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Numerical Prediction on Cookoff Explosion of Explosive
under Strong Confinement
Zhi-Yue Liu1,a and Muhamed Suceska2,b
1
School of Science, Beijing Institute of Technology
Zhongguancun South Avenue No. 5, Beijing 100081, China
Laboratory for Thermal Analysis, Brodarski Institute – Marine Research & Special Technologies
2
Av. V. Holjevca 20, 10000 Zagreb, Croatia
a
[email protected], [email protected]
Keywords: Cookoff explosion, Explosive, Numerical simulation, Finite difference method, Heat
conduction.
Abstract. The cookoff of explosives is of great concern for the safety assurance of explosive
devices in storage, transportation and handling. It may occur in the situation that explosive devices
are subjected to the external heating stimuli such as fire or high-temperature surrounding. In order
to gain ever-increasing knowledge toward the cookoff explosion of explosives, we establish
numerical program to predict the cookoff explosion of explosive in a metal container. The
computational formulation and methods are given in detail. The thermal decomposition and
temperature variation in the interior of explosive were found corresponding to several typical
external heating conditions. The results demonstrate that the method is beneficial to the future study
on this subject.
Introduction
The cookoff of explosives refers to the process of explosion of explosive materials due to local or
bulk heating of explosives under the external heating stimuli such as fire or high-temperature
surrounding. Studies on the occurrence of cookoff explosion as well as its violence are of particular
importance for the safety concern of explosive devices in storage, transportation and handling.
Earlier research was mainly concentrated on the thermal explosion of single explosive under
different heating conditions. The critical heating temperature and the time to explosion were
obtained by both experimental and numerical techniques for some explosives [1-3]. However, in the
practical applications, explosives are often used with or contained in a metallic material. The
influence of such confining material to the cookoff explosion appears by the way of thermal and
mechanical interaction with explosives inside it. Even though a lot of efforts have been devoted to
this subject in the past years [4-6], the unclearness to the whole phenomenon still exists to be
uncovered. Beside the continuing experimental exploration, numerical computation raises more and
more attention among researchers so as to gain better understanding and detailed specifications for
the study. With the aid of computational hardware development, it becomes possible to numerically
simulate the more complex system which would have been conducted only by means of
experiments before. The cookoff explosion of explosives under metallic confinement involves heat
transport by forms of transfer or conduction, chemical decomposition, ignition, deflagration, and
even detonation, as well as their interactions with the confinement. The time related to the whole
process may be quite different, possibly spanning from hours to microseconds corresponding to
various situations. It brings about great challenges to the numerical simulation on the cookoff
process. As a trial study to the subject, this paper presents numerical procedures for predicting the
cookoff explosion of explosive materials confined in the metal container. The external heating from
the end and lateral side of the container are considered to examine the effect of heating types to the
thermal behavior of the explosive. As for the cookoff violence study, it will be left for the future
topics.
Model Assembly Description
The arrangement of energetic material for cookoff test is schematically presented in Fig. 1. It is
composed of a metal tube with the filling of a PBX explosive (95% weight of HMX and 5% other
binders). The tube is 45 cm in length, 60 cm inner diameter and 20 cm wall thickness. Including the
plug parts the whole system is totally 50 cm long. The height of the plug is assumed to be same
dimension of the tube wall thickness. The heating is performed either at one end of the tube or on
the lateral surface, respectively. Owing to the deficiency of the data on the complete material
properties of the explosive mentioned above, the parameters of PBX 9404 explosive is instead used
by consideration that the two explosives have similar main constitutes in the formulation.
Metal tube
Plug
Explosive
Fig. 1 Schematic diagram of energetic material arrangement for cookoff test.
Numerical Method
Governing Equations. The section headings are in boldface capital and lowercase letters. Second
level headings are typed aThe cookoff explosion of energetic material under external heating
condition is of close connection with the thermal behavior of the energetic material. The
temperature, an important thermal parameter, plays a key role in the chemical decomposition
process of energetic material. The temperature variation in a reactive material can be described by
the equation of
.
T
C p
  2T  S .
(1)
t
Where, T is temperature,  is density of material, Cp is heat capacity under constant pressure, is
heat conductivity coefficient, t is time. The last term in the right hand of Eq. 1 indicates the amount
of heat release within a unit time. For inert material this term is equal to zero. For most energetic
materials, the heat release due to chemical reaction may be described by the following equations,
.

S  Q
t
(2)

(  Ea / RT )
 (1   ) Ze
,
t
where Q is the heat release of energetic material per mass,  is the mass conversion of energetic
material due to chemical decomposition, its value varies from 0 to 1 corresponding to the initial
decomposition through full decomposition, Ea is activation energy; Z is pre-exponential factor, R is
universal gas constant. The other symbols are of the same meaning as those in Eq. 1. The second
equation in Eq. 2, in fact, represents the first-order Arrhenius kinetics law equation for chemically
activating materials.
Under two-dimensional cylindrical or planar coordinate system, the Laplacian operator, 2,
appearing in Eq. 1 is usually expressed by the following form,
  2T m T  2T 
2
,
(3)
 T   2 

r r z 2 
 r
where, r and z are the coordinates along the radial and axial directions, respectively, if letting m=1
represent the case of the cylindrical coordinate system; for the case of the planar coordinate system,
there is m=0.
Finite Difference Technique. The finite difference method is used to solve the governing
equations given in the above section. The calculation grid is defined in Fig. 1. The area confined by
a coarse line stands for the material domain and the external cells are virtual computational zones
for boundary conditions setting. The temperature is defined in the center of the cell and the sides of
the cell are of the known coordinate position.
j=Jmax+1
j=Jmax
x
j
O
z
k=Kma
k=K
max
x
+1
k
k=0
k=1
j=1
j=0
Fig. 2 Computational grid definition.
The practical problem described in Fig. 1 involves two materials. In this case, referring to Fig. 3, the
heat flow passing through the interface of different materials may be given by temperature gradient
in either material on both sides of the common interface according to Fourier’s law [7],
T T
T T
qa  a A i a ,
qb  b A b i ,
(4)
xa
xb
where Ti is temperature on the interface, Ta and Tb are temperatures in materials a and b respectively,
A is the interfacial area for heat flow, xa and xb are the distances from the interface, a and b are
heat conductivity. The two heat flow expressions in Eq. 4 should be identical in quantity, so it leads
to the following equivalent heat flow expression using the temperatures in two materials only.
  (xa  xb )
T T
q a b
 A a b .
(5)
b xa  a xb
xa  xb
Considering that the heat flow may experience different material, the difference equations will
be established according to the heat flow balance in a specific element. Referring to Fig. 4 for a
control volume, the heat flow through the radial top and bottom surfaces is given by
T
T
q1  rz  1
, q2  (r  r )z  2
,
(6)
r r r1
r r  r2
however, the heat flow along the axial direction can be trivially given. So, the net heat change in the
volume by heat conduction becomes
 T

 T

T
T
  rzr 2 T
. (7)
q  rz r 2
 r1
 rr  z 2
  z1





r

r

r

z

z
r  r2
r  r1 
r  r2
z  z2
z  z1 


As consequence, the temperature variation in the volume is expressed as follows,


Ti n 1  Ti n
1  1  T
T
1  T
T
  r 2  T

    r 2
 r 1

z 2
  z1
t
  r  r r  r2
r r  r1 
r r r  r2 z 
z z  z2
z z  z1 
(8)
n
(1   )QZ (  Ea / RTi )

e
,
Cp
r2
r
 xa
r1
z
r
Tb
Ti
Ta
r
(b)
)
(a)
 xb
Fig. 3 Heat flow between two materials.
O

z
Fig. 4 Control volume for FDM method.
where t is the time increment in the calculation, the superscripts in temperature denotes the values
before and after the time increment. The partial derivatives with regard to both coordinates in Eq. 8
should be replaced by difference concept like that given in Eq. 5.
Boundary Conditions. For the internal points, the calculation for temperature is accomplished
by the Eq. 8. However, for the zones near the boundaries, the temperature calculation should
include a term representing the heat exchange transfer mechanism. Then in Eq. 8 there is another
term should be covered.
(9)
qex  A(T  T0 ).
where,  is heat exchange coefficient, T is the surface temperature of the test assembly, T0 is the
surrounding temperature, qex is the heat flow.
If temperature is given as a prescribed value on partial boundary, the temperature in the virtual
zone near that boundary is expressed by
(10)
Te  2Ts  Ti ,
where Te, Ts, and Ti are temperatures in the virtual zone, on the boundary, and in the interior zone
respectively.
Time Step Control. In order to hold the stability of the solution of the finite difference equations,
the time step should be limited within a certain range. Here, considering the heat conduction and
heat exchange transfer effects, the time step must be less than
 C p (s ) 2

1
t  min 

(11)
,

2(s /   2) 

where s is the step size of the cell in the calculation. The time step is judged cell by cell and finally
the minimum value is chosen.
Results and Discussion
Two kinds of heating situations were considered in the investigation. One case is that of heating at
one end of the cylindrical casing container with energetic material; the other is the situation that the
heating is located around the middle of the cylindrical casing container. The heating was assumed
with a fixed temperature on the surface of the sample. The container takes the material of steel and
the explosive is assumed as PBX9404. The material parameters [8] used in the calculation are listed
in Table 1.
End Heating. Two kinds of end heating situations were performed. In the first situation, the
heating temperature was 500K, constantly holding at one end of the container. The temperature
around the container was assumed to be at an ambient state. Fig.4 presents the temperature contour
images from computation. The temperature scale is illustrated in the right hand of the figure. The
lines in the figure distinguish the various materials in the system like those shown in Fig. 1. Three
computational snapshots are given here for illustration. It is seen that even though a long period of
time passed the temperature in the explosive does not change so much in magnitude. It indicates
that there is not much chemical decomposition in the explosive sample and can not cause a violent
cookoff consequence in the system. However, when the heating temperature was changed to be
520K, as such shown in Fig. 5, the consequences are very different. In the later phase, the violent
Table 1 Thermally physical and chemical parameters of materials.
Material
Cp
Q
Z


3
(1/s)
(g/cm ) (Cal/g K) (Cal/cm s K) (Cal/g)
Steel
PBX9404
7.896
1.844
0.11
0.24
0.129
1.23x10-3
1.0x103
1.1x1018
Ea
(Cal/mol K)
R
(Cal/mol)
5.0x104
1.987
(a) initial
(b) t=358 seconds
(c) t=1198 seconds
Fig. 4 Temperature contours in the system with 500K external heating temperature at one end.
(a) t=487 seconds
(b) t=547 seconds
(c) t=558 seconds
(d) t=666 seconds
Fig. 5 Temperature contours in the system with 520K external heating temperature at one end.
chemical decomposition appeared, the temperature rises dramatically in the explosive and the fast
reaction spread quickly from the heating end toward the other end. The results provide the
information about the critical heating temperature for the system. It falls within the range of
between 500K and 520K temperature. From Fig. 5, it is also understood that the temperature
gradients are different on the sides of the interface of two materials. The temperature gradient is
caused due to the huge difference on the heat conductivity of steel and explosive. Once the fast
chemical decomposition forms, it rapidly progresses along the explosive material. Here, the heat
conduction mechanism is solely considered, the computation does not represent the actual explosion
of the system. In reality, the practical explosion is much faster than those shown here as the
convective heat transfer mechanism will dominate the process of the chemical decomposition.
Nonetheless, the calculation predicts from one point of view the possibility of cookoff explosion
occurrence. That will help us take some measures to protect or secure the situation.
Lateral Heating. Corresponding to the end heating situation, two sets of heating temperature were
also considered. The heating area was circled along the container with an axial length dimension
same as that in diameter. When heating temperature is imposed to be equal 500K, it is found that
the fast decomposition will appear in the explosive, causing the cookoff occurrence in the system as
shown in Fig. 6. The high temperature region expands toward both two ends of the system. With the
degradation of heating temperature to be 470K, the calculational results are shown in Fig. 7. The
computation shows no appearance of the extremely high temperature region. It means that the
violent decomposition tends to be impossible even after a long heating period. The computational
results indicate that the system will not explode under such heating condition.
Comparing the computations on the two situations of end heating and lateral heating, it can be
seen that the possibility of cookoff explosion of the system is slightly different in terms of the
external heating temperature. In the case of lateral heating, lower surrounding temperature might
lead to cookoff explosion in the system. It means that the system is more vulnerable subjecting to
lateral heating than end heating. However, no much discrepancy on heating temperature range turns
up in the two situations.
(a) t=178 seconds
(b) t=208 seconds
(c) t=215 seconds
(d) t=283 seconds
Fig. 6 Temperature contours in the system with 500K external heating temperature laterally.
(a) t=357 seconds
(b) t=3602 seconds
Fig. 7 Temperature contours in the system with 470Kexternal heating temperature laterally.
Conclusions
The cookoff explosion of the encased explosive had been predicted by numerical techniques. The
governing equations and the computational procedures were given in detail. The study formulated
the heat flow across the interface of different materials. The governing equations were solved by
finite difference method. Two external heating conditions were under investigation. Through the
computation, it was found that the lateral heating is more dangerous for the cookoff explosion
occurrence than the end heating, but there was no huge discrepancy in two forms of heating
condition. The computational results demonstrated that it should be worthy concerning for the
safety of the system either subjecting to lateral heating or end heating.
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