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European Journal of Applied Sciences – Vol. 11, No. 4
Publication Date: August 25, 2023
DOI:10.14738/aivp.114.15364
Partom, Y. (2023). What May Happen After a Thermal Explosion. European Journal of Applied Sciences, Vol - 11(4). 268-275.
Services for Science and Education – United Kingdom
What May Happen After a Thermal Explosion
Y. Partom
18 HaBanim, Zikhron Ya'akov 3094017, Israel
ABSTRACT
Thermal explosion may happen when an explosive is in a temperature field, and
when its boundary temperature is between the critical and ignition temperatures.
Semenov and Frank-Kamenetzky solved the thermal explosion problem
analytically in 1D and for a constant boundary temperature [1,2]. Since the 1960s
thermal explosion problems have been simulated numerically in 3D and for
realistic boundary conditions. In thermal explosion tests it is easier to apply the
boundary temperature gradually, and in this way, they become cookoff tests. When
thermal explosion happens in a small region of an explosive body, the events that
follow, and their overall resultant violence, may be quite diverse, like: 1) a slow
decaying pressure wave; 2) a fast non-decaying wave; 3) a strengthening pressure
wave that builds up to shock initiation and detonation. The outcome of a thermal
explosion depends on: 1) the sensitivity of the explosive; 2) the temperature field
throughout the explosive body at the time of thermal explosion; 3) the geometry of
the explosive body; and 4) the degree of confinement of the explosive body. To
model what may happen after a thermal explosion event, we use our PDSR (=
Pressure Dependent Shear Reaction) together with our TDRR (= Temperature
Dependent Reaction Rate) reactive flow models. For each computational cell these
two models work in sequence. Initially there is a shear reaction handled by PDSR. If
as a result, pressure and temperature there go beyond the threshold for reaction
out of hot spots, TDRR takes over to compute shock initiation and detonation. We
present here computed examples of different outcomes of thermal explosion
events.
INTRODUCTION
Thermal explosion is a dangerous safety event when considering the response of explosives to
heat. A thermal explosion may happen in an explosive device when it is surrounded by a
temperature field that is between what is known as a critical temperature and the explosive
ignition temperature. Ignition temperature is the explosive boundary temperature above
which a self-sustaining combustion wave propagates into the explosive, given that the
combustion products flow away very quickly. Ignition temperature of common explosives is
around 500-600K. When the boundary temperature is above the ignition temperature, a steady
combustion wave propagates into the explosive, as long as the combustion products are
unconfined. Such a combustion wave is usually rather slow, around 1mm/s. When the explosive
boundary temperature is below the ignition temperature, a combustion wave is not formed,
and heat flows into the explosive at a rate determined by heat transfer. The front temperature
of this heat wave may slowly increase because of two reasons:1) slow bulk reaction with an
Arrhenius reaction rate; and 2) slow heat transfer towards the boundary, which decreases as
the heat front moves away from the boundary. Because of the highly nonlinear nature of the
Arrhenius rate equation, whenever the heat front temperature goes above a certain threshold,
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Partom, Y. (2023). What May Happen After a Thermal Explosion. European Journal of Applied Sciences, Vol - 11(4). 268-275.
URL: http://dx.doi.org/10.14738/aivp.114.15364
the reaction process accelerates exponentially, and the temperature at the front increases to
high values in less than a μs. As a result, the explosive there undergoes a fast bulk reaction,
which is known as thermal explosion. For decreasing values of the boundary temperature
below the ignition temperature, the thermal explosion location in the explosive body moves
further away from the boundary. There is therefore a low boundary temperature for which the
thermal explosion distance from the boundary is the furthest. For a spherical explosive body
this would be the center of the sphere. The boundary temperature for which the thermal
explosion location is the furthest from the boundary is known as the critical temperature Tc.
When the boundary temperature is below Tc, a thermal explosion event would not happen. The
phenomenon of thermal explosion in energetic materials was discovered and analyzed in
Russia some 80 years ago by Semenov and Frank-Kamenetzky [1, 2]. As there were no
computers at that time, analysis of this highly nonlinear problem was quite difficult. They were
able to overcome the difficulties by treating the 1D configurations of the problem with
asymptotic approximations. Since the 1960s thermal explosion problems are treated with
computer simulations [3], which make it possible to predict the time and location of thermal
explosions for explosive bodies of various geometries and boundary conditions.
The problem of predicting the time and location of a thermal explosion (known as the Frank- Kamenetzky problem) is quite important for explosive safety considerations. Sometimes it is
enough to know if a thermal explosion is to be expected, and if so, how to avoid it by some
design change. But sometimes it is also important to predict what may happen after a specific
thermal explosion. This part is known as level of violence of a thermal explosion event. In
reality these are two stages of the same process. The first stage (heating stage) includes heat
transfer and slow reaction, and pressure rise is minimal. It ends with a very fast pressure rise
over a small volume of the explosive. In the second stage a weak pressure wave spreads out of
this high-pressure volume, which may subsequently ignite the explosive around it by means of
plastic flow localization and shear band formation [4, 5]. Deflagration fronts then spread out of
the ignited shear bands, similar to deflagration waves out of hot spots. But reaction out of shear
bands (called shear reaction) is much slower and less violent than reaction out of hot spots.
This is mainly because the distance between shear bands is an order of magnitude larger than
the distance between hot spots. On the other hand, ignition threshold of shear bands is much
lower than that of hot spots. Shear bands may ignite from a pressure wave of 0.1-0.2GPa, while
hot spots ignite at 1-2GPa.
The second stage of a thermal explosion process may become rather involved, because the
shear reaction wave may strengthen to become a detonation wave. Such strengthening may
happen in various ways: 1) self-strengthening, when the initial pressurized volume is relatively
high; 2) geometric convergence; 3) reflection from a rigid boundary; and 4) interaction of two
shear waves. This is why transition from shear reaction to shock initiation and detonation
depends strongly on: 1) thermal explosion location; 2) device geometry and 3) its level of
confinement.
From the view point of computer simulations, the two stages of thermal explosion are two
entirely separate problems, to be handled with different computational tools. The first stage
(heat transfer + slow reaction) is slow, and its computer simulation may ignore pressure
variations and use ms time steps. The second stage may be a fast hydrodynamic process, with
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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 4, August-2023
ns time steps and high-pressure waves. This is why the two stages should be treated with
different types of codes, the first stage with a heat transfer code augmented by an Arrhenius
reaction rate, and the second stage with a hydrocode. What needs to be transferred from one
code to the other are the temperature field, the reaction parameter field, and the thermal
explosion location.
In what follows we show results from computed examples of the second stage of thermal
explosion events.
COMPUTED EXAMPLES
The hydrocode we’re using includes two of our previously developed reactive flow models that
work consecutively for each computational cell. The first is a shear reaction model that we call
PDSR (= Pressure Dependent Shear Reaction). We calibrated PDSR for an LX07 like explosive
from our Steven test data. The second is our shock initiation and detonation (or surface burn)
model which we call TDRR (= Temperature Dependent Reaction Rate) [6], that we calibrated
in the past for the same explosive from pop plot data.
The explosive body in our computed examples is a cylinder of 180mm diameter and 380mm
length. We compute only one half of this cylinder, z>0, as it has a symmetry plane at z=0. Around
the explosive we have an aluminum envelop 10mm thick. We assume axial symmetry, and the
computational mesh is 1 cell per mm in both directions. To start stage 2 of the thermal
explosion event we: 1) put an initial temperature T0 over all the explosive cells. T0 represents
an average of the boundary temperature and the somewhat increased temperature near the
thermal explosion location just before the thermal explosion. In the examples that follow we
use T0=350K, and the initial temperature of the explosive at the beginning of the second stage
has been specified accordingly; 2) choose the thermal explosion location cell. In the examples
that follow there are two such locations; 3) let part of the explosive around the cell center
detonate out of the cell center with a programmed burn scheme, and we control the mass of the
explosive that undergoes thermal explosion by changing the radius of the detonating sphere;
and 4) run our combined PDSR-TDRR reactive flow model.
As explained above, the pressure wave spreading out of the thermal explosion is too weak to
cause shock initiation, but it may cause shear initiation, which is handled by PDSR. Reaction
ignition in PDSR depends on two consecutive criterions: 1) shear localization criterion for
starting shear band formation given by:
( )L
PD 0.01GPa / s = (1)
where P=pressure, D=effective plastic strain rate, and L stands for Localization. In the past we
calibrated this localization criterion from simulations on the mesoscale [7], and it may be
generally stated that around a thermal explosion this location criterion is usually met; and 2)
ignition delay (from localization to ignition). Ignition delay (h) depends on the product PD as
well, and we calibrated it in the past from Steven test data [7]. Ignition delay is low for high PD
and high for low PD. Ignition delay in a cell is determined by the coefficient Ah, calibrated from
a preliminary short simulation as explained next. Ah is given by: