Shock Waves (1999) 9: 141 147
Sensitization of two-dimensional detonations in nitromethane
by glass microballoons
E. Bouton1, B.A. Khasainov2, H.N. Presles1, P. Vidal1, B.S. Ermolaev2
1
Laboratoire de Combustion et de D�tonique, UPR 9028 CNRS, F-86960 Futuroscope, France
2
Institute of Chemical Physics, Russian Academy of Sciences, Oulitsa Kossiguina 4, 117977 Moscow, Russia, V334
Received 5 July 1997 / Accepted 13 July 1998
Abstract. Experimental results are reported on charge diameter effect and critical detonation diameter
of nitromethane (NM) gelled by 4 wt.% of PMMA and sensitized by thin-walled (1 �m) monosize glass
microballoons (GMB) of 47 �mor 102 �m size at a constant mass fraction of glass (1%) in the mixture.
The dependence of the detonation velocity on the detonation shock-front total curvature at the explosive
charge axis is presented for steel and PVC detonation tubes of various inner diameters. The predictive
ability of the quasi-one-dimensional hydrodynamic model of detonation-reaction zone in GMB-sensitized
NM is improved by a better description of the hot spot growth stage. Dependencies of detonation velocity
in NM sensitized by different GMBs on the charge diameter are calculated using available data on NM
regression rate at detonation pressures and a reasonable agreement with experimental data is obtained. The
effect of the confinement on the charge diameter effect as well as the dependence of the normal detonation
velocity on the total detonation shock front curvature at the charge axis is also predicted by the model.
However, there are still some difficulties in reproducing the experimental linear correlation between the
critical detonation diameter of GMB-sensitized nitromethane and its reciprocal specific surface area.
Key words: Detonation, Sensitization, Critical diameter, Nitromethane, Glass micro-balloons
1 Introduction mental correlations of critical detonation diameter (as well
as other shock sensitivity characteristics) with initial spe-
cific surface of grains or voids also hold for a wide variety
This work focuses on the properties of two-dimensional
of heterogeneous condensed explosives (Khasainov et al.
steady detonation in a model heterogeneous explosive
1997). Our present goal is: (i) to measure the dependence
made of nitromethane (NM) and thin-walled glass mi-
of the detonation velocity Dn on the shock-front total cur-
croballoons (GMB). The advantage of such heterogeneous
vature Ct at the explosive charge axis for heterogeneous
mixtures is that their microstructure characterized, for ex-
NM-GMB mixtures confined in steel or PVC tubes of var-
ample, by porosity, void size distribution and specific sur-
ious inner diameters, and (ii) to improve the predictive
face area can be varied in a wide range by changing GMB
ability of the proposed model of detonation-reaction zone
size and concentration, and more precisely and easily con-
in GMB-sensitized NM by comparing the calculated re-
trolled than that of porous solid explosives. A practical
sults with the experimental data on the effect of GMBs on
consequence is that the relationship between the micro-
the charge diameter effect curves, the critical detonation
structure of this model explosive and its shock sensitivity
diameters and the dependence of the normal detonation
(e.g. revealed in shock-to-detonation transition or criti-
velocity on the total curvature of the detonation shock
cal detonation diameter tests) can be investigated rigor-
front.
ously. Particularly, we have already observed how various
aspects of the self-sustained detonation process in NM-
GMB mixtures, such as diameter-effect curves and critical
2 Experiments
diameters, depend on the GMB size and mass fraction and
found that the critical diameter varies linearly with the re-
The experiments have been carried out in NM-PMMA-
ciprocal of the initial GMB specific surface area (Presles
GMB mixtures containing 4 wt.% of PMMA (added to
et al. 1995). Then we have observed that similar funda-
increase NM viscosity) and 1 wt% of mono-sized GMBs
with diameter do =47or 102�m. The specific surface area
Correspondence to: P. Vidal
of GMBs and their volume fraction in these mixtures are
An abridged version of this paper was presented at the 16th
5.40 mm-1 or 5.04 mm-1 and 4.2% or 8.6% for do =47 or
Int. Colloquium on the Dynamics of Explosions and Reactive
Systems at Krakow, Poland, from July 27 to August 1, 1997. 102 �m respectively (the GMB wall thickness is close to 1
142 E. Bouton et al.: Sensitization of two-dimensional detonations in nitromethane by glass microballoons
Aluminum coating
Plastrite
Detonator
Fig. 1. Experimental assembly for mon-
diameter observed by
the streak camera
itoring shock front shape near the charge
Glass plate
Detonation tube PMMA barrier
axis
time interval : 100 ns
Fig. 2. An example of a record of a shape of detonation shock
front near the charge axis
�m for GMBs of both kind). The NM-GMB mixtures were
confined by steel or PVC tubes with i.d. ranging from 2.7
Fig. 3. Experimental and calculated diameter effect curves for
mm to 19.6 mm. The tube wall thickness was 2 mm. The
steel confinement
tubes were equipped with three ionization probes located
every 100 ą 0.05 mm. The distance between the initiation
plane and the first probe was at least 10 charge diameters.
The electrical impulses delivered by the ionization probes
at the instant of passage of the detonation front were used
These records were digitized using an Agfa Arcus II scan-
to trigger a Thomson 632 M32 electronic counter (accu-
ner (optical resolution 1200�2400 dpi) and numerically
racy 1 ns) so that the detonation velocity was measured
analyzed to construct the shape of the shock front in the
within 0.5%.
vicinity of the charge axis and at the contact boundary
Figure 1 shows a scheme of the experimental set-up with the confinement. The curvature radii R = 2/Ct of
used to record the shape of a detonation shock front. A 2 the shock meridian at the charge axis were obtained by a
mm-thick glass mirror, covered by a thin aluminum coat- least-square technique using circles to locally approximate
ing obtained by vacuum evaporation, was stuck at the the shape of the shock front. The number of pixels defin-
end of the tube opposite to the plane of detonation initi- ing the arc length on either side of the front-curvature
ation. A Thomson TSN 506-N high-speed streak camera evaluation points was chosen so that the curvature radii
was used to record the interaction of the detonation shock were independent of the arc length. The precision on R
front with the mirror, which was lit up by an electronic obtained with this method cannot be better than 10%.
flash. This interaction destroys the aluminum coating and Experimental points in Fig. 3 show the charge-dia-
leads to a loss of reflectivity of the mirror. The light beams meter effect in form of the dependence of detonation veloc-
are no longer reflected towards the camera and the shape ity on reciprocal charge diameter for both kinds of GMBs
of the detonation front thus appears on the camera record and steel confinement. Figure 4 shows similar results ob-
as a well defined boundary separating a bright area from tained in case of PVC tubes.
a dark one (see Fig. 2). To measure the angles �e between Figure 5 displays the effect of the normal detonation
the detonation fronts and the charge confinement, the end velocity on the angle �e between the detonation front and
of the tube was machine-tooled to the angle �e after suc- the charge confinement. Figure 6 shows the dependence of
cessive approximations over several shots (the closer this the normal detonation velocity on the total curvature of
angle to �e, the higher the accuracy in monitoring �e). the shock front at the charge axis.
charge diameter
E. Bouton et al.: Sensitization of two-dimensional detonations in nitromethane by glass microballoons 143
Fig. 4. Experimental and calculated diameter effect curves for
Fig. 6. Experimental and calculated dependence of normal
PVC confinement
detonation velocity on total curvature
The reactive flow behind the leading shock front is
modeled as a three-component mixture, the GMBs, the
shocked nonreacted NM and its detonation products. The
latter two components are described by their own equa-
tions of state and thermodynamic properties. The usual
assumptions of mechanical equilibrium among the com-
ponents and of isentropic evolution of the shocked non-
reacted NM are also adopted. Glass compressibility is ig-
nored because of the low mass fraction of glass in the
considered mixtures. In contrast to our previous model
(Ermolaev et al. 1995) where we had used HOM equa-
tion of state for NM (Mader 1979), here we have used
more simple equations for the shocked nonreacted NM
(Mie Gr�neisen) and the decomposition products (quasi-
polytropic), because: (i) these equations provide reliable
results both near the CJ detonation pressure and in the
pressure range specific to near-critical detonation condi-
tions (Bouton 1997), and (ii) the present work only aims
at identifying the salient mechanisms of the GMB addi-
Fig. 5. Effect of detonation velocity on angle �e between the tion effect on the reactivity, detonation performance and
detonation shock front and charge confinement
sensitivity of NM.
Estimates show that the Reynolds number for GMBs
under shock loading is sufficiently small to allow for de-
scription of the GMB collapse behind the shock by a visco-
plastic mechanism (Khasainov et al. 1993). Shock energy
3 Model
is mainly localized in the collapsing GMBs and the energy
is dissipated by heat conduction through the GMB wall to
To simulate the properties of two-dimensional steady deto- the adjacent layer of shocked nonreacted NM. The tem-
nations in the considered model heterogeneous explosives, perature of this layer thus increases to a higher value than
we have used as a basis our previous model (Ermolaev that of the NM bulk and local ignition of NM can occur
et al. 1995) which takes into account the visco-plastic hot at the surface of collapsing GMB at lower shock pressures
spot initiation and growth process in the framework of the than of homogeneous NM. This specific process is a par-
macroscopic quasi-one-dimensional hydrodynamic model ticular case of hot-spot initiation mechanism for hetero-
of reaction zone behind the curved detonation front. geneous explosives. We modeled it according to the clas-
144 E. Bouton et al.: Sensitization of two-dimensional detonations in nitromethane by glass microballoons
sical non-adiabatic explosion theory, with an Arrhenius nation regime as maximum values of the total curvature Ct
exothermic decomposition of NM induced by the visco- and the relative detonation velocity deficit 1 - Dn/DCJ,
plastic energy dissipated in glass and NM, and retarded where DCJ denotes the velocity of the plane (Ct = 0)
by the energy losses due to the conductive heat transfer detonation (e.g. Yao and Stewart 1995, for homogeneous
to the colder outer layers of NM around GMBs. gaseous explosives).
The shape of two-dimensional steady fronts is then ob-
After ignition, we assumed that the NM combustion
tained by integrating, for a given translation detonation
proceeds according to the surface burning concept (Apin
velocity, DT , two first-order ordinary differential equa-
1945, Eyring et al. 1949): the chemical decomposition rate
tions in the slope tan � and the position of the shock, from
in heterogeneous NM is taken as the product of the specific
the shock axis towards the edge of the charge. These dif-
surface area As of the flame fronts diverging from GMBs
ferential equations are obtained by combining the (Dn, Ct)
�
and an empirical burning rate rb = bP exp[�T (T - T0)]
relation and the compatibility relationship DT =
(Ermolaev et al. 1995) which depends on the pressure P
Dn/ sin � ensuring that DT is constant along the shock
and on the shocked nonreacted NM temperature T . The
front. The diameter effect DT = f(1/d) is then obtained
normal burning rate parameters b = 2 m/(s�GPa) and
by varying DT and calculating the diameter at which the
� = 1 were extracted from experimental data measured
slope � of the shock equals the boundary value �e. The
by Rice and Foltz (1991), and the temperature sensitiv-
critical diameter and velocity are found when the shock
ity �T was estimated as �T = E/(2RTi2), where E =30
curvature at the charge edge equals the critical value de-
kcal/mole and R are the activation energy and the univer-
fined by the (Dn, Ct) evolution law. The shock angle �e
sal gas constant (8.315 J/mol.K), and Ti is the reaction
at the edge of the charge can be obtained from the anal-
products temperature at the ignition instant. At ignition,
ysis of the oblique shock polars of the explosive and the
the specific surface area of the flame fronts As equals that
confinement. In our work, we used experimental values for
of the GMBs (which however differs from the initial GMB
the angles �e presented in Fig. 5. Since �e is practically
specific surface area Aso due to GMB collapse). Burning
independent on DT , we have used for simplicity the fol-
proceeds at first as a hole burning mechanism and, af-
lowing constant values for these angles: �e =73o and 85o
ter neighboring flames coalesce, as a grain burning mech-
in PVC and steel confinements, respectively.
anism. In most respects, the present model uses practi-
cally the same assumptions and input constants as the
hot-spot model by Khasainov et al. (1993) for describing
4 Results of calculations
the ignition phase of NM due to visco-plastic spherically
symmetric collapse of GMBs in shocked explosive and the
Figures 3 and 4 show the experimental and calculated
hot-spot-growth rate model by Ermolaev et al. (1995) de-
translation detonation velocity DT plotted against the re-
veloped for GMB-sensitized NM. The main contribution
ciprocal of the charge diameter for the two selected GMB
of the present model is an improved description of the hot
sizes and confinements. Each experimental curve presents
spot growth process to provide a smoother transition from
a strong concave part at small diameters, a well known
the hole to grain burning phase. Particularly, we have used
property of most heterogeneous condensed explosives
an approach proposed by Partom (1995) to describe the
(Campbell et al. 1976). Thus, in the vicinity of the critical
evolution of the specific surface area of the flame fronts
diameter, a small change in the charge diameter induces a
induced by hot spots.
large change in the detonation velocity. However, the pre-
The above model of reactive mixture was coupled with cision of the detonation-velocity measurements just above
a steady curved detonation model based on the assump- the critical diameter is significantly higher in PVC tubes
tions of the quasi-one-dimensional detonation shock dy- than in steel tubes because the steel confinement leads
namics theory (Bdzil and Stewart 1988). The model was to rather small critical diameters. Consequently, in spite
closed with the Rankine-Hugoniot conditions at the shock of a qualitative agreement between the experimental and
and the two generalized Chapman-Jouguet (CJ) transonic calculated (DT - 1/d) curves, the experimental precision
conditions at the end of the dependency domain of the is insufficient to conclude that the critical normal detona-
shock (the sonic surface). These latter constraints require tion velocity Dn does not depend on the nature of confine-
that the heat-release rate is compensated by the heat- ment, as it follows from the model. Yet, the experimen-
loss rate caused by the flow divergence. The resulting tal and calculated results show the following unambiguous
first-order ordinary differential system (Ermolaev et al. trends: for a given detonation velocity, the higher the den-
1995, Bouton 1997) describing distribution of flow vari- sity of the confinement, the smaller the charge diameter.
ables along the charge axis between the shock front and Also, the critical diameter is smaller in steel tubes than
the sonic surface of detonation wave reduces to a well in PVC tubes. Another observation is that the two con-
known eigen-value problem which leads to an evolution sidered mono-sized mixtures having the same GMB initial
law for self-sustained detonation shock fronts in the form specific surface have about the same critical diameter for a
of a relationship between the normal detonation velocity given confinement. This feature agrees well with the exis-
Dn and the total shock front curvature Ct for the consid- tence of an empirical linear correlation between the critical
ered explosive and its confinement. For sufficiently state- diameter in PVC tube and the reciprocal of the GMB ini-
sensitive reaction rates, the (Dn, Ct) relation defines crit- tial specific surface As (Presles et al. 1995). Thus, the pro-
ical conditions for the existence of the self-sustained deto- posed model provides a semi-quantitative agreement with
E. Bouton et al.: Sensitization of two-dimensional detonations in nitromethane by glass microballoons 145
Fig. 7. Correlation between the critical detonation diameter
and reciprocal specific surface area of GMBs
Fig. 8. Profiles of pressure P and PŁ, specific surface area
of flame fronts Sf and external (g) and internal (a) radii of
GMBs in a steady plane wave
the experimental trends. Figure 7 shows the experimen-
tal and calculated critical diameters dcr plotted against
the reciprocal of the GMB initial specific surface As for features can be attributed to the fact that the volume
mixtures confined in PVC tubes. The experimental line fraction of the GMBs increases when the GMB diame-
(Presles et al. 1995) indicates that the linear (dcr - A-1) ter is increased at constant GMB mass fraction and wall
s
relationship does not depend on the GMB diameter. The thickness, so that the volume fraction of NM is higher in
calculated lines also indicate a linear (dcr - A-1) correla- mixtures made with smaller GMBs. Furthermore, Fig. 6
s
tion which, contrary to the experimental one, depends on displays a semi-quantitative agreement between the exper-
the GMB diameter, and has a smaller slope than the ex- imental and calculated Curvature-Detonation Velocity re-
perimental line. The modified evolution law for the flame lationships. Their comparison shows that, for a same shock
front area around the GMBs allowed us to obtain a calcu- velocity, detonations in mixtures with smaller GMBs can
lated slope closer to the experimental one than with the sustain larger shock curvatures due to a larger volume
former law (Ermolaev et al. 1995), though a little increase fraction of NM in this mixture. The slopes of the calcu-
of the spacing between the calculated curves associated to lated diameter-effect curves are always greater than the
different GMB sizes must be acknowledged. experimental ones. This is likely to be due to an over-
Figure 6 shows the measured normal detonation ve- simplified detonation-products equation of state that over-
locity plotted against the shock axial total curvature for estimates the detonation-products temperature. This re-
explosive mixtures confined in steel or PVC tubes. As the sults in an under-estimate of the burning-rate tempera-
curvature increases, the velocity decreases from the point ture-sensitivity and, consequently, in an under-estimate of
corresponding to the CJ plane wave, which was found by the heat release rate and curvature. Adjusting the input
extrapolating the diameter effect curves to infinite diam- parameters of the model could help to improve an agree-
eter or zero curvature. The main conclusion of the shock- ment with the experiment. However, calculations also indi-
shape record processing is that, for a given GMB size, and cate that, as the detonation velocity decreases, the shock-
within the experimental accuracy, there exists a unique to-sonic locus distance becomes of the same order of mag-
axial curvature-axial detonation velocity relationship in- nitude as the shock curvature radius, which brings into
dependent of the confinement nature. However, it must question the use we have made of the quasi-one-dimens-
be emphasized that this unique experimental curve only ional detonation shock dynamics theory (Bdzil and
refers to the axial curvature measured above the experi- Stewart 1988). Nevertheless, additional computations
mental critical conditions of existence of the self-sustained have been performed for a larger range of GMB size and
detonation regime because of the loss in resolution of the mass fraction than those studied experimentally. They
front shape records as these critical conditions are ap- show that the larger specific surface area of the heteroge-
proached. Figure 6 also indicates that for a given deto- neous explosive, the smaller its critical diameter and hence
nation velocity, larger curvature corresponds to smaller the higher its shock sensitivity. Importantly, the so-called
GMBs. A similar observation follows from Fig. 3, which effect of shock-sensitivity-reversal with GMB size has been
indicates that, for a given detonation velocity, smaller tube modeled, that is, the decrease of the critical diameter with
diameters correspond to smaller GMBs though both mix- the reduction of the GMB size (at Aso = const.) stops
tures have practically the same specific surface area. These when the GMB size drops to about 10 �m, and further
146 E. Bouton et al.: Sensitization of two-dimensional detonations in nitromethane by glass microballoons
the detonation products respectively and Pg is the gas
pressure in the GMB voids. The profiles of the considered
flow parameters behind the steady planar detonation wave
propagating at velocity D = DCJ = 5785 m/s are shown
versus the distance z = Dt - x in a frame of reference
fixed to the detonation front where z = 0. Figure 8 also
shows profiles of the GMB external and internal radii g
and a and of the specific surface area of the flame fronts
Sf which initially (i.e. at z = 0) is equal to the specific
surface area of the GMBs Aso.
Figure 9 shows, for the same plane wave, profiles of
the specific volumes of the condensed NM (v2) and of its
detonation products (v3) along with profiles of the particle
velocity qn relative to the detonation front and of 1 -
M2 where M = qn/c is the Mach number and c is the
local sound speed. The point where 1 - M2 drops to zero
defines the sonic locus location and the thickness of the
steady detonation wave (7 mm, in the considered case).
Finally, Fig. 10 displays, for comparison, profiles of the
pressure P in NM and of the specific surface area Sf of the
flame fronts, for the above-considered CJ plane wave and,
Fig. 9. Profiles of volume fractions Ć and specific volumes v
of condensed NM (subscript 2) and detonation products (sub- with dashed lines, for the curved detonation near critical
script 3) along with Mach number M and particle velocity qn diameter propagating at velocity 3 mm/�s and curvature
profiles Ct =0.11 mm-1.
As a whole, the proposed hot-spot initiation and
growth model for heterogeneous NM-GMB explosive mix-
tures seems to be a convenient predictive tool for being
incorporated in time-dependent and/or two-dimensional
gas-dynamic code. However, special micro-scale experi-
ments are still required to establish the detailed mech-
anism of hot spot initiation and growth in such heteroge-
neous systems.
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