302 Propellants, Explosives, Pyrotechnics 26, 302 310 (2001)
Physical Model of Explosion Phenomena  Physical Substantiation
of Kamlet s Complaint
Carl-Otto Leiber*
Fraunhofer-Institut für Chemische Technologie (ICT), D 76327 Pfinztal (Germany)
Dedicated to Dr. Axel Homburg on the Occasion of his 65th Birthday
Summary As usual in the science of detonics no cry of indignation
had been heard nor a revolt for a better understanding
It was Dr. Kamlet, 1986, who expressed vividly the deficiencies of
occurred. Whenever such a serious statement applies, the
detonation physics. Since detonation physics is based on first princi-
whole theory and understanding must be intrinsically wrong.
ples his statements at that time were a matter of utmost courage! In his
Therefore, it is appropriate to apply to the current theories an
memory it seems appropriate to try a proof of falsification of this
classical theory in the sense of Karl Popper, and to suggest a new approach of falsification proof in the sense of Karl Popper(2),
approach.
whenever possible. This is done not for the purpose of
Detonics is the science of pressure fronts, and pressure sources.
destruction, but to offer the suggestion of a new basic
Generally, phenomena near a pressure source cannot be described in
approach for a better and more generalized understanding
plane wave terms as the classical theory does. With increasing distance
from the pressure origin plane wave descriptions improve. These of explosion phenomena.
become physically fully correct at distances from the pressure front
where detonics is no longer of interest.
Without a knowledge of the differences between spherical and plane
2. Classical Plane Wave Model of Shock
wave descriptions one cannot estimate the associated errors, nor is
and Detonation
there any real understanding of important phenomena. Therefore, in
any plane wave description of detonics there appear  pathological
phenomena, and it is necessary to invoke several assumptions in order
Starting with the most basic conservation principles one
to arrive at a description that is in accord with real experience.
obtains the following equations in plane wave terms, ignor-
The intent of this contribution is, to point out the basic errors of
plane wave detonics, and the automatic implications of the said ing, as usual, the pressure and particle velocity of the normal
pathological events by using a physically appropriate approach in
state:
terms of a spherical wave description. A consequence of this approach
is the conclusion that explosive phenomena are not restricted to r0us ź rðus upÞð1Þ
energetic materials, but can occur in any dynamically excited two-
phase system, even without any chemical reaction. The considerations
p ź r0usup; ð2Þ
are valid for condensed systems, while for gases other thermoacoustic
and resonance phenomena come into play.
where
r is the density at the pressure p
v ź 1=r is the specific volume at pressure p
1. Kamlet s Complaint
p is the pressure of shock or detonation, if, as
usual, the ambient pressure p0 and velocity
At the end of his professional career Mortimer J. Kamlet(1),
up,0 are ignored.
1986, a worldwide esteemed researcher, courageously sum- us ź D the shock- and detonation velocity, and
marized the state of the art of our science as follows:
up ź W the particle velocities
  The lore of detonation chemistry and physics is a composite of
The quantities indexed with 0 are those of the original state.
sound experimental evidence, anecdotes, apocryphal tales and large
The shock pressure p leads to materials compression,
amounts of misinformation. The misinformation is often not recognized
which is in several terms
as such, arising sometimes from misinterpretation of experimental
results or incorrect measurements and sometimes from deliberate
up up
r v0 us Dr Dv
inaccuracies. In the latter case, it has been convenient to accept and
ź ź resp: ź resp: ź
use certain assumptions or interpretations because they work. Often they r0 v us up r0 us up v0 us
work not because they are intrinsically correct, but rather because they
ð3Þ
contain convenient compensating errors. 
In detonation there is the Chapman-Jouguet (CJ)-condition
for stability
* e-mail: [e-mail]; Formerly member of WIWEB,
D ź up;CJ þ cCJ ð4Þ
Swisttal-Heimerzheim (Germany).
# WILEY-VCH Verlag GmbH, D-69469 Weinheim, 2001 0721-3115/01/0612 0302 $17.50þ:50=0
Propellants, Explosives, Pyrotechnics 26, 302 310 (2001) Physical Model of Explosion Phenomena  Kamlet s Complaint 303
where cCJ is the sound velocity in the CJ-compressed state, These quantities are either determined directly in the material
and is linked therefore, with the Equation of State (EOS) and under consideration, or are derived (based on plane wave
acoustics. assumptions) from the response of a reference material with
From these equations one gets as the impedance of plane exactly known shock properties (impedance methods) that is
waves (Rayleigh line): directly attached to the material under study. Table 1
summarizes the most common methods, see Ref. 3.
pCJ
Z ź ź r0D źðrcÞCJ ð5Þ
The accuracy of the experimental determination of the
up;CJ
velocities is better than 1 2 %. Regardless of the method
This is the basis of the piston model of detonation, see
used, one should be able to determine the detonation pressure
Figure 1, where a planar velocity piston up in motion drives a
of a well-defined explosive with an accuracy of at least 4 % or
planar pressure wave.
better. However, historically various methods have in some
cases yielded very different results. The common   explana-
tions  of these differences stressed experimenters with
2.1 On Non-Ideal Explosives
different levels of experience, different data reduction tech-
niques, different reference materials, explosives from differ-
Since in detonics what seem to be the most basic con-
ent lots manufactured by different means, and so on. The
servation principles are applied, it is commonly believed,
variability due to measurement method was illustrated by
therefore, that this theory should be intrinsically correct. The
Davis and Venable(4), 1970, who compared different
certainty about this is so firm, that any experimentally
methods for the determination of the detonation pressure of
observed deviations from the theoretical expectations are
Comp. B-3 of identical composition with identical reference
attributed to non-ideal explosive behavior or to faults in the
materials by the same experimenters, and found detonation
experiments.
pressures between 26.8 0.6 and 31.2 0.5 GPa. Hollen-
berg, Kleinhanß and Reiling(5), 1981, determined the detona-
tion pressure by X-ray flash techniques of the similar
2.2 On the Experimental Evaluation of the Hugoniots
explosive in Germany to be 19.5 GPa.
of Inert Materials and the Pressure of Detonation
These results give reason to believe that the applied physics
of detonation pressure measurements is not   beyond any
For planar shocks and the piston model of detonation only
doubt  .
two quantities must be determined in order to calculate the
third quantity, provided the original density ro is known.
2.3 On Codes
Very many codes have been developed in the past, as well
as a multitude of appropriate Equations Of State (EOS). The
nature of those EOSs is such that a physical or numerical
construct is taken as the starting point, and any adjustable
parameters are determined from experiments, so that the
results fit. In this way, for example, the most accurate code-
predictions for TNT can be expected by using a TNT-based
EOS. But it was Mader(6) who demonstrated that by using
RDX-data the liquid TNT is described better.
The inadequacy of even modern codes is best demon-
Figure 1. Classical piston model of steady state detonation. The
strated by considering industrial explosives, though some
piston (membrane) of diameter 2a is activated by the CJ-particle
would argue that these are not appropriate examples. In the
velocity up,CJ in order to generate a plane detonation wave which
moves with a detonation velocity D. following discussion, an explosive developed for safe use in
Table 1. Review of the Evaluation of Dynamic Quantities
Experiment Determined parameter Determined parameter Observation by
in the sample in the reference material
Impedance match us or Dus Pins, flash gap, Camera, VISAR
us and ufs
Direct methods:
Shock- and particle velocity D, W Pins, camera, and electromagnetic
us ź D; uplate method (non-conducting
materials only)
Shock velocity and compression D, Dv=v0 Pin, probe, X-ray flash
Shock velocity and pressure us ź D, p Pressure probes
304 Carl-Otto Leiber Propellants, Explosives, Pyrotechnics 26, 302 310 (2001)
mines is calculated with CHEETAH 1.4(7) (the 2.0-Version(8) In acoustics both plane and spherical waves appear. It is
is improved by the introduction of a variable reaction evident that near a pressure source, which can even be a point
kinetics, which provides  adjusting knobs for better para- source, any descriptions in plane wave terms must fail. But
meter fittings of the dominant components). with increasing distance r from the pressure source of radius
Consider a stochiometric mixture of ammonium chloride R, with the limiting case being r )1 , the plane wave
and potassium- or sodium nitrate, and 10 % NG-mixture. description becomes more and more appropriate.
The effects of Kuhn and Käufer(9), 1954, are also consi- The errors of a plane wave description in the vicinity of a
dered. They found, that the addition of very small source even for the simplest case of harmonic motions is
amounts, much below 1 weight %, of substances not neces- shown in Table 3, where the following quantities are used:
sarily considered as either explosives or as inerts, such as
F ź velocity potential
tetryl, dinitrodiphenylamine, and even silicon oil, signifi-
p ź pressure
cantly changes the explosive behavior of the bulk material
x, resp. r distance of the point of observation from the
if the added substances are on the surface of the grains.
source
Their efficiency completely ceases if they are mixed
R ź radius of the spherical source
uniformly into the bulk explosive. The addition of 0.03
r ź density of the medium
weight % tetryl on the surface of the sodium nitrate
c ź sound velocity in this medium
increased the lead block value by 20 to 30 %, and
o ź 2pf (angular frequency)
increased the detonation velocity from 1700 to more than
k ź 2p=l ź o=c (wave number)
2000 m=s. For more details see Ref. 10.
l ź wave length
In Table 2 the experimental values are compared with
CHEETAH 1.4 predictions. These code estimates show that As Table 3 demonstrates, the following statements are true
there are uncovered problems, and further discrepancies are only in the case of spherical waves:
given below.
Quantities strongly dependent on distances appear, and
Therefore, we must carefully examine which assumptions
may rapidly disappear with increasing distance depend-
and which views may be wrong in our detonation insight.
ing on run times. Nevertheless at small distances these
Due to the basic nature of detonation theories this was a
are large, and in the immediate vicinity to the source,
formidable task.
these can be of the order of several times of the corres-
ponding plane wave term. Therefore, these quantities
3. Examination of the Plane Wave Concept are attributed to the near field (NF) in contrast to the far-
field (FF).
All present-day measurements of pressures and Hugoniots Further geometrical quantities appear that are linked with
assume plane waves, where, of necessarity, the energies in the wave number k ź 2p=l ź o=c, or the wavelength l.
compression and flow are equilibrated. In light of the local Since this wavelength characterizes the   dynamics  of
nature of reactions at hot spots, this is surely not adequate. the system under consideraton, there is a geome-
More appropriate in this case would be a description in terms trical=dynamical linkage. The consequence, therefore, is,
of spherical waves, which, unlike plane waves, exhibit near that any relative distance is altered both by the geome-
field terms. According to the above-mentioned CJ-condition trical distance and the   dynamics  . This is also the
an acoustical state in the CJ-compressed condition is reason why the domain of the near field cannot be
assumed, so that we may use acoustical considerations for described only by a distance.
this case.
As can be seen, the streaming velocity of spherical waves
exhibits two terms. The first one corresponds to the pressure,
3.1 Acoustic Description of Pressure Waves which decreases with distances, and defines the energy
density of compression. The second decreases rapidly with
Acoustics is the science of the propagation and attenuation 1=r, and is called the near field term (see above).
of small amplitude pressure waves. Any description of For both plane waves and spherical waves, the description
pressure waves must exhibit in the lowest order acoustical of potential and kinetic energy is equilibrated in the far field.
terms. But in the near field an additional kinetic energy of flow
Table 2. CHEETAH 1.4-Results of Class III Permitted Explosive K 735 With and Without Tetryl, and Experiments
r D pCJ p (GPa) Lead block Mech. energy of
(g=cm3) (m=s) (GPa) Explosion in (cm3) detonation (kJ=cm3)
const. volume
Experiment without tetryl 1.2 1700 70
CHEETAH 1.4 without tetryl 1.2 3912 3.42 0.99 1.702
Experiment with tetryl 1.2 2000 100
CHEETAH 1.4 with tetryl 1.2 3916 3.43 0.99 1.706
Propellants, Explosives, Pyrotechnics 26, 302 310 (2001) Physical Model of Explosion Phenomena  Kamlet s Complaint 305
Table 3. Comparison of Plane and Spherical Harmonic Waves
Quantity Plane wave Spherical wave
@2F @2F @2ðrFÞ @2ðrFÞ
Wave equation ź c2 ź c2
@t2 @x2 @t2 @r2
f1ðct rÞ
General solutions (diverging waves, only): Fðx; tÞ Åºf1ðct xÞ Fðr; tÞ Åº
r
Harmonic solution F(x, t) ź A exp[i(kx-ot)] F(r, t) ź B exp[i(kr 7 ot)]=r
Pressure p ź iroF ź irkcF
Particle velocity up,r ź ikF ikF F=r
Farfield Term (FF) ikF
Nearfield Term (NF) 0 F=r
1
Densities of the potential energy epot. ź rk2F2
2
1
kinetic energy ekin. (FF) ź rk2F2
2
1 F2
kinetic energy ekin. (NF) ź 0 r
2 r2
epot:=ekin:ðFFÞ Åº 1
epot:=ekin:ðNFÞ Åº 1 k2r2
k2r2 ReZ
epot:=ekin:ðtotalÞ Åº 1 ź
1 þ k2r2 rc
k2r2 kr
Impedance Z ź rc rc þ i
1 þ k2r2 1 þ k2r2
u2 k2R2 u2
Power of radiation per unit area N ź rc rc
2 1 þ k2R2 2
results, which always remains in the system. This energy is a streaming energy around the sphere. In other words, most of
resistance of the system against the expansion of the source, the energy is in the near field.
which again is added to the source if this is shrinking. For waves of arbitrary shape, one obtains with the usual
Comparing now the potential energy with the total kinetic procedures the pressure p, the particle velocity up, and the
energy, this ratio varies for small sources or distances as associated impedance Z, as shown in Table 4.
1=k2R2 (resp. 1=k2r2). In the case of small sources kR 1; This table shows, that the shapes of pressure and particle
a multiple of the radiation energy is concentrated in the velocity can be greatly different and can vary in the vicinity
Table 4. Arbitrary Diverging Plane and Spherical Waves
Plane diverging wave Spherical diverging wave
f1ðct rÞ
Fðx; tÞ Åºf1ðct xÞ Fðr; tÞ Åº
r
0
@F @f ðuÞ @u @F r @f ðuÞ @u f
0
p ź r ź r ź rcf p ź r ź ź rc
@t @u @t @t r @u @t r
with u ź ct 7 r one gets
@f ðuÞ 1 @f
0
f ź ź and
@u c @t
Z Z
cpr
0
f ź cf dt ź dt
rc
@F p @F
0
up ź grad F ź ź f ź ur ź grad F ź
@x rc @r
1 1
0
ź f þ f
r r2
Z
p 1
ź þ pdt
rc rr
p p rc
Zarbitrary plane wave ź ź rcZarbitrary spherical wave ź ź
0
up ur 1 þ f =rf
Zarbitrary spherical wave 0
rf r!1
Z ź ź ! 1
0
Zarbitrary plane wave rf þ f
306 Carl-Otto Leiber Propellants, Explosives, Pyrotechnics 26, 302 310 (2001)
of a pressure source due to the pressure=time integral. In the Also at that time, and even after the measurements, the
far field a similarity exists, however. This is described best by controversies lasted over years, and were finally forgotten.)
the relative impedances. Due to the properties of harmonic
waves, their shapes do not vary.
4. Applications in Detonics
Detonation physics is the science of the detonation front
and the origin of pressure. If pressure sources are described in
3.2 On the Physical Applicability of the Piston Model
plane wave terms, serious shortcomings result, which vanish
of Detonation
with increasing distances from them. Therefore, in plane
wave terms detonation science fully and correctly applies at
Due to the constant plane wave impedance of the piston
distances that are fully irrelevant for detonics (far from
model of detonation, one expects that the plane velocity
the front).
piston in motion also radiates plane pressure waves.
The piston model of detonation suggests that a piston
driven membrane radiates plane waves, but Figure 2 evi-
dences a completely different behavior. For a harmonic
4.1 Von-Neumann-Spike, a Theoretical Artifact?
driven plane piston of diameter 2a in motion, the impedances
and pressure distribution as function of ka are shown
In spite of the fact, that experimental work on the von-
according to Stenzel=Brosze(11). For large ka-values, one
Neumann-spike (in Russian, chemical spike) is poor and
gets a real impedance and a serious structured pressure
conflicting, there is the general belief that at the immediate
distribution across the membrane. This result confirms
detonation front there should be an overshoot of pressure of
Huygens principle: All elements of the piston contribute to
about 30 to 40 % above the CJ-pressure. But there are also
the total pressure. Therefore, in a non-axisymmetric system,
measurements without any indication of a von-Neumann
deviations must appear! For small ka-values, however, where
spike.
the real impedance disappears, one gets again plane winds
Since, up to now, all data have been used in combination
according to the plane velocity piston (but no pressure).
with plane wave impedances r0us, or r0 D, in the following
It is to be concluded that plane pistons can never radiate
discussion a guess is made about possible differences in the
plane pressure waves. This includes, that also the piston
vicinity of (harmonic) pressure sources. Therefore, a linear
model of detonation is not physically applicable. (In the
array of harmonic pressure sources is considered.
1980s the publication of these results was refused by six
From the velocity potential of a spherical harmonic source
esteemed journals due to apparent absurdity of the result. But
of the finite size kR:
the well-known acoustician Otto Brosze remembered the
_
V
V eikðr RÞ
very same controversies in the 1920s. At that time a 6 m plane
F ź e iot ð6Þ
4pr 1 ikR
membrane had been attached at the walls of the Reichs-
telegrafenamt in Berlin, and the sound field was measured, one calculates the pressure p, the particle velocity and the
with the result that really no plane pressure fields resulted. corresponding impedance. This source radiates isotropically
Figure 2. Left: Pressure on the plane piston membrane of diameter 2 ka as function of ka. x is the distance from the center of the membrane.
[From Stenzel=Brosze(11) (with kind permission)]. Right: Impedance s of the plane piston membrane of diameter 2 ka.
Propellants, Explosives, Pyrotechnics 26, 302 310 (2001) Physical Model of Explosion Phenomena  Kamlet s Complaint 307
in all directions, but detonic measurements observe only the is not to be excluded that the von-Neumann-spike can be a
components in the direction of the macroscopic wave theoretical artifact.
propagation. Therefore, the components of the radial particle
velocity in this direction are also used.
To demonstrate the difference in the results, in the follow-
4.2 Deflagration-to-Detonation Transition (DDT)
ing a spherical source of size kR ź 1 is used. For this source
one gets from Table 3 an impedance of rc=2, which is half of
The amount of energy liberated in burning does not differ
the value of plane waves. What changes in an array? Eight
too much from that liberated in detonation. This indicates,
identical sources (n ź 8) with synchronous pulsation are
that detonation is not so much a matter of energy, but of
placed equidistant from one an another in an array, as
power transformation. The energy liberated should be half in
shown in Figure 3. According to Huygens principle, the
compression and the other half in flow, if plane wave
macroscopic pressure and particle velocity profiles and the
considerations apply. The transition from burning, where
corresponding impedances are calculated as a function of the
practically the whole energy is liberated in flow, discontinu-
distance from the sources. All geometrical quantities, such as
ously to detonation, where half is in flow and half in
the size and spacing of the sources, the length of the array, and
compression cannot be explained using the piston model
the points of observations, are related to the wave length l.
without making any artificial assumptions.
From
Nevertheless, there are many DDT-models based only on
n n
X X
this view. Roughly speaking these all assume that, in the case
p ź ikrc Fn and up ź grad Fn cos Yn ð7Þ
~ ~ ~
n n n
of combustion, the piston is more or less open. Therefore, all
~ ~
n n
nź1 nź1
fumes can escape (to the rear end) without driving the piston.
one gets the impedance, or the relative impedance (normal- The transition is explained by varieties of successive choking
ized to plane wave expectation) mechanisms of this originally open piston. In this way one
gets a variable ratio of energy in flow and in compression, as
n n
P P
ikrc Fn ik Fn it should be.
~ ~
n n
p Z
~ ~
n n
nź1 nź1
Without any assumptions such a transition results from the
Z ź ź resp: ź
n n
P
up P grad Fn cosYn rc
grad Fn cosYn size of the sources and their arrangements, since the relative
~ ~ ~ ~
n n n n
~ ~
n n
nź1 nź1
impedance is also the ratio of the energy in compression and
ð8Þ
the total energy in flow. This relative impedance greatly
varies from 0 or small values up to 1, and in source
Yn is the angle for the component of the radial value in the
accumulations (see Fig. 3) even in excess of unity)(12).
uniaxial direction, so that up ź ur cos Y. The result is shown
in Figure 3. One notices in this Figure that the relative
impedance in the vicinity of the pressure sources is excessive,
but rapidly drops to the limiting value of 1. Furthermore, this 4.3 Critical Diameter
impedance profile is a measure of the similarity of pressure
and particle velocity profiles. Note that plane wave expecta- Quasistatic pressure sources can radiate their pressure only
tions are identical shapes for pressure and particle velocities. isotropically. For paired sources with increased dynamics
Figure 3 shows, that even in simple cases it cannot be more and more unidirectionality results. A first lobe forma-
expected, that plane wave impedances are present. Further it tion of the directionality results at the critical dimensions.
Figure 3. Half model of a linear 8-source array. The pressure-, particle velocity- and the corresponding relative impedance-profiles as function of
h=l are shown.
308 Carl-Otto Leiber Propellants, Explosives, Pyrotechnics 26, 302 310 (2001)
These are no constants (as it could be believed), but depend leads to a bubble collapse, and if t is the time of shock rise, kR
on the geometrical situation of the sources, which can be in a is altered to
line, area, circle, or even on a circular line. It is interesting to
2pR 2R
kR ź : ð11Þ
note, that the circular line source distribution roughly shows
l tc
about half of the critical diameter compared with a filled up
If the ramp measures the diameter of the bubble, one gets
circular arrangement.
kR ź 1 with a relative impedance of 0.5, which can increase
Such directionality properties are completely out of the
in arrays to >1. This example demonstrates that with
scope of plane waves considerations, but are an immediate
triggering, even when the energetic situation is not altered
consequence of assemblies of spherical pressure sources, and
greatly, the ability of pressure wave radiation changes by 104.
their dynamics. Therefore, critical dimension phenomena
This is also part of DDT, and in the case of two phase
and DDTphenomena are intimately linked(13,14).
liquids, it is the way to get from boiling up to explosions.
Since spherical pressure sources can be (powerfully)
Whereas the triggerability of explosions is a well-recognized
driven by chemical reactions, but also by inertial forces
fact for physical explosions, these circumstances have not
without any chemical decomposition, explosion phenomena
been explored for DDTphenomena of energetic substances.
are not restricted to energetic materials. Any dynamically
These ideas show that it is not possible to stabilize transition
excited two-phase system can explode!
phenomena to such an extent that these can be of practical use.
5.2 Nature of Physical Explosions, [Fuel-Liquid
5. Physical Explosions
Interactions (FLI)]
Depending on its size, kR, a spherical source is a pressure
That there are no gradual transitions, but rather qualitative
wave radiator (Table 3). Therefore kR will be concretized a
jumps, is shown in some accident histories. In the Quebec
bit in the following.
Foundry-Explosion(16) 45 kg of molten steel at 1560 C was
In order to understand   Musical Air-Bubbles and the
dropped into 295 l of water. By calculation only 16 l of water
Sounds of Running Water  Minnaert(15) described properties
evaporated, but the explosion was estimated to have a TNT-
of resonance of bubbles in a liquid by a stiffness=mass model.
equivalency of 5.4 kg. Cratering and explosive devastation
Rewriting his results one gets:
sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi occurred in brick walls up to 53 m away, and more than 6000
c0 r0 K0
windows were broken. It is unlikely that this was the result of
kR ź 3 ź 3 ð9Þ
evaporated water driving a pressure piston, where the gas
c1 r1 K1
phase energy (pV)Gasphase caused cratering. This is even not
where c0 is the sound velocity of bubbles content, and c1 that
possible, if the water evaporates in its own volume, where at
of the medium. The same indices apply for the appropriate
1200 C a pressure of 732 MPa and at 1600 C a pressure of
densities r. With the modulus of compression K ź rc2 and
1220 MPa can resultþ, far from any cratering capability.
the circumstances, that in gases the pressure p is the modulus
If one assumes that the gas phase causes such demolitions
of compression, one gets for usual liquids (like water) at
as noted above, it follows that the severity should decrease
normal conditions:
with decreasing water content. On 19 February 1958 (Rey-
pffiffiffi
nolds Metal Co, Mc Cook, USA(17)), moist or possibly wet
kR ð0; 01 0; 02Þ p ðp in the dimension of barÞ:
scrap aluminum was inserted in a melting furnace. An
ð10Þ
explosion caused six victims, and fourty injured. The
One musical bubble shows the size of kR 0,01, and 10 4 damage amounted to about 1 million US$. In both this
of the total energy only can be radiated by pressure waves.
case and the Quebec Foundry case, quite apparently the
This is the reason why tea water boiling normally behaves
energy of evaporation appeared in the condensed phase as
benignly.
(pV)condensed phase. Due to its low compressibility large
This ability of pressure wave radiation can be changed by
pressures resulted. The explosion damage does not correlate
the ambient pressure p. Just with 10 MPa ambient pressure kR
with the mass of the water available for evaporation.
becomes 0.1, and 10 2 of the total energy is in pressure wave
radiation. Bubble arrangements lead to an increase of
pressure wave radiation that is not additive. Just another
6. Fundamentals of Detonics
tool for varying the ratio is of great importance:
All observable phenomena have a physical basis. With
knowledge of the physical basis of hazardous phenomena,
the safety expert can propose relevant system responses to
5.1 Triggering of Explosions
protect against situations that might arise. Without such
Minnaert s bubbles were freely vibrating systems. In the
case of forced vibrations, the system is governed by the
þ
forcing system. For example, an external triggering shock Kindly Dr. Volk, ICT, calculated these values.
Propellants, Explosives, Pyrotechnics 26, 302 310 (2001) Physical Model of Explosion Phenomena  Kamlet s Complaint 309
knowledge irrelevant testing may result, and testing degen- ciated streaming velocities of Ma < 1, so that only the mass
erates to a chase for the goose with possible big surprises. injection should be the decisive mechanism. In the case of
Historic definitions of explosions involve the evolution of High Velocity Detonation, however, the streaming velocities
gases, so the piston model of detonation, wherein an increase approach domains around Ma ź 1, so that in addition a
in volume must occur, was born. The aforementioned momentum injection should become important.
accidents are not convincing examples of such a model. An Indeed in this domain relative dynamic particle and void
alternative is Minnaert s idea, that a generation of pressure mobilities within a matrix appear. Such occurrences had been
waves results in the condensed phase. shown in experiments(20). If a dynamic activated void over-
takes the shock front in the medium (this corresponds to
0
Ma ź u0 =us >1; with up as the velocity of the void), then
p
a shock compressed void comes by its translative motion
6.1 Mechanisms of Pressure Wave Generation
ahead of the shock front into the fresh medium, where it
expands powerfully into this formerly not compressed
Lord Rayleigh(18) in the last century proposed the idea that
medium, and generates pressure waves again.
pressure waves result by the time-dependent injection of
Due to the large loss power of bubble expansion an onset of
_
mass into a unit volume, rV =V0: This idea is substantiated by
V
chemical decomposition results, and the onset of High
monopole pressure sources described by the spherical wave
Velocity Detonation (in liquids) occurs. Such a void mechan-
equation. In the case of a constant density medium, these
ism has been shown in experiments(21). This criterion of
sources are holes, the volume of which varies with time. In
initiation holds for many liquid explosives, including hydra-
the case of explosives, there is an additional driving force for
zoic acid, see Ref. 22. It is noteworthy that John von
volume variations in the chemical decomposition of the
Neumann remarked in his report on the ZND-theory of
explosive, where the product gases of the decomposition
detonation(23):   The detonation wave initiates the detonation
blow up the holes like balloons. But within this frame inertial
in the neighboring layer of the intact explosive by the
forces also lead to time-variations of volume. There are
discontinuity of material velocity which it produces. This
stiffness=mass-, condensation=evaporation- or sorption=
acts like a vehement mechanical blow ( 1.500 m s 1), and is
desorption-phenomena that can generate time-dependent
probably more effective than high temperature.  This is
volume variations. These latter mechanisms lead to physical
substantiated by a two-phase approach.
explosions without any chemical attributes. The safety-
It must be noted, that these considerations hold for
relevant message of this model is that explosions can be
condensed systems. In gaseous systems other additional
excluded from consideration only in media that always
thermoacoustic driving mechanisms and resonance pheno-
remain homogeneous under any circumstances. This means
mena become important.
that, at least in liquids, an explosion can never be excluded,
since there are in daily life always mechanisms at work that
can cause a liquid to foam up.
In the 20th century the interest on the conversion of the
6.2 Onset of Chemical Reaction and their Completion
energy of flow into energy of compression was renewed in
the context of aerodynamic noises. Lighthill(19) pioneered
The loss power of dynamic activated cracks and voids can
this endeavor, with the result that both momentum- and curl-
be gigantic, up to the order of GW=cm3. It is therefore likely
injections into the unit volume can also generate pressure
that the onset of chemical reaction is non-thermal by bond
sources. Momentum-injection generates a dipole source and
scissions. An indication of this is the observation that, in
is realized by a vibrating string. It is known, however, that
contrast to gases, for condensed explosives the thermal
vibrating strings without resonance amplifications (as in
detonation model does not describe well the detonation
violins) emit only low levels of intensity. The reason for
parameters. A reason for this shortcoming is that the
this is the Mach-number-dependent efficiency of the sources.
maximum expansion velocity of a void is obtained at a
The relative importance of these 3 mechanisms to pressure
radius increase of only about 30 % (Rayleigh bubble). There-
wave radiation is shown in Table 5. One sees that at low
fore the beginning of the expansion does not offer the nature
Mach numbers the effects of mass injection dominate, and
of its driving force. But with the evolution of the chemical
with increasing Mach numbers the momentum and curl-
reaction the classical temperature-dependent reaction
injections become more and more important.
mechanisms appear, which dictate the further progress of
The usual phenomena of explosion, deflagration, and Low
the consumption of the explosive into gaseous products.
Velocity Detonations exhibit Mach numbers of the asso-
Langer and Eisenreich(24) demonstrated that such a thermal
reaction wave resulting from single hot spots develops in a
way very similar to that described above for pressure arrays.
Table 5. Relative Importance of Pressure Generating Mechanisms
The difference, of course, is that the propagation velocities of
Injection of mass Monopole source N / Ma1 u3 !/ u4 pressure and temperature waves are very different. Probably
Injection of momentum Dipole source N / Ma3 u3 !/ u6
these differences can describe the reaction luminosity at the
Injection of curls Quadrupole source N / Ma5 u3 !/ u8
front of detonating systems. This thermal approach explains
(Lighthills u8-  law  )
further the thermodynamic quality of the gases.
310 Carl-Otto Leiber Propellants, Explosives, Pyrotechnics 26, 302 310 (2001)
Int. Pyrotechnics Seminar, Jönköping, Sweden, 24 28 June 1991,
7. Concluding View
pp. 55 56.
(13) C. O. Leiber,   Detonation Model with Spherical Sources K:
Physical Explosions, Low- and Slow Velocity Detona-
Critical Dimension Phenomena Asymptotic Considerations  ,
tions, Double Explosions, Cell Structures of Detonation, 19th Int. Pyrotechnics Seminar, Christchurch, New Zealand,
20 25 February 1994, pp. 40 51.
irregular Detonation Fronts, Dark Waves, DDT, Critical
(14) C. O. Leiber,   Detonation Model with Spherical Sources L:
Diameter Phenomena, Different Pressure and Reaction
Critical Dimension Phenomena near the Sources in a linear
Fronts are irregular phenomena that cannot be explained
Array  , 20th Int. Pyrotechnics Seminar, Colorado Springs, CO,
USA, 25 29 July 1994, pp. 629 642.
within the classical description of detonation. All these
(15) M. Minnaert,   On Musical Air-Bubbles and the Sounds of
phenomena in condensed systems are natural conse-
Running Water  , Phil. Mag. Ser. 7, 16, 235 248 (1933).
quences without any additional assumptions of an assem-
(16) S. G. Lipsett,   Explosions from Molten Materials and Water  ,
bly of monopole and dipole pressure sources.
Fire Technology 2, 118 126 (1966).
(17) L. F. Epstein,   Metal Water Reactions: VII Reactor Safety
Aspects of Metal Water Reactions  , Report GEAP-3335, (1960),
Vallecitos Atomic Lab., GE Co, Pleasanton, CA, USA.
(18) J. W. S. Rayleigh,   The Theory of Sound, Vol. II  , Dover Pub.,
8. References
New York, reprint 1945.
(19) M. J. Lighthill,   The Bakerian Lecture 1961: Sound Generated
(1) M. J. Kamlet,   Lectures on Detonation Chemistry  , Report No.
Aerodynamically  , Proc. Roy. Soc., London A 267, 147 182
1, 9=1986 Naval Surface Weapons Center, Dahlgren, VA, Silver
(1962).
Spring ML, USA.
(20) C. O. Leiber, G. Flajs, R. Hessenmüller, and R. Wild,   Dyna-
(2) K. R. Popper,   Logik der Forschung  , 6th ed., J. C. B. Mohr
misches Verhalten von Diskontinuitäten in einer Festkörper-
(Paul Siebeck) Tübingen 1976.
Matrix  , Zeitschrift für Metallkunde 65(8), 539 541 (1974).
(3) P. Marsh,   LASL Shock Hugoniot Data  , University of Cali-
(21) C. O. Leiber and R. Hessenmüller,   Wie eine Detonation
fornia Press, Berkeley, Los Angeles, London, 1980.
entsteht  , Umschau 84, 507 509 (1984).
(4) W. C. Davis and D. Venable,   Pressure Measurements for
(22) C. O. Leiber,   Detonation Model with Spherical Sources G:
Composition B-3  , 5th Symp. (Int.) on Detonation, August 18 21
Dynamic Void Mobilities, HVD Initiation of Liquid Explosives  ,
1970, Pasadena, CA, ACR-184, Office of Naval Research,
17th Int. Pyrotechnics Seminar, combined with the 2nd Beijing
Department of the Navy, Arlington, VA, pp. 13 21.
Int. Symposium on Pyrotechnics and Explosives, Beijing, China,
(5) G. Hollenberg, H.-R. Kleinhanß, and G. Reiling,   Messung des
October 28 31, 1991, Beijing Institute of Technology Press,
Chapman-Jouguet-Druckes mit Röntgenabsorption  , Zeitschrift
Vol. II, pp. 722 732.
für. Naturforschung 36a, 437 442 (1981).
(23) J. von Neumann,   Report on   Theory of Detonation Waves    ,
(6) Ch. L. Mader,   Numerical Modeling of Detonations  , University
Report OSRD No. 549, (1942). Division B National Defense
of California Press, Berkeley, Los Angeles, London 1979.
Research Committee of the Office of Scientific Research and
(7) L. E. Fried,   CHEETAH 1.39 User s Manual  , Report UCRL-
Development, Section B-1.
MA-117541 Rev. 3, (1996), Lawrence Livermore National
(24) G. Langer and N. Eisenreich,   Hot Spots in Energetic Materials  ,
Laboratory, Livermore, CA, USA.
Propellants, Explosives, Pyrotechnics 24, 113 118 (1999).
(8) L. E. Fried,   CHEETAH 2.0 User s Manual  , Report UCRL-
MA-117541 Rev. 5, (1998), Lawrence Livermore National
Introduction to Acoustics:
Laboratory, Livermore, CA, USA.
E. Skudrzyk,   The Foundations of Acoustics  , Springer, Wien, New
(9) G. Kuhn and H. Käufer,   Über die Beeinflussung des reak-
York, 1971.
tionskinetischen Geschehens bei Sprengstoffen durch Behand-
lung der Kristalloberflächen mit oberflächenaktiven Stoffen  ,
NOBEL-Hefte 20(4), 97 116. (1954). Introduction to Detonics:
(10) C. O. Leiber,   Detonation Model with Spherical Sources M: On P. W. Cooper,   Explosives Engineering  , VCH Pub., New York,
the So Called Non-Ideal Behavior of Explosives  , 23rd Int. Weinheim, Cambridge, 1997.
Pyrotechnics Seminar, Tsukuba, Japan, 30 September 4 October
1997, pp. 441 456.
Acknowledgements
(11) H. Stenzel,   Leitfaden zur Berechnung von Schallvorgängen  , 2.
This paper is devoted to the memory of Dr. Mortimer J. Kamlet (þ),
neubearb. Aufl. von O. Brosze., Springer, Berlin-Göttingen-
and Dr. H. Dean Mallory (þ). Great discussions I owe to Dr. Norbert
Heidelberg 1958.
Eisenreich, ICT, for the nature of the onset and completion of the
(12) C. O. Leiber,   Detonation Model with spherical Sources B:
chemical reactions. Kindly Dr. Ruth M. Doherty, NSWC, Indian Head,
Deflagration Detonation Transition Approach  , 14th Int. Pyro-
edited this paper.
technics Seminar, St. Helier, Jersey, CI, 18 22 September 1989,
pp. 551 562.   Corrections to: Detonation Model with spherical
Sources B: Deflagration Detonation Transition Approach  , 16th (Received March 8, 2001; Ms 2001=027)