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Simple Finite Element modelling applied to the problems in

Simple Finite Element modelling applied to the problems in
Energetic Materials Characterisation: Design of
Pyrotechnic Devices and Explosives Calorimetric Studies

M Braithwaite, N Davies (RMCS, Cranfield University, UK),
W C Evans (Defence Establishment Orchard Hills NSW 2748 (Australia),
& P D Lightfoot (Canadian Explosives Research Laboratory, CANMET Ottawa)

Abstract
Many thermal studies involving
energetic materials are carried out in fixed
geometry and these processes are usually
governed by an energy balance between heat
liberated from reaction and that dissipated
by conduction away from the reaction zone.
Many combustion and thermal explosion
problems applications are defined by similar
mathematical equations, albeit with different
initial and boundary conditions.
Flexible finite element software
employing a high level language input and
with adaptive meshing refinement is now
readily available. This coupled with modern
PCs have made simulations of thermal
reaction processes both straightforward and
requiring little time or testing unlike the
more challenging large finite element codes
of the last 20 years usually requiring main-
frame computers and individual pre-and
post-processors.
This paper will cover two disparate
applications

(i) Pyrotechnic delays.
Pyrotechnic delays can loosely be
categorised into those in which the effects of
gas flow can be ignored and those in which
gas flow influences the delay time. Heat loss
takes place primarily by radiation and
convection from the container walls and by
the release of hot gas in open systems.
Metallic confinement acts as a heat sink but
also as a conduit for improved thermal
conduction to the unreacted pyrotechnic.
This study involves comprehensive
models to simulate both types of pyrotechnic
and will illustrate sensitivities to different
design parameters.

(ii) Calorimetric Studies.
A simulation of an Accelerating Rate
Calorimetry (ARC) is described as part of a
study of onset reactions in ammonium
nitrate decomposition. The ARC involves
comparatively small samples (few grams)
and has a phi factor in excess of 2: near
adiabaticity is obtained via temperature
measurement and heaters governed by
electronic feedback. The determination of
the onset of exothermic reaction in


ammonium nitrate presents a challenge to
the thermal analyst.
Ammonium nitrate chemistry involves a
reversible endothermic reaction, several
irreversible dissociations and phase changes.
The results from the experimental study
were simulated using physical and chemical
property data from the open literature.
These two applications illustrate the use
of this type of software in both teaching,
research and design applications involving
thermal processes in energetic materials.

Introduction
This paper is concerned with the
simulation of reactions in solid-state media
in which, with the exception of the perimeter
of the device, energy transfer is largely
governed by heat conduction. The specific
simulations reported here involve a
deflagration in a pyrotechnic delay column
and thermal explosion in reactive media.
These two systems involve the time
dependent solution of energy and mass

conservation relations with chemical
reaction.
A number of approaches have been used
to simulate thermal explosion and
deflagration ranging from analytic
mathematical descriptions [1], large finite
element codes [2] and bespoke software [3,
4]. The advent of fast personal computers
and mathematical software [5] utilizing high
level languages offers an alternative more
flexible modelling tool for both teaching and
engineering design.

Mathematical Models
The time dependent studies illustrated
here have been carried out in cylindrical
coordinates for exemplification purposes.
The equations solved consist of:
(i)
energy conservation
(ii)
mass conservation
and, where required,
(iii) momentum
conservation

(iv)
equation of state (ideal gas)
(v) thermal
feedback
(ARC)
subject to initial and boundary conditions
discussed in each study. Variables include
local temperatures, pressures, extent of
reaction and gas velocities. Details of the
equations are given later for each
application.

Simulation Software
Ideally, these applications require
software that is flexible and easy to use. It
should be available for common operating
systems platforms with graphical output and
compatibility to other graphics packages.
The solutions of the comparatively simple
models here often have to be extended to
complex geometries and mixed boundary
conditions. Solvers employ finite element or
difference procedures, which require the
setting up of a mesh. Use of adaptive mesh
rezoning in problems whose reaction zone is
small compared with the total geometry is
preferred as this enables more accurate
simulations in a finite computer memory.
Ideally the software should employ a high
level programming language enabling the
description of a simulation in conventional
mathematical notation.
FlexPDE [5] (PDE Solutions Inc) offers
all the above features in a standalone
computer program and has been used in all
the simulations described in this paper. The
software has been previously used by the
present authors in a number of applications
related to this paper and for which analytic
solutions were available [1]: agreement has
always been good.

Applications to Pyrotechnic Delays
A cylindrical delay element in an
aluminium casing has been simulated.
Longitudinal and axial conduction are
included as well as radiative and convective
heat losses on the outer surface of the
casing. The initiation of the pyrotechnic was
introduced by means of a heat pulse. A
similar system has been comprehensively
studied elsewhere [3].
Data on the pyrotechnic were taken from
an Sb/KMnO
4
study

[6] and key parameters
are included in Table 1. This system has
been modelled as a solid phase reaction,
though allowance for air in the porous
pyrotechnic has been included in a constant
gas density process with no gas flow. The
delay element comprised an open cylinder
22mm long and of internal and external
diameters 3.4 and 6.4mm respectively. The
pyrotechnic was assumed to be at 80% of its
maximum density.
Reaction was assumed to take place at
temperatures in excess of a defined ignition
temperature. For the purposes of this study,
physical properties were assumed
temperature invariant





TABLE I : Pyrotechnic Properties

Heat of reaction
9.10
6
J/kg
Reaction order
2/3
Activation energy
20700 J/mol
Arrhenius factor
9.25 s
-1

Ignition temperature 506 K
Heat capacity
540 /kg
-1
K
-1

Thermal
conductivity
0.3 Wm
-1
K
-1

Reactant Density
2300 kgm
-3

Product density
2070 kgm
-3


FlexPDE (v3.01e, PDESolutions Inc) [5]
was used to solve the following equations

2
(
)
[(
(
))
((1
)
(
))
]
(1)
p
g
s
p
p
g
s
p
p
d
kT
q
C
dt
dT
C
dt


=
+


where , , ,
, , , and
p
k
T C
Q

denote
thermal conductivity, fraction of pyrotechnic
unreacted, specific heat (gas or pyrotechnic),
density, temperature, solid fraction and heat
of reaction respectively: subscripts
, and
s g
p refer to reactant pyrotechnic,
reaction products and air.


(
/
)
n
E RT
d
k e
dt =
(2)

where , and E
n a
denote reaction order,
Arrhenius factor and activation energy: no
reaction is allowed below a preset ignition
temperature. Equation (1) is simplified for
the unreactive aluminium casing as Equation
(2) is not required.
The initial conditions are ambient
temperature and no reaction with a heat
pulse into the front end of the pyrotechnic
starting at zero time given by a heat flux
term of the form
2
1
1
2
where
and
t
e are
set at 10
6
and 10 respectively. The boundary
conditions for the whole assembly consist of
standard radiative/ natural convective heat
loss terms for aluminium in cylindrical
geometry. An example program listing is
included in Appendix 1.
Deflagration velocities were determined
by tracking the location of the maximum
temperature history in the central core of the
pyrotechnic. These data were fitted to a
cubic in distance to allow for an accelerating
wave due to preheating of unreacted
pyrotechnic. A typical result is given in
Figure 1 for the case of a reduced (50%)
reaction rate. Some slight acceleration is
observed