations. Note that yn( ) is used in US or UK instead of the notation n,( ) for the spherical Neumann hnction. j,( ) corresponds to a standing wave without a source at the origin, and n,( ) characterizes a standing wave with a source at the origin. The same applies for the Stenzel functions [IV-71 S,( ) and C,( ). Other notations are used in optical science (see Kerker [IV-9]), so the solutions c) and d) show the name Riccati-Bessel functions. We see that the Stenzel functions are equivalent to the Riccati Bessel functions.
Progressive waves are obtained by combining the spherical Bessel and Neumann functions, producing the spherical Hankel functions (spherical Bessel function of the third kind) [IV-81: Combining any of these radial solutions with the time, polar, and azimuth solutions (see Equation (IV-47), one gets the velocity potential (D and pressure p.
59
Spherical/Plane Waves
Asymptotic value
(IV-74)
3 Z
3
(IV-75)
Z
Particle Velocities The particle velocities are obtained by derivations of the wave h c t i o n s @, R, or p. The components are given below. In Stokes-Rayleigh terms:
(IV-76) (see Equation (IV-65). Equation (IV-63) holds for F,(iz). In Stenzel or Rkcati-Bessel terms: The generating functions of S, and C, are:
(IV-77) (IV-78) whereby u n
(4
=
and
cn
vn ( z ) =
(IV-79) (IV-80)
hold. For diverging waves C,( ) contains +i. In terms of the spherical Besselfunctions: For any function of the kind fn(
= jn(
1;
nn( >;or yn( >,hn(’)(1, and hi2’(
the following recurrence relations hold:
(IV-81)
60 Chapter
Stenzel functions
sin
cos z cos z sinz+-
- 1)sinz
- 3Tcos z
3-+ si:z
--1
)
cosz
I
Derivatives of the Stenzel-function
(IV-82) For the derivatives:
(IV-83) one gets:
61
~~
Spherical Bessel functions and their derivatives
~
~~~~~~
~
Spherical Neumann functions and their derivatives
n 0 L
sinz
1
2
[; + +) -
-
,z3 .
for diverging waves nn(z) of the spherical Hankel function, h,'2'(z) is associated with -i.
62 Chapter
Spherical Hankel functions h,,(')(z)= j,(z)
--i e"
0
1
+ i n,(z) and their derivatives
1
j]
Spherical Hankel functions h,"'(z)
0
= j,,(z)
- i n,(z) and their derivatives
-[z-i]
1
--
- i]
I
Correspondencies (identical expressions) (IV-87) and (IV-88)
63 SphericuUPlune
The following relations hold for Stokes and Bessel functions:
(IV-89), (IV-90)
and
(IV-9 l),(IV-92)
or
hold, and for h:')(
):
(IV-95) The following groups of functions appear frequently:
The above equations provide different terms of the solutions of the radial-wave equation for comparison. Many calculations were made on the basis of e-iot,where diverging waves were described by e(+ih-iwt)/r.If one prefers to use the time factor eiwt, i must be replaced by -i and the reverse, must be replaced by &)(kr).
64 Chapter
Radiation Impedance From the pressure = A,
(IV-99)
hk)(kr)
and the particle velocity (IV-100) one gets the impedance Z = p/u, at the sphere's surface r = R. Then (IV- 101)
(IV-102), (IV-103)
For small
values
< (n + 1)/2) the real impedance is: (IV- 104)
and one gets the radiation impedances (kR)2; ( l ~ R ) ~ /and 4 finally (kR)6/81 asymptotically from the 0" to the 2"d order. One is always obtained when becomes large enough to approximate a plane wave, but a maximum value is usually obtained before this limit is reached.
E - SCATTERING OF A PLANE WAVE AT A SPHERE It is important to consider physical processes that can occur when a (shock) pressure wave impinges particles. To simplify we consider a single spherical particle in a medium exposed to a pressure wave.
65 Sphere
This process has been a century old problem, starting with Clebsch, 1863, and Rayleigh [IV-5, IV-61, until Stenzel “-71 solved it in principle. The many effects of the theoretical formulae are obscured by their complexity. That is why some thousands of references on scattering problems exist. The pertinent question is what happens if a plane wave impinges a spherical obstacle? This wave will be distorted, reflected, and scattered. The scattering can cause the intensity behind the sphere to be larger than in front. The macroscopic effects cause the attenuation properties of the wave to become very interdependent. Since spherical waves always exhibit a far- and a near-field, the behavior at the sphere’s surface varies greatly. The sphere can be stimulated to its many possible radiation terms. If it is compressible, like a pore, it will pulsate (order n = 0), if it is moved by the wave, as described in Chapter XIV, it can also radiate (order n = 1). These radiation terms disappear when the sphere is incompressible (n = 0), or if the sphere’s density is identical with that of the surrounding medium (n = l), so that no dipole scattering is possible. The interactions of this moving sphere with the surrounding medium lead to attenuation of the primary wave due to dissipation by viscosity, surface tensions and so forth. Effective cross sections are preferred f o r handy estimates. These cross sections give one value for integral estimates, where the scattered and the incident energy are compared in terms of geometrical areas, see Equ, (IV-45) and Chapter IX. A voluminous graphical review for estimating the effects is given by Mechel “-1 I]. The reflection properties of a sphere in the far-field are described to give insight into the attenuation properties of a pressure wave. The near-field pressure at the surface sphere is considered to estimate the effects of surface coated particles.
Reflection of a Plane Wave at a Rigid Sphere StenzeVBrosze [IV-71 define reflectivity by the ratio of the reflected pressure pr and the incident pressure pi by (IV- 105) where P,( ) are the surface harmonics of the order n and 8 measures the angle of the sphere, where 8 = 0 holds for the incidence. U, and V, are the Stenzel or RiccatiBessel-functions, see Equs. [IV-791, [IV-801. In Figure IV-4 the reflection factors, according to Equ. (IV-105) (limited to the orders n = loo), are plotted for rigid spheres of sizes =O. 1; 1; 3; 4 and 10 as function of the
66 Chapter
direction of the incident wave. The incident wave in Figure IV-5 comes from the nght hand side. The summation from n = 0 to 100 in the leR column involves radial pulsations and also lateral mobilities. the central column (n = 1 to 100) the radial pulsations are excluded, and in the right column (absent lateral mobility) without differences in densities of the medium and the sphere is considered. As can be seen, for small sized spheres 0.5 of the incident energy is asymptotically reflected, but for large spheres 4 times the energy is focussed into the direction of propagation. This leads to the conclusion that small grains attenuate the incoming pressure wave. Large grains enhance waves into the direction of wave propagation, creating unidirectionality. The higher orders are unimportant for small spheres; multipole scattering is important for larger spheres.
This result evidences the remarkable effects of anomalies of large grains in charges.
Pressure on the Surface of a Rigid Sphere The pressure and its corresponding particle velocities at the surface of a rigid sphere are responsible for what happens with surface coatings. The coating will be dispersed off the particle if large differences exist. In addition capillary waves on the surface are created, which lead to a weak radiation into the far field, but act as surface waves at the particle/medium boundary. Also such capillary waves determine the stability of bubbles, see Chapter VII. This pressure pa on the surface of the sphere was calculated by Ballentine, 1928, and is according to Skudrzyk [IV-11, [IV-21
(IV- 106) Figure IV-5 shows these pressure distributions on the sphere for the same cases as in Figure IV-4. For small particles, the surface pressure is isotropic for n = 0 ... 100, and very weak for quadrupole radiators n = 2 ... 100. For increasing particle sizes, the pressure on the surface changes greatly, and storms around the sphere arise, which disperse the surface material into the surrounding medium. A direct explosives application of this result is the Kuhn-Kaufer-selectivity of surfacecoated particles in commercial explosives as well as the stripping-off of surface coatings in aluminized explosives, see Chapter XIV.
Scattering
67
Sphere
Reflection factor of the Sphere
-
I
n = 1,100
n=
a
0
I
n=Lloa
M
0
1
0
I
270
m
I
0
1
1
210
270
3
I
0
I
1
0
I
(I
.. 170
270
4 I
I
270
270
10
Yu
210
I
I
0
170
I
1
I
270
Figure IV-4.
Reflection factors according to Equ. (IV-105) for n = 0 ... 100 (left), n = 1 _..100 (center), and n = 2 ... 100 (right) for rigid spheres of sizes =0.1; 1 ; 3; 4 and 10 as function of the direction of the incident wave. The incident wave comes from the right hand side.
68 Chapter
Surface Pressure on the Sphere (Nearfield) = 1...100
=L O lO w
1
1
1
270
1
= 2-100
M
270
270
w
w
1
1
1
270
170
270
3
M
I
I
270
2 70
4
w
QI
I
270
270
10
w
I
M I"
I
1
270
270
Figure IV-5. Relative pressures on the surface ofthe sphere according to Equ. (IV-106) for n = 0...100 (left), n = 1...100 (center), and n = 2... 100 (right) for rigid spheres of sizes kR = 0.1; 1; 3; 4 and 10 as function of the direction ofthe incident wave. The incident wave comes fiom the right-hand side. The non-uniform surface pressures lead to ,,storms" that disperse the surface coatings into the surrounding medium, and increase the reactivity.
69 Cylindrical
F - CYLINDRICAL WAVES Only plane and spherical waves are discussed here, because description of plane waves addresses infinite homogeneous media far fiom any pressure source, as used in classical detonics. Description of spherical waves can be applied directly to pressure- generating sources on the microscale to construct a microscopic detonation model. However, description of cylindrical waves addresses phenomena in generic dimensions only and is described in detail elsewhere [IV-I], [IV-21, [IV-101, [IV-12] and [IV-18]. Thus, an explosive system can be defined to include its confinement, shape, strength, and deformationalbehavior. The difficulty is that a homogeneous medium is assumed in usual (complex) solutions where no microscopic pressure sources inside the volume are involved. Any mathematical solution in cylindrical waves is so complex that no asymptotic results can be expected. Similar to Equation (IV-47), any standing or progressive harmonic or nonharmonic wave can be constructed by the superposition of the adequate solutions of =
R(r) Z(Z>VCp) T(9,
(IV-107)
where R(r) relates to the cross-section of the cylinder as function of its radius r, Z(z) to the axes z of the cylinder, V(9) to the azimuth angle cp, and T(t) stands for the time dependence. Detonic phenomena in terms of cylindrical waves have not been considered previously. Therefore, some - even tentative - links between such solutions and well-known detonic phenomena are noted below, and will be discussed in detail in the experimental part (Chapters XI, XII, XlX, and others). Standing and progressive waves coexist in the cylinder. Progressive rotating waves may exist in the cylinder, which are waves that propagate along helical surfaces like a screw. Such phenomena are also addressed as spin modes (see evidences of spin detonation in Chapters XI and XII). The complex radiation impedances as quotient of acting pressure and particle velocity vary considerably, explaining perhaps the variety of detonation effects of charges with axial hole (see Chapter XXII).
70
Chapter IV
The influence of change in cross-section is considered, caused by a sudden event or by a cone. This is part of the generation of detonation overdrive in conical geometry, but also of detonation traps used as safety devices, see Chapter XIX. Axial and radial vibrations of the confinement (external cylinder) are also involved. In usual testing of explosives in steel tubes this is of interest if these tubes split (see Chapter XIX). The influence of end caps - or their absence - on a tube is considered.
G - WAVES IN ELASTIC ISOTROPIC MEDIA The isotropic wave properties are sketched in combination with their great manifold. The deduction of the wave equation from basic physics is repeated to demonstrate the exact meaning of the quantities. The approximate character of a continuum can be seen by comparing the results of Chapter XXIII. Any variation in stress shifts the location of any point in the medium. This variation in location is expressed in the coordinates x, y and z by u, v and w. With p as the medium’s density one gets:
(IV- 108)
This Equation holds for any stresdstrain relation of the material. With Hooke’s law, see Chapter XXIII, and the compression A =,& +
AV
+&ZZ
=-
vo
71 in Isotropic
(h+2p)E=
0xr
=
0YY
-
As,
022
=
he,
0yz
=
+hEyy +(h+2P)EW f h Eyy
=
= f
f
2p)Ezz PEyz
=
W Ezx
= oxy
hA+2psxr h A + 2 ~ & ~ = hA+2p~~.
Ezz
=
= =
PExy
2PEyz 2PEZ 2P Exy
(IV-109) Inserting Equation (IV-109) into Equation (IV-108) one gets:
(Iv-110) For the cinematic dsplacements u, v and w one gets for the elongations or contractions: C?U
Eyy =
En=-
dW
-,
Ezz
=-
and for the shears: du
Eyz = -+ - ,
Ezx = -+
?!J
-, ax
du Exy =
+
.
The next components represent the rotation of a rigid body without deformations. With twas a vector with the components u, v and w, % rot m. delivers the components in the direction of the axes: 2m = - - -
=---
i$
az
az’
Differentiating Equation (IV-110) with respect to x, y, and z, one gets [IV-131, [IV-141: (h+2p)V2A=p-a2A
(IV-111)
a
This equation contains a compression. The variations of pressure and volume are in the same direction, so that longitudinal compression waves - called irrotational waves - are described. By elimination of A (i. e. any compression!) in Equ. (IV-110) by a rot-operation, one gets in ti^ or u: p v 2 u - p-,
etc. for v and w. at2
.
(IV-1 12)
This is the description of equivoluminal waves that do not lead to compressions. Such wave types were first described by Navier (1 821) and Poisson (1827).
72 Chapter
Longitudinal Sound Velocity cl From Equation (IV-111) one gets the longitudinal sound velocity c1, also called velocity of dilatation or condensation, or in seismics primary or push (P-wave): p(l + vx1- 2v) .
(IV- 113)
One may puzzle why the shear modulus G is involved. This is because the reference volume is only varied in the direction of wave propagation; it remains constant parallel to the wave fiont. Therefore, the area elements are sheared in direction of wave propagation.
(IV- 113a)
Transverse Sound Velocity c, The shear wave velocity cs, is called velocity of equivoluminal waves, transverse waves, irrotational wave, distortional wave, solenoidal waves, or in seismics the secondary wave or shake. Since polarization is possible, “SV” represents a vertical shake and “SH” a horizontal shake. The wave velocity is:
(IV-114) In hydrodynamics, such as in gases, liquids, or shock hydrodynamic states, the shear modulus G or vanishes, indicating that shear waves are absent.
Bulk Sound Velocity co The bulk sound velocity is related to the modulus of compression K: 2
(IV- 115) Since gases, liquids and hydrodynamic states in shock physics do not exhibit a shear modulus, co = C]
(IV-116)
73
applies. In solids c, is determined experimentally by measuring c1 and c,: c g = \ /C2l
--cs 4 2 3
.
(IV- 117)
These wave types are possible only in an injinite isotropic medium. Compression waves remain compression waves and shear waves shear waves. No interconversion takes place! If these conditions are not given, many important complications occur, and other wave phenomena come into play! Table IV-6. Sound velocities and elastic properties of some materials
Material
Density (gicm’)
Velocities of Sound in m / s
-
Youngs- ShearCompression-. Modulus in Mbar
--- - ---C1
cs
CO
E
G
K
2.7
6100
3100
4940
0.688
0.259
0.659
0.326
Lead
11.34
2200
700
2050
0.160
0.056
0.474
0.44
Iron
7.8
5875
3140
4620
1.99
0.769
1.67
0.30
Copper
8.93
1.23
0.45
1.54
0.37
Plexiglas
1.18
2600
1300
2123
0.053
0.020
0.053
0.333
Polystyrene
1.06
2300
1200
1835
0.040
0.015
0.036
0.313
Windowglass
2.48
6800
3300
5630
0.727
0.270
0.787
0.35
Nitromethane
1.13
1340
0.0303
0.50
TNT
1.624
2480
1340
1940
0.075
0.029
0.061
0.294
TATB
1.876
1980
1160
1460
0.063
0.025
0.040
0.239
PETN
1.75
2980
1640
2300
0.120
0.047
0.093
0.28
1.72
2820
1480
2240
0.099
0.038
0.086
0.3 1
Aluminium
PETN
1340
--- - ----
Sound Velocities and Elastic Properties Conversely the isotropic elastic constants can be determined from the measured sound velocities c1and c, together with the medium’s density p:
(IV-118)
74 Chapter
(IV- 1 19) G=pcZ,
and
(IV- 120) (IV- 12 1)
Also the ratio of the sound velocities used is:
(IV-122)
Some data are given in Table IV-6.
Qualitative Description of possible additional Wave Phenomena The assumption of an infinite extenuated isotropic medium is idealized (in addition to the assumption of an isotropic medium). As one approaches reality, the phenomenon of waves become more complex and multifaceted. This is outlined qualitatively below.
Aelotropic Medium Because a single crystal usually exhibits different elastic properties in different directions, the wave parameters are also different, enabling mixed wave modes (compression and shear) to appear. The direction of wave propagation is different fiom that of energy propagation. We refer to the monograph of Fedorov [IV-I 51 for additional information. Surface waves on the crystal surface must also to be considered. Therefore, it is not possible to realize a plane, uniformly steep wave in a polycrystalline material. Locations of rapid as well as slow wave motion appear, so that phase relations become important; the formerly steep wave splits up, roughens and broadens considerably, and an attenuation results. It is not even possible to give exact estimates of the material's compression, because interconversions between the wave types occurs. A summary of Smith and Stephens [IV-161 addresses such questions. It seems surprising that such important questions have not been addressed before in explosive science. Any molecular-dynamics considerations are therefore unrealistic.
Infinite Isotropic Elastic Medium In a perfectly homogeneous, macroscopic, infinite continuum we only have compression andor stress and shear waves that never convert. This means that no stress occurs in the case of a hypothetical pure compression, and no crack can be opened.
75 Isotropic
Shear waves are equivoluminal waves, and can therefore not open cracks (but only induce them). We have compression (longitudinal) waves cl and shear waves c,, which might become polarized. In addition, we have the bulk wave c 0 that results fi-om CI and c,.
A liquid shows no shear modulus G, and will therefore not exhibit any shear wave; only the bulk sound velocity co is present. The Poisson ratio is v = 0.5, and Youngs modulus E is completely absent. Therefore only one quantity, i. e. the compression modulus K, describes the dynamic behavior of a liquid. Semi-infinite Isotropic Elastic Medium The situation of the possible wave types is completely altered when the same material is considered in a semi-infinite continuum because boundaries come into play. All the waves are partly transmitted and partly reflected at such boundaries, even by a change in sign. Mechanisms are present to convert compression waves into tensile waves, which can open cracks. Only in the case of rectangular incidence of a wave front on the boundary is there no interconversionbetween compression and equivoluminalwaves. In the case of oblique incidence on such a boundary, interconversion between pressure, stress and polarized shear waves is usual. Differences can be observed depending on whether medium I1 is fixed or mobile with respect to medium 1. One wave can split into several wave types at such a boundary. They might be: Reflected waves with possible change in sign, Transmitted waves, Splitting into polarized shear waves, but also Total reflection becomes possible, so that no transmission OCCUTS into the neighboring medium. So a uniform wave in medium I leads to up to four waves in media 1/11. Polarized shear waves are additional phenomena in medium 11.
Surface (Rayleigh) Waves Surface waves (Rayleigh waves) on the medium’s surface are also possible. Their velocity CR is lower than the shear-wave velocity c,, and depends on the Poisson ratio v of the material:
76
Chapter
- = (0,87 + 1,12v) cs
(1+4
.
(IV-123)
If v varies between 0 and 0.5, cR/c,varies between 0.87 and 0.95, and CR is independent of the exciting frequency (no dispersion takes place). The particle velocity is perpendicular to the direction of the wave propagation, and elliptical. Therefore, two components appear - in plane and perpendicular to it. Because these waves spread only into two dimensions, their attenuation is poor. Since they are active only in a thin layer at the surface, their energy density is large so that they exhibit very large amplitudes. The wave amplitude diminishes exponentially with increasing distance from the surface, until the particle velocity is zero at a depth of 0.193 h, and the sign changes at larger distances as attenuation proceeds. Due to the weak attenuation of these waves, in seismics the surface waves at larger distances are dominant, and are responsible for the resulting damage. One may ponder the question, whether such surface waves can be responsible for detonation transfer over unexpected large distances.
Estimate of Dominant Wave Types Since we have now at least three different types of waves, an acoustical estimate is helpfd to evaluate the most dominant wave type. Assuming that - as usually understood - the energy distribution governs initiation, one can apply the theoretical results of Miller and Pursey [IV-171 on energy partition for an oscillating harmonic point source on such a semi-infinite continuum of Poisson ratio v = 0.25. The results are surprising: 67%of the total energy is in the Rayleigh wave, 26%of the total energy is in the shear waves, and only 7%of the total energy is in compression!
Therefore, we conclude that sensitivity can be predominantly governed by surface waves, so that gap-test results in their usual interpretation become questionable. The least important factor in the gap-test, the compression, is determined, see Chapter XII!
Additional Wave Types In addition to Rayleigh's waves, several other, similar types of wave propagation are found in layered boundaries: They are seismic waves, called Lamb [IV-121,
Isotropic Media
Love “-121 or more thoroughly Stonely waves [IV-121. Their common characteristic is that the maximum amplitude is in the contact plane between two media. Other wave types, such as plate, bar, rod and cylindrical waves are present in special geometric configurations of the sample or system under consideration, for example, spinning waves. Pulsations are important in the case of cylindrical waves (see part F), because they might dictate the large fragment patterns of steel tubes seen in steel-tube detonation experiments. Therefore, we conclude that systems vibrations - usually not considered - can dictate detonation transfer, and shortcomings obviously exist in the current description of explosives sensitivity. As a consequence, the surface waves dictate in the macroscopic view any sensitivity aspects of explosive substances, especially in any confinement and in articles (=manufactureditems). This means that the sensitivities of an explosive, and of the same explosive in an ammunition, are basically different, and not linked to each other.
In summary and in combination with the results of Chapter XXIII it appears that the simplifications in the evaluation of detonic experiments are so overwhelming that the ‘measured’results scarcely reflect any real physical quantities. Therefore, these quantities are a matter of convention only, as for example prescribed by rules or laws. It would be worthwhile to consider events in the real world before one extends or continues such legalistic standards.
References [IV-1] E. Skudrzyk, The Foundations of Acoustics, Springer Verlag, Wien-New York, 1971. “-21 “-31 “-41
E. Skudrzyk, Die Grundlagen der Akustik, Springer-Verlag,Wien, 1954. H. Lamb, The Dynamical Theory of Sound, 2nded., 1925, Dover, New York, Reprint 1960 H. Lamb, Hydrodynamics, 6th edition, 1932, Cambridge University Press, Reprint 1974.
“-51
J. W. S. Rayleigh, The Theory of Sound, Vol. 11, Znd Ed. from 189611926, Dover, New York, Reprint 1945.
“-61
J. W. Strutt, Baron Rayleigh, German translation by Fr. Neesen, Die Theorie des Schalls, 2. Band, Chapter XVJI ff., Vieweg, Braunschweig 1880.
“-71
StenzeliBrosze: Leitfaden zur Berechnung von Schallvorgangen,2. Auflage, Springer, BerlidGottingedHeidelberg, 1958.
[IV-8] M. Abramowitz, and I. A. Stegun: Handbook of Mathematial Functions with Formulas, Graphs and Mathematical Tables, 9* ed., US-Dept. of Commerce, NBS, 1970.
“-91
M. Kerker, The Scattering of Light, Academic Press, New York-London, 1969.
[IV-lo] S. Ternkin, Elements of Acoustics, John Wiley, New York, 1981.
78 Chapter IV
“-1
11 F. Mechel, Die Streuung ebener Wellen an Zylindern und Kugeln komplexer Impedanz, Vandenhoeck & Ruprecht, Gottingen, 1966.
[IV-12] K. F. Graff, Wave Motion in Elastic Solids, Oxford University Press, 1975, also Dover Pub., New York, Reprint 1991. “-131 A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4thed., Dover Pub. 1927/1944 reprint. [IV-141 H. Kolsky, Stress Waves in Solids, Dover Pub. New York, 1963, 1957-reprint. “-151 F. I. Fedorov, Theory of Elastic Waves in Crystals, Plenum Press, New York, 1968. “-161 R. T. Smith, and R. W. B. Stephens, Effects of Anisotropy on Ultrasonic Propagation in Solids, Progr. Appl. Mat. Res. 5 (1964), p. 39/64. [lV-17] G. F. Miller and H. Pursey, On the Partition of Energy between elastic waves in a semiinfinite solid,. Proc. Roy. A233 (1955), p. 55/69. Results are found in K. F. Graff, 121. “-181
R. J. Wasley, Stress Wave Propagation in Solids, Marcel Dekker, New York, 1973.
v
PRESSURE SOURCES FOR MODELING
Harmonic Pressure Sources We now evaluate formulations for pressure sources that are or might be used in modeling. There are different requirements for different sources: Finite-sized harmonic sources for analytical investigations and for computer calculations and point sources for asymptotic estimates. But in a multiphase system, arbitrary phases of radiation are required for finite-sized and point sources. A possibility for synchronizing the sources must be attempted. Solution of the spherical-wave equation in terms of the spherical Hankel functions h,(')( ) is useful. Since only radial symmetric vibrations are assumed, the Legendre polynomials P,(cos e), which characterize the dependence fiom the polar angle 8, are not used even though they usually are present in the references:
m=O
By knowing the
Ur
=R
at time t = 0 for r
= R,
we have an initial-value problem and get for
Monopole Source R=
and get
19
80
R
Ao=*=ipo. The appropriate velocity potential is then
If we want to address the pressure-generating mechanism by the definite volume flow , we use V, = 4nR;R,
(V-7)
and get alternative formulations of the identical velocity potential @
Pressure We use the definition P'P-
a@ at
(V-10)
and get * (V-11)
This quantity may be evaluated with respect to a volume-flow or velocity term. Both notations are given. (V- 12) or, if one relates the distance r to a wave length h, so that
B
quantities are used:
If one wants to avoid the imaginary i, it is possible to introduce a constant phase shift 7112 into all the velocity potentials @. Since exp(-i = - i this quantity may be deleted.
81 Modeling Sources
(V-13) The above equations, using Euler's Equation
(V-14)
e"=cosz+isinz, read in the components as follows
I m p =-pc--
30 2h2 1+
'
ko
sink(v -
-
cos k(r r
e
-iwt
.
(V-16)
The modulus is f i times the time average of the real part (see Equ. IV-15), which controls the energy transfer
Particle velocity By definition U r = -grad
@
(V- 18)
and one gets
(V- 19) Similar to the above, the additional expressions are:
(V-20)
(V-21) Equation (V-21), in components, reads as follows:
82
Chapter
sink(r - 4 ) cos k(r - &)
+
1
e-iWf.
times the time average of the real part, is The modulus, again representing calculated according to the previous procedure. In the case that solutions in R are required, the Equality
is to be inserted into the above formulas. In contrast to the pressure, the velocity has two terms, one decreasing with l/r, and the other phase shifted with l/rz, called the near-field term. Therefore, this near field cannot be ignored near the sources. Another situation specific to detonics is with ur: The pressure acts isotropically whereas the velocities show directional properties. All measured quantities relate to the single plane-wave direction of propagation that exist in classical detonics. Therefore, only the components of ur in the direction of propagation are considered; and the particle velocity in detonic terms u p is up = ur cos 8, where 0 is the angle between the direction of wave propagation and the point of observation P (see Fig. As a control, the quotient of p/u, is defined as impedance Z with the well-known expression
83
Modeling Sources
Computer Calculations For computer calculations, the velocity equations are most appropriate because the spherical Hankel fimction is defined by (V-27) Pressure is (V-28)
(V-29)
(V-30)
(V-3 1)
Particle velocity is (V-32) (V-33)
(V-34)
(v-35)
where for a spherical monopole source n
= 0.
84
Chapter V
cos --
sinz ;,(z)=---,
-&)
cosz
>
sinz cosz -nl(z)= -2 ' -n,(z).
3
(V-36) (V-37) (V-38)
For arbitrary orders, the hnctions j,( ) and n,( ) and their derivatives can be generated by recurrence formulas {Equations (IV-82) and (IV-84) - (IV-86), see [IV-I], [IV-S]}, see also *.
Point Sources The properties of a source depend on its size R. For more general considerations it is often usehl and easier to make estimates with point sources. These are defined so that or k ,these 0. But to maintain the source strength definition in the terms of quantities have the originally specified strength e0/2h2 . The Equations (V-13) and (V-21) are most suitable, but no such impedance can be determined because the radiation impedance is a property of a finite source. So we get:
A very short Basic-program for generating the Spherical Bessel and Neumann-Functions and their derivatives jn(x), jn'(x), n,(x), n,'(x): Input: Order n and argument x 10 PRINT "***Spherical Bessel jn(x) and Neumann nn(x)-fct.of order n., and derivatives ***" 20 INPUT "order n = "; n 30 INPUT "argument x = 'I; x 40 jo=SIN(x)/x : no=-COS(x)/x : a=jo : b=no 50 jl=jo/x+no : nl=no/x-jo : aa=j1 : bb-1 60 IF n=O THEN js=-j 1 :ns=-n 1 :GOTO 160:' REM j'o = - j 1 / n'o = - n l 70 jls=jo-2*jl/x : nls=no-2*nl/x 80 IF n=l THEN 150 90 FOR n=l TO n 100 aaa=(2*n+l)*aa/x-a :bbb=(2*n+l)*bb/x-b 110 a=aa : aa=aaa : b=bb : bb=bbb :REM a is the f(n) and aa the f(n+l)-value. 120 NEXT 130 n=n-1 140js=aa+a*n/x : ns=bb+b*n/x 150 PRINT llj ?lnll(!9xll)= lla,,lvn !ln9~(t!xll) llb 160 PRINT V!jff!nV!(!tx!V) = ',js,,"n"'n"("x"> = llnS 170 END Output: jn(x), j'n(x), nn(x), and n'n(x). ~
85 Sources
v
eikr
= -PC--
-iwt
(V-39)
where only the last two terms The components of Equ. (V-39) are
{ ] 2t2 {-
Rep=-pc2:2
Imp=+pc-
:s e
}:;:c
e
characterizes the source strength.
7
(V-40)
,
(V-41)
(V-42) For the particle velocity
(V-43) holds. The factor (kr + i)/kr clarifies the difference fiom plane-wave considerations. The components are
(V-44)
(V-45) Phase Control If we consider many radiators, these will usually not be in phase at any time. Rather they will be vibrating statistically, and vary in strength and phase. But synchronization mechanisms also occur in detonics. Therefore, we need a tool to model statistical sources and successive synchronization. The following procedure was found to be adequate: Stochastic Source Strength The source strength V, is multiplied by a random number of desired random number distribution. Successive synchronization is obtained by multiplication of these random numbers with constant numbers between 1 and 0.
86
Stochastic Source Phase The source phase is characterized by a phase angle Stochastic phases are constructed by random numbers n, between 0 and 1. Then the phase angle is:
(V-46)
x=2nxnr.
Whenever the random numbers are multiplied with a constant between 1 and 0, successive synchronization is obtained, which proceeds as this constant tends to 0. This phase angle is an additive of the exponentials elk(r-RbX
or for point sources
eikr+x
(V-47)
Monopole Cosine Source As demonstrated in Figure IV-2, the radially pulsating source shows different properties in the mode of plane-wave observations, because only "parts of the truth" are investigated. While the pressure p remains isotropic, ur changes because only the plane-wave part is observed in current experimental investigations. Therefore, we get PMonopole cosine - PMonopole
and
~ p = ~ r c o s @ ,
(V-48) (V-49)
but the other transverse component
(V-50)
U t = ursin 0
is beyond primary observation.
RADIATION OF A SHORT MOTION SOURCE As previously outlined, description of the radiating single source in harmonic waves may sometimes be too simple. When there is the assumption that this source radiates from eternity to eternity (fco), no wave shape variations occur. More realistic is the case of a bubble at rest that at a given time sustains a short motion - either expansion or collapse - and then this motion stops.
87
Short
Source
Temkin's Solution [V-1] A spherical, symmetrical source of constant radius R is assumed. At time t = 0 this source starts to expand or collapse during the short time E, and then it stops. We use the velocity potential of this source, [see Equation (IV-33)] and the corresponding particle velocity term [Equation (IV-41)] in another notation. (V-51) (V-52) With the condition
[ :)
+-
f;
r = u,(r,t)
one gets for r = R
[ :)
(V-53)
t--
where f can now be determined exactly. Introducing into Equ. (V-53) gets
=t -
which is identical with (V-55) Integration between - 00 (where u, = 0) and
results in (V-56)
Setting
=
+
one gets (V-57) -m
(V-58)
R/c one
88 Chapter
Equation (V-58) expresses a signal at point r and time t that depends on all signals sent off by the source between the times t = - and t’ = t - (r - R)/c. In other terms, if the source produces a signal at time t, then this signal will be at the location r at the time t + (r - R)/c. Therefore
(V-59) C
is called retarded time, and Equ. (V-58) reads as
(V-60)
A:
Short Constant Source Expansion
Assume that for t < 0 the source is at rest. Then during the time E, this source expands with a constant velocity u = u, and is at rest again for times t > E. Then the motions are: =O
fort&
(V-61) With Equ. (V-60) we get the different velocity potentials: 0, ,and
ecT
I E ) = --
(V-62)
0 &
--e-~f’/~ 0
These become by integration: = 0,
5 t’
E)=
-[I
- e - c r ‘ i Rand ] 2
-cf‘/R
[.
(V-63)
1.
-1
Then the usual acoustical quantities - like pressure p, particle velocities ur, up, ut, condensation etc. - may be determined in the following terms.
89
Pressure: p ( r , t l < o)= 0,
(V-64)
Radial particle velocity: ur(r,tkO)=O, U r (r,O
t'
E) =
(r/ R)2
+ [(r /
(V-65)
> 8) = ([(r/ : - 1).
Ur
(.
/
The other velocities are obtained as outlined for the monopole cosine source. Different Cases
a:
Constant distance r at variable time t
In Equ. (V-59) the distance r = R + c(t - t')
(V-59a)
remains constant. It is appropriate to characterize the variable time by a length ct. Constant time t and variable dktance r
b:
In Equ. (V-59) the time t is fixed, and is characterized as a distance ct = const. c t'
= ct
- (r - R).
(V-59b)
In both cases these quantities are compared with the time of expansion
E,
or EC.
If ct - (r - R) < 0: p=o,
u,
(V-66)
= 0.
If 0 Ic t - (r - R) S c E : exp{ (r-;-4] t ) = pcu
and
(V-67)
90
Chapter V
If c t - (r - R) > c E: (r-R-ct) p ( r , t ) = -pcu
(4
and
(V-68)
Source Expansion Pulse The source surface of radius R remains at rest until a short constant expansion velocity u arises between t = 0 and t = E (within a distance CE). Then the source radius at time t = E becomes R + UE. The surface velocity reduces to 0 at times t > E. Pressure and radial particle velocities are calculated for two cases, where the source of radius R during the time E CE expands, then stops. The expansion velocity is also 1. Figure V-1 shows the expansion pulse for constant distance at varying times, and Figure V-2 the same situations for constant time at varying distance. In the case of a collapse the signs are changed. Curiously, these results are not well known. Nevertheless, this result is known at least since Lamb [V-21 demonstrated that the condensation integral +m
I
sdt = 0
(V-69)
-m
always holds. Dynamic deformations of spherical sources always result in packets of pressure and tension waves, and the reverse. This is why a pressure pulse always leads to cavitation in two-phase media. This phenomenon will be discussed in greater detail later. B:
Short Constant Source Collapse
Since the source expansion or collapse is characterized only by the sign of the source’s surface velocity u, the sign of u must be changed in the above Equations.
91 Short
Source
= const.
ct=0,2
ct = vat
ct.5
5
Figure V-1 This case of constant distance kom the source at variable times represents that at a distance r from the source a pressure and a velocity probe are established, which control the time dependence of pressure and velocity. The pressure pulse is followed by a tensile wave. Near the source are considerable differences between the particle velocity and the pressure. These differences get smaller with increasing distance. A source motion always results both in pressure and tension pulses. This fact is often ignored.
6 r
Figure V-2 The situation of constant time and variable distance may be understood in the following terms: From the whole system at time t or the corresponding distance ct, a snapshot is taken, so that the local pressure and velocity fionts are registered.
References [V-1] S. Temkin, Elements of Acoustics, 1981, John Wiley & Sons, New York, Chichester, Brisbane, Toronto. [V-21 H. Lamb, The Dynamical Theory of Sound, 2nded., 1925, Dover, New York, Reprint 1960
This Page Intentionally Left Blank
VI
RAYLEIGH'S BUBBLEMODEL
Rayleigh's investigation in 1917 [VI-11 concerned itself with the problem of Reynold's description of the sounds emitted by water in a kettle as it comes to boil. Their explanation that this sound was caused by the partial or complete collapse of bubbles as they rise through cooler water was similar to Minnaert's [VI-21. At the same time, the cavitation behind screw propellers had been investigated by Parsons and Cook [VI-31, based on Besant, who in 1859 referred to Cambridge Senate House Problems of 1847. Besant's formulation of the problem was: upon by no
up, "
Due to the technical significance and the scientific challenge, this problem had been considered technically as well as scientifically. We owe scientific milestones Lamb [VI-4], Minnaert [VI-21, Noltingk and Neppiras [VI-51, Plesset and Prosperetti [VI-6], Vokurka [VI-71, and Lauterborn [VI-1 11. These authors investigated stifhesdmass controlled bubbles and the pertinent parts are addressed below. More recently, bubbles based on rectified diffusion have been explored. Cole [VI-81 summarized underwater explosions, Grein [VI-91 reviewed technical cavitation problems, Akulicev [VI-lo] in 1978 edited a monograph on technical and scientific problems of cavitation in cryogenic and boiling liquids, and Lauterborn [VI-1 11 pioneered more modern physical aspects, and arranged the first international symposium on this topic. p = f(p) is the basis making this problem a single-phase event.
Energy Consideration If d is the velocity at the surface of a bubble of radius R at time t, and u is the simultaneous velocity at a distance r > R, then -=-
(VI-1)
93
94 Chapter
The kinetic energy of motion is m
(VI-2) R
with the density of the liquid pm If pa is the pressure at infinite distance, and the potential energy is:
is the initial value of the radius R, then
(VI-3) By equating Equations (VI-2) and (VI-3), Rayleigh obtained the collapse time 't of his bubble
(VI-4)
= 0.91468
J P
This condition leads to some surprises if one considers energy E, time of collapse 't, power of collapse E/z, and also the specific power of collapse for a liquid of density poo= 1 g/cm3,see Table VI-1. The order of magnitude of the specific power of collapse is much larger if one relates to the instantaneous volume V instead of the original volume Vo. Table VI-1.
Demonstration of Energy and Power Situations Pressure 1 bar (0.1 MPa)
1 kbar
10 kbar
Radius 1 cm 1 mm
0.1 mm 10pm 1 mm 0.1 mm 10 pm 1
mm
0.1 mm 10 pm
p v
5
[JI
[nsl
914,700 4 10.' 91,50 4 9,l 4 10~7 914 4 4 10.' 2,890 4 289 4 28.9 4.24 915 4 10~3 91.5 4 9.2
p Vir [W]
(p V/r)/Vo
464 4.64 5 x 10-2
0.1 11 1,lO 11,oo 111 lo3 3.5 x lo7 3 . 5 10' ~ 3 . 5 lo9 ~ 1.1 x lo9 1.1 10" 1.1 x 10"
sX1op 1.5 x lo5 1.5 x lo3 14.7 4 . 6 lo6 ~ 4 . 6 lo4 ~ 4.6~ lo2
[W/cm3]
Power densities appear in the orders of TW/cm3when bubbles are collapsed, increasing as pressure increases and the bubble size decreases.
's Bubble
Deduction of Rayleigh's Equation In order to achieve further simplifications the basic steps for getting Rayleigh's equation are outlined. Euler's equation [VI-11 is used:
_1 _
=---a u
p
du
(VI-5)
u-.
A solution of Equation (VI-5) is (VI-6) so that (VI-7) and (VI-8) whereas (VI-9) Inserting Equation (VI-9) into Equation (VI-5) one gets by integration with assumption of incompressibilityand constant pressure poofor r 3 00 the following Equations. p is the pressure p(r, t) in the medium, so that pa = p - pm is the dynamic (acoustic) pressure (difference) --
p a- - _ _ -
P
P
r
2r4
(VI- 10)
and
(VI-11) Setting r = R one gets for the wall motion the well-known Rayleigh equation, where p' = pa = p - pmis the gas pressure at the bubble wall:
.
(VI-12)
96
We now discuss Vokurka's [VI-7, I] equations. Inserting Equation (VI-12) into Equation (VI-1 l), one gets (VI-13) This equation has three terms: The first term is the gas pressure in the bubble communicating to the liquid. This term is the only component, if linear bubble oscillations occur. The second term results f?om the dynamic motion, and is responsible for non-linear bubble motions. Both the first and second terms depend on lir, so they represent the acoustic field. In other words, pa is governed by
(VI-14) The third term apparently corresponds to the near field (only important in the vicinity of a dynamic bubble) and is responsible for non-linearities. Probably a bit misleadingly, Cole [VI-8] identifies this term as Bernoulli pressure or kinetic wave. He reasons that with Equation (VI-7) 4
PBernouIlr =
(5)
1
.2
1 =-Pu2
2
(VT-15)
holds. In terms of detonics, beside the other term as kinetic field, stagnation pressure in the sense of a water-hammer effect gives a better impression to its meaning.
Any losses such as viscosity, gravity, surface tension, thermal losses of the gas, and radiation losses are ignored here. Some of these losses are considered in later developments of Rayleigh's equation. Lmterborn integrated mainly that of Noltingk and Neppiras [VI-51 with the result that non-linear as well as subharmonic and supraharmonic vibrations appear. These results are not important at this stage because they depend on well-defined initial conditions.
97 's
DEVIN'SBUBBLE Devin's bubble model - in spite of applying to small motions - is most important for the consideration of detonic situations because it includes damping elements; and its relative simplicity allows generalized descriptions. From the Lagrangian equation, Devin [VI-121 considered bubbles of volume V = (4/3)7cR3 in a liquid under the action of a (small) pressure disturbation -p(t). It is appropriate to assign quantities relating to the gas by an accent and to index those relating to the condensed phase by 00. He obtained I,
(VI- 16) For small amplitudes the constants are PCC 4xR
6
b=w
as mass,
(VI- 17)
as spring constant,
(VI- 18)
as damping,
(VI- 19)
where p' is the gas pressure inside the bubble and y' its specific-heatratio. ZitOt.is the total loss of bubble motion, and will be considered later in detail. As usual, one gets the resonance hequency or = 2nf0 as
Minnaert's Resonance Condition [VI-21 wo=&&
Jq
(Vl-20)
For usual liquids, such as water, one gets the resonance condition for air bubbles of pressure p in bar (= 0.1 MPa) as f
300D
[Hz cm
Since the sound velocity of the gas is given by:
(VI-21)
98 Chapter
(VI-22) and the bulk moduli K are defined as
K = pc2.
(VI-23)
The resonance condition, Equation (VI-20), reads as:
(VI-24) where characterizes transformation properties of the bulk moduli of the bubble content and the medium. Similar to Equation (V1-21), for common liquids Z
0.01...0 . 0 2 0
[p' in bar (= 0.1 MPa)]
(VI-25)
holds. Therefore, using Equation (IV-U), an atmospheric bubble releases its energy mostly in flow and very little in compression - boiling tea water sings - and will not explode. But the tendency to explosion increases with increasing static pressure, because (kR)2 also increases with the surrounding static pressure. In open-pored systems we always have p' = pa; closed-pored systems behave differently under pressure. Open-pore systems are pressurized, closed-pored systems are compressed.
Comparison with Rayleigh's Model It is quite clear that a harmonic-motion equation is much more convenient for applications, estimates, and generalizations. Therefore, Devins approach is compared with Rayleigh's equation, which also describes large-amplitude pulses in terms of R, even for shock waves. Obviously an equation linear in V cannot be linear in R. If there is no mass exchange between the bubble wall and the surrounding condensed material, R = holds and one gets for Equations (VI-16) to (Vl-19):
(VI-26)
99
Devin's
Isothermal considerations (y' = 1) and setting & , acoustic term of Rayleigh's Equation (VI-11):
Rii +
= 0 (loss term = 0), lead to the
P' = 0 +-
(VI-29)
P C C
As will be shown later in Equations (VI-36) and (VI-37), a bubble of volume V at rest and &, = 0 collapses in a time w, t = d 2 . Using Equation (VI-20) one gets for this time (VI-30) which compares well for small amplitudes with Rayleigh's time of collapse where A p = p m - p .
(VI-31)
Since the losses are also to be inserted into Devin's solution, this approach is probably more suitable for large amplitudes than the Rayleigh Equation, as analytical solutions and estimates become possible. Note that Rayleigh's Equation is also used for large amplitudes.
Bubble Response to a Velocity Shock [VI-131 If the homogeneous Equation (VI-16) is altered to =0
,
(VI-32)
where
(VI-33) then it is possible to estimate bubble behavior under the action of a velocity shock ur. Without mass transfer by reasons of continuity, the relation between bubble-wall velocity R and radial particle velocity ur is [VI-61:
(VI-34)
This relation does not hold for the case of onset of a chemical reaction at the bubble surface nor for evaporationlcondensationprocesses. Within these limits, we express u, in terms of . Then we can solve Equation (VI-32) for the boundary conditions
100
Chapter
and
I
0 0
> a:
A=-
2
(VI-35)
co=Ja,
(VI-36)
coscot+ A s i n a t ) ,
(VI-37)
coscot - (avo+ Alsinwt},
(VI-38)
P(t)=
A)-wC;,,]sinot
-
(avo+a A ) + ayo]cosot}.
(VI-39)
For a loss-fiee bubble one gets t = x/2 for V(t) = 0, a result that was used above, Equation (VI-30). Apparently the time of collapse depends on damping.
(VI-40) (VI-41) (VI-42)
(VI-43) (VI-44)
cosho't+ A'sinhw't}, =
{
cosh w't + (of
= e-O'{ [w'(o'
v0-
- A')sinhw't
1,
(VI-45)
y o ] c ~ w't ~ h + [co1y0- a ( w ' ~ aA')]sinhw't) ~ . (VI-46)
The actual R and/or the velocity d may be obtained fi-om these Equations. A collapse is obtained only in case I; the bubble creeps to collapse in the other cases (increased losses). Negative volumes appear in calculations because harmonic solutions are used in the volume V; then absolute values must be taken. The physical reason is that a rebound occurs as the volume approaches zero. However, it must kept in mind that these cases
101 s
are of no further interest for the present study because the main losses are fully activated shortly after the beginning of bubble motion. This is shown in Figure VI- 1, where a velocity shock collapses a bubble corresponding to Equation (VI-16) with unity coefficients (m = 1; .b = I; k = l), where the time of collapse is increased correspondingly. As already stated, we barely know of any specified bubble situation. Therefore, we are more interested in generalized versions - in what happens under not well-specified situations,
Figure VI-1. Components of a collapsing bubble pushed by a velocity shock. The loss d corresponds to 6 = 0.25.
Generalized Description of Devin‘s Bubble Equation (VI-16) can be generalized by using nondimensional variables: mot =
generalized time,
(VI-47) (VI-48) (VI-49) (VI-50)
ks
and finally get
102 Chapter
(VI-5 1)
Whenever a jump of the particle velocity is considered, the homogeneous Equation (VI-5 1) with V' and in the case of an applied external constant pressure jump the inhomogeneous Equation (VI-51) with -V, are used, where the final steady-state solution corresponds to this stimulus. This situation is of special interest because stimuli of arbitrary duration and shape can be constructed fi-om this unit-step solution. We consider next the inhomogeneous equation, where the steady-state solution after long times is -V=. Again, there are three cases depending on the loss 6,,,. For the initial conditions:
v0
and
Po
(VI-52)
one gets with the abbreviations A = vo- vm and 6 = Gtot.
(VI-53)
(VI-54) (VI-55)
(VI-56)
(VI-57)
(VI-58) Simply replace v, sin and cos by v', sinh and cosh in Equations (VI-55) to (VI-57).
103 Devin Bubble titot. =
2:
(VI-59)
1,
V~(T)=E-T{
.-.{
(VI-60)
1.
A)r
(VI-6 1)
RELATIONS OF ENERGY AND POWER In general, the power N is defined as the product of a force and a velocity. Rearranging the inhomogeneous Equations (VI-16) and (VI-5 l), and multiplying with the velocity term V or V’ one gets .. . +k,
.
=
-[
d dt 2
+
k
= [p(t)-
and
(VI-62)
(VI-63) L
J
The central terms in the brackets are the lunetic and potential energies of the vibration system. On the right-hand side are two power terms, the added power to the vibrating system p ( t ) V ,resp. ,and the one that is diminished by the power consumed in the damping system. So there are two different counteracting power terms, which depend on the volume velocity and its square. If the exciting term is small compared to the damping term, then the system’s vibrations will be attenuated. The reverse is true if the exciting term is large. A steady state equilibrium is obtained when these contributions balance each other. Depending on the loss 6, regions of a steady state exist where neither attenuation nor amplification occurs.
Energy of the Vibrating System The total energy corresponds to the bracketed term in Equation (VI-63), which changes totally to the potential or totally to the kinetic form dependmg on the time. By considering the potential energy, one gets by using Equation (VI-55):
104 Chapter
three energy terms in all: The first term is a constant, depending on the excitation, the second term is an energy, where an average can be determined, and the third term shows twice the frequency of the vibrational system. Depending on the phase to the exciting system, there are also active and wattless energies at work. The first two terms increase with increasing time, the last term depends only on the vibrational system. Using Equation (VI-54), the time 2n of the period of vibration increases with the loss 6 to (VI-65)
Also, one can express the damping factor exp(-6o/2) by a relaxation time o, this damping factor reads as exp(-do,).
=26.
Then
Experimental Determination of the Loss 6 The ratio of the n. and (n + 1). amplitude of V is: = &r,
/ 2) = e9
,
(VI-66)
or (VI- 67) where 9 is the logarithmic decrement. Equation (VI-66) is an experimental tool to determine the loss from any amplitude (V, V', or V"), that can be derived from the Equations (VI-55) to (VI-57). Whenever serious deviations occur from a straight line a semilogarithmicplot of the amplitude ratios as h c t i o n of the number of cycles, another damping mechanism, 6 # const., 6 = is present. We use the loss 6, determined by the experimental value of the logarithmic decrement in any theories, but only for vibrating systems is 6 < 2.
CAUTION: Due to the quadratic expressions of energy in the single components, the time of energy vibration is half that of the vibrating system, so that the energy ratios of the n. and (n + 1). component cycle (as example of V (only)} is:
105 Bubble
(VI-68) whereas the cycle of the full energy of the vibrational system {(V2+ VI2)/2}, governed by Equation (VI-66) remains unchanged. The significance of the loss 6 can be seen easily from Table VI-2 which lists the full energy loss per original vibration cycle (in %) for values of 6.
6 1.25 1.oo 0.75 0.50 0.25 0.10 0.05 0.01
-
in %
Table Vl-2. Energy loss of vibration cycles in E as function of 6.
100.00 99.93 99.38 96.10 79.47 46.69 26.97 6.09
These values demonstrate that most of the energy is consumed in one vibration cycle, even at a very moderate loss. Therefore, the first motion becomes energetically dominant, and the assumption of a Devin bubble is realistic because all happens during the first collapse or expansion of a bubble. The loss 6 and its resulting damping is a most important factor that dramatically can change the dynamics of the bubble system.
References [VI-11 J. W. S. Rayleigh, On the Pressure Developed in a Liquid During the Collapse of a Spherical Cavity, Phil. Mag 34 (1917), p. 94/98, Scientific Papers, Vol. VI, No 324, p. 504/507. pI-21 M. Minnaert, On Musical Air-Bubbles and the Sounds of Running Water, Phil. Mag. Ser. 7 16 (1933), p. 235/248. [VI-31 C. A. Parsons, and S. S. Cook: Investigations into the Causes of Corrosion or Erosion of Propellers, Trans. Inst. Nav. Arch. 61 (1919). [VI-41 H. Lamb, Hydrodynamics, 6" edition, Reprint from 1932, Cambridge University Press 1974. [VI-51 B. E. Noltingk, and E. A. Neppiras, Cavitation Produced by Ultrasonics, Proc. Phys. SOC. London B 63 (1950), p. 674/ . [VI-61 M. S. Plesset, and A. Prosperetti: Bubble Dynamics and Cavitation, Ann. Rev. Fluid Mech. 9 (1977), p. 145/185.
106 Chapter
[VI-71 K. Vokurka, On Rayleigh's Model of a Freely Oscillating Bubble. I. Basic Relations, Czech. J. Phys. B (1985), p. 28/40, 11. Results, Czech. J. Phys. B (1985), p. 110/120, 111. Limits, Czech. J. Phys. B (1985), p. 121/132. [VI-81 R. H. Cole, Underwater Explosions, Princeton University Press, 1948. vI-91 H. Grein, Kavitation - eine ijbersicht, Lecture at Karman Institute for Fluid Dynamics, Bruxelles, 1973, with 665 citations. Technische Rundschau SULZER, Forschungsheft 1974 [VI-101 V. A. Akulicev, Kavitacia v kriogennych i kipjascich zidkostjach, NAUKA, MOSCOW, 1978. German Translation: Kavitation in kryogenen und siedenden Fliissigkeiten, BundessprachenamtlBICT, edited by C. 0. Leiber. [VI-1 11 W. Lauterborn, and R. Timm: Bubble Collapse Studies at a Million Frames per Second, in W. Lauterbom, Cavitation and Inhomogeneities in Underwater Acoustics, Springer Series in Electrophysics 4, Springer, Berlin, Heidelberg, New York, 1980, p. 42/46. [VI-121 Ch. Devin, jr., Survey of Thermal, Radiation and Viscous Damping of Pulsating Air Bubbles in Water, J. Acoust. Soc. Am. (1959), p. 165411667. [V1-13] K. Magnus, Schwingungen, Teubner Studienbiicher Mechanik, 1976, ISBN 3-519-12302-9.
LOSSESBY VOLUME VARIATIONS
VII
Whereas Rayleigh's bubble model considers the motion in a loss-free medium, the small amplitude Devin-model takes into account damping elements. The examples shown demonstrate the great influence of loss 6 on the dynamic behavior of a bubble. This influence is not simply a gradual one but can change the dynamic behavior qualitatively. It is therefore necessary to define the nature of these losses.
v
We know fiom pressure-wave generation that is essential for pressure-wave emission when controlled by the radiation impedance Z. Such a pressure-wave emission acts on the vibrating bubble as damping because energy is evolved. Since this type of damping is not linked with a conversion of mechanical energy into heat, we call this type of loss Beside this loss which is known as radiation loss must also occur, as we have deduced fiom the experiments in Chapter 11, which stimulate the onset of chemical reactions in energetic materials. In all the following considerations we use acoustical terms for harmonic source motions, even small-argument approximations,because these solutions are tested by acoustical experiments. If the losses are large, they become decisive at early stages. We have at present no idea how to deal with rapid phase transformations at high temperatures and high pressures.
Radiation Loss &ad. of Bubbles
[
ro
Making use of harmonic solutions of Equation (VI- 16) =
v = poeiat
eiaf
iwt
(VII-1)
one notices that bubble compression and bubble wall acceleration are out of phase with the pressure. Omitting the time dependence and inserting the solutions into Equation (V1-16), one gets by rearranging io
Po+-b
.
2 0 0
'
(VII-2)
2 0
107
108
Chapter VN
where is not necessarily, but it can be, the resonance condition 00. The term after the resonance condition, the stifhess OY mass controls the bubble motion and leads to the change in sign of fiequency response finctions H,(R, a), see the following chapters, and the theory of LVD. Taking into account V o = 4 n ~ ~ R = 4 n andusing ~ ~ U r , the equations (VI-17) to (VI-19), one gets
(VII-6) where Z is the radiation impedance: = Pmcm
(VII-7) k2R2 + i W P R =A R r + i w m r = ReZ + i h Z . = Pmcml+k2R2 l+k2R2
Equating the real parts of the Equs. (VII-6) and (VII-7) one gets cm k2R2 1 k2R2 Rr ooRopo ooRo 1+k2R2-koRo l+k2R2’
(VII-8)
or by using the resonant expression, Equation (VI-20)
(VII-9) and in the case of resonance P,ci
3y’p’ p,
= wo with
olc = k:
i a = I j z
3Y’P‘ = + 3y’p’ p, c i + 3Y’P’
12
3Y‘P’
. +
1
(VII-10)
3Y’P‘
This means that the radiation loss can be estimated from the medium’s values p,&, and the properties of the gas content y’p‘ . This radiation loss is independent of frequency and bubble size in the case of resonance! Devin [VII-1] and Nishi [VII-21 have given for resonance hrad. = kR .
(VII-11)
Devin refers to an earlier work of Smith (1 935), which gave the following expression:
109 Losses by Volume
Since the radiation loss depends on the radiating surface, this loss is not very sensitive to the actual shape of the radiator, in contrast to the acting mass m,. Contrary to the radiation loss, this acting mass depends on the shape, and is responsible for the nearfield term.
Radiation Loss
of Cracks
The insensitivity of R, to shape enables us to make estimates on the impedance of an opening or closing crack. Pt
t I
"j
ka=
"
; ;as -E
Approximot ions:
0
0.5
1.0
Figure VII-2. Comparison of the impedances of plane waves (radius of the piston = a), and an impedance corresponding to spherical or -
waves
=a / f i
.
Figure VII-1. Approximations of an opening penny shaped crack with a radiating sphere, and a piston membrane radiating on two sides.
A penny shaped crack of diameter c can be opened by a tensile pulse pt with an appropriate particle velocity up, see Figure VII-1. This situation can be approximated in two ways :
110
Chapter
A spherical radiator of the same radiating surface is evaluated. At the beginning of the opening this penny shaped crack, diameter c, has a radiating area of A = 7ccz/2 (for both sides). On the other hand, a spherical source of radius R shows the surface A = 47cRz, so by equating both areas, an equivalent spherical radiation should be expected by a source of the radius
=
The second option is to use the impedance of a two-sided plane piston, see Equs. (1-5) and (I-6), and Figure VII-1, bottom.
In Figure VII-2 the impedances of a spherical wave radiator and two-sided piston membrane as radiation source are compared.
Dissipative Bubble Losses Up to now several dissipation mechanisms, which convert mechanical energy into heat, have been proposed. In detonics, the generally accepted is Bowden's so called adiabatic hot spot, where an adiabatic compression of the bubble content should intentionally produce heat. But the initiation experiments of Hay and Watson [VII-31 of a mixture of nitroglycerine and EGDN (NGEGDN = 50/50) as a h c t i o n of the voids and their content disproved this view. In the presence of voids the initiating pressure was > 2 kbar, for air and COz bubbles around 1.5 kbar, and for Ar bubbles < 1.5 kbar! This adiabatic hot spot is not always effective; more important can be viscous losses, surface vibrations, and other mechanisms. The reasons for possible failure of the adiabatic mechanism are very simple: No heating is possible if the bubble dimensions compare with the fiee path length of the molecules. On the other hand, heat conductivity lessens adiabatic heating if the bubbles are large. Consequently, there must exist optimal bubble dimensions for an effective 'adiabatic' mechanism. Thermal Loss 6,, Many papers have been written on thermal loss, even before the adiabatic hot spot was proposed. The results of Pfriem (1940) [VII-4], Devin (1959) [VII-11, and Kapustina (1970) [VII-5] agree with each other, whereas the results of Nishi (1975) [VII-21 do not. The results of all approaches are sketched on the following pages. The primed quantities refer to the gas phase and the quantities indexed with liquid state. With the quantities: K' P'C'p
as the thermal difisivity,
00
to the
(VII- 13)
111 by
where K'the heat conductivity, p' the gas density, and cfPthe heat capacity at constant pressure of the bubble content. Pfriem, Devin and Kapustina obtain for the thermal loss in resonance sinhz+sinz 2 coshz-cosz z sinhz-sinz +- z ' coshz-cosz 3(y'-1)
(VII-14)
where (VII-15) Pfriem takes for R Minnaerts resonance radius &, and assumes a constant polytropic index y'. Devin and Kapustina additionally consider the surface tension of the liquid (T, and a variable polytropic index y'la. For the correction of & they use (VII-16) where (VII- 17) (VII-18) This last term contains the polytropic index as a hnction of Under certain circumstances, the value y ' / a can go < 1, which is an error. &. is calculated by iterations with the above formulas.
But Nishi [VII-21uses K'
with the heat capacity at a constant volume V. With
c;
z =d =
(VII-19)
he calculates with Equ. (VII-14) an intermediate value 6, and corrects Minnaert's resonance frequency by
112 Chapter
Jt,
(VII-20)
~ o = o o
where the stifhess correction is E=(l+St.)”
d
sinh d -sin d cash d - cos d
(VII-21)
and
(VII-22) Finally he gets for the thermal loss for the arbitrary driving frequency o
(VII-23) In this solution for the polytropic index (VII-24) holds. Whereas Pfiiem and Kapustina give expressions for the thermal loss only for a resonating bubble, Nishi gives a general expression for all dnving frequencies. All the formulas are small-argument expressions that may be expanded for larger sources. In addition, small and large fiequency approximations are available, not considered above since we have computers available. Nevertheless, they may be useful for estimates with analytical expressions. Since Pfiiem does not change Minnaert’s condition of resonance, his results in the low-fiequency domain are a bit larger than those of Nishi. Pfiiem compares experimental values of air bubbles in water with calculations. In the low-frequency domain, his calculated values are at the lower end of the experimental values. Results for the high-frequency domain are not available at present. Figure VII-4 compares approximately the calculations of PfriemKapustina with those according to Nishi for nitromethane (NM). For the reason of comparison a uniform thermal difisivity D’ had been used.
On the ‘adiabatichot-spot’ Bowden, Mulcahy, Vines, and Yoffe [VII-61 showed the influence of the gas content [air (y’ Y 1.4), nitrogen (y’ = 1.4), ether (y’ = 1.OS), and carbon tetrachloride = 1.13)]
113 by
1s
01
-
Figure VlI-3. Comparison of the calculated losses for an air bubble in NM at atmospheric pressure according to P f i i e d Kapustina and Nishi with uniform thermal diffusivity. (Used data: p = 1 bar; p..= 1.131 g/cm3; = 36.53 dydcm; qm= 0.00613 Poise; c,= 1.322 1O5 c d s ; D’ = 0.2 cm*/s; = 1.4)
on drop-hammer sensitivity, both at normal and reduced ambient pressures. The authors pointed out that the temperature rise according to
(VII-25) may well be sufficient for thermal ignition of the explosive. This argument - in its simplicity - captured the understanding of explosives initiation at that time and even now. The arguments against this adiabatic hot-spot are also simple but convincing: 1.
The temperature rise is much too small to explain thermally induced, rapid explosive decompositions.
2.
When comparing different bubble sizes it becomes clear that
for small bubbles disappears, if the bubble sizes compare with the average free collision path of the molecules. For large bubbles the condition of adiabatics is lost, and the large bubbles behave 1. isothermically so that 3.
cannot be a constant for all pressure and temperature ranges because the defining specific heat capacities C, and Cv show different pressure and temperature dependencies.
114
Chapter
[
=
-.($)
and
(VII-26)
(VII-27)
Evaluation of these Equations (VII-26) and (VII-27) for moderate pressures (up to about 50 bar) can be done with Berthelot’s Equation of State. One gets for argon at 273 K and 50 bar ambient pressure a of 2.33, which may be a reason for the exaggerated high luminosity of shock-pressurized argon. For more details, see Ribaud [VII-7], and Leiber [VII-81. Viscous Loss 6, As the source changes its volume, the surface boundaries are distorted, and rate-
dependent viscous losses are activated {Mallock, 1910, referenced in Devin [VII-11). Devin gives for resonance the expression 8j?frlma
4
rlm
P,
(VII-28) ‘
Nishi gets for an arbitrary driving fi-equency 2
6, =
4 0 0 ~ 1- ~ 4~ 1 ‘, - --
Pm
2
(VII-29)
where qmis the medium’s viscosity, respectively qJpm its kinematic viscosity. The solutions agree well with each other and approximate well the experimental data [111-11, [VII-41, [VII-91. Loss Gaps Figure VII-3 shows the peculiarity that the thermal (adiabatic) hot spot can disappear, and later viscous losses come into play. Thermal and viscous losses were calculated for several liquid explosives with different gas contents in the bubble to determine whether such a loss gap could occur. Two such cases are shown in Figure VII-4. Vapor bubbles in nitromethane (NM) and tetranitromethane (TNM) show loss gaps at fi-equencies between lo5 and lo6 at normal pressures!
115 by
Figure VII-4.
Comparison of the calculated thermal (adiabatic) and viscous losses for air and vapor bubbles in nitromethane (NM), and a vapor bubble in tetranitromethane (TNM) at atmospheric pressure. The data are included in the figure.
Thickness of the Viscous Loss Layer Another interesting question is how the thickness A of the viscous loss layer compares with the actual bubble size. This thickness A was first formulated by Stokes [VII-lo], and is formulated by Temkin and Leung [VII-1 I] with
(VII-30) The ratio AIR increases with decreasing bubble size, so that the 'viscous activated' volume increases, and can under certain circumstances be larger than the bubble volume. In this latter case a 'volume homogeneous' chemical reaction may be initiated. The values for NM are shown in Tables VII-1 and VII-2 [VII-121. Table VII-1. Properties of nitromethane (NM) with air in bubble
Liquid
Gas
P,
1.131 g/cm3
Vapor Pressure p,,
35.65 Torr = 0.0475 bar
Surface tension
36.53 dyn/cm
r,l
0.00613 P
c,
1.322 lo5
p'
1.185
g/cm3
1.4 c1
0.33 lo5
116
Chapter Table VII-2. Thickness of viscous loss layer in NM
f
L.
[Hzl
[cml
[cdsl
[WI
lo2
2.99
299.3
41.5
13.9
0.0042
103
0.299
299.4
13.1
43.9
0.013
f Rres. S
A
lo4
299
299.6
4.15
138.6
0.042
lo5
30.2
302.2
1.31
434.6
0.136
lo6
3.25
325.1
0.415
0.128
0.434
lo7
0.45
453.0
0.131
0.290
1.146
19 Lauterborn correction was used.
Figure VII-5. Spherical oscillations and decay of a single spherical bubble set into motion by a sound field increasing amplitude at about 7 kHz. The holographic framing rate is 66.7 kHz, and the frame size is 2.4 2.0 mm. {Lauterborn and Hentschel [V11-14] with kind permission}.
117 by
Figure VII-6. Interaction of bubbles of different sizes in a sound field of increasing amplitude at about 7 kHz. Note the jet formation of the upper right bubble towards the big one. The holographic framing rate is 66.7 kHz, and the frame size is 4.7 2.9 mm. {Lauterborn and Hentschel [VII-14] with kind permission}.
Other Loss producing Mechanisms Besides the losses discussed above, additional loss mechanisms can be at work that have not yet been quantitativelyformulated. These are addressed here because they are important to explosives initiation.
A bubble surface can acquire surface instabilities, which first appear as a wavy surface with increasing amplitude. In the end this mechanism destroys the bubble. Hullin [VII-131 investigated such phenomena, and Hentschel [VII-141 made a holographic framing camera study, see Figure VII-5. The use of this phenomenon in explosives initiation is a typical chemical view, which probably does not actually apply. The idea is that wavy surfaces lead to instabilities.
118
The resulting droplets are injected into the adiabatic heated gas of this bubble. There droplet/gas/(air?) mixture explodes and initiates the condensed medium! Another, often discussed aspect, is jet formation of a collapsing bubble. No quantitative estimate exists of this effect, but it is generally agreed that it will be most powerful. It is assumed in current models of vapor explosion that this jet formation is a mechanism of melt fragmentation. Figure VII-6 shows that such jet formation can occur between neighboring bubbles, so that the mechanisms for any jetting are more numerous than currently believed. That the suggested losses may really apply is indicated by the pictures with 1 ps interframe distance of Coley and Field [VII-151, [VII-161 (Figure VII-7). They show the building to LVD by a collapsing air bubble in NG. One notes pressure-wave emission as in Fig. 111-1. Also, the chemical reaction starts at the bubble surface, probably within the shear layer. This provides proof of the validity of the suggested loss terms. There seem to be no jetting or surface instabilities. Figure VII-7. Sensitizing effect of an air bubble in a thin NG-film, and buildup to LVD, according to Coley and Field [VII-16]. Inter kame time 1 ps, vertical extent of a kame is 16.6 mm. W is a thin wire, F the film boundary, E the electrode leads. Pressure waves are emitted as the bubble collapses. It appears that the chemical reaction starts at the boundary. (with kind permission of Professor Dr. J. E. Field)
119 Losses by Volume
CALCULATION OF SOME LOSSES Table VII-3. Properties of bubble content at normal conditions
[pW/cm K] [ g/cm3] [J/g K] 260 1.2505 1.0398
[J/g K]
[cm2/s] [cmZ/sl
0.7427
0.200
0.280
Air
1.40
Helium
1.66
1430
0.1787
5.172
3.116
1.547
2.568
Argon
1.66
164
1.783
0.520
0.312
0.177
0.295
Nitrogen
1.40
240
1.251
1.040
0.743
0.185
0.259
Oxygen
1.40
Hydrogen
1.41
1810
0.08987
14.312
10.196
1.407
1.975
Methane
1.30
303
0.7168
2.228
1.712
0.190
0.247
Propane NM-Vapor, 1 bar at 27.3 Torr
1.13
151
2.019
1.668
1.480
0.045
0.051
1.20
135
2.72 0.098
0.823
0.0603 1.65 8
0.0724
8 D vanes with lip Table Properties of some liquid explosives
Po0
lo0
DEGN
[“C] [g/cm3] 20 1.38
[Poise] 0.081
GDN
20
1.488
0.0421
NG
20
1.594
0.36
n-PN
20
1.0573
NM
20
TNM
20
LAN
150
1. TNT
81 85 100
% co0 [dyn/cm] [lo5 c d s ]
PLV [TOIT] 0.0036
1.414
0.038
50.73
1.485
2.5 10-4
0.0069
27.24
ca. 1.1
18.65
1.130
0.00613
37.0
1.313
30
1.64
0.0176
30.34
1.039
8.04
1.050 1.462 1.455 1.443
0.1198 0.109 0.0762
47.0 46.6 45.1
1.61 (?)
0.0139 0.0185 0.0514
120
Chapter
bar
100 bar
I bar
~ ( p )
-
-
100
10
1 r
10
1
03
1
bar
R ( p)
Figure WI-8. Losses ofresonant bubbles in NG at 1 bar (ambient pressure), and 100 bar,. The pointed lines indicate Devin's low-frequency approximation. The radiation loss at ambient pressure is = 0.01, (not shown).
With the in Table VII-3 and VII-4, one can calculate the losses of bubble motions depending on the gas content and the medium, and their dependence on the hydrostatic pressure. With respect to the so-called 'adiabatic' hot spot in Figure VII-8, the losses are calculated for bubbles filled with a high- and a low-y gas in NG. The gases show a
121 by Volume
high and relatively low thermal conductivity. Apparently, the quality of the gas is of little importance. The used quantities are defined before. The losses at 100 bar pressure are shifted into a high-frequency domain, and will be discussed later. The losses for NM are shown in Figure VII-3. Please note that the 'adiabatic' hot spot disappears as the viscous losses appear and become larger. It is our understanding that such losses generally apply. In Figure VII-9, the corresponding losses are shown for pressurized liquid carbon dioxide (involved in a tank car explosion).
I o.i
Q
ambient pressure and -28.43"C.. Minnaert resonance condition is (f x & = 1175 Hz cm). The radiation loss dominates here.
-
If the ambient pressure and the gases vary, it is difficult to make calculational estimates, because the viscosity increases with increasing pressure, depending on the molecular structure. But Kuss [VII-171 has shown that there are exceptions: Gases dissolved in the liquid do not affect the pressure dependence, but may decrease their viscosity up to orders of magnitude. Gases of a more complex molecular structure are more effective than simple ones. Also, Qssolved gases do change the density and compressibility (sound velocity) of the liquids. All these effects make it difficult to make good estimates.
122 Chapter
References [VII-11 Ch. Devin, jr., Survey of thermal, radiation and viscous damping of pulsating air bubbles in water, J. Acoust. SOC.Am. 31 (1959), p. 1654/1667. [VII-21 R. Y, Nishi, The Scattering and Absorption of Sound Waves by a Gas Bubble in a Viscous Liquid, Acustica 33 ( 1975)2,p. 65/74. [VII-3] J. E. Hay, and R. W. Watson: Mechanisms relevant to the initiation of low-velocitydetonations, Ann. N.Y. Acad. Sci., 152 (1968), p. 6211635. [VII-41 H. Pfriem, Zur thermischen Dampfung in kugelsymmetrisch schwingenden Gasblasen, Akust. Z. 5 (1940), p. 202/212. [VII-51 0. A. Kapustina, Gas bubbles in a small amplitude sound field, Sov. Physics Acoustics 15 (1 970), p. 427/438. [VII-6] F. P. Bowden, M. F. R. Mulcahy, R. G. Vines, and A. Yoffe, The detonation of liquid explosives by gentle impact. The effect of minute gas spaces. Proc. Roy. SOC.,London, A 188 (1947), p. 291131 1. [VII-7] M. G. Ribaud, Constantes thermodynamiques des gaz aux tempkratures Clevees. Publ. Scientifiques du Ministere de I’Air, No 266a, Paris, 1961. [VII-8] C. 0. Leiber, Appendix of Approximate quantitative aspects of a hot spot, Part 11: Initiation, Factors of safe handling, reliability and effects of hydrostatic pressure on initiation. J. Haz. Materials 13 (1986), p. 31 1/328. [VII-91 L. van Wijngaarden, One dimensional flow of liquids containing small gas bubbles, Ann. Rev. Fluid Mech. 4 (1972), p. 369/396. [VII-lo] S. Temkin, Personal communication from 02.01.2000. [VII-I I ] S. Temkin, and C.-M. Leung: On the Velocity of a Rigid Sphere in a Sound Wave, J. Sound and Vibration 49 (1976),, p. 75/92. [VII-I2] C. 0. Leiber, Die maximale BlasengroDe als Sicherheitsfaktor bei der Chemie Ing. Technik 50 (1 97Q,, p. 6951697. La grosseur maximale des bulles presentee c o m e facteur de securitC en ce qui concerne l’explosion par resonance de bulles, explosifs 32 ( 1 979)4, p. 208/210. [VII-I 31 Ch. Hullin, Stabilitiitsgrenze pulsierender Luftblasen in Wasser, Acustica 37 (1977), p. 64/73. [VII-14] W. Lauterborn, and W. Hentschel: Cavitation Bubble Dynamics Studied by a High Speed Photography and Holography: Part two, Ultrasonics, March 1986, p. 59/65. [VII-151 G. D. Coley, and J. E. Field: The Sensitisation of Thin Films ofNitroglycerine, Combustion and Flame 21 (1973), p. 335/342. [VII-161 G. D. Coley, and J. E. Field: The Role of Cavities in the Initiation and Growth of Explosion in Liquids, Proc. Roy. Soc. London A 335 (1973), p. 67/86. [VII-I7] E. Kuss, The viscosity of gasioil solutions at high pressures, High Temperatures High Pressures 15 (1983), p. 93/105.
VIII
VARIETY OF INITIATION
MODESBY BUBBLES
In many initiation and ignition mechanisms the most pronounced and important factors are dynamic activated cavities and gas-containing voids. We demonstrated in the preceding chapters how loss mechanisms - from gas heating, and viscous and radiation losses - can change a dynamic bubble qualitatively in many ways. Equation (VI-51) as a seemingly simple vibration equation in combination with variable losses, leads to an understanding of the multiplicity of initiation modes that can also be observed experimentally. The main reason for generalization is to develop a predictive model, see reference [VIII-11. Of particular interest is whether the small and large losses in mobile liquids and plastic solids show different dynamic behavior.
On 'Soft' and 'Hard' Excitations We use Equation (VI-51) to study the action of velocity and external pressure act on a bubble of varying loss 6. The many possible solutions are best shown by a phase plot (Figure VIII-1). The starting condition A is given in terms of V and V'. Then the values of V(z) and V(z) are plotted for the times for varying losses 6 until the final state E is reached. We have two qualitatively very different cases: 1)
'Soft' excitation occurs for sniall losses, where many cycles are needed to reach the final state, and
2)
'Hard' excitation occurs for large loss systems, where the main parameters are definitely settled within a fraction of a cycle.
Assuming that the onset of a chemical reaction in an energetic substance is caused by a given loss, then two situations shown in Figure VIIJ-1 exist. We do not know exactly the condition and time of onset of reaction for a 'soft' excitation. The governing term V' of a loss may increase at later times rather than decrease. Therefore, the loss power may vary with time, and an undetermined system is present. However, a decision whether or not reaction can occur is obtained within a very short time in the case of 'hard' excitations.
123
124 Chapter I v’
V
Figure VITI-1. Phase plot of the solutions to Equation (VI-51). The boundary conditions are given at top right. The locus of start is indicated by A and the final state by E. Note that the cases of bubble dynamics are very different depending on damping,.
SOFTEXCITATIONS Parametric Amplifications (Spontaneous Explosions) There are known spontaneous explosions where no adequate stimulus can easily be determined. Examples of ‘soft’ excitations are physical explosions of liquid gases (retardation of ebullition) and explosives (hydrazoic acid). It is interesting, therefore, to explore whether this simplified bubble model can explain such occurrences. Parametric amplifications are the most powerfd and effective in any amplification mechanisms. Produced by stimuli not usually attributable to an excitation, amplification can take place exponentially in time, except in a system in absolute equilibrium. A prerequisite of such an amplification mechanism is that not all factors in the vibration equation are constants; a time dependence is required. Again we make use of Equation (VI-5 l), and take the homogeneous Equation vrn(7) = 0 , since practically no stimulus is present. A solution of this homogeneous equation is =
,in7
.
(VIII-1)
125 of
Inserting this solution into Equ. (VI-51) one gets
(VIII-2)
v f f +(I+ isfi)v = 0 .
Using 1= h and i6fi = ycos 7’ one gets the well-known Mathieu-type differential equation V” +
+ ycos7’)V = 0 ,
(VIII-3)
which shows stable and unstable solutions depending on the parameters h and y cos . Only for h = 1 and i6R = 0 are stable solutions possible. If i6R increases, the domain of possible unstable solutions also increases accompanied by an exponential pressure amplification, since 6 is time- and (as radiation loss) pressure-dependent. If KI is the exciting frequency and
the resonance frequency, then for
= -= 1
the maximum
output can be expected. If the energy is considered, according to Equation (VI-64) the loss varies with double the exciting frequency. Probability of Resonance The probability of such stimuli depends on the probability of finding Am eigenfrequencies in the range AE/f of the source resonance frequency in a vessel of volume V. Weyl [VIII-21 gave an asymptotic expression of
f 3A ? = 47~-A! Am = 4.irV f h3 f
(VIII-4)
for large vessels. This probability increases with the volume of the vessel and decreasing sound velocity of the medium in it. Such volume dependence of explosion frequencies was observed for explosions of liquid-gas tank cars [VIII-3], see Chapter 111.
How to Avoid Spontaneous Explosions A solution of this problem could be to avoid low-loss systems, and to define a minimum allowable loss. This is not easy because a system’s behavior depends on its dynamic stimulus. The physical reason may be easily seen from the solution of the inhomogeneous equation (VI-51) in the terms of the transmission factor (VIII-5) which never becomes linear when Vo f 0, and Vd # 0. Therefore, nonlinearities are present outside of an equilibrium.
126 Chapter
This is demonstrated in Figure VIII-2, where several variations of Equation (VI-5 1) are compared. For losses of 6 = 0; 2, and 20, the behavior of a bubble of unit volume Vo = 1 is compared for two different pressure steps of infinite duration corresponding to V, = -1, and -10. Also, the initial Vo' was +1; 0, and -1 in each case. Seen fiom the right-hand side, a creeping to collapse may be present for V, = - 1 for 6 = 20, which comes to a real collapse for V, = -10, independent of the initial values of VO'. Scaling laws do not exist.
6=2
6=0
=,v
6 = 20 Figure VIII-2. Stimulus of a bubble system of unit volume but different losses, by an external unit-pressure step corresponding to V, = -1, and the 10-fold unit-pressure step V, = -10 of infinite duration. If the initiation is powerful, systems of high losses collapse; then the influences of the initial conditions V,' = 1, 0, and -1 progressively vanish.
-1
Val-10
Complicated Excitation Modes
*
We have considered cases where a (short-duration) external push (z 0) and a longduration stimulus (z m) acted on a bubble of unit volume Vo = 1. In practice, the durations of the stimuli are between these limits, and a real bubble or a cavitation nucleus may be present that forms a bubble, if expanded. The initial condition of such a cavitation nucleus is Vn = 0.
*
127
Modes
-, 0
-2
*
/
10
6=0
Figure VIII-3. Initial and residual spectrum of harmonic loss-fiee bubble volume vibrations for different durations 5 of excitation. For a 'soft' excitation, the residual motion predominates and is more powerful tban the initial motion. The case of a cavitation nucleus is shown as dashed lines and corresponds to VO= 0. This state disappears for systems with larger losses (see Fig. VIII-4).
We apply Equation (VI-51) to different cases of 'soft' stimuli 6 = 0 and 6 = 1; special situations for 'hard' stimuli do not exist. Optimal times to study effects of 'soft' stimuli are: 1.
Duration of the pressure stimulus is short vibration 271.
2.
Duration of the stimulus equals the time period of volume vibration
3.
Duration of the stimulus is longer than the time period of volume vibration. A neat trick is to excite the system to double value by the increase and decrease of the stimulating pressure pulse. This is most pronounced for = 371.
=
1) compared with the time period of volume
= 271.
The dashed lines in Figures VIII-3 and VIII-4 correspond to values for a cavitation nucleus with the initial condition Vo = 0. The vibration spectrum occurring under the dn-ect action of a stimulus is called initial spectrum and that after cut-off residual spectrum.
128
Chapter
-1
Figure VIII-4. Initial and residual spectra of volume vibrations of a system with a loss 6 = 1. In contrast to Figure VIII-3, only the initial spectrum is of importance.
-1
-1
-
,
Residual
As seen in Figure VIII-3, the residual motion predominates for a low-loss system 6 = 0 and is more powerful than the initial motion. Only in the special case = 27t is the initial spectrum more powerhl than the residual spectrum, which again corresponds to the initial conditions.
Figure VIII-4 shows that for a larger loss 6 = 1 only the initial spectrum is important for the specified excitation.
Dynamic Response of the System The volume term V is sensitive to both static and dynamic pressures. The GV-term is a dynamic pressure radiation source. Therefore, the combination of these two terms (6V' + V) represents a dynamic pressure term. The dynamic response of the stimulated vibration system to the surrounding medium should therefore be governed by these elements. In order to test this assumption, the inhomogeneous Equation (VI-5 1) is applied for a unit pressure jump corresponding to -Vm. The result is shown in Figure VIII-5, where oscillatory waves, an overshot unit pressure jump, and a unit pressure jump are seen, depending on the system's loss.
129
Figure VllI-5. Responses and the dynamic combination of the vibration system, Equ. (V1-51), driven by a unit pressure jump. The unit pressure jump, an overshot jump, or an oscillatory response is obtained, depending on the loss.
:. .... ..... . 5;
1'
..
'....
'..
10
- ._.
..
region
A:
B:
Figure VIII-6. Experimental set-up of Noordzij's experiments (leR side). The oscillatory pressure profile at location A is shown on the right. (Noordzij [VIII-4],with kind permission}.
130 Chapter
Figure VIII-6 shows an experimental setup of Noordzij [VIII-41. Free-rising bubbles of radius = 1 mm have been produced in an aqueous solution of glycerine. This system was stimulated by a bursting diaphragm. The pressure profile at location A, at 20 cm run distance, is shown on the right. The wave shapes changed for longer run distances. Comparison with Figure VIII-5 shows an oscillatory pressure profile.
FREQUENCY EXCITATION MODES Bubble vibrations can be driven by different means. As shown previously, a bubble may be pushed by a velocity shock so that a V'-motion is activated. The volume V changes when the pressure in a bubble is suddenly released or increased. This may occur: a)
By a (static) pressure release or increase of the bubbly system.
b)
Or a bubble blows up due to a chemical reaction. In this case both V' and V are altered.
c)
Or a bubble system may be powerfully excited by an arbitrary pressure shock
d)
Or soft excitations by vibrations may occur, possibly through transport.
This latter excitation is an up-to-now unsolved scientific problem. Transportation of liquid UN Class 1 explosives is forbidden, except for liquid TNT, which is seldom transported. Some liquid explosives, like NM or tetranitromethane (TNM) are in nonexplosives classifications. Therefore, their transportation is allowed. For example, NM is classified as a Class 3 flammable liquid. Except for two tank-car explosions in 1958, transportation in drums is 'safe' according to experience from shipping several 100,000 tons. However, transportation of UN Class 1 NG is strictly forbidden because of many accidents.
What is the difference between NM and NG? Curiously enough, the shock sensitivity to High Velocity Detonation (HVD) of the homogeneous NM is about half that of the homogeneous NG: NM can be initiated by a shock of = 50 kbar, but = 120 kbar are required to initiate NG. NG can be initiated under poorly defined circumstances by vibrations during transportation, whereas NM is considered to be safe. In order to make meaningful estimates, the excitation must be modeled in appropriate parameters. Most general applicability shows solutions in terms of Fourier components. Therefore, Equation (VI-5 1) is considered in fiequency terms.
131
Description of Bubble Dynamics by H( ) [VlII-11 Devin's Equations (VI- 16) or (VI-5 1) hold for the time- and frequency-space, because the corresponding function in the frequency space is obtained by Fourier-transform from any exciting fimction in time, and vice versa. When the vibration system is driven by V,, the corresponding frequency is usually different from that of the vibration system coo in Equations (VI-47) to (VI-49). Therefore, a generalized dimensionless frequency is introduced: (VIII-6) The stimulus of Equation (VI-5 1) is characterized by (VIII-7)
ZQT,
where Vex, characterizes the amplitude of the driving force. The solution of Equation (VI-5 1) in V can be expressed in terms of a frequency response fbnction H(C2, 6): = -,
(VIII-8)
which amplifies or attenuates the input V,. Using this function, the components of Equation (VI-5 1) are: (VIII-9) and
(VIII-10)
.
(VIII- 11)
Different response fbnctions are obtained for different possible excitations with the above relations. Some cases are described as follows: A
Stiffness Activation:
It is assumed that the volume term V, which drives the vibration system, is altered by an internal (or external) pressure change of the bubble. Rewriting Equation (VI-5l), the force balance is: and f
f
v=
.
(VIII-12) (VIII-13)
Inserting Equations (VIII-9) to (VIII-11) into Equation (VIII-13), one finally gets:
132 Chapter
(VIII-14) In Figure VIII-7 the real and imaginary parts of Equation (VIII-14) and the modulus are shown as function of and of the loss 6. It is important to keep in mind that for lowloss systems in resonance Re HA(R, 6) may change its sign, - Im HA(R,6) maximizes, and the amplitude approaches 116.
R1
n-
0-
1
Figure WII-7. Components of the kequency response function H A ( Q 6 ) using Equation (VIII-14). Im H dictates pressure amplification or attenuation, and Re H is linked with the compressibility (volume) of the single source. Note that a change in sign may occur!
Damping Activation:
B
If V' is activated the equation of motion is:
+
=
Sviexc. , and
(VIIJ-15) (VIII-16)
is obtained. In Figure VIII-8 the real and imaginary parts of Equation (VIII-16) and the modulus are shown as functions o f n and the loss 6.
133
Modes
Frequency
0.1
R-
1
x)
n-
Figure VIII-I. Components of the frequency response function H B ( Q6) according to Equation (VIII-16). Im H is now related with the volume, and Re H with the volume flow V'.
Figure VIII-9. Components of the frequency response hnction
6) using Equ. (VI11-19), which is related to V"
134 Chapter
Figure VIII-10. Components of the kequency response function Hc . (Q, 6 ) according to Equation (VIII-20). This function is important for the dynamic losses of a bubble system in the case of transportation.
C
Mass Activation:
A combination of cases A and B is:
(VIII- 17) If the systems vibrations are observed in a vibrating transport vehicle then relative displacements VR= V - V,,,. of the vibration system are observed in the car (V,,, is the stimulus of the transport vehicle), and one gets [VIII-I]: +
=
(VIII- 18)
with the solution for (VIII- 19) The real and imaginary parts of Equation (VIII-l9), and the modulus are shown in Figure VIII-9 as function of R and the loss 6. The solution Equation (VIII- 19) is valid, as specified in all the above cases for the displacement of volume V, where only the exciting term varies. For solution in V' to give H( ) = V'/Vexc,,the V-solution must be differentiated and one gets:
135
(VIII-20) The real and imaginary parts of Equation (VIII-20) and the modulus are shown in Figure VIII-10 as function of and the loss 6. Note that the character of ‘typical’ resonance fimctions is completely lost.
D
Dynamic Pressure Activation:
A solution of the dynamic pressure response (6Vl + V) of the system (Figure VIII-5) is also possible in the above frequency terms by defining another appropriate frequency response function H( ):
)= exciting term
.
(VIII-21)
If we take solutions in this term as standard, then
(VIII-22)
hold. By stifhess (volume) excitation one gets for the response function
(VIII-23) where this response function is not altered when excitation occurs by either V’, the combination of V and V’, or V”. In Figure VIII-11 the real and imaginary parts of Equation (VII-23) and the modulus are shown as fimction of and the loss 6. Low- and high-frequency low-loss systems are amplified in pressure until resonance is reached. With increasing loss this amplification disappears. The limiting condition for -1m H( ) = 1 is 6 = 1.07777 at Q = 1.167407. This shows that mobile liquids can become hazardous that high viscosity is a safety measure. That’sjust the way Nobel did it: He gelatinized NG with nitrocotton (blasting gelatin, BG) for safe transport. That is also the physical reason why the transportation of liquid explosives (UN Class 1, with the exception of liquid TNT) is forbidden.
136
Chapter
3
2 1
lo 8 -1 -2
1
n-
x)
Figure VIII-11. Components of the frequency response function H( ), which dictates pressure amplification in conjunction with Equ. (VIII-23).
Differences between Nitromethane (NM) and Nitroglycerine (NG) Torr for NG at With vapor pressures 30 Torr for NM at room temperature and 2.5 densities of 1.13 and 1.59 g/cm3,respectively, one gets R values of about 188 for NM compared with 0.46 cm/s for NG. This means that at a given transportation stimulus the appropriate varies by a factor of about 400, and is larger for NG than for NM, see Figure VIII-10. The dimensionless frequency 0 gives us for the first time a quantitative measure of transportation dangers of liquid energetic substances.
137 Spots -
HOTSPOTS As outlined previously, power density increases as volume decreases. Hot spots can be obtained in different ways depending on the excitation of the bubbles.
Velocity Shocked Bubble
a) PowerLoss To demonstrate the orders of power loss, resonant bubbles in nitroglycerine (NG) are considered, which are shocked by a velocity blow of ur up = 100 mls. Bubble sizes are expressed by their resonant hequencies according to Equation (VI-20), so that a bubble radius of Ro 0.022 cm is obtained for f = lo4 Equation (VI-62) is used in combination with the Equations (VI-19) and (VI-20). The power loss then is
-
-
(VIII-24)
1
NG (0.1 MPa = 1 bar) up = 100 rn/s
/
Figure WIT-12. Power loss and power loss per unit volume of NG bubbles, shocked by a 100-m/s velocity blow, as function of bubble size (expressed by the resonant kequency), and the total loss 6 at atmospheric pressure.
The actual power loss of a single bubble can be impressive but difficult to determine. We obtain this power loss per unit volume when we relate the power loss to the original bubble volume Vo = 47ch3/3. Power loss and power loss per unit volume are shown as fhctions of bubble size (expressed as resonance hequency) and of loss 6 in Figure VIII-12. The absolute power loss decreases and power-loss density increases up to TW/cm3as bubble size decreases.
138 Chapter
Since the bubble volume also changes, loss power density changes accordingly so that hot spots of power loss appear as the bubble tends to collapse.
b) Hot Spot Let us assume a bubble at rest with volume Vo = 1, which is collapsed at time = 0 by a velocity push corresponding to Vl0 = -1. the dissipated power 6VT2can be calculated in relation to the actual bubble volume V as function of T. The result is shown in Figure VIII-13 in terms of IV'*/V as fimction of 't for varying losses 6. Dissipation hot spots appear in space as well in time. Under favorable conditions systems of a low loss 6,,,. may get larger power loss densities than high-loss systems, because the bubble mobilities are less damped. Note that no additional energy supply is present, because V, = 0.
Figure VIII-13. Power loss per unit volume for a velocity shocked bubble of different loss 6 . In one case low-loss systems attain larger losses than highloss systems due to increased mobility of bubbles. This event also leads to their collapse and create hot spots that are well separated in space and time. Bubbles do not collapse when
6>2.
Hot Spots of Pressure Activated Bubbles Completely different behavior is present when the bubble is pressure activated: Bubbles with loss 6 > 2 are brought to collapse (Figure VIII-14). Additional energy is added as long as the pressure excitation remains active. Figure VIII-14 shows hot-spot distributions in space and time as function of the loss 6, where the initial values are Vo = 1; VI0 = 0, and V, = -1 for 't between 0 and 20. Due to the energy addition, the later hot spots are energetically active - contrary to the case of the velocity shocked bubble - {see the first part on the right-hand side of the Equations (VI-62) and (VI-63)). In general, the number of hot spots increases as the loss 6 decreases, but the time delay decreases with increasing pressure amplitude. The delay of the appearance of hot spots (Figure VIII-14, case 6 = 10) was also observed in initiation experiments on solid explosives by Walker [VIII-51.
139 Spots - Pressure Pulses
6 = 0.3
'OO
L
m
5-
6= 0.15
R
5La 6 = 0.2
00
t: l
6 = 0.25
Figure WIT-14. Hot-spot distribution for a continuous pressure acting on the damped system. These hot spots appear separate in time and space A hot spot appears shortly after application of pressure.
140
Chapter
SHAPE OF PRESSURE PULSES Excitation by Different Pulse Shapes The dnving force always governs the motion of a vibrational system in the steady state. Therefore, the system's behavior usually depends on the shape of this driving force. For an arbitrary stimulus, the system's response can be obtained from Equ. (VI-5 1) by analytical means. However, in practice numerical integration of this equation (such as by the Runge-Kutta-Nystrom procedure) is adequate. In the following the Fourier integral is applied to short-duration pulses compared to the duration of vibration [VIII-6, VIII-71.
A frequency spectrum
is obtained fiom a h c t i o n in time p(t) by
+m
S(O)=
J
(VIII-25)
-02
and the reverse operation frn
(VIII-26) -m
reconstructs the original function. The frequency function Equ. (VIII-25) is complex. By taking the modulus only the phase shift is considered. The spectrum is shown in Figure VIII-15 for various pulses of different shapes. The formulas are noted according to Skudrzyk. If the frequencies f 3 0, then the hnctions 3 1, and the lowfrequency amplitude spectrum depends only on the effective amplitude of the pulse and its duration. For pulses short compared to the duration of their vibration, the stimulus is given by (VIII-27) and is therefore independent of the amplitude and shape of the stimulus. This result is also obtained by numerical integration. Thus, the existence of vibratory systems is verified. To test this statement, systems with larger losses are used to avoid effects of residual vibrations.
141
Spots - Pressure
Figure VIII-15. Some interesting pressure excitation functions
their spectra, see Skudrzyk [VIII-6, VIII-71.
Description of Finite Pressure Rise The procedure is visualized in Figure VIII-16. The original pressure fbnction sl(t) passes through a material of special attenuation behavior. This attenuation is described by the transmission factor which changes the original fimction sl(t) 3 s2(t). This is easily solved with Fourier spectrum. The Fourier-frequency spectrum Al(o) is obtained from sl(t): (VIII-28) We Attenuation in real materials usually increases with increasing frequency f o r assume that all frequencies below a limiting frequency w1are passed, and frequencies > 01 are attenuated to 0 in the medium. This situation can be described by a transmission factor 1 for 0 < w < w = 2n
The received frequency spectrum is:
(VIII-29)
142
Chapter
A2 (0)= G(o) A, (0)=
(I_ 0
for o < o < o , (VIII-30) for
By back transformation of Equation (VIII-30), (VIII-3 1) is obtained, where to is the travel time of the transmitted signal in the medium, and (VIII-32) is the integral sinus. The resultant hnction is shown in Figure VIII- 16, top right. s,N
t-+q
'-
1
t
Figure VIII-16. Description of a finite pressure rise in terms of a frequency dependent transmission function G( w).
Kiipfmiiller {in Kaden [VIII-S]} obtained, by linearization of the central part of the Sifunction with its intersections with pa = 0 and p = p:
143
Hot Spots - Pressure Pulses
z=-.
1
(VIII-33)
2f
For a linear pressure rise, the expression 'c=-
1
(VIII-34)
3f would be better. We can attribute a fiequency f o r to a finite pressure-rise time z, and a wave length h together with the velocity of the wave c or us by c = f h. This expression can be used to relate particle sizes a shock rise length Al, or shock rise time z (Fig. VIII-17).
Figure VIII-17. Particle size kR in relation to the finite shock rise.
kR= 1 PO
p-p)
k R = 10
(VIII-35)
144 Chapter
We show in chapter XIV that particle sizes exist, where in the dynamic activated state no relative particle mobility exists. This case can be used for the estimate of the finite shock rise.
References [VIII-1] K. Magnus, Schwingungen, Teubner, Stuttgart, 1976. [VIII-2] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), p. 441/479. RIII-31 C. 0. Leiber, Approximative quantitative Aspects of a Hot-Spot, J. Haz. Materials 12 (1985), 43164. [VIII-4] L. Noordzij, Shock Waves in Mixtures of Liquids and Air Bubbles, Thesis TH, Twente, 1973. [VIII-51 F. Walker, personal communication. WIII-61 E. Skudrzyk, Die Grundlagen der Akustik, Springer-Verlag, Wien, 1954. PHI-71 E. Skudrzyk, The Foundations of Acoustics, Springer Verlag, Wien-New York, 1971. [VIII-8] see H. Kaden, Impulse und Schaltvorgange in der Nachrichtentechik, Oldenbourg, Munchen, 1957.
Ix
VARIOUS APPROACHES TO DESCRIBE BUBBLE DYNAMIC PHENOMENA
In Chapter I11 we stated a mechanism of pressure-wave generation in very general terms. In Chapter IV the propagation of waves and flow was described in acoustical terms for small amplitudes of harmonic source motions; time dependence was neglected. We demonstrated in Chapter V how complicated the considerations get when a more realistic situation is considered where the source motion becomes more finite and time dependence is introduced. In Chapter VI Rayleigh’s approximate equation for the source motion was presented. Even under given conditions, we do not know the parameters of the active bubbles. Therefore we can present only an overview of bubble dynamics. A generalized view of possible bubble events is again obtained by asymptotic views of small bubble vibrations using Devin’s model, which results fkom first principles. Losses of bubble motion leading to pressure waves and the onset of chemical decomposition are formulated in Chapter VII. Chapter VIII demonstrates that even with a simple model a great variety of initiation modes can be formulated, where the losses alter the behavior. All these considerations have the primary goal to describe the behavior of a multibubble system. In addition to these procedures, there exist further descriptions of bubble dynamics. These are important to know because each view can help us understand specific problems. It becomes evident that the ‘truth’has many aspects, which can be hidden under one perspective but easily seen under another description. These additional descriptions are: Scattering cross sections. Description of bubble dynamics in terms of a quasicontinuum. This is important because the classical detonation model uses a continuous laminar flow model. Shortcomings of the classical view can be demonstrated by comparing the results.
SCATTERING CROSS-SECTIONS Cross-section quantities are useful to know in complicated cases, because calculational results are expressed by one number. As outlined in Chapter IV the scattering-cross section Qs compares the power of the action of a source with that of an incident plane wave. More specific, a spherical 145
146 Chapter
source of arbitrary order can scatter, absorb or both, or backscatter (RADAR scattering). Various cross-sections can be defined, depending on the activities of this source: Qs,.,x
=
Power of a spherical source (with special action) Power of an incident plane wave
2
(IX-1)
which describe the deviations fkom simple geometrical expressions.
For rigid, acoustic hard spheres (particles) [= ;(/CRY
if kR 0.5 ~
Falling weight energy in [J/cm2] > 2.70 0.189 0.0816 0.0306 0.0255 0.006 1
2 2 2 6di,s.s,R up , where up h corresponds to the energy of the falling weight. Calculating the losses of bubbles of various sizes with the data given in Tables VII-3 and VII-4, one gets the dissipative losses (thermal and viscous components). The viscous losses vary more than the thermal.
From Equation (VIII-27), one gets for the loss power
Table Dissipative Losses of Bubbles of varying Size
Radius of Air bubble [cm]
Resonance frequency [Hzl
&h
6,
6diss.R’
10-4
2.83 MHz
0.0185
5.5653
5.58 10.’
0.05
5062
0.0376
0.0111
1.22
0.1
2546
0.0275
0.0056
3.31
0.25
1024
0.0179
0.0022
1.25
0.5
513
0.0129
0.0011
3.5
The experimental and theoretical ratios of 0.1- and 1-cm-diameter air bubbles are > 3 1 and 28.7 respectively, and compare favorably against the model. One can also calculate the scattering cross-sections as a function of the resonance, where bubble
178 Chapter X
sizes are found that do not influence the sensitivity much (< calculation is included in Figure IX-2.
cm). Such a
Influence of Bubble-Content on Sensitivity The sensitivity of a bubble-filled explosive is often interpreted as the action of an adiabatic hot spot. This interpretation was suggested by the early results of Bowden, Mulcahy, Vines and Yoffe in 1947 [X-201. Patent applications were even found for using low-y' gases to lower sensitivity of explosives. Adiabatic heating should be governed by
(X- 10) where T2 and p2 are the final temperatures and pressures in a bubble of initial values T 1 and p1 with a specific heat ratio of y'. Therefore, noble gases should sensitize liquid explosives most strongly, but a gas-less cavity should behave neutrally. No significant difference was found in the dissipative losses for bubbles with varying gas content (see Fig. VII-8). It is therefore important to cite experimental references, which show that adiabatic hot spot consideration is not applicable to sensitivity considerations. Bowden and Yoffe [X-18] point out in various contributions that their suggested adiabatic hot spot does not generally apply! The explosion is not dramatically changed when there is Ar, air or a vacuum bubble in NG. The low-y' gases [such as nC5H12 (y' = 1.08), C2H,(y' = 1.26), or CC14 (y' = 1.13)] should condense when compressed, do not behave as expected with the given y' values. In other experiments, different bubble contents do not lead to a correspondingly different sensitivity. Hay and Watson [X-301 have found that the mixture of nitroglycerine (NG)/ ethyleneglycol dinitrate (EGDN) = shows a threshold to LVD of > 2 kbar in the presence of voids, about 1.5 kbar in the presence of air and carbondioxide filled bubbles, and c 1.5 kbar for argon bubbles. Apparently, the gas has little effect. These arguments were also discussed in Chapter VII.
Other initiation mechanisms We mentioned in Chapter VII other initiation mechanisms caused by bubble instabilities: droplet-gas-air mixtures might explode and initiate a liquid explosive; chemical reactions might occur in the bubbles.
179 Experiments on Sensitivity
Methyl nitrate has a high vapor pressure that causes rapid decomposition and initiation of the liquid. Johannson and Selberg [X-3 11 addressed such a mechanism. One obstacle might be that gas reactions need a finite dimension for the onset of reaction. But when a reaction is started only in the gas phase, entrainment into the liquid is made possible by Taylor instabilities. This means that &me-filled bubbles blown onto a liquid surface are pushed into the formerly homogeneous fluid.
Initiation by Entrainment caused by an Explosive Blast (NM) This experience with methyl nitrate was used with NM, which also has a vapor pressure increasing with temperature in a horizontal 3-in.-diameter steel tube closed at both ends. A combustible NM/air mixture was formed above 60°C but no reaction of the NM happened. An overpressure of about 4 bar or more was obtained by using a hydrogen/oxygen mixture. The NM reacted between 40" and 60°C with increasing temperature [X-321. But beakers filled with NM, at temperatures from 40" to 70"C, could not be initiated with shocks up to 200 bar in shock tubes (coated with NM). The shock pressure at the side walls was 160 bar [X-331, see Chapter XXI. Here a comment is appropriate on the design of experiments: Such experiments look foolish to a practical experimentalist, who knows the insensitivity of NM to HVD, LVD, and even the drop-weight test. But the design of such experiments was developed from a model - with surprising results.
LVD and HVD Sensitivities The model experiments on ceramics (Chapter XV) have shown that differences in the dynamic behavior cannot be detected by quasistatic testing. The experiments of Dempster [X-341 with blasting gelatin (BG) clearly demonstrate that, contrary to 'usual' expectation, shock sensitivities for High Velocity Detonation (HVD) initiation will not parallel the sensitivities determined with the drop weight. The addition of < 10-ym-sized particles sensitizes the initiability to HVD. Adding particles > 10 pm increases its impact and friction sensitivity and LVD (see Table X-6). He designated the ability of sensitization by the amount of powder he had to add to BG to obtain HVD by a No 6 A.S.A. copper-cased detonator (0.25 g tetryl) in a %-in.charge. As an order he estimated 15 x 10' particlesi100 g BG. Such differences can be expected when one considers particle mobility, possible formation of a wake, and pressure-wave attenuation by mobile or fixed particles. Again, we emphasize that dynamic sensitivity data cannot be estimated by quasistatic testing, because the basic mechanisms are very different.
180 Chapter
Table X-6. Results of Dempster IX-341
Explosive composition
Blasting gelatin (BG) NG/NC pure
Dropweight sensitivitv Minimudmaximum height of a 0.5 kg drop-weight in cm for a NO GO/GO.
Detonation event
3 0 - 40
LVD
= 92/8,
BG + lead sulphate 1% 2.5 pm 5% 2.5 pm
BG + barium sulphate 5% 1 5 - 6 0 p m 20% 15 - 60 pm 6% 1 2 p m 1% 6 p m 1% l p m < 0.5 pm
HVD 100 - 140
3- 5 LVD HVD HVD HVD LVD
One might surmise from a drop weight sensitivity to a more general 'sensitivity'. Such conclusions must be strictly avoided, as is clearly demonstrated from the different decomposition products of explosives obtained from detonation, drop-weight impact, and thermal decomposition tests by Bowden and Yoffe [X-191 and shown in Table X-7.
Influence of Static Pressure on Sensitivity From the results of Figure VII-8, one can expect a considerable influence of moderate ambient pressure on the sensitivity of explosives. Two cases must be considered: In open-pore systems the medium as well as the bubble content are pre-pressurized and the void sizes remain constant. In closed-pore systems the bubbles are compressed and smaller. This is why different pressure dependencies occur. Bowden and Yoffe [X-191 found that the efficiency of 100% explosion of NG in the drop-weight test at an impact energy of 5 x lo3 g x cm at 1 bar drops to 0 when the ambient-air or nitrogen pressure increases up to 30 - 20 bar. Gurton [X-351 observed that NG has fewer HVD events when the static pressure is increased to 68 bar, as is the case for crystalline, powdered explosives. Even a LVD initiation is progressively impeded with methane as the ambient pressurizing gas. Thus, the detonation velocity is reduced fiom 7,690 at 1 bar air to 6,670 d s at
181 Experiments on
Table X-7. Fumes of explosives, of different stimulated reaction.
Composition of the h e s [Val.%]
NG
Detonation
32
63.1
4.9
Impact
29.7
2.3
11.7
28.8
22.5
5.0
Thermal decomposition (180°C)
50.3
1.o
2.1
17.2
28.9
0.5
22.8
37
26.7
6.8
PETN Detonation
5.3
Impact
24.3
5.3
9.4
19.1
35.4
6.5
47.6
9.5
1.6
6.3
21.0
2.0
11.3
43.3
18.8
21.3
5.3
22.8
31.0
13.2
8.7
1.8
Thermal decomposition (210°C)
12
1.4
RDX Impact Thermal decomposition (267°C)
22.5
14.3 bar methane, and to 909 m / s at 67.7 bar methane under identical experimental conditions. For tetryl (p=0.9 g/cm3), flaked TNT (p=0.8 g/cm3),NQ (p=0.5 g/cm3)normal LVD have been obtained at ambient pressures of air, ether or pentane. LVD has been suppressed for tetryl at 48 bar, flaked TNT at 41 bar, and NQ at 18 bar with pressurized methane (the only gases used). But initiation of HVD was not impeded by addition of methane up to 68 bar. According to Eitz [X-361, [X-371, a closed-pore gelatinous permitted explosive failed to detonate at 10 bar ambient static pressure of methane, but an open-pore salt explosive detonated completely at 40 bar ambient static methane pressure. He noted that sensitivity to pressurization by nitrogen is more critical than by air or methane. For more details see Reference [X-381.
182 Chapter
In summary, open-pore systems are less sensitive to static pressures, but closed-pore systems are very sensitive; a difference of a few bar can suppress initiation. This will be addressed in greater detail under the heading: Double Explosions in Chapter XIII. Such explosions can occur when the onset of a chemical reaction pressurizes the system so that explosion fails. However, detonation may be initiated when a pressure release occurs by rupture of the confinement (the pile mass 1iJs 08.
References [X-I] E. Bergengren, Alfred Nobel, Bechtle Verlag 1960 (Translation from Swedish). [X-21 F. Trimborn, iiberblick uber die historische Entwicklung des Explosivstoffpriifivesens in Deutschland von den Anfangen bis zur Gegenwart, Nobel Hefte 55 (1989)+ p. 1071159. [X-31 F. Nicklisch, Kooperation von Technik und Recht im Hinblick auf die technische Sicherheit, Chemie-Ing.-Tech. 54 (1 982)~,p. 7421747. [X-3a] W. H. Rodgers, Guerilla Decisionmaking: Judicial Review of Risk Assessments, J. Haz. Mat. 15 (1987), p. 205/217. [X-41 Recommendations on the Transport of Dangerous Goods (Tests and Criteria), Rev. 1, United Nations, February 1989. [X-5] Gesetz uber explosionsgefahrliche Stoffe (Sprengstoffgesetz-Spreng G), 25.08.1969, BGBl I, p. 135811403; 13.09.1976, BGBl I, p.2737 [X-61 Manual of Tests for the Qualification of Explosive Materials for Military Use, NATO, AOP-7, February 1988. [X-71 B. Ausfihrungsanweisung zur Gewerbeordnung f i r das Deutsche Reich, (RGB1. 1900 S. 871) from 1. Mai 1904. and from 22. April 1911 (HMBI. S. 132). 1x431 R. Assheton, History of Explosions on which the American Table of Distances was based, Bureau for the safe Transportation of Explosives and other dangerous Articles, The Institute of the Makers of Explosives, 1930. [X-91 B. J. Thomson, B. G. Freeder, N. L. Heathcote, D. H. Pickering, and T. A. Roberts, Experimental work on the Explosion Hazard of Sodium Chlorate, HSE-Rept. nlIL1HM.EXI8511 (1985). See also [X-9a] M. Steidinger, Die Gefahrklassifizierung von Alkali- und Erdalkalichloraten, Amts- und Mitteilungsblatt der Bundesanstalt fiir Materialforschung und -priifung (BAM) 17 (1 987)3, p. 4931504. [X-9b] C. 0. Leiber,
%a%%S %% , THE-2000,
K6gy6 Kayaku 49 (1988)5, p. 3001304.
[X-101 Poppenberg noted in H. Kast, Die Explosion in Oppau am 21. September 1921 und die Tatigkeit der Chemisch-Technischen Reichsanstalt, Z. ges. SchieB- und Sprengstofies. 20 (1925)11,12,and 21 (1926)l.g. [X-1 11 E. L. Crow, F. A. Davis, M. W. Maxfield, Statistics Manual with Examples taken from Ordnance Development, Dover Pub, New York, 1960, Reprint of NAVORD Rept. 3369 NOTS 948. More recent: [X-1 lb] [X-1 lb] W. J. Dixon, F. J. Massey, jr., Introduction to Statistical Analysis, 4'h ed. McGraw Hill Pub. Co., New York ..., 1983, ISBN 0-07-017073-8.
183
on
[X-121 Wissenschaftliche Tabellen Geigy, Teilband Statistik, 8" Ed. Ciba-Geigy, Base1 1980. [X-131 H.-J. Henning, and R. Wartmann, Stichproben kleinen Umfangs im Wahrscheinlichkeitsnetz, Mittbl. math. Statistik 9 (1 957) p. 16811 [X-141 C. 0. Leiber, Zur Zuverlassigkeit von Testaussagen - Anziindverzugszeit und Fallhammenverte -, Sprengstoffe Pyrotechnik, Schonebeck, 30 (1993)4, p. 16/34. [X-151 W. Strsmsge, Prediction of Extremely Low and High Explosion Frequencies from Sensitiveness Tests, Propellants, Explosives and Pyrotechnics 17 (1992)6, p. 2951297. [X-161 W. Stramsne, Letter communication. [X-171 P. W. Bridgman, The Effect of High Mechanical Stress on Certain Solid Explosives, J. Chem. Physics 15 (1947)~,p. 311/313. [X-181 F. P. Bowden, A. D. Yoffe. Initiation and Growth of Explosion in Liquids and Solids, Cambridge University Press, London, 1952. [X-191 F. P. Bowden, and A. Yoffe. Hot Spots and the Initiation of Explosion, 31d Symp. Combustion and Flame and Explosion Phenomena, Williams & Wilkins, Baltimore, 1949, p. 5511560. [X-201 F. P. Bowden, M. F. R. Mulcahy; R. G. Vines; A. D. Yoffe. The detonation of liquid explosives by gentle impact, Proc. Roy. SOC.London A 188 (1949), p. 291 1311. [X-211 M. M. Chaudhri, and J. E. Field, The role of rapidly compressed gas pockets in the initiation of condensed explosives, Proc. Roy. SOC.London A 340, (1974), p. 1131128. [X-221 S. N. Heavens, and J. E. Field, The ignition of a thin layer of explosive by impact, Proc. Roy. SOC.London A 338, (1974), p. 77/93. [X-231 J. E. Field, G. M. Swallowe, and S. N. Heavens, Ignition mechanisms of explosives during mechanical deformations, Proc. Roy. SOC.London A 382, (1982), p. 23 11244. [X-241 G. M. Swallowe, and J. E. Field, The ignition of a thin layer of explosive by impact; the effect ofpolymer particles, Proc. Roy. SOC.London A 379, (1982), p. 3891408. [X-251 J. E. Field, Hot Spot Ignition Mechanisms for Explosives, Accounts of Chemical Research 25 (1992) I 1, 4891496. [X-261 A. V. Dubovik, V. K. Bobolev, Cuvstvitel'nost' zidkich vzryvcatych sistem k udaru, NAUKA, Moscow, 1978. [X-271 A. V. Dubovik, V. K. Bobolev, Schlagempfindlichkeitflussiger explosionsfahiger Systeme, German Translation of [X-11, edited by C. 0. Leiber, Forschungsbericht aus der Wehrtechnik, BMVg-FBWT 96-3 (1996), Fachinformationszentmmder Bundeswehr, Bonn. [X-28] P. Langen, Email communication, 01.10.2001. [X-291 J. F. Roth, Prtifung der Empfindlichkeit von flussigen und plastischen Sprengstoffen gegen die Implosion von Luftblasen, 36e Congrks International de Chimie Industrielle, Bruxelles, 1966, p. 20122. [X-301 J. E. Hay, R. W. Watson- Mechanisms Relevant to the Initiation of Low Velocity Detonation, Ann. N. Y. Acad. Sci. 152 (1968), p. 621/635. [X-3 11 C. H. Johannson, and H. L. Selberg, The Ignition Mechanism of High Explosives, Appl. Sci. Res. Sect. A, 5 (1955), p. 4391449.
184 Chapter
[X-321 R. Wild, and J. Muller, Verhalten von Nitromethan bei Beaufschlagung mit einer Gasexplosion, BICT Rept. 2.3/2/5799/82 (1 982). [X-331 C. 0. Leiber, P. SteinbeiR, and A. Wagner, On apparent Irregularities of Pressure Profiles in Shock Tubes, Europyro 93, Se Congres International de Pyrotechnie du Groupe de Travail, Strasbourg, 6./1 I.juin 1993, p. 1711178. [X-341 P. B. Dempster, The Effect of Inert Components in the Detonation of Gelatinous Explosives, Disc. Faraday SOC.No 22 (1956), p. 196/202. [X-35] 0. A. J. Gurton, The Role of Gas Pockets in the Propagation of Low Velocity Detonation, Proc. 2"d ONR Symp. on Detonation, 1955, p. 464/482. [X-361 E. Eitz, iiber das Verhalten von Wettersprengstoffen bei Initiierung im anstehenden Gasdruck, Nobel Hefte 28 (l962), p. 62/78. [X-371 E. Eitz, Untersuchungen uber Fragen der SchuRbeeinflussung bei Wettersprengstoffen, Nobel Hefte 31 (196S), p. 1/21. [X-381 C. 0. Leiber, Approximative Quantitative Aspects of a Hot Spot, Part Initiation, Factors of safe Handling, Reliability and Effects of hydrostatic Pressure on Initiation, J. Haz. Mat. 13 (1986), p. 3 11/328.
XI
LOW-
(LVD) AND SLOW-VELOCITY DETONATION (SVD) OF LIQUIDEXPLOSIVES
Low-Velocity Detonation (LVD) proceeds in explosives of usual densities in a velocity range of about 800 to 2500 At the beginning of the last century, Stettbacher [XI-I] recognized that detonations of 7000 m!s and more can be realized. These are today called High-Velocity Detonations With appearance of the classical detonation theory, the paradigm was born that detonation velocity should be univalued in the steady state, corresponding to HVD. Therefore, LVD phenomena were considered to be of a pathological nature, common for liquids but not for solids. Nevertheless, such LVD phenomena are observable, indicating that the theory has serious shortcomings. Researchers who believe in validity of the classical detonation theory, use synonymous expressions for events such as non- or sub-detonative explosions, predetonation transient waves, pressure-wave accompanied combustion, or unsteady detonation. Jacobs [XI-21 presented a review of the early work in 1960. Excellent reviews of LVD have been given, especially for liquids, by Brown and Collins [XI-31 in 1967 and more recently by Dubovik and Bobolev [XI-41 in 1978. A general review was issued in 1973 by Belyaev, Bobolev, Korotkov, Sulimov, and Chuiko [XI-51 in 1973. In the US the word 'Low-Velocity Detonation' has been used for a different situation: If a system shows an actual velocity of 7000 m/s versus a theoretical value of 7700 this behavior is also called Low-Velocity Detonation [XI-61.
LVD governs Safety According to usual understanding of explosives decomposition, most tests address the initial reactions (in the drop weight or friction test, for example) or the ability of HVD to initiate and propagate (in a steel-tube test). We are not considering test arrangements in geometric dimensions below the critical diameter of a detonation. Sometimes, insensitivity was claimed simply because test dimensions were not appropriate for a positive test result. This practice has serious detrimental effects on public safety, because the engineering aspect of HVD detonation dominates in practice. Reliable initiation of HVD within the shortest time or distance by an appropriate high-level stimulus is essential; one is not interested in LVD or any other transient phenomenon. Also a working detonation theory - true or not - for technical predictions is desirable. The explosives safety engineer should have other goals. He should emphasize in his considerations the lowest stimulus that can escalate to an explosion. This would lead to
185
186 Chapter
recognition of real-life hazards. Therefore, all possible transient states during evolution of an explosive event become most important. These two very different aspects, LVD and Slow-Velocity Detonation, SVD, (also named as ‘third detonation velocity’) are now rarely seen. Even so-called safety institutions address only HVD phenomena in their tests. There is a lack of appropriate test methods, because none of the “safety” institutions traditionally perform sufficient research in such areas as LVD, except some decades ago at the US Bureau of Mines [XI-71. The misunderstanding of ‘safety’ is also documented in the United Nations (UN) “Recommendations on the Transport of Dangerous Goods-Tests and Criteria” [XI-81. The thoughtless application of technical detonation aspects to safety by “experts” and institutions will continue to result in casualties, proven by the small number of accidents of chemicals (pure ammonium nitrate or alkaline chlorates and even bleaching powder) that resulted in losses of thousands of lives. None of the conventional tests predict hazards. So, one may ask the value of such a testing, if for example, the cratering of a drum falling from table onto the floor, or the drum catching fre, cannot be predicted [XI-91, [XI-lo]. These are “common” accident stimuli. We have known for a long time that the transition fiom burning to HVD (Deflagrationto-Detonation Transition (DDT)) roughly follows the way indicated in Figure XI-1, both for primary and liquid or solid secondary explosives as well as powders [XI-1 11, [XI-21. possible hazards
/
1-
linear burning
I pulsation 1
time
Figure XI-1. Transition from weak stimuli to HVD. The weakest stimulus leading to an escalation governs the sensitivity. Therefore, the safety engineer should use this plot from !he left-hand side, the application engineer uses it from the right-hand side
-
Reaction may start with linear burning, which becomes irregular or pulsating by any one of many possible mechanisms. Burning pressure increases as the consumed-mass rate increases until onset of LVD, which can hrther transit to HVD [XI-121. Burning results in a dynamic pressure acting as a shock. We have a Shock to Detonation Transition (SDT) when this burning pressure is replaced by external shock. Also, a reverse transition HVD 3 LVD is possible, as will be shown below [XI-131. Since the conditions for LVD and HVD are different, it is not easy to estimate the final response of an explosive system to a given low-level stimulus. Recent works of Sulimov, Ermolaev, Borisov, Korotkov, Khasainov, Khrapovski [XI- 141, and Butcher, Keefe, Robinson, and Beckstead [XI-151, and Sulimov, Ermolaev, Khrapovski [XI- 161 show
187
that transitions from burning to LVD are very complicated and differ from case to case. The above authors investigated mostly macroscopic bulk properties, but the late Dr. H. Dean Mallory recorded simultaneously pictures of the flame fronts and the microscopic pressure development with combined impedance-mirror photography. He showed that the flame front is not locally connected with any pressure fiont. At the beginning of deflagration, singular pressure sources arise ahead the flame front. These pressure sources increase in number and density, and 'cooperate' more and more, until, finally a macroscopic pressure plug appears, see Chapter 11, Figure 11-17.
LVD Stimuli The initiation of LVD is not restricted to weak shocks. Heat, burning, ignition by a hot wire or a squib can initiate LVD after a run distance. Confinement is advantageous but not necessary in each case. Under certain circumstances, a powerful shock, such as caused by mechanical (projectiles) impact can stimulate LVD.
Anomalous Hugoniots As a result of the quasi-continuum approach, we obtained anomal Hugoniots where a pressure increase occurs as the volume increases. In combination with this anomaly and due to the violation of the Becker-Bethe-Weyl relations in this domain, tension shocks become possible. These shocks occur in both liquids and solids. Therefore, the surface waves of the confinement are not solely responsible for cavity or fracture formations: unconfined explosives may also undergo LVD. Considering the dispersive wave velocities these exhibit none, one, or sometimes two regions of pressure transparencies where neither amplification nor attenuation take place. So up to two wave velocities with zero attenuation can be expected. These effects of the inertial motion of the surrounding matrix are called SVD and LVD. In a11 cases, high input pressures attenuate, and low pressures amplify to a stable state. However, no linear relation exists between the pressure and wave velocity. The existence of LVD depends on ambient static pressure according to the pressuredependent losses of the bubbles. Figure XI-2 is an example of an anomalous Hugoniot of porous tetryl in contrast to the dense solid. Figure XI-3 indicates that such a type is not specific of an explosive (liquid deuterium), see the summary [XI-171, but also apply to foams, porous solid materials and powdered beds. It can be seen from Figure XI-2 that less-dense tetryl undergoes a LVD with a pressure of about 5 - 10 kbar where the cusp is observed. Figure XI-3 shows that explosion phenomena in liquids are not restricted to energetic liquids but can be a property of any liquid.
188 Chapter XI
1-
100 kbar
Figure XI-2. Hugoniot of high- and low-density pressed tetryl [XI-171.
Tetryl
So = 1.7
50
0
0.5
0.
0.6
0.7 cm31g 0.8
v-
. kbar
300
-
i
20K
g i d
1
250
.
200
-
Figure XI-3. Hugoniot of liquid deuterium as example of a chemically inert liquid [XI-17].
150.
100
-
50
.
0
0.5
1
2
3 - Y
1
5
189 Liquids
&
SVD and LVD in Liquid Explosives First Observations of an LVD in Liquids Usually Stettbacher [XI-11 is given the honour to have detected the LVD of liquid explosives. However, his real merit is to have been the first to realize that " ... by on (NG) 1400 - 1600 ..." might be a true effect. As shown in Table XI-2, in the years from 1866 to 1913 LVD had been observed between 1,100 and 2,050 m i s for NG. Therefore, it was a big surprise when in 1913 Comey and Holmes observed two velocities: 1,288 and 8,527 Table XI-1 shows velocity windows where no stable velocities are observed; they are governed by the initiation strength and the diameter of the charge. There are two forbidden velocity ranges: one between 2,000 and 7,000 m l s and another one between 800 and 1,200 d s . These two velocity gaps correspond to three types of detonation velocities. The lowest one is called SVD or third detonation velocity; the intermediate is called LVD, where note is taken whether the detonation velocities are above or below the sound velocity c, in the liquid. The third is HVD > 6,000 But Dubovik and Bobolev [XI-41 attribute the SVD to the domain < c,, and LVD to > c,. Table XI-1. Detonation velocities of NG as function of the charge diameter and initiating strength from Clapbam [XI-181
Diameter of the Charge in "
Detonation velocity
of NG in Glass Tubes initiated with
Fulminate No 2 Cap
Fulminate No 6 Cap
Fulminate No 8 Cap
BriskaNo 8 Cap
%
890; 950
810; 890; 1,030
1,350
8,130
%
2,530
1,940; 2,090
1,780
8,700
?4
2,130
1,970; 2,030
1,750
8,250; 8,390
1
2,190
2,020; 2,030
8,130
1%
1,760
1,780; 2,010
8,140
1%
1,910
190
Table XI-2. Some measured detonation velocities of nitroglycerine (NG) [XI-191
Authors
Detonation velocity DWSI
1,100 1,525
Berthelot and Vieille (1891) lead and tin tubes of 3 - 6 mm 1.D." Abel, 30-mm 0 lead tube. Mettegang ( 1 903), 30-mm I. D. iron-tube
2,050 1.125
Blochmann (1906) Comey and Holmes (1 913) Iron tubes 25- and 37.5-mm I. D., 3-m long; initiation by fulminate cap. Same tubes but initiation by special detonators. Kast (1920) Mannesmann tube 30 mm I. D., 5-mm thick, 50 g tetryl Dserschkowitsch and Andreev (1930) Glass tube 22-mm I. D., 2-mm thick, initiation by No. 8 cap. Same tubes but initiation with 10 - 15 g solid (frozen) NG.
1,288 - 8,527 6,700 - 7,600 7.460 1,165 8,750 7,800
Stettbacher (1 930) Naoum and Berthmann (1 93 1) Seamless steel tube 21-mm I. D., 3.5-mm thick, initiation by No 8 cap. Same tubes, but initiation with a 15-g picric acid charge. Seamless steel tube 32-mm 1. D., 3-mm thick, initiation by No 8 cap. Same tubes, but initiation a 50-g picric acid charge. Seam-pressed steel tube 35-mm I. D., 4-mm thick, initiation by No 8 cap. Same tubes, but initiation by a 50-g picric acid charge. Seamless steel tube 51-mm I. D., 3-mm thick, initiation by No 8 cap. Same tubes, but initiation by a 50-g picric acid charge. a
8,750; 8,210 8,290; 8,750 7,010; 7,920 8.340; 8,590 7,820; 5,610; 6,070 8,475; (7,330); 8,680 8,980; 8,270 8,200; 8,540
I.D. - interior diameter
First Observations of SVD (3rdDetonation Velocity) in Liquids LVD
SVD
LVD Transitions
Haeuseler [XI-20] probably first detected in the 1930ies that an SVD or third in addition to the LVD is possible, where the transitions between these velocity regimes had previously been discontinuous. The same hellhydrogen peroxide mixture exhibits HVD of 6,800 an LVD of 2,200 and a SVD of 750 NG and ethylene glycol dinitrate (EGDN) also had 3 regimes with the ratios of the velocities as 1/3/9. In fuel/perchloric acid-mixtures he reported only two velocities (2,400 and -7,000 but he observed only the W D with 7,000 and 8,000 respectively, in fuelhitric acid mixtures and tetranitromethane (TNM). In smear-camera photographs
191 &
straight lines are usually observed, but Figure XI-4 shows irregular behavior of the helhydrogen peroxide mixture. In the center a velocity of 750 m/s (SVD) and at the left and right sides 2,200 m / s (LVD) were noted. This represents an LVD 3 SVD 3 LVD transition. Haeuseler noticed also that a bubble was responsible for this behavior and attributed it to the 'third' detonation velocity. This third velocity was also confirmed by Dubovik and Bobolev [XI-41, and was named Slow-Velocity Detonation, SVD. As mentioned above, they characterize LVD and SVD by the sound velocity of the pure liquid co so that DLvD> CO, and DSVD< CO. In the present terminology SVD corresponds to the lower and LVD the upper pressure transparency points (Figure IX-4).
Figure XI-4. LVD and SVD transitions of a fuel (alcohol)/hydrogen peroxide mixture according to Haeuseler [XI-20]. In the center 750 and left and have been noted. LVD SVD right 2,200 LVD transitions.
LVD
HVD
LVD Transitions
As indicated in Figure XI-1, LVD can discontinuously transit to HVD. Since the properties of HVD and LVD differ significantly, it is of interest to determine if a reverse transition can take place. In classical theory the critical dlameter is a univalued dimension for a given substance at given conditions, below which (HVD) detonation fades out. In cylindrical geometry this dimension is determined by up-and-down experiments, so that this value becomes a mean of many experiments. In the course of critical-diameter determinations of NM, a test arrangement in conical geometry [XI-211 was used. Therefore, a definite critical diameter was obtained in each experiment. Due to the cone's taper there was a permanent overdrive of the detonation, causing a difference of the critical diameter in cylindrical and conical geometry. Contrary to classical theory, no single value for critical diameter has been obtained for a specific test. The values scatter considerably, well beyond experimental error. The detonation faded away at the critical diameter in about 200 (-80%) of experiments. In the other cases, something like burning was observed but in two cases a special effect appeared: In conical geometry and plastic
192 Chapter
Figure XI-5. Demonstration of an 3 LVD 3 fading sequence of detonating NM in conical geometry. On the left are the test assembly and the corresponding witness plate. A wire is plated on the metal plate for HVD but not for In the center and on the right-hand side are shown views of the same process. The latter side was monitored with an electric velocity probe;. the velocities determined with optical and electrical tools vary.
confinement down to the critical diameter of 13.5 mm at room temperature, a nonconstant D of about 6.2 kmJs was measured, at which point it transited to a LVD of D = 1.3 A discontinuous transition to D = 6.25 k d s or more occurred after 12 ps at a diameter of 10 mm, which transited to D = 2.4 km/s; this detonation died out at a diameter of 4 mm (Figure XI-5). This finding indicates that detonation is of a microscopic statistical nature, and cannot be interpreted in terms of a (homogeneous) continuum.
Observations of LVD in Liquids Renewed interest in LVD research was generated by two unexplained tank-carexplosions within six months in Niagara Falls, NY (22.01.1958), and Mt. Pulaski, IL (01.06.1958). NM as a (UN Class 3) flammable liquid had been transported in tank cars. The Department of Transportation (DOT) then allowed hrther transportation only in drums for unexplained reasons. This seemed to be a good decision.
193 &
LVD of Nitromethane (NM)
It was known that neat NM would not undergo LVD, but a mixture of NM/TNM 90/10 would easily do so. That neat NM will not undergo any LVD is based on Chaiken’s theory [XI-221 on two-phase principles combined with changes of the classical approaches of Schall [XI-231 and Cowperthwaite [XI-241, based on a modified classical plane-wave CJ model in a homogeneous or quasi-homogeneous explosive.
So it came as a big surprise [XI-251 when Groothuizen [XI-261, [XI-271 demonstrated that NM can undergo LVD. However, more detailed investigations showed that the existence of LVD in a steel tube of 50 mm i. d., wall thickness A = 10 mm, and 750 mm length depended on the existence of a measuring probe (wire). The velocity scattered between 1,300 and 2,100 m l s . Later Kozak, Kondratjev, Kondrikov, and Starshinov [XI-281 used steel tubes of 10 mm i. d., wall thickness A = 12.5 mm, (inside/outside diameter ratio 0.29), and 250-mm length. Such a ratio appeared favorable to achieve LVD in dense crystalline explosives. Using donor charges of RDX/salt, the initiating pressure was varied between 6 and 70 kbar (5 - 100% RDX); LVD was always obtained. The tubes have not been destroyed, and no LVD 3 HVD transition has been observed. Failure was sometimes observed with initiating charges of 5% RDX. The detonation velocity was observed through 17 2-to-3-mm-diameter bore holes over 250-mm length. Detonation faded away when this diameter was increased to 8.5-mm diameter. A uniform detonation velocity of 2,280 60 m l s was measured in 27 experiments. LVD detonation pressure, determined by an aquarium technique, was 10 kbar. The addition of 2% PMMA (to increase viscosity) did not alter the LVD. The following experiments can only be understood in terms of the HVD criticaldiameter experiments. LVD properties remained when &luting NM with up to 60% CC14,but the initiating pressure increased, Addition of 5% DEA (Diethyl amine) (this sensitizes HVD!) did not influence LVD, but the initiating pressure was somewhat larger. LVD faded away at concentrations of 10 to 13% DEA. Nor did DEA enhance decomposition in NM during combustion under ambient pressure. These findings are important because addition of DEA is considered as a chemical sensitizing agent in HVD research (aci-NM!). The critical diameter of HVD in a thin-walled steel tube was measured as 3.8 mm and the critical diameter of LVD in a steel tube of wall thickness 1 1 - 13 mm was around 5 and 5.5 mm. Replacing the steel tube with a lead tube (I. D. = 10 mm, 0. D. = 36 mm) this tube splits but no LVD is observed. So there is little relationship between the critical diameters of HVD and LVD.
I94 Chapter
LVD of liquid TNT, Isopropyl nitrate, and Ammonium nitrate LVD seems to be quite a common property of liquids. Maybe the initiation of a stable LVD is limited by the critical diameter, but the risk is nevertheless present. Molten TNT (85°C) did not exhibit LVD in 1-in.-diameter pipes. In 2-in.-diameter steel pipes LVD velocity ranged between 900 and 1,100 m/s with corresponding Plexiglas gap lengths between 0.13 and 2 inches [XI-291. Isopropyl nitrate is a good example to study dimensional effects in sensitivity testing. The initiation of HVD depends critically on ambient temperature and static pressure. Pertinent experiments done by Brochet [XI-301resulted in LVD 1,100 - 1,350 m/s in heavy walled steel tubes [XI-271, [XI-311. With respect to ship explosions of liquid explosives: (Texas City, Brest) of ammonium nitrate (AN) the Dutch (TNO) [XI-311 conducted experiments on ANiwater melts between 126" and 228OC that resulted in LVD between 1,000 and 2,200 m / s , but mostly around 1,100 and 1,400
Joyner's last Experiments [XI-32) Hydrazoic acid is a 'dangerous' but a very insensitive substance in the homogeneous state (similar to usual homogeneous liquid explosives like NM) with an HVD initiation pressure of some 50 kbar (Yakoleva and Kurbangalina [XI-33]). This contradictory situation depends on heterogeneity and the associated LVD risk. The experiments of Joyner [XI-321 prove that. His experimental results are listed in Table XI-3. One of these experiments killed him even though he was a skilled experimenter. Table XI-3. Detonation velocities of hydrazoic acid/water solutions (Joyner) [XI-321
HN3 Concentration in d100 g solution
Normality
80 70 67 65 61 60 54 48 38 30 16.7
22.1 18.0 17.2 16.8 15.7 15.4 13.8 12.2 9.8 7.3 4.0
Detonation velocity 8,500 7,300 7,100 1,700 630 384 233 40 2.5 1.o 0.14
195 &
TNMM
-
pLw/DZLVD-Relationship
Many LVD experiments have been performed in the past on liquid explosives. Wellinstrumented experiments were described by Cowperthwaite and Erlich [XI-341 on the system tetranitromethane (TNM)/nitromethane(NM) = 4/5 by weight in a steel tube of %-in. i. d. and a wall thickness of 1/8 inch. Density and sound velocity are estimated at 1.36 g/cm3and 1,170 m / s , respectively. PLVD
\
15
Figure XI-6. LVD of a TNMLVM-system in a steel tube. High input pressures are attenuated and low ones amplified. The apparent detonation velocities give in the lower part. The numbers identify the experiments {Cowperthwaite and Erlich [XI-34]}
k1 \
\
10
t5
-T
1900
I
.
.
,
,
-
Initiation occurred by a variable gap, and pressure in the explosive system was measured at 5 different places with Lagrange pressure gauges. The averaged detonation velocities were calculated fiom the monitored transit times of the waves. The results in Figure XI-6 show that high dynamic input pressures attenuate and low pressures amplify, but in both cases the pressures are of the same order of magnitude. The detonation velocities D do not correspond to the measured pressures p, as one would expect from any plane-wave theory (p/D2is not constant). With a density of pm= 1.36 g/cm3and a sound velocity c, = 1170 m / s (for a mixture of tetranitromethane (TNM)/NM 4/5 by weight) one gets pressures of 7.72 and 6.33 kbar for us/up= 6 (with s 1) and 6.63. The upper value agrees reasonably well with the experimental values in Figure XI-6. Kozak, Kondrikov and Starshinov [XI-351 determined the detonation velocity and the LVD pressure for NM and NG with an initiating shock of 10 - 20 kbar with aquarium technique. The results in Table XI-4 show that the classical expectations
196
Chapter Table XI-4. Relationship between Detonation Velocity and pressure [XI-35]
Explosive NM p o = 1.13 g/cm3
( H W NG po= 1.60 g/cm3
DLVD. k d s l2.13 2.18 2.20 2.28 2.46 2.13 6.30 2.18 2.31 3.12
Shock wave parameter in water us [ k d s l [ M S l 2.35 0.43 2.22 0.36 2.15 0.33 2.28 0.39 2.30 0.40 2.30 0.40 5.50 2.00 2.33 0.42 2.24 0.37 2.16 0.33
PLVD
[GPaI
1.05 0.85 0.77 0.94 0.96 0.96 12.6 1.23 1.07 0.94
do not hold.
Frozen Shocks and Phase Transitions Other viewpoints and experimental results are less well understood: Bolchowitinov [XI-361 saw in the LVD of NG a first-order thermodynamic phase transition and interpreted it as a classical liquidsolid transition. Schaffs [XI-371 found, using dielectric breakthrough by a spark in inert liquids, high dynamic pressures and ’fiozen’shock waves, which he, too, attributed to a first-order phase transition. In spite of his and his coworkers’ “-381, [XI-391 many sophisticated experiments, this liquidcrystal-transition has never been observed. From the view of the microscopic detonation model, Bolchowitinov’s ideas and Schaaffs’ ftozen shock waves are borne out by a liquidfoam thermodynamic phase transition (Chapter X, Figure X-5). The authors actually found a bubbly medium with negative compressibility; therefore, they failed to detect any liquidsolid transition.
Stable and Unstable LVD One can observe (stable) luminous events on a continuous straight line in smear-camera records in HVD, or flash sequences along a straight line. Oscillating patterns can also be observed, but they are unstable. Figure XI-7 shows stable (a) and unstable (b) LVD events of NG/EGDN 50/50 ftom experiments of Watson, Summers, Gibson, and van Dolah [XI-401. More unstable oscillating events in a lead tube [XI-411 are shown in Figure XI-8.
197 &
Figure Smear-camera record of an LVD ofNGiEGDN 50/50 in a Plexiglas tube of 1- h i . d. a Wall thickness ?A in.: Stable LVD.
b Wall thickness 1/16 in.: Unstable LVD. {Watson, et al. [XI-40].}
Table XI-5. LVD experiments of NG/EGDN = 50/50 under different confinements. {Watson et al. [XI-40]).
Wall thickness [in.]
R,Ra
Character of LVD
Lead 1-in. I. D. c = 1,210
1/16 114
0.89 0.80 0.67
unstable unstable unstable
Plexiglas 1-in. I. D. c = 1,840
1/16 1/8 114
0.89 0.80 0.67
unstable stable stable
2,140 1,870
Steel 1-in. I. D. c = 5,200
1/16 118 114
0.89 0.80 0.67
stable stable stable
1,960 1,880 2,110
Aluminium c = 5,000
1/16 114
0.89 0.67
unstable stable
2,040
Confinement: material, 0 [in.], sound velocity
DLVD
Pressure [kbar]
W S I
118
5 - 20
198
Chapter
The properties of the detonating mixture NGEGDN 50150 with pm= 1.55 g/cm3 and co = 1,480 m / s varied considerably with the confining material and its wall thickness (see Table XI-5). The conclusion was that the sound velocity in the confining wall and the thickness are most important, also that the sound velocity in the confinement must be larger than the LVD velocity. This enables the vibrating walls to generate a bubbly liquid ahead of the detonation front, which can enter successively the bubbly liquid. This is the LVD-model o f the US Bureau of Mines (Fig. XI-9). This model has been examined carefully. The wave velocities have been measured for the liquid explosive as well as for the confmement [XI-41]. Schlieren photographs at rectangular prisms indicate that the confinement waves enter the liquid [XI-421, [XI-431. High-frequency cinematography as well as X-ray-flash photographs [XI-441, [XI-451 demonstrate the importance of bubbles. That bubbles really initiate is also shown in various contributions [XI-461, [XI-471. An observed stable LVD in polyethylene confinement seems to contradict the necessity of a cavitating confinement [XI-411. This discrepancy was explained by measuring a wave velocity of 1,900 m/s for polyethylene (in confinement); this is considerably larger than its sound velocity o f 800
199 &
in
Whenever this model applies we observe that whenever the LVD and sound velocities in confinement are different, then the distance between cavitation and the detonation fronts increases with increasing charge length. Therefore, we expect this event to alter the LVD velocity with the length of detonation and finally a failure. However, this has never been observed. On the contrary, bubble accumulation has been observed ahead the LVD front [XI-411. Paterson [XI-481reported experiments on extruded unconfined blasting gelatin of 38-mm 0 up to a length of 5.5 m. He observed LVD velocities - although with some oscillations - in a band from 2,310 to 2,390 m / s . It is most likely that a self-cavitating mechanism exists. The investigations of Dubovik, Goncharov, Denisaev, and Bobolev [XI-491 on LVD of nitroglycerine (NG) show that LVD is much more complex than classical theory indicates. The detonation front of NG in confinement appears to be convex from the fume side. At the wall side DLVDz 2,100 m / s and a pressure of 8 kbar was measured. Light flashes appear, then a dark period of about 25ps follows. In the center of the charge a less luminous reaction occurs, exhibiting DLVD= 1,600 m / s and a pressure of 12 kbar. (Note that p/Dz is not constant!) LVD phenomena, such as an LVD within a LVD in opposite direction can occur, occasionally. The viewpoint that tension pulses can appear due to an anomalous Hugoniot resolves all former difficulties to the complete understanding of LVD. Note that yet another LVD model must be mentioned [XI-421, [XI-501. The confinement wall vibrates and produces a Mach disc in the liquid explosive. Very shortly high pressures (for TNM/NM about 85 kbar) are produced that stimulate the onset of a chemical reaction, which slows to low order. Arguments against this assumption are: A Mach disc can only be produced when the sound velocity in the wall is much larger than that in the liquid. When the ratio of these sound velocities tends to unity, the possibility of Mach disc formation disappears. Furthermore, the existence of a Mach disc is bounded by a circular geometry. 1, 2-DP exhibits a sound velocity of c, = 960 m / s . If the Mach disc model should apply, then the sensitivity of 1,2-DP should increase with increasing sound velocity of the confinement. This applies, but LVD is possible even in lead tubes in noncylindrical geometry, as is seen by the results in Table XI-6 [X1-50], [XI-5 11.
LVD Sensitivity From investigations on 1,2-DP and IBA, Woolfolk [XI-5 I] (Table XI-6) found that the gap-test sensitivity increases as
200 Chapter
1. the sound velocity in confinement increases, 2 . when cylindrical containers are used instead of square ones, 3. when the ratio’s of wall thickness and charge diameter decrease (see Table XI-7), 4. when the length of the charge increases, and 5 . when an indicator plate is used in the experiments. This effect is more pronounced in LVD-sensitive substances (like 1,2-DP) than in less sensitive liquids (like EN and IBA). 6. The presence or absence of experimental tools, like the indicator plate or probes, are of utmost importance (as was outlined for NM). 7. The strength of the confinement is important. 8. Sensitivity is greatest at the limit of LVD stability. Table LVD-Sensitivities
Liquid
Tube Material
Length of Tube
Form of Tube
Sound velocity of tube
RJR,
50% Plexiglas Gap
Lcm]
Wallthickness [cml
i.0
[dsl
[cml
1,2-DP A1
cyl .
5,200
1.27
0.23
0.73
(+)
1,2-DP A1
square
5,200
1.27
0.318
0.67
(+) > 61 - < 91.5
1,2-DP Brass
cyl.
3,700
1.27
0.23
0.73
(+)>95 - < 122
1,2-DP Steel
cyl.
5,100
1.27
0.63
0.50
(+)>91 - < 126
1,2-DP Lead
cyl .
1,200
1.27
0.63
0.50
(+)
>214
95,O
1,2-DP Steel
10.2
cyl.
1.27
0.125
0.84
(-) 91 - 107 (unstable)
1,2-DP Steel
10.2
cyl.
1.27
0.254
0.71
(-) 66 - 122
1,2-DP Steel
10.2
cyl.
1.27
0.63
0.50
(-) 30 -40
1,2-DP Steel
cyl.
1.27
0.63
0.50
(+) 91 - < 126 (-) > 3 0 - < 4 0
EN
Steel
cyl.
1.27
0.23
0.73
(+) > 2.54 - < 5.4 (-)>3.1 - < 3 . 8
IBA
Steel
cyl.
1.27
0.63 0.318
0.50
(+) > 15.5 - < 20.4 (-) > 16 - < 17
IBA
Steel
0.318
0.56
(-) 18.5 -21
IBA
Steel
TBA
Steel
10.2 10.2
cyl .
10.2
cyl.
1.27
0.254
0.71
(-) 18.5 - 21
cyl .
1.27
0.254
0.71
(-) 12
1.2
no LVD
Lead
1,190
1.73
1.8
> 12
1.5
no LVD
Aluminium
5,000
1.73
1.2
> 12
0.1
3.0
Copper
3,810
1.87
1.o
> 12
< 0.1
4.0
Steel
5,200
1.90
0.4
> 12
< 0.1
6.0
Van Dolah, Watson, Gibson, Mason, Ribovich [XI-521 conducted gap tests with Plexiglas barriers on NG/EGDN 50/50, and CAVEA-B110 (nitric acidfuel mixture = Sprengel explosive) for both HVD and LVD. Results showed that the HVD sensitivity depends on the confinement (see Table XI-8). The p,/pi-ratio is the pressure ratio of the transmitted part to the confinement from the Plexiglas in accordance with the corresponding impedances. The sound velocities were calculated using the thin-wire approximation. Cavities or gas bubbles of various content have little influence on the initiating pressure.
Critical Diameters There is no general rule about the ranking of the critical diameters for LVD and HVD. Whereas Seely et al. [XI-53] find for several difluoroaminoalkanes a lower critical
202 Chapter
diameter for LVD, according to Kozak et al. [XI-281 the reverse applies for NM and TNT [XI-291, and the values in Table XI-9. The data in this table XI-9 also demonstrate the importance of the confinement. Table XI-9. Critical Diameters of HVD and LVD [XI-521
Substance
Confining Material
HVD [in.]
LVD [in.] 0.93 - 1.18 1.18 - 1.43
A = 0.035 in.
CAVEA B-1 10
Stainless steel Aluminium
0.12 - 0.24 0.18 - 0.31
NG/EGDN
Aluminium
0.12 - 0.24
The LVD Velocity/Diameter Relationship A velocityidiameter relationship usually exists for HVD of homogeneous explosives, SO that the detonation velocity increases similar to p (l/R) with increasing diameter. Such a relationship does not hold for LVD. Dremin [XI-541 has given the LVD/diameter relationship of NG with increasing diameter in Figure XI-10. Results show that the maximum velocity was obtained at a diameter of about 20 mm in cylindrical glass tubes of wall thickness of -1.5 mm. He explained that a maximum of eigenfiequencies of the empty tube exist at this diameter, which leads to maximum bubble concentration.
3
.
Figure LVD velocity of NG in glass 4
E
I
3
2
n
thicknesses of -1.5 mm. {Dremin "-541 with kind permission}.
I
3
1
i
.-1 z-b
i
203 &
LVD of Liquids under static Pressure Based on the calculations of losses of bubble motion, we have two options: open-, and closed-pore systems: Their results are different. The losses are shown for 1, 100, and 1,000 bar ambient pressure for air and hydrogen bubbles in Figure XI- 11. These losses shift to higher frequencies with increasing pressure.
I
I
1
I
I
1
10000 I
1
I
100
1000
1
1
1000 I
1
10000
I
1
I
1
100 I
I
1000
I
1
I
I
1
I
0.01 I
10 5 100
I
0,l
1 0,5
10 5
1
I
O.1
1 I
t
10 5
I
I
1 05
R ( pm)lbar
R (pml 100bar I R(pm1 0.1 1000 bar
Figure XI-11. Calculation of the losses of air and hydrogen bubbles in NG as function o f ambient pressure. At moderate ambient pressures, the frequencies of a possible activation o f bubble losses move into a high-frequency domain, which cannot be activated by the usual stimuli. At the bottom are shown the corresponding resonance values o f the bubbles as function o f pressure.
204 Chapter
Taking into account the steepness of a pressure stimulus of = 1ps, then a stimulating fkequency of about = 1/(2...3.t) corresponding to = 4 lo5 is active. The losses at ambient static pressures are favorable for initiation, but at 100 bar ambient static pressure no loss can be activated so that initiation fails. The experiments of Gurton [X-81 on NG confirm this (Table XI-lo), [see Figure XI-1 11. Table XI-10. of nitroglycerine (7/16-in. diam) as function of static pressure {Gurton
Fulminate Cap No.
Static pressure [bar1
Gas
Observed LVD [dsl
1
1 .O 11.3 30.3 71.0 1 .O 1.o 14.3 17.7 20.5 32.7 44.3 67.7
Air Methane Methane Methane Air Air Methane Methane Methane Methane Methane Methane
1,410; 1,160; 950 - 1 cap detonated - caps not detonated - caps not detonated 1,260 7,690 6,670 1,205 1,040 1,010 807 909
1 1
1 4 4 4 4 4 4 4 4
The pressure-dependent initiation can be important for safe handling at high pressures, but the risk of depressurization or pressure oscillations may prevent a practical use. This effect is important in accidents. This point of view will be again addressed with the LVD of solids and the case histories of double explosions in Chapter XIII.
Spin Detonation - Jumping Detonation As outlined in Chapter IV-F in generic experiments influences of the actual charge geometry and their confmement is observable. Zimmer [XI-55] observed a spinning detonation in the LVD of NG with the velocity 1,120 m / s in glass cylinders of 10 mm diameter, where the velocity line oscillated in the streak camera view, see Fig XI-12. Such a spinning mode was known to apply for gases, Cook [XI-561 found such behavior for a solid high explosive as well as for ANFO, where the two charges had been encased in cardboard tubes. In order to withstand the detonation pressure, Zimmer used polyethylene containers with wall thickness of between 5 and 10 times the charge diameter of 30 He bisected the container after the experiment and observed spiral grooved indents, see
205 & SVD
Figure XI- 13. The bottom of the container also showed spiral structure, see Figure XI-14. No cavitation was observed in his experiments.
206 Chapter
Figure XI-14: Imprints on the bottom of the polyerthylene container by LVD ofNG. {Courtesy M. Zimmer [XI-55]}
Cook described a behavior for solid explosives (tritonal) that he called jumping detonation: After an LVD run distance of about one charge diameter an HVD appeared ahead of the LVD, and propagated both forward and backwards onto the LVD fi-ont. It is an open question, whether the behavior shown in Figure XI-5 is a liquid analog. A similar situation is shown in Figures XII-4 and XII-5. Aging Properties Blasting gelatin (BG), a mixture of nitrocotton and NG is very insensitive in the homogeneous state. It is an old practice to knead or agitate (in German “walken”) the material before use, so that it gets a “good inhomogeneity”. With time this material homogenizes again and LVD decreases and HVD increases with time, see also [XI-56].
The purpose of experiments is to reach generalized conclusions. For the LVD of liquids it can be stated that: The detonation velocity of HVD is not necessarily constant. LVD must be considered to be a general event. The significance of test results is unclear. The detonation velocity of LVD shows considerable scatter. The properties of LVD depend on the system, and are not necessarily a material property. LVD capability and sensitivity depends strongly on ambient pressure.
207
There are forbidden velocity intervals between LVD and HVD, where no wave propagation has been observed. All results lead to a pessimistic view of any relevant current safety testing of energetic materials. We may state that sensitivity is not a materials property but a systems property.
EXPLOSION TESTSOF THE us BUREAU OF MINES When we compare classical sensitivity tests with the more scientific tests addressing LVD phenomena, there is no relation to actual accidents. From the (static) pressuresensitive LVD viewpoint we surmise that LVD may be at work in double explosions because the first response usually is a weak explosion pressurizing the system. After the pressure release the powerful and catastrophic explosion usually takes place. Nevertheless, the weak stimuli are open to question. - NM- and MMAN-explosions are beyond the scope of 'safety science'.
To help explain these events, Hay and Watson [XI-571 suggested a test (Fig. XI-15) where 1200 ml of the test liquid is in a steel container (10.2 mm i.d., 15 cm length) closed by a polyethylene foil. A bubble screen is developed with pressurized air (0.55 to 1.4 bar). This air is blown in through a PVC-tube with 0.23-cm diameter holes. A projectile of mass 9.4 kg (left) is shot into the container, and the projectile velocity is measured. There is no distinction between explosion or detonation. The averages of the maximum velocity without explosion and the lower velocity with explosions are given in Table XI-1 1. This test shows the sensitivity of substances that were involved in accidents. The disadvantage is that large amounts of liquid are needed, and no quantitative measurements are possible. The authors would prefer an instrumented gap test.
Instrumented Gap Test Ribovich, Watson and Gibson [XI-58], [XI-59] suggested in 1968 an instrumented cardgap test (Figure XI-16). They state that the order of sensitivity corresponds to the values in Table XI- 11, but no actual values are available.
208 Chapter
2.5crn
r15cm--fl_r
Baker _
_
Figure US Bureau of Mines reactivity' test.
~
Air
n
Witness plate Cork standoff Pressure gage Rate probe Acceptor (test) sample Acceptor container
Figure Instrumented gap test of the US Bureau of Mines [XI-%].
Plastic gap Tetryl donor Oscilloscope Trigger Cork base and detonator holder No 8 detonator
The test arrangement has been modified [XI-59] since 1968: A seamless steel tube with a wall thickness A of 0.56 cm, an 0. D. of 4.76 cm, and a length of 40.64 cm is now in use. (Original [XI-58]: A = 0.34 cm, 0. D. = 3.34 cm.) Booster: 5.08-cm diameter, length 5.08 cm, cast pentolite PETN/TNT) (formerly 4.13-cm diameter, length 2.54 of pressed tetryl of density 1.57 g/cm3.) Attenuator of 5.08-cm diameter of cast acryl (formerly 4.13-cm diameter cellulose acetate foils 0.0254-cm thick.)
209 Realistic Explosion Tests
The bubble screen is generated by a perforated plastic tube with 50-kPa pressurized air. The advantage of this arrangement is that only 400 ml liquid is required, and the test results can be monitored. Table XI-11. Sensitivity of some liquids in the US Bureau of Mines Test
Substance
Temperature ["C]
Velocity of the projectile
n-PN
20
91.3f 1.3
AmmonnitrateiWater84/16 AmmonnitrateiWater9515 AmmonnitrateJWater 9713
85 138 150
90 69.6 12.4 48.1 8.0
liquid Ammoniumnitrate
200
38.8f 5.8
MMAN-Solution, 69% MMAN-Solution, 88%
74 74
58.9f 5.8 24.3f 5.6
40 and 20
26.2k 2.7
NOS-365 NM
20
24.1
Otto-I1
20
23.4f 3.2
liquid TNT
85
9.7+ 1.5
2.3
PHYSICAL EXPLOSION The basic pressure generating mechanisms were addressed in Chapter I11 where no chemical decompositions were required. In Chapter IX these mechanisms were described in terms of a quasi-continuum. Chapter X the legal reasons were outlined, and it was explained why such Physical Explosions usually are denied. It was, therefore, most fortunate to come across a rather unique result in hydraulics that was first published in the hydraulics literature in 1983. The researcher's unique findings are presented here. That these are of real concern will be seen in Chapter XIII, where case histories demonstrate real-life occurrences of such phenomena.
210
Cavitation in Large Hydraulic Structures By Eric JLesleighter, Hydraulic Engineering Specialist, Cooma NSW, Australia Cavitation in hydraulic structures is well known. Perhaps the occurrence is, but how well its character? The extent of damage and the associated disruption to operation of such facilities, and the costs of repairs and loss due to outage, are facts of life in numerous high-head water engineering projects. The violence of cavitation with extremely high implosive pressures spells damage to even the hardest materials. When we explore beneficial effects of explosion phenomena, it is difficult to ascribe much of a beneficial nature to cavitation. Nonetheless, knowing more of cavitation might enable better appreciation of shock phenomena. Given the extent of studies which have been made and the number of projects on which major cavitation damage for known reasons has occurred, one could wonder why engineers continue to be “successful” in designing and building structures which have major cavitation problems. Lesleighter [XI-601 in 1983 has described experiences on a high head outlet works for a 180 m high dam which presented a rare opportunity to compare results fiom major instrumentation programs in both physical modeling and investigations and in the field installation. The work broke new ground in discovering scaling cavitation phenomena, but perhaps more importantly, the work documented cavitation implosion damage to stainless steel, consisting of macro-sized denting that had not been previously known.
0
1
1
3
4
YRm
Figure XI-17. Details of the Gate Conduits of the Dartmouth Dam LLOW project in Australia.
21 1
There have been significant developments in aeration of outlet works and spillways to avoid cavitation damage; damage which had been experienced and well-reported on such projects as the Karun Dam spillway in Iran and the Tarbela Dam in Pakistan, and Glen Canyon Dam in the United States, see Chapter XIII.
Dartmouth Dam Project - Modeling and Field Measurements The releases from the Low-Level Outlet Works (LLOW) in the 180 m high Dartmouth Dam project in Australia were planned to range up to 170 m3/s under heads which could be well over 100 m (reservoir full supply level EL 486, gates at EL 3 15, but with back pressure due to submergence of the tailwater tunnel). Figure XI-17 illustrates the gate conduits for the LLOW comprising two conduits, each with a vertical regulating gate, and each discharging into an underground cylindrical expansion chamber. Hydraulic model studies revealed interesting characteristics of cavitation phenomena in the LLOW. The severity of the shock pressures in the prototype, however, was subsequently found to be many times greater than could have been foreseen from those studies. It was in recognition of the general lack of objective information that an extensive instrumentation program was adopted for the LLOW prototype as described by Lesleighter [XI-60]. Six pressure transducers (five quartz and one resistive) were installed flush with stainless steel conduit lining, which had been installed downstream of the vertical slide gates in the outlet works, in positions that had been indicated in the model study to be the area of severest implosion. Signals hom the transducers were captured on a two channel digital oscilloscope capable of a sampling speed of up to 2 x lo6 per second and providing 4 096 samples per capture. The first series of tests revealed huge 'bangs' at irregular and regular intervals which reverberated through the underground valve chamber. Subsequent inspection revealed dents in the stainless steel liner, beyond 1.25 m downstream from the gate slots. The damage consisted not of the microscale, homogeneous pitting normally associated with cavitation damage, but rather large dents (to at least 16 mm long and a fraction of a mm deep), irregular in size and spacing, and evidently associated with irregularly structured vapor cavities, collapsing to produce very high pressures. Figure XI-18 presents photographs of dents which display various characteristics fiom elongated "creases" to rounded shapes, sometimes on the same photograph. Apart fiom the grinding markings, these dents were not present before the testing. One of the photographs shows a section of the stainless steel wall about 350 mm x 250 mm in which numerous dents over the whole area can be seen. On 1 , 2 November 1979 an additional test series followed installation of a 700 MPa range quartz crystal pressure transducer at the downstream end of the conduit (in the
212
Figure XI-18. Macroscopic cavity dents. Left: Numerous dents in a 350 * 250 mm2area of stainless steel. Center and right: Macroscopic dents similar to those usually observed in the microscale, but in dimensions of cm!
area of significant denting). This transducer location TQ22 is shown in Figure XI-17. Figure XI- 19 presents captures at different time bases of pressure transients showing pulses at up to 1,500 MPa. While extending well beyond the manufacturer's stated range of the transducer, the character of the pressures (similar to those experienced at lower magnitude), the fknctional state of the transducer following the tests, and the damage to the stainless steel being consistent with very high pressures, showed that the measurements were genuine. They seem to represent the highest pressures measured in a major hydraulic engineering installation: a "first', on two counts, firstly, the measurement of large magnitudes of cavitation pressure under such conditions, and secondly the irregular occurrence both in time and space of the cavity-implosion damage, of a character quite unlike that normally associated with cavitation.
0
L
0
-
/
I
L-
a
10
Figure XI-19. Measured pressure pulses in the arrangement of Figure XI-17. For details, see the text.
The Dartmouth experiences bring field data from a major engineering facility. In Figure XI-1 9, right, the pulse duration, as defined, in the enlargement is approximately 22 p. The time rate of increase of pressure was approximately 67.0 x lo6 MPds. In
213
fact on the five largest pulses shown on Figure XI-19 the rate of increase (using the raw oscilloscope data) was, from left to right, 72, 59, 68, 68 and 67 (x lo6MPds). Tests made with the injection of up to 1.41 m3/s (atmospheric pressure volume) of air markedly reduced cavitation intensity; this was significantly more air flow than had been predicted as necessary to 'control' cavitation. The model studies had indicated pronounced cavitation would occur in the prototype and this, by the denting and pressures described and the loud, muffled bangs associated with cavity implosion, was borne out. Both the area of worst cavitation and the quantity of air admission to contain the cavitation were not in agreement with model predictions. It would appear, therefore that while the model may have depicted the growth and travel of the large cavities, it dld not simulate the collapse mechanism as far as pressure pulses are concerned in a reliable way. Scale effects and conflicts between model and prototype can normally be kept in check by judicious decisions on the part of the hydraulic model testing engineer. The experiences described above, however, show conflicts with model predictions, part of which we could be aware because we are dealing with a partly non-scaleable phenomenon. The experiences advance the hydraulic engineer's understanding of the cavitation in high-head installations. In the Dartmouth LLOW, we are dealing with a magnitude of cavities, an irregularity of collapse, and a damage characteristic which are all quite different fkom "classical" cavitation; in that respect the experiences are a worthwhile enhancement of our understanding of cavitation. In the context of model/prototype conformity Fujikawa [XI-611 for example, states that the enlargement of a cavity is an isothermal process while its subsequent collapse is adiabatic. Is that the same in the model and in the prototype? Furthermore, is it the same for conditions of limited or incipient cavitation, boundary layer cavitation, and sheared-flow cavitation? It could be expected that it is not, as there are different fluid properties coming into play. While some researchers have suggested that scaling of cavitation might be based upon Weber number similarity, in the LLOW cavitation, we have violent separation from an 84 mm wide gate lip and growth of the bubbles formed there into enormous, irregular cavities in the shear layer; a condition which is unlikely to be influenced significantly by surface tension (Weber number). Much research and numerous theories have been propounded in an effort to understand cavitation phenomena. In the consideration of actual damage fiom cavitation, debate continues regarding the relative roles of the liquid microjet through a collapsing bubble or the shock wave at the instant of rebound of the collapsing bubble as the cause of the damage to the surface. Mitchell and Hammitt [XI-631 mention the shock wave theory as an early explanation of cavitation damage, with counter theories of the rebound from the surface of a collapsed bubble (with corresponding high internal pressure), and the
214
Chapter
microjet as the liquid on the outside of the bubble moves inward at a much higher velocity than the inner surface. They, in presenting analyses of initially spherical bubbles and their development of asymmetry, go on to cite estimates of microjet velocities of up to 500 m / s . They believe the effects are not mutually exclusive. Somewhat in contrast, Ling [XI-611 states that large vapor bubbles, which are "dynamically slower" and nonspherical cannot attain a high concentration of energy on collapse due to premature collision of the bubble wall. He says this accords with a finding of Fujikawa that jet-like collapse of large bubbles does not produce excessive pressure. This brief reference to the literature above indicates that, while major cavitation damage resulting in removal of massive amounts of concrete and even steel lining in hydraulic engineering structures has been reported and studied, much of the practitioner's understanding of cavitation phenomena is based on copious laboratory research on "microscale" bubble formation and damage. It is Lesleighter's view that a large amount of this research is of limited value to the hydraulic design engineer. In contrast the extensive model and prototype study program for the Dartmouth LLOW has shown us cavitation of a different and potentially damaging kind. It has presented objective measurements and cavitation damage in the form of large dents which hitherto, to the author's knowledge, has not been obtained or experienced.
The sheared-flow cavitation, in which cavities swirled and enlarged in a highly turbulent shear layer, displayed anything but spherical cavities. The collapse and damage mechanism in the form of dents of irregular shape (Figure XI-I 8) is a departure from microscale pitting, see Chapter XVIII. The forethought to specify stainless steel lining, and an air admission facility, paid enormous dividends; allowing through additional model work the implementation of relatively convenient modifications to achieve natural air induction and a satisfactory prototype installation.
Conclusions The work described herein represents a significant advance in our understanding of major cavitation in engineering structures. It has dispelled some of our earlier ideas. In this respect, remarks made by Dr. H. T. Falvey of the United States Bureau of Reclamation (personal communication 1986) are pertinent: "Your field investigations continue to be one of the few proofs of the damaging effects of swarms of collapsing cavitation bubbles, ... it makes us think about cavitation phenomena that we never imagined could exist." Measurements of cavitation implosion pressure pulses to 1 500 MPa have been achieved; a finding which has awaited the availability and use of advanced
215 Explosions
instrumentation. Dents in stainless steel, consistent with the pressures measured, have been documented which also throws new light on the ability of large scale cavitation to inflict damage, and cavitation collapse mechanisms quite different fi-om those normally conceived to be associated with cavitation damage.
References [XI-11 A. Stettbacher, Sprengwirkungund chemische Konstitution, Z. ges. SchieR- und Sprengstofies. 13 (1918), p. 2471248. [XI-21 S. J. Jacobs, Non-steady Detonation - A Review of Past Work, 31d Symp. on Detonation, 1960, Office ofNaval Research, ACR-52, p. 784/812. [XI-31 J. A. Brown, and M. Collins: Explosion Phenomena Intermediate Between Deflagration and Detonation, Esso Res. Rept, 1967, DA-49-092-ARO-140. [XI-41 A. V. Dubovik, and V. K. Bobolev: (Cuvstvitelhost' zidkich vzryvcatych sistem k udaru), Schlagempfindlichkeitflussiger explosionsfahigerSysteme Nauka, Moscow, 1978. Also German Translation: Ed. C. 0. Leiber, Schlagempfindlichkeitflussiger explosionsfahiger Systeme, Forschungsbericht aus der Wehrtechnik BMVg-FBWT 96-3 (1996). [XI-51 A. F. Belyaev, V. K. Bobolev; A. 1. Korotkov; A. A. Sulimov and S. V. Chuiko: Transition from Deflagration to Detonation in Condensed Phases, Moscow, 1973; English Translation: Israel Program Scientific Translations, Jerusalem, 1975. [XI-61 J. W. Kury, and H. 0. Hornig: Low Velocity Detonation - High Energy Explosives, Symp. H. D. P., Comportement des milieux denses sous hautes pressions dynamiques, Behaviour of Dense Media Under High Dynamic Pressures, CEA, Saclay 1978, p. 3091320. [XI-71 Review is best given in the 2nd to 6th Detonation Symposia, see W. E. Deal, J. B. Ramsay, A. M. Roach; B. E. Takala: Indexes of the Proceedings for the Ten Symposia on Detonation 1951-93, LA-UR-97-1899 (1997). [XI41 Recommendations on the Transport of Dangerous Goods-Tests and Criteria United Nations, New-York, STISGIAC. 10/11/Rev. 1, 1990, 2ndedition. [XI-91 C. 0. Leiber, HE-2000, Wunsch und Wirklichkeit, 11. Sprengstoffgesprach, 1987, Karl Diehl, Mariahutte. In japanese: K6gy6 Kayaku 49 ( 1988)5,p. 3001304. [XI-101 M. Steidinger, M.: Die Gefahrklassifizierungvon Alkali- und Erdalkalichloraten,Amtsund Mitteilungsblatt BAM 17 (1987)3,p. 4931504. [XI-111 J. Roth, Experiments on the Transition from Deflagration to Detonation, Proc. Conf. Chemistry and Physics of Detonation, 1951, p. 57/67, Office of Naval Res., NSWC MP 87-194. [XI-121 G. B. Kistiakowski, Initiation of Detonation of Explosives, 3rd Symp. Combustion and Flame and Explosion Phenomena, 1948, Williams & Wilkins, Baltimore, 1949, p. 560/565. [XI-131 C. 0. Leiber, Detonation Model with Spherical Sources G Dynamic Void Mobilities, HVD Initiation of Liquid Explosives, Proc. 17* Int. Pyrotechnics Seminar combined with Znd Beijing Int. Symp. Pyrotechnics and Explosives, 1991, Beijing, Vol. 11, p. 722/732, Beijing Institute of Technology Press.
216
Chapter
[XI-14] A. A. Sulimov, B. S. Ermolaev; A. A. Borisov; A. I. Korotkov; B. A. Khasainov; V. E. Khrapovski: On the Mechanism of Deflagration to Detonation Transition in Gas-Permeable High Explosive, 6'h Symp. (Int.) on Detonation, 1976, Office Nav. Res., ACR-221, p. 2501257. [XI-15] A. G. Butcher, R. L. Keefe; N. J. Robinson; M. W. Beckstead: Effects of Igniter and Compaction on DDT Run Up in Plastic Pipes, 6' Symp. (Int.) on Detonation, 1981, Naval Surface Weapon Center, NSWC MP 82-334, p. 1431150. [XI-16] A. A. Sulimov, B. S. Ermolaev; V. E. Khrapovski: Mechanism ofDeflagration-toDetonation Transition in High-Porosity Explosives, Progress in Astronautics and Aeronautics 114 (1988), p. 3221330. [XI-171 S. P. Marsh, LASL Shock Hugoniot Data, University of California Press, Berkeley, 1980. [XI-I 81 Clapham, referenced in J. Taylor: Detonation in Condensed Explosives, Clarendon Press, Oxford, 1952, [XI-I 91 Ph. Naoum, Ad. Berthmann: Die Bestimmung der maximalen Detonationsgeschwindigkeit von Nitroglycerin und Nitroglykol, Z. ges. SchieO- und Sprengstofies. 26 (1 93 1) p. 1881190. [XI-201 E. Haeuseler, Uber die Detonation flussiger Sprengstoffe, insbesondere auf der Basis von Wasserstoffsuperoxyd, Explosivstoffe (1953) p. 64 - 68. [XI-211 H. Badners, C. 0. Leiber: Method for the Determination of the Critical Diameter of High Velocity Detonation by Conical Geometry, Propellants, Explosives, Pyrotechnics 17 ( 1 992), p. 7718 1. [XI-221 R. F. Chaiken, On the Mechanism of Low-Velocity Detonation in Liquid Explosives, Astronautica Acta 17 (1 972) p. 5751587. [XI-231 R. Schall, Die Stabilitat langsamer Detonationen, Z. tech. Physik 6 (1954) p. 4701475. [XI-241 M. Cowperthwaite, Hydrothermodynamics of the Cavitation Model for Low Velocity Detonation, Arch. Proc. Spalania (1974) p. 65/74. [XI-25) W. C. Davis, Notes and Comments on the Round Table Conference on Detonations, Colorado Springs, June 20 - 2 I , 1972, Los Alamos report LA-5368-C (1973). [XI-261 E. W. Lindeijer, Th. M. Groothuizen: Note on the Determination of the Sensitivity to Detonation of Liquids, OECD-Group of Experts on Fire and Explosion Hazards of unstable Substances, Techn. Note TNO-RVO-Ass. 8357 (24. 9. 1970). [XI-271 Th. M. Groothuizen, Bestimmung der industrieller Produkte, Amts- und Mitteilungsblatt der BAM 1 (1 970171)7, p. 1001103. [XI-281 G. D. Kozak, V. V. Kondratjev, B. N. Kondrikov, A. V. Starshinov: Detonation des Nitromethans mit niedriger Geschwindigkeit, Akad Nauk SSSR, Otd. Inst. chim. Fiz. Detonacija, Chernogolovska, 1978, p. 34137. [XI-291 F. C. Gibson, R. W. Watson, J. Ribovich, J. E. Hay: Propellant Ingredient Safety, Quarterly Report October I , 1967 to December 3 1, 1967, US Department of the Interior, Bureau of Mines, Pittsburgh, PA. 1968. [XI-301 C. Brochet, Monopropellant Detonation: Isopropyl Nitrate, Astronautica Acta p. 4191425. [XI-3I ] A. A. Schilperoord, Letter communication from 03.12.1980.
(1970),
217
[XI-321 R. A. Joyner, Work described in J. Taylor: Detonation in Condensed Explosives, Clarendon Press, Oxford, 1952. [XI-331 G. S. Yakovleva, R. Kh. Kurbangalina: Shock wave initiation of liquid hydrazoic acid, in. A. N. Dremin: 6thDetonatsiya Mater. Vses. Simp. Goreniyu Vzryvu, 1980, p. 56/60, CA95(12): 100042~. [XI-34] M. Cowperthwaite, and D. C. Erlich, Investigation of Low-VelocityDetonation Phenomena in Liquid Monopropellantsand Explosives, SRI-Rept. Project PW-23 83 (1974). [XI-35] G. D. Kozak, B. N. Kondrikov, and A. V. Starshinov, Low Velocity Detonation of Liquid Nitrocompounds, 28" International Annual Conference of ICT, June 24 - June 27, 1997, P-61-1. [XI-361 L. G. Bolchowitinov, ijber die Detonation fliissiger Sprengstoffe mit niedriger Geschwindigkeit, Dokl. Akad. Nauk SSSR 130 (1960)s [XI-371 W. Schaaffs, h e r Versteifung und Erstarmng von Stosswellen, Acustica 13 (1963) p. 349/354. [XI-381 W. Schaaffs, and P. Krehl, Rontgenogaphische Untersuchungen iiber das Verhalten kondensierter Materie bei sehr hohen Verdichtungen und Drucken, Z. Naturforsch. 27a (1972) p. 8031808. [XI-391 R. Germer, Untersuchungen zum Problem der Struktur der Kompressionsringe, die beim dielektrischen Funkendurchschlagin Flussigkeiten entstehen, Dissertation TU Berlin 1974. [XI-401 R. W. Watson, C. R. Summers, F. C. Gibson, R. W. van Dolah: Detonations in Liquid Explosives - The Low Velocity Regime, 4'h Symp. ( I t . ) Detonation, 1965, p. 117/125, ACR-126, Washington. [XI-411 R. W. Watson, J. Ribovich, J. E. Hay, R. W. van Dolah: The Stability of Low Velocity Detonation Waves,, 5' Symp. (Int.) Detonation, 1970, p. 81/85, ACR-184, Arlington, VA. [XI-421 A. B. Amster, D. M. McEachern,jr., Z. Pressman: Detonation of NitromethaneTetranitromethaneMixtures: Low and High Velocity Waves, 4' Symp. (Int.) Detonation, 1965, p. 126/134, ACR-126, Washington. [XI-431 F. C. Gibson, C. R. Summers, C. M. Mason, R. W. van Dolah Initiation and Growth of Detonation in Liquid Explosives, 3rdONR-Symp. Detonation, Vol. 2, 1960, p. 4361454, ACR-52. [XI-441 R. W. Watson, The Structure of Low Velocity Detonation Waves, 12'h Symp. (Int.) on Combustion, Pittsburgh, PENN, 1969, p. 7231729, Combustion Institute Pittsburgh. [XI-45] J. E. Hay, R. W. Watson: Mechanisms Relevant to the Initiation of Low Velocity Detonations, Ann. New York Acad. Sci. 152 (1968), p. 621/635. [XI-461 F. P. Bowden, M. P. McOnie: Formation of Cavities and Microjets in Liquids and their Role in Initiation and Growth of Explosion, Proc. Roy. SOC.A 298 (1967), p. 38/50. [XI-471 C. H. Winning, Initiation Characteristicsof Mildly Confined, Bubble-Free Nitroglycerine, 3rdONR-Symp. Detonation, Vol. 2, 1960, p. 4551468, ACR-52. [XI-481 S. Paterson, Referenced in J. Taylor: Detonation in Condensed Explosives, Clarendon Press, Oxford, 1952, p. 161/162.
218 Chapter
[XI491 A. V. Dubovik, A. A. Goncharov, A. A. Denisaev, V. K. Bobolev: Structure of a LOW Velocity Detonation Front in Nitroglycerin, Dokl. Akad. Nauk SSSR, Phys. Chem. 221 (1 975)3, p. 290/293. [XI-SO] R. W. Woolfolk, A. D. Amster: Low Velocity Detonations: Some Experimental Studies and their Interpretation, 121hSymp. (lnt.) on Combustion, Pittsburgh, Penn., 1969, p. 73 1/729, Combustion Institute Pittsburgh. [XI-51] R. W. Woolfolk, Confinement Effects on the Initiation of Low Velocity Detonation in Liquid Explosives. SRI-Rept. [XI-52] R. W. van Dolah, R. W. Watson, F. C. Gibson, C. M. Mason, J. Ribovich: Low Velocity Detonation in Liquid Explosives, Proc. Int. Conf. Sensitivity and Hazards of Explosives, London, 1963, Session 1. [XI-53] L. B. Seely, J. G. Berke, R. Shaw, D. Tegg, and M. Evans, Failure Diameter, Sensitivity and Wave Structure in some bis-Difluoroamino Alkanes, 5" Symp. (Int.) Detonation, 1970, p. 89/98, ACR-184, Arlington, VA. [XI-541 A. N. Dremin, Discussion Note to Ref. [XI-40] Watson, R. W.; J. Ribovich, J. E. Hay, R. W. van Dolah: The Stability of Low Velocity Detonation Waves, 51h Symp. (ht.) Detonation, 1970, p. 81/85, ACR-184, Arlington, VA. [XI-551 M. F. Zimmer, Spin Detonation in Nitroglycerin, Combustion and Flame 12 (1968), p. 114. [XI-56] M. A. Cook, The Science of High Explosives, Facsimile ed. of 1958, Robert E. Krieger Pub., Huntington, 1971. [X1-57] J. E. Hay, R. W. Watson: Initiation of Detonation in Insensitive liquid Explosives by Low Amplitude Compression Waves, 6" Symp. (Int.) on Detonation, Preprints Vol. 11, Suppl. Papers 1976, 61hSymp. (Int.) on Detonation, 1976, p. 115/123. [XI-581 J. Ribovich, R. W. Watson, F. C. Gibson: Instrumented Card-Gap-Test, AIAA Journal 6 (1968), p. 126011263, [XI-591 R. W. Watson, Letter communication from 02.04.1979. [XI-601 E. J. Lesleighter, Cavitation in High Head Gated Outlets - Prototype Measurements and Model Simulation', Proc. 20" Congress Moscow, September 1983. [XI-611 S. Fujikawa, S. C. Ling, Discussion of 'Role of Microair Bubbles on Cavitation' by S. C. Ling, 10" IAHR Symposium on Hydraulic Machinery, Equipment and Cavitation, Tokyo, Vol 1 and 2, 1980. [XI-621 B. R. Parkin, J. W. Holl, 'Incipient Cavitation Scaling Experiments for Hemispherical and 1.5 Caliber Ogive-Nosed Bodies', Report No 7958-264, Ordnance Research Laboratory, Pennsylvania State Univ., May, 1954. [XI-631 T. M. Mitchell, F. G. Hammitt, Asymmetric Cavitation Bubble Collapse', ASME Journal of Fluids Engmeering, March 1973.
LOW VELOCITY DETONATION OF SOLID EXPLOSIVES LVD of Porous Crystalline Explosives First Apin and Bobolev [XII-11 and shortly thereafter Jones and Mitchell [XII-2] reported on LVD phenomena in porous crystalline explosives. NG gun powder (density 1.08 g/cm3),pentrite, picric acid, and RDX of different grain sizes (densities < 1 g/cm3)exhibited detonation velocities between 3,000 and 1,300 m/s in glass tubes (1 to 3 mm I. D.). Granulated TNT and tetryl (densities 1 g/cm3and initiated with a No 6 blasting cap) showed LVD velocities to about 1,100 m / s compared to a possible HVD of > 5,000 m/s. Similar behavior was observed for PETN and RDX, where the velocities were very stable in specified confinement.
-
Some authors compare the detonation velocity to the sound velocity c of the dense material. They see differences when the detonation velocity is lower or higher than c But such differences are not meaningful (as discussed earlier, see Figure IX-4). Also, such supersonic LVDs are present for solid explosives. Vashchenko, Matyushin, Parfenov, Lebedev, and Apin [X11-3] obtained a detonation velocity of 2,300 for tetryl (density 0.95 g/cm3)where only about 30% of the material reacted in the detonation zone. Correspondingly, the heat of detonation was lower and reaction products differed from those in HVD. Typically, 20 - 30% of energy was liberated in LVD compared to that of HVD. In 1958 Cook [XI141 found that, contrary to liquid explosives, no evidence of stable LVD in solid explosives other than for dead-pressed charges in primary explosives. If this finding is real, then no practical use of LVD phenomena of solid explosives is possible. This phenomenon will be discussed in the following sections.
Stability of LVD and Critical Diameters of Porous Systems TNT, RDX, tetryl, and PETN (densities between 0.9 and 1 g/cm3)usually show velocities up to 2,300 m/s above the critical diameter of LVD, with detonation pressures between 8 and 15 kbar. At a certain diameter, which depends on grain size and confinement, a discontinuous or a smooth transition to HVD occurs at detonation velocities > 4,200 m/s and pressures > 65 kbar. No steady transition velocities are found between the lower LVD and the upper HVD values, thus creating a velocity gap. The diameter of transition is proportional to the average grain size for monodisperse
219
220 Chapter
materials. This diameter is about 15 to 25 grain sizes for tetryl and 4 to 8 grain sizes for PETN, but the transition velocity ranges from 2,000 to 2,300 and is independent of the grain sizes [XI-5], [XII-51. This LVD s HVD-transition diameter can correspond to, or be larger than, the critical diameter of HVD. But the critical diameter of LVD is not always lower than the HVD critical diameter. For example, it is greater for RDX. The detonation velocity does not depend on the grain size at the LVD critical diameter. In these experiments the LVD velocity was not dependent on the initiating stimulus when no HVD took place. Steady propagation velocities were observed up to a length of 50 cm (25 diameters) [XI-51, [XII-5]. With PETN, the detonation velocity of LVD decreases when the PETN density is increased from 53.6% (0.95 g/cm3) to 95.6% of TMD (1.70 g/cm3). Up to a density of 82% of TMD, detonation velocity decreases with l / p and then remains constant (see Fig. XII-1). No LVD was observed with PETN at densities > 95.8% of TMD and grain sizes of 1 to 1.25 mm in Plexiglas tubes of 5-mm I. D. [XI-51. D kmis)
08
Figure XII-1. Velocity (W = D) of LVD versus the density of PETN, grain size 1 .O to 1.25 mm, charge diameter 5 mm, Plexiglas confinement. {Belyaev, Bobolev; Korotkov; Sulimov and Chuiko [XI-5]}. I2
'
Po (g/cm')
LVD of Pressurized Porous Explosives This matter has been repeatedly addressed in Chapters X and XI. We concluded that LVD initiation of solid powdered explosives depends on a moderate static pressure increase. Based on Bowden's 'adiabatic' hot-spot idea, Gurton [X-81 investigated in 1955 the detonation behavior of porous nitroguanidine (NQ) ( p = 0.5 g/cm3), tetryl (p = 0.9 g/cm3), and TNT (p = 0.8 g/cm3) under different gases (air, ether and pentane) at atmospheric pressure. He obtained only scattered values of LVD velocities, but observed no influence of the gas at 1 bar on the detonation velocity Gas pressurizing from 1 bar up to 41 bar has only been done with methane. It appeared that the detonation velocities decreased with increasing pressure, and detonation failed above critical static methane gas pressure. These limiting pressures depended on the charge diameter, and were for tetryl between < 27 and < 48 bar, for TNT < 41 bar, and NQ < 14 - 18 bar. The initiation of HVD was not affected by methane up to pressures of 68 bar. But run distance to HVD was influenced by the gas pressure.
22 1 LVD
The pressure dependence of LVD initiation is different for open- and closed-pore systems, as discussed by Leiber [XII-61 in 1986. As a result, a pressurized system is less sensitive to LVD, provided that no pressure release occurs. This fact may explain the occurrence of little understood double explosions. Case histories of double explosions are summarized in Chapter XIIT. It appears that the most serious .. u i i ~ x p c c r ~acxiuciitdi u expiuaiuris wmtt ciuubic cxpiusioiis, w1it;i-eilie iiiitid eveiit W - ~ S usually weak; the second or 3'd report sometimes many seconds later was disastrous. - 'J . . . L . ,
LVD of High-Density Solid Explosives First results on high explosives at high densities were obtained by Gipson and Macek [XII-71 for transition from fast convective burning to a 'full detonation'(DDT). But the Russian investigators Babaitsev, Kondrikov, Paukova, and Tyshevich [XII-8], [XII-91 detected in 1966 that LVD in high-density solid explosives is a separate unique phenomenon. They showed in about 200 experiments that RDX (7 pm)/NaCl (0.2-0.3 mm) 50/50 or 40160) and cast TNT charges (0.95-0.97 relative density), confined in thick-walled steel tubes (9 to 15 mm I. D., 35-36 mm O.D., 15-60 cm long) and initiated by a No. 8 blasting cap directly or via a booster charge, did undergo steady state LVD. The tubes with a small inside diameter were slightly distended, and with a large one sometimes were broken into two pieces along the axis of the tube. The material was entirely consumed, except for some small particles on the inside of the tube. Detonation velocities were observed with a streak camera in a confinement similar to that shown in Figure XII-11. Visibility was greatly increased by gluing tiny crystals of lead azide (1.5-2 mm 0)to the end of the orifices. The LVD velocity remained constant along the charge, and varied within 1.5 to 2.2 km/s depending on the diameter of the channel, space between the orifices, and density of the charge. When the distance between the orifices was increased from 10 to 30 cm for 9-mm and 15-mm channel diameters, the detonation velocities increased from 1.75 to 2.2 km/s and 1.65 to 1.8 kmis, respectively. Neither a velocity increase nor transition to HVD was observed up to a length of 65 charge diameters! This regular effect appeared even though reaction in the LVD wave was incomplete. The most impressive manifestation of this effect was demonstrated in an experiment with a tube containing a long narrow gap (cut with a milling cutter along the generatrix of the tube). The TNT was blown out through the gap, and the LVD wave was not extinguished but proceeded to the very end of the charge. Tyshevich [XII-8a] determined with an aquarium technique the structure of the LVD wave of high-density solid explosives, see Figure XII-2. Two consecutive waves are observed, the first (plastic) wave at about 3-5 kbar initiates the second one, a reaction wave that proceeds at pressures up to 20 kbar.
222
Figure XII-2. Background of the bright screen near the interface at the aquarium experiment. Initiation top right. Top: HVD of cast TNTRDX SO/SO. 1 : Detonation wave; 2: Shock wave in water; 3: Water interface Bottom: LVD of cast TNTiRDX SO/SO. 1: Sound wave in steel (not visible). 2: Sound wave in water. 3: First wave in the charge, where the light points result from the azide explosions at the orifices and are therefore delayed to the originating wave and the corresponding wave (4). -- 4: Second wave in water induced by the first wave, and 6: Moving of the interface is started. A series of waves impacts the interface. One of them, before a dark triangle under the surface (S), appears to be connected to the reactive wave. Since its slope to the horizontal line is larger, velocity and pressure are also increased. { Tyshevich [XII-8a], kind permission Boris Kondrikov}
When cast charges of TNT/RDX or TNTJHMX 50150 were assayed at similar operating conditions, LVD velocity was found to be slightly higher (0.2 - 0.25 kmis) than that of a cast TNT charge. A small increase in detonation rate (0.2-0.3 k d s ) was observed at the end of the 250 to 300-mm-long tube. Acceleration of the LVD wave was more pronounced with cast DINA; its velocity increased from 2.5 to 5 k d s at a distance of about 20 charge diameters, then a LVD 3 HVD transition took place. LVD was also studied in pressed charges, tightly placed in steel tubes of 10-mm I.D., 36-mm O.D., L = 100-120 mm. The above-mentioned boosters of RDX/NaCl ( 7 3 9 2 . 5 to 15/85>initiated LVD in these PETN, RDX, HMX, DKNA, tetryl (relative density 0.96-0.97, except for HMX at 0.92) charges after a small delay. The process propagated at first at a velocity of 1.3-2.0 kmis, then accelerated to 2.5-3 kmis. Beyond that velocity the LVD 3 HVD transition manifested itself with the stronger boosters. Belyaev, Bobolev, Korotkov, Sulimov, and Chuiko [XI-5] report that cast charges of TNT, pentolite, DINA (diethylamindinitrate), and pressed charges of TNT and PETN
223 M,
in steel tubes of 13-mm I. D. and a wall thickness of 10 - 15 mm were ignited. A discontinuous transition to LVD occurred at about 2,000 m i s after some burning. Apparently strong mechanical confinement is important to initiate LVD. In 1969 Obmenin, Balykov, Korotkov, Sulimov, Dubovitskii and Kurkin [XI-51, [XII-101to [XII-121 investigated PETN of 97.5% TMD (1.73 g/cm3) and particle sizes of about 500 pm in steel tubes of 5-mm I. D. and lengths between 150 and 250 mm. Reaction was monitored by observing reaction luminosity through equidistant boreholes of 0.8-mm diameter (assembly very similar to Figure XII-11). The thermally or dynamically stimulated charge transitioned to LVD after a run length of 20 - 30 mm. This transition length depended on the porosity of PETN. This dependence was weaker for plastic-bonded charges. By increasing the wall thickness in steps of 0.5 a unique reproducible relationship was found between the wall thickness and LVD velocity, which depended on the particle-size distribution of the explosive. The detonation velocity transitioned from subsonic to transonic velocity with increasing wall thickness, in relation to the sound velocity c, = 2,250 m/s of dense PETN (Figure XII-3). First it was assumed that this dependence of the detonation velocity on the wall thickness would be a peculiarity of solid explosives, but this same behavior was found later for (liquid) nitromethane by Kozak, Kondratjev, Kondrikov, and Starshinov [XII-131.
Figure XII-3. Velocity of LVD for PETN of 500 pn grains (p = 1.73 g/cm3) in steel confinement of 5-mm I. D. as function of the wall thickness A. {Belyaev,Bobolev,
Strength of Confinement The above result is surprising, because it is assumed in the USA that the impedance of the confinement controls (HVD) detonation. But experiments with confinements of high-strength steel, soft thick lead, or other materials of varying strength, or even of water have shown that the controlling factor is the burst pressure of containment, which is controlled by the material's strength [XI-51, [XII-lo]. Stable LVD proceeds as long as the pressure p for HVD initiation (in the mentioned case 17 kbar) is not exceeded. Therefore. the condition for stable LVD is:
224 Chapter PLVD
2
~ ~ u r s t i nlimit g ofconfinement
< PHVD.
Transition to HVD is controlled by: PI VD
PBurstmglimit of confinement
PHVD.
The initiation pressure of LVD decreases with the strength of confinement. LVD is obtained by ignition as well as by a weak shock.
Explosive/Confinement Interactions and LVD Luminosity The essential dynamic precursor interactions of the confinement with the homogeneous liquid explosive - resulting in its cavitation - are well established for LVD initiation to occur. It seems that the same situation applies to solid explosives because luminous precursors appear at the intcrface. But these precursors are different from those of HVD, as will be discussed later. The confinement shows a larger sound velocity than does the propagation of LVD. This creates wave shifting near the side walls that leads to two luminous fronts (Dubovik, Denisaev, Bobolev [XII-141. But the particle velocity, dctcrmined by the electromagnetic method, has shown only one wave type. It is important to note that powdered explosives in paper confinement can undergo LVD. Obviously, no wave precursors in the confinement are needed [XTI- 151.
Detonation Transfer We are interested in what happens when only one component in a composite charge exhibits LVD. This situation is most interesting for fuzes and for detonation transfer to the booster and from it to the main charge. Some unique experiments of Jones and Cumming [XIl-16] give some insight into this kind of detonation transfer. These authors used TNT (p = I gicm’ and with varying diameter) as acceptor charges, which were boosted by attached donor charges of different detonation velocities DD. The varying detonation velocities of the donor charges were obtained by mixing TNT with AN or by varying the density. The results are shown in Figure XI1-4. Results of the experiments were: 1)
The 7/8-in.-diametcr acceptor charge could not be initiated.
2)
The I -in.-diameter acceptor chargc, initiated by boosters of detonation velocity D D achieved LVD between 1,600 and 3,200 d s , then increased or decreased to 1,700 d s .
No detonation transfer was obtained for Do < 1,600 mis. The acceptor chargc achieved HVD of 3,400
for DD> 3,200
225 L
Explosives
ml5
LOO0
Figure XII-4.
Results of Jones and Cumming for detonation transfer experiments.
YDDSHVDi \f
t
LVD
LV
%
2000
1000
No Initiation T N T (So = 1 g/cm3) 0 1
Charge diameter
1 I/&
inch
li/2
3)
Similar results were obtained for a 1 3/16-in.-diarneter acceptor charge.
4)
The 1%-in.-diameter acceptor charge resulted in HVD at 4,400 m/s with a donor velocity of Do > 1,200 t d ~ .
The experiments do not show initiation gaps between LVD and HVD, but some results from gap tests and transmission of detonation through air indicate that such gaps occur in other experiments. More detailed discussion of these occurrences are discussed in the section entitled Tests.
226 Chapter
Classical Aspects Only negligible shock heating is present in LVD initiation with pressures of 5 - 10 kbar. The presence of local reaction foci, which lead to fluctuating pressures, seems to be generally accepted. Classical shock relations are often used to calculate detonation pressure or compression from experimentally determined wave velocities. There is no proportionality between detonation velocity and pressure for liquid explosives, see Fig. XI-6. Therefore, such experiments should be considered with some reservations.
Explanation of Double Explosions Since even moderate static pressurization prevents LVD initiation, it may be that first reactions can pressurize the system, thereby avoiding direct LVD initiation. The substance is activated dynamically and conditions for LVD initiation become favorable whenever this pressure is released. This explanation also covers the effects of released stemming in commercial blasting, and is consistent with the advice to use betterinitiating devices whenever irregular double explosions are reported in a quarry. Current testing does not address these events. More research on safety relevant matters of detonation is badly needed.
Experiments on Double Explosions and Jump Phenomena A very sensitive ignition mixture of Pb2O3/Si/KC1O4/NChas been boosted on occasion, when the technician reported repeated double explosions with a time difference of about 1 second. This boosting was captured on 2 FASTAX-films at a fi-equency of 3000 pictures/second. The ignition mixture was loosely filled into Plexiglass tubes of 20-mm I. D. and 1-m length. The charge had been boosted at one side with 10 g RDX, initiated by a blasting cap. The high-speed film frames are presented in Figures XII-5 and XII-6. The lines in the drawing correspond to the luminous length in the charge. The booster light is seen over five frames in Figure XII-5, at the left bottom. After a (dark) period of 60 to 70 ms (1 80 pictures), at a distance of 25 cm from the initiated end, reaction started backwards to the initiation side within 90 ms (270 pictures), where a final velocity exceeding 1,500 m l s was reached. Everything was extinguished, no light could be seen. But 700 ms (2,080 pictures) later the same material flashed again, and after about another 15 ms (- 40 pictures) a violent reaction started into the expected forward direction with considerable reaction velocity.
221
4
+-
,
, rn
0.5
m*
Length of tube
,
__c
Figure XII-5. Plot of a FASTAX film (3000 pictureds) of the initiation of an ignition mixture from left to right. The luminous events are plotted as the number of pictures.
Figure XII-6 shows another behavior: With a time delay of about 60 ms (170 pictures) a burning reaction developed both forward and back within about 50 ms about 35 cm distant fiom the initiation site. Then, discontinuously within a short time, acceleration of the reaction velocity occurred on both sides, where velocities of 1,000 m / s or more was obtained rapidly.
228 Chapter
Figure XII-6. Plot of a FASTAX film (3000 picturesis) of the initiation of an ignition mixture J?om left to right. The luminous events are plotted as the number of pictures.
Prof. Yoshida [XII-171 did similar experiments with dry dibenzoylperoxide (BPO) in 50-cm long Plexiglass tubes of 30- or 36-mm I. D. BPO was electrically heated at the top when a deflagration without flames started, then transited aRer few seconds to an explosion. This flameless reaction continued in the remaining substance until it transited to explosion near the bottom. The explosion dented a lead witness plate. When ignition occurred by a fuze head, a deflagration with flames appeared and transited to explosion. The remaining substance also continued its flameless deflagration. Similar phenomena were observed in experiments with granular gun powder, using 30-cm long aluminium tubes of 30-mm I. D.
Proper Reaction Stimulus In sensitivity testing one always uses stimuli that might lead to the most severe event. But the results seen in Figures XII-5 and XII-6 led to the question whether a blasting cap or a match would lead to the most violent reaction.
229
Initiation and Ignition Experiments We were not sure whether an explosive booster would be sufficient to initiate pyrotechnic mixtures. To assure ignition, the pyrotechnic mixture was initiated with a blasting cap in the centre of a glass vessel of 2-1 volume. The luminous contours of 19 frames of a FASTAX film with 3000 picturesh are drawn in Figure XII-7. The view field fills up very slowly in about 6.3 ms. No dramatic results were observed.
Figure X11-7. Development of the reaction of a pyrotechnic mixture initiated with a blasting cap. The luminous contours are shown for the first 19 frames 6.3 ms.
Fig. X11-8.
Development of the reaction of ignition mixture, ignited with a squib. The luminous contours are shown for the first 5 frames 1.6 ms.
A 4-1 glass vessel has been filled with the ignition mixture, and ignition occurred in the center with a squib. Figure XII-8 shows the luminous contours. Within five frames, corresponding to I .6 ms, the reaction covered the view field almost totally. This reaction was unexpectedly severe and violent (this ignition mixture is extremely sensitive to friction). We were lucky to avoid an accident. This ignition mixture is no longer used.
These experiments demonstrate the reality of double explosions in one initiating sequence within a given volume. They demonstrate that a shock is not always the most severe stimulus to establish a violent reaction; burning can be much more effective under certain conditions.
230 Chapter
LVD Spin Detonations in Solids Cook [XII-41 has shown that spin detonation of an ANFO 94/6 mixture is achievable in a cardboard tube. Cast TNT, in a steel tube of 15.5-mm I.D., 34-mm O.D., L = 120 mm, shows a spin mode in LVD [XII-91. The tube was broken along the axis into two almost equal pieces, see Chapter IV. The channel increased in diameter by 3 - 4 mm at the initiation side, the tube took the form of a barrel with furrows towards the other end, and the diameter increased by 5 - 7 mm. The spin-wise furrows with a depth of 0.5 to 1 mm and a step of about 4 cm were clearly seen at the inside surface of the channel, see Figure XII-9, similar to the case ofNG shown in Figure XI-13. Additional references are [XII-9a], [XII-%].
Figure XII-9. Spin mode of cast TNT, for details see the text, compare with Figure XI-13. (Courtesy of Boris Kondrikov)
Further Transients We refer the reader to Cook [XII-41 for additional information on transient phenomena and jump detonation.
LVD OF PYROTECHNICS AND PROPELLANTS Based on the above discussions, LVD is primarily a characteristic of energetic materials, although inert, uniformly sized powders may become explosive under the right conditions, for example in a silo, see Chapter XIII, {Pieper [XII-18]}. LVD aspects have not heretofore been applied to pyrotechnics. Accidents with pyrotechnics have not been severe enough to have been investigated (?!?), so they are not fully understood (see the chlorate explosions, Chapter XIII). Probably the recent disaster of Enschede, NL, 16.05.00, can stimulate us, the experts to revise this situation including the legal paper work.
23 1
Primary Explosives Stresau [XII-191 pressed lead azide and mercury hlminate into thick-walled metal tubes with an i. d./o. d. ratio of about 0.2. Only HVD of about 5,000 m / s was observed below a primary's relative density of 93 - 94%, but the detonation velocity dropped to the LVD level at higher densities, between 1,400 and 1,700 d s (see Figure XII-lo).
I
1
I
I
: I
n
I
1
om
I
8
30001 I
1
*
I I
0 .
a
Figure XII-10. Detonation velocity of mercury fulminate relative to its density in strong confinement. Inside diameter 0.15 in., I. D./O. D. ratio is 0.2. {Stresau [XII-19], and Reference [ma]}.
*L
0
I
*
I
I
. CRYSTAL DENSITY
I I I ' r I
I
I
3
I 4
I
I
5
Loading density (gkm")
Dead Pressing Detonation velocities of 1,700 m / s and lower characterize risks summarized under the heading of "dead pressing", because a second charge may not be initiated to HVD and failure is expected. Such cases will be discussed later.
A real dead pressing is characterized by the fact that, for example, mercury fulminate only burns and is not likely to detonate. It has long been believed that only mercury fulminate can be dead-pressed because it is relatively easy to press to 97% of crystal density at 3,500 bar, but lead azide, lead styphnate, and other primary explosives also show this behavior. Porous and high-density materials exhibit different initiation characteristics, as will be discussed later, so that one may understand that "dead pressing" also depends on the initiating stimulus. Other viewpoints on this phenomenon are described by Dickson, Parry, and Field [XII-201.
LVD of Black Powder Belyaev and Kurbangalina [XII-211 found in 1964 that black powder at densities as low as 0.55 g/cm3undergoes LVD at velocities of 1,300 to 1,500 in 4-mm
232 Chapter
confinement. This is contrary to earlier, classical findings of Kast [XII-221 that reaction of black powder in a steel tube proceeds at 400 m/s but the linear burning rate is much lower. Less-known Problems with Pyrotechnic Delays A black-powder delay cord usually is adjusted to a reaction rate of 120 s/m. This classical delay device is reliable so long as one knows that a reaction "flash over" can annihilate this characteristic. Wehner [XII-231 showed that the delay vanishes completely when the delay cord is pressed in a box clamp; this apparently causes LVD to occur. The properties of pyrotechnic timers definitely depend on the mechanical environment in the components; but this fact is little known, even to experts.
Other Examples of LVD Hershkowitz [XI-31 determined in 1962 - 1964 definite reaction rates of 300 or 900 for the system potassium perchlorate (KP)/Al. Granulated AN coated with NG, tetryl, TNT or HMX undergoes LVD [XI-31, [XII-241, [XIJ-25]. Mixtures of the type 20% PETN, 30 - 55% Al, and potassium or calcium perchlorate or barium peroxide may be adjusted to detonate in a specified velocity range of 1,250 4,200 m i s . The gas production is low [XI-31, [XII-261.
Propellants LVD phenomena of gun propellants, similar to rocket propellants, are possible. It may be that in actuality the Gallwitz factor [XII-271 was an early criterion of such a risk. We were able to detonate a rocket-propellant strand with a steady detonation velocity of 2,350 m/s in the heavy confinement experiments {Obmenin et al. [XII-lo], see Figure XII- 11} . Figure XII-11. Experimental device similar to that ofObmenin et al. [XII-lo]. I. D. / 0. D. = 9.1132 mm, high strength, high toughness steel. Bore holes were used for optical observations.
233
L
Tests
The nonideal HVD of a HMX/AP/Al rocket propellant has been reported by Dick [XII-281. More information can be found in the AGARDograph No. 3 16 [XII-291.
TESTING FOR LVD RISKS As outlined in Chapter XIII, pure chlorates and other chemicals can produce craters on reaction, but do not show any explosive hazard in current testing. It would be challenging to do some typical LVD tests, using the knowledge reported here.
LVD Tube Test A 40-mm barrel with a 116-mm 0. D., filled with the test substance, was used because it forms a thick-walled high-strength tube. In addition, velocity wire probes were inserted but most of them failed. Lead witness plates were initially used to record detonation, but were replaced by lead ingots because the volume of the plate dents could be more easily determined (see Fig. XII-12). The charge was initiated at the top of the barrel or in the first third or center by RDX boosters varying ftom 10 to 50 g. TNT was also found suitable but single blasting caps were not. Table XII-1 shows the results of these tests. Large amounts of the tested material were recovered after the experiment. This test does not directly correspond to an actual accident, where mostly thin-walled drums had been involved. However, for the first time small-scale tests were used on explosion-causing chemicals. Talcum powder and salt dented the witness plate but sand did not. This is caused by the uniform porosity (grain sizes) of salt and talcum powder but sand has a different granular structure. The shocks produced uniform pore collapse, and denting occurred due to hydrodynamic power conversion.
234
235
Table XII-1. Results of some LVD experiments on hazardous chemicals
Substance
Rel. Density
Bar dent [mil
Comments
[%I
Booster/ relative Position
55 60 62 60
50gRDX,top 50gRDX, top 50 gRDX, top 50gRDX, top
35 60 45 45
Note 1)
Potassium chlorate 22 22
50 50
30gTNT, 50gRDX, in upper third.
150 250
Potassium perchlorate
22
60
50gRDX, in upper third.
225 224
Potassium nitrate
22
55
50gRDX, in upper third.
140 125
Bleaching powder
22 23 23 23
80 70 85 80
50gRD X
82 40 35 30
95 Val.% to be inserted into the vessel containing the liquid. This device, known as BLEVEX [XIII-I 171, prevents retardation of ebullition due to heat transfer. Further the cross volume is subdivided to many small cells. While this is an effective tool, due to practical problems, like cleaning, it has no practical application yet.
According to Equation (111-7) the explosion probability also depends on the sound velocities of the liquids. Normal liquids like alcohol have sound velocities around 1,300 m!s and liquefied gases around 800 m!s independent of the volume, explosion probability of liquid gases is at least 4 times that of normal liquids. Since radiation damping also depends on the ambient pressure, see Chapter VII, explosion probability of pressurized liquefied gases is increased further. This estimate compares favorably with results from the study of the tank car project [XIII-1161. For more details, see [XIII-71, and [XIII-1051. In spite of the fact that expanded metal [XIII-117], see Figure XIII-30, is an effective safety device, it has no widespread application due to practical problems like cleaning and difficulty of insertion into big tanks.
Degassing, Foaming up There are many analogues to the above mentioned kitchen examples. They also exhibit properties observable in industrial accidents.
Gushing of Beer Gushing is when a bottle of beer is opened and bubbles releasing carbon dioxide until all beer is gone. In rare cases a bottle explosion can result. A primary prerequisite is the depressurization which in the case of wheat beer is more pronounced due to the higher pressure of carbon dioxide. A hrther condition is a stabilized foam by surfaceactive substances. Thus conditions exist to increase the risk of gushing by such factors as bacteriological influences on the malt, calcium oxalate, metal ions like Fe +++, cleaning agents of‘the bottle. Hop oil, humuline acid, unsaturated fat acids are known to decrease this risk [XIII-1181.
310
Blow-Down Systems - Safety Disks In the chemical process industry as a safety measure blow-down systems or safety disks are in use to allow rapid release of pressurized reactor contents in case of a thermal runaway or emergency. The cross-sectional areas have dimensions for a maximum flow of sound velocity at the orifice. If there occurs a foaming up, the sound velocities decrease drastically, see Figure IX-3, with the result that pressure in the vessel can increase hrther in spite of an open cross section and rupturing of the vessel. In such calculations the assumptions of sound velocities of the pure liquid are usual. There is an indication that the Seveso release was a blow-down event: A burst reactor vessel and an open safety disk!
Degassing of Solids Events can also happen when solid grains are pressurized with gases. If a sudden pressure release occurs, gas desorption is a mean for mass injection into the unit volume leading to generation of a pressure wave. An example of a mishap is the rapid depressurization of coffee after extraction of caffeine with supercritical carbon dioxide.
Degassing of Natural Lakes Tectonic gases can dissolve in any kind of water but their solubility depends on the static pressure. Gas concentration is in deep water much higher compared to the surface layers: the density reduces therefore, so that a (gradually) layering results. Mass transfers can be started by any mechanical or thermal instabilities, which can lead to a spontaneous, even explosive, degassing. The Nios Lake (Cameroon) contained about 4 m3 of dissolved C 0 2 per m3 water. It was estimated that in 1986 150x 1O6 m3 C 0 2 explosively degassed, and still 250x lo6 m3 remained in the water, killing more than 1,700 people (other estimates 4,000) by suffocation [XIII-119]. But other noxious gases can be as important. Other explosive degassings occurred with methane { (Kivu Lake, Monoun Lake, 1984, (killing 37 people), Tanganjika Lake, all in Af?ica, and the Ocracoke in the Gulf of Mexico)). All these eruptions caused many fatalities by suffocation. The occurrence of an explosion-like event was proven by finding a trim line in the soil and foliage caused by a wave 15 above the Monoun lake [XIII-1201. Another effect of degassing is that, by sparkling and foaming up, the original liquid’s density decreases drastically, so that a normal ship (built for normal water buoyancy) sinks into this foam.
31 1 Views
Summary
All these events can be summarized, see the following Table XIII-10. Table XIII-10. Summary on basic mechanisms possibly leading to Explosions.
Basic Mechanism
Cavitation
Stimulation of this mechanism
Possible technical devices, Consequences
Water hammer
All pipe lines,
Joukowski shocks
Chemical Processing, Blow-down Systems.
CondensationiEvaporation
Evaporation,
All kinds of Vapor Explosions
Superheat, Retardation of Ebullition, Drop of static Pressure, Mixing Phenomena Benard Convections)
SorptionDesorption Resonances
Tank farms and vehicles, Storage of liquefied gases, Tubings, Chemical Processing,
External mechanical Shock
In the case of combustible gases a room explosion can follow.
Heat,
Foaming up of liquids, failure in depressurization caused by low sound velocities of bubbly liquids.
Pressure drop, Mixing Phenomena (Benard convections)
Pressure release of External mechanical Shock pressurized condensed (powdered) materials, Carbonized Drinks, Natural Lakes. Phenomenon of Bermuda triangle.
312 Chapter
Progress in Safety by Accidents Many accidents in the past were starting point of specific investigations, but these also led to the development of new and more modern products. In Table XIII-11 some of these occurrences - also some not mentioned in this Chapter - are listed with their consequences. However, there are severe safety risks not yet properly addressed (for example the accidental explosions of insensitive explosives or explosions of non-UN Class 1 substances, and the problems with liquefied gases including the FLI's). Table XIII-11. Progress by Accidents.
Accidents Early accidents with NG and explosives in manufacturing, applications - also in mines -, and transportation 1860 -1 9 10.
Consequences Start of the development of safety technology. Also the risk management was successively adapted by rules and laws. Modeling of risks in tests, mainly by railroad and mining authorities. Foundations of national safety establishments. Probably the first supranational agreements resulted.
AN explosions in the 1920's
Blasting in AN forbidden. Interest in sensitivity problems renewed.
Fire-damp explosion in the pit Hannover 1/2, 1939
Detection of a severe condition in using explosives for fire damp ignition (Kantenschuflbedingung). Start of the development of the German Class-III permitted explosives.
Texas City AN explosion, 1947,
Renewed starting point of the study of AN explosions. First differential thermal analytical (DTA) investigations on explosives started. Development of ANFOs and Slurries.
Experience with drowned ANFO charges. Port Chicago, 1944
President H a r v S. gave an executive order in 1948 to end segregation in the U.S. military. The problem of racism ended for the 50 black "mutineers" finally for one in December 1999, when President Clinton decided to issue a pardon [XIII-~C].
Detonation of 'empty' tubes, Loison, 1952.
Investigation of shock tubes. Later development of shock tubes for regular applications (NONEL).
313 Views
Accidents
Consequences
Delayed Explosion in the pit Minister Stein. 1954155
Deflagrations of permitted explosives resulted in fire damp explosions. Starting point to develop deflagration proof permitted explosives.
Many occurrences with initiators.
Starting point of concern of electrostatic discharges. Development of electrostatic insensitive initiators (U-Zunder).
2 NM explosions within a half year, 1958
Starting point of LVD studies. Recognition of the LVD as a risk.
Further accidents with initiators
Thunderstorms but also stray currents had been found to be casual. Development of electrical highly insensitive initiators (HU-Zunder).
SPERT Reactor, 1962
Starting point for the monitoring of bubbles. Recognition of FLI as a nuclear risk.
John Forrestal carrier accident (1967). Detailed description, see [XIII-121].
Starting point of Insensitive Ammunition development.
Since 1960
More and more remote handling becomes usual practice.
Shell Pemis accident, 1968
Mechanism of rolling over (Benard-convection)first realized as an industrial risk. Also the stability of emulsions is considered.
Carbon-dioxide explosion, 1976.
Starting point of the reconsideration of current theories of explosion and detonation. Address Physical Explosions.
Three Mile Island, 1979
Demonstration that nuclear risks are real and not merely hypothetical. Big social and political changes resulted in their acceptability.
Chemobyl, 1986 1980's
Due to early concern on safety in rocket propulsion, nuclear risks and chemical process industry formalized tools of safety assessments that became common practice. In Germany the legal general clause methods got widespread application, but only to a limited degree on the safety of explosives.
Spontaneous lake degassing C02 and CH4 in the 1980's
Recognition as a major risk, so that in Germany crater lakes are monitored (Laacher crater lake).
3 14
Chapter
References [XIII-11 Jahresbericht der Berufsgenossenschaft Chemie (BG Chemie), Heidelberg, 2000. [XUI-2] B. Hoffmann, Arbeitsschutz und Unfallstatistik, Hauptverband gewerblicher Berufsgenossenschaften, Bonn, 1990. [XlII-2a] A. Brodkorb, BGChemie, Email from 07.01.2002. [XIII-3] M. A. Cook, The Science of High Explosives, Facsimile ed. of 1958, Robert E. Krieger Pub., Huntington, NY, 197 1. [XIU-4] G. S. Biasutti, History of Accidents in the Explosives Industry, CH-1800 Vevey, Corbaz S.A., Montreux. (about 1985) [Xm-4a] C. 0. Leiber, and R. M. Doherty, Review on the Assessment of Safety and Risks, Propellants, Explosives and Pyrotechnics 26 (2001), p. 296/301. [XIII-5] E. Audibert, and L. Delmas, Note sur la deflagration fusante et la double detonation des explosifs brisants, Ann. Min., Paris, 13, Serie 5 (1934), p. 280/306. [XIII-6] R. Sartorius, Contribution a I’etude de la deflagration fusante des explosifs de mine, SthInt. Conf. Leiter grubensicherheitlicher Versuchsanstalten, Dortmund-Derne, 1954, contribution No 17. [XIII-7] C. 0. Leiber, Explosionen von Flussigkeits-Tanks. Empirische Ergebnisse - Typische Unfalle, J. Occ. ACC.3 (1980), p. 21/43. [Xm-8] R. Assheton, History of Explosions on which the American Table of Distances was Based, Including other Explosions of large Quantities of Explosives, Institute of the Makers of Explosives, 1930. [XIII-9] The Port Chicago, CA, Ship Explosion of 17 July 1944, Army-Navy Explosives Safety Board, Washington, DC, 1948, Tech. Paper No 6. [Xm-sa] D. Prouty, Contra Costa County Office of Education, The Port Chicago Desaster, [XlII-9b] Peter Vogel, The Last Wave from Port Chicago, 2001-2002, see also www.portchicaeo.ore. [XIU-9c] G. Miller, Congressman, Press release December 23 1999: Clinton Pardons Freddie Meeks, Port Chicago Sailor, of “Mutiny”. [Xm-101 R. C. Herman, Armed Services Explosives Safety Board, Navy Board of Investigation Official Report from 6. June 195 1 : The Explosion of USS Mount Hood, Seeadler Harbor, Manus Island, 10 November 1944, NTIS: AD 8 12 958. [Xm-1 11 Local Report at the explosion site. [Xm-12] Sprengstofflagerbrand und -detonation in Priim (Brandschutz 911949) in P. Vaulont: Bemerkenswerte Brande und ihre Lehren, W. Kohlhammer, Stuttgart, Berlin, Koln, Mainz, 1970, p. 15/16. [XIu-13] Explosion of BLU-82/B Warheads and related EOD Matters, US Naval Explosive Ordnance Disposal Facility, Indian Head, Maryland, 1978. [XLU-14] H. Kast, Die Explosion in Oppau am 21. September 1921 und die Tatigkeit der Chemisch-Technischen Reichsanstalt, Z. ges. Schien- und Sprengstofkes. 20 (1925)11,12,and21(1926)1- 9 .
315 References
[XIII-151 G. Armislead, jr., Report to John G. Simmonds & Co, Inc., Oil Insurance Underwriters, New York City on The Ship Explosions at Texas City on April 16 and 17, 1947 and their results. [XIII-16] Ville de Brest Archives Municipales: TU-Raport Annuel sur 1'ActivitC du Corps des Sapeurs Pompiers, Annee 1947. [XIII-17] R. S. Egly, Analysis of Nitromethane Accidents, in Symposium on Safety and Handling of Nitromethane in Military Applications, Symposium on Safety and Handling of Nitromethane in Military Applications, Monroe, Louisiana, 14-15 November, 1984. [XIII-lS] Ex Parte 213, Accident No 305.
Interstate Commerce Commission
[XIlI-19] J. Prinz, FAX-communication fi-om 18.11.1991. [XIlI-20] Accident Investigation of BerggewerkschaftlicheVersuchsstrecke, Dortmund-Deme, 1973. [XIII-211 Railroad Accident Report Burlington Northern Inc., MonomethylamineNitrate (MMAN) Explosion, Wenatchee, Washington, August 4, 1974, National Transportation Safety Board, 1976, NTSB-RAR-76-1. [XIU-22] Tadao Yoshida, FAX-communication from 12.02.1991, and 12.01.1992; [XIII-231 Tadao Yoshida, J. Wu; F. Hosoya; H. Hatano; T. Matsuzawa; Y. Wada: Hazard Evaluation of Dibenzoylperoxide(BPO), Proc. 17'h Int. Pyrotechnics Seminar, and 2"d Beijing Int. Symp. Pyrotechnics and Explosives, Vol. 1991, Beijing, p. 9931998. [XIII-24] H. Bartels, Untersuchung eines "Anziindverzogerers",BICT-Rept. 3.417026186, 1986. [Xm-25] Yojiro Mizushima, Letter-communicationfrom 02.02.1991. [Xm-26] Schreibein der BAM an den Bundesminister fir Arbeit und Sozialordnungvom 28. 7. 1959, Az: 4-1287159. [Xm-27] B. J. Thomson, HSE, UK, Letter communication 01.08.1985. [XIII-28] P. Meyer, Kalium-Natriumchlorat - Explosivstoffe ?, Gefahrliche Ladung 30 (1985)7, p. 3011303. [Xm-29] HM Explosives Inspectorate, Special Rept. No 185, HMSO, London. [Xm-30] H. Brunswig, Explosivstoffe, J. A. Barth, Leipzig, 1909, p. 17. [Xm-31] HM Factory Inspectorate incident report ref. 58711479, West Ham 1947. London: HMSO. [XIII-32] Sodium Chlorate Explodes at Howard University. Chem. Eng. News (1952), p. 1416. [Xm-33] Research Report on Fire at Hamilton on 19 March 1969, Lankarshire Fire Brigade. [XIII-34] S. Ruhnau und W. Kelm: Feuer ASCALIA, 8. Alarm im Hamburger Hafen. Brandschutz, Dt. Feuenvehrzeitung 211976, p. 26/3 1. [Xm-35] The fire and explosion at Braehead container depot, Renfrew 4 January 1977, Health & Safety Executive, ISBN 01 1 883220 4. [XllI-36] Sodium Chlorate Explosion, Chemistry in Britain, March 1979, p. 125. [XIlI-37] Chem. Eng. News (1952), p. 3210. [XIII-38] Chemical1 factory and other industrial buildings. Rennes, France. 15.06.1979. Fire Prevention, No 133, p. 52, and HSE Translation No 9549.
[XIII-39] The fire and explosions at River Road, Barking, Essex, 21 January 1980, Health & Safety Executive, 1980, ISBN 0 71 76 0060 2. [XIII-40] The fire and explosions at B&R Hauliers, Salford, 25 September 1982, Health & Safety Executive. [XIII-41] City of Salford, Opinion of Benet Hytner, M.A., Q.C. commissioned by Salford City Council, July 1983. [Xm-42] Sodium chlorate explosion blows hole in safety laws, New Scientist 30.09.1982, p. 892. [Xm-43] Near-disaster as chemicals ‘in transit‘ explode in warehouse, Fire, November 1982, p. 289. [XIII-44] S. Ruhnau, 4. April 1985: Schuppen 74 im Hamburger Hafen, BrandschutdDeutsche Feuenvehr-Zeitung 1Oil 985, p. 383/387. [XIII-45] Explosion nach Lkw-Unfall, Gefahrliche Ladung 34 (1 989)5, p. 2211223. [XLII-46] wwtv.ac-grenobl e/webc tiri eiri soma j !chi mi e:ch lorat .htm . (22.12.0 1). [XIII-47] B. J. Thomson, B. G. Freeder, N. L. Heathcote, D. H. Pickering and T. A. Roberts: Experimental work on the Explosion Hazard of Sodium Chlorate, HSE Res. Rept. January 1985. [XLU-48] M. Steidinger, Die Gefahrklassifizierung von Alkali- und Erdalkalichloraten, Amtsund Mitteilungsblatt der Bundesanstalt f i r Materialforschung und -Priifung (BAM) 17 (1 987)3, p. 493/504. [XIII-49] C. 0. Leiber, Detonation Model with Spherical Sources H: Low Velocity Detonation of solid Explosives - a summary, Proc. 1 sthInt. Pyrotechnics Seminar, Breckenridge, COL, 1992, p. 5631591. [XIII40] A. B. Parker, and N. Watchom: Selbstausbreitung der Zersetzung in Ammonnitrat enthaltenden anorganischen Dungemitteln, J. Sci. Fd. Agric. 16 (1965), p. 355/368. [XIII-511 H. K. Schafer, Thermische Zersetzung von MehrstofFnahrmitteln, 7. BunsenKolloquium, Frankfurt-Hoechst, 29.04.1982. [XIII-52] V. J. Clancey, Daten uber die Explosions- und Feuergefahr von AmmonnitratDungemitteln, RARDE Memorandum (X) 48/63. [XIII-53] H. E. Watts, Bericht der Ammonnitrat Arbeitsgruppe, England, ca. 1948. [XUI-54] B. Kier, and G. Miiller: Erweiterung der Storfall-Datenbank und Erstellung des Handbuchs - StorGlle -, R W T W Essen, 1983 [XIII-55] P. Bockholts, and L. J. B. Koehorst: Handbuch Storfalle - Dokumentation iiber Storfalle in industriellen Anlagen oder mit gefahrlichen Stoffen im Zeitraum von 1981 bis 1986, Erich Schmidt Verlag, Berlin. [XIII-56] Ch. E. Munroe, Die Explosivitat des Ammonnitrats, Z. f i r das ges. SchieB- und Sprengstofhes. 18 (1923)6, p. 61/64. [XIII-571 Many Newspaper reports around 14.01.1991, including “Spiegel”.
317
[XIII-58] A. V. Dubovik, Analysis of the problem for determination of real explosion hazard of low-sensitive explosives and of the methods for testing the explosives on sensitivity to impact. Semenov Institute of Chemical Physics, Russ. Acad. Science, Moskau 1995. [XIII-59] Ming Jun Tang; Qu. Baker; Guo-Shung Zhang, Explosion Hazards in Explosive and Chemical Industries, Proc. 23rdInt. Pyrotechnics Seminar, Tsukuba, Japan, 1997, p. 9251938. UNEP-APELL, Other Accidents involving Ammonium nitrate (Port Neal) from 04.12.2001. www. unepi c.0 [XIU-60] UNEP-APELL, Ammonium nitrate Explosion in Toulouse - France, ~ ~ ~ ~ , . L i n e ~ i ~ . ~ from28.12.2001. [XIII-611 Bericht des 34. Untersuchungsausschusses zur Untersuchung der Ursache des Ungliicks in Oppau, Z. ftir das ges. SchieR- und SprengstofTwes. 19 (1924)3, p. 42/46 and p. 60163. [XIII-62] Elisabeth Lee Wheaton, Texas City remembers, The Naylor Comp., San Antonio, Texas, 1948, 109 p. [XIII-63] K. L. DeMaet, President of Texas City Terminal Railway Company, Texas City, personal communication,November 1984. [XIII-64] Len Spencer, Learning the hard way, Hazardous Cargo Bull., February 1981, p. 22/23. [XIII-65] ANFO-explosion in Kansas City, Missouri, USA, Department of Defense Explosives Safety Board, Letter IWPi102-89. [XIII-66] H. E. Webb, jr., Nitromethane explosion, Chemical & Engineering News (C&EN), March 21, 1977, p. 4. [Xm-67] J. L. Trocino, Drum Rupture Incident at Dynamit Nobel, Angus Defence Systems Rept. 07.08.1989. [XIII-68] 30 Minutes to Desaster, Report of the Accident of May 1, 1991, Angus Chemical Company Nitroparaffins Plant, Sterlington, LA, O.C.A.W. (Oil, Chemical & Atomic Workers Int. Union, AFL-CIO) Local No 4-786. [XIII-691 J. L. Trocino, Handling of Nitromethane in Racing Fuel and Combustion Applications, Symposium on Safety and Handling of Nitromethane in Military Applications, Monroe, Louisiana, 14-15 November, 1984. [XIII-70] R. Loison, Propagation d'une deflagration dans d'huile, Comptes Rendues 234 (1952), p. 512/513.
tube recouvert d'une pellicule
[Xm-7 11 Christmann, iibertragung von Detonationen uber leergelaufene Sprengolleitungen, DNAG-Rept. from 18.11.1971. [XLu-72] S. Wilker, U. Sirringhaus, A. Gupta, H. H. Ehlers, Analyse unverbrannter 25" Annual Meeting ICT, Karlsruhe 1994, Contribution 37. [XIII-73] NATOIAC 258 documents.
318 Chapter XIII
[XIII-74] D. J. Roddy, J. M. Boyce, G. W. Colton, A. L. Dial, Meteor Crater, Arizona, Rim Drilling with Thickness, Structural Uplift, Diameter, Depth, and Mass Balance Calculations, 5thLunar Science Conf. Proc., June 1975. [XIII-75] R. B. Vaile, jr., Pacific Craters and Scaling Laws, J. Geophys. Res. 66 (1961)10, p. 3413/3438. [Xm-76] Fundamentals of protective Design for conventional Weapons, Technical Manual TM 5-855-1, Dept. Army, 3 November 1, 1986. [XIII-77] Mixed Company results with ANFO. [XIU-78] Personal communications of Dr. Kurt Birkle and Dipl.-Phys. Axel M. Quetz, MaxPlanck-Institut f i r Astronomie, Heidelberg, who generously provided the pictures. [XIII-79] David H. Levy, Impact Jupiter : The Crash of Comet Shoemaker-Levy 9, Plenum Publishing Corporation, September 1995, ISBN: 0306450887. [XIII-SO] M. Zimmer, Maare von Geburt an ubenvacht, Frankfurter Allgemeine Zeitung ( F a ) 25.11.1992, No 274, p. N1. [XIII-81] Johanna Jung, Private communication in the 1940’s. [Xm-82] Gefahrliche Mikrowellenherde, Wasser ,,explodierte“, Sicher ist sicher 37 (1 986) p. 3271328. [XIII-83] G. Tissandier, Popular Scientific Recreations, a Storehouse of Instruction and Amusement in which the Marvels of Natural Philosophy, Chemistry, Geology, Astronomy, etc., are explained and illustrated, mainly by means of pleasing Experiments and attractive Pastimes. Translated and enlarged from “Recreations Scientifiques” of Gaston Tissandier (Editor of “La Nature”), probably late 1800’s. [XIII-84] Hearings On Soft Drink Bottles, March 1974, Release # 74-014, US Consumer Product Safety Commission. [XIII-85] F. Mayinger, Wie sind Dampfexplosionen im Lichte neuerer Erkenntnisse beurteilen?, atomwirtschaft, Februar 1982, p. 7418 1. [XIII-86] B. Lafrenz, Physikalische Explosionen, Fb 77 1, Bundesanstalt 1997.
Arbeitsschutz,
[XIII-87] S. G. Lipsett, Explosions from Molten Materials and Water, Fire Technology 2, (1966), p. 1181126. [Xm-88] L. F. Epstein, Metal-Water Reactions: VII. Reactor Safety Aspects of Metal-Water Reactions, Vallecitos Atomic Lab., GE Co, Pleasanton, CA, GEAP-3335 (1960). [XlU-89] H. H. Krause, R. Simon, A. Levy: Smelt-Water Explosions, Fourdrinier Kraft Board Inst. (Battelle), 1973. [XlU-90] S. A. Rydholm, Pulping Processes, Interscience Pub., New York-London-Sydney, 1965. [XlU-9 11 H. Tetmer, Schlackenexplosionen in Siemens-Martin-Stahlwerken, Neue Hutte 4 (1959), p. 3521359. [XlU-92] Frick, Personal Communication 24.09.1982.
319
References
[XIII-93] W. A. Stein, G. Heck, R. Sigel: Beanspruchungen von Kolonnenboden beim plotzlichen Zulauf kalter Flussigkeiten, Vortrag GVC-AusschuD "Thermische Zerlegung von Gas- und Flussigkeitsgemischen", 08.05. 1980, Meersburg. [XIII-94] E. Thiem, Personal Communication. [XIII-95] Reactor Safety Study, Appendix VIII, Physical Processes in Reactor Melt Down Accidents, 1974, WASH-1400. [Xm-96] L. C. Witte, J. E. Cox: Thermal Explosion Hazards, Advances in Nuclear Science and Technology 7 (1973), p. 3291364. [XIII-97] Nuclear Accidents, lnt. Times 4 (1977)i. [Xm-98] W. Leuckel, Personal Communication. [Xm-99] J. Kruschelnizki, Aus Beton wird Beton, PdSU (Presse der Soviet Union) 7 (1987). p. 31. Citation from Stroitelnaja Gazeta, 01.04.1987. [Xm-1001 F. Tolke, Praktische Funktionenlehre, Vol. 1, Elementare und elementare transzendente Funktionen, 2ndEd., Springer, BerlirdGottingeniHeidelberg, 1950. [XUI-1011 Measures to repair damage at Tarbela after tunnel collapse, Water Power & Dam Construction, January 1975, p. 34/36. [XIII-102] Dr. Romano Allione, personal communicationsas a witness of this accident. [XID-l03] VDI-Richtlinie VDI 3783: Ausbreitung von storfallbedingten Freisetzungen Sicherheitsanalyse, Blatt 1: Leichte Case, Mai 1987; Blatt 2: Schwere Case, Juli 1990. [XIII-104] D. R. Bllackmore, J. A. Eyre, G. G. Summers: Dispersion and Combustion Behaviour of Gas Clouds resulting from large Spillages of LNG and LPG on to the Sea, Institute of Marine Engineers and the Royal Institution of Naval Architects, Paper No 29, 1982. [XIII-1051 C. 0 . Leiber, Approximate Quantitative Aspects of a Hot Spot, J. Hazardous Materials 12 ( I985), p. 43/64. [XnI-l06] R. A. Strehlow and W. E. Baker: The Characterization and Evaluation of Accidental Explosions, NASA CR 134779, 1975. [XIII-107] K. Gugan, Unconfined Vapour Cloud Explosions, Inst. Chemical Engineers, 1978. [XIII-1081 Highway Accident Report, Transport Company of Texas, Tractor-Semitrailer (Tank) Collision with Bridge Column and Sudden Dispersal of Anhydrous Ammonia Cargo, 1-610 at Southwest Freeway, Houston, Texas, May 11, 1976, NTSB-HAR-77-1,National Transportation Safety Board. [XIU-l09] Liquified Oxygen tank truck explosion followed by a fire in Brooklyn, NY, May30, 1970, NTSB-HAR-71-6, 1971. [XIII-1101 B. Schnell, Erster zusammenfassenderBericht uber die Untersuchungen zur Aufklarung der Ursache der Explosionskatastrophe vom 28. 7. 1948 bei der B.A.S.F. Stand vom 4. 9. 1948 -, Referat vor dem Parlamentarischen Untersuchungsausschufl des Landtags RheinlancUPfalz. Storch, Bericht uber die Explosionskatastrophe in der Badischen Anilin- und Sodafabrik in Ludwigshafen am 28. Juli 1948, Arbeitsschutz 1, (1949), p. 34/35.
320 Chapter
[XIII-1 1 11 F. Masson, Explosion of a grain silo, BLAYE, France, Summary Report, Ministry for National and Regional Development and the Environment, INERIS-BLAYE-Summary Report-July 1998.
[XIII-I 121 Derailment of Toledo, Peoria and Western Railroad Company’s train No. 20, with resultant fire and tank car ruptures, Crescent City, ILL., June 21, 1970, NTSB-RAR-72-2 (1 972). [XIU-113] Report to the President by the Presidential Commission on the Space Shuttle Challenger Accident, 06.06.1986, Washington.
[XIII-I 141 Brief vom Minister f i r Soziales und Volksgesundheit No 1 vom 26.3.1968 an den Vorsitzenden des Parlaments (der Niederlande). [XLII-115] DuPont Context 7 (1978): 1; New York Times (56) from 25.02.1978. [XIU-I 161 Railroad Tank Car Safety Research and Test Project, Techn. Progress Rept. No 31, (27.02.1976), Ass. American Railroads, Chicago. [Xm-117] R. D. Appleyard, Testing and Evaluation of the explosafe system as a method of controlling the Boiling Liquid Expanding Vapor Explosion (BLEVE), TP 2740, 1980, Vulcan Industrial Packing Ltd., Toronto; Explosion Prevention, Leaflet No E6, February 1980, Explosafe Ltd., Borehamwood, UK. [XIII-118] A. Linemann, Gushing - spontanes ijberschaumen von Flaschenbier, Brauerei Forum 14196, p. 2171219. [XIII-119] K. Tietze, Gefangene Gase in geschichteten Seen, Frankfurter Allgemeine Zeitung, 10.09.1986, No 209, p. 31. - Personal communications (as primary investigator). [XIII-120] L. Gedney, Earth’s Deadly Breath: A Scientific Mystery, Article #785, Alaska Science Forum, September 15, 1986. [XIII-121] I. M. Korotkin, Seeunfalle und Katastrophen von Kriegsschiffen, 5 th ed. Brandenburgisches Verlagshaus, 1991.
XIV DIPOLESCATTERING This section deals with the question of what happens when if a particle or void in a continuum is attacked by a pressure wave. A qualitative answer is simple: Depending on the impedanceis pc or the bulk modulus pc2 of the medium and the particle, scattering and reflections of the impinging wave occur, being different for soft or hard inclusions. No mobility of the inclusion is possible when the strength or viscosity of the matrix is dominant. However, in an inviscid medium - by the conservation of momentum rule - different particle velocities are obtained for volume elements of different densities. For inclusions such as particles or voids, particle velocities different from those of the surrounding medium will be observed. Thus the inclusion acquires a relative:velocity with respect to the surrounding medium. This fact is ignored in classical considerations of explosives because the plane-wave laminar flow continuum approximation of detonation theory is used. Therefore the understanding of the governing mechanisms of porous and heterogeneous, such as commercial and permitted explosives is in classical terms not adequately possible.
Historical Development Acoustics in disperse media has been studied for many years, albeit intermittently. The attenuation properties have been described first probably by Derham [XIV-11 in 1708. Particles may remain on the spot and scatter acoustic energy of the sound field according to dynamic conditions. But it is also possible that they become mobile, and lose their energy by (viscous) drag losses to the surrounding medium. This leads to the most powerfbl attenuation mechanism for pressure waves. Therefore, the attenuation properties of particulate systems are strongly dependent on the dynamic conditions. Also, in a solid matrix cracks are generated by those mobile particles, coagulations can occur due to the Bjerknes forces resulting from the particles' relative velocities. All these effects are important in atmospheric physics, defogging of fumes, criminology, geology, engine combustion, plateau burning, and detonics, where they may sensitize and initiate or, conversely, desensitise and attenuate. This effect is important also in biology, medicine, and fundamental research. Therfore, many authors have devoted much effort to this problem from very different points of view. After Derham's [XIV-11 (1708), and Tyndall's [XIV-21 (1873) first qualitative tests of outdoor attenuation of sound waves by fog, considerations on wave scattering phenomena were reported initially by Stokes and Rayleigh [XIV-31 in 1872. Konig 321
322 Chapter XlV
[XIV-41 in 1891 investigated the particle velocities of tiny particles in a viscid medium in an acoustic field. This work also resolved the Bjerknes forces leading to particle agglomerations. Much energy can be lost by viscous drag (Stokes), because in the mass-activated state the particles exhibit a slip in the surrounding medium. Therefore, particles of suitable size are most effective to attenuate pressure waves. With Konig's work a branching into different directions occurred: Sewell in 1910 [XIV-5] reported on the attenuation of sound by fixed particles, which do not move under pressure-wave attack. Lamb [XIV-61, Urick [XIV-71, Skudrzyk [XIV-81, [XIV-91, Morrison and Stewart [XIV-lo], Temkin and Leung [XIV-1 11, Temkin's monograph [XIV-121, Gregor and Rumpf [XIV- 131, and So0 [XIV-141 (in hydrodynamics) investigated the particle velocity ratio of the particle and the medium explicitly. Epstein and Carhart [XIV-151, Chow [XIV-16], Cole and Dobbins [XIV-171, and many others investigated the acoustic attenuation properties of movable particles in a medium. In spite of the fact that the mobile particles are the attenuators, the velocity ratio usually does not explicitly appear in the mathematical expressions of damping. Mainly Brandt, Freund and Hiedemann [XIV-181 reported in many papers the coagulation of tiny particles in a sound field. In the following we summarize approaches to describe the particle-velocity ratio, related to applications in the dynamically activated condensed state.
Pressure Waves in an Inhomogeneous Medium A
Tiny Particle in an inviscid Medium
The asymptotic description of the acoustically generated particle velocity u' of a fiee mobile particle of zero diameter, with density p' in comparison to that of the medium (up,,,,) with density pm(Equ. (XIV-1) has long been known, see the end of this Chapter. 3
U'P
(XIV- 1)
-
This means that the particle velocity ratio approaches 3 for a point source of zero density p'. If p' > pa, then the velocity of the discontinuity lags that of the matrix. This is momentum transfer combined with an additional acoustic mass.
323 Dipole Scattering
B
Tiny Particle in a viscid Medium
The particle mobility results from counteracting dynamic pressure forces on the particle and the impeding viscid forces of the medium, which fix the particle. The first treatment for gases we owe to Konig. [XIV-4]. If q is the viscosity of the matrix, then for tiny (compared to the wave length rigid spherical particles the following abbreviations should hold, with a Stokes resistance law: (XIV-2)
(XIV-3) 1 + 3b +
9 2
9 3ab + 2
9 9 + -b3 + -b4 2
9 + -b3 2
4
9 + -b4
.
(XIV-4)
4
Skudrzyk [XIV-8], [XIV-91 obtains a similar expression with the parameter of a characteristic frequency fg (XIV-5) where a number (2 is used that relates the mass to the viscous forces. This number is called Roth's number R in detonics,. This number shows a relation to term b of Konig, see Equ. (XIV-2). (XIV-6)
The relative particle velocities are in components: 2b'/P,)+l
.-
Re-
-
+- 1 (XIV-7)
(XIV-8)
324 Chapter
Figure XIV-1. Comparison of different approaches for calculating the particle velocity ratio for a tiny low density interstitial (pYp, = 10.3)as function of the Roth number Solutions are shown from Konig (K), Urick (U), Skudrzyk (S), Temkin and Leung (T-L) for rigid particles. The Morrison and Stewart (M-S) solntion for rigid and soft particles, and the solution of So0 are shown and compared, The transition from viscid to mass-controlled behavior varies by orders of magnitude, depending on the approach. Figure XIV-2. Similar to Figure XIV-I, the relative particle velocities are shown for pYp, = 10-3and p'ip, = 10'. Solutions are shown from Konig (K), Urick (U), Skudrzyk (S), Temkin and Leung (T-L) for rigid particles. The Morrison and Stewart (M-S) solution for rigid and soft particles, and the solution of So0 are shown and compared. The transition from viscid to masscontrolled behavior varies by orders of magnitude, depending on the approach.
Figure XIV-3. Comparison of the solutions in a Gaussian plot for p'/p, = and lo3. Solutions are shown from Konig (K), Urick (U), Skudrzyk (S), Temkin and Leung (T-L) for rigid particles. The Momson and Stewart (M-S) solution for rigid and soft particles, and the solution of So0 are shown and compared.
325 Dipole Scattering
with the modulus and phase angle (XIV-9)
(XIV- 10) The Stokes laminar flow resistance law has been used in both cases, though it is different for rigid and soft particles. We expect dynamic homogeneous behavior for low that of free particles in an inviscid matrix.
values, and for large
values
Many more expressions of the particle velocity ratio are cited in the legends of Figures XIV-1 to XIV-3. Using hydrodynamics, So0 uses the Reynolds number Re directly:
1
U ‘ p - Up,m/
(XIV- 11)
Re = 2R(”p,) instead of Roth’s number the density ratio p’ip, =
Some of the solutions are compared in Figure XIV-1 for
Solutions are shown for p’lpco = and lo3 for additional authors in Figure XIV-2, which, however, do not consider the acoustic mass. A Gaussian plot is given in Figure XIV-3 for later discussions. Skudrzyk’s solution is a Cole circle. As can be seen, the transition regions vary by orders of magnitude depending on the solutions. The solutions of Morrison and Stewart for a rigid and a soft (bubble) particle motion are compared in Figure XIV- 1.
C
Particle of arbitrary Size in an inviscid Medium
Leiber [XIV-l9], IIXIV-201 derived the corresponding velocity ratio for fi-ee particles as a function of particle size for different density ratios according to Lamb [XIV-61. The results of the Equs. (XIV-12 and XIV-13) are shown in Figure XIV-4. {2(p’/pm)+l}{kRsinkR
u‘
R e p =3
(XIV- 12)
Tf/Pm) k R + { 2(P’/pm)+ 1}k2R2 + {2b’/P,)+l}* 2 4
I m p = -3
4
b(p‘/p,)+l}{kRcoskR -sinkR}+(p’/p,)k2R2sinkR (P’/P,)
2 k4
4
(XIV- 13) .
326 Chapter /In
uF4
kR
Figure XIV-4. Free particle velocities in a dynamic field as a function of particle size and density ratio. Dynamic homogeneous behavior obtained for particle sizes corresponding to the condition Re (u'du m) = 1. This may be used to estimate the shock-rise length in the material. Note that the Im-part is negative!
the modulus and the phase angle are given by:
(XIV- 14)
(XIV- 15)
kR =
is a quantity relating the radius R of the particle with dynamics, represented by the wave length h. The asymptotic result of Equation (XIV-1) is again obtained for particles of size kR 3 0. Equations (XIV-12) and (XIV-13) are shown in Figure XIV-4 in a Gaussian plot for different density ratios. As can be seen, the relative particle motion depends on
321 Scattering
Special cases exist for the density ratio range pl/pco< 1, where in the inviscid case This is an experimental u',/u p, = 1 always holds for a specified particle of size tool to evaluate specific dynamic properties, such as the determination of the shock rise length Al. This will be described in more detail later. D
Arbitrarily sized Particle in a compressible, viscid Medium
Temkin and Leung [XIV-111 published their results in 1976 on the relative particle velocity ratio ulP/up,ooof rigid particles activated by a plane harmonic wave in a viscous, non-heat conducting, compressible liquid as a function of the particle size b = kR = = oR/c,, and the cinematic viscosity ( q / ~ ) The ~ . density ratio 6 = poo/p'of the medium and the particle (void) is used as well as a parameter y, which characterizes a ratio of mass and viscous forces; in other terms relating the particle radius R and thickness of the viscous oscillatory layer A (XIV- 16) As before, the dashed quantities relate to the properties of the particle or void, and the m-indexed or unmarked quantities relate to the surrounding matrix. To characterize a dynamic working point independent of radius R of the particle or void, which is not known exactly, we use (y/kRy = l/(kA)2, which depends only on the properties of the fluid (sound velocity c,, cinematic viscosity ( q / ~ )and ~ , pressure rise}.
If the time of a linear pressure rise is f
- 1 / 3 ~and , one has
the appropriate frequency to describe this is
- 212.
-
One gets 1
CmT
(XIV-17)
or the square. The modulus of the particle velocity ratio is (XIV-18) and the phase angle is cp =
-
+
where the auxiliary fhctions are given by
(XIV-19)
328 Chapter
+
=2y2
(XIV-20)
(1+
G = 3(1+ y ) -
(XIV-21)
=
+
(
$)b2 - I]
+
%)i"
-
1)- 36(
%J
(XIV-22)
' and
(XIV-23)
For kR s 0 this solution corresponds to Urick's [XIV-7) expression, and for y 3 that of Leiber [XIV-191, [XIV-201.
to
Figure XIV-5 shows the modulus of the relative particle velocity ~ U ' ~ for / Uthe ~,~~ density ratio 6 = pJp' = lo3 as function of the dynamic working point (y/kR)' for different sizes of the void. The particle velocity ratio increases with an increasing dynamic working point, depending on the void size For future considerations, it is more appropriate to consider the particle velocity ratio at a constant dynamic working point as function of the varying void sizes in the Gaussian area. For the dynamic working points (y/kRy = 5 x lo4 and 2.5 x 10' in Figures XIV-6 and XIV-7, Re(u', /u p,co) and Im(u', /u p,oo) are given as hnction of the variable void size
Figure XIV-5.
'
Relative particle velocity ratio lub /u p.m/ of a source of density ratio 6 = pJp' = 1 as knction of the dynamic working point, and the relative size kR of the source [see Equ. (XlV-lB)].
329 Scattering
e'
=
em
i
(&)'= 50 000 Ub
Re UP-
-1
3
L
Figure XIV-6. Temkin-Leung solution of a source in a matrix of density ratio 6 = pm/p' = lo' as function of its size kR at the dynamic working point ( y / k =~ 5~x lo4.
Size controlled motion
Figure XIV-7.
5 "L
I
I
1
2
I
I
3
Temkin-Leung solution of a source in a matrix of density ratio 6 = p,/p' = 1O3 as function of its size kR at the dynamic working point (y/kR)Z = 2.468 x 10'.
size controlled motion
The maximum attainable particle velocity ratios /u'P/~P,colmax, and (Re uIp /up,co)max. are plotted in Figure XIV-8 as hnctions of the dynamic working point (y/kRy and the corresponding maximum source size (kR),,,
Figure XIV-8. Plot of the maximum attainable particle velocity ratios Re (u',Jup,,) and Re lu',Ju,,
,m
I as function of
the dynamic working point (y/kR)2 for sources of the density ratio 6=p,/p'=103. The corresponding maximum source sizes kR are given on the right-hand side.
330 Chapter
The motion of the bubbles is directed to the reverse side because ~u~p/up,oo~ < 1 for a low dynamic working point. The motion of rigid, nonvibratory particles (such as ions and atoms) is not described here, because they may reach very high velocities that are beyond this dipole scattering mechanism!
Critical Conditions for the Onset of Dynamic Void Mobilities remains in place, so that the two-phase A dynamically activated void of size kR < system behaves dynamic-homogeneously (see Fig. XIV-6). This same void is dynamically almost fully activated for a larger dynamic working point (see Fig. XIV-7). A medium may therefore behave dynamically very differently whether imperfections are above or below a certain critical size. We know that such imperfections (maybe a cavitation nucleus or an interstitial) are mostly beyond control of the experimentalist, mainly when he cannot predict their possible behavior. When a small source increases in size by inertial motion or due to a blow-up by chemical reaction, Re(u',/u,,,) increases up to where approaches the limiting size,
-
which in the example shown in Figure XIV-6 is kR 0.25. When this source expands further, the relative velocity decreases and may even move opposite to the pressure wave. Conversely, when a large source is collapsed by a pressure pulse, its particle velocity increases until the above limiting size is obtained. With further collapse the relative velocity to the matrix decreases until perfectly homogeneous behavior of the two-phase system is reached. Comparing the results for different dynamic working points in Figures XIV-6 and XIV-7 one notices that no similarity solutions exist. From Figure XIV-5 we can see that no such behavior is to be expected for the dynamic working points 1 ... 10. This means that, if the kinematic viscosity of the medium gets very large andor the steepness of the pressure rise is great, this effect disappears completely. The time dependence of drives in a parametric way a "source piston" at a constant dynamic working point, where the sources show uniform sizes and velocities. This acts as an intrinsic dynamic velocity and size filter of the voids, so that a synchronised array is obtained.
33 1
Dipole Scattering
Onset of Dynamic Particle Mobilities p'/poo< 1:
The above conclusions even apply for mobilities of more dense interstitials p' < pa. The case of the mobilities of carbonlgraphite in irodsteel with a density ratio of p'/pm= will considered later. The transition between a dynamic homogeneous (p'/p, and an asymptotic, dynamically activated state [see Equ. is shown in Figure (p'/p,
Figure XIV-9. Relative particle velocity ratio 1u;Ju p,m( of a source of density ratio 6 = p'/pm= 0.25 as hnction of the dynamic working point, and the relative particle size kR [see Equ. (XIV-1S)].
8
p'/pm = 1, and p'/poo> 1:
The case of p'/pm= 1 is shown in Figure medium of the same density appear.
where relative mobilities in a
The mobilities of denser particles are shown in Figures and There is a particular domain where the disperse medium can dynamically separate into parts of different sizes. The degree of separation can be controlled by the particle size at a given dynamic working point and by the variation of the dynamic working point.
332 Chapter
I
1
1
I
I
1
I
,,,
Figure XIV-10. Relative particle-velocity ratio Iu'&,l of a source of density ratio = p'/pm=1 as function of the dynamic working point and the relative particle size [see Equ. (XIV-I 8)].
Figure XIV-11. Relative particle-velocity ratio Iu'&,,l of a source of density ratio 6 = pYp, = 1.35 as function of the dynamic working point and the relative particle size kR [see Equ. (XIV-18)].
These figures address the Ahrens principle of selectivity [XIV-211 for permitted explosives (of the German Class 111, Fig. XIV- 1 1 and XIV- 12). The following options exist for aluminized high explosives (Fig. XIV- 13): the A1 particles remain on the spot, so that a quasi-continuum can react, or the particles move out of reaction zone. In this case there are again two tailoring factors available: a b
the relative particle size and the dynamic working point.
333 Dipole Scattering
Figure XIV-12. Relative particle-velocity ratio lu'Jup,-i of a source of density ratio 6 = p'ip, = 1.85 as function of the dynamic working point and the relative particle size kR [see Equ. (XIV-18)l.
t
I
I
I
I
kR =
I' 0.5
0
I
-1 Figure XlV-13. Relative particle-velocity ratio l ~ ' & ~ , of ~ la source of density ratio 6 = p'ip, = 2 as function of the dynamic working point and the relative particle size kR [see Equ. (XIV-1 S)].
Hydrodynamic Pair Formation: If p'/pco > 1, then a dense particle moves opposite to the pressure wave (with respect to the continuum). This relative mobility may occur by laminar flow, where the matrix closes again behind the particle - or by turbulent flow, where a wake forms
334 Chapter
behind the particle. This wake can interact, as a void, with the pressure wave, so that this void can move ahead of the main matrix. Therefore, a voided matrix can result in a voidless matrix by dynamic attack with a dense particle. The governing factor, whether or not this occurs, is the Reynolds number Re:
(XIV-24) The hydrodynamic character of the particle’s behavior markedly varies with the Reynolds number [XIV-141. Tf
R e < 1:
0
Then a laminar flow is to be expected.
Re = 10:
A curl formation occurs near the particle.
Re = 100:
The vortex size extends to about 1 diameter behind the particle.
Re = 150:
The vortex system begins to oscillate.
Re = 500:
The vortex system leaves the particle, and forms a wake (lower critical Reynolds number)
Hydrodynamic Agglomerations: Particles (or voids) tend to agglomerate in a hydrodynamic field due to Bjerknes forces, so that there is a streaming velocity between them. This phenomenon was already described by Konig [XIV-41. One technical application was the defogging of smokes by ultrasonics, but aggregation of blood corpuscles can also result from intense ultrasonic fields. The antiknock agents based on organo-lead compounds do not form well-defined particles, but can successively agglomerate in large concentrations of > particles/m3 of 10 nm diameter [XIV-221. A self-adjusting mechanism exists in a dynamic field where particles can agglomerate as long as there is a relative streaming velocity between them. This relative velocity disappears (/utP- up, 0) when the motion of the particles becomes viscosity controlled, and no agglomerations can occur. Quite naturally, this effect is also present in dynamic shock interactions. This might be the reason why the tailoring of energetic materials’ behavior is so difficult. The Temkin-Leung solution relates to one bubble in an infinite medium. A correction is necessary in principle when there are many bubbles, according to van Wijngaarden [XIV-231, because the resistance law of one bubble is different from that of a bubble swarm. Also, the error of taking the steady state solution instead of the transient one is small for usual bubbles in usual liquids under usual shock attack {see Ref. [XIV-24]]. Since the drag coefficients cDof particles or voids depend heavily on the Re-number and the properties as well as on concentrations of the interstitials, [XIV- 141 quantitative assessment of general validity of the dynamic particle behavior is not possible.
335 “Non
The considerations in this chapter are useful only to describe serious shortcomings that arise when explosives are considered in a dynamic-homogeneous state. Due to the inhomogeneityvarious many different effects come into play. These effects cannot be resolved by testing beyond the range of the real dynamic environment. Another conclusion is that the usual, solely chemical considerations are not sufficient to predict an explalsives’ behavior. This is odd because the dynamics of particles is also a (usually less,-known)mechanism of reaction kinetics. In the following chapters we will show how the acoustic considerations discussed in this chapters can be used in shock physics. Experimental evidence will also be given that acoustic effects are present and most important in shocked materials.
O N THE SO CALLED NON-IDEAL EXPLOSIVES Numerous codes exist to estimate the properties of high explosives in plane-wave homogeneous material terms. Their results are best for CHNO explosives, where the parameters are not too different. Since these parameters result from a glorified curve fitting method from experimental data, they appear as correlation values. The codes fail therefore in the presence of unusual additional atomic species, like ammonium perchlorate, aluminum, barium nitrate, lead nitrate. The basic reason is that the assumption of a homogeneous material is severely violated. See the values of explosives’ densities in Table XIV-1 and those of potential additives in Table XIV-2. Table XIV-1. Density of explosives
Explosive
Table XIV-2. Density of potential additives
Density [g/cm3]
Additives
Density [g/cm3]
Nitroglycerine (NG)
1.593
Pore (void)
10.~
Nitrocellulose (NC) TNT tetryl
1.66 1.654 1.731
A1 Ammonium chloride Ammonium perchlorate
2.702 1.536 1.95
PETN RDX
1.778 1.806
Barium nitrate Lead nitrate
3.24 4.535
HMX
1.902
NaCl Potassium nitrate
2.165 2.109
336
Chapter
A usual explosive charge shows average densities pmfiom 1.6 to I .8 g/cm3. The for each element with the conservation of momentum with the particle velocity density p‘ and particle velocity u’ p, the conservation of momentum equation u p , pa =
=
p‘
(XIV-25)
should hold. Consequently voids or particles with a different density p‘ f pm will exhibit different particle velocities u’ f up, when these are not fixed by materials cohesive forces, which are excluded by the assumption of a hydrodynamic state. AS demonstrated earlier for particles of density p‘ > pm, their velocity u ’ lags ~ that of the matrix up, , and less dense particles or voids prevail. Obviously, non-homogeneous dynamic behavior of the mixed matrix system results. In the case of shock attack basic principles require dynamic demixing and alteration of the original composition. An additional effect of such dense particles is that they evade the chemical reaction zone, and take away possible reactivity andor energy. Depending on the conditions of a given durable confinement, a secondary (late) reaction becomes possible. In unconfined detonation the fbmes enter air, water, or another non-inert medium, reactions can occur in the volume that are beyond the usual detonic consideration. All these events result fiom first principles, and are ignored in classical considerations. Therefore we cannot expect any physical sense in code calculations beyond engineering estimates. Non-ideal behavior is defined as ‘not by codes-expected real explosive behavior’. Actually the so-called non-ideal behavior of explosives reflects the defects of current detonation models. The term non-ideal is often misused.
Dynamic Voids Behavior Figure XIV-5 shows that dynamically activated voids seldom remain in place in a matrix. Depending on bubble size and the dynamic working point they exhibit velocities relative to the matrix. At low dynamic working points small sources remain on the spot, whereas at high dynamic working points the bubble velocity can exceed that of the shock velocity, and initiation of High Velocity Detonation (HVD) takes place, see Chapter XVI.
Dynamic Dense Particle Behavior Figure XIV- 14 exhibits the Temkin-Leung-solution of aluminum particles in a 1.8 g/cm3-densitymatrix along larger ranges of the dynamic working point y/kR. Large particles always demix, whereas for small particles a transition between dynamic homogeneous behavior and gradual demixing can be adjusted.
337
This mechanism leads to the cases of detonative reactions in the shock front, where the particles remain on the spot, and late postdetonative reactions after the primary detonation, if the particles move backwards. The density ratio, matrix viscosity (toughness), and piarticle sizes are the controlling factors.
1
0.7!
..*.....*... ... n 7E/ "..-
2/ 0.2
C
1.01
I 0.1
I
1
I 10
I 100
I 1'10'
I
I 1.103
PlOL
Y m Figure XIV-14. Dynamic behavior of aluminum particles in working point yikR. {Equation [XIV-18]}.
explosive of assumed density 1.8 g/cm3as function of the dynamic
GERMAN PERMITTED CLASS 111 EXPLOSIVES, AHRENSAND KUHN-KAUFER SELECTIVITY Ahrens [XIV-25, XIV-211 started the unique development of permitted Class I11 explosives in Germany in the 1950's. The goal was to get an explosive that detonates powerfully in a bore hole (confinement), yet is relatively inert when unconfined: A freely suspended detonating explosive charge does not initiate firedamp explosions.
338 Chapter XIV
A sodium chloride/NG 90/10 mixture fulfills the latter requirement, but its working capacity is extremely low. The inert salt quenches any firedamp ignitions. A powerful explosive contains ammonium nitrate but ignites fire damps. Therefore in 1902/06 Bichel [XIV-261 suggested to replace sodium chloride by the relatively chemical inert inverse stochiometric salt pair potassium nitrate ( p = 2.109 g/cm3)/ammoniumchloride (p = 1.536 g/cm3),which intentionally reacts in detonation similar like ammonium nitrate and potassium chloride. This was no technical solution, until Ahrens formulated the idea of a two-step reaction. This idea was that (depending on the proper reaction kinetics) a primary detonation of the sensitizing NG occur, and a late postdetonative reaction of this inverse salt pair can follow in a confining bore hole. This means that in a freely suspended charge the salt constituents of the explosive will not react, rather they act as quenchers of any firedamp ignition. In a confining bore hole, however, this inverse salt pair reacts like ammonium nitrate and potassium chloride. At that time the German explosives industry pioneered this balancing act between safety and power by a weak primary detonation and a late postdetonative reaction. They succeeded by trials and errors supported by Ahrens’ ideas, but an actual working model was long in coming (in the following). It is much easier to understand this behavior of detonation by demixing processes caused by the dynamic particle mobilities {shown in 1979 in honor of Ahrens’ 70thbirthday [XIV-211) than in the terms of any Arrhenius kinetics. The essential dynamics is generated by the detonating sensitizing NG. The dynamical mechanical separation of the inverse salt pair alters their intimacy, and prevents their reactivity. In a freely suspended detonating cartridge the components are dispersed into the surrounding air and quench any firedamp ignitions. In a bore hole this demixing also occurs, but the component’s concentration remains constant in the average volume. Therefore in the volume an additional postdetonative late powerful reaction becomes possible, which is responsible for the final working capacity. The experimentally established dominant influence of the explosive density (usually p = 1.15 - 1.20 g/cm3),the particle sizes and the explosive consistency on proper function are direct governing parameters of the above sketched particle mobilities. Figure and XIV-12 show the possible domains of the particle mobilities for this type of explosive.
Experimental Demonstration of a Selective Detonation In the 1930’s Ahrens [XIV-251 demonstrated by an experiment the selective detonation: It is possible to mix reactive chemicals, such as organic dyes, with explosives. Depending on circumstances, this dye disperses in the air or fully reacts with the explosive in the detonation.
339 Ahrens
Kuhn-Kiiufer Selectivity’s
Effects of Surface Coatings Kuhn and Kaufer [XIV-271 state that a surface-active substance neither acts as an explosive nor as an inert, but that there exists a new unexplained mechanism in detonations. In permitted explosives the addition of tetryl to nitroglycerine evidenced no effects, whereas an intimate diminution of sodium nitrate with tetryl or dinitrodiphenylamine (DNPA) or silicon up to 0.15 wt% changed the explosives characteristics. The addition of 0.03 wt%, and the same grain sizes of the salt components brought optimal results. Such surface-coated material (mostly only the nitrate was covered) evidenced an increase of the lead block volume of 20 to 30%. Even a silicon coating led to an increase of up to 15%. The crushing values according to Hess were affected. The distance of detonation transfer mostly decreased, but for tetryl coating an increase from 5 cm up to 50 cm resulted. The infinite diameter in the case of coatings compared to detonation velocities increased to > 2000 1700 m / s for the uncoated system. The detonation velocities of surface-coated equimolar mixtures of ammonium nitrate and salt showed no effects. In classical terms there is little understanding of the effects of surface-coated explosives by inert waxes, and non of the importance of tetryl coatings of salts used in explosives, as example in permitted explosives. Since this author entered the field of explosives at the late stage of the development of Class I11 permitted explosives, he is aware of the discussions on the effects of surface coatings of potassium or sodium nitrate by tetryl, and dinitrodiphenylamine by about 0.03 wt% of the nitrate. The originators relied on their experimental results, and the others (including this author) remembered that fiom the thermochemical viewpoint an amount of 0.02 % in the total explosive was ‘nothing’. Accordingly, in later developments surface coatings were not applied. Theory of Surface Coatings In Figure IV-5 (Chapter IV) there are shown surface pressures on a (freely suspended) sphere as hnction of its relative size For small particles the near-field pressure on the sphere’s surface is low and relatively uniform over the surface. As the particle size increases more and more directionalities come out and the pressure increases. As a consequence “storms” of the particle velocities arise on the spheres’ surface. By this mechanism we get in addition to the dynamic demixing process a selectivity with respect to interactions of surface-coated materials. For small particles effects of surface-coatings disappear, but for larger particles effects of surface coatings become significant. These effects lead to a powerful dispersion of any thin surface coatings of the grains into the surrounding medium, so that an increased reactivity results. In the case of wax-coated explosive grains, two cases exist: For thin coatings the surface material disperses and contributes positively to reactivity. For thick coatings the
340 Chapter
effects are those of a vibration insulating layer, so that such a coating can also cancel such an effect. A further application can be to strip off oxide layers from metal particles.
New Aspects These presented dynamic demixing processes of the constituents open the doors wide for tailoring explosive properties, which are far removed fiom any classical code considerations. Reversibly, such effects are the reason why an HE estimate fails completely in the case of commercial mixed explosives. Table XIV-3 compares the CHEETAH predictions with the reality of a permitted Class I11 explosive (K-735). Table XIV-3. CHEETAH 1.4-results of Class I11 permitted explosive K 735 with and without tetryl, and experiments D m [dS]
p
[m31
Exp. without tetryl CHEET without tetryl Exp. with tetryl CHEET with tetryl
PCJ [GPa]
p [GPa] explosion in const. volume
lead block [cm31
mech. energy of detonation @J/cm3]
70
1.2 1.2
1,700 3,912
3.42
0.99
1.2 1.2
2,000 3,916
3.43
0.99
- 1.702
100 - 1.706
CASCADED COLLISIONS There are also less-known particle effects that can be used for finite sized samples. For example Curran and Winkler [XIV-281 used a sequence of elastic collisions to produce velocity gains in dynamic experiments. The effects are vividly demonstrated also in a physical toy "Astroblaster", where four balls of progressively smaller diameter resting on top of one another are dropped to the floor, and the smallest one jumps many times the dropping height.
341 Cascaded
Central Elastic Impacts The general case :shown in Figure XIV-15 is considered, where perfect elastic conditions are assumed. On the left the larger mass ml strikes with velocity v1another (smaller) mass m2 at rest or in motion with velocity v2,o. The velocities of ml, v1' and m2, v2 after collision are of interest. If the arrangement is cascaded by n masses one gets with the abbreviation 4, = and the conservations of momentum and energy the ultimate velocity v,, of the nthmass m,.
ml
m2
ml -b
v2,o
v1'
+f, V1
Before impact
m2
Figure XIV-15. Configuration of the central elastic impact. This arrangement can also be cascaded, where m2 continues to push another (smaller) mass m3, etc.
v2
After Impact
Winkler [XIV-291 considered these cases with arbitrary velocities v2,0, (v2,o = 0, or - v1 or arbitrary) and a chain of n masses, also including elastic losses. The condition for continuous impacts in the n-mass chain is q, = f < 1 , otherwise a decoupling occurs. He obtains for the case of Figure XIV-15:
(XIV-26) and for a chain of'n bodies
(XIV-27) For the special case v2,o = 0 he gets for 2 and n consecutive impacts
(XIV-28) with the asymptotic results q 3 0 v, = 2" vl, so we get for a mass chain v2 = 2vl; v3 = 4 v1............. Figure XIV-16 presents the situation for varying mass ratio's q. The difference in velocity between the two masses after collision remains constant. =v*.
(XIV-29)
342
Elasticlnl~act Masses
rest
9 8
6
9 5
5
4
3
' 2
>'
1 0
-1 -2
0,001
0,Ol
0,1
10
1
100
loo0
q = WWl Figure XIV-16. Velocity ratios of the final element and the original velocity of the first element as function of mass ratio q (which is the same for the whole cascade).
If, however, v2, = - v,, as it is in the "Astroblaster" or if a large mass moves against an expanding bubble or moves into a swarm of particles one gets
i=k
with the asymptotic results q 0 v, = (2" -l)v,, so we get for a mass chain v2 = 3vl; v3 = 7 v, ............. It is amazing to note that with an initial velocity of v1 = 5 m / s by a n = 11-chain (hypothetically) the velocity of escape with 11.3 km/s could be almost obtained. Winkler demonstrated, however that this is really hypothetical, since even with a mass ratio of 0.01 a mass of 1 g needs 13 bodies in a chain with a resulting mass of the 1St one of ml = 10" kg! Such multicollision phenomena demonstrate that by particle effects high and very high velocities in the gas phase beyond the particle velocities of the fumes are realistic. These cascaded particle velocities explain the expansion (by 1,OOO%) and acceleration of the shaped-charge jet with a high gradient of the velocities fkom the tip velocity of
343
about 9 km/s down to the main mass of the slug with 300 to 1,000 Typically only about 10 to 20% of the liner mass "goes" into the high speed jet (see the before mentioned mass ratios), not formerly understood [XIV-301. In classical relations the ultimate velocity umxof a body accelerated by an explosive charge is given by the maximum expansion velocity of the reaction products, where c is the sound velocity in the compressed state. Using the classical relation c = r D + 1) , where D is the detonation velocity and r the polytrope of expansion one gets (XIV-31)
so that for an exploisive of about D 8 km/s and r 3 umaxis around 6 km/s. Velocities in excess of this quantity need special effects, see [XIV-311, and Chapter XXII. References [XIV-I] W. D. Derham, Experimenta & Observationes de Soni Motu, aliisque ad id attinentibus, factae, Phil. Trans. Roy. SOC.London 26 (1708), p. 2/36. [XIV-2] J. Tyndall, The Science of Sound, New York, Phil. Libr. Inc. (1964), p. 3051376. [XW- 31 J. W. S. Rayleigh, The Theory of Sound, Vol.
1886, (95 296, 334, 3 3 9 , Dover
Pub., New York, reprint 1945. [XIV-41 W. Konig, (1891)3, p. 353/370.
Untersuchungen, Ann. Phys. Chem. NF 42
[XIV-5] C. J. T. Sewell, The Extinction Of Sound In a Viscous Atmosphere by Small Obstacles of Cylindrical and Spherical Form, Phil. Trans. Roy. SOC.A 210 (1910), p. 2391270. [XW-6] H. Lamb, Hydrodynamics, 6thed., Cambridge University Press, 1932, $298. [XIV-7] R. J. Urick, The Absorption Of Sound In Suspensions Of Irregular Particles, J. Acoust. SOC.Am. 20 (1948), p. 283/289. [XIV-81 E. Skudrzyk, Die Grundlagen der Akustik, Springer-Verlag, Wien, 1954. [XIV-9] E. Skudrzyk, The Foundations of Acoustics, Springer Verlag, Wien-New York, 1971. [XIV-101 F. A. Morrison, and M. B. Stewart: Small Bubble Motion In An Accelerated Liquid, Trans. ASME, J. Appl. Mech. (1976), p. 399/403. [XIV-111 S. Temkin., and C.-M. Leung: On The Velocity Of A Rigid Sphere In A Sound Wave, J. Sound and Vibration 49 (1976)1, p. 75/92 (there is an error in print.) Without error: [XIV-141. [XIV-121 S. Temkin, Elements of Acoustics, John Wiley, New York, 1981. [XIV-13] W. Gregor, and H. Rumpf The Attenuation of Sound In Gas-Solid Suspensions, Powder Technology 15 (1 976), p. 4 3 6 1. [XIV-l4] S. L. Soo, Fluid Dynamics of Multiphase Systems, Blaisdell Pub., Waltham, MASS (1 967).
344 Chapter XIV
[XIV-I 51 P. S. Epstein, and R. R. Carhart: The Absorption of Sound In Suspensions and Emulsions. I: Water Fog In Air, J. Acoust. SOC.Am 25 (1953)3, P. 553/565. [XIV-16] J. C. F. Chow, The Attenuation and Dispersion ofAcoustic Waves in a Two-Phase Medium, Thesis, Brown University 1969. [XW-17] J. E. Cole, and R. A. Dobbins, Measurements of The Attenuation of Sound by a Warm Air Fog, J. Atmospher. Sc. 28 (1971)2, p. 202/209. [XIV-18] 0. Brandt, H. Freund, and E. Hiedemann: Zur Theorie der akustischen Koagulation, Kolloid Z. 77 (1936)1, p. 103/115. [XIV-l9] C. 0. Leiber, Dynamic Particle Motion In Materials As A Consequence Of The Finite Shock Rise, 5'h Int. Conf. High Energy Rate Fabrication, 1975, Denver, CO, p. 1.5.1 1.5.16. [XIV-20] C. 0. Leiber, Shock Augmentation By Bubble Flow, J. appl. phys. 47 (1976)9, p. 3971/3978. [XIV-2 I] C.O. Leiber, Zur Ahrens' Selektivitat, Ansatz zu einer Theorie gewerblicher Sprengstoffe, Nobel-Hefie 45 (1979)3, p. 65/76, Correction: Nobel-Hefie 45 (1979)4, p. 167. [XIV-22] J. B. Homer, and T. R. Hurle: Shock Tube Studies on the Decomposition of Tetramethyl-Lead and the Formation of Lead Oxide Particles, Proc. Roy. SOC.London A 327 (1972), p. 61/79. [XIV-23] L. van Wijngaarden, Hydrodynamic Interaction Between Gas Bubbles in a Liquid, J. Fluid Mech. 77 (1976), p. 27/44. [XW-241 S. Temkin, On The Response of a Sphere to an Acoustic Pulse, J. Fluid Mech. 54 (1972)2, p. 339/349. [XN-251 H. Ahrens, Die Bedeutung des selektiven Verhaltens der Detonationswelle im Gebiet der Wettersprengstoffe, Explosivstoffe 4 (1956), p. 102/109. [XzV-26] C. E. Bichel, Verfahren zur Herstellung wettersicherer Sprengstoffe, DRP 175 391, 1902/1906. [XIV-27] G. Kuhn, and H. Kaufer: iiber die Beeinflussung des reaktionskinetischen Geschehens bei Sprengstoffen durch Behandlung von Kristalloberflachen mit oberflachenaktiven Stoffen, NOBEL-Hefie 20 ( 1954)4, p. 97/116. [XIV-28] S. Winkler, Personal communication. [XIV-29] S. Winkler, Unpublished Results, "Multiple elastische StoDe", and "Der zentrale elastische Stow fiom March 2002. [XIV-30] M. Held, in Hazard Studies for Solid Propellant Rocket Motors, AGARDograph No. 3 16, 1990. [XN-311 L. A. Merzhievskii, V. M. Titov, Yu. I. Fadeenko, G. A. Shvestsov, High-speed Launching of Solid Bodies, Journal Combustion, Explosion and Shock Waves, 23 (1987)s p. 5761589.
xv
FINITESHOCK RISE
Dipole scattering was described in acoustic terms in chapter XIV. Description of the characteristics of 'shock' in acoustic terms is necessary with respect to the applicability of these results to shock phenomena,. In classical understanding [XV- 11 the shock rise length approaches atomic or molecular distances; therefore, particle displacement by shock attack cannot be expected. Experiments described in this chapter show that particle and void displacements do occur. This allows us to discuss classical shock-wave theory. Reservations and opposition against these considerations arise fiom the suggested acoustic treatmenl of shock and detonation phenomena. Opposing arguments are that acoustics is a science of small density variations in linear terms, and shocks deal with finite density variations in nonlinear terms. While it is true that shock phenomena are explained in nonlinear terms, acoustic phenomena are not limited to linear theory. It is well-known that expansions (Taylor series for example) are possible for finite functions, where the first derivative is of an acoustic nature. Consequently, whenever additional 'acoustic' quantities in this first derivative come into play, these can never be negated by later nonlinear terms. However, if one neglects the compression-dependent quantities, the laws of shock physics and acoustics show an isomorphism. The difference is that in acoustics the quantities are almost constants, in shock physics they are compression diependent.
Plane-Wave "Exact" Shock Relations From the continuity of mass, one gets for plane-wave assumptions with original density poat ambient pressure po, the density p at high pressure p along with the shock velocity us and the particle velocity up P(us -
= Pous
(XV-1)
leading to the density or specific volume (v = Up) ratio
(XV-2) and the relative compression (condensation) Ap/po = (p - po)/po, or Av/vo = (vg - v)/vo
346 Chapter XV
AP Po
or us-up
AV - u p
(XV-3)
us
The conservation of momentum leads to = POUSUP
(XV-4)
if the ambient pressure po is small compared to the shock pressure p. It has been a long-accepted rule that a linear relationship exists between the shock (U,) and particle velocity (up) in this plane-wave relationship. us = A -t B u p 2
(XV-5)
where A should be 1.2 times the bulk sound velocity co, and B may be approximately 1.7 [XV-21. The Hugoniots of liquids have in the past been easily obtained by determination of the bulk sound velocity and one or two high pressure points.
Description of a Finite Pressure Rise As shown in Chapter VIII, one can describe the dimensionless particle size finite shock rise in terms of a shock-rise length Al, or shock rise time (Figure VIII- 17) by
for a
(XV-6)
(XV-7)
In the preceding chapter it was shown that particle sizes exist, where in the dynamic activated state no relative particle mobility exists. This case can be used for the estimate of finite shock rise.
341
SHOCK-ACTIVATED PARTICLE MOBILITIES Dipole scattering has been described in acoustic terms. With respect to the applicability of the results to shock phenomena, it will be shown that the acoustic phenomena also apply in shock physics. Direct experiments to demonstrate this behavior in explo,sivescannot be performed because the energetic materials would decompose. Therefore, we assume that the dynamic behavior of inert materials also applies to high explosives, for which dipole scattering can be shown only indirectly.
Experiments on Dynamic Particle Mobility [XV-21, [XV-31 We know from chapter XIV that particles in a shocked material show different dynamic properties depending on the density ratio p'/poo,their size, and the dynamic working point or strength of the matrix: They may be fixed on the spot for various reasons. They move in or opposite to the shock direction, depending on their size and the density ratio p'lp,.
They can agglomerate. Particles p' > poocan produce hydrodynamic pairs. Cast and spheroildal cast iron are good substances to demonstrate dynamic particle mobility, because the particle sizes and shapes of graphite vary greatly. Also, the density ratio of carbon and iron is p'/poo 0.25. This leads to a relative, asymptotic
-
particle motion from shock attack u'p/up,m= 2 according to Equation (XIV-1). The condition dp/u,,, = 1 can always be achieved in an inviscid medium by an appropriate particle size when p'Ipm 1. According to Equation (XIV-12) particle size is 1.735 for the density ratio p'lp, = 0.25. Larger particles are lagging and the smaller predominate. The smaller ones disintegrate the solid-state matrix due to their mobility.
-
Cast iron of constant carbon content has been used when particle sizes, shapes, and distributions were varied. [XV-41 Figure XV-1 shows a dynamically activated graphite particle in shocked cast iron, which drives the cracks like a moving wedge, so that the cast iron disintegrates. The direction of shock attack is shown by an arrow. Another
348
Chapter
picture is taken to show that the particle in the circle of Figure XV-1 really moved. Figure XV-2 shows vibration or striation patterns of this graphite particle, which are absent in static fracture of cast iron. The shock energy is dissipated very rapidly by the particle motion. Usually the whole samples of normal cast iron (with small graphite particles) are fragmented by shock attack into tiny pieces. The particles’ mobilities are the reason why cast iron will not ‘survive’ shock experiments.
Figure XV-1. Shocked cast iron, where the graphite particle (in the circle) dnves the crack. That this particle really has moved is shown in Figure XV-2, which represents a micrograph of the iron surface of the upper left (in the circle) near the graphite particle.
Figure XV-2. The surface of the iron (Fe, dark in Figure XV-I) shows vibration structures produced by the mobility of the graphite particle C. This structure is completely absent in statically broken cast iron. The arrow indicates the shock direction.
349
Entirely different behavior was found for spheroidal cast iron, where the sample was not destroyed, even in intimate contact with an explosive (Fig. XV-3).
Figure XV-3. Shocked spheroidal cast iron of same composition as the cast iron of Fig. XV-I. The high explosive was attached to the upper side. The shock direction is indicated by the arrow.
In Figure XV-4 one spherite remains in place, another one moves opposite to the shock direction, and a small fragment moves in direction of the shock. No disintegration of the sample occurred, but the spherites vibrated very powerfully, due to the scattering of the shock energy, as can be seen in Figure XV-5. This vibration pattern is completely absent in static fiacture.
Figure XV-4. Shocked spheroidal cast iron of same composition as the cast iron of Fig. XV-1. Upper left a larger spherite breaks into a larger piece, moving opposite to the shock direction (arrow), and a small one moving in the shock direction. The spherite in the center remains in place, but is deformed due to its pressure reflecting properties corresponding to kR - 1.735.
350 Chapter XV
Figure View into a broken graphite spherite, where the vibration patterns have absorbed the pressure energy. Statically loaded spherites do not exhihit such a structure.
Figure Spheroidal cast iron somewhat below the frame of Figure X V - 3 . The spheroids did not move.
The Shock Rise One can estimate the shock-rise length A1 = us for full shock attack, where is the rise time of the shock and us its velocity, from the size of the locally fixed particle. For kR = 2nR/h, one obtains by Fourier transform the expression = 2nR/(2..3)z
With
=
-
us 2.5
x
us = 2.5
x
WAl
{see Equs. (XV-6) - (XV-8)).
1.735 for spheroidal cast iron, one obtains for the shock-rise length
351 Shock activated Particle Mobilities
A1 - 0.72
x
(2P:).
One gets a shock-rise length of -1 8 pm with 2R = 25pm. This quantity compares favorably with the results reported fiom other methods [XV-51, [XV-61. A1 compares with tens of pm, but not with atomic distances [XV-71, [XV-81. Figure XV-6 is a picture of the shocked spheroidal cast iron a little distance below that of Figure XV-3. Clearly it can be seen that all the spheroids remain on their spot. The dynamic particle mobility was also investigated in another way: As sketched at the left side of Figure XV-7, an aluminium rod was used, its upper and lower surfaces optically polished. Tungsten (W)-powder (-1 pm particle diameter) was inserted between the two piolished surfaces. As expected, the W impacts only occurred in the upper part of the rod. The flow was laminar, as indicated by the Reynolds number, so that the holes closed again or impacts occurred. Agglomerations were also seen (Fig. XV-8). More favorable for explosives are singular impacts, see Fig. XV-9.
Figure XV-I. Hydrodynamic pair formation by a dynamic, activated tungsten particle. 1 p' -1 9.3 g/cm3 in the boundary between the aluminium pieces of densityp, part only. The downstream impacts were produced by cavities.
tungsten particles of density
- 2.7 gkm3 impacted the upstream
352 Chapter XV
Figure XV-8. Dynamic impacts of tungsten particles on the upper side of the rod (Fig. Impacts closed again or occurred. Also, agglomerations had taken place.
Figure XV-9. Dynamic impacts of tungsten particles on the upper side of the rod (Fig. Impacts closed again or occurred. Also, agglomerations had taken place to a extent.
Hydrodynamic Pair Formation Assume that dense particles acquire a relative velocity lulP- ~ ~opposite , ~ tol the shock direction: When this relative velocity is low, the trail closes as in a laminar flow. When this velocity gets larger, a wake is formed that is driven like a void in the shock direction (Fig. XV-10). This produces a hydrodynamic-pair formation of wedges, as seen in Figure XV-7, where a tungsten particle impacted the upstream side, and the resulting cavity the downstream side. The matrix was aluminium. That the hole at the downstream side was really produced by a void was established by the similarity of cavity impacts on metals (cavitation holes).
353
That even a tiny void can act like a rigid particle, and drive cracks is shown in Figure XV-10, where a void in molybdenum has driven this trail. Molybdenum was chosen, since the sample exhibited voids of nearly identical size uniformly distributed in the volume. These results indicate that the dynamic behaviour of any condensed material strongly depends on the mobilities of the voids and particles. Some dramatic effects will be outlined in later chapters.
Figure XV-10. Dynamic trail of a void in molybdenum. The void did not close despite the high dynamic pressure of 375 kbar. However, such a void completely disappears by hot forging. The direction of the shock is indicated by the arrow.
Memory Effect
- (Materials Never Forget Shocked States)
The dynamic mobilities of particles and voids are memories of shocked states lasting forever, if no counteracting events occur. Thus we have a historic shock indicator, which may be used in accident investigations, criminology, and geology. Some examples follow:
Liquid Carbon-dioxide Tank Car Explosion, 1976 A liquid carbon-dioxide tank car explosion occurred in 1976 in Haltern, FRG [XV-91 (see Fig. XV-11).
354 Chapter XV
We have two hypotheses for this accident: Rupture caused by materials embrittlement, [XV-8] and A bubble-resonance explosion of the pressurized, liquid carbon dioxide. This represents a physical explosion.
Figure XV-11. Liquid pressurized carbon-dioxide tank car explosion of Haltem, 02.09.1976.
Figure XV-12. Electron micrographs of the fracture surface of a sample of the tank-car plate at a distance of 300 - 400 m. The pore elongations indicate an earlier shock attack.
The first hypothesis is the officially adopted version. However, there are strong physical arguments for the second mechanism. Debris from the tank-car plate were found at distances of more than 400 m. The explanations for such distances were that gases built up from propelling gas evolution by vaporisation. However, liquid carbon dioxide shows a transition to the solid state when expanded. Therefore, this distance and fragmentation is only possible by shock attack from the liquid medium. Estimates are given in Reference [XV-91 and Figure XV- 12 {electron micrographs of the fracture surface of such a debris sample (from Haltern accident) at a distance of 300 400 m} . The pore elongations similar to those in Figure XV- 10 indicate that a shock attack has really taken place.
355
Shock
The Decline of the Minoan Culture The minoan culture on Crete disappeared at the time of the Santorin (Thera) volcanic activity (- 1400 B.C.) [XV-111, it was wiped out over a radius of about 120 km. Sir Evans in 1927 believed that a catastrophic earthquake was the reason, Marinatos in 1939 reported links to the Thera volcanic explosion with associated tsunami waves. Wunderlich in 1972 [XV- 121 favored processes of cultural evolution. He offered bituminous alabaster samples of Knossos palace (Fig. XV-13) as proof. This alabaster is a geological temperature indicator, because bitumen disappears between 60°C and 120°C, leaving only white burned gypsum. Therefore, this bituminous alabaster sample from the Knossos palace indicates that there had been no fire and no classical catastrophy, usually associated with tire. But this sample in Figure XV- 13 also shows signs of a shock attack, which resulted in almost unidirectional particle displacements. So we have in this bituminous alabaster sample, in addition to the geologic temperature indicator, a tool to prove a shock wave attack, even after some 1000 years.
Figure XV-13. Bituminous alabaster sample from the Knossos palace, which show indications of a probable shock attack, and that temperature did never exceed 60 - 120°C. Picture taken from the late Wunderlich [XV-I 21.
Different Dynamic Fracture Modes [XV-131 As will be outlined later, the opening of cracks in a dense quasi-homogeneousmaterial depends on the presence of tensile pulses. But Figure XV-1 introduces the possibility that other fi-acture mechanisms might also be at work. The dynamically activated graphite particle dlrives a crack, not by tension but by dynamic pressure. Taking into account the result of Figure XV-10, voids or bubbles can also become mobile; they disintegrate the solid matrix like moving wedges. It is important to demonstrate that different dynamic failure mechanisms exist for single crystals and dense and porous materials which cannot be detected by more static
356 Chapter XV
testing. Examples are given for the Hugoniot Elastic Limit (HEL) of sapphire in different orientations, (Fig. XV- 14) where the cumulative fi-equencyin Gaussian coordinates is plotted versus the HEL. As can be seen, no Gaussian distribution applies.
.E
Bf
-1
Figure XV-14. Cumulative frequencies of the Hugoniot Elastic Limit (HEL) of sapphire single crystals in different crystal directions. As can be seen no Gaussian distribution exists.
9
5
M-
10 -
1
i I
1
1
m
KO
HE1
in
Kbar
The HELs are plotted for dense and porous alumina, beryllium oxide, and boron carbide in Gaussian coordinates, where similar and dissimilar slopes (variances) can be detected (Fig. XV- 15). The porosities are listed in Table XV- 1. The dynamic fi-acture behavior differs in the two columns by a statistical confidence level of 99%, but is the same within each column. The limiting porosity between the two failure modes is between 0.8 and 2%. Note that the same fracture mode is present for porous materials of different composition in the left part. To determine such differences by more 'static' testing, no meaningfd differences were found by measuring the bending strength (Fig. XV-16).
Figure Cumulative ftequencies of the Hugoniot Elastic Limit (HEL) for different types of brittle polycrystalline ceramics. With a confidence level of 99% the HEL-determining mechanism is different for the highdensity samples Lucalox and hot-pressed Carborundum.
cows
11
100
150
HEL in Kbar
351
Shock activated Particle Mobilities
x E 999 Wesgo A1995
Figure XV-16. The static bend-strength mechanism is, with a statistical significance of 99%, the same for high density porous materials.
10
LO
45
bend strength
kplmm2
Table XV-1. Materials of different dynamic fracture modes.
Material Coors AD 85 Diamonite P-3142-1 Wesgo A1-995 Boron carbide Beryllium oxide
Material
Porosity [%I
Carborundum hot pressed
0.8
Lucalox
0.2
Porosity [%] 6.6 5.5
4.0 2 5.6
In summary, the dynamic fracture mechanisms differ for single crystals, for high density polycrystalline materials, and for porous materials. For the latter, materialspecific properties show only weak influences. Such differences cannot be detected by static tests.
References [XV-11 R. Courant, and K. 0. Friedrichs: Supersonic Flow and Shock Waves, Springer, New York, Heidelberg, Berlin, Appl. Math. Sciences, Vol. 21, 1948, 1976. [XV-21 G. Flajs, and C. 0. Leiber: Failure of Usual K-Statement in the Case of Dynamic Attack on Heterogeneous Materials, Int. J. Fracture 10 (1974)3, p. 4481451. [XV-3] G. Flajs, R. l-lessenmiiller, C. 0. Leiber, R. Wild: Dynamisches Verhalten von Diskontinuitaten in einer Festkorper-Matrix, Z. Metallkunde. 65 (1974)8, p. 5391541. [XV-4] Prof. Rudiger from Krupp Forschungsinstitut, Essen, kindly supplied these samples.
358 Chapterm
[XV-5] M. A. Mogilevskij, Metallographic Method of Assessing the Thickness of a Shock Wave Front, Fizika gor. vzryva 9 (19736, p. 905/909. Translation: Combustion, Explosion, Shock Waves 9 (1973)6, p. 795/798. [XV-61 S. Winkler, S.: Messung von Profilen schwacher StoDwellen, Verhandl. DPG (VI) 11 (1976)2, p. 136/137. LXV-71 C. 0. Leiber, Dynamic Particle Motion in Materials as a Consequence of the Finite Shock Rise, 5'h Int. Conf. on High Energy Rate Fabrication, Denver, CO, 24/26 June 1975, Conf. Proc. 1.4.111.4.10. C. 0. Leiber, Shock Augmentation by Bubble Flow, J. Appl. Phys. 47 (1976)9, p. 397113978, [XV-91 C. 0. Leiber, Explosionen von Flussigkeits-Tanks. Empirische Ergebnisse - typische Unfalle, J. Occ. ACC.3 (1980), p. 21/43. [XV-101 J. Ziebs, and K. Naseband: Prtifzeugnis der BAM vom 20. 7. 1977. [XV-I I] D. L. Page, The Santorini Volcano and the Destruction of Crete, London, 1970. [XV-121 H. G. Wunderlich, Wohin der Stier Europa trug, 2. Auflage, Rowohlt, Reinbek, 1972. [XV-131 C. 0. Leiber, Vergleich der StoRbruchmechanismenbei dichter und poroser Keramik sowie bei Einkristallen, Planseeber. Pulvermetallurgie 22 (1974)1, p. 41147.
XVI VOIDPRECURSORS We showed in the preceding chapters that a void or bubble can collapse in a pressure field or behave as a rigid particle, so that it might be driven by the dynamic pressure in the shock direction (see Figures XV-7, and XV-10). The maximum attainable particle velocity of such a void amounts to = 3 up, (Chapter XIV), where an internal synchronization mechanism has also been outlined, so that pressure driven bubbles appear at the front in the same size and phase so that a bubble piston of uniform size and velocity is activated [see the Figures (XIV-6) and (XIV-7)]. This bubble piston dominates the material’s shock behavior above a certain pressure level, where this piston overtakes the shock front of the matrix material, then expands strongly into the low-pressure medium. As pointed out in Chapters VI, and VII the corresponding and most powerful dissipative losses of bubble dynamics force the onset of chemical reaction. Voids are only collapsed by pressure below a definite limiting pressure. This is a very general mechanism, difficult to establish in inert materials but easy in reactive materials because such a mechanism leads to onset of chemical reaction or to HVD. It is conceivable that collapsing bubbles relate to Low Velocity Detonations (LVD) (see Chapter VIII), and expanding bubbles ahead of the shock front to High Velocity Detonation (HVD). The precursors from the high pressure side are messengers between the reacted and the unreacted explosive, passing information to the unreacted region that something is about to happen. A messenger of this kind is not predicted by the classical detonation model, but such information is transmitted by fluctuations of the sound velocity in the reacted medium in more modern modifications.
Precursors in Detonation The idea that precursors may be active in explosion processes is quite old. As early as 1883, Berthelot [XVI-I] suggested that, in gas detonations, hot gas molecules from the reacted side penetrate into the unreacted fresh gas and initiate it. Schuster [XVI-21 was the first to describe the basis of early detonation theory in the terms of Riemann’s shock waves [XVI-3]. This proved that shock waves are the fastest possible events. This finding also (discredited later suggestions of precursors. It is ironic that Schuster was the one to discuss Lord Rayleigh’s reservations against Riemanns waves and pointed out the approximate character of the plane-wave assumption. He stated:
360 Chapter
to
TOApin's [XVI-41 paradigm that ejecta of the hot reaction products should induce the further chemical decomposition in condensed explosives, the opposition pointed out that this mechanism would fail in the transmission of detonation through inert barriers. Cook's heat pulse seemed to address the same phenomena [XVI-51. His experiments were not understood with respect to said paradigm, and had therefore been ignored. Since we now know that shock precursors do exist, this situation changes. From the experience with detonation of cavitated charges, discovered by Ahrens and Woodhead {summary [XVI-6]}, we now know that the detonation velocity is not the fastest event. In the following, we consider the dynamically activated voids in the unreacted twophase system or hot spots of the beginning chemical reaction. Here, hydrodynamic events lead to initiation of the chemical reaction of HVD, and not the reverse!
Luminosity ahead of the Detonation Front Detonation luminosity has been investigated for a long time. The phenomenon is discussed by Johansson and Persson [XVI-71, but they do not mention precursors. Fosse [XVI-s] (Fig. XVI-l), van den Berghe [XVI-91, and Held [XVI-101 have found light ahead of the detonation zone of porous explosives. The interpretation has been in terms of luminous air shocks or scattering of detonation luminosity in the unreacted material, but Leiber [XVI-1 11 determined in 1970 that luminosity precedes the detonation zone. Spaulding [XVI- 121 substantiated, in experiments with porous explosives, existence of luminous ejecta in unreacted, low-density material against a solid at stagnation pressures of 10 - 20 kbar. Held [XVI-131 did not find luminous precursors for detonating homogeneous NM.
Pressure Precursors Spaulding [XVI-121 determined pressure precursors with quartz gauges, Hessenmiiller and Leiber [XVI- 141 evaluated X-ray densitographs of detonating low density PETN (courtesy of Dr. Jamet, ISL) by a photographic isodensitographic technique. These results were later confirmed by computer-aided picture analysis [XVI-151. Pressure precursors appeared in both cases, as seen in Figure XVI-2. Because cylindrical charges were used, the precursors are only visible in the center of the charge. Excellent color prints can be made fi-om the photographic densitographs. An example is the cover of this book.
361 Precursors
Figure XVI-1. Luminosity ahead of the detonation front of ANFO of 50-mm diameter in water [XW-81. The shock front is seen in the water. (Courtesy of Dr. FossC.)
HVD Initiation of Liquid Explosives The macroscopic limiting condition for bubble precursors ahead of the shock front and for onset of HVD are plane-wave Hugoniots of the unreacted homogeneous explosive: HVD:
(XVI- 1)
or in terms of the relative compression HVD:
Av -=--20.33. vo
(XVI-2)
us
LVD pressure generated a compression of about Av u LVD: -=--P=0.15.*.0.17. YO
us
(XVI-3)
3 62 Chapter
Figure XVI-2. Evaluation of an X-ray photograph of detonating porous PETN. Top: X-ray photograph of detonating porous PETN (Courtesy Dr. Jamet.)
Center: Photographic isodensities of the above picture.
Bottom: Computer-aided picture analysis
363 Precursors
Therefore, gaps in icompression, pressure and detonation velocity must exist between SVD, LVD, and HVD phenomena. A transition can only occur discontinuously, SO we HVD transitions, and also the reverse, expect only discontinuous LVD HVD 3 LVD. Furthermore, the cited macroscopic conditions are only asymptotic approximations of microscopic events under favorable conditions if any discontinuities appear. These lead in a pressure field to pressure and tension waves; the latter leads to two events: onset of cavitation, and an originally homogeneous material becomes heterogeneous. Therefore, we have to expect considerable scatter in initiation experiments. But none of these events can be understood by the classical shock heating process of initiation. Some difficulties are: The shock velocity of detonating explosives is the detonation velocity D. The corresponding particle velocity W is W = D/( l- + 1) when a F-model is used. Kamlet, Short, and Defourneaux [XVI-16] obtained from classical code calculations, correlations between r and the initial density pothat are independent of chemical composition:
r=-0'655 +0.702+1.107po
and
(XVI-4)
Po r=i.9+0.6~,.
(XVI-5)
For usual explosives r > 2 always holds. Then the relative compression at the CJ detonation pressure pcJ Av
-
-
1
i+r
(XVI-6)
is smaller than the necessary compression for initiation.
Nitromethane (NM): The experimental particle (up) and shock (us) velocities of unreacted NM are plotted in Figure XVI-3 as function of the shock pressure p [XVI-171 to [XVI-20]. The limit of the onset of HVD ~ i='3 x~ , corresponds ~ ~ to . a pressure p = 50 kbar. Walker and Wasley [XVI-2 11 found in their experiments minimum initiation pressures of -5 1 kbar for > 40-ps-long pressure pulses. Fauquignon and Moulard [XVI-221 reported initiation data of 53-kbar amplitude with a duration of up to 1.5 ps in hot-spot cascades for (even sensitized) NM. Campbell, Davis and Travis [XVI-231 reported in their classic paper initiation pressures of -85 kbar, depending on the induction time. This pressure corresponds closely to the onset of shock opacity [XVI-241.
364 Chapter mm '
1
1
I
NO-HVO- lnitiotron HVD- lnitiotim I
- LVO
* possible
-Detonation pressure
roo
50 P-
150 kbor
Figure XVI-3. Summary of the experimental plane-wave Hugoniots of unreacted NM, and their scatter according to References [XVI-I 71 to [XVI-20]. Onset of HVD starts at 5 3u,,. No HVD initiation was possible in liquids below Q , 5~ 3up,,, but was observed in solids [XVI-25]. A pressure- and velocity-gap exists between HVD and LVD phenomena.
Detonation velocities as well as diameter, ambient temperature, and pressure derivatives have been reported by many researchers. The detonation pressures reported range from 127 to 148 kbar, but few original experimental data have been published. Davis and Ramsay [XVI-26] have given sets of the free-surface velocities of Dural plates; their results are: po= 1.133 g/cm3 [23"C], D = 6.299 k d s , pcJ = 134 4 kbar, with r = 2.36 W = 1.875 km/s should hold. We get a relative compression (AV/V.)~~ = 0.298, compared to a compression at the onset of HVD of the unreacted NM of 0.33. This detonation compression corresponds to compression of the unreacted NM at 34.3 kbar.
*
NitromethaneIAcetone Mixture 84/16: One expects as lowest initiation pressure of HVD p > 68 kbar with po= 1.08 g/cm3 and the Hugoniot us = 1.892 + 1.697 up [XVI-27]. The maximum unreacted shock pressure, applied to a 4-mm-thick sample, was 84 kbar [(Av/vO)CJ = 0.3521. The detonation is characterized by: detonation velocity D = 6.00 km/s, particle velocity W = 1.68 determined by the electromagnetic method, and corresponding detonation pressure pcJ = 109 kbar [ ( A V / V ~=)0.281. ~ ~ This detonation compression corresponds to compression of the unreacted material at 39.3 kbar.
NitromethaneIAcetone Mixture 75/25: With p. = 1.05 g/cm3 and the Hugoniot us = 1.915 + 1.768 up, one expects as lowest initiation pressure of HVD p > 76 kbar [XVI-271. The maximum shock pressure applied to a 4-mm thickness undetonated sample was 83.5 kbar [(Av/vO)cj= 0.341)]. The detonation data are: D = 5.75 km/s, W = 1.47 km/s, pcJ = 89 kbar, and
365 Precursors
(Av/vJCJ = 0.256. This detonation compression agrees with the compression of the unreacted material at 32.9 kbar. Liquid TNT: With po= 1.473 g/cm3between 80.7 and 82.3"C, and the Hugoniot us = 2.145 + 1.571 up, one expects as lowest initiation pressure of HVD p > 100 kbar [XVI-281. The 10-mmthick sample detonated at shock pressures > 110 kbar. Campbell et al. [XVI-231 reported a pressure of 125 kbar with an induction time of 0.7 ps [(AV/V&~ = 0.357)]. They obtained, with the zero-thickness free-surface velocity of 2024 aluminum, PCJ = 171.8 kbar, corresponding to W = 1.791 km/s, and the corresponding compression is (Av/v&.~= 0.271 with r = 2.685. The detonation data at 93°C are: D = 6.600 kmls and po= 1.455 glcm3.This detonation compression corresponds to compression of the unreacted material at 55.7 kbar [XVI-29].
Problems with 1:heproper Hugoniots The accuracy of Hugoniot fits is very limited. This situation is illuminated for
Nitroglycerine (NG): With po = 1.596 g/cm3 Dremin et al. [XVI-301 specified the Hugoniot as us = 2.24 + 1.66 up. One expects as lowest initiation pressure of W D p > 134 kbar, but the authors found initiation of a 25-mm-thick sample at 120 kbar, but not at 110 kbar. Recalculations of the single experimental data by least-squares leads to the form: us = 2.292 + 1.624 up,
but if one takes into account the sound velocity c, al., [XVI-311 one gets
=
1.485 km/s fi-om Kurbangalina et
us = 1.540 + 2.164 up. We get pressures from 132.8 to 162 kbar for udup = 3. Even better is the situation for nitromethane, where we have 4 sets of data, and their combinations.
3 66 Chapter
Table XVI-1. HVD Initiation Data.
Substance
Density
co
Ref.
HVD Initiation [kbar] Estimated exp. Ref.
[m31
WSI
Nitromethane (NM)
1.124
1.294
[XVI-321
> 48
> 50
[XVI-211
Tetranitromethane (TNM)
1.636
1.039
[XVI-33]
> 45
> 70
[XVI-38]
Methyl nitrate
1.207
1.167
[XVI-311
> 42
Nitroglycerine (NG)
1.596
1.485
[XVI-311
> 90
> 110
[XVI-301
Glycol dinitrate
1.488
1.414
[XVI-311
> 76
Liquid TNT, 82°C
1.455
1.37
[XVI-321
> 70
Hydrazine (99.8%)
1.008
2.108
[XVI-311
> 115
Hydrazoic acid
1.122
1.284
[XVI-311
> 47
1,l Dinitroethane
1.355
1.270
[XVI-341
> 56
1,l Dinitropropane
1.254
1.390
[XVI-341
> 62
2,2 Dinitropropane
1.221
1.240
[XV1-34]
> 48
i-Propyl nitrate
1.036
1.100
[XVI-35]
> 32
> 110 [XVI-28]
Rule of Thumb From known sound velocity c, one may estimate the Hugoniot by the equation: u,=1.2~c,+1.7~,. Data are compiled in Table XVI- 1. From this table we can establish a rank-order for HVD initiation. This could be usefd for transport regulations whenever HVD criteria would apply. But this is not the case. For example, NG, which is too dangerous to be transported, is much less sensitive to HVD initiation than NM, which can be transported as a flammable liquid. Hydrazoic acid is so dangerous that the experimentator rarely survives his experiments. Nevertheless, hydrazoic acid behaves in its HVD initiation like an insensitive homogeneous explosive { Yakoleva and Kurbangalina. [XVI-36]}, its very low critical diameter of 0.08 mm may be responsible for an easy LVD s HVD transition [XVI-371. This seemingly conflicting situation is easily understood when one considers the analogous initiating pressure of bubbly NG: -2 kbar for LVD and > 110 kbar for HVD [XVI-381.
361 Precursors
Therefore, we have sound reasons for reservations against so-called safety tests that address only HVD, as is usually done for transportation of hazardous materials.
Stability of the Precursor Mechanism The activation mechanism of precursors works for any shock-to-detonation transition (SDT) when the entering shock is sufficiently powerful. Due to parametric amplification of bubble migration by their size variation, LVD s HVD and HVD LVD transitions also become possible. Statistical scatter in HVD experiments is to be expected because of microscopic mechanisms. This is also reflected in the critical diameters in conical geometry [XVI-39], see Chapter XI, Fig. XI-5. Deflagration to detonation transitions (DDT) may be understood in the following terms: Deflagration means a blowup of single reaction sources, which will produce pressure waves according to combustion dynamics, which in turn create a pressure field. Within this pressure field, successively the following expanding reaction pockets will be driven into the direction of pressure-wave propagation, and detonation may develop, see Figures XIV-6 and XIV-7. This is the first time that an intrinsic mechanism of DDT in a substance is suggested, which shows no place for any principal exclusion of DDT. This is very important for safety assessments, because experts now believe that cases exist in which DDT does not occur - or a stable DDT should exist which could be used for technical processes.
Sensitization by dense Inerts, Hydrodynamic Pair Formation Initiation of open-pore (commercial) explosives depends on ambient hydrostatic pressure. Even some bars of ambient gas pressure prevent initiation. Therefore, socalled sensitizers ;ire added to seismic explosives to make initiation possible at high ambient pressures in seismic applications. Such sensitizers are inerts of relatively high density (> 2.8 g/cm3), such as barium sulphate, lead sulphate, iron oxide, and so on, ranging in diameter from 0.5 up to 10 pm. These act by hydrodynamic pair formation (see Fig. XV-8). Dempster [XVI-401 determined the minimum particle diameter, just sensitizing blasting gelatine to a high velocity detonation (HVD). Leiber [XVI-411 considered this in terms of Konig’s solution for dynamic particle behavior in a viscid medium. The tiny particles get mobility if the fixing viscous forces are overcome by the dynamic pressure. Then these particles develop a void trail leading to onset of HVD. The viscosity of this matrix was calculated as a test with the minimum effective particle, since all parameters and its resistance law in the matrix were known. The viscosity of liquid nitroglycerine V G ) was obtained from back calculations.
368 Chapter 274'
References [XVI-I] M. Berthelot, Sur la force de matieres explosives d aprCs la thermochimie, Vol. I, 3rd ed., Gauthier-Villars, Paris, 1883. [XVI-2] A. Schuster, Note to H. B. Dixon, Bakerian Lecture: On the Rate of Explosion in Gases, Phil. Trans. Roy. SOC.London A 84 (1893), p. 1521154. [XVI-3) B. Riemann, iiber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Gotting. Nachr. 1859, p. 1921196; Gotting. Abh. VIU, p. 243/265; Fortschritte der Physik XV, p. 123/130. [XVI-41 Apin, cited according to F. A. Baum, K. P. Stanyukovich and B.I. Shekter: Fizika vzryva (Physics of an Explosion), Moscow, 1959. [XVI-5] M. A. Cook, The Science of High Explosives, Robert E. Krieger, Huntington, 1971. [XVI-61 Annex in Hazard Studies for Solid Propellant Rocket Motors, AGARDograph NO 3 16, 1990, AGARD-AG-3 16.
[XVI-71 C. H. Johansson, and P. A. Persson, Detonics of High Explosives, Academic Press, London-New York, 1970. [XVl-8] C,. Fosse, Influence de I'amorGage sur la pression exercee dans l'eau par les explosifs industriels, Comm. XXXVIe Congres Int. Chimie Industrielle, Bruxelles, 1966, Gr. X - S. 24 - 121, p. 28/32. [XVI-91 R. vanden Berghe, Contribution de la photographie ultra-rapide a I'etude des Detonations, Comm. XXXVIe Congres Int. Chimie Industrielle, Bruxelles, 1966, Gr. X - S. 24 - 152, p. 48/51. [XVI-lo] M. Held, Struktur der Detonationsfiont, Explosivstoffe 17 (1969)11-12,p. 2411249.
[XVI-111 C. 0. Leiber, Beeinflussung der Initiiemng durch Diskontinuitaten, Jahrestagung 1970 des ICT, p. 71/90. [XVI-121 R. L. Spaulding, Precursors in Detonations in porous Explosives, 7th Symp. (Int.) on Detonation, Annapolis, 1981, NSWC MP 82-334, p. 877/886. [XVI-131 M. Held, Letter communication, december 1990. [XVI-141 R. Hessenmuller, and C. 0. Leiber: Wie eine Detonation entsteht, Umschau 84 (1984)17, p. 5071509. [XVI-15] W. Miiller, BICT, contributed this result, see C. 0. Leiber, Physikalisch-chemische Prinzipien der Slurries, Gliickauf Forschungshefte 45 (19846, p. 287/295, J. Ind. Explosives SOC.,Japan, (Kogyo Kayaku) 46 (1985)5, p. 270/284. English: C. 0. Leiber, and R. M. Doherty, Physical and Chemical Principles of Slurries, in 0. Machacek, Application of Demilitarized Gun and Rocket Propellants in Commercial Explosives, NATO Science Series, Kluwer Acad. Publishers, Dordrecht, Boston, London, 2000, p. 911109. [XVI-16] M. J. Kamlet, and J. M. Short: The Chemistry of Detonations. VI: A "Rule for Gamma" as a Criterion for Choice among Conflicting Detonation Pressure Measurements, Combustion and Flame 38 (1980), p. 221/230. [XVI-17] B. G. Craig, in M. Van Thiel, A. S. Kusubov and A. C. Mitchell, Compendium of Shock Wave Data, Vol. 2, Suppl. 1, Lawrence Radiation Laboratory, Livermore, 1967. [XVI-I 81 B. G. Craig, in S. P. Marsh, LASL Shock Hugoniot Data, University of California Press, 1980.
369 Precur~svrs
[XVI-191 V. S. Ilyukin, P. F. Pokhil, 0. K. Rozanov and N. S. Shvedova in M. Van Thiel, A. S. Kusubov, A. C. Mitchell, Compendium of Shock Wave Data, Vol. 2, Suppl. 1, Lawrence Radiation Laboratory, Livermore, 1967. [XVI-20] D. R. Hardesty, On the Index of Refraction of Shock-CompressedLiquid Nitromethane, J. Appl. Phys. 47 (1976)5, p. 199411998. [XVI-211 F. E. Walker, and R. J. Wasley, Initiation of Nitromethane with Relatively Long Duration, Low Amplitude Shock Waves, Combustion and Flame (1970), p. 233/246. [XVI-221 C. Fauquignon, and H. Moulard, Shock Sensitivity of Nitromethane with Well Defined Hot-Spot Distribution, Acta Astronautica (1 978), p. 1035/1040. [XVI-231 A. W. Campbell, W. C. Davis, J. R. Travis: Shock Initiation in Liquid Explosives, Phys. Fluids 4 (1961)4, p. 498/510. [XVI-241 0. B. Yakusheva, V. V. Yakushev and A. N. Dremin, The Opacity Mechanism of Shock-CompressedOrganic Liquids, High Temperatures-High Pressures 3 (1971), p. 2611266. [XVI-251 C. 0. Leiber, Die Detonation als reaktive Mehrphasenstromung, Rheol. Acta 14 (1975), p. 921100. English: A Simple Model for the Initiation of Chemical Systems, in J. M. Jenkins and J. R. White, Proc. Int. Conf. Res. Primary Explosives, Vol. 1, ERDE, Waltham Abbey, 1975, contr. 3-1/3-17. [XVI-261 W. C. Davis, and J. B. Ramsay: Detonation Pressures of PBX-9404, Composition B, PBX-9502, and Nitromethane, 7thSymp. (Int.) Detonation, Annapolis, 1981, p. 53 11539, NSWC MP 82-334. [XVI27] A. N. Dremin, 0. K. Rozanov, Measurement of the Shock Adiabats of Mixtures of Nitromethane with Acetone, Izv. Akad Nauk SSSR, Seriya Khimicheskaya, 1964, No 8, p. 151311514. [XVI-28] W. B. Garn, Determination of the Unreacted Hugoniot for Liquid TNT, J. Chem. Phys. 30 (1959)3, p. 819/822. [XVI-29] W. B. Garn, Detonation Pressure of Liquid TNT, J. Chem. Phys. 32 (1960)3, p. 653/655. [XVI-30] A. N. Dremin, 0.K. Rozanov, S. D. Savrov, V. V. Jakushev: Shock Initiation in Nitroglycerine, Fiz. gor. vzryva 3 (1967)1,p. 11118. [XVI-3I] R. Kh. Kiurbangalina,Yu. A. Kustov, G. S. Yakoleva: Velocity of Ultrasound in some liquid explosive Nitroesters and Nitrohydrogens, Russian Ultrasonics (19793, p. 93/98. [XVI-321 B. 0. Reese, L. B. Seely, E. Shaw, D. Tegg: Coefficient of Thermal Expansion of Nitromethane aind Four J. Chem. Eng. & Eng. Data (1970)1,p. 1401142. [XVI-331 Landolt-Bornstein, Vol. 5, W. Schaaffs: Molekularakustik, Springer, Berlin, Heidelberg, New York, 1967. [XVI-34] C. M. Tamer, E. Shaw, M. Cowperthwaite: Detonation Failure Diameter Studies of four liquid Nitroalkanes, J. Chem. Phys. 64 (1976)6, p. 266512673, [XVI-35] Th. Groothuizen, H. J. Pasman, A. A. Schilperoord, LVD in Some Liquids, Jahrestagung ICT, 1976, p. 371/386.
370 Chapter
[XVI-361 G. S. Yakoleva, and R. Kh. Kurbangalina: Shock Wave Initiation of Liquid Hydrazoic Acid, in A. N. Dremin: 6th Detonatsiya Mater. Vses. Simp. Goreniyu Vzryvu, 1980, p. 56/60, CA95( 12): 1 0 0 0 4 2 ~ [XVI-371 G. S. Yakoleva, and R. Kh. Kurbangalina: Critical diameter of a stable detonation of liquid hydrazoic acid, Fiz. Gor. vzryva 12 (1976)5, p. 7741775, CA86(10): 57636s. [XVI-38] A. V. Dubovik, and V. K. Bobolev: Schlagempfindlichkeit flussiger explosionsfahiger Systeme, Nauka, Moskau, 1978. [XVI-39] H. Badners, and C. 0. Leiber, Methode zur Ermittlung des kritischen Durchmessers der Schnellen Detonation in konischer Geometrie, BICT-Rept. 2 10/7703/88, 12. Sprengstoffgesprach, 4./6. 10. 1988, UnterluD. [XVI-401 P. B. Dempster, The Effect of Inert Components in the Detonation of Gelatinous Explosives, Disc. Faraday Soc. No 22 (1956), p. 196/202. [XVI-41] C. 0. Leiber, Sensibilisierung und Phlegmatisierung von Sprengstoffen, Rheol. Acta 14 (1975)1, p. 85/91.
XVII ALTERATIONS OF HUGONIOTS BY BUBBLE FLOW Most Hugoniots in u,/up-formare believed to be linear for dense materials, so that two experimental points and the bulk sound velocity co are adequate to determine the Hugoniot. Two deviations are possible: 1) In the low-pressure region a liquid can become bubbly, altering the sound velocity, as outlined in Chapter IX. For precursors one should expect deviations too, at least at conditions where an intrinsic generated void can overtake the shock velocity, so that us < u1P'
(XVII-1)
In this case, a dynamic activated void, coming from the high-pressure side, overtakes the shock velocity, and penetrates the shock zone, then it expands into the low-pressure medium ahead the shock zone. These void precursors lead under certain conditions to pressure precursors and negative compressibility,where the volume increases with increasing pressure. Therefore, the oncoming shock enters not the originally specified medium, but a cavitated one with different properties and also a pressure different from ambient. Cavitation luminosity results, because of the powerful stimulation of the void expansion in the low-pressure medium, and they collapse again. By most powerful dissipative losses Srom the bubble expansion phase-locked ionizing events, and onset of chemical reaction results, until this void is caught again by its stimulating shock. Inevitably, the Hugoniot of this medium must be changed in pressure domains where u& 5 3. Therefore, it is most doubtfbl whether Hugoniots can be extrapolated beyond the experimental limits. Additional reciprocal effects exist that shock affects bubbles and particles, and also the reverse that the shape of the shock is altered by the presence of these interstitials, especially its attenuation properties. According to this view, neither the high-pressure nor the low-pressure side can be specified exactly when bubbles and precursors are expected. In addition, onset of the void mobility depends on specific constellations, mostly beyond control of the experimentalist, so that considerable statistical scatter can be expected, leading to very different results from author to author, and from experiment to experiment. We have lost, therefore, the physical basis of classical shock physics and Hugoniots become doubtful when plane-wave impedances are used for experimental data reduction. Keeping these principal shortcomings in mind, we assume in the following that planewave Hugoniots should hold approximately. As "corrections" we only consider the effects of the extra. particle velocities uTp as an offset to the plane-wave shocks. Also it may be that ulp as the fastest event may play the role of the shock velocity us. 37 1
372 Chapter
Such specific questions outside of the current considerations can only be discussed for a well-defined homogeneous material, for which many experimental controversial results from various authors are available. Such results we demonstrate for carbon tetrachloride because this liquid does not explode and carbon tetrachloride has been investigated over a long time period by various authorities. We partly follow a summary given by Shefield [XVII-11 in his thesis.
Hugoniots of Liquids Carbon Tetrachloride Carbon tetrachloride melts at -22.9"C at atmospheric pressure, and at +12,1"C at 1000 bar, so that it freezes at moderate static high pressures [XVII-21. (Boiling occurs at 77°C at normal conditions). The density at 20°C is 1.594 g/cm3, and the density variation with temperature dp/dt is - 195 x g/cm3 "C. The sound velocity is relatively low, c = 938 and the sound velocity variation with temperature is about -3.1 "C. Its viscosity is 0.97 CP (20°C). According to the low-pressure alterations of the Hugoniots of bubbly liquids (Chapter IX) one can expect a low-pressure Hugoniot cusp for carbon tetrachloride for up 0.2465, us 1.48 [ k d s ] at a pressure of 5.8 1 kbar.
-
-
With Dick's original low-pressure Hugoniot us = 1.1 1 + 1.67 up, one can expect for the range of conditions u,/up = 3:
up = 0.835;
us = 2.50 at a pressure p = 33.3 kbar.
u,/up = 2.5:
up = 1.337;
us = 3.34 at a pressure p = 71.3 kbar.
high-pressure alterations which are caused by precursors. As early as 1939 Schardin [XVII-31 found by shot experiments on carbon tetrachloride an opacity arising, which he attributed to a partial solidification (similar in water). By shots of 1200 m/s velocity a weak Mach wave could be observed, which is overtaken by a spherical shock wave, which appeared to be opaque - and was interpreted as a solidification wave - of constant velocity of 1300 m/s. Doring and Burkhardt [XVII-41 in 1944 described calculations to verify this. Walsh and Rice [XVII-51 found more recently (1957), in a pressure range from 10 to 170 kbar, a slight opacity at 60 to 80 kbar that increased to complete opacity at 130 to 170 kbar. Doran and Ahrens [XVII-61 stated in 1963 that carbon tetrachloride is a good electrical conductor at 170 kbar. Cook and Rogers [XVII-7] in 1963 did not observe transitions in shock velocity between 12 to 129 kbar, nor did Dick [XVII-8] between 28 to 170 kbar a few months later. They found that above 70 kbar CC14 becomes conductive and is a good
373 by
conductor at 120 kbar. This was confirmed in 1968 by Mitchell and Keeler [XVII-91. They measured cclnductivity increases of mho/cm at 65 kbar up to -1 mho/cm at 160 kbar. Dick [XVII-101 did find a break in the Hugoniot at 165 kbar (us/up 2.17) in 1970. However, Voskoboinikov and Bogomolev [XVII-1 11 in 1968 had made no mention of any transition between 80 and 200 kbar.
-
But Yushko, Krishkevich, and Kormer [XVII-121 and Kormer [XVII-131 in 1968 did find a sudden anomalous increase in reflectivity at pressures > 300 kbar, during investigation of the optical reflectivity at the shock front. For a compression of p/po 2.2 (u,/u, 1.83), reflectivity increases up to 18% compared to an expected value of about 3%#.They discuss this event as possible anomal increase of the refractive index or as metallization. Kormer also found between 200 and 300 kbar anomalous effects, which had been interpreted as energy consuming phase transitions or as chemical breakdown with the formation of free carbon. Yakusheva, Yakushev and Dremin [XVII-141 in 1971 found arising shock opacities for benzene, carbon tetrachloride, acetone, dichlorethane, and nitromethane (NM) at pressures of 135, 80, 100, 120, and 85 kbar, respectively. And Lysne [XVII-151 described a nonlinear velocity relationship between 2 to 12 kbar, and pointed out that no freezing should have occurred.
-
I -
-
Other Liquids
Dick [XVII-lo], [XVII- 17 to 191 also investigated the Hugoniots us, and upof other liquids (carbon disulfide, and benzene), see the Appendix to this Chapter, Table XVII-4 to XVII-6. These values are plotted in the terms us versus up,p versus (us/up),and p versus v/vo. The pressures are given as a secondary measure according to p = po x us x up. The inverse of the relative compression is also calculated from us/up in Figures XVII-I to XVII-3, where the markings in the plots indicate the authors, also listed (M) in the Tables. In contrast to carbon tetrachloride, more pronounced irregularities appear (see Figs. XVII-2 and XVII-3). Together with the dipole scattering mechanism one may therefore conclude that in general there exists no linear velocity relationship for liquids. The densities, thermal expansions, sound velocities, and their variation with temperature of the investigated liquids are given in Table XVII-1. That the evaluation of a proper Hugoniot may become cumbersome is demonstrated by comparing the linear velocity relationships of various authors for CC14 in Table XVII-2. The values for Hugoniots of CC1, in the terms of us = A + B up and the quadratic Hugoniot are listed, and the limit of validity is also noted. The quadratic Hugoniot may be taken over all the values within the experimental limits, and no cusp exists: us = 1.136 + 1 . 7 9 6 ~ 0.0712~: ~ .
(XVII-2)
3 74 Chapter Table XVII-1. Properties of the Liquids investigated
Substance
t ["C]
P [glcm3]
dpldt g cm-3R ' ]
C
dcldt
[ds]
[m s-l K-'I
[cP]
Carbon tetrachloride
20
1.594
- 195
938
- 3.1
0.968
Carbon Disulfide
20
1.264
- 152
1157
- 3.25
0.363
Benzene
20
0.878
- 106
1324
- 4.6
0.650
Table XVII-2. Various Hugoniots of Carbon tetrachloride
B
A
Author
rZ
on us
Validityin pressure [kbar]
Cook and Rogers [XVII-7]
1.296 0.061
1.480 f0.050
0.997
0.062
12 - 129
Lysne [XVII-151
0.985 k0.058
2.275 rtr 0.223
0.937
0.066
2 - 12 (Quadratic Hugoniot fits better)
Dick [XVII-101
1.464 0.047
1.424 0.017
0.995
0.129
26 - 633
Dick [XVII-lo], [XVII-161
1.593 0.093
1.461 rtr 0.060
0.992
0.083
28 - 180
Mitchell and Keeler [XVU-9]
1.622 0.067
1.441 0.038
0.994
0.037
70 - 165
Walsh and Rice [XVLI-5], [XVII-I 61
1.559
1.473
Over all
1.402 0.037
1.462 0.017
74 - 171 (2 points) 0.991
0.171
1 - 633
Qualitative Explanation of Hugoniot Effects Up to now most of these effects have been "explained" as liquidlsolid phase transitions {for example, water 3 ice VII [XVII-13]), chemical decomposition, polymerization, or metallization. No solidification has been verified, and few two-wave configurations have been found, as would be characteristic for a classical phase transition. Only few authors report two-wave configuration in the shock reflection techniques. The dipole scattering of voids easily explains the seemingly conflicting phenomena. For carbon tetrachloride really "unexpected" high-pressure effects are observed at pressures p > 33 kbar, corresponding to UJU, 5 3. Actually, the experimental scatter is between 30 and 60 kbar. Small voids penetrate into the original medium ahead of the shock zone, where they expand and form a bubble screen, followed by onset of opacity.
375 Alferafions Hugoniofs by Bubble Flows
..
I'
98-
76-
.
'i
x
54-
8
32-
.*:
*. c6H6
'08
*
1-
'I+ 6
P Ikborl
*'
1
c6
I'
"I'
zoo.
..:
100
"I
I...
.
** '*?*a I .
*
3.:
'.; 'i*I
+*-.
0.1
0.L
0.5
07
1
VlV,
Figure XVII-1. Hugoniots of Carbon Tetrachloride (Data of Table XVII-4, Appendix)
Figure XVII-2. Hugoniots of benzene (Data of Table XVII-5, Appendix)
316
Chapter
, Y
Figure XVII-3. Hugoniots for Carbon Disulfide (Data of Table XVII-6, Appendix)
1
I.
5
I
0
f,
0.1
0.h
0.5 vlv,
0.6
017
i c
Bond scission occurs because of losses of bubble expansion within the viscous layer A, leading to an increase in electrical conductivity and onset of the chemical reaction (when possible). When the voids expand further against the oncoming shock, a local negative compressibility results, where the volume increases as the pressure increases. In Fig. XVII-1 two such domains are found, one at about 65 kbar and the other at 165 kbar. This latter point was found by Dick [XVII-lo], but the first cusp is the result of measurements of other authors.
377 Energy Barriers
If optical observations are directed to a bubbly layer, it is obvious that an increase in reflectivity occurs as this layer foams up. This effect is most pronounced for large bubbles, which penetrate each other. The condition p/po 2.2 corresponds to about u$up 1.83, and the (kR)mx,-valueapproaches the value of large voids 0.6 (see Fig. XIV-8).
-
-
-
In summary, all these events and their scatter can be explained by the same working mechanism of bubble flow. This is a new conclusion based on the studies described above.
ENERGY BARFUERS Apparently there exists an intrinsic mechanism that compression may not increase as the shock pressurle increases. Walker [XVII-20] pointed out that a Mach barrier or a heat barrier exists in transonic motion, and that such a barrier is also present for shock waves in detonation. Considering the results on compression, such a mechanism seems to be of a more general nature. This will also be seen in the case of porous graphite, described later. A compression Avlv, = up/u, can be obtained both by static and dynamic means. As we know, the group velocity originally was specified by Stokes in hydrodynamics. Steverding [XVII-211 has given arguments that up is a group velocity and us a phase velocity. Therefore, we may apply the Reynolds-Rayleigh wave theorem [XVII-221 to [XVII-241 to the above velocity ratio u p _ group velocity - hydrodynamic energy flux us phase velocity hydrodynamic energy content '
(XVII-3)
which represents the ratio of the hydrodynamic energy flux in the time z and the hydrodynamic enlergy content in the column of length us x z. The usual case is that the ratio of Equation (XVII-3) is less than unity, so that the hydrodynamic energy flux never exceeds the hydrodynamic energy content. This system can always deliver energy. For ulP > us we get in the "bubble piston" a group velocity exceeding the phase velocity, and the ratio is larger than unity. This means that the hydrodynamic energy flux exceeds the hydrodynamic energy content. This is a catastrophe for the shocked system, because tlhe system's energy content is exhausted very rapidly due to the larger
378
Chapter
energy flux,. Roughly speaking, the system delivers more hydrodynamic energy than it has. This situation is good for attenuation purposes with inert materials, which cannot activate firther energy sources. However, if there are chemical energetic materials, the hydrodynamic energy deficiency is compensated for by the chemical energy release. This is a very general formal description of High Velocity Detonation (HVD) [XVII-25].
Model of a Shock-Pressure Barrier The formal argument discussed above will be substantiated by an active working mechanism: High pressure voids penetrate the shock zone under given conditions, and expand into the low-pressure medium ahead. By the basic pressure generation mechanism of a mass flow rate pov per unit volume, this expanding void is a pressure source, which balances the oncoming shock by its pressure radiation. The primary shock creates its own resistance. This pressure source may be described by the inhomogeneous wave equation. For convenience we use the harmonic solution of the wave equation for small sources, described in Chapter V, Equs. (V-7) and (V-l2), in the terms of V =4ZR2d =4nR2~,
p=w
and
=
kc,
pve'k('-R)
4nr(lWe get a simple (point) source when the void is small, R definite volume flow, and the amplitude is: PV 4nr
(XVII-4) (XVII-5)
0. The source shows a
(XVII-6)
One gets for r = R and up = u, p = oRpOup.
(XVII-7)
For porous materials with bubble precursors (u& 5 3) we expect a Hugoniot of the form pou, =
(XVII-8)
319
Energy Barriers
Figure XVII-4 demonstrates that this form applies for all conditions of u,lup, and no pronounced cusps or breaks appear.
10Figure XVII-4.
8-
!
7-
I
tI
6-
*' *
**
"Hugoniots" of carbon tetrachloride (+), carbon disulfide (*), and benzene the terms (po up) versus the pressure p in kbar.
The "Hugoniots", determined from points u,lup < 3, are compared with those over the h l l range; the pressure p is taken in kbar, so that the density is in glcm3 and the velocity in k d s : Table XVll-3.
= 0.1259
cs2: C6Hh
poup
p0up
xP
~ " rz ~= 0.9966 ~
Full range
= 0.1368 x
p0625x rz = 0.9926
Precursors only
= 0.1 103 x
poaE1 r2 = 0.9938
Full range
= 0.1053 x
po6329
r2= 0.9974
Precursors only
= 0.0997 x PO 6442
P = 0.9978
Full range
Model of an Expanding Rayleigh Bubble The dynamics of bubble expansion or collapse can be described with Rayleigh's equation. Lamb /[XVII-23]has given solutions for the maximum velocity u, of an expanding bubble with internal gas pressure p of the specific heat ratio y, where p is the matrix density and v = l l p its specific volume. (XVII-9)
in
380 Chapter
If the initial bubble radius is R,, then the radius R of this bubble for its maximum expansion velocity is:
(
;)3y-3
(XVII- 10)
For values of 1, 1.66, 2, and 3, the bubble radius increases between 40, 29, 26, and 20% for its maximum expansion velocity. Since the pressure dependencies of the specific heats are different for very high-pressure gases, > 1.66 becomes possible [XVII-26]. There are two non-classical options to represent porous Hugoniots. Comparison with experiments determines which one gives the best representations.
Hugoniots of Porous Solid Media and Foams Hugoniots of materials of varying porosity, such as foams and porous metals [XVII-27] differ greatly. Figure XVII-5 shows the Hugoniots of all kinds of graphites (with densities from 2.21 to 0.27 g/cm3 for low-density graphite foams in a pressure/ relative compression (p/(v/vo)) plot [XVII-27], [XVII-281. Only those Hugoniots are shown where the voids overtook the shock front that is u,/uD< 3 [XVII-29].
kbar
1
Figure XVII-5.
500
p/(v/v,) plots for yaphites different densities, from 2.23 to low-density foams with p = 0.27 P/cm', where bubbles may overtake the shock front. Experimental values of McQueen and Marsh [XVII271, [XVII-28] are used.
The data of Figure XVII-5 are plotted in Lamb's terms in Figure XVII-6, and in terms of pressure shielding in Figures XVII-7 and XVII-8, Figure XVII-8 giving the best results.
381
Energy Barriers 0r
Figure XVII-6. Plot of the Hugoniots of Figure XVII-5 in Lamb's terms:
t :: P
=
>
resp.
32 -
9
Figure XWI-7. Plot of the Hugoniots of XVII-5 in terms of pressure
a
I:
shielding
The
deviations at the lower end are caused by the high- and lowdensity foams.
? L & 3 2 1 kbar 1000
Finally in Fig. XVII-9 the model of the pressure barrier has been applied to 537 available data poiints of graphite (top). Below the plot in the terms of a pressure barrier is shown. The indicated deviations result from low density foams. It is interesting that the deviations of such a 'general' Hugoniot are not too large, if each particle velocity is considered. In Figure XVII-9 all experimental values [XVII-281 have been used (also those us/up> 3).
3 82 Chapter
Figure XVII-8. Plot of the Hugoniots of Figure XVII-5 in terms of pressure shielding for densities 1.54 ... 2.2 g/cm3.
1
loo0
Figure XVll-9. 02
08
04
"Hugoniots" of graphites of different porosity for arbitrary condition All available 537 data points were used.
10
VIVO
On the top these data are shown in the classical arrangement, and below in the unified picture of a pressure bamer.
o
+
,
,
,
,
,
0
I
1000
p in kbar
383 Energy Barriers
Pressure Determination by X-Ray Flash Techniques Compression is determined directly by the X-ray flash technique. A corresponding pressure is usually concluded from the application of the plane-wave shock equations. This method is considered to be the most "basic''. The conditions for the onset of the bubble overtaking the shock front can be given in classical plane-wave terms as the inverse of the relative compression as the velocity ratio usfup= v,/Av < 3. 1
I
I
I
I
I
I
1
1
density in glcrn3
7-
-
I
I
I
100 200 300
I
I
I
I
I
I
Figure XVII-10. u& = v,,/Av values as function of pressure for graphites of different porosities [XVII-28]. Compression plateaus are obtained at high pressures. Only limiting compression is independent ffom the applied shock pressure. Only classical plane-wave considerations are applied in this case.
500 600 700 800 (kbarl
In Figure XVII-10 the graphite data [XVII-281 (initial densities 1.54 .. 2.2 g/cm3) are plotted in these terms versus the pressure p [XVII-29]. Compression plateaus are obtained, so that any pressure determinations by X-rays can only lead to a minimum pressure. Because: onset of High Velocity Detonation (HVD) is linked to the condition ush, = 3, we have reasons to doubt the reliability of detonation pressures determined with X-ray techniques.
9
Figure XVII-11. In the case of compression plateaus, the shock velocities become a linear function of the pressure p, independent from the materials poi-osity. The same experimental values are used as in Figure XVII-10 {McQueen and Marsh [XWI-28]}. H10 200 300 400
600 700 800 900 p lkborl
384
As a result of such a defined pressure barrier of expanding bubbles, a constant compressibility is to be expected. Figure XVII-11 demonstrates that this constant compressibility is real, because the shock velocity increases linearly with pressure, quite independent fiom materials porosity [XVII-30].
Comparison with Classical Calculations The Gruneisen equation of state is often used to derive properties from dense materials to the porous state, where it is assumed that for varying densities p and temperatures T, the product pT remains constant, r being the Gruneisen coefficient. But in spite of the fact that the zero-pressure room-temperature Gruneisen coefficient of diamond is r = 2.6, r = 1.5 is far more representative for the high temperatures of porous graphites [XVII-281. With the assumption that
is only a hnction of the specific volume
P=I'pore
(XVII-11)
the Hugoniots of porous materials are estimated by
(XVII- 12)
where PH, v, and v,* represent the pressure of the material at its crystal density, initial volume of the crystal density material and the initial volume of the porous material, respectively. From Equation (XVII-12) it is not possible to calculate the pressure of the porous Hugoniot when that volume is less than its initial volume at crystal density. Therefore, Equation (XVII- 13) was introduced [XVII-321 after several simplifications according to Zel'dovich and Raizer [XVII-3 I]:
(XVII- 13)
where K,, is cold compressibility given by the bulk sound velocity c, =
.
(XVII-14)
Experimental
Although Equation XVII-14 seems to successfully estimate the shock pressure of lowdensity porous materials, the accuracy is not satisfactory compared with the results obtained using Equation (XVII-12), especially when calculating the shock velocity. The description of‘the porous Hugoniots in terms of Lamb or pressure shielding appeared to be much simpler and more accurate than this classical description [XVII-301. In principle only one measured point is necessary for a porous material to determine POUP =
f(P).
But there are some basic, general reservations against the use of these Hugoniots: All measured data of porous Hugoniots are treated by plane-wave assumptions, which have been demonstrated to fail. Therefore, there is no sound basis for a thorough physical discussion.
APPENDIX TO CHAPTER XVII: SUMMARY OF EXPERIMENTAL HUGONTOTS Table XVII-4. Summary of the Hugoniots of Carbon tetrachloride Po ‘Up p [gicm’] [ M s ] [kmis] [kbar]
1.590 1.590 1.590 1.590 1.590 1.582 1.582 1.582 1.582 1.582 1.582 1.650 1.645 1.667 1.590 1.577 1.571
4.200 3.290 2.850 2.180 1.930 1.380 1.690 1.880 1.180 1.250 1.430 1.520 1.820 1.660 2.320 2.270 2.470
Ushp
1.930 128.8 2.176 1.360 71.1 2.419 1.100 49.8 2.590 0.605 20.9 3.603 0.390 11.9 4.948 0.178 3.8 7.752 0.337 9.0 5.014 0.418 12.4 4.497 0.103 1.9 11.456 0.136 2.6 9.191 0.207 4.6 6.908 0.195 4.8 7.794 0.336 10.0 5.416 0.263 7.2 6.311 0.720 26.5 3.222 0.840 30.0 2.702 0.830 32.2 2.975
T IKI ? ? ? ? ?
297 297 297 297 297 297 263 265 254 295 295 295
Authors
Cook & Rogers [XVII-7] Cook & Rogers [XVII-7] Cook&Rogers [XVII-7] Cook & Rogers [XVII-7] Cook &Rogers [XVII-7] Lysne [XVII-15] Lysne [XVII-151 Lysne [XW-15] Lysne [XVII-15] Lysne [XVlI-15] Lysne [XVII-15] Lysne [XVII-15] Lysne [XVII-15] Lysne [XVII-15] Dick [XVII-101 Dick [XVII-101 Dick [XVII-lo]
M
0 0 0 0 0
+ + + + + + + + + * * *
386 Chapter Po us up P [g/cm3] [km/s] [km/s] [kbar]
1.586 1.594 1.596 1.571 1.606 1.591 1.598 1.577 1.571 1.606 1.580 1.571 1.571 1.586 1.596 1.574 1.610 1.580 1.588 1.571 1.571 1.584 1.598 1.582 1.580 1.586 1.588 1.588 1.598 1.580 1.584 1.576 1.584 1.594 1.603 1.604 1.607 1.582 1.577 1.590 1.596 1.554 1.545 1.538
2.910 2.950 3.280 3.320 3.460 3.440 3.500 3.740 3.860 4.080 4.070 4.270 4.520 4.660 4.710 4.880 5.340 5.210 5.720 5.690 6.130 6.440 6.800 6.720 6.780 7.130 7.550 7.960 8.060 8.240 8.260 2.510 4.600 3.370 3.320 3.970 3.450 4.920 4.850 3.510 4.750 4.480 4.030 3.720
1.030 1.040 1.250 1.330 1.330 1.360 1.450 1.610 1.690 1.730 1.770 1.970 2.070 2.100 2.150 2.360 2.550 2.620 2.950 3.060 3.220 3.440 3.640 3.690 3.770 4.050 4.400 4.580 4.740 4.740 4.840 0.710 2.070 1.180 1.190 1.580 1.210 2.320 2.235 1.325 2.170 1.980 1.670 1.460
47.5 48.9 65.4 69.3 73.9 74.4 81.0 94.9 102.4 113.3 113.8 132.1 146.9 155.2 161.6 181.2 219.2 215.6 267.9 273.5 310.0 350.9 395.5 392.2 403.8 457.9 527.5 578.9 610.5 617.1 633.2 28.0 150.8 63.3 63.3 100.6 67.0 180.5 170.9 73.9 164.5 137.8 103.9 83.5
ushp
2.825 2.836 2.624 2.496 2.601 2.529 2.413 2.322 2.284 2.358 2.299 2.167 2.183 2.219 2.190 2.067 2.094 1.988 1.938 1.859 1.903 1.872 1.868 1.821 1.798 1.760 1.715 1.737 1.700 1.738 1.706 3.535 2.222 2.855 2.789 2.512 2.85 1 2.120 2.170 2.649 2.188 2.262 2.413 2.547
T [KI
Authors
M
* 295 Dick[XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-lo] * 295 Dick [XVII-lo] * 295 Dick [XVE-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII- 101 * 295 Dick [XVII-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-lO] * 295 Dick [XVII-lo] * 295 Dick [XVII-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-lo] * 295 Dick [XVII-lo] * 295 Dick [XVII-lo] * 295 Dick [XVII-101 * 295 Dick [XVII-101 * 295 Dick [XVII-101 302.1 Dick [XVII-101, [XVU-16] 298.1 Dick [XVII-lo], [XVU-16] X X 293.1 Dick [XVII-lo], [XVII-16] 289.1 Dick [XVII-lo], [XVU-16] 288.1 Dick [XVII-lo], [XVn-16] 286.1 Dick [XVII-101, [XVU-161 299.1 Dick IXVII-101, [XVU-161 298.1 Walsh & Rice[XVII-5], [XVII-16] x 295.1 Walsh & Rice [XVII-5], [XVII-161 x + 291.1 Mitchell & Keeler [XVU-9] + 3 1I . 1 Mitchell & Keeler [XVU-9] + 3 15.1 Mitchell & Keeler [XVII-9] + 3 17.1 Mitchell & Keeler [XVU-9]
387 Experimental Hugoniots
[g/cm3] [ k d s ] [ W s ] [kbar] 1.560 1.616 1.568 1.554
3.820 4.330 3.970 4.760
1.560 92.9 1.900 132.9 1.580 98.3 2.170 160.5
[KI 2.448 2.278 2.512 2.193
308.1 Mitchell & Keeler [XVII-g] 283.1 Mitchell & Keeler [XVII-9] 305.1 Mitchell & Keeler [XvII-9] 311.1 Mitchell &Keeler [XVn-9]
+ + + +
Table XVII-5. Summary of the Hugoniots of Benzene
po us up p [g/cm3] [ W s ] [ W s ] [kbar]
0.877 0.869 0.870 0.870 0.875 0.879 0.880 0.866 0.885 0.877 0.881 0.869 0.869 0.885 0.871 0.870 0.870 0.875 0.880 0.868 0.887 0.871 0.876 0.870 0.870 0.874 0.881 0.872 0.871 0.875 0.876 0.876 0.881
2.780 2.720 2.960 3.310 3.440 3.470 3.850 3.890 4.050 4.050 4.090 4.380 4.520 4.790 4.770 5.000 5.280 5.460 5.520 5.710 6.000 5.930 6.170 6.220 6.430 6.820 7.230 7.160 7.250 7.660 8.240 8.610 8.910
0.770 0.890 0.890 1.050 1.120 1.120 1.360 1.450 1.450 1.480 1.590 1.770 1.850 1.900 1.940 2.160 2.290 2.330 2.370 2.610 2.860 2.920 3.340 3.440 3.660 3.920 4.150 4.200 4.290 4.610 4.990 5.210 5.380
18.7 21.0 22.9 30.2 33.7 34.1 46.0 48.8 51.9 52.5 57.2 67.3 72.6 80.5 80.6 93.9 105.1 111.3 115.1 129.3 152.2 150.8 180.5 186.1 204.7 233.6 264.3 262.2 270.9 308.9 360.1 392.9 422.3
uhp
3.610 3.056 3.325 3.152 3.071 3.098 2.830 2.682 2.793 2.736 2.572 2.474 2.443 2.521 2.458 2.314 2.305 2.343 2.329 2.187 2.097 2.030 1.847 1.808 1.756 1.739 1.742 1.704 1.689 1.661 1.651 1.652 1.656
T [KI
Authors
ca295 Dick [XVII-101 ca295 Dick [XVn-lO] ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVlI-lo] ca295 Dick [XvII-101 ca295 Dick [XVII-lo] ca295 Dick [XVlI-lo] ca295 Dick [XVU-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XW-101 ca295 Dick [XVlI-101 ca295 Dick [XVII-101 ca295 Dick [XW-101 ca295 Dick [XW-101 ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XW-101 ca295 Dick [XVII-101
M
* *
*
* * * *
* *
*
* * *
*
*
* * * * *
*
* * * *
* * * * * *
* *
388 Chapter
pa up P [glcm3] [ M s ] [ k d s ] [kbar]
0.871 0.874 0.885 0.885 0.885 0.885 0.885 0.885 0.877 0.877 0.874 0.879 0.884 0.881 0.884 0.895 0.877 0.887 0.872 0.866 0.883
8.820 8.970 4.590 4.590 3.160 2.770 2.470 1.970 5.420 5.370 3.350 5.310 5.990 5.900 4.650 5.880 5.740 4.050 5.800 5.660 4.100
5.420 5.510 1.920 1.880 0.980 0.607 0.560 0.280 2.320 2.280 0.930 2.290 2.620 2.530 1.740 2.510 2.530 1.320 2.550 2.470 1.448
416.3 431.9 77.9 76.3 27.4 14.8 12.2 4.8 110.2 107.3 27.2 106.8 138.7 131.5 71.5 132.0 127.3 47.4 128.9 121.0 52.4
ushp
1.627 1.627 2.390 2.441 3.224 4.563 4.410 7.035 2.336 2.355 3.602 2.318 2.286 2.332 2.672 2.342 2.268 3.068 2.274 2.291 2.83 1
T [KI
Authors
M
* ca295 Dick [XVII-lo] ca295 Dick [XVII-lo] * ? Cook &Rogers [XVII-7] 0 ? Cook & Rogers [XVII-71 0 ? Cook & Rogers [XVII-71 0 ? Cook & Rogers [XVII-7] 0 ? Cook & Rogers [XVII-7] 0 ? Cook & Rogers [XVII-7] 0 + 295.1 Dick [XVII-lo], [XVII-16] 295.1 Dick [XVLI-lo], [XVIl-16] + 296.1 Dick [XVII-lo], [XVII-16] + 293.1 Dick [XVII-lo], [XVII-16] + 288.1 Dick [XVII-lo], [XVII-16] + 291.1 Dick [XVII-lo], [XVII-16] + 288.1 Dick [XVII-lo], [XVII-16] + 278.1 Dick [XVII-lo], [XVII-16] + 295.1 Dick [XVII-lo], [XVII-161 + 286.1 Dick [XVII-lo], [XVU-16] + 299.1 Dick [XVII-lo], [XVII-161 + 305.1 Walsh & Rice [XVII-5], [XVII-16] x 289.1 Walsh & Rice [XVII-51, [XVII-I 61 x
Table XVII-6. Summary of the Hugoniots of Carbon disulfide
us up P [glcm3] [ k d s ] [ k d s ] [kbar] pa
1.260 1.249 1.251 1.251 1.257 1.263 1.264 1.245 1.272 1.260 1.266 1.249 1.249 1.272 1.253 1.251
2.470 2.410 2.590 2.940 3.060 3.090 3.390 3.430 3.470 3.470 3.510 3.530 3.550 3.650 3.620 3.780
0.750 23.3 0.860 25.8 0.860 27.8 1.010 37.1 1.070 41.1 1.080 42.1 1.310 56.1 1.390 59.3 1.400 61.7 1.420 62.0 1.520 67.5 1.720 75.8 1.810 80.2 1.870 86.8 1.910 86.6 2.130 100.7
4 u p
3.293 2.802 3.011 2.910 2.859 2.861 2.587 2.467 2.478 2.443 2.309 2.052 1.961 1.951 1.895 1.774
T [KI
Authors
ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-lo] ca295 Dick [XVII-lo] ca295 Dick [XW-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-lo] ca295 Dick [XVII-101 ca295 Dick [XVII-101
M
* *
*
* * * *
* * *
* * * * *
*
389 Experimental
po us u~p p [g/cm3] [ W s ] [knds] [kbar]
1.257 1.264 1.248 1.275 1.253 1.258 1.251 1.251 1.255 1.266 1.254 1.253 1.257 1.258 1.258 1.266 1.253 1.255 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.260 1.243
4.180 4.200 4.400 4.860 4.800 5.230 5.200 5.680 6.040 6.460 6.360 6.440 6.730 7.340 7.640 7.840 7.980 8.090 3.830 3.750 3.630 3.290 3.180 2.700 1.910 1.900 1.650 2.810 2.380 3.220 3.280 2.730 2.450 2.700 3.170 2.900 2.880 2.920 2.790 3.070 2.980 4.320
2.280 2.330 2.560 2.770 2.830 3 200 3 310 3 480 3 720 3.920 3.980 4.060 4.370 4710 4 930 5 090 5 090 5 180 1.280 1.460 1.120 1.210 1080 0630 0300 0 280 0.190 0.871 0.577 1.070 1 170 0 800 0 664 0 770 0.965 0.869 0.928 0.933 0.829 0 995 0 901 2412
119.7 123.6 140.5 171.6 170.2 210.5 215.3 247.2 281.9 320.5 317.4 327.6 369.6 434.9 473.8 505.2 508.9 525.9 61.7 68.9 51.2 50.1 43.2 21.4 7.2 6.7 3.9 30.8 17.3 43.4 48.3 27.5 20.4 26.1 38.5 31.7 33.6 34.3 29.1 38.4 33.8 129.5
~ s h p
1.833 1.802 1.718 1.754 1.696 1.634 1.570 1.632 1.623 1.647 1.597 1.586 1.540 1.558 1.549 1.540 1.567 1.561 2.992 2.568 3.241 2.719 2.944 4.285 6.366 6.785 8.684 3.226 4.124 3.009 2.803 3.412 3.689 3.506 3.284 3.337 3.103 3.129 3.365 3.085 3.307 1.791
T [KI
Authors
M
* ca295 Dick [XVII-lo] * ca295 Dick [XVII-lo] * ca295 Dick [XW-101 * ca295 Dick [XW-101 * ca295 Dick [XW-101 * ca295 Dick [XVII-lo] * ca295 Dick [XVII-lo] * ca295 Dick [XVII-101 * ca295 Dick [XVLI-101 * ca295 Dick [XVII-lo] * ca295 Dick [XVII-lo] * ca295 Dick [XVII-101 * ca295 Dick [XVII-lo] * ca295 Dick [XVII-lo] * ca295 Dick [XVII-101 * ca295 Dick [XVII-101 * ca295 Dick [XVII-101 * ca295 Dick [XVII-lo] + ? Cook & Rogers [XVII-71 + ? Cook & Rogers [XVII-7] + ? Cook & Rogers [XVII-7] + ? Cook & Rogers [XVII-71 + ? Cook & Rogers [XVII-7] + ? Cook & Rogers [ X n - 7 ] + ? Cook & Rogers [XW-7] + ? Cook & Rogers [XVrr-7] + ? Cook & Rogers [XVII-7] RT Sheffield [XVII-9] RT Sheffield [XVII-9] RT Sheffield [XW-9] RT Sheffield [xVrr-9] RT Sheffield [XVII-9] RT Sheffield [XVII-9] RT Sheffield [XVII-9] RT Sheffield [XVII-9] RT Sheffield [XW-9] RT Sheffield [XVII-9] RT Sheffield [XVII-9] RT Sheffield [XVII-9] RT Sheffield [XW-9] RT Sheffield [XVII-9] 306.1 Walsh & Rice [XW-5], [XVII-161 x
390 Chapter
References [XVII-11 S. A. Shefield, Shock Induced Reaction in Carbon Disulfide, Thesis, Washington State University, Pullman, WA, 1978. [XVII-2] S. D. Hamann, Physico-Chemical Effects of Pressure, Butterworths, London, 1957 [XVII-3] H. Schardin, Comm. 5'h Sess. Dt. Akad. Luftfahrtforsch., 23. 6. 1939, Discussion note in Probleme der Detonation, 1940, Dt. Akad. Luftfahrtforsch., p. 85/88. [XVII-4] W. Doring, and G. Burkhardt: Beitrage zur Theorie der Detonation, Berlin-Gatow, Gottingen, Posen, 1944, p. 55158. [XVII-5] J. M. Walsh, and M. H. Rice: Dynamic Compression of Liquids from Measurements of Strong Shock Waves, J. Chem. Phys. 26 (1957), p. 8151823. [XVIId] D. G. Doran, and T. J. Ahrens: Electrical Effects of Shock Waves: Conductivity in CsI and KI. Thermoelectric Measurements in Metals, Rept. Stan. Res. Inst. PGU-4100, 1963, AD 423342. [XVII-7] M. A. Cook, and L. A. Rogers: Compressibility of Solids and Liquids at High Pressures, J. Appl. Phys. 34 (1963), p. 233012336. [XVII-8] R. D. Dick, Shock Compression of Liquid Carbon Tetrachloride and Benzene, Bull. Am. Phys. SOC.9 (1964), p. 547. [XVII-9] A. C. Mitchell, and R. N. Keeler: Technique for Accurate Measurement of the Electrical Conductivity of Shocked Fluids, Rev. Sci. Instr. 39 (1968), p. 5131522. [XVII-101 R. D. Dick, Shock Wave Compression of Benzene, Carbon Disulfide, Carbon Tetrachloride, and Liquid Nitrogen, J. Chem. Phys. 52 (1970)12,p. 602116032. [XVII-111 I. M. Voskoboinikov, and V. M.Bogomolev: Measurement of Temperatures of Shock-Compressed CC14 and of a CC14-C6H6 Solution, JETP Letters 7 (1968), p. 264/266. [XVII-121 K. B. Yushko, G. V. Krishkevich, and S. B. Kormer: Change in the Refractive Index of a Liquid Compressed by a Shock Wave. Anomalous Optical Properties of Carbon Tetrachloride, JETP Letters 7 (1968), p. 7/10. [XVII-13] S. B. Kormer, Optical Study of the Characteristics of Shock Compressed Condensed Dielectrics, Sov. Physics USPEKHI I (1968)2, p. 2291254. [XVII-14] 0. B. Yakusheva, V. V. Yakushev and A. N. Dremin, The Oacity Mchanism of Sock-Cmpressed Oganic Lquids, High Temperatures-High Pressures 3 (1971), p. 2611266. [XVII-15] P. C. Lysne, Equation of State of Liquid CC14 to 16 kbar - a Comparison of Shock and Static Experiments, J. Chem. Phys. 55 (l971), p. 524215246, [XVTI-16] M. van Thiel, Compendium of Shock Wave Data, Vol. I , Vol. 1 - Suppl. 1, V O ~2,. Vol. 2 - Suppl. 1, Lawrence Radiation Laboratory, 1967, UCRL-50108. [XVII-l6a] P. Marsh, LASL Shock Hugoniot Data, University of California Press, Berkeley, Los Angeles, London, 1980. [XVII-17] R. D. Dick, Shock Compression Data For Liquids. I. Six Hydrocarbon Compounds, J. Chem. Phys. 71 ( 1 979)~,p. 320313212. [XVII-1 81 R. D. Dick, and G. I. Kerley: Shock Compression Data For Liquids. II. Condensed Hydrogen And Deuterium, J. Chem. Phys. (1980)10,p. 526415271,
391
[XVII-191 R. D. Dick, Shock Compression Data For Liquids. m. Substituted Methane Compounds, Ethylene Glycol, Glycerol, And Ammonia, J. Chem. Phys. 74 (1981)7, p. 405314061. [XVII-201 F. E. Walker, Description of a Shock-Wave Velocity Barrier, Propellants and Explosives 6 (1981), p. 15/16. [XVII-21] B. Steverding, Quantization of Stress Waves and Fracture, Mater. Sci. Eng. 9 (1972), p. 1851189. [XVII-221 Lord Rayleigh, On Progressive Waves, Proc. London Math. Soc. 9 (1877), p. 21126, Scientific Papers, Vol. I, Dover Pub. New York, 1964, p. 322/327. [XVII-23] H. Lamb, Hydrodynamics, 6'h ed., Cambridge University Press, 1932. [XVII-24] A. Sommerfeld, Mechanik der deformierbaren Medien, Akad. Verlagsgesellschaft, Leipzig, 1945. [XVII-25] C. 0 . Leiiber, Basic Differences Between Dynamic And Static High Pressure Effects, High Temperatures - High Pressures 9 (1977), p. 573/574. [XW-26] C. 0 . Leiber, Approximative Quantitative Aspects of a Hot Spot, Part Initiation, Factors of Safe Handling, Reliability And Effects Of Hydrostatic Pressure On Initiation, J. Haz. Mat. 13 (1 986), p. 31 11328 (Appendix). [XVII-271 R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz and W. J. Carter: The Equation of State of Solids from Shock Wave Studies in High Velocity Impact, ed. by Ray Kinslow, Academic Press, 1970, p. 293 ff. [XVII-28] R. G. McQueen, and S. P. Marsh: Hugoniots of Graphites of Various Initial Densities and the Equation of State of Carbon, HDP-Symposium 1967, Paris: Behavior of Dense Media under High Dynamic Pressures, Gordon & Breach, New York, 1968, p. 2071216. [XVII-29] C. 0 . Leiber, Alteration of the Hugoniots by Bubble Flow, J. Appl. Phys. 46 (1975)5, p. 230612307, [XVII30] M. Araki, and C. 0 . Leiber, Linear Relation between Shock Velocity and Pressure in Relation to the Grheisen Parameter, 1991 Topical Conference on Shock Compression of Condensed Matter, Williamsburg, VA, USA, 17120 06. 1991, in S. C. Schmidt, R. D. Dick, J. W. Forbes, D. G. Taskers: Shock Compression of Condensed Matter 1991, p. 5471550, Elsevier Science Pub. Amsterdam, 1992. [XVII-3I] Ya. B. Zel'dovich, and Yu. P. Raizer (edited by W. D. Hayes and R. F. Probstein): Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. Academic Press, New York and London, 1967, p. 6851715. [XVII-32] G. A. Simons, and H. H. Legner: An Analytic Model For The Shock Hugoniot In Porous Materials, J. Appl. Phys. 53 (1982)2, p. 9431947.
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XVIII
CRITICAL DIMENSIONS
In the present microscopic detonation model many well-known but in classical terms little-understood effects have been resolved as a single source or a bubbly quasicontinuum. Another approach includes consideration of cooperating pressure sources. These pressure sources can most effectively be driven by chemical reactions, where gases blow up the reaction sites in a condensed medium. Other possible drivers might be purely physical mechanisms. The onset of chemical reaction is governed by losses of bubble motion in liquids and by crack formation in solids. A challenging problem from the safety aspects is that of the critical diameter, or better (in the present view) that of critical dimensions, which is not involved in classical detonation models as a necessary constituent. An excellent review on critical phenomena was given by Donna Price [XVIII-I].
In cylindrical geometry below the critical diameter, an existing detonation fades out or fails to propagate when initiation is attempted. It is erroneously believed that the critical diameter is a system constant. Also erroneously, one uses the critical diameter of cylindrical geometry in classical terms to answer questions of thin-film detonations. The cylindrical critical diameter varies by orders of magnitude, fiom tens of pm (hydrazoic acid) up to tens of centimeters for slurry explosives. This large range leads to significant uncertainties of so-called safety tests. Whenever small-scale samples are smaller than the critical diameter, by testing an insensitivity is inferred that actually does not exist. The UN tests for insensitive materials are typical examples of such dangerous misconceptions promulgated from results of safety tests. Burning of an explosive is possible below the critical diameter, where an isotropic low pressure results, but a unidirectional high pressure appears in detonation. Thus this process is intimately related to the Deflagration-to-Detonation Transition (DDT) phenomenon. To clarify the present difficulties of understanding these phenomena, we will discuss the classical aspects and problems of critical-diameter phenomena and some unanswered questions in the following sections.
393
394 Chapter
Classical Aspects of Critical Phenomena There is only one basic approach in the classical plane-wave model of detonation that of Khariton [XVIII-2], who argued that any side rarefactions of velocity c fiom the periphery would not disturb the chemical reaction before its completion within the time &hcm. of the ongoing chemical reaction. This would order the critical dimension in any geometry to CT,hem., so that the critical diameter (XVIII- 1) Belyaev and Kurbangalina [XVIII-3] used this approach to calculate the temperature dependence of the critical diameter by varying T,hem with reaction kinetics approaches. Dremin and Trofimov [XVIII-41 modified this approach by considering a twodimensional flow pattern of the corner turning test. But to get remarkable agreement of the experimental critical diameter they needed the following data: initial density, Hugoniot of the initial explosive, detonation velocity and pressure, sound velocity in the original explosive, sound velocity in the CJ-compressed state, the velocities of reaction inhomogeneities, the detonation velocity in the shock-compressed state, and the induction time of the chemical reaction.
Facts and Problems about Critical Diameters As a highlight of the Dremin-Trofimov-mode1 Enig and Petrone [XVIII-51 calculated the critical diameters of nitromethane (NM) and liquid TNT at 85°C. But Kurbangalina and Patronova [XVIII-61 found that this critical diameter in general varies greatly (by several orders of magnitude) with the confinement. So the critical diameters of liquid TNT at 85°C are in steel confinement: 3.8 mm; in glass 60 mm and in paper 90 mm. It was also found that the power of initiation should be meaningful, but this influence is cancelled by using heavy confinement (thick-walled steel cylinders) [XVIII-7], [XVIII-81. No correlation was found between the critical diameter and shock temperature (and the chemical reaction time ~ , h ~ ~as. was ), expected [XVTII-31. Tarver, Shaw and Cowperthwaite [XVIII-9] compared experimental values of the critical diameters of fluid nitroalkanes with the calculated Dremin-Trofimov values. Large deviations were found in cases where the detonation parameters used for calculation had not been experimentally determined.
395
Nor is there a general rule of the critical diameters for LVD and HVD. Results of Seely, Berke, Shaw, Tegg and Evans [XVIII-lo] demonstrate that for several difluoroaminoalkanes in general, LVD critical diameters are lower than those of HVD. Kozak, Kondratjev, Kondrikov and Starshinov [XVIII-1 11 found that the reverse applies for nitromethane (NM). The HVD critical diameter in steel confinement is, at 3.8 mm, lower than the LVD critical diameter, at 5 - 5.5 mm. This difference is much more pronounced for CAVEA B-1 10 {nitric acidhe1 solution (oil)}, where in aluminium confinement of wall-thickness 0.9 the HVD critical diameter is about 4.6 - 7.9 mm compared to the LVD-values of 30 - 36 mm {van Dolah, Watson, Gibson, Mason and Ribovich [XVIII-121). Another unexpected effect occurs: Detonation fails when the confinement is deformed only slightly. This proves that the strength of this confinement is more important than its material (impedance) and mass [XVIII-61, [XVIII-1 11. Up to now it was believed that the critical diameter was a constant for a specific system, so that it !jhould be a univalued reliable safety measure. But in cylindrical geometry the critical diameter is determined by an UP-and-DOWN method, SO that scattering in critical-diameter values cannot be resolved. It was also commonly believed that detonation should fade away below the critical diameter. But each experiment delivers a critical diameter in conical geometry. Therefore, such a method is of particular inlerest to study the scattering of critical diameter and its nature. The results from more than 100 experiments in conical geometry are that the detonation faded out in aboul 80% of all cases at the critical diameter, which had considerable scatter. Transition fiom detonation to burning occurred in most of the other 20% of experiments at this critical diameter. But in two cases at the critical diameter, at about 15 mm at room temperature transitions s LVD =$ HVD 3 LVD 3 Fading out occurred and faded at ca. 5.5 mm very low diameter (at room temperature), even in soft plastic confinement [XVIII- 131 (see Figure XI-5). In gas detonation the (fiee) critical diameter is sometimes determined in a unique way: The gas detonation in a confined tube enters an infinite volume. As the diameter of this tube decreases to the critical diameter (assumed without confinement) detonation does not proceed into the infinite volume. Cell formation has been simultaneously observed. Whereas some authors [XVIII- 141, [XVIII- 151 have found between 8 and 13 cells at the critical diameter, other authors [XVIII-161 had evidence that no general specific number existed. But a relation between the number of cells and a critical diameter has been obtained. The side rarefactions dictate critical dimensions in classical terms, but this concept fails for thin-film detonations and the detonation of 'empty' tubes. In modern codes critical-diameter phenomena are entered by the empirical pop-plot data. Obviously, the experiences and approaches described above do not contain any insights into qualitative differences between isotropic burning and unidirectional detonation events or transition phenomena.
396 Chapter
Microscopic Detonation Model The pressure sources within the volume are activated by external or internal pressure, so that they radiate pressure waves (see Figure XVIII-1). Their radiation is amplified or attenuated, and net production is observed at the right surface. This volume property can be reduced to a surface property by Green’s theorem. The general solution of such a problem is given by the Helmholtz-Huygens-integral, which simplifies to the for a plane surface. All this is not exactly tractable, so that suitable simplifications are desirable.
Figure XVIII-1. Microscopic Detonation Model: The piston (left) activates the pressure sources in the volume, which in turn amplify or attenuate the input pressure. The result of the volume properties can be observed in the area of observation at the right-hand side.
Depending on the volume concentration of the sources and their dynamic stimulation, pressure transparency increases, and the sources in the deep volume prepare the sources in the right plane of observation. This evolution dies out in the volume below a certain concentration of the pressure sources. Since one pressure source (hot spot) radiates always isotropically, and detonation shows a directivity of pressure emission, many hot spots are needed to produce a detonation if they radiate pressure waves unidirectionally. Several source configurations are of practical interest: a linear array of sources (thin-film detonation), a rectangular array of sources (detonation of a prism), a circular array of sources (cylindrical detonation), an array on the circle line (periphery), (shock tube, NONELe-effects).
The number of reaction centers necessary to get a unidirectional pressure is important. This task addresses also the problem of cell numbers in relation to a critical diameter. These problems are too complicated to be solved in general terms. Therefore, asymptotic solutions are applied by assuming spherical harmonica1 waves, which at present are hrther simplified by point sources. NONEL (= non electrical initiation) is the name of the first commercial product using this effect.
391
Dimensions
Coupling of Geometric Quantities with Dynamics We know from Bridgman's [XVIII-171 experiments of that detonation is not a matter of static high pressure but of its dynamical nature. Therefore, any detonation model needs dynamic characterization. As done in Chapter IV, all geometric quantities, like distances or source sizes, are related to dynamics, expressed in terms of wave length or the wave number k = 2.nIh; so that dimensionless quantities result that characterize a dynamic system. Such a dynamic component has not been considered before now in detonation science. By Fourier integral, (see Chapter VIII), we can relate the shock rise time 'rhockrlse to a frequency for a wave length clf. Each geometric quantity is related to the wave length h or k. or
b1-Z
h = ( 2 . . . 3 ~ ~ h ~ ~ kor ~ i ~ ~ ) c
2. 3
27c shockrise
2 Tshockrisec
(XVIII-2)
Harmonic Pressure Source The harmonic pressure p" of a finite-sized spherical source, of radius Ro in a medium of density p and sound velocity c with the corresponding original volume velocity Po= 47c~g& , depends on the distance rlh to the point of observation P (see Figure XVIII-2). (XVIII-3) For an asymptotic generalization, we use point sources with Ro 0, but take into account the given finite volume strength and get without the time dependence: (XVIII-4) with a constant phase shift of90". This single spherical source always radiates isotropically.
398 Chapter
Linear Arrays of Harmonic Pressure Sources In a linear array, (see Figure XVIII-2) each source contributes, depending on its individual strength, phase and distance, to the total pressure p at the point of observation P. It is assumed for simplicity reasons that each source shows an identical strength and phase.
Figure XVIII-2. Configuration of a linear source array. The path of summation is indicated for finite-sized sources on the righthand side.
h =
(t)+(t)' -
2
2( ;)(&
2( ;)(E
-
I)(
I)($)
+ {(E - I)($)}
(XVIII-5)
9t')+ { >'
and one gets: (XVIII-6)
399 Dimensions
Asymptotic Considerations A Thermodynamic Approximation All sources (of the same strength and phase, see above) of the array show an identical infinitely large distance rlh to the point of observation P, then
(XVIII-7) holds. Accordingly the total volume strength of n sources is represented by a single giant source of volume strength . Since h is not an additive quantity, it is evident that within this asymptotic approximation the dynamic property of the giant source is different fiom that of the individual sources. We call this a thermodynamic approximation, because chemical kinetics addresses only the gas evolution rate of n reaction sources (hot spots) { Po} at an infinite distance.
Explosion (Fraunhofer) Approximation
B
Explosions are usually not observed at infinite distances ti-om reacting sources, but are observed at some finite distance or even at the detonation fiont. Therefore, the varying distances of Equation (XVIII-5) must be used, which dictate both the strength and phases of the sources at the point of observation P. But it is cumbersome to get analytical expressions for the sum of Equation (XVIII-6), so further approximations are used.
(:)
If -
[Y), -
which means that the point of observation P is at a distance of ten
times or more of the diameter of the source array, then the bracketed ( { }) term in Equation (XVIII-5) can be ignored. A constant distance r'lh is used for the strength, and an approximated, varying distance from the source to P is used for the phase relations. (XVIII-6a)
400 Chapter
With the approximation for small arguments
+
= 1 n/2
one gets:
(XVIII-Sb)
Inserting into Equ. (6a) one gets: (XVIII-8)
The second bracketed expression is the directivity fbnction D(9), and can be evaluated according to the basic relation: n
n
(XVIII-9) Using the abbreviation
e = x(Ad/h)cos 9
one gets (XVIII-10)
Since the modulus of
=
-1 1 = 12sin
lzl
XI ,Equ. (XVIII-10) firther simplifies to (XVIII- 1 1)
so the directivity pressure ID1 depends on the angle between the array and the (varying) point of observation, and on the number of sources.
C
Fresnel Approximation
In the Fresnel approximation the ignored terms in the distance are considered. However, it seems that this approximation does not contribute to more specific insights to present-day detonic consideration. Exact solutions will be more important when the point of observation moves in the line or area of the pressure sources.
40 1
Dimensions
Infinitely Dense Linear Arrays of Harmonic Pressure Sources If the number n of the sources in the linear array increases, one gets a densely packed linear array with a length of (TI - 1) With the abbreviation 8= cos 9 one gets fi-om Equ. (XVIII-11): sin no sin lim -- -nsine
n a m
-
sinn cos n.(!/e/h)cos3
(XVIII-12) '
The transition from isotropic radiation behavior of the linear array to lobe formation is shown in Figure XVIII-3, left, as varies from 0 to 1 or more.
The Minimum Number of Pressure Sources One pressure souirce always shows isotropic radiation. So the question arises on how many pressure sources are needed in a linear array to get a lobe formation? This is easily answered by the application of Equations (XVIII-11) and (XVIII-12). Just two sources can produce a lobe formation (see Fig.XVIII-3, right), where for n = 2 the side lobes (not shown in Figure XVIII-3, right) are as pronounced as the central lobe. Additional sources reduce and shift these side lobes, but do not contribute much more to directivity, provided the length = 1 is given.
Lobe Formation in other Geometric Configurations The origin of the directivity fimction (or radiation factor) was originally developed step by step for a linear array, since optical and acoustical tools were rarely available to detonation scientists. Nevertheless such problems have been considered and solved for a century in opticrs [XVIII-18] and acoustics [XVIII-191. The directivity functions are given in Table XVIII- 1 for the above-mentioned geometric, densely packed geometries according to Stenzel [XVIII-20], [XVIII-211. The lowest dynamic conditions of lobe formation are given on the right side. The directivity functions are shown in Figure XVIII-4 for the densely packed circular area and ring. The results are spectacular: a lobe formation is obtained for a circular ring at about half the diameter of that of a densely packed circular area. Another surprise is that in the rectangular prism only the extent of one dimension is decisive for lobe formation (see Table XVIII- 1. Application to Detonation Problems The commonality of all densely packed configurations is that the directivity function /DI for or = 0 is always isotropic, and the intensity of pressure-wave
402
Densely packed Linear Array
Single Sources on the Linear Arrav
L:
ID1
-
IDI
Figure XVIII-3. Directivity hnctions of a linear array. Lefi: as function ofUh , the transition from isotropic to unidirectional radiation is seen as varies from 0 to 2 1 A first lobe formation occurs for = 1. Right: for the condition = 1, the directivity is plotted versus the source numbers n. Two sources can lead to a lobe formation, but the side lobes are as pronounced as the main lobe. These side lobes are reduced and tend more to 90" as the number of the sources n increases.
radiation is zero
0
) as is the case of a benign combustion.
As
or
((all)increases, pressure and anisotropy of the pressure radiation increase. If with 3 1 in the linear array a lobe formation is obtained where the direction of the high pressure is perpendicular to the source array, and no pressure is present in the line of the array. This is characteristic and attributable to detonation phenomena, so that the domain 0 < 5 1 can characterize DDT. Pressure-wave radiation as well as unidirectionality increase in this domain. However, when increases further, pressure and directionality increase, but minor side lobes also appear. This is the first time that a basic mechanism is presented in acoustical terms, where pressure-wave radiation as well as directivity of DDT is explained without assumptions.
Using Equ. (XVIII-2) together with the conditions of the first lobe formation, one gets for a linear array:
(XVIII- 13)
403
Densely Packed Circular Area
1.0
1.0
Figure XVIII-4. Directivity functions of a densely packed circular area (left), and a densely packed circular ring (right). The conditions for lobe formation are for the ring about half of what they are for the area!
and for a cylindrical, or shock tube (NONEL) detonation:
(XVIII- 14) These relations aire similar to Khariton's expression (l = physical background.
but with a very different
One can also see that the conditions of the critical diameter may be obtained for a fixed geometry by increasing the dynamics. Therefore, the facts that critical diameters depend on initiation, triggering, and the stage of overdrive (see the case of conical geometry) can be understood qualitatively. Depending on the circumstances, the dimensions may scatter as the dynamic situations vary. The unclear situation of the critical cell numbers at the critical diameter is also resolved: A minimum number of 2 - 3 cells is necessary when the dimensions of the assembly comply with the dynamics. Therefore, the number of cells over the critical diameter can be constant, or nearly so in similar situations.
404
Chapter
Table XVIII-I. Directivity Functions of densely-packed Configurations. Condition = 0 --ID/ = 0
I p - '=
a
a
1
=0.7655
__ - 1 2197
In thin-film detonation, the minimum film thickness should correspond to about half of the critical diameter (in cylindrical geometry) according to classical theory. Watson, Ribovich, and Gibson [XVIII-221 have shown that thin films of NG/EGDN 50150 show ideal detonation velocities, not sensitive to dilution. LVD only was observed for a film thickness of 0.8 mm, but in the range of 1.6 to 3.2 mm both HVD and LVD have occurred. The minimum LVD-film thickness is about 0.13 mm according to Ribovich [XVIII-231, compared to the critical diameter in cylindrical geometry (in Al-tubes, wall thickness 0.9 mm) of 3 - 6 mm determined by van Dolah et al. [XVIII-12]. This classical, conflicting behavior is easily explained by the reasoning presented here. The NONEL shock tube [XVIII-241, [XVIII-25] is a plastic tube of 3 - 6 mm 0. D. and 1 - 1.5 mm I. D. covered on the inside with 15 - 25 mg/m, corresponding to 0.32 mg/cm2 of PETN or HMX, and having a minimum film thickness of 1.9 pm. A shock can travel through this tube in spite of the fact that the critical diameter of LVD of PETN in steel confinement is less than 5 mm [XVIII-26]. A classical interpretation of this event does not exist. Other situations, for example that a detonation can die away as the confinement widens, cannot be explained by the present asymptotic considerations. This behavior is satisfactorily explained by making specific calculations in the line or area of the pressure sources. This will be outlined in the following.
CRITICAL-I)IMENSION PHENOMENA NEAR THE SOURCES IN A
LINEARARRAY Actually, effects at the detonation front are studied in detonation physics. Therefore, phenomena at and/or near the reaction sources must be considered separately. In this context, we are interested in why and how the source characteristics change from subcritical to supercritical source arrangements. This question cannot be answered in general terms. We have to expand the point sources to finite-sized sources, because the source impedance, as a measure of the ratio of energy density in compression and flow, is a property of finite-sized sources, see Equ. (IV-44). Harmonic solutions are used to avoid additional time effects. It is evident that now special situations must be considered.
Further Description of the Harmonica1 Spherical Source Beside the description of the pressure source by Equation (XVIII-3), see also Equ. (V-9) the radial particle velocity u, is given by
(XVIII-15) -
.
(kR)2 kr + i kR+ i (kr)2
The common factor
ik(r-R)
v
is usefd when one considers point sources (R
0), because
2h2
it characterizes the source strength. However, this quantity is not appropriate in considerations of the isolated effects of varying both the geometric and dynamic properties. Since (XVIII-16) one may use alternatively as a common quantity R , the bubble-wall velocity. Assuming the continuity of mass (no mass exchange between the medium and the bubble content) one gets R=Ur,
where u, is an external stimulating quantity.
(XVIII-17)
406
The quotient of Equations (XVIII-3) and (XVIII- 15) is the impedance Z = - = pc-
kr
kr
=
(XVIII- 18)
which holds for each distance r.
Equation (IV-44) dictates the ratio of energy density E in compression and flow, and addresses DDT. The value at the source surface is obtained by using r = R. As can be seen, the plane-wave value Z = pc is obtained for r 3 00, so that any assumption of plane-wave impedance near the source is incorrect. The above equations, using Euler’s Equation
(XVIII-(20)
=cosz+isinz
read in the components as follows R e p = -pc--
v0
cos k(r -
2h2 1 Imp =
{kR
+ sink(r -
sink(r-R) rih
2h2 1 + (kR)2
(XVIII-2 1)
rlh
rlh
-
cosk(r-R) rlh
(XVIII-22)
and the modulus, which is half of the time average of the real part
(XVIII-23)
for kR 3 0 one gets IpI
= pc--
v
2h2
1
(XVIII-24)
(4
Equation (XVIII-15), in components, reads as follows: cos k(r -
Reur =
sin k(r -
+
2h2 1+(kR)2
(XVIII-25) and (XVIII-26)
407
and the modulus, representing half of the time average of the real part, is calculated according to the previous procedure. R
In contrast to the pressure, the velocity has two terms: one dropping with l/r and the other phase shifted with l/rz, which is called near-field term. Therefore, this near field cannot be ignored near the sources. Another situation specific to detonics is with u,: the pressure acts isotropically but the velocities show directional properties. In classical detonics only one (plane wave) direction of propagation is present, so all measured quantities relate to this plane-wave entity. In detonics only the components of u, in the direct ion of propagation are considered; and the particle velocity u p is up = u, cos 8,
(XVIII-27)
where 8 is the angle between the direction of wave propagation and the point of observation P (see Figure IV-2).
Transition from a Linear Array to an Areal Array Whereas in Figure XVIII-2 only a linear array was considered for vector values like the velocities, some aidditional aspects come into play that are absent in scalar pressure. Figure XVIII-5 slhows possible distributions of the sources in an area. Additional&, the azimuth cp must be considered for diameter profiles in three dimensions.
Figure XVIII-5. Possible source configurations
408
So the angles are (XVIII-28)
With respect to computer calculations, it is better to determine cos 0 indirectly from sin 0:
and (XVIII-3 0)
Monopole-Cosine Source: The pressure as a scalar remains isotropic, but the radial velocity u, is connected to the plane-wave particle velocity by u p = ur cos 0 cos cp . The velocity potential for the derivation of this particle velocity reads cos 8 x cos cp. Ignoring the transverse component of the velocity, we obtain also
case C O S T
up =
(XVIII-3 1)
and 1 d@
1
r
r
(XVIII-32)
According to this the velocity equations for an area array read n
n
30 (2
as function of temperature, see Figure XIX-15 ~~
NM1EDA = 99.995/0.005 Vol.% +23"C, fresh
10.5
NM1EDA = 99.99510.005 Vol.% -25"C, fresh
13.8
Cap-test as
313
0/5
of tests
439 Critical Diameter(s)
Nitromethane
Figure XIX-15. Critical diameter as function of the temperature of the system NMEDA = 99.9Y0.05.
NM/EDA = 99.95/0.05 VOl%
?
..-
-10
-4
0
8
10
.a
so
M
so
Temperature ["C]
Critical Diameter of the NWMethanol System The critical diameter relationship of the NM/methanol system in conical geometry at + 19°C is given in Figure XIX-16. Where transportation is concerned our UN experts always refer to room temperature, and do not consider temperature variations. Such temperature variations are shown in Figures XIX- 17 and XIX- 18. Beyond any experimental error, the types of temperature variations vary greatly. __
m7
i
*'
NM/methanol (+ 19OC)
-1,
Hi
*.
--"' 22
Figure XIX-16. Critical diameter in conical geometry
s-4
-mi
g 2 ,J .E,o 1
E ' ._ 8 -
.I
1
_-___.__ LIO.0
e . 0
e5.0
e7.s
-~7-
00.0
0L.S
Concentration of NM [Volo/~]
m.0
T
07.9
-
7
100.0
440
Chapter
NM/methanol = 80/20 Vol%
,
4
,
,
,
ma
M
ma
,
,
,
,
70
?a
#a
,
Figure Temperature dependence the critical diameter of NM/methanol 80/20 system.
,
Temperature ["C]
Figure Temperature dependence the critical diameter of NM/methanol 85/15 system.
Critical Diameter of the NWi-Propyl Nitrate System As shown even NE as an explosive liquid can increase the critical diameter of NM. i-Propyl nitrate (i-PN) exhibits at +23"C; +60°C and +80°C a critical diameter > 30 mm. One should expect, therefore that addition of i-PN to NM should increase its critical diameter. But Figure XIX-19 shows that this is not the case. On the contrary i-PN is only a little bit sensitizing, but at a limit of > 25% i-PN the critical diameter increases steeply to > 30 mm.
44 1 Critical Diameter(s)
Nitromethane
so-=-
--
NM/Isopropylnitrate (IPN) (+ 2 O O C ) W
.-
Figure XIX-19. Critical diameter of the NM/i-PN system at room temperature.
I*
7-
w
a0
*,
Concentration of NM [Vol0h]
Critical Diameters of NM as Function of its Dilution and Confinement Kurbangalina and Patronova (1976) [XIX- 161 investigated d,, of NM as a hnction of dilution and confinement. Their results are summarized in Table XIX-9. For the experiments in steel tubes, the wall thickness A in mm, the ratio of inside/outside radius R,/R, of the cylinders, the experimental results, and whether a detonation (Det.) had been observed, or not (-), are indicated. An influence of the strength of the confinement was found. The fading out of detonation was accompanied by a dilatation of the tube. It was not possible to decrease the critical diameter repeatedly by increasing the wall thickness A and wall strength. Usually a wall thickness of 3 to 5 mm is most efficient for a decreased critical diameter. Such an effect also appears when the glass tubes are covered with metal foils.
The aci-Form of NM Van Dolah, Herickes, Ribovich, and Damon (1958) [XIX- 171 investigated the shock sensitivity of NM-systems. The shock sensitivity of neat NM increases with the temperature, so the 50% gap value is 4.5 and 6.0 mm for 0°C and 40°C respectively. By the addition of 5% EDA there increases this gap value by 11.6 mm, and 4 mm by the addition of 5% strong acids.
442 Chapter
Table XIX-9. Critical Diameters (HVD) of NM as a Function of Dilution (acetone = Ac) and Confinement
NM/Ac=92/8 ( p = 1.10)
3.5
NM/Ac = 90110 (p = 1.09)
4.6 5 5 5 4.2 4.2
4 3 2 12.0 17.8
0.38 0.45 0.56 0.15 0.11
Det. Det.
17 18.1 18.1 18.1 17.1 15.0
5.0 4.5 4.0 12.0 30.0
0.64 0.67 0.69 0.42 0.20
Det. Det.
NM/Ac= 81.3/18.7 (p = 1.05)
27
42
12
33
54
12.5
136
220
~
12.9
Det.
This increase in sensitivity is sometimes attributed to the formation of aci-nitromethane. Whereas the normal form of NM is: CH3N02that of the aci-form is CHzNOzH. The reactions of pure NM are: K,
CH3N02f) CHzNO-z Abbreviation: CH2N0-*+ Abbreviation:
=
10."
(N) f) (A-) + f)
CHzN0-2H (AI+
K ~ lo3 =
* (AH)
It seems that the aci-form should be unstable, and therefore NM must be sensitized if the concentration of the aci-form increases. Conclusion: Any substance, which can not produce an aci-form would not be influenced by such additives like strong acids and amines. If this is wrong, additives can in general alter critical diameters and shock sensitivities. Therefore it is of interest to know whether the aci-form is crucial for sensitization or not.
443 of
The hypothesis th,at the aci-form will increase sensitivity is rejected for the following reasons: Acids and bases sensitize NM, so the aci-form cannot be responsible for any increase in sensitivity! Also other liquids, which do not exhibit an aci-form can be sensitized by the addition of DEA [such as a mixture of NM with nitropropane (NP)]. Introducing NP does not change the mechanism of the influence of DEA on the detonation of NM. Presumably, the influence of DEA on the detonation of NP is similar to that on detonation of NM [XE-181. In extensive falling weight experiments no increase of sensitivity was found for NM with increased concentration of the aci-form [XIX-19]. The sensitizers do not increase the linear burning velocity [XIX-20]. The sensitizers do not increase the sensitivity to LVD initiation. At concentrations above 10 to 15% they behave more as inert additives [XIX-20]. Since acids as well as bases sensitize, the aci-form cannot be responsible for any sensitivity variations. In general, the sensitizers increase HVD sensitivity and critical diameters only and not LVD.
The gap values found were correlated with the above-found species concentration. A relatively good linear correlation was found only for the gap sensitivity in mm and the logarithm of the total ion concentrations [XIX-191. In summary this author believes that any critical diameters or sensitivities can neither be predicted nor understood in classical terms. The statistical scatter in the effects, their dependence on confinement, and also the correlation of gap sensitivity with total ion concentration (ions are cavitation nuclei) indicate towards validity of the proposed model of critical diameter. It is difficult to suggest appropriate experiments that would make a result decisive. With respect to safety considerations the result is unsatisfactory, since anything becomes possible under certain circumstances and 'safe' safety margins vanish.
Application of Detonation Traps An ingenious application of all the properties of critical-diameter phenomena evaluated in Chapter XVIII, historically going back probably to Medard. A detonation in a pipe of supercritical diameter can be extinguishes by reducing the cross section to subcritical diameter and expandable tubes. An increase of the cross-section to
444
Chapter XIX
supercritical diameter reduces the particle velocities. Further expansion possibilities are additional safety measures. Angus Chemical Corporation uses such detonation traps in the transfer of NM in pipes. A well-proven description is given in Figure XIX-20. Also plastic materials can be used to advantage.
Figure XIX-20. Proven detonation trap of Angus Chemical Corporation [XIX-2 11 (with kind permission)
445 ofNitromethane
References [XIX-11 H. Eying, R. E. Powell, G. H. Duffey, R. B. Parlin: The Stability of Detonation, Chem. Rev. 45 (1949)1, p. 691181. [XM-21 H. Badners and C. 0. Leiber: Method for the Determination of the Critical Diameter of High Velocity Detonation by Conical Geometry, Propellants, Explosives, Pyrotechnics 17 (1992)2, p. 77/81. [XM-31 A. W. Campbell, M. E. Malin; T. E. Holland: Detonation in homogeneous explosives, 2"d ONR Symp. Detonation, 1955, p. 336/359. [XM-41 W. C. Davis, Craig; Ramsay, Phys. Fluids 8 (1965), 2169 cited from J. E. Enig; F. J. Petrone: The Failure Diameter Theory of Dremin, Symp. (Int.) Detonation 1970, p. 99/104. [XIX-5] following Campbell et al., cited in C. H. Johannson and P. A. Persson, Detonics of High Explosives, Academic Press, London, 1970. [XIX-6] H. Presles, Ch. Brochet: Detonation cylindrique divergente du Nitromethane, Astronautica Acta 17 (1972), p. 5671573. [XIX-7] N. Kaplan, S.A. Johnson; R.C. Sill; L.H. Peebles: Thermal and Shock Sensitivity of Nitromethane, Progress Rept. No 1-35, Jet Propulsion Laboratory, CALTEC, 1945. Referenced according to Military and Commercial Applications of Nitromethan-Based Explosive Systems, International Minerals & Chemical Corporation, Nitroparaffin Div., Des Plaines, [XIX-8] Manual of tests for the qualification of explosive materials for military use, Allied Ordnance Publication, AOP-7, February 1988. [XM-91 A. Makovky, L. Lenji: Nitromethane, J. Chem. Phys. 64 (1958), p.2665/2673. [XlX-101 Commercial Solvents Co., New York:. Storage and Handling of Nitromethane NP Series, TDS No 2, 2"d edition. [XM-1 I] B. N. Kondrikov, G.D. Kozak; V.M. Raikova; A.V. Starshinov: Detonation of Nitromethane, Dokl. Akad. Nauk SSSR 233 (1977)3,p.402/405. [XIX-12] R. Engellke, W.L. Earl; C.M. Rohlfing: Production of the Nitromethane Aci Ion by UV Irradiation: Its Effect on Detonation Sensitivity, J. Phys. Chem. 90 (1986)4, p. 545l547. [XM-13] R. Engelke, W.L. Earl; C.B. Storm: Detonation Sensitivity of Liquid Nitromethane Doped with Methyl Nitrite, Propellants, Explosives and Pyrotechnics 13 (1988), p. 189/190. [XM-141 M. Kusakabe, S. Fujiwara: Effects of liquid Diluents on Detonation Propagation in Nitromethane, 6" Symp. (Int.) Detonation, ACR-221, Office of Naval Research, Arlington, VA, 1976, p. 133/142. [XM-151 C. Brochet, H. Presles; R. Cheret: Detonation Characteristics of some liquid mixtures of Nitromethane and Chloroform or Bromoform, Proc. 15th Symp. Int. Combustion, Combustion Inst. Pittsburgh, PA, 1975, p. 29/40.
446
Chapter
[XIX-16] R. Ch. Kurbangalina, L. I. Patronova: Einflulj einer Stahlhulse auf den kritischen Durchmesser der Detonation kondensierter Sprengstoffe, Fiz. gor. vzryva 12 (1976)4, p.6431647. [XIX-17] R. W. van Dolah, J. A. Herickes, J. Ribovich, and G. H. Damon: Shock Sensitivity Studies of Nitromethane Systems, XXXIe Congrks International de Chimie Industrielle, Liege, 1958, p. 1211126.
[XIX-18] B.N. Kondrikov, G.D. Kozak, A.V. Starshinov, Effect of amines on critical detonation diameter of nitrocompounds, Chemical Physics of Condensed Explosive Systems, Mendeleev Institute of Chemical Technology, Moscow, 1979, v. 104, p. 91-95. [XM-191 U. Ticmanis, K. Mehlhose, W. Rudo, P. Langen and C. 0. Leiber: Determines ACINitromethane Sensitivity?, Contribution to DEA 1060, Liquid Propellant Technology, BRL, Aberdeen, ML, March 1983 [XM-201 G. D. Kozak, V. V. Kondratjev, B. N. Kondrikov, and A. V. Starshinov: LVD ofNM, in Akad. Nauk SSR, Otd. Inst. chim. Fiz. Detonacija, Chernogolovka, ed. by A. N. Dremin, 1978, p. 34/37. [XIX-2 11 Physical Properties and recommended handling, shipping, and storage precautions for Nitromethane, Bulletin JLTN-3, Joseph L. Trocino & Associates, Sherman Oaks. Also Angus Technical Data Sheet, TDS No 3 (4" edition).
xx
SMOOTH AND ROUGH PRESSURE FRONTS,
DARKWAVESAND DDT Introduction Arguments were expressed in Chapter I that the classical plane-wave model of volume homogeneous detonation only holds approximately, and plane detonation waves with plane impedances cannot exist. This is established by the unique experimental work of Mallory, Chapter 11, who demonstrated the multiple character of detonation fronts. In Chapter XVIII the varying pressure profiles were shown for sub- and super-critical diameters, which resulted also in a link to the DDT. Therefore it is of interest to demonstrate the construction of the observed detonation profiles by the assembly of the single pressure sources, that is to reduce the macroscopic observations to the basic principle of pressure-wave generation.
Modeling of Pressure Fronts In order to describe the variety of pressure fronts in Chapter 11, we consider arrays of pressure sources {see Fig. XVIII-2 (linear array), and Fig. XVIII-5 (areal array)} and use Huygens principle to sum up the contributions of the n single pressure sources at the point of observation P. A notation different from that used in Chapter XVIII is demonstrated below; leads to the same results (see Chapter IV). n
(XX-1) F=l
ii=l
The particle velocities are (XX-2)
(xx-3)
441
448
Chapter
" 1 ut = - c - - ~ c o s ( P I
7
(XX-4)
n =1
with the corresponding impedances Z = pfu, respectively. The solution of the spherical wave equation, in terms of the spherical Hankel-finctions h,,,(')() and Legendre polynomial P,(cos where 8 is the angle between the polar axis and the direction of motion, is in spherical coordinates
(XX-5) E=O
and the velocity T
(XX-6) We can determine the appropriate coefficients A, if we know the velocity at the source boundary. Only two pure modes are realistic in real life: Monopole Source, m = 0:
a)
A radially symmetric source surface motion, R = u , independent of the angle 8, is only possible for Po(cos = 1. Therefore, (XX-7)
(XX-8)
aO=
v
&k(r-R)
-iwt
4nr I-ikR
b)
(XX-9)
Monopole-Cosine Source:
In the case of a "plane-wave expectation" we observe the properties of a monopole source in an assumed uniaxial direction and ignore the other components. The pressure as a scalar remains isotropic, but the radial velocity u, is connected to the plane-wave particle velocity by up = u, cos 8. The velocity potential for derivation of this particle velocity reads cos 8. Ignoring the transverse component of the velocity, we get in addition to
449 Smooth
rough Pressure fronts
(XX- 10) also
Ut =
~
1 a@ - -=
1
ine
(XX-11)
(see Chapter IV).
c)
Dipole Source, m = 1:
If we directly define the source’s surface motion as
cos e , we get
(XX- 12) with the appropriate pressure, radial and transverse velocities, and the corresponding impedances. It appears that in cases b) and c) ut is out of phase with the pressure {see Equs. (XX-1) to (XX-4)) and carresponds to the near field of the particle velocity up, If one wants to avoid the imaginary i, it is possible to introduce a constant phase shift 7c/2 into all the velocity potentials Since exp(-in/2) = - i this quantity may be deleted. In the following, we only consider the moduli of the quantities, but split the impedance into its real and imaginary parts because the real part controls the energy partition. Further, all pressure sources have the same phase and strength in this consideration. The physical reason, why this applies in the case of HVD was outlined in Chapter XIV.
Application to a Linear Array We now use this adgorithm for a linear, equidistant source array, where all sources show the same strength and phase. By varying the parameters of Equation (XX-l), we model the transition from smooth to rough fronts in Figures XX-1 to XX-5. Figure 11-14 shows the results of Mallory and Greene [XX-11 with very strange patterns in concentrated NM/acetone 80/20 systems: On the left we see a very smooth front and on the right a more structured front. These regimes are divided by a dark wave. From the above results we know that conditions of smooth and rough fronts exist. It is reasonable to suggest that in the case of darkness nothing happens, which means that in the regime of dark waves pressure sources are absent. This assumption has been used to calculate Figure XX-6, intended to model qualitatively the behavior shown in Figure 11-14,
450
Chapter
Figure Pressure distribution of 60 harmonic-driven spherical pressure sources of size kR = 0.5; diameter of the array 6h (supercritical array). The sources foam up, and penetrate themselves.
Figure Pressure distribution of 30 harmonic-driven spherical pressure sources of size kR = 1; diameter of the array 6 1 (supercritical). The sources foam UP.
Adlh=O,Z kR.1.0
A?-----
n=2O A d l A = 0.3 kR :0.9625
IS
Figure XX-3. Pressure distribution of 20 harmonic-driven spherical pressure sources of size kR = 0.9425; diameter of the array 6h (supercritical). The sources are at the limit of foaming.
45 1
Figure Pressure dstribution of 12 harmonic-driven spherical = 0.5; pressure sources of size diameter 6h (supercritical). The sources are well separated.
n=12
AdlX = 0.5 kR = 1.0
Adlh =
= 1.5
Figure Pressure distribution of 12 harmonic-driven spherical pressure sources of size kR = 1; diameter of the array 6h. Comparing Figure XX-5 with Figure XX-4, one notices that effects of source spacing cannot be compensated for by the size of the sources.
The relative size of the sources and their distance is shown at the right-hand side ofFigures XX-1 to
xx-5.
During development of the present ideas, a paper was presented by Leiber [XX-21 based on arrays of point sources. In discussion Professor Dremin referred to optical observations of the pressure fiont in inert solids, liquids, and detonating high explosives, which had been made in Russia [XX-31, [XX-41. A mirror-like reflection of visible light was obtained in most cases, indicating that the shock fionts were smooth. He suggested that this implied that if bubbles did exist they had to be small compared to the wave length of the light used. This is not necessarily true according to our present calculation. In preparing Figures XX-1 and XX-2 we used a model of selfpenetrating expanding reaction clusters. In that case relatively large sources may be present, which nevertheless cannot be seen by optical means. Numerous calculations show that the character of an individual source in such an array can only be seen if the distance between the sources, A d a , increases and the size of the sources decreases.
452 Chapter
Figure XX-6. An attempt to model the dark wave seen in Figure 11-14, as a test of the present concepts. The dark waves are due to the local absence of pressure sources, and the smoothhough transition is realized by the source spacing. All parameters are indicated in the figure.
On the other hand, the pressure front smoothes more and more as the sources penetrate each other like foam.
Deflagration Detonation Transition Approach In the following we focus our interest on energy partition problems to understand the transition fiom the benign to the hazardous behavior of bubbles, mainly in assemblies. At this stage, the interaction terms between neighboring sources in arrays is considered. This is a shortcoming because conphase bubble motions influence the others. In a hture effort such interactions must be taken into consideration. We refer in Chapter IV to the fact that the ratio of the potential energy density (coupled with pressure-wave emission), and kinetic-energy density of flow is always unity for plane waves. In terms of a spherical wave at distance r = R at the surface of a monopole source (bubble) this ratio is due to the near (or kinetic) field Epotential Etotal kinetic
-
k2
l+k2R2
-
ReZ PC
(XX-13) '
The power of pressure-wave radiation per unit area is controlled by impedance at the source surface. Re Zlpc is the ratio of the spherical and plane-wave power of pressurewave emission. For a single source this ratio varies from very small to unity depending on its size.
Explosion risks are linked to pressure-wave emission. Therefore the ratio (Equ. (XX-13)) is relevant to safety. We now consider this ratio for a bubble in a usual liquid together with the factors that may alter it. for a bubble is given by Minnaert's resonance frequency [XX-51, (see Chapter VI). From Equation (VI-25) ofthe kR z {0.01~~~0.02}& under normal conditions it is seen that only about system's energy is released by compression. Practically the total energy is in the flow. This bubble always behaves benign. For single bubbles, !&increases with the static surrounding pressure p or by an imposed change of its collapse time. This explains the experimental fact that explosions can be triggered by a pressure shock, which does not usually lead directly to initiation. Also, "triggering" is a well-known phenomenon for vapor explosions (FLI) not predicted by current theories. The influence of triggering was never explored for onset of DDT in explosives.
Impedances of Source Arrays As demonstrated in Figures XVIII-6 to XVIII-18, pressure and particle velocity do not correspond to each other near the front. However, the similarities increase with increasing distance hlh. This behavior is reflected by the Re Zlpc-profile, which approaches the plane-wave value of one with increasing distance hlh. The singlesource quantity Re Zlpc increases fi-om its original value < 1 in an array up to larger than unity. The cooperative effects for the development of pressure-wave emission in an array become evident. The impedance of an array is different from that of a single component. These alterations are not easy to predict, nor is there an asymptotic tool for estimates, because the impedance is a specific property of a volume source and does not exist for point sources. As new example in Figure XX-7 the impedance of an n = 8 source array is Re Zlpc = 1... 1.6 depending on the locus of the point of observation P, whereas for a single source of size kR = 1 we have an impedance value of 0.5 only.
We consider an n = 20 source linear array to get an idea of possible impedance changes in a linear array. We compress and expand this array by varying The corresponding source sizes remain fixed to = 0.1 and 1, respectively. Their individual impedances Re Zlpc are 0.01 and 0.5. The impedances at a fixed locus of observation at the lothsource ofthe array are given in Figures XX-8 and XX-9 as As can be seen, the real impedance varies in this assembly up to function of about 0.2 and 1.5 in the maximum, respectively!
454
Chapter
Figure Modeling of an n = 8 source linear array in pressure, particle velocity and impedance as function of
0.L
= 0.016
Im PC
h.0 h
kR’O.1
Figure Impedance variation of a small source (kR = 0.1) array. The real impedance vanes from 0.01 to a maximum of about 0.2 depending on the source spacing.
I;lo
“ 2 0
0.1
I
0.1
I
I
-
455
rough Pressure
p= I
kR=l i -10
ec
5 -%
Figure XX-9. Impedance variation of a kR = 1 source array. The real impedance varies from 0.5 up to a maximum of about 1.5 depending on the source spacing.
0.5
I
I
I
-
I
I
1.0
1.5
The source sizes are variable according to our assumed model of detonation. Therefore, the impedances of an n = 20 source array with constant source spacing = 0.05 are given in Figure XX-10 as a hnction of the varying source size kR. The locus of observation is at the lothsource and between the lothand 1lthsource. Jumps occur for the impedance of growing sources, but these jumps disappear or become smaller for shrinking sources. Therefore, large fluctuations of the energy partitions occur at the front so that no continuous transition to a final result can be expected. One may speculate on the influence of the numbers of sources in an array where the constituents show low impedance. Calculations indicate that in this case the impedance increases little, even for very large source numbers. A bubble under normal conditions with kRL 0.01 has an impedance of -0.0001. An n = 10.000 bubble array with spacing A d h = 0.0025 has an impedance of -0.02. This might be the reason why even swarms of 'normal' bubbles behave benignly.
-
Next we consider the question how the impedance varies under conditions of constant source size = 1, constant source spacing Adh= 0.05, but increasing number of sources (constant source configuration, increasing diameter). (n - 1) = dh is the diameter of the source array, where = 1 characterizes the critical diameter of such a linear source array. The impedance as function of the multiples of this critical diameter is plotted in Figure XX-11. Contrary to the classical detonation theory, this impedance slowly approaches an asymptotic value that characterizes independence
456
Chapter
kR = Figure XX-10. Impedance variation of an n = 20 source array of constant source spacing as function of the varying source sizes.
---_ n = 20
Adlh =
hlX= 0
I
0.5
1.0
1.5
u ec
Figure XX-11. Impedance of a regular source array with increasing numbers of sources (in multiples of the critical diameter).
from the charge geometry. An impedance larger than that of plane waves is obtained. This means that more energy is released by pressure waves than by flow. On the other hand, a subcritical diameter of detonation is characterized by a predominant energy release by flow. One may puzzle on the hrther interpretation of these results. It may be that the excess impedance characterizes such things as the von Neumann spike. Variation of the impedance with increasing diameter of the array may describe the geometry dependence (time dependence) of "steady" detonation.
Pressure
Discussion of Impedance Results As known, proper explanation of DDT processes is a weak point of all plane-wave theories. Since the early approaches of Kistiakowski [XX-61 and Ubbelohde [XX-71 all subsequent approaches used the "porous piston". The classical piston became porous with variable permeability for back venting. In an open piston the hmes vent backwards, resulting in classical burning without pressure-wave radiation. Different assumed mechanisms choke this porosity with increasing flow velocity of the combustion products: pore collapse, fusion andor comminution of the particles, increasing hydrodynamic resistance of gas permeation by exceptional increase of reaction rate, and so on. According to Kistiakowsky, the violence of combustion leads to "the formation of the detonation wave by a shock wave running ahead of the flame front.." This is a link between DDT and Shock to Detonation Transition (SDT), where the "burning pressure" corresponds to a shock. The reaction clusters are considered pressure sources in the theory presented here. Their radiative power increases with reaction rate, ambient pressure, and cooperative effects of an appropriate array of the reaction centers. An internal or external pressure rise transforms the ability of the bubbles to radiate pressure waves. Energy release occurs initially by flow, but energy partition shifts more and more to pressure-wave radiation, as "dynamics" and pressure increase. Under circumstances (Re Z/pc > 1) the latter becomes larger than the energy of flow. There are many ways to alter the impedance of a single source and a source array: The size and impedance of a single source may be increased by ambient static pressure or an external pressure triggering. It is important to realize that this external pressure shock remains in amplitude much below the critical level of direct initiation. In chemically reactive systems this value is also altered by the motion of forced voids, induced by the rate of chemical decomposition. It is unlikely to stabilize a DDT process, as experience has shown. In this model we have hrther explained the triggering of vapor explosions or bubble resonance explosions of liquid gases and the transition from benign to hazardous twophase flow, depenlding on the circumstances. Bubbly liquids and fluidized beds behave benignly under some conditions and dangerously under others. Chemical-energy release is favored when no other energy sources are present. In summary: DDT and the problems of the critical diameter are intimately linked.
458
CELL AND HERRINGBONE STRUCTURES IN DETONATION Introduction The existence of pressure precursors in detonation has been shown in Chapter XVI. The classical expanding bubble model of Rayleigh was used in all these cases. Such a mechanism have many consequences: Onset of High Velocity Detonation (HVD) in liquid explosives can occur as these bubbles powerfully expand into the low-pressure medium. This expanding bubble screen is a natural barrier to the oncoming original shock, leading to unique Hugoniot formulations, to regions of constant compression and compressibility under certain conditions. The shock is propagated by the expanding bubble screen so that the measured values of the particle velocity correspond to the (maximum) bubble expansion velocity.
It is provocative to use such a basic view to explain complicated problems of detonation, such as the cell and herring-bone structure of detonation in gases and condensed explosives.
Cell Patterns Many scientists believe that shock-wave structures like those indicated by soot imprints would be associated only with (chemical) detonation events. But if one considers Antolik’s [XX-81 first observations of soot imprints of high-voltage discharges in 1875, reasons are given to doubt such statements. The early observations of Campbell and Woodhead [XX-91 and Campbell and Finch [XX-101 show that gas detonations do not have a homogeneous structure in the volume. The internal beauty of detonation structures was resolved by Shchelkin and Troshin [XX-1 11 and researchers in the early 1950’s. For example, Figure XX- 12 shows stroboscopic laser Schlieren photographs perpendicular to the direction of the moving detonation wave in a gas mixture of 4 H 2 + 3 0 2 , at room temperature at initial pressure of 87.3 Torr in a prism of 1-in. x 1%-in. crosssections and the corresponding soot imprint on the side wall [XX-12].
459
and herringbone Structures ofDetonation
Figure XX-12. Stroboscopic laser Schlieren photographs of a detonation wave and the soot imprint recorded on the wall. The medium was a 4 + 3 0 2 mixture contained in a 1 in. x 1'/z in. cross-section tube and maintained initially at a pressure of 87.3 Tom and room temperature. The view is perpendicular to the moving wave. {With kind permission of A. K. Oppenheim.[XX-12].}
These observed cell patterns may become more or less regular in different experiments. High regularity is usually accompanied by low initial gas pressures and diluted detonating system, as is exemplified by the regular soot patterns of Strehlow [XX-131. This is shown as Figure XX-13, the soot pattern of a detonating 50% (2 H2 + + 50% Ar mixture at 65 Torr initial pressure in a cross section of 1% in. x 3% in..
460
Chapter
Figure XX-13. Soot imprints in juxtaposition with a detonating 50% (2 H2 + 0 2 ) + 50% Ar mixture at 65 Torr initial pressure in a cross section of 1% in. (upper) x 3% in. (lower) [XX131. (Kind permission of John Strehlow, (Prof Roger Strehlow passedaway+ 11.12.1990).
The detonating surface roughens in these experiments (see Figure XX-12). Shchelkin [XX-141 suggested the analogy of condensed explosives behavior with gas detonations. H. Dean Mallory [XX-151 observed such fiont roughening for detonating diluted nitromethane (NM). Further nonuniformity in NM detonation was found earlier by Campbell, Holland, Malin, and Cotter [XX-16], and by Dremin, Rozanov and Trofimov [XX-171.
TOobserve structures of detonating condensed high explosives, various techniques have been applied. Howe, Frey, and Melani [XX-181 used optical observations but Urtiew, Kusubov and Duff [XX-191 and many others attached a witness plate to a charge. Regular cell patterns, observed with detonating gases, were mostly absent, but more or less regular herringbone structures were observable. Figure XX-14 shows such a herring-bone structure of detonating nitromethane (NM) attached to a plastic tube.
Figure XX-14. Herring-bone structure of detonating nitromethane attached to a plastic tube.
46 1 of
Figure XX-15. View of the detonating surface of a CzHz/02 10190 mixture at 300 Torr initial pressure. {By courtesy of Held [XX-20].)
Figure C2H2+ 2.5 0 2 (low pressure 0.03 up to 0.2 bar) experiments of Soloukhin [XX-21]. His interpretation was: (a) Detonation destroyed, only transverse waves remain in the central area of the diverging wave until the rarefaction waves arrive; @) Detonation is re-established in the center after some transverse waves collided. The rationale based on the microscopic detonation model is outlined in the text.
The rough detonation front in gases is seen in Figure XX-15, which represents a view of the detonating surface of a C2H2/0210/90 mixture at 300 Torr initial pressure {Held [XX-201) corresponds with Mallory’s impedance-mirror pictures of detonating NM, see Chapter 11.
462 Chapter
The method of Soloukhin [XX-2 11 on the determination of the critical diameter mentioned in Chapter XVIII is presented in Figure XX- 16. He concludes that detonation will not proceed into the semi infinite volume below the (unconfined) critical diameter of the tube due to side rarefactions. He correlated with this critical diameter a critical number of cells. We give another explanation: The pressure profile is parabolic in the subcritical case but shows a belt type in the supercritical case. In the case of reinitiation in the greater volume (right), the stimulus of the pressure sources depends on the particle-velocity profile, which in the supercritical case also shows a belt, so that detonation can proceed. This detonation fades out in the subcritical case, since the detonation must die due to the parabolic profile (see Figures XVIII-19 and XVIII-20). That is why detonations fail or proceed.
Modeling of Cell and Herring-Bone Patterns Ignoring Antolik’s results, complicated theoretical work has been done by many authors to calculate cell spacings in terms of reaction chemistry. A further problem is that the mechanism for generation of the soot imprints is not yet conclusively resolved. Mach stem interactions are discussed, but streaming velocities may also be considered. Within the scope of the microscopic detonation model with spherical sources, the reaction chemistry will be forced by the dissipative losses of bubble dynamics and motion, not by the usual thermal processes. Therefore, detonation structure is primarily governed by bubble dynamics and only indirectly by the resulting chemical reaction. To test this idea, the model of an expanding bubble ahead of the shock fi-ont is assumed without any hrther artificial assumptions. This void overtakes the shock velocity, and expands forward into the low-pressure medium (Chapters XVI and XVII) like a Rayleigh bubble (Chapters V and VI). It again is stopped in its motion by the follow-on pressure pulse. This is simplified to the model that the high-pressure bubble expands with constant velocity and then stops. Such a single source was described in Chapter V (Radiation of a short motion source). Detonation is the result of a number n of cooperating pressure sources. Only sources in the area of observation are considered, whereby their properties are governed by the deep volume sources. The geometric situation in this area is sketched in Figure XVIII-5. The point of observation P with the coordinates xp and yp moves over the diameter of the circle to obtain profiles of acoustic quantities, and for each point a Huygens summation is performed: The pressure p and radial particle velocity u r of an assemble of pulsating sources do not depend on the angles, and therefore a direct summation of the contributions from the single sources is possible. The 3 possible alternatives in the time conditions together with the appropriate distances have to be checked in each step (see Equations V-61 ff):
463
(XX- 14) m=l
(XX- 15) m=l
Since the plane-wave particle velocity uponly considers the velocity component in the line of propagation, the following procedure holds: m=n U p , P = C u r , m ( r m , tcos ) . 0,
(XX- 16)
.cos(pm .
m=l
The finite sources of radius R show acoustic quantities only at their surface, but not inside. Therefore, the path of the point of observation moves along the centerline of the circle, and on its source surface when a source is present (see Figure XVIII-2, right side). Consequently, the minimum distance r becomes R in the above Equations (XX- 14)(XX-16) at the location of the sources XS, ys. Accordingly, the distance h from the source line also varies.
(XX- 17)
I f R > h or R > J ( X ~ - X ~ ) ~ + Y ;
and
r = d(xs - xp)2 +
+ h2
If r I R then r = R is used.
(XX- 18)
The angles 8 and cp are obtained as indicated in Figure XVIII-5. Example This situation is simulated in the following discussion by consideration of 13 identical rapidly expanding sources (described in Figure V-1) with an expansion of CE = 0.1, which are regularly situated in a semicircle. The pressure profile is calculated at the front of the array as a function of the time from the top for a semicircle of diameter 1 by Huygens principle. The time varied from ct = 0 to 0.5. Note that calculations have been done also outside the semicircular charge. Each pressure source behaves identically and synchronously. Figure XX- 17 shows the results.
464
1
.
4
0
0.5
0.75
Figure XX-17. Results of the Huygens summation from the regular distributed sources in the upper semicircle in terms of pressure. The cells and herring-bone structure are visible.
Due to the many undercuts of the profiles, a direct plot of the quantities is not advisable; rather a hardware model is made, which shows the various structures from different sites (see Figure XX-18). In Figures XX-I 7 and XX-18 near the front, regular cell patterns of relatively small amplitude variations are seen, but the herring-bone structure shows large variations in amplitudes behind this front. It is clear that the cell structure comes out at relatively small pressures, like in gas detonations, and vanishes completely for high detonation pressures of condensed materials. In this latter case the herring-bone structure remains visible. Both structures are linked in this approach. Further, the structures in pressure and velocity are quite similar so that it is not possible to resolve the quantity responsible for cell painting. As shown earlier, the plane transverse wave effects are in addition intrinsic properties of the chosen model.
465 of
Figure XX-18. View of the various functions from different sides.
466
0
0.5
0.75
P
Figure XX-19. Structure of 13 rapidly ( E = 0.1) collapsing sources that occur in LVD, where tension pulses at the front dominate (see also Figure Xx-20). Compare the proposed pressure structure of HVD, where bubble expansion occurs (Figure XX-17).
The cell dimension in terms of ct orders to the expansion quantity C E of the single pressure source (see Figure XX-17). This applies for HVD, where the bubble expands ahead of the shock zone. If bubble is collapsed as in the case of LVD, the results of Figures V-1 and V-2 apply also when the sign is changed, see Chapter V. Therefore in bubble collapse a tension pulse appears first, followed by a pressure rise as it is for a source arrangement: First a tension front appears and later a pressure increase arises (see Figures XX-19 and xx-20).
461 of
Figure XX-20. Left is shown the profile of HVD with expanding bubbles and right, LVD with collapsing bubbles. The horizontal line indicates the limit of tension waves. A tension wave is leading the LVD, and much later some pressure comes out.
No structure will be visible in soot imprints, since these structures are in the tension domain (see Figure XX-20). The structured tension domains represent tension shocks, which result in local cavitation and fracture. (Note that negative compressibility exists for LVD so that, according to the Becker-Bethe-Weyl-relations, a tension shock can result.) Cavities, lessential for LVD to occur, can be produced by an intrinsic process of self-cavitation without any vibrational Confinement. Indeed, experiments have shown that LVD is possible without any confinement.
PROPAGATION OF DETONATION The detonation profiles shown in this Chapter are not stationary but move into the direction of the shock. There are unique model experiments by a two-dimensonal gel technique pioneered by Dear, Field and Walton [XX-221, [XX-231 to demonstrate this. A gel layer made by dissolving 12 wt% gelatine in water at 330 K - finally is placed in a suitable shape between spaced glass blocks. A striker is shot by a gas gun onto this assembly, and the shock is monitored by a Schlieren technique. For details, see [XX-221. ~
468 Chapter
Figure XX-21. Gelatine experiment of Dear, Field and Walton [XX-22]. Three cavities of 3 mm 0 perpendicular to a shock wave, S. J is the jet in the first cavity and S’ the shock produced by the rebound of the first cavity. The step-by-step sequence of collapse has a inter-kame time of
5 ps. (Courtesy of Prof Field, Cambridge.)
The conclusion drawn from the authors [XX-22] are: 21
1, 2,
0.26
of
2 2
3,
469 of
Figure XX-22. Rectangular array of nine cavities, 3 mm 0, collapsed by a shock wave S. Interkame time 4.25~~. (Courtesy of Prof. Field, Cambridge.)
7 of
by no
of
by ”
470
Chapter
The investigations resolved the dominant influence of bubble sizes and their distances. Further the collapse of one bubble emits pressure waves which in turn collapse the next one. This is seen in Figure XX-22, which shows a rectangular array of nine bubbles: The picture is consistent with the view that the stimulating shock collapses the first row which in turn emits pressure waves collapsing the next row and SO on.
References [XX-I] H. D. Mallory, G. A. Greene, Luminosity and Pressure Aberrations in Detonating Nitromethane Solutions, J. Appl. Phys. 40 (1969), p. 493314938, [XX-2] C. 0. Leiber, The Structure of Detonation within the Bubble Dynamic Model, Symposium H.D.P. 1978, Conf. Proc. p. 1831194. [XX-3] S. B. Kormer, Optical Study of the Characteristics of Shock-Compressed Condensed Dielectrics, Sov. Phys. USPEKHI 11 (1968), p. 2291254. [XX-4] A. N. Dremin, S. D. Savrov, V. S. Trofimov, K. K. Shvedov, Detonation Waves in Condensed Media, NAUKA, Moscow 1970. [XX-5] M. Minnaert, On Musical Air Bubbles and the Sound of Running Water, Phil. Mag. Ser. 7, 16 (1933), p. 2351248. [XX-6] G. B. Kistiakowski, Initiation of Detonation of Explosives, 3rdSymposium on Combustion and Flame and Explosion Phenomena, Williams & Wilkins, Baltimore, 1949, p. 5601565. [XX-7] A. R. Ubbelohde, Transition from Deflagration to Detonation: The Physico-Chemical Aspects of Stable Detonation, 3rdSymposium on Combustion and Flame and Explosion Phenomena, Williams & Wilkins, Baltimore, 1949, p. 5661571. [XX-81 K. Antolik: Das Gleiten elektrischer Funken, Ann. Physik, Ser. 2, 154 (1875), p. 14/37.
[XX-9] C. Campbell, D. W. Woodhead: Striated Photographic Records of Explosion Waves, Chem. SOC.130 (1927), p. 157211578, [XX-lo] C. Campbell, A. C. Finch: Striated Photographic Records of Explosion Waves. Part An Explanation of the Strioe, J. Chem. SOC.131 (1927), p. 209412106. [XX-111 K. I. Shchelkin, Ya. K. Troshin: Gasdynamics of Combustion, Mono Book COT., Baltimore, 1965. [XX-l2] A. K. Oppenheim: Introduction to Gasdynamics of Explosions, Udine 1970, Springer, Wien, New York, 1972, p. 24.
[=-I31 R. A. Strehlow, A. A. Adamczyk, R. J. Stiles: Transient Studies of Detonation Waves, Astronautica Acta 17 (l972), p. 5091527. [XX-l4] K. I. Shchelkin: Zh. Eksperim. i Teor. Fiz. 36 (1959), p. 600. [XX-I 51 H. D. Mallory: Turbulent Effects in Detonation Flow: Diluted Nitromethane, J. Appl. Phys. 38 (1967), p. 530215306, [XX-16] A. W. Campbell, T. E. Holland, M. E. Malin, T. P. Cotter: ..Nature 178 (1956), p. 38.
47 1
[XX-17] A. N. Dremin, 0. K. Rozanow, V. S. Trofimov: On the Detonation ofNitromethane, Combustion and Flame 7 (1963), p. 1531162. [XX-18] P. Howe, R. Frey, G. Melani: Observations Concerning Transverse Waves in Solid Explosives, Combustion Science and Techn. 14 (1976), p. 63/74. [XX-19] P. A. Urtiew, A. S. Kusubov, R. E. Duff Cellular Structure of Detonation in Nitromethane, Combustion and Flame 14 (1970), p. 117/122. [XX-20] M. Held, Struktur der Detonationsfront in verschiedenen Gas-Luftgemischen, Jahrestagung ICT, 1973, p. 279/297. [XX-21] Soloukhin, R. I.: Multiheaded Structure of Gaseous Detonation, Combustion and Flame 10 (1966), p. 51/58. [XX-22] J. P. Dear, J. E. Field, A. J. Walton, Gas compression and jet formation in cavities collapsed by a shock wave, Nature 332 (1988) No 6164, p. 5051508. [XX-23] J. Dear, J. Field, A Study of the Collapse of Cavities using Two-Dimensional Gelatine Configurations, Int. Symposium on Cavitation, April 1986, Sendai, Japan, p. 89/94.
This Page Intentionally Left Blank
XXI SHOCK TUBES It was Loison [XU-11probably who noticed in 1952 that oil-covered tubes with an high inside pressure of air and oxygen detonated. Remarkably the flash point of oil had little influence on the effect. This observation was apparently the starting point of shock tube studies. Explosive shock tubes, TLX compounds (Thin Layer Explosive) or as a registered trade mark NONELO (NON Electrical initiation device) are used for ignition or initiation transfer. They are plastic tubes of 1 - 2 mm I. D.. The inside area is covered with 10 - 30 mg/m of a reactive material (HMX mixed with Al) corresponding to an area density of 0.5 mg/cm*, see Figure XXI-I .
Figure XXI-1. Electronmicrograph of the inside of a NONEL tube. (Semi-lateral cut, the grains are HMX and Al). (Courtesy G. Konig, BICT).
Initiation of Shocks Initiation is effected only by shocks, whereas fire, burning or mechanical impacts are not effective. Such initiating shocks can be produced by initiators or squibs, or by the output of a similar device so that the transfer of detonation is similar to the action of fluidics, see Figure XXI-2.
473
474
Chapter
Figure XXI-2. Shock distributor using the principles of fluidics. If for example the lower tube is initiated the resulting shock in the metallic distributor initiates the other 3 lines.
Bundles of None1 can be initiated by one cap by detonation transfer. If initiation occurs at one end, a shock wave travels with surprisingly constant velocity of about 1,800 to 2,000 m / s through this tube, and flashes at the other end, see Figure XXI-3, firing ignition devices or detonators. The shock pressure is much below the bursting strength of the tube due to the low reactive mass. It is possible, therefore, to use such devices inside an explosive charge! On the other side, fi-om the results of charges with an axial hole (Chapter XXII) we know that a fast detonation can drive detonation of a substance with a low detonation rate. If one introduces strands of such a substance with a low detonation velocity into such a shock tube, a much increased and powerful detonation takes place. Such phenomena are known as ‘linear ignition’ in ITLX devices. Due to the increased mass involved the resulting pressure is increased. These lines must be metal-cased tubes of proper strength to protect the surrounding media.
Extrapolation to Larger Dimensions The advantage of basic studies on phenomena is the recognition of unknown or ignored risks. NM was used in the Tactical Explosive System (TEXS) program to produce antitank ditches in the field. Very large quantities of NM must be fed by delivery tubes into the tube systems. It was therefore important to make the systems safe since experiments showed that warm NM could be initiated by an explosive blast of some bar amplitude. As the investigations of the NONEL problem, and the TEXS project timely coincided, the NONEL problem was extrapolated to TEXS safety: Whenever an
415
Shock Tubes
"empty" with NM inside wetted tube should show NONEL effects, this would be a serious safety problem in the field. Later it appeared that such risks are present in general, see Chapter XIII. In order to test shock-tube effects in larger dimensions, plastic tubes, inside wetted with NM, of 6 m length, and 50 mm I. D. were used. Comparative tests were done with a steel tube. In both cases a booster charge of 40 g plastic PETN was applied at one end. Figure XXI-3 left shows the optical luminous impressions using the NMcoated plastic tube. The right hand-side shows the ejecta of the reaction products of the NM-coated steel tube (open camera technique).
Figure XXI-3. Left: Luminous events inside the NM-wetted plastic tube, Right: Luminous events as shock leaves the end of a NM-wetted steel tube, both tubes were initiated with a 40 g PETN charge. (Work performed by Alfied Wagner)
On the Mechanism of Shock Tubes We know that the technical system of None1 works with some reliability. But it appeared that there is some sensitivity to illumination in bright sun. This effect can be avoided by using black tubes. Another concern is the reproducibility and accuracy of the shocks produced. Experimental work on the mechanism were done on enlarged systems where the diameter of the tube was 3 cm and more [XXI-2] - [XXI-51. From high speed photographic studies the conclusion was drawn that the explosive blow disperses the deposit from the walls into the volume, where it reacts more or less homogeneously in the volume. It is questionable, whether such a mechanism also works in smalldiameter tubes. Reasons may be found that reaction can take place at the walls of the tube. Accordingly different thermohydrodynamic approaches exist: Quasivolumehomogeneous detonation, where detonation is fed from energetic sources in the volume (like a dust explo;sion),and that detonation may be fed by diffusion flames from the wall sides as Fujitsuna and Tsuge [XXI-61 suggested.
476 Chapter
There was only one direct experimental work on the dimensions of Nonel: Gaodi and Jingsong [XXI-71 performed measurements of the detonation electric effect (DEE). They concluded that there exist steady and non-steady states, and the run distance to a steady state can be as long as 50 to 90 cm. Further, the authors stated that the DEE-steady state wave profiles (electrical amplitudehime) are very similar for the same tube at different locations, but vary in different experiments of the same type. Since there is an interrelationship between DEE effects and the detonation mechanism, this can be quite different in different tubes. Based on large-scale system studies, it is not clear whether there is a reaction of the dispersed material in the volume of the tube, less likely in small diameters, or whether reactions at the side walls feed such effects that raises the following questions: 1)
Are there different detonation mechanisms at work ?
2)
Is it possible that detonation can be fed from reactions at the wall sides?
3)
What are the influences of homogeneous and less homogeneous coatings of the wall sides ?
4)
Is there a theoretical reason for a critical diameter?
5)
Why is the critical diameter of Low Velocity Detonation of PETN in a cylindrical charge larger (= 5 mm) in steel confinement than that of a film detonation in a plastic tube? This question is still answerable with the nature of the critical diameter in Chapter XVIU, see Figure XVIII-4.
6)
Is the Nonel mechanism an up to now not considered industrial risk? See Chapter XIII.
Suggested Physical Model of NONEL As an elementary step of explosion or detonation we assume that an explosive particle bums to gaseous products using the air in the tube. This combustion process occurs in pockets with the volume variation in time V = 4 x R 2 R , in turn causing mass injection p V into the surrounding medium of density p, and leading to a pressure wave
emission. The radial velocity d corresponds to the particle velocity u, of the surrounding medium if mass-transfer between the source and the adjacent medium is absent (which actually is never realized). The geometry of this model is shown in Figure XXI-4, so that the Equations of Chapter V, {see the Equations (XX-1) and (XX-10) in the previous Chapter} can be applied with the expressions for the distances with the notation of Figure XXI-4:
(XXI-1)
Shock Tubes
I
c2
centrol line of observolion
A - l i n e of observation i f hlh =
Figure XXI-4. Geomehy of the suggested NONEL model, Care must be taken that the line of observation at the place of a source is on the surface of the sphere, see right hand side. cp L angle of the sources,
wall
9 = 90" - L axis of the array and the distance r,A.
i +N
Figure XXI-4 shows sources in a circular ring that contribute in pressure and particle velocity to the point of observation P, which moves, in varying distances hlh = const. over the diameter, so that macroscopic pressure and particle-velocity waves are obtained by using Huygens principle. Since finite-sized sources are considered that radiate from their surface only, the summation path is around the source surface in each case, and the minimum distance amounts to this radius of the source of size kR=27cRlA., see Figure XXI-4, right-hand side. If the source has a directional character sin 6 resp. cos 8 has to be considered. Again for the reasons of simplicity it is assumed that these reaction sites work in an acoustic harmonic mode, so that the solutions become time independent. Uniform velocity potentials of harmonic pulsating sources were used with the same phases and amplitudes, but very different areal distributions. These were: 1
Equidistance on the circle line, Diameter Relationship, Scaling Properties.
2 3 4 5 6
Statistical distribution on the circle line. Various statistical distributions in the cross area near the circle. Various statistical distributions in the cross area near the center. Various homogeneous statistical distributions in the cross area. Regular distribution in the cross area.
Equidistant Sources on a Circular Ring Critical-DimensionPhenomena
For a densely packed circular ring with Fraunhofer approximation, the transition from isotropic to direclional pressure radiation can be described, see Chapter XVIII. A first lobe formation is obtained for the dimension = 0.7655, see Figure XVIII-4 and Table XVITT-1.
478
Chapter 273
90" Figure XXI-5. Directional function of a densely packed circular ring, see also Figure XVIII4 right-hand side. The condition = 0.7655 is (beside film detonations) the lowest condition of a first lobe formation for all known configurations. The number of lobes and side lobes corresponds to the number of pressure peaks in the exact solutions, see Figures XXI-6 and XXI-7. The minor lobes for = 1.5 and 3 shown in the Figure on the negative side (for better visibility) are in reality placed on the positive axis.
For n equally spaced sources according to Stenzel/Brosze [XVIII-2 11 the directional function ID1 is given by
(XXI -2)
For moderate values of of infinite sources =
cos 9 one gets for n 2 4 sources the directivity function Again this function is shown in Figure XXI-5.
If 3 0, almost isotropic very low pressure radiation results, for = 0.7655, pressure and particle velocity increase and show a unidirectionality without any minor lobes. Above this value pressure amplitudes increase accordingly and in addition minor side lobes appear. Their numbers increase as the diameter increases. The number of these minor lobes corresponds to the numbers of pressure peaks in the exact solution, see Figures XXI-6 and XXI-7. In Figure XXl-6 the profiles of a monopole cosine circular array are shown for = 1.5, where the relative impedance is magnified greatly so that the propagation velocity correspondingly increases. The degree of variation as hnction of the diameter is sketched in Figure XXI-8. One can see that below the critical diameter pressure and propagation velocities get very low, and energy density is almost in flow.
419
F?
^i
IUI
10
0
Figure XXM. Pressure, impedance. and particle velocity (up) half profiles of 2 I equidistant monopole mine sources in a spherical of = 1.5 (about double the critical 3 peaks over the diameter in Figure XXI-5 as the main lobe with two minor lobes. IPI 20
Figure XXI-7. Pressure semi-profile of a circular ring with dianmer ( O h )= 3.0. For the number of the maxima, compare with Figure XXIJ.
10
0
Scaling Parameters A pronounced influence of the charge diameter on steady state detonation exists in condensed detonics. In Figure XXI-9 the central pressure of a constant source density circular array of different diameters is shown as a hnction of the distance In this scale pronounced geometrical influences of are present, which disappear if the distance hlh is related to the charge diameter Therefore in this case an appropriate scaling parameter is the diameter of this tube.
480
'
6-
' 5-
=01
=" kR=02566 n.42 l 0 c r l kR=O1
n = 4 2 1co O 3 9 O c r I
kR=
n
lcn
Figure XXI-8. Relative impedance of supercritical, critical, and subcritical diameters circular-ring arrays. In the supercritical case the energy release is in compression and the velocity is largely increased. In the subcritical case the energy release IS predominantly in flow.
-
4
1
h/@
20 \
10
'.
\, '%.
'. . 0s hlk
10
0
5
1 hlA
multiples of @ / A
Figure XXI-9. Central pressure of a constant source density array o f different A diameters as function of proper scaling parameter is the diameter-related distance h/0 (right-hand side).
Effects of the Source Distribution in the Circular Area In Figure XXI-10 and XXI-11 the results of calculational experiments are shown for the macroscopic pressure waves over half of the diameter of a tube with 40 harmonic sources in various statistical distributions in the area and at or near the side walls, see the above mentioned cases (1) - (6). As a result very different pressure profiles appear even for the same properties of the constituting pressure sources. As indicated in Figure XXI-5 pressure peaks corresponding to the lobes are observed if the diameter of the source array is large enough. The pressure profile shows a normal profile in the center, at the tube side after the decay again a peak arises, so that very unusual profiles appear. With increasing dynamics related to diameter more and more peaks appear. This means that by varying the dynamics at a geometrical fixed
48 I Shock
Figure XXI-10.
Pressure profiles of 40 harmonic pressure sources of size kR = I in various distributions in a circle of diameter = 2.0 up to the distance = 10.
diameter the pressure profiles change greatly. Figure XXI-11 also indicates that the side wall pressure can be larger than the central pressure.
482 Chapter
Figure XXI-11. Pressure profiles of 40 harmonic pressure sources of size kR = 1 in various distributions in a circle of diameter = 2.0 up to the distance = 10. No secondary pressure peak appears at upper left since the effective diameter got small.
In the course of many computer experiments it seems not possible, conversely to deduce fiom the macroscopic wave property any microscopic situation. It is possible, therefore that the different DEE situations observed are caused by identical mechanisms.
483 Shock
Experiments on Shock Tubes Wagner [XXI-s] conducted many experiments in large shock tubes, see Figure XXI-3. To demonstrate that the measured shocks are not primarily the result of booster charge the shock profiles, of an uncoated plastic tube and a NM-coated plastic tube, both initiated with a 20-g PETN charge are compared in Figure XXI-12. The shock in the uncoated tube was relatively low and attenuated rapidly, the shock in the NM-coated tube was much larger and the integral output was 22 times greater.
Pw.D?uctdwer Owehmrtisg b e i
: :
14.332 . p i c 3.79 i i e c
Figure XXI-12. Comparison of the shock outputs of an empty plastic tube and a NM-coated one, both boosted with a 20 g PETN charge. Note the different scales.
To determine the shock velocity and pressure a steel tube of 50-mm I. D., and 4.5-m length was used. At 3 sides, and 1-m distance respectively, pressure transducers were inserted. In 9 experiments 6 positive results were obtained, where the average shock velocities ranged between 1,430 and 2,100 m/s. The scatter in a single experiment did not exceed 200 nds. The differences in pressure, pressure profile, impulse, and positive pulse durations at the side walls of an uncoated and an NM-wetted tube are shown in Figure XXI-12. The shapes of the pressure profiles of NM-wetted tubes were usually like those in Figure XXI- 11, right side, which correspond to the calculations shown in Figure XXI-10. There are also exceptions. Many experiments, some of them shown in Figure XXI-13, evidenced many pressure amplitudes and pressure shapes. These results demonstrate that the different DEEeffects may correspond also to the situations in pressure profiles. We conclude that the different macroscopic pressure profiles prevent an engineering application, where reproducible effects are required. Nevertheless the basic microscopic mechanisms may be the same. But a serious problem remains: Usually a risk is realized or ignored by a
484
In the axis of the tube
at the side wall of the tube
La
I#.
7z.
r4. X. 4D.
11. 14. li. I. D. 4. -16.
..... J ..... J ......1 .....I ..... I ..... .....J ...... DTwkMttlmg bat : r-IIu,,*
z
I'mMI:
1.43 m .u Is*):8
.Ian
. . . . . . . . . . . . . . . .I......I ..... I ..... ..... J . . , _ . . frvchnrtl.g
bet :
P.M
DISC
X-Qbq: 1 I-RLmi: .wI t l l l t ¶
Figure XXI-13. Manifolds of possible pressure profiles in NM-coated steel tubes. It appears that the pressure at the wall side can be larger than in the axis. The pressure measurements were made simultaneously in the center and at the wall side in single experiments fiom top to bottom. (A. Wagner)
485
few experiments only. The results and variations in pressure amplitudes, as fiom some bars up to more than 200 bar at same conditions demonstrate that the evaluation of such NONEL risks is a very delicate problem. For examples, see Chapter XIII.
Initiation of NM by Explosive Blasts See Chapter X.
SURFACE DETONATIONS The shock tube experiments in the previous section are based on the concept of mass injection into the unit volume, see Chapter 111, where a combustible reacts with the surrounding atmosphere. Sometimes such types are also called Since this mechanism is not bounded to any geometrical situation, such types are also to ble expected in other geometries, for example on a wick or another surface.
Wick Detonation In the course of an accident investigation Plewinsky et al. [XXI-91, [XXI-101 soaked a cotton wick with (TMDS) { (CH3)2HSi-0SiH(CH3)2,boiling point 71°C) and inserted this into a Plexiglass cylinder of 3-cm I. D. and 1-m long, filled with up to 40 bar oxygen. After ignition with an incandescent wire they observed with an image converter and streak camera partly steady detonations between 400 to 1500 m/s. It appeared that the velocity changed within the same experiment. No correlation with ambient oxygen pressure was found.
Detonation on a. Porous Surface Plewinsky et al. [XXI-lo] also investigated quartz sand soaked with TMDS or dodecane (CI2Hz6)in a pressurized oxygen atmosphere. Depending on oxygen at the low pressure higher thlan 10 bar, velocities were observed between 948 pressure end to 1 180 m/s at a pressure of 30 bar for TMDS. But no detonation was observable for dodecane between 20 and 40 bar ambient oxygen pressure - in contrast to Lyamin [XXI-1 11, who got a detonation above an oxygen pressure of 5 bar.
486 Chapter
Detonation on the Surface Plewinsky et al. [XXI-12] also found one- and two-dimensional surface detonations of TMDS between 700 and 1100 at oxygen pressures of 25 - 30 bar.
References [XXI-11 R. Loison, Propagation dime dkflagration dans un tube recouvert d'une pellicule d'huile, C.R. 234 (1952), p. 512/513. [XXI-2] E. K. Dabora, K. W. Ragland, J. A. Nicholls, A Study of heterogeneous Detonations, Astronautica Act., 12 (1966). p. 9/16. [XXI-31 A. A. Borisov, S. M. Kogarko. A. V. Lyubimov, Ignition of Fuel Films behind Shock Waves in Air and Oxygen, Combustion and Flame, 12 (1967). p. 4651468. [XXI-41 K. W. Ragland, J. A. Nicholls, Two Phase Detonation of a liquid Layer. AIAA-J. 7 (1969). p. 859/863. [XXI-5] M. A. Nettleton, Shock Wave Chemistry in Dusty Gases and Fogs: A Review, Combustion and Flame 28 (1977). p. 3/16. [XXI-6] Y. Fujitsuna, S. Tsuge, On Detonation Waves supported by DifFusion Flames. I: The equivalent Chapman-Jouguet condition, 14'h Symp. (Int.). Combustion, Pittsburgh. 1973, p. 1265/1275. [XXI-7] Xie Gaodi and Liu Jingsong: Detonation Electric Effect of the Nonel-tube, Proc. lothInt. Pyrotechnics Seminar, Karlsruhe, 1985, p. 74-1174-10. [XXI-8] C. 0. Leiber, P. SteinbeilJ, and A. Wagner, On apparent Irregularities of Pressure Profiles in Shock Tubes, Europyro 93, 5e Congres International de Pyrotechnie du Groupe de Travail, Strasbourg, 6./ll.juin 1993, p. 171/178. [XXI-9] B. Plewinsky, Heterogene Detonationen und indirekte Zundvorgange, PTBMitteilungen 100 (1 990)4, p. 266/270. [XXI-lo] B. Plewinsky, W. Wegener, K.-P. Herrmann, Heterogeneous Detonation Along a Wick, in A. L. Kuhl, J. R. Bowen, J.-C. Leyer, A. Borisov, Dynamics of Explosions, Vol. 114 of Progress in Astronautics and Aeronautics, AIAA, Washington, DC, 1988, ISBN 0-930403-47-9. [XXI-1 11 G. A. Lyamin, Heterogeneous Detonation in a rigid porous medium, Fiz. Gor. Vzryva 20 (1984)6, p. 1341138.
[XXI-12] B. Plewinsky, W. Wegener, K.-P. Herrmann, Surface Detonations and Indirect Ignition Processes, in A. L. Kuhl, J.-C. Leyer, A. A. Borisov, W. A. Sirignano, Dynamics of Detonations and Explosions: Detonations, Vol. 133 of Progress in Astronautics and Aeronautics, AIAA, Proc. of 12IhInt. Colloquium on Dynamics of Explosions and Reactive Systems, Ann Arbor, Mich., July 1989.
DETONATION PHENOMENA IN CHARGES WITH AN
AXIALCAVITY Selle in 1932 [XXII-I] began to investigate detonation phenomena of charges with an axial cavity or axially cavitated cylinders, in some respect related to shaped charges. Ahrens [XXII-21 iin Germany continued this work horn 1938 until 1945 at CTR. Simultaneously and independently this matter was investigated by Woodhead [XXII-3] and Titman [XXII-41 in England. Together they published their results in 1965. Both parties conducted detailed investigations without any theory or model as a guidance. The detonic behavior of cavitated charges - either internally as an axial cavity or externally as an air gap, i. e., a gap between a cartridge and the bore hole - is considerably different hom that of a homogeneous charge. Significant increases or decreases of "sensitivity" are observed, and these changes are not unidirectional. Some experimental facts from small-scale experiments on high and commercial explosives are sketched in the following sections to bring out the considerable gap in knowledge for judging relevant safety problems in the real world. The critical-diameter conditions of full cylindrical charges outlined in Chapter XVIII do not hold in this case. Mallory demonstrated this in experiments in 1987 [XXII-5]. Full cylinders of propellants do not detonate, but cavitated ones in smaller diameters do. At the end of the 11950's Lavrentyev in Novosibirsk, Russia, focussed the interest on these phenomena for obtaining tools for high-speed launching of particles for the study of meteorite impacts [XXII-6], [XXII-7].
Experiments Cylinders with an axial cavity usually show increased plate penetration compared with a full-sized cylinder. Accordingly Kast's brisance value [XXII-8] is increased or decreased as a hnction of the sensitivity of the explosive, the size of the cavity, see Figure XXII-1, and the length of the charge. The relative crushing value is largest for an insensitive explosive like TNT and decreases as sensitivity increases in the order: TNT > picric acid > tetryl> PETN/wax > RDX > PETN. This value depends on the length of the charges with axial cavities. There is an almost constant crushing of copper cylinders for different lengths of h l l charges; but this value varies greatly, up to a factor of six, for cavitated charges between 4 and 40 cm length.
487
488 Chapter
Figure XXII-I. Kast crushing values of TNT charges of 80-nun length and 2 I -mm 0 (density 1.55 g/cm3)as function of the diameter of the cylindrical axial cavity. {Ahrens [XXII-2] with kind permission.}
Zagumennov et al., 1969, see Ref. [XXII-61, found that the primary shock wave is extinguished at a charge length of greater than 200 calibers, "there is found in the region between its front and the detonation front a second shock wave, which appears at approximately the 1OOth caliber. When the second shock wave is extinguished a third is formed (in the 160-210 caliber range)". This cavity acts also ballistically. A 5-mm 0 steel ball at the end of a 80-cm long charge of TNT of 2 I-mm 0. D. and having a 4-mm diameter cavity reached a velocity between 4,200 and 4,500 m/s [XXII-21. Leiber (1967, not published) has not been able to reproduce this effect with short charges and a mass of some grams. But Titov et al. 1968, [XXII-61 accelerated bodies in the dimensions of mm (1 - 2 g steel spheres) up to 3 - 12 k d s , thereby using explosive masses of 10 - 100 kg. Velocities up to 14 k d s were obtained for Nichrome spheres of a 80 - I00 pm fraction into vacuum. Also liquid hydrogen is inserted into the cavity as a high-speed driver. The detonation velocity of cavitated charges DH was determined optically from reaction luminosity. Apparently this value is relatively independent of density, is constant, but changes from experiment to experiment (see Table XXII-1). DHincreases as charge asymmetries increase. This velocity is lower when the end of the charge is open than when it is closed. Another behavior in detonation velocity DK is obtained for mixed charges (full cylinders in combination with axial-cavity cylinders). These effects disappear when the cavity is filled with water {Kirsch, Papineau-Couture, Winkler [XXII-9]}. Contrary to homogeneous high explosives, where usually DH> D, for commercial explosives the opposite behavior of DH < D was observed. The differences in crushing (Kast) do not correlate with the value (DH- D)/D, where D is the detonation velocity of the full cylinder (without cavity), but the detonation transit times are additive. However, when the charge is composed of both full cylinders and cylinders with cavities, the detonation velocity depends on density (see Table XXII-l), and the transit times are not additive.
489
Table XXII-1. Detonation velocities of full cylinders (0.D. 21 mm) and axially cavitated charges (1. D. 4 mm) and mixed charges according to Ahrens [XX11-2]
TNT
PETN
Mixed charge DK [ d s l
Cylinder h l l D WI
Cylinder with axial cavity DH [m/s1
1.44 1.50
6,490 6,690
7,000 6,920
1.55
6,800
7,060
1.40 1.50
7,100 7,480
8,450 8,680
9,480 9,800
1.55
7,630
8,720
9,860
1.60 1.66
7,780
8,570
9,880
7,960
8,580
10,210
Density wcm31
7,160
Zwischcnlage
Figure XXII-2. Left: Arrangement of the charges with foils, details see the text.. Center: Detonation velocity of axially cavitated PETN charges, see left, of density p = 1.5 g/cm3as function of the area density 6p of the inert barrier. On the right side these barrier are noted in German, as Ahrens presented. (With kind permission.)
490
When the inside of the cavity is covered with lead foil and the ends are open, the detonation velocity DK of PETN (density 1.5 g/cm3) drops to 8,320 m / s compared to But DK = 9,120 m / s when only the ends of the PETN charge are DH = 8,680 covered with lead foil. When the entire cavity is covered with lead foil, one gets DH = 7,650 m / s compared to D = 7,480 m i s for a full cylinder. With only three quarters of the circumference of the cavity covered, the detonation velocity is DH = 8,110 m/s on the uncovered side and 7,390 m i s on the covered side. Ifthis charge (4 columns of PETN, density 1.5 g/cm3, 0. D. 21 mm, cavity diameter 4 mm, length per column 40 mm, see Figure XXII-2) is periodically interrupted with foils of inert materials, the detonation velocity D depends on the area density of this barrier material. If this area density p < 0.005 g/cmZ,no influence is observed, this means that the detonation velocity D Hhas been observed. If p > 0.008 g/cm2, DH jumps to a value of 9,900 m/s. The usual value of D His obtained again for p = 0.1 g/cm2, and the detonation velocity D of the full cylinder is obtained when p > 3 g/cm2. This effect depends on the distances of the disks, and is greatest at 40 mm; a decrease is obtained again above this value (see Figure XXII-2). Ahrens described the results in this range by the formula =
-
1210 ~ o g ( ~ pin)m/s
6p = ,-0.0019 2
where DZ is the detonation velocity with foils between the cavitated cylinders, 6 is the thickness of the obstacle in cm and p its volume density. The foils were inert materials like paper, cellophane, aluminum, mica, steel, copper, lead, and gold. Surprisingly the effects did not depend on the kind of the inert barrier. Luminous Phenomena A first luminosity with a velocity of 9,470 m / s was emitted at the end of a TNT charge with an axial cavity compared to the TNT detonation velocity D = 7,030 A bit later an additional luminous component was released with a longer run distance. Both events depend on the charge length, sensitivity, and brisance of the explosive. Sometimes a predetonation is induced. Very large velocities are obtained in vacuum (see Table XXII-2), where the differences up to 50 ps are measured between the first and second flash. These luminous ejecta can correspond to Cook’s [XXII-101 heat pulse that leads to initiation. For a more extensive summary, with some attempts of explanations, the reader is referred to Johansson and Persson [XXII-1 11.
49 1
Charges with an axial Cavity
Table XXIIl-2. Velocities of the luminous events leaving the cavitated charge in vacuum according to Ahrens [XXII-2].
Density
Cavity LengtWDiam.
DH
First fast flash
Slow second flash
wm31
[dml
[dsl
W S I
[dsl
TNT
1.55
4013 20013
17,040 - 15,400 17,080 - 14,980
13,160 - 8,180 12,480 - 5,060 18,380
PETN
1.50
80010
80016
7,760 8,600 8,470
18,580
13,900 - 6,300 15,700 13,420 - 4,900 23,340
Precursors Obviously the observed precursors that induce initiation of neighboring charges with a larger detonation velocity than for each individually, do not correspond to those discussed in Chapters XIV and XVI. The difference is that conservation of momentum requires that the velocities of the ejecta increase as their density decreases. This is true, but in Chapter XIV an additional virtual acoustic mass of the displaced volume was present, which limited the velocity, even for zero-density particles, to at most ulP= 3 up, This;prerequisite is absent for axially cavitated charges, therefore the precursor velocities are not limited by such acoustical mass. The dependence on the area mass of the foils indicates that momentum transfer is essential.
No explanation of the effects shown in Figure XXII-2, and Table XXIII-2 exist at presence. One aplproach may be to consider cascaded collisions of particles as described in Chapter XIV. Within this view such super velocities of tiny particles and the luminous events can be approached.
492
References [XXII-11 H. Selle, probably not published results, but starting point for Ahrens’ work 1938 1945 at CTR, referenced in [XXII-2]. [XXII-2] H. Ahrens; h e r den Detonationsvorgang bei zylindrischen Sprengstoffladungen mit axialer Hohlung, Explosivstoffe 13 (1965)5, p. 1241134, Explosivstoffe 13 (1965)6, p. 1551164, Explosivstoffe 13 (1965)7, p. 1801198, 2. communication (commercial explosives): Explosivstoffe 13 (1965)10,p. 2671276, Explosivstoffe 13 (1965)11, p. 295/309. [XXII-3] D. W. Woodhead, Velocity of Detonation of a Tubular Charge of Explosive, Nature 160 (1 947), p. 644. [XXII-4] D. W. Woodhead and H. Titman: Detonation Phenomena in a Tubular Charge of Explosive, Explosivstoffe 13 (1965)5, p. 1131123, Explosivstoffe 13 (1965)6, p. 1411155. [XXn-5] H. D. Mallory, Personal communication. [XXII-6] L. A. Merzhievskii, V. M. Titov, Yu. I. Fadeenko, G. A. Shvetsov, High-speed Launching of solid Bodies, J. Combustion, Explosion, Shock Waves 23 (1987)5, p. 576/589. [XXII-7] L. A. Merzhievskii, V. M. Titov, High-speed Collision, , J. Combustion, Explosion, Shock Waves 23 (1987)5, p. 5891604.
[XXII-s] H. Kast, Spreng- und Zundstoffe, Vieweg, Braunschweig, 192 1. Bekanntmachung der Priifiorschriflen f i r Sprengstoffe, Zundmittel und Sprengzubehor, vom 21. August 1974, Beilage Bundesanzeiger Nr. 161 vom 30. August 1974. [XXII-9] M. Kirsch, G. Papineau-Couture, C. A. Winkler: The Detonation Velocity of Axially Cavitated Cylinders of Cast DNA, Can. J. Res. 26, Sec. B (1948)5, p. 435/440. [XXII-101 M. A. Cook, The Science of High Explosives, Robert E. Krieger Pub., Huntington, NY, 1971. [XXII-111 C. H. Johannson, and P. A. Persson: Detonics of High Explosives, Academic Press, London, New York, 1970.
XXIII
MICROSCOPIC AND MACROSCOPIC PROPERTIES OF SOLIDS WithSiegfried Haussuhl
Preceding the origin of explosion and detonation phenomena was a mass injection into the unit volume of an otherwise isotropic medium (liquids). The mass injection is realized by dynamically activated voids, bubbles or cavities. The latter correspond to a break-up of the liquid. Therefore the apparent analog of cavitation in liquids is fracture in solids. This is the baseline for the following considerations. Isotropic liquids are characterized by the density and sound velocity c 0 (or the corresponding bullk modulus K), homogeneous solids in their most simple form exhibit two indedendent properties that are reflected in the corresponding sound velocities and associated moduli. Therefore the wave properties get more complex even in this simple approximation. Actually such seemingly isotropic solids are composed of anisotropic single crystallites often greatly varying their properties with their direction. A statistical average of a continuum emerges from a statistical volume average without preferences. This continuum applies to large constructions like a bridge but in the microscale the properties of the composing elements get noticeable. Those aspects appear, for example, if a pressure rise is compared with the dimensions of a crystallite. Since the continuium is based on the coherence of all volume elements, different properties in different directions cause stresses by variations in temperature or stress, so that stresses in the microscale appear. The continuum remains intact as long as the coherence strength can compensate the microstresses. If microcracks are caused, the altered matrix can get very different properties. All such phenomena are important in the behavior of detonics as well as in the related mechanical material behavior. Properties like pressability, ductility, proper mock evaluations and optimizing the pairing of different materials in a composite, aging and rough handling aspects depend on it. A note on the historical development illustrates the long way to those aspects [XXIII-11: Assume first a perfectly rigid homogeneous material - nothing happens, not even compression. Nevertheless this view from before Galilei's time is even nowadays applicable to somle mechanical aspects. It was 1678, when Hooke (1635-1703), and independently from him, Mariotte (1680) detected the linear law of elasticity: 'ut tensio 493
494 Chapter
sic vis'. This brought in the concept of the homogeneous body, but much time was required to generalize Hooke's approach, Young (1807) related the elastic deformation with the modulus of elasticity E, and Navier (182 1) on a molecular basis found wave motions, where only 1 constant was used, Cauchy (1 822) derived the modern form of waves with 2 constants for isotropic bodies from Hooke's law. Cauchy (1828) extended these considerations to anisotropic media and believed that 15 constants were necessary, but Green (1 827) found with elastic energy considerations that 21 independent constants were adequate. The latter applies to single crystals of greatly varying symmetry. Even at present the understanding of single crystals is far outside the explosives community, since daily life experience was born fiom isotropic bodies. Another reason is the great complexity of single crystal elasticity, so that engineering results cannot be obtained easily. The development of the insights into waves started probably with Newton and paralleled the evolution of the science of elasticity. Young (1 801) introduced for vibrations of small amplitude the superposition principle, which is in use in Fourier transforms. Riemann (1 859) introduced the shock in gases, which was later adopted by Becker (1915 and 1920) for condensed materials. Similar problems as in elasticity appear in the field of elastic waves. Always the most simple and from daily life obvious cases were considered, mostlyplane compression waves a But in this book reasons are given that neither the continuum nor the plane compression waves are adequate for a full understanding of detonic events. Single-crystal elastic properties in their general form are addressed below and the differences to an assumed isotropic behavior are presented. The averaging procedures of single-crystal elasticity towards a continuum are shown. The possible effects of single-crystal elastic anisotropy in a polycrystalline matrix are discussed. The multitude of wave types is indicated in preferential directions but the consequences were not outlined in detail. Then the quasicontinuum is addressed - always with reference to shortcomings. Finally fracture dynamics in a homogeneous continuum describes the "initiation laws" found in experiments. From this isomorphic description we conclude that initiation of detonation in solids is a matter of fracture (in accordance with Chapter 111).
495 Properties Single
A: PROPERTIES OF SINGLE CRYSTALS Introduction In classical view detonation is mostly a chemical event, but in this treatment it is primarily a mechanical hydrodynamic process. Therefore, it is of interest to show links between chemistry and mechanics. One such link is the elastic constants, which represent the binding forces of the molecules. Usually, solid materials are described by two elastic constants, one thermal expansion coefficient, and one single constant for other properties, all of which are isotropic. Even the number of elastic constants reduces to one in the hydrodynamic state. Due to the aelotropic nature of substances, this does not apply in the mesoscale. Through our experience with isotropic materials, we have acquired a description of material behavior that actually can fail. Often one believes that transport and other properties, the Griineisen constant for example, are specific material properties. Actually they are not. Due to extensive experience with magnetism on the basis with single crystals, it is known that electric and magnetic properties are, strongly directionally dependent, anisotropic. To get an idea for the possible complications in the microscale and the wave propagation processes, the following review gives the evaluation of measured single crystal properties. All procedures described are classical ones, with the reference to Voigt [XXIII-21, Hearmon [XXIII-31- [XXIII-51,Nye [XXIII-61, and Haussiihl [XXIII-71, where additional citations can be found.
On Directional Properties Whereas scalar properties (zero-rank tensors) like temperature, density, specific heat, entropy, and volume do not show any directional dependence, electric, magnetic and elastic fields do so. Stresses and strains must be represented by symmetric second-rank tensors with 6 components. If an isotropic elastic behavior is assumed, then stresses and strains always show the same directions. In a vector/vector-relationship (secondrank tensor with 9 components) or the relation of a second-rank tensor with a scalar like temperature or hydrostatic pressure, one gets the thermal expansion or uniaxial compressibilityin a specified direction, or related to the volume, where again one gets a scalar property. Finally, if two second-rank tensors are related, one gets a fourth-
496
Chapter xyI(I
rank tensor as in the case of the generalized Hooke's law. In the latter case the elastic properties largely depend on the directions, and hrthermore, a given stress in a specified direction results in strains in general different directions.
Orientations, Choice of Axes Any crystal is characterized uniquely by its unit cell with the cell constants a, b, and c together with the angles between the corresponding axes = L (b, c), p = L (a, c), and y = L (a, b), which form a 'natural' system of coordinates. These axes can also be described by vectors with a joint origin, and one gets a, b, and C. In the most general case none of the axes have equal length nor are the angles equal, which can be different from 90". Using the MILLERindices (hkl) any crystal plane (area) can be characterized by intersecting the axes at a/ h, blk and cll. This is the crystallographic description of crystal lattices, obtained a. e. from X-ray measurements. In the following we prefer to use the symbols al, a2, a3 for the crystallographic basic vectors a, b, c and ctl, a2,a 3 for the corresponding angles y. For many applications the Cartesian reference system (also known as crystal-physical system) with the basic vectors ei (i = 1, 2, 3) is most appropriate, and will be used hereafter. Table XXIII-1 gives the definition of the vectors which can be calculated with the help of Table XXIII-2. Table XXIII-1. Axes Convention e3
II a3
e2
a**-
II a2* a3
Xal
az* isperpendicularto(010)
el = e2 x e3
vector product V=a~(a2xa3)
V = volume of the unit cell Table XXIII-2. Basic crystallographic lattice and reciprocal lattice
CrystalSystem triclinic
Basic lattice
Reciprocal Lattice
al # az # a3 # al
. ak sin J
f
+
#
= ula2u3sinq s i n q s i n q
coscL2 cosa3
for i, j, k, i, j ...cyclic * COSajcosak-cosaj cosaj = sinaj sinak
V* = 1IV always
491
CrystalSystem
Basic lattice
Reciprocal Lattice
a1 + a2 + a3 + a1
monoclinic
.
= a 3 = 90",
02.
+ 90"
*
=
sin
*
1 s i n ~ ~
a3 =-
V = a1 a2 a3 sin a 2
*
trigonal
V = a12 a3 sin 1x3
trigonal,
a1 = a2 = a3,
rhombohedral
120" > = a: ,/(I
cubic
*
*
*
+ 90"
3C0sz
a1 = a2 = as,
V = a13
=90",
= a2* = 90",
=
*
1
=a3 =
*
a1
+ 2 cos3 =
a3* = 60"
*
sina, sinal
= a2= a3
~
I
cos= 90"
2
=
1 2cos(a1/2)
al* = a2* = a3* = l/al al* =
= a3* = 900
Thermal Expansion The components (xi, of the tensor of thermal expansion are defined by E.. = 'J
1 '
AT
7
(XXIII-1)
where ~ i are j the c'omponentsof the deformation tensor. The aij describe the deformation of tht: medium upon variation of temperature AT in a linear approximation. The main components al1, and a33 give the longitudinal deformation along the Cartesian basic vectors ei upon a temperature variation of A T = 1 K.
For the definition of the linear compressibility under hydrostatic pressure an analogous relation is valid (XXIII-2)
498 Chapter
Where Ap is the variation of the hydrostatic pressure. The extreme values of thermal expansion and the corresponding directions can be obtained by calculation of the eigenvalues and the appropriate eigenvectors of the matrix
[a:
a12
cc;
1:;
a13
]
. The same procedure can be used for other second-rank tensors
like the linear compressibility, see Equ. (XXIII-2).
Generalized Hooke's Law Hooke (1 635-1703), and independently fiom him, Mariotte (1 680) detected the generalized linear law of elasticity: 'ut tensio sic vis'. A single crystal usually shows different elastic properties in each direction. The anisotropic properties are tensor properties, so that the stresses (3 are interrelated with the strains E by a tensor function. For small deformations each compound of the stress tensor components of the strain tensor E, and vice versa. = Cijk/ Ekl
and
E i j = Sijkl O k l .
is proportional to the
(XXIII-3)
(XXIII-4)
i, j, k and 1represent 1, 2 or 3. Cijkl are the components of the elasticity tensor, and sijkl are the components of the elastic compliance tensor. In the general case such a 4'h rank tensor shows 34 = 8 1 independent components. This number reduces to 36 due to the properties = oji and E~ = c j i . Taking into account the reversibility of the elastic deformation process, we obtain the symmetry relations cqkl= C k / q and Sqk/ = Sk/ij
, which reduce the number of independent components to 21 (Green, 1827).
When the crystal's symmetry increases, the number of constants decreases until, in the cubic case three and for an elastic isotropic medium, two constants will describe the elastic properties completely. Usually the four indices are represented by two, but a 4" rank tensor is present. The quantities cu and si are called elastic constants and elastic compliances, respectively. In this transcription the following rules apply: ii= i ij
and
9 - i - j for i + j.
Example: c , , 3 ~ ~cI2 and c12323 c ~ ~ .
(XXIII-5) (XXIII-6)
499
Properties Single
Hearmon and Haussuhl point out that in a transformation of the coordinates this should be done only by wing the 4 indices. The following relations hold: s..
s.... 7
i,9-k-13
(XXIII-7) siikl
9-i-j,9-k-l
(XXIII-8)
3
4sijkl
(XXIII-9)
9
and the reverse: forp,q=1,2or3
Sijkl =
ykl
4 s .ykl . =s
and
(XXIII-10)
for p = 1 , 2 or 3 and q = 4 , 5 or 6 .
(XXIII-11)
for p, q = 4, 5 or 6.
(XXIII- 12)
With some exceptions the notations for the (stifhess) constants are given in c, and for the compliances in s.
Stress Components In Cartesian notation for the components we have: 0 1 1 = C l l E l l + C12E22 + C13E33 + Cl4E23 + W E 3 1 + C16E12 0 2 2 =C21Ell+C:!2E22+C23E33+C24E23+C25E31 0 3 3 =C31E11+C?,2E22+C33E33fC34E23
+C26E12
+C35&31 +C36&12
0 2 3 = C 4 1 & 1 l + C 4 2 & 2 2 + C 4 3 E 3 3 + C 4 4 & 2 3 + C 4 5 & 3 1 +C46E12
(XXIII-13)
O31 = C51 El 1 + C52 E22 + C53 E33 + C54 E23 + C55 E31+ C56E12 0 1 2 = C61 El 1 + C62 E22 + C63 E33 + C64 E23 + C65 E31+ C66E12
Since cu = cji the stresses oij are related with the strains EM by 21 elastic constants, if only a linear approximation is considered. By rotation of the reference system the constants ck change, which will be outlined later. The same holds for the compliances, where s k = ski: El1 =sllO1l
+S12~22+S13033+S14023+S15031+S16012
E22 = S21 O11 'S22
0 2 2 +S230 3 3 + S 2 4 0 2 3
+S25031+
S26012
E33 = S31°11 + S 3 2 0 2 2 + S 3 3 0 3 3 + S 3 4 0 2 3 + S 3 5 0 3 1 + S 3 6 0 l 2 E23 = S41°11 + S 4 2 0 2 2 + S 4 3 0 3 3 + S 4 4 0 2 3 + S 4 5 0 3 1 + S 4 6 0 1 2 E31 = S5l
1 'S52022
+S53033+S54023 +S55031 +S56012
El2 = s61°11 + S 6 2 0 2 2 + S 6 3 0 3 3 + S 6 4 0 2 3 + S 6 5 0 3 1 + S 6 6 0 1 2
(XXIII-14)
500
Interconversion between the Constants c and the Compliances s C12
C22
C23
C24
C25
C26
C13
C23
C14
C24
C33
C34
C35
C36
C34
C44
C45
C46
C25
C35
C45
C55
C56
and
Sl1
S12
S13
S14
S15
S16
S12
S22
S23
S24
S25
$26
S13
S23
S33
S34
S14
S24
5‘34
S15
S25
S16
S26
5‘3.5
S36
S45
3’46
S56
S66
S56
S45 S36
S46
represent the elastic constants and compliances in the form of matrices. By inverting the matrices one gets the other corresponding quantities where, as pointed out above, the rules of the four indices must apply. In representation with two indices, this interconversion fi-om cpqto spq,and the reverse is done by
(-
(- 1) P + ~ A S p q
l)p’q Acpy
and
Ac
SPq =
cpq=
h
(XXIII- 15)
’
where Acpqis the minor determinant of cij, where the pthline and the qthcolumn are cancelled. A similar relationship holds for s.
Physical Significance of the Quantities Compliancesspw4 Six stress components are present, if in Equation (XXIII-13) only the strain €1 1 exists and the others are zero. If in Equation (XXIII-14) only the stress is present, then the corresponding strain along the direction el of the stress is
With
s1 = 1
/ I~ ~
the elastic modulus in the direction of el is defined, which is usually designated as Young’s modulus. Similar relations hold for the e2 and e3 directions. A compression stress 022 exists if there is an elongation c l l =s12022 and one gets
sI2=-- v21 Eli
Since s12= tizl, one gets s12=
-=
=-%
E22
EII
,where v is the Poisson ratio.
€1 1
and the reverse:
501
~ 1= 4 ~ 4 1interrelates a shear stress with an elongation in the el direction. A similar relationship holds for ~ 2 ,4 and so forth. Finally s44interrelates a shear stress with a shear strain.
In general the following holds: spp (-s....,,,,) (p = 1, 2 or 3) relate strains and stresses in the same directions (s qq = l/Eqq) (longitudinal component). spq(= s;jJ) (p $: q, p, q = 1, 2 or 3) relate strains with a perpendicular stress (Poisson ratio). (spq= -vqplEqq = -vpq/Epp) (transverse interaction coefficients). an elongation with a shear a) spq(= 2 siig) (p = 1, 2 or 3; q = 4, 5 or 6) relate stress, and the reverse. Or relate in the same plane an extensional strain with a shear stress (coupling coefficients). b) spq(= 2 SuJk) (p = 1, 2 or 3; q = 4, 5 or 6) relate an elongation with a shear stress in and the reverse. Or relate in the perpendicular plane an extensional strain with a shear stress (coupling coefficients). spp(= 4 slJg)(p = 4, 5 or 6) relate a shear stress with a shear strain in the same plane (spp= UG,) (shear components).
spq(= 4 s,jJk) (p f q, p, q = 4, 5 or 6) relate a shear stress with a shear strain in a perpendicular plane (coupling coefficients).
Rotation of the axes will change all values. The coupling coefficients appear in crystals of lower symmetry, and are less important, since they are usually an order of magnitude smaller than the average longitudinal components {Haussiihl [XXIII-1 l]}. St@msses cpppq Similar properties hold for cpqvalues. However, c1 is not the Young’s modulus, since the single strain til I is associated with at least six stresses. The stiffnesses are related with wave propagation phenomena, and Haussiihl [XXIII- 121 typified these as 1)
cpp (p = 1, 2, 3) longitudinal elastic stifbess,
2)
cp, (p # q; p, q = 1 , 2 or 3) transverse interaction coefficients,
3)
cpp (p = 4, 5 or 6) elastic shear stiffness,
502 Chapter
4)
cpq (p f q; p = 1, 2 .... 6; q = 1, 2 .... 6) coupling coefficients in crystals of lower symmetry with minor importance, since usually these are an order of magnitude smaller than the average longitudinal components corresponding to the situation with the compliances spq {Haussuhl [XXIII-ll]}. For the monoclinic R-HMX, this result was tested for approximation purposes with the finding that it is important to make the cdsik-conversion with the fidl tensor. Then the coupling coefficients can be ignored. Completely wrong results are obtained if the c&k-conversion is performed in the higher orthorhombic symmetry. The reason for this is that in the tensor conversion the coupling coefficients greatly influence the shear values, s ~s55,, s66.
Elastic Constants in different Crystal Systems Triclinic, 21 Constants in c and s, 6 in Thermal Expansion: C11
c12
C13
C14
C15
Cl6
C22
C23
C24
C25
C26
C33
C34
C35
C36
C44
C45 C55
S11
S12
S13
314
S15
S16
S22
S23
S24
325
S26
5’33
3’34
S35
S36
C46
345
S46
C56
S55
S56
C66
(XXIII-16)
366
The sti%ess/compliance conversion or the opposite is obtained by the inversion of the full written matrix in ck and sk, also for the following cases. For systems of orthorhombic and higher symmetry simple expressions are available, see Voigt [XXIII-21 or Hearmon [XXIII-5]. Monoclinic, 13 Constants in c and s, 4 in Thermal Expansion: There are 3 different notations, but only the following corresponds to the use of a Cartesian system:
(XXIII- 17) C44
C55
C46
0
S46
0
s55
0
C66
S66
503 Properties Single
Orthorhombic, 9 Constants in c and s, 3 in Thermal Expansion: S11 812 s13 0 0 0 C11 C12 C13 0 0 0 s22 s23 0 0 c22 c23 0 0 0 c33
0 c44
0 0
0 0 0
c55
0
s55
s66
c66
Trigonal (without mirror planes), 7 Constants in c and s, 2 in Thermal Expansion: C11
C12 C11
C13 C13 c33
C14 -C14
C15
S11
0 0
-ClS
0
0 0
c44
S13
S14
811
$13 s33
-S14
-815
0
0 0
- C15
c44
1
0
5'12
Szpl
0 -2S15
C14
h1
2314 %ll
-512)
- c12)
(XXIII-19) Trigonal (with mirror planes), 6 Constants in c and s, 2 in Thermal Expansion: C11
C12 C11
C13 C13 c33
0 0 0 0
C14 -C14
0 c44
c44
0 0 0 0 1
S11
S12 S11
S13 S13 5-33
S14 -Sl4
0
C14
0 0 0 0
0 0 0 0 2S14 2(Sl1 - s12)
7 (Cll - c12)
(XXIII-20) Tetragonal (no mirror plane containing the fourfold axis), 7 Constants in c and s, 2 in Thermal Expansion: C11
C12
C13
C16
C11
C13 c33
-C16
0 c44
0 0 c44
0 0 0 C66
S11
S12
S13
S11
S13 s33
0 0 0 s44
0
S16 -S16
0 0
0 0 0 5'66
(XXIII-21)
504 Chapter
Tetragonal (with mirror planes containing the fourfold axis) and s, 2 in Thermal Expansion: C11
0
Cl2
0 0 0 0
ell
0 c44
0 0
S11
0 0 0
S12 s11
0 0
0 0 0 0
0 0 0 0 0
544
,6
Constants in c
(XXIII-22)
566
C66
Hexagonal, 5 Constants in c and s, 2 in Thermal Expansion: CI 1
C12 C11
Cl3
0 0 0
0 0 0 0 0
0 0 0 0
C44
C44
s11
s12
S13
0 544
0
C44
0
1
0 0
0
0
c44
Isotropic, 2 Constants in c and s, 1 in Thermal Expansion: Cll
c12
c12
CII
c12
0
0 0 0
0 0 0
~(C11-cl2)
0
0
0 0
Cl 1
1
SII
s12 SII
s12 s12
0 0
0 0
0 0
(XXIII-23)
505
Properties
Lam6 Constants of the Isotropic Body The above c-matrix reads with the Lam6 constants and their inverse as h h+2p
h+2p
0 h 0 h+2p 0 P
0 0 0 O
0 0 0 0
sl+tS2
s1
0 0 0
s1
s1 + i s 2
s1
s1 ++s2
-
0 0 0 0
0 0 0 0 0
-
P O P
s2
Not m bulk modulus
Not in bulk modulus
(XXIII-26) so that h = C l 2 = C13 = C21 = C23 = C31 = C32 C44
s1 = S12 = S13 =
s2 = s44 = S55 = S66 = 2(s,,
= C55 = C66
+ 2P = c11 = c 2 2 =
(s1
+
)= s11 =
=
(XXIII-27) the combination I@ + 2p), resp. on the right-hand side (sl
+
identifies stresses
and strains in the: same direction, and h, resp. sl the perpendicular components. The conversions are: s2=-
1
P 1
h
(XXIII-28)
a=--2P (%+ 3 4
This is the description for a hypothetical isotropic body. By comparison of the tensors of the real material constituents, one can estimate the degree of simplifications by any averaging procesises described hereafter. Hydrodynamic State If G = = 0 applies, as in the case of a liquid or gas where no shear stia e ss is present, one gets for the hydrodynamic state, with Poisson’s ratio v = 0.5 only one meaningful elastic constant K = h. The matrix inversion leads to a singularity (s2 =
506
Chapter
(XXIII-29)
Rotated Elastic Constants of Single Crystals
el’
e2’
e3’
el
UI I
u2 I
u31
e2
u12
u22
u32
e3
u13
u23
u33
The following relations hold for these direction cosines in an orthogonal coordinate system, which often reduce the complexity of the expressions considerably.
j=l
i=l
2.y r=l
3 uik
=o
x u yUQ
=o
(XXIII-30)
j=l
With i = 1, 2, 3
The rotated elastic compliances are obtained according to the tensor transformation: s!. qkl
ukk*
Si*j*k*l* (summation o v e r i * , j * , k * , l * h r n l t o 3 )
(XXIII-31)
for example (XXIII-32)
507
Anisotropy of Young's Modulus Derived from Hooke's law in the uniaxial experiment, Young's modulus is defined by Ell = l/sll. Or in the orthogonal rotated case, = 1/dll. The transformation yields for the direction e', = ull el + u12 ez + ul3 e3: 4 1 ='ill1 +2u;1
4 =u11 '1111 + 4 2 s2222 + 4 3 '3333 +
1."2
('1122
+2'1212)+2u?1
'3331 + 4 u ? 3 u12 '3332
+4u?1 u12 u13 '('1123
(s1133 +2s1313)+2u;2
?.3
(s2233 +2s2323)+
u13 '1113 + 4 u ? 2 u11 '2221 +4u?2 u13 '2223 +
+ 4 U ? l u12 81112 +4$1 +4u?3 '11
1."3
+
f2s1213)+4U?2
u l l u13 (82213 +2s1223)+4u:3
'11
'12
('3312
+2s1323)'
(XXIII-33) The rotated Young's modulus E(ull, u12,~ 1 3=) l/slllare obtained from the rotated compliances. The simplest cases are: UIl =
1:
u12= 1:
S'I1 = s11, sII1=
~ 1 =3 1: ~ ' 1 = 1
sZ2,and
~33.
It is useful to search simultaneously for the maximum and minimum values in these procedures. In general only the isotropic body shows an isotropic Young's modulus.
Anisotropy of Axial Compression Using Hooke's law, a hydrostatic pressure, p, results in compressional strains ~11,~22, ~33.Since shears do not lead to any variations in volume, the uniaxial compressibilities are:
s33=
=
= (s33+ '13 + s 2 3 ) .
(XXIII-36)
(Pressure is a negative tension, therefore the values are usually negative.) By adding these equations one gets the volume compressibility,
508
(XXIII-37) which is rotational invariant, whereas this does not apply for and Sjj.! For calculating the rotated linear compressibilities, sr1= -(stl + ~ ‘ 1 +2 ~ ~ 1 3etc., ) the rotated components sI11, sll2and sg13are calculated by the transformation (XXIII-31). For example we find with e’l = u l l + uI2 + ~ 1 3 S;, =
Sll
+~ : 2 S22 +~ : 3 S33 + 2 U I
1 ~ 1 S2l 2
+2 UI
1 ~ 1 3
+2 ~
1 ~2 1 S32 3
(XXIII-38)
(XXIII-39) (XXIII-40) (XXIII-41)
[
The components sijform a second-rank tensor Sii
3
‘zSl/kk k=l
1
. For u11 = 1, etc. one
again obtains the expressions of Equations (XXIII-34) to (XXIII-36). One can use the ratio of the linear compressibility S’ {Equation (XXIII-38)J and one third of the volume compressibility d 3 as a measure of anisotropy, since this corresponds to an averaged linear compressibility, see Equ. (XXIII-37). A percentage can also be given (XXIII-42) In the calculational procedures one should note that u 13 remains constant for rotations around
A negative linear compressibility is also possible, as for tellurium. In this case, one axis expands under hydrostatic pressure but the total volume shrinks. Sometimes such behavior is also found in thermal expansion. The consequences of such an anomaly are a catastrophe for the polycrystalline system if a variation of temperature or external stress takes place. These anomalous uniaxial compressibilities can easily be evaluated in most cases in the main axes, An example of this is ammonium tetraoxalate dihydrate.
509
Primary explosives also exhibit such properties, which would explain their pronounced sensitivity to heat and shock. But such questions have not been discussed before in the field of explosives [XXIII-131.
Anisotropy of the General (Shear) Modulus Contrary to the simplified isotropic system, the shear modulus becomes much more complicated in crystals of lower symmetry, see the heading Physical significance of spppq.The rotated s'66 leads therefore not to a shear modulus in the original sense. Therefore this rotated modulus is named as general modulus.
Irrotational Properties In general, there are only very few quantities that are rotational invariants. These include the volume compressibility defined by
K = -___
the thermal volume
expansion, the heat capacity, and the entropy. But the axial compressibilities and linear thermal expansions are not! Only in the assumed isotropic body and in the cubic class of materials are uniaxial compression and thermal expansion always isotropic. This note is important, since many technical materials are composed from cubic materials (Al, Fe, Cu, W...) from which we gained our experience.
Treatment of the Elastic Tensor In order to get a survey on compressibilities, Young's and shear modulus, a rotation of the reference systlem is performed. In this procedure, the maximum and minimum values of the quantities can also be determined. 9
ede2
Rotation of el towards e2 by an angle cp around the axis e3:
510
Chapter
e3/ el
Rotation of e3 towards el by an angle cp around the axis ez: e'l
e'z
e'3
el e2 e3
e3/ ez
Rotation of e3 towards e2 by an angle cp around the axis el:
e'~
e'z
sin cp
el
e'3
cos cp u21
u31 =-
1
ull
e2
u12 = -~
cp
u22 =--
cos cp
1 u32
e3
U13 = COS
cp
u23 =
- sin cp
u33 = 0
511
ELASTIC PROPERTIES OF THE POLYCRYSTALLINE AGGREGATE Technical materials are usually polycrystalline, which means they are constituted from an assembly of single crystals. Since the orientation of these single crystals shows no preference, the elastic properties of polycrystalline materials may appear like an isotropic specimen in the macroscopic sense. However, deviations are possible, if a preference of orientation of the single crystals is caused by a technical process. This might be a wanted or an unwanted event. Also, such events become possible in the preparation of explosives, so that TATB may become anisotropic when pressed into charges. For obtaining averaged isotropic elastic constants of a polycrystalline material without texture, the classical averages according to Voigt (1910) [XXIII-21, assume uniform deformation in the macroscopic assembly as upper bounds are given, and those of Reuss [XXIII-141(1929) assume uniform stresses. According to Neerfeld [XXIII-151 (1947) {or also Hill [XXIII-161, 1952) the arithmetic means of the Voigt and Reuss values describe experimental values well. {Other averaging procedures, of which there are many, see Kri‘iner [XXIII-171 (1958), the most modern are that of HashinShtrikman [XXIIl[-18],which reduce the upper Voigt band, are not used here.} The quantities are characterized by the appropriate index V, R or N depending on the averaging procedure used, see Table XXIII-3.
Solid Isotropic Body - Comparison of the Voigt and Reuss Procedures Only 9 constants from the set of 21 are used. All contribute to obtain the Young’s modulus E and the shear modulus G. For the compression modulus K the single crystal shears 3C and 3C’, respectively, are not needed because shears do not lead to volume variations. Below, the isotropic elastic constants in use are summarized and also their conversions along with their development from K and G of the Voigt/Reuss averaging. At the right-hand side the physical meaning is given. Neerfeld Mean Average The Voigt and Reuss procedures give different values. The Young’s modulus E and the shear modulus G are usually larger in the Voigt than in the Reuss procedure. The experimental values are between these extremes. Neerfeld suggests the use of the
512 Chapter
averages of sI and % s2 from the Voigt and Reuss procedures. But use of the arithmetic or geometric averages of E, G, and K also lead to good results. Table XXlll-3: Voigt/Reuss averaging procedure
(XXIII-43)
System
Voigt-Parameters
Triclinic monoclinic orthorhombic
3A I +~ 2+ 2 ~ 3 3B = c23 + c13+ c12 3 c = c44 + c55 + C66
Tetragonal , 716 constants
3A = 2 I + c33 3B = 2 c13 + ~ 1 3 c = 2 c44 + c66
Trigonal, 716 constants Hexagonal Cubic
1)
The bulk modulus K = h +
3A' 3B' 3c' 3A' 3B' 3C'
2
= S I I+ ~ 2 + 2 ~ 3 3 =~ 2+ 3 si3 + s12 = s44 + s55 =2 SII
+ s66
+~ 3 3
2 ~ 1 +3 ~ 1 = 2 s44 + S66 =
2
3A = 2 cI1+ cj3 3B = 2 ~ 1 +3 ~ 1 2 3 c = 2 c44 + (CII - c12)/2
3A' 3B' 3C'
=2 S I ~ ~ 3 3
3A = 3 3B = 3 ~ 1 3c=3c44
3A' 3B' 3C'
= 3 SII
Isotropic Bulk modulus
=---
Isotropic Shear modulus
3
Reuss-Parameters
G,
=
2
+
=2 ~ 1+ 3 ~ 1 2 =2
=
s44 + 2 (Sll - Sl2)
3~
1
K -
3
1 2
= 3 s44
-
+ 3C)
( A-
5
=
5
2 2(1+v)G = - -= 3 3(1-2v)- 3(1-2v)
-'
1
and the
~
compressibility as K = 1/K, where p is the (isotropic) hydrostatic pressure, and
AV/V, the relative compression.
(XXIU-44)
The rigidity or shear modulus G = =
~
3(1-2v)K 2 ( 1 + v ) = 2(1+v)
1
-023
==)E23 (XXrn-45)
3)
Young's modulus
9KG 3K+G
= --
G(3h+2G) - p(3h+2p)-
- E l - ;-
Poisson ratio v =
O11
(uniaxial
(XXIU-46)
tension); 4)
1
3K - 2G
h
h
% (XXIII-47) El 1
513
Lam6-constants are h =
5)
=G =
2v
-
2 -G 3
=
-
2 3
=
~
2vG 2vp vE and (1 - 2v) - (1 - 2v) - (1 + v)(l - 2v) ~
which corresponds to the shear modulus G.
(XXIII-48)
The corresponding constants (based on the compliance) are
1 h v -_ 2p (2p+3h)=
s1=
-
l
1 andIs 6G
1 2p
1 2G
=-=-=--
v+l
. (XXIII-49)
Comparison with the sik-tensors of the isotropic medium is useful to recognize their meaning:
The s2values are not needed for estimation of the bulk modulus.
Peculiarities of Elastic Constants Rule of Thumb
In many solid materials h = holds. With Equ. (XXIII-47) v = 0,25 results. Most solids exhibit values o f v between 0.2 and 0.4. For v = 0.25 K = -5p 3
and
=
=
5 2
5 3
(XXIII-5 1) 5 2
= -G
hold.
(XXIII-52)
514
Chapter
Ideal Gases, Liquids, Hydrodynamic States in Shock Physics Ideal gases and liquids do not resist to any shear movement. This is characterized by = = 0, and v = 0,5 results. Only one meaningful elastic constant remains as the bulk modulus K = h.
Interconversion of the Technical Elastic Constants The different sets of elastic constants in technical use need only two basic independent quantities. Tthe interconversions are given in Table XXIII-4 Table XXIII-4. Interconversions of the Technical Elastic Constants
Given
h
E
P
V
K
G
Needed h
(1 - 2v)h 2v
P=
s1=
1 9K
V
1 2
2 3
2vG (1 - 2v)
2 3
v+l
2=
E=
1 6G
1 2G
+ 2G)
E
+
h 2(h +
V=
G
2v)K 2(1+ v)
2(1+ v)G
K
2(1+ v)
K=
2 h+-p 3
2v)
+ G)
2v)
Polyctystalline Elastic Properties
Evaluation and Comparison of Elastic Properties, Mixing Properties By using the Voi@’Reuss averages one can compare different substances by a first view. But a general overview on elasticity is necessary to compare substances of different crystal classes, or also to suggest an elastic mock. Haussiihl [XXIII-19] suggested the following procedure: The elastic tensor exhibits two independent scalar invariants:
+
= cll + c22+ c33+ 2c12 2c13+
, where 1 = -,
(XXIII-53)
see above, and = c,,
+ c~~+ c~~+ 2c4, + 2c5, + 2 c 6 6
(XXIII-54)
The combination +
- (‘11
+ c22 + ‘33 + c12
18
c13
+ c23 + c44 + ‘55 + ‘66)
(XXIII-55)
9
is also a scalar invariant with more details than in I (9K). It is the arithmetic mean of all important stifhesses of the crystal under consideration. Haussiihl defined a quantity with MV as the molecular volume, MW the molecular weight, L Loschmidt number, p density A4w
(XXIII-5 6)
LP
Chemical Structure - Elasticity Relationship Elastic properties of simple systems were calculated by quantum-theoreticalmodels. Currently we want to find correlations between constituents and elasticity within one isotypic group of crystals. The above mentioned S-value, which addresses specific lattice energies and interaction potentials, was almost constant for a great number of isotypic ionic crystals {Haussiihl [XXIII-19]}. Further there is an additivity rule, so that values for component mixtures are obtained correctly. It is this author’s hypothesis that the rotated Young’s modulus body can be a picture of projections of the bonding strength of the chemical groups in the unit cell. Since bonds in chemical systems are not isotropic, the natural consequence must be anisotropic elastilc behavior.
516
Chapter
Characterization of the Anisotropy in Polycrystalline Media By averaging procedures up to 21 constants are condensed to 2, so considerable deviations occur if uniform strains or uniform stresses are considered. These differences result in different moduli fiom the different averaging procedures, and can be expressed by the following measures of polycrystalline anisotropy [XXIII-20]: 1
Bulk modulus anisotropy: A, * = 100
-
in%.
Since the Voigt averaging
procedure gives the upper limit, this value is usually positive. Cubic crystals exhibit no anisotropy in compression, so this value should be 0 or small due to the differences of the Cik to Sik inversions.
2
Shear modulus anisotropy:
in%
= 100
.
Measures of Microscopic Anisotropies The single crystal anisotropy in shear A, is best characterized for crystals by A ~ = % ; when
c44=c66
C66
Rotating the general modulus their ratio Gmax/Gmin.
2c44 . (c* 1 - c12)
oneuses A ~ =
(XXIII-57)
one can determine also the GmaX and G,,,-values, and
The anisotropy in compression is given by Equation (XXIII-42) in different directions A c = -.
(XXIII-58)
K
The extreme values can be determined by rotation. The Young’s modulus anisotropy is best characterized by determination of the maximum and minimum values of the body, and one gets Ernax.
(XXIII-59)
Emin.
This system has the disadvantage that only crystals of a distinct class can be compared. But it is also possible, to use the Voigt/Reuss averages as a standard, and to compare them with the extremes of the single crystal properties.
517 Appendix:
Appendix: Analytical Formula of Interconversions Cik
Sik
for some Cases
Interconversion: ((compliancesare obtained by the exchange of clkwith s&and the reverse.)
Orthorhombic, 9 Constants in c and s, 3 in Thermal Expansion s33 -
sg
s
=
'13
= ('12
C23
=
S3,-sLj
(.I,
=
c22
('13 c12
S
6 1 1 s22 -s t ) c33=
'23
-'33
1
'12) c44
S
('13
'23 -'22
'13)
c55
=544
1 =-
S
'12
-'I1
555
(XXIIIA- 1)
1
'23
c66=-
566
S
where s = XI1
s22 s 3 3 + 2 h 2 S l 3
- s22
s t - s 1 1 $3
-s33
4,
(XXIIIA-2)
Trigonal, 6 Constants in c and s, 2 in Thermal Expansion c 1 1 + c12
s33
=-
c11 -c12
s44
=-
XI
S13
(XXIIIA-3)
=
Tetragonal, 6 Constants in c and s, 2 in Thermal Expansion 1
s33
CI1
+ c12 = -
c 1 1 - c12
=
c33
=
X 1 s13
x 1
1
c44
where
+
(SU ~
c13=
=-
1 Chh = S 66
(XXIIIA-5)
518 Chapter
(XXIIIA-6) Hexagonal, 5 Constants in c and s, 2 in Thermal Expansion CII+CIz=-
s33
XI
(XXIIIA-7) 1
=-
(XXIIIA-8) Cubic, 3 Constants in c and s, 1 in Thermal Expansion 1
s12
+ s12) C1l
=-Ti
=
C44
=-
(XXIIIA-9)
544
c12
where x 3 = ( S I I -s12&11
+2S,,)=S:l+Sll
s12-2s:2.
(XXIIIA-10)
Isotropic, 2 Constants in c and s, 1 in Thermal Expansion CII
=-
(Sll + s12)
where x 3 = (Sll-SIZ~Sll+2S12)= s:1+SllSl2-2St
= S12(s:l+S11-2S12).
(XXIIIA-12)
519 Elastic Data
ELASTIC DATAOF PETN, RDX, s-HMX AND AP In the following the data sheets of the now known single elastic properties in full tensor notation of explosives are given. The thermal behavior, and some related wave velocities are also included. For comparison the tetragonal (with 6 constants) PETN, the orthorhombic RDX and ammonium perchlorate (AP) (with 9 constants), the monoclinic S-HMX (with 13 constants) and its temperature dependence are listed. In the case of AP there are two sources, where one (Landolt-Bornstein) made the mistake of exchanging [loo] with [OlO] direction (exchange of the axes a and b). This example also indicates the accuracy of single-crystal measurements, where the Haussiihl measurements are preferred. The bold-type figures in the following tables are the primary and directly determined quantities.
520 Chapter
Table XXIII-5.
PETN Single Crystale [XXIII-21J p = 1.773 g/cm3, tetragonal, 6 constants, MW = 316.1447 (1 GPa = 1E+9 N/m2= 1E+10 dyn/cmz = 10 kbar)
Tensor invariants
the dimensions of Haussuhl (C: (Id’ Nm-’1; S: (I@” Nm])
C = 0.8987 c11= c33 = c44 = c66 = c12 = cl3 =
S = 266.05
in 1E+ 10 dyn/cm* 17.180 12.140 5.030 3.930 5.430 7.480
SII =
s33 = s44 = S6h =
s12 = sl3 =
in 1E-1 3 cm2/dyn 79.888 139.066 198.807 254.453 - 5.219 - 46.007
Voigt/Reussaverages of the Elastic Propertes and wave velocities
V
Voigt 0.0970 0.0454 0.1178 0.2976
Neerfeld 0.0964 0.0439 0.1143 0.3024
Reuss 0.0958 0.0424 0.1108 0.3073
Lam6 constants h in Mbar p in Mbar
0.0667 0.0454
0.0671 0.0439
0.0676 0.0424
-2.526 11.016
-2.646 11.396
-2.774 11.802
2338 2980 1600 1483
233 1 2955 1572 1460
2324 2930 1545 1435
K in Mbar G in Mbar E in Mbar
SI
in Mbar-‘
% s2 in Mbar-‘
sound velocities co in m i s CI in m i s cs in m/s CR in m / s
Thermal expansion:
Temperature not specified, room temperature.
52 1 PETN Table XXIII-6.
Measures ofAnisotropy of PETN: Volume compressibility
KV~I.
relative uniaxial compressibilities Scxx= 3 K ~ I C~~ , ,/~ .
from single crystal data 1.044 10." Scii = 0.824 S c 2 2 = 0.824 S c 3 3 = 1.352
I
from VoigtlReuss data 1.037 10." (Neerfeld) f
Rotated relative uniaxial compressibilty ScJSc, (maximum) = 1.64 in (el - ez)-plane in (el - e3)-plane in (ez - e3)-plane in plane perpendicular to (el + ez)
S, S,, S, S,,
= 0.824
1.352 = 1.352 = 1.352 =
S,, S, S,, S,,
=
0.824
= 0.824 = 0.824 = 0.824
S,,IS, S,,/S,,n S,,/S,,, S,,/S,
1.oo 1.64 = 1.64 = 1.64 = =
Rotated Young's modulus (1 f s, ) in Mbar E,,,,,JE,,,i,, (maximum) = 1.89 in (el - ez)-plane in (el - e3)-plane in (ez - e3)-plane in plane perpendicular to (el + ez)
Em,, = 0.125 Em,, = 0.136 Em,, = 0.136 Em,, = 0.1 19
Em, = 0.099 Em,, = 0.072 Em, = 0.072 Em, = 0.072
from single crystal data
E,,/E,, = 1.26 EmaX/E,,, = 1.89 baX/E,,,, = 1.89 E,,IE,, = 1.66
from VoigtlReuss data
Shear anisotropy
Rotated General (Shear modulus) (1/s6,j in Mbar GmrudGmin (maximum) = 1.74 in (el - ez)-plane in (el - e3)-plane in (e2 e3)-plane in plane perpendicular to (el + ez) ~
Thermal expansion
Gmax= 0.0503 G, = 0.0503 G, = 0.0503 = 0.0503
G,
= 0.0503
=
= 0.0322
=
= 0.0322
= 0.0301
1.00 1.56 = 1.56 = 1.67
522 Chapter
P E T N , Young's modulus in comparison
Mbar 14 12 10
008
330
~
\.-.30
,,-*----
~
Figure XXIII-1 Comparison of rotated Young's moduli in different planes of PETN, compared with Voigt and Reuss averages.
300
~
,
006-
.
00402
00
~
270
0.04
0.08 0.10 0.12
; '
002~
-
240
~
14
210
-. -.
~- - .
150
180
Mbar ,
0.05
_- - -..
30
P E T N General modulus In comparison
-
Figure XXIII-2 Comparison of the rotated General (Shear) moduli in different planes of PETN, compared with Voigt and Reuss averages.
04
0.02 01
- 270 001
-
02
-
03
-
~
240
04 05 210
- _ _~-~-
_-' 150
180
,- -.
14 121.0
P E T N Comparison of axial compressibilities
-
1
300
04-
,
02-
00
270
0 2 -
'
04-
10
.
240
Figure XXIII-3 Comparison of the rotated uniaxial relative compressibilities in different planes of PETN, compared with 1/3 of the volume compressibility.
523
Table XXIII-7.
Selected Sound velocities in the tetragonal Single Crystal PETN (Pure Modes) Longitudinal (compression) waves:
A possible longitudinal wave is:
Transversal (equivoluminal)waves: Possible transversal (shear) waves are:
Longitudinal and transverse waves in combination: Possible combination waves are:
524 Chapter
Table XXIII-8.
RDX Single Crystal (2OOC) [XXIII-22 and 22al p = 1.80 g/cm3, orthorhombic, 9 constants, MW = 222.1207 (1 G P a = 1E+9N/m2=
c11= c22 = c33 = c44 = c55 = c66 = = c13 = c23 = CIZ
10kbar)
in 1 E+ 10 dyn/cm2 19.602 25.020 17.928 4.068 5.166 6.912 8.208 5.904 5.814
SII =
s22 = s33 =
SM= s5s =
s66 =
Sl2 = sl3 = sl3 =
in 1E-1 3 cm2/dyn 62.713 47.866 63.970 245.821 193.573 144.676 - 17.060 - 15.120 - 9.905
Tensor invariants in the dimensions of Haussuhl (C: /IdoNm-']; S: /l(r2°Nml) C = 1.096
S = 224.51
VoigYReussaverages of the Elastic Properties and wave velocities
V
Voigt 0.1 138 0.0607 0.1546 0.2735
Neerfeld 0.1 122 0.0590 0.1SO6 0.2764
Reuss 0.1106 0.0573 0.1465 0.2792
Lam6 constants h in Mbar in Mbar
0.0733 0.0607
0.0729 0.0590
0.0725 0.0573
% s 2 in Mbar-'
-1.769 8.236
-1.835 8.475
-1.906 8.729
sound velocities co in m / s CI in cs in m/s CR in m / s
2514 3289 1836 1696
2497 3256 1810 1673
2479 3223 1784 1649
K in Mbar G in Mbar E in Mbar
sl
in Mbar-'
Thermal expansion (at room temperature) = 86 K-' a22= 24 K-'
=
8 1 10-6K-'
525
Table XXlll-9.
Measures ofAnkotropy of RDX (2OOC): Volume compressibility
KV~I
relative uniaxial compressibilities scxx= 3 Kxx/ ~~~l
from single crystal data 9.0380 Scii = 1.0135 Sc22 = 0.6938 Sc33 = 1.2927
from VoigtIReuss data 8.9117 (Neerfeld)
Rotated relative uniaxial compressibilty ScJSc, (maximum) = 1.86 in (el ez)-plane in (el - e3)-plane in (ez - e3)-plane in plane perpendicular to (el + ez) ~
S,,, S, S, S,,
S,, S,,, S,, S,,,
1.014 1.293 = 1.293 = 1.293 = =
= 0.694
S,,,/S,,,
=
1.014 = 0.694 = 0.854
S,,,IS,, S,,/S, S,,/S,,
=
=
1.46 1.28 = 1.86 = 1.51
,
Rotated Young's ,modulus(1/s, ) in Mbar E r n a m i (maximum) n = 1.78 in (el ez)-plane in (el - e3)-plane in (e2 - e3)-plane in plane perpendicular to (el + ez) ~
ha,= 0.2089 Em, E,, Em,
= 0.1595 =
0.2089
= 0.1809
Em, = 0.1595 Em, = 0.1379 Em,,= 0.1 176 = 0.1270
E,,/E,,, E,,/E, E,,/E,, E,,IE,,
=
1.31 1.16 = 1.78 = 1.42 =
from single crystal data Shear anisotropy
in (el ez)-plane in (el - e3)-plane in (ez - e,)-plane in plane perpendicular to (el + ez) ~
A,
G,,,, G,, G, G,,
c44 = -= 0.589
= 0.069 1
G,
= 0.0691
G,,IG,
=
= 0.0637
G, G,,, G,,,
= 0.0517
G,,/G, G,,,/G,, G,,/G,,
=
= 0.0760 =
0.0693
= 0.0407 = 0.0455
I .00 1.23 = 1.87 = 1.52
Rotated general Shear modulus (1/s66)in Mbar GrnJGrnin (maximum) = 2.1 724
526 Chapter RDX Youngs modulus
Mbar
-
0 25
Figure XXIII-4 Comparison of rotated Young's moduli in different planes of RDX. compared with Voigt and Reuss averages.
330
0 20 0 15
~
300
0 10 0 05 000 -270
0 05 0 10 0 15
240
0 20 210
0 25
~~
.
150
180
Mbar
o
0 10
RDX
General modulus
330 G
0 08 0
Figure
300
60
0 04 0 02
000
270
90
Comparison of the rotated General (Shear) moduli in different planes of RDX, compared with Voigt and Reuss averages.
0 02 0 04 0 06
' 20
240 ~
0 06 150
210
0 10
0
/-\
1 2
30
RDX uniaxial compressibilities
10
08
300
Figure XXIII-6 Comparison of the rotated uniaxial relative compressibilities in different planes of RDX, compared with 1/3 of the volume compressibility.
06 04
~
0.2 -
0.2
8
270
00 ~
06
240
10 1 2 -
210
-Jc 160
150
521
Table XXIII-10.
O-HMX Single Crystal, 26OC [XXIII-231 p = 1.903 g/cm3,monoclinic, 13 constants, MW = 296.1609 (1 GPa = 1E+9 N/m2 = 1E+10 dyn/cm2 = 10 kbar)
c11=
c22 = c33 = c44 = CSS =
c66 = c1z = c13 =
c15 = c23 = c2s = c35 = c46 =
in 1 E+10 dyn/cm* 20.800 26.900 17.600 2.900 6.600 3.800 4.000 13.000 0.600 6.600 - 1.500 0.100 3.000
s11 =
s22 = s33 =
s44 =
= S66 = s12 = s13 = s15 =
s23 = s25 =
s35 = S46 =
in 1E- I3 crn2/dyn 89.829 41.639 113.229 1881.200 154.255 1435.600 2.808 - 67.367 - 6.507 - 17.743 9.477 0.376 -.1485.100
Tensor invariants in the dimensions of Haussiihl (C: (Id' Nm-9; S: /1lrZo Nm]) C = 1.136
S = 293.4
Voigfleuss averages of the Elastic Properties and wave velocities K in Mbar G in Mbar E in m a r V
Lam6 constants h in Mbar in Mbar
s1 in Mbar-' % s 2 in Mbar-' sound velocities co in m/s CI in cs in CR in m / s
Voigt 0.1250 0.0544 0.1425 0.3100
Neerfeld 0.1249 0.0336 0.0898 0.3802
Reuss 0.1249 0.0128 0.0371 0.4504
0.08873 0.0544
0.1025 0.0336
0.1 163 0.0128
- 2.175
- 4.071
- 12.134
9.191
14.882
39.071
2563 3222 1691 1571
2562 2974 1308 1247
2561 273 1 820 777
528 Chapter
Table XXIII-11.
Measures of Anisotropy ofJ-HMX (26OC): Volume compressibility
KV~I.
relative uniaxial compressibilities Scxx= 3 K ~ K ~~, , I . /
from single crystal data 8.009 10.’’ S C I ~0.9465 sc22 = 1.0002 Sc33 = 1.0532
I
I
from Voigt/Reuss data 8.005 (Neerfeld)
A‘ = 1 *
0
0
s = 0.0587%
I
Rotated relative uniaxial compressibilty Scdx, max/Scdx, min (maximum) = 1. I79 in (el eZ)-plane in (el - e3)-plane in (e2 - e3)-plane in plane perpendicular to (el + e2) ~
S,,, = 1.0002 Smax = 1.082 Smax = 1.053 Smax = 1.073
Rotated Young’s modulus (l/s,I ) in (el ez)-plane in (el - e3)-plane in (ez - %)-plane in plane perpendicular to (el + ez) ~
p - HMX, - Oktogen Monoclinic 13/4 Space Group: PZl/c 2 GHsNsOs Mol-Wt. = 296.156640
Em,, = 0.2402 Emax= 0.1946 Emax= 0.2402 Emax= 0.16 13
S,,, S,, S,,” S,,
= 0.946 = 0.918 =
1.0002
= 0.954
Mbar E&h,,
Reciprocal Lattice
a, = 6,5380 (8) A b, = 11,054 (2) A c, = 8,702 (2) A = 124,44 O V = 5 18.668 A3
a* = 0,18546 b* = 0,09046 c* = 0,13934 p* = 55,56” = y* = 900
I ~ 9 6 7o/cm3
v*= 1
=
(maximum) = 17.925
Em,, = 0.0254 Em,,= 0.0883 Em, = 0.0200 Em,, = 0.0134
Lattice
n=
1.057 1.179 = 1.053 = 1.125 =
S,,,/S,,, S,,,/S,,, sma,/s,,,
I w3 A-3
E,,,/E,, E,,,IE,,, E,,,/E,,, E,,,IE,,,
9.453 2.203 = 12.029 = 12.002
=
=
Thermal expansion in 10-5K-’(room temp.) -0.29 11.6
2.30 2.68 volume 13.1
529
Figure XXILI-7. Comparison of rotated Young's moduli in different planes of HMX (26"C), compared with Voigt and Reuss averages.
15 10
W
25 1
Mbar
30
300
0 06
compar~sono f
the general modulus
- - ,
330
z6'c
R
0
0101
80
0 04
\ 000
270
Figure XXIII-8. Companson of the rotated General (Shear) moduli in different planes of D-HMX, compared w t h Voigt and Reuss averages
I
1
0 02
Oo2: 004
1
0
120
240
0 10
I50
i
180
R-HMX Comparison of 3oaxial cornpressibilities
0
--
1 2 4
Figure XXIII-9. Comparison of the rotated uniaxial relative compressibilities in different planes of D-HMX, compared with 1/3 of the volume compressibility.
210
-..
~~~
180
150
Selected Sound Velocities of the monoclinic D-HMX (26OC), (Pure Modes) Possible longitudinal (compression)waves: Ir,, =
d"
CII = 3759.7
P
Possible transversal (equivoluminal)waves: cl* =
is Jc; -
=
1234.5 m / s
=1413.1 m / S
c12 = -
~,3=1831.5m/~ 3,4 =
c14 = 407.8 ~ 1= 5 1826.7
cl4 = 428.9 m/s
Possible combination waves (longitudinaland transversal waves): = 3308.1 m / s
elt2 1858.7 m / s 1
CIt1,2 =
ci13 = 3341.2 m / s cI14 = 1862.2
*
*
cI1+ c33 + 2c55 2c15 2c35 4
(ell -c33 2\
*2C15
4
T2C35)2
+ (c13 + c55
c15
* I2 c35
ci15 = 3510.7 m / s cll6 = 1273
53 1
Table XXIII-12.
D-HMX Single Crystal, 107OC [XXIII-231 p = 1.877 g/cm3,monoclinic, 13 constants, MW = 296.1609 (1 GPa= 1E+9N/m2=1E+10dyn/cm2= lokbar)
c11= c22 = c33 = c44 = css = c66 = c12 = cl3 = c15 = c23 = CZS =
c35 = c46 =
in 1E,+10 dyn/cmz 18.700 25.800 16.100 2.700 6.100 3.500 3.800 12.000 0.400 6.500 - 1.400 0.100 2.900
SII = s22 =
s33 = s44 = s55 =
s66 = s12 = sl3 =
s15 = s23 = s25 =
s35 = s46 =
Tensor invariants in the dimensions of Haussuhl (C:
in 1E- 13 cmz/dyn 103.119 43.976 129.212 3365.000 166.587 2596.000 4.371 - 78.596 - 4.470 - 2 1.075 10.152 - 1.801 2788.000
Nrn-7;5:
C = 1.058
Nmj)
S=277.11
VoigZUeuss averages of the Elastic Properties and wave velocities
V
Voigt 0.1169 0.0501 0.1316 0.3124
Neerfeld 0.1168 0.0288 0.0769 0.3904
Reuss 0.1167 0.0075 0.0222 0.4683
Lam6 constants ilin Mbar in Mbar
0.0835 0.0501
0.0976 0.0288
0.1116 0.0075
s1 in
- 2.374
- 4.827
- 21.13
9.973
17.340
66.24
2496 3129 1634 1519
2494 2862 1214 1165
2493 2599 634 602
K in Mbar G in Mbar E in Mbar
Mbar-'
% s2 in Mbar-'
sound velocities co in
in m i s cs in m i s CR in CI
532
Table XXIII-13.
Measures ofAnBotropy ofJ-HMX (I0 7'C) Volume compressibility
K V ~ ~ .
relative uniaxial compressibilities Sell = 3 K i l l K ~ ~
~
from single crystal data 8.571 SCII= 1.0114 Sc22 = 0.9546 . Sc33 = 1.0340
I
from Voigt/Reuss data 8.563 10." (Neerfeld)
Ac
= 100-
= 0.0909??
Rotated relative uniaxial compressibilty Scdx, max/Scdx, min (maximum) = 1.1445 in (el - ez)-plane in (el - e3)-plane in (e2 - e3)-plane in plane perpendicular to (el + ez)
S,,, S,, S,, S,,,
in (el ez)-plane in (el - e3)-plane in (e2 - e3)-plane in plane perpendicular to (el + e2)
Em,, = 0.2274 Em,, = 0.1773 Em,, = 0.2274 Em,, = 0.1508
~
= = = =
S,, S,,, S,,, S,,
1.011 1.092 1.034 1.063
= 0.954
S,,IS, S,,IS,,
= 0.955
S,,,/S,,
= 0.955
=
S,,IS,,
0.954
from Voigt/Reuss data
I
c44 A, =-=0.7714 Chh
=
E,,,IE,,, = 15.650 E,,,IE,, = 2.291 = 19.887 E,,,/E,, EmaJEmm = 20.320
Em, = 0.0145 Em,, = 0.0774 Em,, = 0.01 14 Em,, = 0.0074
from single crystal data Shear anisotropy
1.060 1.144 = 1.083 = 1.1 14 =
A,? = 100*
= 73.82%
c Rotated Young's modulus (I/s,, ) in Mbar E d m i n(maximum) = 30.730 Rotatedgeneral Shear modulus (I/&) in Mbar GJG,,,, in (el - ez)-plane in (el - e3)-plane in (e2 - e3)-plane in plane perpendicular to (el + e2)
G,,, G,, G,,, G,,
= 0.0723 = 0.0600 = 0.0464 = 0.0368
G,,, G,,, G,,, G,,,
= 0.0039 = 0.0257 = 0.0030 = 0.0041
(maximum) = 32.86 G,,,/G,, G,,,/G,, G,,,IG,, G,,,/G,,,
18.764 2.338 = 15.627 = 8.884 = =
Thermal expansion There are 4 thermal expansion coefficients to be considered, which are unknown for this temperature.
Selected Sound Velocities of the monoclinic D-HMX (107OC), (Pure Modes) Possible longitudinal (compression) waves:
Possible transversal (equivoluminal) waves: Ct* =
=
Jc;
-
t:
-
=
1199 m / s
cy ~ 1 3 6 &S 6
ct3 = 1792 m / s ct4 = 303.2 m / s
= 1788 m/s ci4 = 326.4 m / s
Possible combination waves (longitudinal and transversal waves): C,*l
= 3 157 m / s = 1801 m / s
clt5 = 3371 &S elt6 = 1196 &S
534
Table XXIII-14.
Ammonium perchlorate (AP) Single Crystal [XXIII-25] p = 1.949 g/cm3, orthorhombic, 9 constants, MW = 117.491 (1 GPa = 1E+9 N/m2 = 1E+10 dyn/cm2 = 10 kbar) c11= c22 = c33 = c44 = c55 = c66 = c12 = c13 = c23 =
in 1 E+1 I dyn/cm2 2.297 2.356 3.012 0.469 0.584 0.964 1.660 0.735 1.033
s11 =
s22 = s33 =
sa= s55 =
s66= s12 =
513 =
s23 =
Tensor invariants in the dimensions of Haussiihl (C: /IdnNm-9; S: C = 1.46
in IE-13 cm2/dyn 88.702 93.839 39.077 213.220 171.233 103.734 - 62.389 - 0.248 - 16.959
NmJ)
S = 145.8
VoigdReussaverages of the Elastic Properties and wave velocities K in Mbar G in Mbar E in Mbar Lamb constants h in Mbar in Mbar
in Mbar-' Yi s2 in Mbar-'
SI
sound velocities c, in m/s CI in m l s cs in m l s in m / s
Thermal expansion a l l= 79 = 49 = 60
Voigt 0.1613 0.0686 0.1802 0.312
Neerfeld 0.1608 0.0624 0.1656 0.328
Reuss 0.1602 0.0562 0.1509 0.243
0.1156 0.0686
0.1192 0.0624
0.1227 0.0562
- 1.741
- 1.980
- 2.272
7.290
8.014
8.898
2877 360 1 1876 I744
2872 3538 1788 A 1667
2867 3473 1698 1586
K-' K-' K-'
AP
Table XXIII-15.
Measures ofAnisotropy ofAP (Haussuhl): Volume compressibility
KV~I.
relative uniaxial compressibilities Sell = 3 BLl1i K~~~
from single crystal data 6.243 lo-’’ Sell = 1.253 Sc22 = 0.696 Sc33 = 1.051
Rotated relative uniaxial compressibilty S C ~ mar, /Scd4 in (el - e2)-plane in (el e3)-plane in (e2 e3)-plane in plane perpendicular to (el + e2) ~
~
S, S, S,, S,,
1.253 = 1.253 = 1.051 = 1.051 =
S, S,,= S, S,
I
from Voigt/Reuss data 6.220 10-l’ (Neerfeld) = 100-
= 0.359%
(maximum) = 1.800
= 0.696
sm,/s,,, = 1.800
1.051 = 0.696 = 0.975
S,,iS,,, S,,,/S,, S,,/S,,
1.192 1.509 = 1.078 = =
Rotated Young’s modulus (l/sl ) in Mbar E d m i n(maximum) = 2.40 in (el - ez)-plane in (el - e3)-plane in (e2 - e3)-plane in plane perpendicular to (el + e2)
Em,, = 0.248 Em,, = 0.256 Em,, = 0.256 Em, = 0.256
Em,, = 0.107 E,i, = 0.1 13 E,in = 0.107 Emin= 0.157
Em,/Emi,, = 2.326 E,,xiE,,, = 2.270 E,,/E,i,, = 2.401 EmaXiEmi,, = 1.628
from single crystal data Shear anisotropy
=-c44 = 0.486 %6
= 100-
= 9.93%
Rotatedgeneral Shear modulus (l/s66)in Mbar G,,,JG,,,, (maximum) = 3.18 in (el - ez)-plane in (el - e3)-plane in (e2 e3)-plane in plane perpendi.. cular to (el + ez) ~
Gmax = 0.0964 G,, = 0.0780 G,, = 0.0599 G,, = 0.1035
Gmi, 0.0325 Gmin= 0.0584 G,;l = 0.0469 G, = 0.0520 1
Gm,iGmi,, = 2.963 GmmiGmin = 1.335 G,,iG,,, = 1.278 Gm,iGmi,, = 1.989
536
- ~~-
0 0 25
AP Rotated Moduli
20 015
2
010
E
0.05
Figure XXIII-10. Comparison of the rotated Young’s moduli and general shear moduli in different planes of
(3 OM:
270
005010-
0.15
~
0 25 210 180
AP Rotated uniaxial relative compressibilities
0 ‘ 4
--
12
l o
I
0 20 O
240
~
\
Figure XXIII-11. Comparison of the rotated uniaxial relative compressibilities in different planes of AP.
531
AP Table XXIII-16.
Ammonium perchlorate (AP) Single Crystal [XXIII-26] p = 1.95 g/cm3, orthorhombic, 9 constants, MW = 117.491 (Note: a and b axes wrongly chosen IIlOO] and [OIO] are here interchanged!) (1 GPa = 1E+9 N/mZ= 1E+l 0 dyn/cmz = 10 kbar) c11=
c22 = c33 = c44 = c55 =
c66 =
c12 = cl3 = c23 =
in 1E+11 dynicmz 2.510 2.460 3.150 0.660 0.470 1.030 1.630 1.150 0.760
s11 =
s22 =
s33 = s44
=
s55 = s66 =
= = 523 = 512 513
Tensor invariants in the dimensions of Haussiihl (C:
in 1E- 13 cmZ/dyn 78.00 71.00 38.00 152.00 213.00 97.00 - 46.00 - 17.00 0.00
Nm-’1; S: [10-20Nm])
C = 1.53
S = 153.6
Voigmeuss averages of the Elastic Properties and wave velocities Voigt 0.169 0.074 0.193 0.3 1
Neerfeld 0.166 0.068 0.180 0.32
Reuss 0.164 0.063 0.167 0.33
0.120 0.074
0.123 0.068
0.126 0.063
1.60 6.78
- 1.78
- 1.99
% s2 in Mbar-’
7.32
7.96
sound velocities co in CI in m / s cs in m i s in m i s
2943 3702 1944 1800
292 1 3634 1871 1740
2899 3564 1795 1670
K in Mbar G in Mbar E in Mbar Lam6 constants h in Mbar in Mbar SI
in Mbar-’
Thermal expansion
Not available for this set.
~
538 Chapter
Table
Measures ofAnkotropy ofAP (LB): (Note: a and b axes wrongly changed) Volume compressibility
from single crystal data 6.1 lo-'' S C I=~0.7377 SCZZ = 1.230 S~33= 1.033
KV~I.
relative uniaxial compressibilities Sell = 3 K I ~ / K V ~ ~ .
Rotated relative uniaxial compressibilty S Q ~ -/Scd4 in (el ez)-plane in (el - e3)-plane in (ez - e3)-plane in plane perpendicular to (el + e2) ~
S,,, = 0.738 S,, = 0.738 S,,, = 1.033 S m m = 0.984
1.230 1.033 = 1.230 = 1.033
Smax =
S,, S,, S,,,
mi,,
=
I
from VoigtlReuss data 6.0 1 10-12(Neerfeld)
I' A
= 100-
= 1.49%
(maximum) = 1.67 S,,,/S,, S,,,/S,, S,,,/S,, S,,,/S,,
1.67 1.40 = 1.19 = 1.05
=
=
Rotated Young's modulus (l/s,,) in Mbar E d r n i , (maximum) , = 2.05 in (el ez)-plane in (el - e3)-plane in (ez - e3)-plane in plane perpendicular to (el + e2) ~
Em,, = 0.260 Em,, = 0.263 Em,, = 0.263 Em,, = 0.263
Em, = 0.128 Em, = 0.125 Em, = 0.140 Em,, = 0.165
E,,,/E,, E,,,/E,,, Em,,/Em, E,,/E,,
= 2.03 = 2.1 1 =
=
1.88 1.59
from single crystal data Shear anisotropy A, =
= 0.641
Rotatedgeneral Shear modulus (1/s66) in M a r GAGrni,,(maximum) = 3.18 in (el ez)-plane in (el - e3)-plane in (ez e3)-plane in plane perpendicular to (el + ez) ~
~
Gmax =
G,,, G,,
0.103
= 0.067 = 0.092
G m a x = 0.107
Ankotropy of thermal expansion
Not present.
G,," = 0.042 G,,, = 0.047 G m m = 0.066 G,,, = 0.055
= 2.48
G,/Gm,n
1.42 1.39 = 1.95
=
=
Gmax/Gmm
539
Properties
Estimating
MACROSCOPIC APPEARANCE OF THE MESOSCALE CHARACTER Knowing single crystalline properties opens many doors. The properties of polycrystalline material can be estimated from the properties of single crystals by averaging processes. This is interesting from the economic aspects: At least 10 to 20 kg are needed to characterize a new energetic material. This can be extremely expensive. If there is the chance to make a single crystal from a few grams, many properties of the final product can be estimated. One even can estimate the multitude of polymorphs, and the workability of the material. Material pairings would indicate whether special problems are to be expected. Another question is the finding of appropriate mocks. A material’s anisotropy is of importance in possible deviations of the assumed homogeneous macroscopic material. This can be macroscopic in textures, or in the microscale when the geometric dimensions of the material compare with the shock-rise length. In this latter case also aspects of initiation stand out. A link between chemical and physical characteristics is given by the structure and anisotropy of mechanical and thermal properties.
ESTIMATES OF POLYCRYSTALLINE PROPERTIES Sound Velocities From the averaged data for single crystals the corresponding polycrystalline sound velocities are obtained as well as estimates of the Hugoniots. The sound velocities from the Neerfeld/Hill procedure for PETN, RDX and AP are compared with experimental values [XXIII-27], [XXIII-28] in Table XXIII-18.
Polycrystalline Hugoniot Estimates There are linear shock- (us) and particle velocity (up) (in kmis or mm/ps) relationships of the form:
540
Chapter
Table XXIII-18. Comparison of the polycrystalline Sound Velocities of PETN, RDX, and
density wcm31
PETN Average pol ycrystal exp.
RDX Average
CI
[ds]
cc [ d s ]
co [ d s ]
E [GPa]
G [GPa]
K [GPa]
V
1.774
2,955
1,573
2,331
11.43
4.39
9.64
0.3024
1.77 1.75 1.72
2,980 2,820
1,640 1,480
2,320 2,300 2,240
12 9.9
4.7 3.8
9.3 8.6
0.28 0.31
1.8
3,257
1,810
2,497
15.06
5.90
11.22
0.28
18.02 16.56
6.83 6.24
16.64 16.08
0.32 0.33
2.650
polycrystal exp. 1.8 AP Average [XXIII-25] [XXIII-26]
1.95 1.949
3,634 3,538
polycrystal exp.
1.95 1.90
2,180
1,871 1,788
2,921 2,872 2,840
Table XXIII-19. Hugoniot Estimates and Experiments
Material density wcm31 PETN
1.774
co
Estimated Hugoniot
[MSI
2.331
Experimental or ‘valid’ Hugoniot Ref. [XXTu-27], [XXIII-28], [XXIII-29]
u,, -2.33+1.7up = 2.320
1.774
+ 2.61 u p -0.38
u 2p
+
= 2.32
1.774 1.77
RDX O-HMX
=2.811+1.73uP + u , ~24.195 = 2.42
1.80 1.799
2.497
1.903
2.562
u,
2.50 + 1.7 u p us = 2.78
u,
236 + 1,7 u p = 2.75
1.891 1.95
+ 1.91 u p + 1.9 u p + 2.30 u p 4 u p 5 0.58
= 2.901
2.921
u,,
22.92 + 1.7 u p
+ 2.058 u p
54.1
541
-
55 4 5
, 1
1
/
4 0
....
35
P
30
3
p = I ,774
............ p = 1.77
25
p
20 1 5 1 0 0
us
Figure XXIII-12. Comparison of the Hugoniot estimate from single-crystal data with experiments on polycrystalline PETN at various densities.
.
= 1.72
I
02
04
06
lip
[km/s] or [ m m l ~ s ]
08
=A+s up
i n
(XXIII-60)
where often A is assumed to correspond with the bulk sound velocity co. For such estimates one can use an averaged value s 1.7, see Table XXIII-19. Actually the experimentally determined slope varies for explosives from 0.66 to 4! Figure XXIII-12 indicates that such crude estimates are good enough for the case of PETN.
Ductility and Pressability of Explosive Powders Ductile materials are preferred for manufacturing pressed materials. It would be helphl, therefore, to get indications when a material is ductile. For example, tungsten (W) and diamond are very hard, lead (Pb), gold (Au) or silver (Ag) are ductile. Figure XXIII-13 shows, that W has nearly isotropic Young’s and shear moduli. Contrary to this the shear modulus is largest for Pb where Young’s modulus is smallest, and vice versa. This situation enables internal flaws to be formed in a matrix. Pressability of powders depends on many factors. Of importance are ductility and the elastic stress conditions. Assuming a (triaxial) compression of the powders, the final pressure release occurs uniaxially. This is also true for a critical crystallite configuration in the polycrystalline matrix where maximum and minimum Young’s moduli are opposite. Their stress difference must be balanced by the coherence strength of the polycrystalline matrix. It becomes obvious that the pressability of powders (without additives for stabilization) decreases with increasing anisotropy of the Young’s modulus and decreasing ductility. It is noteworthy that D-HMX seems to be pressable in a conventional way. Figure XXIII-14 shows some old experimental results [XXIII-301. A coherent matrix of the polycrystalline aggregate is often not realized in a phase transition during compression.
542
330
40
8
_r
-
1L
Lead
~
-
-
20:
c
"
40 50 -
Figure XXIll-13. Comparison of the E and G moduli of cubic lead (top),
\'
.-
minimum
' I
'"1
Figure XXIII-14. Pressability of High Explosive Powders [MII-301.
Thermoshock Resistance of the Matrix Leiber [XXIII-3 11 calculated for the case of high alumina the thermoshock resistance. The results of calculation fkom thermoelastic constants and thermal expansions were close to experimental values. The approach was: The polycrystalline matrix was a critical configuration of single crystallites where the maximum product of the thermal expansion and Young's modulus EX^)^^^ is very close to minimum (Exa)fin. The resulting thermal stress A(stheml = T EX^)^^ - EX^)&^) AT must be balanced by the
543
sample's tensile strength. If the thermal stresses are larger than this tensile strength, crack formation occurs, and the sample weakens or is destroyed. A volume increase occurs in the intermediate range.
Hugoniot Elastic Limit (HEL) Quite a similar procedure was used to estimate the Hugoniot elastic imit (HEL). EL is the pressure that separates the elastic state and the onset of the plastic wave. It was found that the Young's modulus anisotropy governs this limit for brittle and ductile materials [XXIII-32]. As a result ,
I
Dimensional Instabilities of a Matrix Dimensional instabilities occur if there are samples of different structures in different directions, and the changes in temperature are relatively moderate. For example, in Figure XXIII- 15 different permanent dimensional changes resulting fiom thermocycles (1 Wmin) between 233 and 338 K are shown for a cast Octol85/15 sample in the axial and radial directions [XXIII-331.
0.8
5m
3
I'
~
I .
0.4-
......-. .'
Structural instabilities of Octol in different axial- and radial-sample
544 Chapter
~
330
30c
02
on
2 c
ni n2 03 24.
S-HMX
Aluminum
n5
on
ni
no
2
:C
0
h
AP
Polystyrene as assumed binder
Figure XXIII-16. In-scale comparison of the Young’s moduli of potential ingredients of a mixture. The scales are given in Mbar (0.8 Mbar = 80 GPa).
Structural Compatibilities Rocket propellant composites or explosives are never pure substances, but are composites of ingredients chosen for efficiency or technical manufacturing reasons. Often such energetic materials contain aluminum with high moduli in combination with medium-moduli oxidizers and low-moduli explosive components and binders. In Figure XXIII-16 are shown the Young’s moduli of aluminum, ammonium perchlorate, HMX and polystyrene (as assumed binder). It is obvious that environmentally induced stresses can damage such a material severely [XXIII-341. This discussion looks to be of academic interest because only a few materials are involved but there are general applications, as example there are few chemically based
545 Properties
differences between sodium and potassium nitrate, but great ones in mechanical behavior. It is possible, therefore, to optimize mixtures not only from the chemical aspect but also from the mechanical one in some cases, see Table XXIII-20. Table XXTTI-20. Comparison of sodium and potassium nitrate
NaNO3 KNO3
2.257 2.109
28 16.4
2.07 3.60
31 19.2
2.88
3,626
12.5
11.89
1.76
2,790
19.6
7.60
Aging and Detonic Sensitivity Properties Aging is not simply a matter of chemical but also of structural stability. The structural stability can be related with chemical stability in the case of a fracture-induced chemical alteration of the material. In transportation and storage, smaller or larger variations of stresses or temperatures occur usually with resultant internal stress variations in the samples. Even if these stresses are elastic, with time inelastic irreversible debonding or dewetting can occur. Consequently, the density of the matrix decreases with time, the matrix becomes porous, and in the extreme case, the structural stability is lost. Therefore the detonic properties are coupled with the porosity, shock sensitivity, critical diameters, detonation velocity and pressure, and even the Gurney constant. There are cases where moderate temperature cycles increase the porosity by as much as lo%, see Figure XXIII-15! The proneness for sensitivity alterations depends therefore on the anisotropic properties of the material and the environmentally induced stresses. In the case of temperature variations, an approximation AO = , can be used for estimates of thermally maximal induced internal stresses. For example, for an aluminized, bonded composite mixture (of different materials) with the values for aluminium K-' and polystyrene with E = 3.46 MPa and E = 75.6 MPa and = 2.3 K'' one obtains about 1.5 MPa K = 1.5 N/mm2 K for the internal stress =7 [XXIII-341. It is obvious that thermal shocks are severe impacts.
546 Chapter
Catastrophes for the Polycrystalline Matrix The common physical basis of the above-described phenomena is that in dense polycrystalline materials the single crystalline components are held together by their coherence strength, which defines the strength of the material. This strength compensates for the internal (micro)strains, which can be reduced or even eliminated in metals by annealing processes, which in turn enable shear and flaws. If the internal strength is surpassed by the internal stresses, internal crack formation occurs by debonding or dewetting. The result: Pore formations combined with a decrease in density. More drastic problems appear when the axial thermal expansion and/or axial compressibility become negative (anomalous behavior), but the volume property always remains normal. Examples are tellurium (mechanical and thermal), B-HMX (thermal), y-HMX (thermal), and thallium azide at 113°C (thermal), for which normal behavior is observed at room temperatures. Table XXIII-21. Comparison of HE-data and possible mock materials
density S co [gicm3] [10-20Nml [ d s ]
K [GPa]
E [GPa]
G [GPa]
ScmaJ
Scmin
Emin
GmaJ Gmi,
PETN
1.772
266.05
2331
9.64
30.2
11.4
1.64
1.89
1.67
RDX
1.80
224.51
2497
11.2
27.64
15.1
1.86
1.78
2.17
8-HMX
1.903
293.4
2562
12.5
9.0
3.4
1.18
17.93 19.88
1.949
145.8
2872
16.1
16.6
6.24
1.80
2.40
3.18
KCI
1.984
111.53
3014
18,O
24.2
9.5
1.00
2.27
2.67
Melamine
1.573
229.19
3061
14.7
20.0
7.8
12.10
5.48
5.35
Rhamnose 1.457
324.69
3356
16.4
16.3
6.1
-8.0
2.14
2.69
Sugar
554.44
3123
15.5
21,2
8.3
3.11
2.18
1.69
1.588
EmaJ
On Mock Explosives Mocks are used as non-explosive model substances. Depending on the specified properties, there are different mocks in use, i. e. those for thermal or mechanical (strength) properties. But, the mechanical-strength mocks cannot model structural incompatibilities or wave properties. Therefore it is appropriate to find inert materials with single-crystal elastic properties similar to the explosive components. Since these properties are unique fingerprints of the substances, one can do this only with knowledge of all possible crystal sets very approximately. In Table XXIII-21 are
541
compared the densities, sound velocities co, and mechanical elastic properties (Neerfeld averages) and the maximum anisotropic ratios of explosives and current mock suggestions KC1 and sugar and other possible substances. This Table shows that there is really no optimum mock. Especially the bulk moduli and corresponding bulk sound velocities do not match. Based on dipole scattering, the density ratio is best for KC1, but this shows a high ductility. Melamine can be a mock for the anisotropy of Young’s modulus of l3-HMX. For the study of the effects of anomal compressibilities Rhamnose (a sugar C6HI2O5* H20) is a good material. Sugare (Saccharose C12H22011) is much less compliant than the usual explosives, and anisotropies are (beside l3-HMX) stronger. In the course of investigating the static and dynamic behavior of a composite explosive KS-32 (HTPB/HMX 15/85), Corley [XXIII-36]used a mock material (HTPBISugar 17.42B2.58) and developed a nonlinear viscoelastic dynamic material EOS. If the conclusions drawn fiom Table XXIII-2 1 hold, then the dynamic properties of the real KS-32 should be “better” than predicted by the mock for compression but “worser” for Young’s modulus.
Textures Whereas polycrystalline isotropic behavior results from collections of crystals assembled as a statistical average without preferential crystal orientations, there exist cases where there is a directional preference in the crystal assemblage. These can be created intentionally (textured transformer laminations) or unintentionally (deep drawing of metals, the so-called earrings) by technical manufacturing processes. Up to now only one effect in the field of high explosives was noted: TATB shows a structure similar to graphite, and preferential orientations are obtained by extrusion processes. The detonation velocities differ somewhat, depending on the direction in the macroscopic piece. In a field, where an increase of the detonation velocity by some % is considered as a success, it seems appropriate to give such phenomena more attention.
The elastic data of sugar (monoclinic) are not available in current tables, and were measured by Haussuhl [XXIII-35]. The values (in 10’’ N/m2)are: c11= 2.071; c22 = 3.016; c33 = 3.053; c44 - 0.817; C55 = 0.816; c66 = 1.027; C12 = 1.132; c13 = 0.868; cl5 = - 0.392; c23 = 1.143; c25 = 0.139; c35 = -0.178; c46 = 0.030, at density p = 1.588 g/cm3.
548 Chapter
CONCLUSIONS FROM SINGLE CRYSTAL DATA Polymorphs It is an indication of structural instability when any moduli vary greatly with direction or temperature, and an easy phase transition is anticipated. This means that the possible existence of polymorphs can be expected. The above Figures XXIII-7, and XXIII-8 for IJ-HMX support this conclusion, where HMX exhibits the phases listed in the Table XXIII-22. In Table XXIII-23 the polymorphs of AN are listed. Table XXIII-22. Polymorphs of HMX [XXI11-24]
Phase
stability [K]
U-HMX P-HMX
376-435 RT-376
Y-HMX 6-HMX
435 to melt.
Phase transitions
+ 0) = 116 "C ( u + P ) = 116°C -+y) = 154 "C (P + y) = 154 "C + 6) = 193 - 201 "C (y+6)=167-183"C (P + 6) = 167 - 182 "C
(a
Since this is a general rule, we can also conclude that a material with many polymorphs will exhibit large anisotropies in elasticity or possess large temperature derivatives of the elastic constants. In addition the thermal expansions are also related to elastic anisotropies, further greatly varying thermal expansions in different axes and volume changes can also be expected. Such substances are lead azide, ammonium nitrate, and CL 20, for example.
Volume Variations by Phase Transitions First-order phase transitions are usually accompanied by volume changes, see Figure IX-5. Conversely, volume changes are often attributed to phase transitions. Olinger, Roof and Cady [XXIII-391 compressed RDX, see Figure XXIII-17, and detected a high-pressure phase transition to RDX-I11 not observed in normal cases.
549 from
Table XXIII-23. Polymorphs of AN with the axes a, b, and c [XXIII-37], [XXIII-38]
("C) 2OoC):338 b (T
